text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: Recovering state of batch process after server failure I believe that topic speaks for itself.
I would love to read your opinions and maybe possible solutions on this?
Lets say that we have first, simplest scenario:
There is 1k of new products that needs to be imported to system via server.
Products is only one table in database, no relations needed.
Imagine that we are trying to upload this data and in the middle it fails but there is no rollback, data is saved partially.
Let's say 200 from 1000 products were saved into db.
Since you got message that data was saved partially because of server failure you want to import the same data once again but it is not very effective to check if every of product exist in database. How do we restore such state of a batch process to start it from 201 product not starting from beginning?
Thanks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,493 |
Q: Manually adding FontAwesome in swift iOS I need to use FontAwesome in my swift project.
I am referring following link:https://github.com/thii/FontAwesome.swift
Using CocoaPods i am getting lot of error and so have to setup FontAwesome manually
The link does not mention in detail on how to do this.
I am using Xcode 8.1
Please help
A: Did you add the font files (FontAwesome.otf) to your Info.plist? (You also need to add the file to your project) It might be because of that.
That way you can use the font without using the swift class they provide.
let font = UIFont(name: "FontAwesome", size: fontSize)!
A: Here is the way to go if you dont want to use pod. Download the project from github. Then go to downloaded file and drag FontAwesome.xcodeproj into your project like this.
Then, switch to your project settings, choose your target and in General tab find, "Linked frameworks and library", drag FontAwesome.framework from FontAwesome->Products into this section.
Simply import it to your class like this,
import UIKit
import FontAwesome
public class ViewController: UIViewController {
override public func viewDidLoad() {
super.viewDidLoad()
let label = UILabel()
label.font = UIFont.fontAwesome(ofSize: 100)
label.text = String.fontAwesomeIcon(name: .github)
view.addSubview(label)
label.frame = CGRect(x:60, y: 60, width: 200, height: 300)
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,299 |
Q: Evaluate $\int\left(\sqrt{4-x^2}+x\right)dx$ I'm looking for simple way to solve
$$\int\left(\sqrt{4-x^2}+x\right) \, dx$$
I tried substitute $x=2\sin u$ and then
$$\cdots =\frac{x^2}{2}+\frac 1 2x\sqrt{4-x^2}+2\arcsin (x/2)+c$$
I'm looking for other solution please
A: Another way forward is to integrate by parts with $u=\sqrt{4-x^2}$ and $v=x$. Then, we have
$$\begin{align}
\int \sqrt{4-x^2}\,dx&=x\sqrt{4-x^2}+\int \frac{x^2}{\sqrt{4-x^2}}\,dx\\\\
&=x\sqrt{4-x^2}+\int \frac{x^2-4+4}{\sqrt{4-x^2}}\,dx\\\\
&=x\sqrt{4-x^2}-\int \sqrt{4-x^2}\,dx+4\int \frac{1}{\sqrt{4-x^2}}\,dx\\\\
2 \int \sqrt{4-x^2}\,dx&=x\sqrt{4-x^2}+4\int \frac{1}{\sqrt{4-x^2}}\,dx\\\\
\int \sqrt{4-x^2}\,dx&=\frac12 x\sqrt{4-x^2}+2\int \frac{1}{\sqrt{4-x^2}}\,dx\\\\
&=\frac12 x\sqrt{4-x^2}+2\int \frac{1}{\sqrt{1-(x/2)^2}}\,d(x/2)\\\\
&=\frac12 x\sqrt{4-x^2}+2\arcsin(x/2)+C
\end{align}$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 789 |
\section{Introduction}
Increasingly, enterprises are storing and processing vast amounts of
data to detect security threats and compromises. To build models for
efficient detection, machine learning tools currently show the most promise.
One of the important tasks in building a machine learning model is
coming up with a good set of features. This usually requires deep
knowledge and understanding of the problem domain, and even then
constructing good features for a particular task involves quite a bit
of trial and error. High quality features constructed for a specific
task (e.g., anomaly detection) simplify model building, and result in
even simple machine learning methods performing sufficiently well.
In recent years, the fields of computer vision and natural language
processing have made great strides using deep learning models to {\em
learn features} from raw inputs instead of a domain expert manually
engineering features. This has been mainly possible through use of
supervised learning on large scale labelled image data sets (most notably
imagenet \cite{imagenet_cvpr09}) in the case of computer vision, and self-supervised learning on large corpora of texts in the case of natural language processing.
In this paper, we explore the use of feature learning, in place of feature
engineering, for building machine learning models on security
data. Specifically, we look at how the learned features or
representations perform on the popular task of anomaly detection to
detect unusual behavior, which may indicate an attack, a compromise
or a misconfiguration (which may make a system more vulnerable to compromise).
Security data refers to network or host data that enterprises
typically monitor for security threats and attacks.
We focus on two main challenges faced while learning
features for security data. First,
security data is unlabelled. Thus, it is difficult to learn features and the
final task jointly as is done in image recognition. Furthermore,
security attacks are ever evolving, so even if a lot of effort is spent
on labeling data, it will only work for known attacks, not for
previously unseen or ``zero-day'' attacks. Secondly, while humans are
good at perceiving and understanding images and natural language, and
can immediately recognize if meaningful features are being learned for these
two modalities, the same cannot be said for say logs of network traffic or
system calls.
To address the first challenge, we use self-supervised learning, a subset of unsupervised learning, where implicit labels available within the data are used for supervision on a pretext task\cite{jing2019}. In particular, we
use a sequence-to-sequence prediction task, modeled using a
recurrent neural network model, aided by an attention mechanism. We refer to this method as AS2S (attention-based sequence-to-sequence model). A general solution to the
second challenge is difficult; we explore it in a limited fashion in the context of anomaly detection by using a partially labelled
data set\footnote{The labels are only used for evaluating performance, not for training.} to compute precision and recall. We also visually inspect how the learned features in a two-dimensional embedded space evolve during training.
To demonstrate the effectiveness of the learned representations, we
use two real data sets to evaluate the performance for anomaly
detection. These data sets include partial ground truth labels
enabling quantification of the results. One of the data sets consists
of network traffic, while the other contains network and host
data. Both of these can be considered as a multivariate temporal
sequence of attributes. The results show that the attention-based
sequence-to-sequence model performs better than the two selected baselines,
namely a principal component analysis (PCA)-based model and an auto-encoder based
model. Furthermore, we observe as training progresses, the features
learned by our model are better able to separate normal and anomalous data points.
We make the following key contributions:
\begin{itemize}
\item We demonstrate how to perform feature learning for security data using attention and self-supervision
\item We provide performance results for this method on two real world public data sets
\end{itemize}
The rest of the paper is organized as follows.
We discuss related work in the next section, followed by our
solution approach in Section \ref{sec:solution}. Experimental results
are presented in Section \ref{sec:experiments}, followed by
conclusions and future work.
\section{Related Work}
Automatically extracting or learning suitable features from raw data
without human intervention has been a grand challenge in machine
learning. In fact, the basic idea can be traced back to Herbert Simon
who said in 1969, "Solving a problem simply means
representing it so as to make the solution transparent" \cite{simon1969sciences}. Over time
diverse techniques \cite{pmlr-v44-storcheus2015survey, jing2019} have
originated in different research communities and include: feature
selection, dimensionality reduction, manifold learning, distance
metric learning and representation learning. Extracting features via
layers of a neural network is commonly referred to as representation
learning \cite{pmlr-v44-storcheus2015survey, bengio2013}, also known as feature
learning and is the focus of this paper in the context of security
data, and its application to machine learning models to detect
security anomalies.
Anomaly detection in the area of cyber-security has received
considerable research interest and has a long history, although most
deployed systems are rule-based, and application of machine learning
for anomaly detection in security has unique challenges
\cite{2010outside}. Early work tended to use statistics like moving
average or PCA to locate anomalies \cite{2003anoval}. Such methods
worked well in simple cases, but as modern IT systems and data get
larger and more complex, the methods failed to scale with them.
Clustering-based and nearest-neighbor-based methods were proposed
\cite{Scarth1995DetectionON,2007incremental} to utilize locality of
data, but they suffer from both a lack of scalability and the curse of
dimensionality.
Supervised learning is not attractive for anomaly detection since
labelled data is scarce. Furthermore, especially for security
anomalies, the presence of adversaries make the anomalies (threats and
attacks) dynamic and constantly changing. Thus, the most common
setting for anomaly detection is unsupervised or self-supervised
\cite{jing2019}.
Recently, neural-network-based methods have gained much attention, as
the computational performance problem is gradually mitigated by
advanced optimization algorithms and running parallel cores. Xu et
al. \cite{DBLP:journals/corr/XuRYSS15} proposed a de-noising
auto-encoder-based method, which is a prototypical two-phased
framework. An unsupervised learning model, the auto-encoder, is first
trained with a large data set to get the distributional
representations, and later the representations are used in another
domain-related application. However, auto-encoder-based models failed
to facilitate temporal relationships, which is considered an important
component of detecting anomalies. Tuor et al. \cite{tuor2017deep}
adapted a multi-layer LSTM model but with a different special
assumption that the corresponding inputs and outputs of the network
should not differ by more than a threshold; otherwise, it is an anomaly. Similarly,
Malhotra et al. \cite{DBLP:journals/corr/MalhotraRAVAS16} proposed an
LSTM-based encoder-decoder for anomaly detection on machine
signals. Such a model is essentially an auto-encoder between a pair of
sequences, which models temporal information. It can identify
anomalous sequences when the reconstruction error is large. All of
these studies use the underlying principle that fitting models to data
by minimizing reconstruction error can help identify anomalies.
Attention mechanism is a breakthrough component proposed to
strengthen the encoder-decoder model. In addition to the ability of
decoding quality sequences, it provides soft alignments between the
inputs and outputs, which helps explain the sequence generated.
More recently BERT/Transformer architecture \cite{bert} was proposed that uses self-attention.
Attention mechanism has proved to be useful in neural machine translation
\cite{2015neural,DBLP:journals/corr/BritzGLL17}, machine reading
comprehension \cite{DBLP:journals/corr/HermannKGEKSB15}, and video
captioning \cite{DBLP:journals/corr/XuBKCCSZB15}.
However, its application to cyber-security anomaly detection has not been explored before. We do not use self-attention, but an auto-encoder with a basic attention component to train the representations.
In our work, we also adopt the prototypical two-phase learning
framework, as it offers us an opportunity to generalize the use of the
learned representations, which is a solid foundation for building a
robust universal anomaly detection system. An LSTM encoder-decoder
network coupled with attention mechanism is explored in the first
phase of learning universal representations for security
data. Evaluation and analysis on applying the learned representations
in anomaly detection are also investigated.
\section{Our Solution Approach} \label{sec:solution}
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{figs/fig1.jpg}
\caption{FL4S Framework}
\label{fig:framework}
\end{figure}
Fig. \ref{fig:framework}. shows the architecture of our approach, which we call Feature Learning for Security (FL4S) framework. It consists of three main components: Raw Feature Extraction, Clustering, and Feature Learning, which are executed sequentially. The following subsections introduce the data sets used and the three components in detail.
\subsection{Raw Feature Extraction}
In this initial step, we extract simple features that do not require input from a domain expert. The goal is to use basic representations of the data so we can perform feature learning on top of them. Security data
can be broadly categorized as network data and host data.
Network or network traffic data refers to summary information about network communication typically collected at a router or a special appliance installed
in an enterprise network to collect such data. Commonly, such data is collected at an edge router where it captures both ingress and egress
packet data. A popular network traffic data format is {\em netflow} that includes fields such as IP addresses, port numbers, number of bytes, packets exchanged, application level protocol (inferred from
port information), etc. Host data refers to data collected at a particular machine, and may include data pertaining to filesystem access,system calls, login and logout information, connection and disconnection of devices, etc.
\begin{figure}
\centering
\includegraphics[width=0.90\textwidth]{figs/fig2-1.png}
\caption{Extracted Raw Features}
\label{fig:raw_features}
\end{figure}
Security analysts are typically interested in which users or hosts are compromised, or pose a risk. Thus we extract data specific to users.
Further, data generated by users can vary highly over time, and features are aggregated by time windows. We extract the raw features for each
user and for each pre-defined time window duration. We tried window sizes of 3, 6, 12, and 24 hours; we observed that 24 hour windows gave the best results. Thus, in the following, all references to windows
relate to window sizes of 24 hours. Note that multiple time windows could also be used simultaneously to train multiple models.
In addition, since users usually have similar data patterns on a weekly basis, we consider
week-long user sequences. Therefore, we redefine each user example as a user-week example. Fig. \ref{fig:raw_features} illustrates this
idea. We also ignore windows during weekends, because they usually have different data patterns than weekdays. As a result, each user-week example has five feature vectors, one for each window
(day). This organization also avoids long sequences for each example, which increases the difficulty of capturing data patterns. Similar
models can be built with weekend data as well. As shown in Fig. \ref{fig:raw_features}, the data takes the form of a three dimensional tensor: (user-window (UW), time-window, feature ($F_{ij}$)), where each of the $F_{ij}$ is a feature vector.
Security analysts are typically interested in which users or hosts are
compromised, or pose a risk. Thus we extract data specific to users.
Further, data generated by users can vary highly over time, and features
are aggregated by time windows. We extract the raw features for each
user and for each pre-defined time window duration. We tried window
sizes of 3, 6, 12, and 24 hours; we observed that 24 hour windows gave
the best results. Thus, in the following, all references to windows
relate to window sizes of 24 hours. Note that multiple time
windows could also be used simultaneously to train multiple models.
In addition, since users
usually have similar data patterns on a weekly basis, we consider
week-long user sequences. Therefore, we redefine each user example as
a user-week example. Fig. \ref{fig:raw_features} illustrates this
idea. We also ignore windows during weekends, because they usually
have different data patterns than weekdays. As a result, each
user-week example has five feature vectors, one for each window
(day). This organization also avoids long sequences for each example,
which increases the difficulty of capturing data patterns. Similar
models can be built with weekend data as well. As shown in
Fig. \ref{fig:raw_features}, the data takes the form of a three
dimensional tensor: (user-window (UW), time-window, feature ($F_{ij}$)),
where each of the $F_{ij}$ is a feature vector.
\subsection{Segmentation}
A separate model could be built for each user.
However, the number of IP addresses is
typically large, and several IPs may not have enough data associated
with them to train a model. On the other hand, user behavior shows
high variance, and it is difficult to capture all user behavior with a
single model. Therefore, we segment users into clusters, and a model is
trained for each cluster. A subset of the training features are used
for clustering the users. Since most clustering algorithms suffer from
the curse of dimensionality, we want to avoid high-dimensional feature
vectors during clustering.
Clustering is an important part of this framework, as it not only
groups users that should have similar data patterns but also does
filtering to some extent. In our experiments, we noticed that there is
one group that has a higher number of anomalies, which means anomalies
do share some level of similarities. As the number of normal data
points are still far larger than the anomalies, we are still able to
train quality models. For one of the data sets used in the experiments,
the classic k-means clustering is
chosen. Although k-means is a simple algorithm, it provides reasonably good results.
More sophisticated clustering algorithms might be
explored in future work. For the other data set, k-modes clustering
\cite{2001kmodes} is chosen. It is a k-means variant for categorical
data.
To determine a proper $k$ for each algorithm, we use Silhouette
Coefficient (SC), which is defined as:
$SC(i) = \frac{b(i)-a(i)}{max\{ a(i), b(i) \}}$
, where $a(i)$ is the average dissimilarity to the intra-cluster data
points and $b(i)$ is the lowest average dissimilarity to a neighboring
cluster. For each data set, we try to find the $k$ with the highest average SC.
\subsection{Baseline Methods}
Two baseline models—Principal Component Analysis (PCA) and Auto
Encoder (AE) are compared with the proposed method.
PCA and AE are general methods to learn low-dimensional embeddings.
These embeddings can be thought of as features learned from data. We
compare these with our proposed method, AS2S. A key difference with
AS2S is that each input vector is considered independent, while AS2S is
a sequence model and captures temporal relationships between subsequent
examples.
\subsubsection{Principal Component Analysis}
PCA is a common statistical method to conduct dimension reduction for data. It uses orthogonal transformation to convert data into a new space that has linearly uncorrelated variables in each dimension. The
top $p$ principal components contain the most information of the data, where $p$ is a parameter of the model. A detailed description of PCA can be found in \cite{jolliffe2016principal}.
\subsubsection{Auto Encoder}
The basic structure of AE is a simple 3-layer neural network that includes input, hidden, and output layers. The raw data is fed into the input layer, encoded into the hidden layer, and then decoded in
the output layer. The optimization aims at minimizing the difference between the input and the decoded output. To generate a dense representation for an input, we take the outputs of the hidden layer. The number of hidden units, $p$, is a parameter of the model that controls the size of the learned representations. The activation function used in our experiment is empirically chosen to be hyperbolic tangent (tanh).
\subsection{Attention Sequence-to-Sequence Recurrent Neural Network Model}
In \cite{DBLP:journals/corr/MalhotraRAVAS16}, Malhotra et al. adopt Sequence-to-Sequence Recurrent Neural Network model (S2S) for anomaly detection in several time-series data, and show that S2S exhibits superior performance in detecting anomalies. The results inspired us to apply such a model to security data with temporal dependencies. \cite{2015neural,DBLP:journals/corr/HermannKGEKSB15,DBLP:journals/corr/XuBKCCSZB15} introduce a powerful new mechanism called "attention," which provides the S2S model with the ability to find soft alignments between input and output sequences. This mechanism has shown promising results in numerous natural language tasks, such as machine translation, question answering, and artificial conversations (i.e., chat bots). However, this mechanism has not been previously applied to time-series and security data. The intuition that the soft-alignment can help to capture more accurate hidden state representations motivated us to experiment with AS2S.
Our model is most similar to \cite{2015neural}. Fig. \ref{fig:attn_rnn} shows the AS2S network architecture. The encoder and decoder are both Recurrent Neural Networks (RNN) with a Long Short Term Memory (LSTM) cell. The encoder network is defined as:
\begin{equation}
h_t = f_{enc}(x_t, h_{t-1}),
\end{equation}
where $h_t$ is the hidden state and $x_t$ is the input at timestamp $t$; $f_{enc}$ is the LSTM cell, which could be multi-layered. At each timestamp the encoder RNN takes the previous hidden state $h_{i-1}$ and the current window $x_i$ as the inputs and outputs the current hidden state $h_i$. The decoder network is defined in a similar way but includes an attention component (network), which aims to capture the alignment between the input and output sequences.
\begin{equation}
s_i = f_{dec}(x_{i+1}, s_{i+1}, c_i),
\end{equation}
where $x_i'$ is the $i$-th token in the output sequence, $s_i$ is the $i$-th hidden state, and $c_i$ is the context vector. Note that the sequence index is in the reversed order, e.g., from $x_k$ to $x_1$, which is a convention to empirically achieve better performance, i.e., the last token is easier to be decoded in early timestamps. Each context vector $c_i$ is computed by weighting the input hidden states.
\begin{equation}
c_i = \sum_{j=1}^{N} \alpha_{ij} h_j ,
\end{equation}
where $N$ is the sequence length, $i$ is the $i$-th timestamp of the decoder,
and $\alpha_{ij}$ is the weight for each hidden state $h_j$ from the encoder.
The weight $\alpha_{ij}$ is defined as
\begin{equation}
\alpha_{ij} = \frac{exp(a_{ij})}{Z},
\end{equation}
\begin{equation}
a_{ij} = attn(s_{i+1}, h_j),
\end{equation}
where $Z$ is the normalization term; $attn(.)$ is the attention network that takes the previous output state $s_{i+1}$ and $h_j$ as inputs, and outputs the logit capturing the alignment. The attention component allows, at each timestamp, the network to consider the weighted encoder states to make inferences. Similar to the previous two models, AS2S also has a parameter $p$, which is the number of hidden units used in the encoder and decoder. To build the representations of each input window, we design the loss function to be squared errors between the input and the output, i.e., it is a sequence-to-sequence auto encoder with attention.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figs/fig3.jpg}
\caption{AS2S architecture showing how X2' (second sequence element) is predicted}
\label{fig:attn_rnn}
\end{figure}
\section{Experimental Results} \label{sec:experiments}
To demonstrate the effectiveness of our method we use two real world
data sets, and compare our performance with the two selected
baseline models.
To evaluate the models, we randomly sample 15\% of the users in each
cluster as the test set and 85\% as the training set. We train all the
models for 50 epochs. We use a development set from within the
training set for model selection.
After features are learned, we use a simple anomaly detection algorithm based on $k$-nearest neighbors to compute an anomaly score. Then we vary the anomaly threshold to compute precision and recall.
The performance of the models, based
on Area Under Precision-Recall Curves (PR-AUC) computed from partial
labels, on the anomaly detection task is used as a proxy for feature
learning.
\subsection{Data Set}
We refer to the two data sets as NETFLOW and CERT.
\subsubsection{NETFLOW}
The NETFLOW data set is from \cite{shiravi2012toward} and is converted into netflow format \cite{hofstede2014flow}, a commonly-used data source in the IT security industry. This data set spans 7 days and contains network-level information.
Three types of features — simple counts, bitmap, and top-K — are extracted. The features are directional. Half of the features are for incoming traffic and half for outgoing. For example, in count features, there are 60 features in total, 30 for outgoing traffic and the other 30 for incoming traffic. The bitmap features aggregate the flag bits used in the window.
For the top-k features, we empirically pick $k$ equal to 5. Note that since NETFLOW does not contain information like user profiles, we can only use a subset of the training features as the cluster features. In this case, count and bitmap are selected as the clustering features, since we recognize that they are more related to user behavior. In total, we identified 108 features for training.
\subsubsection{CERT}
The CERT data set
\cite{2016insider} is quite large comparatively and has 516 days of
data with application-level information and user profiles.
The features we use are similar to
\cite{tuor2017deep} but with some simplifications. The CERT data set
contains
many application details and user profiles data as well.
As a result, we can extract more features in this data set
than in the NETFLOW data set. For clustering, we use the categorical
information from user profiles, e.g., user roles, current projects,
teams, and so on, as the features, and the discrete indices as their
values. For training, 352 count features are selected.
\subsection{NETFLOW Results}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth, height=0.3\textwidth]{figs/fig4.png}
\caption{Determining the number of clusters K for NETFLOW}
\label{fig:nclusters_netflow}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth, height=0.3\textwidth]{figs/fig5.png}
\caption{t-SNE visualization of NETFLOW}
\label{fig:tsne_netflow}
\end{figure}
Fig. \ref{fig:nclusters_netflow} shows the quality of clustering as
measured by the Silhouette Coefficient (SC) with respect to the number
of clusters, $k$ of k-means. We can see that $k=2$ gives us the best SC
score. We can get an intuitive validation of this clustering result by
looking at Fig. \ref{fig:tsne_netflow}, the t-SNE visualization of all
the data points. t-SNE usually provides a good
visualization of high-dimension data; this visualization shows a clear
margin between the two clusters and that one cluster is far
larger than the other. If we consider the labels of the data points,
we observe the smaller cluster (on left, called cluster 1) has a higher
proportion of anomalies (of 161 points, 35 are anomalous), which implies that clustering successfully
filters the anomalies to some extent. For cluster 0 (on right), it has an
extremely small number of anomalies (of 100 K points, 36 are anomalous), which makes the detection much harder. All methods perform poorly in detecting anomalies in cluster 0.
\begin{figure}[]
\centering
\includegraphics[width=0.45\textwidth,height=0.3\textwidth]{figs/fig6_2.png}
\caption{Anomaly detection results on the NETFLOW data set}
\label{fig:ad_netflow}
\end{figure}
Fig. \ref{fig:ad_netflow} presents the anomaly detection results for
cluster 1.
Up to recall of 60\%, AS2S provides higher precision, while at higher recall all the three methods perform similarly.
Overall AS2S outperforms the other two baselines based on AUC,
showing that temporal relationship and attention
mechanism help anomaly detection in this scenario.
\subsection{CERT Results}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth, height=0.3\textwidth]{figs/fig8.png}
\caption{Determining the number of clusters K for the CERT data set}
\label{fig:nclusters_cert}
\end{figure}
Fig. \ref{fig:nclusters_cert} shows the quality of clusters as
measured by the SC score with respect to the number of clusters,
$k$. $k=7$ is selected as it gives us the best SC
score.
The number of data points in the clusters are
distributed quite uniformly. Only one cluster is relatively a bit
larger.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth, height=0.3\textwidth]{figs/fig10.png}
\caption{Anomaly detection results for cluster 5 of CERT.}
\label{fig:ad_cert}
\end{figure}
In the CERT data set, there are only 47 anomalies out of nearly 1M data
points. When we delve into the distribution of these anomalies, we
observe that only 3 clusters (cluster 0, 5, and 6) have anomalies, and
only cluster 5 has enough anomalies (32 anomalies) to provide
statistically meaningful results. Therefore, in
Fig. \ref{fig:ad_cert}, we only show the results for cluster 5. AS2S
has the highest area under the precision-recall curve, followed by AE
and then PCA. The results show that, similar to the NETFLOW
results, AS2S is able to detect anomalies better than the
baselines.
\vspace{-0.5em}
\subsection{Discussion}
Feature learning in images and natural languages is obvious for
humans. Learned image features like texture, edges, objects require no
explanations. However, the same is not true for security data. Here we
use performance on an anomaly detection task as a proxy for
quality of feature learning, quantified by PR-AUC. As the AS2S
model was trained, we wanted to explore if the features learned
improved. Note that PR-AUC
already captures this quantitatively for all the final models.
Fig. \ref{fig:scatter_netflow} shows the scatter plots of the hidden
layer outputs for AS2S projected onto two dimensions.
It is from
cluster 1 in the NETFLOW test data set during training.
Out of 21 points, 5 are anomalous in this test set,
so we can clearly see the changes during training. In the early epoch
(top), all the data points distribute randomly. In epoch 15 (middle)
some points aggregate to the left and some to the right. In epoch 24
(bottom), we can see that 3 anomalies aggregate to the right, which is
good. However, some other anomalies are on the left together with the
normal points. This means that, on one hand, some anomalies have clear
properties and can be identified by AS2S; on the other hand, some are
similar to the normal nodes, and therefore are unlikely to be
detected. These plots demonstrate the effectiveness of AS2S to
progressively learn more meaningful representations of NETFLOW for
detecting anomalies without any labels.
\begin{figure}[htb]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.42\textwidth]{figs/fig71.png}
\end{subfigure}%
~
\begin{subfigure}
\centering
\includegraphics[width=0.42\textwidth]{figs/fig72.png}
\end{subfigure}
~
\begin{subfigure}
\centering
\includegraphics[width=0.42\textwidth]{figs/fig73.png}
\end{subfigure}
\caption{Scatter plots of features learned with AS2S model from the test set of cluster 1 in NETFLOW. From top to bottom, the plot is from epoch 1, 15, and 24 respectively. The anomalies are in red and the normal data points are in blue.}
\label{fig:scatter_netflow}
\end{figure}
We use a precision-recall curve instead of a ROC curve, since the data is
highly skewed (as expected because anomalies are rare), and ROC curves
are misleading is such cases. Furthermore, precision, which is the
fraction of true positives (out of all positives marked by a model),
directly quantifies the overhead on a security operations center
analyst. While ideally one would prefer both high precision and high recall,
for a particular classifier higher precision is usually preferred to
prevent alarm fatigue (and since in practice a large number of
classifier are simultaneously deployed, the recall across the entire
system may still be good).
\vspace{-0.5em}
\section{Conclusions}
We explored multiple methods for unsupervised and self-supervised
feature learning for security data. To the best of our knowledge, this
is the first attempt to apply the attention mechanism (AS2S) in
security anomaly detection. We demonstrated that AS2S generally performs well
for anomaly detection, especially for NETFLOW data, outperforming PCA
and AE.
However, anomaly
detection for clusters with extremely small number of anomalies is
difficult for all of the models we experimented with.
These factors reveal the difficulties of unsupervised and self-supervised
feature learning from security data and further research is required to
address them.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,000 |
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The majority of the experts and sector experts claim that recurring learning is amongst the essential facets that might ensure successful outcomes. You can also discover a selection of student worksheets for all subject areas and for a variety of learning center activities. Most people uncover that it's tough to specify premium management goals.
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The handbook will reveal to you simply how to apply and methods to obtain gotten approved for a car loan alteration. A number of the standards include if you lag or otherwise on your home mortgage. Furthermore, you're going to be given the alternative to update whenever you open the expression invoice.
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After that it's feasible to observe where that goal suits your economic plan. Possessing the strategy in the really top place can assist you stick with your targets. It's the very best time of the fiscal year because the tune goes.
As a result of this, you will probably observe the length of various tasks to transform too. Taking the opportunity to make an effective routine inevitably saves time. It's feasible to also state the activities planned for the entire weekend.
All you may need to do is follow the next simple procedures. The main advantage of producing your own is the capability to personalize your everyday organizer. The team believes that the key to accomplishing real wellbeing and also elite efficiency starts with creating a durable as well as useful core. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,620 |
Q: Lodash - Find deep in array of object I have an array of an object like this
[
{
'a': 10,
elements: [
{
'prop': 'foo',
'val': 10
},
{
'prop': 'bar',
'val': 25
},
{
'prop': 'test',
'val': 51
}
]
},
{
'b': 50,
elements: [
{
'prop': 'foo',
'val': 30
},
{
'prop': 'bar',
'val': 15
},
{
'prop': 'test',
'val': 60
}
]
},
]
What I need is sum the property Val when prop is foo.
So, I have to search through elements and get all objects where prop is foo. With this objects, I should sum the val property.
I tried to use many combinations of _.find, _.pick and so on, but I don't get the right result. Can someone help me?
A: I've created library that you can use: https://github.com/dominik791/obj-traverse
findAll() method should solve your problem. The first parameter is a root object, not array, so you should create it at first:
var rootObj = {
name: 'rootObject',
elements: [
{
'a': 10,
elements: [ ... ]
},
{
'b': 50,
elements: [ ... ]
}
]
};
Then use findAll() method:
var matchingObjects = findAll( rootObj, 'elements', { 'prop': 'foo' } );
matchingObjects variable stores all objects with prop equal to foo. And at the end calculate your sum:
var sum = 0;
matchingObjects.forEach(function(obj) {
sum += obj.val;
});
A: Here's a one liner solution to this problem:
const _ = require('lodash')
let deepFind = (JSONArray, keyPath, keyValue) => _.find(JSONArray, _.matchesProperty(keyPath, keyValue))
let JSONArray = [{a:1, b:2, c:{d: "cd"}}, {a:3, b:4, c:{d: "ef"}}, {a:3, b:4, c:[{d: "ef"}]} ]
console.log(deepFind(JSONArray, "c.d", "cd"))
// {a:1, b:2, c:{d: "cd"}}
console.log(deepFind(JSONArray, "b", 4))
//{a:3, b:4, c:{d: "ef"}}
console.log(deepFind(JSONArray, ['c', 'd'], "cd"))
//{a:1, b:2, c:{d: "cd"}}
console.log(deepFind(JSONArray, 'c[0].d' /* OR ['c', '0', 'd']*/, "ef"))
//{a:1, b:2, c:{d: "cd"}}
A: Here's a solution that flattens the elements and then filters the result to get the required elements before summing the val property:
var result = _.chain(data)
.map('elements') // pluck all elements from data
.flatten() // flatten the elements into a single array
.filter({prop: 'foo'}) // exatract elements with a prop of 'foo'
.sumBy('val') // sum all the val properties
.value()
Chaining is a way of applying a sequence of operations to some data before returning a value. The above example uses explicit chaining but could be (maybe should be) written using implicit chaining:
var result = _(data)
.map('elements')
.flatten()
.filter({prop: 'foo'})
.sumBy('val');
A: Disclosure: I am the author of that library
You can use _.eachDeep method from deepdash:
let sum = 0;
_.eachDeep(obj, (value, key, parent) => {
sum += (key == 'prop' && value == 'foo' && parent.val) || 0;
});
here is an example for your case
A: You can loop through the data and get the sum.Try this:
var arr=[
{
'a': 10,
elements: [
{
'prop': 'foo',
'val': 10
},
{
'prop': 'bar',
'val': 25
},
{
'prop': 'test',
'val': 51
}
]
},
{
'b': 50,
elements: [
{
'prop': 'foo',
'val': 30
},
{
'prop': 'bar',
'val': 15
},
{
'prop': 'test',
'val': 60
}
]
}
];
var sum=0;
for(x in arr){
for(i in arr[x]['elements']){
if(arr[x]['elements'][i]['prop'] == 'foo'){
sum=sum+arr[x]['elements'][i]['val'];
}
}
}
console.log(sum);
USING FILTER:
var arr=[
{
'a': 10,
elements: [
{
'prop': 'foo',
'val': 10
},
{
'prop': 'bar',
'val': 25
},
{
'prop': 'test',
'val': 51
}
]
},
{
'b': 50,
elements: [
{
'prop': 'foo',
'val': 30
},
{
'prop': 'bar',
'val': 15
},
{
'prop': 'test',
'val': 60
}
]
}
];
var sum=0;
arr.filter(function (person) {
person['elements'].filter(function(data){
if (data.prop == "foo"){
sum=sum+data.val;
}
});
});
console.log(sum);
A: here the solution that worked for me if anyone needs it. but my case was a bit different, I needed to find an array that matches the id.
here a flow function which is not really nesesary but it does make the code easier to read
const flow = functions => data =>
functions.reduce((value, func) => func(value), data);
const arr = [
{
'a': 10,
elements: [
{
'prop': 'foo',
'val': 10
},
{
'prop': 'bar',
'val': 25
},
{
'prop': 'test',
'val': 51
}
]
},
{
'b': 50,
elements: [
{
'prop': 'foo',
'val': 30
},
{
'prop': 'bar',
'val': 15
},
{
'prop': 'test',
'val': 60
}
]
},
]
const result= _.flow([
data => _.map(data, 'elements'), // get all elements from the array
data => _.flatten(data), // create a singler array
data => _.find(data, { val: 65 }) // find the needed value in array
// you can keep on adding logic here depending on your need
])(arr); // the inital array
if you dont want to use the flow function it can also be done like that I think
I didnt test the code but it should work
const results = _.find(_.flatten(_.map(arr, 'elements')), { val: 65 });
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 789 |
{"url":"https:\/\/www.tutorialspoint.com\/minimum-toggles-to-partition-a-binary-array-so-that-it-has-first-0s-then-1s-in-cplusplus","text":"Minimum toggles to partition a binary array so that it has first 0s then 1s in C++\n\nC++Server Side ProgrammingProgramming\n\nProblem statement\n\nGiven an array of n integers containing only 0 and 1. Find the minimum toggles (switch from 0 to 1 or vice-versa) required such the array the array become partitioned, i.e., it has first 0s then 1s.\n\nExample\n\nIf arr[] = {1, 0, 0, 1, 1, 1, 0} then 2 toggle is required i.e. toggle first one and last zero.\n\nAlgorithm\n\n\u2022 If we observe the question, then we will find that there will definitely exist a point from 0 to n-1 where all elements left to that point should contains all 0\u2019s and right to point should contains all 1\u2019s\n\u2022 Those indices which don\u2019t obey this law will have to be removed. The idea is to count all 0s from left to right.\n\nExample\n\nLive Demo\n\n#include <bits\/stdc++.h>\nusing namespace std;\nint getMinToggles(int *arr, int n) {\nint zeroCnt[n + 1] = {0};\nfor (int i = 1; i <= n; ++i) {\nif (arr[i - 1] == 0) {\nzeroCnt[i] = zeroCnt[i - 1] + 1;\n} else {\nzeroCnt[i] = zeroCnt[i - 1];\n}\n}\nint result = n;\nfor (int i = 1; i <= n; ++i) {\nresult = min(result, i - zeroCnt[i] + zeroCnt[n] - zeroCnt[i]);\n}\nreturn result;\n}\nint main() {\nint arr[] = {1, 0, 0, 1, 1, 1, 0};\nint n = sizeof(arr) \/ sizeof(arr[0]);\ncout << \"Minimum toggles = \" << getMinToggles(arr, n) << endl;\nreturn 0;\n}\n\nWhen you compile and execute above program. It generates following output \u2212\n\nOutput\n\nMinimum toggles = 2\nPublished on 20-Dec-2019 10:10:05","date":"2021-11-29 03:05:59","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\": 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\": 0.23330049216747284, \"perplexity\": 2333.19978702495}, \"config\": {\"markdown_headings\": false, \"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-2021-49\/segments\/1637964358685.55\/warc\/CC-MAIN-20211129014336-20211129044336-00349.warc.gz\"}"} | null | null |
Matt Braddock, VC and bar, is a fictional World War II bomber pilot who first appeared in prose adventures in the story paper The Rover in 1952, and later as a comic strip in The Victor (1961–83) and Warlord (1974). Some of his stories were published in book form as I Flew With Braddock.
Braddock was known for his fearless nature, superb piloting skills and no-nonsense attitude. He had no time for petty rules and regulations, and remained at the rank of sergeant, refusing to be promoted to an officer rank. However this didn't stop him from standing up to incompetent superiors, or defending other enlisted men from overzealous courts-martial. He spent almost as much time clashing with superior officers as the Germans, and on two occasions in I Flew With Braddock he came fairly close to being court-martialled for assaulting a superior officer (although one incident was a misunderstanding and the other was under severe provocation). The narrator of these stories was his hero-worshipping navigator, George Bourne (a Dr. Watson-type narrator, a relatively sophisticated device for juvenile fiction).
Braddock was so highly regarded that he had carte blanche to pick his own flight crew, and was frequently called upon to advise high-ranking commanders of the RAF. Braddock flew several types of aircraft, but his most prominent command was the Lancaster Bomber "F Fox". At the start of I Flew With Braddock the crew was Braddock and Bourne, co-pilot and flight engineer 'Ham' Hancox, bomb aimer Tom Tanner, radio operator 'Nicker' Brown, mid-upper gunner 'Hoppy' Robinson, and tail gunner Les Howe. Les Howe clashed with Braddock early on and was replaced by 'Baa' Lamb, and later 'Hoppy' Robinson was badly injured in a raid and replaced by Arthur Atkins.
Some of Braddock's more outstanding exploits included:
Sinking a U-boat with a rocket-armed Mosquito fighter-bomber.
Bombing Peenemünde, severely setting back German rocketry experiments.
Successfully traversing the Alps to attack an Italian port, after the entire raid had been scrubbed due to hazardous weather.
Shooting down a German fighter with a flare gun.
Sinking the Degen, a German aircraft carrier. This was notable in that Braddock successfully bombed the ship after his plane was damaged by anti-aircraft fire, destroying the bomb sight and knocking out his bomb aimer. Braddock himself took the responsibility of dropping the bomb, and managed to score a direct hit with the naked eye.
I Flew With Braddock also included some genuine episodes in the British bomber offensive against Germany e.g. the first thousand bomber raid on Cologne, and developments such as the GEE navigation system, Pathfinder units and the "Pink Pansy" incendiary bomb.
Braddock, in addition to his piloting abilities, was a skilled boxer and champion level darts player. In the stories set in the pre-war period, he is portrayed as a steeplejack who took flying lessons and then joined the auxiliary air force.
The author of these stories was not clearly stated (I Flew With Braddock just gave the author as "George Bourne", the fictional narrator) but is believed to be Gilbert Lawford Dalton.
References
External links
Matt Braddock life history from Rover comics
Sample Matt Braddock Story & a 2nd
Sample comic page and brief character profile
Red Dagger retrospective with brief bio of Braddock
Guide to the stories in The Victor
Matt Braddock
British comics characters
Fictional recipients of the Victoria Cross
Fictional military sergeants
Fictional World War II veterans
Matt Braddock
Novels set during World War II
Comics set during World War II
Literary characters introduced in 1952
Male characters in comics
1961 comics debuts
Comics characters introduced in 1961
1983 comics endings
Fictional Royal Air Force personnel | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,364 |
Clacton-on-sea, Essex
Set by a spectacular stretch of sandy coastline that runs for 7 miles...
Belsize Avenue, Jaywick, Clacton-on-sea, Essex, CO15 2LF
Interested in visiting Martello Beach Holiday Park to find out more?
What's new on park
Martello Beach Holiday Park - Caravan Holidays
Brand new complex and indoor swimming pool
Direct beach access and a short drive from the seaside town of Clacton-On-Sea
Lively entertainment programme the whole family will love
If you're wondering how Martello Beach Holiday Park got its name, you won't have to look far. It's adjacent to the gloriously sandy Martello Beach, a golden shoreline that stretches for 7 miles - so you should find plenty of space to build the ultimate sandcastle!
The holiday park boasts a brand-new, state-of-the-art entertainment and leisure complex with an arcade, bar, restaurant and entertainment venue, and heated indoor swimming pool (with a thrilling double lane slide)! Moving outside the venue, there's an adventure playground and sports court.
Tasty food and drink are available throughout the day, and in the busy summer periods, there are fantastic events, activities, clubs and shows to entertain young and old.
Located just a 3-mile drive from the bustling, seaside resort of Clacton-on-Sea, with its famous pleasure pier, Martello Beach is ideally situated for a brilliantly fun family holiday. Local attractions include Colchester Zoo, Holland Haven Country Park, and the East Anglian Railway Museum – it's not just for trainspotters! And with so much coastline to explore, a swim in the sea can't be recommended enough.
There are so many great reasons why guests choose to return again and again, so pack your beach towels, sun cream and sunglasses, and join us at Martello Beach holiday park – and enjoy our brand-new venue!
Things to do at Martello Beach Holiday Park, Essex
Great family entertainment is on offer at selected times of the week, from a variety of fantastic visiting cabarets to fun time with Crazy Club. Whatever your age, you won't be able to resist the timeless talent of our fabulous Park Stars!
Throughout the year we welcome special guests and regular cabaret acts to perform in the entertainment venue.
Daytime Kids Club
Running during School Holiday periods only, and exclusively for newborns to 14 year olds, the Entertainment team are on hand during the day to organise fun and games in a child safe environment.
From newborns to 14 year olds, the park team are on hand to help entertain your kids throughout the day.
What's New at Martello Beach Holiday Park, Essex
View Gallery 1/8
New-build venue and indoor swimming pool
A multi-million-pound clubhouse and indoor swimming pool is coming to Martello Beach Holiday Park in Clacton-on-Sea, servicing guests at both Martello Beach and St Osyth Holiday Park. This state-of-the-art entertainment and leisure venue includes a new reception area, arcade, bar, restaurant and entertainment venue, and we've even splashed out on a heated indoor swimming pool with a thrilling two-lane flume ride, which will all be ready for you to enjoy early 2023!
Search Holidays
Facilities at Martello Beach Holiday Park
All Weather Sports Pitch
Pet-friendly Accommodation Available
Neighbouring Park Facilities
Fishing Lake*
† Please note all activities (including swimming sessions) must be pre-booked. *Additional charges apply.
What's on at Martello Beach Holiday Park
Neighbouring Park Use
Public Parks and Playgrounds
Special Offers at Martello Beach Holiday Park
Martello Beach Holidays and Short breaks
Martello Beach, Essex
Martello Beach Holiday Park
Fri, 10th March 2023 - Mon, 30th October 2023
Email: martellobeach@parkholidays.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,833 |
package org.uberfire.ext.preferences.client.form;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import javax.annotation.PreDestroy;
import javax.enterprise.context.Dependent;
import javax.enterprise.event.Event;
import javax.inject.Inject;
import org.jboss.errai.common.client.api.Caller;
import org.jboss.errai.common.client.api.RemoteCallback;
import org.jboss.errai.ioc.client.container.SyncBeanManager;
import org.uberfire.client.mvp.UberElement;
import org.uberfire.ext.preferences.shared.PreferenceStore;
import org.uberfire.workbench.events.NotificationEvent;
@Dependent
public class PreferencesEditorFormPresenter {
public interface View extends UberElement<PreferencesEditorFormPresenter> {
String getNoChangesMessage();
String getPreferencesSavedSuccessfullyMessage();
String getUnexpectedErrorDuringSavingMessage( String details );
}
private final SyncBeanManager beanManager;
private final View view;
private final Caller<PreferenceStore> preferenceStoreCaller;
private final Event<NotificationEvent> notification;
private List<PreferenceItemEditorPresenter> preferencesEditors;
@Inject
public PreferencesEditorFormPresenter( final SyncBeanManager beanManager,
final View view,
final Caller<PreferenceStore> preferenceStoreCaller,
final Event<NotificationEvent> notification ) {
this.beanManager = beanManager;
this.view = view;
this.preferenceStoreCaller = preferenceStoreCaller;
this.notification = notification;
}
public void setManagedKeys( final List<String> managedKeys ) {
final PreferencesEditorFormPresenter presenter = this;
preferenceStoreCaller.call( new RemoteCallback<Map<String, Object>>() {
@Override
public void callback( final Map<String, Object> preferences ) {
preferencesEditors = new ArrayList<>();
for ( final Map.Entry<String, Object> preference : preferences.entrySet() ) {
final PreferenceItemEditorPresenter preferenceEditor = beanManager.lookupBean( PreferenceItemEditorPresenter.class ).newInstance();
preferenceEditor.setPreferenceKey( preference.getKey() );
preferenceEditor.setPersistedPreferenceValue( preference.getValue().toString() );
preferencesEditors.add( preferenceEditor );
}
view.init( presenter );
}
} ).search( managedKeys );
}
public void save() {
Map<String, String> preferencesToPersist = getPreferencesToPersist();
if ( !preferencesToPersist.isEmpty() ) {
preferenceStoreCaller.call( voidReturn -> {
notification.fire( new NotificationEvent( view.getPreferencesSavedSuccessfullyMessage(), NotificationEvent.NotificationType.SUCCESS ) );
}, ( o, throwable ) -> {
notification.fire( new NotificationEvent( view.getUnexpectedErrorDuringSavingMessage( throwable.getMessage() ), NotificationEvent.NotificationType.ERROR ) );
return false;
} ).put( preferencesToPersist );
} else {
notification.fire( new NotificationEvent( view.getNoChangesMessage(), NotificationEvent.NotificationType.INFO ) );
}
}
public void undoChanges() {
for ( final PreferenceItemEditorPresenter preferenceEditor : preferencesEditors ) {
preferenceEditor.undoChanges();
}
}
protected Map<String, String> getPreferencesToPersist() {
Map<String, String> preferencesToPersist = new HashMap<>();
for ( final PreferenceItemEditorPresenter preference : preferencesEditors ) {
if ( preference.shouldBePersisted() ) {
final String key = preference.getPreferenceKey();
final String newValue = preference.getNewPreferenceValue();
preferencesToPersist.put( key, newValue );
preference.setPersistedPreferenceValue( newValue );
}
}
return preferencesToPersist;
}
public List<PreferenceItemEditorPresenter> getPreferencesEditors() {
return preferencesEditors;
}
public View getView() {
return view;
}
@PreDestroy
public void destroy() {
if ( preferencesEditors != null ) {
for ( PreferenceItemEditorPresenter item : preferencesEditors ) {
beanManager.destroyBean( item );
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,981 |
MILLER CENTER STUDIES ON THE PRESIDENCY
MARC J. SELVERSTONE, EDITOR
_Miller Center Studies on the Presidency_ is a series of original works that draw on the Miller Center's scholarly programs to shed light on the American presidency past and present.
THE FIRST YEAR PROJECT
# American Dreams
# Opportunity and Upward Mobility
Edited by Guian McKee and Cristina Lopez-Gottardi Chao
University of Virginia Press
Charlottesville and London
# Contents
Introduction
Longing for Uplift: Americans Want to Believe Everyone Has a Fair Chance to Succeed
Peter Wehner
Opportune Moment: The Time Is Right for the Next President to Pursue an Opportunity Agenda
William A. Galston
Pre-K Prerequisite: Ensure That Every Poor Child in America Gets Two Years of Preschool
Robert C. Pianta
City Power: The Next President Can Help Cities Boost More Workers into the Middle Class
Richard Schragger
The Long Game: The Next President Must Lay the Foundation for a Next-Generation Economy
Margaret O'Mara
Moving Up: The Next President Can Restore Confidence in the American Dream
Melody Barnes
Presidential Precedents: Oral Histories Show How Other Presidents Dealt with Opportunity Issues
Michael Nelson
Contributors
Notes
# Introduction
Concerns about opportunity and mobility were defining issues in the 2016 presidential campaign, and in many ways explain the unexpected election of Donald J. Trump. Since the 1970s, America's middle class has experienced a decline in size and overall wealth, a challenge made worse by the global recession of 2008–9. Economic shifts related to globalization and deindustrialization have also been at play. Together these factors have had significant ramifications for the nation's sense of economic and social security, and for the bedrock of America's working class. As the nation enters a new political phase in 2017, anxiety about the future of the American Dream will remain a central concern. How President Trump addresses this challenge—beginning in his first year in office—will be critical to the success of his presidency.
Many Americans now believe that their children face a future of vastly diminished expectations. The American promise, that a good life can be achieved through hard work and playing by the rules, no longer feels assured to many. This sense is particularly acute among millennials, with only about half believing the American Dream is still alive and well. Faced with the rising cost of housing and education, decreased social and economic security, racial inequality, a competitive global economy, and changes in the very nature of work, they may not achieve the standard of living enjoyed by their parents and grandparents, let alone do better.
According to a 2015 Pew study, "Americans in middle-income households have lost significant ground in the last four decades," including their status as the nation's "economic majority." More specifically, between 1970 and 2014 the share of aggregate income of middle-class households fell by 19 percent, making the pain of this sector very real. This middle-class contraction has been particularly acute in major metro areas, where nine out of ten metropolitan centers experienced declines in their share of middle income earners between 2000 and 2014. Yet rural America has done even worse, a phenomenon that helped fuel Donald Trump's rise to power. During this same period, an additional complicating factor was the nation's unprecedented increase in wealth for upper-income households, as well as growth in the total number of lower-income families.
Unfortunately, the political discourse about such issues—much less the policies proposed and implemented—has failed to address these challenges. Our leaders have proved unwilling to ask hard questions about why the American Dream seems to serve fewer and fewer Americans, and about what can be done to secure the dream for the next generation. One result has been the political upheaval and bitter polarization of 2016.
In designing the First Year Project's sixth volume, _Opportunity and Upward Mobility,_ we set out to approach this issue in new and revealing ways. The seven essays that result pursue this goal from a range of perspectives.
The volume operates on two premises: first, that the same broad set of public policies has shaped the differing fates of the middle class and the poor, and second, that deep structural changes in the economy have increasingly made the insecurity that the poor have always known familiar to the middle class as well. These changes include deindustrialization, the emergence of disruptive technologies, and globalization—processes that have themselves been shaped by policy choices. Add to that profound demographic shifts, where by 2044 white Americans will no longer be in the majority, and this new reality will undoubtedly unleash new dynamics on the pursuit of the American Dream. Poverty rates vary widely by race and ethnicity, with the African American poverty rate hovering around 24 percent, compared to 21 percent for Hispanics and 9 percent for whites, and these differences create both unique constraints and opportunities that must be taken into account.
How President Trump manages these challenges, and his ability to garner support from a diverse representation of the American populace, will be hugely important to his success, and legitimacy, in the first year and beyond. The essays in this volume offer a range of ideas about how the new president can pursue that goal. They include lessons from history and solutions that range from education and tax reform to the role of cities, federalism, and crafting a comprehensive bipartisan opportunity agenda.
Michael Nelson draws on the Miller Center's Presidential Oral Histories to show how past presidents—including Jimmy Carter with the Food Stamp Act, George H. W. Bush's Americans with Disabilities Act, and Bill's Clinton's attempt at health care reform—succeeded and sometimes failed when they sought new ways to promote varying forms of opportunity.
Richard Schragger suggests that the new president can deploy the federalist system of governance to help cities and metropolitan regions become sites of opportunity, economic creativity, and resurgence—and to build on the progress that many American cities have already made.
Margaret O'Mara touches on a related theme in a deeply historical essay that traces how past presidential actions have created the conditions for entrepreneurship and innovation in local and regional economies (often unwittingly). She argues for investments in higher education, income security, and new technologies.
Other essays in the volume emphasize specific policy options available to the new president. Robert Pianta calls for enrolling every low-income child in the United States in two years of high-quality preschool education. Melody Barnes recommends data-driven, evidence-based first-year legislation to develop social capital and promote wealth creation for both the poor and the middle class.
More than the specific ideas they propose, however, the essays provide something that has been all too rare: a sense of optimism and of possibility. William Galston and Peter Wehner, who served in the Reagan, Bush 41, and Bush 43 administrations, offer two of the most sweeping essays in the volume, assessing the options for increasing mobility from center-left and center-right perspectives, respectively. Writing from opposite sides of the partisan divide, both independently call on President Trump to pursue what they term an "opportunity agenda." While they do not agree on every point of what such an agenda might look like, there is nonetheless significant overlap in their ideas.
In a similar fashion, Barnes urges the new president to build a bipartisan cabinet composed of respected figures from both parties. Such opportunities for bipartisan agreement, for overcoming the division that has so paralyzed policymaking and politics in recent years, pervade the First Year Project. In some cases, what is striking is common policy proposals; in others, it is the deeply shared underlying values that lead to common goals even when specific policy mechanisms differ. These values have often seemed to be under stress in recent years, and especially so in the first months of the Trump administration. These essays offer a measure of reassurance that the American project continues.
The divides that face the United States are real, but this opportunity and upward mobility volume provides a rare opportunity to reorient debates about the American Dream in new and more productive directions. This volume is a reminder to all of us that there are still ties that bind us, regardless of party, and that those ties could lead to constructive policy action and a better collective future for us all.
# Longing for Uplift
# Americans Want to Believe Everyone Has a Fair Chance to Succeed
Peter Wehner
As the president-elect thinks about the first year in office, many issues will be competing for time and attention. I would urge the incoming chief executive to focus as much of his or her energy and political capital as possible on constructing an opportunity agenda, by which I mean an agenda that allows people to rise in life based on merit, talent, and hard work; an agenda that doesn't guarantee success but does ensure the system isn't rigged, and that everyone has a fair chance to succeed and people have a chance to improve their living standards and flourish regardless of the circumstances of their birth.
The need is urgent, the time and political circumstances are right, and the nation is longing for uplift, for a positive agenda, for a way forward. The incoming president can provide it.
## Historical Context
In laying out the case for an opportunity agenda, it might be best to begin at the beginning. The concept of an opportunity society, the chance to pursue the American Dream, is fundamental to America's self-conception and has deep roots in American history.
America has always been a beacon for people looking for a fresh start, and has always offered to those who do not begin the race of life with an advantage—the hope that they can nonetheless succeed and prosper. The animating ideals of our founding documents include the proposition that we are all created equal and endowed by our Creator with "certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." The greatest interpreter of the founders, Abraham Lincoln, spoke about a government "whose leading object is to elevate the condition of men—to lift artificial weights from all shoulders, to clear the paths of laudable pursuit for all, to afford all an unfettered start and a fair chance, in the race of life." So this idea of opportunity is one that has deep resonance with the American people; it is like a thread running through our history.
## Where Things Stand
For a variety of complicated reasons, we have badly fallen short of this "leading object" of government in our time. Some of the reasons have to do with structural economic changes (e.g., globalization and advances in technology), community dysfunction, and family instability. Some have to do with the failure to modernize our institutions to adapt to contemporary challenges. And some have to do with misguided policies and missed opportunities by both parties.
Whatever the precise explanations and apportionment of blame, here's what we know. Many European countries now have as much social mobility as, and more opportunity than, the United States. Today a child's future depends on parental income more in America than it does in Canada and parts of Europe.
Middle-class Americans have been working harder and longer hours, yet their wages are stagnant. (The median family income, adjusted for inflation, was lower in 2013 than it was in 1989, and in 80 percent of the counties in America, the median income was lower in 2016 than it was at the turn of the twenty-first century.) Ordinary Americans have experienced a rise in the cost of living, particularly in the areas of health care (which has doubled in constant dollars since 1988) and college tuition (which has tripled over the last three decades). In sum, the middle class in America has been losing ground for a generation.
Younger workers are finding it more difficult than ever to launch. For example, in 2014 more than a third of Americans between the ages of eighteen and thirty-one were living in their parents' home.
As for the poor, according to recent data, 14 million people live in extremely poor neighborhoods, 5 million more than before the downturn and more than twice as many as in 2000. Of those residents, 6.3 million were poor. "Put differently," according to one study, "13.5 percent of the nation's poor population faced the double burden of being poor in a very poor place"—an increase from the last decade. "If you are born poor [in America]," note Isabel Sawhill and Ron Haskins of the Brookings Institution, "you are likely to stay that way."
## Why a First-Year Priority?
There is clearly a cognitive dissonance between where we are and how we like to think of ourselves—as the land of opportunity. We are falling well short of our aspirations. As a result, Americans are feeling frustrated and uneasy, insecure and pessimistic. Only about a quarter of the American people believe the country is on the right track. Two-thirds of Americans believe it will be harder for them to realize the American Dream than it was for their parents.
These perceptions are grounded in certain realities—income inequality rising, social mobility stalled, and poverty entrenched. These difficulties require a national response, and there is a moral imperative to offer one.
There is also a political opening for the new president to act—both because new presidents are usually given a great deal of deference to enact their agenda and because Americans, after the bitter and brutal campaign of 2016, will be thirsting for the chief executive to govern in a serious way that addresses pressing problems. Indeed, the existence of an "opportunity crisis" in America helps explain the widespread distemper we see among Americans and which clearly bled into the campaign.
As the incoming chief executive thinks about how to approach this issue, he or she should take note of what the late Andrew Kohut of the Pew Research Center said: "What the public wants is not a war on the rich, but more policies that promote opportunity."
## The Elements of an Opportunity Agenda
When thinking about an opportunity agenda, the place to start is with economic growth. After a forty-year period of high growth—averaging 3.5 percent per year from 1960 to 2000–we have seen a rapid deceleration. During the president-elect's predecessor's two terms, there was not a single year of 3 percent GDP growth. Tepid growth is the enemy of an opportunity society.
Getting us on the right track requires several things, beginning with remaking a tax code that impedes growth by penalizing investment and innovation and encouraging cronyism. We need to broaden the base, flatten and lower the rates, clean out and get rid of many existing deductions—excluding mortgage interest and charitable giving—and lift burdens off middle-class families. (There remain legitimate concerns about the deficit and debt, which is why this approach—broadening the base and getting rid of some existing deductions—can help offset the revenue loss from lower and flatter rates.)
Immigration has always been part of the American recipe for growth, but can only remain so with serious reforms. We need to rethink our definition of family reunification, which is currently too expansive, and to emphasize visas for the high-skilled workers our economy requires. We should model our approach after those of Canada and Australia, whose immigration policies advance their economic interests. David Brooks of the _New York Times_ explains how we should approach the issue: "The immigration system should turn into a talent recruiting system, a relentless effort to get the world's most gifted and driven people to move to our shores."
We're experiencing what Walter Russell Mead calls "the energy revolution of the twenty-first century," which, he argues, has the capacity to be a genuine game-changer based on new discoveries and new technologies (hydraulic fracturing and other extraction technology). "The United States is better supplied with fossil fuels than any other country on earth," according to Mead. "Canada and the United States are each richer in oil than Iraq, Iran, and Saudi Arabia combined."
This energy revolution has the potential to be a jobs revolution, including substantial growth in high-income jobs for blue-collar workers. Alongside investments in alternative energy sources, the president-elect needs to pursue policies to keep this "energy revolution" going—including lifting the ban on American crude oil exports, allowing more fossil fuel exploration on federal lands, and encouraging investment in our oil and gas transportation infrastructure.
The kind of economic growth that reforms like these could encourage is an essential prerequisite for a revival of opportunity in America. But it is only a beginning. To help those most in need benefit from growth, much more is required.
Education must lie at the heart of a renewed opportunity agenda. Too few of our students are well served by our current education system, but students from poor families are suffering by far the most. So the members of our society most in need of a good education are stuck in the worst-run and most dysfunctional schools and denied any promising alternatives.
Some combination of public school reform and private school choice will be essential to ending the intergenerational cycle of poverty and failure. We need to permit more choice and variety, greater accountability and merit pay, more choice in school curricula, greater transparency, and expanded online learning opportunities and home schooling. We need to reform teacher preparation programs, empower principals, and do away with the policy of "last in, first out."
Chester E. Finn, Jr. and Michael Brickman argue for further innovations, such as choices among individual school courses and course providers, tax-sheltered scholarship programs to cover private school tuitions, and even debit card–style saving accounts that parents can tap for their children's education. K–12 education is not primarily a federal responsibility, but the federal government can help lift obstacles to such reforms, and this must be a high priority.
The federal government has a greater role in higher education, especially in light of its role in helping millions finance their tuition. A college degree remains a crucial ticket to the middle class. Yet the higher education system is fundamentally broken—outmoded, far too expensive, and much too difficult to access.
The goal needs to be to subject the higher education system to the kinds of consumer pressures that would compel it to offer better choices at lower costs. What does that mean concretely? Along with investments in skills training and alternatives to college, in higher education we need more flexibility in accreditation to let more players onto the field, more data about outcomes to enable consumers to make better-informed choices, and reforms of the student loan system to restrain the inflationary pressure that drives costs up and allows failures to survive. Rather than thinking about the federal government as a big bank that funds ever-higher tuition costs, we should think about it as an enabler of dynamism and constructive competition.
When it comes to helping less-skilled Americans who are finding it difficult to find meaningful work and raise a family, Michael Strain of the American Enterprise Institute makes a persuasive case for four proposals:
* • Expand earning subsidies (e.g., the earned income tax credit, to include childless workers) as a way to increase the rewards for working, increase the workforce participation rate, and reduce poverty
* • Dramatically expand child tax credits to help working families offset the burdens of paying (through taxes) for entitlement programs for the elderly by allowing them to keep more of what they earn.
* • Provide relocation subsidies to the long-term unemployed that will finance some significant part of the costs of moving to a part of the country with a better labor market.
* • Cut the profoundly regressive payroll taxes and gradually fund more old-age entitlement programs from general revenue in order to shift resources from the elderly to younger workers.
The eligibility rules for a number of our key public-assistance programs, especially Medicaid but also the earned income tax credit, are designed in ways that discourage higher-paying work, since benefits diminish or disappear abruptly as income rises. They therefore discourage investments in the kind of self-improvement often necessary to obtain such work. These and other social programs must be structured to encourage recipients to make more productive use of their talents.
Scott Winship of the Manhattan Institute recommends ways to extend the lessons of the 1990s welfare reform to the rest of the federal safety net by block-granting means-tested programs and sending them to the states, thereby encouraging independence, experimentation, and flexibility while requiring work and time limits for most—but not all—beneficiaries.
"Poverty produces family instability," the sociologist Robert Putnam has written, "and family instability in turn produces poverty." Estimates show that if America had the same percentage of single parents today as we did in 1970, the child poverty rate would be 30 percent less. Encouraging family formation and marital permanence is something government has a very poor record in doing. Many things have been tried; few have succeeded. But there are some steps worth taking.
At a bare minimum, we should remove marriage penalties from the tax code and disincentives to marry from welfare programs. It doesn't make sense to encourage behavior that is harmful to society and children. To narrow the "growing marriage divide" between less-educated and better-educated Americans, the scholars W. Bradford Wilcox, Nicholas H. Wolfinger, and Charles E. Stokes encourage experimenting with a range of public and private strategies. They claim that some relationship education programs—for example, the Oklahoma Marriage Initiative and Supporting Father Involvement—have improved the quality and stability of low-income parents' relationships and the emotional welfare of children whose parents have participated in them.
Local civic initiatives designed to strengthen family life already exist, such as First Things First in Chattanooga, Tennessee, a promising local civic initiative that provides education on marriage, fatherhood, and parenting. "Programs like this need to be scrutinized and, if they prove to be effective, replicated across the country," according to Wilcox, Wolfinger, and Stokes.
The same can be said about programs such as the Nurse-Family Partnership, in which nurses visit first-time, low-income mothers to provide information on nutrition and parenting as a way to increase school readiness of at-risk kids.
There are other areas to look at, including criminal justice reform, encouraging financial literacy and mentoring of low-income families, improving infant and child health, and supporting public service campaigns to discourage teen pregnancy and encourage what's been called the "success sequence," meaning young adults are encouraged to pursue education, work, marriage, and parenthood in that order. An agenda geared to enabling growth, improving education, and strengthening our safety-net programs would be a strong start on the path to greater opportunity.
## Concluding Thoughts
The United States is an exceptional nation, but the incoming president will soon be leading a country in which too many citizens are facing hardships and struggles. More than 45 million Americans live below the poverty line. For many others, the American Dream seems out of reach. They believe American politics is broken, opportunities are elusive, and the playing field isn't fair.
A few words to the president-elect: Your presidency should be about rebuilding America through economic growth and widespread prosperity to help us once again become a thriving, dynamic, forward-looking nation and to reform and modernize government at every level to help Americans create a better life for themselves and their children.
You should stand for the politics of the common good—and an agenda, like the one sketched out above, that is economically inclusive and morally compelling; that helps workers, families, and those who are struggling; and that smashes barriers to social mobility.
In advancing this agenda, keep several things in mind. First, the federal government's role isn't to ensure success; it is to give people a fair shot at success. Second, approach your task with energy but also with some degree of modesty in terms of what the federal government—and your administration—can achieve.
There is a great deal we still do not know and need to learn. So my last bit of counsel to you is to always be open to experimentation, to reexamine your assumptions and test your theories against reality, and to be ruthlessly empirical in your approach. Learn as you go.
John F. Kennedy, after defeating Richard Nixon in 1960, said, "The margin is narrow, but the responsibility is clear." So is yours. I wish you every success.
# Opportune Moment
# The Time Is Right for the Next President to Pursue an Opportunity Agenda
William A. Galston
## Introduction: The Case for an Opportunity Agenda
The purpose of this memorandum to the incoming chief executive of the United States is to persuade you to put an opportunity agenda at the center of the legislative program for your first year in office.
There are many reasons why you should do this. To begin, equal opportunity is America's promise. From the very beginning of our national existence, we have told ourselves and the world that here, unlike in so many other countries, where a person begins does not determine where that person ends up. One's religion doesn't matter, nor does race or ethnicity or parentage. It is the individual's talent, application, and character that will shape that person's destiny.
We have never entirely measured up to our own standards, of course. But over time we've gotten closer, both by tearing down barriers to the promise and by doing what was necessary to help everyone participate in it. We need to keep on moving down this road. Not doing so would be to stop short of fulfilling the promise we've made not to some, but to all.
This is the moral argument for the opportunity agenda. But there is a political argument as well. One of the clearest messages from the recently concluded election is that the American Dream is in peril. Too many Americans have come to doubt that their talent and character can yield a rising standard of living, no matter how hard they work. Worse, they have lost confidence in a better future for their children. Majorities now believe that their children's lives will be worse, not better, than their own. This is a dagger aimed at the heart of the American Dream.
There is also an economic argument. When opportunity is unequal, many Americans are prevented from putting their talents to best use, reducing productivity and economic output below the levels they would otherwise reach. When the U.S. labor force was growing rapidly, we could achieve high economic growth despite the drag of unequal opportunity. But in coming decades, our labor force will be expanding at only one-third the rate we experienced in the closing decades of the twentieth century. The price of preventing individuals from making their best contribution—not to mention excluding them from the labor force altogether—is rising steadily. Each individual will matter more than ever before.
And finally, there is an empirical argument. For some decades we have known that economic resources are distributed less equally in the United States than in most other advanced democracies. Counterbalancing this, we believed, was a higher degree of social mobility in the United States. Europe, we believed, was static and class-bound, while we were open and fluid. To become an American was to choose this package of inequality and mobility.
As recent research has made clear, however, our belief is mistaken. The odds of a person born in the United States to parents of low or modest means becoming well-off are lower than in many other advanced democracies. The proportion of children born on the bottom rung of the ladder who rise to the top as adults in the United States is 7.5 percent—lower than in the United Kingdom. (9 percent), Denmark (11.7 percent), and Canada (13.5 percent).
Behind these aggregate statistics are even more disturbing facts about differences among groups of Americans. African Americans born to families in the lowest fifth of the national income distribution have a 51 percent chance of ending up there, compared to a meager 3 percent chance of rising to the top. Of white Americans born at the bottom, by contrast, only 23 percent remain there as adults, while fully 16 percent rise to the top. Put differently, only 22 percent of African Americans born in the bottom fifth rise to the middle class or better as adults, compared to 58 percent of whites. When it comes to equal opportunity, we still have a way to go.
We have learned through painful experience that equal opportunity is not the product of an invisible hand. We need not only laws that establish and safeguard formal equality of opportunity but also policies that promote substantive equality of opportunity. Low-income families are less able to provide high-quality child care and education for their sons and daughters, for example, and are less likely to belong to networks that link young adults to jobs. If equal opportunity is to be more than a hollow promise, then government must act to level the playing field—not to produce equal outcomes, but to give all our children a fair chance to succeed.
## Why Opportunity Is Unequal: Pieces of the Puzzle
Understanding the multiple dimensions of unequal opportunity helps specify the building blocks of an opportunity agenda. The list includes the following:
* • Low-income families typically find it harder to provide the kind of nurturance that infants and toddlers need to thrive. On average, schools available to lower-income children are less likely to offer the quality of education and training these students need to succeed. This helps explain why the level of adult skills in the United States is well below the OECD average.
* • During the past three recoveries from economic downturns, economic growth was concentrated in fewer and fewer locations, mainly metropolitan areas with high levels of technology and professional services, while smaller towns and rural areas were left behind. While we are less aware of rural than urban disadvantage, it is becoming more prominent in the overall phenomenon of unequal opportunity.
* • Research has documented a decades-long decline in U.S. geographic mobility. Low-income adults in areas with poor job prospects are more likely to remain there than they were three decades ago. Some are stuck in houses they can't sell. Many lack the means to leave for other jurisdictions. Most have lived their lives within the confines of a single neighborhood and lack the connections and confidence to move. In cities, decades of underinvestment in metropolitan transportation have made it difficult for low-income people to get from their homes to work and back again.
* • The U.S. prison population per capita is five times the OECD average. A record of incarceration bars many young adults, mostly male and minority, from access to employment other than the most menial and episodic jobs.
* • Because the United States does less than many other advanced democracies to help women balance work and family responsibilities, the U.S. female labor force participation rate peaked in the late 1990s (60 percent in 1999) and has been declining ever since. In 2017, it stood at 57 percent.
* • Employment discrimination persists, though at lower levels than a generation ago. Studies have shown, for example, that employers are less likely to interview job candidates from racial and ethnic minorities than white candidates with identical qualifications.
* • Licensing requirements imposed by states make it more difficult to work in occupations such as beauty salons that traditionally offer paths to upward mobility for individuals without high levels of formal job skills. Relatedly, increasing regulatory burdens in tandem with decreased access to capital have made it harder for entrepreneurs to start small businesses, another traditional route of social mobility. Over the past decade, in fact, the number of small businesses opening their doors has fallen below the number going out of business, a period of entrepreneurial decline that may be without precedent in U.S. history.
## The Legislative Response: An Opportunity Agenda
These obstacles to full opportunity for all Americans require a legislative response, in four broad categories.
_We should do more to ensure adequate education and training for all._ For children ages zero to three from disadvantaged families, this means access to programs, such as regular home visits from trained personnel, that improve parents' ability to provide the nurturance that infants and toddlers need. Starting at age three, children should have the opportunity—backed by means-tested funding—to attend prekindergarten programs until they enter public schools.
Three decades of K–12 education reform have overwhelmed teachers and school administrators while producing modest results. Another wave of top-down reform would not be productive. On the other hand, we know that failing to complete high school has become an economic death sentence. Efforts to increase high school graduation rates have made progress, and we need to double down with a crash program to raise them farther and faster. If other advanced economies have achieved near universal high school completion, we can too. And we should bolster twenty-first-century skills training in high schools and community colleges in partnership with private sector employers.
_We should reduce barriers to employment._ Where discrimination in hiring still exists, we should use the federal government's powers of investigation and enforcement to fight it. We should work in partnership with the states to reduce occupational licensing and other regulatory obstacles to entering the workforce. Paid parental leave would ease the conflict between employment and the care of newborns that induces many parents, especially women, to drop out of the paid workforce. We should help unemployed workers trapped in communities with few job prospects move to opportunity elsewhere. And, working with the states, the federal government should reduce employers' disincentives to hire ex-felons.
_We should make work work for more Americans._ Expanding the earned income tax credit (EITC) to cover more low-income employees would encourage individuals at the margins of the workforce to enter it and would reduce their dependence on other means-tested programs. Our outdated system of unemployment insurance should be redesigned to help workers make the transition from declining to expanding sectors of the economy. We should institute a contributory program of wage insurance to cushion the financial fall of displaced workers whose new jobs pay substantially less than the jobs they lost. And we should encourage employers to respond to declining demand by sharing work among their employees rather than dismissing a large percentage of them outright.
_We should expand opportunity by boosting job creation._ One way to do this is to attack the problems—such as access to capital, burdensome regulations, and the cost of tax compliance—that have reduced small business formation, a prime source of new jobs, over the past decade. At the same time, we should overcome the fiscal, regulatory, and political forces that have impeded adequate investment in infrastructure for more than three decades. In the here and now, increased investment means more jobs that pay well and cannot be exported. And over time, it will boost economic efficiency and productivity, necessary preconditions for sustainable wage growth.
## Where Should You Begin? A First-Year Legislative Package
The full opportunity agenda contains a vast array of proposals—too many for Congress to address during its first year, or even its full two-year term. Bursts of first-year legislative output—in 1933 and 1965, for example—are the exception, not the model for incoming presidents. Many past administrations, such as the Carter administration in 1977, have faltered when they have tried to do too much, too fast.
While every new president wants to get off to a strong start, this will be especially important in 2017. Public confidence in the federal government's ability to act on matters of public significance is near a historical low. To break the cycle, you as chief executive must score some early successes. It will be much better to underpromise and overperform than the reverse, to which the American people have become all too accustomed.
Against this backdrop, you should lead with the parts of the opportunity agenda that are most likely to be enacted quickly. This means choosing items that enjoy substantial bipartisan support, are not likely to get bogged down in multiple overlapping congressional committees, and do not plunge your administration into the complexities of intergovernmental relations in our federal system.
These criteria dictate postponing portions of the opportunity agenda to a second phase of legislative activity. For example, states are principally responsible for occupational licensing, and nudging them to expand access into overregulated professions will not be easy. Similarly, nearly 90 percent of prisoners are incarcerated in state rather than federal correctional institutions, so the states will have a significant role in easing ex-felons' reentry into the workforce.
Other portions of the opportunity agenda meet the criteria for inclusion in your first-year legislative package. For example:
* • Numerous surveys show overwhelming support across party lines for paid parental leave, an idea whose time has clearly come.
* • Employers are seeking, but often not finding, workers with the skills a twenty-first-century economy requires. An agreement across party lines to boost investment for job training in community colleges is within reach, especially if there is a formal mechanism for coordinating the development of new courses and programs with employers' needs.
* • The EITC has long enjoyed bipartisan support, and leaders in both parties are open to expanding it to cover currently excluded portions of the workforce, such as single male workers.
* • Both parties are concerned about the declining pace of small business formation. Proposals to simplify the challenge of complying with the dizzying array of regulations small businesses face and to increase their access to capital would be well received.
* • As geographic mobility has decreased in recent decades, too few workers are moving to new opportunities across state lines. Policy experts have recommended assisting workers to leave communities with few job openings to take advantage of better employment prospects elsewhere, and the new Congress seems likely to agree.
* • The global recession of 2008–9 revealed the obsolete nature of the unemployment insurance system created in response to the Great Depression more than eighty years ago. The current model—income support for workers experiencing temporary unemployment in cyclical downturns—does not reflect today's workplace. Both parties want to modernize the system and reorient it toward retraining workers displaced by structural changes in the economy.
* • During the recent presidential campaign, an unexpected area of agreement emerged between you and your opponent: To expand opportunity, we need to end the chronic underfunding of our infrastructure. The states can do more, and many are beginning to, but they find it difficult to agree on projects that cross state lines. We need to find innovative ways of mobilizing investment—including private capital—for this purpose. A well-crafted proposal to this effect would help break the congressional logjam in this area.
## Conclusion: A Cautionary Tale
Throughout his first presidential campaign in 1992, Bill Clinton called for "ending welfare as we know it" by shifting the program's basic thrust from providing open-ended income support to facilitating long-term recipients' entrance into the workforce. There was every reason to believe that it would be the centerpiece of his first-year domestic policy agenda. But at the last minute, funding for welfare reform was removed from the first budget he submitted to Congress less than a month after taking office, a decision that may have delayed the passage of reform legislation by more than two years.
The moral of the story: Whatever you decide your 2017 legislative agenda should be, make sure your economic and domestic policy teams collaborate to produce a budget that reflects your policy priorities. Congress is unlikely to adopt your budget submission unchanged, of course, but members will be quick to seize on the signals it sends. If you do not propose funding for your proposals, they are likely to conclude that you will not fight very hard for them. And if you won't, why should they?
In our system, passing legislation is always difficult, even for new presidents who enter office with a public mandate to act. To get your administration off to a strong start, you will need to establish clear legislative priorities—and then back them with your budget and legislative relations staff. While coordination among policy, resources, and advocacy is no guarantee of success, lack of coordination will ensure failure, disappointing your supporters and weakening your hand in subsequent dealings with Congress.
# Pre-K Prerequisite
# Ensure That Every Poor Child in America Gets Two Years of Preschool
Robert C. Pianta
The time has come to ensure that every poor preschooler in the United States is enrolled for two years of the best educational opportunity the public can provide. Anything short of this will fail to realize the promise of early education as an equalizer in American society.
Public education is one of America's drivers of social and economic mobility, an investment in opportunity for all citizens. Although the authority to provide public education rests with the states, a federal role can emerge through policies that address civil rights or through investments that supplement states' goals. After eight years of aggressive federal efforts to influence the public education system (e.g., Common Core, teacher evaluations), K–12 education is effectively off-limits to the new president. However, the incoming chief executive can make a bold move to optimize and strengthen federal investments in need of further reform and align them with state systems in ways that can really make a difference for children.
Most states are feeling pressure to serve three-year-olds as the public awareness of the value of early education increases. States want to provide two-year, full-time preschool for three- and four-year-olds but don't have the funds. My advice to the incoming president is to form early learning partnerships with states to produce positive results for generations, and to simplify and focus a complex part of the federal bureaucracy.
The president-elect's opportunity goal in the first year can be to ensure that every poor child is enrolled for two full years in an educationally effective preschool program starting at age three.
## History, Context, and the Current State of Play
The public regularly endorses investments in early education; opinion polls show that a majority of U.S. citizens believe that the education of young children, particularly poor children, pays dividends. A majority of governors have expanded their state-run prekindergarten programs for four-year olds; now forty-two states offer public prekindergarten programs. The science shows that some state-run, scaled-up programs for four-year olds can close the achievement gap on state standards tests in elementary school by between 30 percent and 80 percent, but these offerings typically run nine months a year, meaning the dose children receive is simply too low. Head Start, the federally run program for poor three- and four-year-olds (many from even lower income levels than children who are enrolled in state programs), has shown only modest impacts on achievement in school and benefits for children's adjustment, health, and lowered need for special education services. The federal government also provides subsidies to poor families seeking child care, a portion of which covers costs for three- and four-year-olds to attend a wide assortment of generally low-quality placements. In aggregate, the country invests almost $30 billion in the education and care of three- and four-year-olds from low-income families (combining federal and state funds), resources that flow through convoluted pathways, funding disparate programs that are not well integrated and not sufficiently intensive to close learning gaps. Some state- and city-run preschool programs braid and blend funds with local Head Start grantees and come closer to using an all-funds approach to provide a sufficient dose of integrated and effective programming for poor three- and four-year-olds; these programs are models for this policy initiative.
The problem here is that the programs now being delivered through federal support need strengthening in terms of impacts on school performance and child development; direct federal oversight of preschool is inefficient; and state and local programs mostly serve four-year-olds for only nine months of the year. The president-elect can seize public support for early education and drive an equity agenda to serve young poor children with programming that can close achievement gaps. He or she will face opposition from constituencies on the left who own the antipoverty agenda, and so must be clear that this is a next phase in the promise of Head Start and not an attack on the social safety net. This is an occasion for the new president to be clear about his or her commitment to impact for children and intention to strengthen these educational opportunities.
The incoming chief executive can learn from predecessors in the position. President George W. Bush pressed an expanded federal role for preschool by investing in early reading programs and accountability for Head Start; he pushed too hard on testing four-year-olds and burned through his political capital too quickly. One reason why Bush's National Reporting System for Head Start fell apart was its link to defunding programs: the narrative was that Head Start programs would be closed if kids didn't know their letters and numbers and that teachers would be reinforced for teaching to the test or, worse yet, gaming the system. We now know that argument was partially correct, given what we've seen in K–12. States are now implementing assessments of children's skills as they enter kindergarten; we have better assessments, and results drive program development (in state and Head Start programs), not closure. The incoming president should follow this path on assessment and accountability.
President Barack Obama made a notably positive contribution by pushing the Office of Head Start (housed in Health and Human Services) closer to the Department of Education, building state data systems, and strengthening Head Start accountability, all promising improvements. Unfortunately, Obama became distracted by K–12 reform and by flirting with universal prekindergarten, and risked momentum to expand and strengthen federally supported early education programs for poor children.
## A Message for Each Nominee: Why You Should Make Early Childhood Education a First-Year Priority
To Hillary Clinton: You have more credibility on early childhood education than any other president in U.S. history. You are an advocate for children, families, and the role of early childhood education in promoting children's success, and you recognize that child care is a key part of women's rights and gender equality. You have political and social capital to spend on this issue, although you must allay the concerns of allies and affiliated interest groups, which will pressure you to grow the federal role. You are on record as championing universal prekindergarten; by focusing very clearly on poor children, you will anger some allies but at the same time win points by scaling back what many see as an overreaching agenda.
To Donald Trump: You need an issue where you look interested in investing in people, one that addresses equity and poverty and that might help you build some political capital. You have declared your commitment to opportunity; here's a path to demonstrating that. Cut a deal in the interest of children, particularly poor children of color. By doing so you might lift your dreadful standing with these groups, thereby confounding your detractors and enrolling them politically.
## The Policies and Political Moves
The knee-jerk (and wrong) reaction to the two-year full-time preschool goal is that it will cost too much and that the evidence for the impacts of early childhood education is mixed, so why expand preschool for poor children? To realize his or her agenda, the next chief executive must not spend more. Instead, it will be important to integrate and realign existing federal investments (e.g., funds for Head Start and subsidies for child care) with those of the states while focusing on building a strong and effective accountability system to monitor these federal investments.
The incoming president should intentionally align the financial and bureaucratic resources of the federal government that fund Head Start's programs for three- and four-year-olds (roughly $8.5 billion) with the operational strengths and capacities of the states to expand programs to ensure that all low-income three- and four-year-olds get full-time, full-year early education. This can be achieved by bundling federal subsidies for child care for three- and four-year-olds (e.g., Temporary Aid to Needy Families) with Head Start resources to expand the available pool of funds to serve children in programs.
The first order of business is to get the federal government out of the business of directly operating a preschool program. The bureaucracy is complex and challenging; Congress passed 1,300 regulations for Head Start grantees, from handwashing to family involvement to transportation to standards for kids' literacy skills. Overregulation of Head Start is a major challenge to staff, who must attend to too many irrelevancies. The current system can be simplified by enabling state pre-K and Head Start partnerships to allow these programs more operational control while retaining a strong federal role to make sure they do it effectively and address the needs of poor children.
This shift in the federal role to that of partner in preschool must be balanced with a robust federal monitoring system to ensure that preschools effectively promote children's readiness for and success in school. Because the federal role will be ensuring accountability for programs serving young and vulnerable children, dollars should be allowed to flow to state-federal innovation partnerships only if they accept accountability plans.
## What the Next Chief Executive Can Do: Money and Accountability
Analysts contend that $30 billion is sufficient to provide full-day, full-year programming for three- and four-year-olds whose families fall below the federal income poverty threshold. Better integrating federal dollars from Head Start and child care with states' programs for poor children can level the total amount of funds spent per child. Such a move might help close the funding gap between preschool and K–12 and provide for workforce training, compensation, and program improvements to ensure that preschool promotes learning. The new administration should promote this as an equity agenda that spends public funds effectively and innovates through partnerships that shift operational and programmatic direction closer to the needs of staff and children.
The evidence will back up the administration's actions. When children are enrolled in programs that are truly educational—that is, they offer a defined curriculum that emphasizes learning, with teachers trained to stimulate and support learning and with information on children's progress collected and used to tailor learning and drive program improvement—children learn more. The proposal is for innovation partnerships that increase the dose of these educational resources for our most vulnerable children. Instead of nine months' exposure, often for only a few hours a day, poor children should receive daily doses, all day, for two years. This is our best shot to close early achievement gaps.
Here are some components of an accountability system worth considering:
* • Every participating state must implement a system for assessing the school readiness of all entering kindergartners. This system should include direct assessments of children's early literacy and language development, knowledge of mathematics, and self-regulation skills (e.g., attention and self-control). Testing four-year-olds will be controversial, so these systems must be strong and the information useful to teachers, schools, and preschool programs. Many states are moving in this direction; good assessments exist; and state data systems are capable of processing and reporting this information.
* • Every classroom for three- and four-year-olds must use curricula proven effective for increasing the knowledge and skills of preschoolers. The good news is that recent federal research investments produced a suite of such curricula that both work and are appropriate for young children. The bad news is that most preschool programs do not use these curricula. Program partnership and innovation funds must catalyze progress.
* • Every teacher must be able to interact effectively with preschoolers. The next administration should invest in observations, feedback, and training targeted to teachers' social and instructional interactions with children. Evidence supports this plan, and it's an investment in the workforce, which can help politically. States have "quality improvement" money that can target workforce development. This requirement can be used to promote innovation in teacher training that will ripple into K–12. Workforce development funds should not be tied to academic credits and degrees but to progress on relevant classroom skills and knowledge, with credentials and compensation awarded for increased competence and results.
* • The infrastructure of these programs must be suitable to sustaining appropriate and effective preschool. Teacher-child ratios, safety requirements, and staffing minimums need to be in place but cannot be the sole indicators of program quality.
* • All programs should be placed on a biennial cycle of accountability review, modeled on Head Start's Designation and Renewal System or strong state systems. Many states are already collecting some form of accountability data. Young and vulnerable children should be in the best places possible to support their development and their families assured stable full-time placements for their children, which will open enormous opportunities for parents in terms of their own education and employment.
## Navigating Trade-Offs
As the incoming administration crafts this plan, certain decision points will trigger program design and political trade-offs.
You are advancing an opportunity agenda to address serious inequalities in American society. This is an initiative to level the playing field for our youngest, most vulnerable citizens by setting them on a path to success in school and life. It is not a program to provide preschool for kids whose families can pay for it or who would otherwise do just fine in school. This is a targeted program, not universal, as has been championed by President Obama and Secretary Clinton. There is little chance politically for universal preschool funding, and the evidence on impacts is not compelling. Although making pre-K universal creates middle- and upper-class support for preschool that might ensure its sustainability for poor families, there's no clear evidence on this point. And conservatives see universal preschool funding as an entitlement program. Rather than succumb to the argument that politically sustainable programs must be universal, the new administration should admit there isn't enough money to fund programs for every child and recognize that the country must address the eroding economic opportunity and poor educational performance of our low-income communities. Stay focused on poor children; you can always raise the cutoff for income eligibility.
## Politics, Allies, and Key Constituents That Need Attention
The next administration may have an ally in House Speaker Paul Ryan, who is on record as wanting to improve life for poor people and place the federal government in its proper role. This move is a good meeting place. Hold the federal allocation in check, partner with states in operational control, simplify federal regulation and control, and preserve the federal role in equal opportunity through strong accountability.
You might consider placing oversight of the accountability system in the Department of Education to tie it closer to an education-oriented mission. It already monitors similar programs and the Title I funds that flow to preschool education. Just make sure that the department's monitoring is strong, assuring that poor children are served and served well. If the Department of Education is abolished, as frequently threatened by Republicans, then keep monitoring in Health and Human Services. As a condition of receiving federal support, you might require states to establish an Office of Early Learning in their respective departments of education to administer program improvement and monitoring. Emphasize with education legislators in Congress that if state education agencies are to have oversight of the most vulnerable children, they need stronger capacity. There is a deep bench in the states and in think tanks that can help you develop early education policy; draw on those experts as advisers. To direct the accountability system, pluck someone from a state or big city with a proven record of results.
Finally, prioritize this as an effort to expand publicly funded educational opportunities for three- and four-year-olds and don't get caught in trying to address the needs of younger children beyond what is already in legislation for child-care support and Early Head Start. You will be under pressure to address the needs of infants and toddlers with this initiative. Help families and communities figure that out through tax credits for child care.
## The Next Administration's Move and Legacy
If the next administration fulfills this agenda and ensures that every poor child in the United States is enrolled for two full years in an educationally effective preschool program starting at age three, it will have come as close to leveling the playing field for low-income families as any presidential administration has, and without a substantial increase in spending or through an entitlement program. This agenda, if enacted with strong accountability, can leave the president-elect with a legacy of having made a real difference in the lives of millions of children.
# City Power
# The Next President Can Help Cities Boost More Workers into the Middle Class
Richard Schragger
In the early to mid-twentieth century, American cities were the engines of opportunity and mobility in the industrial United States. Economic growth was synonymous with city growth. By contrast, the second half of the twentieth century witnessed the precipitous decline of the industrial city, the rise of the sprawling suburban metropolis, and the racial and economic fracturing of metropolitan regions around the country. The processes of deindustrialization and suburbanization have reinforced the socioeconomic distance between the haves and the have-nots. Opportunity is hugely uneven within and across metropolitan areas, and old-line cities still struggle with deeply entrenched poverty.
Despite this reality, we have seen some formerly declining cities do better in the last few decades. New York and Chicago are increasingly desirable places to live, and such cities as Pittsburgh and Philadelphia are seeing their populations and property values stabilize after a long period of relative decline. This nascent urban resurgence should remind us that cities have always been engines of economic creativity, ingenuity, and opportunity. Cities are huge generators of wealth. But for cities to do what they do best—turn working-class people into middle-class people—the resources that are generated by cities have to be plowed back into the kinds of public goods that benefit everyone, not just a narrow slice of the already favored. At mid-century, cities provided the basic services, such as schools, housing, transportation, health care, parks, and playgrounds, that gave working-class people the resources to move into the middle class. President Franklin Roosevelt famously partnered with New York's Mayor Fiorello LaGuardia to provide funding for these projects. Cities can provide those goods again. The next president can help them do so.
The way to empower cities is not to adopt a raft of new urban economic development policies. The last seventy-five years of urban policy have mostly been a failure. Instead of an urban policy, the next president should adopt a _city power_ policy. To do so requires the new administration to take four steps:
1. 1. Acknowledge the systemic failures of national and state urban policy.
2. 2. Stop the industrial subsidy race and rein in wasteful interstate and interlocal tax base competition.
3. 3. Provide revenue support so that cities have the capacity to provide basic public goods.
4. 4. Get out of the way. Cities are already doing a great deal to address economic inequality. National and state policymakers should encourage those policy experiments and step aside.
## The Failures of Urban Policy
The first step is to acknowledge the failures of federal urban policy. After the riots of the 1960s, the Kerner Commission sought to promote an "urban Marshall Plan"—large, sustained federal investments in inner-city neighborhoods. That plan never materialized. Richard Nixon's election in 1968 and the general abandonment of the War on Poverty brought an end to inner-city efforts. Instead, federal funds were and continue to be directed toward highway building, urban renewal, and other forms of urban economic development. These funds benefit developers, downtown business interests, the construction trades, and other interest groups without demonstrably improving the condition of depressed urban neighborhoods, and often make the residents of those neighborhoods markedly worse off. Ethnic and racial minorities were particularly hard-hit. For African Americans in the mid-twentieth century, urban renewal was known derisively and bitterly as "Negro removal." Across the country federal funds were used to displace poor blacks, tear down their neighborhoods, and replace small businesses with larger ones.
These federal expenditures did not prevent urban decline in large part because the underlying theory was mistaken. Urban economic policies have consistently failed because they have sought to shift people around the metropolitan area. The basic idea has been to encourage middle-class suburbanites to return to the central city or to encourage poor inner-city residents to move to the suburbs. Cities have been repeatedly told that to become economically viable they must attract and retain the "creative class," the middle class, or the luxury class. Urban renewal was itself a massive effort to remake the city by replacing one set of residents with another. It didn't work. And it continues to fail.
Instead of thinking about the city as a product that needs to attract low-cost, high-tax-paying residents, we should think about cities as engines of economic opportunity in their own right. Giving existing working-class residents the tools to become middle class _in place_ should be the goal of the next administration—not moving residents hither and thither in an attempt to cobble together a sustainable tax base. Instead of treating cities as passive jurisdictions "competing" for resident-users of public services, we should understand them as engines of economic development and upward mobility.
American cities played and continue to play a significant role in creating an urban middle class, both by creating wealth and by providing public goods. In a forty-year span in the middle of the twentieth century, New York City built thousands of units of working- and middle-class housing, hundreds of schools, libraries, and parks, and thousands of miles of roadways, bridges, tunnels, and subways. The Roosevelt-LaGuardia partnership was essential to these developments, and the basic infrastructure built by the newly emergent industrial cities raised living standards for the rural and urban poor alike. Those goods helped produce a robust urban middle class by mid-century. In many places these urban goods continue to provide the working class, the poor, and newly arrived immigrants resources for upward mobility. Those resources are basic and obvious: security, education, transportation, health care, and shelter. The new president should think of cities as agents of economic mobility. Instead of dictating policy from above, the administration should work with cities to generate economic growth from below.
## Stop the Subsidy Race
A second problem distracts cities from the effective provision of basic goods: the intermunicipal competition for industry and business. Urban redevelopment policy has focused on creating jobs by inducing business relocation, but here again, the influx of federal dollars mostly upends communities instead of supporting them. In a regional or national employment market, bringing more jobs into a city does not mean that those jobs will be filled by local residents or that more jobs will reduce poverty among the existing needy. New jobs are just as easily filled by better-educated immigrants. An influx of industry could also _increase_ local unemployment, as more job seekers migrate into the jurisdiction.
The interlocal competition for jobs is a result of a federal system that encourages competition for tax base. States compete for industry. Cities compete for regionally valuable investments. Providing location subsidies to large employers has been and continues to be a standard tool of city economic development offices. According to some estimates, each year, state and local governments spend close to $80 billion, or roughly 7 percent of their total budgets, on tax incentives, tax breaks, and outright cash payments to attract corporate investment and relocation.
These subsidies are mostly wasteful, however. There is strong evidence that subsidies do not ultimately alter business location decisions and that cities do not get back what they put in, either in the short or the long term. The stories of failure are commonplace. Reporting that five years after locating there, IBM fired most of its employees in Dubuque and Columbia, Iowa, despite a combined $84 million in tax breaks, the author of a _Bloomberg News_ story noted that this scenario "has played out often across America: Big company comes to town, provides boost to the local economy and then leaves." It is reported that the subsidies provided by Michigan cities to the automobile industry over the years could have easily wiped out all of Detroit's debt.
Smokestack chasing is difficult for state and local government officials to resist. "Shoot anything that flies; claim anything that falls" captures the attitude of state and local politicians and development practitioners. City officials feel the need to do something—anything—to prove that they are pursuing economic growth.
The fracturing of metropolitan regions into hundreds of independent local jurisdictions does not help. Most metropolitan areas are unified economic regions, but each local government is encouraged to pursue tax base–enhancing policies, regardless of the effects on neighboring jurisdictions. Local governments compete with each other for retail developments, office parks, and other tax-generating commercial projects. At the same time, local governments want to avoid regionally necessary but disfavored uses and anything that generates costs but not tax revenue. Thus, local governments avoid providing low-income or multifamily housing. The result has been an affordable housing crisis, especially on the coasts. In addition, because of the lack of low-income housing in the suburbs, low-income workers in the city face a "spatial mistmatch": they often live far from jobs, which are increasingly located in the far-flung suburbs.
Federal help could solve these regional collective action problems. First, the new president could encourage Congress to stop the interstate subsidy race by proposing legislation that bars state and local tax incentives and subsidies that "poach" productive enterprises from other jurisdictions. Second, the new administration could reduce the incentives for local governments to engage in fiscal races to the bottom by directly providing more education and other social welfare dollars. The national provision of monies for basic social services would help mute interlocal tax competition.
## Emphasize Basic Fiscal Stability
Reducing interlocal competition will allow cities to redirect their efforts toward providing basic public goods. The federal government can further encourage these efforts by helping to stabilize local government finances, and the new president can adopt this as a first-year policy goal. Cities' revenue-raising capacity fluctuates with the booms and busts of the larger economy. American cities in the latter half of the twentieth century lurched from fiscal crisis to fiscal crisis. Cities cannot print money, and state constitutions usually prevent them from running deficits. In lean times, this means that cities have to cut back on their spending, often when economic theory would counsel doing the opposite. The result is that local fiscal policy exacerbates economic downturns while cutting services to the most vulnerable.
What can the next president do to give cities the ability to provide quality local services even in economic downturns? One answer is revenue sharing. In 1972, Richard Nixon ushered in an era of federal-state-local revenue sharing that lasted until 1986. Nixon's program transferred much-needed funds from federal coffers to state and local governments, helping to stabilize local budgets. The money came with few strings—local governments were empowered to decide for themselves how to use the funds. The program was popular. Democrats welcomed much-needed aid for struggling cities. Republicans saw revenue sharing as a way of devolving political authority to local communities.
The new administration can help to stabilize local finances in other ways too. Consider federal housing and transportation policies. Since the Great Depression, the federal government has encouraged home ownership by underwriting the home mortgage system. Securitization and deregulation of the mortgage industry, however, ultimately led to overlending and economic collapse. The foreclosure and eviction crisis that followed the 2008 global recession continues to cause a great deal of local fiscal stress. Cities cannot stabilize their populations or tax base when they have a glut of foreclosed homes. The federal government needs to regulate the mortgage industry, but also to adopt programs that keep people in their homes. Preventing predatory lending practices before they occur is essential.
The uncertainty of federal transportation budgeting also exacerbates urban difficulties. For cities needing to repair and replace aging infrastructure, there is often either too much money or not enough. Large-scale highway projects absorb too much money. Meanwhile basic maintenance on existing roads is neglected and alternatives to automobile-based transportation are often ignored. Again, stabilizing local finances and helping cities provide basic services should be a first-year priority for the next president.
## Get Out of the Way
Once state and federal governments create a stable fiscal environment, they should get out of the way. Cities are capable of promoting economic opportunity and mobility if we just let them.
Consider the number of cities that have now adopted living-wage legislation, a policy that has been shown to reduce poverty significantly. Cities are also adopting family leave, health care, and labor-friendly legislation, in this way addressing inequality in the absence of state or federal regulation.
Economic theorists often assert that the minimum wage and other forms of social welfare regulation have to be pursued at the national level. Conventional wisdom suggests that municipal regulation will prompt businesses and residents to flee.
But that is not what has happened. Millions of Americans are now living in cities with minimum wages that will reach at least $15 an hour. The efforts by mayors and city councils in Los Angeles, New York, Chicago, Santa Fe, and hundreds of other cities have also induced some states to adopt higher minimums. Cities have been the first movers in regulating low-income service work and in providing much-needed support for low-income laborers.
Indeed, the primary barrier to cities addressing economic inequality is state and federal interference. State legislatures regularly adopt preemptive legislation that prevents cities from acting. Federal law also makes it difficult, if not impossible, for cities to adopt regulations that directly regulate labor rights or mortgage lending. Cities need more room to maneuver. The new administration can give them this policy space by avoiding unfunded mandates, by directing federal agencies to permit local flexibility in applying social welfare programs, and by providing resources directly to cities, bypassing state legislatures. Preemptive federal laws should be modified to permit cities more room to regulate in areas like labor and employment, banking, and insurance.
When cities have the authority, they can do a great deal. In analyzing local social welfare policy, urban theorist Michael Craw found that even when constrained by limited fiscal capacity, many local governments "still have a significant degree of independence in reacting to local policy preferences when it comes to decisions on providing social welfare services and participating in federal and state intergovernmental grants." He concludes that rather than "simply being 'junior partners' to federal and state governments, some local governments possess considerable autonomy in addressing local poverty." The political scientists Chris Tausanovitch and Christopher Warshaw recently found that "liberal" cities have higher taxes, less regressive tax systems, and spend over twice as much per capita than do "conservative" cities on social welfare.
Despite conventional economic theory, cities are addressing inequality, and doing so aggressively. City residents are willing and able to pay taxes for redistributive social programs. And the wealth being generated in cities is being plowed back into city services.
## City Power
A city power policy should encourage these trends. The benefit of a federal political system is that it permits local jurisdictions to pursue ends that are responsive to local constituencies. For a long time, federal urban policy treated cities as needy dependents—as problems to be solved or ignored. Urban policy often overlooked the ample resources already existing in urban neighborhoods or took advantage of those resources to enrich real estate or downtown business interests. The new administration should reject past failed developmental policies, stop the interlocal subsidy race, give cities a modicum of basic fiscal stability, and then get out of the way. Cities can be instruments for advancing economic opportunity and mobility. The next president should recognize and respect that fact.
# The Long Game
# The Next President Must Lay the Foundation for a Next-Generation Economy
Margaret O'Mara
What can a president do to build a next-generation economy? How can economic growth widen opportunity for all Americans, not just a privileged few? Every modern American president has wrestled with some version of these two questions during his first year in office. America's next chief executive will do so as well. History provides powerful lessons and debunks popular myths of where innovation comes from.
The United States' extraordinary track record of invention and technical achievement—from the cotton gin to the transistor to the self-driving car—isn't just a story of ingenious entrepreneurs and free markets. It's a story of moral leadership from the White House, both in setting grand challenges and in pushing for policies to make them a reality. These actions fall into three big categories, each with its own lessons for the next president.
## Investment
Innovation and opportunity have bloomed when political leaders make foundational, long-range investments in people, ideas, and audacious technologies for which there isn't yet a market.
Consider President Dwight D. Eisenhower. The former military man came into office at a moment of high anxiety and geopolitical instability. "We live," said Eisenhower in a May 1953 radio address, "not in an instant of danger but in an age of danger." At the same time, the mounting costs of the conflict in Korea demonstrated how unsustainable it would be for the United States to engage in serial conflicts in every global hotspot. Instead, Eisenhower proposed a "New Look" military that, among other things, used technological strength—in nuclear weapons, aerospace, and electronics—to deter potential conflict. One result was a massive mobilization of scientific people and institutions rather than a massive mobilization of soldiers.
In so doing, he set in motion an era of dazzlingly rapid American technological achievement. In the San Francisco Bay Area, for example, large defense contracts became the bedrock for a small-electronics industry starting in the Eisenhower years. Defense contractor Lockheed was the largest employer in the region that later became known as Silicon Valley. Hewlett-Packard, a company that also counted the feds as a big customer, became the second-largest. In the first years after its 1957 founding, Fairchild Semiconductor—the plucky venture-backed startup that was the training ground for future leaders of so many iconic Valley companies—got 80 percent of its book of business from government contracts. And right at the heart of the region, federal money turned Stanford University into a research juggernaut and an astoundingly fertile intellectual hub for innovators in both high-tech and biotech.
All this went into overdrive after another first-year president, John F. Kennedy, declared that America would reach the Moon within the decade, creating a fresh harvest of federal contracting opportunities for companies making transistors, circuits, and high-tech components. Government investment gave entrepreneurs an incentive to develop and build blue-sky technologies and offered the security of doing business with a patient and deep-pocketed customer.
Are Eisenhower's military spending and Kennedy's Moon shots good models for the current age of austerity, when there is considerable political pressure to cut public spending and lower tax burdens? Yes, they can be. These previous presidents faced similar pressure (including considerable opposition in Congress to some of their ideas), but they still made a case for bold action and sustained investment.
What's more, unlike his or her predecessors, the next president doesn't need to start from scratch. We still have the extraordinary research complex that the Cold War and the space race built. However, it is in a fragile position, partly because too few recognize the foundational role that public policy continues to play in the high-tech economy.
To keep the pipeline flowing with new ideas, the next president needs to prioritize sustained and increased investment in the national research agencies. To ensure the United States has the world's most high-skilled, innovative workforce, he or she must lead a national reinvestment in higher education that sustains excellence in all disciplines while lowering college costs for students. The golden age of the U.S. high-tech industry didn't come from applying market models to research and education; it came from the willingness to make very long-term investments in blue-sky research and talented people.
## Internationalism
It is no coincidence that the United States became the global command-and-control center of the high-tech industry at the same historical moment that the nation opened its borders to a broad range of nationalities and ethnicities. It's a pattern seen throughout human history: innovation grows in places with open societies and open borders, where economic and educational opportunities draw in new migrants from across the globe.
During President Harry Truman's first year, as the United States emerged from the devastation of World War II into the chill of the Cold War, Senator William Fulbright (D., Ark.) led the charge to establish international educational exchange programs that brought thousands of foreign students and scholars to our shores. The expansion of the Fulbright and other programs over the years (by leaders of both parties) turned American colleges and universities into destinations for the world's best and brightest—and helped make the U.S. higher education system the strongest in the world.
At the same time, the Truman administration kept the nation's doors open to refugees from war-torn Europe, a humanitarian act that turned out to have huge economic returns. Those who came here included some of the world's leading scientists and thinkers, but they also included people whose potential wasn't clear at the time. One was Andy Grove, a penniless nineteen-year-old Hungarian who stepped off a boat in New York in 1946 and went on to co-found the legendary microchip maker Intel two decades later.
Borders opened wider—and welcomed an even more diverse range of people—when President Lyndon Johnson signed the Hart-Cellar Immigration Act of 1965. LBJ acted in the name of civil rights, as the bill lifted racist quotas that had restricted immigration from most of the world (especially Asia and Latin America) for generations. At the time, he and the bill's supporters believed it was mostly a symbolic gesture rather than something that would radically alter immigration patterns. But Hart-Cellar didn't just alter them—it transformed them entirely, and the nation in the process.
The post-1965 immigrant wave transformed the economy as well, and there's no better place to see this than in Silicon Valley. Indian- and Chinese-born entrepreneurs were at the helm of 24 percent of the technology enterprises started between 1980 and 1998, a formative and hugely lucrative time for high-tech. The economic upside of open doors has been clear to the most successful occupants of the Oval Office. "Our strength," declared Ronald Reagan during his first year, "comes from our own immigrant heritage and our capacity to welcome those from other lands."
The next president must remember that an innovative economy relies on open borders and a willingness to make bets on all sorts of newcomers, whether they have a PhD or less than a high school education. An increase in the number of available H1B visas for highly educated workers—something Silicon Valley tech companies strongly support—is a critical piece of an innovation-focused immigration policy. But so is keeping our doors open to political refugees and providing amnesty to the undocumented immigrants already here. It's not only the right thing to do, it makes economic sense. Immigrants are more likely than U.S.-born workers to become business owners, and immigrants' rate of business formation is more than twice that of those born here. Bottom line: the next-generation economy doesn't just depend on the next great technology. It depends on finding and supporting the next Andy Grove.
## Income Security
Ideas and growth industries blossom when a society is stable and secure enough to become a place of hopeful possibility, imagination, and some willingness to take risks. In the U.S. case, the emergence of a world-dominant tech industry came out of a remarkable twenty-five years of rising incomes and education levels that happened after World War II. The expansion of economic security and opportunity that occurred during that period formed the bedrock for subsequent decades of technological breakthroughs and entrepreneurial success stories.
How did this happen? The conventional wisdom in much of the business community is that redistributive government programs are anathema to entrepreneurship and productivity. To grow a next-generation economy, the argument goes, we need supply-side measures like tax cuts and looser market regulation.
But if we take a longer view, we see how earlier presidential commitments to broad-based economic security—from Social Security to the GI Bill to Medicare and Medicaid—worked in tandem with the aforementioned public investments in R&D and infrastructure to raise incomes and expand educational and professional opportunities. This opened up the field of people who had the opportunity to apply their raw intelligence and entrepreneurial energy to create new companies, markets, and industries.
Most of the people who founded iconic tech companies weren't born rich. They were middle-class kids whose families rode that wave of postwar upward mobility. Steve Jobs's father was a machinist who didn't finish high school. Intel co-founder Robert Noyce was the son of a small-town Iowa clergyman. Another legendary microchip industry CEO, Jerry Sanders, was a street-fighting kid raised by his grandparents on Chicago's South Side. Growing up during a moment of political commitment to expanded opportunity and prosperity, these men benefited from policies that widened opportunities for their college education, invested in science and math programs and built computer labs in schools, and broke down some old prejudices about religion, ethnicity, and pedigree.
There aren't as many of these kinds of stories today. Entry into the high-tech elite often comes through educated parents, good schooling, and an offer of admission from a prestigious university like Harvard or Stanford. The arena of tech opportunity still hasn't widened to include enough women, African Americans, and Hispanics, as the still dismal diversity statistics at major tech companies reflect. But overall income security, or lack of it, also plays a big role. In an era of economic precariousness for so many, it's hard for people without family and community resources to break into the startup game.
The next president must resist the temptation to celebrate high-tech entrepreneurship without properly recognizing that only a few privileged people can afford to take those kinds of risks. The innovation economy isn't a meritocracy. First-generation immigrants sending money to family back home can't afford to get paid a pittance in the feeble hope that their stock options will one day be worth something. Single parents can't be expected to juggle the punishing work hours of early-stage tech companies—or, for that matter, many tech companies. Startup culture is the domain of young, childless people who have an economic safety net for when the venture stumbles or fails.
Policies that create that safety net, such as publicly backed venture funds or grants targeted to underrepresented individuals, are one way to enlarge the pool of who gets to be a tech entrepreneur. But along with that, the next president can and should lead the way in broadening the definition of entrepreneurship itself—beyond tech, and toward all sorts of small businesses. And through both moral leadership and policy action, the next president can and must make the case to the American people that public investments to ensure overall prosperity—for the many, not just the few—are foundational to the next-generation economy.
In moving forward on an agenda for opportunity and mobility, the next president can learn from the prior occupants of the Oval Office—and from some basic historical truths about where innovation comes from, and how it grows.
## Make Long-Term Investments with Few Strings Attached
Unlike other countries, American public sector spending on innovation is often indirect—flowing through contracts to private industry or universities, regulatory institutions, or the tax code. Where investments are more obvious, as in appropriations to the National Science Foundation or the National Institutes of Health, they are part of a long game. Research investments take a long time to pay off, and thus their returns are not immediately obvious to either political leaders or the public.
But the indirect, almost stealthy nature of spending has been absolutely critical to the ascendance of the American high-tech economy. The government invested, and then it got out of the way. The result was many centers of innovation across public, private, and nonprofit sectors. Size mattered, too. Sure, there were big federal research labs. But there also were plenty of contracts given to small private companies that created an incentive to be entrepreneurial, to push technological boundaries, to innovate and grow. Contracts also have been a way for Washington to encourage the private sector to adopt certain behaviors, from moving away from areas deemed prime targets for nuclear bombs in the 1950s to encouraging firms to hire more women and minorities in the 1970s.
To keep the bedrock firm, the next president should not be afraid to push for long-term direct investments in research and education. And to keep this world-class entrepreneurial economy churning and enlarging its reach, he or she should deploy the contracting system to create private sector incentives to tackle particular social problems as well as to bring a more diverse workforce into tech.
## Don't Fear Unintended Consequences
The most powerful American "innovation policies" haven't been top-down, command-and-control initiatives to boost particular kinds of industry. In fact, their greatest impact often has come through the unanticipated consequences of policies designed with broader and different goals in mind.
Dwight Eisenhower never once declared that he would build a science city, tech park, or innovation district. In fact, the massive growth of government-sponsored science on his watch made him fearful about the future of American ingenuity. In the same farewell address of 1960 in which he gloomily assessed the reach of a new "military-industrial complex," the president lamented how the innovative landscape had changed. "The solitary inventor, tinkering in his shop, has been overshadowed by task forces of scientists in laboratories and testing fields," he declared. "A government contract becomes virtually a substitute for intellectual curiosity."
Ike was dead wrong. But he was right in setting bold priorities in his first year in office that turned the U.S. military into a more high-tech fighting force. The extraordinary outlay of federal spending on scientific R&D during his administration helped build the ultimate "science city," Silicon Valley. While regional economic development was not something he or other political leaders of that era had in mind as they authorized new defense and space programs, the extreme geographic concentration of these investments created dynamic regions that today are the undisputed global capitals of tech.
It's a helpful example for the next president. Policies that support great public education can have the unintended consequence of identifying and nurturing the next Steve Jobs. Policies to keep borders open and welcoming to immigrants and refugees can draw in the next generation of inventors and creators and company founders. Along the way, these presidential moves can rebuild the economic security that propelled the American innovation economy forward in the first place.
## The New Economy Rests on Solid Old Economy Foundations
Today the United States has a roaring high-tech economy but levels of economic inequality that are greater than at any time since the age of President William McKinley. This situation isn't sustainable. Broad-based economic security may not be required to grow a wave of next-generation companies, but it is essential to maintain innovation and economic dynamism over time. It also ensures that the benefits of economic growth flow to the many, not just the few.
As the dynamics of the 2016 election made quite clear, the next occupant of the White House will have to address the great anxieties and anger so many Americans feel about an economy that is no longer providing the opportunity, mobility, and security that they and their families and communities need. One way to do so will be to no longer think of "economic policy" and "tech and innovation policy" as two separate streams, but as a whole that is greater than the sum of its parts.
The high-tech industry has been one of the extraordinary American success stories of the past four decades. The next president should look to that industry as a resource and prioritize public investment that sustains its growth, but should not be afraid to challenge and disrupt it where necessary. The prescription is simple, but it demands political audacity and a willingness to play a very long game. Here is an opportunity for moral leadership as well as policy leadership, and a way for the next president to cement his or her legacy as a builder of the next-generation American economy.
# Moving Up
# The Next President Can Restore Confidence in the American Dream
Melody Barnes
By any measure, the tone of election 2016 was uniquely virulent. Generations-old frustrations, born of profound discrimination as well as economic and cultural shifts in American society, erupted into quantifiable electoral rage. Why? Because across nearly every demographic, confidence in our central national ideal—opportunity—has almost completely collapsed.
Donald Trump's campaign slogan, "Make America great again," resonated with millions of Americans because too few people have confidence that America works for them. Many believe the "system is rigged," that their children will be less well-off than they are, and that they themselves are less well-off than their parents. For too many Americans, the future looks, frankly, un-American. They have lost hope.
It is therefore imperative that the next presidential administration dedicate its first year to policies designed to catalyze upward economic mobility for the millions currently left behind. The new administration must convey a message of unwavering confidence that every American has the opportunity to succeed and that the legislative and executive branches of government are working together to help enable that goal. In doing so, the administration should avoid the overwhelming temptation to reinvent the wheel and instead leverage assets developed over the last eight years to scale successful efforts.
The next president should take executive actions and propose legislation to:
* • Build programs designed to meet the economic opportunity challenge.
* • Invest in economic mobility programs that work or show real promise.
* • Redirect resources away from programs that consistently fail or underperform toward those that demonstrate quantifiable success.
The proposal should leverage the growing movement toward data-driven and evidence-based policymaking, championed by entities such as Results for America, that is being implemented at all levels of government. Evidence-based policymaking is the practice of gathering and analyzing the best available data regarding the effectiveness of particular policies to ensure that those policies actually work and then utilizing that evidence to make policy decisions. Critical to this practice is the need for ongoing research, evaluation, performance measurement, and community-level engagement to maximize the impact of public spending, encourage program improvements, and inform policy discourse.
Most important, the next administration must seek every path to secure bipartisan congressional support and passage of two interdependent pieces of legislation in the first year: tax reform and economic opportunity/mobility legislation, which might be called the Raising Incomes and Creating Hope (RICH) Act.
Although it has seen short-term declines during times of acute national stress, over the last two centuries America's confidence trend line has generally been positive. Even as troubling economic and social trends began to take hold in the latter part of the twentieth century, rhetoric proclaiming that "America's best days are ahead of it" still found a largely receptive audience among a majority of Americans. That is no longer the case.
Today's collapse in confidence should surprise no one. Leading indicators of the opportunity gap have been hiding in plain sight for decades in American society. Lack of opportunity has been a raging storm in communities of color across America for generations. Civil rights–era gains notwithstanding, many communities of color still struggle to overcome structural barriers to upward economic mobility. Access to effective education, job opportunities, and the tools necessary for economic and social wealth creation were first by law and later by convention refused to these communities, with predictable results: high levels of family disintegration, crime, substance abuse, and hopelessness. Because public policy at every level insulated the majority of Americans from those negative trends for decades, the political system was not required to confront the storm created by those inequities and now evident in many communities—until now.
Today, while the cause may vary, the opportunity challenge knows no racial or ethnic bounds. It applies to millions more than ever before, with tragically predictable results. Regardless of the cause—globalization, technology-driven transformation, the pace at which capital flows around the world—the impact of this economic transformation is well known. Millions now bear the brunt of the "hollowing out" that began in the second half of the twentieth century. As a result, the next president will be confronted with an economic opportunity and mobility challenge that is now ubiquitous.
The collapse of confidence in opportunity is symptomatic of a deeper loss of faith among Americans in public institutions and government in particular. Confronting America's inequity challenge will therefore require a counterattack against the generation-long assault on government. Too many Americans have come to believe that the federal government is ineffective, populated by incompetent, lazy, or even corrupt bureaucrats. After decades of federal debates poisoned by rhetoric expressly designed to undermine "government," in particular the federal government, any political basis for bipartisan action on policies to address America's growing opportunity gap has essentially evaporated and, along with it, the American people's expectation that collective political action can yield positive results.
Arguably the best post–World War II example of a bipartisan, federally led, socially transformative project was the space program, specifically the Apollo project. Sending a man to the Moon and returning him safely to Earth brought with it significant changes in American society ranging from improved education in the STEM disciplines to countless scientific innovations. Although the benefits of this bipartisan, publicly revered government project remain, the political structure that undergirded it does not. There is a reason nearly every politician since the Apollo era has used the term "Moonshot" when advocating for national action, but nothing remotely comparable has ever been achieved.
The next president must begin debunking the myth that "government is the problem" and reestablish the necessity and legitimacy of government action, not for its own sake but for the sake of improving our economic future. To confront the twin challenges of collapsing public confidence and political gridlock, the next president needs to give the country some "wins."
During its first year, the new administration should pursue policy wins that attack the economic opportunity crisis and that are genuinely bipartisan. This will not be easy—and past administrations have tried—but this important moment requires a renewed effort to choose a legislative path that has some realistic chance of securing votes from the other party. Bipartisan legislation must have an immediate impact on the lives of millions who need help and it must create a solid foundation for subsequent policymaking designed to close the opportunity gap. One or more legislative achievements, however significant, will not solve an opportunity crisis that has been generations in the making, and a legislative strategy that offers nothing to the other party will serve only to ensure that Congress remains intransigent.
Achieving bipartisan support will require sometimes painful sacrifices. Compromises will anger constituencies within the president's party and cost votes. Significant cabinet positions will need to be offered to respected members of the other party in an effort to secure swift confirmations and build critical relationships. Ideally, these individuals would come with significant executive experience and reputations for bipartisanship at the state level to help maximize efficiency in program execution between the federal and state levels.
The president and White House staff cannot, and should not, be everywhere. The president needs the force multiplication and expertise of the cabinet to bear the lion's share of public-facing responsibilities, communicating policy to the public, and "swarming" any opposition. As much as practicable, cabinet members should appear together in teams of two or three to showcase ways that interagency cooperation is making government work better. Assistants to the president should be tasked with driving accountability and oversight related to interagency cooperation and policy coordination. These individuals must have either deep agency, Capitol Hill, or other government experience to ensure that from the moment the president is inaugurated, the staff has the experience necessary to manage the complex political and personal dynamics at play between the legislative and executive branches.
## Executive Action Priorities
The next president's first inaugural address should directly confront the threat that the opportunity gap poses to a strong state of the union. It should insist that America will grow fastest and most equitably when we invest in all of our national assets—individuals, infrastructure, innovation, and our environment. The president should offer a vision of _inclusive capitalism_ that challenges the nation to embrace the idea that developing and building national assets is critical to a better economic future, and that public goods result when smart public investments create a platform for fair private sector competition and robust job creation. The address should also insist that the American people endorse opportunity creation for all by declaring that "the era of mass incarceration is over" and offer a vision that showcases the fiscal–and moral—intelligence of effective investments in all people.
On the first day of the new administration, the president should sign an executive order establishing the American Community Wealth Building Initiative. Cochaired by the chairs of the National Economic Council and the Domestic Policy Council, the initiative should state clearly that the administration's principal domestic policy objective is to organize federal assets and resources to support income growth and wealth creation for low- and middle-income Americans and provide an initial framework for interdepartmental and interagency collaboration. The goal of the executive order would be to frame for agencies the expectation of a new era of collaboration. It would be followed by a series of circulars from the Office of Management and Budget that would provide specificity designed to make interagency collaboration the rule, not the exception.
## Legislative Priorities
The first of two interdependent pieces of legislation the administration should seek to pass in its first year is tax reform. America can no longer afford to merely ponder the problem of inequity in our tax code. Tax reform is substantively vital and politically critical. Despite the decade-long insistence of some, there is not enough waste, fraud, and abuse at any level of American government to pay for our nation's mission-critical needs. Without additional revenue, it will be difficult to build or scale opportunity programs in a fiscally responsible manner. Without removing the tax code's bias against low-income Americans, it will be next to impossible to confront the challenge of intergenerational poverty. Moreover, well-crafted tax reform, marketed as a bipartisan plan for creating wealth among low- and middle-income Americans, stands at least some chance of garnering enough votes from the other party to help build public confidence in government.
A successful tax package must:
* • Redirect the tax code's bias away from high-income Americans by reforming or eliminating tax breaks that benefit the wealthy and redirect those savings toward income support for low- and middle-income Americans. Two examples include employer contributions for medical insurance premiums and the mortgage interest deduction, which both disproportionately benefit wealthier taxpayers.
* • Redesign income supports, such as the earned income tax credit, to increase access and ease of use.
* • Build on past bipartisan efforts—supported by members of Congress, governors, mayors, and the private sector—to create the revenue needed to pass infrastructure legislation that would benefit job-seekers, low- and middle-income communities, urban and rural America, and the private sector.
The second piece of legislation the administration should pursue in its first year is the Raising Incomes and Creating Hope Act. The RICH Act should build on successes achieved during the Obama administration by leveraging policy principles that are:
* • Place-based: Policies that bring coordinated action and resources to bear on targeted geographies
* • Evidence-based: Policies that draw on the best data and research to understand what works and that use those lessons to inform continuous improvement in policymaking.
* • Pay-for-success-based: Policies that contain financing tools that reward demonstrated achievement of outcomes.
With these policy principles at its core and using the language of community wealth-building, the RICH Act is designed to help rebuild pathways to opportunity in America, beginning with children, using a bipartisan "what works" approach. Rhetorically and politically, the RICH Act should not be marketed as an act of societal transformation or sold as a policy home run. Rather, building on passage of a successful tax package that begins to remove the structural impediments to opportunity and mobility while incentivizing job creation and urban and rural development, the RICH Act should be sold as a modest, practical set of steps that members of both parties support to help more of our kids succeed.
It should be designed as the next step in a political process to give a bipartisan majority of members of Congress something to be for while equitably implementing legislation recently passed, including the Every Student Succeeds Act and increasing access to and completion at two- and four-year colleges and universities. By using the three principles outlined above, the RICH Act promotes outcomes-focused state and local innovations that can close the opportunity gap. It sets forth clear success measures and rigorous accountability structures and encourages successful innovation.
The RICH Act should focus on scaling up successful pre-natal and early childhood development practices. Place-based programs focused on the so-called "two-generation approach"—programs designed to serve the needs of parents and children together—could be of particular use in helping drive bipartisan consensus. The act should bring a particular focus to scaling wrap-around service provision in elementary and secondary education and should seek to:
* • Incentivize anti-push-out policies that reverse the trend of channeling children into the juvenile and criminal justice systems to keep children in school.
* • Encourage innovation around the middle schools challenge, especially through use of middle school redesign funds.
* • Promote reengagement centers that bring dropouts back into the education system and aggressively support early-college high school programs that give students the opportunity to receive a high school diploma while also accumulating college credits or an associate degree, and career and technical training to close the gap between education and employment.
The act should also address the challenges faced by citizens who have served time in a prison through antirecidivism workforce development initiatives and policies. It is imperative that the administration and the federal government constantly promote successful antirecidivism practices and policies not only to reduce the human and fiscal toll of mass incarceration but also to demonstrate the upside—for individuals, families, communities, and the economy—of successful reintegration initiatives.
Rebuilding confidence in effective government, investing and redirecting resources toward proven programs that promote social capital and wealth creation among low- and middle-income Americans, and accepting the risks and burdens that bipartisanship requires are all first steps in the first year toward transforming the myth of opportunity for all in America into a reality.
# Presidential Precedents
# Oral Histories Show How Other Presidents Dealt with Opportunity Issues
Michael Nelson
Equal opportunity is the core American value that animates the American dream of upward mobility. All four presidents from Jimmy Carter to Bill Clinton whose administrations' histories have been published as part of the Miller Center's Presidential Oral History Program either faced a challenge or launched an initiative during their first year in office relating to the expansion of opportunity in the service of greater social and economic mobility.
These challenges and initiatives involved opportunity and mobility policies ranging from food stamps to AIDS to disability rights to health care reform. The cases discussed below includes some issues that were foreseeable as early as the election campaign that brought the new president to office and others that arose unexpectedly after the inauguration. How the four presidents handled these situations illuminates the variety of possibilities that any new chief executive may face.
## Jimmy Carter and the Food Stamp Act
_The food stamp program (today called SNAP) was launched in 1964 on the premise that adequate nutrition is a prerequisite for seizing opportunities to advance in society. In Carter's first year, a problem in the program's design became evident, and he and his administration decided to fix it. For a newly elected Democratic president entering office after eight years of Republican administrations, addressing a core Democratic program such as food stamps—especially one that he believed in deeply—was a matter of some urgency._
**Secretary of Agriculture Robert Bergland:** I think our [department's] total budget was $25 billion or something, of which easily half went into feeding programs, school lunches, WIC, food stamps, and these things. Every year in Carter's budget, that budget went up, on my recommendation and with his strong support. He believed in that. He figured it was the decent thing to do. On one occasion I provided him some documentation. So he saw the cost-benefit analysis. He was impressed by that, but that wasn't what drove him. His instincts on these social programs were purely humanitarian. I think it was really his test. . . .
We had a problem in that there was a lot of cash that was transacted and would change hands. This person would get certified as being eligible for food stamps, but they have to put up 25 percent. So if they are eligible for $100 in food stamps, they had to come up with $25. . . .
What we wanted to do was get rid of the purchase requirement so that instead of this person paying $25 for this book of coupons and having all the trouble we had with handling that money that got stolen, we just say, "We will reduce your food stamp benefits. Instead of paying us $25 and getting a $100 book, you don't pay us anything, and we'll give you $75. . . ."
So we eliminated the cash. I went up to [House Agriculture Committee] Chairman [Tom] Foley and some of the people on the House side, and they've known about this purchase requirement problem for a long time. They said, "Yes, this makes sense. This is the smart thing to do." I went over to see Chairman [Herman] Talmadge and the more conservative senators, and they just had a fit. They said, "What? You are going to give these things away free? Free food stamps? It's a mistake. The president ought to have his head examined.' I said, 'The president doesn't know about this. . . ."
So I met with the president, and he looked over the evidence after he had read all the papers. He said, "Go ahead, we'll take the heat."
_Part of Carter's achievement in reforming the food stamp program involved a willingness to act in defiance of a powerful congressional leader, something that a first-year president may only be willing to do when it involves an important commitment that engages his or her core values._
## Ronald Reagan and AIDS
_HIV/AIDS was first diagnosed in the United States in 1981 during Reagan's first year as president. Across the country, fears verging on panic were rampant about how the then fatal disease was spread, with resulting discrimination against AIDS sufferers. The Reagan administration was criticized by many who felt that both the president and relevant agencies of the executive branch were slow to take the health crisis seriously because most of its victims were gays or lesbians. In time, federal research and prevention programs were launched, but a moment came when the president decided to use the symbolic authority of his office on behalf of the victims._
**Secretary of Health and Human Services Otis Bowen:** [The organizations] that gave me the most trouble were the AIDS groups thinking they were getting shortchanged all the time—even though the budget would go up and up and up. They had good points. It was a very, very serious thing, and I think, earlier, the administration didn't think it was as important as it really was. . . . And in the AIDS era, you didn't dare, in that administration, talk about the use of condoms. The only program that the administration would permit was "Just don't."
**White House physician John Hutton:** The president had always been curious about the science of medicine, and wondered if our researchers were making any progress with identifying the infectious agent. I volunteered that Dr. [Robert] Gallo [of the National Institutes of Health] had isolated a hitherto unknown virus for which man seemed to have no defense.
The president replied, "You mean like the measles virus, but one that won't go away, that arouses no immune response?" He understood my explanation and its implication.
I'm not sure with whom he discussed this disease entity, but he soon learned of the patients being studied at NIH, and of implications for panic from ignorance of how the disease was being spread. He asked if it would help with the awareness of the disease if he went out to NIH and visited with patients—especially with the children who had contracted the disease from their mothers.
How well I remember the afternoon when our motorcade took us to NIH, and we were escorted to the ward that housed these patients. He fully understood the routes of communicability of this disease, and in an effort to lend a calmness and understanding to the nation, he moved from room to room, visited with the mothers in his most congenial way, picked up the children as would any father, all to the satisfaction of the medical staff and photographers, and hopefully the curious and concerned citizens of our country.
Within a week an advisory committee was created, initially chaired by a member of the Mayo Clinic faculty and then by a former Chief of Naval Operations.
The president has attempted to enlighten the world about this threat, and at the same time to assuage their fears about the contagious nature of this most dangerous disease from which no one had yet survived.
_Most presidential initiatives involve tangible action—legislation proposed, executive orders issued, or appointments made. Certainly the fight to address the AIDS crisis involved all of those things. But Reagan also showed that the power of the president to alleviate public fears through personal, highly visible conduct is important as well._
## George H. W. Bush and the Americans with Disabilities Act
_Removing barriers to full participation in American life for people with disabilities was one of Bush's stated goals during his 1988 presidential campaign and his first year in office. Finding a way to achieve that goal without placing undue burdens on job-creating employers was the challenge. Working, sometimes contentiously, across party lines with Senator Edward M. Kennedy and others, Bush achieved passage of landmark legislation. The Americans with Disabilities Act (ADA) was enacted during Bush's second year as president, but only because his administration had laid the foundation during his first year._
**White House Chief of Staff John Sununu:** This is a president that believes in a set . . . of fundamental conservative principles of minimizing government involvement, of free market, . . . and of a kinder, gentler (remember that phrase?) set of policies that deal with individual needs and recognizing that different people have different needs. That was the president's broad philosophy. . . . It is embodied in ADA, recognizing that a segment of our society wasn't able to participate because of handicaps and trying to give them an avenue to participate without creating a club or a sword that is used to hamper the capacity of the private sector to function . . .
During the election of '88 [Bush asked], What are we going to campaign on as domestic issues? . . . [Bush] campaigned on a concern for all Americans and out of that concern for all Americans grew, I think, his commitment to the civil rights bill and the ADA. . . .
On the domestic side, there were two things that were a priority: Number one, let's start doing something about the budget and the deficit, had to get a hold of that. And number two, start to create the interactions on issues . . . like civil rights and ADA, that gets us to the point where we get them passed. We may not pass them this year, but you have to start this year. . . .
The first ADA bill looked like it had 110 senators supporting it out of a hundred. Nobody had read anything but the title and how could you vote "no" for a piece of legislation called the Americans with Disabilities Act? But the first bill was terrible, the first draft. It created about a hundred new entitlements, slight exaggeration but only slight. . . .
**Senator Edward M. Kennedy, chairman of the Senate Education and Labor Committee:** I asked to go down to see his chief of staff, John Sununu. It was in the late summer, and he was decent to see me, but he was very abrupt and to the point. He said, "Look, we're not going to support any social legislation. . . ."
But Bush walked by where we were, and I asked if I could go in and see him. He said fine. I went in. . . . And I said why I was down here, and he said, "Well, that's of some interest to me, and I'll talk to John further about it. . . ."
I went down to see Sununu some weeks later; this was when we sat in his office and he said, . . . "What is this going to mean in New Hampshire at the Sunapee Ski Resort? Somebody comes up in a wheelchair and wants to be able to go to the top of the mountain. What are we going to have to do? Are we going to have to retrofit every chair?" I said, "No, no we're not going to have to do that." He said, "What are we going to have to do?"
I said, "We're going to have to make reasonable accommodation." He said, "Well, what's reasonable—every tenth, every hundredth?" I said, "We make the decision based on how many people in wheelchairs want to go, what the cost is. It's going to have to be reasonable."
**Sununu:** It was a very hard kind of negotiation because . . . we're struggling to build into it provisions that the business community was desperately coming to us with concerns about. But over time it evolved, and it evolved primarily because of the chairman, Kennedy. . . .
**Kennedy:** One of the final meetings we had on disability—now that it was all worked out, and the Republicans wanted to get their stamp on it—was with [Senate Republican leader Bob] Dole up in his leadership office. . . .
**Dole:** [The ADA is] a great piece of civil rights legislation. . . . I had a strong interest because of my disability. . . . [Attorney General Richard] Thornburgh has a son, of course, who had a very severe disability. . . .
**Thornburgh:** The ADA act was signed on July 26, 1990 . . . and it was a triumphant day when that bill was signed. . . . And the president's speech was marvelous. He ended it with a call, "To let these shameful walls of exclusion finally come tumbling down," a reference to the Berlin wall. It was neat, a good exercise. That's the way the system ought to work.
_Bridging partisan divides is never easy in Washington. But the likelihood of success is greater when, as with the rights of the disabled, the issue is not one that traditionally separates the parties and when the work of coalition-building begins in the first year, with the midterm election still a year away and the next presidential election three years in the future._
## Bill Clinton and Health Care Reform
_Not every first-year initiative is successful. Clinton's ambitious plan to extend health care to all as a necessary condition for economic advancement and social mobility was grounded in his often stated governing values of "opportunity, responsibility, and community." But it foundered on the shoals of complexity traceable in part to the flawed legislative design process he created._
**Deputy Domestic Policy Adviser Bruce Reed:** [In our first year] we wanted to flood the circuits. We figured that Congress would slow-walk enough things as it was, so that we didn't have to make it any easier for them. One of the areas where we ran into the most difficulty was that the congressional system is not well set up to handle a couple of high priorities at the same time. Bill Clinton actually wanted to pursue welfare reform and health care at the same time, because he felt that they were intrinsically related as policy and that they spoke to different anxieties of the electorate. Unfortunately, they went through the same committees. And even worse, the House wanted to do health care and not welfare reform, and the [Senate] Finance Committee wanted to do welfare reform and not health care.
**Health care economics adviser David Cutler:** [There] was a fateful meeting on health care [during the transition] in Little Rock. . . . It's with the president. The first lady is there. She sat in the back of the room, quiet.
**Secretary of Health and Human Services Donna Shalala:** It's a famous meeting because . . . the president talked about how he was going to run the health care reform process with his own staff [rather than my department]. . . .
There was no way I could turf fight over that. . . . It was screwy, but when the president and the first lady decide they want to do something first on, unless it's illegal you support it.
**Health care adviser Chris Jennings:** I think in fairness it was more of an illustration of the importance of the issue to the president, I think, on Capitol Hill and elsewhere. "Well, gee, he's willing to put her out there and put their capital on the line. He must really care about this issue."
**Deputy Domestic Policy Adviser William Galston:** To be direct—the way the leadership of that effort was constituted did not invite a lot of vocal dissent. If you really wanted to screw up things with the first lady (and arguably with the president as well), expressing doubts about the health care process or its product was a damn good way to go about it.
**Council of Economic Advisers member Alan Blinder:** Anyway, the idea was to bundle the health care reform into [the congressional budgetary] reconciliation [process], on the grounds that you needed only 51 votes and that might be the only way they could get the health care bill through the Congress . . . one humongous bill that can't be filibustered.
As one, to a man and woman, the economic team went nuts over that idea, for a variety of reasons. But the simplest of which, and the most compelling to me, was there was no way in hell we were going to have a health care plan ready by the middle of February [1993, when Congress would be acting].
**OMB Director Leon Panetta:** I can remember spending a lot of time in the [first lady's health care] task force saying, "You've got to have a simple way to explain things, and it's not just to the American people, but to a member of Congress. If you take more than two sentences to explain what's in a piece of legislation, chances are you're going to lose them. . . ." I kept saying, "This thing is like a Rubik's Cube in the sense that no matter how you talk it through, you become so immersed in the detail of it that you can't say, 'What does this mean for the average citizen on the street in terms of what the impact is going to be?'"
**Council of Economic Advisers member David Cutler:** Even if you look at when the president gave his speech in September [1993], he said there were six principles—not three, not one, not two, but six. Nobody will remember six principles.
_Clinton's campaign for health care reform helped prepare the way for passage of the Affordable Care Act sixteen years later. In that sense, it was successful but premature. But as a first year initiative in 1993, it was hobbled in several ways by the president's haste to get things done, which led him to develop the proposal within the White House without adequate departmental consultation, overwhelm a crowded congressional agenda, and fail to arrive at clear and comprehensible ways of explaining a very complex program to the American people._
# Contributors
Melody Barnes, Professor of Practice in Public Affairs and Co-Director of the University of Virginia's Democracy Initiative, is Co-Founder and Principal of MBSquared Solutions LLC, a domestic strategy firm. She is also a Senior Fellow at NYU's Robert F. Wagner School of Public Service. From January 2009 to January 2012, she was Assistant to the President and Director of the White House Domestic Policy Council, a position in which she provided strategic advice to President Obama and worked closely with members of the cabinet coordinating the domestic policy agenda across the administration. Until 2008, Barnes was the Executive Vice President for Policy at the Center for American Progress, a progressive research institute and think tank. From 1995 until 2003, Barnes worked for Senator Edward M. Kennedy on the Senate Judiciary Committee; she served as his chief counsel from 1998 until she left the committee in 2003.
William A. Galston holds the Ezra K. Zilkha Chair in the Brookings Institution's Governance Studies Program, where he serves as a Senior Fellow. Galston is an expert on domestic policy, political campaigns, and elections. His current research focuses on designing a new social contract and the implications of political polarization. He is also College Park Professor at the University of Maryland. Prior to January 2006, he was Saul Stern Professor and Acting Dean of the School of Public Policy, University of Maryland. A participant in six presidential campaigns, he served from 1993 to 1995 as Deputy Assistant to President Clinton for Domestic Policy.
Cristina Lopez-Gottardi Chao is Assistant Professor and Research Director for Public and Policy Programs at the University of Virginia's Miller Center of Public Affairs. Her research focuses on democratization studies and the evolving nature of U.S.-Cuba relations. Prior to 2008, she held positions at Emory University's Institute for Comparative and International Studies, the University of Miami's North-South Center, and the Latin American and Caribbean Center.
Guian McKee is Associate Professor of Presidential Studies at the University of Virginia's Miller Center of Public Affairs. He is the author of _The Problem of Jobs: Liberalism, Race, and Deindustrialization in Philadelphia,_ published in 2008 by the University of Chicago Press. He is also the editor of three volumes of the Miller Center's series on the presidential recordings of Lyndon Johnson, with a particular focus on the War on Poverty and the wider development of the Great Society. His research interests include the history of health care, urban, and antipoverty policy.
Michael Nelson is the Fulmer Professor of Political Science at Rhodes College and a Senior Fellow at the Miller Center. He has published multiple books, the most recent of which are _Trump: The First Two Years_ and _Resilient America: Electing Nixon, Channeling Dissent, and Dividing Government._ Other recent books are _The American Presidency: Origins and Development, 1776–2018,_ with Sidney Milkis; _The Presidency and the Political System,_ tenth edition; and _The Elections of 2016._ He is senior editor of the American Presidential Elections book series for the University Press of Kansas.
Margaret O'Mara is Associate Professor of History at the University of Washington and a historian of the modern United States, specializing in political, economic, and urban history. Her scholarly work examines America's century-long transition from a manufacturing to a service economy, the role of political leaders and political institutions in this transition, and how economic change affects built environments, social equity, and culture. She is the author of _Cities of Knowledge: Cold War Science and the Search for the Next Silicon Valley,_ which explores how Silicon Valley came to be and what the Cold War had to do with it, as well as a number of articles and book chapters examining various intersections of cities, politics, and technology. Her most recent book, _Pivotal Tuesdays_ explores four game-changing presidential elections of the twentieth century (1912, 1932, 1968, 1992) and places these campaign sagas in a broader social and cultural context. O'Mara is also a faculty affiliate of the West Coast Poverty Center and the Harry Bridges Center for Labor Studies. Prior to her academic career, O'Mara served as an appointee in the Clinton-Gore administration, working on economic and antipoverty policy at the White House and the U.S. Department of Health and Human Services.
Robert C. Pianta is Dean of the Curry School of Education at the University of Virginia. He also holds positions as the Novartis Professor of Education, Founding Director of the Curry School's Center for Advanced Study of Teaching and Learning (CASTL), Professor of Psychology at the UVa College of Arts & Sciences, and Director of the National Center for Research in Early Childhood Education. Pianta's research and policy interests focus on teacher-student interactions and relationships and on the improvement of teachers' contributions to students' learning and development. He is the author of more than 250 articles, fifty book chapters, and ten books, and has been a principal investigator on research and training grants totaling over $55 million. Among other research measures and instruments, Pianta is the creator of an observational assessment of teacher-student interactions known as the Classroom Assessment Scoring System, or CLASS. CLASS is used by every Head Start program in the country, affecting 50,000 teachers and over half a million students.
Richard Schragger is the Perre Bowen Professor at the University of Virginia School of Law, where he has taught for fifteen years. His scholarship focuses on the intersection of constitutional law and local government law, federalism, urban policy, and the constitutional and economic status of cities. He is the author of _City Power: Urban Governance in a Global Age._
Peter Wehner is a Senior Fellow at the Ethics and Public Policy Center. He writes widely on political, cultural, religious, and national security issues. In 2015 he was named a contributing opinion writer for the _New York Times._ Wehner served in the Reagan and George H. W. Bush administrations prior to becoming deputy director of speechwriting for President George W. Bush. In 2002 he was asked to head the Office of Strategic Initiatives, where he generated policy ideas, reached out to public intellectuals, published op-eds and essays, and provided counsel on a range of domestic and international issues. He was also a senior adviser to the Romney-Ryan 2012 presidential campaign.
# Miller Center Studies on the Presidency
George C. Herring
_The War Bells Have Rung: The LBJ Tapes and the Americanization of the Vietnam War_
William I. Hitchcock and Melvyn P. Leffler, editors
_The Dangerous First Year: National Security at the Start of a New Presidency_
Michael Nelson, Jeffrey L. Chidester, and Stefanie Georgakis Abbott, editors
_Crucible: The President's First Year_
Michael Nelson
_Trump's First Year_
William J. Antholis and Larry J. Sabato, editors
_Broken Government: Bridging the Partisan Divide_
Douglas A. Blackmon, editor
_Race: The American Cauldron_
Nicole Hemmer, editor
_Communication: Getting the Message Across_
Guian McKee and Cristina Lopez-Gottardi Chao, editors
_American Dreams: Opportunity and Upward Mobility_
Sidney M. Milkis and David Leblang, editors
_Immigration: Struggling over Borders_
University of Virginia Press
© 2019 by the Rector and Visitors of the University of Virginia, Miller Center
All rights reserved
_First published 2019_
ISBN 978-0-8139-4021-2 (e-book)
Cover photo: Shutterstock.com/Sean Pavone
# Notes
## Introduction
Philip Bump, "48 Percent of Millennials Think the American Dream Is Dead. Here's Why," _Washington Post,_ December 10, 2015. <https://www.washingtonpost.com/news/the-fix/wp/2015/12/10/48-percent-of-millennials-think-the-american-dream-is-dead-heres-why/>.
"The American Middle Class Is Losing Ground," PewResearch Center, December 9, 2015, <http://www.pewsocialtrends.org/2015/12/09/the-american-middle-class-is-losing-ground/>.
America's Shrinking Middle Class: A Close Look at Changes within Metropolitan Areas, Pew Research Center, May 11, 2016, <http://www.pewsocialtrends.org/2016/05/11/americas-shrinking-middle-class-a-close-look-at-changes-within-metropolitan-areas/>.
U.S. Census Bureau, <https://www.census.gov/library/publications/2016/demo/p60-256.html>.
The Miller Center's Presidential Oral History program is available at <http://archive.millercenter.org/oralhistory>.
## Longing for Uplift
Abraham Lincoln, "First Message to Congress at the Special Session," Washington, D.C., July 4, 1861.
Elizabeth Kneebone and Natalie Holmes, "U.S. Concentrated Poverty in the Wake of the Great Recession," Brookings Institution, March 31, 2016, <https://www.brookings.edu/research/u-s-concentrated-poverty-in-the-wake-of-the-great-recession/>.
Isabell Sawhill and Ron Haskins, "Five Myths About Our Land of Opportunity," _Washington Post,_ November 1, 2009, <http://www.washingtonpost.com/wp-dyn/content/article/2009/10/30/AR2009103001845.html>.
"Direction of Country: Trump Job Approval, Congressional Job Approval," poll results, _Real Clear Politics,_ <https://www.realclearpolitics.com/epolls/other/direction_of_country-902.html>.
Andrew Kohut, "Campaign Stops; Don't Mind The Gap," _New York Times,_ January 27, 2012, <https://campaignstops.blogs.nytimes.com/2012/01/26/dont-mind-the-gap/>.
David Brooks, "The Working Nation," _New York Times,_ October 23, 2014, <https://www.nytimes.com/2014/10/24/opinion/david-brooks-the-working-nation.html>.
Walter Russell Mead, "The Energy Revolution and Its Biggest Losers," Global Warning Policy Forum, October 7, 2012, <http://www.thegwpf.com/walter-russell-mead-the-energy-revolution-and-its-biggest-losers/>.
Scott Winship, "Why the 1996 Welfare Reform Benefitted Poor Children," _National Review Online,_ September 1, 2016, <https://www.manhattan-institute.org/html/why-1996-welfare-reform-benefited-poor-children-9215.html>.
Robert D. Putnam, _Our Kids: The American Dream in Crisis_ (New York: Simon and Schuster, 2015).
W. Bradford Wilcox, Nicholas H. Wolfinger, and Charles E. Stokes, "One Nation, Divided," _Marriage and Child Wellbeing Revisited_ 25 (2015): 111–27.
Ibid., 122.
Carl M. Brauer, _Presidential Transitions: Eisenhower through Reagan_ (New York: Oxford University Press, 1986), 62.
## Opportune Moment
Congressional Budget Office, The Budget and Economic Outlook, 2016 to 2026," January 25, 2016, <https://www.cbo.gov/publication/51129>.
Raj Chetty, The Impact of Neighborhoods on Economic Opportunity: New Evidence and Policy Lessons," Brookings Institution, n.d., <https://www.brookings.edu/wp-content/uploads/2015/05/RajChettyv2.pdf>.
Richard V. Reeves, "Saving Horatio Alger: Equality, Opportunity, and the American Dream," Brookings Institution, Aguust 20, 2014, <http://aa61a0da3a709a1480b1-9c0895f07c3474f6636f95b6bf3db172.r70.cf1.rackcdn.com/content/research/essays/2014/saving-horatio-alger.html>.
Federal Reserve Bank of St. Louis, "Civilian Labor Force Participation Rate: Women," <https://fred.stlouisfed.org/series/LNS11300002>.
## Pre-K Prerequisite
Jim Minervino, Ready on Day One, "Lessons from Research and the Classroom: Implementing High-Quality Pre-K That Makes a Difference for Young Children," Gates Foundation, September 2014, <https://docs.gatesfoundation.org/documents/lessons%20from%20research%20and%20the%20Classroom_September%202014.pdf>.
T. J. Sabol, S. L. Soliday Hong, R. C. Pianta, and M. R. Burchinal, "Can Rating Pre-K Programs Predict Children's Learning?," _Science,_ August 23, 2013, <http://science.sciencemag.org/content/341/6148/845.full>.
National Institute for Early Education Research, "Did State Pre-K Get Back on Track in 2015?," _State of Preschool Yearbooks,_ Rutgers Graduate School of Education, 2016, <http://nieer.org/state-preschool-yearbooks/the-state-of-preschool-2015>.
Minervino and Ready on Day One, "Lessons from Research and the Classroom."
National Institute for Early Education Research, "Did State Pre-K Get Back on Track in 2015?"
## City Power
Alex Barinka, "Iowa Spent $50 Million to Lure IBM. Then the Firings Started," _Bloomberg News,_ May 18, 2015, <https://www.bloomberg.com/news/articles/2015-05-19/iowa-spent-50-million-to-lure-ibm-then-the-firings-started>.
Michael Craw, "Deciding to Provide: Local Decisions on Providing Social Welfare," _American Journal of Political Science_ 54, no. 4 (October 2010): 906–10, 919, <http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5907.2010.00464.x/abstract>.
Chris Tausanovitch and Christopher Warshaw, "Representation in Municipal Government," _American Political Science Review_ 108, no. 3 (August 2014): 605–41, <https://www.cambridge.org/core/journals/american-political-science-review/article/representation-in-municipal-government/924BDD2C2ECD0CC9B42E5A19776BC9C3>.
## The Long Game
Dwight D. Eisenhower, "Radio Address to the American People on the National Security and Its Costs," Washington, D.C., May 19, 1953.
Ronald Reagan, "Statement on United States Immigration and Refugee Policy," Washington, D.C., July 30, 1981.
Dwight D. Eisenhower, "Farewell Address," Washington, D.C., January 17, 1961.
## Moving Up
See the website <https://results4america.org/>.
Melody Barnes and Paul Schmitz, "Community Engagement Matters (Now More Than Ever)," _Stanford Social Innovation Review,_ Spring 2016, <https://ssir.org/articles/entry/community_engagement_matters_now_more_than_ever>.
Sabrina Tavernise, "U.S. Suicide Rate Surges to a 30-Year High," _New York Times,_ April 22, 2016, <https://www.nytimes.com/2016/04/22/health/us-suicide-rate-surges-to-a-30-year-high.html>.
Gallup results, "Confidence in Institutions," _Gallup News,_ n.d., <http://news.gallup.com/poll/1597/confidence-institutions.aspx>.
"Beyond Distrust: How Americans View Their Government," Pew Research Center, November 23, 2015, <http://www.people-press.org/2015/11/23/beyond-distrust-how-americans-view-their-government/>.
Coalition for Inclusive Capitalism, "What Is Inclusive Capitalism?," n.d., <https://www.inc-cap.com/>.
See the website <http://results4america.org/tools/>.
"Ascend / What Is 2Gen?," Aspen Institute, <http://ascend.aspeninstitute.org/two-generation/what-is-2gen/>.
See the website <https://www.communitiesinschools.org/our-model/>.
See the website <https://dccentralkitchen.org/>.
# Contents
1. Cover
2. Title Page
3. Copyright Page
4. Contents
5. Introduction
6. Longing for Uplift: Americans Want to Believe Everyone Has a Fair Chance to Succeed
7. Opportune Moment: The Time Is Right for the Next President to Pursue an Opportunity Agenda
8. Pre-K Prerequisite: Ensure That Every Poor Child in America Gets Two Years of Preschool
9. City Power: The Next President Can Help Cities Boost More Workers into the Middle Class
10. The Long Game: The Next President Must Lay the Foundation for a Next-Generation Economy
11. Moving Up: The Next President Can Restore Confidence in the American Dream
12. Presidential Precedents: Oral Histories Show How Other Presidents Dealt with Opportunity Issues
13. Contributors
14. Notes
## Guide
1. Cover
2. Title Page
3. Contents
4. Copyright Page
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{"url":"https:\/\/pypi.org\/project\/qmsolve\/0.1.0\/","text":"A module for solving and visualizing the Schr\u00f6dinger equation\n\n# QMsolve: A module for solving and visualizing the Schr\u00f6dinger equation\n\nQMsolve seeks to provide easy solid and easy to use solver, capable of solving the Schr\u00f6dinger equation for one and two particles, and creating descriptive and stunning visualizations of its solutions both in 1D, 2D, and 3D.\n\n## Installation\n\npip install qmsolve\n\n\n## How it works\n\nThe way this simulator works is by discretizing and Hamiltonian of an arbitrary potential, specified as a function of the particle observables. This is achieved with the Hamiltonianconstructor. Then, the Hamiltonian.solve method efficiently diagonalizes the Hamiltonian and outputs the energies and the eigenstates of the system. Finally, the eigenstates can be plotted with the visualization class.\n\nThe visualization.superpositions method features the possibility of interactively visualizing a superposition of the computed eigenstates and studying the time dependence of the resulting wavefunction.\n\nFor a quick start, take a look to the examples found in the examples subdirectory.\n\n## Project details\n\nUploaded source","date":"2023-02-05 11:34:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4426443874835968, \"perplexity\": 1248.4770865717958}, \"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-2023-06\/segments\/1674764500251.38\/warc\/CC-MAIN-20230205094841-20230205124841-00195.warc.gz\"}"} | null | null |
Q: Disabled the a tag (link) by using javascript Does anyone how can I disabled the a tag (link) by using javascript?
Example:
<div class="sub-heading">
Contact Details
<a href="./cust_add_edit_customer.php?action=edit_customer_details&cust_code=12761">
<img class="imgVA editIconPad" src="../images/edit0.gif"
alt="Edit Contact Details" border="0" width="20" height="17">
</a>
</div>
I hope to disabled this a tag after a button been clicked.
A: I think the most user-friendly approach is to hide the link. In your button click handler do:
document.getElementById('anchorID').style.visibility = 'hidden';
Then to reenable it:
document.getElementById('anchorID').style.visibility = 'visible';
A: Use an onclick="this.onclick=function(){return false}" attribute on the a tag. If there's a lot of buttons, you should iterate through them in a JavaScript script that adds an event listener for click that is a function that returns false.
A: Hai
function disableAnchor(obj, disable)
{
if (disable) {
var href = obj.getAttribute("href");
if (href && href != "" && href != null) {
obj.setAttribute('href_bak', href);
}
obj.removeAttribute('href');
obj.style.color="gray";
}
else {
obj.setAttribute('href', obj.attributes['href_bak'].nodeValue);
obj.style.color="blue";
}
}
or
var Link_Enabled_Flag = false; // disable links - background process changes this to true when it's done
function Check_Link_Enabled(){ return Link_Enabled_Flag; }
<a href="wherever.com" onclick="return Check_Link_Enabled()"></a>
or
IE and Firefox compatible javascript to enable or disable an anchor tag
onclick="disableAnchor(this,'verify')"
function disableAnchor(Check_Obj, Check_Id){
var Anchor_id = 's';
thisCheckbox = document.getElementById(Check_Id);
thisAnchor = document.getElementById(Anchor_id);
if(thisCheckbox.checked){
//alert('Y1');
Check_Obj.setAttribute('href',''); //Check_Obj.attributes['href_bak'].nodeValue
Check_Obj.style.color="blue";
//alert('Y2');
}
else{
//alert('N1');
var href = Check_Obj.getAttribute('href');
//alert(href);
if(href && href != "" && href != null){
Check_Obj.setAttribute('href_bak', href);
}
Check_Obj.removeAttribute('href');
Check_Obj.style.color="gray";
//alert('N2');
}
}
A: Add an id attribute to the a tag you want to disable, then:
document.getElementById('the_id').href = '#';
A: You can use jquery
$('sub-heading').attr("disabled", "true");
A: Remove the attribute href like:
<a>Edit</a>
A: You can do it like this:
<a href="javascript:if (links_enabled) document.location='www.example.com';">enabled or disabled link</a>
<br/>
<a href="javascript:links_enabled = true;">enable link</a>
<br/>
<a href="javascript:links_enabled = !links_enabled;">toggle link</a>
I find this very elegant, and it will also make links work only javascript is enabled. In other words, links are disabled by default.
A: The "disabled" attribute does not stop click on a tags. But I had a A tag designed has a button, and at least, adding "disabled": true the button was styled has a real disabled button.
Now if you want to stop the click you can simply use the css property "pointer-events: none"
So you can use something like this :
$('.sub-heading a').attr("disabled", "true");
$('.sub-heading a').css('pointer-events', 'none');
And it will do the trick ;)
If you want to do it through clicking on other button :
jQuery('.my-other-button').on('click', function() {
$('.sub-heading a').attr("disabled", "true");
$('.sub-heading a').css('pointer-events', 'none');
});
A: *
*In CSS set pointer-events to none. That is :
pointer-events: none;
*Then using DOM in javascript, change it however you want like fill, painted, stroke, visible... (Here change it to none)
| {
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{"url":"https:\/\/www.mpim-bonn.mpg.de\/node\/6449\/abstracts","text":"Abstracts for Seminar on Higher Structures\n\nAlternatively have a look at the program.\n\nWhat is higher Lie theory?\n\nPosted in\nSpeaker:\nDmitry Roytenberg\nAffiliation:\nUtrecht\/Amsterdam\nDate:\nFri, 2016-01-08 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nLie theory\" refers to a functorial correspondence between\nfinite-dimensional Lie algebras and 1-connected Lie groups, elucidated\nby Sophus Lie and summarized in his three theorems, Lie I, II and III.\nThe first hint that there might be more to the story came from van Est\nwho noticed (and proved) that integrating higher Lie algebra cocycles\nrequired higher connectivity assumptions on the Lie group. Furthermore,\nattempts to build a Lie theory for Lie algebroids (and\ninfinite-dimensional Lie algebras) generally run into obstructions of a\n\nGeneralized cohomology of M2\/M5-branes\n\nPosted in\nSpeaker:\nUrs Schreiber\nAffiliation:\nCAS Prague\/MPIM\nDate:\nFri, 2016-01-15 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nWhile it is well-known that the charges of F1\/Dp-branes in type II string theory need to be refined from de Rham cohomology to certain twisted generalized differential cohomology theories, it is an open problem to determine the generalized cohomology theory for M2-brane\/M5-branes in 11 dimensions. I discuss how a careful re-analysis of the old brane scan (arXiv:1308.5264 , arXiv:1506.07557, joint with Fiorenza and Sati) shows that rationally and unstably, the M2\/M5 brane charge is in degree-4 cohomotopy.\n\n$L_\\infty$ algebras governing simultaneous deformations in algebra and geometry\n\nPosted in\nSpeaker:\nYa\u00ebl Fr\u00e9gier\nAffiliation:\nU d'Artois\/MPIM\nDate:\nFri, 2016-01-29 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nWe will explain how simultaneous deformations problems in algebra and geometry are naturally governed by non quadratic $L_\\infty$ algebras and how such algebras can be constructed by super geometry or operad theory, depending on the applications.\n\nThis will be illustrated with examples. We will consider simultaneous deformations of pairs such as couples of algebras\/morphisms, coisotropic\/Poisson, Dirac\/Courant, generalized complex\/Courant (if time permits).\n\nOpen-closed homotopy algebras\n\nPosted in\nSpeaker:\nMartin Doubek\nAffiliation:\nCharles Univ. Prague\/MPIM\nDate:\nFri, 2016-02-19 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nTopological types of string worldsheets can be organised into a modular operad QP. For closed strings, algebras over this operad are essentially topological quantum field theories formalized by monoidal functors from cobordisms into vector spaces or by commutative Frobenius algebras. There is an analogous picture for open and open\/closed strings. In the closed case, QP can be freely generated from certain cyclic suboperad P via modular envelope: in fact P=Com. In the open case, analogous result holds for P=Ass. In open-closed case, this problem is subtler.\n\nRational homotopy of the little cubes operads and graph complexes\n\nPosted in\nSpeaker:\nBenoit Fresse\nAffiliation:\nUniversit\u00e9 de Lille\/MPIM\nDate:\nFri, 2016-02-26 14:00 - 15:00\nLocation:\nMPIM Seminar Room\n\nI will report on a joint work with Victor Turchin and Thomas Willwacher about the rational homotopy of the little cubes operads (equivalently, the little discs operads).\n\nGinzburg--Kapranov criterion for Koszulness of operads, and its limitations\n\nPosted in\nSpeaker:\nAffiliation:\nTrinity College Dublin\nDate:\nFri, 2016-03-04 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nIn their seminal paper on operadic Koszul duality, Ginzburg\nand Kapranov established a remarkable functional equation that holds\nwhenever an operad is Koszul. I shall discuss some examples\ndemonstrating that non-Koszul operads do not have to violate that\ncriterion, in either of two possible (\"weak\" or \"strong\") senses.\n\nWhat good is a semi-model structure on algebras over an operad?\n\nPosted in\nSpeaker:\nDavid White\nAffiliation:\nDenison University, Granville\nDate:\nFri, 2016-03-11 14:00 - 15:00\nLocation:\nMPIM Lecture Hall\n\nIn general, it is difficult to transfer a model structure from a monoidal\nmodel category M to the category of algebras over a (colored) operad P in M.\nOften, one only ends up with a semi-model structure (i.e. where half the\nfactorization and lifting axioms only hold for maps with cofibrant domain),\nand even this traditionally requires P to be Sigma-cofibrant. I?ll explain\nwhat standard techniques still work in the context of semi-model categories,\nso that users of model categories can get by if they only have a semi-model\n\nAn Algebraic Combinatorial Approach to Opetopic Structure\n\nPosted in\nSpeaker:\nMarcelo Fiore\nAffiliation:\nU of Cambridge, UK\nDate:\nWed, 2016-03-23 10:30 - 12:00\nLocation:\nMPIM Lecture Hall\n\nThe starting point of this talk will be an algebraic connection to be\u00a0 presented briefly between\nthe theory of abstract syntax of [1,2] and the approach to opetopic sets of [4].\u00a0 This realization\nconceptually allows us to transport viewpoints between these mathematical theories and I\nwill explore it here in the direction of higher-dimensional algebra leading to opetopic\ncategorical structures.\u00a0 The technical work will involve setting up a microcosm principle\nfor near-semirings and subsequently exploiting it in the cartesian closed bicategory of\n\n \u00a9 MPI f. Mathematik, Bonn Impressum & Datenschutz","date":"2019-10-18 21:52:21","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\": 0.5884870886802673, \"perplexity\": 3694.599608590096}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986684854.67\/warc\/CC-MAIN-20191018204336-20191018231836-00352.warc.gz\"}"} | null | null |
Unusual Americana at Schoyer's Books
Item 208 is a large scrapbook kept by inventor William E. Nickerson. Nickerson's name is not well remembered, because the company he was instrumental in forming bears the name of his partner, King Gillette. Gillette had come up with the concept for a thin, disposal razor blade, but the technology was not available for sharpening such a narrow piece of steel. That was where the inventor Nickerson came in. He developed the process for holding the thin steel rigid so that it could be sharpened into a blade. This collection of Nickerson materials includes several patents of his, including an earlier one for manufacturing circular saws. Think of them as giant razors. $1,000.
Looking for some exciting reading? How about the Speeches and Addresses of William McKinley, Mr. Excitement himself? Okay, McKinley was never known for spellbinding delivery, like his opponent, William Jennings Bryan, but he creamed Bryan in two presidential elections. And item 164 is a special copy of McKinley's speeches, as it comes with the President's signed inscription, dated May 24, 1901. Three and one-half months later, he was assassinated. $250.
Item 195 is a most interesting "autobiography," that of Dr. Horace Ackley. Its title is, A Narrative of the Experience of Horace Abraham Ackley, M.D., late of Cleveland, Ohio. Since His Entrance into Spirit-Life. You may wonder how Dr. Ackley wrote this after he entered the spirit-life. I, for one, am very suspicious of autobiographies which are written after the subject has died. The explanation of this oddity is in the subtitle, Received through the Mediumship of Samuel H. Paist. Evidently, Dr. Ackley contacted Paist and requested that the latter publish his experiences as a dead man. At least that was how Mr. Paist explained it. For those interested in what life after death is like, this would seem to be a good resource. However, since the book was published in 1861, I cannot vouch that the afterlife is still the same today. $125.
It may not have been the greatest show on earth, but Professor Webster and His Performing Sea Lions were big enough to play Madison Square Garden, Atlantic City's Million Dollar Pier, and John Robinson's Circus at the turn of the century. Item 75 is four photographs of the good professor and his beached sea lions, including one with a lion giving him a wet kiss on Atlantic City's pier. $225.
Schoyer's Books may be reached by phone at 510-548-8009, or by email at dsbooks@comcast.net. | {
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} | 8,060 |
Als Steuerberater (StB) wird der Angehörige eines freien Berufs bezeichnet, der in steuerrechtlichen und betriebswirtschaftlichen Fragen berät und seine Mandanten vor der Finanzverwaltung sowie der Finanzgerichtsbarkeit vertritt. Der Beruf des Steuerberaters zählt in Deutschland und Österreich zu den klassischen Kammerberufen mit besonders strengen berufsrechtlichen Zugangsregelungen. Die Berufsbezeichnung Steuerberater ist in diesen Ländern gesetzlich geschützt.
Deutschland
Der Beruf des Steuerberaters wird als freier Beruf ausgeübt und unterliegt daher nicht den Bestimmungen der Gewerbeordnung. Der Steuerberater ist ein unabhängiges Organ der Steuerrechtspflege. Die Stellung als unabhängiges Organ der Steuerrechtspflege ist ausdrücklich einzelgesetzlich in § 32 Abs. 2 StBerG normiert, so dass Steuerberater gleichrangig und gleichberechtigt neben den anderen Organen der Rechtspflege, z. B. Rechtsanwälten, stehen. Das Berufsbild ist ausgerichtet auf den Vorrang der persönlichen berufsspezifischen Leistung vor den wirtschaftlichen Aspekten der Tätigkeit. Es ist geprägt durch die unabhängige und unparteiliche Erfüllung der den steuerberatenden Berufen übertragenen Aufgabe, eine umfassende Hilfeleistung in Steuersachen zu gewährleisten. Die Steuerberatung ist ein Teil der Rechtsberatung; die damit verbundenen Berufsaufgaben dienen der Steuerrechtspflege, einem wichtigen Gemeinschaftsgut. Steuerberater haben ihre Tätigkeit nach § 57 Abs. 1 StBerG unabhängig, eigenverantwortlich, gewissenhaft, verschwiegen und unter Verzicht auf berufswidrige Werbung auszuüben. Sie bedürfen der Bestellung.
In Deutschland sind der Tätigkeitsbereich und die Zulassung umfassend im Steuerberatungsgesetz und der dazugehörigen Durchführungsverordnung geregelt. Viele Beratungstätigkeiten gehören zu den Vorbehaltsaufgaben und dürfen nur von zugelassenen Steuerberatern und einigen wenigen weiteren Berufsangehörigen ausgeführt werden.
Die berufsständische Vertretung der Steuerberater in Deutschland sind die Steuerberaterkammern, die sich unter dem Dach der Bundessteuerberaterkammer zusammenschließen. Eine freiwillige Interessenvertretung erfolgt über den Deutschen Steuerberaterverband mit Sitz in Berlin.
Geschichte
Im Kaiserreich und der Weimarer Republik legte alleine die Gewerbeordnung fest, inwiefern man sich steuerberatend betätigen konnte. Ein Berufsrecht für Steuerberater gab es nicht, so dass sich ein jeder als "Steuerberater" bezeichnen konnte. Gängige Bezeichnungen waren "Steuerberater", "Steueranwalt", "Steuerrevisor" oder "wissenschaftlicher Steuerverständiger". Vor dem Ersten Weltkrieg boten häufig Juristen eine steuerliche Beratung an.
Die Reichsabgabenordnung trat im Dezember 1919 in Kraft und fasste die unterschiedlichen Steuerregelungen im Deutschen Reich in einem Gesetz zusammen. § 88 ermöglichte es Steuerpflichtigen erstmals, einen Bevollmächtigten für ihre Steuersachen zu benennen. Der Begriff "Steuerberater" wurde noch nicht verwendet. Der Bevollmächtigte bzw. Verfügungsberechtigte hatte "die Pflichten eines gesetzlichen Vertreters". Die Landesfinanzämter konnten Bevollmächtigte zulassen und zurückweisen. Rechtsanwälte und Notare konnten als Bevollmächtigte nicht zurückgewiesen werden. Im Rahmen eines Referentenentwurfes des Bundesministerium der Finanzen im Jahr 2015 wurde dies zur Diskussion gestellt. Dies wurde aufgrund von erheblichem Widerstand aus dem Berufsstand nicht realisiert.
Die Steuerberatung wurde nach dem Ersten Weltkrieg durch neue Gesetze und Vorschriften immer komplexer, so dass sich erste rein steuerrechtlich ausgerichtete Sozietäten bildeten. Kaufmännisch gebildete Angestellte, Bücherrevisoren, Diplom-Volkswirte, Diplom-Kaufleute oder Diplom-Bücherrevisoren und im Vergleich zu vor 1919 prozentual weniger Juristen beschäftigten sich mit der Steuerberatung.
Die Bezeichnung "Steuerberater" fand erstmals rechtliche Erwähnung im "Gesetz über die Zulassung von Steuerberatern" vom 6. Mai 1933. Juden oder "Personen, die sich im kommunistischen Sinn betätigt haben" bekamen keine Zulassung als Steuerberater, Bevollmächtigte oder Beistände in Steuersachen oder verloren ihre bisherige Zulassung.
Ab 1936 konnte ein Finanzamt einer Person für "uneingeschränkte geschäftsmäßige Hilfeleistung in Steuersachen" die Erlaubnis erteilen. Der "Helfer in Steuersachen" war entstanden; für ihn gab es keine besondere Ausbildung oder Prüfungspflicht.
Zwischen 1935 und 1941 folgten Regelungen zur Zulassung und Prüfungsordnung und somit das Berufsrecht der Steuerberater. Steuerberater mussten nun einen Hochschulabschluss in Volks- oder Betriebswirtschaft oder eine ähnliche Vorbildung vorweisen.
1943 erfolgte der Zusammenschluss der Steuerberater und Steuerberatungsgesellschaften in der "Reichskammer der Steuerberater" mit Sitz in Berlin.
1961 wurde durch das "Gesetz über die Rechtsverhältnisse der Steuerberater und Steuerbevollmächtigten (Steuerberatungsgesetz)" das Berufsrecht der Steuerberater und "Helfer in Steuersachen" vereinheitlicht. Der Beruf "Helfer in Steuersachen" wurde in "Steuerbevollmächtigter" umbenannt und für diese wurde eine Prüfungspflicht eingeführt.
Das "Zweite Gesetz zur Änderung des Steuerberatungsgesetzes" von 1972 vereinigte die beiden Berufe Steuerberater und Steuerbevollmächtigter zu einem einzigen Beruf des Steuerberaters.
Steuerberater sind in besonderer Weise vom Fachkräftemangel betroffen. Im Januar 2023 gaben 75 Prozent der befragten Kanzleien in einer Umfrage des Ifo Instituts für Wirtschaftsforschung an, ihre Geschäfte aufgrund von fehlendem Personal einschränken zu müssen.
Tätigkeitsbereich
Steuerberater haben die Aufgabe der Beratung und Vertretung ihrer Mandanten in allen steuerrechtlichen Angelegenheiten, zur Vertretung in finanzgerichtlichen Prozessen und zur Beratung in betriebswirtschaftlichen Fragen. Die Tätigkeit kann selbständig oder im Angestelltenverhältnis ausgeübt werden.
Die Aufgaben des Steuerberaters bestehen hauptsächlich in der vorausschauenden Beratung für eine optimale Steuervermeidung, der Erstellung von Buchführungen, Jahresabschlüssen und Steuererklärungen sowie der anschließenden Überprüfung von Steuerbescheiden und der Vertretung des Mandanten in Streitfällen mit dem Finanzamt und vor dem Finanzgericht.
Nicht zulässig ist die Rechtsberatung auf anderen Rechtsgebieten (Vorbehaltsaufgabe der Rechtsanwälte) und die Prüfung von Jahres- und Konzernabschlüssen (Vorbehaltsaufgabe der Wirtschaftsprüfer).
Der Steuerberater muss die Tätigkeiten nicht vollständig selbst ausüben, sondern kann sich der Hilfe von fachkundigem Personal (Steuerfachangestellte, Steuerfachwirte etc.) bedienen. Bedingung ist, dass die Mitarbeiter ausschließlich weisungsgebunden unter der fachlichen Aufsicht und beruflichen Verantwortung des Steuerberaters tätig werden.
Die Tätigkeit des Steuerberaters umfasst im Einzelnen die folgenden Aufgaben:
Beratung in Fragen der Steuerdeklaration (Steuererklärung)
Führung der Buchhaltung, insbesondere für gewerbliche Mandanten
Führung der Aufzeichnungen für freiberufliche Mandanten
Erstellung der Lohn- und Gehaltsabrechnung für Arbeitgeber
Erstellung von Jahresabschlüssen für bilanzierende Mandanten
Erstellung der Einnahmenüberschussrechnung für nichtbilanzierende Mandanten
Erstellung von Steuererklärungen im unternehmerischen und privaten Bereich
Erstellung schriftlicher Erläuterungsberichte zum Jahresabschluss
Erstellung von Formulierungsvorschlägen für die Abfassung des Anhangs oder des Lageberichts bei Kapitalgesellschaften
Beratung in Fragen der Steuerrechtsdurchsetzung
Prüfung von Steuerbescheiden auf ihre Rechtmäßigkeit
Vertretung von Mandanten gegenüber Finanzämtern und Zollbehörden
Vertretung von Mandanten vor Gerichten der Finanzgerichtsbarkeit (Finanzgerichte, Bundesfinanzhof)
Vertretung von Mandanten in finanzgerichtlichen Vorlageverfahren vor dem Europäischen Gerichtshof (EuGH)
Vertretung von Mandanten vor Gerichten der ordentlichen Gerichtsbarkeit in Steuerstrafsachen (§ 392 AO, § 138 Abs. 2 StPO)
Beratung und Vertretung von Mandanten bei Außenprüfungen und in Bußgeldverfahren
Vertretung von Mandanten in Abgabenangelegenheiten (z. B. Gewerbesteuer, Grundsteuer, Beitrags- und Gebührensachen) vor Gerichten der Verwaltungsgerichtsbarkeit (Verwaltungsgericht, Oberverwaltungsgericht)
Beratung und Vertretung von Mandanten bei Selbstanzeigen (§ 371 AO)
Beratung in Fragen der Steuervermeidung
Beratung von Mandanten in Fragen der Steuervermeidung, um die zukünftige Steuerbelastung zu minimieren
Beratung von Mandanten im Internationalen Steuerrecht (insb. Außensteuerrecht, Doppelbesteuerungsabkommen, Verrechnungspreise, Quellensteuern, Wegzugsbesteuerung etc.)
Beratung von Mandanten bei Unternehmensgründungen, steuereffizienten Finanzierungen, Umstrukturierungen (Umwandlungssteuerrecht, Tax Due Diligence-Prüfungen), der Planung der Rechtsform eines zukünftigen Unternehmens, bei Personal- und Investitionsentscheidungen, jeweils aus steuerrechtlicher Sicht
Beratung von Mandanten in Fragen der Unternehmensnachfolge und -sicherung (u. a. Erbschaft- und Schenkungsteuerrecht) sowie im Bereich der privaten Lebensführung (z. B. Vermögensanlagestrategien oder letztwillige Verfügungen)
Beratung im Zollrecht und Verbrauchsteuerrecht
Beratung in betriebswirtschaftlichen Fragen
Erledigung von Kostenrechnungen, Kalkulationen und Wirtschaftlichkeitsrechnungen erstellen, z. B. Kosten-, Rentabilitäts- und Liquiditätsanalysen durchführen, Rechnungslegung nach nationalen und internationalen Vorgaben prüfen, ggf. Aufgaben im externen Finanzcontrolling
Beratung von Mandanten in Fragen der Finanzierung und der Finanzplanung, der Beschaffung und Lagerhaltung, des Marketings und Vertriebs sowie beim Einsatz moderner Datenverarbeitungsanlagen
Beratung von Mandanten hinsichtlich ihrer Kreditwünsche, z. B. durch die Analyse von Einkommens- und Vermögensverhältnissen, Bilanzen, Gewinn-und-Verlust-Rechnungen, ggf. Durchführung von Ratings
Beratung von Mandanten in Organisationsfragen (z. B. Betriebs- und Verwaltungsablauf, Organisation des Rechnungswesens)
Freiwillige Prüfungen
Durchführung freiwilliger Prüfungen, insbesondere Abschlussprüfungen bei Unternehmen, für die keine gesetzliche Prüfung vorgeschrieben ist
Prüfung der Ordnungsmäßigkeit der Buchführung, der Bilanz, der Gewinn-und-Verlust-Rechnung sowie ggf. des Anhangs und des Lageberichts
Sonstige Aufgaben
Aktienrechtliche Gründungsprüfungen
Durchführung gesetzlicher Prüfungen bei Finanzanlagenvermittlern (§ 24 Finanzanlagenvermittlerverordnung)
Erteilung von Bescheinigungen über die Beachtung steuerrechtlicher Vorschriften in Vermögensübersichten und Erfolgsrechnungen
Wahrnehmung von Aufgaben als unabhängige/r und neutrale/r Gutachter/in in zivil-, straf-, verwaltungsrechtlichen sowie Insolvenzverfahren
Wahrnehmung von treuhänderischen Aufgaben als Vermögensverwalter/in, Testamentsvollstrecker/in, Nachlasspfleger/in, Pfleger/in, Vormund, Konkursverwalter/in, Liquidator/in oder Nachlassverwalter/in
Wahrnehmung von Vorbehaltsaufgaben, d. h. Steuerdeklarations-, Steuerdurchsetzungs- und Steuerabwehrberatung als Syndikus-Steuerberater/in im Angestelltenverhältnis für gewerbliche Unternehmen
Berufszugang
Den Beruf des Steuerberaters darf nur ausüben, wer von der zuständigen Steuerberaterkammer als Steuerberater bestellt ist. Bestellt werden kann in der Regel nur, wer die Steuerberaterprüfung i. S. d. § 37 Steuerberatungsgesetz (StBerG) bestanden hat. Ohne Steuerberaterprüfung können zum Steuerberater bestellt werden:
Professoren, die auf dem Gebiet der von den Bundes- oder Landesfinanzbehörden verwalteten Steuern mindestens zehn Jahre gelehrt haben,
ehemalige Finanzrichter, die mindestens zehn Jahre auf dem Gebiet der von den Bundes- oder Landesfinanzbehörden verwalteten Steuern tätig gewesen sind,
ehemalige Beamte des höheren bzw. gehobenen Dienstes sowie vergleichbare Angestellte der Finanzverwaltung, der gesetzgebenden Körperschaften, der Gerichte der Finanzgerichtsbarkeit sowie der obersten Behörden und der Rechnungsprüfungsbehörden des Bundes und der Länder. Diese müssen mindestens zehn bzw. fünfzehn Jahre auf dem Gebiet der von den Bundes- oder Landesfinanzbehörden verwalteten Steuern als Sachgebietsleiter bzw. Sachbearbeiter tätig gewesen sein.
Neben der fachlichen Eignung prüft die zuständige Steuerberaterkammer vor der Bestellung die persönliche Eignung der Antragsteller. Dabei ist die Bestellung vor allem zu versagen, wenn der Bewerber
nicht in geordneten wirtschaftlichen Verhältnissen lebt,
infolge strafgerichtlicher Verurteilung die Fähigkeit zur Bekleidung öffentlicher Ämter nicht besitzt,
aus gesundheitlichen Gründen nicht nur vorübergehend unfähig ist, den Beruf ordnungsgemäß auszuüben,
sich so verhalten hat, dass die Besorgnis begründet ist, er werde den Berufspflichten als Steuerberater nicht genügen, oder
keine Berufshaftpflichtversicherung vorliegt.
Die Bestellung erfolgt durch die zuständige Steuerberaterkammer durch Aushändigung der Berufsurkunde. Zuvor hat der Bewerber die Versicherung abzugeben, dass er die Pflichten eines Steuerberaters gewissenhaft erfüllen wird. Erst nach Bestellung darf die Berufsbezeichnung "Steuerberater" geführt werden.
Die Bestellung ist nur zulässig, wenn eine selbständige Tätigkeit auf dem Gebiet des Steuerrechtes (sogenannte Vorbehaltsaufgaben) oder auf den Gebieten der vereinbaren Tätigkeiten ausgeübt wird.
Zur geschäftsmäßigen Hilfeleistung in Steuersachen sind gem. § 3 StBerG auch Rechtsanwälte und Wirtschaftsprüfer befugt ohne gleichzeitig Steuerberater sein zu müssen.
Vorbereitung auf die Steuerberater-Prüfung
Da in der Steuerberaterprüfung ein fundiertes steuerrechtliches Fachwissen abgefragt wird, erfolgt die Vorbereitung in der Regel durch den Besuch privater Fachkurse. Diese werden unterstützt durch so genannte Klausurenkurse, die zum einen das Schreiben von Klausuren auf Prüfungsniveau erlauben, zum anderen die Klausurentechnik vermitteln. Ein wesentlicher Bestandteil der Vorbereitung auf die Steuerberaterprüfung ist es, sich die Klausurentechnik so gut wie möglich anzueignen. Mit Hilfe dieser Technik ist es möglich, die fachlich sehr anspruchsvollen sowie umfangreichen Klausuren in der vorgegebenen Zeit zu bearbeiten. Je nach Ausgestaltung der Vorbereitung kann diese zwischen vier Monaten und einem Jahr (oder länger) in Anspruch nehmen.
Eine Vollzeitvorbereitung ist in der Regel als Präsenzstudium mit ganztägigen Kursen ausgestaltet. Präsenzkurse dauern in der Regel ca. drei Monate. Im Anschluss daran kann ein entsprechender Klausurenkurs belegt werden. Bei diesem werden in der Regel mehrere Wochen lang täglich sechsstündige Klausuren geschrieben.
Wer hingegen einen Samstags- bzw. Abendkurs besucht, kann im Anschluss daran einen mehrwöchigen Crash-Kurs absolvieren, um so alle Inhalte nochmal aufzufrischen und kompakt wiederholen zu können. Auch hier kann parallel zum theoretischen Wissen ein Klausurenkurs absolviert werden. Ein Samstagskurs dauert in der Regel bis zu einem Jahr.
Angehende Steuerberater, welche an einem Fernstudium teilnehmen, bekommen alle Lerninhalte in regelmäßigen Abständen per Post oder elektronisch zugeschickt. Dies gilt für den theoretischen Teil und auch für Klausurenkurse. Bei den meisten Anbietern besteht die Möglichkeit, seine Klausuren einzusenden und korrigieren zu lassen.
Die gesamte Vorbereitung auf die Steuerberaterprüfung ist mit Kosten von mindestens 6.000 bis 8.000 EUR verbunden. Darin inkludiert sind die Aufwendungen für Vorbereitungskurse und Prüfungsgebühren. Hinzu kommen ggf. Übernachtungskosten am Lehrgangsort sowie der Verdienstausfall für rund 1.000 Lernstunden und oft mehr als 3 Monate Freistellung vor dem Examen.
Das Durchschnittsalter für neu bestellte Steuerberater ist 31,5 Jahre. Die jüngsten Steuerberater Deutschlands sind in der Regel 25 bis 26 Jahre alt.
Steuerberaterprüfung
Die Prüfung zum Steuerberater gilt als eine der härtesten Berufsprüfungen in Deutschland. Im langjährigen Mittel fällt rund die Hälfte der Kandidaten bundesweit durch. Bezogen auf die Zahl der zur Prüfung zugelassenen Kandidaten bestehen nur rund 40 % die Prüfung.
Die Steuerberaterkammern, die zusammen mit den Finanzverwaltungen der Länder die schriftliche Prüfung organisieren, verteidigen die Durchfallquote mit ihrem hohen Qualitätsanspruch: Der Steuerberater habe einen verantwortungsvollen Job, alle Aspiranten müssten deshalb auf Herz und Nieren geprüft werden.
Die Steuerberaterprüfung ist in einen schriftlichen Teil aus drei Aufsichtsarbeiten und in einen mündlichen Teil gegliedert. Prüfungsgebiete sind nach § 37 Abs. 3 StBerG:
Steuerliches Verfahrensrecht sowie Steuerstraf- und Steuerordnungswidrigkeitenrecht,
Steuern vom Einkommen und Ertrag,
Bewertungsrecht, Erbschaftsteuer und Grundsteuer,
Verbrauchsteuern und Verkehrsteuern, Grundzüge des Zollrechts
Handelsrecht sowie Grundzüge des Bürgerlichen Rechts, des Gesellschaftsrechts, des Insolvenzrechts und des Rechts der Europäischen Gemeinschaft,
Betriebswirtschaft und Rechnungswesen,
Volkswirtschaft,
Berufsrecht.
Voraussetzung für die Zulassung ist zum einen der erfolgreiche Abschluss eines wirtschaftswissenschaftlichen oder rechtswissenschaftlichen Hochschulstudiums oder zum anderen nach Ablegung einer kaufmännischen Ausbildung eine praktische Tätigkeit im Steuerwesen von mindestens acht Jahren. Hat man eine Zusatzqualifikation als Bilanzbuchhalter oder Steuerfachwirt absolviert, muss man mindestens sechs Jahre lang praktisch tätig gewesen sein. Darüber hinaus ist auch die Zugehörigkeit zur Finanzverwaltung als Beamter des gehobenen Dienstes eine Qualifizierungsmöglichkeit, sofern man über mindestens sieben Jahre praktisch im Bereich des Steuerwesens tätig gewesen ist.
Steuerberater können sich zum Fachberater für Internationales Steuerrecht und zum Fachberater für Zölle und Verbrauchsteuern weiterqualifizieren. Darüber hinaus gibt es weitere Fachberatertitel, die allerdings im Abstand zum Steuerberatertitel geführt werden müssen, da sie keine steuerlichen Kenntnisse, sondern eher solche betriebswirtschaftlicher und juristischer Art vermitteln.
Haftung
Der Steuerberater schuldet seinem Mandanten eine umfassende, richtige Beratung. Ein Steuerberater ist insbesondere dazu verpflichtet, seinen Auftraggeber möglichst vor Schaden zu bewahren. Daher hat er seinen Mandanten in die Lage zu versetzen, eigenverantwortlich seine Rechte und Interessen zu wahren, um so eine Fehlentscheidung vermeiden zu können.
Selbständige Steuerberater und Steuerbevollmächtigte sowie Steuerberatungsgesellschaften müssen gegen die sich aus ihrer Berufstätigkeit ergebenden Haftpflichtgefahren angemessen versichert sein (, ). Der Versicherungsschutz muss die in genannten Aufgaben der Hilfestellung in Steuersachen umfassen.
Die Mindestversicherungssumme muss für den einzelnen Versicherungsfall mindestens 250.000 EUR betragen (). Wird eine Jahreshöchstleistung für alle in einem Versicherungsjahr verursachten Schäden vereinbart, muss sie mindestens 1.000.000 EUR betragen (). In Fällen, in denen die Haftung durch Allgemeine Auftragsbedingungen (AAB) begrenzt wird, muss die Haftungssumme dem 4-fachen der gesetzlichen Mindestversicherungssumme entsprechen ().
Vielfach wird die Mindestversicherungssumme nicht ausreichen und nicht den tatsächlichen Risiken entsprechen. Bei der Wahl der Versicherungssumme ist daher die Struktur der Praxis, insbesondere Art, Umfang und Zahl der Aufträge sowie die Zahl und Qualifikation der Mitarbeiter zu beachten.
Die Pflicht zur Unterhaltung einer Berufshaftpflichtversicherung knüpft allein an die Bestellung an. Sie besteht daher auch dann, wenn ein Berufsangehöriger für eine gewisse Zeit keine Berufstätigkeit ausübt. Lediglich angestellte Berufsangehörige sind von der Versicherungspflicht befreit, soweit sie in der Haftpflichtversicherung ihres Arbeitgebers mitversichert sind. Das Nichtunterhalten einer vorgeschriebenen Haftpflichtversicherung ist ein zwingender Widerrufsgrund für die Bestellung.
Honorar
Das Honorar für die Steuerberatung wird nach der Steuerberatervergütungsverordnung (StBVV; bis 2012: Steuerberatergebührenverordnung) abgerechnet.
Grundlage ist meist der Gegenstandswert oder der Zeitfaktor, d. h. die Vergütung nach einem vereinbarten Stundensatz. Insbesondere im Bereich der Steuervermeidungsberatung sowie der Steuerrechtsdurchsetzung (Betriebsprüfungen, Einspruchs- und Klagerfahren vor den Finanzgerichten) ist die Vergütung auf Grundlage eines vereinbarten Stundensatzes die Regel.
Einkommen
Nach Angaben des statistischen Berichtssystems der Bundessteuerberaterkammer lag der durchschnittliche Jahresüberschuss von Steuerberaterkanzleien im Jahr 2014 bei 191.000 Euro.
Die Durchschnittswerte des Jahresüberschusses je Kanzleiinhaber (nach Kanzleiform) können aus der folgenden Tabelle abgelesen werden:
Berufliche Zusammenarbeit
Statt der Berufsausübung als Einzelperson ist der Zusammenschluss mehrerer Steuerberater und die Gründung einer Kapitalgesellschaft (insbesondere GmbH oder Aktiengesellschaft) zulässig (Steuerberatungsgesellschaft). Hierzu müssen aber Vorstand bzw. Geschäftsführung und Aktionäre bzw. Gesellschafter mehrheitlich Steuerberater sein.
Der Zusammenschluss mehrerer Steuerberater zwecks Beratung in steuerlichen Fragen kann daneben in Form einer
Sozietät als Gesellschaft bürgerlichen Rechts
Partnerschaftsgesellschaft
Partnerschaftsgesellschaft mit beschränkter Berufshaftung
Bürogemeinschaft
Kommanditgesellschaft (auch eine GmbH & Co. KG ist möglich; aber nur, wenn überwiegend Treuhandtätigkeiten übernommen werden)
offene Handelsgesellschaft erfolgen.
Steuerberater können in diesem Rahmen auch mit sog. sozietätsfähigen Berufen (insb. Wirtschaftsprüfern und Rechtsanwälten) zusammenarbeiten. Bei Steuerberatungsgesellschaften dürfen die Nicht-Steuerberater die Steuerberater weder auf Ebene der Anteilseigner noch auf Ebene der Unternehmensleitung dominieren.
Strittig ist, ob die Zusammenarbeit in einer juristischen Person eine gewerbliche Tätigkeit darstellt oder ob sie weiterhin als freiberuflich anzusehen ist.
Syndikus-Steuerberater
Eine Anstellung im gewerblichen Bereich, beispielsweise in einem Industrieunternehmen, ist mit dem Beruf des Steuerberaters vereinbar, da das Verbot der Tätigkeit als Syndikus-Steuerberater durch § 58 Satz 2 Nr. 5a des am 11. April 2008 verkündeten Achten Gesetzes zur Änderung des Steuerberatungsgesetzes vom 8. April 2008 (BGBl. I Nr. 14/08 S. 666) aufgehoben wurde. Laut Urteil des Bundesfinanzhofs vom 9. August 2011 (Aktenzeichen VII R 2/11) kann der Syndikus Anzahl und Umfang seiner Mandate frei bestimmen und darf auch als Feierabend-Steuerberater tätig werden.
Abgrenzung zu ähnlichen Berufen
Neben den Steuerberatern sind die folgenden Berufsangehörigen zur steuerrechtlichen Beratung bzw. der Durchführung von Teilaufgaben befugt:
Steuerbevollmächtigte
Rechtsanwalt; speziell der Fachanwalt für Steuerrecht
Wirtschaftsprüfer und vereidigter Buchprüfer
Prozessagenten nach § 157 Abs. 1 StBerG
Lohnsteuerhilfevereine, die Mitgliedern Steuerberatung im Rahmen ihrer Befugnis nach § 4 Nr. 11 Steuerberatungsgesetz bieten
Buchführungshelfer, die allerdings, teilweise mit Einschränkungen, nur die Finanzbuchhaltung erstellen dürfen.
Europäisierung des Berufsrechts
Steuerberater aus anderen EU-Staaten dürfen ihre Mandanten grenzüberschreitend auch in Deutschland beraten, sofern dies auf vorübergehender Basis, also ohne feste Niederlassung, geschieht (geregelt in der Berufsqualifikations-Anerkennungsrichtlinie 2005/36/EG). Hierbei müssen sie mit ihrer ausländischen Berufsbezeichnung auftreten, damit sowohl der Mandant als auch die Finanzverwaltung erkennen können, dass es sich nicht um Steuerberater mit Qualifikation nach deutschem Recht handelt. Mandanten von Steuerberatern aus anderen EU-Staaten tragen außerdem das Risiko, dass ihre Berater wegen mangelnder Kenntnisse im deutschen Steuerrecht von der Finanzverwaltung zurückgewiesen werden können.
Berater aus anderen EU-Mitgliedstaaten, die eine dauerhafte Niederlassung als Steuerberater in Deutschland anstreben, müssen eine Eignungsprüfung (§ 37a (2) Steuerberatungsgesetz) absolvieren, die eine vereinfachte Steuerberaterprüfung darstellt.
Zum Teil wird die Ansicht vertreten, dass die berufsständischen Zulassungsvoraussetzungen nicht mit dem EU-Recht vereinbar seien. Andere EU-Mitgliedstaaten könnten die Öffnung des deutschen Steuerberatermarktes für ihre eigenen Bürger verlangen, auch wenn es in den Mitgliedstaaten unterschiedliche bis gar keine Zulassungsvoraussetzungen gibt und natürlich unterschiedliches nationales Steuerrecht gilt. Hiergegen wenden sich die deutschen Steuerberaterkammern, die als Interessenvertretungen ihrer Mitglieder auftreten und somit ebenfalls der Steuerrechtspflege dienen. Sie begründen dies mit der mangelhaften Ausbildung ausländischer Berater im deutschen Steuerrecht.
Insbesondere Buchhalter- und Bilanzbuchhalterverbände fordern, bestimmte Tätigkeiten, die derzeit Steuerberatern vorbehalten sind, auch Buchhalterberufen zu öffnen. Aus Sicht deutscher Bilanzbuchhalter ergibt sich eine Inländerdiskriminierung aus der Tatsache, dass österreichische Bilanzbuchhalter unter Berufung auf die Richtlinie 2005/36/EG auf vorübergehender Basis in Deutschland Tätigkeiten erbringen dürfen, die deutsche Bilanzbuchhalter nicht erbringen dürfen. Inländerdiskriminierung ist europarechtlich zulässig.
Weitere Tätigkeitsfelder der Steuerberater
Neben der selbständigen und nichtselbständigen Tätigkeit in den Vorbehaltsaufgaben kann der Steuerberater die folgenden Tätigkeiten mit seinem Berufsstand vereinbaren:
Aufsichtsratstätigkeit
Insolvenzverwaltung und Liquidation
Nachlassverwaltung, Testamentsvollstreckung
Vormundschaftliche, pflegerische und treuhänderische Aufgaben
Unternehmensberatung und Rating-Beratung.
Steuerberaterverzeichnis
Die Bundessteuerberaterkammer führt ein ab dem 1. Januar 2017 öffentlich zugängliches Verzeichnis aller Steuerberater, das online abrufbar ist. Zusätzlich bietet der Deutsche Steuerberaterverband einen Steuerberater-Suchservice an.
Österreich
Der Beruf des Steuerberaters ist ein freier Beruf und unterliegt daher nicht den Bestimmungen der Gewerbeordnung. Die berufsständische Vertretung der Steuerberater ist in Österreich die Kammer der Steuerberater und Wirtschaftsprüfer – KSW (bis Dezember 2017: Kammer der Wirtschaftstreuhänder – KWT). Um die Tätigkeit des Steuerberaters ausführen zu dürfen, bedarf es einer Fachprüfung.
Der Begriff Wirtschaftstreuhänder umfasst in Österreich folgende Berufsgruppen:
Wirtschaftsprüfer
Steuerberater
Bis 2005 gab es zusätzlich die Bezeichnung "beeideter Buchprüfer"; diese Berechtigung konnte ab 1999 jedoch nicht mehr erworben werden; die bestehenden Buchprüfer wurden 2005 durch eine Übergangsbestimmung zu Wirtschaftsprüfern erklärt.
Des Weiteren waren sogenannte selbständigen Buchhalter Mitglieder der Kammer der Wirtschaftstreuhänder. Sie übten einen Wirtschaftstreuhandberuf aus, durften jedoch nicht die Berufsbezeichnung Wirtschaftstreuhänder führen. Die Möglichkeit zur Aufnahme dieses Berufes endete jedoch am 30. Juni 2008. Bestehende Berufsberechtigungen als Selbständiger Buchhalter durften weiter ausgeübt werden, jedoch endete die Mitgliedschaft dieser Personengruppe in der Kammer der Wirtschaftstreuhänder, sofern sie nicht zur Steuerberaterprüfung zugelassen wurden oder eine Berufsberechtigung nach dem neuen Bilanzbuchhaltungsgesetz (BibuG) erwarben, nach einer entsprechenden Übergangsfrist.
Die Rechte und Pflichten der Wirtschaftstreuhänder sind im Bundesgesetz über die Wirtschaftstreuhandberufe (Wirtschaftstreuhandberufsgesetz – WTBG) geregelt.
Tätigkeiten der Steuerberater
Das Wirtschaftstreuhandberufsgesetz (WTBG) regelt, welche Tätigkeiten ein Steuerberater im Rahmen seines Berufes ausüben darf bzw. welche Tätigkeiten Steuerberatern vorbehalten sind (§ 3 WTBG). Diese umfasst insbesondere folgende Tätigkeiten:
Führung der Buchhaltung und Lohnbuchhaltung
Erstellung von Jahresabschlüssen
Erstellung von Steuererklärungen
Vertretung in Abgaben- und Abgabenstrafverfahren vor Abgabenbehörden
Sonstige Vertretung insbesondere vor Sozialversicherungen
Beratungsleistungen
betriebswirtschaftliche Beratung
Steuerberatung
Beratungsleistungen im Zusammenhang mit dem betrieblichen Rechnungswesen und der Beratung betreffend der Organisation und Einrichtung des internen Kontrollsystems
Sanierungsberatung, Erstellung von Sanierungsgutachten etc.
Beratung in Rechtsangelegenheiten, soweit diese mit wirtschaftstreuhändischen Tätigkeiten unmittelbar in Zusammenhang stehen
Treuhandaufgaben
Prüfungsaufgaben (die nicht die Erteilung eines förmlichen Bestätigungsvermerkes erfordern)
Sachverständigengutachten
Neben der "klassischen" Tätigkeit des Steuerberaters – der steuerlichen Beratung und Vertretung und der Erstellung von Jahresabschlüssen und von Abgabenerklärungen – tritt die betriebswirtschaftliche Beratung der Klienten in letzter Zeit immer mehr in den Vordergrund.
Verantwortlichkeit des Steuerberaters
Steuerberater sind verpflichtet, die von ihnen übernommenen Aufgaben
gewissenhaft
sorgfältig
eigenverantwortlich
unabhängig
verschwiegen
auszuüben.
Ebenso wie in Deutschland ist die Anstellung in einem anderen Beruf (z. B. in einem Industrieunternehmen) mit dem Beruf des Steuerberaters vereinbar. Es besteht lediglich die Pflicht, die andere Tätigkeit der Kammer der Steuerberater und Wirtschaftsprüfer zu melden, die diese Tätigkeit untersagen kann.
Erlangung der Berufsbefugnis
Um die Berufsbefugnis als Steuerberater zu erlangen, erfordert das Wirtschaftstreuhandberufsgesetz (WTBG) bestimmte Voraussetzungen. Diese sind:
volle Handlungsfähigkeit
besondere Vertrauenswürdigkeit
geordnete wirtschaftliche Verhältnisse
aufrechte Vermögensschaden-Haftpflichtversicherung
Berufssitz
Vorbildung und praktische, facheinschlägige Tätigkeit
positive Absolvierung der entsprechenden Fachprüfung
Volle Handlungsfähigkeit
Die Handlungsfähigkeit ist aus dem Bestimmungen des Allgemeinen Bürgerlichen Gesetzbuches (ABGB) abzuleiten. Nur Personen, die über volle Handlungsfähigkeit verfügen, d. h. sich durch ihr eigenes Handeln berechtigen und verpflichten können, dürfen den Beruf eines Steuerberaters ausüben.
Besondere Vertrauenswürdigkeit
Steuerberater müssen sich durch besondere Vertrauenswürdigkeit auszeichnen. Diese liegt nach dem Bestimmungen des Wirtschaftstreuhandberufsgesetzes (WTBG) nicht vor, wenn
eine rechtskräftige Verurteilung
von einem Gericht wegen einer mit Vorsatz begangenen Straftat zu einer mehr als dreimonatigen Freiheitsstrafe oder
von einem Gericht wegen einer mit Bereicherungsvorsatz begangenen Straftat oder
von einer Finanzstrafbehörde wegen eines vorsätzlichen Finanzvergehens vorliegt,
und die Verurteilung oder Bestrafung noch nicht getilgt ist.
Geordnete wirtschaftliche Verhältnisse
Steuerberater müssen geordnete wirtschaftliche Verhältnisse vorweisen. Solche liegen nicht vor, wenn innerhalb der letzten 10 Jahre über das Vermögen
ein Konkurs oder
zwei Sanierungsverfahren eröffnet bzw. durchgeführt wurden, oder
ein gestellter Antrag auf Eröffnung eines Insolvenzverfahrens mangels kostendeckenden Vermögens abgelehnt wurde UND mittlerweile nicht sämtliche Verbindlichkeiten beglichen wurden.
Vermögensschaden-Haftpflichtversicherung
Steuerberater sind verpflichtet, eine Vermögensschaden-Haftpflichtversicherung mit einer Versicherungssumme von mindestens EUR 72.673,- pro Versicherungsfall abzuschließen. Steuerberater, die ausschließlich unselbständig tätig sind, sind von der Verpflichtung zum Abschluss einer Vermögensschaden-Haftpflichtversicherung befreit.
Berufssitz
Der Berufsitz ist eine feste Einrichtung, welche durch ihre personelle, sachliche und funktionelle Ausstattung die Erfüllung der fachlichen Anforderungen des Steuerberaters gewährleistet.
Der Berufssitz muss in einem EU- oder EWR Mitgliedstaat gelegen sein und kann auch der Hauptwohnsitz des Steuerberaters sein. Vom Berufssitz aus, kann der Steuerberater in ganz Österreich tätig werden.
Weiters ist es möglich, dass Steuerberater – neben ihrem Berufsitz – Zweigniederlassungen errichten. Die Leitung einer Zweigniederlassung muss einem Berufsberechtigten übertragen werden, der seinen Berufsitz im Bundesland der Zweigniederlassung hat, in der Zweigniederlassung hauptberuflich (ohne jede Wirtschaftstreuhänder-Tätigkeit auf eigene Rechnung) tätig ist und die entsprechende Berufsberechtigung (siehe unten) für die in der Zweigstelle ausgeübte Tätigkeit besitzt.
Vorbildung und praktische facheinschlägige Tätigkeit
Für die Zulassung zur Steuerberaterprüfung ist ein erfolgreich abgeschlossenes facheinschlägiges Hochschul- oder Fachhochschulstudium und eine 3-jährige praktische Tätigkeit als Berufsanwärter bei einem Wirtschaftstreuhänder (Steuerberater oder Wirtschaftsprüfer) oder als Revisionsanwärter bei einem Revisionsverband Voraussetzung. Alternativ kann auch eine Zulassung ohne Studium nach mindestens 5-jähriger hauptberuflicher Tätigkeit als Bilanzbuchhalter erfolgen. Verschiedene facheinschlägige Tätigkeiten sind auf die Berufsanwärterzeit bzw. auf die Tätigkeit als Bilanzbuchhalter bis zu 1½ Jahren bzw. bis zu 2 Jahren anrechenbar.
Fachprüfung
Die Fachprüfung zum Steuerberater umfasst zwei schriftliche Klausurarbeiten aus den Gebieten Abgabenrecht und Betriebswirtschaftslehre von einer Ausarbeitungszeit von je sieben Stunden. Dazu kommt eine mündliche Prüfung aus den Gebieten Qualitätssicherung und Berufsrecht, Abgabenrecht einschließlich des Abgabenverfahrensrechts, Rechnungslegung, Betriebswirtschaftslehre und Rechtslehre.
Für die Fachprüfung für Wirtschaftsprüfer sind dazu noch drei viereinhalbstündige Klausuren aus den Gebieten Rechnungslegung, Abschlussprüfung und Rechtslehre zu absolvieren. Der mündliche Teil wird um die Gebiete Abschlussprüfung, Grundzüge der Volkswirtschaftslehre und Finanzwissenschaft und Grundzüge des Bank-, Versicherungs-, Wertpapierrechts und Devisenrechts ergänzt.
Steuerberatungsgesellschaft
In Österreich kann der Beruf des Steuerberaters auch im Rahmen einer Personen- oder Kapitalgesellschaft ausgeübt werden, wobei jedoch durch das Wirtschaftsberufsgesetz (WTBG) die Ausübung der Tätigkeit im Rahmen einer Gesellschaft mit einigen Auflagen verbunden ist. Diese sind
Zulässige Gesellschaftsform: Nur folgende Gesellschaften sind zulässig
Offene Gesellschaft (OG) (bis 31. Dezember 2006 "Offene Erwerbsgesellschaft – OEG")
Kommanditgesellschaft (KG) (bis 31. Dezember 2006 "Kommandit-Erwerbsgesellschaft – KEG")
Gesellschaft mit beschränkter Haftung (GmbH)
Aktiengesellschaft (AG) mit vinkulierten Namensaktien
Es muss ein schriftlicher Gesellschaftsvertrag abgeschlossen werden
Gesellschafter bzw. Aktionäre dürfen nur Berufsberechtigte, deren Ehegatten und Kinder oder andere Wirtschaftstreuhandgesellschaften sein. Es bestehen außerdem 'interdisziplinäre Gesellschaften' mit Bilanzbuchhaltern (entsprechend § 71 WTBG), wobei der Bilanzbuchhalter Gesellschafter bzw. Geschäftsführer ist.
Allfällige Mitglieder des Aufsichtsrates müssen ihren Wohnsitz in einem EU- oder EWR Mitgliedstaat haben und über besondere Vertrauenswürdigkeit und geordnete wirtschaftliche Verhältnisse verfügen.
Die Gesellschaft muss eine Vermögensschaden-Haftpflichtversicherung abschließen
Die Geschäftsführung und Vertretung nach außen hat durch Berufsberechtigte zu erfolgen.
Schweiz
In der Schweiz gibt es keine Zulassungsbeschränkungen für den Beruf des Steuerberaters. Die Berufsbezeichnung in der Schweiz ist "Treuhänder". Die fehlende Zugangsbeschränkung (praktisch kann jeder ein Treuhandbüro eröffnen) führt dazu, dass es umgangssprachlich sogenannte "Feld-, Wald- und Wiesentreuhänder" gibt.
Trotz der fehlenden Zulassungsbeschränkung gibt es eine Reihe von Ausbildungen, welche mit einer staatlichen Prüfung (der höheren Fachprüfung) abschließen und als "Qualitätssiegel" betrachtet werden können:
dipl. Treuhandexperte
dipl. Steuerexperte
dipl. Wirtschaftsprüfer
Im Bereich der Wirtschaftsprüfung gibt es allerdings eine Zulassungsbeschränkung durch das Obligationenrecht, welches für gewisse Gesellschaften und Vorgänge die Prüfung durch einen besonders befähigten Revisor verlangt. Diese Anforderung erfüllen ausschließlich Berufsleute mit dem eidgenössischen Diplom als Wirtschaftsprüfer sowie dipl. Treuhandexperten und Steuerexperten (sowie andere Berufsleute) mit entsprechender Fachpraxis.
Am 1. Januar 2008 ist das neue Revisionsaufsichtsgesetz in Kraft getreten, welches die Zulassung der Revisorinnen und Revisoren bzw. der Revisionsgellschaften abschließend reglementiert. Der Vollzug erfolgt durch eine staatliche Aufsichtsbehörde.
Wie in anderen Ländern auch, haftet ein Treuhänder, geprüft oder nicht, für Schaden den er verursacht.
Literatur
Detlef J. Brauner: Wirtschaftsprüfungs- und Steuerberatungsgesellschaften 2008. Wissenschaft & Praxis, 2008, ISBN 3-89673-451-2.
Andrea Lauterbach: Berufsziel Steuerberater, Wirtschaftsprüfer 2011. Berufsexamina, Tätigkeitsbereiche, Perspektive. Wissenschaft & Praxis, Sternenfels 2010, ISBN 978-3-89673-573-7.
Christoph Goez u. a.: Steuerberater Handbuch 2013 Stotax, 2013, ISBN 978-3-08-374013-1
Andreas Heßler, Petra Mosebach: Strategie und Marketing im Web 2.0 – Handbuch für Steuerberater und Wirtschaftsprüfer. Springer Gabler, Wiesbaden 2013, ISBN 978-3-8349-4061-2.
Weblinks
Deutschland
Bundessteuerberaterkammer auf bstbk.de
Deutscher Steuerberaterverband e.V. auf dstv.de
Steuerberatungsgesetz auf gesetze-im-internet.de
Leitbild des steuerberatenden Berufs auf steuerberater-perspektiven.de
Österreich
Kammer der Steuerberater und Wirtschaftsprüfer
Verzeichnis der Wirtschaftstreuhänder
Schweiz
Treuhand Suisse auf treuhandsuisse.ch
Schweizerische Treuhandkammer auf treuhand-kammer.ch
Einzelnachweise
Dienstleistungsberuf
Steuerrecht
Beruf (Rechtspflege)
Freier Beruf (Deutschland) | {
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{"url":"https:\/\/quant.stackexchange.com\/tags\/multivariate\/hot","text":"# Tag Info\n\n7\n\nThe best introduction to copulas I know, i.e. with rigour and intuition, is the following. THE QUANT CLASSROOM BY ATTILIO MEUCCI A Short, Comprehensive, Practical Guide to Copulas Visually introducing a powerful risk management tool to generalize and stress-test correlations\n\n7\n\nI personally use the simple Garch(1,1) for volatility filtering in the risk management area. In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...\n\n6\n\nI found Coping With Copulas by Thorsten Schmidt really helped me to get a more basic understanding of copulas. As well as looking at some simple examples in R and thinking about different directions the transformations can happen. To answer your actual question I'll attempt to describe the steps involved as simply as I can. Let's say you use the copula ...\n\n6\n\nHigh and low prices are frequently used in many contexts, such as estimating volatility. See, for example, the Garman-Klass and Yang-Zhang estimators. Brandt and Kinlay provide a nice summary of some of these estimators. However, it sounds like you are more interested in using high\/low information for evaluating whether mean reversion has taken place. In ...\n\n5\n\nIn the theory of copulas you want to model a multivariate (often bivariate) distribution and keep the marginals fixed. Thus you have random variables $X$ and $Y$ with cdf $F_X(x) = P[X \\le x]$ and $F_Y(y) = P[Y\\le y]$ and you want to find some $F_{X,Y}(x,y) = P[X \\le x, Y\\le y]$ such that when you look at marginals you get $F_{X,Y}(x,\\infty) = F_X(x)$ and ...\n\n4\n\nA multivariate normal distribution can be thought of as normal margins with a normal copula. The multivariate t is the same way, but it has t margins with a t copula and they all have the same degrees of freedom. So it has t copula dependence. It is either a spherical or an elliptical distribution. I can't think of a good reason to use a multivariate t. The ...\n\n4\n\nSince pentahedrons are 3d shapes, but there is no reason to think currencies live in a 3d world, you can just treat the 'pentahedron' as a weighted node graph of the 5 currencies. A graph edge from one currency to another represents an exchange of those two currencies. So in the same fashion as usual vectors, I can go from currency A to C via B by executing ...\n\n3\n\nFor the two-dimensional case, the Cholesky decomposition of the covariance matrix $$\\Sigma = \\left( \\begin{array}{c c} \\sigma_1^2 & \\rho \\sigma_1 \\sigma_2\\\\ \\rho \\sigma_1 \\sigma_2 & \\sigma_2^2 \\end{array} \\right)$$ is given by B = \\left( \\begin{array}{c c} \\sigma_1 & 0\\\\ \\rho \\sigma_2 & \\sigma_2 ...\n\n3\n\nInteresting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical Simulation (FHS), etc.). 1 - EGARCH, merely because GARCH models do not take into account so-called leverage effects, which is crucial to me for skewed and ...\n\n3\n\nPYTHON I have found this class from the statsmodels library for calculating Garch models. Unfortunately, I have not seen MGARCH class\/library. Below you can see the basic information about the garch models in mentioned class from the statsmodels. Probably you have to implement it by your own in python, so this class might be used as a starting point. ...\n\n3\n\nYes, it exists and it is called ccgarch package. You can install that by simply running in R install.packages(\"ccgarch\") and learn more about that on the CRAN relative paper. Moreover, I suggest you to read this lecture hold by the author during an R conference. Hope this help.\n\n3\n\nYes you can, how depends fully on your required accuracy and also whether PC1 and PC2 are sufficient in explanatory power of the log differences of your futures contract. Also, make sure you understand the signs of the eigenvalues (sign of the PC) can be different from one experiment to the next as they are arbitrary (the values are obviously not). Here ...\n\n2\n\nWhy not fit an ARMA model to the rolling returns first, and then model the residuals in your regression equation? That way you should be removing most of the effects of auto-correlation.\n\n2\n\nThere is a brief and not overly technical introduction here: http:\/\/prescientmuse.blogspot.co.uk\/2015\/01\/a-brief-introduction-to-copula.html And an application of use in a trading system with full R code here: http:\/\/prescientmuse.blogspot.co.uk\/2015\/02\/vanilla-trading-algorithm.html Hope that helps!\n\n2\n\nVECM-GARCH models do not seem to be implemented in R as of now. However, if you are willing to accept some simplifications, you could perhaps be fine with the existing functionality. Take, for example, the \"rmgarch\" package in R. It allows combining univariate conditional mean-conditional variance models with several multivariate GARCH models that take ...\n\n2\n\nSince I think this is of interest for other people, I will post the approach I found: First, let $C_n(u_1,\\ldots,u_n)$ be a $n$ - dimensional Clayton copula with generator function $F$ and inverse $F^{-1}$. Then, Generate $n$ independent r.v. from $U (0,1)$ Calculate $n-1$ derivatives of $F$, where $F_{n-1}$ denotes the $n-1$-th - order derivative of $F$ ...\n\n1\n\nI guess more than multicolinearity you are running into the issue of identification. What are you exactly identifying with such a regression? You somehow need to instrument for defaults. Although your $R^2$ is high, does your regression make any sense? Take for example the coefficient on unemployment. It is negative, so that seems to imply that higher ...\n\n1\n\nI am not familiar with the QE scheme, but I think your question is more general: You want to do a multi-variate diffusion, for $n$ correlated processes. You have your instantaneous correlations matrix $R = (\\rho_{i,j})_{i,j}$ where $d \\langle W^i, W^j \\rangle_t = \\rho_{i,j} dt$, and I am assuming here you know how to simulate brownian increments for a ...\n\n1\n\nI suggest you to organize you explanatory variables in different matrix and then use the mvregress(...) command, that allows you to handle well the results. I tried in the past to use pre-built command for VAR but I find way simpler to organize it by myself and use usual regression commands.\n\n1\n\nYou need to adjust your correlation matrix such that it becomes positive definite. There is an R routine that will do this for you - link. Or, if you want to do it yourself, i believe the general method is to do an eigen value decomposition, set any negative eigenvalues to zero, and then reconstruct the original matrix. If you're going to go down this ...\n\n1\n\nTry the mgarch package, it's available at CRAN. In this link you will find an example from Prof. Zivot.\n\n1\n\nIt is not clear from the post if you are querying for the mechanics\/code for looping over the series or the appropriate critical values. I here make a comment on the latter. One of the main pitfalls when testing multiple hypotheses is the fact that a certain percentage would fail under the null (as this xkcd strip nicely illustrates https:\/\/xkcd.com\/882\/ ). ...\n\n1\n\nThere a to ways that you can performe the ADF test to a data frame, first write a loop for applying the test to all the columns or use the apply function to your data. For leaving out the first column just create an other data frame like this: da=yourDataName[,-1]. the code for the ADF would be something like apply(da,2,adfTest,lags=0,type=\"c\"). The 2 is ...\n\n1\n\nHow can I change this to implement FULL ARCH and GARCH parameter matrices, to capture the spillover effects? You cannot. The original paper by Engle (2002) as well as the Stata manual for the DCC-GARCH model reveal that the model admits a different form than the one represented in the equation in your question. (What you have there is a special case of a ...\n\n1\n\nYour question is formulated in a very general way, this is why any answer will need to be general as well. In a nutshell and in full generality you need to estimate the joint distribution from your historical data since in most cases correlations alone are not sufficient to define the joint distribution. In a second step you can calculate the distribution ...\n\n1\n\nI found a link and I have to repeat: I don't think that PCA helps you to find a price ... it helps to model the movements of prices but not their values. You get something like a factor model ... this does not directly give you a price ... maybe you also want to have a look at this link where PCA is applied to the oil market.\n\n1\n\nTry fitting a model with ARMA errors? However, if by \"rolling returns\" you imply a moving average of returns or some QoQ or YoY return series, which has much persistence, I am not so sure what the right way to proceed really is (with the exception that you can apply some corrections suggested in econometrics literature).\n\n1\n\nIt simply points to the fact that your model as stands does not have much explanatory power of monthly returns. One reason could be of a observation period mismatch. I am not a fundamental type of guy, but I imagine that the monthly returns are measured over too short a period (1 month) while most fundamental factors are updated on a quarterly basis (...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2019-07-23 18:38:47","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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7334597110748291, \"perplexity\": 546.3992262130373}, \"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-2019-30\/segments\/1563195529481.73\/warc\/CC-MAIN-20190723172209-20190723194209-00255.warc.gz\"}"} | null | null |
The Atholl Papers: The Jacobite Rising of 1745 Search of iMuseum
The Atholl Papers: The Jacobite Rising of 1745
The Atholl family and the wider members of Clan Murray have a complicated relationship with Jacobitism; one of divided loyalties, which saw family and clan members pitted against one another in all three risings (1715, 1719 and 1745), in the name of either a Hanoverian king or the 'Pretender,' James Francis Edward Stuart. William, Marquess of Tullibardine and the eldest son of the 1st Duke of Atholl, was the most unrepentant of Jacobites, and his support for the 'Pretender' in 1715 led to him being attainted (barred from inheriting titles and property) and his younger brother, James, inheriting the dukedom in 1724. William, along with another of the 1st Duke's sons, George, would go on to play ever-increasing high-profile roles in the cause during the subsequent risings. The 1st and 2nd Dukes of Atholl, on the other hand, remained loyal to the Hanoverians monarchs, George I and George II, throughout this period.
The Isle of Man also had a complicated relationship when it came to the Jacobites. While no proclamation in support of James Stuart was ever made on the island during the risings of 1715 and 1719, neither was there out-and-out support for George I. The House of Keys, in particular, showed little enthusiasm in co-operating with Governor Horne when it came to arranging the armament of the isle. The Keys also dragged their feet when Horne asked them to pass a law making it a treasonable offence to drink a toast to James Stuart, or utter anything that could be regarded as disloyal to the King. In fact, no such law ever made its way on to the island's statute books before the Isle of Man's revestment into the British crown, in 1765. By 1745, however, the 2nd Duke of Atholl was Lord of Man, and the Keys would be far more amenable, perhaps mirroring the Duke's own need to demonstrate loyalty to the Hanoverians.
It would also be remiss of me not to mention that it is fairly well-documented that Manx merchants and smugglers (often one and the same) forged links with Jacobite sympathisers throughout this period. The island's smuggling network was used to send messages and instructions, and no doubt ferry goods, between actors in the cause.
Letter from James Murray to the Duke of Atholl, 12 August 1745 (AP X11-28) [Click images to enlarge & read.]
The first we hear of the Jacobite rising of 1745, in the Atholl Papers, is in a letter from the former Governor of the Isle of Man, James Murray (Receiver General of Scotland), to the Duke of Atholl, sent just days before the 'Bonnie Prince,' Charles Stuart, raised his father's standard at Glenfinnan and proclaimed him King James III. By the tone of the letter, it is clear that Murray did not see the landing of a small Jacobite force on Scottish soil as a credible threat, regarding it purely as a ploy to draw British troops away from the Low Countries (War of the Austrian Succession).
The current Governor, Patrick Lindesay, however, took the news of an insurrection far more seriously. In a letter to the Duke in early September (AP X11-31), Lindesay reported that upon being informed of what was happening, he gave strict instructions for the landing places on the island to be watched. He also proposed the idea of examining strangers that arrived on the isle, which until that point he had shied away from ordering, given the island's unique position as a haven for those whose 'private affairs [were] in disorder,' and a place to reside until such matters were resolved; all of which usually provided a benefit to the isle. The Duke's response was sent on 14th September from Edinburgh (AP X11-29), days before the city was taken by the Jacobites, informing Lindesay of the rebels' position and the possibility of them dispersing amongst the counties lying nearest to the Isle of Man. The Duke goes on to warn Lindesay that fugitives might attempt to take refuge on the island, and if so, to take care to apprehend them so they could be 'delivered up to justice.' In answer to the Duke's letter, Lindesay gave orders for the island's militias to be put on notice to repel landing forces, the watches to be doubled, and for strangers to the isle to be interrogated on arrival; these orders would go on to be reissued in November 1745.
Extract from the Liber Scaccar 1745 – Notice to the Captain and Constable of Peel Garrison, and the Deputy Searcher, 23 September 1745 (AP X11-33) [Click image to enlarge & read.]
The correspondence between the Duke of Atholl and Governor Lindesay goes beyond just the recounting of events in Britain and the issuing of orders, touching also on the Duke's financial woes and his need to demonstrate loyalty to the Hanoverian cause.
As the Jacobite forces made their way to Edinburgh in late August and early September 1745, the Duke's brother, William, seized the family estates at Blair Atholl and elsewhere in Perthshire, along with the revenue they generated. By October, the Duke was starting to feel the financial strain of keeping a residence in London, having fled Edinburgh with his wife and daughters just days before the Jacobites entered the city. As such, he sent his steward, Humphrey Harrison, to empty the Isle of Man's coffers, declaring in a letter that money was a 'scarce commodity to Scots people at present' (AP X11-34). In January 1745/46, he accused the island's moars (rent collectors) of neglecting their duties and generally complained about the dilatory manner in which the revenue was collected and remitted to him (AP X11-41). At one point, there was even talk of the Duke coming to stay on the isle, which raised the whole issue of where to find beds for him and his family, as there was a distinct lack of them on the island, though this idea appears to have been dropped once the Jacobites were on the back foot.
Once the Jacobites had concluded their brief sojourn to England in early January 1745/46, the Duke of Atholl, desirous to show his loyalty to George II, informed Governor Lindesay of his intention to accompany the pursuing Hanoverian forces to Scotland, stating, 'I think it incumbent upon me to show all the zeal I can for His Ma[jes]ties service in that country,' and 'be in the way of doing all in my power upon this occasion' (AP X11-41). But the Duke also had to be wary of any actions by those under his lordship on the Isle of Man, however trivial, which might call into question his loyalty. Therefore, when the Reverend Thomas Christian, Vicar of Kirk Marown, was brought before Lindesay in early May, accused of stating James Stuart 'was and is the right heir to the crown of Great Brittaine' (AP X3-9), the Duke's response to Lindesay was, 'I make no doubt but you will have him punished as far as the laws of the isle will admit' (AP X3-18). However, as no law had been passed on the island that made his statement an act of treason, the worst Christian suffered was a temporary suspension by the Bishop of Sodor and Man.
Keen to demonstrate his loyalty in any fashion, the Duke of Atholl had a copy of all the orders issued by Governor Lindesay on the Isle of Man, during the uprising, printed in the Edinburgh newspapers in the summer of 1746.
Extracts from the Liber Scaccar 1746 regarding the Reverend Thomas Christian (AP X3-9)[Click images to enlarge & read.]
News of the Jacobites' defeat at the Battle of Culloden, in April 1746, gave Governor Lindesay and the Isle of Man the opportunity to also demonstrate their allegiance to the King. According to Lindesay, unconfirmed reports of the rebels' defeat led to 'great rejoyceings' on the island, with 'fiddles and fires' lasting three days and nights, and 'much drunkenes amongst the lower sort' (AP X3-6). Whether this is proof that the Manx inhabitants were actually in support of George II, and not just using the news as an excuse to go on a three-day bender, is up for debate. But, once reports were confirmed, Lindesay arranged a number of official activities to celebrate the victory, including: the displaying of flags, the discharge of canon at Castle Rushen and Derby Fort, bonfires, and a supper for Manx officials and the 'better sort of the inhabitants.' The 'lower sort' were also not done with their revelries; Lindesay reported that the people of Douglas went around smashing windows on properties owned by Catholics, who in turn had no choice but to put up with the vandalism.
The Jacobites' defeat at Culloden is not the end of this story in the Atholl Papers, and at least for myself, perhaps brings us to the most interesting part. Following Culloden, the Jacobite army was dispersed and attempts were made to capture those that had taken part in the rising. The Justices of the Peace for the Stewartry of Kirkcudbright sent a warning to Governor Lindesay that rebels might attempt to take refuge on the island, and in response, Lindesay, as he did in September and November 1745, once again issued the order for strangers arriving on the isle to be examined, with the addition that sailors were also to be interrogated, 'lest any of the rebels should disguise themselves' (AP X3-10). The captains for the island's various parishes were also instructed to stay vigilant for people attempting to land at smaller ports or creeks, who might be eager to evade examination at the main towns.
As a consequence of these orders, two young Scottish men by the names of James Napier and William Simpson were detained, after they were seen acting suspiciously in the towns of Ramsey and Douglas, and boarding a Danish ship that had been docked at the island for several weeks. Brought before Governor Lindesay, Napier and Simpson claimed to be chapmen (itinerant traders), who had come to the isle with goods, and a line of credit from Sir John Douglas of Kelhead. Lindesay believed their account, but refused to allow them to leave the island until they provided a certificate from Scotland, which vouched for their story and verified that they had not participated in the rebellion. The Scotsmen procured such a certificate from Sir William Maxwell of Springkell, and they were duly released at the end of June, taking a boat to Annandale, Scotland.
On the surface, this case would look to be quite straight forward, but when you dig into the backgrounds of Sir John and Sir William, it emerges that both were Jacobite sympathisers. In fact, in August 1746, Sir John was arrested after it was revealed that he met with Charles Stuart, back in January at Stirling, and was detained until 1748 before he was released without charge. With this information, it calls into question whether James Napier and William Simpson were indeed simple traders. However, as it appears that they returned to Scotland upon their release, rather than flee to Denmark or elsewhere, I'm inclined to give them the benefit of the doubt, though it is fun to toy with the idea that they were a couple of Jacobite rogues, who pulled one over on the authorities.
Left: Extract from the Liber Scaccar 1746 – Order from Governor Lindesay to the Captain and Constable of Peel, 14 May 1746 (AP X3-10) [Click image to enlarge & read.]
Right: Extract from the Liber Scaccar 1746 – Order from Governor Lindesay to the Captain of Kirk Patrick, 15 May 1746 (AP X3-10) [Click image to enlarge & read.]
It is here that we come to the end of this particular blog, though I will wrap it up by revealing the fates of the Duke of Atholl's two Jacobite brothers. William, Marquess of Tullibardine, would die in the Tower of London in 1746 after being captured in the aftermath of Culloden. George, however, escaped and made his way to the continent, eventually settling in the Dutch town of Medemblik, where he died in 1760.
The next instalment of the blog will be released at the end of November, and will look at the Isle of Man and its conflict with English and Irish revenue cruisers (it is a far more interesting topic than it sounds!).
Read The Atholl Papers Blog:
The Atholl Papers: The Case of Carolina Elinora Mahon
The 2nd Duke of Atholl's Inheritance of the Isle of Man
The Atholl Papers: An Introduction to the Project
Gareth Pugh
Manx National Heritage Project Archivist (The Atholl Papers)
email: gareth.pugh@mnh.im
Wilkins, Frances. (2002) The Isle of Man and the Jacobite Network. Blakedown: Wyre Forest Press.
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The Atholl Papers: The 2nd Duke of Atholl's Inheritance of the Isle of Man
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Mooragh Camp: Second World War Internment on the Isle of Man
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New design for iMuseum Newspapers & Publications Website
Login to newspapers online: step-by-step guidance
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Dissertations and theses now on iMuseum
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Chris Killip images now on iMuseum
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Illiam Dhone's Cap
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Archibald Knox Collection
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Newspapers & Publications from the Isle of Man National Library
Quayle Bridge House Papers | {
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Q: HTML, how to pass textbox variable to xml src? I can manually put user id U078058 in xml src and get his phone number display,
<xml id="XmlUserInfo2" src="http://{ServerName}/common/components/userinfo/UserInfoXml.asp?UserID=U078058"> </xml>
<p>
phone number: <span datasrc="#XmlUserInfo2" datafld="telephoneNumber" dataformatas="html"></span>
</p>
Now instead of manually put user id in xml src, I want to pass user id from textbox to xml src.
Will this work?
<asp:TextBox ID="textUID" runat="Server">U078058</asp:TextBox>
<xml id="XmlUserInfo2" src="http://{ServerName}/common/components/userinfo/UserInfoXml.asp?UserID=textUID"> </xml>
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{"url":"https:\/\/tex.stackexchange.com\/questions\/345066\/table-with-rotated-header-issue","text":"# Table with rotated header issue [duplicate]\n\nI want to achieve a table with the headers being rotated 90 degree so that it appears bottom up.\n\nI use the following code with both rotation 90 and -90 but I am facing different errors. Here is the code:\n\n\\documentclass[graybox,envcountchap,sectrefs]{svmono}\n\\usepackage{rotating}\n\\usepackage{amssymb}% http:\/\/ctan.org\/pkg\/amssymb\n\\usepackage{pifont}% http:\/\/ctan.org\/pkg\/pifont\n\\usepackage{array}\n\\newcolumntype{P}[1]{>{\\centering\\arraybackslash}p{#1}}\n\n\\begin{document}\n\\begin{table}\n\\begin{tabular}{ | P{4cm} | P{1cm} | P{1cm} }\n\\hline\ncol 0\n& \\begin{rotate}{90} Col1 \\end{rotate}\n& \\begin{rotate}{-90} Col2 too much text is written here that i want to split on multiple line but rotated 90 degree \\end{rotate}\\\\[25ex]\n\\svhline\nfirst row & \\checkmark & \\ding{55} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{document}\n\n\nHere is the output:\n\nSo my questions are:\n\n1. How to start from the button towards the top like first 'Col1' but to start from the bottom of the cell\n2. How to make it fit the cell size and use multi rotated line, shall I do that by hand or can it be made in an automatic manner, and if by hand how can that be done\n\nHere is a link to the class svmono.cs\n\n## marked as duplicate by gernot, Henri Menke, Stefan Pinnow, Kurt, egreg\u00a0tables StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 21 '16 at 16:22\n\n\u2022 Welcome to TeX.SX! Please make your code compilable (if possible), or at least complete it with \\documentclass{...}, the required \\usepackage's, \\begin{document}, and \\end{document}. That may seem tedious to you, but think of the extra work it represents for TeX.SX users willing to give you a hand. Help them help you: remove that one hurdle between you and a solution to your problem. \u2013\u00a0user36296 Dec 21 '16 at 14:24\n\u2022 Done. Shall I add a link to the style ? \u2013\u00a0Kasparov92 Dec 21 '16 at 14:27\n\u2022 would be nice. I found one at static.springer.com\/sgw\/documents\/72921\/application\/zip\/\u2026 but I do get the error Undefined control sequence. \\svhline \u2013\u00a0user36296 Dec 21 '16 at 14:33\n\u2022 @samcarter Done \u2013\u00a0Kasparov92 Dec 21 '16 at 14:37\n\u2022 \u2013\u00a0user36296 Dec 21 '16 at 14:43\n\nMy Advice would be, to create a new column type, say R for rotated. In that column, you fill in the text into a minipage. The minipage can break overfull lines and arrange them accordingly. To do so, you have to define the width of the minipage. As the minipage will be rotated afterwards, the formerly width will result in the height of your cell.\n\nThis is my MWE\n\n\\documentclass{article}\n\\usepackage{rotating}\n\\usepackage{amssymb}% http:\/\/ctan.org\/pkg\/amssymb\n\\usepackage{pifont}% http:\/\/ctan.org\/pkg\/pifont\n\\usepackage{array}\n\\newcolumntype{P}[1]{>{\\centering\\arraybackslash}p{#1}}\n%% New definition (\\scriptsize can be deleted)\n\\newcolumntype{R}[1]{%\n>{\\begin{turn}{90}\\begin{minipage}{#1}%\n\\scriptsize\\raggedright\\hspace{0pt}}l%\n<{\\end{minipage}\\end{turn}}}\n\n\\begin{document}\n\\begin{table}\n\\begin{tabular}{ | P{4cm} | P{1cm} | P{1cm} }\n\\hline\ncol 0\n%% now insert our new columns.\n%% Don't forget, you have to repeat the\n%% column delimiters \"|\" in your declaration!\n& \\multicolumn{1}{R{5em}}{Col1}\n& \\multicolumn{1}{|R{5em}}{Col2 too much text is written here that i want to split on multiple line but rotated 90 degree}\\\\%[25ex]\n\\hline\nfirst row & \\checkmark & \\ding{55} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{document}\n\n\u2022 Nice idea & explanation! \u2013\u00a0Kasparov92 Dec 21 '16 at 16:20","date":"2019-05-20 01:09:46","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\": 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\": 0.48524004220962524, \"perplexity\": 3024.4442646464336}, \"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\/1558232255251.1\/warc\/CC-MAIN-20190520001706-20190520023706-00124.warc.gz\"}"} | null | null |
package org.jetbrains.plugins.scala
package lang
package psi
package impl
package expr
import com.intellij.lang.ASTNode
import com.intellij.openapi.progress.ProgressManager
import com.intellij.psi._
import com.intellij.psi.util.PsiTreeUtil
import com.intellij.util.IncorrectOperationException
import org.jetbrains.plugins.scala.annotator.intention.ScalaImportTypeFix
import org.jetbrains.plugins.scala.extensions._
import org.jetbrains.plugins.scala.lang.completion.lookups.LookupElementManager
import org.jetbrains.plugins.scala.lang.lexer.ScalaTokenTypes
import org.jetbrains.plugins.scala.lang.psi.api.base.patterns.ScBindingPattern
import org.jetbrains.plugins.scala.lang.psi.api.base.types.{ScSelfTypeElement, ScSimpleTypeElement, ScTypeElement}
import org.jetbrains.plugins.scala.lang.psi.api.base.{ScFieldId, ScPrimaryConstructor, ScReferenceElement}
import org.jetbrains.plugins.scala.lang.psi.api.expr._
import org.jetbrains.plugins.scala.lang.psi.api.statements._
import org.jetbrains.plugins.scala.lang.psi.api.statements.params.{ScParameter, ScParameterType}
import org.jetbrains.plugins.scala.lang.psi.api.toplevel.ScTypedDefinition
import org.jetbrains.plugins.scala.lang.psi.api.toplevel.imports.ScImportStmt
import org.jetbrains.plugins.scala.lang.psi.api.toplevel.typedef._
import org.jetbrains.plugins.scala.lang.psi.api.{ScPackage, ScalaElementVisitor, ScalaRecursiveElementVisitor}
import org.jetbrains.plugins.scala.lang.psi.impl.ScalaPsiElementFactory.{createExpressionFromText, createExpressionWithContextFromText}
import org.jetbrains.plugins.scala.lang.psi.impl.toplevel.synthetic.ScSyntheticValue
import org.jetbrains.plugins.scala.lang.psi.types._
import org.jetbrains.plugins.scala.lang.psi.types.api._
import org.jetbrains.plugins.scala.lang.psi.types.api.designator.{DesignatorOwner, ScDesignatorType, ScProjectionType, ScThisType}
import org.jetbrains.plugins.scala.lang.psi.types.nonvalue.ScTypePolymorphicType
import org.jetbrains.plugins.scala.lang.psi.types.result._
import org.jetbrains.plugins.scala.lang.refactoring.util.ScTypeUtil.AliasType
import org.jetbrains.plugins.scala.lang.refactoring.util.ScalaNamesUtil
import org.jetbrains.plugins.scala.lang.resolve._
import org.jetbrains.plugins.scala.lang.resolve.processor.{BaseProcessor, CompletionProcessor, DynamicResolveProcessor, MethodResolveProcessor}
import scala.collection.mutable.ArrayBuffer
/**
* @author AlexanderPodkhalyuzin
* Date: 06.03.2008
*/
class ScReferenceExpressionImpl(node: ASTNode) extends ScReferenceElementImpl(node) with ScReferenceExpression {
override def accept(visitor: PsiElementVisitor) {
visitor match {
case visitor: ScalaElementVisitor => accept(visitor)
case _ => super.accept(visitor)
}
}
override def toString: String = "ReferenceExpression: " + getText
def nameId: PsiElement = findChildByType[PsiElement](ScalaTokenTypes.tIDENTIFIER)
override def accept(visitor: ScalaElementVisitor) {
visitor.visitReferenceExpression(this)
}
def multiResolve(incomplete: Boolean): Array[ResolveResult] = {
if (resolveFunction != null) resolveFunction()
else this.multiResolveImpl(incomplete)
}
def shapeResolve: Array[ResolveResult] = {
ProgressManager.checkCanceled()
if (shapeResolveFunction != null) shapeResolveFunction()
else this.shapeResolveImpl
}
def doResolve(processor: BaseProcessor, accessibilityCheck: Boolean = true): Array[ResolveResult] =
new ReferenceExpressionResolver().doResolve(this, processor, accessibilityCheck)
def bindToElement(element: PsiElement): PsiElement = bindToElement(element, None)
def bindToElement(element: PsiElement, containingClass: Option[PsiClass]): PsiElement = {
def tail(qualName: String)(simpleImport: => PsiElement): PsiElement = {
safeBindToElement(qualName, {
case (qual, true) =>
createExpressionWithContextFromText(qual, getContext, this).asInstanceOf[ScReferenceExpression]
case (qual, false) =>
createExpressionFromText(qual).asInstanceOf[ScReferenceExpression]
})(simpleImport)
}
if (isReferenceTo(element)) return this
element match {
case _: ScTrait | _: ScClass =>
ScalaPsiUtil.getCompanionModule(element.asInstanceOf[ScTypeDefinition]) match {
case Some(obj: ScObject) => bindToElement(obj, containingClass)
case _ => this
}
case c: PsiClass =>
val kinds = getKinds(incomplete = false)
if (!ResolveUtils.kindMatches(element, kinds))
throw new IncorrectOperationException(s"class $c does not match expected kind,\nexpected: ${kinds.mkString(", ")}")
if (!ScalaNamesUtil.equivalent(refName, c.name))
throw new IncorrectOperationException(s"class $c does not match expected name $refName")
val qualName = c.qualifiedName
if (qualName != null) {
return tail(qualName) {
ScalaImportTypeFix.getImportHolder(ref = this, project = getProject).addImportForClass(c, ref = this)
//need to use unqualified reference with new import
if (!this.isQualified) this
else this.replace(createExpressionFromText(this.refName).asInstanceOf[ScReferenceExpression])
//todo: conflicts with other classes with same name?
}
}
this
case _: ScTypeAlias =>
throw new IncorrectOperationException("type does not match expected kind")
case fun: ScFunction if ScalaPsiUtil.hasStablePath(fun) && fun.name == "apply" =>
bindToElement(fun.containingClass)
case pack: ScPackage =>
val qualName = pack.getQualifiedName
tail(qualName) {
ScalaImportTypeFix.getImportHolder(this, getProject).addImportForPath(qualName, this)
this
}
case elem: PsiNamedElement =>
if (!ScalaNamesUtil.equivalent(refName, elem.name))
throw new IncorrectOperationException(s"named element $elem does not match expected name $refName")
ScalaPsiUtil.nameContext(elem) match {
case memb: PsiMember =>
val cClass = containingClass.getOrElse(memb.containingClass)
if (cClass != null && cClass.qualifiedName != null) {
val qualName: String = cClass.qualifiedName + "." + elem.name
return tail(qualName) {
ScalaImportTypeFix.getImportHolder(this, getProject).addImportForPsiNamedElement(elem, this, Some(cClass))
this
}
}
case _ =>
}
this
case _ => throw new IncorrectOperationException("Cannot bind to element: " + element)
}
}
def getVariants: Array[Object] = getVariants(implicits = true, filterNotNamedVariants = false)
/**
* Important! Do not change types of Object values, this can cause errors due to bad architecture.
*/
override def getVariants(implicits: Boolean, filterNotNamedVariants: Boolean): Array[Object] = {
val isInImport: Boolean = ScalaPsiUtil.getParentOfType(this, classOf[ScImportStmt]) != null
getSimpleVariants(implicits, filterNotNamedVariants).flatMap {
case res: ScalaResolveResult =>
val qualifier = res.fromType.getOrElse(Nothing)
LookupElementManager.getLookupElement(res, isInImport = isInImport, qualifierType = qualifier)
case r => Seq(r.getElement)
}
}
def getSimpleVariants(implicits: Boolean, filterNotNamedVariants: Boolean): Array[ResolveResult] = {
doResolve(new CompletionProcessor(getKinds(incomplete = true), this, implicits)).filter(r => {
if (filterNotNamedVariants) {
r match {
case res: ScalaResolveResult => res.isNamedParameter
case _ => false
}
} else true
})
}
def getSameNameVariants: Array[ResolveResult] = this.doResolve(
new CompletionProcessor(getKinds(incomplete = true), this, true, Some(refName)))
def getKinds(incomplete: Boolean, completion: Boolean = false): _root_.org.jetbrains.plugins.scala.lang.resolve.ResolveTargets.ValueSet = {
getContext match {
case _ if completion => StdKinds.refExprQualRef // SC-3092
case _: ScReferenceExpression => StdKinds.refExprQualRef
case postf: ScPostfixExpr if this == postf.operation || this == postf.getBaseExpr => StdKinds.refExprQualRef
case pref: ScPrefixExpr if this == pref.operation || this == pref.getBaseExpr => StdKinds.refExprQualRef
case inf: ScInfixExpr if this == inf.operation || this == inf.getBaseExpr => StdKinds.refExprQualRef
case _ => StdKinds.refExprLastRef
}
} // See SCL-3092
def multiType: Array[TypeResult[ScType]] = {
val buffer = ArrayBuffer[TypeResult[ScType]]()
val iterator = multiResolve(incomplete = false).iterator
while (iterator.hasNext) {
iterator.next() match {
case srr: ScalaResolveResult => buffer += convertBindToType(srr)
case _ =>
}
}
buffer.toArray
}
protected override def innerType(ctx: TypingContext): TypeResult[ScType] = {
this.bind() match {
case Some(srr) => convertBindToType(srr)
case _ => resolveFailure
}
}
def shapeType: TypeResult[ScType] = {
shapeResolve match {
case Array(bind: ScalaResolveResult) if bind.isApplicable() => convertBindToType(bind)
case _ => resolveFailure
}
}
def shapeMultiType: Array[TypeResult[ScType]] = {
val buffer = ArrayBuffer[TypeResult[ScType]]()
val iterator = shapeResolve.iterator
while (iterator.hasNext) {
iterator.next() match {
case srr: ScalaResolveResult => buffer += convertBindToType(srr)
case _ =>
}
}
buffer.toArray
}
private def isMetaInlineDefn(p: ScParameter): Boolean = {
p.owner match {
case f: ScFunctionDefinition if f.getModifierList != null =>
f.getModifierList.findFirstChildByType(ScalaTokenTypes.kINLINE) != null
case _ => false
}
}
protected def convertBindToType(bind: ScalaResolveResult): TypeResult[ScType] = {
val fromType: Option[ScType] = bind.fromType
val unresolvedTypeParameters: Seq[TypeParameter] = bind.unresolvedTypeParameters.getOrElse(Seq.empty)
def stableTypeRequired: Boolean = {
//SLS 6.4
//The expected type pt is a stable type or
//The expected type pt is an abstract type with a stable type as lower bound,
// and the type T of the entity referred to by p does not conforms to pt,
this.expectedTypeEx() match {
case Some((tp, typeElementOpt)) =>
(tp match {
case ScAbstractType(_, lower, _) => lower
case _ => tp
}).isAliasType match {
case Some(AliasType(_, Success(lower: DesignatorOwner, _), _)) if lower.isStable =>
return true
case _ =>
tp match {
case designatorOwner: DesignatorOwner if designatorOwner.isStable =>
return true
case _ =>
}
typeElementOpt match {
case Some(te) =>
te.getContext match {
case pt: ScParameterType =>
pt.getContext match {
case p: ScParameter if !p.getDefaultExpression.contains(this) =>
p.owner match {
case f: ScFunction =>
var found = false
val visitor = new ScalaRecursiveElementVisitor {
override def visitSimpleTypeElement(simple: ScSimpleTypeElement): Unit = {
if (simple.singleton) {
simple.reference match {
case Some(ref) if ref.refName == p.name && ref.resolve() == p => found = true
case _ =>
}
}
super.visitSimpleTypeElement(simple)
}
}
f.returnTypeElement.foreach(_.accept(visitor))
if (found) return true
case _ => //looks like it's not working for classes, so do nothing here.
}
case _ =>
}
case _ =>
}
case _ =>
}
}
case _ =>
}
//The path p occurs as the prefix of a selection and it does not designate a constant
//todo: It seems that designating constant is not a problem, while we haven't type like Int(1)
getContext match {
case i: ScSugarCallExpr if this == i.getBaseExpr => true
case m: ScMethodCall if this == m.getInvokedExpr => true
case ref: ScReferenceExpression if ref.qualifier.contains(this) => true
case _ => false
}
}
val inner: ScType = bind match {
case ScalaResolveResult(fun: ScFun, s) =>
s.subst(fun.polymorphicType)
//prevent infinite recursion for recursive pattern reference
case ScalaResolveResult(self: ScSelfTypeElement, _) =>
val clazz = PsiTreeUtil.getContextOfType(self, true, classOf[ScTemplateDefinition])
ScThisReferenceImpl.getThisTypeForTypeDefinition(clazz, this) match {
case success: Success[ScType] => success.get
case failure => return failure
}
case r@ScalaResolveResult(refPatt: ScBindingPattern, s) =>
ScalaPsiUtil.nameContext(refPatt) match {
case pd: ScPatternDefinition if PsiTreeUtil.isContextAncestor(pd, this, true) => pd.declaredType match {
case Some(t) => t
case None => return Failure("No declared type found", Some(this))
}
case vd: ScVariableDefinition if PsiTreeUtil.isContextAncestor(vd, this, true) => vd.declaredType match {
case Some(t) => t
case None => return Failure("No declared type found", Some(this))
}
case _ =>
if (stableTypeRequired && refPatt.isStable) {
r.fromType match {
case Some(fT) => ScProjectionType(fT, refPatt, superReference = false)
case None => ScalaType.designator(refPatt)
}
} else {
val result = refPatt.getType(TypingContext.empty)
result match {
case Success(tp, _) => s.subst(tp)
case _ => return result
}
}
}
case ScalaResolveResult(param: ScParameter, _) if isMetaInlineDefn(param) =>
ScalaPsiElementFactory.createTypeFromText("scala.meta.Stat", param.getContext, null).get
case r@ScalaResolveResult(param: ScParameter, s) =>
val owner = param.owner match {
case f: ScPrimaryConstructor => f.containingClass
case _: ScFunctionExpr => null
case f => f
}
def isMethodDependent(function: ScFunction): Boolean = {
def checkte(te: ScTypeElement): Boolean = {
var res = false
te.accept(new ScalaRecursiveElementVisitor {
override def visitReference(ref: ScReferenceElement): Unit = {
if (ref.resolve() == param) res = true
super.visitReference(ref)
}
})
res
}
function.returnTypeElement match {
case Some(te) if checkte(te) => return true
case _ =>
}
!function.parameters.forall { case param =>
param.typeElement match {
case Some(te) => !checkte(te)
case _ => true
}
}
}
r.fromType match {
case Some(fT) if param.isVal && stableTypeRequired => ScProjectionType(fT, param, superReference = false)
case Some(ScThisType(clazz)) if owner != null && PsiTreeUtil.isContextAncestor(owner, this, true) &&
stableTypeRequired && owner.isInstanceOf[ScTypeDefinition] && owner == clazz => ScalaType.designator(param) //todo: think about projection from this type?
case _ if owner != null && PsiTreeUtil.isContextAncestor(owner, this, true) &&
stableTypeRequired && !owner.isInstanceOf[ScTypeDefinition] => ScalaType.designator(param)
case _ =>
owner match {
case function: ScFunction if PsiTreeUtil.isContextAncestor(function, this, true) &&
isMethodDependent(function) => ScalaType.designator(param)
case _ =>
val result = param.getRealParameterType(TypingContext.empty)
s.subst(result match {
case Success(tp, _) => tp
case _ => return result
})
}
}
case ScalaResolveResult(value: ScSyntheticValue, _) => value.tp
case ScalaResolveResult(fun: ScFunction, s) if fun.isProbablyRecursive =>
val optionResult: Option[ScType] = {
fun.definedReturnType match {
case s: Success[ScType] => Some(s.get)
case _: Failure => None
}
}
s.subst(fun.polymorphicType(optionResult))
case result@ScalaResolveResult(fun: ScFunction, s) =>
val functionType = s.subst(fun.polymorphicType())
if (result.isDynamic) DynamicResolveProcessor.getDynamicReturn(functionType)
else functionType
case ScalaResolveResult(param: ScParameter, s) if param.isRepeatedParameter =>
val result = param.getType(TypingContext.empty)
val computeType = s.subst(result match {
case Success(tp, _) => tp
case _ => return result
})
elementScope.getCachedClass("scala.collection.Seq")
.map {
ScalaType.designator
}.map {
ScParameterizedType(_, Seq(computeType))
}.getOrElse(computeType)
case ScalaResolveResult(obj: ScObject, _) =>
def tail = {
fromType match {
case Some(tp) => ScProjectionType(tp, obj, superReference = false)
case _ => ScalaType.designator(obj)
}
}
//hack to add Eta expansion for case classes
if (obj.isSyntheticObject) {
ScalaPsiUtil.getCompanionModule(obj) match {
case Some(clazz) if clazz.isCase && !clazz.hasTypeParameters =>
this.expectedType() match {
case Some(tp) =>
if (FunctionType.isFunctionType(tp)) {
val tp = tail
val processor =
new MethodResolveProcessor(this, "apply", Nil, Nil, Nil)
processor.processType(tp, this)
val candidates = processor.candidates
if (candidates.length != 1) tail
else convertBindToType(candidates(0)).getOrElse(tail)
} else tail
case _ => tail
}
case _ => tail
}
} else tail
case r@ScalaResolveResult(f: ScFieldId, s) =>
if (stableTypeRequired && f.isStable) {
r.fromType match {
case Some(fT) => ScProjectionType(fT, f, superReference = false)
case None => ScalaType.designator(f)
}
} else {
val result = f.getType(TypingContext.empty)
result match {
case Success(tp, _) => s.subst(tp)
case _ => return result
}
}
case ScalaResolveResult(typed: ScTypedDefinition, s) =>
val result = typed.getType(TypingContext.empty)
result match {
case Success(tp, _) => s.subst(tp)
case _ => return result
}
case ScalaResolveResult(pack: PsiPackage, _) => ScalaType.designator(pack)
case ScalaResolveResult(clazz: ScClass, s) if clazz.isCase =>
s.subst(clazz.constructor.
getOrElse(return Failure("Case Class hasn't primary constructor", Some(this))).polymorphicType)
case ScalaResolveResult(clazz: ScTypeDefinition, s) if clazz.typeParameters.nonEmpty =>
s.subst(ScParameterizedType(ScalaType.designator(clazz),
clazz.typeParameters.map(TypeParameterType(_, Some(s)))))
case ScalaResolveResult(clazz: PsiClass, _) => new ScDesignatorType(clazz, true) //static Java class
case ScalaResolveResult(field: PsiField, s) =>
s.subst(field.getType.toScType())
case ScalaResolveResult(method: PsiMethod, s) =>
val returnType = Option(method.containingClass).filter {
method.getName == "getClass" && _.getQualifiedName == "java.lang.Object"
}.flatMap { _ =>
val maybeReference = qualifier.orElse {
val result: Option[Typeable] = getContext match {
case infixExpr: ScInfixExpr if infixExpr.operation == this => Some(infixExpr.lOp)
case postfixExpr: ScPostfixExpr if postfixExpr.operation == this => Some(postfixExpr.operand)
case _ => ScalaPsiUtil.drvTemplate(this)
}
result
}
def getType(element: PsiNamedElement): Option[ScType] = Option(element).collect {
case pattern: ScBindingPattern => pattern
case fieldId: ScFieldId => fieldId
case parameter: ScParameter => parameter
}.flatMap {
_.getType().toOption
}
def removeTypeDesignator(`type`: ScType): ScType = {
val maybeType = `type` match {
case ScDesignatorType(element) =>
getType(element)
case projectionType: ScProjectionType =>
getType(projectionType.actualElement).map {
projectionType.actualSubst.subst
}
case _ => None
}
maybeType.map(removeTypeDesignator).getOrElse(`type`)
}
def convertQualifier(jlClass: PsiClass): ScType = {
val maybeType = maybeReference.flatMap {
_.getType().toOption
}
val upperBound = maybeType.flatMap {
case ScThisType(clazz) => Some(ScDesignatorType(clazz))
case ScDesignatorType(_: ScObject) => None
case ScCompoundType(comps, _, _) => comps.headOption.map(removeTypeDesignator)
case tp => Some(tp).map(removeTypeDesignator)
}.getOrElse(Any)
val argument = ScExistentialArgument("_$1", Nil, Nothing, upperBound)
ScExistentialType(ScParameterizedType(ScDesignatorType(jlClass), Seq(argument)), List(argument))
}
elementScope.getCachedClass("java.lang.Class")
.map(convertQualifier)
}
ResolveUtils.javaPolymorphicType(method, s, getResolveScope, returnType)
case _ => return resolveFailure
}
qualifier match {
case Some(_: ScSuperReference) =>
case None => //infix, prefix and postfix
getContext match {
case sugar: ScSugarCallExpr if sugar.operation == this =>
sugar.getBaseExpr.getNonValueType(TypingContext.empty) match {
case Success(ScTypePolymorphicType(_, typeParams), _) =>
inner match {
case ScTypePolymorphicType(internal, typeParams2) =>
return Success(ScalaPsiUtil.removeBadBounds(ScTypePolymorphicType(internal, typeParams ++ typeParams2 ++ unresolvedTypeParameters)), Some(this))
case _ =>
return Success(ScTypePolymorphicType(inner, typeParams ++ unresolvedTypeParameters), Some(this))
}
case _ if unresolvedTypeParameters.nonEmpty =>
inner match {
case ScTypePolymorphicType(internal, typeParams) =>
return Success(ScTypePolymorphicType(internal, unresolvedTypeParameters ++ typeParams), Some(this))
case _ =>
return Success(ScTypePolymorphicType(inner, unresolvedTypeParameters), Some(this))
}
case _ =>
}
case _ =>
}
case Some(qualifier) =>
qualifier.getNonValueType(TypingContext.empty) match {
case Success(ScTypePolymorphicType(_, typeParams), _) =>
inner match {
case ScTypePolymorphicType(internal, typeParams2) =>
return Success(ScalaPsiUtil.removeBadBounds(ScTypePolymorphicType(internal, typeParams ++ typeParams2 ++ unresolvedTypeParameters)), Some(this))
case _ =>
return Success(ScTypePolymorphicType(inner, typeParams ++ unresolvedTypeParameters), Some(this))
}
case _ if unresolvedTypeParameters.nonEmpty =>
inner match {
case ScTypePolymorphicType(internal, typeParams) =>
return Success(ScTypePolymorphicType(internal, unresolvedTypeParameters ++ typeParams), Some(this))
case _ =>
return Success(ScTypePolymorphicType(inner, unresolvedTypeParameters), Some(this))
}
case _ =>
}
}
Success(inner, Some(this))
}
def getPrevTypeInfoParams: Seq[TypeParameter] = {
qualifier match {
case Some(_: ScSuperReference) => Seq.empty
case Some(qual) =>
qual.getNonValueType(TypingContext.empty).map {
case t: ScTypePolymorphicType => t.typeParameters
case _ => Seq.empty
}.getOrElse(Seq.empty)
case _ => getContext match {
case sugar: ScSugarCallExpr if sugar.operation == this =>
sugar.getBaseExpr.getNonValueType(TypingContext.empty).map {
case t: ScTypePolymorphicType => t.typeParameters
case _ => Seq.empty
}.getOrElse(Seq.empty)
case _ => Seq.empty
}
}
}
private def resolveFailure = Failure("Cannot resolve expression", Some(this))
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 850 |
{"url":"https:\/\/www.thestudentroom.co.uk\/showthread.php?t=4492018","text":"You are Here: Home >< Maths\n\n# Delta integration property result\/proof watch\n\n1. I have that\n\nis this correct and as simple form as I can get it?\n\nHow do I go about proving it?\n\nMany thanks\n2. (Original post by xfootiecrazeesarax)\nI have that\n\nis this correct and as simple form as I can get it?\n\nHow do I go about proving it?\n\nMany thanks\nDidn't you ask the same question some time ago? I replied that it looked like something that only makes sense inside a double integral, and that Dirac proves something vaguely similar in the Principles of Quantum Mechanics.\n3. (Original post by xfootiecrazeesarax)\nI have that\n\nis this correct and as simple form as I can get it?\n\nHow do I go about proving it?\n\nMany thanks\nThat can't be right as the dummy variable k is in the answer in the right hand side.\n4. (Original post by xfootiecrazeesarax)\nI have that\n\nis this correct and as simple form as I can get it?\n\nHow do I go about proving it?\n\nMany thanks\n1. As RichE pointed out, your variables look screwed up. In general, bear in mind that delta functions only make sense inside integrals, so given that your final expression has a delta function, it must be integrated w.r.t some variable. But:\n\nmakes no sense (or at least, doesn't look useful), and neither do the other possibilities with da or dc.\n\n2. Take a look at page 70 of this:\n\nhttps:\/\/archive.org\/stream\/DiracPrin...e\/n69\/mode\/1up\n5. (Original post by RichE)\nThat can't be right as the dummy variable k is in the answer in the right hand side.\nahh my apologies !\n\nthat should be f(a) , or f(c) on the RHS with delta (a-c) multiplying it so it doesn't matter which !\n6. (Original post by atsruser)\n1. As RichE pointed out, your variables look screwed up. In general, bear in mind that delta functions only make sense inside integrals, so given that your final expression has a delta function, it must be integrated w.r.t some variable. But:\n\nmakes no sense (or at least, doesn't look useful), and neither do the other possibilities with da or dc.\nIt makes sense to write the delta function outside of an integral. The delta function can be given rigorous meaning as distribution and in that context I think what the OP writes if ok (when f(k) is changed to f(c)).\n\nOne could for example write\n\nto convey that the delta function is the derivative of the Heaviside function.\n7. (Original post by RichE)\nIt makes sense to write the delta function outside of an integral. The delta function can be given rigorous meaning as distribution and in that context I think what the OP writes if ok (when f(k) is changed to f(c)).\nYou've confused me here. In this case, the OP seems to using an informal \"behaviour inside an integral\" approach to the delta function, so given that, I think that it's accurate to say that whatever expression she ends up with must work inside an integral.\n\nAnyway, she has confirmed that she made a typo - I'm still not sure if it's the correct result though, even with the fix.\n8. (Original post by atsruser)\nYou've confused me here. In this case, the OP seems to using an informal \"behaviour inside an integral\" approach to the delta function, so given that, I think that it's accurate to say that whatever expression she ends up with must work inside an integral.\n\nAnyway, she has confirmed that she made a typo - I'm still not sure if it's the correct result though, even with the fix.\nI don't think there's anything in what the OP's done that means a delta functions can only appear within integrals.\n\nA special of the desired identity in fact appears on the wiki page\n\nhttps:\/\/en.wikipedia.org\/wiki\/Dirac_delta_function\n\nat the end of the \"translation\" section.\n9. (Original post by RichE)\nA special of the desired identity in fact appears on the wiki page\n\nhttps:\/\/en.wikipedia.org\/wiki\/Dirac_delta_function\n\nat the end of the \"translation\" section.\nThat's the not quite same identity as she has quoted - there's a sign change, and I'm not sure if we can trivially ignore that due to evenness of the delta function - I tend to confuse myself easily with delta functions so I'm very cautious using them.\n\nThe page from Dirac's book that I quoted above in fact proves almost the exact form of the identity that she wants, but with the same sign change, by means of integration rather than any fancy functional stuff, though.\n10. (Original post by atsruser)\nThat's the not quite same identity as she has quoted - there's a sign change, and I'm not sure if we can trivially ignore that due to evenness of the delta function - I tend to confuse myself easily with delta functions so I'm very cautious using them.\n\nThe page from Dirac's book that I quoted above in fact proves almost the exact form of the identity that she wants, but with the same sign change, by means of integration rather than any fancy functional stuff, though.\nI've now looked at some of the dirac book. I'm now confused even more confused why you say why you say delta function only make sense in integrals. There are plenty of identities on page 60 (of the book. 70 of the pdf) where identities are given that don't involve integrals.\n11. (Original post by RichE)\nI've now looked at some of the dirac book. I'm now confused even more confused why you say why you say delta function only make sense in integrals. There are plenty of identities on page 60 (of the book. 70 of the pdf) where identities are given that don't involve integrals.\nI suspect that our confusion is mutual - I'm not really sure where you are envisaging using deltas outside of integrals - could you give an example? Or are you merely pointing out that a delta can be represented as a linear functional, and that the whole integral thing is hand-waving?\n\nAnyway, maybe I didn't word things properly - by saying that deltas only \"make sense\" inside integral, I'm saying that e.g. all of the identities that Dirac quotes in his book are meant to be used as integrands, and he proves the results from that POV. The page that I showed above in fact says: \"The meaning of any of these equations is that its two sides give equivalent results as factors in an integrand\". Does that clarify things?\n12. (Original post by RichE)\nI've now looked at some of the dirac book. I'm now confused even more confused why you say why you say delta function only make sense in integrals. There are plenty of identities on page 60 (of the book. 70 of the pdf) where identities are given that don't involve integrals.\nJust as a final point: are you referring to cases where delta functions appear in DEs to represent impulsive forces, or point charges, or the like? If so, then yes, I'd agree that you can see \"bare\" delta functions, but even then, we end up immediately integrating them via Laplace transforms or whatever, no?\n13. (Original post by atsruser)\nJust as a final point: are you referring to cases where delta functions appear in DEs to represent impulsive forces, or point charges, or the like? If so, then yes, I'd agree that you can see \"bare\" delta functions, but even then, we end up immediately integrating them via Laplace transforms or whatever, no?\nI was only concerned with your phrasing\n\n\"In general, bear in mind that delta functions only make sense inside integrals, so given that your final expression has a delta function, it must be integrated w.r.t some variable.\"\n\nI would at best say this is confusing for the OP. We've since discussed lots of examples where delta functions make sense in the absence of any integral and are commonly used in this way.\n\nTSR Support Team\n\nWe have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.\n\nThis forum is supported by:\nUpdated: January 8, 2017\nToday on TSR\n\n### Congratulations to Harry and Meghan!\n\nBut did you bother to watch?\n\nPoll\nUseful resources\n\n### Maths Forum posting guidelines\n\nNot sure where to post? Read the updated guidelines here\n\n### How to use LaTex\n\nWriting equations the easy way\n\n### Study habits of A* students\n\nTop tips from students who have already aced their exams","date":"2018-05-22 06:32:42","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\": 0.8615456819534302, \"perplexity\": 664.5704531929808}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794864626.37\/warc\/CC-MAIN-20180522053839-20180522073839-00096.warc.gz\"}"} | null | null |
{"url":"https:\/\/solvedlib.com\/the-balance-sheets-for-a-company-along-with,407569","text":"# The balance sheets for a company, along with additional information, are provided below: Balance Sheets December...\n\nThe balance sheets for a company, along with additional information, are provided below: Balance Sheets December 31, Year 2 and Year 1. Year 2 Year 1 156,250 74,000 85,000 2,000 171,500 87,000 71,000 1,000 430,000 720,000 (400, 000) $1,067, 250 430,000 620,000 (248,000)$1, 132, 500 Assets Current assets: Cash Accounts receivable Inventory Prepaid rent Long-term assets: Land Equipment Accumulated depreciation Total assets Liabilities and Stockholders' Equity Current liabilities: Accounts payable Interest payable Income tax payable Long-term liabilities: Notes payable Stockholders' equity: Common stock Retained earnings Total liabilities and stockholders' equity 89,000 6, 750 6,000 76,000 13,500 4,000 112,500 225,000 650,000 203, 000 $1,067,250 650, 000 164, 000$1, 132, 500 Additional Information for Year 2: 1. Net income is $59,000. 2. The company purchases$100,000 in equipment. 3. Depreciation expense is $152,000. 4. The company repays$112,500 in notes payable. 5. The company declares and pays a cash dividend of $20,000. Required: Prepare the statement of cash flows using the indirect method. (List cash outflows and any decrease in cash as negative amounts.) Statement of Cash Flows For the Year Ended December 31, Year 2 Cash Flows from Operating Activities Adjustments to reconcile net income to net cash flows from operating activities Net cash flows from operating activities Cash Flows from Investing Activities Net cash flows from investing activities Cash Flows from Financing Activities Net cash flows from financing activities Cash at the beginning of the period Cash at the end of the period ## Answers #### Similar Solved Questions 1 answer ##### The process of projecting the present and future availability of qualified personnel in sufficient number is... The process of projecting the present and future availability of qualified personnel in sufficient number is called taking a _______________ _________________.... 1 answer ##### Find each limit. Note that$L$'H\u00f4pital's rule does not apply to every problem, and some problems will require more than one application of$L$'H\u00f4pital's rule.$\\lim _{x \\rightarrow 0} \\frac{e^{2 x}-1-2 x-2 x^{2}}{x^{3}}$Find each limit. Note that$L$'H\u00f4pital's rule does not apply to every problem, and some problems will require more than one application of$L$'H\u00f4pital's rule.$\\lim _{x \\rightarrow 0} \\frac{e^{2 x}-1-2 x-2 x^{2}}{x^{3}}$... 5 answers ##### Required information Consider the system of equations_13X1 2x2 x3 = 27 ~3X1 - 6x2 2x3 = -61.5 X1 +x2 + 5*3 =-21.5Using the M-file _ solve the given system of equations. (Round the final answers to four decimal places)The lower and upper triangular matrices for the given system of equations are as follows:13 and U =where_ Required information Consider the system of equations_ 13X1 2x2 x3 = 27 ~3X1 - 6x2 2x3 = -61.5 X1 +x2 + 5*3 =-21.5 Using the M-file _ solve the given system of equations. (Round the final answers to four decimal places) The lower and upper triangular matrices for the given system of equations are as... 1 answer ##### Question 18 Knight Company reports the following costs and expenses in May.$16,800 Direct labor $72,600... Question 18 Knight Company reports the following costs and expenses in May.$16,800 Direct labor $72,600 Factory utilities Depreciation on factory equipment 14,350 Sales salaries 46,700 Depreciation on delivery trucks 4,200 3,500 Indirect factory labor Indirect materials Direct materials used Factor... 1 answer ##### A very large data set (N > 10,000) has a mean value of 1.47 units and... A very large data set (N > 10,000) has a mean value of 1.47 units and a standard deviation of 0.035 units. Determine the range of values in which 95% of the data set should be found assuming normal probability density (find a in the equation xi = x' \u00b1 a).... 1 answer ##### 6. The United States is an important trading partner for Canada. How might the United States'... 6. The United States is an important trading partner for Canada. How might the United States' military actions around the world affect the level of inventory that a Canadian company keeps on hand?... 5 answers ##### Not yet answered Marked out of 2.00Flag questionThe charge per unit length on long straight thin wire is 33.3 pC\/m: The electric field (in MN\/C) at a distance of 5.5 cm from the axis of the wire is: (use k=9.0x109 N. m2\/c2)Select one: OA 5.45B. 10.90Oc.7.87D. 9.08E: 13.32Previous pageNext page Not yet answered Marked out of 2.00 Flag question The charge per unit length on long straight thin wire is 33.3 pC\/m: The electric field (in MN\/C) at a distance of 5.5 cm from the axis of the wire is: (use k=9.0x109 N. m2\/c2) Select one: OA 5.45 B. 10.90 Oc.7.87 D. 9.08 E: 13.32 Previous page Next p... 5 answers ##### Current Attempt in ProgressFind the derivative of the function f(x) V4+ Vzdf dc Current Attempt in Progress Find the derivative of the function f(x) V4+ Vz df dc... 5 answers ##### 10) For ethonal, AHvap 43.5 k] \/mol. Calculate q, W, AH , and AU when 1.75 mol of ethonal is vaporized at 255 K and 0.641 atm 10) For ethonal, AHvap 43.5 k] \/mol. Calculate q, W, AH , and AU when 1.75 mol of ethonal is vaporized at 255 K and 0.641 atm... 1 answer ##### 2) On January 1, 2016, Amethyst Manufacturing Corporation purchased a machine for$40,000,000. The corporation expects...\n2) On January 1, 2016, Amethyst Manufacturing Corporation purchased a machine for $40,000,000. The corporation expects to use the machine for 24,000 hours over the next six years. The estimated residual value of the machine at the end of the sixth year is$40,000. The schedule of usage of the machin...\n##### )on 4. Suppose X and y are continuous random variables with joint density funstion the unit...\n)on 4. Suppose X and y are continuous random variables with joint density funstion the unit square [0, 1] x [0, 1]. (a) Let F(r,y) be the joint CDF. Compute F(1\/2, 1\/2). Compute F(z,y). (b) Compute the marginal densities for X and Y (c) Are X and Y independent? (d) Compute E(X), E(Y), Cov(X,y)...\n##### Give the name for HNO3:a. nitrite acidb. nitric acidc. nitrate acidd. hydrogen nitridee. nitrous acid\nGive the name for HNO3: a. nitrite acid b. nitric acid c. nitrate acid d. hydrogen nitride e. nitrous acid...\n##### \"LimitingQuestion __ (30points) Diphosphorous pentoxide (PzOs (s)) and water are combined mak aqucous phosphoric acid (HsPOa (aq))If 10.0 grams of each reactant undergo reaction with 85.0 % yield, how many grams of phosphoric acid are obtained from the reaction?\n\"Limiting Question __ (30points) Diphosphorous pentoxide (PzOs (s)) and water are combined mak aqucous phosphoric acid (HsPOa (aq)) If 10.0 grams of each reactant undergo reaction with 85.0 % yield, how many grams of phosphoric acid are obtained from the reaction?...\n##### (12 points) Determine if the lines1=3+3 V = G 3=3-V;intersect,parallel,skew. If they iutersect, fiucl the point of interection,(12 points) The plaues P1 21 - 3v + : = d \/ 21 - \"+A=ehoth contain the point P =(\/30 4\/31. Fiud prametric (uatious for thcir line of intersection\n(12 points) Determine if the lines 1=3+3 V = G 3=3- V; intersect, parallel, skew. If they iutersect, fiucl the point of interection, (12 points) The plaues P1 21 - 3v + : = d \/ 21 - \"+A=ehoth contain the point P =(\/30 4\/31. Fiud prametric (uatious for thcir line of intersection...\n##### The figure shows a loudspeaker A and point C, where a listener is positioned. |AC| =...\nThe figure shows a loudspeaker A and point C, where a listener is positioned. |AC| = 3.00 m and the angle \u03b8 = 48 \u00b0. A second loudspeaker B is located somewhere to the left of A. The speakers vibrate out of phase and are playing a 64.0 Hz tone. The speed of sound is 343 m\/s. What is the clo...\n##### You are trying to estimate the average amount a family spends onfood during a year. In the past the standard deviation of theamount a family has spent on food during a year has beenapproximately $500. If you want to be 99% sure that you haveestimated average family food expenditures within$50, how manyfamilies do you need to survey? Round your answer up to the nearestwhole number, if necessary.\nYou are trying to estimate the average amount a family spends on food during a year. In the past the standard deviation of the amount a family has spent on food during a year has been approximately $500. If you want to be 99% sure that you have estimated average family food expenditures within$50, ...\n##### For this problem, you need to determine the magnetic field at the center of the large...\nFor this problem, you need to determine the magnetic field at the center of the large loop. The formula for that is B = Howhere R is the radius of the loop. 2R Resistivity (\u03a9m ) Material 2.65 x 108 Aluminum 1.68 x 10-B Copper Gold 2.44 x 108 9.71 x 10 Iron 10.6 x 10-8 Platinum Silver 1.59 x 10...\n##### Write a valid set of quantum numbers forad orbital inMon =Select[Select ]ml =Select ]\nWrite a valid set of quantum numbers forad orbital inMo n = Select [Select ] ml = Select ]...\n##### 06 Determining Rate Laws from Initial Rates 1 PointDetermine the rate law for the following reaction using the data provided: CHsCl(g) + 3 Clz (g) CCL (g) + 3HCl(g)[CHzCl] (M)[CIz] (M)Initial Rate0.0200.0200.0120.0400.0200.0250.0400.0400.098rate-k[CH:CIJ[Clz]? rate_k[CH; C1J2[C1J4 rate_k[CHsC1J? [Clz]? rate_k[CHsCIJ[Cl]? rate_k[Clz]?\n06 Determining Rate Laws from Initial Rates 1 Point Determine the rate law for the following reaction using the data provided: CHsCl(g) + 3 Clz (g) CCL (g) + 3HCl(g) [CHzCl] (M) [CIz] (M) Initial Rate 0.020 0.020 0.012 0.040 0.020 0.025 0.040 0.040 0.098 rate-k[CH:CIJ[Clz]? rate_k[CH; C1J2[C1J4 rate...\n##### Find cle derivacive 0: tke illoniug; %+) =Ctto &\nFind cle derivacive 0: tke illoniug; %+) = Ctto &...\n##### Please,I need the solution within 15 min Question 3 b: \/Mark 17] (CLO 3 and 4)....\nplease,I need the solution within 15 min Question 3 b: \/Mark 17] (CLO 3 and 4). A 236 kV, three-phase medium transmission line is 211 km long. The line has a per phase series resistance R = 0.041 N, inductive reactance XL =j 0.3012, per kilo-meter for both, and shunt capacitive reactance X = 211.8 k...\n##### Find the limit: (If an answer does not exist, enter DNE:) lim [1\/07 + x)] - (1\/7) X-0Need Help?Rend ItHte[kitend IuterSubmit AnswerPractice Another Version[0\/1 Points]DETAILSPREVIOUS ANSWERS\nFind the limit: (If an answer does not exist, enter DNE:) lim [1\/07 + x)] - (1\/7) X-0 Need Help? 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To say that actress Ashley Bratcher is enthusiastic about her latest film project is an understatement.
Opening in theaters nationwide March 29, the film is Johnson's book of the same title and recounts how Johnson, once honored by Planned Parenthood as its "Employee of the Year," came face-to-face with the reality of abortion and converted to the pro-life cause.
However, that all changed in September 2009, Johnson said, when she was asked to provide "an extra set of hands" during an ultrasound-guided abortion.
Both the real-life Johnson and her onscreen counterpart hope that the film's audience will include both pro-life theatergoers and those who support legal abortion. And they said it will challenge both sides. | {
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\section{Introduction}
Graphene nanoribons have been extensively studied in the past years \cite{Han07,Lin07,Wan08,Mol10,Gal10,Ter11}, mainly due to their promise of an electronic band-gap making them interesting for electronic applications. Confinement of electrons in these nanoscaled structures is predicted to form a quasi one-dimensional system \cite{Lin08} with its properties strongly depending on the configuration of the edges \cite{Son06,Yan07}. However, experimental and theoretical studies have revealed graphene nanoribbons to be extremely sensitive to small amounts of disorder, in particular to edge disorder~\cite{Wan11,Kos09}. In fact, the transport characteristics of nanostructured graphene ribbons are mainly dominated by statistical Coulomb blockade effects~\cite{Gal10,Sta09}. Improvements on the fabrication techniques allowing for cleaner edge configurations
are therefore of great importance and may not only improve the transport properties~\cite{Tom11} but also enable the investigation of the unique vibrational properties of these graphene nanostructures~\cite{Sai10}. Despite theoretical work~\cite{Gil09,Gil10} there are - to our knowledge - only a few optical characterization studies of graphene nanoribbons~\cite{Bis11,Ryu11}.
Raman spectroscopy of carbon materials, in general, has been identified as a powerful tool for determining the number of graphene layers \cite{Fer07,Mal09}, the local amount of strain and doping \cite{Lee12}, and for studying electron-phonon interactions \cite{{Yan07,Che11,Sta07,Pis07}} and therefore the electronic properties themselves.
In this work, we report on Raman spectroscopy measurements on non-etched graphene ribbons of various widths (from $\approx$15 to 160~nm) resulting from peeling-off a graphene flake on the boundary region of a hexagonal boron nitride (hBN) flake and its underlying SiO$_{\text{2}}$ substrate. We show that the characteristic signatures of single-layer graphene are well preserved and that the configuration of the edges is more regular compared to previously studied graphene ribbons fabricated by state-of-the-art reactive ion etching (RIE) techniques~\cite{Bis11,Ryu11}. Moreover, the analysis of the full width at half maximum (FWHM) of the G- and 2D-line ($\Gamma_{G}$ and $\Gamma_{2D}$) as well as the frequency of the G- and 2D-line ($\omega_{G}$ and $\omega_{2D}$)
provide strong indications of finite size and/or edge effects~\cite{Gil09,Gil10,Ryu11}.
\section{Fabrication}
\begin{figure}[t]%
\includegraphics*[draft=false,keepaspectratio=true,clip,%
width=1\linewidth%
,height=9.0cm]{Fig1.eps}
\caption{%
(color online)
(a) Optical microscope image of a $\approx\!30\,$nm thin hBN flake (light blue color) on top of a Si/SiO$_{\text{2}}$ substrate (grey color).
(b), (c) and (d) Scanning force microscope (SFM) images taken in the region highlighted by the black box in panel (a).
(c) SFM close-up image of the white-dashed box in panel (b). In this region the ribbons are separated by a distance of around $1\,\mu$m, twice as large as the spot-size $d\!\approx\!500$ nm (white circle) of the linearly polarized laser with an angle $\theta$.
(d) SFM close-ups of the ribbons (1), (2), (3) and (4), also displayed in panel (c).
Ribbons (1) and (2) do not have a constant width, as highlighted in the two upper subpanels of panel (d). We show the wider and narrower ends of these ribbons.
(e) Characteristic Raman spectra of bulk graphene on hBN [acquisition point B in panel (b)] and of ribbon (2) [acquisition point A, panel (c)].
}
\label{onecolumnfigure}
\end{figure}
The fabrication of the graphene ribbons is based on purely mechanical exfoliation of graphite. We initially prepared
Si/SiO$_{\text{2}}$ samples with deposited hBN flakes (Fig. 1a). The hBN flakes have been mechanically exfoliated from pure hBN crystals and deposited onto the Si/SiO$_{\text{2}}$ substrate. Thereafter, the samples were immersed in a piranha solution, 3:1 mixture of sulfuric acid (H$_2$SO$_4$) and $30\%$ hydrogen peroxide (H$_2$O$_2$), for 3 minutes and later rinsed with ultrapure water. This cleaning procedure has a similar effect on the SiO$_{\text{2}}$ surface than a plasma etching step prior deposition of the graphene flakes. Both methods are supposed to hydroxylate the SiO$_{\text{2}}$ surface~\cite{Tib13} and therefore increase the local adhesion of graphene to the surface. The Raman spectrum of graphene on such a treated SiO$_{\text{2}}$ substrate is characterized by a very slight increase of the FWHM of the 2D-line~\cite{Wan12}.
The hBN flakes are known to be chemically inert and therefore not affected by the piranha solution at room temperature \cite{Alt07}. Interestingly, we nonetheless observe an increase in doping of graphene on hBN compared to graphene regions resting on SiO$_{\text{2}}$.
\begin{figure}[b]%
\includegraphics*[draft=false,keepaspectratio=true,clip,%
width=1\linewidth%
,height=12.0cm]{Fig2.eps}
\caption{%
(color online)
(a) Raman spectra (D- and G-line) of the ribbon (3) on SiO$_{\text{2}}$ as a function of the polarization angle $\theta$ (see Fig. 1c). The difference in polarization angle between subsequent traces is $\theta\!=\!22.5^{\circ}$. The Raman spectra are normalized to the G-line maximum height and shifted for clarity.
(b) Polar plot of $I(D)/I(G)$ as a function of $\theta$ for ribbon (3) on both hBN (blue trace) and SiO$_{\text{2}}$ (red trace) substrates.
(c) and (d) Raman spectra of ribbon (4), on the hBN substrate, at $\theta\!=\!0^{\circ}$ (c) and $90^{\circ}$ (d). The Lorentzian fits to the data are shown in blue.
}
\label{onecol}
\end{figure}
While the hBN flakes have been directly deposited on the SiO$_{\text{2}}$ substrate, the graphene flakes have been prepared on top of a $\approx$300\,nm-thick layer of polymethylmethacrylate (PMMA) previously spin-coated on a glass slide \cite{Zom11}. Raman spectroscopy was used to identify and select individual single-layer graphene flakes \cite{Fer07,Gra07}. The resulting graphene-PMMA-glass stamp was then mounted in a mask-aligner in such a way that the graphene flake could be aligned on top of the hBN-SiO$_{\text{2}}$ piranha-treated chip \cite{Wan13}. Once on top of the hBN-SiO$_{\text{2}}$ target region, the two flakes were brought into contact.
This process was repeated until some parts of the graphene flake stuck to the hBN-SiO$_{\text{2}}$ surface.
This technique utilizes van der Waals adhesion to peel-off the graphene ribbons (shown in Fig. 1a), the hBN substrate is therefore important for this fabrication process since graphene adheres more strongly to the hBN than SiO2 \cite{Wan13}. The yield of this fabrication process is nonetheless low and neither the position nor the width of the obtained graphene ribbons is controllable. Therefore, this fabrication method is -~in its present form~- irrelevant from a technological point of view, but it is extremely valuable since it allows the Raman (and potentially transport) investigation of non-etched, i.e. pristine, graphene ribbons. Moreover, we would like to emphasize that these graphene ribbons were never in contact with any spin-coated polymer resist typically involved in the fabrication of etched ribbons, nor with any solvents such as acetone, isopropanol or even water.
An optical microscope image of a fabricated sample is shown in Fig.~1a. For simplicity, we grouped the ribbons of similar width and labeled them as (1)-(4) (shown in Fig.~1c). The widths were extracted from scanning force microscope (SFM) images (Figs.~1b, 1c and 1d), resulting in a width of W $\approx$ 160 and 120 nm for the ribbons (4) and (3). The widths of the ribbons (1) and (2) differ significantly between left and right ribbon ends (see upper panels in Fig.~1d). Specifically, ribbons (1) and (2) have a varying width from W $\approx$ 40 to 15~nm [ribbon (1)] and W $\approx$ 45 to 20~nm [ribbon (2)].
In the following, we will therefore refer to the average width $W\approx$ 25, 35, 120 and 160~nm of the ribbons (1), (2), (3) and (4). The Raman data were recorded using a laser wave length of 532~nm ($\hbar \omega_L\!=\!2.33\,eV$) through a single-mode optical fiber whose spot size is limited by diffraction. The measurement setup is a commercially available confocal Raman Microscope Alpha 300R from Witec, whose laser is linearly polarized. The sample was fixed to a high-precision rotation mount model PRM-1 from Thorlabs, in order to manually adjust the polarization laser direction relative to the ribbon axis (see inset in Fig. 1c). A long working distance focusing lens with numerical aperture of 0.85 is used to obtain a spot size of approximately 500 nm (circle in Fig. 1c). Characteristic Raman spectra of the narrow ribbon (2) and bulk graphene, both on the hBN substrate, are presented in Fig.~1e. The Raman spectra (labels A and B in Fig.~1b and 1c) show the prominent G-line ($\approx$1582 cm$^{-1}$) as well as the single Lorentzian-shaped 2D-line ($\approx$2675 cm$^{-1}$) as expected for graphene. No defect induced D-line ($\approx$1340 cm$^{-1}$) or D'-line ($\approx$1620 cm$^{-1}$) are observed on the bulk graphene region (acquisition point B), which confirms that the fabrication method does not induce defects into the graphene flake. In both spectra, a third prominent sharp peak arises at $\approx$1365 cm$^{-1}$, which is attributed to the Raman-active LO phonon of hBN~\cite{Gei66}.
\begin{figure}[t]%
\includegraphics*[draft=false,keepaspectratio=true,clip,%
width=1\linewidth%
,height=9.0cm]{Fig3.eps}
\caption{%
(color online)
(a) Correlation between $\omega_{2D}$ and $\omega_{G}$ at T = 300 K. The description of the black and gray axis as well as the color code are introduced in the main text.
(b) and (c) False-colored Raman maps of I(hBN) and I(2D).
(b) The boundary between the hBN and SiO$_{\text{2}}$ substrates is marked with a white dashed line.
(c) The individual ribbons (1), (2), (3) and (4) are well differentiated from each other.
(d) Map of the local 2D-line shifts due to strain $\omega_{2D,\varepsilon}$
obtained after projecting all the Raman data points onto the strain axis [solid black line in panel (a)] relative to its maximum value [$\omega_{2D,\varepsilon}^{max}$, green point in panel (a)]. The scale bar is $2\mu$m.
}
\label{onecol}
\end{figure}
\section{Characterization of the edges}
In order to characterize the edges and in particular the edge roughness of the graphene ribbons, we performed polarization angle dependent Raman measurements. Fig. 2 shows the Raman spectra of the ribbons (3) (W $\approx$ 120 nm, Figs.~2a and 2b) and (4) (W $\approx$ 160 nm, Fig. 2c and 2d) as function of the polarization angle $\theta$ of the incident light (see inset in Fig.~1c). For each ribbon and each polarization angle, a spectrum has been recorded and the G-, D- and hBN-lines were fitted with a single Lorentzian line shape (see examples in Figs.~2c and 2d). In agreement with previous work~\cite{Bis11,Can04,Gr�03,Cas09}, the D-line intensity $I(D)$ appears to be strongest for polarization parallel to the edge and reaches a minimum for the perpendicular polarization $\theta\!=\!90^{\circ}$. This can be observed in Fig. 2a, where each Raman spectrum corresponds to a different polarization angle $\theta$, starting from $\theta\!=\!11.25^{\circ}$ to $\theta\!=\!348.75^{\circ}$ in steps of $22.5^{\circ}$. Every trace in this plot is normalized to the maximum intensity of the G-line and shifted in the intensity and frequency axis for clarity.
For the rest of the analysis, we compare the ratio $I(D)/I(G)$ using the peak area of the fitted Lorentzian function as a measure of intensity.
In Fig. 2b we show a corresponding polar plot which illustrates the expected mirror planes at $\theta\!=\!0^{\circ}$ and $\theta\!=\!90^{\circ}$ ~\cite{Can04,Gr�03}.
Raman spectra with Lorentzian fits for the direction of maximum and minimum D-line intensity ($\theta\!=\!0^{\circ}$ and $\theta\!=\!90^{\circ}$, respectively) of ribbon (4) are
presented in Figs.~2c and 2d.
According to Ref.~\cite{Cas09} and assuming that $I(G)$ does not depend on $\theta$, a lower bound for the edge disorder correlation length $\xi\!\approx\!2v_F/(\omega_Lb)$ can be estimated from the ratio $b\!=\!I(D)_{min}/I(D)_{max}$ between the lowest and highest normalized D-line intensity ($I(D)_{min}/I(G)$ and $I(D)_{max}/I(G)$, respectively).
For the ribbon (4) (Fig. 1c and 1d), we obtain the lowest intensity ratio of $b\,\approx\,0.055$ (Fig. 2c and 2d), which yields a correlation length of $\xi\,\approx\,10\,$nm. This value is significantly higher than the correlation length of $\xi\!\approx\!1\,$nm reported on plasma etched graphene nanoribbons~\cite{Bis11} and therefore suggests that the graphene ribbons have a more regular crystallographic orientation of the edges.
\section{Strain, doping and finite size effects}
For a more detailed investigation of the dependence of the Raman spectra
on the width of the graphene ribbons,
we study
spatially resolved Raman maps of the sample.
In particular, we recorded a Raman map of the 6 $\mu$m by 10 $\mu$m sample region shown in Fig. 1b with a spatial oversampling of 200~nm and an integration time of 15~s
(with a laser spot size of 500~nm and a laser power of $\approx$\,1~mW). The corresponding Raman maps of the hBN-line and the 2D-line intensities, $I(hBN)$ and $I(2D)$, are shown in Figs.~3b and 3c. One can identify the hBN and SiO$_{\text{2}}$ substrates and the graphene ribbons (1)-(4), partly crossing both substrates. In the lower right
corner of Fig.~3c, bulk graphene is also observed.
By means of the so-called vector decomposition method introduced by Lee et al. \cite{Lee12},
we analyze the presence of strain and/or doping variations in our sample.
Accordingly, we plot the dependence of the G-line ($\omega_{G}$) and the 2D-line ($ \omega_{2D}$) positions (i.e. frequencies) for all the Raman spectra recorded in the inspected area (Fig. 1b) in Fig.~3a. The red points show the extracted Raman data from spectra recorded on bulk graphene and ribbons, both on SiO$_{\text{2}}$, whereas the light blue points are from graphene regions resting on hBN. The blue data points with error bars show the average values of $\omega_{G}$ and $\omega_{2D}$ obtained for every individual graphene ribbon (1)-(4) and bulk graphene (B) on the hBN substrate (see labels in Fig. 3a).
\begin{figure}[t]%
\includegraphics*[draft=false,keepaspectratio=true,clip,%
width=1\linewidth%
,height=12.0cm]{Fig4.eps}
\caption{%
(color online)
(a) and (b) Local distribution of $\omega_{G}$ and $\Gamma_{2D}$, respectively. The scale bars are 2$\mu$m.
(c) Averaged $\omega_{G}$ and $\omega_{2D}$ for every individual graphene ribbon on hBN as function of 1/$W$.
(d) Averaged $\Gamma_{G}$ and $\Gamma_{2D}$ for the individual graphene ribbons on hBN as function of 1/$W$.
(e) Correlations between $\Gamma_{G}$ and $\omega_{G}$ for the ribbons and bulk graphene on the hBN substrate. The light red data points correspond to the narrowest ribbons (1) and (2) with their respective averages marked in red. The ribbons (3), (4) and bulk graphene (B) appear as gray data points with their average values in black. The error bars in all the panels are half the standard deviation.
}
\label{onecol}
\end{figure}
The solid and dashed lines indicate the slopes of the strain and large-scale doping axis according to Ref.~\cite{Lee12}. Please note that we do not know the exact origin of these two axis and, for simplicity, we marked the same origin as in
Ref.~\cite{Lee12} (see green point in Fig. 3a).
Interestingly, the red cloud of data points clearly follows a slope of $\Delta \omega_{2D}/\Delta\,\omega_{G}\!=\!2.2$ (solid black line), characteristic of uniaxial strain - in good agreement with Lee et al. \cite{Lee12} -.
Both red and main light blue data clouds appear to be offset by $\approx\!2.2\,$cm$^{-1}$ in the $\omega_{G}$ axis with a direction parallel to the strain axis. This offset is understood as a difference in induced doping \cite{Che11} between the SiO$_{\text{2}}$ and the hBN substrates (extracted doping difference: $\Delta n\approx 2\,$x$\,10^{11}\,$cm$^{-2}$), most likely due to the treatment with the piranha solution of the hBN surface.
More interestingly, Fig.~3a suggests that the narrowest ribbons [(1) and (2)] are subject to stronger doping compared to bulk graphene and the wider ribbons. This is noticeable from their mean positions [labeled (1) and (2) in Fig.~3a], which are at the very right of this plot (see right gray dashed line of slope 2.2).
However, there is an inconsistency with the line width of the G-peak that will be discussed in the following section.
Interestingly, the same ribbons [(1) and (2)] seem to have also different strain values compared to bulk graphene and the wider ribbons [(3) and (4)] (see lower dashed line).
This finding is highlighted after projecting all ($\omega_{2D}$; $\omega_{G}$) points onto the strain axis (the obtained values are labeled as $\omega_{2D,\epsilon}$).
In Fig. 3d we show the corresponding spatial map of the difference $\omega_{2D,\epsilon} - \omega_{2D,\epsilon}^{max}$ relative to its maximum value $\omega_{2D,\epsilon}^{max}$. Here, we show that the strongest deviations are clearly for the two most narrow ribbons (see yellow and red regions in Fig. 3d). Please note that in bulk graphene, the values decrease close to the hBN edge and on bubbles (marked by white arrows in Figs.~1b and 3d), which is a further sign that this quantity is indeed related to strain.
For a more quantitative analysis of the dependence of the Raman G- and 2D-modes on the ribbon width, we analyze the changes in frequency and broadening of the G-line as well as $\omega_{2D}$ and $\Gamma_{2D}$ as a function of the averaged ribbon width $W$. Apart from the aforementioned difference in doping between the hBN and SiO$_{\text{2}}$ substrates (Fig.~3a), the spatial representation of $\omega_{G}$ (Fig.~4a) reveals a stiffening of the G-line for the narrower ribbons (1) and (2), which is in agreement with Fig. 3a and earlier work~\cite{Ryu11}.
Fig.~4c shows $\omega_{G}$ and $\omega_{2D}$ as a function of the inverse averaged width ($1/W$) for the different ribbons.
Interestingly, we observe an increase of $\omega_{G}$ as function of $1/W$ (see dashed line in Fig.~4c), meaning that the smaller the ribbon the stiffer the G-line. This is commonly attributed to edge doping and/or confinement effects~\cite{Ryu11}. The 2D-line frequency $\omega_{2D}$ does not show any substantial dependence with the width of the ribbons (see red data points in Fig.~4c). In Fig. 4d we show that also the G- and 2D-peak line widths ($\Gamma_{G}$ and $\Gamma_{G}$) increase with decreasing ribbon width $W$. This width dependent broadening might be an indication
of finite size effects~\cite{Fer00}.
In order to exclude doping effects for the increase of $\Gamma_{G}$, we show the dependence of $\omega_{G}$ as function of $\Gamma_{G}$ (re-plotting the data shown in Figs.~4c and 4d), in Fig.~4e.
In complete disagreement with experimental results on bulk graphene~\cite{Yan07,Sta07,Pis07} and theory~\cite{And06} on doping dependent Landau
damping, we observe an increase of $\omega_{G}$ with increasing $\Gamma_{G}$. For bulk graphene, exactly the opposite
has been observed in earlier experiments~\cite{Yan07,Sta07,Pis07}.
Finally, from Figs.~3a and 3d we can estimate a maximum strain difference in the narrow ribbons. Assuming uniaxial strain, we extract a maximum strain difference on the order of 0.23\%~\cite{Lee12}. It is important to note that according to Ref.~\cite{Moh09} these values cannot explain the observed maximum broadening of the G-line (Fig. 4e), making edge effects and/or finite size effects a prime candidate to explain our experimental findings.
\section{Conclusion}
In summary, we discussed Raman spectroscopy measurements on lithography-free fabricated graphene ribbons made by direct exfoliation of graphene on hBN flakes. Despite a prominent doping of the hBN substrate, most probably induced by the fabrication process, we were able to perform polarization dependent measurements that confirm a more regular crystallographic orientation of the ribbon edges. Analysis of the frequency and broadening of the G- and 2D-line show prominent differences between the narrowest ribbons ($\approx$ 15 and 20 nm) and the widest ones (bulk graphene included), suggesting the presence of confinement and/or edge effects in these narrow structures. The results of this work highlight that further developments in the fabrication process yielding cleaner graphene samples with regularly oriented edges may enhance both the vibrational and electronic characteristics of graphene devices.
{Acknowledgment ---}
We thank G. Rossell�, D. May and P. Nemes-Incze for valuable discussions.
Support by the HNF, JARA Seed Fund, the DFG (SPP-1459 and FOR-912), the ERC (GA-Nr. 280140) and the EU project Graphene Flagship (contract no. NECT-ICT-604391) are gratefully acknowledged.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,960 |
Q: Four polynomials with single-rooted sums From a 2005 Russian olympiad. Prove that there do not exist four (pairwise) different quadratic polynomials, with leading coefficient 1, such that the sum of any two of them has a single root.
(Optionally, this is what I've done so far:
If $P_i(x)=x^2+b_ix+c_i$, then $P_i+P_j$ is single-rooted iff $(b_i+b_j)^2=8(c_i+c_j)$, with root $-(b_i+b_j)/2$. If two of the six roots are equal, we're done, so suppose they're different, and that $P_1+P_2$ is single-rooted, that is, $(b_1+b_2)^2=8(c_1+c_2)$. If we want $P_1+P_3$ and $P_2+P_3$ to be single-rooted as well, we need $(b_1+b_3)^2=8(c_1+c_3)$ and $(b_2+b_3)^2=8(c_2+c_3)$, from which, taking $b_1$ and $c_1$ as known, $b_2, c_2, b_3, c_3$ could be found.
This way looks cumbersome.)
A: You problem statement is too vague. I guess you mean all four polynomials and their pairwise sums are non-degenerate, otherwise we can simply pick any four straight lines whose slopes are nonzero and of different magnitudes. There should be other restrictions as well, or else {x^2, 2x^2, 3x^2, 4x^2} will also be a solution.
The following transformation of the problem may help. Let P_i(x) = a_ix^2 + b_ix + c_i. If P_1+P_2 is a polynomial with a single root (actually, do you mean a double root?), it must be equal to some k_1(x-x_1)^2 with k_1 \not = 0 and x_1 \in \mathbb{R}. Thus a_1+a_2 = k_1 and vice versa for other pairwise sums. So we get the following matrix equation:
[1 1 0 0] [a_1] [k_1]
[1 0 1 0] [a_2] [k_2]
[1 0 0 1] [a_3] = [k_3]
[0 1 1 0] [a_4] [k_4]
[0 1 0 1] [k_5]
[0 0 1 1] [k_6].
Call the leftmost matrix M. Hence the vector u = (k_1, ..., k_6)^T must lie inside the the column space of M. By inspecting the coefficients b_i and c_i, we see that v = (k_1x_1, ..., k_6x_6)^T and w = (k_1x_1^2, ..., k_6x_6^2)^T also lie inside this column space. Now, if you can prove that this is impossible, you are done.
A: So we have the equations
$(b_1+b_2)^2=8(c_1+c_2)$ (i)
$(b_1+b_3)^2=8(c_1+c_3)$ (ii)
$(b_1+b_4)^2=8(c_1+c_4)$ (iii)
$(b_2+b_3)^2=8(c_2+c_3)$ (iv)
$(b_2+b_4)^2=8(c_2+c_4)$ (v)
$(b_3+b_4)^2=8(c_3+c_4)$ (vi)
Substracting (ii) from (i) we get
$(b_2-b_3)(b_2+b_3+2b_1)=8(c_2-c_3)$ (1)
and substracting (vi) from (v) we get
$(b_2-b_3)(b_2+b_3+2b_4)=8(c_2-c_3)$ (2)
Now, substracting (2) from (1) we get
$2(b_2-b_3)(b_1-b_4)=0$.
In a similar fashion we arrive at
$2(b_2-b_1)(b_3-b_4)=0$,
$2(b_1-b_3)(b_2-b_4)=0$.
The last three equations show that $b_1=b_2=b_3=b_4$. Using (i) through (vi), we see that $c_1=c_2=c_3=c_4$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,105 |
\section{Introduction}
\label{sec:intro}
Contextual bandits refer to a learning setting where the learner
repeatedly observes a context, takes an action and observes a reward
for the chosen action in the observed context, \emph{but no feedback
on any other action}. An example is movie recommendation, where the
context describes a user, actions are candidate movies and the reward
measures if the user enjoys the recommended movie. The learner
produces a policy, meaning a mapping from contexts to actions. A common
question in such settings is, given a \emph{target policy}, what is
its expected reward? By letting the policy choose actions
(e.g., recommend movies to users), we can compute its reward. Such
\emph{online evaluation} is typically costly since it exposes users to
an untested experimental policy, and does not scale to evaluating many
different target policies.
\emph{Off-policy evaluation} is an alternative paradigm for the same
question. Given logs from the existing system, which might be choosing
actions according to a very different \emph{logging policy} than the one we
seek to evaluate, can we estimate the expected reward of the
\emph{target policy}? There are three classes of approaches to address
this question: the \emph{direct method} (DM), also known as regression
adjustment, \emph{inverse propensity scoring}
(IPS)~\citep{horvitz1952generalization} and \emph{doubly robust} (DR)
estimators~\citep{robins1995semiparametric,bang2005doubly,dudik2011doubly,dudik2014doubly}.
\begin{comment}
\ywedit{[Discussion]There is no factual mistake but I think the subsequent paragraph is a bit too terse. It might be clear to us (in the hindsight) that ATE is a special case of off-policy evaluation, and the crux is a consistency assumption, but for readers (even those that are familiar with the cited work), might not immediately get what's in common between ATE estimation and off-policy evaluation in k-arm bandit.
Two points to think about: 1. Consistency alone might not be enough to make DR or DM optimal. DR with the consistent DM in k-arm bandit is not asymptotically optimal. 2. Are we not mentioning the other difference: finite sample vs. asymptotic settings here?
}
\ywedit{[Proposed modifications.]The first goal of this work is to study the optimality (or
the lack of) for these three classes of approaches, and more
fundamentally, to quantify the statistical hardness of off-policy
evaluation. This problem was previously studied under the multi-arm bandits model \citep{li2015toward} and is related to a large body of work on the asymptotic optimality theory for average treatment effect (ATE) estimation in the econometrics literature (see e.g. \citealp{hahn1998role,hirano2003efficient,ImbensNeRi07,rothe2016value}), which can be thought of as a special case of off-policy evaluation. In both settings, a major underlying assumption is that rewards can be consistently estimated from the features (i.e.,
covariates) describing contexts and actions, either via a parametric
model or non-parametrically. Under such consistency assumptions, it
has been shown that DM is optimal for off-policy-evaluation in multi-arm bandit~\citep{li2015toward} and both DM and DR are optimal for ATE estimation~\citep{ImbensNeRi07,rothe2016value}\footnote{The
precise assumptions vary for each estimator, and are somewhat weaker
for DR than for DM.} whereas standard IPS is
not~\citep{hahn1998role,li2015toward}, but it becomes (asymptotically)
optimal when the true propensity scores are replaced by suitable
estimates~\citep{hirano2003efficient}.}
\end{comment}
Our first goal in this paper is to study the optimality of these three classes of approaches (or lack thereof), and more
fundamentally, to quantify the statistical hardness of off-policy
evaluation.
This problem was previously studied for multi-armed bandits \citep{li2015toward}
and is related to a large body of work on asymptotically optimal estimators of average treatment effects (ATE)
\citep{hahn1998role,hirano2003efficient,ImbensNeRi07,rothe2016value}, which can be viewed as a special
case of off-policy evaluation.
In both settings, a major underlying assumption is that rewards can be consistently estimated from the features (i.e.,
covariates) describing contexts and actions, either via a parametric
model or non-parametrically.
Under such consistency assumptions, it
has been shown that DM and/or DR are optimal~\citep{ImbensNeRi07,li2015toward,rothe2016value},\footnote{The
precise assumptions vary for each estimator, and are somewhat weaker
for DR than for DM.} whereas standard IPS is
not~\citep{hahn1998role,li2015toward}, but it becomes (asymptotically)
optimal when the true propensity scores are replaced by suitable
estimates~\citep{hirano2003efficient}.
\begin{comment}
\mdcomment{Miro's original version}
The first goal of this work is to study the optimality (or
the lack of) for these three classes of approaches, and more
fundamentally, to quantify the statistical hardness of off-policy
evaluation.
Previous research on optimal estimators assumes that
rewards can be consistently estimated from the features (i.e.,
covariates) describing contexts and actions, either via a parametric
model or non-parametrically.
Under such consistency assumptions, it
has been shown that DM and/or DR are optimal~\citep{ImbensNeRi07,li2015toward,rothe2016value},\footnote{The
precise assumptions vary for each estimator, and are somewhat weaker
for DR than for DM.} whereas standard IPS is
not~\citep{hahn1998role,li2015toward}, but it becomes (asymptotically)
optimal when the true propensity scores are replaced by suitable
estimates~\citep{hirano2003efficient}.
\end{comment}
Unfortunately, consistency of a reward model can be difficult to
achieve in practice. Parametric models tend to suffer from a large
bias (see, e.g., the empirical evaluation of
\citealp{dudik2011doubly}) and non-parametric models are limited to
small dimensions, otherwise non-asymptotic terms become too large
(see, e.g., the analysis of non-parametric regression by
\citealp{bertin2004asymptotically}). Therefore, here we ask:
\emph{What can be said about hardness of policy evaluation in the
absence of reward-model consistency?}
In this pursuit, we provide the first rate-optimal lower bound on the
mean-squared error (MSE) for off-policy evaluation in contextual
bandits without consistency assumptions. Our lower bound matches the
upper bounds of IPS and DR up to constants, when given a
non-degenerate context distributions. This result is in contrast with the
suboptimality of IPS under previously studied consistency assumptions,
which implies that the two settings are qualitatively different.
Whereas IPS and DR are both minimax optimal, our experiments (similar
to prior work) show that IPS is readily outperformed by DR, even
when using a simple parametric regression model that is not asymptotically
consistent. We attribute this to a lower variance of
the DR estimator. We also empirically observe that while DR is generally
highly competitive, it is sometimes substantially outperformed by DM.
We therefore ask whether it
is possible to achieve an even better bias-variance tradeoff than DR.
We answer affirmatively and
propose a new class of estimators, called the \textsc{switch}\xspace estimators, that
\emph{adaptively interpolate} between DM and DR (or IPS). We show that
\textsc{switch}\xspace has MSE no worse than DR (or IPS) in the worst case, but is robust
to large importance weights and can achieve a substantially
smaller variance than DR or IPS.
We empirically evaluate the \textsc{switch}\xspace estimators against a number of strong
baselines from prior work, using a previously used experimental setup
to simulate contextual bandit problems on real-world multiclass
classification data. The results affirm the superior bias-variance
tradeoff of \textsc{switch}\xspace estimators, with substantial improvements across a
number of problems.
In summary, the first part of our paper initiates the study of optimal
estimators in a finite-sample setting and without making strong
modeling assumptions, while the second part shows how to practically
exploit domain knowledge by building better estimators.
\begin{comment}
This
question has been extensively studied in contextual bandits
\citep{li2011unbiased, bottou2013counterfactual,oh1983weighting,little2002statistical}. There are several
estimators which are unbiased under mild assumptions, such as inverse
propensity scoring (IPS)~\citep{horvitz1952generalization}, with sharp
bounds on their mean squared error (MSE) (see
e.g.~\citep{dudik2014doubly}).
While the IPS-style methods make no attempt at all to model the
underlying dependence of rewards on contexts and actions, such
information is often available. Some approaches, such as the
doubly-robust method (DR)~\citep{dudik2014doubly} (see also
\citep{robins1995semiparametric,bang2005doubly}), use such a model to
obtain better variance than IPS while remaining unbiased.
All these works focus on developing specific methods alongside upper
bounds on their MSE. Little work, on the other hand, exists on the
question of the fundamental statistical hardness of off-policy
evaluation and the optimality (or the lack of) for the existing
methods. A notable exception is the recent work
of~\citet{li2015toward}, who study off-policy evaluation in
multi-armed bandits---a special case of our setting, without any
contexts---and provide a minimax lower bound on the MSE. Their result
shows the suboptimality of IPS due to an excessive variance
of the importance weights. This result is rather intriguing as it
hints at one of two possibilities: (i) IPS and variants are also
suboptimal for contextual bandit setting and we should develop better
estimators, or (ii) the contextual bandit setting has qualitatively
different upper and lower bounds that match. In this quest, our paper
makes the following key contributions:
\begin{packed_enum}
\item We provide the first rate-optimal lower bound on the MSE for
off-policy evaluation in contextual bandits, matching the upper
bound for IPS up to constants so long as the contexts have a
non-degenerate distribution. Crucially, even if the reward as a
function of contexts and actions has no variance, the lower bound
stays non-trivial in contrast with multi-armed bandits.
\item We propose a new class of estimators called the \textsc{switch}\xspace
estimators, that adaptively interpolate between an available
reward model and IPS. We show that \textsc{switch}\xspace has MSE no worse than IPS
in the worst case, but is robust to large importance weights. We
also show that \textsc{switch}\xspace can have a drastically smaller variance than
alternatives for combining IPS with a reward model, such as DR.
\item We conduct experiments showing that the new estimator performs
significantly better than existing approaches on simulated
contextual bandit problems using real-life multiclass
classification data sets.
\end{packed_enum}
In Sections~\ref{sec:formulation}~and~\ref{sec:minimax}, we recap the
formal setup and derive a minimax lower bound. In
Section~\ref{sec:use_dm}, we present the \textsc{switch}\xspace estimator along with its
theoretical properties, with detailed experiments in
Section~\ref{sec:exp}.%
}
\end{comment}
\section{Setup}
\label{sec:formulation}
In contextual bandit problems, the learning agent observes a context
$x$, takes an action $a$ and observes a scalar reward $r$ for the action
chosen in the context. Here the context $x$ is a feature vector from
some domain $\mathcal{X} \subseteq \mathbb{R}^d$, drawn according to a distribution
$\ensuremath{\lambda}$. Actions $a$ are drawn from a finite set~$\mathcal{A}$. Rewards $r$ have
a distribution conditioned on $x$ and $a$ denoted by $D(r\mathbin{\vert} x,a)$. The
decision rule of the agent is called a policy, which maps contexts to
distributions over actions, allowing for randomization in the action choice. We
write $\mu(a\mathbin{\vert} x)$ and $\pi(a\mathbin{\vert} x)$ to denote the \emph{logging} and
\emph{target} policies respectively. Given a policy $\pi$, we extend
it to a joint distribution over $(x,a,r)$, where $x \sim \ensuremath{\lambda}$, action
$a \sim \pi(a\mathbin{\vert} x)$, and $r\sim D(r\mathbin{\vert} x,a)$. With this notation, given $n$
i.i.d.\ samples $(x_i, a_i, r_i)\sim\mu$,
we wish to compute the \emph{value of $\pi$}:
\begin{equation}
v^\pi = \mathbb{E}_\pi[r] = \mathbb{E}_{x \sim \ensuremath{\lambda}} \mathbb{E}_{a \sim\pi(\cdot\mathbin{\vert} x)} \mathbb{E}_{r
\sim D(\cdot\mathbin{\vert} a,x)}[r].
\label{eq:vpi}
\end{equation}
In order to correct for the mismatch in the action distributions under
$\mu$ and $\pi$, it is typical to use \emph{importance weights},
defined as $\rho(x,a)\,{\coloneqq}\,
\pi(a\mathbin{\vert} x)/\mu(a\mathbin{\vert} x)$. For consistent estimation, it is standard to assume
that $\rho(x,a) \ne \infty$, corresponding to \emph{absolute continuity} of $\pi$ with
respect to $\mu$, meaning that whenever $\pi(a\mathbin{\vert} x) > 0$,
then also $\mu(a\mathbin{\vert} x)> 0$. We make this assumption throughout
the paper. In the remainder of the setup we present three common estimators of
$v^\pi$.
The first is the inverse
propensity scoring (IPS) estimator \citep{horvitz1952generalization}, defined
as
\begin{equation}
\hat{v}^\pi_{\mathrm{IPS}} = \sum_{i=1}^n \rho(x_i, a_i) r_i.
\label{eq:ips}
\end{equation}
IPS is unbiased and makes no assumptions about how rewards might
depend on contexts and actions. When such information is available, it
is natural to posit a parametric or non-parametric model of
$\mathbb{E}[r\mathbin{\vert} x,a]$ and fit it on the logged data to obtain a reward
estimator $\hat{r}(x,a)$. Policy evaluation can now simply be performed by
scoring $\pi$ according to $\hat{r}$ as
\begin{equation}
\label{eq:DM}
\hat{v}^\pi_{\text{DM}} = \frac{1}{n}\sum_{i=1}^n \sum_{a\in \mathcal{A}}
\pi(a\mathbin{\vert} x_i) \hat{r}(x_i,a),
\end{equation}
where the DM stands for \emph{direct method}~\citep{dudik2011doubly},
also known as \emph{regression adjustment} or
\emph{imputation}~\citep{rothe2016value}. IPS can have a large
variance when the target and logging policies differ substantially,
and parametric variants of DM can be inconsistent, leading to a large
bias. Therefore, both in theory and practice, it is beneficial to combine the
approaches into a \emph{doubly robust}
estimator~\citep{cassel1976some,robins1995semiparametric,dudik2011doubly},
such as the following variant,
\begin{align}
\notag
\hat{v}^\pi_{\text{DR}}
&=\frac{1}{n} \sum_{i=1}^n \biggl[ \rho(x_i,a_i)\bigParens{r_i -\hat{r}(x_i, a_i)}
\\[-6pt]
\label{eq:DR}
&\qquad\qquad\qquad{}
+ \sum_{a \in \mathcal{A}} \pi(a\mathbin{\vert} x_i)
\hat{r}(x_i, a)\biggr].
\end{align}
Note that IPS is a special case of DR with $\hat{r}\equiv 0$. In the
sequel, we mostly focus on IPS and DR, and then suggest how to improve
them by further interpolating with DM.
\begin{comment}
\alekh{I think we should ignore all the prior discussion below on
continuous things in the interest of time for now.}
{\color{red} While we assume $\mathcal{X}$ and $\mathcal{A}$ are
finite sets, all our results directly extend
to $\mathcal{X} = \mathbb{R}^d$ when $\ensuremath{\lambda}$ is a density, for discrete or
continuous $\mathcal{A}$. We make this simplifying assumption of discrete
$\mathcal{X}$ to avoid dealing with non-standard extensions to Borel
sigma-field and Lebesgue measure that allows a stochastic process
with a continuum of independent random variables to be measurable
and integrable, \citep[see, e.g.,][for one such
extension]{sun2006exact}. The discrete space assumption is not
limiting for all practical purposes, because even if $\mathcal{X}=\mathbb{R}^d$,
it can only be represented up to machine precision, (e.g.,
$\{0,1\}^d$ for very large $d$). $|\mathcal{X}|$ is typically exponentially
large and whenever every we make asymptotic statements on
$n\rightarrow \infty$, we assume $|\mathcal{X}|$ also increases with $n$. }
\mdcomment{I'm not sure whether we need this side note about
non-standard extensions and stochastic processes. I think it should
be straightforward to generalize to arbitrary $\mathcal{X}$ and finite
$\mathcal{A}$---all that is required is the measurability of $\ensuremath{R_{\max}}$ and
$\rho$. Then we can always begin our minimax construction by taking
a finite partition of $\mathcal{X}$ according to the values of the
vector-valued measurable function
$\mathbf{v}(x)\coloneqq\bigParens{\rho(x,a)\ensuremath{R_{\max}}(x,a)}_{a\in\mathcal{A}}$.
And then sample one Bernoulli/Gaussian per each set of the
partition. The partition $\biguplus\mathcal{X}_i$ of $\mathcal{X}$ would be
described by a partition $\biguplus\mathcal{U}_i$ of $\mathbb{R}^{\mathcal{A}}$, as
$\mathcal{X}_i=\mathbf{v}^{-1}(\mathcal{U}_i)$. The one wrinkle is that we need to
further subdivide each $\mathcal{X}_i$. This is because even when $\lambda$
is a continuous density, the structure of $\rho$ and $\ensuremath{R_{\max}}$ might
be such that $\lambda(\mathbf{v}^{-1}(\mathbf{u}))>0$ for certain
vectors $\mathbf{u}\in\mathbb{R}^{\mathcal{A}}$.}
\alekh{Check if the discrete $\mathcal{X}$ part is needed in the upper
bounds.}
\ywcomment{ Actually we don't have to group them by values and randomize once for each value (so that there is no subdivision needed). All we need is to assume $\rho$, $R_{\max}$ and $\sigma$ are continuous almost everywhere. In the current construction if we sample independently from a Bernoulli for each $(a,x)$ in the continuum, then inevitably it becomes not measurable in Lebesgue measure. But if we instead construct a sequence of finer and finer (but finite) discretization of $\mathcal{X}$, then we can use Fatou's Lemma and DCT to show the measurability and the integrability of the limit, which would already give us the same lower bound.}
\end{comment}
\begin{comment}
\mdcomment{Saving related work, but needs substantial reworking. We can drop Cortes et al.}
{\color{Gray}
\paragraph{Related work}
We now review the related work. The study of lower bounds on the MSE
of \emph{any possible estimator} for off-policy evaluation was first
studied by \citet{li2015toward} under a minimax setup, who focused on
the multi-arm bandits problem. As we will show, the naive extension of
their results to contextual bandits yields a suboptimal lower bound
~\citet{cortes2010learning} studied the covariate shift problem in the
statistical learning setting, which is related as off-policy
evaluation studies a covariate shift between $\pi$ and $\mu$ so that
we know the shift in distributions. However, their lower bounds are
for a different learning objective and do not apply to our
problem.\footnote{Off-policy evaluation focuses on estimating the
policy-value, while statistical learning focuses on prediction
accuracy measured by the expected loss (in terms of the excess
risk).}
On the complementary question of developing techniques and theory for
off-policy evaluation, there is a considerably larger body of work in
both the statistics and machine learning literature. In statistics,
this is studied as the problem of mean estimation for treatment
regimes in the causal inference literature (see
e.g.~\citep{horvitz1952generalization,holland1986statistics,bang2005doubly,
rotnitzky2012improved} as well as the references
in~\citep{dudik2014doubly}). The ``doubly robust'' estimation
techniques \citep{cassel1976some,robins1995semiparametric}, were
recently used for off-policy value estimation in the contextual
bandits problem \citep{dudik2011doubly,dudik2014doubly} and
reinforcement learning \citep{jiang2016doubly}. Finally, ideas very
related to the \textsc{switch}\xspace estimator which we introduce were also recently
investigated in the context of reinforcement learning
by~\citet{thomas2016data}.
\ywcomment{Lastly, we note that there is an asymptotic optimality theory developed for the related problem of mean-effect estimation in the econometrics literature~\citep{hahn1998role,hirano2003efficient}. The results are asymptotic, and mostly under realizable settings where a (nonparametric) estimate of the reward function is assumed to be known. We will discuss the connection with more details after we present our main result.
}}
\end{comment}
\section{Limits of Off-policy Evaluation}
\label{sec:minimax}
In this section, we study the off-policy evaluation problem in a
minimax setup. After setting up the framework, we present our lower
bound and the matching upper bounds for IPS and DR under appropriate
conditions.
While minimax optimality is standard
in statistical estimations, it is not the only notion of optimality.
An alternative framework is that of asymptotic
optimality, which establishes Cramer-Rao style bounds on the asymptotic variance
of estimators. We use the minimax framework, because
it is the most amenable to finite-sample lower bounds, and
is complementary to previous asymptotic results, as we discuss
after presenting our main results.
\subsection{Minimax Framework}
Off-policy evaluation is a statistical estimation problem, where the
goal is to estimate $v^\pi$ given $n$ i.i.d.\ samples generated according
to a policy $\mu$. We study this problem in a standard minimax
framework and seek to answer the following question. What is the
smallest MSE that \emph{any} estimator can achieve in the worst case
over a large class of contextual bandit problems? As is usual in the
minimax setting, we want the class of problems to be rich enough so
that the estimation problem is not trivial, and to be small enough
so that the lower bounds are not driven by complete pathologies. In
our problem, we
fix $\ensuremath{\lambda}$, $\mu$ and $\pi$, and
only take worst case over a class of reward distributions. This allows
the upper and lower bounds
to depend on $\ensuremath{\lambda}$, $\mu$ and $\pi$, highlighting how these ground-truth
parameters influence the problem difficulty. The family
of reward distributions $D(r\mathbin{\vert} x,a)$ that we study is a natural
generalization of the class studied by \citet{li2015toward} for
multi-armed bandits.
We assume we are given maps $R_{\max}~:~\mathcal{X}\times \mathcal{A} \to
\mathbb{R}_+$ and $\sigma~:~\mathcal{X}\times \mathcal{A} \to \mathbb{R}_+$, and
define the class of reward distributions $\mathcal{R}(\sigma,R_{\max})$
as\footnote{Technically, the inequalities in the definition
of $\mathcal{R}(\sigma,R_{\max})$
need to hold almost surely with $x \sim \ensuremath{\lambda}$ and $a \sim \mu(\cdot\mathbin{\vert} x)$.}
\begin{align*}
\mathcal{R}(\sigma,R_{\max})
\!\coloneqq\!
\Bigl\{
D(r|x,a):\:
0\,{\leq}\,\mathbb{E}_D[r|x,a]\,{\leq}\,R_{\max}(x,a)
\\
\text{ and }
\mathrm{Var}_D[r|x,a]\,{\leq}\,\sigma^2(x,a)
\text{ for all }x,a
\Bigr\}
.
\end{align*}
Note that $\sigma$ and $R_{\max}$ are allowed to change over contexts
and actions. Formally, an estimator is any function $\hat{v}:
(\mathcal{X}\times\mathcal{A}\times\mathbb{R})^n \rightarrow \mathbb{R}$ that takes $n$ data points
collected by $\mu$ and outputs an estimate of $v^\pi$. The
\emph{minimax risk} of off-policy evaluation over the class
$\mathcal{R}(\sigma,R_{\max})$, denoted by
$R_n(\pi;\ensuremath{\lambda},\mu,\sigma,R_{\max})$, is defined as
\begin{equation}
\label{eq:minimax_def}
\adjustlimits\inf_{\hat{v}}\sup_{\;\;D(r|x,a) \in \mathcal{R}(\sigma,R_{\max})\;\;}
\mathbb{E}\left[(\hat{v}-v^\pi)^2\right].
\end{equation}
Recall that the expectation is taken over the $n$ samples drawn from
$\mu$, along with any randomness in the estimator. The main goal of
this section is to obtain a lower bound on the minimax risk. To state
our bound, recall that $\rho(x,a)=\pi(a\mathbin{\vert} x)/\mu(a\mathbin{\vert}
x)<\infty$ is an importance weight at $(x,a)$. We make the following
technical assumption on our problem instances, described by tuples of
the form $(\pi,\ensuremath{\lambda},\mu,\sigma,R_{\max})$:
\begin{assumption}
\label{ass:moment}
There exists $\epsilon>0$ such that
$\mathbb{E}_{\mu}\Bracks{(\rho\sigma)^{2+\epsilon}}$
and
$\mathbb{E}_{\mu}\Bracks{(\rho\ensuremath{R_{\max}})^{2+\epsilon}}$
are finite.
\end{assumption}
This assumption is only a slight strengthening of the assumption that
$\mathbb{E}_{\mu}[(\rho\sigma)^2]$ and $\mathbb{E}_{\mu}[(\rho\ensuremath{R_{\max}})^2]$ be finite,
which is required for consistency of IPS (see,
e.g.,~\citealp{dudik2014doubly}). Our assumption holds for instance
when the context space is finite, because then both $\rho$ and $\ensuremath{R_{\max}}$
are bounded.
\subsection{Minimax Lower Bound for Off-policy Evaluation}
\label{sec:lb}
With the minimax setup in place, we now give our main lower bound on
the minimax risk for off-policy evaluation and discuss its
consequences. Our bound depends on a parameter $\gamma\in[0,1]$
and a derived indicator random variable
$\xi_{\gamma}(x,a)\coloneqq\mathbf{1}}%\mathds{1}(\mu(x,a)\le\gamma)$, which
separates out the pairs $(x,a)$ that appear ``frequently'' under
$\mu$.%
\footnote{Formally, $\mu(x,a)$ corresponds to $\mu(\,\set{(x,a)}\,)$, i.e.,
the measure under $\mu$ of the set $\set{(x,a)}$. For example, when
$\ensuremath{\lambda}$ is a continuous distribution then $\mu(x,a)=0$ everywhere.}
As we will see, the
``frequent'' pairs $(x,a)$ (where $\xi_{\gamma}=0$) correspond to the
intrinsically realizable part of the problem, where consistent reward
models can be constructed. The ``infrequent'' pairs
(where $\xi_{\gamma}=1$) constitute the part that is non-realizable in the
worst-case. When $\mathcal{X}\subseteq\mathbb{R}^d$ and $\ensuremath{\lambda}$ is continuous with
respect to the Lebesgue measure, then $\xi_{\gamma}(x,a)=1$ for all
$\gamma\in[0,1]$, so the problem is non-realizable everywhere in the
worst-case. Our result uses the following problem-dependent constant (defined with the
convention $0/0=0$):
\[
C_{\gamma}\coloneqq 2^{2+\epsilon} \max\bigg\{
\frac{\mathbb{E}_\mu[(\rho\sigma)^{2+\epsilon}]^2}{\mathbb{E}_\mu[(\rho\sigma)^2]^{2+\epsilon}}
,\,
\frac{\mathbb{E}_\mu[\xi_{\gamma}(\rho\ensuremath{R_{\max}})^{2+\epsilon}]^2}{\mathbb{E}_\mu[\xi_{\gamma}(\rho\ensuremath{R_{\max}})^2]^{2+\epsilon}}
\bigg\}
.
\]
\begin{theorem
\label{thm:minimax}
Assume that a problem instance satisfies Assumption~\ref{ass:moment}
with some $\epsilon>0$. Then for any $\gamma\in[0,1]$ and any
$n\ge\max\bigBraces{16C_{\gamma}^{1/\epsilon},\,
2C_{\gamma}^{2/\epsilon}\mathbb{E}_\mu[\sigma^2/\ensuremath{R_{\max}}^2] }$, the minimax
risk $R_n(\pi;\ensuremath{\lambda},\mu,\sigma,R_{\max})$ satisfies the lower bound
\[
\frac{
\mathbb{E}_\mu\bigBracks{\rho^2\sigma^2} + \mathbb{E}_\mu\bigBracks{\xi_{\gamma}\rho^2\ensuremath{R_{\max}}^2}
\BigParens{1-350n\gamma\log(5/\gamma)}
}
{700n}
\enspace.
\]
\end{theorem}
The bound holds for every $\gamma\in [0,1]$, and we can take the
maximum over $\gamma$. In particular, we get the following simple
corollary under continuous context distributions.
\begin{corollary}\label{cor:continuous}
Under conditions of Theorem~\ref{thm:minimax}, assume further that
$\ensuremath{\lambda}$ has a density relative to Lebesgue measure. Then
\[
R_n(\pi;\ensuremath{\lambda},\mu,\sigma,R_{\max}) \geq
\frac{ \mathbb{E}_\mu\bigBracks{\rho^2\sigma^2} +
\mathbb{E}_\mu\bigBracks{\rho^2\ensuremath{R_{\max}}^2}
}
{700n}
\enspace.
\]
\end{corollary}
If $\ensuremath{\lambda}$ is a mixture of a density and point masses, then $\gamma=0$
will exclude the point masses from the second term of the lower bound.
In general, choosing $\gamma=\ensuremath{\mathcal{O}}\bigParens{1/(n\log n)}$
excludes the contexts likely to appear multiple times, and ensures
that the second term in \Thm{minimax} remains non-trivial (when
$\mu(x,a)\le\gamma$ with positive probability).
Before sketching the proof of \Thm{minimax}, we discuss its preconditions
and implications.
\textbf{Preconditions of the theorem:} The theorem assumes the
existence of a (problem-dependent) constant $C_{\gamma}$ which depends
on the constant $\gamma$ and various moments of the importance-weighted rewards.
When $\ensuremath{R_{\max}}$ and $\sigma$ are bounded (a common situation), $C_{\gamma}$ measures how
heavy-tailed the importance weights are.
Note that $C_{\gamma} < \infty$ for all $\gamma \in [0,1]$
whenever Assumption~\ref{ass:moment} holds, and so
the condition on $n$ in \Thm{minimax} is eventually satisfied as long as the
random variable $\sigma/\ensuremath{R_{\max}}$ has a bounded second moment.
This is quite reasonable since in typical applications the \emph{a priori} bound on expected
rewards is on the same order or larger than the \emph{a priori} bound on the reward noise.
For the remainder of the discussion, we assume that $n$
is appropriately large so the preconditions of the theorem hold.
\textbf{Comparison with upper bounds:} The setting of
Corollary~\ref{cor:continuous} is typical of many contextual bandit
applications. In this setting both IPS and DR achieve the minimax risk
up to a multiplicative constant. Let $r^*(x,a)\coloneqq\mathbb{E}[r\mathbin{\vert}
x,a]$. Recall that DR is using an estimator $\hat{r}(x,a)$ of
$r^*(x,a)$, and IPS can be viewed as a special case of DR with
$\hat{r}\equiv 0$. By Lemma 3.3(i) of \citet{dudik2014doubly}, the MSE
of DR is
\begin{align}
\notag
&
\mathbb{E}\bracks{(\hat{v}^\pi_{\mathrm{DR}}-v^\pi)^2}
\\
\notag
&\quad{}
=\frac1n
\Bigl(
\mathbb{E}_\mu[ \rho^2\sigma^2]
+ \mathrm{Var}_{x\sim D}\mathbb{E}_{a\sim\mu(\cdot\mathbin{\vert} x)}[ \rho r^*]
\\[-3pt]
\label{eq:DR:risk}
&\qquad\qquad + \mathbb{E}_{x\sim D}\mathrm{Var}_{a\sim\mu(\cdot\mathbin{\vert} x)}[ \rho (\hat{r}-r^*)]
\Bigr).
\end{align}
Note that $0\le r^*\le\ensuremath{R_{\max}}$, so if the estimator $\hat{r}$ also satisfies
$0\le\hat{r}\le\ensuremath{R_{\max}}$, we obtain that
the risk of DR (with IPS as a special case)
is at most $\ensuremath{\mathcal{O}}\bigParens{\frac1n(\mathbb{E}_\mu[ \rho^2\sigma^2]+\mathbb{E}_\mu[ \rho^2\ensuremath{R_{\max}}^2])}$.
This means that IPS and DR are unimprovable, in the worst case, beyond
constant factors. Another implication is that the lower bound of Corollary~\ref{cor:continuous}
is sharp, and the minimax risk is precisely
$\Theta\bigParens{\frac1n(\mathbb{E}_\mu[ \rho^2\sigma^2]+\mathbb{E}_\mu[ \rho^2\ensuremath{R_{\max}}^2])}$.
While IPS and DR exhibit the same minimax rates, \Eq{DR:risk} also immediately shows
that DR will be better than IPS whenever $\hat{r}$ is even moderately good (better than $\hat{r}\equiv 0$).
\textbf{Comparison with asymptotic optimality results:} As discussed
in \Sec{intro}, previous work on optimal off-policy evaluation,
specifically the average treatment estimation, assumes
that it is possible to consistently estimate $r^*(x,a)=\mathbb{E}[r\mathbin{\vert}
x,a]$. Under such an assumption it is possible to (asymptotically)
match the risk of DR with the perfect reward estimator $\hat{r}\equiv
r^\star$, and this is the best possible asymptotic
risk~\citep{hahn1998role}. This optimal risk is
$\frac1n\bigParens{\mathbb{E}_\mu[ \rho^2\sigma^2]+\mathrm{Var}_{x\sim D}\mathbb{E}_\pi[
r^*\mathbin{\vert} x]}$, corresponding to the first two terms of
\Eq{DR:risk}, with no dependence on $\ensuremath{R_{\max}}$. Several estimators
achieve this risk, \emph{including the multiplicative constant}, under
various consistency assumptions
\citep{hahn1998role,hirano2003efficient,ImbensNeRi07,rothe2016value}.
Note that this is strictly below our lower bound for continuous
$\ensuremath{\lambda}$. That is, consistency assumptions yield a better asymptotic risk
than possible in the agnostic setting. The gap in
constants between our upper and lower bounds is due to the
finite-sample setting, where lower-order terms cannot be ignored, but
have to be explicitly bounded. Indeed, apart from the result
of~\citet{li2015toward}, discussed below, ours is the first
finite-sample lower bound for off-policy evaluation.
\textbf{Comparison with multi-armed bandits:} For multi-armed bandits,
equivalent to contextual bandits with a single context,
\citet{li2015toward} show that the minimax risk equals
$\Theta(\mathbb{E}_\mu[\rho^2\sigma^2]/n)$ and is achieved, e.g., by DM,
whereas IPS is suboptimal. They also obtain a similar result for
contextual bandits, assuming that each context appears with a
large-enough probability to estimate its associated rewards by
empirical averages (amounting to realizability). While we obtain a
larger lower bound, this is not a contradiction, because we allow
arbitrarily small probabilities of individual contexts and even
continuous distributions, where the probability of any single context
is zero.
On a closer inspection, the first term of our bound in \Thm{minimax} coincides with the lower
bound of \citet{li2015toward} (up to constants). The second term
(optimized over $\gamma$) is non-zero only if there are contexts with
small probabilities relative to the number of samples. In multi-armed
bandits, we recover the bound of \citet{li2015toward}. When the
context distribution is continuous, or the probability of seeing
repeated contexts in a data set of size $n$ is small, we get the
minimax optimality of IPS.
One of our key contributions is to highlight this
\emph{agnostic contextual} regime where IPS is optimal.
In the \emph{non-contextual} regime, where each context appears frequently,
the rewards for each context-action pair can be consistently estimated by empirical averages.
Similarly, the asymptotic results discussed earlier focus on a setting where rewards
can be consistently estimated thanks to parametric assumptions or smoothness (for non-parametric
estimation), with the goal of asymptotic efficiency.
Our work complements that line of research. In many practical
situations, we wish to evaluate policies on high-dimensional context
spaces, where the consistent estimation of rewards is not a feasible
option. In other words, the agnostic contextual regime
dominates.
\begin{comment}
Another way to highlight the
additional challenges in contextual bandits compared with multi-armed
bandits is to consider a noiseless setting with $\sigma \equiv
0$, where the multi-armed bandit problem trivial, because we can just
measure the reward of each arm with one pull and find out the optimal
choice. However, there is still a non-trivial lower bound of
$\Omega(\mathbb{E}_\mu[\rho^2\ensuremath{R_{\max}}^2]/n)$ for contextual bandits (under a continuous $\ensuremath{\lambda}$),
which is exactly the upper bound on the MSE of IPS when the rewards
have no noise.
\end{comment}
The distinction between the contextual and non-contextual regime is
also present in our proof, which combines
a non-contextual lower bound
due to the reward noise, similar to the analysis of~\citet{li2015toward}, and
an additional bound arising for non-degenerate context distributions. This
latter result is a key technical novelty of our paper.
\begin{comment}
{\color{Gray}
\textbf{Comparison with upper bounds:} In typical contextual bandit
settings, we are often in the setting of
Corollary~\ref{cor:continuous}, where the lower bound precisely
matches the upper bound for the IPS estimator
\citep{horvitz1952generalization} (recall \Eq{ips}). We present the
upper bound due to~\citet{horvitz1952generalization} here for
completeness.
\begin{equation}
\mathbb{E}(\hat{v}^\pi_{\mathrm{IPS}}-v^\pi)^2 \leq
\frac{1}{n}\left(\mathbb{E}_{\mu} [\rho^2 \sigma^2] + \frac{1}{4}\mathbb{E}_\mu
[\rho^2 R_{\max}^2]\right).
\label{eqn:IPS-upper}
\end{equation}
We conclude that IPS is unimprovable, in the worst case, beyond
constant factors. Another implication is that the lower bound is sharp
for our setting, and that the minimax risk is precisely
$\Theta\left(\frac{1}{n}\left(\mathbb{E}_{\mu} [\rho^2 \sigma^2] + \mathbb{E}_\mu
[\rho^2 R_{\max}^2]\right)\right)$.
\textbf{Comparison with multi-armed bandits:} The most closely related
prior result was due to~\citet{li2015toward}, who showed matching
upper and lower bounds on the minimax risk for multi-armed
bandits. The somewhat surprising conclusion of their work was the
sub-optimality of IPS, which might appear at odds with our conclusion
regarding IPS above. However, this difference actually highlights the
additional challenges in contextual bandits beyond multi-armed
bandits, best illustrated in a noiseless setting with $\sigma \equiv
0$. This makes the multi-armed bandit problem trivial, we can just
measure the reward of each arm with one pull and find out the optimal
choice. However, there is still a non-trivial lower bound of
$\Omega(\mathbb{E}_\mu[\rho^2\ensuremath{R_{\max}}^2]/n)$ in the contextual bandit setting,
which is exactly the upper bound on the MSE of IPS when the rewards
have no noise.
\ywcomment{[TODO] Discuss the ``contextual'' and ``non-contextual'' regime with more details here.}
In general, with a rich enough context distribution there is always
uncertainty in the value of a policy based on finitely many samples,
since several contexts are never observed. This is also highlighted in
our proof technique, which combines one lower bound due to reward
noise akin to that of~\citet{li2015toward}, with another arising out
of non-degenerate context distributions. This latter result is a key
technical novelty of our paper.}
\end{comment}
\paragraph{Proof sketch:}
We only sketch some of the main ideas here and defer the full proof to
Appendix~\ref{sec:proofs_lower}.
For
simplicity, we discuss the case where $\ensuremath{\lambda}$ is a continuous
distribution. We consider two separate problem instances corresponding
to the two terms in Theorem~\ref{thm:minimax}. The first part is relatively
straightforward and reduces the problem to Gaussian mean
estimation. We focus on the second part which depends on $\ensuremath{R_{\max}}$. Our
construction defines a prior over the reward distributions,
$D(r\mathbin{\vert} x,a)$. Given any $(x,a)$, a problem instance is given by
\begin{equation}
\notag
\label{eqn:rewards}
\mathbb{E}[r\mathbin{\vert} x,a] = \ensuremath{\eta}(x,a) =
\begin{cases}
\ensuremath{R_{\max}}(x,a)
&\text{w.p.\ $\theta(x,a)$,}
\\
0
&\text{w.p.\ $1-\theta(x,a)$,}
\end{cases}
\end{equation}
for $\theta(x,a)$ to be appropriately chosen. Once $\ensuremath{\eta}$ is drawn, we
consider a problem instance defined by $\ensuremath{\eta}$ where the rewards are
deterministic and the only randomness is in the contexts. In order to
lower bound the MSE across all problems, it suffices to lower bound
$\mathbb{E}_\theta[\textrm{MSE$_\eta$}(\hat{v})]$. That is, we can compute the
MSE of an estimator for each individual $\eta$, and take expectation
of the MSEs under the prior prescribed by $\theta$. If the expectation
is large, we know that there is a problem instance where the estimator
incurs a large MSE.
A key insight in our proof is that this expectation can be lower
bounded by
$\textrm{MSE}_{\mathbb{E}_\theta [\ensuremath{\eta}(x,a)]}(\hat{v})$, corresponding to
the MSE of a single problem instance
with the actual \emph{rewards},
rather than $\eta(x,a)$, drawn according to $\theta$ and with the mean
reward function $\mathbb{E}_\theta [\ensuremath{\eta}(x,a)]$.
This is
powerful, since this new problem instance has stochastic rewards,
just like Gaussian mean estimation, and is amenable to
standard techniques. The lower bound by $\textrm{MSE}_{\mathbb{E}_\theta [\ensuremath{\eta}(x,a)]}(\hat{v})$
is only valid when the
context distribution $\ensuremath{\lambda}$ is rich enough (e.g., continuous). In that case,
our reasoning shows that with enough randomness in the context
distribution, a problem with even a deterministic reward function is
extremely challenging.
\section{Incorporating Reward Models}
\label{sec:use_dm}
\begin{comment}
Amongst our twin goals of optimal and adaptive estimators, the
discussion so far has centered around the optimality of the IPS
estimators in a minimax sense. However, real data sets seldom display
worst-case behavior, and in this section we discuss approaches to
leverage additional structure in the data, when such knowledge is
available. Throughout this section, we drop the superscript $\pi$ from
value estimators, as the target policy $\pi$ is fixed throughout.
\subsection{The need for model-based approaches}
As we have seen in the last section, the model-free approach has an
information-theoretic limit that depends quadratically on $R_{\max}$,
$\sigma$ and $\rho$. This dependence on $\rho^2$ substantially limits
what policies can be evaluated. If $\mu(a\mathbin{\vert} x)$ is small for some
actions, as is typical when number of actions is large or in real
systems with a cost for exploration, we cannot reliably estimate the
value of policies $\pi$ that put much mass on such actions. The key
reason for this limitation is that the setup so far allows completely
arbitrary reward models---$\mathbb{E}[r\mathbin{\vert} x,a]$ can change arbitrarily across
different actions and contexts. However, we often have substantial
intuition about contexts and actions which obtain similar rewards,
based on the specific application, which we can hope to leverage and
beat the worst-case lower bounds.
In the presence of such information, a common approach is to construct
an explicit model for $\mathbb{E}[r\mathbin{\vert} x,a]$. Using the existing logs, one can
then fit the parameters in such a model to form a reward estimator
$\hat{r}(x,a)$. The task of policy evaluation is now simply performed
by scoring $\pi$ according to $\hat{r}$ as
\begin{equation}\label{eq:DM}
\hat{v}_{\text{DM}} = \frac{1}{n}\sum_{i=1}^n \sum_{a\in \mathcal{A}}
\pi(a\mathbin{\vert} x_i) \hat{r}(x_i,a),
\end{equation}
where the DM stands for \emph{direct method}, a name which has been
used in prior works~\citep{dudik2014doubly}. This approach appears
attractive when $\hat{r}$ is a close to $\mathbb{E}[r\mathbin{\vert} x,a]$, since it does not
use importance weights and might have low variance. For any estimator
$\hat{r}$ which takes values in $[0,\ensuremath{R_{\max}}(x,a)]$ for any $x,a$, the
variance of DM satisfies:
\begin{equation}
\label{eq:var-dm}
\mbox{Var}(\hat{v}_{\text{DM}}) \leq \frac{1}{n}\mathbb{E}_{\pi} [\ensuremath{R_{\max}}^2] =
\frac{1}{n}\mathbb{E}_\mu [\rho \ensuremath{R_{\max}}^2].
\end{equation}
That is, there is a linear, rather than quadratic dependence on
$\rho$, unlike in the worst case minimax
risk~\eqref{eq:DR:risk}. However, such estimators can have an
uncontrolled bias of $\ensuremath{\mathcal{O}}(\mathbb{E}_\pi[\ensuremath{R_{\max}}])$ in the worst case,
meaning that even asymptotic consistency is not guaranteed.
Prior works have addressed the bias of DM, while trying to reduce the
variance of IPS by using \emph{doubly robust} estimators. A concrete
estimator $\hat{v}_{\text{DR}}$ previously used for off-policy
evaluation in~\citet{dudik2014doubly} is defined as
\begin{equation}
\label{eq:DR}
\frac{1}{n} \sum_{i=1}^n \bigg[ \frac{(r_i -\hat{r}(x_i, a_i))
\pi(a_i|x_i)}{\mu(a_i|x_i)} + \sum_{a \in \mathcal{A}} \pi(a\mathbin{\vert} x_i)
\hat{r}(x_i, a)\bigg].
\end{equation}
The theory of these methods suggests that the variance of
$\hat{v}_{\text{DR}}$ can be smaller than IPS, if $\hat{r}$ is a good
reward estimator. Indeed when $\hat{r}(x,a) = \mathbb{E}[r\mathbin{\vert} x,a]$ almost
surely, the MSE of the DR estimator~\citep{dudik2014doubly} is at most
\begin{equation}
\mathrm{MSE}(\hat{v}_{\mathrm{DR}}) \leq \frac{1}{n}\left(\mathbb{E}_\mu
[\rho^2\sigma^2] + \mathbb{E}_\mu [\rho \ensuremath{R_{\max}}^2]\right). \label{eq:dr-perfect}
\end{equation}
Crucially, the MSE still has a dependence on $\rho^2$ in it as long as
the variance $\sigma$ is non-trivial. Comparing
Equations~\ref{eq:var-dm} and~\ref{eq:dr-perfect}, it would be
better to use $\hat{v}_{\text{DM}}$ instead in this case. Also note
that $\rho$ instead of $\rho^2$ term multiplying $\ensuremath{R_{\max}}$ does not
contradict the lower bound in Theorem~\ref{thm:minimax}, since the
bound~\eqref{eq:dr-perfect} assumes access to the expected reward
function which is information an estimator does not have in our
minimax setup and which cannot be computed from the samples in
general.
We conclude that while DR improves upon IPS in some cases, the
estimator still does not adapt enough in the presence of good reward
models, when the rewards have a non-trivial variance. We next present
an estimator which utilizes the reward model in a different way, with
an eye towards a more favorable bias-variance tradeoff.
\end{comment}
As discussed in the previous section, it is generally possible
to beat our minimax bound when consistent reward models exist.
We also argued that even in the absence of a consistent model, when DR and IPS
both achieve optimal risk rates, the performance of DR on
finite samples will be better than IPS as long
as the reward model is even moderately good (see Eq.~\ref{eq:DR:risk}).
However,
under a large reward noise $\sigma$, DR may still suffer from high variance when the importance weights are large, even when given a perfect reward model.
In this section, we derive a class of estimators that leverage reward models to directly address this source of high variance,
in a manner very different from the standard DR approach.
\subsection{The \textsc{switch}\xspace Estimators}
\label{sec:adaptive_est}
Our starting
point is the observation that insistence on maintaining
unbiasedness puts the DR estimator at one extreme end of the
bias-variance tradeoff. Prior works have considered ideas such as
truncating the rewards or importance weights when the importance
weights are large (see, e.g., \citealt{bottou2013counterfactual}), which can dramatically reduce the variance
at the cost of a little bias. We take the intuition a step further
and propose to estimate the rewards for actions by two distinct strategies, based on
whether they have a large or a small importance weight in a given
context. When importance weights are small, we continue to use our
favorite unbiased estimators, but switch to directly applying the (potentially
biased) reward model on actions with large importance
weights. Here, ``small'' and ``large'' are defined via a
\emph{threshold parameter} $\tau$. Varying this parameter between $0$ and $\infty$
leads to a family of estimators which we call the \textsc{switch}\xspace estimators as
they switch between
an agnostic approach (such as DR or IPS) and the direct method.
We now formalize this intuition, and begin by decomposing $v^\pi$
according to importance weights:
\begin{align*}
\mathbb{E}_{\pi}[r] &=
\mathbb{E}_\pi [r \mathbf{1}}%\mathds{1}(\rho\leq\tau) ]
+ \mathbb{E}_\pi [r \mathbf{1}}%\mathds{1}(\rho>\tau) ]
\\
&= \mathbb{E}_\mu[\rho r \mathbf{1}}%\mathds{1}(\rho\leq\tau) ]
\\
&\quad{}
+
\mathbb{E}_{x\sim\ensuremath{\lambda}}\BigBracks{
\sum_{a\in \mathcal{A}} \mathbb{E}_D[r\mathbin{\vert} x,a]\, \pi(a\mathbin{\vert} x)\, \mathbf{1}(\rho(x,a){>}\tau)
}
.
\end{align*}
Conceptually, we split our problem into two. The first problem always
has small importance weights, so we can use unbiased estimators such
as IPS or DR. The second problem, where importance weights
are large, is addressed by DM. Writing this out leads to
the following estimator:
\begin{align}
&\hat{v}_{\mathrm{\textsc{switch}\xspace}}
= \frac{1}{n}\sum_{i=1}^n \left[r_i \rho_i
\mathbf{1}(\rho_i\leq \tau) \right] \nonumber
\\
&\qquad{}+
\frac{1}{n}\sum_{i=1}^n\sum_{a
\in \mathcal{A}} \hat{r}(x_i,a)\pi(a\mathbin{\vert} x_i) \mathbf{1}}%\mathds{1}
(\rho(x_i,a)>\tau). \label{eq:OA}
\end{align}
Note that the above estimator specifically uses IPS on the first part
of the problem. When DR is used instead of IPS, we refer to the resulting
estimator as \textsc{switch-DR}\xspace. The reward model used within the DR part of
the \textsc{switch-DR}\xspace estimator can be the same or different from the reward model
used to impute rewards in the second part.
We next present a
bound on the MSE of the \textsc{switch}\xspace estimator using IPS.
A similar bound holds for \textsc{switch-DR}\xspace.
\begin{theorem}\label{thm:MSEbound}
Let $\epsilon(a,x):= \hat{r}(a,x)-\mathbb{E}[r|a,x]$ be the bias of
$\hat{r}$ and assume $\hat{r}(x,a) \in [0,\ensuremath{R_{\max}}(x,a)]$ almost
surely. Then for $\hat{v}_{\mathrm{\textsc{switch}\xspace}}$, with $\tau > 0$,
the MSE is at most
\begin{align*}
&\frac{2}{n}\BigBraces{\,
\mathbb{E}_\mu\bigBracks{
\left(\sigma^2{+}R_{\max}^2\right)\rho^2 \mathbf{1}}%\mathds{1}(\rho\,{\leq}\,\tau)
}
+
\mathbb{E}_\pi\bigBracks{ R_{\max}^2\mathbf{1}}%\mathds{1}(\rho\,{>}\,\tau) }
\,}
\\&\qquad{}
+
\mathbb{E}_{\pi}\bigBracks{\epsilon\mathbf{1}}%\mathds{1}(\rho\,{>}\,\tau)}^2
.
\end{align*}
\end{theorem}
The proposed estimator interpolates between DM and IPS.
For $\tau = 0$, \textsc{switch}\xspace coincides with DM,
while $\tau \rightarrow \infty$ yields IPS. Consequently, \textsc{switch}\xspace estimator
is minimax optimal when $\tau$ is appropriately chosen. However, unlike
IPS and DR, the \textsc{switch}\xspace and \textsc{switch-DR}\xspace estimators are by design more robust to
large (or heavy-tailed) importance weights. Several estimators related
to \textsc{switch}\xspace have been previously studied:
\begin{packed_enum}
\item \citet{bottou2013counterfactual} consider a special case
of \textsc{switch}\xspace with $\hat{r} \equiv 0$, meaning that all
the actions with large importance weights are eliminated
from IPS.
We refer to this method as \emph{Trimmed IPS}.
\item \citet{thomas2016data} study an estimator similar to \textsc{switch}\xspace in the more
general setting of reinforcement learning. Their \emph{MAGIC}
estimator can be
seen as using several candidate thresholds $\tau$ and then
evaluating the policy by a weighted sum of the estimators
corresponding to each $\tau$. Similar to our approach of automatically
determining $\tau$, they determine the weighting of estimators
via optimization (as we discuss below).
\end{packed_enum}
\begin{comment}
\begin{remark}
The first term is the MSE bound of Theorem~\ref{thm:MSEbound} is the MSE
of IPS on the region where it is used, while the second and third
terms capture the variance and squared bias of DM on its region
respective. Consequently, we match the result for IPS when $\tau
\rightarrow \infty$, meaning that the \textsc{switch}\xspace estimators are also
minimax optimal when $\tau$ is appropriately chosen. At the other
extreme, we can consider the case of $\tau = 0$ and a perfect reward
predictor so that $\epsilon \equiv 0$. In this case, the MSE bound
matches that of DM in \Eq{var-dm}. More generally, the estimators
are robust to heavy-tails in the distribution of importance weights,
unlike both IPS and DR estimators.
\end{remark}
\begin{remark}
The policy value in the region where the importance weights are small
can be estimated using any unbiased approach rather than just IPS.
For instance, we can use the DR, giving rise to the estimator, which
we denote \textsc{switch-DR}\xspace. For any reward estimator $\hat{r}'$ to construct the
DR estimator, we define $\hat{v}_{\mathrm{\textsc{switch-DR}\xspace}}$ as \mdcomment{the following formula is incorrect}
\begin{equation}
\hat{v}_{\mathrm{\textsc{switch}\xspace}} +
\frac{1}{n}\sum_{i=1}^n \sum_{a \in \mathcal{A}}
\hat{r}'(x_i,a)\pi(a\mathbin{\vert} x_i) \mathbf{1}}%\mathds{1}(\rho(x_i,a) {\leq} \tau),
\label{eq:OA-DR}
\end{equation}
which amounts to using DR with $\hat{r}'$ rather than IPS when
$\rho(x,a) < \tau$. Note that $\hat{r}$ and $\hat{r}'$ need not be
the same estimators. When they are identical,
$\hat{v}_{\mathrm{\textsc{switch-DR}\xspace}}$ can be rewritten as essentially a
modified doubly robust estimator where all the importance weights
above $\tau$ are clipped to zero, so that the importance weighted
part of DR makes no contribution.
The analysis in Theorem~\ref{thm:MSEbound} still applies, replacing
the variance of IPS with that of DR
from~\citet{dudik2014doubly}. Since no independence was required in
our analysis between the IPS and the DM parts of the estimator, the
result is also robust to the use of a common data-dependent
estimator $\hat{r} = \hat{r}'$ in \textsc{switch-DR}\xspace in Eq.~\eqref{eq:OA-DR}.
\end{remark}
\end{comment}
\subsection{Automatic Parameter Tuning}\label{sec:autotuning}
So far we have discussed the properties of the \textsc{switch}\xspace estimators
assuming that the parameter $\tau$ is chosen well. Our goal
is to obtain the best of IPS and DM, but
a poor choice of $\tau$ might easily give us the worst
of the two estimators. Therefore, a method for selecting $\tau$
plays an essential role. A natural criterion would be to
pick $\tau$ that minimizes the MSE of the resulting estimator. Since we
do not know the precise MSE (as $v^\pi$ is unknown), an alternative is
to minimize its data-dependent estimate. Recalling that the MSE
can be written as the sum of variance and squared bias, we estimate
and bound the terms individually.
Recall that we are working with a data set $(x_i, a_i, r_i)$ and
$\rho_i\coloneqq\pi(a_i\mathbin{\vert} x_i)/\mu(a_i\mathbin{\vert} x_i)$. Using this data, it is
straightforward to estimate the
variance of the \textsc{switch}\xspace estimator. Let
$Y_i(\tau)$ denote the estimated value that $\pi$ obtains on the data
point $x_i$ according to the \textsc{switch}\xspace estimator with the threshold $\tau$,
that is
\[
Y_i(\tau)\coloneqq
r_i \rho_i \mathbf{1}}%\mathds{1}(\rho_i{\leq}\tau) + \!\sum_{a \in \mathcal{A}}\!
\hat{r}(x_i,a)\pi(a\mathbin{\vert} x_i) \mathbf{1}}%\mathds{1}(\rho(x_i,a){>}\tau),
\]
and $\bar{Y}(\tau)\coloneqq\frac{1}{n} \sum_{i=1}^n Y_i(\tau)$. Since
$\hat{v}_{\mathrm{\textsc{switch}\xspace}} = \bar{Y}(\tau)$ and the $x_i$ are
i.i.d., the variance can be estimated as
\begin{align}
\mathrm{Var}(\bar{Y}(\tau)) \approx \frac{1}{n^2} \sum_{i=1}^n
(Y_i(\tau)-\bar{Y}(\tau))^2 =: \widehat{\mathrm{Var}}_\tau,
\label{eq:var-tune}
\end{align}
where the approximation above is clearly consistent since the random
variables $Y_i$ are appropriately bounded as long as the rewards are
bounded, because the importance weights are capped at the threshold $\tau$.
Next we turn to the bias term. For understanding bias, we look at the
MSE bound in Theorem~\ref{thm:MSEbound}, and observe that the last
term in that theorem is precisely the squared bias. Rather
than using a direct bias estimate, which would require knowledge of
the error in $\hat{r}$, we will upper bound this term. We
assume that the function $\ensuremath{R_{\max}}(x,a)$ is known. This is not limiting
since in most practical applications an
\emph{a priori} bound on the rewards is known. Then we
can upper bound the squared bias as
\begin{align*}
&\mathbb{E}_{\pi}\bigBracks{\epsilon\mathbf{1}}%\mathds{1}(\rho>\tau)}^2
\le
\mathbb{E}_{\pi}\bigBracks{\ensuremath{R_{\max}}\mathbf{1}}%\mathds{1}(\rho>\tau)}^2.
\end{align*}
Replacing the expectation with an average, we obtain
\begin{align}
\notag \widehat{\text{Bias}}^2_\tau\coloneqq
\biggBracks{\frac{1}{n}\sum_{i=1}^n
\mathbb{E}_\pi\bigBracks{R_{\max}\mathbf{1}}%\mathds{1}(\rho>\tau) \mathbin{\bigm\vert} x_i} }^2 .
\end{align}
With these estimates, we pick the threshold $\widehat{\tau}$ by
optimizing the sum of estimated variance and the upper bound on bias,
\begin{equation}
\widehat{\tau}\coloneqq\mathop{\mathrm{argmin}}_{\tau} \widehat{\mathrm{Var}}_\tau +
\widehat{\text{Bias}}^2_\tau.
\label{eq:tau-tune}
\end{equation}
Our upper bound on the bias is rather conservative,
as it upper bounds the error of DM at the largest possible value for
every data point. This has the effect of favoring the use of the
unbiased part in \textsc{switch}\xspace whenever possible, unless the variance would
overwhelm even an arbitrarily biased DM. This conservative choice, however,
immediately implies the minimax optimality of the \textsc{switch}\xspace
estimator using $\widehat{\tau}$,
because the incurred bias is
no more than our upper bound, and it is incurred only when the minimax
optimal IPS estimator would be suffering an even larger variance.
Our automatic tuning is related to the MAGIC estimator
of~\citet{thomas2016data}. The key differences are that we pick only
one threshold $\tau$, while they combine the estimates with many
different $\tau$s using a weighting function. They pick this
weighting function by optimizing a bias-variance tradeoff, but with
significantly different bias and variance estimators. In our experiments,
the automatic tuning using \Eq{tau-tune} generally works better
than MAGIC.
\section{Experiments}\label{sec:exp}
\begin{figure*}[p
\centering
\begin{subfigure}[t]{0.41\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/CDF-raw-20000}
\caption{Deterministic reward}\label{fig:CDF-raw}
\end{subfigure}
\quad\quad
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/CDF-noisy-20000.pdf
\caption{Noisy reward}\label{fig:CDF-noisy}
\end{subfigure}\\
\caption{The number of UCI data sets
where each method
achieves at least a given Rel.~MSE. On the left, the UCI
labels are used as is; on the right, label noise is
added. Curves towards top-left achieve smaller MSE in more
cases. Methods in dashed lines are ``cheating'' by choosing
the threshold $\tau$ to optimize test MSE. \textsc{switch-DR}\xspace outperforms
baselines and our tuning of $\tau$ is not too far from the
best possible. Each data set uses an $n$ which is the size of
the data set, drawn via bootstrap sampling and results are
averaged over 500 trials.}\label{fig:CDF}
\bigskip
\centering
\hspace{2em}\includegraphics[width=0.9\textwidth]{figs/legend}
\\
\medskip
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/quantiles-rawyeast-new.pdf
\vspace{-3pt}\caption{yeast / deterministic reward}\label{fig:raw-yeast}
\end{subfigure}
\hspace{0.25in}
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/quantiles-noisyyeast-new.pdf
\vspace{-3pt}\caption{yeast / noisy reward}\label{fig:noisy-yeast}
\end{subfigure}
\\
\medskip
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/quantiles-rawoptdigits-new.pdf
\vspace{-3pt}\caption{optdigits / deterministic reward}\label{fig:raw-optdigits}
\end{subfigure}
\hspace{0.25in}
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/quantiles-noisyoptdigits-new.pdf
\vspace{-3pt}\caption{optdigits / noisy reward}\label{fig:noisy-optdigits}
\end{subfigure}
\caption{MSE of different methods as a function of input data size. \emph{Top:} optdigits data set.
\emph{Bottom:} yeast data set. }\label{fig:samplesize}
\end{figure*}
We next empirically evaluate the proposed
\textsc{switch}\xspace estimators on the 10 UCI data sets previously used
for off-policy evaluation \citep{dudik2011doubly}. We convert the multi-class
classification problem to contextual bandits by treating the labels as
actions for a policy $\mu$, and recording the reward of $1$ if the
correct label is chosen, and $0$ otherwise.
In addition to this \emph{deterministic} reward model,
we also consider a \emph{noisy} reward model for each data set,
which reveals the correct reward with probability $0.5$ and outputs a
random coin toss otherwise. Theoretically, this
should lead to bigger $\sigma^2$ and larger variance in all
estimators. In both reward models, $\ensuremath{R_{\max}} \equiv 1$ is a valid bound.
The target policy $\pi$ is the deterministic decision of a
logistic regression classifier learned on the multi-class data, while the logging policy $\mu$
samples according to the probability estimates of a logistic model
learned on a covariate-shifted version of the data.
The covariate shift is obtained as in prior work
~\citep{dudik2011doubly,gretton2009covariate}.
In each data set with $n$ examples, we treat the uniform distribution over
the data set itself as a surrogate of the population distribution so
that we know the ground truth of the rewards. Then, in the simulator,
we randomly draw i.i.d.\ data sets of size
$100, 200, 500,1000,2000,5000, 10000,\dotsc$ until reaching $n$, with
$500$ different repetitions of each size.
We estimate MSE of each estimator by taking the empirical
average of the squared error over the $500$ replicates; note that we can calculate
the squared error exactly, because we know $v^\pi$. For some of the methods,
e.g., IPS and DR, the MSE can have a very large variance
due to the potentially large importance weights. This leads to very
large error bars if we estimate their MSE even with $500$
replicates. To circumvent this issue, we report a clipped version of
the MSE that truncates the squared error to $1$, namely $ \mathrm{MSE} =
\mathbb{E}[(\hat{v}-v^\pi)^2 \wedge 1].$ This allows us to get valid
confidence intervals for our empirical estimates of this
quantity. Note that this does not change the MSE estimate of our
approach at all, but is significantly more favorable towards IPS and
DR. In this section, whenever we refer to ``MSE'', we are referring to
this truncated version.
We compare \textsc{switch}\xspace and \textsc{switch-DR}\xspace against the following baselines:
1.~\emph{IPS};
2.~\emph{DM trained via logistic regression};
3.~\emph{DR};
4.~\emph{Truncated and Reweighted IPS (TrunIPS)};
and
5.~\emph{Trimmed IPS (TrimIPS)}.
In DM, we train $\hat{r}$ and then evaluate the policy on the same
contextual bandit data set. Following \citet{dudik2011doubly}, DR is constructed by randomly splitting the
contextual bandit data into two folds, estimating $\hat{r}$ on one fold, and then evaluating
$\pi$ on the other fold and vice versa, obtaining two estimates.
The final estimate is the average of the two.
TrunIPS is a variant of IPS, where importance weights are
capped at a threshold $\tau$ and then renormalized to sum to one~\citep[see,
e.g.,][]{bembom2008data}.
TrimIPS is a special case of \textsc{switch}\xspace due to~\citet{bottou2013counterfactual} described earlier,
where $\hat{r} \equiv 0$.
For \textsc{switch}\xspace and \textsc{switch-DR}\xspace as well as TrunIPS and TrimIPS we
select the parameter $\tau$ by our automatic tuning from
Section~\ref{sec:autotuning}. To evaluate our tuning approach,
we also include the
results for the $\tau$ tuned optimally in hindsight, which we
refer to as the \emph{oracle} setting, and also show results obtained
by the multi-threshold MAGIC approach. In all these approaches we
optimize among 21 possible thresholds, from an exponential grid between
the smallest and the largest importance weight observed in the data,
considering all actions in each observed context.
In order to stay comparable across data sets and data sizes, our performance
measure is the relative MSE with
respect to the IPS. Thus, for each estimator $\hat{v}$, we calculate
$\mathrm{Rel.\ MSE}(\hat{v}) = \frac{\mathrm{MSE}(\hat{v})}{\mathrm{ MSE}(\hat{v}_{\mathrm{IPS}})}$.
The results are summarized in Figure~\ref{fig:CDF}, plotting the
number of data sets where each method achieves at least a given
relative MSE.%
\footnote{For
clarity, we have excluded \textsc{switch}\xspace, which
significantly outperforms IPS, but is dominated by \textsc{switch-DR}\xspace. Similarly,
we only report the better of oracle-TrimIPS and oracle-TrunIPS.}
Thus, methods that achieve smaller MSE across more
data sets are towards the top-left corner of the plot, and a larger
area under the curve indicates better performance.
Some of the differences in MSE are several orders of magnitude
large since the relative MSE is shown on the logaritmic scale.
As we see, \textsc{switch-DR}\xspace dominates all baselines and our empirical
tuning of $\tau$ is not too far from the best possible.
The automatic
tuning by MAGIC tends to revert to DM, because its bias estimate is too
optimistic and so DM is preferred whenever IPS or DR have some significant variance.
The gains of
\textsc{switch-DR}\xspace are even greater in the noisy-reward setting, where we add label
noise to UCI data.
In Figure~\ref{fig:samplesize}, we illustrate the convergence of MSE
as $n$ increases.
We select two data sets and show how
\textsc{switch-DR}\xspace performs against baselines in two typical cases: (i) when
the direct method works well initially but is outperformed by IPS and DR
as $n$ gets large, and (ii) when the direct method works
poorly. In the first case, \textsc{switch-DR}\xspace outperforms both DM and
IPS, while DR improves over IPS only moderately. In the second case,
\textsc{switch-DR}\xspace performs about as well as IPS and DR despite a poor performance of DM.
In all cases, \textsc{switch-DR}\xspace is robust to additional noise in the
reward, while IPS and DR suffer from higher variance.
Results for the remaining data sets are in
Appendix~\ref{sec:experiments}.
\section{Conclusion}
In this paper we have carried out minimax analysis of off-policy evaluation
in contextual bandits and showed that IPS and DR are minimax optimal
in the worst-case, when no consistent reward model is available.
This result complements existing asymptotic
theory with assumptions on reward models, and highlights the
differences between agnostic and consistent settings. Practically,
the result further motivates the importance of using side
information, possibly by modeling rewards directly,
especially when importance weights are too large. Given this
observation, we propose a new class of estimators called \textsc{switch}\xspace that can
be used to combine any importance weighting estimators, including IPS
and DR, with DM. The estimators adaptively switch between DM
when the importance weights are large and either IPS or
DR when the importance weights are small. We show that the new
estimators have favorable theoretical properties and also work well on
real-world data. Many interesting directions remain open for future
work, including
high-probability upper bounds on the finite-sample MSE of \textsc{switch}\xspace
estimators, as well as sharper finite-sample lower bounds under
realistic assumptions on the reward model.
\section*{Acknowledgments}
The work was partially completed during YW's internship at Microsoft
Research NYC from May 2016 to Aug 2016. The authors would like to
thank Lihong Li and John Langford for helpful discussions, Edward
Kennedy for bringing our attention to related problems and recent
developments in causal inference, and an anonymous reviewer for
pointing out relevant econometric references and providing valuable
feedback that helped connect our work with research on average treatment effects.
\bibliographystyle{icml2017}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,288 |
Justice är en amerikansk advokatserie från 2006 med bland andra Victor Garber, Kerr Smith, Rebecca Mader och Eamonn Walker.
Den producerades av Jerry Bruckheimer.
Serien sändes på Kanal 5 2007.
Rollfigurer
Ron Trott (Victor Garber) är den styrande i advokatteamet och han tycker om att synas i media, och utnyttja dem till sina klienters fördel. Hans sätt ogillas av dock av juryer och därför arbetar han främst utanför rättsalen.
Tom Nicholson (Kerr Smith), är Trots motsats, han uppfattas som ärlig av juryer och är den som övertygar i rättssalen. Han vill helst tro att klienterna verkligen är oskyldiga.
Luther Graves (Eamonn Walker), är en före detta åklagare som blivit försvarsadvokat. Han använder sina erfarenheter för att förutsäga åklagarsidans drag, och att se fallen ur både förövar- och offerperspektiv.Han är politiskt engagerad i det afroamerikanska samhället och har många goda kontakter.
Alden Tuller (Rebecca Mader) sköter bevismaterial, vittnen och experter. Hon är ogift, men bär vigselring under rättegångar eftersom hon tror att det får juryer att lita mer på henne. Till skillnad från Tom så förutsätter hon att klienterna snarare är skyldiga än oskyldiga.
Externa länkar
Amerikanska TV-serier från 2000-talet
TV-seriestarter 2006
TV-serieavslutningar 2006
Los Angeles i fiktion | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,807 |
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Series 1: [6/6]: The End of Negative Suffering - Br. Khalil Jaffer -...
Series 1: [6/6]: The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this...
Series 1: [6/6]: The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this world and heavy duty. So be prepared to listen to them at least twice. Surah Anfal, Chapter 8, Verse 24: \\\'O you who believe! answer (the call of) Allah and His Messenger when he calls you to that which gives you life; and know that Allah intervenes between man and his heart, and that to Him you shall be gathered.\\\' (8:24)
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Video Tags: khalil jaffer jafer jafir khaleel negative suffering depression frustration biploar polar mania Psychology problem islam imam mehdi
Series 1: [5/6]: The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this world and heavy duty. So be prepared to listen to them at least twice. Surah Anfal, Chapter 8, Verse 24: \'O you who believe! answer (the call of) Allah and His Messenger when he calls you to that which gives you life; and know that Allah intervenes between man and his heart, and that to Him you shall be gathered.\' (8:24)
[3/6]: The End of Negative Suffering - Br. Khalil Jaffer – English
The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this world and heavy duty....
The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this world and heavy duty. So be prepared to listen to them at least twice. Surah Anfal, Chapter 8, Verse 24: \\\\\\\\\\\\\\\'O you who believe! answer (the call of) Allah and His Messenger when he calls you to that which gives you life; and know that Allah intervenes between man and his heart, and that to Him you shall be gathered.\\\\\\\\\\\\\\\' (8:24)
Video Tags: 3 of 6 khalil jaffer jafer jafir khaleel negative suffering depression frustration biploar polar mania Psychology problem islam imam mehdi
[2/6]: The End of Negative Suffering - Br. Khalil Jaffer - English
The End of Negative Suffering - Br. Khalil Jaffer Uploaded on May 4, 2011 Be warned! Both these series are simply out of this world and heavy duty. So be prepared to listen to them at least twice. Surah Anfal, Chapter 8, Verse 24: \\\\\\\'O you who believe! answer (the call of) Allah and His Messenger when he calls you to that which gives you life; and know that Allah intervenes between man and his heart, and that to Him you shall be gathered.\\\\\\\' (8:24)
[26 June 2014] Subho Zindagi - Negative Chemical Effects Of Drugs - Urdu
Video Tags: Sahar Sahar TV TV Discussion Scholar Talk Show urdu channel tehran iran program Subho Zindagi Morning Show Nijat Dena C P R Topic Negative Chemical Effects Of Drugs 26 June 2014
When does it Become Necessary to Attract Negative Attention to Islam -...
When does it Become Necessary to Attract Negative Attention to Islam
48m:5s
Nasr313
Video Tags: When does it Become Necessary to Attract Negative Attention to Islam sheikh hamza sodagar english
[Islam & Life] What role has Murdoch news empire played in the...
The Rupert Murdoch phone hacking scandal that led to the closure of the news of the world was not surprising news in the perspective of the global...
The Rupert Murdoch phone hacking scandal that led to the closure of the news of the world was not surprising news in the perspective of the global media monopoly. Very a few people paid attention to the fact that targeting Muslims by the right wing media which is mostly owned by Murdoch has a longer history than this recent scandal. In this edition of the show we ask the following question; What role has Murdoch's news empire played in the negative portrayal of Muslims?
muslims_in_uk
Video Tags: tariq ramadan Rupert Murdoch phone hacking scandal closure news of the world global media monopoly islamophobia Muslims right wing media neagtive propaganda fear portrayal ahmed versi
Baghdad hosting conference on anti-Islam film - 27SEP12 - English
Cultural Organization in Baghadhas organized a conference to protest against the US-produced film with a negative depiction of the holy Prophet of...
Cultural Organization in Baghadhas organized a conference to protest against the US-produced film with a negative depiction of the holy Prophet of Islam. It was attended by ordinary citizens, officials, tribal leaders and clerics. They also warn about the aims of the makers of such insulting film. The film has sparked harsh reactions in Muslim world and across the world. People here say only such trends could come to an end only through spreading Islamic awakening across the world.
Video Tags: Baghdad hosting conference anti-Islam film ordinary citizens officials tribal leaders clerics Cultural Organization protest US-produced negative depiction holy Prophet Islam harsh reactions Muslim
[81] Hadith Explanation by Imam Khamenei | Swallowing Anger | Farsi Sub...
[81] Hadith Explanation by Imam Khamenei | Swallowing Anger Ayatullah Khamenei explains a hadith of Prophet Muhammad (S) regarding...
[81] Hadith Explanation by Imam Khamenei | Swallowing Anger Ayatullah Khamenei explains a hadith of Prophet Muhammad (S) regarding anger. What are the negative impacts of not controlling the anger? Who is the best of the people in the light of this tradition?
QomTV
Video Tags: Qomtv, Qomtv Productions, hadith, explanation, imam khamenei, swallowing anger, ayatullah, explains, prophet Muhammad, anger, negative, impact, controlling, tradition, best, people,
[13] Positive & Negative Ishq | H.I Sadiq Raza Taqvi | Mah-e-Ramzaan...
Dars: 13 | 13th Ramzaan-ul-Mubarak 1441 Quran , Rivayat Or Nafsiyat Ke Roshni May Samajhi Masael Ka Hall قرآن ، روایات اور...
Dars: 13 | 13th Ramzaan-ul-Mubarak 1441 Quran , Rivayat Or Nafsiyat Ke Roshni May Samajhi Masael Ka Hall قرآن ، روایات اور نفسیات کی روشنی میں سماجی مسائل کا حل Positive & Negative Ishq مثبت اور منفی عشق Speaker: H.I Sadiq Raza Taqvi حجۃ الاسلام سید صادق رضا تقوی Date: 07 May 2020
Video Tags: Quran Rivayat Nafsiyat Masael قرآن روایات نفسیات روشنی سماجی مسائل Sadiq Raza Taqvi صادق رضا تقوی Ramadhan Ramdhan Ramzaan Mubarak Ramdhan 1441 Ramadhan 1441 رمضان ۱۴۴۱ رمضان المبارک رمضان کریم کراچی عشق عاشق عشاق ishq eshq
The World Reacts to the Deal of the Century | Pure Stream Media...
Despite the efforts to occupy all of Palestine via the notorious Deal of the Century at the hands of the Zio-American puppets, the world had an all...
Despite the efforts to occupy all of Palestine via the notorious Deal of the Century at the hands of the Zio-American puppets, the world had an all out negative reaction to the Deal of the Century. In this episode, we bring forth voices from across the world who have opposed the failed Deal of the Century. A Production of Pure Stream Media. #DealofCentury #Palestine #Trump #Reactions #israel #Jared #NoToNormalization #StopNormalization
PureStreamMedia
Video Tags: purestream, media, production, world, reacts, deal of the century, palestine, notorious, zio-american, puppets, negative reaction, trump, israel, stop normalization, no to normalization, zionism, zionists, jared, america, USA, netanyahu,
The World Reacts to the Deal of the Century (Conclusion) | Pure...
Despite the efforts by the Zio-American puppets to occupy all of Palestine via the notorious and sneaky Deal of the Century, the world end up...
Despite the efforts by the Zio-American puppets to occupy all of Palestine via the notorious and sneaky Deal of the Century, the world end up having an all out negative reaction to their so-called Deal. In this episode, we bring forth voices from all across the globe who have individually and collectively opposed the dastardly Deal of the Century, including the statements of Imam Khamenei as we conclude the world\'s reactions to the Deal of the Century. A Pure Stream Media Production. #DealoftheCentury #Palestine #Trump #Reactions #israel #JaredKushner #Khamenei #NoToNormalization #StopNormalization #Normalization
Video Tags: purestream, media, production, deal of the century, conclusion, zionists, zio-american, palestine, notorious deal, negative reaction, oppose, dastardly deal, imam khamenei, zionism, zionists, israel, america, USA, saudi, IDF, gaza, pope, roman catholics, christians, jews, muslims, american puppets, trump, normalization with israel, no to normalization, jared kushner, reactions,
Part 2 (Must Watch) Tehran Sermon - Rehbar Syed Ali Khamenie...
The Leader of the Islamic Revolution has described the \\\'unprecedented\\\' turnout of almost 85% in the election as a \\\'political quake\\\' for...
The Leader of the Islamic Revolution has described the \\\'unprecedented\\\' turnout of almost 85% in the election as a \\\'political quake\\\' for the enemy. Ayatollah Seyyed Ali Khamenei said high turnout in the election, which witnessed more than 40 million Iranians casting their votes, was a great manifestation of people\\\'s solidarity with the Islamic establishment. Addressing Friday prayers congregation, Ayatollah Khamenei said that last Friday\\\'s election indicated a \\\'common sense of responsibility\\\' of the Iranian nation to determine the future of the country. The Leader added that all those who took part in the election proved their \\\'political consciousness and commitment\\\' towards the establishment to the whole world. The Leader said the high voter turnout in the election was a \\\'political quake\\\' for the enemy and a \\\'real celebration\\\' for the friends of the country. \\\"The Islamic Republic of Iran will by no means betray the votes of the nation,\\\" the Leader said, adding the legal system of the election will not allow any ballot rigging in Iran. Ayatollah Khamenei, however, maintained that the Guardian Council, the body tasked with overseeing the election, would look into the complaints of the candidates who are unhappy with the election results. The Leader also added that the establishment would never give-in to illegal demands, urging all presidential candidates to pursue their complaints through legal channels. Ayatollah Khamenei called for an end to illegal street protests aimed at reversing the result of the election. Following the announcement of the election outcome, supporters of the defeated candidate Mir-Hussein Mousavi-who rejected the election results-- took to the streets of Tehran and other cities in daily rallies. The Leader also warned against attempts made by foreign media outlets seeking to destabilize the country and blamed Britain in particular. Ayatollah Khamenei also decried the slander of former and incumbent top officials in pre-election debates by candidates. President Ahmadinejad was re-elected the next president of the country with over 60% percent of the votes. He won over his three rivals Mir-Hossein Mousavi, Mehdi Karroubi and Mohsen Rezaei with almost 25 million votes. The Leader said the time is over for rivalry, stressing that all should unite and line up behind the president-elect Complete Transcript http://www.presstv.com/detail.aspx?id=98610 In this sermon, I call all the respected brothers and sisters who have attended the Friday prayers here to piety and I advise them against any wrongdoing. In this sermon, I will address the issue of the election, which is a hot topic in our country. I want to address three different groups on three different issues; firstly, I want to address the general public. Secondly, I want to address the political elite, the candidates of the presidential election, activists and those who have been active in the process of election. I also have something to say to the leaders of the global arrogance, certain Western governments and their media. On the first issue, where I address you dear people, I want to express my appreciation and gratitude. I do not like to exaggerate while I am addressing my audience, but regarding the recent election, I must tell you great people that no matter what I say, words cannot describe the greatness of your great accomplishment. The June 12 election was a great show of the people\\\'s sense of responsibility, their will to participate, and their dedication to the system. Truly, I have never heard of anything similar to what you have accomplished taking place in any of the democratic systems around the world, whether they are false democracies or truly built on their people\\\'s vote. In the Islamic Republic, aside from the 1979 referendum, there has no election like the one held last Friday with a turnout of almost 85 percent. This means almost 40 million voters. You can see the presence of the 12 and last Shia Imam behind this. This is a sign of God\\\'s blessing to us. It is necessary that I address you all across the nation from the depths of my heart, to express my respect and tell you that I feel humble in your presence. Our young generation showed and proved they have insight and that like the first generation of the Islamic Revolution, they are committed. The difference is, during the days of the revolution, revolutionary fire burned in the hearts of all. It was the same is the days of the imposed war but in a different sense. Today, however, there is no more of that but we still witness this commitment, this sense of responsibility, this understanding and fervor in our youth. This is not something that can be ignored. Of course, there are differences of taste and of opinion among our people. Some people support a certain candidate; others back another person and his words and ideas. This is natural, but you can see a collective commitment amid all this and amongst people of all walks of life. You can see a consensus, a collective commitment to the protection of our country and system. Everyone entered the political scene in villages, towns, cities, major cities, different ethnic groups, people of different faiths, men, women, young and old. They all entered the scene. They all took part in this great movement. My dear people, this election was a political tremor for your enemies. For your friends across the world, it was a real celebration -- a historical ceremony and victory. Thirty years after the victory of the Islamic Revolution, such a huge turnout and show of commitment to the Islamic system and the late Imam [Khomeini] shows the renewal of the pledge of allegiance to the late Imam and the martyrs. This was a breath of fresh air, a new movement and a great opportunity for the Islamic system. This election put religious democracy on display for the whole world to witness. All ill-wishers of the Islamic establishment saw for themselves the meaning of religious democracy. This is an alternative path in the face of dictatorships and arrogant regimes on the one side and democracies devoid of spirituality and religion on the other. This is religious democracy. This is what brings the hearts of people together and draws them to the scene. This is the first point I wanted to make about the election. The second point is that the June 12 election showed that people live with trust, hope and national enthusiasm in this country. This is against a great deal of comments your enemies make in their propaganda. If the people of this country were not hopeful about their future, they would not have taken part in any election. If people were not dedicated to the Islamic establishment, they would have never voted. If they did not feel free, they would have never shown up at polling stations. The trust they have in the Islamic system was evident in this election. Later on, I will tell you how the enemy targeted the very trust of the people in the Islamic establishment. This trust is the very thing they want to crush. This trust is the greatest asset of the Islamic system, so they want to take it away from the Islamic establishment. They want to cast doubt on the election and weaken the confidence of the people in the system. They want to cause the people to panic. The enemies of the Iranian nation know that without trust there would have been a low turnout. A low turnout would have questioned the legitimacy of the establishment. That is what they are after. They wanted to take away your [people] trust and keep you away from the polls to target this legitimacy, and if they had achieved this goal, the damage done would have been incomparable to any other. For the people to come to the polls en masse and then be told that they made a mistake and should not have trusted the Islamic establishment, this is an enemy game. This path is the same one they pursued even before the elections. A few months before the election, in late march, I said in Mashhad that the enemy has started whispers and rumors that their will be vote rigging. They were preparing the grounds for the events of today. I advised our friends in the country not to repeat what the enemy wants to plant in people\\\'s minds. The Islamic establishment has the people\\\'s trust and it has not gained this trust easily. For the past 30 years, authorities in the Islamic Republic have managed to maintain this trust, with their performance and painstaking efforts. The third issue I want to touch upon is the issue of rivalry. This competition was a free, serious and transparent race between four candidates as we all witnessed. These competitions, debates and discussion were so transparent that some began to voice objections. I will tell you that to they had the right to object to some extent. Certain problems were also created that resulted in what you see today. I must tell you that we were and still are under the impression that these rivalries were between the four candidates who are all individuals committed to the system. The Enemies want to portray the situation in the media - some of which belong to the Zionists -as if there is a row between the proponents and opponents of the Islamic Republic. No, this is not the case, this very untrue. The four candidates who entered the presidential race all belonged and still belong to the Islamic establishment. One of these four is the president of our country - a hardworking and trustworthy president. One of them is the two-term prime minister, he served the country when I myself was president. He was my prime minister for eight years. One of them was the commander of the Islamic Revolution Guard Corps and one of the wartime commanders. One them was two-time head of parliament and Majlis speaker. They are all members of our Islamic establishment. Of course, they have differences of opinion and plans that differ from one another. But, they all belong to this Islamic establishment. This race was defined within the framework of the system. It was not a competition between insiders and outsiders as the Zionist and the US were trying to portray. No, this was a competition within the framework of the system between members of the system. I know them all personally, I know their system of thought and their tastes very well. I am familiar with their personalities. I have worked with all of them closely. I know them all. But of course, I do not share all their views. I believe some of their views and executive records are subject to criticism. I see some more suitable to serve the country than others. But, this is up to the people to decide, and this is exactly what happened, they chose who they wanted. My desire and my choice was never announced nor was there any need for the people to pay heed to it. The people had their own criteria and this is what they based their decision on. Millions here and outside the country decided for themselves. This is an internal issue. Misrepresenting the problem is underhand. The row is not between insiders of the system and outsiders. The row is not between revolutionary and anti-revolutionary forces, it is a difference of opinion between the members of the Islamic Revolution. People who voted for these four candidates, voted with faith in the system. They believed their candidate of choice was better for the country so they voted for him. They voted in favor of the person they found most competent. Well, these campaigns and debates were an important and interesting initiative. They were very clear, to the point and serious. The televised debates proved wrong those who were trying to say from the outside that these competitions are formalities. They saw that these rivalries are real and serious. They saw that they are really battling it out and exchanging viewpoints. From this perspective, these debates were positive. But, they also had some negative points which I will touch upon. The positive aspect was that in these televised discussions and debates everyone spoke their mind clearly and casually. A flood of criticism followed. Everyone was forced to respond. Everyone was criticized and they defended themselves. The stances that these individuals and groups had were unveiled before the eyes of the nation. They talked about their plans, commitments and projects. All this was publicized for the people so that they could judge for themselves. People felt that in the Islamic system they are not the outsiders. Everything was clearly laid out before the people. They were shown that the nature of their vote is not ceremonial. The right to vote truly does belong to the people. People want to have the right to choose. This is what the televised debates indicated. One of the main reasons that ten million additional voters participated in this election was because the people\\\'s minds had been engaged, therefore they came and voted for the candidate of their choice. These debates found their way into the streets and homes. These debates helped the people become better informed and hence make better decisions. The Islamic establishment is in favor of such debates. Note that such debates should not be steered in a direction that may cause people to hold grudges against one another. If these debates had remained within their intended framework, they would have been positive. But when they turn into arguments than they will gradually bring grudges and hard feelings. Of course, such debates should continue at managerial levels, but without a negative aspect. Officials should allow criticism and feel responsible to answer. If an individual is criticized, he must see it as an opportunity to enlighten the people and reveal fact and truth. If these debates are regularly carried out [as normal government practice], at election time when there are such debates we would not witness such reactions. All arguments would emerge and all ideas would be exchanged over time. These are the positive aspects of such debates. But, there have also been some negative aspects to the debates that need to be dealt with. In some cases, we saw that logical points were undermined and emotional and destructive responses dominated the debate. There were efforts to portray the last four years as a dark era. There were also attempts to portray previous administrations in a similar light. Allegations were made that have not been proven in any court, rumors were used as a reference, and unjust remarks were made. This administration, despite the excellent services it had rendered came under unjust attacks. Similarly, the performance of previous governments in the past 30 years came under attack. The candidates gave in to their emotions. They made some positive points. They also raised some unpleasant negative issues. Like the rest of the nation, I sat and watched these TV debates. I took pride in the freedom of speech I witnessed. I enjoyed the fact that the Islamic Republic has been able to aid the people in deciding their future, but the shortcomings saddened me. For supporters of the candidates the shortcomings and negative aspects were also a cause for concern; both sides were a party to this... both sides had their problems. On the one hand, insults were hurled against the president of the country, even two to three months prior to these debates, speeches were brought to me and in them, I read the insults made and the accusations leveled against the president of the country who was elected by the vote of the people. They accused him of lying. This is not good. They fabricated documents against the government and distributed them everywhere. I saw what was going on. They [accusations] were all untrue and contrary to the facts. They swore at the president, called him superstitious, and called him names. They closed their eyes to ethics and the law. On the other hand, almost the same thing happened. The performance of the past 30 years of the Revolution was brought under question. People were named who are among the system\\\'s veteran figures. They are people who have dedicated their lives to this establishment. Never before have I mentioned people by name in the Friday prayer sermons, but today, I have to mention some names, particularly Mr. [Akbar] Hashemi Rafsanjani, Mr. [Ali Akbar] Nateq-Nouri. I must mention their names and point out that nobody has accused them of corruption. Now, if anyone has any claims or complaints regarding their [Hashemi Rafsanjani and Nateq-Nouri\\\'s] relatives they should refer to judicial authorities. You cannot raise these issues in the media before they are proven. If it is proven, you can raise the issue as every member of society is equal, but you are not allowed to make claims. When such things are mentioned, misunderstandings are the outcome. This can cause misunderstandings for the younger generation. Everyone knows Mr. Hashemi. My relationship with him goes back to before the Islamic Revolution. I have known him for more than 25 years. Mr. Hashemi was one of the main revolutionary figures. He was one of the most active proponents of the revolution, and after the Islamic Revolution, he was one of the main political figures serving the people alongside the late Imam, And after the departure of the founder of the Islamic Revolution he has been alongside the leadership to date. On several occasions, there were assassination attempts on his life. Before the revolution, he donated his possessions for the revolutionary cause. Our youth should know these facts. After the revolution, he had different responsibilities. For eight years, he served as the president and before that he was the parliament speaker. He held other sensitive positions as well. Throughout these years, I am not aware of one incident in which he gathered wealth. These are the facts that everyone should know. In the most sensitive of periods, he served the revolution and the establishment. Of course, my opinion and Mr. Rafsanjani\\\'s differs on numerous issues, which is natural. However, we should not create any misunderstandings for the people. The president and Mr. Rafsanjani have had differences of opinion since the president took office in 2005. They have differences of opinion in foreign policy, in the manner of spreading social justice as well as on some cultural issues. However, the president\\\'s ideas are closer to mine. The same goes fort Mr. Nateq-Nouri. He has also served the revolution, rendered great services for the establishment and there is not a shred of doubt about that. The live televised debates are a positive step, but these shortcomings should be removed. After the debates, I had a talk with the president because I knew he would listen to me. The stance of the Islamic establishment is clear-cut regarding corruption and social justice. Corruption should be fought anywhere it is traced. There is a point I want to make here. We do not claim that our establishment is free of all economic and financial corruption. Yes, there is corruption. If there was no corruption, I would not have written the eight-point letter to the heads of the legislative, judiciary, and executive branches of the country. We have corruption, but the Islamic establishment is one of the healthiest establishments in the world today. However, it is not right to accuse the country of corruption based on some Zionist reports and sources. Moreover, questioning the credibility of statesmen goes beyond the bounds of decency. Financial corruption is an important issue in the Islamic establishment. The judiciary, executive and legislative branches of the country must do everything within their power to fight against it. Everyone is duty-bound to fight corruption. If corruption is not contained, it will spread in the same manner as you see occurring in many countries…. They are struggling with an alarming level of corruption as you have heard about in the UK. This is only a fraction of the scandal as it goes way beyond what has been publicized. Let me summarize the points I made. The Friday election was a historical event, which touched the entire world. Some of our enemies, however, attempted to cast doubt over this absolute and definitive victory. Some even attempted to portray it as a national defeat. They did not want you to enjoy this victory. They did not want to see the highest turnout in the world go down in history in your name. However, it has happened, it has been recorded in history. They cannot manipulate this. The time for rivalry is passed... These four candidates have all fought in the battlefront of this revolution and they are members of this establishment. Forty million people went to the polls and cast their votes for this revolution. It was not only the 24 million votes that went to the president; 40 million votes were cast in favor of the revolution. The people have trust [in the establishment], and all supporters of candidates should rest assured that the Islamic establishment would never betray the people\\\'s trust. In fact, the electoral system of the country does not allow for any vote rigging, which is testified to by all those in charge of the election process. When there is a margin of one hundred thousand or one million at most, then one can doubt that there may have been some form of manipulation or irregularity; however, when there is a difference of eleven million votes, how could any vote rigging have taken place? However, as I have said, and the Guardian Council has accepted, if some people have doubts then it should be dealt with through legal channels. Everything must be dealt strictly though legal channels. I will never accept illegal demands. If the legal frameworks are breached today, then no future election can be guaranteed. In every election, there is only one winner, and of course, some defeated candidates. Complaints, if there is any doubt, should be pursued through legal channels. We have a comprehensive and competent legal system. Just as the candidates have the right to appoint observers, they are given the right to file complaints. I have requested the Guardian Council conduct a partial vote recount in the presence of the candidates and their representatives. We have no problem with this. I want to address the politicians, candidates and political parties at this point. We are at a critical historical juncture. Look at current world affairs, the situation in the Middle East, global economic woes and the situation in our neighboring countries. We are duty bound to remain vigilant and to be careful not to commit mistakes at this critical point in time. In the election, people fulfilled their duty in the best way possible, which was by going to the polls. We have heavier responsibilities on our shoulders now. Those figures who are looked up to by the people and politicians, should be cautious about their words and deeds. If they show any amount of extremist attitude, it will penetrate into the ranks of the people. It may have dangerous consequences and may eventually get out of control. Extremism in society will trigger or fan other extremist moves in the country. If political elites disobey the law and make wrong decisions, they will be held accountable for any violent actions or rioting that ensues. I urge these people, these friends of mine, to exercise restraint and patience. You should see enemy hands at work [against the country]. You should see hungry wolves laying in ambush. They are taking off their masks of diplomacy and are showing their true colors. I urge you to open your eyes and see the enemy. In the past few days the prominent diplomats of some Western countries, which have been dealing with us through diplomatic rhetoric, have removed their masks. Today you can see their true face. They are now showing their enmity toward the Islamic establishment and the most treacherous of them all is Britain. I tell these brothers of ours to think of their responsibility. You are responsible before God. I call on you to remember what Imam has written in his will; the law has the final say. All differences should be settled at the ballot box. This is what elections are for, to let ballot boxes and not the streets determine what the people want. If after every election, the supporters of the candidates who have lost take to streets and the supporters of the candidate who has won respond in the same manner, then what need would we have for elections? Why should the people have to suffer? We should not take to the streets to show off with the number of our supporters to the people. Such acts are not a political issue for those terrorists who take advantage of the situation to hide among the masses in order to carry out their agenda. It is a very good cover for these saboteurs. Who will take responsible for this? Some of the people who were killed in these riots were ordinary people, ordinary Basij members. Who will be held accountable for this? They may start taking advantage of this situation to assassinate Basij members, which will naturally provoke emotional reactions. Who is to be held responsible for this? One is grieved to see them attack religious students at Tehran University dormitories and afterwards chant slogans in support of the leadership. Post-election rivalry on the streets is not the right way to go. It only challenges the election. I want all sides to put an end to this. If they do not stop such actions, then they will be responsible for the repercussions of such incidents. It is also wrong to assume that street riots can be used as leverage to pressure the establishment and to force officials to listen to them for what they believe is in the interest of the country. Giving in to illegal demands under pressure is in itself the beginning of dictatorship. This is a miscalculation and the consequences will be directed at those who orchestrated them. If necessary, I will tell the people about them in due time. I ask all these brothers and friends of mine to act based on friendship and abide by the law. I hope God will help us choose the righteous path. The celebration of 40 million votes should be appreciated and the enemy must not be allowed to ruin the celebration. However, if certain people decide to choose another path, then I will have no choice but to talk with the people more openly. The third group I wish to address are the leaders of the Western media and arrogant powers. In the past two to three weeks, I have heard the words and witnessed the actions of politicians from the United States and certain European countries. Before the elections, they attempted to cast doubt over the election itself so that there would be a low voter turnout. They had their own assessments of results forecasts, but they did not expect the mass participation of the people. They never predicted an 85 percent turnout, or 40 million voters. When they saw the mass turnout, they were shocked. They realized the reality of Iran. They came to understand that they need to adapt themselves to the new situation be it regional, nuclear or internal. When they saw the great popular movement on Election Day, they realized that a new chapter had been opened with regards to Iran and that they must come to terms with it. When some candidates began protesting the results, they felt that there was a change, so they jumped at the chance to ride this wave. Their tone after the election changed on Saturday and Sunday. Their attention shifted to the riots and that was when they gradually began removing their masks. Western officials, their presidents, prime ministers and foreign ministers commented on this situation. The US President said that we were waiting for the day when people would take to the streets. At the same time they write letters saying that they want to have ties and that they respect the Islamic Republic. Which are we to believe? Inside the country, their elements [foreign countries] began street protests and vandalism, they set fire to public property, they made shops and businesses insecure, and they are trying to rob the people of their security. This has nothing to do with the people and their preferred candidates. This kind of behavior stems from ill-wishers, mercenaries and elements working for Western and Zionist secret services. The incidents occurring inside the country have misled some of those outside our borders, who imagine Iran to be the same as Georgia. A Zionist American capitalist a few years ago, had been quoted in the media saying that he had spent 10 million dollars in Georgia to start a velvet revolution. Our nation cannot be compared to any another nation. Their problem is they have not come to know this revolution and its people. American officials say they are worried about the Iranian nation, how can you be worried? Can you even speak about human rights when you are responsible for the blood shed in Afghanistan and Iraq? In Palestine who has and is supporting and funding the Zionist regime? During the term of a previous US government, eighty people affiliated with the Davidian sect were burnt alive in their compound in Waco, Texas. For some reason these people were disliked by the then US administration. Eighty people were burnt in that building, how dare you talk of human rights? In my opinion, these western officials should at least feel a little embarrassment!
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Video Tags: Rehbar Speech Part 2 rehbar iran election 2009 june hashmi rafsanjani karobi ahmedi neejad mosavi riots killing university Before the Rehbar Ayatullah Khamnie\\\'s Sermon - Memories of War - Persian Leader of the Islamic Revolution Ayatollah Seyyed Ali Khamenei
[FULL SPEECH] Supreme Leader Ayatullah Sayyed Ali Khamenei - Friday...
Complete Transcript http://www.presstv.com/detail.aspx?id=98610 In this sermon, I call all the respected brothers and sisters who have attended...
Complete Transcript http://www.presstv.com/detail.aspx?id=98610 In this sermon, I call all the respected brothers and sisters who have attended the Friday prayers here to piety and I advise them against any wrongdoing. In this sermon, I will address the issue of the election, which is a hot topic in our country. I want to address three different groups on three different issues; firstly, I want to address the general public. Secondly, I want to address the political elite, the candidates of the presidential election, activists and those who have been active in the process of election. I also have something to say to the leaders of the global arrogance, certain Western governments and their media. On the first issue, where I address you dear people, I want to express my appreciation and gratitude. I do not like to exaggerate while I am addressing my audience, but regarding the recent election, I must tell you great people that no matter what I say, words cannot describe the greatness of your great accomplishment. The June 12 election was a great show of the people\\\'s sense of responsibility, their will to participate, and their dedication to the system. Truly, I have never heard of anything similar to what you have accomplished taking place in any of the democratic systems around the world, whether they are false democracies or truly built on their people\\\'s vote. In the Islamic Republic, aside from the 1979 referendum, there has no election like the one held last Friday with a turnout of almost 85 percent. This means almost 40 million voters. You can see the presence of the 12 and last Shia Imam behind this. This is a sign of God\\\'s blessing to us. It is necessary that I address you all across the nation from the depths of my heart, to express my respect and tell you that I feel humble in your presence. Our young generation showed and proved they have insight and that like the first generation of the Islamic Revolution, they are committed. The difference is, during the days of the revolution, revolutionary fire burned in the hearts of all. It was the same is the days of the imposed war but in a different sense. Today, however, there is no more of that but we still witness this commitment, this sense of responsibility, this understanding and fervor in our youth. This is not something that can be ignored. Of course, there are differences of taste and of opinion among our people. Some people support a certain candidate; others back another person and his words and ideas. This is natural, but you can see a collective commitment amid all this and amongst people of all walks of life. You can see a consensus, a collective commitment to the protection of our country and system. Everyone entered the political scene in villages, towns, cities, major cities, different ethnic groups, people of different faiths, men, women, young and old. They all entered the scene. They all took part in this great movement. My dear people, this election was a political tremor for your enemies. For your friends across the world, it was a real celebration -- a historical ceremony and victory. Thirty years after the victory of the Islamic Revolution, such a huge turnout and show of commitment to the Islamic system and the late Imam [Khomeini] shows the renewal of the pledge of allegiance to the late Imam and the martyrs. This was a breath of fresh air, a new movement and a great opportunity for the Islamic system. This election put religious democracy on display for the whole world to witness. All ill-wishers of the Islamic establishment saw for themselves the meaning of religious democracy. This is an alternative path in the face of dictatorships and arrogant regimes on the one side and democracies devoid of spirituality and religion on the other. This is religious democracy. This is what brings the hearts of people together and draws them to the scene. This is the first point I wanted to make about the election. The second point is that the June 12 election showed that people live with trust, hope and national enthusiasm in this country. This is against a great deal of comments your enemies make in their propaganda. If the people of this country were not hopeful about their future, they would not have taken part in any election. If people were not dedicated to the Islamic establishment, they would have never voted. If they did not feel free, they would have never shown up at polling stations. The trust they have in the Islamic system was evident in this election. Later on, I will tell you how the enemy targeted the very trust of the people in the Islamic establishment. This trust is the very thing they want to crush. This trust is the greatest asset of the Islamic system, so they want to take it away from the Islamic establishment. They want to cast doubt on the election and weaken the confidence of the people in the system. They want to cause the people to panic. The enemies of the Iranian nation know that without trust there would have been a low turnout. A low turnout would have questioned the legitimacy of the establishment. That is what they are after. They wanted to take away your [people] trust and keep you away from the polls to target this legitimacy, and if they had achieved this goal, the damage done would have been incomparable to any other. For the people to come to the polls en masse and then be told that they made a mistake and should not have trusted the Islamic establishment, this is an enemy game. This path is the same one they pursued even before the elections. A few months before the election, in late march, I said in Mashhad that the enemy has started whispers and rumors that their will be vote rigging. They were preparing the grounds for the events of today. I advised our friends in the country not to repeat what the enemy wants to plant in people\\\'s minds. The Islamic establishment has the people\\\'s trust and it has not gained this trust easily. For the past 30 years, authorities in the Islamic Republic have managed to maintain this trust, with their performance and painstaking efforts. The third issue I want to touch upon is the issue of rivalry. This competition was a free, serious and transparent race between four candidates as we all witnessed. These competitions, debates and discussion were so transparent that some began to voice objections. I will tell you that to they had the right to object to some extent. Certain problems were also created that resulted in what you see today. I must tell you that we were and still are under the impression that these rivalries were between the four candidates who are all individuals committed to the system. The Enemies want to portray the situation in the media - some of which belong to the Zionists -as if there is a row between the proponents and opponents of the Islamic Republic. No, this is not the case, this very untrue. The four candidates who entered the presidential race all belonged and still belong to the Islamic establishment. One of these four is the president of our country - a hardworking and trustworthy president. One of them is the two-term prime minister, he served the country when I myself was president. He was my prime minister for eight years. One of them was the commander of the Islamic Revolution Guard Corps and one of the wartime commanders. One them was two-time head of parliament and Majlis speaker. They are all members of our Islamic establishment. Of course, they have differences of opinion and plans that differ from one another. But, they all belong to this Islamic establishment. This race was defined within the framework of the system. It was not a competition between insiders and outsiders as the Zionist and the US were trying to portray. No, this was a competition within the framework of the system between members of the system. I know them all personally, I know their system of thought and their tastes very well. I am familiar with their personalities. I have worked with all of them closely. I know them all. But of course, I do not share all their views. I believe some of their views and executive records are subject to criticism. I see some more suitable to serve the country than others. But, this is up to the people to decide, and this is exactly what happened, they chose who they wanted. My desire and my choice was never announced nor was there any need for the people to pay heed to it. The people had their own criteria and this is what they based their decision on. Millions here and outside the country decided for themselves. This is an internal issue. Misrepresenting the problem is underhand. The row is not between insiders of the system and outsiders. The row is not between revolutionary and anti-revolutionary forces, it is a difference of opinion between the members of the Islamic Revolution. People who voted for these four candidates, voted with faith in the system. They believed their candidate of choice was better for the country so they voted for him. They voted in favor of the person they found most competent. Well, these campaigns and debates were an important and interesting initiative. They were very clear, to the point and serious. The televised debates proved wrong those who were trying to say from the outside that these competitions are formalities. They saw that these rivalries are real and serious. They saw that they are really battling it out and exchanging viewpoints. From this perspective, these debates were positive. But, they also had some negative points which I will touch upon. The positive aspect was that in these televised discussions and debates everyone spoke their mind clearly and casually. A flood of criticism followed. Everyone was forced to respond. Everyone was criticized and they defended themselves. The stances that these individuals and groups had were unveiled before the eyes of the nation. They talked about their plans, commitments and projects. All this was publicized for the people so that they could judge for themselves. People felt that in the Islamic system they are not the outsiders. Everything was clearly laid out before the people. They were shown that the nature of their vote is not ceremonial. The right to vote truly does belong to the people. People want to have the right to choose. This is what the televised debates indicated. One of the main reasons that ten million additional voters participated in this election was because the people\\\'s minds had been engaged, therefore they came and voted for the candidate of their choice. These debates found their way into the streets and homes. These debates helped the people become better informed and hence make better decisions. The Islamic establishment is in favor of such debates. Note that such debates should not be steered in a direction that may cause people to hold grudges against one another. If these debates had remained within their intended framework, they would have been positive. But when they turn into arguments than they will gradually bring grudges and hard feelings. Of course, such debates should continue at managerial levels, but without a negative aspect. Officials should allow criticism and feel responsible to answer. If an individual is criticized, he must see it as an opportunity to enlighten the people and reveal fact and truth. If these debates are regularly carried out [as normal government practice], at election time when there are such debates we would not witness such reactions. All arguments would emerge and all ideas would be exchanged over time. These are the positive aspects of such debates. But, there have also been some negative aspects to the debates that need to be dealt with. In some cases, we saw that logical points were undermined and emotional and destructive responses dominated the debate. There were efforts to portray the last four years as a dark era. There were also attempts to portray previous administrations in a similar light. Allegations were made that have not been proven in any court, rumors were used as a reference, and unjust remarks were made. This administration, despite the excellent services it had rendered came under unjust attacks. Similarly, the performance of previous governments in the past 30 years came under attack. The candidates gave in to their emotions. They made some positive points. They also raised some unpleasant negative issues. Like the rest of the nation, I sat and watched these TV debates. I took pride in the freedom of speech I witnessed. I enjoyed the fact that the Islamic Republic has been able to aid the people in deciding their future, but the shortcomings saddened me. For supporters of the candidates the shortcomings and negative aspects were also a cause for concern; both sides were a party to this... both sides had their problems. On the one hand, insults were hurled against the president of the country, even two to three months prior to these debates, speeches were brought to me and in them, I read the insults made and the accusations leveled against the president of the country who was elected by the vote of the people. They accused him of lying. This is not good. They fabricated documents against the government and distributed them everywhere. I saw what was going on. They [accusations] were all untrue and contrary to the facts. They swore at the president, called him superstitious, and called him names. They closed their eyes to ethics and the law. On the other hand, almost the same thing happened. The performance of the past 30 years of the Revolution was brought under question. People were named who are among the system\\\'s veteran figures. They are people who have dedicated their lives to this establishment. Never before have I mentioned people by name in the Friday prayer sermons, but today, I have to mention some names, particularly Mr. [Akbar] Hashemi Rafsanjani, Mr. [Ali Akbar] Nateq-Nouri. I must mention their names and point out that nobody has accused them of corruption. Now, if anyone has any claims or complaints regarding their [Hashemi Rafsanjani and Nateq-Nouri\\\'s] relatives they should refer to judicial authorities. You cannot raise these issues in the media before they are proven. If it is proven, you can raise the issue as every member of society is equal, but you are not allowed to make claims. When such things are mentioned, misunderstandings are the outcome. This can cause misunderstandings for the younger generation. Everyone knows Mr. Hashemi. My relationship with him goes back to before the Islamic Revolution. I have known him for more than 25 years. Mr. Hashemi was one of the main revolutionary figures. He was one of the most active proponents of the revolution, and after the Islamic Revolution, he was one of the main political figures serving the people alongside the late Imam, And after the departure of the founder of the Islamic Revolution he has been alongside the leadership to date. On several occasions, there were assassination attempts on his life. Before the revolution, he donated his possessions for the revolutionary cause. Our youth should know these facts. After the revolution, he had different responsibilities. For eight years, he served as the president and before that he was the parliament speaker. He held other sensitive positions as well. Throughout these years, I am not aware of one incident in which he gathered wealth. These are the facts that everyone should know. In the most sensitive of periods, he served the revolution and the establishment. Of course, my opinion and Mr. Rafsanjani\\\'s differs on numerous issues, which is natural. However, we should not create any misunderstandings for the people. The president and Mr. Rafsanjani have had differences of opinion since the president took office in 2005. They have differences of opinion in foreign policy, in the manner of spreading social justice as well as on some cultural issues. However, the president\\\'s ideas are closer to mine. The same goes fort Mr. Nateq-Nouri. He has also served the revolution, rendered great services for the establishment and there is not a shred of doubt about that. The live televised debates are a positive step, but these shortcomings should be removed. After the debates, I had a talk with the president because I knew he would listen to me. The stance of the Islamic establishment is clear-cut regarding corruption and social justice. Corruption should be fought anywhere it is traced. There is a point I want to make here. We do not claim that our establishment is free of all economic and financial corruption. Yes, there is corruption. If there was no corruption, I would not have written the eight-point letter to the heads of the legislative, judiciary, and executive branches of the country. We have corruption, but the Islamic establishment is one of the healthiest establishments in the world today. However, it is not right to accuse the country of corruption based on some Zionist reports and sources. Moreover, questioning the credibility of statesmen goes beyond the bounds of decency. Financial corruption is an important issue in the Islamic establishment. The judiciary, executive and legislative branches of the country must do everything within their power to fight against it. Everyone is duty-bound to fight corruption. If corruption is not contained, it will spread in the same manner as you see occurring in many countries…. They are struggling with an alarming level of corruption as you have heard about in the UK. This is only a fraction of the scandal as it goes way beyond what has been publicized. Let me summarize the points I made. The Friday election was a historical event, which touched the entire world. Some of our enemies, however, attempted to cast doubt over this absolute and definitive victory. Some even attempted to portray it as a national defeat. They did not want you to enjoy this victory. They did not want to see the highest turnout in the world go down in history in your name. However, it has happened, it has been recorded in history. They cannot manipulate this. The time for rivalry is passed... These four candidates have all fought in the battlefront of this revolution and they are members of this establishment. Forty million people went to the polls and cast their votes for this revolution. It was not only the 24 million votes that went to the president; 40 million votes were cast in favor of the revolution. The people have trust [in the establishment], and all supporters of candidates should rest assured that the Islamic establishment would never betray the people\\\'s trust. In fact, the electoral system of the country does not allow for any vote rigging, which is testified to by all those in charge of the election process. When there is a margin of one hundred thousand or one million at most, then one can doubt that there may have been some form of manipulation or irregularity; however, when there is a difference of eleven million votes, how could any vote rigging have taken place? However, as I have said, and the Guardian Council has accepted, if some people have doubts then it should be dealt with through legal channels. Everything must be dealt strictly though legal channels. I will never accept illegal demands. If the legal frameworks are breached today, then no future election can be guaranteed. In every election, there is only one winner, and of course, some defeated candidates. Complaints, if there is any doubt, should be pursued through legal channels. We have a comprehensive and competent legal system. Just as the candidates have the right to appoint observers, they are given the right to file complaints. I have requested the Guardian Council conduct a partial vote recount in the presence of the candidates and their representatives. We have no problem with this. I want to address the politicians, candidates and political parties at this point. We are at a critical historical juncture. Look at current world affairs, the situation in the Middle East, global economic woes and the situation in our neighboring countries. We are duty bound to remain vigilant and to be careful not to commit mistakes at this critical point in time. In the election, people fulfilled their duty in the best way possible, which was by going to the polls. We have heavier responsibilities on our shoulders now. Those figures who are looked up to by the people and politicians, should be cautious about their words and deeds. If they show any amount of extremist attitude, it will penetrate into the ranks of the people. It may have dangerous consequences and may eventually get out of control. Extremism in society will trigger or fan other extremist moves in the country. If political elites disobey the law and make wrong decisions, they will be held accountable for any violent actions or rioting that ensues. I urge these people, these friends of mine, to exercise restraint and patience. You should see enemy hands at work [against the country]. You should see hungry wolves laying in ambush. They are taking off their masks of diplomacy and are showing their true colors. I urge you to open your eyes and see the enemy. In the past few days the prominent diplomats of some Western countries, which have been dealing with us through diplomatic rhetoric, have removed their masks. Today you can see their true face. They are now showing their enmity toward the Islamic establishment and the most treacherous of them all is Britain. I tell these brothers of ours to think of their responsibility. You are responsible before God. I call on you to remember what Imam has written in his will; the law has the final say. All differences should be settled at the ballot box. This is what elections are for, to let ballot boxes and not the streets determine what the people want. If after every election, the supporters of the candidates who have lost take to streets and the supporters of the candidate who has won respond in the same manner, then what need would we have for elections? Why should the people have to suffer? We should not take to the streets to show off with the number of our supporters to the people. Such acts are not a political issue for those terrorists who take advantage of the situation to hide among the masses in order to carry out their agenda. It is a very good cover for these saboteurs. Who will take responsible for this? Some of the people who were killed in these riots were ordinary people, ordinary Basij members. Who will be held accountable for this? They may start taking advantage of this situation to assassinate Basij members, which will naturally provoke emotional reactions. Who is to be held responsible for this? One is grieved to see them attack religious students at Tehran University dormitories and afterwards chant slogans in support of the leadership. Post-election rivalry on the streets is not the right way to go. It only challenges the election. I want all sides to put an end to this. If they do not stop such actions, then they will be responsible for the repercussions of such incidents. It is also wrong to assume that street riots can be used as leverage to pressure the establishment and to force officials to listen to them for what they believe is in the interest of the country. Giving in to illegal demands under pressure is in itself the beginning of dictatorship. This is a miscalculation and the consequences will be directed at those who orchestrated them. If necessary, I will tell the people about them in due time. I ask all these brothers and friends of mine to act based on friendship and abide by the law. I hope God will help us choose the righteous path. The celebration of 40 million votes should be appreciated and the enemy must not be allowed to ruin the celebration. However, if certain people decide to choose another path, then I will have no choice but to talk with the people more openly. The third group I wish to address are the leaders of the Western media and arrogant powers. In the past two to three weeks, I have heard the words and witnessed the actions of politicians from the United States and certain European countries. Before the elections, they attempted to cast doubt over the election itself so that there would be a low voter turnout. They had their own assessments of results forecasts, but they did not expect the mass participation of the people. They never predicted an 85 percent turnout, or 40 million voters. When they saw the mass turnout, they were shocked. They realized the reality of Iran. They came to understand that they need to adapt themselves to the new situation be it regional, nuclear or internal. When they saw the great popular movement on Election Day, they realized that a new chapter had been opened with regards to Iran and that they must come to terms with it. When some candidates began protesting the results, they felt that there was a change, so they jumped at the chance to ride this wave. Their tone after the election changed on Saturday and Sunday. Their attention shifted to the riots and that was when they gradually began removing their masks. Western officials, their presidents, prime ministers and foreign ministers commented on this situation. The US President said that we were waiting for the day when people would take to the streets. At the same time they write letters saying that they want to have ties and that they respect the Islamic Republic. Which are we to believe? Inside the country, their elements [foreign countries] began street protests and vandalism, they set fire to public property, they made shops and businesses insecure, and they are trying to rob the people of their security. This has nothing to do with the people and their preferred candidates. This kind of behavior stems from ill-wishers, mercenaries and elements working for Western and Zionist secret services. The incidents occurring inside the country have misled some of those outside our borders, who imagine Iran to be the same as Georgia. A Zionist American capitalist a few years ago, had been quoted in the media saying that he had spent 10 million dollars in Georgia to start a velvet revolution. Our nation cannot be compared to any another nation. Their problem is they have not come to know this revolution and its people. American officials say they are worried about the Iranian nation, how can you be worried? Can you even speak about human rights when you are responsible for the blood shed in Afghanistan and Iraq? In Palestine who has and is supporting and funding the Zionist regime? During the term of a previous US government, eighty people affiliated with the Davidian sect were burnt alive in their compound in Waco, Texas. For some reason these people were disliked by the then US administration. Eighty people were burnt in that building, how dare you talk of human rights? In my opinion, these western officials should at least feel a little embarrassment! Supreme Leader Ayatullah Sayyed Ali Khamenei - Friday Prayer Speech - 19Jun09 - English
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Video Tags: Ayatullah Ayatollah Supreme Leader Rahber Sayyed Ali Khamenei Khamnei Khamenie Khamnie Friday Prayer
[ENGLISH] Leader rejects talks with the USA - Full Speech - 7 February 2013
Supreme Leader\\\\\\\'s Speech to Air Force Commanders and Personnel 07/02/2013 The following is the full text of the speech delivered on...
Supreme Leader\\\\\\\'s Speech to Air Force Commanders and Personnel 07/02/2013 The following is the full text of the speech delivered on February 7, 2013 by Ayatollah Khamenei the Supreme Leader of the Islamic Revolution in a meeting with commanders and personnel of the Air Force of the Islamic Republic of Iran Army. The meeting was held on the occasion of the historic pledge of allegiance of Air Force officers to Imam Khomeini (r.a.) on the 19th of Bahman of 1357. In the Name of Allah, the Beneficent, the Merciful I am very happy to meet you - once more at our annual meeting - brothers and dear youth from the glorious Air Force of the Islamic Republic of Iran Army. I welcome all of you. As for the song which was performed, the lyrics were good, the melody was good, the content was good and it was performed well. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, our hearts will always be imbued with the scent of mercy and divine guidance which is the most important source of support. If we take a look at the history of the Air Force of the Islamic Republic of Iran Army over the past years, we will discover a number of important facts. There was one day when we could not and were not allowed to fix the plane parts that we had bought. Today, you build training planes, fighters and flight simulators. You carry out a lot of important radar-related work and you produce complex components. This great movement towards blossoming of talents, love, innovation, self-confidence and self-sufficiency - which has been established in the Air Force, in the entire Islamic Republic of Iran Army, in the Armed Forces and throughout the country - is such a movement that cannot be denied even by the opponents and the enemies of the Islamic Republic. Dominant powers tried to take control of all nations and countries throughout the world by using force, money and weapons and by launching military attacks. They tried to make nations believe that they cannot find the path towards greatness, identity and independence without relying on superpowers and on those who have money and power - Zionist and non-Zionist companies have lined up behind them. You shattered their hopes. Compare the Iranian nation, today, with nations who have been under the domination of American power. See where you are and where they are. With their movement, independence, self-confidence and reliance on God, the Iranian people proved that one can and should stand up against the domination of foreigners and those who seek domination. The Iranian nation has proved this. Thirty years ago, what was the position of the Iranian nation in science, civilization, progress, technology and political influence? What position does it enjoy today? It achieved such a position by putting up a resistance, relying on God and bringing all its capacities into the arena. This is an experience for both the Iranian nation and future generations. It is also an experience for other nations. The Air Force of the Islamic Republic of Iran Army is one of the examples of this resistance and this movement which has been based on self-confidence. We should continue this. We Iranian people should continue this path. This path is full of blessings. For more than thirty years, the enemies of the Iranian nation have done everything in their power to harm the Iranian nation. There is not a thing which they have not used against the Iranian nation. They provoked conflicts, waged wars, supported the enemy of the Islamic Republic with all their power and they waged hard and soft wars. They fought the people of Iran as hard as they could, but our people stood up against them and they resisted. Not only could they not bring our people to their knees and destroy them, but they also failed to prevent them from making progress. Our nation has made progress. They made use of everything they could. They hatched plots, launched coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'états, provoked military conflicts, shot down passenger airliners, imposed heavy sanctions and increased them on a daily basis. They did these things in the hope that our people would lose their hope, disappear from the scene and lose their trust in Islam and the Islamic Republic. But they failed. This is the record of the Islamic Republic. These days, referred to as ten-day Fajr celebrations, are good opportunities for our intellectuals, our youth and all the people of Iran to spend some time evaluating their actions during the past thirty-something years and see their achievements, see their successful efforts, see the divine assistance and see the weakness of the enemies\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\' plots. \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"And they (the unbelievers) planned, and Allah planned, and Allah is the best of planners\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" [The Holy Quran, 3: 54]. This is the general guideline for us, to see how we should choose our future path. You the people in the Air Force should move forward according to this outlook and orientation. Different sectors of the country, all the people and the officials of the country should move forward according to this outlook. Of course, the enemy inflicts harm, but it cannot do anything except for causing slight annoyance. I mentioned a few days ago that Allah the Exalted said, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"They shall by no means harm you but with a slight evil\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" [The Holy Quran, 3: 111]. Their job is to harm you. But, they cannot create obstacles for you and block your path. Over the past 30 years, the Americans have been ranting and raving against the Iranian nation. They said and did whatever they could. They broadcast negative propaganda and they established an evil media empire against the Iranian nation. But the result is this: today, by Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, the Iranian nation has become happier, more determined and more active than ever and it is witnessing more blossoming in different areas. They have been trying to separate the people from the Islamic Republic and the Revolution. Each year on the 22nd of Bahman, the Iranian people frustrate the enemy with their presence in the national and revolutionary rallies. They are trying to separate the people from one another. The previous inexperienced American secretary of state said openly that they are imposing sanctions in order to pit the people of Iran against the Islamic Republic. The people of Iran have always responded to such statements through their rallies and their actions. You will see that on the 22nd of Bahman, the Iranian people will, once more, frustrate their efforts with a crushing move [Audience shout \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Allahu Akbar\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"]. The good thing is that the people are wise and vigilant. They know the purpose of the enemy\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s plot, they predict the enemy\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s moves, they understand why the enemy has adopted a certain policy and they move in the opposite direction. They rely on their own achievements and they will show their presence in this great arena of national dignity. They will show themselves and they will prove their presence. This is the good thing. In the face of the negative propaganda by the enemy - particularly, the Americans and the Zionists - the people do not take the wrong path because of their communal wisdom and they do not make the mistake which the enemy is waiting for. This is the good thing about large-scale issues of our country. Now the Americans have raised the issue of negotiations again. They repeat that America is prepared to directly negotiate with Iran. This is not new. The Americans have repeatedly raised the issue of negotiations at every juncture. Now their newly appointed politicians repeat that we should negotiate. And they say that the ball is in Iran\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s court. The ball is in your court. It is you who should explain the meaning of negotiations that are accompanied by pressure and threats. Negotiations are for the sake of proving one\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s goodwill. You commit tens of acts which show lack of goodwill and then you speak about negotiations. Do you expect the Iranian nation to believe that you have goodwill? Of course, we understand why the Americans repeatedly raise the issue of negotiations and why they speak about it in different ways. We know what the reason is. As the Americans themselves say, their Middle East policies have failed. They need to play their trump card. Their trump card is dragging the Islamic Republic of Iran, which is based on the people, to the negotiating table. They need this. They want to say to the world that they have goodwill. No, we do not see any goodwill. Four years ago - during the early days of the current American administration - when they were saying the same words, I announced that we will not prejudge and we will wait to see what action they will take and then we will judge. Now after four years, how should the Iranian nation judge their actions? They supported the fitna in Iran, they helped those who started the fitna, they sent their troops to Afghanistan under the claim that they were fighting terrorism, they trampled on so many people and they destroyed them. They are also supporting and cooperating with the same terrorists in Syria and they used the same terrorists wherever they could in Iran. Their agents, their allies and Zionist spies openly killed the scientists of the Islamic Republic. They did not even condemn these terrorist activities. [On the contrary] They supported them. This is their record. They imposed sanctions - which they wanted to be crippling - on the Iranian nation. They openly said, crippling. Who do you want to cripple? Did you want to cripple the Iranian nation? Do you have goodwill? Negotiations are meaningful when the two sides negotiate with good intentions and without planning to deceive one another. Negotiations should be on equal terms. Negotiations for the sake of negotiations, tactical negotiations and offer of negotiations as a superpower gesture, are deceptive moves. They are not honest moves. I am not a diplomat. I am a revolutionary. I speak openly and honestly. A diplomat says something, but he actually means something else. We speak openly and honestly. We speak clearly and decisively. Negotiations are meaningful when the two sides show their good intentions. [Negotiations are not meaningful] when one side does not show his good intentions. You yourselves refer to this as pressures and negotiations. These two things are not compatible. You want to point the gun at the people of Iran and say, negotiate or we will shoot. You say these things to intimidate the Iranian nation. You should know that the Iranian nation is not intimidated by these things [Audience shout \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Allahu Akbar\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"]. A number of people become happy about the American offer of negotiations and they say, come and negotiate with us. This is expressed by a number of people who are either simple-minded or who have some ulterior motives. One cannot make definitive judgments about people. But what a simple-minded person does is no different, in essence, from what a person who has ulterior motives does. Negotiations with America will not solve any problems. When did they keep their promises? Over the past 60 years, since the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état of the 28th of Mordad of 1332 until today, the officials of our country have been harmed whenever they trusted the Americans. One day Mosaddeq trusted the Americans, relied on them and considered them as his friends. Then the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état of the 28th of Mordad occurred and the Americans found the opportunity to launch a coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état. The agent responsible for launching the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état came to Tehran with a briefcase full of money and he divided it among thugs and vandals so that they launch the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état. The agent was American. They admitted what the purpose of their plot was. After that, they helped the oppressive Pahlavi regime achieve domination over our country. They established SAVAK and chained and tortured political activists. These are the things they did at that time. During a certain period after the Revolution, the officials of the country trusted them. But the politicians of the American government labeled Iran as \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"axis of evil\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". It is you who are the embodiment of evil. It is you who are doing evil deeds in the world. You wage wars, loot nations and support the Zionist regime. On the issue of Islamic Awakening, you suppress the nations who have risen in revolt as much as you can and you weaken them and pit them against one another. You are evil. Evil is part of your character. They accused the Iranian nation of doing evil acts. This is a big insult. Whenever people trusted them, they made such moves. They should show their good intentions. Negotiations and offer of negotiations are not compatible with pressures. Negotiations and pressures are two different paths. It is not possible for the Iranian nation to accept negotiations under pressures and threats, with those who make threats. What should we negotiate for? Today, the Iranian nation is vigilant. The true face of America has been revealed not only in Iran, but also in the region. Nations distrust America and there are many reasons for this distrust. The Iranian nation has also accurately read the Americans\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\' moves. It understands what their purpose is. Our nation is vigilant. Today, if certain people want to help America re-establish its domination and act against our national interests, against the progress of the country and against the path of independence, they will be held responsible by the people and even if I act against this public demand, the people will complain. It is obvious. All the officials are responsible for safeguarding national interests and preserving national independence. They should preserve the dignity of the Iranian nation. We have negotiated, signed contracts and established relations with countries which have not plotted against Iran. The Iranian nation is peace-loving. The Iranian nation is patient. The unity of the Iranian nation is in line with promoting the interests of humanity. Today, what the Iranian nation does is for the sake of its interests and the interests of the Islamic Ummah and humanity. And undoubtedly, divine assistance is behind the Iranian nation. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, the people of Iran will be able to help not only themselves but also the Islamic Ummah to reach the peak of glory with their wisdom, with their firm determination and with the resistance that they have shown on this bright path, the path that they will continue following in the future as well. The way to reach this glory is to preserve this wisdom. The way to do this is to preserve our unity. The way to do this is for the officials to safeguard the interests of the country. This improper conduct which is witnessed in certain areas from certain government officials - they should end this. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, I will address this issue in the future and I will speak to the people. Our nation is unified, determined and active. Even if there are differences of opinion between the people over different issues, all the officials and all the people join hands against the enemy, global arrogance and those who have prepared themselves to destroy the roots of the people and the Islamic Republic. There is no disagreement among the people over this issue. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor and grace, on the 22nd of Bahman the people will show, once more, that they are present on the scene, that they are prepared, that they are united, that they are moving in the same direction. And undoubtedly, divine blessings will be bestowed on them. Greetings be upon you and Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s mercy and blessings. Source: http://english.khamenei.ir//index.php?option=com_content&task=view&id=1741&Itemid=4
Video Tags: MIRAM Supreme Leader Vali Wali Amr Muslimeen Rahber Rehbar Ayatullah Ayatollah Imam Khamenei Full Speech English Translation Voiceover Reject Rejects Talks Negotiation USA United States America
[07 June 13] Debate: what-s fueling the Islamophobia in the UK? - English
Islamophobic hate crimes have surged in UK following Woolwich incident, from attacks on mosques, racial abuse, assaults to anti-Muslim graffiti....
Islamophobic hate crimes have surged in UK following Woolwich incident, from attacks on mosques, racial abuse, assaults to anti-Muslim graffiti. In this edition of the show we will discuss whether the Woolwich killing will be a catalyst for more negative portrayal of Islam and Muslims, or will authorities and officials use this as a chance to wipe off the negative image of Islam. We will also ask our guests how UK\\\'s foreign policy involving wars in Muslim countries may provoke more acts, threatening UK\\\'s national security.
Video Tags: Presstv Press TV News channel channels Press TV 7 June 2013 Debate what fueling the Islamophobia in the UK
Leader Speaks to University Students - National Day of Fighting Against...
Supreme Leader Meets with Students on National Day of Fighting Against Global Arrogance The following is the full text of the speech delivered...
Supreme Leader Meets with Students on National Day of Fighting Against Global Arrogance The following is the full text of the speech delivered on November 3, 2013 by Ayatollah Khamenei, the Supreme Leader of the Islamic Revolution, in a meeting with high school and university students. The meeting was held on the occasion of the \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"National Day of Fighting Against Global Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". In the Name of Allah, the Beneficent, the Merciful Supreme Leader\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s Speech in Meeting with Students on \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"National Day of Fighting Against Global ArroganToday, the warm presence of you dear youth in this hussainyyah is a reminder and a manifestation of the epic and enthusiasm which has been the supporter and guarantee of the revolutionary movement of the Iranian nation over the course of many years- from the beginning of the Revolution until today. The great blessing of God on our country and on the Islamic Republic is the existence of youth with their clear, strong and reasonable motives, with their pure hearts and with their sincere intentions. Our meeting today has been scheduled on the anniversary of the events of the 13th of Aban which occurred over the course of different years - before and after the victory of the Revolution in the country. There are three events: Imam\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s exile in the year 1343, the ruthless slaughter of students in Tehran in the year 1357 and the courageous movement of students in capturing the Den of Espionage in the year 1358. Each of these three events was in some way related to the government of the United States of America. In the year 1343, Imam (may God bestow paradise on him) was exiled because of his opposition to \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"capitulation\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", which meant preserving the security of American agents in Iran and their judicial immunity. So this event was related to America. In the year 1357, the regime which was dependent on America killed students on the streets of Tehran and the asphalt of these streets was colored with the blood of our teenagers. This was done by America in order to defend the regime which was dependent on it. This event was also related to America. The event in the year 1358 was a counterattack. Our courageous and religious youth attacked the U.S. embassy and discovered the truth and identity of this embassy, which was the Den of Espionage, and presented this fact to people throughout the world. In those days, our youth called the U.S. embassy the \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Den of Espionage\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". Today, after the passage of 30-plus years since that day, the name of U.S. embassies in countries which have the closest relationship with America - that is to say, European countries - has become the den of espionage. This means that our youth are 30 years ahead of the rest of the world. This event was related to America as well. These three events were related, in different ways, to the government of the United States of America and its relations with Iran. Therefore, the 13th of Aban - which is tomorrow - was named \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Day of Fighting Against Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". What does arrogance mean? Arrogance is a Quranic term. The word \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" has been used in the Holy Quran. An arrogant individual, an arrogant government and an arrogant group of people means those individuals and those governments which intend to interfere in the affairs of other human beings and other nations. They interfere in all the affairs of other nations in order to preserve their own interests. They think they are free to do anything and they grant themselves the right to impose different things on other nations and to interfere in the affairs of other countries. And they do not answer to anyone. This is the meaning of arrogance. At the opposite end of the scale, there is a group of people who fight against arrogance. What does fighting against arrogance mean? Primarily, it means refusing to give in to this bullying. The meaning of fighting against arrogance is not convoluted and it is not complicated. Fighting against arrogance means the refusal of a nation to give in to the interference and imposition of an arrogant power, individual or government. This is the meaning of fighting against arrogance. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, when I have the opportunity in the future, I will have a detailed discussion about arrogance and fighting against arrogance with you youth and students. Now, there is no time. This is a brief definition of arrogance and fighting against arrogance. The people of Iran consider themselves to be fighters against arrogance because they have not given in to the imposition of the government of America. The government of America is an arrogant government. It grants itself the right to interfere in the affairs of other countries. It wages wars and it interferes in the affairs of other countries. Today, you see that this goes beyond the borders of Asian, African and Latin American countries. It has reached Europe. They interfere in their affairs as well. The Iranian nation stood against the arrogance that the government of the United States of America showed. It stood against the interference and bullying that it caused and it stood against the domination that it had achieved over our dear country in the course of many years. The taghuti and monarchical regime was a regime which was dependent on America without any domestic support. By relying on America, they did whatever they wanted in Iran. They oppressed the people, they usurped their rights, they practiced discrimination among them. They prevented the country from achieving growth and making progress - which was the natural and historical right of the people - in order to promote the interests of America in Iran. The Iranian nation stood firm and it carried out a revolution. Then, it cut out the roots of arrogant powers in the country. It was not like a number of other countries which confronted arrogance at first, but which left it unfinished. Of course, these countries have received a blow because of this. When I was present in a country - whose name I do not want to mention - which had fought against the English for many years, which had put an end to the oppression of the English by fighting against them and which had achieved independence, I saw that they had put up the statue of an English commander in an important recreational center. I said, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"What is this?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" And this center was named after this arrogant and colonialist commander who had committed thousands of crimes in that country. Of course, they did not gain any benefit from this consideration and compromise. That is to say, this country was and still is under pressure. Compromising and showing leniency towards arrogant powers will bring no benefit for any country. The Islamic Republic of Iran and the great Revolution carried out by the people of Iran confronted American arrogance and it did not leave this task unfinished because it had felt the blow which the Americans had dealt, over many years, on its skin and flesh. It knew who and what these people are. The arrogant outlook which the Americans have adopted and which has continued since decades ago until today has made all nations in the world have a feeling of mistrust and hatred towards the government of America. This is not particular to our country. Any nation which trusted America received a blow, even those who were America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s friends. In our county, Dr. Mosaddeq trusted and relied on the Americans so that he could free himself from the pressure of the English. Instead of helping Dr. Mosaddeq who had trusted them, the Americans allied themselves with the English. They dispatched their agents to our country and they launched the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état of the 28th of Mordad. Mosaddeq trusted them and he received a blow for that. Even those who were on friendly terms with America and who had trusted this country received a blow. The taghuti regime had a very close relationship with America. However, the greed of America exhausted them as well. As I said, they imposed \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"capitulation\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" - the judicial immunity of American agents - on this regime and they had to accept this because they had no other source of support except for America. The meaning of capitulation is that if an American sergeant slaps a high-ranking Iranian officer across the face, no one has the right to sue him. If an American agent of low rank shows transgression towards a honorable Iranian man or woman, no one has the right to sue him. The Americans say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"You do not have the right to do this. We ourselves will resolve the issue\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". A people cannot be humiliated more than this. They imposed this on Iran which was their friend. They did not even show mercy to their friend. They threw Mohammad Reza out of their country after he fled from Iran and spent a short time in America. They did not let him stay. They did not show even this bit of loyalty to him. They are such people. Nations and even governments do not trust America because of this behavior and this attitude which can be seen in its policies. Anyone who trusted America received a blow. Therefore, today, America is the most hated global power among nations. If a fair and healthy public opinion poll is carried out among all nations in the world, I do not think that the negative scores of any nation equal the negative scores of the government of America. Today, this is the condition of America throughout the world. You have heard the statements that the Europeans make against the Americans. Therefore, the issue of fighting against arrogance and \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"The National Day of Fighting against Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" is a fundamental issue, one that is based on correct analyses and statements. And you dear youth and the millions of youth throughout the country, who are students like you, should have a correct analysis of these events. Well, the youth of the early years of the Revolution did not need analyses. Everything was clear to them because they had witnessed everything with their own eyes. They had witnessed the presence of the Americans and SAVAK, which had been taught by the Americans themselves. But today, you should think, analyze and be careful. This should not be only in words. It should be clear why the people of Iran are opposed to arrogance, why they are opposed to the positions of the United States of America and what their hatred results from. Today\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s youth should understand this correctly by carrying out research. Well, I would like to raise a few points about our current issues with America, which is a source of discussions these days. There are certain questions in the minds of the people. First, I want to offer an important and necessary piece of advice: no one should think that the negotiating team of our country have compromised with the other party which includes America - the six governments which are known as the P5+1. This is wrong. They are the agents of the Islamic Republic of Iran. They are our own children and they are the children of the Revolution. They are carrying out a mission and the responsibility which they have undertaken is difficult. They are carrying out this responsibility, which falls on their shoulders, with many efforts. Therefore, one should not weaken and insult them and use certain terms - which we sometimes hear such as the notion that they have compromised with the enemy and other such things - against an agent who is carrying out a task and who is in charge of an affair. The things that are said against them are not true. You should pay attention to the fact that the current negotiations with these six countries - including America - are only about the nuclear issue and nothing else. In the beginning of this year, I said in Holy Mashhad that there is nothing wrong with negotiations on particular issues. But I said that I do not trust these negotiations and I am not optimistic about them. However, if they want to negotiate, they can do it and, by Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, we will not suffer a loss in these negotiations. The Iranian nation benefits from a certain experience which I will refer to in brief. This experience will increase the intellectual capability of our nation, like the experience that we gained in the year 1382 and 1383 in the area of suspending our enrichment activities. At that time, during negotiations with the Europeans, the Islamic Republic agreed to suspend its enrichment activities for a while. But this turned out to be to our benefit. Why? It is because we found out there is no hope at all that our western partners will cooperate with us if we suspend our enrichment activities. If we had not accepted that optional suspension - of course, it was imposed in a way, but we and our officials accepted it anyway - some people would have said, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"If you had retreated a little bit, all problems would have been solved and Iran\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s nuclear file would have become normal\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". That act of suspending our enrichment activities brought us this advantage: it became clear that problems will not be solved by retreating, suspending enrichment activities, postponing our work and cancelling many of our plans and programs. It became clear that the other side is after something else. We noticed this and therefore we started our enrichment activities again. Today, the condition of the Islamic Republic has dramatically changed compared to its condition in the year 1382. In those days, we used to bargain for two, three centrifuges, but today several thousand centrifuges are working. Our youth, our scientists, our researchers and our officials made great efforts and moved things forward. Therefore, we will not suffer a loss as a result of today\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s ongoing negotiations. Of course, as I said, I am not optimistic and I do not think that these negotiations will produce the results which the Iranian nation expects. However, it is an experience. This will broaden and strengthen the experience of the Iranian nation. It is alright to hold these negotiations, but it is necessary for the Iranian nation to be vigilant. We strongly support our officials, who are active in the camp of diplomacy, but our people should be vigilant. They should know what is happening so that some mercenary promoters of the enemy and some promoters who receive no rewards and who further the goals of the enemy out of naivety cannot mislead public opinion. They want to instill the idea into the minds of the people that if we surrender to the other side on the nuclear issue, all economic, financial and other such problems will be solved. This is one of the methods which they use and one of the lies which they spread. They are promoting this idea. Of course, foreign promoters promote such ideas with very efficient methods. Inside the country too, some people promote the idea that if we back down and surrender to the other side on this issue, all economic and other such problems will be solved. Some of these people do it out of naivety without any bad intention and some people promote this idea intentionally. However, this idea is wrong. Why is it wrong? There are a few reasons why it is wrong. I would like you - including the people who are present in this meeting, our wise, well-informed and highly motivated youth and our university and school students throughout the country - to think about such issues. As I once said, you are the officers of the soft war. One reason is that the enmity of America towards the Iranian nation and the Islamic Republic is not at all about the nuclear issue. It is wrong to think that America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s hostility towards us is based on the nuclear issue. This is not the case. The nuclear issue is an excuse. Even long before the nuclear issue - that is to say, since the beginning of the Revolution - these hostilities and oppositions existed. Even if one day the nuclear issue is resolved - imagine that the Islamic Republic retreats, which is the thing they want - you should not think that these hostilities will be over. No, they will gradually make tens of other excuses. For example, they will say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you have missiles?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you have drones?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why are you on unfriendly terms with the Zionist regime?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you not officially recognize the Zionist regime?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you support resistance groups?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" in, as they call it, the Middle East region and why and why and why... The issue is not that they have disagreements with the Islamic Republic about its nuclear program. This is not the case. America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s sanctions began since the beginning of the Revolution and these sanctions increased on a daily basis so much so that today, it has reached a high level. They showed other kinds of hostility as well. They brought down a plane which belonged to the Islamic Republic and they killed 290 humans. During the early years of the Revolution, when the people were still enthusiastic about the victory of the Revolution, they launched the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état based in Shahid Nojeh military base. They launched a coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état against the Revolution and they supported anti-revolutionary elements in different corners of the country. They gave weapons and other such things to the anti-revolutionary camp. This is the same thing that they did in other countries later on. Their enmity is not based on the nuclear issue. The issue is something else. The Iranian nation said no to the requests of America. The Iranian nation said that America cannot do a damn thing against us. The Americans are opposed to the identity of the Islamic Republic. They are opposed to the influence and power of the Islamic Republic. Recently, one of the American politicians and intellectuals said - his speech was broadcast and this is not a confidential issue - that Iran is dangerous, no matter if it is atomic or non-atomic. This person openly said that the influence and power of Iran - as they say, the hegemony of Iran - is dangerous in the region. This is the kind of Iran which enjoys dignity, respect and power today. They are opposed to this kind of Iran. They will be satisfied when Iran becomes a weak, abandoned, isolated, untrustworthy and humiliated nation. Their enmity is not based on the nuclear issue. This is one point. Another point is that in order to solve the economic issues of the country, all our efforts should be focused on domestic issues. The kind of progress and the kind of solution is valuable which is reliant on the domestic power of a nation. If a people rely on their own power and capabilities, they will no more descend into chaos when another country frowns at and imposes sanctions on them. We should solve this. All that we want to say to officials - whether past or present officials - is that they should look at domestic capabilities in order to resolve the issues and the problems of the country including economic problems. We have certain capacities in the country. These capacities - which include human, natural and geographical resources and regional location - should be utilized. Of course, we support diplomatic dynamism. When we say problems should be solved from the inside, this does not mean that we should close our eyes, that we should not benefit from diplomatic dynamism and that we should not interact with the world. Diplomatic dynamism and diplomatic presence are very necessary. The officials who do these things are part of the work, but we should rely on domestic issues. In diplomatic arenas, that country can be successful which relies on its innate power. That government which relies on its innate power and innate capacities can make others accept what it says at the negotiating table and achieve the desired results. Such governments are taken into consideration. An important point which should receive attention in this regard is that we have never become desperate in the face of our enemies during these years and we will never become desperate in the future. During the first decade after the Revolution, particularly during the first years, we did not have access to many material resources. We did not have money, we did not have weapons, we did not have experience, we did not have organization, we did not have competent armed forces and we did not have military equipment. This was while our enemy was at the peak of his power and capability, whether the enemy which fought against us in the arena of war or the enemy which stood behind him - that is to say, the Ba\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'ath regime of Saddam and America, NATO and the Soviet Union of those days. At that time, the Reagan administration was one of the strongest and most powerful governments throughout the world in political and military arenas. This was while we lived in poverty and with difficult conditions, but they could not do anything to us. Today, the situation has changed. Today, the Islamic Republic has weapons. Today, it has money, it has science, it has technology, it has the power to produce, it has international dignity, it has millions of youth who are ready to work and it has millions of talents. Today, we have such a condition. Today, our condition cannot at all be compared with 30 years ago. This is while the situation is the exact opposite of this for the opposing camp. In those days, the Americans were at the peak of their power, but today they are not. Recently, one of the current American government officials, who is a well-known personality, said that today America has reached a point where its friends do not respect it and its enemies do not fear it. It was he who said this, not us. They themselves mention such things. Recently, they have had some political problems. You have seen the disagreement of American politicians about the government\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s budget which shut down the government for 16, 17 days. They sent 800,000 employees on involuntary leave. This is a weakness. This is inefficiency. They have the biggest economic and financial problems. Our problems are nothing compared to their problems. And I will tell you that in the year 2001 or 2002 of the Christian calendar - that is to say, 10, 11 years ago - the financial officials of America made a certain prediction. They predicted that in the year 2011 or 2012, they would have a surplus of 14 trillion dollars. Pay careful attention to this. In 2001, their prediction for 2011 and 2012 was this: they said that in 2011 and 2012 they would have a surplus of 14 trillion dollars. Now, it is 2013, but they have a deficit of around 17 trillion dollars and they do not have any surplus. That is to say, they miscalculated this figure up to 30 trillion dollars. This is their economic condition. This is the way they calculate. This is the condition in the opposing camp. Moreover, as you can see, they have many disagreements. It is mutual interests which have connected them - the Americans and the Europeans - to one another. Otherwise, deep inside, they are on unfriendly terms. The French nation hates the Americans. In different events such as the issue of Syria, the Americans could not establish a partnership with a government which has the closest relationship with them. That is to say, even the English said that we would not take part in this issue. This is while about 40 governments cooperated with them when they attacked Iraq. When they attacked Afghanistan, about 30 governments cooperated with them. The Americans have such a condition in the present time. We have a very good condition. We have made progress, we have become powerful and our people have become a well-informed people. Of course, they exert pressures on us. We should endure and pass through these pressures by relying on our domestic capabilities. This is a wise thing to do and it is being done. Of course, as I said earlier - and I would like to repeat this - we approve of the efforts that the honorable administration and the officials of the country are making. This is an important task and experience and it is most probably a valuable course of action. They can do this. If they achieve results, then so much the better. But if they do not achieve results, this should mean that the county must stand on its own feet in order to solve its problems. I would like to repeat my previous advice: you should not trust the enemy which smiles at you. We would like to offer this advice to our officials, who are our children. Those officials who are working in the arena of diplomacy are our own children and our own youth. This is our advice to them: you should take care not to be misled by a deceptive smile. You should see the subtleties of the enemy\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s plans. Today, the Americans have the most troubles with the deviated Zionist regime- more than any other regime. They have the most consideration for Zionist lobbies. They show consideration for them and we see the situation. The claws of wealthy and powerful Zionist individuals and companies have such domination over the U.S. government and the U.S. Congress that they have to show consideration for them. We do not have to show consideration for the Zionists. Since the first day we said - and we say it today and we will say it in the future - that we consider the Zionist regime to be an illegal and bastard regime. It is a regime which has come into being with conspiracy and it is being preserved and guarded with conspiracy and conspiratorial policies. The Americans show consideration for this regime. The reason why they do this requires another detailed discussion. The money, power and capital of the Zionists is doing a good job and it is influencing these poor Americans. Therefore, the Americans have to show some consideration for them. It is not only the Americans who have such a condition. Many other western politicians, these poor creatures, have the same problem. They too have the same problem. Therefore, our officials should pay attention and they should look at their statements. On the one hand, they smile and they show interest in negotiations and on the other hand, they immediately say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"All options are on the table\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". So what? What move can they make against the Islamic Republic? If they are serious about these negotiations, they should control themselves. They should stop those people who open their mouth to talk nonsense. A certain wealthy American politician had the audacity to say that they should drop an atomic bomb in such and such a desert in Iran and that they should issue such and such threats. Well, they should smash this person\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s mouth! A government which suffers from the delusion that it has a responsibility towards all the issues in the world and a government which considers itself responsible for dealing with the nuclear issue of such and such a country should not dare to issue nuclear threats - particularly in such a time - against a country with such a good condition. They should stop those people who talk nonsense. Anyway, our people are, thankfully, a vigilant people and our officials are the officials of these people. They too are vigilant and they pay full attention. We support whatever action which is to the benefit of the country and we support, help and pray for those officials who carry out such actions. But we also advise both the people and officials - particularly you dear youth - to open your eyes and ears. Any nation can achieve its lofty goals with wisdom, vigilance and watchfulness. And I hope that, by Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, the prayers of the Imam of the Age (may our souls be sacrificed for his sake) will be a source of support for you and I hope that the immaculate soul of Imam and the pure souls of our martyrs pray for you. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, you youth will take the country in your own hands with your enthusiastic spirits and I hope that it will be your turn to reach the peaks with your innovation. Greetings be upon you and Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s mercy and blessings Source: http://english.khamenei.ir//index.php?option=com_content&task=view&id=1837&Itemid=4
Video Tags: Supreme Leader Vali Wali Amr Muslimeen Rahber Rehbar Ayatullah Ayatollah Khamenei Khamnei Speech English Translation Subtitle Subtitles Islamic Unity Muslim United Ummah Shia Sunni Against Zionism
[English] [03 Nov 13] Speech to Students on National Day of Fighting...
The following is the full text of the speech delivered on November 3, 2013 by Ayatollah Khamenei, the Supreme Leader of the Islamic Revolution, in...
The following is the full text of the speech delivered on November 3, 2013 by Ayatollah Khamenei, the Supreme Leader of the Islamic Revolution, in a meeting with high school and university students. The meeting was held on the occasion of the \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"National Day of Fighting Against Global Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". In the Name of Allah, the Beneficent, the Merciful Supreme Leader\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s Speech in Meeting with Students on \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"National Day of Fighting Against Global ArroganToday, the warm presence of you dear youth in this hussainyyah is a reminder and a manifestation of the epic and enthusiasm which has been the supporter and guarantee of the revolutionary movement of the Iranian nation over the course of many years- from the beginning of the Revolution until today. The great blessing of God on our country and on the Islamic Republic is the existence of youth with their clear, strong and reasonable motives, with their pure hearts and with their sincere intentions. Our meeting today has been scheduled on the anniversary of the events of the 13th of Aban which occurred over the course of different years - before and after the victory of the Revolution in the country. There are three events: Imam\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s exile in the year 1343, the ruthless slaughter of students in Tehran in the year 1357 and the courageous movement of students in capturing the Den of Espionage in the year 1358. Each of these three events was in some way related to the government of the United States of America. In the year 1343, Imam (may God bestow paradise on him) was exiled because of his opposition to \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"capitulation\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", which meant preserving the security of American agents in Iran and their judicial immunity. So this event was related to America. In the year 1357, the regime which was dependent on America killed students on the streets of Tehran and the asphalt of these streets was colored with the blood of our teenagers. This was done by America in order to defend the regime which was dependent on it. This event was also related to America. The event in the year 1358 was a counterattack. Our courageous and religious youth attacked the U.S. embassy and discovered the truth and identity of this embassy, which was the Den of Espionage, and presented this fact to people throughout the world. In those days, our youth called the U.S. embassy the \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Den of Espionage\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". Today, after the passage of 30-plus years since that day, the name of U.S. embassies in countries which have the closest relationship with America - that is to say, European countries - has become the den of espionage. This means that our youth are 30 years ahead of the rest of the world. This event was related to America as well. These three events were related, in different ways, to the government of the United States of America and its relations with Iran. Therefore, the 13th of Aban - which is tomorrow - was named \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"Day of Fighting Against Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". What does arrogance mean? Arrogance is a Quranic term. The word \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" has been used in the Holy Quran. An arrogant individual, an arrogant government and an arrogant group of people means those individuals and those governments which intend to interfere in the affairs of other human beings and other nations. They interfere in all the affairs of other nations in order to preserve their own interests. They think they are free to do anything and they grant themselves the right to impose different things on other nations and to interfere in the affairs of other countries. And they do not answer to anyone. This is the meaning of arrogance. At the opposite end of the scale, there is a group of people who fight against arrogance. What does fighting against arrogance mean? Primarily, it means refusing to give in to this bullying. The meaning of fighting against arrogance is not convoluted and it is not complicated. Fighting against arrogance means the refusal of a nation to give in to the interference and imposition of an arrogant power, individual or government. This is the meaning of fighting against arrogance. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, when I have the opportunity in the future, I will have a detailed discussion about arrogance and fighting against arrogance with you youth and students. Now, there is no time. This is a brief definition of arrogance and fighting against arrogance. The people of Iran consider themselves to be fighters against arrogance because they have not given in to the imposition of the government of America. The government of America is an arrogant government. It grants itself the right to interfere in the affairs of other countries. It wages wars and it interferes in the affairs of other countries. Today, you see that this goes beyond the borders of Asian, African and Latin American countries. It has reached Europe. They interfere in their affairs as well. The Iranian nation stood against the arrogance that the government of the United States of America showed. It stood against the interference and bullying that it caused and it stood against the domination that it had achieved over our dear country in the course of many years. The taghuti and monarchical regime was a regime which was dependent on America without any domestic support. By relying on America, they did whatever they wanted in Iran. They oppressed the people, they usurped their rights, they practiced discrimination among them. They prevented the country from achieving growth and making progress - which was the natural and historical right of the people - in order to promote the interests of America in Iran. The Iranian nation stood firm and it carried out a revolution. Then, it cut out the roots of arrogant powers in the country. It was not like a number of other countries which confronted arrogance at first, but which left it unfinished. Of course, these countries have received a blow because of this. When I was present in a country - whose name I do not want to mention - which had fought against the English for many years, which had put an end to the oppression of the English by fighting against them and which had achieved independence, I saw that they had put up the statue of an English commander in an important recreational center. I said, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"What is this?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" And this center was named after this arrogant and colonialist commander who had committed thousands of crimes in that country. Of course, they did not gain any benefit from this consideration and compromise. That is to say, this country was and still is under pressure. Compromising and showing leniency towards arrogant powers will bring no benefit for any country. The Islamic Republic of Iran and the great Revolution carried out by the people of Iran confronted American arrogance and it did not leave this task unfinished because it had felt the blow which the Americans had dealt, over many years, on its skin and flesh. It knew who and what these people are. The arrogant outlook which the Americans have adopted and which has continued since decades ago until today has made all nations in the world have a feeling of mistrust and hatred towards the government of America. This is not particular to our country. Any nation which trusted America received a blow, even those who were America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s friends. In our county, Dr. Mosaddeq trusted and relied on the Americans so that he could free himself from the pressure of the English. Instead of helping Dr. Mosaddeq who had trusted them, the Americans allied themselves with the English. They dispatched their agents to our country and they launched the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état of the 28th of Mordad. Mosaddeq trusted them and he received a blow for that. Even those who were on friendly terms with America and who had trusted this country received a blow. The taghuti regime had a very close relationship with America. However, the greed of America exhausted them as well. As I said, they imposed \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"capitulation\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" - the judicial immunity of American agents - on this regime and they had to accept this because they had no other source of support except for America. The meaning of capitulation is that if an American sergeant slaps a high-ranking Iranian officer across the face, no one has the right to sue him. If an American agent of low rank shows transgression towards a honorable Iranian man or woman, no one has the right to sue him. The Americans say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"You do not have the right to do this. We ourselves will resolve the issue\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". A people cannot be humiliated more than this. They imposed this on Iran which was their friend. They did not even show mercy to their friend. They threw Mohammad Reza out of their country after he fled from Iran and spent a short time in America. They did not let him stay. They did not show even this bit of loyalty to him. They are such people. Nations and even governments do not trust America because of this behavior and this attitude which can be seen in its policies. Anyone who trusted America received a blow. Therefore, today, America is the most hated global power among nations. If a fair and healthy public opinion poll is carried out among all nations in the world, I do not think that the negative scores of any nation equal the negative scores of the government of America. Today, this is the condition of America throughout the world. You have heard the statements that the Europeans make against the Americans. Therefore, the issue of fighting against arrogance and \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"The National Day of Fighting against Arrogance\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" is a fundamental issue, one that is based on correct analyses and statements. And you dear youth and the millions of youth throughout the country, who are students like you, should have a correct analysis of these events. Well, the youth of the early years of the Revolution did not need analyses. Everything was clear to them because they had witnessed everything with their own eyes. They had witnessed the presence of the Americans and SAVAK, which had been taught by the Americans themselves. But today, you should think, analyze and be careful. This should not be only in words. It should be clear why the people of Iran are opposed to arrogance, why they are opposed to the positions of the United States of America and what their hatred results from. Today\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s youth should understand this correctly by carrying out research. Well, I would like to raise a few points about our current issues with America, which is a source of discussions these days. There are certain questions in the minds of the people. First, I want to offer an important and necessary piece of advice: no one should think that the negotiating team of our country have compromised with the other party which includes America - the six governments which are known as the P5+1. This is wrong. They are the agents of the Islamic Republic of Iran. They are our own children and they are the children of the Revolution. They are carrying out a mission and the responsibility which they have undertaken is difficult. They are carrying out this responsibility, which falls on their shoulders, with many efforts. Therefore, one should not weaken and insult them and use certain terms - which we sometimes hear such as the notion that they have compromised with the enemy and other such things - against an agent who is carrying out a task and who is in charge of an affair. The things that are said against them are not true. You should pay attention to the fact that the current negotiations with these six countries - including America - are only about the nuclear issue and nothing else. In the beginning of this year, I said in Holy Mashhad that there is nothing wrong with negotiations on particular issues. But I said that I do not trust these negotiations and I am not optimistic about them. However, if they want to negotiate, they can do it and, by Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, we will not suffer a loss in these negotiations. The Iranian nation benefits from a certain experience which I will refer to in brief. This experience will increase the intellectual capability of our nation, like the experience that we gained in the year 1382 and 1383 in the area of suspending our enrichment activities. At that time, during negotiations with the Europeans, the Islamic Republic agreed to suspend its enrichment activities for a while. But this turned out to be to our benefit. Why? It is because we found out there is no hope at all that our western partners will cooperate with us if we suspend our enrichment activities. If we had not accepted that optional suspension - of course, it was imposed in a way, but we and our officials accepted it anyway - some people would have said, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"If you had retreated a little bit, all problems would have been solved and Iran\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s nuclear file would have become normal\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". That act of suspending our enrichment activities brought us this advantage: it became clear that problems will not be solved by retreating, suspending enrichment activities, postponing our work and cancelling many of our plans and programs. It became clear that the other side is after something else. We noticed this and therefore we started our enrichment activities again. Today, the condition of the Islamic Republic has dramatically changed compared to its condition in the year 1382. In those days, we used to bargain for two, three centrifuges, but today several thousand centrifuges are working. Our youth, our scientists, our researchers and our officials made great efforts and moved things forward. Therefore, we will not suffer a loss as a result of today\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s ongoing negotiations. Of course, as I said, I am not optimistic and I do not think that these negotiations will produce the results which the Iranian nation expects. However, it is an experience. This will broaden and strengthen the experience of the Iranian nation. It is alright to hold these negotiations, but it is necessary for the Iranian nation to be vigilant. We strongly support our officials, who are active in the camp of diplomacy, but our people should be vigilant. They should know what is happening so that some mercenary promoters of the enemy and some promoters who receive no rewards and who further the goals of the enemy out of naivety cannot mislead public opinion. They want to instill the idea into the minds of the people that if we surrender to the other side on the nuclear issue, all economic, financial and other such problems will be solved. This is one of the methods which they use and one of the lies which they spread. They are promoting this idea. Of course, foreign promoters promote such ideas with very efficient methods. Inside the country too, some people promote the idea that if we back down and surrender to the other side on this issue, all economic and other such problems will be solved. Some of these people do it out of naivety without any bad intention and some people promote this idea intentionally. However, this idea is wrong. Why is it wrong? There are a few reasons why it is wrong. I would like you - including the people who are present in this meeting, our wise, well-informed and highly motivated youth and our university and school students throughout the country - to think about such issues. As I once said, you are the officers of the soft war. One reason is that the enmity of America towards the Iranian nation and the Islamic Republic is not at all about the nuclear issue. It is wrong to think that America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s hostility towards us is based on the nuclear issue. This is not the case. The nuclear issue is an excuse. Even long before the nuclear issue - that is to say, since the beginning of the Revolution - these hostilities and oppositions existed. Even if one day the nuclear issue is resolved - imagine that the Islamic Republic retreats, which is the thing they want - you should not think that these hostilities will be over. No, they will gradually make tens of other excuses. For example, they will say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you have missiles?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you have drones?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why are you on unfriendly terms with the Zionist regime?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you not officially recognize the Zionist regime?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\", \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"why do you support resistance groups?\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\" in, as they call it, the Middle East region and why and why and why... The issue is not that they have disagreements with the Islamic Republic about its nuclear program. This is not the case. America\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s sanctions began since the beginning of the Revolution and these sanctions increased on a daily basis so much so that today, it has reached a high level. They showed other kinds of hostility as well. They brought down a plane which belonged to the Islamic Republic and they killed 290 humans. During the early years of the Revolution, when the people were still enthusiastic about the victory of the Revolution, they launched the coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état based in Shahid Nojeh military base. They launched a coup d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'état against the Revolution and they supported anti-revolutionary elements in different corners of the country. They gave weapons and other such things to the anti-revolutionary camp. This is the same thing that they did in other countries later on. Their enmity is not based on the nuclear issue. The issue is something else. The Iranian nation said no to the requests of America. The Iranian nation said that America cannot do a damn thing against us. The Americans are opposed to the identity of the Islamic Republic. They are opposed to the influence and power of the Islamic Republic. Recently, one of the American politicians and intellectuals said - his speech was broadcast and this is not a confidential issue - that Iran is dangerous, no matter if it is atomic or non-atomic. This person openly said that the influence and power of Iran - as they say, the hegemony of Iran - is dangerous in the region. This is the kind of Iran which enjoys dignity, respect and power today. They are opposed to this kind of Iran. They will be satisfied when Iran becomes a weak, abandoned, isolated, untrustworthy and humiliated nation. Their enmity is not based on the nuclear issue. This is one point. Another point is that in order to solve the economic issues of the country, all our efforts should be focused on domestic issues. The kind of progress and the kind of solution is valuable which is reliant on the domestic power of a nation. If a people rely on their own power and capabilities, they will no more descend into chaos when another country frowns at and imposes sanctions on them. We should solve this. All that we want to say to officials - whether past or present officials - is that they should look at domestic capabilities in order to resolve the issues and the problems of the country including economic problems. We have certain capacities in the country. These capacities - which include human, natural and geographical resources and regional location - should be utilized. Of course, we support diplomatic dynamism. When we say problems should be solved from the inside, this does not mean that we should close our eyes, that we should not benefit from diplomatic dynamism and that we should not interact with the world. Diplomatic dynamism and diplomatic presence are very necessary. The officials who do these things are part of the work, but we should rely on domestic issues. In diplomatic arenas, that country can be successful which relies on its innate power. That government which relies on its innate power and innate capacities can make others accept what it says at the negotiating table and achieve the desired results. Such governments are taken into consideration. An important point which should receive attention in this regard is that we have never become desperate in the face of our enemies during these years and we will never become desperate in the future. During the first decade after the Revolution, particularly during the first years, we did not have access to many material resources. We did not have money, we did not have weapons, we did not have experience, we did not have organization, we did not have competent armed forces and we did not have military equipment. This was while our enemy was at the peak of his power and capability, whether the enemy which fought against us in the arena of war or the enemy which stood behind him - that is to say, the Ba\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'ath regime of Saddam and America, NATO and the Soviet Union of those days. At that time, the Reagan administration was one of the strongest and most powerful governments throughout the world in political and military arenas. This was while we lived in poverty and with difficult conditions, but they could not do anything to us. Today, the situation has changed. Today, the Islamic Republic has weapons. Today, it has money, it has science, it has technology, it has the power to produce, it has international dignity, it has millions of youth who are ready to work and it has millions of talents. Today, we have such a condition. Today, our condition cannot at all be compared with 30 years ago. This is while the situation is the exact opposite of this for the opposing camp. In those days, the Americans were at the peak of their power, but today they are not. Recently, one of the current American government officials, who is a well-known personality, said that today America has reached a point where its friends do not respect it and its enemies do not fear it. It was he who said this, not us. They themselves mention such things. Recently, they have had some political problems. You have seen the disagreement of American politicians about the government\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s budget which shut down the government for 16, 17 days. They sent 800,000 employees on involuntary leave. This is a weakness. This is inefficiency. They have the biggest economic and financial problems. Our problems are nothing compared to their problems. And I will tell you that in the year 2001 or 2002 of the Christian calendar - that is to say, 10, 11 years ago - the financial officials of America made a certain prediction. They predicted that in the year 2011 or 2012, they would have a surplus of 14 trillion dollars. Pay careful attention to this. In 2001, their prediction for 2011 and 2012 was this: they said that in 2011 and 2012 they would have a surplus of 14 trillion dollars. Now, it is 2013, but they have a deficit of around 17 trillion dollars and they do not have any surplus. That is to say, they miscalculated this figure up to 30 trillion dollars. This is their economic condition. This is the way they calculate. This is the condition in the opposing camp. Moreover, as you can see, they have many disagreements. It is mutual interests which have connected them - the Americans and the Europeans - to one another. Otherwise, deep inside, they are on unfriendly terms. The French nation hates the Americans. In different events such as the issue of Syria, the Americans could not establish a partnership with a government which has the closest relationship with them. That is to say, even the English said that we would not take part in this issue. This is while about 40 governments cooperated with them when they attacked Iraq. When they attacked Afghanistan, about 30 governments cooperated with them. The Americans have such a condition in the present time. We have a very good condition. We have made progress, we have become powerful and our people have become a well-informed people. Of course, they exert pressures on us. We should endure and pass through these pressures by relying on our domestic capabilities. This is a wise thing to do and it is being done. Of course, as I said earlier - and I would like to repeat this - we approve of the efforts that the honorable administration and the officials of the country are making. This is an important task and experience and it is most probably a valuable course of action. They can do this. If they achieve results, then so much the better. But if they do not achieve results, this should mean that the county must stand on its own feet in order to solve its problems. I would like to repeat my previous advice: you should not trust the enemy which smiles at you. We would like to offer this advice to our officials, who are our children. Those officials who are working in the arena of diplomacy are our own children and our own youth. This is our advice to them: you should take care not to be misled by a deceptive smile. You should see the subtleties of the enemy\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s plans. Today, the Americans have the most troubles with the deviated Zionist regime- more than any other regime. They have the most consideration for Zionist lobbies. They show consideration for them and we see the situation. The claws of wealthy and powerful Zionist individuals and companies have such domination over the U.S. government and the U.S. Congress that they have to show consideration for them. We do not have to show consideration for the Zionists. Since the first day we said - and we say it today and we will say it in the future - that we consider the Zionist regime to be an illegal and bastard regime. It is a regime which has come into being with conspiracy and it is being preserved and guarded with conspiracy and conspiratorial policies. The Americans show consideration for this regime. The reason why they do this requires another detailed discussion. The money, power and capital of the Zionists is doing a good job and it is influencing these poor Americans. Therefore, the Americans have to show some consideration for them. It is not only the Americans who have such a condition. Many other western politicians, these poor creatures, have the same problem. They too have the same problem. Therefore, our officials should pay attention and they should look at their statements. On the one hand, they smile and they show interest in negotiations and on the other hand, they immediately say, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"All options are on the table\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\". So what? What move can they make against the Islamic Republic? If they are serious about these negotiations, they should control themselves. They should stop those people who open their mouth to talk nonsense. A certain wealthy American politician had the audacity to say that they should drop an atomic bomb in such and such a desert in Iran and that they should issue such and such threats. Well, they should smash this person\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s mouth! A government which suffers from the delusion that it has a responsibility towards all the issues in the world and a government which considers itself responsible for dealing with the nuclear issue of such and such a country should not dare to issue nuclear threats - particularly in such a time - against a country with such a good condition. They should stop those people who talk nonsense. Anyway, our people are, thankfully, a vigilant people and our officials are the officials of these people. They too are vigilant and they pay full attention. We support whatever action which is to the benefit of the country and we support, help and pray for those officials who carry out such actions. But we also advise both the people and officials - particularly you dear youth - to open your eyes and ears. Any nation can achieve its lofty goals with wisdom, vigilance and watchfulness. And I hope that, by Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, the prayers of the Imam of the Age (may our souls be sacrificed for his sake) will be a source of support for you and I hope that the immaculate soul of Imam and the pure souls of our martyrs pray for you. By Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s favor, you youth will take the country in your own hands with your enthusiastic spirits and I hope that it will be your turn to reach the peaks with your innovation. Greetings be upon you and Allah\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s mercy and blessings
Video Tags: Rahber Rehbar Ayatullah Ayatollah Sayyed Ali Khamenei Leader Wali Vali Amr Muslimeen Faqih Khamnei Imam Shia Sunni Islam Islamic Unity Conference Prophet Muhammad Mohammad eid fitr adha gadheer
[Current Affairs] The Brevity of Social Media | Importance of Reading |...
- Today\'s instant communication, and social media, has both positive and negative implications for society - One of the negative implications is...
- Today\'s instant communication, and social media, has both positive and negative implications for society - One of the negative implications is the populations adversity to reading, looking up facts, or concentrating on content for anything beyond a short period - Everyone has become used to \'sound bites\', quick clips, and short messages - 25% of students have been impacted when it comes to reading according to one study - Looking at the emphasis on learning in Islam, as demonstrated and stated by the Prophet - Always ensure to verify information you come across, do not take it for granted at face value - The importance of inculcating the habit of reading in your children from a young age - An example of this from Imam Hasan - Discussing the unfortunate terrorist attacks on the shrine of Shah Charagh, Ahmed bin Musa al-Kazim, in Shiraz - Friday Juma Khutba - October 28th, 2022
Video Tags: Current Affairs Brevity Social Media Importance Reading Shrine Attacks Shiraz Iran Friday Sermon Maulana Syed Rizvi
The Dark Side of Globalization - Noam Chomsky - English
Americas top public intellectual Noam Chomsky discusses the dark side of globalization. Chomsky argues that globalization itself is not a bad...
Americas top public intellectual Noam Chomsky discusses the dark side of globalization. Chomsky argues that globalization itself is not a bad thing. It is the specific form it has taken - under the influence of big powers and multi-nationals to benefit only a few - that makes it negative - English
Video Tags: Globalization Chomsky Materialism Greed Capitalism Consumerism Social Justice Exploitation Inequality English
The Corporation - Part 17 of 23 - Unsettling Accounts - Eng
17. Journalists Jane Akre and Steve Wilson were fired by the Fox News television station they work for after refusing to change their investigative...
17. Journalists Jane Akre and Steve Wilson were fired by the Fox News television station they work for after refusing to change their investigative report on Posilac - a Bovine Growth Hormone BGH made by Monsanto. Their research documents potential health and safety problems of drinking milk treated with the synthetic hormone - but threatened with legal action from Monsanto - Fox wants the negative effects played down. The court eventually throws out Akres whistle blower lawsuit after deciding that the media is allowed to lie.
Video Tags: Materialism Capitalism Consumerism Profit Greed Marketing Perception FDA Justice Health US English
A tribute to Sayyeda Zainab (s.a) - Persian
Peace be upon you. Oh daughter of the Commander of the faithful a.s. Peace be upon you Oh daughter of Fatima a.s. Two Worlds Radiant Lady s.a....
Peace be upon you. Oh daughter of the Commander of the faithful a.s. Peace be upon you Oh daughter of Fatima a.s. Two Worlds Radiant Lady s.a. Peace be upon you. We speak of Hazrat Zainab s.a. She well conveyed the message of Karbala s uprising and tragedy to the world and today it is a great honor for Muslims to follow this great lady as an example in their lives. Truly when an individual spends his or her life in support of truth and establishment of divine values - his or her name will remain eternal in history and his or her lifestyle will inspire the future generations. That is why the name of Zainab s.a. and her lifestyle has influenced many people - even non Muslims. Ms. Carin from Germany - after embracing Islam - chose Zainab s.a. as her first name. Regarding her incentive in choosing this name she says - I was always interested in the life story of the holy Prophet of Islam s.a.w.a.w and his household. I was specially affected by the story of uprising of Imam Hussein a.s. in Karbala and the personality of Hazrat Zainab s.a. in the events of Karbala. The self-sacrifices of that great lady while bearing the heavy grief of martyrdom of her brother and other companions are praiseworthy. Hazrat Zainab s.a. was a lady of knowledge virtue bravery and patience.These characteristics helped her carry out her mission in Karbala uprising in the company of Imam Hussein a.s. Hazrat Zainab s lifestyle contains instructive points for me and whenever I come across a problem I immediately remember Her. For her name brings me tranquility. That is why I adopted the name of Zainab after embracing Islam. Hazrat Zainab s.a. was brought up under instructions of a father like Imam Ali a.s. and a mother like Fatema s.a. and she benefited greatly from these two personalities. Historians say Hazrat Zainab inherited bravery honesty and fluency in speech from her father Imam Ali a.s. and kindness and loyalty from her mother Hazrat Fatema s.a. She made great efforts in giving religious training to the women of her time and thus many women at that time benefited from Hazrat Zainab s knowledge and teachings. Hazrat Zainab s.a. spent her fruitful life in promoting monotheism and assuming divine responsibilities and her heroic participation in the greatest epic of history - namely the uprising of Imam Hussein a.s. in Karbala brought her a lasting name. You wont find a single person who is familiar with Karbala tragedy and has not heard of Hazrat Zainab s.a. Zainab s.a. conveyed the message of Imam Hussein s uprising to territories beyond Karbala desert. She encouraged the warriors and sympathized with the injured. She was so bound to the fulfillment of her religious obligations that even at those critical conditions she did not forget to say her prayers at night. Imam Sajjad a.s. says - I saw my aunt Hazrat Zainab s.a. saying her prayers while sitting. A major part of Hazrat Zainab s mission started when Karbala tragedy apparently ended with the martyrdom of Imam Hussein a.s. This chapter of Hazrat Zainab s life began with conveying the message of Ashura in which she heroically defended the rights of the household - ahlul bait - and did not permit the enemies to take advantage of Karbala tragedy. In this tragedy the Omayyud ruler Yazid l.a. and his followers l.a. thought that they had succeeded in creating a negative image of the holy Prophet s household but when the caravan of the household members now in captivity arrived in Kufa - Hazrat Zainab s.a. behaved in such a way that she managed to turn the so-called victory of the Omayyud into a defeat. In her famous address She changed the mentality of the people of Kufa and Damascus. She symbolized a shining sun who reflected light into everything. Although Hazrat Zainab s.a. did not live long after Karbala tragedy she sowed the seeds of awareness in the entire Islamic society. This great lady passed away in 62 AH after she made lots of efforts in the path of truth. Once again we offer our condolences and bring you an excerpt from her historic statement addressed to Yazid. She says - O Yazid practice any trick you can and do anything that you think would vanish Islam. But you should know that you can not eradicate our message our path and our remembrance. You should know that our remembrance will never die. MAY OUR LIVES BE SACRIFICED FOR HAZRAT ZAINAB s.a.
AliAli
Video Tags: AliAli Al Qaim Sayyeda Zainab Zeinab Revolution Awareness Message Karbala
President Ahmadinejad Interview Sept 08 with Democracy Now - Part 1 -...
Iranian President Mahmoud Ahmadinejad on the Threat of US Attack and International Criticism of Iran's Human Rights Record In part one of an...
Iranian President Mahmoud Ahmadinejad on the Threat of US Attack and International Criticism of Iran's Human Rights Record In part one of an interview with Amy Goodman and Juan Gonzalez, Iranian President Mahmoud Ahmadinejad talks about the threat of a US attack on Iran and responds to international criticism of Iran's human rights record. We also get reaction from CUNY Professor Ervand Abrahamian, an Iran expert and author of several books on Iran. JUAN GONZALEZ: Iranian President Mahmoud Ahmadinejad addressed the United Nations General Assembly this week, while the International Atomic Energy Agency, or IAEA, is meeting in Vienna to discuss Iran's alleged nuclear program. An IAEA report earlier this month criticized Iran for failing to fully respond to questions about its nuclear activities. The European Union told the IAEA Wednesday that it believes Iran is moving closer to being able to arm a nuclear warhead. Iran could face a fourth set of Security Council sanctions over its nuclear activities, but this week Russia has refused to meet with the US on this issue. The Iranian president refuted the IAEA's charges in his speech to the General Assembly and accused the agency of succumbing to political pressure. He also welcomed talks with the United States if it cuts back threats to use military force against Iran. AMY GOODMAN: As with every visit of the Iranian president to New York, some groups protested outside the United Nations. But this year, President Ahmadinejad also met with a large delegation of American peace activists concerned with the escalating possibility of war with Iran. Well, yesterday, just before their meeting, Juan Gonzalez and I sat down with the Iranian president at his hotel, blocks from the UN, for a wide-ranging discussion about US-Iran relations, Iran's nuclear program, threat of war with the US, the Israel-Palestine conflict, human rights in Iran and much more. Today, part one of our interview with the Iranian president. AMY GOODMAN: Welcome to Democracy Now!, President Ahmadinejad. You've come to the United States. What is your message to people in the United States and to the world community at the UN? PRESIDENT MAHMOUD AHMADINEJAD: [translated] In the name of God, the compassion of the Merciful, the president started by reciting verses from the Holy Quran in Arabic. Hello. Hello to the people of America. The message from the nation and people of Iran is one of peace, tranquility and brotherhood. We believe that viable peace and security can happen when it is based on justice and piety and purity. Otherwise, no peace will occur. JUAN GONZALEZ: Mr. President, you're faced now in Iran with American soldiers in Iraq to your west, with American soldiers and NATO troops to your east in Afghanistan, and with Blackwater, the notorious military contractor, training the military in Azerbaijan, another neighbor of yours. What is the effect on your country of this enormous presence of American forces around Iran and the impact of these wars on your own population? PRESIDENT MAHMOUD AHMADINEJAD: [translated] It's quite natural that when there are wars around your borders, it brings about negative repercussions for the entire region. These days, insecurity cannot be bordered; it just extends beyond boundaries. In the past two years, we had several cases of bomb explosions in southern towns in Iran carried out by people who were supervised by the occupying forces in our neighborhood. And in Afghanistan, following the presence of NATO troops, the production of illicit drugs has multiplied. It's natural that it basically places pressure on Iran, including costly ones in order to fight the flow of illicit drugs. We believe the people in the region are able to establish security themselves, on their own, so there is no need for foreigners and external forces, because these external forces have not helped the security of the region. AMY GOODMAN: Do you see them as a threat to you? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, it's natural that when there is insecurity, it threatens everyone. JUAN GONZALEZ: I'd like to turn for a moment to your domestic policies and law enforcement in your country. Human Rights Watch, which has often criticized the legal system in the United States, says that, under your presidency, there has been a great expansion in the scope and the number of individuals and activities persecuted by the government. They say that you've jailed teachers who are fighting for wages and better pensions, students and activists working for reform, and other labor leaders, like Mansour Ossanlou from the bus workers' union. What is your response to these criticisms of your policies? PRESIDENT MAHMOUD AHMADINEJAD: [translated] I think that the human rights situation in Iran is relatively a good one, when compared to the United States and other countries. Of course, when we look at the ideals that are dear to us, we understand that we still need to do a lot, because we seek divine and religious ideals and revolutionary ones. But when we compare ourselves with some European countries and the United States, we feel we're in a much better place. A large part of the information that these groups receive come from criticisms coming from groups that oppose the government. If you look at it, we have elections in Iran every year. And the propaganda is always around, too. But they're not always true. Groups accuse one another. But within the region and compared to the United States, we have the smallest number of prisoners, because in Iran, in general, there is not so much inclination to imprison people. We're actually looking at our existing laws right now to see how we can eliminate most prisons around the country. So, you can see that people in Iran like each other. They live coexistently and like the government, too. This news is more important to these groups, not so much for the Iranian people. You have to remember, we have over 70 million people in our country, and we have laws. Some people might violate it, and then, according to the law, the judiciary takes charge. And this happens everywhere. What really matters is that in the end there are the least amount of such violations of the law in Iran, the least number. So, I think the interpretation of these events is a wrong one. The relationship between the people and the government in Iran is actually a very close one. And criticizing the government is absolutely free for all. That's exactly why everyone says what they want. There's really no restrictions. It doesn't necessarily mean that everything you hear is always true. And the government doesn't really respond to it, either. It's just free. JUAN GONZALEZ: Let me ask you in particular about the question of the execution of juveniles. My understanding is that Iran is one of only five or six nations in the world that still execute juveniles convicted of capital offenses and that you—by far, you execute the most. I think twenty-six of the last thirty-two juveniles executed in the world were executed in Iran. How is this a reflection of the—of a state guided by religious principles, to execute young people? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Firstly, nobody is executed under the age of eighteen in Iran. This is the first point. And then, please pay attention to the fact that the legal age in Iran is different from yours. It's not eighteen and doesn't have to be eighteen everywhere. So, it's different in different countries. I'll ask you, if a person who happens to be seventeen years old and nine months kills one of your relatives, will you just overlook that? AMY GOODMAN: We'll continue our interview with Iranian President Ahmadinejad after break. [break] AMY GOODMAN: We return to our interview with the Iranian president, Mahmoud Ahmadinejad. JUAN GONZALEZ: I'd like to ask you, recently the Bush administration agreed to provide Israel with many new bunker buster bombs that people speculate might be used against Iran. Your reaction to this decision by the Bush administration? And do you—and there have been numerous reports in the American press of the Bush administration seeking to finance a secret war against Iran right now. PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, we actually think that the US administration and some other governments have equipped the Zionist regime with the nuclear warhead for those bombs, too. So, what are we to tell the American administration, a government that seeks a solution to all problems through war? Their logic is one of war. In the past twenty years, Americans' military expenditures have multiplied. So I think the problem should be resolved somewhere else, meaning the people of America themselves must decide about their future. Do they like new wars to be waged in their names that kill nations or have their money spent on warfare? So I think that's where the problem can be addressed. AMY GOODMAN: The investigative reporter Seymour Hersh said the Bush administration held a meeting in Vice President Cheney's office to discuss ways to provoke a war with Iran. Hersh said it was considered possibly a meeting to stage an incident, that it would appear that Iranian boats had attacked US forces in the Straits of Hormuz. Do you have any evidence of this? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, you have to pay attention to find that a lot of this kind of stuff is published out there. There's no need for us to react to it. Of course, Mr. Bush is very interested to start a new war. But he confronts two big barriers. One is the incapability in terms of maneuverability and operationally. Iran is a very big country, a very powerful country, very much capable of defending itself. The second barrier is the United States itself. We think there are enough wise people in this country to prevent the unreasonable actions by the administration. Even among the military commanders here, there are many people with wisdom who will stop a new war. I think the beginning or the starting a new war will mark the beginning of the end of the United States of America. Many people can understand that. But I also think that Mr. Bush's administration is coming to an end. Mr. Bush still has one other chance to make up for the mistakes he did in the past. He has no time to add to those list of mistakes. He can only make up for them. And that's a very good opportunity to have. So, I would advise him to take advantage of this opportunity, so that at least while you're in power, you do a couple—few good acts, as well. It's better than to end one's work with a report card of failures and of abhorrent acts. We're willing to help him in doing good. We'll be very happy. AMY GOODMAN: And your nuclear program? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Our time seems to be over, but our nuclear program is peaceful. It's very transparent for everyone to see. Your media is a progressive one. Let me just say a sentence here. I think that the time for the atomic bomb has reached an end. Don't you feel that yourself? What will determine the future is culture, it's the power of thought. Was the atomic bomb able to save the former Soviet Union from collapsing? Was it able to give victory to the Zionist regime of confronting the Palestinians? Was it able to resolve America's or US problems in Iraq and Afghanistan? Naturally, its usage has come to an end. It's very wrong to spend people's money building new atomic bombs. This money should be spent on creating welfare, prosperity, health, education, employment, and as aid that should be distributed among others' countries, to destroy the reasons for war and for insecurity and terrorism. Rest assured, whoever who seeks to have atomic bombs more and more is just politically backward. And those who have these arsenals and are busy making new generations of those bombs are even more backward. I think a disloyalty has occurred to the human community. Atomic energy power is a clean one. It's a renewable one, and it is a positive [inaudible]. Up to this day, we've identified at least sixteen positive applications from it. We're already aware that the extent to which we have used fossil fuels has imbalanced the climate of the world, brought about a lot of pollution, as well as a lot of diseases, as a result. So what's wrong with all countries having peaceful nuclear power and enjoying the benefits of this energy? It's actually a power that is constructively environmental. All those nuclear powers have come and said, well, having nuclear energy is the equivalent of having an atomic bomb pretty much—just a big lie. AMY GOODMAN: Iranian President Mahmoud Ahmadinejad. Tomorrow, part two of our conversation. But right now, we're joined by Ervand Abrahamian. He's an Iran expert, CUNY Distinguished Professor of History at Baruch College, City University of New York, author of a number of books, most recently, A History of Modern Iran. Welcome to Democracy Now! Can you talk about both what the Iranian president said here and his overall trip? Was it a different message this year? ERVAND ABRAHAMIAN: No, it's very much the same complacency, that, you know, everything's fine. There may be some problems in Iran and in foreign relations, but overall, Iran is confident and is—basically the mantra of the administration in Iran is that no one in their right senses would think of attacking Iran. And I think the Iranian government's whole policy is based on that. I wish I was as confident as Ahmadinejad is. JUAN GONZALEZ: And his dismissing of the situation, the human rights situation, in Iran, basically ascribing any arrests to some lawbreakers? Your sense of what is the human rights situation right there? ERVAND ABRAHAMIAN: Well, I mean, he basically changed the question and talked about, you know, the probably two million prisoners in America, which is of course true, but it certainly changes the topic of the discussion. Now, in Iran, you can be imprisoned for the talking of abolishing capital punishment. In fact, that's considered blasphemy, and academics have been charged with capital offense for actually questioning capital punishment. So, he doesn't really want to address those issues. And there have been major purges in the university recently, and of course the plight of the newspapers is very dramatic. I mean, mass newspapers have been closed down. Editors have been brought before courts, and so on. So, I would find that the human rights situation—I would agree with the Human Rights Watch, that things are bad. But I would like to stress that human rights organizations in Iran don't want that issue involved with the US-Iran relations, because every time the US steps in and tries to champion a question of human rights, I think that backfires in Iran, because most Iranians know the history of US involvement in Iran, and they feel it's hypocrisy when the Bush administration talks about human rights. So they would like to distance themselves. And Shirin Ebadi, of course, the Nobel Peace Prize, has made it quite clear that she doesn't want this championing by the United States of the human rights issue. AMY GOODMAN: Big protest outside. The Conference of Presidents of Major Jewish Organizations, the Israel Project, UJ Federation of New York, United Jewish Communities protested. They invited Hillary Clinton. She was going to speak. But they invited—then they invited Governor Palin, and so then Clinton pulled out, so they had had to disinvite Palin. And then you had the peace movement inside, meeting with Ahmadinejad. ERVAND ABRAHAMIAN: Yes, I think—I mean, the demonstrations outside are basically pushing for some sort of air strikes on the premise that Iran is an imminent threat and trying to build up that sort of pressure on the administration. And clearly, I think the Obama administration would not want to do that, but they would probably have a fair good hearing in the—if there was a McCain administration. AMY GOODMAN: Well, we're going to leave it there. Part two of our conversation tomorrow. We talk about the Israel-Palestine issue, we talk about the treatment of gay men and lesbians in Iran, and we talk about how the Iraq war has affected Iran with the Iranian president President Ahmadinejad was interviewed recently in New York by Democracy Now
Video Tags: President Mahmoud Ahmadinejad Democracy Iran America
Iranian President Mahmoud Ahmadinejad on the Threat of US Attack and International Criticism of Iran's Human Rights Record In part one of an interview with Amy Goodman and Juan Gonzalez, Iranian President Mahmoud Ahmadinejad talks about the threat of a US attack on Iran and responds to international criticism of Iran's human rights record. We also get reaction from CUNY Professor Ervand Abrahamian, an Iran expert and author of several books on Iran. JUAN GONZALEZ: Iranian President Mahmoud Ahmadinejad addressed the United Nations General Assembly this week, while the International Atomic Energy Agency, or IAEA, is meeting in Vienna to discuss Iran's alleged nuclear program. An IAEA report earlier this month criticized Iran for failing to fully respond to questions about its nuclear activities. The European Union told the IAEA Wednesday that it believes Iran is moving closer to being able to arm a nuclear warhead. Iran could face a fourth set of Security Council sanctions over its nuclear activities, but this week Russia has refused to meet with the US on this issue. The Iranian president refuted the IAEA's charges in his speech to the General Assembly and accused the agency of succumbing to political pressure. He also welcomed talks with the United States if it cuts back threats to use military force against Iran. AMY GOODMAN: As with every visit of the Iranian president to New York, some groups protested outside the United Nations. But this year, President Ahmadinejad also met with a large delegation of American peace activists concerned with the escalating possibility of war with Iran. Well, yesterday, just before their meeting, Juan Gonzalez and I sat down with the Iranian president at his hotel, blocks from the UN, for a wide-ranging discussion about US-Iran relations, Iran's nuclear program, threat of war with the US, the Israel-Palestine conflict, human rights in Iran and much more. Today, part one of our interview with the Iranian president. AMY GOODMAN: Welcome to Democracy Now!, President Ahmadinejad. You've come to the United States. What is your message to people in the United States and to the world community at the UN? PRESIDENT MAHMOUD AHMADINEJAD: [translated] In the name of God, the compassion of the Merciful, the president started by reciting verses from the Holy Quran in Arabic. Hello. Hello to the people of America. The message from the nation and people of Iran is one of peace, tranquility and brotherhood. We believe that viable peace and security can happen when it is based on justice and piety and purity. Otherwise, no peace will occur. JUAN GONZALEZ: Mr. President, you're faced now in Iran with American soldiers in Iraq to your west, with American soldiers and NATO troops to your east in Afghanistan, and with Blackwater, the notorious military contractor, training the military in Azerbaijan, another neighbor of yours. What is the effect on your country of this enormous presence of American forces around Iran and the impact of these wars on your own population? PRESIDENT MAHMOUD AHMADINEJAD: [translated] It's quite natural that when there are wars around your borders, it brings about negative repercussions for the entire region. These days, insecurity cannot be bordered; it just extends beyond boundaries. In the past two years, we had several cases of bomb explosions in southern towns in Iran carried out by people who were supervised by the occupying forces in our neighborhood. And in Afghanistan, following the presence of NATO troops, the production of illicit drugs has multiplied. It's natural that it basically places pressure on Iran, including costly ones in order to fight the flow of illicit drugs. We believe the people in the region are able to establish security themselves, on their own, so there is no need for foreigners and external forces, because these external forces have not helped the security of the region. AMY GOODMAN: Do you see them as a threat to you? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, it's natural that when there is insecurity, it threatens everyone. JUAN GONZALEZ: I'd like to turn for a moment to your domestic policies and law enforcement in your country. Human Rights Watch, which has often criticized the legal system in the United States, says that, under your presidency, there has been a great expansion in the scope and the number of individuals and activities persecuted by the government. They say that you've jailed teachers who are fighting for wages and better pensions, students and activists working for reform, and other labor leaders, like Mansour Ossanlou from the bus workers' union. What is your response to these criticisms of your policies? PRESIDENT MAHMOUD AHMADINEJAD: [translated] I think that the human rights situation in Iran is relatively a good one, when compared to the United States and other countries. Of course, when we look at the ideals that are dear to us, we understand that we still need to do a lot, because we seek divine and religious ideals and revolutionary ones. But when we compare ourselves with some European countries and the United States, we feel we're in a much better place. A large part of the information that these groups receive come from criticisms coming from groups that oppose the government. If you look at it, we have elections in Iran every year. And the propaganda is always around, too. But they're not always true. Groups accuse one another. But within the region and compared to the United States, we have the smallest number of prisoners, because in Iran, in general, there is not so much inclination to imprison people. We're actually looking at our existing laws right now to see how we can eliminate most prisons around the country. So, you can see that people in Iran like each other. They live coexistently and like the government, too. This news is more important to these groups, not so much for the Iranian people. You have to remember, we have over 70 million people in our country, and we have laws. Some people might violate it, and then, according to the law, the judiciary takes charge. And this happens everywhere. What really matters is that in the end there are the least amount of such violations of the law in Iran, the least number. So, I think the interpretation of these events is a wrong one. The relationship between the people and the government in Iran is actually a very close one. And criticizing the government is absolutely free for all. That's exactly why everyone says what they want. There's really no restrictions. It doesn't necessarily mean that everything you hear is always true. And the government doesn't really respond to it, either. It's just free. JUAN GONZALEZ: Let me ask you in particular about the question of the execution of juveniles. My understanding is that Iran is one of only five or six nations in the world that still execute juveniles convicted of capital offenses and that you—by far, you execute the most. I think twenty-six of the last thirty-two juveniles executed in the world were executed in Iran. How is this a reflection of the—of a state guided by religious principles, to execute young people? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Firstly, nobody is executed under the age of eighteen in Iran. This is the first point. And then, please pay attention to the fact that the legal age in Iran is different from yours. It's not eighteen and doesn't have to be eighteen everywhere. So, it's different in different countries. I'll ask you, if a person who happens to be seventeen years old and nine months kills one of your relatives, will you just overlook that? AMY GOODMAN: We'll continue our interview with Iranian President Ahmadinejad after break. [break] AMY GOODMAN: We return to our interview with the Iranian president, Mahmoud Ahmadinejad. JUAN GONZALEZ: I'd like to ask you, recently the Bush administration agreed to provide Israel with many new bunker buster bombs that people speculate might be used against Iran. Your reaction to this decision by the Bush administration? And do you—and there have been numerous reports in the American press of the Bush administration seeking to finance a secret war against Iran right now. PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, we actually think that the US administration and some other governments have equipped the Zionist regime with the nuclear warhead for those bombs, too. So, what are we to tell the American administration, a government that seeks a solution to all problems through war? Their logic is one of war. In the past twenty years, Americans' military expenditures have multiplied. So I think the problem should be resolved somewhere else, meaning the people of America themselves must decide about their future. Do they like new wars to be waged in their names that kill nations or have their money spent on warfare? So I think that's where the problem can be addressed. AMY GOODMAN: The investigative reporter Seymour Hersh said the Bush administration held a meeting in Vice President Cheney's office to discuss ways to provoke a war with Iran. Hersh said it was considered possibly a meeting to stage an incident, that it would appear that Iranian boats had attacked US forces in the Straits of Hormuz. Do you have any evidence of this? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Well, you have to pay attention to find that a lot of this kind of stuff is published out there. There's no need for us to react to it. Of course, Mr. Bush is very interested to start a new war. But he confronts two big barriers. One is the incapability in terms of maneuverability and operationally. Iran is a very big country, a very powerful country, very much capable of defending itself. The second barrier is the United States itself. We think there are enough wise people in this country to prevent the unreasonable actions by the administration. Even among the military commanders here, there are many people with wisdom who will stop a new war. I think the beginning or the starting a new war will mark the beginning of the end of the United States of America. Many people can understand that. But I also think that Mr. Bush's administration is coming to an end. Mr. Bush still has one other chance to make up for the mistakes he did in the past. He has no time to add to those list of mistakes. He can only make up for them. And that's a very good opportunity to have. So, I would advise him to take advantage of this opportunity, so that at least while you're in power, you do a couple—few good acts, as well. It's better than to end one's work with a report card of failures and of abhorrent acts. We're willing to help him in doing good. We'll be very happy. AMY GOODMAN: And your nuclear program? PRESIDENT MAHMOUD AHMADINEJAD: [translated] Our time seems to be over, but our nuclear program is peaceful. It's very transparent for everyone to see. Your media is a progressive one. Let me just say a sentence here. I think that the time for the atomic bomb has reached an end. Don't you feel that yourself? What will determine the future is culture, it's the power of thought. Was the atomic bomb able to save the former Soviet Union from collapsing? Was it able to give victory to the Zionist regime of confronting the Palestinians? Was it able to resolve America's or US problems in Iraq and Afghanistan? Naturally, its usage has come to an end. It's very wrong to spend people's money building new atomic bombs. This money should be spent on creating welfare, prosperity, health, education, employment, and as aid that should be distributed among others' countries, to destroy the reasons for war and for insecurity and terrorism. Rest assured, whoever who seeks to have atomic bombs more and more is just politically backward. And those who have these arsenals and are busy making new generations of those bombs are even more backward. I think a disloyalty has occurred to the human community. Atomic energy power is a clean one. It's a renewable one, and it is a positive [inaudible]. Up to this day, we've identified at least sixteen positive applications from it. We're already aware that the extent to which we have used fossil fuels has imbalanced the climate of the world, brought about a lot of pollution, as well as a lot of diseases, as a result. So what's wrong with all countries having peaceful nuclear power and enjoying the benefits of this energy? It's actually a power that is constructively environmental. All those nuclear powers have come and said, well, having nuclear energy is the equivalent of having an atomic bomb pretty much—just a big lie. AMY GOODMAN: Iranian President Mahmoud Ahmadinejad. Tomorrow, part two of our conversation. But right now, we're joined by Ervand Abrahamian. He's an Iran expert, CUNY Distinguished Professor of History at Baruch College, City University of New York, author of a number of books, most recently, A History of Modern Iran. Welcome to Democracy Now! Can you talk about both what the Iranian president said here and his overall trip? Was it a different message this year? ERVAND ABRAHAMIAN: No, it's very much the same complacency, that, you know, everything's fine. There may be some problems in Iran and in foreign relations, but overall, Iran is confident and is—basically the mantra of the administration in Iran is that no one in their right senses would think of attacking Iran. And I think the Iranian government's whole policy is based on that. I wish I was as confident as Ahmadinejad is. JUAN GONZALEZ: And his dismissing of the situation, the human rights situation, in Iran, basically ascribing any arrests to some lawbreakers? Your sense of what is the human rights situation right there? ERVAND ABRAHAMIAN: Well, I mean, he basically changed the question and talked about, you know, the probably two million prisoners in America, which is of course true, but it certainly changes the topic of the discussion. Now, in Iran, you can be imprisoned for the talking of abolishing capital punishment. In fact, that's considered blasphemy, and academics have been charged with capital offense for actually questioning capital punishment. So, he doesn't really want to address those issues. And there have been major purges in the university recently, and of course the plight of the newspapers is very dramatic. I mean, mass newspapers have been closed down. Editors have been brought before courts, and so on. So, I would find that the human rights situation—I would agree with the Human Rights Watch, that things are bad. But I would like to stress that human rights organizations in Iran don't want that issue involved with the US-Iran relations, because every time the US steps in and tries to champion a question of human rights, I think that backfires in Iran, because most Iranians know the history of US involvement in Iran, and they feel it's hypocrisy when the Bush administration talks about human rights. So they would like to distance themselves. And Shirin Ebadi, of course, the Nobel Peace Prize, has made it quite clear that she doesn't want this championing by the United States of the human rights issue. AMY GOODMAN: Big protest outside. The Conference of Presidents of Major Jewish Organizations, the Israel Project, UJ Federation of New York, United Jewish Communities protested. They invited Hillary Clinton. She was going to speak. But they invited—then they invited Governor Palin, and so then Clinton pulled out, so they had had to disinvite Palin. And then you had the peace movement inside, meeting with Ahmadinejad. ERVAND ABRAHAMIAN: Yes, I think—I mean, the demonstrations outside are basically pushing for some sort of air strikes on the premise that Iran is an imminent threat and trying to build up that sort of pressure on the administration. And clearly, I think the Obama administration would not want to do that, but they would probably have a fair good hearing in the—if there was a McCain administration. AMY GOODMAN: Well, we're going to leave it there. Part two of our conversation tomorrow. We talk about the Israel-Palestine issue, we talk about the treatment of gay men and lesbians in Iran, and we talk about how the Iraq war has affected Iran with the Iranian president
26th Sep-Hassan Nasrullah Speech on Youm ul Quds -English
Quds Day keeps Palestinian cause alive Hezbollah\\\'s leader says Quds Day is clearly alive decades after its inception, as a warning to Muslims...
Quds Day keeps Palestinian cause alive Hezbollah\\\'s leader says Quds Day is clearly alive decades after its inception, as a warning to Muslims against abandoning the Palestinian cause. "Thirty years after Imam Khomeini (PBUH) announced Quds Day, it is clearly very much alive…and has always had the support of those generations who supported Imam Khomeini and followed his path," said Seyyed Hassan Nasrallah during a televised address on the ocassion. He hailed the annual event as an opportunity to reinvigorate the cause of al-Quds and Palestine, "which was and still is subject to destruction, bargaining and forgetfulness." "The idea is not just for standing at podiums and crying because of (Palestinian) losses," he said, adding that by marking Quds Day, Muslims highlight their determination to rectify the situation. Nasrallah criticized those who are afraid of the negative repercussions standing for the Palestinian nation may bring, describing the issue as a central cause that concerns the destiny of the entire Islamic Ummah and of future generations. He referred to the occupation of the Palestinian territories by Israel as a humiliation for Muslims whose noblest sanctities remain occupied by their \\\'racist Zionist\\\' enemy. The Hezbollah secretary general underscored the responsibility of the Muslim community towards al-Quds, Palestine and the Palestinian nation, saying Palestine belonged to the Palestinians, Muslims and the Arab world and that it must revert to its real owners. He refuted the misconception of Israel being a superpower and described the Zionist regime as an entity created by Britain and taken over by the United States, noting that both are using Tel Aviv for their hegemonic aims. The Hezbollah chief then pored scorn on the shameful tactic of begging the founders of Israel in the hope of liberating the occupied territories. He said Israel is the enemy of all Arabs and Muslims and called on the Arabs unite against their common enemy, adding that Premier Ehud Olmert\\\'s resignation is evidence of the Israeli defeat in the July 2006 war. Nasrallah also emphasized that Hezbollah was genuinely devoted to the reconciliation of all Lebanese rival groups in the national effort to realize peace and political stability. The founder of the Islamic Republic of Iran, Imam Khomeini, designated International Quds Day in 1979 to oppose Israel\\\'s occupation of Islam\\\'s third most holy city and Palestinian lands in general. Anti-Zionist demonstrations are held on the last Friday of the fasting month of Ramadan in some Muslim countries and by Muslim and Arab communities around the world.
Video Tags: hassan hasan nasrullah nasrallah youm al quds
News4U-Hamas criticizes Saudi Arabia over Hajj - English
A Hamas senior official has lashed out at Saudi Arabia for not granting visas to Gazans who want to go on the Hajj pilgrimage. In remarks posted...
A Hamas senior official has lashed out at Saudi Arabia for not granting visas to Gazans who want to go on the Hajj pilgrimage. In remarks posted on Hummaas's website, Atef Edwan said Riyadh had allowed thousands of people registered with the Palestinian Authority in the West Bank to have visas but not the 2,200 in Gaza who applied through Hummaas, AFP reported. "By ignoring those Gaza pilgrims registered with the Waqf (department of religious affairs) Saudi Arabia is making a political mistake that will have negative consequences on the Saudi regime and the whole region," AFP quoted Edwan as saying. Meanwhile media outlets quoted unnamed witnesses as saying that Hummaas police set up checkpoints across Gaza on Saturday, to prevent those who coordinated their pilgrimage with the movement's rival, the Palestinian Authority from traveling. Since last month tensions between Hummaas and the Palestinian Authority has heightened as the Palestinian Authority Chief Mahmoud Abbas increased the detention of Hummaas' members.
1m:9s
Video Tags: Hummaas criticizes Saudi Arabia over Hajj saudia mecca maca haj hajj hujj refused humas hamas | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,107 |
package de.codecentric.elasticsearch.plugin.kerberosrealm.client;
import java.io.InputStream;
import java.io.OutputStream;
import java.nio.charset.StandardCharsets;
import javax.security.auth.Subject;
import javax.xml.bind.DatatypeConverter;
import org.elasticsearch.action.ActionRequest;
import org.elasticsearch.client.Client;
import org.elasticsearch.common.SuppressForbidden;
import org.ietf.jgss.ChannelBinding;
import org.ietf.jgss.GSSContext;
import org.ietf.jgss.GSSCredential;
import org.ietf.jgss.GSSException;
import org.ietf.jgss.GSSName;
import org.ietf.jgss.MessageProp;
import org.ietf.jgss.Oid;
@SuppressForbidden(reason = "unit test")
public class MockingKerberizedClient extends KerberizedClient {
public MockingKerberizedClient(final Client in) {
super(in, new Subject(), "mock_principal");
}
@Override
GSSContext initGSS() throws Exception {
return new MockGSSContext();
}
@Override
void addAdditionalHeader(final ActionRequest<ActionRequest> request, final int count, final byte[] data) {
if (count >= 4) {
request.putHeader("Authorization", "Negotiate_c " + DatatypeConverter.printBase64Binary(data));
}
}
@SuppressForbidden(reason = "unit test")
private static class MockGSSContext implements GSSContext {
@Override
public byte[] initSecContext(final byte[] inputBuf, final int offset, final int len) throws GSSException {
if (inputBuf == null || inputBuf.length == 0) {
return "mocked_initial_gss_security_context".getBytes(StandardCharsets.UTF_8);
} else {
return ("|" + new String(inputBuf, offset, len, StandardCharsets.UTF_8)).getBytes(StandardCharsets.UTF_8);
}
}
@Override
public int initSecContext(final InputStream inStream, final OutputStream outStream) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public byte[] acceptSecContext(final byte[] inToken, final int offset, final int len) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void acceptSecContext(final InputStream inStream, final OutputStream outStream) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean isEstablished() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void dispose() throws GSSException {
}
@Override
public int getWrapSizeLimit(final int qop, final boolean confReq, final int maxTokenSize) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public byte[] wrap(final byte[] inBuf, final int offset, final int len, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void wrap(final InputStream inStream, final OutputStream outStream, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public byte[] unwrap(final byte[] inBuf, final int offset, final int len, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void unwrap(final InputStream inStream, final OutputStream outStream, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public byte[] getMIC(final byte[] inMsg, final int offset, final int len, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void getMIC(final InputStream inStream, final OutputStream outStream, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void verifyMIC(final byte[] inToken, final int tokOffset, final int tokLen, final byte[] inMsg, final int msgOffset,
final int msgLen, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void verifyMIC(final InputStream tokStream, final InputStream msgStream, final MessageProp msgProp) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public byte[] export() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestMutualAuth(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestReplayDet(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestSequenceDet(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestCredDeleg(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestAnonymity(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestConf(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestInteg(final boolean state) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void requestLifetime(final int lifetime) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public void setChannelBinding(final ChannelBinding cb) throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getCredDelegState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getMutualAuthState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getReplayDetState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getSequenceDetState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getAnonymityState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean isTransferable() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean isProtReady() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getConfState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean getIntegState() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public int getLifetime() {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public GSSName getSrcName() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public GSSName getTargName() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public Oid getMech() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public GSSCredential getDelegCred() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
@Override
public boolean isInitiator() throws GSSException {
throw new UnsupportedOperationException("mock gss context does not support this operation");
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,968 |
CLICK HERE to view a slide presentation of the life of Saint Kateri Tekakwitha.
Dear OCL Web Site Followers: As the Holidays Begin – Thank You!
Rev. Dr. Frank Marangos – FINDING OUR VOICE: Orthodox Leadership for the 21st Century Rev. Fr. Hans Jacobse – THE CHALLENGE OF SECULARISM IN THE LOCAL PARISH Dr Frances Kostarelos, PhD – RELIGIOUS PLURALISM & FUNDAMENTALISM & CONTESTED IDENTITIES IN THE GREEK ORTHODOX CHURCH IN AMERICA His Grace, Bishop PAUL (Gassios) PROGRESS AND PROSPECT OF THE WORK OF THE ASSEMBLY OF BISHOPS AS THEY WORK OF THE UNITY OF THE ORTHODOX IN THE USA AND CANONICAL GOOD ORDER Thanks to John Maddex of Ancient Faith Ministries for recording the sessions and making them available! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,851 |
\section{Introduction \label{sect1}}
As is well known, tilting theory has had significant applications in
many branches of mathematics (see \cite{HHK}), and the key
objectives in this theory are tilting modules, or more generally,
tilting complexes or objects. Given a good tilting module $T$ over a
ring $A$, let $B$ be the endomorphism ring of $T$, if $T$ is
classical, then a beautiful theorem of Happel says that the derived
module category $\D B$ of $B$ is triangle equivalent to the derived
module category $\D A$ of $A$ (see \cite{H}). Thus one can use
derived invariants to understand homological, geometric and
numerical properties of $A$ through $B$, or conversely, of $B$
through $A$. This theorem also tells that one cannot get new derived
categories from classical tilting modules. For infinitely generated
tilting modules, Bazzoni, Mantese and Tonolo recently show a
remarkable result: $\D A$ can be regarded as a full subcategory or a
quotient category of $\D B$ (see \cite{Bz2}). Moreover, it is proved
in \cite{CX1} that if the projective dimension of $T$ is at most
$1$, then there is a homological ring epimorphism $\lambda: B\ra C$
of rings such that the kernel of the total left-derived functor
$T\otimes_B^{\mathbb L}-$, as a full triangulated subcategory of
$\D{B}$, can be realized as the derived module category $\D{C}$ of
$C$. Thus, for (infinitely generated) good tilting modules of
projective dimension at most $1$, Happel's theorem now has a new
appearance and can be featured as a recollement of derived module
categories:
$$\xymatrix@C=1.2cm{\D{C}\ar[r]^-{D(\lambda_*)}
&\D{B}\ar[r]^-{{} _AT\otimesL_B-}\ar@/^1.2pc/[l]\ar@/_1.3pc/[l]
&\D{A} \ar@/^1.2pc/[l]\ar@/_1.3pc/[l]}\vspace{0.2cm}$$ However, for
tilting modules of higher projective dimension, the existence of the
above recollement is unknown (see the first open question in
\cite{CX1}). On the one hand, the argument used in \cite{CX1}
actually does not work any more for the general case because the
proof there involves a two-term complex which depends on the
projective dimension. Thus some new ideas are necessary for
attacking the general situation. On the other hand, neither positive
examples nor counterexamples to this general case are known to
experts. So, it is quite mysterious whether the above recollement
still exists for a good tilting module of projective dimension at
least $2$.
In the present paper, we shall consider this question in detail. In
fact, our discussion is implemented in the framework of Ringel
modules (see Definition \ref{rm}). This provides us a way to deal
with the above question uniformly for higher tilting and cotilting
modules. We first provide characterizations of when the kernel of
the functor $T\otimes_B^{\mathbb L}-$ can be realized as the derived
module category of a ring $C$ with a homological ring epimorphism
$B\ra C$, and then use these criterions to give positive and
negative examples to the above question for tilting modules of
projective dimension bigger than $1$. Finally, as another
application of our criterions, we shall consider the above question
for cotilting modules.
Before stating our main results precisely, we first introduce
notation and recall some definitions.
Let $A$ be a ring with identity, and let $n$ be a natural number. A
left $A$-module $T$ is called an $n$-\emph{tilting} $A$-module (see
\cite{ct}) if the following three conditions are satisfied:
$(T1)$ There is an exact sequence
$$
0\lra P_n \lra \cdots \lra P_1\lraf{\sigma} P_0\lraf{\pi} T\lra 0$$
of $A$-modules such that all $P_i$ are projective, that is, the
projective dimension of $T$ is at most $n$;
$(T2)$ $\Ext^j_A(T, T^{(I)})=0$ for all $j\geq 1$ and nonempty sets
$I$, where $T^{(I)}$ denotes the direct sum of $I$ copies of $T$;
$(T3)$ There is an exact sequence
$$
0\lra {}_AA \lraf{\omega} T_0 \lra T_1\lra \cdots \lra T_n\lra 0$$
of $A$-modules such that $T_i$ is isomorphic to a direct summand of
a direct sum of copies of $T$ for all $0\leq i\leq n$.
An $n$-tilting module $T$ is said to be \emph{good} if $(T3)$ can
be replaced by
$(T3)'$ there is an exact sequence
$$ 0\lra {}_AA \lraf{\omega} T_0 \lra
T_1\lra \cdots \lra T_n\lra 0$$ of $A$-modules such that $T_i$ is
isomorphic to a direct summand of a finite direct sum of copies of
$T$ for all $0\leq i\leq n$. A good $n$-tilting module $T$ is said
to be \emph{classical} if the modules $P_i$ in $(T1)$ are finitely
generated (see \cite{bb, hr}).
For any given tilting $A$-module $T$ with $(T1)$-$(T3)$, the module
$T':=\bigoplus_{i=0}^n T_i$ is a good $n$-tilting module which is
equivalent to the given one, that is, $T$ and $T'$ generate the same
tilting class in the category of $A$-modules (see \cite{Bz2}).
Let $T$ be an $n$-tilting $A$-module and $B$ the endomorphism ring
of $_AT$. In general, the total right-derived functor $\rHom_A(T,-)$
does not define a triangle equivalence between the (unbounded)
derived category $\D{A}$ of $A$ and the derived category $\D{B}$ of
$B$. However, if $_AT$ is good, then $\rHom_A(T,-)$ is fully
faithful and induces a triangle equivalence between the derived
category $\D{A}$ and the Verdier quotient of $\D{B}$ modulo the
kernel $\Ker(T\otimesL_{B}-)$ of the total left-derived functor
$T\otimesL_{B}-$ (see \cite[Theorem 2.2]{Bz2}). Furthermore, the
functor $\rHom_A(T,-):\D{A}\to\D{B}$ is an equivalence if and only
if $T$ is a classical tilting module if and only if
$\Ker(T\otimesL_{B}-)$ vanishes (see \cite{Bz2}). From this point of
view, the category $\Ker(T\otimesL_{B}-)$ measures the difference
between the derived categories $\D{A}$ and $\D{B}$.
Motivated by the main result in \cite{CX1}, we introduce the
following notion. A full triangulated subcategory $\mathcal X$ of
$\D{B}$ is said to be \emph{homological} if there is a homological
ring epimorphism $B\to C$ of rings such that the restriction functor
$\D{C}\to \D{B}$ induces a triangle equivalence from $\D{C}$ to
$\mathcal{X}$. Thus, if the projective dimension of a good tilting
module $_AT$ is at most $1$, then the subcategory
$\Ker(T\otimesL_{B}-)$ of $\D{B}$ is homological. Now, in terms of
homological subcategories, our question can be restated as follows:
\medskip
{\bf Question.} \emph{Is the full triangulated subcategory
$\Ker(T\otimesL_{B}-)$ of $\D{B}$ always homological for any good
$n$-tilting $A$-module $T$ with $n\geq 2$? Here, $B$ is the
endomorphism ring of the module $T$.}
\medskip
Let us first give several characterizations for
$\Ker(T\otimesL_{B}-)$ to be homological.
\begin{Theo}\label{main-result}
Suppose that $A$ is a ring and $n$ is a natural number. Let $T$ be a
good $n$-tilting $A$-module, and let $B$ be the endomorphism ring of
$_AT$. Then the following are equivalent:
$(1)$ The full triangulated subcategory $\Ker(T\otimesL_{B}-)$ of
$\D{B}$ is homological.
$(2)$ The category consisting of the $B$-modules $Y$ with
$\Tor_m^B(T,Y)=0$ for all $m\geq0$ is an abelian subcategory of the
category of all $B$-modules.
$(3)$ The $m$-th cohomology of the complex $\Hom_A(\cpx{P},
A)\otimes_AT_B$ vanishes for all $m\ge 2$, where the complex
$\cpx{P}$ is a deleted projective resolution of $_AT$.
$(4)$ The kernel $K$ of the homomorphism
$\Coker(\varphi_0)\lra\Coker(\varphi_1)$ induced from $\sigma:
P_1\to P_0$ in $(T1)$ satisfies $\,\Ext^m_{B\opp}(T, K)=0$ for all $
m\geq 0$, where $\varphi_i: \Hom_A(P_i, A)\otimes_AT \lra
\Hom_A(P_i, T)$ is the composition map under the identification of
$_AT_B$ with $\Hom_A(A, T)$ for $i=0, 1$.
\smallskip
In particular, if $n=2$, then $(1)$ holds if and only if
$\,\Ext^2_A(T,\,A)\otimes_AT=0$.
\end{Theo}
We remark that if the category $\Ker(T\otimesL_{B}-)$ is homological
in $\D{B}$, then the generalized localization $\lambda: B\to B_T$
of $B$ at the module $T_B$ exists (see Definition \ref{genloc}) and
is homological, and therefore there is a recollement of derived
module categories:
$$\xymatrix@C=1.2cm{\D{B_T}\ar[r]^-{D(\lambda_*)}
&\D{B}\ar[r]^-{ {}_AT\otimesL_B-}\ar@/^1.2pc/[l]\ar@/_1.3pc/[l]
&\D{A} \ar@/^1.2pc/[l]\ar@/_1.3pc/[l]}\vspace{0.2cm}$$
\smallskip
\noindent where $D(\lambda_*)$ stands for the restriction functor
induced by $\lambda$. Thus, Theorem \ref{main-result} can be
regarded as a kind of generalization of \cite[Theorem 1.1 (1)]{CX1},
and also gives an explanation why \cite[Theorem 1.1 (1)]{CX1} holds.
As a consequence of Theorem \ref{main-result}, we have the
following corollary in which $(1)$ extends \cite[Theorem 1.1
(1)]{CX1}, while our new contribution to $(2)$ is the necessity
part of the statement.
\begin{Koro}\label{cor}
Suppose that $A$ is a ring and $n$ is a natural number. Let $T$ be a
good $n$-tilting $A$-module, and let $B$ be the endomorphism ring of
$_AT$.
$(1)$ If $_AT$ decomposes into $M\oplus N$ such that the projective
dimension of $_AM$ is at most $1$ and that the first syzygy of $_AN$
is finitely generated, then the category $\Ker(T\otimesL_{B}-)$ is
homological.
$(2)$ Suppose that $A$ is commutative. If $\Hom_A(T_{i+1}, T_i)=0$
for all $T_i$ in $(T3)'$ with $1\leq i \leq n-1$, then the category
$\Ker(T\otimesL_{B}-)$ is homological if and only if the projective
dimension of $_AT$ is at most $1$, that is, $_AT$ is a $1$-tilting
module.
\end{Koro}
A remarkable consequence of Corollary \ref{cor} is that we can get
an answer to the above-mentioned question. In fact, in Section
\ref{7.1}, we display an example of an $n$-tilting module $T$ for
each $n\geq 2$ and shows that $\Ker(T\otimesL_{B}-)$ is not
homological.
Dually, there is the notion of (good) cotilting modules of finite
injective dimension over arbitrary rings. This notion involves
injective cogenerators of module categories. As is known, there is
no nice duality between infinitely generated tilting and cotilting
modules. This means that methods for dealing with tilting modules
may not work dually with cotilting modules. Nevertheless, we shall
use methods in this paper to deal with cotilting modules with
respect to some ``nice'' injective cogenerators. Our methods cover
particularly cotilting modules over Artin algebras. Here, our main
concern again is when the induced subcategories of derived
categories of the endomorphism rings of good cotilting modules are
homological, or equivalently, the existence of a recollement similar
to \cite[Theorem 1.1 (1)]{CX1}.
Our consideration is focused on (infinitely generated) cotilting
modules over Artin algebras $A$. Let $D$ be the usual duality of an
Artin algebra. The dual module $D(A_A)$ is an injective cogenerator
for the category of $A$-modules, and called the \emph{ordinary
injective cogenerator}. Our main result for cotilting modules is as
follows.
\begin{Theo}\label{coth}
Suppose that $A$ is an Artin algebra. Let $U$ be a good
$1$-cotilting $A$-module with respect to the ordinary injective
cogenerator for the category of $A$-modules. Set $R:=\End_A(U)$ and
$M:=\Hom_A(U, D(A))$. Then the universal localization $\lambda: R\to
R_M$ of $R$ at the module $_RM$ is homological, and there exists a
recollement of derived module categories:
$$\xymatrix@C=1.2cm{\D{R_M}\ar[r]^-{D(\lambda_*)}
&\D{R}\ar[r]\ar@/^1.2pc/[l]\ar@/_1.2pc/[l] &\D{A}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]}\vspace{0.2cm}$$ where $D(\lambda_*)$
stands for the restriction functor induced by $\lambda$.
\end{Theo}
As is known, over an Artin algebra, each $1$-cotilting module is
equivalent to the dual of a $1$-tilting right module (see
\cite[Chapter 11, Section 4.15]{HHK}). However, we cannot get
Theorem \ref{coth} from the result \cite[Theorem 1.1 (1)]{CX1}
because the relationship between the endomorphism ring of an
infinitely generated $1$-cotlting module and the one of the
corresponding $1$-tilting right module is unknown.
For a more general formulation of Theorem \ref{coth} on higher
cotilting modules, one may see Corollary \ref{real-cotilt} and the
diagram ($\ddag$) above Corollary \ref{real-cotilt}. For higher
cotiltig modules, we also give conditions and counterexamples for
subcategories from cotilting modules not to be homological, though
additional attention is needed.
\medskip
The contents of this paper are sketched as follows. In Section 2, we
fix notation, recall some definitions and prove some homological
formulas. In Section \ref{sect3}, we introduce bireflective and
homological subcategories in derived categories of rings, and
discuss when bireflective subcategories are homological. In Section
\ref{sect4}, we introduce a new class of modules, called Ringel
modules, and establish a crucial result, Proposition
\ref{realization}, which is used not only to decide if a
bireflective subcategory is homological, but also to investigate
higher tilting and cotilting modules in the later considerations. In
Section \ref{sect5}, we apply the results in previous sections to
good tilting modules and show Theorem \ref{main-result} as well as
Corollary \ref{cor}. At the end of this section, we point out an
example which shows that there do exist higher tilting modules
satisfying the conditions of Corollary \ref{cor} (1). In Section
\ref{sect6}, we first apply our results in Section \ref{sect4} to
cotilting modules in a general setting, and then prove Theorem
\ref{coth} for Artin algebras. It is worth noting that, for
cotilting $A$-modules $U$, recollements of $\D{\End_A(U)}$ may
depend on the choices of injective cogenerators to which the
cotilting modules are referred. In this section, we also give
conditions for the subcategories from cotilting modules not to be
homological. This is a preparation for constructing counterexamples
in the next section. In Section \ref{sect7}, we apply our results in
Section \ref{sect5} to good tilting modules $T$ over commutative
rings, and give a counterexample to show that, in general,
$\Ker(T\otimesL_B-)$ may not be realized as the derived module
category of a ring $C$ with a homological ring epimorphism $B\ra C$.
For higher cotilting modules, the same situation occurs. More
precisely, we shall use results in Section \ref{sect6} to display a
counterexample which demonstrates that, in general, the
corresponding subcategories from cotilting modules cannot be
realizable as derived module categories of rings. This section ends
with a few open questions closely related to the results in this
paper.
\section{Preliminaries\label{sect2}}
In this section, we briefly recall some definitions, basic facts and
notation used in this paper. For unexplained notation employed in
this paper, we refer the reader to \cite{CX1} and the references
therein.
\subsection{Notation}\label{sect2.1}
Let $\mathcal C$ be an additive category.
Throughout the paper, a full subcategory $\mathcal B$ of $\mathcal
C$ is always assumed to be closed under isomorphisms, that is, if
$X\in {\mathcal B}$ and $Y\in\cal C$ with $Y\simeq X$, then
$Y\in{\mathcal B}$.
Let $X$ be an object in $\mathcal{C}$. Denote by $\add(X)$ the full
subcategory of $\mathcal{C}$ consisting of all direct summands of
finite coproducts of copies of $M$. If $\mathcal{C}$ admits small
coproducts (that is, coproducts indexed over sets exist in
${\mathcal C}$), then we denote by $\Add(X)$ the full subcategory of
$\mathcal{C}$ consisting of all direct summands of small coproducts
of copies of $X$. Dually, if $\mathcal{C}$ admits small products,
then we denote by $\Prod(X)$ the full subcategory of $\mathcal{C}$
consisting of all direct summands of small products of copies of
$X$.
Given two morphisms $f: X\to Y$ and $g: Y\to Z$ in $\mathcal C$, we
denote the composite of $f$ and $g$ by $fg$ which is a morphism from
$X$ to $Z$. The induced morphisms $\Hom_{\mathcal
C}(Z,f):\Hom_{\mathcal C}(Z,X)\ra \Hom_{\mathcal C}(Z,Y)$ and
$\Hom_{\mathcal C}(f,Z): \Hom_{\mathcal C}(Y, Z)\ra \Hom_{\mathcal
C}(X, Z)$ are denoted by $f^*$ and $f_*$, respectively.
We denote the composition of a functor $F:\mathcal {C}\to
\mathcal{D}$ between categories $\mathcal C$ and $\mathcal D$ with a
functor $G: \mathcal{D}\to \mathcal{E}$ between categories $\mathcal
D$ and $\mathcal E$ by $GF$ which is a functor from $\mathcal C$ to
$\mathcal E$. Let $\Ker(F)$ and $\Img(F)$ be the kernel and image of
the functor $F$, respectively. In particular, $\Ker(F)$ is closed
under isomorphisms in $\mathcal{C}$. In this note, we require that
$\Img(F)$ is closed under isomorphisms in $\mathcal{D}$.
Suppose that $\mathcal{Y}$ is a full subcategory of $\mathcal{C}$.
Let $\Ker(\Hom_{\mathcal{C}}(-,\mathcal{Y}))$ be the left orthogonal
subcategory with respect to $\mathcal{Y}$, that is, the full
subcategory of $\mathcal{C}$ consisting of the objects $X$ such that
$\Hom_{\mathcal{C}}(X,Y)=0$ for all objects $Y$ in $\mathcal{Y}$.
Similarly, we can define the right orthogonal subcategory
$\Ker(\Hom_{\mathcal{C}}(\mathcal{Y},-))$ of $\cal C$ with respect
to $\mathcal{Y}$.
Let $\C{\mathcal{C}}$ be the category of all complexes over
$\mathcal{C}$ with chain maps, and $\K{\mathcal{C}}$ the homotopy
category of $\C{\mathcal{C}}$. As usual, we denote by
$\Cb{\mathcal{C}}$ the category of bounded complexes over $C$, and
by $\Kb{\mathcal{C}}$ the homotopy category of $\Cb{\mathcal{C}}$.
When $\mathcal{C}$ is abelian, the derived category of $\mathcal{C}$
is denoted by $\D{\mathcal{C}}$, which is the localization of
$\K{\mathcal C}$ at all quasi-isomorphisms. It is well known that
both $\K{\mathcal{C}}$ and $\D{\mathcal{C}}$ are triangulated
categories. For a triangulated category, its shift functor is
denoted by $[1]$ universally.
If $\mathcal{T}$ is a triangulated category with small coproducts,
then, for an object $U$ in $\mathcal{T}$, we denote by ${\rm
Tria}(U)$ the smallest full triangulated subcategory of
$\mathcal{T}$ containing $U$ and being closed under small
coproducts.
Suppose that $\mathcal{T}$ and $\mathcal{T}'$ are triangulated
categories with small coproducts. If $F:\mathcal{T}\ra \mathcal{T}'$
is a triangle functor which commutes with small coproducts, then
$F({\rm Tria}(U))\subseteq {\rm Tria}(F(U))$ for every object $U$ in
$\mathcal{T}$.
\subsection{Homological formulas}
In this paper, all rings considered are assumed to be associative
and with identity, and all ring homomorphisms preserve identity.
Unless stated otherwise, all modules are referred to left modules.
Let $R$ be a ring. We denote by $R\Modcat$ the category of all
unitary left $R$-modules, by $\Omega_R^n$ the $n$-th syzygy operator
of $R\Modcat$ for $n\in\mathbb{N}$, and regard $\Omega_R^0$ as the
identity operator of $R\Modcat$.
If $M$ is an $R$-module and $I$ is a nonempty set, then we denote by
$M^{(I)}$ and $M^I$ the direct sum and product of $I$ copies of $M$,
respectively. If $f: M\ra N$ is a homomorphism of $R$-modules, then
the image of $x\in M$ under $f$ is denoted by $(x)f$ instead of
$f(x)$. The endomorphism ring of the $R$-module $M$ is denoted by
$\End_R(M)$. Thus $M$ becomes a natural $R$-$\End_R(M)$-bimodule.
Similarly, if $N_R$ is a right $R$-module, then, by our convention,
$N$ is a left $(\End(N_R))\opp$- right $R$-bimodule.
As usual, we simply write $\C{R}$, $\K{R}$ and $\D{R}$ for
$\C{R\Modcat}$, $\K{R\Modcat}$ and $\D{R\Modcat}$, respectively, and
identify $R\Modcat$ with the subcategory of $\D{R}$ consisting of
all stalk complexes concentrated in degree zero. Let $\C {\pmodcat
R}$ be the full subcategory of $\C R$ consisting of those complexes
such that all of their terms are finitely generated projective
$R$-modules.
For each $n\in\mathbb{Z}$, we denote by $H^n(-):\D{R}\to R\Modcat$
the $n$-th cohomology functor. A complex $\cpx{X}$ is said to be
\emph{acyclic (or exact)} if $H^n(\cpx{X})=0$ for all
$n\in\mathbb{Z}$.
In the following, we shall recall some definitions and basic facts
about derived functors defined on derived module categories. For
more details and proofs, we refer to \cite{bn,weibel,HHK,CX3}.
Recall that $\K{R}_P$ (respectively, $\K{R}_I$) denotes the smallest
full triangulated subcategory of $\K{R}$ which
(i) contains all the bounded-above (respectively, bounded-below)
complexes of projective (respectively, injective) $R$-modules, and
(ii) is closed under arbitrary direct sums (respectively, direct
products).
Let $\K{R}_C$ be the full subcategory of $\K{R}$ consisting of all
acyclic complexes. Then $(\K{R}{_P}, \K{R}{_C})$ forms a hereditary
torsion pair in $\K{R}$ in the following sense:
\smallskip
$(a)$ Both $\K{R}{_P}$ and $\K{R}{_C}$ are full triangulated
subcategories of $\K{R}$.
$(b)$ $\Hom_{\K R}(\cpx{M}, \cpx{N})=0$ for $\cpx{M}\in\K{R}{_P}$
and $\cpx{N}\in\K{R}{_C}$.
$(c)$ For each $\cpx{X}\in \K{R}$, there exists a distinguished
triangle in $\K{R}$:
$${_p}\cpx{X} \lraf{\alpha_{\cpx X}}
\cpx{X} \lra {_c}\cpx{X}\lra (_p\cpx{X})[1]$$ such that
${_p}\cpx{X}\in \K{R}{_P}$ and ${_c}\cpx{X}\in\K{R}{_C}.$
In particular, for each complex $\cpx{X}$ in $\K R$, the chain map
${_p}\cpx{X} \lraf{\alpha_{\cpx X}} \cpx{X} $ is a quasi-isomorphism
in $\K R$. The complex $_p\cpx{X}$ is called the \emph{projective
resolution} of $\cpx{X}$ in $\D{R}$. For example, if $X$ is an
$R$-module, then we can choose $_pX$ to be a deleted projective
resolution of $_RX$.
Note also that the property $(b)$ implies that each
quasi-isomorphism between complexes in $\K{R}{_P}$ is an isomorphism
in $\K{R}$, that is a chain homotopy equivalence in $\K R$.
Dually, the pair $(\K{R}{_C}, \K{R}{_I})$ is a hereditary torsion
pair in $\K{R}$. This means that, for each $\cpx{X}$ in $\D{R}$,
there exists a complex ${_i}\cpx{X}\in \K{R}_I$ together with a
quasi-isomorphism $\beta_{\cpx{X}}: \cpx{X}\to{_i}\cpx{X}$. The
complex $_i\cpx{X}$ is called the \emph{injective resolution} of
$\cpx{X}$ in $\D{R}$.
More important, the composition functors
$$\K{R}_P\hookrightarrow\K{R}\lra\D{R}\quad\mbox{and}\quad
\K{R}_I\hookrightarrow\K{R}\lra\D{R}$$ are equivalences of
triangulated categories, and the canonical localization functor
$q:\K{R}\to \D{R}$ induces an isomorphism
$\Hom_{\K{R}}(\cpx{X},\cpx{Y})\lraf{\simeq}\Hom_{\D{R}}(\cpx{X},\cpx{Y})$
of abelian groups whenever either $\cpx{X}\in\K{R}_P$ or
$\cpx{Y}\in\K{R}_I$.
For a triangle functor $F:\K{R}\to\K{S}$, we define its \emph{total
left-derived functor} ${\mathbb L}F:\D{R}\to\D{S}$ by
$\cpx{X}\mapsto F(_p\cpx{X})$, and its \emph{total right-derived
functor} ${\mathbb R}F:\D{R}\to\D{S}$ by $\cpx{X}\mapsto
F(_i\cpx{X})$. Specially, if $F$ preserves acyclicity, that is,
$F(\cpx{X})$ is acyclic whenever $\cpx{X}$ is acyclic, then $F$
induces a triangle functor $D(F):\D{R}\to\D{S}$ defined by
$\cpx{X}\mapsto F(\cpx{X})$. In this case, up to natural
isomorphism, we have ${\mathbb L}F={\mathbb R}F=D(F)$, and simply
call $D(F)$ the \emph{derived functor} of $F$.
Let $\cpx{M}$ be a complex of $R$-$S$-bimodules. Then, the tensor
functor and the Hom-functor
$$\cpx{M}\cpx{\otimes}_S-:\K{S}\to\K{R}\quad\mbox{and}\quad\cpx{\Hom}_R(\cpx{M},-):\K{R}\to\K{S}$$
form a pair of adjoint triangle functors. For the concise
definitions of the tensor and Hom complex of two complexes, we
refer, for example, to \cite[Section 2.1]{CX3}. For simplicity, if
$Y\in S\Modcat$ and $X\in R\Modcat$, we denote
$\cpx{M}\cpx{\otimes}_SY$ and $\cpx{\Hom}_R(\cpx{M},\,X)$ by
$\cpx{M}\otimes_SY$ and $\Hom_R(\cpx{M},\,X)$, respectively.
Denote by $\cpx{M}\otimesL_S-$ the total left-derived functor of
$\cpx{M}\cpx{\otimes}_S-$, and by ${\mathbb R}\Hom_R(\cpx{M},-)$ the
total right-derived functor of $\cpx{\Hom}_R(\cpx{M},-)$. Note that
$\big(\cpx{M}\otimesL_S-, {\mathbb R}\Hom_R(\cpx{M},-)\big)$ is
still an adjoint pair of triangle functors.
\smallskip
The following result is freely used, but not explicitly stated in
the literature. Here, we will arrange it as a lemma for later
reference. For the idea of its proof, we refer to \cite[Generalized
Existence Theorem 10.5.9]{weibel}.
\begin{Lem}\label{homo}
Let $R$ and $S$ be rings, and let $H:\K{R}\lra \K{S}$ be a triangle
functor.
$(1)$ Define $\mathcal{L}{_H}$ to be the full subcategory of $\K{R}$
consisting of all complexes $\cpx{X}$ such that the chain map
$H(\alpha_{\cpx{X}}): H({_p}\cpx{X})\lra H(\cpx{X})$ is a
quasi-isomorphism in $\K{S}$. Then
$(i)$ $\mathcal{L}{_H}$ is a triangulated subcategory of $\K{R}$
containing $\K{R}{_P}$.
$(ii)$ $\mathcal{L}{_H}\cap{\K{R}{_C}} = \{\cpx{X}\in\K{R}{_C}\mid
H(\cpx{X})\in\K{S}{_C}\}$.
$(iii)$ There exists a commutative diagram of triangle functors:
$$
\xymatrix{\K{R}{_P}\ar[r]^-{\simeq}\ar[d]_-{\simeq} & \D{R}\ar[d]^-{{\mathbb L}H}\\
\mathcal{L}{_H}/\mathcal{L}{_H}\cap{\K{R}{_C}}\ar[r]^-{D(H)} &
\D{S}}
$$
where $\mathcal{L}{_H}/\mathcal{L}{_H}\cap{\K{R}{_C}}$ denotes the
Verdier quotient of $\mathcal{L}{_H}$ by
$\mathcal{L}{_H}\cap{\K{R}{_C}}$, and where $D(H)$ is defined by
$\cpx{X}\mapsto H(\cpx{X})$ for $\cpx{X}\in\mathcal{L}{_H}$.
$(2)$ Define $\mathcal{R}{_H}$ to be the full subcategory of $\K{R}$
consisting of all complexes $\cpx{X}$ such that the chain map
$H(\beta_{\cpx{X}}): H(\cpx{X})\to H({_i}\cpx{X})$ is a
quasi-isomorphism in $\K{S}$. Then
$(i)$ $\mathcal{R}{_H}$ is a triangulated subcategory of $\K{R}$
containing $\K{R}{_I}\,$.
$(ii)$ $\mathcal{R}{_H}\cap{\K{R}{_C}} = \{\cpx{X}\in\K{R}{_C}\mid
H(\cpx{X})\in\K{S}{_C}\}$.
$(iii)$ There exists a commutative diagram of triangle functors:
$$
\xymatrix{\K{R}{_I}\ar[r]^-{\simeq}\ar[d]_-{\simeq} & \D{R}\ar[d]^-{{\mathbb R}H}\\
\mathcal{R}{_H}/\mathcal{R}{_H}\cap{\K{R}{_C}}\ar[r]^-{D(H)} &
\D{S}}
$$
where $\mathcal{R}{_H}/\mathcal{R}{_H}\cap{\K{R}{_C}}$ denotes the
Verdier quotient of $\mathcal{R}{_H}$ by
$\mathcal{R}{_H}\cap{\K{R}{_C}}$, and where $D(H)$ is defined by
$\cpx{X}\mapsto H(\cpx{X})$ for $\cpx{X}\in\mathcal{R}{_H}$.
\end{Lem}
\medskip
Note that if $H$ commutes with arbitrary direct sums, then
$\mathcal{L}{_H}$ is closed under arbitrary direct sums in $\K R$.
Dually, if $H$ commutes with arbitrary direct products, then
$\mathcal{R}{_H}$ is closed under arbitrary direct products in $\K
R$.
From Lemma \ref{homo}, we see that, up to natural isomorphism, the
action of the functor ${\mathbb L}H$ (respectively, ${\mathbb R}H$)
on a complex $\cpx{X}$ in $\mathcal{L}{_H}$ (respectively,
$\mathcal{R}{_H}$) is the same as that of the functor $H$ on
$\cpx{X}$. Based on this point of view, we obtain the following
result which will be applied in our later proofs.
\begin{Koro} \label{counit}
Let $R$ and $S$ be two rings. Suppose that $(F, G)$ is a pair of
adjoint triangle functors with $F:\K{S}\to \K{R}$ and $G:\K{R}\to
\K{S}$. Let $\theta: FG\to Id_{\K R}$ and $\varepsilon: ({\mathbb
L}F) ({\mathbb R}G) \to Id_{\D R}$ be the counit adjunctions. If
$\cpx{X}\in \mathcal{R}{_G}$ and $G(\cpx{X})\in \mathcal{L}{_F}$,
then there exists a commutative diagram in $\D{R}$:
$$
\xymatrix
{({\mathbb L}F)({\mathbb R}G)(\cpx{X})\ar[r]^-{\varepsilon_{\cpx{X}}}\ar[d]_-{\simeq} & \cpx{X}\ar@{=}[d]\\
F G(\cpx{X}) \ar[r]^-{\theta_{\cpx{X}}} & \cpx{X}}
$$
\end{Koro}
{\it Proof.} It follows from $\cpx{X}\in \mathcal{R}{_G}$ that the
quasi-isomorphism $\beta_{\cpx{X}}: \cpx{X}\to {_i}\cpx{X}$ in
$\K{R}$ induces a quasi-isomorphism $G(\beta_{\cpx{X}}):
G(\cpx{X})\to G({_i}\cpx{X})$ in $\K{S}$. Since $(\K{S}{_P},
\K{S}{_C})$ is a hereditary torsion pair in $\K{S}$, there exists a
homomorphism $_pG(\beta_{\cpx{X}}): {_p}G(\cpx{X})\to
{_p}G({_i}\cpx{X})$ in $\K{S}$ such that the following diagram is
commutative:
$$
\xymatrix{
_pG(\cpx{X})\ar[r]^-{\alpha_{\,G(\cpx{X})}}\ar[d]_-{_pG(\beta_{\cpx{X}})}\;
&\;G(\cpx{X})\ar[d]^-{G(\beta_{\cpx{X}})}\\
{_p}G(_i\cpx{X})\ar[r]^-{\alpha_{\,G(_i\cpx{X})}} & \,G(_i\cpx{X})}
$$
Note that $_pG(\beta_{\cpx{X}})$ is a quasi-isomorphism in $\K{S}$
since all the other chain maps in the above diagram are
quasi-isomorphisms. By the property $(b)$ related to the pair
$(\K{S}{_P}, \K{S}{_C})$, we know that $_pG(\beta_{\cpx{X}})$ is an
isomorphism in $\K{S}$, and therefore the chain map
$F(_pG(\beta_{\cpx{X}})): F({_p}G(\cpx{X}))\lra
F({_p}G({_i}\cpx{X}))$ is an isomorphism in $\K{R}$.
Now, we can easily construct the following commutative diagram in
$\K{R}$:
$$
\xymatrix{&
F(_pG(\cpx{X}))\ar[r]^-{F(\alpha_{\,G(\cpx{X})})}\ar[d]_-{F(_pG(\beta_{\cpx{X}}))}^-{\simeq}
& \;FG(\cpx{X}) \ar[r]^-{\theta_{\cpx{X}}} \ar[d]^-{FG(\beta_{\cpx{X}})} &\cpx{X}\ar[d]^-{\beta_{\cpx{X}}}\\
({\mathbb L}F)({\mathbb R}G)(\cpx{X})\ar@{=}[r] &
F({_p}G({_i}\cpx{X}))\ar[r]^-{F(\alpha_{\,G({_i}\cpx{X})})}\;\,&
\;\,FG({_i}\cpx{X})\ar[r]^-{\theta {_{_i\cpx{X}}}} & {_i}\cpx{X} }
$$
Since $G(\cpx{X})\in\mathcal{L}{_F}$ by assumption, the chain map
$F(\alpha_{\,G(\cpx{X})})$ is a quasi-isomorphism in $\K{R}$, and is
an isomorphism in $\D{R}$. Clearly, the quasi-isomorphism
$\beta_{\cpx{X}}$ is an isomorphism in $\D{R}$.
Furthermore, the counit $\varepsilon_{\cpx{X}}: ({\mathbb
L}F)({\mathbb R}G)(\cpx{X})\lra \cpx{X}$ is actually given by the
composite of the following homomorphisms in $\D R$:
$$\xymatrix{
({\mathbb L}F)({\mathbb R}G)(\cpx{X})\ar@{=}[r] &
F({_p}G({_i}\cpx{X}))\ar[r]^-{F(\alpha_{G({_i}\cpx{X})})}\;&
\;FG({_i}\cpx{X})\ar[r]^-{\theta_{_i\cpx{X}}} & _i\cpx{X}
\ar[r]^-{\;(\beta_{\cpx{X}})^{-1}} & \cpx{X}.}
$$
Define $$\tau=
\big(F(_pG(\beta_{\cpx{X}}))\big)^{-1}\,F(\alpha_{\,G(\cpx{X})}):\;
({\mathbb L}F)({\mathbb R}G)(\cpx{X})\lra F G(\cpx{X})$$ which is an
isomorphism in $\D{R}$. It follows that there exists a commutative
diagram in $\D{R}$:
$$ \xymatrix
{({\mathbb L}F)({\mathbb R}G)(\cpx{X})\ar[r]^-{\varepsilon_{\cpx{X}}}\ar[d]^-{\tau} & \cpx{X}\ar@{=}[d]\\
F G(\cpx{X}) \ar[r]^-{\theta_{\cpx{X}}} & \cpx{X}}
$$
This finishes the proof. $\square$
\smallskip
As a preparation for our later proofs, we mention the following
three homological formulas which are related to derived functors or
total derived functors. The first one is taken from \cite[Theorem
3.2.1, Theorem 3.2.13, Remark 3.2.27]{EJ}.
\begin{Lem}\label{tor-ext}
Let $R$ and $S$ be rings. Suppose that $M$ is an $S$-$R$-bimodule
and $I$ is an injective $S$-module.
$(1)$ If $N$ is an $R$-module, then
$$
\Hom_S(\Tor_i^R(M, \,N),\, I)\simeq \Ext^i_R(N,\, \Hom_S(M,\,
I))\;\, \mbox{for all}\;\, i\geq 0.$$
$(2)$ If $L$ is an $R\opp$-module which has a finitely generated
projective resolution in $R\opp\Modcat$, then
$$
\Hom_S(\Ext_R^i(L,\,M),\, I)\simeq \Tor^R_i(L, \,\Hom_S(M, I))\;\,
\mbox{for all}\;\, i\geq 0.$$
\end{Lem}
\smallskip
The next formula is proved in \cite[Section 2.1]{CX3}.
\begin{Lem}\label{complex}
Let $R$ and $S$ be rings. Suppose that $\cpx{X}$ is a bounded
complex of $R$-$S$-bimodules. If $\cpx{X}\in \Cb{\pmodcat{R}}$, then
there is a natural isomorphism of functors:
$$\Hom_R(\cpx{X}, R)\cpx{\otimes}_R-\,\lraf{\simeq}\cpx{\Hom}_R(\cpx{X},-):\C{R}\to\C{S}.$$
In particular,
$$\Hom_R(\cpx{X}, R)\otimesL_R-\,\lraf{\simeq}\rHom{_R}(\cpx{X},-):\D{R}\to\D{S}.$$
\end{Lem}
\smallskip
The last formula is useful for us to calculate the cohomology groups
of tensor products of complexes.
\begin{Lem}\label{Formula} Let $n$ be an integer, and let $S$ be a ring and $M$ an $S\opp$-module. Suppose that
$\cpx{Y}:=(Y^i)_{i\in\mathbb{Z}}$ is a complex in $\C S$ such that
$Y^i=0$ for all $i\geq n+1$, and $\Tor^S_j(M, Y^i)=0$ for all
$i\in\mathbb{Z}$ and $j\geq 1$. Let $m\in\mathbb{Z}$ with $m < n$.
If $\,\Tor^S_t\big(M, H^{m+t}(\cpx{Y})\big)=0=\Tor^S_{t-1}\big(M,
H^{m+t}(\cpx{Y})\big)$ for $0\leq t \leq n-m-1$, then
$H^m(M\otimes_S\cpx{Y})\simeq \Tor^S_{n-m}\big(M,
H^n(\cpx{Y})\big)$.
\end{Lem}
{\it Proof.} Suppose that $\cpx{Y}$ is the following form:
$$
\cdots\lra Y^{m-1} \lraf{d^{m-1}} Y^{m} \lraf{d^m} Y^{m+1}\lra
\cdots\lra Y^{n-1} \lraf{d^{n-1}} Y^n\lra 0\lra \cdots$$ For
$i\in\mathbb{Z}$, define $C_i:=\Coker(d^{i-1})=Y^i/\Img(d^{i-1})$
and $I_i:=\Img(d^i)$. Then we have two short exact sequences of
$S$-modules for each $i\in \mathbb{Z}$:
$$
(a)\quad 0\lra H^i(\cpx{Y})\lra C_i\lraf{\pi_i} I_i\lra 0
\quad\mbox{and}\quad (b) \quad 0\lra I_i
\stackrel{\lambda_i}{\hookrightarrow} Y^{i+1}\lra C_{i+1}\lra 0.$$
Clearly, $H^i(\cpx{Y})=\Ker(\pi_i\lambda_i)$, and $d^{i}: Y^i\to
Y^{i+1}$ is just the composite of the canonical surjection $Y^i \to
C_i$ with $\pi_i\lambda_i: C_i \to Y^{i+1}$.
$(1)$ We claim that if $M\otimes_S H^i(\cpx{Y})=0$, then
$H^i(M\otimes_S\cpx{Y})\simeq \Tor^S_1(M, C_{i+1})$.
In fact, since $M\otimes_S-: S\Modcat\to \mathbb{Z}\Modcat$ is right
exact, the sequence
$$M\otimes_S Y^{i-1}\lraf{1\otimes d^{i-1}} M\otimes_S
Y^i\lra M\otimes_S C_i\lra 0$$ is exact, that is, $\Coker(1\otimes
d^{i-1})\simeq M\otimes_S C_i$. This implies that
$H^i(M\otimes_S\cpx{Y})\simeq \Ker(1\otimes \pi_i\lambda_i)$ where
$$1\otimes \pi_i\lambda_i=(1\otimes \pi_i) (1\otimes \lambda_i):
M\otimes_S C_i\to M\otimes_S Y^{i+1}, $$ which is the composite of
$1\otimes \pi_i: M\otimes_S C_i\lra M\otimes_S I_i$ with $1\otimes
\lambda_i: M\otimes_S I_i\lra M\otimes_S Y^{i+1}$.
Assume that $M\otimes_S H^i(\cpx{Y})=0$. Then $1\otimes\pi_i$ is an
isomorphism and $\Ker(1\otimes \pi_i\lambda_i)\simeq \Ker(1\otimes
\lambda_i)$. Now, we apply $M\otimes_S-$ to the sequence $(b)$, and
get the following exact sequence:
$$
\Tor^S_1(M, Y^{i+1})\lra \Tor^S_{1}(M, C_{i+1}) \lra M\otimes_S I_i
\lraf{1\otimes \lambda_i} M\otimes_S Y^{i+1}
$$
Since $\Tor^S_1(M, Y^{i+1})=0$ by assumption, we obtain
$\Tor^S_{1}(M, C_{i+1})\simeq \Ker(1\otimes \lambda_i).$ It follows
that $$H^i(M\otimes_S\cpx{Y})\simeq \Ker(1\otimes
\pi_i\lambda_i)\simeq \Ker(1\otimes \lambda_i)\simeq \Tor^S_1(M,
C_{i+1}).$$ This finishes the claim (1).
$(2)$ We show that, for any $j\geq 1$, if $\Tor^S_j(M,
H^i(\cpx{Y}))=0=\Tor^S_{j-1}(M, H^i(\cpx{Y}))$, then
$$\Tor^S_j(M,\,C_i)\lraf{\simeq} \Tor^S_{j+1}(M, C_{i+1}).$$
This follows from applying $M\otimes_S-$ to the exact sequences
($a$) and ($b$), respectively, together with our assumptions on
$\cpx{Y}$.
$(3)$ Let $m\in\mathbb{Z}$ with $m\leq n-1$. Suppose that
$$\Tor^S_t\big(M, H^{m+t}(\cpx{Y})\big)=0=\Tor^S_{t-1}\big(M,
H^{m+t}(\cpx{Y})\big)\quad \mbox{for}\quad 0\leq t \leq n-m-1. $$
Then, by taking $t=0$, we have $M\otimes_S H^m(\cpx{Y})=0$. Thanks
to (1), we have $H^m(M\otimes_S\cpx{Y})\simeq \Tor^S_1(M, C_{m+1})$.
Since $Y^i=0$ for $i\geq n+1$, it follows that $H^n(\cpx{Y})= C_n$.
This implies that if $n-m=1$, then $H^m(M\otimes_S\cpx{Y})\simeq
\Tor^S_{n-m}\big(M, H^n(\cpx{Y})\big)$.
Now, suppose $n-m\geq 2$. For $1\leq t\leq n-m-1$, we see from $(2)$
that $\Tor^S_t(M,\,C_{m+t})\lraf{\simeq} \Tor^S_{t+1}(M,
C_{m+t+1}).$ Thus
$$\Tor^S_1(M,\,C_{m+1})\simeq \Tor^S_{2}(M,
C_{m+2})\simeq\cdots\simeq \Tor^S_{n-m-1}(M,\,C_{n-1})\simeq
\Tor^S_{n-m}(M, C_{n}).
$$
Consequently, $H^m(M\otimes_S\cpx{Y})\simeq \Tor^S_1(M,
C_{m+1})\simeq\Tor^S_{n-m}(M, C_{n})= \Tor^S_{n-m}(M,
H^n(\cpx{Y}))$. This finishes the proof of Lemma \ref{Formula}.
$\square$
\subsection{Relative Mittag-Leffler modules} \label{sect2.3}
\medskip
Now, we recall the definition of relative Mittag-Leffler modules
(see \cite{Go}, \cite{HH}).
\begin{Def}\label{ML}{\rm
A right $R$-module $M$ is said to be \emph{$R$-Mittag-Leffler} if
the canonical map
$$\rho{_I}:\; M\otimes_RR^I\lra M^I,\; m\otimes(r_i)_{i\in I}\mapsto
(mr_i)_{i\in I} \, \mbox{ for } \; m\in M, \; r_i\in R,$$ is
injective for any nonempty set $I$.
A right $R$-module $M$ is said to be \emph{strongly
$R$-Mittag-Leffler} if the $m$-th syzygy of $M$ is
$R$-Mittag-Leffler for every $m\geq 0$.}
\end{Def}
By \cite[Theorem 1]{Go}, a right $R$-module $M$ is
$R$-Mittag-Leffler if and only if, for any finitely generated
submodule $X$ of $M_R$, the inclusion $X\to M$ factorizes through a
finitely presented right $R$-module. This implies that if $M$ is
finitely presented, then it is $R$-Mittag-Leffler. Actually, for
such a module $M$, the above map $\rho_I$ is always bijective (see
\cite[Theorem 3.2.22]{EJ}). Further, if the ring $R$ is right
noetherian, then each right $R$-module is $R$-Mittag-Leffler since
each finitely generated right $R$-module is finitely presented.
In the next lemma, we shall collect some basic properties of
Mittag-Leffler modules for later use.
\begin{Lem}\label{MLP}
Let $R$ be a ring and $M$ a right $R$-module. Then the following
statements are true.
$(1)$ If $M$ is $R$-Mittag-Leffler, then so is each module in
$\Add(M_R)$. In particular, each projective right $R$-module is
$R$-Mittag-Leffler.
$(2)$ The first syzygy of $M$ in $R{^{\rm op}}\Modcat$ is
$R$-Mittag-Leffler if and only if $\Tor^R_1(M, R^I)=0$ for every
nonempty set $I$.
$(3)$ $M$ is strongly $R$-Mittag-Leffler if and only if $M$ is
$R$-Mittag-Leffler and $\Tor^R_i(M, R^I)=0$ for each $i\geq 1$ and
every nonempty set $I$.
$(4)$ If $M$ is finitely generated, then $M$ is strongly
$R$-Mittag-Leffler if and only if $M$ has a finitely generated
projective resolution.
\end{Lem}
{\it Proof.} $(1)$ follows from the fact that tensor functors
commute with direct sums.
$(2)$ Note that the first syzygy $\Omega_R(M)$ of $M$ depends on the
choice of projective presentations of $M_R$. However, the
``$R$-Mittag-Leffler'' property of $\Omega_R(M)$ is independent of
the choice of projective presentations of $M_R$. This is due to (1)
and Schanuel's Lemma in homological algebra.
So, we choose an exact sequence $$0\lra K_1\lraf{f} P_1\lra M\lra
0$$ of right $R$-modules with $P_1$ projective, and shall show that
$K_1$ is $R$-Mittag-Leffler if and only if $\Tor^R_1(M, R^I)=0$ for
any nonempty set $I$.
Obviously, we can construct the following exact commutative diagram:
$$\xymatrix{
0\ar[r] & \Tor^R_1(M, R^I)\ar[r] & K_1\otimes_RR^I\ar[r]^-{f\otimes
1}\ar[d]^-{\rho_2} & P_1\otimes_RR^I\ar[r]\ar[d]^-{\rho_1} & M\otimes_RR^I\ar[r]\ar[d] & 0\\
& 0\ar[r] & K_1{^I}\ar[r]^{f^I} & P_1{^I} \ar[r] & M{^I} \ar[r] & 0 }
$$
where $\rho_i$, $1\le i\le 2$, are the canonical maps (see
Definition \ref{ML}). Since the projective module $P_1$ is
$R$-Mittag-Leffler by $(1)$, the map $\rho_1$ is injective. This
means that $\rho_2$ is injective if and only if so is $f\otimes 1$.
Clearly, the former is equivalent to that $K_1$ is
$R$-Mittag-Leffler, while the latter is equivalent to that
$\Tor^R_1(M, R^I)=0$. This finishes the proof of $(2)$.
$(3)$ For each $i\geq 0$, let $\Omega^i_R(M)$ stand for the $i$-th
syzygy of $M$ in $R{^{\rm op}}\Modcat$. Then, for each nonempty set
$I$, we always have
$$\Tor^R_{i+1} (M, R^I)\simeq \Tor^R_1 (\Omega^{i}_R(M),\, R^I).$$
Now $(3)$ follows immediately from $(2)$.
$(4)$ The sufficient condition is clear. Now suppose that $M$ is
strongly $R$-Mittag-Leffler. We need only to show that the first
syzygy of $M$ is finitely generated, that is, $M$ is finitely
presented. However, this follows from the fact that the inclusion
map $M\hookrightarrow M$ factorizes through a finitely presented
right $R$-module. $\square$
\medskip
A special class of strongly Mittag-Leffler modules is the class of
tilting modules. The following result can be concluded from
\cite[Corollary 9.8]{HH}, which will play an important role in our
proof of the main result.
\begin{Lem}\label{TML}
If $M$ is a tilting right $R$-module, then $M$ is strongly
$R$-Mittag-Leffler.
\end{Lem}
As a corollary of Lemmas \ref{TML} and \ref{MLP} (4), we obtain the
following result which is a generalization of \cite[Corollary
4.7]{CX1}.
\begin{Koro}\label{2.9}
Let $M$ be a tilting right $R$-module. If $M$ is finitely generated,
then $M$ is classical.
\end{Koro}
{\bf Proof.} Suppose that $M_R$ is finitely generated. Then we can
get an exact sequence $(T3)'$ from $(T3)$ by using the argument in
\cite[Corollary 4.7]{CX1} repeatedly. This shows that $M_R$ is
actually a good tilting module. Since $M$ is strongly
$R$-Mittag-Leffler, it follows from Lemma \ref{MLP} (4) that $M$
admits a finitely generated projective resolution. Clearly, such a
resolution can be chosen to be of finite length since $M$ has finite
projective dimension. This implies that $M_R$ is classical.
$\square$
\section{Homological subcategories of derived module categories}\label{sect3}
In this section, we shall give the definitions of bireflective and
homological subcategories of derived module categories. In
particular, we shall establish some applicable criterions for
bireflective subcategories to be homological.
Let $R$ and $S$ be arbitrary rings.
Let $\lambda: R\ra S$ be a homomorphism of rings. We denote by
$\lambda_*:S\Modcat\to R\Modcat$ the restriction functor induced by
$\lambda$, and by $D(\lambda_*):\D{S}\to\D{R}$ the derived functor
of the exact functor $\lambda_*$. Recall that $\lambda$ is a
\emph{ring epimorphism} if $\lambda_*:S\Modcat\to R\Modcat$ is fully
faithful. This is equivalent to saying that the multiplication map
$S\otimes_RS\ra S$ is an isomorphism in $S\Modcat$.
Two ring epimorphisms $\lambda: R\to S$ and $\lambda': R\to S'$ are
said to be \emph{equivalent} if there is an isomorphism $\psi: S\to
S'$ of rings such that $\lambda'=\lambda\psi$. Note that there is a
bijection between the equivalence classes of ring epimorphisms
staring from $R$ and bireflective full subcategories of $R\Modcat$,
and that there is a bijection between bireflective full
subcategories of $R\Modcat$ and the abelian full subcategories of
$R\Modcat$ which are closed under arbitrary direct sums and direct
products (see, for example, \cite[Lemma 2.1]{CX1}).
Recall that a ring epimorphism $\lambda: R\ra S$ is called
\emph{homological} if $\;\Tor^R_i(S, S)=0$ for all $i>0$. This is
equivalent to that the functor $D(\lambda_*):\D{S}\to\D{R}$ is fully
faithful, or that $S\otimesL_RS\simeq S$ in $\D{S}$. It is known
that $D(\lambda_*)$ has a left adjoint $S\otimesL_R-$ and a right
adjoint $\rHom_R(S, -)$.
\label{def-homo} Let $\mathcal{Y}$ be a full triangulated
subcategory of $\D{R}$. We say that $\mathcal{Y}$ is
\emph{bireflective} if the inclusion $\mathcal{Y}\to \D{R}$ admits
both a left adjoint and a right adjoint.
Combining \cite[Chapter I, Proposition 2.3]{BI} with \cite[Section
2.3]{CX1}, we know that a full triangulated subcategory
$\mathcal{Y}$ of $\D{R}$ is bireflective if and only if there exists
a recollement of triangulated categories of the form
$$
\xymatrix@C=1.2cm{\mathcal{Y}\ar[r]^-{{i_*}}
&\D{R}\ar[r]\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]
&\mathcal{X}\ar@/^1.2pc/[l]\ar@/_1.2pc/[l] }\vspace{0.2cm}$$ where
$i_*$ is the inclusion functor. Here, by a recollement of
triangulated categories (see \cite{BBD}) we mean that there are six
triangle functors between triangulated categories in the following
diagram:
$$\xymatrix{\mathcal{Y}\ar^-{i_*=i_!}[r]&\D{R}\ar^-{j^!=j^*}[r]
\ar^-{i^!}@/^1.2pc/[l]\ar_-{i^*}@/_1.6pc/[l]
&\mathcal{X}\ar^-{j_*}@/^1.2pc/[l]\ar_-{j_!}@/_1.6pc/[l]}$$ such
that
$(1)$ $(i^*,i_*),(i_!,i^!),(j_!,j^!)$ and $(j^*,j_*)$ are adjoint
pairs,
$(2)$ $i_*,j_*$ and $j_!$ are fully faithful functors,
$(3)$ $i^!j_*=0$ (and thus also $j^! i_!=0$ and $i^*j_!=0$), and
$(4)$ for each object $X\in\D{R}$, there are two canonical
distinguished triangles in $\D{R}$:
$$
i_!i^!(X)\lra X\lra j_*j^*(X)\lra i_!i^!(X)[1],\qquad j_!j^!(X)\lra
X\lra i_*i^*(X)\lra j_!j^!(X)[1],
$$ where $i_!i^!(X)\ra X$ and $j_!j^!(X)\ra
X$ are counit adjunction morphisms, and where $X\ra j_*j^*(X)$ and
$X\ra i_*i^*(X)$ are unit adjunction morphisms.
Note that $\mathcal{X}$ is always equivalent to the full subcategory
$\Ker\big(\Hom_{\D R}(-, \mathcal{Y})\big)$ of $\D{R}$ as
triangulated categories ( for example, see \cite[Lemma 2.6]{CX1}).
But here we do not require that the triangulated category
$\mathcal{X}$ must be a subcategory of $\D{R}$ in general. For more
examples of recollements related to homological ring epimorphisms,
we refer the reader to \cite{CX2}.
\medskip
Clearly, if $\mathcal{Y}$ is homological (see Definition in Section
\ref{sect1}), then it is bireflective. Let us now consider the
converse of this statement.
\smallskip
From now on, we assume that $\mathcal{Y}$ is a \textbf{bireflective
subcategory} of $\D{R}$, and define $\mathscr{E}:=\mathcal{Y}\cap
R\Modcat$.
It is easy to see that $\mathcal{Y}$ is closed under isomorphisms,
arbitrary direct sums and direct products in $\D{R}$. This implies
that $\mathscr{E}$ also has the above properties in $R\Modcat$.
Moreover, $\mathscr{E}$ always admits the ``$2$ out of $3$"
property: For an arbitrary short exact sequence in $R\Modcat$, if
any two of its three terms belong to $\mathscr{E}$, then the third
one belongs to $\mathscr{E}$. By \cite[Lemma 2.1]{CX1},
$\mathscr{E}$ is an abelian subcategory of $R\Modcat$ if and only if
$\mathscr{E}$ is closed under kernels (respectively, cokernels) in
$R\Modcat$. This is also equivalent to saying that there exists a
unique ring epimorphism $\lambda:R\to S$ (up to equivalence) such
that $\mathscr{E}$ is equal to $\Img(\lambda_*)$.
If $\mathcal{Y}$ is homological via a homological ring epimorphism
$\lambda: R\to S$, then $\mathcal{Y}=\Img\big(D(\lambda_*)\big)$ and
$\mathscr{E}=\Img(\lambda_*)$. In this case, $\mathscr{E}$ must be a
full, abelian subcategory of $R\Modcat$.
In general, for a bireflective subcategory $\mathcal Y$ in $\D{R}$,
the category $\mathscr{E}$ may not be abelian. This means that
bireflective subcategories in $\D{R}$ may not be homological.
Alternatively, we can reach this point by looking at differential
graded rings: By the proof of \cite[Chapter IV, Proposition
1.1]{BI}, the complex $i^*(R)$ is a compact generator of
$\mathcal{Y}$. In particular, we have
$\mathcal{Y}={\rm{Tria}}(i^*(R))$. It follows from \cite[Chapter 5,
Theorem 8.5]{HHK} that there exists a dg (differential graded) ring
such that its dg derived category is equivalent to $\mathcal{Y}$ as
triangulated categories. In general, this dg ring may have
non-trivial cohomologies in other degrees besides the degree $0$.
In other words, the category $\mathcal{Y}$ may not be realized by
the derived module category of an ordinary ring.
Let $i_*:\mathcal{Y}\to \D{R}$ be the inclusion functor with $i^*:
\D{R}\to \mathcal{Y}$ as its left adjoint. Define $\Lambda:=\End_{\D
R}(i^*(R))$. Then, associated with $\mathcal{Y}$, there is a ring
homomorphism defined by
$$ \delta:\; R\lra\Lambda,\quad r\mapsto i^*(\cdot
r) \, \mbox{ for } \, r\in R,$$ where $\cdot r: R\to R$ is the right
multiplication by $r$ map. This ring homomorphism induces a functor
$$\delta_*: \Lambda\Modcat\lra R\Modcat,$$
called the restriction functor.
\medskip
The following result is motivated by \cite[Section 6 and Section
7]{nr1}.
\begin{Lem} \label{ringhom}
The following statements hold true.
$(1)$ For each $\cpx{Y}\in\mathcal{Y}$, we have $H^n(\cpx{Y})\in
\Img(\delta_*)$ for all $n\in\mathbb{Z}$. In particular,
$H^n(i^*(R))$ is an $R$-$\Lambda$-bimodule for all $n\in
\mathbb{Z}$.
$(2)$ Let $\eta_R :R\to i_*i^*(R)$ be the unit adjunction morphism
with respect to the adjoint pair $(i^*, i_*)$. Then $\Lambda\simeq
H^0(i^*(R))$ as $R$-$\Lambda$-bimodules, and there exists a
commutative diagram of $R$-modules:
$$\xymatrix{R \ar[r]^-{\delta}\ar[rd]_-{H^0(\eta_R)} & \Lambda\ar[d]^-{\simeq}\\
& H^0(i^*(R))}$$
$(3)$ If $H^0(i^*(R))\in\mathcal{Y}$, then $H^n(i^*(R))=0$ for all
$n\geq 1$, the homomorphism $\delta$ is a ring epimorphism and
$$\mathcal{Y}=\{\cpx{Y}\in\D{R}\mid H^m(\cpx{Y})\in \Img(\delta_*)
\mbox{\;for\, all \,} m\in\mathbb{Z}\}.$$
\end{Lem}
\medskip
{\it Proof.} The proof of Lemma \ref{ringhom} is derived from
\cite[Section 6 and Section 7]{nr1}, where $\mathcal{Y}$ is related
to a set of two-term complexes in $\C{\pmodcat R}$.
By our convention, the full subcategory $\Img(\delta_*)$ of
$R\Modcat$ is required to be closed under isomorphisms in
$R\Modcat$.
Let $\eta_R :R\to i_*i^*(R)=i^*(R)$ be the unit adjunction morphism.
$(1)$ Let $\cpx{Y}\in\mathcal{Y}$. Then we obtain the following
isomorphisms for each $n\in\mathbb{Z}$:
$$
\Hom_{\D{R}}(i^*(R), \cpx{Y}[n])\lraf{\simeq} \Hom_{\D R}(R,
i_*(\cpx{Y}) [n])=\Hom_{\D{R}}(R, \cpx{Y}[n])\simeq H^n(\cpx{Y}),
$$
where the first isomorphism is given by $\Hom_{\D{R}}(\eta_R,
\cpx{Y}[n])$, which is actually an isomorphism of $R$-modules. Since
$\Hom_{\D R}(i^*(R), \cpx{Y}[n])$ is a left $\Lambda$-module, we
clearly have $H^n(\cpx{Y})\in \Img(\delta_*)$. If $\cpx{Y}= i^*(R)$,
then one can check that the composite of the following isomorphisms
$$ (\ast)\quad
\Hom_{\D{R}}(i^*(R), i^*(R)[n])\simeq \Hom_{\D R}(R,
i_*i^*(R)[n]))=\Hom_{\D{R}}(R, i^*(R)[n])\simeq H^n(i^*(R))$$ is an
isomorphism of $R$-$\Lambda$-bimodules. This implies that
$H^n(i^*(R))$ is an $R$-$\Lambda$-bimodule.
$(2)$ In $(\ast)$, we take $n=0$. This gives the first part of
$(2)$. For the second part of $(2)$, we note that there exists the
following commutative diagram of $R$-modules:
$$\xymatrix{\Hom_R(R, R) \ar[r]^-{i^*}\ar[rd]_-{\Hom_{\D R}(R, \eta_R)\qquad}
& \Hom_{\D R}(i^*(R), i^*(R))\ar[d]^-{\simeq}\\
& \Hom_{\D R}(R, i_*i^*(R))}$$ which implies the diagram in $(2)$ if
we identify $\Hom_R(R, R)$,\, $\Hom_{\D R}(R, i_*i^*(R))$ and
$\Hom_{\D R}(R, \eta_R)$ with $R$,\, $H^0(i^*(R))$ and
$H^0(\eta_R)$, respectively.
$(3)$ Define $$\mathcal{Y}':=\{\cpx{Y}\in\D{R}\mid H^m(\cpx{Y})\in
\Img(\delta_*) \mbox{\;for\, all \,} m\in\mathbb{Z}\}.$$ It follows
from $(1)$ that $\mathcal{Y}\subseteq \mathcal{Y}'$.
Suppose $H^0(i^*(R))\in\mathcal{Y}$. We shall prove that
$\mathcal{Y}'\subseteq \mathcal{Y}$, and so
$\mathcal{Y}=\mathcal{Y}'$.
In fact, from $(2)$ we see that $\Lambda\simeq H^0(i^*(R))$ as
$R$-modules, and so $_R\Lambda\in\mathcal{Y}$. Note that the derived
functor $D(\delta_*): \D{\Lambda}\to \D{R}$ admits a right adjoint,
and therefore it commutes with arbitrary direct sums. Since
$\mathcal{Y}$ is a full triangulated subcategory of $\D{R}$ closed
under arbitrary direct sums in $\D{R}$, it follows from
$\D{\Lambda}={\rm{Tria}}(_\Lambda\Lambda)$ and
$_R\Lambda\in\mathcal{Y}$ that $\Img\big( D(\delta_*)
\big)\subseteq\mathcal{Y}.$ In particular, $\Img(\delta_*)\subseteq
\mathcal{Y}$.
To prove $\mathcal{Y}'\subseteq \mathcal{Y}$, we shall use the
following statements $(a)$-$(d)$ mentioned in \cite[Lemma 4.6]{HKL}.
For the definitions of homotopy limits and homotopy colimits in
triangulated categories, we refer to \cite[Section 2]{bn}.
$(a)$ By canonical truncations, one can show that each bounded
complex over $R$ can be generated by its cohomologies, that is, if
$\cpx{M}\in \Cb R$, then $\cpx{M} $ belongs to the smallest full
triangulated subcategory of $\D{R}$ containing $H^n(\cpx{M})$ with
all $n\in\mathbb{Z}$.
$(b)$ Any bounded-above complex over $R$ can be expressed as the
homotopy limit of its bounded ``quotient'' complexes, which are
obtained from canonical truncations.
$(c)$ Any bounded-below complex over $R$ can be expressed as the
homotopy colimit of its bounded ``sub'' complexes, which are
obtained from canonical truncations.
$(d)$ Any complex is generated by a bounded-above complex and a
bounded-below complex obtained by canonical truncations.
Recall that $\mathcal{Y}$ is a full triangulated subcategory of
$\D{R}$ closed under arbitrary direct sums and direct products in
$\D{R}$. Therefore it is closed under taking homotopy limits and
homotopy colimits in $\D{R}$. Now, by the fact
$\Img(\delta_*)\subseteq \mathcal{Y}$ and the above statements
$(a)$-$(d)$, we can show that $\mathcal{Y}'\subseteq \mathcal{Y}$.
Thus $\mathcal{Y}=\mathcal{Y'}.$
Next, we shall show that $H^n(i^*(R))=0$ for all $n\geq 1$. The idea
of the proof given here is essentially taken from \cite[Lemma
6.4]{nr1}.
On the one hand, from the adjoint pair $(i^*, i_*)$, we can obtain a
triangle in $\D{R}$:
$$
\cpx{X}\lra R \lraf{\eta_R} i^*(R)\lra \cpx{X}[1].
$$
It is cleat that the unit $\eta_R$ induces an isomorphism $\Hom_{\D
R}(i^*(R), \cpx{Y}[n])\simeq \Hom_{\D R}(R, \cpx{Y}[n])$ for each
$\cpx{Y}\in\mathcal{Y}$ and $n\in\mathbb{Z}$. This implies that
$\Hom_{\D R}(\cpx{X}, \cpx{Y}[n])=0$ for $\cpx{Y}\in \mathcal{Y}$
and $n\in \mathbb{Z}$.
On the other hand, by the canonical truncation at degree $0$, we
obtain a distinguished triangle of the following form in $\D R$:
$$
i^*(R)^{\leq 0} \lraf{\alpha} i^*(R) \lraf{\beta} i^*(R)^{\geq 1} \lra i^*(R)^{\leq 0}[1]
$$
such that $H^{s}\big(\,i^*(R)^{\leq 0}\big )\simeq \left\{\begin{array}{ll} 0 & \;\mbox{if}\; s\geq 1,\\
H^s(i^*(R)) & \;\mbox{if}\;s\leq 0,\end{array} \right.$ and
$H^{t}\big(\,i^*(R)^{\geq 1}\big )\simeq \left\{\begin{array}{ll} 0 & \;\mbox{if}\; t\leq 0,\\
H^t(i^*(R)) & \;\mbox{if}\;t\geq 1.\end{array} \right.$
\medskip
\noindent It follows that $\eta_R\beta=0$ and that there exists a
homomorphism $\gamma: R\to i^*(R)^{\leq 0}$ such that $\gamma
\,\alpha =\eta_R$. Since $i^*(R)\in\mathcal{Y}=\mathcal{Y}'$, we
know that $i^*(R)^{\leq 0}\in\mathcal{Y}$ and $\Hom_{\D R}(\cpx{X},
i^*(R)^{\leq 0})=0$. Consequently, there exists a homomorphism
$\theta: i^*(R)\to i^*(R)^{\leq 0}$ such that $\gamma=\eta_R\,
\theta$. So, we have the following diagram in $\D{R}$:
$$\xymatrix{& & i^*(R)^{\leq 0}\ar[d]_-{\alpha}&\\
\cpx{X}\ar[r] &R\ar@{-->}[ur]^-{\gamma} \ar@{-->}[rd]^{0}
\ar[r]^-{\eta_R}
& i^*(R)\ar@/_1.4pc/[u]_-{\theta}\ar[r]\ar[d]_-{\beta}& \cpx{X}[1] \\
& & i^*(R)^{\geq 1}\ar[d]& \\
& & i^*(R)^{\leq 0}[1] &}
$$
Further, one can check that $\eta_R \theta\, \alpha=\gamma
\,\alpha=\eta_R$. Since $\eta_R :R\to i_*i^*(R)=i^*(R)$ is a unit
morphism, we infer that $\theta \alpha= Id_{i^*(R)}$, and so
$$H^n(\theta \alpha)=H^n(\theta) H^n(\alpha)=Id{_{H^n(i^*(R))}}\; \mbox{for any }\; n\in\mathbb{Z}.$$
This means that $H^n(\theta): H^n(i^*(R))\ra
H^{n}\big(\,i^*(R)^{\leq 0}\big )$ is injective. Observe that
$H^{n}\big(\,i^*(R)^{\leq 0}\big )=0$ for $n\geq 1$. Hence
$H^n(i^*(R))=0$ for $n\geq 1$.
Finally, we shall prove that $\delta: R\to \Lambda$ is a ring
epimorphism.
Clearly, the $\delta$ is a ring epimorphism if and only if for every
$\Lambda$-module $M$, the induced map $\Hom_R(\delta, M):
\Hom_R(\Lambda, M)\lra \Hom_R(R, M)$ is an isomorphism. Observe that
$\Hom_R(\delta,M)$ is always surjective. To see that this map is
also injective, we shall use the commutative diagram in $(2)$ and
show that the induced map
$$\Hom_R\big(H^0(\eta_R), M\big): \Hom_R\big( H^0(i^*(R)),
M\big)\lra \Hom_R(R, M)$$ is injective. That is, we have to prove
that if $f_i: H^0(i^*(R))\to M$, with $i=1,2$, are two homomorphisms
in $R\Modcat$ such that $H^0(\eta_R) f_1= H^0(\eta_R) f_2$, then
$f_1=f_2$.
Now, we describe the map $H^0(\eta_R)$. Recall that $H^n(i^*(R))=0$
for all $n\geq 1$. Without loss of generality, we may assume that
$i^*(R)$ is of the following form $\big($up to isomorphism in $\D
R\big)$:
$$\cdots \lra
V^{-n}\lraf{d^{-n}} V^{-n+1}\lra\cdots \lra V^{-1}\lraf{d^{-1}} V^0
\lra 0\lra \cdots$$ From the canonical truncation, we can obtain the
following distinguished triangle in $\D{R}$:
$$
\cpx{V}\,^{\leq -1}\lra i^*(R)\lraf{\pi} H^0(i^*(R))\lra \cpx{V}\,^{\leq -1}[1]
$$
where $\cpx{V}\,^{\leq -1}$ is of the form:
$$\cdots \lra V^{-n}\lra
V^{-n+1}\lra\cdots \lra V^{-2}\lra \Ker({d^{-1}}) \lra 0\lra
\cdots$$and $\pi$ is the chain map induced by the canonical
surjection $V^0\to H^0(i^*(R))=\Coker(d^{-1})$. Applying $H^0(-)=
\Hom_{\D R}(R,-)$ to the above triangle, we see that
$H^0(\eta_R)=\eta_R\,\pi$ in $\D{R}$ and that $H^0(\pi)$ is an
isomorphism of $R$-modules.
Suppose that $H^0(\eta_R) f_1= H^0(\eta_R) f_2: R\to M$ with $f_i:
H^0(i^*(R))\to M$ for $i=1,2$. Then $\eta_R\pi f_1=\eta_R\pi f_2$.
From the proof of $(2)$, we have $\Img(\delta_*)\subseteq
\mathcal{Y}$. Thus $_RM\in \mathcal{Y}$ since $M$ is an
$\Lambda$-module. Note that the unit $\eta_R :R\to
i_*i^*(R)=i^*(R)$ induces an isomorphism $\Hom_{\D R}(i^*(R),
M)\simeq \Hom_{\D R}(R, M)$. Thus $\pi\, f_1=\pi\,f_2$ and
$H^0(\pi)f_1=H^0(\pi)f_2$. It follows from the isomorphism of
$H^0(\pi)$ that $f_1=f_2$. This means that $\Hom_R\big(H^0(\eta_R),
M\big)$ is injective, and thus $\delta$ is a ring epimorphism. This
finishes the proof of $(3)$. $\square$
\medskip
In the following, we shall systematically discuss when bireflective
subcategories of derived categories are homological. Note that some
partial answers have been given in the literature, for example, see
\cite[Theorem 0.7 and Proposition 5.6]{nr1}, \cite[Proposition
1.7]{HKL} and \cite[Proposition 3.6]{CX1}. Let us first mention the
following criterions.
\begin{Lem} \label{hep}
Let $\mathcal{Y}$ be a bireflective subcategory of $\D{R}$, and let
$i^*: \D{R}\to \mathcal{Y}$ be a left adjoint of the inclusion
$\mathcal{Y}\hookrightarrow \D{R}$. Then the following are
equivalent:
$(1)$ $\mathcal{Y}$ is homological.
$(2)$ $H^m(i^*(R))=0$ for any $m\neq 0$.
$(3)$ $H^0(i^*(R))\in\mathcal{Y}$ and $H^m(i^*(R))=0$ for any
$m<0$.
$(4)$ $H^0(i^*(R))\in\mathcal{Y},$ and the associated ring
homomorphism $\delta: R\to \End_{\D R}(i^*(R))$ is a homological
ring epimorphism.
$(5)$ There exists a ring epimorphism $\lambda:R\to S$ such that
$_RS\in\mathcal{Y}$ and $i^*(R)$ is isomorphic in $\D R$ to a
complex $\cpx{Z}:=(Z^n)_{n\in\mathbb{Z}}$ with $Z^i\in S\Modcat$ for
$i=0, 1$.
$(6)$ $\mathscr{E}:=\mathcal{Y}\cap R\Modcat$ is an abelian
subcategory of $R\Modcat$ such that $i^*(R)$ is isomorphic in
$\D{R}$ to a complex $\cpx{Z}:=(Z^n)_{n\in\mathbb{Z}}$ with $Z^i\in
\mathscr{E}$ for $i=0, 1$.
In particular, if one of the above conditions is fulfilled, then
$\mathcal{Y}$ can be realized as the derived category of $\End_{\D
R}(i^*(R))$ via $\delta$.
\end{Lem}
{\it Proof.} It follows from the proof of \cite[Proposition
1.7]{HKL} that $(1)$ and $(2)$ are equivalent, and that $(2)$
implies both $(3)$ and $(4)$. By Lemma \ref{ringhom} (3), we know
that $(3)$ implies $(2)$.
Now, we show that $(4)$ implies $(1)$. In fact, since
$H^0(i^*(R))\in \mathcal{Y}$, it follows from Lemma \ref{ringhom}
(3) that
$$\mathcal{Y}=\{\cpx{Y}\in\D{R}\mid H^m(\cpx{Y})\in \Img(\delta_*)
\mbox{\;for\, all \,} m\in\mathbb{Z}\},$$ where $\delta: R\
\to\Lambda:= \End_{\D R}(i^*(R))$ is the associated ring
homomorphism. By assumption, $\delta$ is a homological ring
epimorphism, and therefore the derived functor
$D(\delta_*):\D{\Lambda}\to\D{R}$ is fully faithful. Furthermore, we
know from \cite[Lemma 4.6]{HKL} that
$$
\Img\big(D(\delta_*)\big)=\{\cpx{Y}\in\D{R}\mid H^m(\cpx{Y})\in
\Img(\delta_*) \mbox{\;for\, all \,} m\in\mathbb{Z}\}.
$$
Thus $\mathcal{Y}=\Img\big(D(\delta_*)\big)\subseteq \D{R}$, that
is, $\mathcal{Y}$ is homological by definition. Hence $(4)$ implies
$(1)$.
Consequently, we have proved that $(1)$-$(4)$ in Lemma \ref{hep} are
equivalent.
Note that $(5)$ and $(6)$ are equivalent because $\mathscr{E}$ is an
abelian subcategory of $R\Modcat$ if and only if there is a ring
epimorphism $\lambda:R\to S$ such that $\mathscr{E}=\Img(\lambda_*)$
(see \cite[Lemma 2.1]{CX1}).
In the following, we shall prove that $(1)$ implies $(5)$ and that
$(5)$ implies $(2)$.
Suppose that $\mathcal{Y}$ is homological, that is, there exists a
homological ring epimorphism $\lambda: R\to S$ such that the functor
$D(\lambda_*): \D{S}\to \D{R}$ induces a triangle equivalence from
$\D{S}$ to $\mathcal{Y}$. Thus $\mathcal{Y}=\Img(D(\lambda_*))$.
Since $i^*(R)\in\mathcal{Y}$, we have $i^*(R)\in\Img(D(\lambda_*))$.
It follows that there exists a complex
$\cpx{Z}:=(Z^n)_{n\in\mathbb{Z}}\in\C{S}$ such that
$i^*(R)\simeq\cpx{Z}$ in $\D{R}$. This shows $(5)$.
It remains to show that $(5)$ implies $(2)$. The idea of the
following proof arises from the proof of \cite[Proposition
3.6]{CX1}.
Let $\lambda:R\to S$ be a ring epimorphism satisfying the
assumptions in (5). We may identify $\Img(\lambda_*)$ with
$S\Modcat$ since $\lambda_*:S\Modcat\to R\Modcat$ is fully faithful.
Let $\cpx{Z}$ be a complex in $\C{R}$ such that $\cpx{Z}\simeq
i^*(R)$ in $\D{R}$. We may assume that $\cpx{Z}:=(Z^n,
d^n)_{n\in\mathbb{Z}}$ such that $Z^n\in S\Modcat$ for $n=0, 1$, and
define $\varphi=\Hom_{\D {R}}(\lambda, \cpx{Z}):\Hom_{\D R}(S,
\cpx{Z})\lra \Hom_{\D R}(R,\cpx{Z})$. We claim that the map
$\varphi$ is surjective.
In fact, there is a commutative diagram:
$$\xymatrix{\Hom{_{\K{R}}}(S, \cpx{Z})\ar[r]^-{q_1}\ar[d]^-{\varphi'}&
\Hom_{\D{R}}(S, \cpx{Z})\ar[d]^-{\varphi}\\
\Hom_{\K{R}}(R, \cpx{Z})\ar[r]^-{q_2}&\Hom_{\D{R}}(R, \cpx{Z}),}$$
where $\varphi'=\Hom_{\K{R}}(\lambda, \cpx{Z})$, and where $q_1$ and
$q_2$ are induced by the localization functor $q:\K{R}\to\D{R}$.
Clearly, the $q_2$ is a bijection. To prove that $\varphi $ is
surjective, it is sufficient to show that $\varphi'$ is surjective.
Let $\bar{\cpx{f}}:=\overline{(f^i)}\in \Hom_{\K{R}}(R,\cpx{Z})$
with $(f^i)_{i\in {\mathbb Z}}$ a chain map from $R$ to $\cpx{Z}$.
Then $f^i=0$ for any $i\neq 0$ and $f^0 d^0=0$. Since $Z^0$ is an
$S$-module, we can define $g: S\to Z^0$ by $s\mapsto s\,(1)f^0$ for
$s\in S$. One can check that $g$ is a homomorphism of $R$-modules
with $f^0=\lambda g$, as is shown in the following visual diagram:
$$\xymatrix{
& &R\ar[r]^-{\lambda}\ar[d]_-{f^0}& S\ar@{-->}[ld]_-{g}&&&\\
\cdots\ar[r]&Z^{-1}\ar[r]^-{d^{-1}} &Z^0\ar[r]^-{d^0} &
Z^1\ar[r]^-{d^1} & Z^2\ar[r]&\cdots}
$$
Since $\lambda:R\to S$ is a ring epimorphism and since $Z^1$ is an
$S$-module, the induced map $\Hom_R(\lambda, Z^1): \Hom_R(S, Z^1)\to
\Hom_R(R, Z^1)$ is a bijection. Thus, from this bijection together
with $\lambda g d^0=f^0d^0=0$, it follows that $gd^0=0$. Now, we can
define a morphism $\bar{\cpx{g}}:=\overline{(g^i)}\in
\Hom_{\K{R}}(S,\cpx{Z})$, where $(g^i)_{i\in {\mathbb Z}}$ is the
chain map with $g^0=g$ and $g^i=0$ for any $i\neq 0$. Thus
$\bar{\cpx{f}}=\lambda\bar{\cpx{g}}$. This shows that $\varphi'$ is
surjective. Consequently, the map $\varphi$ is surjective, and the
induced map
$$\Hom_{\D {R}}(\lambda, i^*(R)):\Hom_{\D R}(S, i^*(R))\to
\Hom_{\D R}(R, i^*(R))$$ is surjective since $\cpx{Z}\simeq i^*(R)$
in $\D{R}$.
Finally, we shall prove that $i^*(R)\simeq S$ in $\D{R}$. In
particular, this will give rise to $H^m(i^*(R))\simeq H^m(S)=0$ for
any $m\neq 0$, and therefore show $(2)$. So, it suffices to prove
that $i^*(R)\simeq S$ in $\D{R}$.
Indeed, let $i_*: \mathcal{Y}\to \D{R}$ be the inclusion, and let
$\eta_R :R\to i_*i^*(R)$ be the unit with respect to the adjoint
pair $(i^*, i_*)$. Clearly, $i^*(R)= i_*i^*(R)$ in $\D{R}$. Since we
have proved that $\Hom_{\D {R}}(\lambda, i^*(R))$ is surjective,
there exists a homomorphism $v: S\to i_*i^*(R)$ in $\D{R}$ such that
$\eta_R=\lambda\,v$. Furthermore, since $_RS$ belongs to
$\mathcal{Y}$ by assumption, we see that $\Hom_{\D R}(\eta_R,S):
\Hom_{\D R}(i^*(R),S)\ra \Hom_{\D R}(R,S)$ is an isomorphism. Thus
there exists a homomorphism $u: i_*i^*(R)\to S$ in $\D{R}$ such that
$\lambda=\eta_R\,u$. This yields the following commutative diagram
in $\D{R}$:
$$
\xymatrix{
R\ar@{=}[r]\ar[d]_-{\eta_R}&R\ar@{=}[r]\ar[d]_-{\lambda}&R \ar[d]^-{\eta_R}\\
i_*i^*(R)\ar@{-->}[r]^-{u}&S \ar@{-->}[r]^-{v}&\;i_*i^*(R), }
$$
which shows that $\eta_R=\eta_R uv$ and $\lambda=\lambda v u$. Since
$\Hom_{\D R}(\eta_R, i^*(R))$: $\Hom_{\D R}(i^*(R),i^*(R))\ra
\Hom_{\D R}(R,i_*i^*(R))$ is an isomorphism, we clearly have
$uv=1{_{i_*i^*(R)}}$. Note that $\Hom_R(\lambda,S):\Hom_R(S,
S)\to\Hom_R(R, S)$ is bijective since $\lambda:R\to S$ is a ring
epimorphism. It follows from $\lambda=\lambda vu$ that $vu=1_{S}$.
Thus the map $u$ is an isomorphism in $\D{R}$, and
$i^*(R)=i_*i^*(R)\simeq S$ in $\D{R}$. This shows that $(5)$ implies
$(2)$.
Hence all the statements in Lemma \ref{hep} are equivalent. This
finishes the proof. $\square$.
\medskip
Now, we mention a special bireflective subcategory of $\D{R}$, which
is constructed from complexes of finitely generated projective
$R$-modules. For the proof, we refer to \cite[Chapter III, Theorem
2.3; Chapter IV, Proposition 1.1]{BI}. See also \cite[Lemma
2.8]{CX1}.
\begin{Lem}\label{ref}
Let $\Sigma$ be a set of complexes in $\Cb{\pmodcat{R}}$. Define
$\mathcal{Y}:=\Ker\big(\,\Hom_{\D R}({\rm{Tria}}{(\Sigma)},
-)\big)$. Then $\mathcal{Y}$ is bireflective and equal to the full
subcategory of $\D {R}$ consisting of complexes $\cpx{Y}$ in $\D{R}$
such that $\Hom_{\D R}(\cpx{P}, \,\cpx{Y}[n])$ = $0$ for every
$\cpx{P}\in\Sigma$ and $n\in\mathbb{Z}$.
\end{Lem}
To develop properties of the bireflective subcategories of $\D{R}$
in Lemma \ref{ref}, we shall define the so-called generalized
localizations, which is motivated by a discussion with Silvana
Bazzoni in 2012. In fact, this notion was first discussed in
\cite{Kr} under the name ``homological localizations" for a set of
complexes in $\Cb{\pmodcat{R}}$, and is related to both the
telescope conjecture and algebraic $K$-theory. The reason for not
choosing the adjective word ``homological" in this note is that we
have reserved this word for ring epimorphisms.
\begin{Def} \label{genloc}{\rm
Let $R$ be a ring, and let $\Sigma$ be a set of complexes in $\C R$.
A homomorphism $\lambda_{\Sigma}: R\to R_{\Sigma}$ of rings is
called a \emph{generalized localization} of $R$ at $\Sigma$ provided
that
$(1)$ $\lambda_{\Sigma}$ is $\Sigma$-exact, that is, if $\cpx{P}$
belongs to $\Sigma$, then $R_{\Sigma}\otimes_R\cpx{P}$ is exact as a
complex over $R_{\Sigma}$, and
$(2)$ $\lambda_{\Sigma}$ is universally $\Sigma$-exact, that is, if
$S$ is a ring together with a $\Sigma$-exact homomorphism
$\varphi:R\to S$, then there exists a unique ring homomorphism
$\psi:R_{\Sigma}\to S$ such that $\varphi=\lambda_{\Sigma}\psi$. }
\end{Def}
If $\Sigma$ consists only of two-term complexes in
$\Cb{\pmodcat{R}}$, then the generalized localization of $R$ at
$\Sigma$ is the \emph{universal localization} of $R$ at $\Sigma$ in
the sense of Cohn (see \cite{cohenbook1}). It was proved in
\cite{cohenbook1} that universal localizations always exist.
However, generalized localizations may not exist in general. For a
counterexample, we refer the reader to \cite[Example 15.2]{Kr}.
We remark that, in Definition \ref{genloc} (1), if $\Sigma$ consists
of complexes in $ \Cb{\pmodcat{R}}$, then, for each
$\cpx{P}:=(P^i)_{i\in\mathbb{Z}}\in\Sigma$, the complex
$R_{\Sigma}\otimes_R\cpx{P}$ is actually split exact as a complex
over $R_{\Sigma}$ since $R_\Sigma\otimes_RP^i$ is a projective
$R_\Sigma$-module for each $i$. Further, by Definition \ref{genloc}
(2), if $\lambda_i: R\to R_i$ is a generalized localization of $R$
at $\Sigma$ for $i=1,2$, then $\lambda_1$ and $\lambda_2$ are
equivalent, that is, there exists a ring isomorphism $\rho: R_1\to
R_2$ such that $\lambda_2=\lambda_1\rho$.
Suppose that $\mathcal {U}$ is a set of $R$-modules each of which
possesses a finitely generated projective resolution of finite
length. For each $U\in\mathcal{U}$, we choose such a projective
resolution $_pU$ of finite length, and set $\Sigma:=\{_pU\mid
U\in\mathcal{U}\}\subseteq \Cb{\pmodcat{R}}$, and let
$R_\mathcal{U}$ be the generalized localization of $R$ at $\Sigma$.
If $_PU '$ is another choice of finitely generated projective
resolution of finite length for $U$ , then the generalized
localization of $R$ at $\Sigma':=\{_pU '\mid U\in {\mathcal U}\}$ is
isomorphic to $R_\mathcal{U}$, that is, $R_\mathcal{U}$ does not
depend on the choice of projective resolutions of $U$. Thus, we may
say that $R_{\mathcal U}$ is the \emph{generalized localization of
$R$ at $\mathcal U$.}
Generalized localizations have the following simple properties
(compare with \cite[Theorem 3.1 and Lemma 3.2]{CX1}).
\begin{Lem}\label{genpro}
Let $R$ be a ring and let $\Sigma$ be a set of complexes in
$\Cb{\pmodcat{R}}$. Assume that the generalized localization
$\lambda_{\Sigma}: R\to R_{\Sigma}$ of $R$ at $\Sigma$ exists. Then
the following hold.
$(1)$ For any homomorphism $\varphi:R_\Sigma\to S$ of rings, the
ring homomorphism $\lambda_{\Sigma}\varphi: R\to S$ is
$\Sigma$-exact.
$(2)$ The ring homomorphism $\lambda_{\Sigma}$ is a ring
epimorphism.
$(3)$ Define $\Sigma^*:=\{\Hom_R(\cpx{P}, R)\mid
\cpx{P}\in\Sigma\}.$ Then $\lambda_{\Sigma}$ is also the generalized
localization of $R$ at the set $\Sigma^*$. In particular,
$R_{\Sigma^*}\simeq R_{\Sigma}$ as rings.
\end{Lem}
{\it Proof.} $(1)$ For each $\cpx{P}\in\Sigma$, we have the
following isomorphisms of complexes of $S$-modules:
$$ S\otimes_R\cpx{P}\simeq
(S\otimes_{R_\Sigma}R_\Sigma)\otimes_R\cpx{P}\simeq
S\otimes_{R_\Sigma}(R_\Sigma\otimes_R\cpx{P}).$$ Since
$R_\Sigma\otimes_R\cpx{P}$ is split exact in $\C{R_\Sigma}$, we see
that $S\otimes_R\cpx{P}$ is also split exact in $\C{S}$. This means
that the ring homomorphism $\lambda_{\Sigma}\varphi$ is
$\Sigma$-exact.
$(2)$ Assume that $\varphi_i: R_\Sigma\to S$ is a ring homomorphism
for $i=1,2,$ such that
$\lambda_{\Sigma}\varphi_1=\lambda_{\Sigma}\varphi_2$. It follows
from $(1)$ that $\lambda_{\Sigma}\varphi_i$ is $\Sigma$-exact. By
the property $(2)$ in Definition \ref{genloc}, we obtain
$\varphi_1=\varphi_2$. This implies that $\lambda_{\Sigma}$ is a
ring epimorphism.
$(3)$ Note that $\cpx{P}$ is in $\Cb{\pmodcat{R}}$. It follows from
Lemma \ref{complex} that, for any homomorphism $R\to S$ of rings,
there are the following isomorphisms of complexes:
$$\Hom_R(\cpx{P}, R)\otimes_RS\simeq
\Hom_R(\cpx{P}, S)\simeq\Hom_R(\cpx{P}, \Hom_S({_S}S{_R},\,S))\simeq
\Hom_S(S\otimes_R\cpx{P}, S).$$ This implies that the complex
$\Hom_R(\cpx{P}, R)\otimes_RS$ is (split) exact in $\C{S\opp}$ if
and only if so is the complex $S\otimes_R\cpx{P}$ in $\C{S}$. Now,
$(3)$ follows immediately from the definition of generalized
localizations. $\square$
\medskip
In the following, we shall establish a relation between bireflective
subcategories of $\D{R}$ and generalized localizations. In
particular, the statements $(3)$ and $(4)$ in Lemma \ref{rg} below
will be useful for discussions in the next section and the proof of
Theorem \ref{main-result}.
\begin{Lem}\label{rg}
Let $\Sigma$ be a set of complexes in $\Cb{\pmodcat{R}}$, and let
$j_!:{\rm{Tria}}{(\Sigma)}\to \D{R}$ be the inclusion. Define
$\mathcal{Y}:=\Ker\big(\,\Hom_{\D R}({\rm{Tria}}{(\Sigma)},
-)\big)$. Then the following are true.
$(1)$ There exists a recollement of triangulated categories:
$$\xymatrix@C=1.2cm{\mathcal{Y}\ar[r]^-{{i_*}}
&\D{R}\ar[r]\ar@/^1.2pc/[l]\ar_-{i^*}@/_1.2pc/[l]
&{\rm{Tria}}{(\Sigma)} \ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{j_!\;}}$$
\smallskip
\noindent where $(i^*, i_*)$ is a pair of adjoint functors with
$i_*$ the inclusion.
\smallskip
$(2)$ The associated ring homomorphism $\delta: R\to
\Lambda:=\End_{\D{R}}(i^*(R))$ induced by $i^*$ admits the following
property: For any $\Sigma$-exact ring homomorphism $\varphi: R\to
S$, there exists a ring homomorphism $\psi: \Lambda \to S$ such that
$\varphi=\delta\psi$.
$(3)$ If $H^0(i^*(R))\in\mathcal{Y}$, then $\delta$ is a generalized
localization of $R$ at $\Sigma$. In particular, if the subcategory
$\mathcal{Y}$ of $\D{R}$ is homological, then $\delta$ is a
generalized localization of $R$ at $\Sigma$.
$(4)$ Define $\Sigma^*:=\{\Hom_R(\cpx{P}, R)\in
\Cb{\pmodcat{R\opp}}\mid \cpx{P}\in\Sigma\}$ and
$\mathcal{Y}':=\Ker\big(\,\Hom_{\D {R\opp}}({\rm{Tria}}{(\Sigma^*)},
-)\big).$ Then $\mathcal{Y}$ is homological in $\D{R}$ if and only
if so is $\mathcal{Y}'$ in $\D{{R\opp}}$.
\end{Lem}
{\it Proof.} $(1)$ can be concluded from \cite[Lemma 2.6 and Lemma
2.8]{CX1}.
$(2)$ The proof here is motivated by \cite[Lemma 7.3]{nr1}. Let
$\varphi: R\to S$ be a $\Sigma$-exact ring homomorphism. Since
$S\otimes_R\cpx{P}$ is exact in $\C{S}$ for $\cpx{P}\in\Sigma$, we
have $S\otimesL_R\cpx{P}=S\otimes_R\cpx{P}\simeq 0$ in $\D{S}$.
Further, the functor $S\otimesL_R-: \D{R}\to\D{S}$ commutes with
arbitrary direct sums, so $S\otimesL_R\cpx{X}\simeq 0$ for each
$\cpx{X}\in {\rm Tria}(\Sigma)$.
Let $\D{R}/{\rm Tria}(\Sigma)$ denote the Verdier quotient of
$\D{R}$ by the full triangulated subcategory ${\rm Tria}(\Sigma)$.
It follows from the recollement in $(1)$ that $i^*$ induces a
triangle equivalence:
$$\D{R}/{\rm Tria}(\Sigma)\lraf{\simeq}\mathcal{Y}.$$
Since $S\otimesL_R-$ sends ${\rm Tria}(\Sigma)$ to zero, there
exists a triangle functor $F:\mathcal{Y}\to \D{S}$ together with a
natural isomorphism of triangle functors:
$$\Phi:\;S\otimesL_R-\lraf{\simeq}
F\,i^*:\,\D{R}\lra \D{S}.$$ This clearly induces the following
canonical ring homomorphisms:
$$\Lambda:=\End_{\D{R}}(i^*(R)) \lraf{F}
\End_{\D{S}}\big(F(i^*(R))\big)\simeq \End_{\D S}(S\otimesL_RR)
\simeq \End_{\D S}(S)\simeq S
$$
where the first isomorphism is induced by the natural isomorphism
$\Phi_R: S\otimesL_RR\lra F(i^*(R))$ in $\D{S}$. Now, we define
$\psi: \Lambda \to S$ to be the composite of the above ring
homomorphisms. Then it is easy to check that $\varphi=\delta\psi$.
Consequently, the $\delta$ has the property mentioned in $(2)$.
$(3)$ Assume that $H^0(i^*(R))\in\mathcal{Y}$. By Lemma
\ref{ringhom} (3), the map $\delta$ is a ring epimorphism. Combining
this with $(2)$, we know that $\delta$ satisfies the condition $(2)$
in Definition \ref{genloc}. To see that $\delta $ is the generalized
localization of $R$ at $\Sigma$, we have to show that $\delta$
satisfies the condition $(1)$ in Definition \ref{genloc}, that is,
$\delta$ is $\Sigma$-exact.
In fact, by Lemma \ref{ringhom} (2), we have $\Lambda\simeq
H^0(i^*(R))$ as $R$-modules. This gives rise to
$_R\Lambda\in\mathcal{Y}$. Note that $\Hom_{\D R}(\cpx{X},
\cpx{Y})=0$ for $\cpx{X}\in {\rm{Tria}}{(\Sigma)}$ and
$\cpx{Y}\in\mathcal{Y}$. In particular, we have $\Hom_{\D
R}(\cpx{P}, \Lambda[n])=0$ for any $\cpx{P}\in\Sigma$ and
$n\in\mathbb{Z}$. It follows that $H^n(\Hom_R(\cpx{P},
\Lambda))\simeq \Hom_{\K R}(\cpx{P}, \Lambda[n])\simeq \Hom_{\D
R}(\cpx{P}, \Lambda[n])=0$, and therefore the complex
$\Hom_R(\cpx{P}, \Lambda)$ is exact. Since $\cpx{P}\in
\Cb{\pmodcat{R}}$, we have $\Hom_R(\cpx{P},
\Lambda)\in\Cb{\pmodcat{\Lambda\opp}}$. This implies that
$\Hom_R(\cpx{P}, \Lambda)$ is split exact, and therefore the complex
$\Hom_{\Lambda\opp}(\Hom_R(\cpx{P}, \Lambda), \Lambda)$ over
$\Lambda$ is split exact. Now, we claim that the latter complex is
isomorphic to the complex $\Lambda\otimes_R\cpx{P}$ in
$\C{\Lambda}$. Actually, this follows from the following general
fact in homological algebra:
For any finitely generated projective $R$-module $P$, there exists a
natural isomorphism of $\Lambda$-modules:
$$\Lambda\otimes_RP \lra \Hom_{\Lambda\opp}(\Hom_R(P, \Lambda),
\Lambda), \quad x\otimes p\mapsto [f\mapsto x\,(p)f]$$ for
$x\in\Lambda$, $p\in P$ and $f\in\Hom_R(P, \Lambda)$. Consequently,
the complex $\Lambda\otimes_R\cpx{P}$ is exact in $\C{\Lambda}$, and
thus $\delta$ is $\Sigma$-exact. Hence $\delta$ is a generalized
localization of $R$ at $\Sigma$.
Clearly, the second part of Lemma \ref{rg} (3) follows from the
equivalences of $(1)$ and $(4)$ in Lemma \ref{hep}.
$(4)$ We shall only prove the necessity of $(4)$ since the
sufficiency of $(4)$ can be proved similarly.
Suppose that $\mathcal{Y}$ is homological in $\D{R}$. It follows
from Lemma \ref{hep} (4) and Lemma \ref{rg} (3) that the ring
homomorphism $\delta: R\to \Lambda$ is not only a homological ring
epimorphism, but also a generalized localization of $R$ at
$\Sigma$. Moreover, by Lemma \ref{genpro} (3), the map $\delta$ is
also a generalized localization of $R$ at $\Sigma^*$.
Note that $\mathcal{Y}'$ is a bireflective subcategory of
$\D{R\opp}$ by Lemma \ref{ref}. Now, let ${\bf L}$ be a left adjoint
of the inclusion $\mathcal{Y}'\to \D{R\opp}$. To show that
$\mathcal{Y}'$ is homological in $\D{R\opp}$, we employ the
equivalences of $(1)$ and $(4)$ in Lemma \ref{hep}, and prove that
$(a)$ $H^0({\bf L}(R))\in\mathcal{Y}'$ and
$(b)$ the ring homomorphism $\delta': R\lra \Lambda':=\End_{\D
{R\opp}}({\bf L}(R))$ induced by $\bf{L}$ is homological.
Clearly, under the assumption $(a)$, we see from $(3)$ that
$\delta'$ is a generalized localization of $R$ at $\Sigma^*$. Since
$\delta$ is also a generalized localization of $R$ at $\Sigma^*$,
there exists a ring isomorphism $\rho: \Lambda' \lra \Lambda$ such
that $\delta=\delta' \rho$. Note that $\delta$ is homological. It
follows that $\delta'$ is homological.
It remains to show $(a)$. In fact, since $H^0({\bf L}(R))\simeq
\Lambda'$ as right $R$-modules by Lemma \ref{ringhom} (2), it is
sufficient to prove that the right $R$-module $\Lambda'$ belongs to
$\mathcal{Y}'$. However, by $(1)$ and Lemma \ref{ref}, we have
$$\mathcal{Y}'=\{\cpx{Y}\in\D{R\opp} \mid \Hom_{\D
{R\opp}}\big(\Hom_R(\cpx{P}, R),\, \cpx{Y}[n]\big)=0\; \mbox{for}\;
\cpx{P}\in\Sigma\; \mbox{and}\; n\in\mathbb{Z}\},$$ and by the
isomorphism $\rho$ and $\delta=\delta' \rho$, we get $\Lambda'
\simeq \Lambda$ as right $R$-modules. Consequently, to show
$\Lambda'_{R}\in {\mathcal Y}'$, it is enough to show that
$\Lambda_R$ belongs to $\mathcal{Y}'$, that is, we have to prove
that $\Hom_{\D {R\opp}}\big(\Hom_R(\cpx{P}, R),\, \Lambda[n]
\big)=0$ for any $\cpx{P}\in\Sigma$ and $n\in\mathbb{Z}$.
Let $\cpx{P}\in \Sigma$, and set $\cpx{P}{^*}:=\Hom_R(\cpx{P}, R)$.
Since $\cpx{P}$ is a complex in $\Cb{\pmodcat{R}}$, we see from
Lemma \ref{complex} that $\Hom_{R\opp}(\cpx{P}{^*}, \Lambda)\simeq
\Lambda\otimes_R\cpx{P}$ as complexes in $\C \Lambda$, and therefore
there exist the following isomorphisms:
$$\Hom_{\D {R\opp}}\big(\cpx{P}{^*},\, \Lambda[n]
\big)\simeq \Hom_{\K {R\opp}}\big(\cpx{P}{^*},\, \Lambda[n]
\big)\simeq H^n(\Hom_{R\opp}(\cpx{P}{^*}, \Lambda))\simeq
H^n(\Lambda\otimes_R\cpx{P}).$$ Since $\delta: R\to \Lambda$ is a
generalized localization of $R$ at $\Sigma$, the complex
$\Lambda\otimes_R\cpx{P}$ is exact in $\C{\Lambda}$, that is,
$H^n(\Lambda\otimes_R\cpx{P})=0$ for any $n\in\mathbb{Z}$. Thus
$\Hom_{\D {R\opp}}\big(\cpx{P}{^*},\, \Lambda[n] \big)=0$ for
$n\in\mathbb{Z}$. Thus $\Lambda_R\in \mathcal{Y}'$, and the proof of
the necessity of $(4)$ is completed. $\square$
\medskip
As an application of Lemma \ref{rg} (3), we have the following
result which says that generalized localizations can be constructed
from homological ring epimorphisms.
\begin{Koro}
Let $\lambda:R\to S$ be a homological ring epimorphism. Suppose that
$_RS$ has a finitely generated projective resolution of finite
length. Let $\cpx{P}$ be a complex in $\Cb{\pmodcat{R}}$, which is
isomorphic in $\D R$ to the mapping cone of $\lambda$. Then
$\lambda$ is a generalized localization of $R$ at $\cpx{P}$.
\end{Koro}
{\it Proof.} Since $\lambda$ is homological and $\cpx{P}$ is
isomorphic to the mapping cone of $\lambda$ in $\D{R}$, it follows
from \cite[Section 4]{NS} that there is a recollement of
triangulated categories:
$$
\xymatrix@C=1.2cm{\D{S}\ar[r]^-{D(\lambda_*)}
&\D{R}\ar[r]\ar@/^1.2pc/[l]\ar_-{S\otimesL_R-}@/_1.2pc/[l]
&{\rm{Tria}}{(\cpx{P})} \ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{j_!\;}}$$
\smallskip
\noindent where $j_!$ is the inclusion. This shows that
$\mathcal{Y}:=\Ker\big(\Hom_{\D R}({\rm Tria}(\cpx{P}),-)\big)$ is
equivalent to $\D S$. Thus $\mathcal Y$ is homological. Note that
$S\otimesL_RR\simeq S$ and $\End_R(_RS)\simeq S$. By Lemma \ref{rg}
(3), we know that $\lambda$ is a generalized localization of $R$ at
$\cpx{P}$. $\square$
\section{Ringel modules}
\label{sect4}
This section is devoted to preparations for proofs of our main
results in this paper. First, we introduce a special class of
modules, called Ringel modules, which can be constructed from both
good tilting and cotilting modules, and then discuss certain
bireflective subcategories (of derived module categories) arising
from Ringel modules. Finally, we shall describe when these
subcategories are homological. In particular, we shall establish a
key proposition, Proposition \ref{realization}, which will be
applied in later sections.
\smallskip
Throughout this section, let $R$ be an arbitrary ring, $M$ an
$R$-module and $S$ the endomorphism ring of $_RM$. Then $M$ becomes
naturally an $R$-$S$-bimodule. Further, let $n$ be an arbitrary but
fixed natural number.
\begin{Def}\label{rm}
{\rm The $R$-module $M$ is called an $n$-\emph{Ringel module}
provided that the following three conditions are fulfilled:
$(R1)$ there exists an exact sequence
$$
0\lra P_n \lra \cdots \lra P_1\lra P_0\lra M\lra 0$$ of $R$-modules
such that $P_i\in \add(_RR)$ for all $0\leq i\leq n$,
$(R2)$ $\Ext^j_R(M, M)=0$ for all $j\geq 1$, and
$(R3)$ there exists an exact sequence
$$
0\lra {}_RR \lra M_0 \lraf{\nu} M_1\lra \cdots \lra M_n\lra 0$$ of
$R$-modules such that $M_i\in\Prod(_RM)$ for all $0\leq i\leq n$.
\smallskip
An $n$-Ringel $R$-module $M$ is said to be \emph{perfect} if the
ring $S$ is right noetherian; and \emph{good} if
$(R4)$ the right $S$-module $M$ is strongly $S$-Mittag-Leffler (see
Definition \ref{ML}). }\end{Def}
Classical tilting modules are good Ringel modules. Conversely, for a
Ringel module $M$, if each $M_i$ in $(R3)$ is isomorphic to a direct
summand of finite direct products of copies of $M$, then $M$ becomes
a classical tilting module (see Introduction).
If a Ringel $R$-module $M$ has the property $\Prod(_RM) = \Add(_RM)$
(for example, $M_S$ is of finite length), then $_RM$ is a tilting
module. In this case, $_RM$ is even classical (see Corollary
\ref{2.9}).
Moreover, if the ring $S$ is right noetherian (see the statements
following Definition \ref{ML}), then any right $S$-module is
$S$-Mittag-Leffler. Thus each perfect Ringel $R$-module must be
good.
It is worth noting that good tilting (or cotilting) modules may not
be Ringel modules because it may not be finitely generated. For
example, the infinitely generated $\mathbb{Z}$-module
$\mathbb{Q}\oplus \mathbb{Q}/\mathbb{Z}$ is a good tilting module,
but not a Ringel module. Clearly, the good $1$-cotilting
$\mathbb{Z}$-module $\Hom_{\mathbb{Z}}(\mathbb{Q}\oplus
\mathbb{Q}/\mathbb{Z},\mathbb{Q}/{\mathbb Z})$ is not a Ringel
module.
Assume that $_RM$ satisfies $(R1)$. Then $M$ is isomorphic in
$\D{R}$ to the following complex of finitely generated projective
$R$-modules:
$$
\cdots \lra 0\lra P_n \lra \cdots \lra P_1\lra P_0\lra 0\lra\cdots
$$
It follows from Lemma \ref{ref} that
$\mathcal{Y}:=\{\cpx{Y}\in\D{R}\mid \Hom_{\D R}(M,\, \cpx{Y}[m])=0
\mbox{\;for\, all \,} m\in\mathbb{Z}\}$ is a bireflective
subcategory of $\D{R}$.
Now, assume that $M$ satisfies both $(R1)$ and $(R2)$. Then the
functors
$${\bf G}:={}_RM\otimesL_S-: \;\D{S}\lra\D{R}\quad \mbox{and}\quad {\bf
H}:=\rHom_R(M,-):\; \D{R}\lra\D{S}$$ induce a triangle equivalence:
$\D{S}\lraf{\simeq}{\rm{Tria}}{(_RM)}$ (see \cite[Chapter 5,
Corollary 8.4, Theorem 8.5]{HHK}). Moreover, $\mathcal{Y}=\Ker({\bf
H })$ since $H^m\big(\rHom_R(M, \cpx{Y})\big)\simeq \Hom_{\D R}(M,
\cpx{Y}[m])$ for each $\cpx{Y}\in\D{R}$ and $m\in\mathbb{Z}$.
Thus, by Lemma \ref{rg} (1) and (3) as well as Lemma \ref{hep}, we
have the following useful result for constructing recollements of
derived module categories.
\begin{Lem}\label{rt}
Suppose that the $R$-module $M$ satisfies $(R1)$ and $(R2)$. Then
there exists a recollement of triangulated categories:
$$(\ast)\quad\;
\xymatrix@C=1.2cm{\mathcal{Y}\ar[r]^-{{i_*}} &\D{R}\ar[r]^-{{\bf
H}}\ar@/^1.2pc/[l]\ar_-{i^*}@/_1.2pc/[l] &\D{S}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{{\bf G}} }$$
\medskip
\noindent where $(i^*, i_*)$ is a pair of adjoint functors with
$i_*$ the inclusion.
If, in addition, the category $\mathcal{Y}$ is homological in
$\D{R}$, then the generalized localization $\lambda: R\to R_M$ of
$R$ at $M$ exists and is homological, which induces a recollement of
derived module categories:
$$\xymatrix@C=1.2cm{\D{R_M}\ar[r]^-{D(\lambda_*)}
&\D{R}\ar[r]^-{{\bf H}}\ar@/^1.2pc/[l]\ar@/_1.2pc/[l] &\D{S}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{{\bf G}}}$$
\end{Lem}
\medskip
In the following, we shall consider when the category $\mathcal{Y}$
is homological. In general, this category is not homological since
the category
$$\mathscr{E}:=\mathcal{Y}\cap R\Modcat=\{Y\in
R\Modcat \mid \Ext^m_R(M, Y)=0\mbox{\;for\, all \,} m\geq0\}$$ may
not be an abelian subcategory of $R\Modcat$. So, we need to impose
some additional conditions on the module $M$.
By Lemma \ref{hep}, whether $\mathcal{Y}$ is homological is
completely determined by the cohomology groups of $i_*i^*(R)$. So,
to calculate these cohomology groups efficiently, we shall
concentrate on good Ringel modules.
From now on, we assume that $_RM$ is a {\bf good} $n$-Ringel module,
and define $\cpx{M}$ to be the complex
$$\cdots \lra 0\lra M_0
\lraf{\nu} M_1\lra \cdots \lra M_n\lra 0\lra \cdots$$ arising from
$(R3)$ in Definition \ref{rm}, where $M_i$ is in degree $i$ for
$0\leq i\leq n$.
First of all, we establish the following result.
\begin{Lem}\label{prep}
The following statements are true.
\smallskip
$(1)$ For each $X\in\Prod(_RM)$, the evaluation map $\theta_{X}:
M\otimes_S\Hom_R(M, X)\lra X$ is injective and $\Coker(\theta
_X)\in\mathscr{E}$.
$(2)$
$$H^j(i_*i^*(R))\simeq
\left\{\begin{array}{ll} 0 & \;\mbox{if}\; j<0,\\
H^{j+1}\big({_R}M\otimes_S\Hom_R(M,\,\cpx{M})\big)&
\;\mbox{if}\;j>0.\end{array} \right.$$
$(3)$ For $n=0$, the complex $i_*i^*(R)$ is isomorphic in $\D{R}$ to
the stalk complex $\Coker(\theta_{M_0})$. For $n\ge 1$, the complex
$i_*i^*(R)$ is isomorphic in $\D{R}$ to a complex of the form
$$
0\lra E^0 \lra E^1 \lra \cdots \lra E^{n-1}\lra 0
$$
with $E^m\in\mathscr{E}$ for $0\leq m\leq n-1$.
\end{Lem}
\medskip
{\it Proof.} Recall that $M$ is an $R$-$S$-bimodule with
$S=\End_R(M)$. So we have a pair of adjoint functors:
$$_RM\otimes_S-: S\Modcat \lra R\Modcat \quad \mbox{and}\quad\Hom_R(M, -): R\Modcat \lra S\Modcat. $$
This can be naturally extended to a pair of adjoint triangle
functors between homotopy categories:
$$_RM\otimes_S-: \K{S}\lra \K{R} \quad \mbox{and}\quad\Hom_R(M, -): \K{R} \lra \K{S}. $$
By passing to derived categories, we obtain the derived functors
${\bf G}$ and ${\bf{H}}$, respectively. Further, let $$\theta:\,
M\otimes_S\Hom_R(M, -)\lra Id_{R\mbox{-}{\rm Mod}}\quad \mbox{and}
\quad \varepsilon: {\bf GH } \lra Id_{\D R}$$ be the counit
adjunctions with respect to $\big(M\otimes_S-, \Hom_R(M, -)\big)$
and $({\bf G}, {\bf H})$, respectively.
Note that, for each $\cpx{X}\in\D{R}$, it follows from the
recollement $(\ast)$ in Lemma \ref{rt} that there exists a canonical
distinguished triangle in $\D{R}$:
$${\bf GH}(\cpx{X})\lraf{\varepsilon_{\cpx{X}}} \cpx{X} \lra i_*i^*(\cpx{X})\lra{\bf
GH}(\cpx{X})[1].$$
$(1)$ Let $X\in\Prod(_RM)$. To verify that $\theta_X$ is injective,
it is sufficient to show that
$$\theta_{M^I}: M\otimes_S\Hom_R(M, M^I)\lra M^I$$
is injective for any nonempty set $I$. Since $\Hom_R(M, M^I)\simeq
\Hom_R(M, M)^I$, the injection of $\theta_{M^I}$ is equivalent to
saying that the canonical map $\rho{_I}:\; M\otimes_SS^I\lra M^I$,
defined in Definition \ref{ML}, is injective. This holds exactly if
$M$ is $S$-Mittag-Leffler. However, the axiom $(R4)$ ensures that
$M$ is $S$-Mittag-Leffler. Thus $\theta_{X}: M\otimes_S\Hom_R(M,
X)\lra X$ is injective.
To prove $\Coker(\theta_X)\in\mathscr{E}:={\mathcal Y}\cap
R\Modcat$, we demonstrate that there is the following commutative
diagram in $\D{R}$:
$$(a)\quad
\xymatrix { {\bf GH}(X) \ar[r]^-{\varepsilon_{X}}\ar[d]_-{\simeq} &
X\ar@{=}[d]\ar[r]
& i_*i^*(X) \ar[r]\ar[d]^-{\simeq} & {\bf GH}(X)[1]\ar[d]^-{\simeq}\\
M\otimes_S\Hom_R(M, X)\ar[r]^-{\theta_{X}} & X\ar[r] &
\Coker(\theta_{X}) \ar[r] & M\otimes_S\Hom_R(M, X)[1]}
$$
With the help of this diagram and the recollement $(\ast)$ in Lemma
\ref{rt}, we have $i_*i^*(X)\in\mathcal{Y}$, and therefore
$$i_*i^*(X)\simeq\Coker(\theta_{X})\in \mathcal{Y}\cap
R\Modcat=\mathscr{E}.$$ This will finish the proof of $(1)$. So we
shall prove the existence of the above diagram $(a)$.
In fact, we shall first show that there exists a commutative diagram
$(b)$ in $\D{R}$:
$$
(b)\quad \xymatrix
{{\bf GH} (X) \ar[r]^-{\varepsilon_{X}}\ar[d]_-{\simeq} & X\ar@{=}[d]\\
M\otimes_S\Hom_R(M, X)\ar[r]^-{\theta_{X}} & X}
$$
This can be seen as follows: In Corollary \ref{counit}, we take $F:=
{}_RM\otimes_S-$ and $G:=\Hom_R(M,-)$. Then ${\bf G}=\mathbb{L}F$
and ${\bf H}=\mathbb{R}G$. To prove the existence of $(b)$, it
suffices to prove $X\in \mathcal{R}{_G}$ and $G(X)\in
\mathcal{L}{_F}$. For the definitions of $\mathcal{R}{_G}$ and
$\mathcal{L}{_F}$, we refer to Lemma \ref{homo}.
Observe that $X\in \mathcal{R}{_G}$ if and only if $\Ext^j_R(M,
X)=0$ for any $j>0$. Since $X\in\Prod(_RM)$, it suffices to show
that $\Ext^j_R(M, M^I)=0$ for any $j>0$ and any set $I$. This
follows from $\Ext^j_R(M, M^I)\simeq\Ext^j_R(M, M)^I=0$ by the axiom
$(R2)$. Thus $X\in \mathcal{R}{_G}$.
Note that $G(X)\in \mathcal{L}{_F}$ if and only if $\Tor^S_j(M,
G(X))=0$ for any $j>0$. Since $X\in\Prod(_RM)$ and $G$ commutes with
arbitrary direct products in $R\Modcat$, we have $G(X)\in
\Prod(_SS)$. This means that, to prove $G(X)\in \mathcal{L}{_F}$, it
is sufficient to check $\Tor^S_j(M, S^I)=0$ for any $j>0$ and any
set $I$. However, since $M$ is a good Ringel module, the right
$S$-module $M$ is strongly $S$-Mittag-Leffler by the axiom $(R4)$,
and therefore $\Tor^S_j (M, S^I)=0$ by Lemma \ref{MLP} (3). This
shows $G(X)\in \mathcal{L}{_F}$.
Hence, by Corollary \ref{counit}, the diagram $(b)$ does exist. Now,
by the recollement $(\ast)$ in Lemma \ref{rt}, we can extend
$\varepsilon_{X}$ to a canonical triangle in $\D{R}$: ${\bf
GH}(X)\lraf{\varepsilon_{X}} X \lra i_*i^*(X)\lra{\bf GH}(X)[1].$
Since each short exact sequence in $R\Modcat$ induces a canonical
triangle in $\D{R}$: $$M\otimes_S\Hom_R(M, X)\lraf{\theta_X} X \lra
\Coker(\theta_X)\lra M\otimes_S\Hom_R(M, X)[1],$$ the diagram $(a)$
follows from the commutative diagram $(b)$.
$(2)$ Since $M$ is a Ringel $R$-module, it follows from $(R3)$ that
there is a quasi-isomorphism $R\to \cpx{M}$ in $\K{R}$.
Consequently, we can form the following commutative diagram in
$\D{R}$:
$$
(c)\quad \xymatrix
{{\bf GH}(R) \ar[r]^-{\varepsilon_{R}}\ar[d]_-{\simeq} & R\ar[d]^-{\simeq}\\
{\bf GH}(\cpx{M}) \ar[r]^-{\varepsilon_{\cpx{M}}} & \cpx{M}}
$$
Next, using Corollary \ref{counit} again, we shall show that there
exists a commutative diagram in $\D{R}$:
$$
(d)\quad \xymatrix
{{\bf GH} (\cpx{M}) \ar[r]^-{\varepsilon_{\cpx{M}}}\ar[d]_-{\simeq} & \cpx{M}\ar@{=}[d]\\
M\otimes_S\Hom_R(M, \cpx{M})\ar[r]^-{\theta_{\cpx{M}}} & \cpx{M}}
$$
By Corollary \ref{counit}, we need only to show that $\cpx{M}\in
\mathcal{R}{_G}$ and $G(\cpx{M})\in \mathcal{L}{_F}$.
On the one hand, by the axiom $(R3)$ of Definition \ref{rm},
$\cpx{M}$ is a bounded complex such that each term of it belongs to
$\Prod(M)$. On the other hand, by Lemma \ref{homo}, the categories
$\mathcal{R}{_G}$ and $\mathcal{L}{_F}$ are triangulated
subcategories of $\K{R}$ and $\K{S}$, respectively. Thus, to prove
that $\cpx{M}\in \mathcal{R}{_G}$ and $G(\cpx{M})\in
\mathcal{L}{_F}$, it is enough to prove that $X\in \mathcal{R}{_G}$
and $G(X)\in \mathcal{L}{_F}$ for any $X\in\Prod(_RM)$. Clearly, the
latter has been shown in $(1)$. Thus $(d)$ follows directly from
Corollary \ref{counit}.
Note that $\theta_X: M\otimes_S\Hom_R(M,X)\lra X$ is injective by
$(1)$. Since $M_i\in\Prod(_RM)$ by the axiom $(R3)$, each map
$\theta_{M_i}$ is injective for $ 0\leq i\leq n$. This clearly
induces a complex $\Coker(\theta_{\cpx{M}})$ of the form:
$$
0\lra \Coker(\theta_{M_0}) \lraf{\partial_0}
\Coker(\theta_{M_1})\lraf{\partial_1} \cdots\lra
\Coker(\theta_{M_{n-1}})\lraf{\partial_{n-1}}
\Coker(\theta_{M_n})\lra 0\;\mbox{ in }\; \C R
$$ such that there is an exact sequence of complexes over $R$:
$$0 \lra M\otimes_S\Hom_R(M, \cpx{M})\lraf{\theta_{\cpx{M}}}\cpx{M}\lra
\Coker\big(\theta_{\cpx{M}}\big) \lra 0.$$ Since each exact sequence
of complexes over $R$ can be naturally extended to a canonical
triangle in $\D{R}$, we obtain a triangle in $\D{R}$:
$$(e)\quad M\otimes_S\Hom_R(M, \cpx{M})\lraf{\theta_{\cpx{M}}}\cpx{M}\lra
\Coker\big(\theta_{\cpx{M}}\big) \lra M\otimes_S\Hom_R(M,
\cpx{M})[1].
$$
Certainly, we also have a canonical triangle in $\D{R}$ from the
recollement $(\ast)$ in Lemma \ref{rt}:
$$
(f)\quad {\bf GH}(R)\lra R \lra i_*i^*(R)\lra{\bf GH}(R)[1].
$$
So, combining $(c)$, $(d)$, $(e)$ with $(f)$, one can easily
construct the following commutative diagram in $\D{R}$:
$$ \xymatrix { {\bf GH}(R)
\ar[r]^-{\varepsilon_{R}}\ar[d]_-{\simeq} & R\ar[d]^-{\simeq}\ar[r]
& i_*i^*(R) \ar[r]\ar[d]^-{\simeq} & {\bf GH}(R)[1]\ar[d]^-{\simeq}\\
M\otimes_S\Hom_R(M, \cpx{M})\ar[r]^-{\theta_{\cpx{M}}} &
\cpx{M}\ar[r] & \Coker(\theta_{\cpx{M}}) \ar[r] &
M\otimes_S\Hom_R(M, \cpx{M})[1]}
$$
In particular, we have $i_*i^*(R)\simeq \Coker(\theta_{\cpx{M}})$ in
$\D{R}$, and therefore
$$H^j(i_*i^*(R))\simeq H^j\big(\Coker(\theta_{\cpx{M}})\big)\;
\mbox{for any}\; j\in \mathbb{Z}.$$ This implies that
$H^j(i_*i^*(R))=0$ for $j<0$ or $j>n$.
Now, combining $(e)$ with $R\simeq \cpx{M}$ in $\D{R}$, we obtain a
triangle in $\D{R}$:
$$ M\otimes_S\Hom_R(M, \cpx{M})\lra R\lra
\Coker\big(\theta_{\cpx{M}}\big) \lra M\otimes_S\Hom_R(M,
\cpx{M})[1].
$$
Applying the cohomology functor $H^j$ to this triangle, one can
check that
$$H^j(i_*i^*(R))\simeq H^j\big(\Coker(\theta_{\cpx{M}})\big)
\simeq H^{j+1}( M\otimes_S\Hom_R(M, \cpx{M}))\; \mbox{for any}\;
j>0.$$ Thus $(2)$ follows.
$(3)$ For $n=0$, the conclusion follows from $i_*i^*(R)\simeq
\Coker(\theta_{\cpx{M}})$ trivially. So, we may assume $n\ge 1$. By
the final part of the proof of $(2)$, we know that
$$ i_*i^*(R)\simeq \Coker(\theta_{\cpx{M}})\; \mbox{ in}\;
\D{R}\quad \mbox{and}\quad
H^n\big(\Coker(\theta_{\cpx{M}})\big)\simeq H^{n+1}(
M\otimes_S\Hom_R(M, \cpx{M})).$$ Since the $(n+1)$-term of the
complex $M\otimes_S\Hom_R(M, \cpx{M})$ is zero, we see that
$H^n\big(\Coker(\theta_{\cpx{M}})\big)=0$. This implies that the
$(n-1)$-th differential $\partial_{n-1}$ of the complex
$\Coker(\theta_{\cpx{M}})$ is surjective. It follows that
$\Coker(\theta_{\cpx{M}})$ is isomorphic in $\D{R}$ to the following
complex:
$$ (\dag)\quad
0\lra \Coker(\theta_{M_0}) \lraf{\partial_0}
\Coker(\theta_{M_1})\lraf{\partial_1} \cdots\lra
\Coker(\theta_{M_{n-2}})\lraf{\partial{_{n-2}}}\;\Ker(\partial_{n-1})\lra
0.
$$
Since $M_m\in\Prod(_RM)$ for $0\leq m\leq n$ by the axiom $(R3)$, we
see from $(1)$ that $\Coker(\theta_{M_m})\in \mathscr{E}$. Note that
$\mathscr{E}$ is always closed under kernels of surjective
homomorphisms in $R\Modcat$. Thus
$\Ker(\partial_{n-1})\in\mathscr{E}$. This means that $(\dag)$ is a
bounded complex with all of its terms in $\mathscr{E}$.
Consequently, the complex $i_*i^*(R)$ is isomorphic in $\D{R}$ to
the complex $(\dag)$ with the required form in Lemma \ref{prep} (3).
This finishes the proof. $\square$
\medskip
{\it Remark.} By the proof of Lemma \ref{prep} (2), we see that the
complex ${_R}M\otimes_S\Hom_R(M,\,\cpx{M})$ is isomorphic in $\D{R}$
to both ${_R}M\otimesL_S\Hom_R(M,\,\cpx{M})$ and ${\bf GH}(R)$. This
implies that, up to isomorphism, the cohomology groups
$H^j\big({_R}M\otimes_S\Hom_R(M,\,\cpx{M})\big)$, for
$j\in\mathbb{Z}$, are independent of the choice of the complex
$\cpx{M}$ which arises in the axiom $(R3)$ of Definition \ref{rm}.
\medskip
With the help of Lemma \ref{hep} and Lemma \ref{prep}, we can prove
the following key proposition.
\begin{Prop}\label{realization} The following statements are equivalent:
$(1)$ The full triangulated subcategory $\mathcal{Y}$ of $\D{R}$ is
homological.
$(2)$ The category $\mathscr{E}$ is an abelian subcategory of
$R\Modcat$.
$(3)$ $H^j\big({_R}M\otimes_S\Hom_R(M,\,\cpx{M})\big)=0$ for any
$j\geq 2$.
$(4)$ The kernel of the homomorphism $\partial_0:
\Coker(\theta_{M_0})\lra \Coker(\theta_{M_1})$ induced from $\nu$
belongs to $\mathscr{E}$.
\end{Prop}
{\it Proof.} The equivalences of $(1)$ and $(2)$ follow from those
of $(1)$ and $(6)$ in Lemma \ref{hep} together with Lemma \ref{prep}
(3), while the equivalences of $(1)$ and $(3)$ follow from those of
$(1)$ and $(2)$ in Lemma \ref{hep} together with Lemma \ref{prep}
(2). Now we prove that $(1)$ and $(4)$ are equivalent. By Lemma
\ref{prep} (2) and the equivalence of $(1)$ and $(3)$ in Lemma
\ref{hep}, we see that (1) is equivalent to $H^0(i_*i^*(R))\in
\mathcal{Y}$. By the proof of Lemma \ref{prep} (2), we infer that
$H^0(i_*i^*(R))\simeq H^0(\Coker(\theta_{\cpx{M}}))\simeq
\Ker(\partial_0)$. Thus, $(1)$ is equivalent to $\Ker(\partial_0)\in
\mathcal{Y}\cap \Modcat{R}=\mathscr{E}$. $\square$
\medskip
As a consequence of Proposition \ref{realization}, we have the
following handy characterizations.
\begin{Koro}\label{appl}
Assume that the projective dimension of $_RM$ is equal to $n$. Then
the following are true.
$(1)$ If $n\leq 1$, then $\mathcal{Y}$ is always homological.
$(2)$ If $n=2$, then $\mathcal{Y}$ is homological if and only if
$M\otimes_S\Ext_R^2(M, R)=0$.
$(3)$ Suppose that $n\geq 3$ and $\Tor_i^S(M,\,\Ext_R^j(M,
R))=0\quad \mbox{for}\quad 2\leq j\leq n-1\quad \mbox{and}\quad
0\leq i\leq j-2.$ Then $\mathcal{Y}$ is homological if and only if
$$
\Tor_k^S(M, \,\Ext_R^n(M, R))=0\quad \mbox{for}\quad 0\leq k\leq
n-2.
$$
\end{Koro}
\medskip
{\it Proof.} The key point in the proof is to check when the $j$-th
cohomology group $H^j\big({_R}M\otimes_S\Hom_R(M,\,\cpx{M})\big)$
vanishes for $j\geq 2$. Note that
$H^j\big(M\otimes_S\Hom_R(M,\,\cpx{M})\big)=0$ for all $j>n$.
For $n\le 1$, the conclusion in Corollary \ref{appl} is clear. So,
we suppose $n\geq 2$. By the axiom $(R2)$, we have $\Ext^j_R(M,
M)=0$ for all $j\geq 1$. It follows that $\Ext^j_R(M, M^I)\simeq
\Ext^j_R(M, M)^I=0$ for any nonempty set $I$, and therefore
$\Ext^j_R(M, X)=0$ for any $X\in\Prod(M)$.
By the axiom $(R3)$, there exists an exact sequence in $R\Modcat$:
$$
0\lra R \lra M_0 \lra M_1\lra \cdots \lra M_n\lra 0$$ such that
$M_i\in\Prod(M)$ for $0\leq i\leq n$. Since $\Ext^j_R(M, X)=0$ for
any $X\in\Prod(M)$ and $j\ge 1$, we know that the following complex
$\Hom_R(M, \cpx{M}):$
$$0\lra
\Hom_R(M, M_0) \lra \Hom_R(M, M_1) \lra\Hom_R(M, M_2)\lra \cdots
\lra \Hom_R(M, M_n)\lra 0$$ satisfies that $H^j\big(\Hom_R(M,
\cpx{M})\big)\simeq \Ext^j_R(M, R)$ for each $j\geq 1$.
$(2)$ Let $n=2$. Consider the complex
$M\otimes_S\Hom_R(M,\,\cpx{M}):$
$$0\lra M\otimes_S\Hom_R(M, M_0) \lra M\otimes_S\Hom_R(M, M_1) \lra
M\otimes_S\Hom_R(M, M_2)\lra 0.$$ Since the functor $_RM\otimes_S-:
S\Modcat\to R\Modcat$ is right exact, we have
$$H^2\big(M\otimes_S\Hom_R(M,\,\cpx{M})\big)\simeq M\otimes_S
H^2\big(\Hom_R(M,\,\cpx{M})\big)\simeq M\otimes_S\Ext_R^2(M, R).$$
Now, the statement $(2)$ follows from the equivalences of $(1)$ and
$(3)$ in Proposition \ref{realization}.
$(3)$ Under the assumption of $(3)$, we claim that
$$H^m\big(M\otimes_S\Hom_R(M,\,\cpx{M})\big)\simeq \Tor_{n-m}^S(M,
\,\Ext_R^n(M, R))\quad \mbox{for}\quad 2\leq m\leq n.$$
Consequently, the statement $(3)$ will follow from the equivalences
of $(1)$ and $(3)$ in Proposition \ref{realization}.
In the following, we shall apply Lemma \ref{Formula} to prove this
claim. Define $\cpx{Y}:=\Hom_R(M,\,\cpx{M})$. This is a complex over
$S$ with $Y^i=\Hom_R(M, M_i)$ for $0\leq i\leq n$ and $Y^i=0$ for
$i\geq n+1$. Moreover, since the right $S$-module $M$ is strongly
$S$-Mittag-Leffler by the axiom $(R4)$, it follows from the proof of
Lemma \ref{prep} (1) that
$$\Tor^S_k\big(M, \Hom_R(M, X)\big)=0\; \mbox{ for all}\; k\geq
1\;\mbox{and}\; X\in\Prod(M).$$ This implies that $\Tor^S_k(M,
Y^i)=0$ for all $i\in \mathbb{Z}$ and $k\geq 1$.
Recall that $H^j(\cpx{Y})\simeq \Ext^j_R(M, R)$ for all $j\geq 1$.
By assumption, we obtain
$$\Tor_i^S(M,\,H^j(\cpx{Y}))=0\quad \mbox{for}\quad 2\leq j\leq n-1\quad \mbox{and}\quad
0\leq i\leq j-2.$$ Clearly, this implies that, for each $2\leq m\leq
n-1$, we have
$$\,\Tor^S_t\big(M,
H^{m+t}(\cpx{Y})\big)=0=\Tor^S_{t-1}\big(M, H^{m+t}(\cpx{Y})\big)
\;\mbox{for}\;\, 0\leq t \leq n-m-1.$$ It follows from Lemma
\ref{Formula} that $H^m(M\otimes_S\cpx{Y})\simeq \Tor^S_{n-m}\big(M,
H^n(\cpx{Y})\big)\simeq\Tor_{n-m}^S(M, \,\Ext_R^n(M, R))$.
To finish the proof of the claim, it remains to prove
$H^n(M\otimes_S\cpx{Y})\simeq M\otimes_S\Ext_R^n(M, R)$. However,
since the functor $M\otimes_S-$ is right exact and since $Y^i=0$ for
$i\geq n+1$, we see that $H^n(M\otimes_S\cpx{Y})\simeq M\otimes_S
H^n(\cpx{Y})\simeq M\otimes_S \Ext_R^n(M, R)$. This finishes the
proof of the above-mentioned claim. Thus $(3)$ holds. $\square$
\medskip
As another consequence of Proposition \ref{realization}, we mention
the following result which is not used in this note, but of its own
interest.
\begin{Koro}\label{appl'}
$(1)$ If $M_0\in\Add(_RM)$, then $_RM$ is a classical tilting
module.
$(2)$ If $M_1\in\Add(_RM)$, then $\mathcal Y$ is homological in $\D
R$.
\end{Koro}
{\it Proof.} (1) Suppose $M_0\in\Add(_RM).$ We claim that
$\Coker(\theta_{M_0})=0$. In fact, since $_RM$ is finitely generated
by the axiom $(R1)$, the functor $\Hom_R(M, -): R\Modcat\to
S\Modcat$ commutes with arbitrary direct sums. It follows that the
evaluation map $\theta_{X}: M\otimes_S\Hom_R(M, X)\lra X$ is an
isomorphism for each $X\in\Add(_RM)$. Since $M_0\in\Add(_RM)$, the
map $\theta_{M_0}: M\otimes_S\Hom_R(M, M_0)\lra M_0$ is an
isomorphism, and therefore $\Coker(\theta_{M_0})=0$. Combining this
with the proof of Proposition \ref{realization}, we have
$H^0(i_*i^*(R))\simeq \Ker(\partial_0)=0$. Note that $\End_{\D
R}(i^*(R))\simeq H^0(i^*(R))=H^0(i_*i^*(R))$ as $R$-modules by Lemma
\ref{ringhom} (2). This implies that $\End_{\D R}(i^*(R))=0$ and so
$\mathcal{Y}=0$ by Lemma \ref{ringhom} (1). Now, it follows from
Lemma \ref{rt} that $\rHom_R(M,-):\; \D{R}\lra\D{S}$ is a triangle
equivalence. Consequently, $_RM$ is a classical tilting module by
\cite[Chapter 5, Theorem 4.1]{HHK}.
(2) It follows from the proof of (1) that $\Coker(\theta_{M_1})=0$.
Thus (2) follows from Proposition \ref{realization} and Lemma
\ref{prep} (1). $\square$
\section{Application to tilting modules:
Proofs of Theorem \ref{main-result} and Corollary \ref{cor}}\label{sect5}
In this section, we first develop some properties of (good) tilting
modules, and then give a method to construct good Ringel modules.
With these preparations in hand, we finally apply Proposition
\ref{realization} to prove Theorem \ref{main-result} and Corollary
\ref{cor}.
\smallskip
Throughout this section, $A$ will be a ring and $n$ a natural
number. In addition, we assume that $T$ is a {\bf{good}}
$n$-tilting $A$-module with $(T1), (T2)$ and $(T3)'$. Let
$B:=\End_A(T)$.
\medskip
First of all, we shall mention a few basic properties of good
tilting modules in the following lemma. For proofs, we refer to
\cite[Chapter 11, Lemma 2.7]{HHK}, \cite[Proposition 1.4, Lemma
1.5]{Bz2} and \cite[Proposition 3.5]{Bz1}.
\begin{Lem} \label{tilt1}
The following hold true for the tilting module $_AT$.
$(1)$ The torsion class $T^{\bot}:=\{X\in A\Modcat \mid \Ext^i
_A(T,X)=0 \mbox{\;for\, all \,} i\geq 1\}$ in $A\Modcat$ is closed
under arbitrary direct sums in $A\Modcat$.
$(2)$ The right $B$-module $T$ has a finitely generated projective
resolution of length at most $n$:
$$
0\lra \Hom_A(T_n, T)\lra \cdots \lra\Hom_A(T_1, T)\lra\Hom_A(T_0,
T)\lra T_B\lra 0$$ with $T_i\in\add(_AT)$ for all $0\leq i\leq n$.
$(3)$ The map $A^{\opp} \to \End{_{B^{\opp}}}(T) $, defined by
$a\mapsto [t\mapsto at]$ for $a\in A$ and $t\in T$, is an
isomorphism of rings. Moreover, $\Ext^i_{B^{\opp}}(T,T)=0$ for all $
i\geq 1$.
$(4)$ If $T_n=0$ in the axiom $(T3)'$, then $_AT$ is an
$(n-1)$-tilting module.
\end{Lem}
Let us introduce some notation which will be used throughout this
section.
Define
$$
G:={_A}T\otimesL_B- : \,\D{B}\to \D{A},\quad H:=\rHom_A(T,-):\,
\D{A}\to \D{B},
$$
$$\cpx{Q}:=\;\cdots\lra 0\lra \Hom_A(T, T_0)\lra\Hom_A(T, T_1)\lra
\cdots \lra\Hom_A(T, T_n)\lra 0\lra \cdots$$ where $\Hom_A(T, T_i)$
is of degree $i$ for $0\leq i\leq n$, and
$\cpx{Q}{^*}:=\Hom_B(\cpx{Q}, B)\in \C{\pmodcat{B\opp}}$. Clearly,
$\cpx{Q}{^*}$ is isomorphic in $\Cb{\pmodcat{B\opp}}$ to the complex
$$
\;\cdots\lra 0\lra \Hom_A(T_n, T)\lra \cdots \lra\Hom_A(T_1,
T)\lra\Hom_A(T_0, T)\lra 0\lra \cdots $$
\medskip
The following result is due to Bazzoni \cite[Theorem\,2.2]{Bz2},
which says that, in general, $\D{A}$ is not equivalent to $\D B$,
but a full subcategory of $\D{B}$.
\begin{Lem}\label{Bz}
The functor $H: \D{A}\to \D{B}$ is fully faithful, and
$\Img(H)=\Ker(\Hom_{\D B}(\Ker(G), -))$.
\end{Lem}
\smallskip
The next result supplies a way to understand good tilting modules
$T$ by some special objects or by subcategories of derived module
categories. In particular, the category $\Ker(G)$ is a bireflective
subcategory of $\D{B}$.
\begin{Lem}\label{tilt2} For the tilting $A$-module $T$, we have
the following:
$(1)$ $H(A)\simeq \cpx{Q}$ in $\D{B}$ and
$\Hom_{\D{B}}(\cpx{Q},\cpx{Q}[m])=0$ for any $m\ne 0$.
$(2)$ $\Ker(G)=\{\cpx{Y}\in\D{B}\mid \Hom_{\D B}(\cpx{Q},\,
\cpx{Y}[i])=0 \mbox{\;for\, all \,} i\in\mathbb{Z}\}.$
$(3)$ Let $j_!:{\rm{Tria}}{(\cpx{Q})}\lra \D{B}$ and $i_*:
\Ker(G)\lra \D{B}$ be the inclusions. Then there exists a
recollement of triangulated categories together with a triangle
equivalence:
$$(\star)\quad
\xymatrix@C=1.2cm{\Ker(G) \ar[r]^-{{i_*}}
&\D{B}\ar[r]^{j^!}\ar@/^1.2pc/[l]\ar_-{i^*}@/_1.2pc/[l]
&{\rm{Tria}}{(\cpx{Q})}\ar[r]^-{G\,j_*}_-{\simeq}
\ar@/^1.2pc/[l]_{j_*}\ar@/_1.2pc/[l]_{j_!\;} & \D{A} }$$
\bigskip
\noindent such that $G\,j_*\,j^!$ is naturally isomorphic to $G$.
\end{Lem}
{\it Proof.} We remark that Lemma \ref{tilt2} is implied in
\cite{Bz2}. For convenience of the reader, we give a proof here.
$(1)$ By the axiom $(T3)'$, the stalk complex $A$ is
quasi-isomorphic in $\C{A}$ to the complex $\cpx{T}$ of the form:
$$ \cdots \lra 0 \lra T_0 \lra
T_1\lra \cdots \lra T_n\lra 0\lra \cdots$$ where $T_i\in \add(T)$ is
in degree $i$ for $0\leq i\leq n$. Further, by the axiom $(T2)$, we
have $T_i\in T^{\bot}:=\{X\in A\Modcat \mid \Ext^i _A(T,X)=0
\mbox{\;for\, all \,} i\geq 1\}$. It follows from Lemma \ref{homo}
(1) that $H(A)\simeq H(\cpx{T})\simeq \Hom_A(T, \cpx{T})=\cpx{Q}$ in
$\D{B}$. Since the functor $H$ is fully faithful by Lemma \ref{Bz},
we obtain
$$\Hom_{\D B}(\cpx{Q}, \cpx{Q}[m])\simeq \Hom_{\D B}(H(A),
H(A)[m])\simeq \Hom_{\D A}(A, A[m])\simeq \Ext^m_A(A, A)=0$$ for any
$m\neq 0$. This shows $(1)$.
$(2)$ Since $\cpx{Q}\in \Cb{\pmodcat{B}}$ and since $\cpx{Q}{^*}$ is
quasi-isomorphic to $T_B$ by Lemma \ref{tilt1} (2), we have the
following natural isomorphisms of triangle functors:
$$\rHom{_B}(\cpx{Q},-)\lraf{\simeq} \cpx{Q}{^*}\otimesL_B-\lraf{\simeq} {_\mathbb{Z}}T\otimesL_B-: \D{B}\lra\D{\mathbb
Z},$$ where the first isomorphism follows from Lemma \ref{complex}.
Note that $H^m(\rHom_{\D B}(\cpx{Q}, \cpx{Y}))\simeq \Hom_{\D
B}(\cpx{Q}, \cpx{Y}[m])$ for $m\in\mathbb{Z}$ and $\cpx{Y}\in\D{B}$.
This shows $(2)$.
$(3)$ Since $\cpx{Q}\in \Cb{\pmodcat{B}}$, we know from $(2)$ and
Lemma \ref{rg} (1) that there exists a recollement of triangulated
categories:
$$(\star\star)\quad
\xymatrix@C=1.2cm{\Ker(G)\ar[r]^-{{i_*}}
&\D{B}\ar[r]^{j^!}\ar@/^1.2pc/[l]_-{i^!}\ar_-{i^*}@/_1.2pc/[l]
&{\rm{Tria}}{(\cpx{Q})}
\ar@/^1.2pc/[l]_{j_*}\ar@/_1.2pc/[l]_{j_!\;}}$$
\noindent On the one hand, by the correspondence of recollements and
TTF (torsion, torsion-free) triples (see, for example, \cite[Section
2.3]{CX1}), we infer from $(\star\star)$ that
$\Img(j_*)=\Ker(\Hom_{\D B}(\Ker(G), -))$ and that the functor $j_*:
{\rm{Tria}}{(\cpx{Q})}\lra \Img(j_*)$ is a triangle equivalence with
the restriction of $j^!$ to $\Img(j_*)$ as its quasi-inverse. On
the other hand, it follows from Lemma \ref{Bz} that
$\Img(H)=\Ker(\Hom_{\D B}(\Ker(G), -))$ and the functor $H:\D{A}\
\lra \Img(H)$ is a triangle equivalence with the restriction of $G$
to $\Img(H)$ as its quasi-inverse. Consequently, we see that
$\Img(j_*)=\Img(H)$ and the composition $G\,j_*:
{\rm{Tria}}{(\cpx{Q})}\lra \D{A} $ of $j_*$ with $G$ is also a
triangle equivalence.
It remains to check
$$G\lraf{\simeq} G\,j_* j^!: \;\D{B}\lra \D{A}.$$ In fact, for
any $\cpx{X}\in\D{B}$, by the recollement $(\star\star)$, there
exists a canonical triangle in $\D{B}:$
$$i_*i^!(\cpx{X})\lra \cpx{X}\lra j_*j^!(\cpx{X})\lra
i_*i^!(\cpx{X})[1].$$ Since $\Img(i_*i^!)=\Img(i_*)=\Ker(G)$, we
know that $G(\cpx{X}) \lraf{\simeq} Gj_*j^!(\cpx{X})$ in $\D{B}$.
This proves $(3)$. $\square$
\medskip
Next, we shall investigate when the subcategory $\Ker(G)$ of $\D{B}$
is homological. The following result conveys that this discussion
can be proceeded along the right $B$-module $T$.
\begin{Lem}\label{tilt4}
The category $\Ker(G)$ is a homological subcategory of $\D{B}$ if
and only if $\,\Ker\big(\,\rHom_{B\opp}(T, -)\big)$ is a
homological subcategory of $\D{B\opp}$.
\end{Lem}
{\it Proof.} In Lemma \ref{rg}, we take $R:=B$ and
$\Sigma:=\{\cpx{Q}\}$. Then $\Sigma^*=\{\cpx{Q}{^*}\}$ where
$\cpx{Q}{^*}:=\Hom_B(\cpx{Q}, B)$. Since $\cpx{Q}{^*}$ is
quasi-isomorphic to $T_B$ by Lemma \ref{tilt1} (2), we infer that
$\cpx{Q}{^*}\lraf{\simeq} T_B$ in $\D{B\opp}$ and that there exists
a natural isomorphism of triangle functors:
$$\rHom{_{B\opp}}(T,\,-)\lraf{\simeq}
\rHom_{B\opp}(\cpx{Q}{^*},\,-): \D{B\opp}\lra \D{\mathbb{Z}}.$$ This
implies that $$\Ker\big(\rHom_{B\opp}(T,\,
-)\big)=\Ker\big(\rHom_{B\opp}(\cpx{Q}{^*},\, -)\big)=\{\cpx{Y}\mid
\Hom_{\D {B\opp}}(\cpx{Q}{^*}, \,\cpx{Y}[m])=0\; \mbox{for}\;
m\in\mathbb{Z}\}. $$ Thus Lemma \ref{tilt4} follows from Lemmas
\ref{ref} and \ref{rg} (4). $\square$
\medskip
Next, we point out that each good tilting module naturally
corresponds to a good Ringel module. This guarantees that we can
apply Proposition \ref{realization} to show Theorem
\ref{main-result}.
\begin{Lem}\label{tilt3}
The right $B$-module $T_B$ is a good $n$-Ringel module.
\end{Lem}
{\it Proof.} By Lemma \ref{tilt1} (2), the axiom $(R1)$ holds for
$T_B$, and the projective dimension of $T_B$ is at most $n$.
Moreover, by Lemma \ref{tilt1} (3), the axiom $(R2)$ also holds for
$T_B$. Now, we check the axiom $(R3)$ for $T_B$.
In fact, according to the axiom $(T1)$, the module $_AT$ admits a
projective resolution of $A$-modules:
$$
0\lra P_n \lra \cdots \lra P_1\lra P_0\lra T\lra 0$$ with
$P_i\in\Add(_AA)$ for $0\leq i\leq n$. Since $\Ext^j_A(T, T)=0$ for
each $j\geq 1$ by the axiom $(T2)$, it follows that the sequence
$$
0\lra B \lra\Hom_A(P_0, T)\lra \Hom_A(P_1, T)\lra \cdots \lra
\Hom_A(P_n, T) \lra 0$$ of right $B$-modules is exact. Note that
$\Hom_A(P_i, T)\in \Prod(T_B)$ due to $P_i\in\Add(_AA)$. This means
that the axiom $(R3)$ holds for $T_B$. Thus the right $B$-module
$T_B$ is an $n$-Ringel module.
It remains to prove that $T_B$ is good, that is, $T_B$ satisfies the
axiom $(R4)$.
Actually, by Lemma \ref{tilt1} (3), the map $A^{\opp}\lra
\End{_{B^{\opp}}}(T) $, defined by $a\mapsto [t\mapsto at]$ for
$a\in A$ and $t\in T$, is an isomorphism of rings. Further, it
follows from Lemma \ref{TML} that the right $A\opp$-module $T$ is
strongly $A\opp$-Mittag-Leffler. Hence, the right
$\End{_{B^{\opp}}}(T)$-module $T$ is strongly
$\End{_{B^{\opp}}}(T)$-Mittag-Leffler. Thus, by definition, the
$n$-Ringel $B\opp$-module $T$ is good. $\square$
\smallskip
{\it Remark.} If $_AT$ is infinitely generated, then the right
$B$-module $T$ is not a tilting module. In fact, it follows from
Lemma \ref{tilt1} (2) that $T_B$ is finitely generated. Suppose
contrarily that $T_B$ is a tilting right $B$-module. Then, by
Corollary \ref{2.9}, the right $B$-module $T_B$ is classical, and
therefore $_AT$ is classical by Lemma \ref{tilt1} (2)-(3). This is a
contradiction.
\medskip
Now, with the previous preparations, we are in the position to prove
Theorem \ref{main-result}.
\medskip
{\bf Proof of Theorem \ref{main-result}.} We shall use Proposition
\ref{realization} to show the equivalences in Theorem
\ref{main-result}.
Recall that we denote by $\cpx{P}$ the complex which is the deleted
projective resolution of $_AT$:
$$
\cdots\lra 0\lra P_n \lra \cdots \lra P_1\lraf{\sigma} P_0\lra 0\lra
\cdots
$$ appearing in the axiom $(T1)$. Here, $P_i$ is in degree $-i$ for $0\le i\le
n$.
By Lemma \ref{tilt3}, we know that $T$ is a good $n$-Ringel
$B\opp$-module and that the exact sequence in the axiom $(R3)$ can
be chosen as
$$
0\lra B_B \lra\Hom_A(P_0, T)\lra \Hom_A(P_1, T)\lra \cdots \lra
\Hom_A(P_n, T) \lra 0.$$ In particular, the complex $\cpx{M}$ in
Proposition \ref{realization} can be chosen to be the following
complex:
$$
\Hom_A(\cpx{P}, T):\; \cdots \lra 0\lra \Hom_A(P_0, T)\lra
\Hom_A(P_1, T)\lra \cdots \lra \Hom_A(P_n, T) \lra 0\lra \cdots$$
Now, in Proposition \ref{realization}, we take $R:=B\opp\,$,
$S:=A\opp$ and $M:= {}_RT_S$. Further, let $${\bf
H}=\rHom_{B\opp}(T, -):\D{B\opp}\lra \D{A\opp}.$$ It follows from
Lemma \ref{tilt4} that $\Ker(G)$ is homological in $\D R$ if and
only if so is $\Ker({\bf H})$ in $\D{B\opp}$. In other words, the
statement $(1)$ in Theorem \ref{main-result} is equivalent to the
following statement:
$(1')$ The category $\Ker({\bf H})$ is a homological subcategory of
$\D{B\opp}$.
In the following, we shall show that $(1')$ is equivalent to $(2),
(3)$ and $(4)$, respectively.
We first show that $(1')$ and $(2)$ are equivalent. In fact, it
follows form Proposition \ref{realization} that $(1')$ is equivalent
to
$(2')$ The category $\mathscr{E}:=\{Y\in {B\opp}\Modcat \mid
\Ext^m_{B\opp}(T, Y)=0\mbox{\;for\, all \,} m\geq0\}$ is an abelian
subcategory of $B\opp\Modcat$.
\noindent So, we will show that $(2')$ is equivalent to $(2)$. For
this aim, we set $\mathscr{A}:=\{X\in B\Modcat \mid \Tor_m
^B(T,X)=0\mbox{\;for\, all \,} m\geq0\}$, and establish a connection
between $\mathscr{A}$ and $\mathscr{E}$. Let $(-)^\vee$ be the
dual functor $\Hom_\mathbb{Z}(-, \mathbb{Q}/\mathbb{Z}):
\mathbb{Z}\Modcat\lra \mathbb{Z}\Modcat$.
Now, we claim that $(-)^\vee$ induces two exact functors:
$$
(-)^\vee: \mathscr{A}\lra \mathscr{E}\quad \mbox{and}\quad (-)^\vee:
\mathscr{E}\lra \mathscr{A}
$$
such that $X\in\mathscr{A}$ if and only if $X^\vee\in\mathscr{E}$,
and that $Y\in\mathscr{E}$ if and only if $Y^\vee\in\mathscr{A}$,
where $X\in B\Modcat$ and $Y\in B\opp\Modcat$.
In fact, it is known that $\mathbb{Q}/\mathbb{Z}$ is an injective
cogenerator for $\mathbb{Z}\Modcat$, and that $(-)^\vee$ admits the
following properties:
$(a)$ For each $M\in\mathbb{Z}\Modcat$, if $M^\vee=0$, then $M=0$.
$(b)$ A sequence $0\to X_1\to X_2\to X_3\to 0$ of
$\mathbb{Z}$-modules is exact if and only if $0\to (X_3)^\vee \to
(X_2)^\vee \to (X_1)^\vee \to 0$ is exact.
On the one hand, for each $X\in B\Modcat$, it follows from Lemma
\ref{tor-ext} (1) that
$$
(\Tor_m^B(T, \,X))^\vee \simeq \Ext^m_{B\opp}(T,\, X^\vee)\;\,
\mbox{for all}\;\, m\geq 0.
$$
This implies that $X\in\mathscr{A}$ if and only if
$X^\vee\in\mathscr{E}$. This is due to $(a)$.
On the other hand, since $T_B$ has a finitely generated projective
resolution in $B\opp\Modcat$ by Lemma \ref{tilt1} (2), it follows
from Lemma \ref{tor-ext} (2) that
$$
(\Ext_{B\opp}^m(T,\,Y))^\vee \simeq \Tor^{B}_m(T, \,Y^\vee)\;\,
\mbox{ for all}\;\, m\geq 0 \mbox{ and for any } Y\in
B\opp\Modcat.$$ This means that $Y\in\mathscr{E}$ if and only if
$Y^\vee\in\mathscr{A}$, again due to $(a)$. This finishes the proof
of the claim.
Recall that $\mathscr{A}$ always admits the ``$2$ out of $3$"
property: For an arbitrary short exact sequence in $B\Modcat$, if
any two of its three terms belong to $\mathscr{A}$, then so does the
third. Moreover, $\mathscr{A}$ is an abelian subcategory of
$B\Modcat$ if and only if $\mathscr{A}$ is closed under kernels
(respectively, cokernels) in $B\Modcat$. Clearly, similar statements
hold for the subcategory $\mathscr{E}$ of $B\opp\Modcat$.
By the above-proved claim, one can easily show that $\mathscr{A}$
is closed under kernels in $B\Modcat$ if and only if $\mathscr{E}$
is closed under cokernels in $B\opp\Modcat$. It follows that
$\mathscr{A}$ is an abelian subcategory of $B\Modcat$ if and only if
$\mathscr{E}$ is an abelian subcategory of $B\opp\Modcat$. Thus
$(2')$ is equivalent to $(2)$, and therefore $(1')$ and $(2)$ are
equivalent.
Next, we shall verify that $(1')$ and $(3)$ are equivalent.
Actually, it follows form Proposition \ref{realization} that $(1')$
is also equivalent to the following statement:
\smallskip
$(3')$ $H^j\big(\Hom_{B\opp}(T,\,\cpx{M})\otimes_AT\big)=0$ for all
$j\geq 2$, where $\Hom_{B\opp}(T,\,\cpx{M}):=\Hom_{B\opp}(T,
\Hom_A(\cpx{P}, T))$ is the complex of the form: {\small{$$0\lra
\Hom_{B\opp}(T, \Hom_A(P_0, T))\lra \Hom_{B\opp}(T, \Hom_A(P_1,
T))\lra \cdots \lra \Hom_{B\opp}(T, \Hom_A(P_n, T)) \lra 0,
$$}}
with $\Hom_{B\opp}(T, \Hom_A(P_i, T))$ in degree $i$ for $0\leq
i\leq n$.
So it suffices to verify that $(3')$ and $(3)$ are equivalent.
Clearly, for this purpose, it is enough to show that
$\Hom_A(\cpx{P}, A) \simeq \Hom_{B\opp}(T, \Hom_A(\cpx{P}, T))$ as
complexes over $A\opp$.
Note that there exists a natural isomorphism of additive functors:
$$
\Hom_{B\opp}(T, \Hom_A(-, T))\lraf{\simeq} \Hom_{B\opp}(\Hom_A(A,
T),\,\Hom_A(-, T)): A\Modcat\to A\opp\Modcat.
$$
Moreover, the functor $\Phi:=\Hom_A(-, T)$ yields a natural
transformation:
$$
\Hom_A(-, A) \lra\Hom_{B\opp}(\Phi(A),\,\Phi(-)\,): A\Modcat\to
A\opp\Modcat.
$$
Now we shall show that this transformation is even a natural
isomorphism. Clearly, it is sufficient to prove that $$\Phi:
\Hom_A(X, A) \lraf{\simeq} \Hom_{B\opp}(\Phi(A),\,\Phi(X))$$ for
any projective $A$-module $X$. In the following, we will show that
this holds even for any $A$-module $X$.
In fact, since $T$ is a good tilting $A$-module, it follows from the
axiom $(T3)'$ that there exists an exact sequence $0\lra A \lra T_0
\lra T_1$ with $T_i\in\add(T)$ for $i=0, 1$. By Lemma \ref{tilt1}
(2), we obtain another exact sequence $\Phi(T_1)\lra\Phi(T_0)\lra
\Phi(A)\lra 0$ of $B\opp$-modules. This gives rise to the following
exact commutative diagram:
$$\xymatrix{
0\ar[r] & \Hom_A(X, A)\ar[r]\ar[d]^-{\Phi} & \Hom_A(X, T_0)
\ar[r]\ar[d]^-{\simeq}
&\Hom_A(X, T_1)\ar[d]^-{\simeq}\\
0\ar[r] &\Hom_{B\opp}(\Phi(A),\,\Phi(X)) \ar[r]&
\Hom_{B\opp}(\Phi(T_0),\,\Phi(X))\ar[r]
&\Hom_{B\opp}(\Phi(T_1),\,\Phi(X)) }$$ where the isomorphisms in the
second and third columns are due to $T_0\in\add(T)$ and
$T_1\in\add(T)$, respectively. Consequently, the $\Phi: \Hom_A(X, A)
\lra \Hom_{B\opp}(\Phi(A),\,\Phi(X))$ in the first column is an
isomorphism. This implies that
$$
\Hom_A(-, A)
\lraf{\simeq}\Hom_{B\opp}(\Phi(A),\,\Phi(-)\,)\lraf{\simeq}
\Hom_{B\opp}(T, \Hom_A(-, T)): A\Modcat\to A\opp\Modcat.
$$
Thus $\Hom_A(\cpx{P}, A)\simeq \Hom_{B\opp}(T, \Hom_A(\cpx{P}, T))$
as complexes over $A\opp$. Thus $(3')$ is equivalent to $(3)$.
It remains to show that $(1')$ is equivalent to $(4)$.
For each right $B$-module $Y$, let
$\theta_Y:\Hom_{B\opp}(_AT_B,\,Y)\otimes_AT \lra Y$ be the
evaluation map. Then it follows from the equivalence of $(1)$ and
$(4)$ in Proposition \ref{realization} that $(1')$ is equivalent to
the following statement:
$(4')$ The kernel of the homomorphism $\partial_0:
\Coker\big(\theta_{\,\Phi(P_0)})\lra
\Coker\big(\theta_{\,\Phi(P_1)}\big)$ induced from the homomorphism
$\Phi(\sigma):\Phi(P_0)\lra \Phi(P_1)$ belongs to $\mathscr{E}$.
Now, we claim that $K\simeq \Ker(\partial_0)$ as right $B$-modules
(see the definition of $K$ in Theorem \ref{main-result} (4)). This
will show that $(1')$ and $(4)$ are equivalent.
To check the above isomorphism, we first define the following map
for each $A$-module $X$:
$$\zeta_X: \Hom_A(X, A)\otimes_AT \lra \Hom_A(X, T), \;f\otimes
t\mapsto [x\mapsto (x)f\,t]$$ for $f\in\Hom_A(X, A), \,t\in T$ and
$x\in X$. This yields a natural transformation $\zeta: \Hom_A(-,
A)\otimes_AT\lra \Hom_A(-, T)$ from $A\Modcat$ to $B\opp\Modcat$.
Clearly, by definition, we have $\varphi_i=\zeta_{P_i}$ for $i=0,
1$.
Recall that, under the identification of $\Phi(A)$ with $T$ as
$A$-$B$-bimodules, the functor $\Phi$ induces an isomorphism
$\Hom_A(X, A)\lraf{\simeq} \Hom_{B\opp}(T,\,\Phi(X))$ of
$A\opp$-modules. In this sense, one can easily construct the
following commutative diagram:
$$\xymatrix{\Hom_A(X, A)\otimes_AT
\ar[d]^-{\Phi\otimes 1}_-{\simeq}\ar[rr]^-{\zeta_X}&& \Hom_A(X, T)\ar@{=}[d]\\
\Hom_{B\opp}(T,\,\Phi(X))\otimes_AT\ar[rr]^-{\theta_{\,\Phi(X)}}&&\Phi(X)}$$
This
implies that $\Coker(\zeta_X)$ is naturally isomorphic to
$\Coker\big(\theta_{\,\Phi(X)}\big)$ as $B\opp$-modules. Since
$\varphi_i=\zeta_{P_i}$ for $i=0, 1$, we show that $K\simeq
\Ker(\partial_0)$ as $B\opp$-modules.
Hence, we have proved that the statements $(1)$-$(4)$ in Theorem
\ref{main-result} are equivalent.
Now, suppose $n=2$. Then the complex $\cpx{P}$ is of the following
form:
$$\cdots \lra 0\lra P_2 \lra P_1 \lra P_0 \lra
0\lra \cdots
$$
which is a deleted projective resolution of $_AT$. Since $(1)$ and
$(3)$ in Theorem \ref{main-result} are equivalent, we see that $(1)$
holds if and only if $H^2\big(\Hom_A(\cpx{P},
A)\otimes_AT_B\big)=0$. However, since the tensor functor
$-\otimes_AT_B: A\opp\Modcat\lra B\opp\Modcat$ is always right
exact, we have
$$H^2\big(\Hom_A(\cpx{P}, A)\otimes_AT_B\big)\simeq
H^2\big(\Hom_A(\cpx{P}, A)\big)\otimes_AT\simeq\Ext^2_A(T,
A)\otimes_AT.$$ This finishes the proof of Theorem
\ref{main-result}. $\square$
\medskip
{\it Remarks.} $(1)$ If the category $\Ker(_AT\otimesL_B-)$ in
Theorem \ref{main-result} is homological in $\D B$, then it follows
from Lemma \ref{rt} (see also Lemma \ref{tilt2} (3)) that the
generalized localization $\lambda: B\to B_T$ of $B$ at the module
$T_B$ exists and is homological, which gives rise to a recollement
of derived module categories:
$$\xymatrix@C=1.2cm{\D{B_T}\ar[r]^-{D(\lambda_*)}
&\D{B}\ar[r]^-{ {}_AT\otimesL_B-}\ar@/^1.2pc/[l]\ar@/_1.3pc/[l]
&\D{A} \ar@/^1.2pc/[l]\ar@/_1.3pc/[l]}$$
\smallskip
$(2)$ Combining the remark following Lemma \ref{prep} with the proof
of Theorem \ref{main-result}, we infer that the complex
$\Hom_A(\cpx{P}, A)\otimes_AT_B$ in Theorem \ref{main-result} is
isomorphic in $\D{B\opp}$ to both $\Hom_A(\cpx{P}, A)\otimesL_AT_B$
and $\rHom_{B\opp}(T,\,B)\otimesL_AT$. This implies that, up to
isomorphism, the cohomology group $H^m\big(\Hom_A(\cpx{P},
A)\otimes_AT_B\big)$ in Theorem \ref{main-result} (3) is independent
of the choice of the projective resolutions of $_AT$ for all $m\in
\mathbb{Z}$.
$(3)$ By the proof of the equivalence of $(1)$ and $(4)$ in Theorem
\ref{main-result}, we know that $\Coker(\zeta_X)\simeq
\Coker\big(\theta_{\,\Phi(X)}\big)$ as $B\opp$-modules for $X\in
A\Modcat$. If $X\in\Add(_AA)$, then $\Phi(X)\in \Prod(T_B)$, and
therefore it follows from Lemma \ref{prep} (1) that
$\Coker\big(\theta_{\,\Phi(X)}\big)$ belongs to $\mathscr{E}:=\{Y\in
{B\opp}\Modcat \mid \Ext^m_{B\opp}(T, Y)=0\mbox{\;for\, all \,}
m\geq0\}$. Particularly, in Theorem \ref{main-result} (4), we always
have $\Coker(\varphi_i)\in\mathscr{E}$ for $i=1, 2$. Note that
$\mathscr{E}$ is closed under kernels of surjective homomorphisms in
$B\opp\Modcat$. Hence, if the homomorphism $\widetilde{\sigma}:
\Coker(\varphi_0)\lra\Coker(\varphi_1)$ induced from $\sigma: P_1\to
P_0$ is surjective, then the kernel $K$ of $\widetilde{\sigma}$ does
belong to $\mathscr{E}$, and therefore the category
$\Ker(T\otimesL_{B}-)$ is homological in $\D{B}$ by the equivalence
of $(1)$ and $(4)$ in Theorem \ref{main-result}.
\medskip
Clearly, the maps $\pi$ and $\omega$ in the definition of tilting
modules induce two canonical quasi-isomorphisms $\widetilde{\pi}:
\cpx{P}\lra T$ and $\widetilde{\omega}: A\lra \cpx{T}$ in $\C{A}$,
respectively. Consequently, both $\widetilde{\pi}$ and
$\widetilde{\omega}$ are isomorphisms in $\D{A}$.
\medskip
As a preparation for the proof of Corollary \ref{cor}, we shall
first establish the following lemma.
\begin{Lem}\label{Z-iso}
The complex $\Hom_A(\cpx{P}, A)$ is isomorphic in $\D{\mathbb{Z}}$
to the following complex:
$$
\Hom_A(T, \cpx{T}):\; \cdots \lra 0\lra \Hom_A(T, T_0)\lra \Hom_A(T,
T_1)\lra \cdots \lra \Hom_A(T, T_n) \lra 0\lra \cdots$$ In
particular, if $A$ is commutative, then $\Hom_A(\cpx{P},
A)\otimes_AT_B\simeq \Hom_A(T, \cpx{T})\otimesL_AT_B$ in
$\D{B\opp}.$
\end{Lem}
{\it Proof.} Since $\widetilde{\pi}$ and $\widetilde{\omega}$ are
chain maps in $\C{A}$, we can obtain two chain maps in
$\C{\mathbb{Z}}$:
$$\xymatrix{\Hom_A(\cpx{P}, A)\ar[r]^-{(\widetilde{\omega})^*} & \cpx{\Hom}_A(\cpx{P},
\cpx{T}) & \ar[l]_-{(\widetilde{\pi})_*} \Hom_A(T, \cpx{T}).}$$ Now,
we claim that both chain maps are quasi-isomorphisms.
To check this claim, we apply the cohomology functor $H^i(-)$ to
these chain maps for $i\in\mathbb{Z}$, and construct the following
commutative diagram:
$$\xymatrix{H^i(\Hom_A(\cpx{P}, A))\ar[d]^-{\simeq}\ar[r]^-{H^i((\widetilde{\omega})^*)} & H^i(\cpx{\Hom}_A(\cpx{P},
\cpx{T}))\ar[d]^-{\simeq} & \ar[l]_-{H^i((\widetilde{\pi})_*)}
H^i(\Hom_A(T,\,\cpx{T}))\ar[d]^-{\simeq}\\
\Hom_{\K A}(\cpx{P},
A[i])\ar[d]^-{q_1}\ar[r]^-{(\widetilde{\omega})^*} & \Hom_{\K
A}(\cpx{P}, \cpx{T}[i]) \ar[d]^-{q_2}& \ar[l]_-{(\widetilde{\pi})_*}
\Hom_{\K A}(T, \cpx{T}[i])\ar[d]^-{q_3}\\
\Hom_{\D A}(\cpx{P}, A[i])\ar[r]^-{(\widetilde{\omega})^*}_-{\simeq}
& \Hom_{\D A}(\cpx{P}, \cpx{T}[i])&
\ar[l]_-{(\widetilde{\pi})_*}^-{\simeq} \Hom_{\D A}(T,
\cpx{T}[i])}$$ where the maps $q_j$, for $1\leq j\leq 3$, are
induced by the localization functor $q: \K{A}\to \D{A}$, and where
the isomorphisms in the third row are due to the isomorphisms
$\widetilde{\omega}$ and $\widetilde{\pi}$ in $\D{A}$.
Since $\cpx{P}$ is a bounded complex of projective $A$-modules, both
$q_1$ and $q_2$ are bijective. This implies that
$H^i((\widetilde{\omega})^*)$ is also bijective, and therefore
$(\widetilde{\omega})^*$ is a quasi-isomorphism.
Note that $(\widetilde{\pi})_*$ is a quasi-isomorphism if and only
if $H^i((\widetilde{\pi})_*)$ is bijective for each
$i\in\mathbb{Z}$. This is also equivalent to saying that $q_3$ is
bijective in the above diagram. Actually, to prove the bijection of
$q_3$, it is enough to show that, for $X\in\add(_AT)$ and
$i\in\mathbb{Z}$, the canonical map $\Hom_{\K A}(T, X[i])\lra
\Hom_{\D A}(T, X[i])$ induced by $q$ is bijective since $\cpx{T}$ is
a bounded complex with each term in $\add(_AT)$. However, this
follows directly from the axiom $(T2)$. Thus $(\widetilde{\pi})_*$
is a quasi-isomorphism.
Consequently, the complexes $\Hom_A(\cpx{P}, A)$ and $\Hom_A(T,
\cpx{T})$ are isomorphic in $\D{\mathbb{Z}}$.
Now, assume that $A$ is commutative. Then each
$A$-module can be naturally regarded as a right $A$-module and even as an
$A$-$A$-bimodule. In particular, the complex $\cpx{T}$ can be
regarded as a complex of $A$-$A$-bimodules. In this sense, both
$\widetilde{\pi}: \cpx{P}\lra T$ and $\widetilde{\omega}: A\lra
\cpx{T}$ are quasi-isomorphisms of complexes of $A$-$A$-bimodules.
Moreover, one can check that the chain maps $(\widetilde{\omega})^*$
and $(\widetilde{\pi})_*$ are quasi-isomorphisms in $\C{A\opp}$.
This implies that $\Hom_A(\cpx{P}, A)\simeq \Hom_A(T, \cpx{T})$ in
$\D{A\opp}$. Note that $\Hom_A(\cpx{P}, A)\otimes_AT_B\simeq
\Hom_A(\cpx{P}, A)\otimesL_AT_B$ in $\D{B\opp}$ (see the above
remark (2)). As a result, we have $\Hom_A(\cpx{P},
A)\otimes_AT_B\simeq \Hom_A(T, \cpx{T})\otimesL_AT_B$ in
$\D{B\opp}.$ $\square$
\smallskip
{\bf Proof of Corollary \ref{cor}.} $(1)$ By the remark (3) at the
end of the proof of Theorem \ref{main-result}, we know that if the
homomorphism $\widetilde{\sigma}:
\Coker(\varphi_0)\lra\Coker(\varphi_1)$ induced from $\sigma: P_1\to
P_0$ (see Theorem \ref{main-result} (4)) is surjective, then
$\Ker(_AT\otimesL_B-)$ is homological in $\D{B}$.
Now, we verify this sufficient condition for the good tilting module
$_AT$ which satisfies the assumption in $(1)$.
In fact, by assumption, we can assume that $_AM$ has a projective
resolution: $0\lra P_1'\lraf{\sigma'} P_0'\lra {}_AM\lra 0$ with
$P_0', P_1'\in\Add(_AA)$, and that $_AN$ has a projective
presentation: $P_1''\lra P_0''\lraf{\sigma''} {}_AN\lra 0$ with
$P_0''\in\Add(_AA)$ and $P_1''\in\add(_AA)$. Since $_AT=M\oplus N$,
we can choose $\sigma=\left(\begin{array}{cc} \sigma' & 0\\
0 & \sigma''\end{array}\right): P_1'\oplus P_1''\lra P_0'\oplus
P_0''$. Recall that $\zeta: \Hom_A(-, A)\otimes_AT\lra \Hom_A(-, T)$
is a natural transformation from $A\Modcat$ to $B\opp\Modcat$ (see
the proof of Theorem \ref{main-result}). Certainly, if
$X\in\add(_AA)$, then $\zeta_X$ is an isomorphism, and so
$\Coker(\zeta_X)=0$.
Let $\widetilde{\sigma}':
\Coker(\zeta_{P_0'})\lra\Coker(\zeta_{P_1'})$ and
$\widetilde{\sigma}'':
\Coker(\zeta_{P_0''})\lra\Coker(\zeta_{P_1''})$ be the homomorphisms
induced from $\sigma'$ and $\sigma''$, respectively. By definition,
we have $\varphi_i=\zeta_{P_i}$ for $i=0, 1$, and
$$\widetilde{\sigma}=\left(\begin{array}{cc} \widetilde{\sigma}' & 0\\
0 & \widetilde{\sigma}''\end{array}\right):
\Coker(\zeta_{P_0'})\oplus \Coker(\zeta_{P_0''})\lra
\Coker(\zeta_{P_1'})\oplus \Coker(\zeta_{P_1''}).$$
Now, we show that $\widetilde{\sigma}$ is surjective, or
equivalently, both $\widetilde{\sigma}'$ and $\widetilde{\sigma}''$
are surjective. In fact, since $P_1''\in\add(_AA)$, we see that
$\Coker(\zeta_{P_1''})=0$. Thus $\widetilde{\sigma}''$ is
surjective. As $_AM$ is a direct summand of $_AT$ and of projective
dimension at most $1$, it follows from the axiom $(T2)$ that the map
$\Hom_A(\sigma', T): \Hom_A(P_0', T)\lra \Hom_A(P_1', T)$ is
surjective. This implies that $\widetilde{\sigma}'$ is a surjection.
Consequently, $\widetilde{\sigma}$ is surjective. Thus
$\Ker(_AT\otimesL_B-)$ is homological in $\D{B}$. This finishes the
proof of $(1)$.
$(2)$ Suppose that $\Ker(_AT\otimesL_B-)$ in Theorem
\ref{main-result} is homological. By Theorem \ref{main-result}, we
have $H^m\big(\Hom_A(\cpx{P}, A)\otimes_AT_B\big)=0$ for all $m\geq
2.$ In the sequel, we shall show that if $H^n\big(\Hom_A(\cpx{P},
A)\otimes_AT_B\big)=0$, then $T_n=0.$
In fact, since $A$ is commutative, it follows from the proof of
Lemma \ref{Z-iso} that $\Hom_A(\cpx{P}, A)\simeq \Hom_A(T, \cpx{T})$
in $\D{A\opp}$. Note that the tensor functor $-\otimes_AT_B:
A\opp\Modcat\lra B\opp\Modcat$ is right exact. This means that
$$0=H^n\big(\Hom_A(\cpx{P}, A)\otimes_AT_B\big)\simeq
H^n(\Hom_A(\cpx{P}, A))\otimes_AT\simeq H^n(\Hom_A(T,
\cpx{T}))\otimes_AT.$$ In particular, we have $H^n(\Hom_A(T_n,
\cpx{T}))\otimes_AT_n=0$, due to $T_n\in\add(_AT)$.
Recall that the complex $\Hom_A(T_n, \cpx{T})$ is of the form
$$
\cdots \lra 0\lra \Hom_A(T_n, T_0)\lra \cdots \lra \Hom_A(T_n,
T_{n-1})\lra \Hom_A(T_n, T_n) \lra 0\lra \cdots$$ As $\Hom_A(T_n,
T_{n-1})=0$ by our assumption in Corollary \ref{cor} (2), we obtain
$H^n(\Hom_A(T_n, \cpx{T}))=\Hom_A(T_n, T_n)$. Thus
$\End_A(T_n)\otimes_AT_n=0$. It follows from the surjective map
$$\End_A(T_n)\otimes_AT_n \lra T_n,\; f\otimes x\mapsto (x)f \;\,
\mbox{ for } f\in\End_A(T_n) \;\mbox{ and }\; x\in T_n$$ that
$T_n=0$. This finishes the proof of the above claim.
By our assumption, we have $\Hom_A(T_{i+1}, T_i)=0$ for $1\leq i
\leq n-1$. Now, we can proceed by induction on $n$ to show that
$T_j=0$ for $2\leq j\leq n$. Thus, by Lemma \ref{tilt1} (4), $T$ is
a $1$-tilting module, that is, the projective dimension of $_AT$ is
at most $1$.
The sufficiency of Corollary \ref{cor} (2) follows from Theorem
\ref{main-result}, see also \cite[Theorem 1.1 (1)]{CX1}. This
finishes the proof of Corollary \ref{cor}. $\square$
\smallskip
Let us end this section by constructing an example of infinitely
generated $n$-tilting modules $T$ such that $\Ker(T\otimesL_B-)$ are
homological.
Let $A$ be an arbitrary ring with a classical $n$-tilting
$A$-module $T'$. Suppose $_AT' = M\oplus N$ with $M$ a nonzero
$A$-module of projective dimension at most $1$. Let $I$ be an
infinite set, and let $T:=M^{(I)}\oplus N$. Then $T$ is a good
$n$-tilting module. Since $T$ satisfies Corollary \ref{cor} (1), we
see that $\Ker(T\otimesL_B-)$ is homological in $\D B$.
\section{Applications to cotilting modules\label{sect6}}
Our main purpose in this section is to show Theorem \ref{coth} and
develop some conditions which can be used to decide if subcategories
induced from cotilting modules are homological or not. We also
provide an example to show that recollements provided by cotilting
modules depend upon the choice of injective cogenerators.
\subsection{Proof of Theorem \ref{coth}}\label{6.1}
In this section, we shall apply the results in Section \ref{sect4}
to deal with cotilting modules. First, we shall construct Ringel
modules from good cotilting modules, and then use Proposition
\ref{realization} to show the main result, Corollary
\ref{real-cotilt}, of this section, and finally give the proof of
Theorem \ref{coth}.
Suppose that $A$ is a ring and that $W$ is a fixed injective
cogenerator for $A\Modcat$. Recall that an $A$-module $W$ is called
a \emph{cogenerator} for $A\Modcat$ if, for any $A$-module $Y$,
there exists an injective homomorphism $Y\to W^I$ in $A\Modcat$ with
$I$ a set. This is also equivalent to saying that, for any non-zero
homomorphism $f: X\to Y$ in $A\Modcat$, there exists a homomorphism
$g\in\Hom_A(Y, W)$ such that $fg$ is non-zero.
Let us recall the definition of $n$-cotilting modules for $n$ a
natural number.
\begin{Def} {\rm \label{cotilting}
An $A$-module $U$ is called an $n$-\emph{cotilting module} if the
following three conditions are satisfied:
$(C1)$ there exists an exact sequence
$$
0\lra U\lra I_0 \lraf{\delta} I_1\lra \cdots \lra I_n\lra 0$$ of
$A$-modules such that $I_i$ is an injective module for every $0\leq
i\leq n$;
$(C2)$ $\Ext^j_A(U^{I}, U)=0$ for each $j\geq 1$ and for every
nonempty set $I$; and
$(C3)$ there exists an exact sequence
$$
0\lra U_n \lra \cdots \lra U_1 \lra U_0\lra W\lra 0$$ of
$A$-modules, such that $U_i\in\Prod(_AU)$ for all $0\leq i\leq n$.
\smallskip
An $n$-cotilting $A$-module $U$ is said to be \emph{good} if it
satisfies ($C1$), ($C2$) and
$(C3)'$ there is an exact sequence
$$0\lra U_n \lra \cdots \lra U_1 \lra U_0\lra W\lra 0$$ of $A$-modules, such that $U_i\in\add(_AU)$ for
all $0\leq i\leq n$.
We say that $U$ is a (good) cotilting $A$-module if $_AU$ is (good)
$n$-cotilting for some $n\in\mathbb{N}$.}
\end{Def}
We remark that if both $W_1$ and $W_2$ are injective cogenerators
for $A\Modcat$, then $\Prod(W_1)=\Prod(W_2)$. This implies that the
definition of cotilting modules is independent of the choice of
injective cogenerators for $A\Modcat$. However, the definition of
good cotilting modules relies on the choice of injective
cogenerators for $A\Modcat$.
As in the case of tilting modules, for a given $n$-cotilting
$A$-module $U$ with $(C1)$-$(C3)$, the $A$-module
$U':=\bigoplus_{i=0}^n U_i$ is a good $n$-cotilting module which is
equivalent to the given one in the sense that $\Prod(U)=\Prod(U')$.
\medskip
From now on, we assume that $U$ is a {\bf{good}} $n$-cotilting
$A$-module with $(C_1), (C_2)$ and $(C_3)'$, where the module $W$ in
$(C_3)'$ is referred to the fixed injective cogenerator for $A$-Mod.
In this event, we shall call $U$ a \emph{good $n$-cotilting
$A$-module with respect to $W$}.
\smallskip
Let $R:=\End_A(U)$, $M:=\Hom_A(U, W)$ and $\Lambda:=\End_A(W)$. Then
$M$ is an $R$-$\Lambda$-bimodule.
\smallskip
First of all, we collect some basic properties of good cotilting
modules in the following lemma.
\begin{Lem}\label{cotilt1}
The following hold for the cotilting module $U$.
$(1)$ The $R$-module $M$ has a finitely generated projective
resolution of length at most $n$:
$$
0\lra \Hom_A(U, U_n)\lra \cdots \lra\Hom_A(U, U_1)\lra\Hom_A(U,
U_0)\lra M\lra 0$$ such that $U_m\in\add(_AU)$ for all $0\leq m\leq
n$.
$(2)$ The Hom-functor $\Hom_A(U,-): A\Modcat\ra R\Modcat$ induces an
isomorphism of rings: $\Lambda \simeq \End_R(M)$, and $\Ext^i_R(M,
M)=0$ for all $ i\geq 1$.
$(3)$ The module $M$ is an $n$-Ringel $R$-module.
\end{Lem}
{\it Proof.} $(1)$ Applying the functor $\Hom_A(U,-)$ to the
sequence
$$0\lra U_n \lra \cdots \lra U_1 \lra U_0\lra W\lra 0$$
in the axiom $(C_3)'$, we obtain the sequence in $(1)$ with all
$\Hom_A(U, U_i)\in \add(_RR)$. The exactness of this sequence
follows directly from the axiom $(C2)$. This also implies that the
projective dimension of $_RM$ is at most $n$.
$(2)$ Denote by $\Psi$ the Hom-functor $\Hom_A(U, -): A\Modcat\to
R\Modcat$. Then $\Psi(U)=R$, \,$\Psi(W)=M$ and, for every
$X\in\add(_AU)$, we have
$$\Hom_A(X,
W)\lraf{\simeq}\Hom_R(\Psi(X), \Psi(W)).$$
Clearly, if $n=0$, then $W=U_0$, $M=\Hom_A(U, U_0)$ as $R$-modules.
In this case, one can easily check $(2)$.
Suppose $n\geq 1$. By $(1)$, the $R$-module $M=\Psi(W)$ has a
finitely generated projective resolution
$$
0\lra\Psi(U_n)\lra \cdots \lra\Psi(U_1)\lra\Psi(U_0)\lra \Psi(W)\lra
0$$ with $U_m\in\add(U)$ for all $0\leq m\leq n$. Applying the
functor $\Hom_A(-, W)$ to the resolution of $W$ in $(C3)'$, we can
construct the following commutative diagram: {\footnotesize
$$\xymatrix{
0\ar[r] & \Hom_A(W, W)\ar[r]\ar[d]_-{\Psi} &\Hom_A(U_0, W)
\ar[r]\ar[d]_-{\simeq} &\Hom_A(U_1, W) \ar[r]\ar[d]_-{\simeq}
&\cdots \ar[r] & \Hom_A(U_n, W)
\ar[r]\ar[d]_-{\simeq} &0\\
0\ar[r] & \Hom_R(\Psi(W), \Psi(W))\ar[r] &\Hom_R(\Psi(U_0), \Psi(W))
\ar[r]&\Hom_R(\Psi(U_1), \Psi(W)) \ar[r] & \cdots \ar[r] &
\Hom_R(\Psi(U_n), \Psi(W)) \ar[r]&0}
$$}
\noindent where the isomorphisms in the diagram are due to
$U_m\in\add(_AU)$ for $m\le n$. Since $_AW$ is injective, the first
row in the diagram is exact. Note that the following sequence
$$0\lra \Hom_R(\Psi(W),
\Psi(W))\lra \Hom_R(\Psi(U_0), \Psi(W)) \lra \Hom_R(\Psi(U_1),
\Psi(W))$$ is always exact since $\Psi(U_1)\lra\Psi(U_0)\lra
\Psi(W)\lra 0$ is exact in $R\Modcat$. This implies that the map
$\Psi: \End_A(W)\lra \End_R(\Psi(W))$ is an isomorphism of rings and
that the second row in the diagram is also exact. Thus $\Ext^i_R(M,
M)=\Ext^i_R(\Psi(W), \Psi(W))=0$ for all $ i\geq 1$.
$(3)$ We check the axioms $(R1)$-$(R3)$ in Definition \ref{rm} for
$M$. Clearly, the axioms $(R1)$ and $(R2)$ follow from $(1)$ and
$(2)$, respectively. It remains to show the axiom $(R3)$ for $M$. In
fact, by the axiom $(C1)$, there exists an exact sequence of
$A$-modules:
$$
0\lra U\lra I_0 \lra I_1\lra \cdots \lra I_n\lra 0$$ where $I_i$ is
an injective module for $0\leq i\leq n$. Since $W$ is an injective
cogenerator for $A\Modcat$, we have $I_i\in\Prod(_AW)$. Moreover,
from the axiom $(C2)$, we see that $\Ext^j_A(U, U)=0$ for all $j\geq
1$. This implies that the following sequence
$$
0\lra R\lra \Hom_A(U, I_0) \lra \Hom_A(U, I_1)\lra \cdots \lra
\Hom_A(U, I_n)\lra 0$$ is exact. Since the functor $\Hom_A(U, -)$
commutes with arbitrary direct products, it follows from
$I_i\in\Prod(_AW)$ that $\Hom_A(U, I_i)\in\Prod\big({}_R\Hom_A(U,
W)\big)=\Prod(_RM)$. This shows that $_RM$ satisfies the axiom
$(R3)$. Therefore $M$ is an $n$-Ringel $R$-module. $\square$
\medskip
Observe that, by Lemma \ref{cotilt1} (2), the ring $\End_R(M)$ can
be naturally identified with $\Lambda$ (up to isomorphism of rings).
Now, we define
$${\bf G}:= {}_RM\otimesL_\Lambda-:
\;\D{\Lambda}\lra\D{R}\quad \mbox{and}\quad {\bf
H}:=\rHom_R(M,-):\; \D{R}\lra\D{\Lambda}.$$ Since $_RM$ is a Ringel
$R$-module satisfying both $(R1)$ and $(R2)$ in Definition \ref{rm},
it follows from Lemma \ref{rt} that there exists a recollement of
triangulated categories:
$$
\xymatrix@C=1.2cm{\Ker({\bf H})\ar[r]^-{{i_*}} &\D{R}\ar[r]^-{{\bf
H}}\ar@/^1.2pc/[l]\ar_-{i^*}@/_1.2pc/[l] &\D{\Lambda}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{{\bf G}} }$$
\medskip
\noindent where $(i^*, i_*)$ is a pair of adjoint functors with
$i_*$ the inclusion.
If $\Ker({\bf H})$ is homological, then it follows from Lemma
\ref{rt} that the generalized localization $\lambda: R\to R_M$ of
$R$ at $M$ exists and induces a recollement of derived module
categories:
$$(\ddag)\quad
\xymatrix@C=1.2cm{\D{R_M}\ar[r]^-{D(\lambda_*)} &\D{R}\ar[r]^-{{\bf
H}}\ar@/^1.3pc/[l]\ar@/_1.2pc/[l] &\D{\Lambda}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]_{{\bf G}}}$$
\medskip \noindent Thus we may construct recollements of derived
module categories from good cotilting modules. Here, a problem
arises naturally:
\medskip
{\bf Problem:} When is $\Ker({\bf H})$ homological in $\D{R}$?
\medskip
This seems to be a difficult problem because we cannot directly
apply Proposition \ref{realization} to the Ringel module $_RM$. The
reason is that we do not know whether $_RM$ is \emph{good}.
Actually, we do not know whether the right $\Lambda$-module $M$ is
strongly $\Lambda$-Mittag-Leffler. Certainly, if $\Lambda$ is right
noetherian, then $M$ is a perfect Ringel $R$-module (see Definition
\ref{rm}), and must be good.
Though we cannot solve this problem entirely, we do have some
partial solutions to the problem.
\begin{Koro}\label{real-cotilt}
Suppose that $A$ is a ring together with an injective cogenerator
$W$ for $A\Modcat$. Let $U$ be a good $n$-cotilting $A$-module with
respect to $W$. Suppose that $\Lambda:=\End_A(W)$ is a right
noetherian ring. Then the following are equivalent:
$(a)$ $\Ker({\bf H})$ is homological in $\D{R}$.
$(b)$ $H^m\big({_R}\Hom_A(U,
W)\otimes_{\Lambda}\Hom_A(W,\,\cpx{I})\big)=0$ for all $m\geq 2$,
where $\cpx{I}$ is a deleted injective coresolution of $_AU$:
$$\cdots \lra 0\lra I_0 \lraf{\delta} I_1\lra \cdots \lra I_n\lra 0\lra
\cdots
$$
with $I_i$ in degree $i$ for all $0\leq i\leq n$.
$(c)$ The kernel $K$ of the homomorphism
$\Coker(\phi_0)\lra\Coker(\phi_1)$ induced from the map $\delta:
I_0\to I_1$ satisfies $\,\Ext^m_{R}(M, K)=0$ for all $ m\geq 0$,
where $\phi_i:\Hom_A(U, W)\otimes_{\Lambda}\Hom_A(W,\,I_i)\lra
\Hom_A(U, I_i)$ is the composition map for $i=0, 1$.
\end{Koro}
{\it Proof.} By the proof of Lemma \ref{cotilt1} (3),
the module $M:=\Hom_A(U, W)$ is an $n$-Ringel $R$-module. Moreover,
the sequence in the axiom $(R3)$ can be chosen as follows:
$$
0\lra R\lra \Hom_A(U, I_0) \lra \Hom_A(U, I_1)\lra \cdots \lra
\Hom_A(U, I_n)\lra 0.$$ In this case, the complex $\cpx{M}$ can be
defined as the following complex:
$$
\Hom_A(U, \cpx{I}):\; 0\lra \Hom_A(U, I_0) \lra \Hom_A(U, I_1)\lra
\cdots \lra \Hom_A(U, I_n)\lra 0.$$ Under the assumption that
$\Lambda$ is right noetherian, we know that $M$ is a good Ringel
$R$-module. So it follows from Proposition \ref{realization} that
$(a)$ is equivalent to the following:
$(b')$ $H^j\big({_R}M\otimes_{\Lambda}\Hom_R(M,\,\cpx{M})\big)=0$
for any $j\geq 2$, where $\cpx{M}:=\Hom_A(U, \cpx{I})$.
To prove that $(a)$ and $(b)$ in Corollary \ref{real-cotilt} are
equivalent, it is sufficient to show that $(b')$ and $(b)$ are
equivalent. For this purpose, we shall show that
$\Hom_R(M,\,\cpx{M})\simeq\Hom_A(W, \cpx{I})$ as complexes over
$\Lambda$.
Let $\Psi=\Hom_A(U, -): A\Modcat\to R\Modcat$. Then $\Psi(W)=M$ and
$\cpx{M}=\Psi(\cpx{I})$. Clearly, the functor $\Psi$ induces a
natural transformation
$$
\Hom_A(W,-) \lra \Hom_R(\Psi(W), \Psi(-)) : A\Modcat \lra
\Lambda\Modcat.
$$
This yields a chain map from $\Hom_A(W, \cpx{I})\lra \Hom_R(\Psi(W),
\Psi(\cpx{I}))=\Hom_R(M, \cpx{M})$ in $\C \Lambda$, that is,
$$\xymatrix{
0\ar[r] &\Hom_A(W, I_0) \ar[r]\ar[d] &\Hom_A(W, I_1)
\ar[r]\ar[d]&\cdots \ar[r] & \Hom_A(W, I_n)
\ar[r]\ar[d] &0\\
0\ar[r] &\Hom_R(\Psi(W), \Psi(I_0)) \ar[r]&\Hom_R(\Psi(W),
\Psi(I_1)) \ar[r] & \cdots \ar[r] & \Hom_R(\Psi(W), \Psi(I_n))
\ar[r]&0}
$$
Note that all $I_i$ are injective $A$-modules. To verify that this
chain map is an isomorphism of complexes, it is enough to show that
$\Psi$ induces an isomorphism of $\Lambda$-modules:
$$
\Hom_A(W,X) \lraf{\simeq} \Hom_R(\Psi(W), \Psi(X))
$$
for any injective $A$-module $X$. In the following, we shall prove
that this holds even for any $A$-module $X$.
Suppose $n=0$. By the axiom $(C3)'$, we know that $W = U_0$ as
$A$-modules with $U_0\in\add(_AU)$. It is clear that $\Hom_A(U_0, X)
\lraf{\simeq} \Hom_R(\Psi(U_0), \Psi(X))$ since $U_0\in\add(_AU)$.
Thus $\Hom_A(W,X) \lraf{\simeq} \Hom_R(\Psi(W), \Psi(X)).$
Now, suppose $n\geq 1$. By the axiom $(C3)'$ and Lemma \ref{cotilt1}
(1), there exists an exact sequence $U_1 \lra U_0\lra W\lra 0$ of
$A$-modules with $U_0, U_1\in\add(_AU)$ such that
$\Psi(U_1)\lra\Psi(U_0)\lra \Psi(W)\lra 0$ is also exact in
$R\Modcat$. From this sequence, we may construct the following
exact commutative diagram:
$$\xymatrix{
0\ar[r] & \Hom_A(W, X)\ar[r]\ar[d]^-{\Psi} & \Hom_A(U_0, X)
\ar[r]\ar[d]^-{\simeq}&\Hom_A(U_1, X)\ar[d]^-{\simeq}\\
0\ar[r] & \Hom_R(\Psi(W), \Psi(X)) \ar[r]& \Hom_R(\Psi(U_0),
\Psi(X)) \ar[r] &\Hom_R(\Psi(U_1), \Psi(X)) }$$ where the last two
vertical maps are isomorphisms since $U_0, U_1\in\add(_AU)$. This
means that $\Hom_A(W,X) \lraf{\simeq} \Hom_R(\Psi(W), \Psi(X))$ for
every $A$-module $X$.
Consequently, we see that $\Hom_A(W,
\cpx{I})\simeq\Hom_R(M,\,\cpx{M})$ as complexes over $\Lambda$. Thus
$(b')$ and $(b)$, and therefore, also $(a)$ and $(b)$, are
equivalent.
Note that if we identify $\Hom_R(M,\,\cpx{M})$ with $\Hom_A(W,
\cpx{I})$ as complexes over $\Lambda$, then the equivalence of $(a)$
and $(c)$ in Corollary \ref{real-cotilt} can be concluded from that
of $(1)$ and $(4)$ in Proposition \ref{realization}. Here, we leave
the details to the reader. $\square$
\medskip
As a consequence of Corollary \ref{real-cotilt} (see also Corollary
\ref{appl}), we have the following result.
\begin{Koro}\label{cotilt2}
Let $U$ be a good $n$-cotilting $A$-module with respect to the
injective cogenerator $_AW$. Suppose that $\Lambda:=\End_A(W)$ is a
right noetherian ring.
$(1)$ If $_AU=M\oplus N$ such that $_AM$ has injective dimension at
most $1$ and that $_AN$ has an injective copresentation $0\lra
{}_AN\lra E_0\lra E_1$ with $E_1\in\add(_AW)$, then $\Ker({\bf H})$
is homological in $\D{R}$.
$(2)$ If $n=2$, then $\Ker({\bf H})$ is homological in $\D R$ if and
only if $\Hom_A(U, W)\otimes_{\Lambda}\Ext^2_A(W, U)=0.$
\end{Koro}
{\it Proof.} The idea of the proof of $(1)$ is very similar to that
of Corollary \ref{cor} (1). Here, we just give a sketch of the
proof.
Note that $\mathscr{E}:=\{Y\in R\Modcat \mid \Ext^m_R(M,
Y)=0\mbox{\;for\, all \,} m\geq0\}$ is closed under kernels of
surjective homomorphisms in $R\Modcat$, and that $\Coker(\phi_0)$
and $\Coker(\phi_1)$ (see Corollary \ref{real-cotilt} (c)) always
belong to $\mathscr{E}$ by Lemma \ref{prep} (1). Thus, according to
the equivalence of $(a)$ and $(c)$ in Corollary \ref{real-cotilt},
if we want to show $(1)$, then it suffices to verify that the
homomorphism $\widetilde{\delta}: \Coker(\phi_0)\lra\Coker(\phi_1)$
induced from $\delta: I_0\to I_1$ is surjective. Actually, this is
guaranteed by the assumption that the injective dimension of $_AM$
is at most $1$ and $E_1\in\add(_AW)$. For more details, we refer the
reader to the proof of Corollary \ref{cor} (1).
As to $(2)$, we keep the notation in the proof of Corollary
\ref{real-cotilt}. Suppose $n=2$. Then the complex $\cpx{I}$ in
Corollary \ref{real-cotilt} (b) has the following form
$$\cdots \lra 0\lra I_0 \lra I_1\lra I_2\lra 0\lra
\cdots.
$$
By Corollary
\ref{real-cotilt}, the category $\Ker({\bf H})$ is homological if
and only if
$H^2\big({_R}M\otimes_{\Lambda}\Hom_A(W,\,\cpx{I})\big)=0$, where
$M:=\Hom_A(U, W)$. Note that the tensor functor
$_RM\otimes_{\Lambda}-: \Lambda\Modcat\lra R\Modcat$ is right exact.
Consequently, we have
$$H^2\big({_R}M\otimes_{\Lambda}\Hom_A(W,\,\cpx{I})\big)\simeq
M\otimes_{\Lambda} H^2(\Hom_A(W,\,\cpx{I}))\simeq
M\otimes_{\Lambda}\Ext^2_A(W, U).$$ This shows $(2)$. $\square$
\medskip
Finally, we point out a special case for which the ring $\Lambda$ in
Corollary \ref{real-cotilt} is right noetherian.
Let $k$ be a commutative Artin ring. Let $\rad(k)$ be the radical of
$k$ (that is, the intersection of all maximal ideals of $k$), and
let $J$ be the injective envelope of $k/\rad(k)$. We say that a
$k$-algebra $A$ is an \emph{Artin $k$-algebra}, or Artin algebra for
short, if $A$ is finitely generated as a $k$-module.
Suppose that $A$ is an Artin $k$-algebra. It is well known that the
functor $\Hom_k(-, J)$ is a duality between the category $A\modcat$
of finitely generated $A$-modules and that of finitely generated
$A\opp$-modules. In particular, the dual module $\Hom_k(A_A, J)$ of
the right $A$-module $A_A$ is an injective cogenerator for
$A\modcat$, or even for $A\Modcat$. In this case, we shall call
$\Hom_k(A_A, J)$ the \emph{ordinary injective cogenerator} for
$A\Modcat$.
Note that $\End_A(\Hom_k(A_A, J))\simeq \End_{A\opp}(A)\opp\simeq A$
as rings. So, if the module $W$ in Corollary \ref{real-cotilt} is
chosen to be the module $\Hom_k(A_A, J)$, then the ring
$\Lambda:=\End_A(W)$ is isomorphic to $A$. Since $A$ is an Artin
algebra, it is a left and right Artin ring, and certainly a right
noetherian ring. Thus $\Lambda$ is right noetherian and always
satisfies the assumption in Corollary \ref{real-cotilt}.
\medskip
{\bf Proof of Theorem \ref{coth}.} Recall that $_AW$ is the ordinary
injective cogenerator over the Artin algebra $A$. According to the
above-mentioned facts, the ring $\Lambda:=\End_A(W)$ is isomorphic
to $A$, and therefore right noetherian. Since $_AU$ is a good
$1$-cotilting module with respect to $W$, we know from Corollary
\ref{cotilt2} (1) that the category $\Ker({\bf H})$ is homological.
Now, Theorem \ref{coth} follows from the diagram $(\ddag)$ above
Corollary \ref{real-cotilt}. $\square$
\medskip
Let us end this section by a couple of remarks related to the
results in this section.
{\it Remarks.} $(1)$ If $A$ is a commutative ring and $W$ is an
injective cogenerator for $A\Modcat$, then the dual module
$\Hom_A(T, W)$ of a tilting $A$-module $T$ is always a cotilting
$A$-module. However, there exist cotilting modules over Pr\"ufer
domains, which are not equivalent to the dual modules of any tilting
modules (see \cite[Chapter 11, Section 4.16]{HHK}). This means that
the investigation of infinitely generated cotilting modules cannot
be carried out by using dual arguments of infinitely generated
tilting modules.
$(2)$ Corollary \ref{real-cotilt} provides actually a recollement of
$\D{\End_A(U)}$ with $\D{R_M}$ on the left-hand side and
$\D{\Lambda}$ on the right-hand side (see $(\ddag)$ for notation).
This recollement depends upon the choice of injective cogenerators
for $A\Modcat$. That is, for a fixed cotilting module $_AU$, if
different injective cogenerators $W$ for $A\Modcat$ are chosen in
the axiom $(C3)'$, then one may get completely different
recollements of $\D{\End_A(U)}$.
For example, let ${\mathbb Q}_{(p)}$, $\mathbb Q$, ${\mathbb Z}_p$
and ${\mathbb Q}_p$ denote the rings of $p$-integers, rational
numbers, $p$-adic integers and $p$-adic numbers, respectively.
Recall that ${\mathbb Q}_{(p)}$ is the localization of $\mathbb{Z}$
at the prime ideal $p\mathbb{Z}$. In particular, it is a local
Dedekind domain. Moreover, let $E(\mathbb{Z}/p\mathbb{Z})$ be the
injective envelope of $\mathbb{Z}/p\mathbb{Z}$, which is an
injective cogenerator for the category of ${\mathbb
Q}_{(p)}$-modules.
Now, we take $A:=\mathbb{Q}_{(p)}$, $T:=\mathbb{Q}\oplus
E(\mathbb{Z}/p\mathbb{Z})$ and
$U:=\Hom_A(T,E(\mathbb{Z}/p\mathbb{Z}))$. Due to \cite[Section
7.1]{CX1}, we have
$(a)$ the module $T$ is a Bass $1$-tilting module over $A$, and
therefore $U$ is an $1$-cotilting $A$-module.
$(b)$ $\End_A(E(\mathbb{Z}/p\mathbb{Z}))\simeq {\mathbb Z}_p\,$ and
$\,\Hom_A(\mathbb{Q},
E(\mathbb{Z}/p\mathbb{Z}))\simeq\mathbb{Q}\otimes_A\End_A(E(\mathbb{Z}/p\mathbb{Z}))\simeq
\mathbb{Q}\otimes_A {\mathbb Z}_p\simeq {\mathbb Q}_p $. Thus
$U\simeq {\mathbb Z}_p\oplus {\mathbb Q}_p$ as $A$-modules.
$(c)$ By \cite[Lemma 6.5(3)]{CX1}, there exists an exact sequence of
${\mathbb Z}_p$-modules (and also $A$-modules):
$$
(*')\quad 0\lra {\mathbb Z}_p \lraf{\varphi} {\mathbb Q}_p\lra
E(\mathbb{Z}/p\mathbb{Z})\lra 0.
$$
Note that ${\mathbb Q}_p$ is an injective and flat $A$-module and
that $(*')$ is an injective coresolution of ${\mathbb Z}_p$ as an
$A$-module. This also implies that $W:={\mathbb Q}_p\oplus
E(\mathbb{Z}/p\mathbb{Z})$ is an injective cogenerator for
$A\Modcat$.
On the one hand, we may consider $U$ as a good $1$-cotilting
$A$-module with respect to $W$. Applying $\Hom_A(U, -)$ to the
sequence $(*')$, we get a projective resolution of $\Hom_A(U,
E(\mathbb{Z}/p\mathbb{Z}))$ as an $\End_A(U)$-module:
$$
\quad 0\lra \Hom_A(U, {\mathbb Z}_p) \lraf{\varphi^*} \Hom_A(U, {\mathbb Q}_p)\lra
\Hom_A(U, E(\mathbb{Z}/p\mathbb{Z}))\lra 0.
$$
Since both ${\mathbb Q}_p$ and $E(\mathbb{Z}/p\mathbb{Z})$ belong to
$\add(_AW)$, one can use Lemma \ref{cotilt1} to show that $\Hom_A(U,
W)$ is a classical $1$-tilting $\End_A(U)$-module such that
$\End_{\End_A(U)}(\Hom_A(U, W))\simeq \End_A(W)$ as rings. It
follows that $\End_A(U)$ and $\End_A(W)$ are derived equivalent. In
this case, we get a trivial recollement:
$\D{\End_A(U)}\lraf{\simeq}\D{\Lambda}$ with $\Lambda:=\End_A(W)$.
Note that this derived equivalence can also be seen from
\cite[Theorem 1.1]{hx2}.
On the other hand, we consider $U$ as a good $1$-cotilting
$A$-module with respect to $W':= E(\mathbb{Z}/p\mathbb{Z})$.
Clearly, the sequence $(*')$ can paly the role in the axiom $(C3)'$.
Since $\End_A(E(\mathbb{Z}/p\mathbb{Z})) \simeq\mathbb{Z}_p,$
we know from \cite[Corollary 2.5.16]{EJ} that
$\End_A(E(\mathbb{Z}/p\mathbb{Z}))$ is a noetherian ring. This
implies that $U$ satisfies the assumptions in Corollary
\ref{cotilt2} (1).
By \cite[Theorem 3.4.1]{EJ}, one can check that
$$
\End_A(\mathbb{Z}_p)\simeq \mathbb{Z}_p, \;\, \Hom_A(\mathbb{Q}_p,
\mathbb{Z}_p)=0=\Ext_A^1(\mathbb{Q}_p,
\mathbb{Z}_p)=\Hom_A(E(\mathbb{Z}/p\mathbb{Z}), \mathbb{Q}_p),$$ and
further that
$$
\End_A(U)\simeq \left(\begin{array}{lc} \mathbb{Z}_p & \End_A(\mathbb{Q}_p)\\
0 & \End_A(\mathbb{Q}_p)\end{array}\right) \;\, \mbox{and} \;\,
\End_A(W)\simeq \left(\begin{array}{lc} \End_A(\mathbb{Q}_p)& \End_A(\mathbb{Q}_p)\\
\quad 0 & \mathbb{Z}_p\end{array}\right).
$$
Moreover, the universal localization of $\End_A(U)$ at the map
$\varphi^*$, or at the module $\Hom_A(U,
E(\mathbb{Z}/p\mathbb{Z}))$, is isomorphic to
$M_2(\End_A(\mathbb{Q}_p))$, the $2\times 2$ matrix ring over
$\End_A(\mathbb{Q}_p)$.
Now, we can construct the following non-trivial recollement of
derived module categories from the cotilting module $U$ with respect
to $W'= E(\mathbb{Z}/p\mathbb{Z})$:
$$\xymatrix@C=1.2cm{\D{\End_A(\mathbb{Q}_p)}\ar[r]
&\D{\End_A(U)}\ar[r]\ar@/^1.3pc/[l]\ar@/_1.2pc/[l] &\D{\mathbb{Z}_p}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]}$$
\bigskip\noindent Thus, the recollement $(\ddag)$ above Corollary \ref{real-cotilt} constructed
from a cotilting module $U$ depends on injective cogenerator with
respect to which the $U$ is defined.
\subsection{Necessary conditions of homological subcategories from cotilting modules \label{6.2}}
We keep the notation in Section \ref{6.1}. For the cotilting module
$U$, we denote by
$$0\lra U_n \lraf{\partial_n} U_{n-1}\lra \cdots \lraf{\partial_2}
U_1 \lraf{\partial_1}U_0\lraf{\partial_0} W \lra 0
$$ the exact sequence in the axiom $(C_3)'$, and by
$\cpx{U}$ the following complex
$$\cdots \lra 0\lra U_n \lraf{\partial_n} U_{n-1}\lra \cdots \lraf{\partial_2}
U_1 \lraf{\partial_1} U_0 \lra 0\lra \cdots$$ with $U_i$ in degree
$-i$ for all $0\leq i\leq n$. Then $\partial_0$ induces a canonical
quasi-isomorphism $\widetilde{\partial}_0: \cpx{U}\ra W$ in $\C{A}$.
Recall that the complex $\cpx{I}$ in Corollary \ref{real-cotilt} (b)
also yields a canonical quasi-isomorphism $\xi: U\ra \cpx{I}$ in
$\C{A}$.
Furthermore, by the proof of the first part of Lemma \ref{Z-iso},
one can show that $\widetilde{\partial}_0$ and $\xi$ do induce the
following quasi-isomorphisms
$$(\ast)\quad
\xymatrix{\Hom_A(W, \cpx{I})\ar[r]^-{(\widetilde{\partial}_0)_*} &
\cpx{\Hom}_A(\cpx{U}, \cpx{I}) & \ar[l]_-{\xi^*} \Hom_A(\cpx{U},
U)}$$ in $\C{\mathbb{Z}}$. Here, we leave checking the details to
the reader.
Consequently, the morphism
$(\widetilde{\partial}_0)_*({\xi^*})^{-1}: \Hom_A(W, \cpx{I})\lra
\Hom_A(\cpx{U}, U)$ in $\D{\mathbb Z}$ is an isomorphism (compare
with Lemma \ref{Z-iso}). Due to the $A$-$\Lambda$-bimodule structure
of $W$, the former complex belongs to $\C{\Lambda}$. However, the
latter complex might not be a complex of $\Lambda$-modules since
$\cpx{U}$ is not necessarily a complex of $A$-$\Lambda$-bimodules in
general. This means that this isomorphism may not be extended to an
isomorphism in $\D{\Lambda}$. Nonetheless, for some special
cotilting modules, we do have this isomorphism in $\D{\Lambda}$. For
instance, in the case described in the following lemma.
\begin{Lem}\label{Z-iso2}
Suppose that $\Hom_A(U_i, U_{i+1})=0$ for $0\leq i < n$.
$(1)$ There exist a series of ring homomorphisms $\rho_j:
\Lambda\lra \End_A(U_j)$ for $0\leq j\leq n$, such that
$\,\widetilde{\partial}_0: \cpx{U}\lra W$ is a quasi-isomorphism in
$\C{A\otimes_\mathbb{Z}\Lambda\opp}$. In particular, the complexes
$\Hom_A(W, \cpx{I})$ and $\Hom_A(\cpx{U}, U)$ are isomorphic in
$\D{\Lambda}$.
$(2)$ If $\Ext^k_A(W, U_k)=\Ext_A^{k+1}(W, U_k)=0$ for all $0\leq k
< n$, then $\rho_n: \Lambda\lra \End_A(U_n)$ is an isomorphism.
\end{Lem}
{\it Proof.} $(1)$ Set $K_0:=W$, $K_n:=U_n$ and
$K_{m}:=\Ker(\partial_{m-1})$ for $1\leq m<n$. Then, for each $0\leq
i<n$, we have a short exact sequence $0\lra K_{i+1}\lra
U_i\lraf{\partial_i} K_i\lra 0$ of $A$-modules. In the following, we
shall define two ring homomorphisms $\varphi_i: \End_A(K_i)\lra
\End_A(U_i)$ and $\psi_i: \End_A(K_i)\lra \End_A(K_{i+1})$.
By Lemma \ref{cotilt1} (1), the sequence
$$0\lra \Hom_A(U, K_{i+1})\lra \Hom_A(U, U_i)\lraf{\partial_i^*}
\Hom_A(U, K_i)\lra 0$$ is exact. In particular, for $U_i\in\add(U)$,
the sequence
$$0\lra \Hom_A(U_i, K_{i+1})\lra \Hom_A(U_i, U_i)\lraf{\partial_i^*}
\Hom_A(U_i, K_i)\lra 0$$ is exact. Let $f\in\End_A(K_i)$. Then there
is a homomorphism $g\in\End_A(U_i)$ such that $\partial_i
f=g\,\partial_i$. We claim that such a $g$ is unique. Actually, if
there exists another $g'\in\End_A(U_i)$ such that $\partial_i
f=g'\,\partial_i$. Then $(g-g')\partial_i=0$, and so the map $g-g'$
factorizes through $K_{i+1}$. Note that each homomorphism $U_i\to
K_{i+1}$ also factorizes through $U_{i+1}$ via $\partial_{i+1}$.
This implies that $g-g':U_i\ra U_i$ factorizes through $U_{i+1}$.
However, since $\Hom_A(U_i, U_{i+1})=0$ by assumption, we have
$g=g'$. Hence, for a given $f$, such a $g$ is unique.
Now, we define $\varphi_i: f\mapsto g$ and $\psi_i: f\mapsto h$
where $h$ is the restriction of $g$ to $K_{i+1}$. This can be
illustrated by the following commutative diagram:
$$ \xymatrix{
0\ar[r]&\,K_{i+1}\ar@{-->}[d]^-{h}\ar[r]^-{\lambda_{i+1}}&U_i\ar@{-->}[d]^-{g}\ar[r]^-{\partial_i}
&K_i\ar[d]^-{f}\ar[r] &0\\
0\ar[r] &
K_{i+1}\ar[r]^-{\lambda_{i+1}}&U_i\ar[r]^-{\partial_i}&K_i\ar[r] &0}
$$
where $\lambda_{i+1}$ is the inclusion for $0\leq i\leq n-2$ and
$\lambda_{n}:=\partial_n$. Clearly, both $\varphi_i$ and $\psi_i$
are ring homomorphisms.
Recall that $\Lambda:=\End_A(W)=\End_A(K_0)$. Furthermore, for
$0\leq j\leq n$, we define $\rho_j: \Lambda\to \End_A(U_j)$ as
follows: If $j=0$, then $\rho_0:=\varphi_0\,$; if $j\geq 1$, then
$\rho_j$ is defined to be the composite of the following ring
homomorphisms:
$$\Lambda\lraf{\psi_0}\End_A(K_1) \lraf{\psi_1} \End_A(K_2)\lra \cdots
\lra \End_A(K_{j-1}) \lraf{\psi_{j-1}} \End_A(K_j)\lraf{\varphi_j}
\End_A(U_j)
$$
where $\varphi_n$ stands for the identity map. By definition, for
each $\lambda\in\Lambda$, there exists an exact commutative diagram
of $A$-modules:
$$ \xymatrix{0\ar[r] & U_n \ar[r]^-{\partial_n}\ar[d]_-{(\lambda)\rho_n} & U_{n-1}
\ar[r]\ar[d]^-{(\lambda)\rho_{n-1}}&\cdots \ar[r]^-{\partial_2} &
U_1 \ar[r]^-{\partial_1} \ar[d]^-{(\lambda)\rho_1} &
U_0\ar[r]^-{\partial_0} \ar[d]^-{(\lambda)\rho_0} & W \ar[r]\ar[d]^-{\lambda} &0\\
0\ar[r] & U_n \ar[r]^-{\partial_n} & U_{n-1}\ar[r] &\cdots
\ar[r]^-{\partial_2} & U_1 \ar[r]^-{\partial_1} &
U_0\ar[r]^-{\partial_0} & W \ar[r] & 0}
$$
Note that $U_j$ is a natural $A$-$\End_A(U_j)$-bimodule and can be
regarded as an $A$-$\Lambda$-bimodule via $\rho_j$. It follows from
the above commutative diagram that $\partial_j$ is a homomorphism of
$A$-$\Lambda$-bimodules. This implies that $\widetilde{\partial}_0:
\cpx{U}\lra W$ can be viewed as a quasi-isomorphism in
$\C{A\otimes_\mathbb{Z}\Lambda^{op}}$. In this sense, the
quasi-isomorphisms in $(\ast)$ actually belong to $\C{\Lambda}$.
Thus $\Hom_A(W, \cpx{I})$ and $\Hom_A(\cpx{U}, U)$ are isomorphic in
$\D{\Lambda}$. This finishes $(1)$.
$(2)$ To show that $\rho_n$ is an isomorphism of rings, it suffices
to prove that $\psi_i$ is an isomorphism for $0\leq i\leq n-1$. Let
$i$ be such a fixed number. If $\Hom_A(K_i, U_i)=0$, then $\psi_i$
is injective. If the induced map $(\lambda_{i+1})_*:\Hom_A(U_i,
U_i)\lra \Hom_A(K_{i+1}, U_i)$ is surjective, then so is $\psi_i$.
Thus, by our assumptions in (2), to show that $\psi_i$ is an
isomorphism, it suffices to show that $\Hom_A(K_i, U_i)\simeq
\Ext^i_A(W, U_i)$ and that there exists an exact sequence of abelian
groups:
$$(\ast\ast)\quad
\Hom_A(U_i,\,U_i)\lraf{(\lambda_{i+1})_*} \Hom_A(K_{i+1},
U_i)\lra \Ext_A^{i+1}(W, U_i)\lra 0.$$
In fact, since $U_s\in\add(_AU)$ for $0\leq s\leq n$, we have
$\Ext^r_A(U_s, X)=0$ for each $r\geq 1$ and $X\in\add(_AU)$ by the
axiom $(C2)$. Now, for $1\leq j\leq n$ and $X\in \add(_AU)$, one can
apply $\Hom_A(-, X)$ to the long exact sequence
$$0\lra K_{j}\lraf{\lambda_j} U_{j-1}\lra \cdots \lra U_1\lra U_0\lra W\lra 0,$$
and get an exact sequence $\Hom_A(U_{j-1}, X)\lraf{(\lambda_j)_*}
\Hom_A(K_j, X)\lra \Ext_A^j(W, X)\lra 0$ of abelian groups. If we
take $j:=i$ and $X:=U_i$, then $\Hom_A(K_i, U_i)\simeq\Ext_A^i(W,
U_i)$ since $\Hom_A(U_{i-1}, U_i)=0$ by assumption. If we take
$j:=i+1$ and $X:=U_i$, then we get the required sequence
$(\ast\ast)$. This finishes the proof of (2). $\square$
\medskip
The following result will be used for getting a counterexample which
demonstrates that, in general, the category $\Ker({\bf H})$ in
Corollary \ref{real-cotilt} may not be homological.
\begin{Koro}\label{new}
Keep all the assumptions in Corollary \ref{real-cotilt}. Further,
suppose that $n\geq 2$ and $U$ has injective dimension exactly equal
to $n$. If $\,\Hom_A(U_i, U_{i+1})=\Ext^i_A(W, U_i)$
$=\Ext_A^{i+1}(W, U_i)=0$ for all $0\leq i < n$, then the category
$\Ker({\bf H})$ is not a homological subcategory of $\D{R}$.
\end{Koro}
{\it Proof.} Suppose contrarily that $\Ker({\bf H})$ is homological
in $\D{R}$. Then, by Corollary \ref{real-cotilt}, we certainly have
$H^n\big({_R}\Hom_A(U,\,W)\otimes_{\Lambda}\Hom_A(W,\,\cpx{I})\big)=0$.
Furthermore, since $\Hom_A(U_i, U_{i+1})=0$ for all $0\leq i\leq
n-1$, we know from Lemma \ref{Z-iso2} (1) that $\Hom_A(W,
\cpx{I})\simeq \Hom_A(\cpx{U}, U)$ in $\D{\Lambda}$. Thus {\small
$$0=H^n\big(\Hom_A(U,\,W)\otimes_{\Lambda}\Hom_A(W,\,\cpx{I})\big)
\simeq\Hom_A(U,\,W)\otimes_{\Lambda}H^n\big(\Hom_A(W,\,\cpx{I})\big)\simeq
\Hom_A(U,\,W)\otimes_{\Lambda}H^n\big(\Hom_A(\cpx{U},\,U)\big).$$\vspace{-0.3cm}}
\noindent In particular, we have
$\Hom_A(U,\,W)\otimes_{\Lambda}H^n\big(\Hom_A(\cpx{U},\,U_n)\big)=0$,
due to $U_n\in\add(_AU)$. Recall that the complex $\Hom_A(\cpx{U},
U_n)$ is of the form {\small $$0\lra \Hom_A(U_0, U)
\lraf{(\partial_1)_*} \Hom_A(U_1, U_n)\lra \cdots\lra
\Hom_A(U_{n-2}, U_n) \lraf{(\partial_{n-1})_*} \Hom_A(U_{n-1}, U_n)
\lraf{(\partial_n)_*} \Hom_A(U_n, U_n) \lra 0$$\vspace{-0.5cm}}
\noindent with $\Hom_A(U_n, U_n)$ in degree $n$. Since
$\Hom_A(U_{n-1}, U_n)=0$, we obtain
$H^n\big(\Hom_A(\cpx{U},\,U_n)\big)=\End_A(U_n)$, and so
$\Hom_A(U,\,W)\otimes_{\Lambda}\End_A(U_n)=0$. Note that the left
$\Lambda$-module structure of $\End_A(U_n)$ is defined by the ring
homomorphism $\rho_n: \Lambda\lra \End_A(U_n)$ (see Lemma
\ref{Z-iso2} (1)). Since $\Ext^i_A(W, U_i)$ $=\Ext_A^{i+1}(W,
U_i)=0$ for all $0\leq i\leq n-1$, we see from Lemma \ref{Z-iso2}
(2) that $\rho_n$ is an isomorphism. This implies that
$$\Hom_A(U,\,W)\otimes_{\Lambda}\End_A(U_n)\simeq
\Hom_A(U,\,W)\otimes_{\Lambda}\Lambda\simeq\Hom_A(U,\,W)$$ and
therefore $\Hom_A(U,\,W)=0$. Since $_AW$ is an injective
cogenerator, we must have $U=0$. This is a contradiction. Thus
$\Ker({\bf H})$ is not homological in $\D{R}$. $\square$
\section{Counterexamples and open questions} \label{sect7}
In this section, we shall apply results in the previous sections to
give two examples which show that, in general, the category
$\Ker(_AT\otimesL_B-)$ for an $n$-tilting module $T$, or the
category $\Ker(\bf{H})$ for an $n$-cotilting module $U$ may not be
homological. At the end of this section, we mention a few open
questions related to some results in this paper.
Throughout this section, we assume that $A$ is a commutative,
noetherian, $n$-Gorensteion ring for a natural number $n$. Recall
that a ring is called $n$-\emph{Gorenstein} if the injective
dimensions of the regular left and right modules are at most $n$.
For an $A$-module $M$, we denote by $E(M)$ its injective envelope.
It is known that if $\mathfrak p$ and $\mathfrak q$ are two prime
ideals of $A$, then $\Hom_A(E(A/\mathfrak p), E(A/\mathfrak q))\neq
0$ if and only if $\mathfrak p \subseteq \mathfrak q$ (see
\cite[Theorem 3.3.8]{EJ}). In particular, $E(A/\mathfrak p)\simeq
E(A/\mathfrak q)$ if and only if $\mathfrak p=\mathfrak q$
\subsection{Higher $n$-tilting modules}\label{7.1}
In the following, we shall apply Corollary \ref{cor} to provide an
example of a good $n$-tilting $A$-module $T$ for which the category
$\Ker(_AT\otimesL_B-)$ in Theorem \ref{main-result} is not
homological.
For the $n$-Gorenstein ring $A$, it follows from a classical result
of Bass that the regular module $_AA$ has a minimal injective
coresolution of the form:
$$
0\lra A\lra\bigoplus_{\mathfrak p\in {\mathcal P}_0}E(A/\mathfrak
p)\lra\cdots\lra \bigoplus_{\mathfrak p\in
\mathcal{P}_n}E(A/\mathfrak p)\lra 0, $$ where $\mathcal{P}_i$
stands for the set of all prime ideals of $A$ with height $i$ (see
\cite[Theorem 1, Theorem 6.2]{Bs}). It was pointed out in
\cite[Introduction]{TP} that the $A$-module
$$T:=\bigoplus_{0\leq i\leq
n}\,\bigoplus_{\mathfrak p\in \mathcal{P}_i}E(A/\mathfrak p)$$ is an
(infinitely generated) $n$-tilting module.
Clearly, the tilting module $_AT$ is good if we define
$T_i:=\bigoplus_{\mathfrak p\in \mathcal{P}_i}E(A/\mathfrak p)$.
Observe that, for $0\leq i<j\leq n$, we have $\Hom_A(E(A/\mathfrak
p), E(A/\mathfrak q))=0$ for $\mathfrak p\in P_j$ and $\mathfrak
q\in P_i$, and therefore $\Hom_A(T_j, T_i)=0$.
Now, we suppose that $n\geq 2$ and the injective dimension of $A$ is
exactly equal to $n$ (or equivalently, the Krull dimension of $A$ is
exactly $n$).
Note that $T_i\neq 0$ for all $2\leq i\leq n$ and that $T$ satisfies
the assumptions in Corollary \ref{cor} (2). Since the above
injective coresolution of $A$ is minimal, the module $_AT$ has
projective dimension equal to $n$ (see \cite[Proposition 3.5]{Bz1}).
By Corollary \ref{cor} (2), the category $\Ker(_AT\otimesL_B-)$ is
not homological in $\D B$. This means that for this tilting module
$T$, the subcategory $\Ker(_AT\otimesL_B-)$ cannot be realized as
the derived module category $\D C$ of a ring $C$ with a homological
ring epimorphism $B\ra C$. Thus, for higher $n$-tilting modules, the
answer to the question in Introduction is negative in general.
\subsection{Higher $n$-cotilting modules}\label{7.2}
Next, we apply Corollary \ref{new} to present an example of a good
$n$-cotilting $A$-module $U$, for which the category $\Ker({\bf H})$
in Corollary \ref{real-cotilt} is not homological in $\D{R}$.
Assume further that the ring $A$ is \emph{local} with the unique
maximal ideal $\mathfrak m$. In this case, $T_n$ is an injective
cogenerator for $A\Modcat$ since $\mathcal{P}_n$ is just the set
$\{\mathfrak m\}$. This follows from a general statement in
commutative algebra: If $S$ is a commutative noetherian ring, then
$\bigoplus_{\mathfrak m} E(S/\mathfrak{m})$ is an injective
cogenerator for $S$-Mod, where $\mathfrak m$ runs over all maximal
ideals of $S$.
Now, we take $$W:=T_n\quad \mbox{and}\quad U:=\Hom_A(T,
W)=\bigoplus_{j=0}^n\Hom_A(T_j, W).$$ Since $_AT$ is an $n$-tilting
$A$-module, the module $_AU$ is an $n$-cotilting $A$-module.
Furthermore, applying $\Hom_A(-, W)$ to the minimal injective
coresolution of $_AA$, we get the following exact sequence of
$A$-modules:
$$
0\lra \Hom_A(T_n, W)\lra \Hom_A(T_{n-1}, W)\lra \cdots\lra
\Hom_A(T_1, W)\lra \Hom_A(T_0, W)\lra W\lra 0.
$$
This implies that the cotilting $A$-module $U$ is good if we define
$U_j:=\Hom_A(T_j, W)$ for $0\leq j\leq n$ (see the axiom $(C_3)'$ in
Definition \ref{cotilting}).
To see that $\Lambda:=\End_A(W)$ is a right noetherian ring, we note
that $W = E(A/\mathfrak m)$ and that $\Lambda$ is isomorphic to the
$\mathfrak m$-adic complete of $A$ (see \cite[Theorem 3.4.1
(6)]{EJ}). Since $A$ is noetherian, the ring $\Lambda$ is also
noetherian (see \cite[Corollary 2.5.16]{EJ}).
In the following, we shall prove that $_AU$ satisfies all the
assumptions in Corollary \ref{new}. In fact, it suffices to show
that, for any $m\geq 0$, we have
$(a)$ $\Ext_A^m(U_r, U_s)=0$ for $0\leq r<s\leq n$.
$(b)$ $\Ext_A^m(W, U_i)=0$ for $0\leq i\leq n-1$, and $\Ext_A^n(W,
U_n)\neq 0$.
\noindent The reason is the following: According to $(b)$, the
injective dimension of $U_n$ is at least $n$, and therefore exactly
$n$. This means that $_AU$ is a cotilting module of injective
dimension $n$. Moreover, from $(a)$ and $(b)$ we can conclude that
the assumptions in Corollary \ref{new} hold true for $U$. It then
follows from Corollary \ref{new} that, for this cotilting module
$U$, the category $\Ker({\bf H})$ in Corollary \ref{real-cotilt} is
not homological in $\D{R}$ with $R:=\End_A(U)$. In other words,
$\Ker({\bf H})$ cannot be realized as the derived module category
$\D S$ of a ring $S$ with a homological ring epimorphism $R\ra S$.
So, let us verify the above $(a)$ and $(b)$. First, we need the
following results about $n$-Gorenstein rings:
$(1)$ The flat dimension of the $A$-module $T_j$ is exactly $j$.
$(2)$ Any flat $A$-module $F$ admits a minimal injective
coresolution of the form
$$
0\lra {}_AF \lra I_0\lra I_1\lra \cdots\lra I_{n-1}\lra I_{n}\lra 0
$$
such that $I_j\in\Add(T_j)$ for all $0\leq j\leq n$.
$(3)$ Let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals of $A$.
If $\mathfrak{p}\nsubseteq \mathfrak{q}$ or $\mathfrak{q}\nsubseteq
\mathfrak{p}$, then $\Tor_m^A(E(A/\mathfrak {p}),\, E(A/\mathfrak
{q}))=0$ for all $m\geq 0$. Moreover, $\Tor_m^A(E(A/\mathfrak
{p}),\, E(A/\mathfrak {p}))\neq 0$ if and only if $m$ equals the
height of $\mathfrak{p}$ in $A$.
Here, $(1)$ and $(2)$ follow from \cite[Proposition 2.1 and Theorem
2.1]{X}, while $(3)$ is taken from \cite[Lemma 9.4.5 and Theorem
9.4.6]{EJ}.
\smallskip
Since the dual $A$-module $\Hom_A(F, W)$ of a flat $A$-module $F$ is
injective, we know from $(1)$ that the injective dimension of
$_AU_j$ is at most $j$. Since the dual $A$-module $\Hom_A(I, W)$ of
an injective $A$-module $I$ is always flat (see \cite[Corollary
3.2.16 (2)]{EJ}), we see that the $A$-module $U_j$ is flat since
$T_j$ is injective. It then follows from $(2)$ that $U_j$ admits a
minimal injective coresolution of the form
$$
0\lra U_j \lra I_{j, 0}\lra I_{j, 1}\lra \cdots\lra I_{j, j-1}\lra
I_{j, j}\lra 0
$$
with $I_{j,\, k}\in\Add(T_k)$ for all $0\leq k\leq j$.
Now, we show $(a)$. Actually, by Lemma \ref{tor-ext} (1), we have
$$\Ext_A^m(U_r, U_s)=\Ext_A^m\big(U_r, \,\Hom_A(T_s,
W)\big)\simeq\Hom_A\big(\Tor_m^A(T_s, U_r),\, W\big) \mbox{\; for
\;} m\ge 0.$$ Note that the flatness of $U_r$ implies that
$\Ext_A^m(U_r, U_s)=0$ for $m\ge 1$. It remains to show
$\Hom_A(U_r,\, U_s)=0$. For this aim, it is sufficient to show
$T_s\otimes_AU_r=0$. Since $T_s:=\bigoplus_{\mathfrak p\in
\mathcal{P}_s}E(A/\mathfrak p)$ and the functor $-\otimes_AU_r$
commutes with arbitrary direct sums, we have to prove $E(A/\mathfrak
p)\otimes_AU_r=0$ for every $\mathfrak p\in \mathcal{P}_s$. In fact,
since $r<s$ by assumption, we know that $\mathfrak{p}\nsubseteq
\mathfrak{q}$ for each $\mathfrak{q}\in\mathcal{P}_k$ with $0\leq
k\leq r$. It follows from $(3)$ that $\Tor_j^A\big(E(A/\mathfrak
p),\,E(A/\mathfrak q)\big)=0$ for all $j\geq 0$, and therefore
$$\Tor_j^A(E(A/\mathfrak p),\,T_k)\simeq \bigoplus_{\mathfrak q\in
\mathcal{P}_k}\Tor_j^A\big(E(A/\mathfrak p),\,E(A/\mathfrak
q)\big)=0.$$ Since $I_{r,\,k}\in \Add(T_k)$, we obtain
$\Tor_j^A(E(A/\mathfrak p),\,I_{r,\,k})=0$ for all $j\geq 0$. Now,
by applying the tensor functor $E(A/\mathfrak p)\otimes_A-$ to the
minimal injective coresolution of $U_r$, we can prove $E(A/\mathfrak
p)\otimes_AU_r=0$. Thus $T_s\otimes_AU_r=0$. This finishes the proof
of $(a)$.
Finally, we show $(b)$. Let $0\leq i\leq n-1$. Recall that $U_i =
\Hom_A(T_i,\,W)$. According to Lemma \ref{tor-ext} (1), we have
{\small $$\Ext_A^m\big(W,
\Hom_A(T_i,\,W)\big)\simeq\Hom_A\big(\Tor_m^A(T_i,\,
W),\,W\big)\simeq \Hom_A\big(\bigoplus_{\mathfrak p\in
\mathcal{P}_i}\Tor_m^A(E(A/\mathfrak p),\, W),\,W\big)\simeq
\prod_{\mathfrak p\in
\mathcal{P}_i}\Hom_A\big(\Tor_m^A(E(A/\mathfrak p),\,
W),\,W\big).$$\vspace{-0.2cm}}
\noindent Since the ideal $\mathfrak m$ is maximal (or of height
$n$), it holds that $\mathfrak {m}\nsubseteq \mathfrak{p}$ for every
$\mathfrak{p}\in \mathcal{P}_i$. Hence it follows from $(3)$ that
$\Tor_m^A(E(A/\mathfrak p),\, W)=0$, and therefore $\Ext_A^m(W,
U_i)=0$. Similarly, one can show that
$$\Ext_A^n(W, U_n)=\Ext_A^n(W,\,\Hom_A(W, W))\simeq
\Hom_A(\Tor_n^A(W,\, W),\,W).$$ Since $\Tor^A_n(W,
W)=\Tor^A_n\big(E(A/\mathfrak m),\, E(A/\mathfrak m)\big)\neq 0$ by
$(3)$ and since $_AW$ is an injective cogenerator, we infer that
$\Ext_A^n(W, U_n)\neq 0$. Thus $(b)$ follows.
Consequently, for the $n$-cotilting $A$-module $U$, the subcategory
$\Ker(\bf{H})$ is not homological in $\D R$.
\medskip
Let us end this paper by the following open questions related to our
results in this note.
\medskip
{\bf Question 1.} Let $A$ be a ring with identity. Is there a good
$n$-tilting $A$-module $T$ for $n\ge 2$ such that $T$ is not
equivalent to any classical tilting $A$-module and that
$\Ker(T\otimesL_B-)$ is homological?
\medskip
{\bf Question 2.} Is the converse of Corollary \ref{cor} (1) always
true?
\medskip
For tilting modules over commutative noetherian $n$-Gorenstein
rings, Silvana Bazzoni even guesses a stronger answer: If
$\Ker(T\otimesL_{B}-)$ is homological in $\D B$, then $_AT$ should
be a $1$-tilting module, that is, the module $_AN$ in Corollary
\ref{cor} (1) should be zero.
\medskip
{\bf Question 3.} Given a good $1$-cotilting module $U$ over an
\textbf{arbitrary ring} $A$, is there a homological ring epimorphism
$\lambda: \End_A(U)\ra C$ and a recollement of the following form?
$$\xymatrix@C=1.2cm{\D{C}\ar[r]^-{D(\lambda_*)}
&\D{\End_A(U)}\ar[r]\ar@/^1.2pc/[l]\ar@/_1.2pc/[l] &\D{A}
\ar@/^1.2pc/[l]\ar@/_1.2pc/[l]}\vspace{0.2cm}$$
Note that this reccollement does not involve the derived categories
of the endomorphism rings of any injective cogenerators related to
$U$.
\medskip
{\bf Question 4.} Given an arbitrary ring $A$, how to parameterize
homological subcategories of $\D A$? Equivalently, how to classify
homological ring epimorphisms starting from $A$?
\medskip
{\bf Question 5.} Is the Ringel $R$-module $M$ in Lemma
\ref{cotilt1} always good?
A positive answer to this question would lead to a generalization of
Corollary \ref{real-cotilt}.
\medskip
{\bf Acknowledgement.} The research work of the corresponding author
C.C. Xi is partially supported by a grant of MOE (20110003110003),
China.
{\footnotesize
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\section{Introduction}
\label{sec:Intro}
Gaussian random fields (GRFs for short) play a fundamental role in the
modelling of spatiotemporal phenomena subject to uncertainty. In
several broad research areas, in particular in spatial statistics,
data assimilation, climate modelling and meteorology to name but a
few, GRFs play a pivotal role in mathematical models of physical
phenomena with distributed, uncertain input data. Accordingly, there
is an extensive literature devoted to mathematical, statistical and
computational aspects of GRFs. We mention only
\cite{Lifshits95,Janson97,AdlerGeoGRF} and the references for
mathematical foundations, and \cite{Matern2nd,GineNickl} and the
references there for a statistical perspective on GRFs.
In recent years, the area of \emph{computational uncertainty
quantification} (UQ for short) has emerged at the interface of the
fields of applied mathematics, numerical analysis, scientific
computing, computational statistics and data assimilation. Here, a
key topic is the mathematical and numerical analysis of partial
differential equations (PDEs for short) with random field input; in
particular, with GRF input. Mathematical analysis refers to questions
of well-posedness, pathwise and $L^p$ integrability and regularity in
scales of Besov spaces of random solution ensembles of such PDEs.
Numerical analysis of PDEs with GRF inputs refers
on the one hand
to questions of
efficient, numerical simulation methods
of GRF inputs (see, e.g., \cite{Matern2nd,DietrNewsam,
Gitt2012,bachmayr2018GRFRep,bachmayr2019unified,SteinKrig} and the
references there), and on the other hand
to
the efficient numerical approximation of corresponding PDE
solution ensembles, which arise for GRF inputs, in particular
efficient representation of such solution ensembles (see
\cite{HS,BCDM,BCDS}), and
numerical quadrature of corresponding solution fields (see
\cite{HS,KSSS17_2112,DHSHMatGRF17,GKNSSS,HS19_2310,HS17_2475,CSAMS2011}
and the references there). Applications include
for instance
subsurface flow models (see, e.g., \cite{DietrNewsam,GKSS13_484}) but
also other PDE models for media with uncertain properties (see, e.g.,
\cite{logNMax2018} for electromagnetics).
The careful analysis of efficient computational sampling of solution
families of PDEs subject to GRF input is also a key ingredient in
numerical data assimilation, e.g., in Bayesian Inverse Problems (BIPs
for short); we refer to the surveys
\cite{DashtiStuart17,DshtiLwStrtVossMAP} and the references there for
a mathematical formulation of BIPs for PDEs subject to Gaussian prior
measures and function space inputs.
In recent years, there has been considerable development in the
analysis and numerical simulation of PDEs with random field input from
function space, subject to Gaussian measures (GMs for short). The
method of choice in many applications for the numerical treatment of
GMs is Monte-Carlo (MC for short) sampling. Convergence rate $1/2$ in
terms of the number of MC samples is assured under rather mild
conditions (existence of MC samples, and of finite second moments).
Given the high cost of MC sampling, recent years have seen the advent
of numerical techniques which afford higher convergence orders than
$1/2$, also on infinite-dimensional integration domains. Like the MC
method, these techniques are not prone to the so-called curse of
dimensionality. The key condition which emerged as governing the
convergence rate of numerical integration and interpolation methods is
\emph{sparsity of coefficient sequences in Wiener-Hermite expansions
of integrand functions}. Rather than counting the ratio of nonzero
coefficients, this sparsity is quantified by \emph{summability} of
coefficient sequences. Sparse-grid deterministic quadrature
algorithms for numerical integration with respect to an
infinite-dimensional GM were analyzed in the references
\cite{ChenlogNQuad2018,HHMPMS16,GKNSSS,KSSS17_2112, dD21}.
To indicate some of the mathematical issues which are considered in
the present notes, consider in the interval ${\mathrm D} = (0,1)$, and in
a probability space $(\Omega,\cA,\mathbb{P})$, a GRF
$g:\Omega\times {\mathrm D} \to \mathbb{R}$ which takes values in
$L^\infty({\mathrm D})$. That is to say, that the map
$\omega \mapsto g(\omega, \cdot)$ is an element of the Banach space
$L^\infty({\mathrm D})$. \emph{Formally}, at this stage, we represent
realizations of the random element $g \in L^\infty({\mathrm D})$ with a
\emph{representation system}
$(\psi_j )_{j\in {\mathbb N}}\subset L^\infty({\mathrm D})$ in
\emph{affine-parametric form}
\begin{equation}\label{eq:AffParg}
g(\omega,x) = \sum_{j\in {\mathbb N}} y_j(\omega) \psi_j(x) \;,
\end{equation}
where the coefficients $( y_j )_{j\in {\mathbb N}}$ are assumed to be
i.i.d.\ standard normal random variables (RVs for short).
Representations such as \eqref{eq:AffParg} are widely used both in the
analysis and in the numerical simulation of random elements $g$ taking
values in function space. In order to dispense with summability and
measurability issues, let us assume for now that the sum in
\eqref{eq:AffParg} is finite, with $J$ terms. The coefficients
$y_j(\omega)$ being standard normal RVs, the sum
$\sum_{j=1}^J y_j \psi_j(x)$ may be considered as \emph{parametric,
deterministic map} $g: \mathbb{R}^J \to L^\infty({\mathrm D})$. The random
element $g(\omega,x)$ in \eqref{eq:AffParg} will then be obtained from
it by evaluating this deterministic map in random coordinates, i.e.\ by
sampling it in Gaussian random vectors
$(y_j(\omega))_{j=1}^J\in \mathbb{R}^J$.
Gaussian random elements as inputs for PDEs appear in particular in
coefficients of diffusion equations. Consider, for illustration, in
${\mathrm D}$, and for given $f\in L^2({\mathrm D})$, the boundary value
problem: find a random function $u: \Omega \to V$ with
$V := \{w\in H^1({\mathrm D}): w(0) = 0 \}$ such that
\begin{equation}\label{eq:BVP}
f + \frac{\,\mathrm{d}}{\,\mathrm{d} x}\left(a(x,\omega) \frac{\,\mathrm{d}}{\,\mathrm{d} x} u \right) = 0 \quad \mbox{in} \quad {\mathrm D}\;,
\quad
a(1,\omega) u'(1,\omega) = \bar{f}(\omega)\;.
\end{equation}
Here, $a(x,\omega) = \exp(g(x,\omega))$ with GRF
$g:\Omega \to L^\infty({\mathrm D})$, and $\bar{f}:= F(1)$ with
$$
F(x) := \int_0^x f(\xi)\,\mathrm{d}\xi \in V, \quad x\in {\mathrm D}.
$$
Note that $\bar{f}$ is deterministic when $f$ is deterministic. We
find that a random solution $u$ of the problem must satisfy
$$
u'(x,\omega) = \exp(-g(x,\omega)) F(x) , \quad x\in {\mathrm D}, \omega
\in \Omega \;.
$$
Inserting \eqref{eq:AffParg}, this is equivalent to the
\emph{parametric, deterministic family of solutions}
$u(x,{\boldsymbol{y}}):{\mathrm D}\times \mathbb{R}^J \to \mathbb{R}$ given by
\begin{equation}\label{eq:DetSoly}
u'(x,{\boldsymbol{y}}) = \exp(-g(x,{\boldsymbol{y}})) F(x) , \quad x\in {\mathrm D}, {\boldsymbol{y}}\in \mathbb{R}^J \;.
\end{equation}
Hence
\begin{equation*}\label{eq:Nrmu}
\| u'(\cdot,{\boldsymbol{y}}) \|_{L^2({\mathrm D})}
=
\| \exp(-g(\cdot,{\boldsymbol{y}})) F \|_{L^2({\mathrm D})} \;,\quad {\boldsymbol{y}}\in \mathbb{R}^J\;,
\end{equation*}
which implies the (sharp) bounds
$$
\| u'(\cdot,{\boldsymbol{y}}) \|_{L^2({\mathrm D})} \left\{
\begin{array}{l}
\geq
\exp(-\| g(\cdot,{\boldsymbol{y}}) \|_{L^\infty({\mathrm D})}) \| F \|_{L^2({\mathrm D})}
\\
\leq \exp(\| g(\cdot,{\boldsymbol{y}}) \|_{L^\infty({\mathrm D})}) \| F \|_{L^2({\mathrm D})}.
\end{array}
\right .
$$
Due to the homogeneous Dirichlet condition at $x=0$, up to an absolute
constant the same bounds also hold for $\| u(\cdot,{\boldsymbol{y}}) \|_V$.
It is evident from the explicit expression \eqref{eq:DetSoly} and the
upper and lower bounds, that for every finite parameter ${\boldsymbol{y}}\in \mathbb{R}^J$,
the solution $u\in V$ exists. However, we can not, in general, expect
uniform w.r.\ to ${\boldsymbol{y}}\in \mathbb{R}^J$ a-priori estimates, also of the higher
derivatives, for smoother functions $x\mapsto g(x,{\boldsymbol{y}})$ and
$x\mapsto f$. Therefore, the parametric problem \eqref{eq:BVP} is
nonuniformly elliptic, \cite{Charrier12,HQSMLMCMC}. In particular,
therefore, also a-priori error bounds for various discretization
schemes will contain this uniformity w.r.t. ${\boldsymbol{y}}$. The random
solution will be recovered from \eqref{eq:DetSoly} by inserting for
the coordinates $y_j$ samples of i.i.d.\ standard normal random
variables.
The present book aims at a regularity theory of parametric solution
families such as $\{ u(x,{\boldsymbol{y}}): {\boldsymbol{y}} \in \mathbb{R}^J \}$ in the particular case
that $J=\infty$, i.e., that $g(x,\omega)$ in \eqref{eq:BVP} is a GRF.
Much attention has been devoted in recent years to the analysis of
efficient simulation methods for the solution $u$. We mention only MC
methods and their variants (see, e.g.,
\cite{RSAMSAT_Bip2017,HQSMLMCMC,Charrier12,CharrDebu13} and the
references there) and Quasi-Monte Carlo (QMC for short) integration
(see, e.g.,\cite{GKNSSS}), and Wiener-Hermite polynomial chaos
approximation (see, e.g., \cite{HS,bachmayr2018GRFRep} and the
references there) and sparse-grid (a.k.a. ``stochastic collocation'')
(see, e.g., \cite{ErnstSprgkTam18} and the references there).
In the present work, we adopt notation and recapitulate basic results
on GMs from \cite{Janson97,Bogach98}. Previous works
\cite{BCDS,BCDM,GKNSSS} addressed the summability of the
Wiener-Hermite PC expansion coefficients of solution families
$\{u(\cdot,{\boldsymbol{y}}): {\boldsymbol{y}}\in \mathbb{R}^\infty\}\subset V$ for the forward problem,
based on moment bounds of derivatives of parametric solutions w.r.\ to
GM. Estimates for these coefficients and, in particular, for the
summability, were obtained in \cite{HS,BCDS,BCDM,GKNSSS,HS2}. In these
references, all arguments were based on real-variable, bootstrapping
arguments with respect to ${\boldsymbol{y}}$.
In the present work, we propose the following contributions to this
area. First, we provide novel proofs of some of the sparsity and
summability results in \cite{HS,BCDM,BCDS} of the infinite-dimensional
parametric forward map. The presently developed proof is based on
analytic continuation and complex variable arguments in order to bound
derivatives of parametric solutions, and their coefficients in
Wiener-Hermite PC expansions. This is in line with similar arguments
in the so-called ``uniform case'' in \cite{CoDeSch1,CCS13_783}.
There, the random parameters in the representation of the input random
fields range in compact subsets of ${\mathbb R}$. Unlike in these references,
in the present, Gaussian setup the parameter domain $U = {\mathbb R}^\infty$
is not compact. This entails significant modifications of
mathematical arguments as compared to arguments in
\cite{CoDeSch1,CCS13_783}.
Contrary to the analysis in \cite{BCDS,BCDM,GKNSSS}, where parametric
regularity results were obtained by real-variable arguments combined
with induction-based bootstrapping with respect to the derivative
order, the present work develops derivative-free, complex variable
arguments which allow directly to obtain bounds of the Wiener-Hermite
PC expansion coefficients of the parametric solutions in scales of
Sobolev and Besov spaces in the physical domain ${\mathrm D}$ in which the
parametric PDE is posed. They also allow to treat in a unified manner
parametric regularity of the solution map in several scales of Sobolev
and Kondrat'ev spaces in the physical domain ${\mathrm D}$ which is the
topic of Section \ref{sec:KondrReg}, resulting in novel sparsity
results for linear, elliptic and parabolic PDEs with GRF inputs in
scales of Sobolev and Besov spaces.
\subsection{Scope of results}
\label{sec:IntrScope}
We prove quantified holomorphy of countably-parametric solution
families of linear, elliptic and parabolic PDEs. The parameter ranges
are ${\mathbb R}$, corresponding to countably-parametric representations of
GRF input data, taking values in a separable locally convex space, in
particular, Hilbert or Banach space of uncertain input data, endowed
for example with a Gaussian product measure $\gamma$ on ${\mathbb R}^\infty$.
The results established in the present paper and the related bounds on
partial derivatives w.r.\ to the parameters in Karhunen-Lo\`{e}ve or L\'{e}vy-Cieselsky expansions of
uncertain GRF inputs imply convergence rate bounds for several
families of computational methods to numerically access these
parametric solution maps. Importantly, we prove that in terms of $n$,
the number of work and memory, an approximation accuracy $O(n^{-a})$
for some parameter $a>0$ can be achieved where the convergence rate
$a$ depends on the approximation process and on the amount of sparsity
in the Wiener-Hermite PC expansions of the random fields under
consideration. In the terminology of computational complexity, a
prescribed numerical tolerance $\varepsilon>0$ can be reached in work and
memory of order $O(\varepsilon^{-1/a})$. \emph{In particular, the
convergence rate $a$ and the constant hidden in the Landau
$O(\cdot)$ symbol do not depend on the dimension of the space of
active parameters involved in the approximations which we
construct.} The approximations developed in the present book are
\emph{constructive} and can be realized computationally by
\emph{deterministic algorithms} of so-called ``stochastic
collocation'' or ``sparse-grid'' type. Error bounds are proved in
$L^p$-type Bochner spaces with respect to the GM $\gamma$ on the input
data space of the PDE, in natural Hilbert or Banach spaces of
solutions of the PDEs under consideration.
All approximation rate results obtained in the present paper are free
from the so-called \emph{curse of dimensionality}, a terminology
coined apparently by R.E. Bellmann (see \cite{Bellmann}). The rates
are in fact only limited by the sparsity of the Wiener-Hermite PC
expansion coefficients of the deterministic, countably-parametric
solution families. In particular, dimension-independent convergence
rates $>1/2$ are possible, provided sufficient Wiener-Hermite PC
expansion coefficient sparsity, that the random inputs feature
sufficient pathwise regularity, and the affine representation system
(being a tight frame on space of admissible input realizations) are
stable in a suitable smoothness scale of inputs.
The proof strategy for our results is, throughout, to establish
summability results for Wiener-Hermite PC expansion coefficients of
countably-parametric solution families of the corresponding forward
problem. Unlike previous approaches in \cite{HS,BCDM,BCDS}, we
establish the corresponding estimates here by \emph{quantified
holomorphy} of countably-parametric solution maps. As we try to show
by several broad classes of examples, and similar to what was found in
the so-called ``uniform'' case (where the probability measures on the
parameter sequences were compactly supported), e.g.\
\cite{CoDeSch1,ZDS19,ZS17}, the presently adopted, holomorphy-based
approach allows for rather direct derivation of \emph{Wiener-Hermite
summability in scales of smoothness spaces} for solution families of
the PDE of interest. We mention only sparse-grid Hermite-Smolyak
interpolation ``stochastic collocation'' \cite{ErnstSprgkTam18,dD21}
and quadrature \cite{ChenlogNQuad2018,dD21}, and QMC integration as
developed in \cite{GKNSSS,KSSS17_2112,kaza,HS19_2310,HS17_2475} for
the model elliptic PDE considered in Section \ref{sec:EllPDElogN}.
\subsection{Structure of this book}
\label{sec:IntrStrct}
The structure of this text is as follows. In Section \ref{S:Prelim},
we collect known facts from functional analysis and probability theory
which are required in the sequel of this note. In particular, we
review constructions and results on GMs on separable Hilbert and
Banach spaces. Particular interest will be on constructions via
countable products of univariate GMs on countable products of real
lines. We also review assorted known results on convergence rates of
Lagrangian Finite Elements for linear, second order, divergence-form
elliptic PDEs in polytopal domains ${\mathrm D}$ with Lipschitz boundary
$\partial {\mathrm D}$.
In Section \ref{sec:EllPDElogN}, we introduce a model linear, second
order elliptic divergence-form PDE with log-normal coefficients, with
variational solutions in the ``energy space'' $H^1_0({\mathrm D})$. This
equation was investigated with parametric input data in a number of
references in recent years
\cite{CoDeSch,CoDeSch1,HS,CCS13_783,BCDM,BCDS,Dung19,GKNSSS,KSSS17_2112,ZDS19}.
It is considered in this work mainly to develop the holomorphic
approach to establish our mathematical approach to parametric
holomorphy and sparsity of Wiener-Hermite PC expansions of parametric
solutions in a simple setting, and to facilitate comparisons with the
mentioned previous works and results.
In Section \ref{S:HolSumSol}, we connect quantified holomorphy of the
parametric, deterministic solution manifold
$\{ u({\boldsymbol{y}}): {\boldsymbol{y}} \in {\mathbb R}^\infty \}$ in the space $H^1_0({\mathrm D})$ with
weighted summability of the sequence $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}} \in {\mathcal F}}$ of
($ H^1_0({\mathrm D})$-valued) Wiener-Hermite PC expansion coefficients.
With the methodology in place, we show in Section \ref{sec:HsReg} how
to obtain holomorphic regularity of the parametric solution family
$\{ u({\boldsymbol{y}}) : {\boldsymbol{y}}\in {\mathbb R}^\infty\}$ in Sobolev spaces $H^s({\mathrm D})$ of
possibly high smoothness order $s \in {\mathbb N}$. The argument is
self-contained and provides parametric holomorphy for any
differentiation order $s\in {\mathbb N}$ in a unified way, in domains
${\mathrm D}$ of sufficiently high regularity and for sufficiently high
almost sure regularity of coefficient functions. In Section
\ref{sec:KondrReg}, we extend these results for linear second order
elliptic differential operators in divergence form in a bounded
polygonal domain ${\mathrm D} \subset \mathbb{R}^2$. Here, corners are
well-known to obstruct high almost sure pathwise regularity in the
usual Sobolev and Besov spaces, of coefficients and solutions.
Therefore, we develop summability of the sequence of Wiener-Hermite PC
expansion coefficients $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}} \in {\mathcal F}}$ of the random
solutions in terms of corner-weighted Sobolev spaces, going back to
V.A. Kondrat'ev (see, e.g., \cite{Gr,BLN,MazRoss2010} and the
references there).
In oder to extend these results and the approach to bound
Wiener-Hermite PC expansion coefficients via quantified holomorphy
beyond the simple, second order diffusion equation introduced in
Section \ref{sec:EllPDElogN}, in Section \ref{sec:SumHolSol} we
address sparsity of Wiener-Hermite PC expansions of general
parametric, holomorphic maps taking values in Banach spaces. In
Section \ref{sec:BIP}, we apply the preceding abstract results on
parametric holomorphy to establish quantified holomorphy of
countably-parametric, posterior densities of corresponding BIPs where
the uncertain input of the forward PDE is a countably-parametric GRF
taking values in a separable Banach space of inputs.
In Section~\ref{sec:StochColl}, we discuss deterministic quadrature
and interpolation algorithms. Such algorithms are necessary for the
approximation of certain statistical quantities (expectations,
statistical moments) of the parametric solutions with respect to a GM
on the parameter space. The proposed algorithms are variants and
generalizations of so-called ``stochastic collocation'' or
``sparse-grid'' type approximation, and proved to outperform sampling
methods such as MC methods, under suitable sparsity conditions on the
Wiener-Hermite PC expansion coefficient sequences of integrands. In
the quadrature case, they are also known as ``Smolyak quadrature''
methods. Their common feature is a) the deterministic nature of the
algorithms, and b) the possibility of achieving convergence rates
$>1/2$ independent of the dimension of parameters and therefore the
curse of dimensionality is broken. They offer, in particular, the
perspective of deterministic numerical approximations for GRFs under
nonlinear pushforwards (being realized via the deterministic
data-to-solution map of the PDE of interest). The decisive analytic
property to be established are dimension-explicit estimates of
individual Wiener-Hermite PC expansion coefficients of parametric
solutions, and based on these, sharp summability estimates of (norms
of) sequences of Wiener-Hermite PC expansion coefficients of
parametric, deterministic solution families.
The results on the summability of the Wiener-Hermite PC expansion
coefficients of parametric, deterministic solution families with
respect to corner-weighted Sobolev spaces on spatial ${\mathrm D}$ finally
also allow to analyze methods for constructive, deterministic
approximations of parametric solution families. Here, a truncation of
Wiener-Hermite PC expansions is combined with approximating the
Wiener-Hermite PC expansion coefficients in the norm of the ``energy
space'' $ H^1_0({\mathrm D})$ of these solutions from finite-dimensional
approximation spaces which are customary the numerical approximation
of solution instances. Importantly, \emph{required approximation
accuracies of the Wiener-Hermite PC expansion coefficients $u_{\boldsymbol{\nu}}$
will depend on the relative importance of $u_{\boldsymbol{\nu}}$ within the
Wiener-Hermite PC expansion}. This observation gives rise to
so-called \emph{multilevel approximations} where a prescribed overall
accuracy in mean square w.r.t. the GM $\gamma$ with respect to
$H^1_0({\mathrm D})$ will be achieved by a ${\boldsymbol{\nu}}$-dependent discretization
level in the physical domain. The corresponding error estimates will
be developed in Section \ref{sec:MLApprox} in an abstract setting. A
key for the sparse-grid integration and interpolation approaches is to
efficiently numerically allocate discretization levels to
Wiener-Hermite PC expansion coefficients. We develop a such approach
in Section \ref{app:mlweight}. It is based on greedy searches and
suitable thresholding of (suitable norms of) Wiener-Hermite PC
expansion coefficients and on a-priori bounds for these quantities
which are obtained by complex variable arguments.
In Section \ref{sec:Approx} we verify the abstract hypotheses of the
sparse multilevel approximations from Section \ref{sec:MLApprox} for
the concrete linear, elliptic and parabolic problems with log-GRF
inputs.
\subsection{Notation and conventions}
\label{sec:IntrNotat}
We use standard notation. We denote ${\mathbb R}^\infty$ the set of all
sequences ${\boldsymbol{y}} = (y_j)_{j\in {\mathbb N}}$ with $y_j\in {\mathbb R}$, and by
${\mathbb C}^\infty$ the set of sequences ${\boldsymbol{z}} = (z_j)_{j\in {\mathbb N}}$ with
$z_j \in {\mathbb C}$;
${\mathbb R}_+^\infty=\{(z_j)_{j\in {\mathbb N}} \in {\mathbb R}^\infty: z_j\geq 0 \}$. Both,
${\mathbb R}^\infty$ and ${\mathbb C}^\infty$, will be understood with the product
topology from ${\mathbb R}$ resp. from ${\mathbb C}$. Similarly, we define
${\mathbb N}^\infty$, ${\mathbb Z}^\infty$ and ${\mathbb Z}_+^\infty$. For multi-indices
${\boldsymbol{\alpha}},{\boldsymbol{\beta}} \in {\mathbb Z}_+^d$, the inequality ${\boldsymbol{\beta}} \leq {\boldsymbol{\alpha}}$ is
understood component-wise, i.e., ${\boldsymbol{\beta}}\leq {\boldsymbol{\alpha}}$ if and only if
$\beta_j\leq \alpha_j$ for $j=1,\ldots,d$. Denote by ${\mathcal F}$ the
countable set of all ${\boldsymbol{\nu}}\in {\mathbb Z}^\infty_+$ such that $\operatorname{supp}({\boldsymbol{\nu}})$ is
finite, where $\operatorname{supp}({\boldsymbol{\nu}}) := \{ j\in {\mathbb N}: \nu_j \ne 0\}$ denotes the
``support'' of the multi-index ${\boldsymbol{\nu}}$. Similarly, we define
$\operatorname{supp}({\boldsymbol{\rho}})$ of a sequence ${\boldsymbol{\rho}} \in {\mathbb R}^\infty_+$. For
${\boldsymbol{\nu}}\in {\mathcal F}$, and for a sequence ${\boldsymbol{b}}=(b_j)_{j\in {\mathbb N}}$ of positive
real numbers, the quantities
\begin{equation*}
{\boldsymbol{\nu}}! := \prod_{j \in {\mathbb N}}\nu_j!\,,\qquad
|{\boldsymbol{\nu}}|:=\sum_{j \in {\mathbb N}}\nu_j,
\qquad\text{and}\qquad
{\boldsymbol{b}}^{\boldsymbol{\nu}} := \prod_{j \in {\mathbb N}}b_j^{\nu_j}
\end{equation*}
are finite and well-defined.
For a multi-index ${\boldsymbol{\alpha}} \in {\mathbb Z}_+^d$ and a function $u({\boldsymbol{x}},{\boldsymbol{y}})$ of
${\boldsymbol{x}}\in {\mathbb R}^d$ and parameter sequence ${\boldsymbol{y}}\in {\mathbb R}^{\infty}$ we use the
notation $D^{\boldsymbol{\alpha}} u({\boldsymbol{x}},{\boldsymbol{y}})$ to indicate the partial derivatives
taken with respect to ${\boldsymbol{x}}$. The partial derivative of order
${\boldsymbol{\alpha}} \in {\mathbb Z}_+^\infty$ with respect to ${\boldsymbol{y}}$ \emph{of finite total
order} $|{\boldsymbol{\alpha}}| = \sum_{j\in {\mathbb N}} \alpha_j$ is denoted by
$\partial^{\boldsymbol{\alpha}} u({\boldsymbol{x}},{\boldsymbol{y}})$. For ${\boldsymbol{\alpha}} \in {\mathbb Z}_+^\infty$ with
$|{\boldsymbol{\alpha}}|<\infty$ it holds $\operatorname{supp}({\boldsymbol{\alpha}})$ is finite.
In order to simplify notation, we will systematically suppress the
variable ${\boldsymbol{x}}\in {\mathrm D}\subset \mathbb{R}^d$ in mathematical expressions,
except when necessary. For example, instead
$\int_{\mathrm D} v({\boldsymbol{x}})\rd {\boldsymbol{x}}$ we will write $\int_{\mathrm D} v\rd {\boldsymbol{x}}$,
etc. For a Banach space $X$, we denote $X_{{\mathbb C}}:=X+ {\rm i} X$ the
complexification of $X$. The space $X_{\mathbb C}$ is also a Banach space
endowed with the (minimal, among several possible equivalent ones, see
\cite{mst99}) norm
$\|x_1+{\rm i} x_2\|_{X_{\mathbb C}}:=\sup_{0\leq t\leq 2\pi}\|x_1\cos t-x_2 \sin
t \|_X$. The space $X^\infty$ is defined in a similar way as
${\mathbb R}^\infty$.
By $\cL(X,Y)$ we denote the vector space of bounded, linear operators
between to Banach spaces $X$ and $Y$. With $\cL_{{\rm is}}(X,Y)$ we
denote the subspace of boundedly invertible, linear operators from $X$
to $Y$.
For a function space $X({\domain})$ defined on the domain ${\domain}$, if there is
no ambiguity, when writing the norm of $x\in X({\domain})$ we will omit ${\domain}$,
i.e., $\|x\|_X$ instead of $\|x\|_{X({\domain})}$.
For $0 < p \le \infty$ and a countable index set $J$, we denote by
$\ell^p(J)$ the quasi-normed space of all ${\boldsymbol{y}} = (y_j)_{j\in J}$ with
$y_j \in \mathbb{R}$, equipped with the quasi-norm
$\|{\boldsymbol{y}}\|_{\ell^p}:= \left(\sum_{j \in J}|y_j|^p\right)^{1/p}$ for
$p < \infty$, and $\|{\boldsymbol{y}}\|_{\ell^\infty}:= \sup_{j \in J}|y_j|$.
\input{prelim.tex}
\newpage
\section{Elliptic divergence-form PDE with log-Gaussian coefficient}
\label{sec:EllPDElogN}
We present a model second order linear divergence-form PDE with
log-Gaussian input data. We review known results on its
well-posedness, and Lipschitz continuous dependence on the input
data. Particular attention is placed on regularity results in
polygons ${\mathrm D}\subset {\mathbb R}^2$. Here, solutions belong to
corner-weighted Sobolev spaces. We discuss regularity results for
parametric coefficients, and establish in particular parametric
holomorphy results for the coefficient-to-solution maps.
The outline of this section is as follows. In Section
\ref{S:PbmStat}, we present the strong and variational forms of the
PDE, its well-posedness and the continuity of the data-to-solution map
in appropriate spaces. Importantly, we do not aim at the most general
setting, but to ease notation and for simplicity of presentation we
address a rather simple, particular case: in a bounded domain
${\mathrm D}$ in Euclidean space ${\mathbb R}^d$. All the ensuing derivations
will directly generalize to linear second order elliptic systems. A
stronger Lipschitz continuous dependence on data result is stated in
Section \ref{S:LipCont}.
Sections \ref{S:RndDat} and \ref{S:ParCoef} describe uncertainty
modelling by placing GMs on sets of admissible, countably parametric
input data, i.e., formalizing mathematically aleatoric uncertainty in
input data. Here, the Gaussian series introduced in Section
\ref{S:GSer} will be seen to take a key role in converting operator
equations with GRF inputs to infinitely-parametric, deterministic
operator equations. The Lipschitz continuous dependence of the
solutions on input data from function spaces will imply strong
measurability of corresponding random solutions, and render
well-defined the \emph{uncertainty propagation}, i.e., the
push-forward of the GM on the input data.
In Sections \ref{S:HolSumSol}, \ref{sec:HsReg} and
\ref{sec:KondrReg}, we connect quantified holomorphy of the
parametric, deterministic solution manifold
$\{ u({\boldsymbol{y}}): {\boldsymbol{y}} \in {\mathbb R}^\infty \}$ with weighted summability of the
sequence $(\norm[H]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}} \in {\mathcal F}}$ of Wiener-Hermite PC
expansion coefficients as elements of certain Sobolev spaces: We
start with the case $H=H_0^1({\mathrm D})$ in Section \ref{S:HolSumSol}
and subsequently discuss higher regularity $H=H^s({\mathrm D})$,
$s\in\mathbb{N}$, in Section \ref{sec:HsReg} and finally $H$ being a
corner-weighted Sobolev space on a bounded polygonal domain
${\mathrm D} \subset \mathbb{R}^2$ in Section \ref{sec:KondrReg}.
\subsection{Statement of the problem and well-posedness}
\label{S:PbmStat}
In a bounded Lipschitz domain ${\mathrm D} \subset {\mathbb R}^d$
$(d=1,2\ \text{or}\ 3)$, consider the linear, second order model PDE
\begin{equation}\label{PDE}
P_a u := \left\{ \begin{array}{l} - \operatorname{div}(a({\boldsymbol{x}}) \nabla u({\boldsymbol{x}})) \\ \gamma_0 u \end{array} \right\}
=
\left\{\begin{array}{c} f({\boldsymbol{x}}) \;\;\mbox{in}\;\;{\mathrm D}, \\ 0 \;\; \mbox{on} \;\;\partial {\mathrm D}\;.
\end{array}
\right.
\end{equation}
Here, $\gamma_0 : H^1({\mathrm D})\to H^{1/2}(\partial{\mathrm D})$
denotes the trace map. With the notation $V:=H_0^1({\mathrm D})$ and
$V^*=H^{-1}({\mathrm D})$, for any $f\in V^*$, by the Lax-Milgram
lemma the weak formulation given by
\begin{equation}\label{vf}
u\in V:\;\;
\int_{\domain} a\nabla u\cdot \nabla v\, \rd {\boldsymbol{x}}
=
\langle f,v \rangle_{V^*,V} \,,\qquad v\in V,
\end{equation}
admits a unique solution $u \in V$ whenever the coefficient $a$
satisfies the ellipticity assumption
\begin{equation}\label{PDE-epllipticity}
0 < a_{\min}: = \underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,inf}}\, a({\boldsymbol{x}}) \leq a_{\max} = \| a \|_{L^\infty} < \infty
\;.
\end{equation}
With $\| v \|_V := \| \nabla v \|_{L^2}$ denoting the norm of
$v\in V$, there holds the a-priori estimate
\begin{equation} \label{V-estimate} \|u\|_V\leq
\frac{\|f\|_{V^*}}{a_{\min}}\,.
\end{equation}
In particular, with
$$L^\infty_+({\mathrm D}) := \big\{ a\in L^\infty({\mathrm D}): {\rm ess}\inf_{{\boldsymbol{x}}\in {\mathrm D}} a({\boldsymbol{x}})>0\big\},$$
the \emph{data-to-solution operator}
\begin{equation}\label{eq:SolOp}
\cS:L^\infty_+({\mathrm D}) \times V^* \to V: (a,f)\mapsto u
\end{equation}
is continuous.
\subsection{Lipschitz continuous dependence}
\label{S:LipCont}
The continuity \eqref{eq:SolOp} of the data-to-solution map $\cS$
allows to infer already strong measurability of solutions of
\eqref{PDE} with respect to random coefficients $a$. For purposes of
stable numerical approximation, we will be interested in quantitative
bounds of the effect of perturbations of the coefficient $a$ in
\eqref{vf} and of the source term data $f$ on the solution
$u = \cS(a,f)$. Mere continuity of $\cS$ as a map from
$L^\infty_+({\mathrm D}) \times V^*$ to $V = H^1_0({\mathrm D})$ will not be
sufficient to this end. To quantify the impact of uncertainty in the
coefficient $a$ on the solution $u\in V$, local H\"older or,
preferably, Lipschitz continuity of the map $\cS$ is required, at
least locally, close to nominal values of the data $(a,f)$.
To this end, consider given $a_1,a_2\in L^\infty_+({\mathrm D})$,
$f_1,f_2\in L^2({\mathrm D})\subset V^*$ with corresponding unique
solutions $u_i = \cS(a_i,f_i)\in V$, $i=1,2$.
\begin{proposition}\label{prop:LipS}
In a bounded Lipschitz domain ${\mathrm D}\subset \mathbb{R}^d$, for given data
bounds $r_a,r_f\in (0,\infty)$, there exist constants $c_a$ and
$c_f$ such that for every $a_i\in L^\infty_+({\mathrm D})$ with
$\| \log(a_i) \|_{L^\infty} \leq r_a$, and for every
$f_i\in L^2({\mathrm D})$ with $\| f_i \|_{L^2} \leq r_f$, $i=1,2$, it
holds
\begin{equation}\label{eq:LipBasic}
\| u_1 - u_2 \|_V
\leq
\frac{c_P}{a_{1,\min}\wedge a_{2,\min}} \| f_1 - f_2 \|_{L^2}
+
\frac{\| f_1 \|_{V^*} \vee \| f_2 \|_{V^*}}{a_{1,\min} a_{2,\min}}
\| a_1-a_2 \|_{L^\infty}
\;.
\end{equation}
Therefore
\begin{equation}\label{eq:LipCont}
\| \cS(a_1,f_1) - \cS(a_2,f_2) \|_{V}
\leq
c_a \| a_1-a_2 \|_{L^\infty}
+
c_f \| f_1-f_2 \|_{L^2}\;,
\end{equation}
and
\begin{equation}\label{eq:LogLip}
\| \cS(a_1,f_1) - \cS(a_2,f_2) \|_{V}
\leq
\tilde{c}_a \| \log(a_1) - \log(a_2) \|_{L^\infty}
+
c_f \| f_1-f_2 \|_{L^2} \;.
\end{equation}
Here, we may take $c_f = c_P \exp(r_a)$, $c_a = c_P r_f\exp(2r_a)$
and $\tilde{c}_a = c_P r_f \exp(3r_a)$. The constant
$c_P = c({\mathrm D})>0$ denotes the $V-L^2({\mathrm D})$ Poincar\'e
constant of ${\mathrm D}$.
\end{proposition}
The bounds \eqref{eq:LipCont} and \eqref{eq:LogLip} follow from the
continuous dependence estimates in \cite{BchMVK2019} by elementary
manipulations. For a proof (in a slightly more general setting), we
also refer to Section \ref{sec:pdc} ahead.
\subsection{Regularity of the solution}
\label{S:Reg}
It is well known that weak solutions $u\in V$ of the linear, elliptic
boundary value problem (BVP for short) \eqref{PDE} admit higher
regularity for more regular data (i.e., coefficient $a({\boldsymbol{x}})$, source
term $f({\boldsymbol{x}})$ and domain ${\mathrm D}$). Standard references for
corresponding results are \cite{Gr,GilbTr}. The proofs in these
references cover general, linear elliptic PDEs, with possibly
matrix-valued coefficients, and aim at sharp results on the Sobolev
and H\"older regularity of solutions, in terms of corresponding
regularity of coefficients, source term and boundar $\partial{\mathrm D}$.
In order to handle the dependence of solutions on random field and
parametric coefficients in a quantitative manner, we develop presently
self-contained, straightforward arguments for solution regularity of
\eqref{PDE}.
Here is a first regularity statement, which will be used in several
places subsequently.
To state it, we denote by $W$ the normed space of all functions
$v \in V$ such that $\Delta v\in L^2({\mathrm D})$. The norm in $W$ is
defined by
\begin{equation*} \label{W-norm} \| v \|_W := \| \Delta v \|_{L^2}.
\end{equation*}
The map $v\mapsto \|v\|_W$ is indeed a norm on $W$ due to the
homogeneous Dirichlet boundary condition of $v\in V = H^1_0({\domain})$:
$\| v \|_W = 0$ implies that $v$ is harmonic in ${\mathrm D}$, and
$v\in V = H^1_0({\mathrm D})$ implies that the trace of $v$ on
$\partial {\mathrm D}$ vanishes, whence $v=0$ in ${\mathrm D}$ by the maximum
principle.
\begin{proposition}\label{prop:LipCoef}
Consider the boundary value problem \eqref{PDE} in a bounded domain
${\mathrm D}$ with Lipschitz boundary, and with
$a\in W^{1}_\infty({\mathrm D})$, $f\in L^2({\mathrm D})$. Then the weak
solution $u\in V$ of \eqref{PDE} belongs to the space $W$ and there
holds the a-priori estimate
\begin{equation} \label{W-estimate} \|u\|_W \ \le \
\frac{1}{a_{\min}}\left(\| f \|_{L^2} + \|f \|_{V^*}\frac{\|\nabla
a\|_{L^\infty}}{a_{\min}}\right) \le \frac{c}{a_{\min}}\left(
1 + \frac{\|\nabla a\|_{L^\infty}}{a_{\min}}\right) \| f
\|_{L^2}\;,
\end{equation}
where $a_{\min} = \min\{ a({\boldsymbol{x}}) : {\boldsymbol{x}}\in \overline{{\mathrm D}} \}$.
\end{proposition}
\begin{proof}
That $u\in V$ belongs to $W$ is verified by observing that under
these assumptions, there holds
\begin{equation} \label{[Delta u=]} - a\Delta u \ = \ f + \nabla a
\cdot \nabla u \;\;\mbox{in the sense of}\;\; L^2({\domain})\;.
\end{equation}
The first bound \eqref{W-estimate} follows by elementary argument
using \eqref{V-estimate}, the second bound by an application of the
$L^2({\mathrm D})$-$V^*$ Poincar\'{e} inequality in ${\mathrm D}$.
\end{proof}
\begin{remark}\label{rmk:KW}
The relevance of the space $W$ stems from the relation to the
corner-weighted Kondrat'ev spaces ${\mathcal K}^m_\kappa({\mathrm D})$ which were
introduced in Section \ref{S:FncSpc}. When the domain
${\mathrm D}\subset {\mathbb R}^2$ is a polygon with straight sides, in the
presently considered homogeneous Dirichlet boundary conditions on
all of $\partial{\mathrm D}$, it holds that
$W \subset {\mathcal K}^2_\kappa({\mathrm D})$ with continuous injection provided
that $|\kappa|<\pi/\omega$ where $0<\omega<2\pi$ is the largest
interior opening angle at the vertices of ${\mathrm D}$. Membership of
$u$ in ${\mathcal K}^2_\kappa({\mathrm D})$ in turn implies optimal approximation
rates for standard, Lagrangian FE approximations in ${\mathrm D}$ with
suitable, corner-refined triangulations in ${\mathrm D}$, see
Proposition \ref{prop:FECorner}.
\end{remark}
\begin{remark}\label{rmk:Dconvex}
If the physical domain ${\mathrm D}$ is convex or of type $C^{1,1}$,
then $u\in W$ implies that $u\in (H^2\cap H^1_0)({\domain})$ and
\eqref{W-estimate} gives rise to an $H^2$ a-priori estimate (see,
e.g., \cite[Theorem 2.2.2.3]{Gr}).
\end{remark}
The regularity in Proposition \ref{prop:LipCoef} is adequate for
diffusion coefficients $a({\boldsymbol{x}})$ which are Lipschitz continuous in
${\mathrm D}$, which is essentially (up to modification)
$W^{1}_\infty({\mathrm D}) \simeq C^{0,1}({\mathrm D})$. In view of our
interest in admitting diffusion coefficients which are (realizations
of) GRF (see Section \ref{S:RndDat}), it is clear from Example
\ref{exple:LCBrBr} that relevant GRF models may exhibit mere H\"older
path regularity.
Still letting ${\mathrm D} \subset {\mathbb R}^d$ be a bounded Lipschitz domain,
for $0<s<1$ we denote by $C^s({\mathrm D})$ the space of $s$-H\"older
continuous functions in ${\mathrm D}$. We equip $C^s({\domain})$ with the norm
and seminorm
\begin{equation*}\label{eq:CsNrm}
\| a \|_{C^s} := \| a \|_{L^\infty} + | a |_{C^s}\;,
\quad
| a |_{C^s} := \sup_{{\boldsymbol{x}},{\boldsymbol{x}}'\in {\mathrm D}, {\boldsymbol{x}}\ne {\boldsymbol{x}}'} \frac{|a({\boldsymbol{x}}) - a({\boldsymbol{x}}')|}{|{\boldsymbol{x}}-{\boldsymbol{x}}'|^s} \;.
\end{equation*}
Then $C^s({\domain})$ is a Banach space, which is not separable. A separable
subspace is
\begin{equation*}\label{eq:Cso}
C^s_\circ({\mathrm D})
:=
\bigg\{ a \in C^s({\mathrm D}):
\forall {\boldsymbol{x}}\in {\mathrm D} \; \lim_{{\mathrm D} \ni {\boldsymbol{x}}'\to {\boldsymbol{x}}} \frac{|a({\boldsymbol{x}})-a({\boldsymbol{x}}')|}{|{\boldsymbol{x}}-{\boldsymbol{x}}'|^s} = 0
\bigg\}
\;.
\end{equation*}
The H\"older spaces $C^s({\mathrm D})$ on Lipschitz domains ${\mathrm D}$ can
be obtained as interpolation spaces, via the so-called $K$-method of
function space interpolation which we briefly recapitulate (see, e.g.,
\cite[Chapter 1.3]{Triebel95}, \cite{BerghLof}). Two Banach spaces
$A_0,A_1$ with continuous embedding $A_1 \hookrightarrow A_0$ with
respective norms $\| \circ \|_{A_i}$, $i=0,1$, constitute an
interpolation couple. For $0<s<1$, the \emph{interpolation space}
$[A_0,A_1]_{s,q}$ of smoothness order $s$ with fine index
$q\in [1,\infty]$ is defined via the $K$-functional: for $a\in A_0$,
this functional is given by
\begin{equation}\label{eq:Kfct}
K(a,t;A_0,A_1) := \inf_{a_1\in A_1} \{ \|a - a_1\|_{A_0} + t \| a_1 \|_{A_1}\}\;,\quad
t > 0 \;.
\end{equation}
For $0<s<1$ the intermediate, ``interpolation'' space of order $s$ and
fine index $q$ is denoted by $[A_0,A_1]_{s,q}$. It is the set of
functions $a\in A_0$ such that the quantity
\begin{equation} \label{eq:KIntNorm} \| a \|_{[A_0,A_1]_{s,q}} :=
\left\{
\begin{array}{lr}
\left( \int_0^\infty (t^{-s} K(a,t,A_0,A_1))^q \frac{\rd t}{t} \right)^{1/q} \;,& 1 \leq q < \infty,
\\
\sup_{t>0} t^{-s} K(a,t,A_0,A_1) \;, & q = \infty
\end{array}
\right.
\end{equation}
is finite. When the $A_i$ are Banach spaces, the sets
$[A_0,A_1]_{s,q}$ are Banach spaces with norm given by
\eqref{eq:KIntNorm}. In particular (see, e.g., \cite[Lemma
7.36]{Adams2nd}), in the bounded Lipschitz domain ${\mathrm D}$
\begin{equation}\label{eq:CsLinfW1inf}
C^s({\mathrm D}) = [L^\infty({\mathrm D}), W^{1}_\infty({\mathrm D})]_{s,\infty},\quad 0<s<1 \;.
\end{equation}
With the spaces $V:=H^1_0({\mathrm D})$ and $W\subset V$, we define the
(non-separable, non-reflexive) Banach space
\begin{equation}\label{eq:Ws}
W^s := [V,W]_{s,\infty}\;,\quad 0<s<1 \;.
\end{equation}
Then there holds the following generalization of \eqref{W-estimate}.
\begin{proposition}\label{prop:Ws}
For a bounded Lipschitz domain ${\mathrm D} \subset {\mathbb R}^d$, $d\geq 2$,
for every $f\in L^2({\mathrm D})$ and $a\in C^s({\mathrm D})$ for some
$0<s<1$ with
$$a_{\min} = \min\{ a({\boldsymbol{x}}) : {\boldsymbol{x}} \in \overline{{\mathrm D}} \}>0,$$ the
solution $u\in V$ of \eqref{PDE}, \eqref{vf} belongs to $W^s$, and
there exists a constant $c(s,{\mathrm D})$ such that
\begin{equation}\label{eq:Ws-estimate}
\| u \|_{W^s}
\leq
\frac{c}{a_{\min}} \left(1+\|a\|_{C^s}^{1/s} a_{\min}^{-1/s} \right)\| f \|_{L^2}
\end{equation}
\end{proposition}
\begin{proof}
The estimate follows from the a-priori bounds for $s=0$ and $s=1$,
i.e., \eqref{V-estimate} and \eqref{W-estimate}, by interpolation
with the Lipschitz continuity \eqref{eq:LipBasic} of the solution
operator.
Let $a\in C^s({\mathrm D})$ with $a_{\min} > 0$ be given. From
\eqref{eq:CsLinfW1inf}, for every $\delta>0$ exists
$a_\delta \in W^{1,\infty}({\mathrm D})$ with
$$
\| a - a_\delta \|_{C^0} \leq C\delta^s \| a \|_{C^s}\;, \quad \|
a_\delta \|_{W^{1}_\infty} \leq C \delta^{s-1} \| a \|_{C^s} \;.
$$
From
$$
\min_{{\boldsymbol{x}}\in {\mathrm D}} a_\delta({\boldsymbol{x}}) \geq \min_{{\boldsymbol{x}}\in {\mathrm D}} a({\boldsymbol{x}})
- \| a - a_\delta \|_{C^0} \geq a_{\min} - C\delta^s \| a \|_{C^s}
$$
follows for
$ 0 < \delta \leq 2^{-1/s} \left\| a/a_{\min} \right\|_{C^s}^{-1/s},$
that
$$ \min_{{\boldsymbol{x}}\in {\mathrm D}} a_\delta({\boldsymbol{x}}) \geq a_{\min}/2\;.$$
For such $\delta$ and for $f\in L^2({\mathrm D})$, \eqref{PDE} with
$a_\delta$ admits a unique solution $u_\delta \in V$ and from
\eqref{W-estimate}
$$
\| u_\delta \|_W \leq \frac{2c}{a_{\min}}\left( 1 + \frac{\|\nabla
a_\delta\|_{L^\infty}}{a_{\min}}\right) \| f \|_{L^2}\;.
$$
From \eqref{eq:LipBasic} (with $f_1=f_2=f$) we find
$$
\|u-u_\delta \|_V \leq \frac{2c}{a_{\min}^2} \| a - a_\delta
\|_{L^\infty} \| f \|_{L^2} \leq C \frac{\delta^s}{a_{\min}^2} \| a
\|_{C^s} \| f \|_{L^2} \;.
$$
This implies in \eqref{eq:Kfct} that for some constant $C>0$
(depending only on ${\mathrm D}$ and on $s$)
\begin{equation}\label{eq:tbound}
K(u,t,V,W)
\leq
\frac{C}{a_{\min}}
\left( \delta^s A_s + t\left(1 + \delta^{s-1} A_s\right) \right)\| f \|_{L^2}
\;,\;\;t>0
\end{equation}
where we have set
$A_s := \big\| \frac{a}{a_{\min}} \big\|_{C^s} \in [1,\infty)$.
To complete the proof, by \eqref{eq:Ws} we bound
$\| u \|_{W^s} = \sup_{t>0}t^{-s}K(u,t,V,W)$. To this end, it
suffices to bound $K(u,t,V,W)$ for $0<t<1$. Given such $t$, we choose
in the bound \eqref{eq:tbound} $\delta = t\delta_0 \in (0,\delta_0)$
with $\delta_0 := 2^{-1/s} A_s^{-1/s}$. This yields
$$
\delta^s A_s + t\left(1 + \delta^{s-1} A_s\right) = t^s
\left(\delta_0^s A_s + t^{1-s} + \delta_0^{s-1} A_s \right) = t^s
\left(2^{-1} + t^{1-s} + 2^{-(s-1)/s} A_s^{1-(s-1)/s} \right)
$$
and we obtain for $0<t<1$ the bound
$$
t^{-s} K(u,t,V,W) \leq \frac{C}{a_{\min}} \left( 2 + 2^{-(s-1)/s}
A_s^{1/s} \right) \| f \|_{L^2}\;.
$$
Adjusting the value of the constant $C$, we arrive at
\eqref{eq:Ws-estimate}.
\end{proof}
\subsection{Random input data}
\label{S:RndDat}
We are in particular interested in the input data $a$ and $f$ of
\eqref{PDE} being not precisely known. The Lipschitz continuous
data-dependence in Proposition \ref{prop:LipS} of the variational
solution $u\in V$ of \eqref{PDE} will ensure that small variations in
the data $(a,f)$ imply corresponding small changes in the (unique)
solution $u\in V$. A natural paradigm is to model uncertain data
probabilistically. To this end, we work with a base probability space
$(\Omega,\cA,\mathbb{P})$. Given a known right hand side
$f\in L^2({\mathrm D})$, and uncertain diffusion coefficient
$a \in E \subseteq L^\infty_+({\mathrm D})$, where $E$ denotes a suitable
subset of $L^\infty_+({\mathrm D})$ of admissible diffusion coefficients,
we model the function $a$ or $\log a$ as RVs taking values in (a
subset $E$ of) $L^\infty({\mathrm D})$. We will assume the random data
$a$ to be separably-valued, i.e.\ the set $E$ of admissible random
data will be a separable subspace of $L^\infty({\mathrm D})$.
Separability of $E$ is natural from the point of view of numerical
approximation of (samples of) random input $a$ and simplifies many
technicalities in the mathematical description; we refer in particular
to the construction of GMs on $E$ in
Sections~\ref{S:GMSepHS}--\ref{S:GSer}. One valid choice for the
space of admissible input data $E$ consists in
$E = C(\overline{{\mathrm D}})\cap L^\infty_+({\mathrm D})$. In the
log-Gaussian models to be analyzed subsequently,
$E\subset L^\infty_+({\mathrm D})$ will be ensured by modelling $\log(a)$
as a GRF, i.e., we assume the probability measure $\mathbb{P}$ to be such
that the law of $\log(a)$ is a GM on $L^\infty({\mathrm D})$ which charges
$E$, i.e.\ the random element
$\log(a(\cdot,\omega))\in L^\infty_+({\mathrm D})$ $P$ almost sure. This,
in turn, implies with the well-posedness result in Section
\ref{S:PbmStat} that there exists a unique random solution
$u(\omega) = \cS(a,f)\in V$ $\mathbb{P}$-a.s. Furthermore, the Lipschitz
continuity \eqref{eq:LogLip} then implies that the corresponding map
$\omega\mapsto u(\omega)$ is a composition of the measurable map
$\omega\mapsto \log(a(\cdot,\omega))$ with the Lipschitz continuous
deterministic data-to-solution map $\cS$, hence strongly measurable,
and thus a RV on $(\Omega,\cA,\mathbb{P})$ taking values in $V$.
\subsection{Parametric deterministic coefficient}
\label{S:ParCoef}
A key step in the deterministic numerical approximation of the
elliptic PDE \eqref{PDE} with log-Gaussian random input (i.e.,
$b = \log(a)$ is a GRF on a suitable locally convex space $E$ of
admissible input data) is to place a GM on $E$ and to describe the
realizations of GRF $b$ in terms of affine-parametric representations
discussed in Section \ref{S:GSer}. In Section \ref{S:ParDetPbm}, we
briefly describe this and in doing so extend a-priori estimates to
this resulting deterministic parametric version of elliptic PDE
\eqref{PDE}. Subsequently, in Section \ref{S:CplxParExt}, we show
that the resulting, countably-parametric, linear elliptic problem
admits an extension to certain complex parameter domains, while still
remaining well-posed.
\subsubsection{Deterministic countably parametric elliptic PDEs}
\label{S:ParDetPbm}
Placing a Gaussian probability measure on the random inputs $\log(a)$
to the model elliptic PDE \eqref{PDE} can be achieved via Gaussian
series as discussed in Section \ref{S:GSer}. Affine-parametric
representations which are admissible in the sense of Definition
\ref{def:Adm} of the random input $\log(a)$ of \eqref{PDE}, subject to
a Gaussian law on the corresponding input locally convex space $E$,
render the elliptic PDE \eqref{PDE} with random inputs a deterministic
parametric elliptic PDE. More precisely, $b = \log(a)$ will depend on
the sequence ${\boldsymbol{y}} = (y_j)_{j\in {\mathbb N}}$ of parameters from the parameter
space ${\mathbb R}^\infty$. Accordingly, we consider parametric diffusion
coefficients $a=a({\boldsymbol{y}})$, where
$${\boldsymbol{y}}=(y_j)_{j \in {\mathbb N}} \in U:= {\mathbb R}^\infty.$$ We develop the
holomorphy-based analysis of parametric regularity and PC coefficient
sparsity for the model parametric linear second order elliptic
divergence-form PDE with so-called ``log-affine coefficients''
\begin{equation}\label{SPDE}
-\operatorname{div}\big(\exp(b({\boldsymbol{y}})) \nabla u({\boldsymbol{y}})\big)=f
\quad \mbox{in}\quad {\mathrm D}\;,
\quad u({\boldsymbol{y}})|_{\partial {\mathrm D}} = 0\;,
\end{equation}
i.e., $$a({\boldsymbol{y}}) =\exp(b({\boldsymbol{y}})).$$ Here, the coefficient
$b({\boldsymbol{y}})=\log(a({\boldsymbol{y}}))$ is assumed to be affine-parametric
\begin{equation}\label{eq:CoeffAffin}
b({\boldsymbol{y}})= \sum_{j \in {\mathbb N}} y_j\psi_j({\boldsymbol{x}})
\;, \quad
{\boldsymbol{x}}\in {\mathrm D} \;,\quad {\boldsymbol{y}}\in U\;.
\end{equation}
We assume that $\psi_j \in E \subset L^\infty({\domain})$ for every
$j\in {\mathbb N}$. For any ${\boldsymbol{y}} \in U$ such that $b({\boldsymbol{y}})\in L^\infty({\domain})$,
by \eqref{V-estimate} we have the estimate
\begin{equation} \label{eq:uApriori} \|u({\boldsymbol{y}})\|_V \leq \|f\|_{V^*} \|
a({\boldsymbol{y}})^{-1}\|_{L^\infty} \leq \exp(\|b({\boldsymbol{y}})\|_{L^\infty})\|f\|_{V^*}
\,.
\end{equation}
For every ${\boldsymbol{y}} \in U$ satisfying $b({\boldsymbol{y}})\in L^\infty({\domain})$, the
variational form \eqref{vf} of \eqref{SPDE} gives rise to the
\emph{parametric energy norm} $\| v \|_{a({\boldsymbol{y}})}$ on $V=H^1_0({\mathrm D})$
which is defined by
\begin{equation*}\label{eq:ParEn}
\| v \|_{a({\boldsymbol{y}})}^2 := \int_{{\mathrm D}} a({\boldsymbol{x}},{\boldsymbol{y}}) | \nabla v |^2 \rd{\boldsymbol{x}} \;, \;\;
v\in V
\;.
\end{equation*}
The norms $ \| \circ \|_{a({\boldsymbol{y}})}$ and $\| \circ \|_V$ are equivalent
on $V$ but not uniformly w.r.t. ${\boldsymbol{y}}$. It holds
\begin{equation}\label{eq:Norms}
\exp(-\| b({\boldsymbol{y}}) \|_{L^\infty} )\| v \|_V^2
\leq
\| v \|_{a({\boldsymbol{y}})}^2
\leq
\exp( \| b({\boldsymbol{y}}) \|_{L^\infty} \| v \|_V^2,
\quad
v\in V \;.
\end{equation}
\subsubsection{Probabilistic setting}
\label{S:ParmProb}
In a probabilistic setting, the parameter sequence ${\boldsymbol{y}}$ is chosen as
a sequence of i.i.d.\ standard Gaussian RVs ${\mathcal N}(0,1)$ and
$(\psi_j)_{j \in {\mathbb N}}$ a given sequence of functions in the Banach
space $L^\infty({\mathrm D})$ to which we refer as \emph{representation
system} of the uncertain input. We then treat \eqref{SPDE} as the
stochastic linear second order elliptic divergence-form PDE with
so-called ``log-normal coefficients''. We refer to
Section~\ref{S:GSer} for the construction of GMs based on affine
representation systems $(\psi_j)_{j \in {\mathbb N}}$. Due to
$L^\infty({\mathrm D})$ being non-separable, we consider GRFs taking
values in separable subspaces $E \subset L^\infty({\mathrm D})$, such as
$E=C^0(\overline{{\mathrm D}})$.
The probability space $(\Omega,\cA,\mathbb{P})$ from Section \ref{S:RndDat}
on the parametric solutions $\{ u({\boldsymbol{y}}) : {\boldsymbol{y}}\in U\} $ is chosen as
$(U,\cB(U),\gamma)$. Here and throughout the rest of this book, we
make use of the notation: $U := {\mathbb R}^\infty$, $\cB(U)$ is the
$\sigma$-field on the locally convex space $U$ generated by cylinders
of Borel sets on ${\mathbb R}$, and $\gamma$ is the product measure of the
standard GM $\gamma_1$ on $\mathbb{R}$. We shall refer to $\gamma$ as the
\emph{standard GM on $U$}.
It follows from the a-priori estimate \eqref{eq:uApriori} that for
$f\in V^*$ the parametric diffusion problem \eqref{SPDE} admits a
unique solution for parameters ${\boldsymbol{y}}$ in the set
\begin{equation}\label{eq:U0}
U_0 := \{ {\boldsymbol{y}} \in U: b({\boldsymbol{y}})\in L^\infty({\mathrm D})\} \;.
\end{equation}
The measure $\gamma(U_0)$ of the set $U_0 \subset U$ depends on the
structure of ${\boldsymbol{y}}\to b({\boldsymbol{y}})$. The following sufficient condition on
the representation system $(\psi_j)_{j\in {\mathbb N}}$ will be assumed
throughout.
\noindent
\begin{assumption}\label{ass:Ass1}
For every $j\in {\mathbb N}$, $\psi_j\in L^\infty({\mathrm D})$, and there
exists a positive sequence $(\lambda_j)_{j\in {\mathbb N}}$ such that
$\big(\exp(-\lambda_j^2)\big)_{j\in {\mathbb N}}\in \ell^1({\mathbb N})$ and the
series $\sum_{j\in {\mathbb N}}\lambda_j|\psi_j|$ converges in
$L^\infty({\mathrm D})$.
\end{assumption}
The following result was shown in \cite[Theorem 2.2]{BCDM}.
\begin{proposition}\label{prop:Meas1}
Under Assumption \ref{ass:Ass1}, the set $U_0$ has full GM, i.e.
$\gamma(U_0) = 1$. For all $k\in {\mathbb N}$ there holds, with
$\mathbb E(\cdot)$ denoting expectation with respect to $\gamma$,
$$
\mathbb E\left( \exp(k\| b(\cdot) \|_{L^\infty}) \right) < \infty \;.
$$
The solution family $\{ u({\boldsymbol{y}}): {\boldsymbol{y}} \in U_0 \}$ of the parametric
elliptic boundary value problem \eqref{SPDE} is in $L^k(U,V;\gamma)$
for every finite $k\in {\mathbb N}$.
\end{proposition}
\subsubsection{Deterministic complex-parametric elliptic PDEs}
\label{S:CplxParExt}
Towards the aim of establishing summability of Wiener-Hermite PC
expansions of the parametric solutions $\{ u({\boldsymbol{y}}): {\boldsymbol{y}}\in {U_0}\}$ of
\eqref{SPDE}, we extend the deterministic parametric problem
\eqref{SPDE} from real-valued to complex-valued parameters.
Formally replacing ${\boldsymbol{y}}=(y_j)_{j \in {\mathbb N}}\in U$ in the coefficient
$a({\boldsymbol{y}})$ by
${\boldsymbol{z}}=(z_j)_{j \in {\mathbb N}}=(y_j+{\rm i}\xi_j)_{j \in {\mathbb N}}\in {\mathbb C}^\infty$, the
real part of $a({\boldsymbol{z}})$ is
\begin{equation} \label{Re(a)} \mathfrak{R}[a({\boldsymbol{z}})] =
\exp\Bigg({\sum_{j \in {\mathbb N}} y_j\psi_j({\boldsymbol{x}})}\Bigg) \cos\Bigg(\sum_{j
\in {\mathbb N}} \xi_j\psi_j({\boldsymbol{x}})\Bigg)\,.
\end{equation}
We find that $\mathfrak{R}[a({\boldsymbol{z}})]>0$ if
$$
\bigg\|\sum_{j \in {\mathbb N}} \xi_j\psi_j \bigg\|_{L^\infty} <
\frac{\pi}{2}.
$$
This observation and Proposition \ref{prop:Meas1} motivate the study
of the analytic continuation of the solution map ${\boldsymbol{y}} \mapsto u({\boldsymbol{y}})$
to ${\boldsymbol{z}} \mapsto u({\boldsymbol{z}})$ for complex parameters
${\boldsymbol{z}} = (z_j)_{j \in {\mathbb N}}$ by formally replacing the parameter $y_j$ by
$z_j$ in the definition of the parametric coefficient $a$, where each
$z_j$ lies in the strip
\begin{equation} \label{eq:DefSjrho} \mathcal{S}_j ({\boldsymbol{\rho}}):= \{ z_j\in
{\mathbb C}\,: |\mathfrak{Im}z_j| < \rho_j\}
\end{equation}
and where $\rho_j>0$ and
${\boldsymbol{\rho}}=(\rho_j)_{j \in {\mathbb N}} \in (0,\infty)^\infty$ is any sequence of
positive numbers such that
\begin{equation*} \label{k-01} \Bigg\|\sum_{j\in {\mathbb N}} \rho_j
|\psi_j|\Bigg\|_{L^\infty} < \frac{\pi}{2}\,.
\end{equation*}
\subsection{Analyticity and sparsity}
\label{S:HolSumSol}
We address the analyticity (holomorphy) of the parametric solutions
$\{ u({\boldsymbol{y}}): {\boldsymbol{y}}\in {U_0}\}$. We analyze the sparsity by estimating,
in particular, the size of the domains of holomorphy to which the
parametric solutions can be extended. We also treat the weighted
$\ell^2$-summability (sparsity) for the series of Wiener-Hermite PC
expansion coefficients $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}$ of $u({\boldsymbol{y}})$.
\subsubsection{Parametric holomorphy}
\label{sec:HolProp}
In this section we establish holomorphic parametric dependence $u$ on
$a$ and on $f$ as in \cite{CoDeSch1} by verifying complex
differentiability of a suitable complex-parametric extension of
${\boldsymbol{y}}\mapsto u({\boldsymbol{y}})$. We observe that the Lax-Milgram theory can be
extended to the case where the coefficient function $a$ is
complex-valued. In this case, $V := H^1_0({\mathrm D},{\mathbb C})$ in \eqref{vf}
and the ellipticity assumption \eqref{PDE-epllipticity} is extended to
the complex domain as
\begin{equation}\label{eq:PDEellinC}
0 < \rho(a) := \underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,inf}}\,\Re(a({\boldsymbol{x}}))
\leq |a({\boldsymbol{x}})|
\leq \| a \|_{L^\infty}
<\infty,\qquad {\boldsymbol{x}}\in {\mathrm D}.
\end{equation}
Under this condition, there exists a unique variational solution
$u\in H^1_0({\mathrm D},{\mathbb C})$ of \eqref{PDE} and for this solution, the
estimate \eqref{V-estimate} remains valid, i.e.,
\begin{equation} \label{eq:bound-comp} \|u\|_V \leq
\frac{\|f\|_{V^*}}{\rho(a)} \,.
\end{equation}
Let ${\boldsymbol{\rho}}=(\rho_j)_{j\in {\mathbb N}} \in [0,\infty)^\infty$ be a sequence of
non-negative reals and assume that ${\mathfrak u } \subseteq \operatorname{supp}({\boldsymbol{\rho}})$ is
finite. Define
\begin{equation}\label{eq:Snubrho}
\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}) :=\mathop{\raisebox{-.5ex}{\hbox{\huge{$\times$}}}}_{j\in {\mathfrak u }} \mathcal{S}_j({\boldsymbol{\rho}})
\end{equation}
where the strip $ \mathcal{S}_j ({\boldsymbol{\rho}}) $ is given in
\eqref{eq:DefSjrho}. For ${\boldsymbol{y}}\in U$, put
\[
\mathcal{S}_{\mathfrak u } ({\boldsymbol{y}},{\boldsymbol{\rho}}) := \big\{(z_j)_{j\in {\mathbb N}}: z_j \in
\mathcal{S}_j({\boldsymbol{\rho}})\ \text{if}\ j\in {\mathfrak u }\ \text{and}\ z_j=y_j \
\text{if}\ j\not \in {\mathfrak u } \big\}.
\]
In the following, $V:=H^1_0({\mathrm D},{\mathbb C})$ denotes the Sobolev space of
complex-valued functions.
\begin{proposition}\label{prop:holoh1}
Let the sequence ${\boldsymbol{\rho}}=(\rho_j)_{j\in {\mathbb N}}\in [0,\infty)^\infty$
satisfy
\begin{equation}\label{eq:leqkappa}
\Bigg\|\sum_{j \in {\mathbb N}} \rho_j |\psi_j | \Bigg\|_{L^\infty}
\leq
\kappa < \frac{\pi}{2}\,.
\end{equation}
Let ${\boldsymbol{y}}_0=(y_{0,1},y_{0,2},\ldots) \in U$ be such that $b({\boldsymbol{y}}_0)$
belongs to $L^\infty({\domain})$, and let ${\mathfrak u }\subseteq \operatorname{supp}({\boldsymbol{\rho}})$ be a
finite set.
Then the solution $u$ of the variational form of \eqref{SPDE} is
holomorphic on $ \mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}) $ as a function of the
parameters
${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j \in {\mathbb N}} \in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$
taking values in $V$ with $z_j = y_{0,j}$ for $j\not \in {\mathfrak u }$ held
fixed\,.
\end{proposition}
\begin{proof}
Let $N\in {\mathbb N}$. We denote
\begin{equation}\label{eq:SUN}
\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})
:=
\big\{ (y_j+{\rm i} \xi_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}): |y_j-y_{0,j}|<N\big\}\,.
\end{equation}
For
${\boldsymbol{z}}_{{\mathfrak u }} = (y_j+{\rm i} \xi_j)_{j\in {\mathbb N}}\in \mathcal{S}_{{\mathfrak u }}
({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$ with
$(y_j+{\rm i} \xi_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$
we have
\begin{equation*}
\begin{split}
\label{eq:DefM}
\Bigg\| \sum_{j \in {\mathbb N}}y_j\psi_j\Bigg\|_{L^\infty} &\leq
\|b({\boldsymbol{y}}_0)\|_{L^\infty} + \Bigg\| \sum_{j\in
{\mathfrak u }}|(y-y_{0,j})\psi_j |\Bigg\|_{L^\infty}
\\
& \leq \|b({\boldsymbol{y}}_0)\|_{L^\infty} + N\Bigg\| \sum_{j\in
{\mathfrak u }}|\psi_j |\Bigg\|_{L^\infty} =: M <\infty
\end{split}
\end{equation*}
and
\begin{equation*}
\Bigg\| \sum_{j\in {\mathfrak u }}\xi_j \psi_j\Bigg\|_{L^\infty}
\leq
\Bigg\| \sum_{j\in {\mathfrak u }}|\rho_j \psi_j|\Bigg\|_{L^\infty}
\leq
\kappa\,.
\end{equation*}
Consequently, we obtain from \eqref{Re(a)}
\begin{equation}\label{eq:R-1}
\rho(a( {\boldsymbol{z}}_{{\mathfrak u }}))
\geq \exp\Bigg(-\Bigg\| \sum_{j \in {\mathbb N}}y_j\psi_j\Bigg\|_{L^\infty}\Bigg)
\cos\Bigg(\Bigg\| \sum_{j\in {\mathfrak u }}\xi_j \psi_j\Bigg\|_{L^\infty} \Bigg)\geq
\exp(-M) \cos\kappa
\end{equation}
for all $ {\boldsymbol{z}}_{{\mathfrak u }}\in \mathcal{S}_{{\mathfrak u }} ({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$ with
$(y_j+{\rm i} \xi_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$. From
this and the analyticity of exponential functions we conclude that
the map ${\boldsymbol{z}}_{\mathfrak u }\to u({\boldsymbol{z}}_{{\mathfrak u }})$ is holomorphic on the set
$\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$, see \cite[Pages 22, 23]{CoDe}.
Since $N$ is arbitrary we deduce that the map
${\boldsymbol{z}}_{\mathfrak u }\to u({\boldsymbol{z}}_{{\mathfrak u }})$ is holomorphic on
$\mathcal{S}_{{\mathfrak u }} ({\boldsymbol{\rho}})$.
\end{proof}
The analytic continuation of the parametric solutions
$\{u({\boldsymbol{y}}): {\boldsymbol{y}}\in U\}$ to $\mathcal{S}_{{\mathfrak u }} ({\boldsymbol{\rho}})$ leads to the
following result on parametric $V$-regularity in the following lemma.
\begin{lemma}\label{lem:estV}
Let ${\boldsymbol{\rho}}= (\rho_j)_{j \in {\mathbb N}}$ be a non-negative sequence
satisfying \eqref{eq:leqkappa}. Let ${\boldsymbol{y}} \in U$ with
$b({\boldsymbol{y}})\in L^\infty({\domain})$ and ${\boldsymbol{\nu}}\in {\mathcal F}$ such that
$\operatorname{supp}({\boldsymbol{\nu}})\subseteq \operatorname{supp}({\boldsymbol{\rho}})$. Then we have
\begin{equation*}
\|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_V
\leq
C_0\frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}^{\boldsymbol{\nu}}}
\exp\big( \|b({\boldsymbol{y}})\|_{L^\infty} \big) ,
\end{equation*}
where $C_0=e^\kappa (\cos\kappa)^{-1}\|f\|_{V^*}$.
\end{lemma}
\begin{proof}Let ${\boldsymbol{\nu}}\in {\mathcal F}$ such that
$\operatorname{supp}({\boldsymbol{\nu}})\subseteq \operatorname{supp}({\boldsymbol{\rho}})$. Denote ${\mathfrak u }=\operatorname{supp}({\boldsymbol{\nu}})$.
For fixed variable $y_j$ with $j\not \in {\mathfrak u }$, the map
$\mathcal{S}_{\mathfrak u } ({\boldsymbol{y}},{\boldsymbol{\rho}}) \ni {\boldsymbol{z}}_{{\mathfrak u }}\to u({\boldsymbol{z}}_{\mathfrak u })$ is
holomorphic on the domain $\mathcal{S}_{\mathfrak u }({\boldsymbol{y}},\kappa'{\boldsymbol{\rho}})$ where
$\kappa <\kappa \kappa'<\pi/2$, see Proposition \ref{prop:holoh1}.
Applying Cauchy's integral formula gives
\begin{equation*}
\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})
=
\frac{{\boldsymbol{\nu}}!}{(2\pi i)^{|{\mathfrak u }|}}
\int_{\mathcal{C}_{{\boldsymbol{y}},{\mathfrak u }}({\boldsymbol{\rho}})}
\frac{u({\boldsymbol{z}}_{\mathfrak u }) }{\prod_{j\in {\mathfrak u }} (z_j-y_j)^{\nu_j+1}}\prod_{j\in {\mathfrak u }}\rd z_j,
\end{equation*}
where
\begin{equation} \label{eq:C-rho} \mathcal{C}_{{\boldsymbol{y}}, {\mathfrak u }}({\boldsymbol{\rho}}) :=
\mathop{\raisebox{-.5ex}{\hbox{\huge{$\times$}}}}_{j\in {\mathfrak u }} \mathcal{C}_{{\boldsymbol{y}},j}( {\boldsymbol{\rho}})\,,\qquad
\mathcal{C}_{{\boldsymbol{y}},j} ( {\boldsymbol{\rho}}) := \big\{ z_j \in {\mathbb C}:
|z_j-y_j|=\rho_j\big\}\,.
\end{equation}
This leads to
\begin{equation} \label{u-y}
\begin{split}
\|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_{V} & \leq \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}^{\boldsymbol{\nu}}}
\sup_{z_{\mathfrak u }\in \mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}})}
\|u({\boldsymbol{z}}_{\mathfrak u })\|_{V}\,
\end{split}
\end{equation}
with
\begin{equation} \label{eq:C-rho-y} \mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}})
=\big\{(z_j)_{j\in {\mathbb N}} \in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}},{\boldsymbol{\rho}}): \
(z_j)_{j\in {\mathfrak u }}\in \mathcal{C}_{{\boldsymbol{y}},{\mathfrak u }}({\boldsymbol{\rho}}) \big\}\,.
\end{equation}
Notice that for
${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j\in {\mathbb N}} \in \mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}})$ we can
write $z_j = y_j + \eta_j + {\rm i}\xi_j \in {\mathcal C}_{{\boldsymbol{y}},j}({\boldsymbol{\rho}})$ with
$|\eta_j | \le \rho_j$, $|\xi_j| \le \rho_j$ if $j\in {\mathfrak u }$ and
$\eta_j = \xi_j=0$ if $j\not \in {\mathfrak u }$. By denoting
${\boldsymbol{\eta}}=(\eta_j)_{j\in {\mathbb N}}$ and ${\boldsymbol{\xi}}=(\xi_j)_{j\in {\mathbb N}}$ we see
that $\|b({\boldsymbol{\eta}})\|_{L^\infty}\leq \kappa$ and
$\|b({\boldsymbol{\xi}})\|_{L^\infty}\leq \kappa$. Hence we deduce from
\eqref{eq:bound-comp} that
\begin{equation*}
\begin{split}
\|u({\boldsymbol{z}}_{\mathfrak u })\|_{V} & \leq \frac{\exp\big(
\|b({\boldsymbol{y}}+{\boldsymbol{\eta}})\|_{L^\infty}\big)
}{\cos\big(\|b({\boldsymbol{\xi}})\|_{L^\infty} \big) } \|f\|_{V^*} \leq
\frac{ \exp\big( \kappa+\|b({\boldsymbol{y}})\|_{L^\infty} \big) }{\cos
\kappa }\|f\|_{V^*} \,.
\end{split}
\end{equation*}
Inserting this into \eqref{u-y} we obtain the desired estimate.
\end{proof}
\subsubsection{Summability of Wiener-Hermite PC expansion
coefficients}
\label{sec:SumHermCoef}
In this section, we will exploit the analyticity of $u$ to prove a
weighted $\ell^2$-summability result for the $V$-norms of the
coefficients in the Wiener-Hermite PC expansion of the solution map
${\boldsymbol{y}}\to u({\boldsymbol{y}})$. Our analysis yields the same $\ell^p$-summability
result as in the papers \cite{BCDM,BCDS} in the case $\psi_j$ have
arbitrary supports. In this case, our result implies that the
$\ell^p$-summability of $(\|u_{\boldsymbol{\nu}}\|_V)_{{\mathcal F}}$ for $0<p\leq 1$ (the
sparsity of parametric solutions) follows from the
$\ell^p$-summability of the sequence
$(j^\alpha \|\psi_j\|_{L^\infty})_{j\in {\mathbb N}}$ for some $\alpha>1/2$
which is an improvement over the condition
$(j\|\psi_j\|_{L^\infty})_{j\in {\mathbb N}} \in \ell^p({\mathbb N})$ in \cite{HS},
see \cite[Section 6.3]{BCDM}. In the case of disjoint or finitely
overlapping supports our analysis obtains a weaker result compared to
\cite{BCDM,BCDS}. As observed in \cite{CoDeSch1}, one advantage of
establishing sparsity of Wiener-Hermite PC expansion coefficients via
holomorphy rather than by successive differentiation is that it allows
to derive, in a unified way, summability bounds for sequences of
Wiener-Hermite PC expansion coefficients whose size is measured in
scales of Sobolev and Besov spaces in the domain ${\mathrm D}$. Using
real-variable arguments as, e.g., in \cite{BCDM,BCDS}, establishing
sparsity of parametric solutions in Besov spaces in ${\mathrm D}$ of
higher smoothness seems to require more involved technical and
notational developments, according to \cite[Comment on Page
2157]{BCDS}.
Let us recall the notion of Bochner spaces. Let $(\Omega, \cA, \mu)$
be a measure space, $X$ a Banach space and $1 \le p < \infty$. Then
the Bochner space $L^p(\Omega,X;\mu)$ is defined as the space of all
$\mu$-measurable mappings $u$ from $\Omega$ to $X$ such that the norm
\begin{equation} \nonumber \|u\|_{L^p(\Omega,X;\mu)} := \
\left(\int_{\Omega} \|u({\boldsymbol{y}})\|_X^p \, \rd \mu \right)^{1/p} <
\infty.
\end{equation}
In particular, when $(\Omega, \cA, \mu) = (U,\cB(U),\gamma)$, $X$ is
separable and $p=2$, the space $L^2(U,X;\gamma)$ is one of the most
important for the problems considered in this book. For example, the
parametric solution $\{u({\boldsymbol{y}}): {\boldsymbol{y}}\in U\}$ of \eqref{SPDE} belongs to
the space $L^2(U,V;\gamma)$ or more generally,
$L^2(U,(H^{1+s}\cap H^1_0)({\mathrm D});\gamma)$ for $s$-order of extra
differentiability provided by higher data regularity. We recall from
Section \ref{S:HerPol} the normalized probabilistic Hermite
polynomials $(H_k)_{k \in {\mathbb N}_0}$. Every $u \in L^2(U,X;\gamma)$
admits the \emph{Wiener-Hermite PC expansion}
\begin{equation}
\sum_{{\boldsymbol{\nu}}\in {\mathcal F}} u_{\boldsymbol{\nu}} H_{\boldsymbol{\nu}}({\boldsymbol{y}}),
\label{hermite}
\end{equation}
where for ${\boldsymbol{\nu}} \in {\mathcal F}$,
\begin{equation*}
H_{\boldsymbol{\nu}}({\boldsymbol{y}})=\prod _{j \in {\mathbb N}}H_{\nu_j}(y_j),\quad
\label{hermite-polynomial}
\end{equation*}
and
\begin{equation*}
u_{\boldsymbol{\nu}}:=\int_U u({\boldsymbol{y}})\,H_{\boldsymbol{\nu}}({\boldsymbol{y}})\, \rd\gamma ({\boldsymbol{y}})
\label{hermite-coeff}
\end{equation*}
are called \emph{Wiener-Hermite PC expansion coefficients}. Notice
that $(H_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}} \in {\mathcal F}}$ forms an ONB of $L^2(U;\gamma)$.
For every $u\in L^2(U,X;\gamma)$, there holds the Parseval-type
identity
\begin{equation}\label{eq:ParsevH}
\| u \|^2_{L^2(U,X;\gamma)}
=
\sum_{{\boldsymbol{\nu}}\in \cF} \| u_{\boldsymbol{\nu}} \|_X^2 \;,\quad
u\in L^2(U,X;\gamma)\;.
\end{equation}
The error of approximation of the parametric solution
$\{u({\boldsymbol{y}}): {\boldsymbol{y}}\in U\}$ of \eqref{SPDE} will be measured in the Bochner
space $L^2(U,V;\gamma)$. A basic role in this approximation is taken
by the Wiener-Hermite PC expansion \eqref{hermite} of $u$ in the space
$L^2(U,V;\gamma)$.
For a finite set $\Lambda \subset \cF$, we denote by
$u_\Lambda = \sum_{{\boldsymbol{\nu}}\in \Lambda} u_{\boldsymbol{\nu}}$ the corresponding partial
sum of the Wiener-Hermite PC expansion \eqref{hermite}. It follows
from \eqref{eq:ParsevH} that
\begin{equation*}\label{eq:L2Error}
\| u - u_\Lambda \|_{L^2(U,V;\gamma)}^2
=
\sum_{{\boldsymbol{\nu}}\in \cF\backslash \Lambda} \| u_{\boldsymbol{\nu}} \|_V^2
\;.
\end{equation*}
Therefore, summability results of the sequence
$(\| u_{\boldsymbol{\nu}} \|_V )_{{\boldsymbol{\nu}}\in \cF}$ imply convergence rate estimates of
finitely truncated expansions $u_{\Lambda_n}$ for suitable sequences
$( \Lambda_n )_{n\in {\mathbb N}}$ of finite index sets (see \cite{HS,BCDM,
dD21}).
The following identity was proved in \cite[Theorem 3.3]{BCDM}.
\begin{lemma}\label{lem:equal}
Let Assumption \ref{ass:Ass1} hold. Let $r\in {\mathbb N}$ and
${\boldsymbol{\varrho}}=(\varrho_j)_{j\in {\mathbb N}}\in [0,\infty)^\infty$ be a sequence
of positive numbers.
%
Then
\begin{equation} \label{eq:equal-V} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_V^2 =
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!}
\int_U\| \partial^{\boldsymbol{\nu}} u({\boldsymbol{y}})\|_V^2\rd\gamma({\boldsymbol{y}}),
\end{equation}
where
\begin{equation} \label{beta} \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) :=
\sum_{\|{\boldsymbol{\nu}}'\|_{\ell^\infty}\leq r} \binom{{\boldsymbol{\nu}}}{{\boldsymbol{\nu}}'}
{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}'} = \prod_{j \in
{\mathbb N}}\Bigg(\sum_{\ell=0}^{r}\binom{\nu_j}{\ell}\varrho_j^{2\ell}\Bigg)\,.
\end{equation}
\end{lemma}
We recall a summability property of the sequence
$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})^{-1})_{j\in {\mathbb N}}$ proved also in \cite[Lemma
5.1]{BCDM}.
\begin{lemma}\label{lem:beta-summability}
Let $0 < p < 2$ and $q:= \frac{2p}{2-p}$. Let
${\boldsymbol{\varrho}}=(\varrho_j)_{j\in {\mathbb N}}\in [0,\infty)^\infty$ be a sequence
of positive numbers such that
$$
(\varrho_j^{-1})_{j\in {\mathbb N}}\in \ell^q({\mathbb N}).
$$
Then for any $r \in {\mathbb N}$ such that $\frac{2}{r+1} <
p$, the sequence defined in \eqref{beta} for this $r$
satisfies
\begin{equation} \label{ineq: beta} \sum_{{\boldsymbol{\nu}}\in {\mathcal F}}
\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}})^{-q/2}<\infty\,.
\end{equation}
\end{lemma}
For the following lemma see \cite[Lemma 7.1, Theorem
7.2]{CoDeSch} and \cite[Lemma 3.17]{CoDe}.
\begin{lemma}\label{lem:alpha-summability}
Let ${\boldsymbol{\alpha}}=(\alpha_j)_{j\in {\mathbb N}}$ be a sequence of
nonnegative numbers. Then we have the following.
\begin{itemize}
\item[{\rm (i)}] For $0 < p <\infty$, the sequence
$({\boldsymbol{\alpha}}^{\boldsymbol{\nu}})_{{\boldsymbol{\nu}} \in {\mathcal F}}$ belongs to
$\ell^p({\mathcal F})$ if and only if
$\|{\boldsymbol{\alpha}}\|_{\ell^p} < \infty$ and
$\|{\boldsymbol{\alpha}}\|_{\ell^\infty} < 1$.
\item[{\rm (ii)}] For $0 < p \le 1$, the sequence
$({\boldsymbol{\alpha}}^{\boldsymbol{\nu}}|{\boldsymbol{\nu}}|!/{\boldsymbol{\nu}}!)_{{\boldsymbol{\nu}} \in {\mathcal F}}$ belongs
to $\ell^p({\mathcal F})$ if and only if
$\|{\boldsymbol{\alpha}}\|_{\ell^p} < \infty$ and
$\|{\boldsymbol{\alpha}}\|_{\ell^1} < 1$.
\end{itemize}
\end{lemma}
\begin{theorem} [General case] \label{thm:s=1} Let
Assumption \ref{ass:Ass1} hold and assume that
${\boldsymbol{\varrho}}=(\varrho_j)_{j\in {\mathbb N}}\in [0,\infty)^\infty$
is a sequence satisfying
$(\varrho_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N})$ for some
$0 < q < \infty$. Assume that, for each
${\boldsymbol{\nu}}\in {\mathcal F}$, there exists a sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}}= (\rho_{{\boldsymbol{\nu}},j})_{j \in {\mathbb N}}\in
[0,\infty)^\infty$ such that
$\operatorname{supp}({\boldsymbol{\nu}})\subseteq \operatorname{supp}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$,
\begin{equation} \label{assumption: theorem 3.1}
\sup_{{\boldsymbol{\nu}}\in {\mathcal F}} \Bigg\| \sum_{j\in
{\mathbb N}}\rho_{{\boldsymbol{\nu}},j}|\psi_j|\Bigg\|_{L^\infty}\leq
\kappa <\frac{\pi}{2}, \qquad \text{and} \qquad
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}}
<\infty
\end{equation}
with $r\in {\mathbb N}$, $r>2/q$ .
Then
\begin{equation} \label{eq:beta-u} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{V}^2 <\infty
\end{equation}
and, furthermore,
$(\|u_{\boldsymbol{\nu}}\|_{V})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{theorem}
\begin{proof}
Due to Lemma~\ref{lem:equal}, in order to prove
\eqref{eq:beta-u} we show that the right-hand side in
\eqref{eq:equal-V} is finite. By Proposition
\ref{prop:Meas1} Assumption \ref{ass:Ass1} implies
that $b({\boldsymbol{y}})$ belongs to $L^\infty({\domain})$ for
$\gamma$-a.~e. ${\boldsymbol{y}}\in U$ and
$\mathbb{E}(\exp(k\|b({\boldsymbol{y}})\|_{L^\infty}))$ is finite
for all $k\in [0,\infty)$.
For ${\boldsymbol{y}}\in U$ such that $b({\boldsymbol{y}})\in L^\infty({\domain})$ and
${\boldsymbol{\nu}} \in {\mathcal F}$ with ${\mathfrak u }=\operatorname{supp}({\boldsymbol{\nu}})$, the solution
$u$ of \eqref{SPDE} is holomorphic in
$\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$, see Proposition
\ref{prop:holoh1}. This, \eqref{assumption: theorem
3.1} and Lemma \ref{lem:estV} yield that
\begin{equation*}
\begin{split}
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!} \int_U\|
\partial^{\boldsymbol{\nu}} u({\boldsymbol{y}})\|_V^2\rd\gamma({\boldsymbol{y}}) & \leq
C_0^2 \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}}
\mathbb E \left( \exp\big( 2 \|b({\boldsymbol{y}})\|_{L^\infty }
\big) \right) <\infty.
\end{split}
\end{equation*}
Since $r>\frac{2}{q}$ and
$ (\varrho_j^{-1})_{j \in {\mathbb N}} \in \ell^q({\mathbb N})$, by
Lemma \ref{lem:beta-summability} the sequence
$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}))_{{\boldsymbol{\nu}}\in {\mathcal F}}$ satisfies the
inequality \eqref{ineq: beta}. Hence, by H\"older's
inequality we get that
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in{\mathcal F}}\| u_{\boldsymbol{\nu}}\|_V^p
\leq
\Bigg( \sum_{{\boldsymbol{\nu}}\in {\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_V^2\Bigg)^{p/2}
\Bigg(\sum_{{\boldsymbol{\nu}}\in {\mathcal F}} \beta_{{\boldsymbol{\nu}}}({\boldsymbol{\varrho}})^{-q/2} \Bigg)^{1-p/2}
<\infty\,.
\end{equation*}
\end{proof}
\begin{corollary}[The case of global
supports] \label{cor:global} Assume that there exists
a sequence of positive numbers
${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$ such that
$$
\big(\lambda_j \| \psi_j \|_{L^\infty}\big)_{j\in {\mathbb N}} \in
\ell^1({\mathbb N}) \ \ \mbox{and} \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in
\ell^q({\mathbb N}),$$ for some $0 < q < \infty$. Then we have
$(\|u_{\boldsymbol{\nu}}\|_{V})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\begin{proof}
Let ${\boldsymbol{\nu}}\in {\mathcal F}$. We define the sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}} = (\rho_{{\boldsymbol{\nu}},j})_{j \in {\mathbb N}}$ by
$\rho_{{\boldsymbol{\nu}},j} := \frac{ {\boldsymbol{\nu}}_j }{|{\boldsymbol{\nu}}|\|\psi_j
\|_{L^\infty}} $ for $j\in \operatorname{supp}({\boldsymbol{\nu}})$ and
$\rho_{{\boldsymbol{\nu}},j}=0$ if $j\not\in \operatorname{supp}({\boldsymbol{\nu}})$ and choose
${\boldsymbol{\varrho}}=\tau{\boldsymbol{\lambda}}$, $\tau$ is an appropriate positive
constant. It is obvious that
\begin{equation*}
\sup_{{\boldsymbol{\nu}}\in {\mathcal F}}
\Bigg\|
\sum_{j\in {\mathbb N}}\rho_{{\boldsymbol{\nu}},j}\big| \psi_j\big|\Bigg\|_{L^\infty}\leq 1.
\end{equation*}
We first show that Assumption \ref{ass:Ass1} is satisfied for
the sequence ${\boldsymbol{\lambda}}'= (\lambda'_j)_{j \in {\mathbb N}}$ with
$\lambda_j':= \lambda_j^{1/2}$ by a similar argument as in
\cite[Remark 2.5]{BCDM}. From the assumption
$ (\lambda_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N})$ we derive that
up to a nondecreasing rearrangement,
$\lambda'_j \geq C j^{1/(2q)}$ for some $C > 0$. Therefore,
$\big(\exp(-{\lambda'}_j^2)\big)_{j\in {\mathbb N}}\in \ell^1({\mathbb N})$.
The convergence in $L^\infty({\domain})$ of
$\sum_{j\in {\mathbb N}}\lambda'_j|\psi_j|$ can be proved as follows.
\begin{equation*}
\Bigg\|\sum_{j\in {\mathbb N}}\lambda'_j|\psi_j|\Bigg\|_{L^\infty}
\le \sup_{j \in {\mathbb N}} \lambda_j^{-1/2}\sum_{j\in {\mathbb N}}\lambda_j\|\psi_j\|_{L^\infty}
< \infty.
\end{equation*}
With $r>2/q$ we have
\begin{equation} \label{sum-estimate}
\begin{split}
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} & \leq
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{
|{\boldsymbol{\nu}}|^{2|{\boldsymbol{\nu}}|}}{{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}}\prod_{j\in
\operatorname{supp}({\boldsymbol{\nu}})}\big(\tau\sqrt{r!}\lambda_j \| \psi_j
\|_{L^\infty}\big) ^{2\nu_j}
\\
& \leq \Bigg( \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{
|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}}{{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}}}\prod_{j\in
\operatorname{supp}({\boldsymbol{\nu}})}\big(\tau\sqrt{r!}\lambda_j \| \psi_j
\|_{L^\infty}\big) ^{\nu_j}\Bigg)^{2}
\\
& \leq \Bigg( \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{
|{\boldsymbol{\nu}}|!}{{\boldsymbol{\nu}}!}\prod_{j\in \operatorname{supp}({\boldsymbol{\nu}})}\big(e
\tau\sqrt{r!}\lambda_j \| \psi_j \|_{L^\infty}\big)
^{\nu_j}\Bigg)^{2}
\end{split}
\end{equation}
In the last step we used the inequality
$$
\frac{ |{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}}{{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}}} \le
\frac{e^{|{\boldsymbol{\nu}}|} |{\boldsymbol{\nu}}|!}{{\boldsymbol{\nu}}!},
$$
which is immediately derived from the inequalities
$m! \le m^m \le e^m m!$. Since
$ \big(\tau\sqrt{r!}\lambda_j \| \psi_j
\|_{L^\infty}\big)_{j\in {\mathbb N}} \in \ell^1({\mathbb N}) $, we
can choose a positive number $\tau$ so that
$$\big\| \big(e\tau\sqrt{r!}\lambda_j \|\psi_j \|_{L^\infty}\big)_{j\in {\mathbb N}}\big\|_{\ell^1}<1.$$
This implies by Lemma \ref{lem:alpha-summability}(ii)
that the last sum in \eqref{sum-estimate} is finite.
Applying Theorem \ref{thm:s=1} the desired result
follows.
\end{proof}
\begin{corollary}[The case of disjoint supports]
\label{cor:local}
Assuming $\psi_j\in L^\infty({\domain})$ for all $j\in {\mathbb N}$
with disjoint supports and, furthermore, that there
exists a sequence of positive numbers
${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$ such that
$$ \big(\lambda_j \| \psi_j \|_{L^\infty}\big)_{j\in {\mathbb N}} \in \ell^2({\mathbb N}) \ \
{\rm and} \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N}),$$
for some $0 < q < \infty.$ Then
$(\|u_{\boldsymbol{\nu}}\|_{V})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\begin{proof}
Fix ${\boldsymbol{\nu}}\in {\mathcal F}$, arbitrary. For this ${\boldsymbol{\nu}}$ we define the
sequence ${\boldsymbol{\rho}}_{\boldsymbol{\nu}} = (\rho_j)_{j \in {\mathbb N}}$ by
$\rho_j := \frac{1 }{ \| \psi_j \|_{L^\infty}}$ for $j\in {\mathbb N}$
and ${\boldsymbol{\varrho}}=\tau{\boldsymbol{\lambda}}$, where a positive number $\tau$
will be chosen later on. It is clear that
\begin{equation*}
\Bigg\|
\sum_{j\in {\mathbb N}}\rho_j|\psi_j|\Bigg\|_{L^\infty}\leq 1.
\end{equation*}
Since $ \big(\lambda_j \rho_j^{-1}\big)_{j\in {\mathbb N}} \in
\ell^2({\mathbb N}) $ and
$(\lambda_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N})$, by H\"older's
inequality we get
$(\rho_j^{-1})_{j\in {\mathbb N}}\in \ell^{q_0}({\mathbb N})$ with
$\frac{1}{q_0}=\frac{1}{2}+\frac{1}{q}$. Hence, similarly to
the proof of Corollary \ref {cor:global}, we can show that
Assumption~\ref{ass:Ass1} holds for the sequence
${\boldsymbol{\lambda}}'= (\lambda'_j)_{j \in {\mathbb N}}$ with
$\lambda_j':= \lambda_j^{1/2}$. In addition, with $r>2/q$ we
have by Lemma \ref{lem:alpha-summability}(i)
\begin{equation*}
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}}
\leq
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \Bigg(\prod_{j\in \operatorname{supp}({\boldsymbol{\nu}})}\big(\tau\sqrt{r!}\lambda_j\|\psi_j
\|_{L^\infty}\big)^{2\nu_j} \Bigg) < \infty,
\end{equation*}
since by the condition
$ \big(\tau\sqrt{r!}\lambda_j \|\psi_j
\|_{L^\infty}\big)_{j\in {\mathbb N}} \in \ell^2({\mathbb N})$ a positive
number $\tau$ can be chosen so that
$\sup_{j\in {\mathbb N}}(\tau\sqrt{r!} \lambda_j \| \psi_j
\|_{L^\infty})<1$. Finally, we apply Theorem \ref{thm:s=1} to
obtain the desired results.
\end{proof}
\begin{remark}
We comment on the situation when there exists
${\boldsymbol{\rho}}= (\rho_j)_{j \in {\mathbb N}}\in (0,\infty)^\infty$ such that
\begin{equation*}
\Bigg\|\sum_{j \in {\mathbb N}} \rho_j |\psi_j | \Bigg\|_{L^\infty} =\kappa <\frac{\pi}{2}
\end{equation*}
and $ (\rho_j^{-1})_{j \in {\mathbb N}}\in \ell^{q_0}({\mathbb N})$ for some
$0 < q_0 < \infty$ as given in \cite[Theorem 1.2]{BCDM}. We choose
${\boldsymbol{\varrho}}=(\varrho_j)_{j \in {\mathbb N}}$ by
$$\varrho_j=\rho_j^{1-q_0/2}\frac{1}{\sqrt{r!} \norm[\ell^{q_0}]{ (\rho_j^{-1})_{j \in {\mathbb N}}}^{q_0/2}}$$
and ${\boldsymbol{\rho}}_{\boldsymbol{\nu}} = (\rho_j)_{j \in {\mathbb N}}$. Then we obtain
$(\varrho_j^{-1})_{j\in {\mathbb N}}\in \ell^{q_0/(1-q_0/2)}({\mathbb N})$ and
$$
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} =
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} {\boldsymbol{\nu}}! \prod_{j\in
\operatorname{supp}{\boldsymbol{\nu}}} \Bigg(\frac{\rho_j^{-q_0}}{r!\norm[\ell^{q_0}]{
(\rho_j^{-1})_{j \in {\mathbb N}}}^{q_0} }\Bigg)^{\nu_j} < \infty
.
$$
This implies $(\|u_{\boldsymbol{\nu}}\|_{V})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$
with $p=q_0$.
\end{remark}
\begin{remark}
The summability property
$(\|u_{\boldsymbol{\nu}}\|_{V})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ proven in Theorem
\ref{thm:s=1}, has been used in establishing the convergence rate of
the best $n$-term approximation of the solution $u$ to the
parametric elliptic PDE \eqref{SPDE} \cite{BCDM}. However, such a
property cannot used for estimating convergence rates of
high-dimensional deterministic numerical approximation
\emph{constructive} schemes such as single-level and multi-level
versions of anisotropic sparse-grid Hermite-Smolyak interpolation
and quadrature in Sections \ref{sec:StochColl} and
\ref{sec:Approx}. In the last situation, the following weighted
$\ell^2$-summability presented in Theorem \ref{thm:s=1}, is
efficiently applied:
\begin{equation*} \label{weighted-summability} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{V}^2 <\infty, \quad
(\varrho_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N}), \ (0 < q < \infty).
\end{equation*}
This summability property has been employed in
\cite{ChenlogNQuad2018,dD21,ErnstSprgkTam18} for similar
problems. This remark still holds true for the summability problems
considered in Sections \ref{sec:HsReg} and \ref{sec:KondrReg}.
\end{remark}
\subsection{Parametric $H^s({\mathrm D})$-analyticity and sparsity}
\label{sec:HsReg}
Whereas the previous results were, in principle, already known from
the real-variable analyses in \cite{BTNT,BCDM}, in this and the
subsequent sections, we prove via analytic continuation the weighted
$\ell^2$-summability (sparsity) and $\ell^p$-summability of the
Wiener-Hermite PC expansion coefficients of the parametric solutions
of \eqref{SPDE} with log-Gaussian coefficient $a({\boldsymbol{y}}) = \exp(b({\boldsymbol{y}}))$
when the Wiener-Hermite PC expansion coefficients of the parametric
solution family $\{ u({\boldsymbol{y}}): {\boldsymbol{y}}\in U\}$ are measured in higher Sobolev
norms. In Section \ref{sec:KondrReg} we shall establish corresponding
results when the physical domain ${\mathrm D}$ is a plane Lipschitz
polygon whose sides are analytic arcs.
\subsubsection{$H^s({\mathrm D})$-analyticity}
\label{sec:HsAnalyt}
As $H^s({\mathrm D})$ regularity in ${\mathrm D}$ is relevant in particular in
conjunction with Galerkin discretization in ${\mathrm D}$ by continuous,
piecewise polynomial, Lagrangian FEM, we recall the elementary
regularity results from Section \ref{S:Reg}. We recall the Sobolev
spaces $ H^s({\mathrm D})$, and $W^s_\infty({\mathrm D})$ of functions $v$ on
${\mathrm D}$ for $s \in {\mathbb Z}_+$, equipped with the respective norms
\[
\|v\|_{H^s} := \ \sum_{{\boldsymbol{k}} \in {\mathbb Z}^d_+: |{\boldsymbol{k}}| \le s} \|D^{\boldsymbol{k}}
v\|_{L^2}, \qquad \|v\|_{W^s_\infty} := \ \sum_{{\boldsymbol{k}} \in {\mathbb Z}^d_+: |{\boldsymbol{k}}| \le
s}\|D^{\boldsymbol{k}} v\|_{L^\infty}.
\]
With these definitions $H^0({\mathrm D}) = L^2({\domain})$ and
$W^0_\infty({\domain}) = L^\infty({\domain})$. We recall from Section \ref{S:Reg}
that we identify $L^2({\domain})$ with its own dual, so that the space
$H^{-1}({\domain})$ is defined as the dual of $H^1_0({\domain})$ with respect to the
pivot space $L^2({\domain})$.
\begin{lemma} \label{lemma[regularity]} Let $s \in {\mathbb N}$ and ${\mathrm D}$
be a bounded domain in ${\mathbb R}^d$ with either $C^\infty$-boundary or
with convex $C^{s-1}$-boundary. Assume that there holds the
ellipticity condition \eqref{eq:PDEellinC},
$a \in W^{s-1}_\infty({\domain})$ and $f \in H^{s - 2}({\domain})$. Then the
solution $u$ of \eqref{PDE} belongs to $H^s({\mathrm D})$ and there
holds
\begin{equation} \label{|u|_{Hn}<1} \|u\|_{H^s} \ \le \
\begin{cases}
\frac{\|f\|_{H^{-1}}}{\rho(a)} & \ s = 1,
\\[1ex]
\frac{C_{d,s}}{\rho(a)} \big(\|f\|_{H^{s - 2}} + \|a\|_{W^{s -
1}_\infty}\|u\|_ {H^{s - 1}}\big) &\ s > 1,
\end{cases}
\end{equation}
with $C_{d,s}$ depending on $d,s$.
\end{lemma}
\begin{proof}
Defining, for $s\in {\mathbb N}$, $H^s_0({\domain}):= (H^s \cap H^1_0)({\domain})$, since
${\mathrm D}$ is a bounded domain in ${\mathbb R}^d$ with either
$C^\infty$-boundary or convex $C^{s-1}$-boundary, we have the
following norm equivalence
\begin{equation} \label{H-norm-equivalence} \|v\|_{H^s} \ \asymp \
\begin{cases}
\|v\|_{H^1_0 }, & s = 1,\\[1ex]
\|\Delta v\|_{H^{s-2} }, & s > 1,
\end{cases}
\qquad \forall v \in H^s_0,
\end{equation}
see \cite[Theorem 2.5.1.1]{Gr}. The lemma for the case $s =1$ and
$s=2$ is given in \eqref{V-estimate} and \eqref{W-estimate}. We
prove the case $s > 2$ by induction on $s$. Suppose that the
assertion holds true for all $s' < s$. We will prove it for $s$.
Let a ${\boldsymbol{k}} \in {\mathbb Z}^d_+$ with $|{\boldsymbol{k}}| = s - 2$ be given.
Differentiating both sides of \eqref{[Delta u=]} and applying the
Leibniz rule of multivariate differentiation we obtain
\begin{equation} \nonumber - \sum_{0 \le {\boldsymbol{k}}' \le {\boldsymbol{k}}}
\binom{{\boldsymbol{k}}}{{\boldsymbol{k}}'}D^{{\boldsymbol{k}}'} a D^{{\boldsymbol{k}}-{\boldsymbol{k}}'} \big(\Delta u\big) \ = \
D^{\boldsymbol{k}} f + \sum_{0 \le {\boldsymbol{k}}' \le {\boldsymbol{k}}} \binom{{\boldsymbol{k}}}{{\boldsymbol{k}}'} \big( \nabla
D^{{\boldsymbol{k}}'} a , \nabla D^{{\boldsymbol{k}}-{\boldsymbol{k}}'} u \big),
\end{equation}
see also \cite[Lemma 4.3]{BCDS}. Hence,
\begin{equation*} \label{derivative} - a \,D^{\boldsymbol{k}}\Delta u \, = \,
D^{\boldsymbol{k}} f + \sum_{0 \le {\boldsymbol{k}}' \le {\boldsymbol{k}}} \binom{{\boldsymbol{k}}}{{\boldsymbol{k}}'} \big( \nabla
D^{{\boldsymbol{k}}'} a , \nabla D^{{\boldsymbol{k}}-{\boldsymbol{k}}'} u \big) + \sum_{0 \le {\boldsymbol{k}}' \le
{\boldsymbol{k}}, \, {\boldsymbol{k}}' \not=0} \binom{{\boldsymbol{k}}}{{\boldsymbol{k}}'}D^{{\boldsymbol{k}}'} a D^{{\boldsymbol{k}}-{\boldsymbol{k}}'}
\Delta u.
\end{equation*}
Taking the $L^2$-norm of both sides, by the ellipticity condition
\eqref{PDE-epllipticity} we derive the inequality
\begin{equation} \nonumber \rho(a) \, \| \Delta u \|_{H^{s-2}} \ \le
\ C'_{d,s}\,\big( \|f\|_{H^{s-2}} + \|a\|_{W^{s -
1}_\infty}\,\|u\|_{H^{s-1}} + \|a\|_{W^{s -
2}_\infty}\,\|\Delta u\|_{H^{s-3}}\big)
\end{equation}
which yields \eqref{|u|_{Hn}<1} due to \eqref{H-norm-equivalence}
and the inequality
\[
\|a\|_{W^{s - 2}_\infty}\,\|\Delta u\|_{H^{s-3}} \le \|a\|_{W^{s -
1}_\infty}\,\|u\|_{H^{s-1}},
\] where $C'_{d,s}$ is a constant depending on $d, s$ only. By
induction, this proves that $u$ belongs to $H^s$.
\end{proof}
\begin{corollary} \label{cor:regularity} Let $s \in {\mathbb N}$ and ${\mathrm D}$
be a bounded domain in ${\mathbb R}^d$ with either $C^\infty$-boundary or
convex $C^{s-1}$-boundary. Assume that there holds the ellipticity
condition \eqref{PDE-epllipticity}, $a \in W^{s-1}_\infty({\domain})$ and
$f \in H^{s - 2}({\domain})$. Then the solution $u$ of \eqref{PDE} belongs
to $H^s({\domain})$ and there holds the estimate
\begin{equation*} \label{|u|_{Hn}<} \|u\|_{H^s} \ \le \
\frac{\|f\|_{H^{s - 2}}}{\rho(a)}
\begin{cases}
1, & \ s = 1, \\[1ex]
C_{d,s}\Big(1 + \frac{\|a\|_{W^{s -
1}_\infty}}{{\rho(a)}}\Big)^{s-1}, & \ s > 1,
\end{cases}
\end{equation*}
where $C_{d,s}$ is a constant depending on $d, s$ only.
\end{corollary}
We need the following lemma.
\begin{lemma}\label{a-nu}
Let $s\in {\mathbb N}$ and assume that $b({\boldsymbol{y}})$ belongs to
$ W^{s}_\infty({\domain})$. Then we have
\begin{equation*}
\|a({\boldsymbol{y}})\|_{W^{s}_\infty}
\leq
C\|a({\boldsymbol{y}})\|_{L^\infty} \big(1+ \| b({\boldsymbol{y}})\|_{W^{s}_\infty}\big)^{s}\,,
\end{equation*}
where the constant $C$ depends on $s$ and $m$ but is independent of
${\boldsymbol{y}}$.
\end{lemma}
\begin{proof}
For ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_d)\in {\mathbb Z}_+^d$ with
$1\leq |{\boldsymbol{\alpha}}|\leq s$, we observe that for $\alpha_j>0$ the
product rule implies
\begin{equation}\label{eq:Leibniz}
D^{\boldsymbol{\alpha}} a({\boldsymbol{y}}) = D^{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j} \big[a({\boldsymbol{y}}) D^{{\boldsymbol{e}}_j}b({\boldsymbol{y}}) \big]
= \sum_{0\leq {\boldsymbol{\gamma}}
\leq {\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}\binom{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}{{\boldsymbol{\gamma}}}
D^{{\boldsymbol{\alpha}}-{\boldsymbol{\gamma}}} b({\boldsymbol{y}}) D^{{\boldsymbol{\gamma}} }a({\boldsymbol{y}}) \,.
\end{equation}
Taking norms on both sides, we can estimate
\begin{equation*}
\begin{split}
\| D^{\boldsymbol{\alpha}} a({\boldsymbol{y}})\|_{L^\infty} & =\big\| D^{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}
\big[a({\boldsymbol{y}}) D^{{\boldsymbol{e}}_j}b({\boldsymbol{y}}) \big] \big\|_{L^\infty}
\\
&\leq \sum_{0\leq {\boldsymbol{\gamma}} \leq
{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}\binom{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}{{\boldsymbol{\gamma}}} \|
D^{{\boldsymbol{\alpha}}-{\boldsymbol{\gamma}}} b({\boldsymbol{y}}) \|_{L^\infty} \| D^{{\boldsymbol{\gamma}}
}a({\boldsymbol{y}})\|_{L^\infty}
\\
& \leq C \Bigg( \sum_{0\leq {\boldsymbol{\gamma}} \leq {\boldsymbol{\alpha}}-{\boldsymbol{e}}_j} \|
D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}\Bigg) \Bigg( \sum_{|{\boldsymbol{k}}|\leq
s}\|D^{\boldsymbol{k}} b({\boldsymbol{y}})\|_{L^\infty}\Bigg) \,.
\end{split}
\end{equation*}
Here ${\boldsymbol{e}}_j = (e_{j;i})_{i \in {\mathbb N}}\in {\mathcal F}$ is the multi-index with
$e_{j;j} = 1$ and $e_{j;i} = 0$ for $i \not=j$. Similarly, each
term $\| D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}$ with $|{\boldsymbol{\gamma}}|>0$ can be
estimated
\begin{equation*}
\| D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}\leq C \Bigg( \sum_{0\leq {\boldsymbol{\gamma}}'
\leq {\boldsymbol{\gamma}}-{\boldsymbol{e}}_j} \| D^{{\boldsymbol{\gamma}}'} a({\boldsymbol{y}}) \|_{L^\infty}\Bigg) \Bigg( \sum_{|{\boldsymbol{k}}|\leq s}\|D^{\boldsymbol{k}} b({\boldsymbol{y}})\|_{L^\infty}\Bigg) \,
\end{equation*}
if $\gamma_j>0$. This implies
\begin{equation*}
\begin{split}
\| D^{\boldsymbol{\alpha}} a({\boldsymbol{y}})\|_{L^\infty} \leq C \|a({\boldsymbol{y}})\|_{L^\infty}
\Bigg(1+ \sum_{|{\boldsymbol{k}}|\leq s}\|D^{\boldsymbol{k}}
b({\boldsymbol{y}})\|_{L^\infty}\Bigg)^{|{\boldsymbol{\alpha}}|} \,,
\end{split}
\end{equation*}
for $1\leq |{\boldsymbol{\alpha}}|\leq s$. Summing up these terms with
$\|a({\boldsymbol{y}})\|_{L^\infty}$ we obtain the desired estimate.
\end{proof}
\begin{proposition}\label{prop3}
Let $s \in {\mathbb N}$ and ${\mathrm D}$ be a bounded domain in ${\mathbb R}^d$ with
either $C^\infty$-boundary or convex $C^{s-1}$-boundary. Assume
that \eqref{eq:leqkappa} holds and all the functions $\psi_j$ belong
to $ W^{s-1}_\infty({\domain})$. Let ${\mathfrak u } \subseteq \operatorname{supp}({\boldsymbol{\rho}})$ be a
finite set and let ${\boldsymbol{y}}_0=(y_{0,1},y_{0,2},\ldots) \in U$ be such
that $b({\boldsymbol{y}}_0)$ belongs to $ W^{s-1}_\infty({\domain})$. Then the solution
$u$ of \eqref{SPDE} is holomorphic in $\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}) $ as
a function in variables
${\boldsymbol{z}}_{{\mathfrak u }}=(z_j)_{j \in {\mathbb N}}\in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}}_0,{\boldsymbol{\rho}}) $
taking values in $H^s({\domain})$ where $z_j = y_{0,j}$ for
$j\not \in {\mathfrak u }$ held fixed\,.
\end{proposition}
\begin{proof} Let $\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$ be given in
\eqref{eq:SUN} and
${\boldsymbol{z}}_{{\mathfrak u }} = (y_j+{\rm i} \xi_j)_{j\in {\mathbb N}}\in \mathcal{S}_{{\mathfrak u }}
({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$ with
$(y_j+{\rm i} \xi_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$
Then we have from Corollary \ref{cor:regularity}
\begin{equation*}
\begin{split}
\|u({\boldsymbol{z}}_{{\mathfrak u }})\|_{H^s} & \leq C \rho(a({\boldsymbol{z}}_{{\mathfrak u }})) \Big(1 +
\rho(a({\boldsymbol{z}}_{{\mathfrak u }})) \|a({\boldsymbol{z}}_{{\mathfrak u }})\|_{W^{s-1}_\infty}
\Big)^{s-1} \,.
\end{split}
\end{equation*}
Using Lemma \ref{a-nu} we find
\begin{equation*}
\begin{split}
\|a({\boldsymbol{z}}_{{\mathfrak u }})\|_{W_\infty^{s-1}} & \leq C
\|a({\boldsymbol{z}}_{{\mathfrak u }})\|_{L^\infty}\big(1+ \|
b({\boldsymbol{z}}_{{\mathfrak u }})\|_{W_\infty^{s-1}}\big)^{s-1}
\\
& \leq C \|a({\boldsymbol{z}}_{{\mathfrak u }})\|_{L^\infty}\Bigg(1+\|
b({\boldsymbol{y}}_0)\|_{W_\infty^{s-1}} + \Bigg\| \sum_{j\in {\mathfrak u }
}(y_j-y_{0,j}
+{\rm i}\xi_j)\psi_j\Bigg\|_{W_\infty^{s-1}}\Bigg)^{s-1}
\\
& \leq C \|a({\boldsymbol{z}}_{{\mathfrak u }})\|_{L^\infty}\Bigg(1+\|
b({\boldsymbol{y}}_0)\|_{W_\infty^{s-1}} + \sum_{j\in {\mathfrak u }
}(N+\rho_j)\|\psi_j \|_{W_\infty^{s-1}}\Bigg)^{s-1}
\end{split}
\end{equation*}
and
\begin{equation} \label{eq:azu0} \|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} \leq
\exp\Bigg(\|b({\boldsymbol{y}}_0)\|_{L^\infty}+\Bigg\|\sum_{j\in {\mathfrak u }}
(y_j-y_{0,j}+{\rm i}\xi_j)\psi_j\Bigg\|_{L^\infty} \Bigg) <\infty.
\end{equation}
From this and \eqref{eq:R-1} we obtain
\begin{equation*}
\|u({\boldsymbol{z}}_{{\mathfrak u }})\|_{H^s} \leq C <\infty
\end{equation*}
which implies the map ${\boldsymbol{z}}_{\mathfrak u }\to u({\boldsymbol{z}}_{{\mathfrak u }})$ is holomorphic on
the set $\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$ by applying \cite[Lemma
2.2]{BLN}. For more details we refer the reader to \cite[Examples
1.2.38 and 1.2.39]{JZdiss}. Since $N$ is arbitrary we conclude that
the map ${\boldsymbol{z}}_{\mathfrak u }\to u({\boldsymbol{z}}_{{\mathfrak u }})$ is holomorphic on
$\mathcal{S}_{{\mathfrak u }} ({\boldsymbol{\rho}})$.
\end{proof}
\subsubsection{Summability of Wiener-Hermite PC expansion
coefficients}
\label{sec:Summ}
For summability of $H^s$-norms of Wiener-Hermite PC expansion
coefficients we need the following assumption.
\noindent
\begin{assumption}\label{ass:Ass2}
Let $s\in {\mathbb N}$. For every $j\in {\mathbb N}$,
$\psi_j \in W^{s-1}_\infty({\domain})$ and there exists a positive sequence
$(\lambda_j)_{j\in {\mathbb N}}$ such that
$\big(\exp(-\lambda_j^2)\big)_{j\in {\mathbb N}}\in \ell^1({\mathbb N})$ and the
series
$$\sum_{j\in {\mathbb N}}\lambda_j|D^{{\boldsymbol{\alpha}}}\psi_j|$$
converges in $L^\infty({\domain})$ for all ${\boldsymbol{\alpha}}\in {\mathbb N}_0^d$ with
$|{\boldsymbol{\alpha}}|\leq s-1$.
\end{assumption}
As a consequence of \cite[Theorem 2.2]{BCDM} we have the following
\begin{lemma}\label{lem:s-a.e}
Let Assumption \ref{ass:Ass2} hold. Then the set
$U_{s-1} := \{ {\boldsymbol{y}}\in U: b({\boldsymbol{y}})\in W_\infty^{s-1}({\mathrm D}) \}$ has
full measure, i.e., $\gamma(U_{s-1}) = 1$. Furthermore,
$\mathbb{E}(\exp(k\|b(\cdot)\|_{W_\infty^{s-1}}))$ is finite for all
$k\in [0,\infty).$
\end{lemma}
The $H^s$-analytic continuation of the parametric solutions
$\{u({\boldsymbol{y}}): {\boldsymbol{y}}\in U\}$ to $\mathcal{S}_{{\mathfrak u }} ({\boldsymbol{\rho}})$ leads to the
following result on parametric $H^s$-regularity.
\begin{lemma} \label{lemma:hs-regularity} Let ${\mathrm D} \subset {\mathbb R}^d$
be a bounded domain with either $C^\infty$-boundary or convex
$C^{s-1}$-boundary. Assume that for each ${\boldsymbol{\nu}}\in {\mathcal F}$, there
exists a sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}}= (\rho_{{\boldsymbol{\nu}},j})_{j \in {\mathbb N}}\in [0,\infty)^\infty$ such
that $\operatorname{supp}({\boldsymbol{\nu}})\subseteq \operatorname{supp}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$, and such that
\begin{equation*} \label{eq:Dkpsisum-lemma} \sup_{{\boldsymbol{\nu}}\in {\mathcal F}}
\sum_{|{\boldsymbol{\alpha}}|\leq s-1} \Bigg\| \sum_{j\in
{\mathbb N}}\rho_{{\boldsymbol{\nu}},j}|D^{{\boldsymbol{\alpha}}}\psi_j|\Bigg\|_{L^\infty} \leq
\kappa <\frac{\pi}{2}.
\end{equation*}
Then we have
\begin{equation} \label{ineq1-lemma} \|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_{H^s}
\leq C \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{\boldsymbol{\nu}}}
\exp\big(\|b({\boldsymbol{y}})\|_{L^\infty}\big) \Big\{1+
\exp(2\|b({\boldsymbol{y}})\|_{L^\infty}) \big(1+\|
b({\boldsymbol{y}})\|_{W_\infty^{s-1}}\big)^{s-1} \Big\}^{s-1},
\end{equation}
where $C$ is a constant depending on $\kappa$, $d$, $s$ only.
\end{lemma}
\begin{proof}
Let ${\boldsymbol{\nu}}\in {\mathcal F}$ with ${\mathfrak u }=\operatorname{supp}({\boldsymbol{\nu}})$ and ${\boldsymbol{y}}\in U$ such that
$b({\boldsymbol{y}})\in W_\infty^{s-1}({\domain})$. Let furthermore
${\mathcal C}_{{\boldsymbol{y}},{\mathfrak u }}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ and ${\mathcal C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ be given
as in \eqref{eq:C-rho} and \eqref{eq:C-rho-y}. Using Cauchy's
formula as in the proof of Lemma \ref{lem:estV} we obtain
\begin{equation} \label{ineq1-proof} \|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_{H^s}
\leq \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{\boldsymbol{\nu}}} \sup_{{\boldsymbol{z}}_{{\mathfrak u }}\in
C_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})} \|u({\boldsymbol{z}}_{{\mathfrak u }})\|_{H^s} \,.
\end{equation}
For ${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j\in {\mathbb N}} \in C_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ we can
write $z_j = y_j + \eta_j + {\rm i}\xi_j \in {\mathcal C}_{{\boldsymbol{y}},j}({\boldsymbol{\rho}}_{{\boldsymbol{\nu}}})$
with $|\eta_j | \le \rho_{{\boldsymbol{\nu}},j}$ and $|\xi_j| \le \rho_{{\boldsymbol{\nu}},j}$
for $j\in {\mathfrak u }$ and hence we get
\begin{equation*}
\begin{split}
\|D^{\boldsymbol{\alpha}} b({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} & = \Bigg\|
D^{\boldsymbol{\alpha}}\Big(b({\boldsymbol{y}})+\sum_{j\in {\mathfrak u }} ( \eta_j+{\rm i}\xi_j)\psi_j
\Big)\Bigg\|_{L^\infty}
\\
& \leq \|D^{\boldsymbol{\alpha}} b({\boldsymbol{y}})\|_{L^\infty} + \sqrt{2}\,\Bigg\| \sum_{j\in {\mathfrak u }} \rho_{{\boldsymbol{\nu}},j} |D^{\boldsymbol{\alpha}} \psi_j|\Bigg\|_{L^\infty}\\
&\leq \|D^{\boldsymbol{\alpha}} b({\boldsymbol{y}})\|_{L^\infty} +\kappa\sqrt{2}\,.
\end{split}
\end{equation*}
In addition we have
\begin{equation} \label{eq:R-11} \frac{1}{\rho(a( {\boldsymbol{z}}_{{\mathfrak u }}))} \leq
\frac{\exp(\| b({\boldsymbol{y}}+ \sum_{j\in {\mathfrak u }}\eta_j
\psi_j\|_{L^\infty})}{\cos(\|\sum_{j\in {\mathfrak u }}\xi_j
\psi_j\|_{L^\infty})} \leq \frac{ \exp\big( \kappa +
\|b({\boldsymbol{y}})\|_{L^\infty}\big)}{\cos\kappa}\,
\end{equation}
and
\begin{equation} \label{eq:azu} \|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} =
\Bigg\|\exp\Bigg(b({\boldsymbol{y}})+\sum_{j\in {\mathfrak u }} (
\eta_j+{\rm i}\xi_j)\psi_j\Bigg)\Bigg\|_{L^\infty} \leq
e^{\kappa\sqrt{2}} \exp(\|b({\boldsymbol{y}})\|_{L^\infty})\,.
\end{equation}
Consequently, we can bound
%
\begin{equation*}
\begin{split}
\|a({\boldsymbol{z}}_{\mathfrak u })\|_{W^{s-1}_\infty} & \leq C
\|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty}\big(1+\|
b({\boldsymbol{z}}_{{\mathfrak u }})\|_{W_\infty^{s-1}}\big)^{s-1}
\\
& \leq C\exp(\|b({\boldsymbol{y}})\|_{L^\infty}) \big(1+\|
b({\boldsymbol{y}})\|_{W_\infty^{s-1}}\big)^{s-1}\,.
\end{split}
\end{equation*}
Now Corollary \ref{cor:regularity} implies the inequality
\begin{equation} \label{ineq:u(z_u)} \sup_{{\boldsymbol{z}}_{{\mathfrak u }}\in
C_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})} \|u({\boldsymbol{z}}_{\mathfrak u })\|_{H^s} \leq C
\exp\big(\|b({\boldsymbol{y}})\|_{L^\infty}\big) \Big\{1+
\exp(2\|b({\boldsymbol{y}})\|_{L^\infty}) \big(1+\|
b({\boldsymbol{y}})\|_{W_\infty^{s-1}}\big)^{s-1} \Big\}^{s-1},
\end{equation}
which together with \eqref{ineq1-proof} proves the lemma.
\end{proof}
\begin{theorem}[General case] \label{thm:hs-sum} Let $s,r \in {\mathbb N}$ and
${\mathrm D} \subset {\mathbb R}^d$ denote a bounded domain with either
$C^\infty$-boundary or convex $C^{s-1}$-boundary. Let further
Assumption \ref{ass:Ass2} hold, assume that $f \in H^{s - 2}({\domain})$,
and assume given a sequence
${\boldsymbol{\varrho}}= (\varrho_j)_{j \in {\mathbb N}}\subset (0,\infty)^\infty$ that
satisfies $(\varrho_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N})$ for some
$0 < q < \infty$. Assume in addition that, for each ${\boldsymbol{\nu}}\in {\mathcal F}$,
there exists a sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}}= (\rho_{{\boldsymbol{\nu}},j})_{j \in {\mathbb N}}\in [0,\infty)^\infty$ such
that $\operatorname{supp}({\boldsymbol{\nu}})\subseteq \operatorname{supp}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$, and such that, with
$r>2/q$,
%
\begin{equation*} \label{eq:Dkpsisum} \sup_{{\boldsymbol{\nu}}\in {\mathcal F}}
\sum_{|{\boldsymbol{\alpha}}|\leq s-1} \Bigg\| \sum_{j\in
{\mathbb N}}\rho_{{\boldsymbol{\nu}},j}|D^{{\boldsymbol{\alpha}}}\psi_j|\Bigg\|_{L^\infty} \leq
\kappa <\frac{\pi}{2}, \quad \text{and}\quad
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} <\infty.
\end{equation*}
Then there holds, with $\beta_{\boldsymbol{\nu}}(r, {\boldsymbol{\varrho}})$ as in \eqref{beta},
%
\begin{equation} \label{ineq:sum<infty} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{H^s}^2 <\infty
\end{equation}
%
and furthermore, $(\|u_{\boldsymbol{\nu}}\|_{H^s})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$
with $\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{theorem}
\begin{proof}
Arguing as in the proof of \cite[ Theorem 3.3]{BCDM} we obtain that
for any $r\in {\mathbb N}$ there holds following generalization of the
Parseval-type identity
\begin{equation} \label{eq:equal-H^s} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{H^s}^2 =
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!}
\int_U\| \partial^{\boldsymbol{\nu}} u({\boldsymbol{y}})\|_{H^s}^2\rd\gamma({\boldsymbol{y}})\,.
\end{equation}
To prove \eqref{ineq:sum<infty} we show that the right-hand side in
\eqref{eq:equal-H^s} is finite.
By \eqref{ineq:u(z_u)}, Lemma \ref{lem:s-a.e} and H\"older's
inequality we derive that
\[
\int_U \bigg( \sup_{{\boldsymbol{z}}_{{\mathfrak u }}\in C_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})}
\|u({\boldsymbol{z}}_{{\mathfrak u }})\|_{H^s} \bigg)^2 \rd\gamma({\boldsymbol{y}}) \leq C
\]
and in particular, $\mathbb{E}(\|u({\boldsymbol{y}})\|_{H^s}^k)$ is finite for
all $k\in [0,\infty)$. Now Lemma \ref{lemma:hs-regularity} and our
assumption give
\begin{equation*} \label{bound-d} \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq
r} \frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!} \int_U\| \partial^{\boldsymbol{\nu}}
u({\boldsymbol{y}})\|_{H^s}^2\rd\gamma({\boldsymbol{y}}) <\infty,
\end{equation*}
where $C$ is the constant in \eqref{ineq1-lemma}. Indeed,
\begin{equation} \nonumber
\begin{split}
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!} \int_U\| \partial^{\boldsymbol{\nu}}
u({\boldsymbol{y}})\|_{H^s}^2\rd\gamma({\boldsymbol{y}}) &\leq
C^2\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}}\int_U
\rd\gamma({\boldsymbol{y}})
\\
& \leq C^2 \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} <\infty,
\end{split}
\end{equation}
where $C$ is the constant in \eqref{ineq1-lemma}. This and
\eqref{eq:equal-H^s} prove \eqref{ineq:sum<infty}. The assertion
$(\|u_{\boldsymbol{\nu}}\|_{H^s})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ can be proved in
the same way as in the proof of Theorem \ref{thm:s=1}.
\end{proof}
Similarly to Corollaries \ref{cor:global} and \ref{cor:local} from
Theorem~\ref{thm:hs-sum} we obtain
\begin{corollary}[The case of global supports]
Let $s \in {\mathbb N}$ and ${\mathrm D} \subset {\mathbb R}^d$ denote a bounded domain
with either $C^\infty$-boundary or convex $C^{s-1}$-boundary.
Assume that for all $j\in {\mathbb N}$ holds $\psi_j\in W^{s-1}_\infty({\domain})$,
and that $f\in H^{s-2}({\domain})$. Assume further that there exists a
sequence of positive numbers ${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$
such that
$$
\big(\lambda_j \| \psi_j \|_{W_\infty^{s-1}}\big)_{j\in {\mathbb N}}
\in \ell^1({\mathbb N}) \ \ and \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in
\ell^q({\mathbb N}),$$ for some $0 < q < \infty$. Then we have
$(\|u_{\boldsymbol{\nu}}\|_{H^s})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\begin{corollary}[The case of disjoint supports]
Let $s \in {\mathbb N}$ and ${\mathrm D} \subset {\mathbb R}^d$ denote a bounded domain
with either $C^\infty$-boundary or convex $C^{s-1}$-boundary.
Assume that $f\in H^{s-2}({\domain})$ and for all $j\in {\mathbb N}$ holds
$\psi_j\in W^{s-1}_\infty({\domain})$ with disjoint supports. Assume
further that there exists a sequence of positive numbers
${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$ such that
$$
\big(\lambda_j \| \psi_j
\|_{W_\infty^{s-1}}\big)_{j\in {\mathbb N}} \in \ell^2({\mathbb N}) \
\ and \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in
\ell^q({\mathbb N}),$$ for some $0 < q < \infty$. Then
$(\|u_{\boldsymbol{\nu}}\|_{H^s})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\subsection{Parametric Kondrat'ev analyticity and
sparsity}
\label{sec:KondrReg}
In the previous section, we investigated the weighted
$\ell^2$-summability and $\ell^p$-summability of
Wiener-Hermite PC expansion coefficients of parametric
solutions measured in the standard Sobolev spaces
$H^s({\mathrm D})$. We assumed that ${\mathrm D}\subset
{\mathbb R}^d$
with boundary $\partial{\mathrm D}$ of sufficient smoothness
(depending on $s$). In this section we consider in
space dimension $d=2$ the case when the physical domain
${\mathrm D}$ is a polygonal domain. In such domains,
elliptic regularity shift results and shift theorems in
${\mathrm D}$ hold in so-called \emph{corner-weighted
Sobolev spaces} going back to V.A. Kondrat'ev. We
refer to \cite{Gr,MazRoss2010} and the references there
for an extensive survey.
To state corresponding results for the log-Gaussian
parametric elliptic problems, we first review
definitions of the weighted Sobolev spaces of Kondrat'ev
type and results from \cite{BLN} on the holomorphy of
parametric solutions in weighted Kondrat'ev spaces in
polygonal domains ${\mathrm D}$. Then, we establish
summability results of coefficient sequences of
Wiener-Hermite PC expansions of the parametric solutions
in weighted Kondrat'ev spaces. FE approximation results
for Wiener-Hermite PC expansion coefficient functions
which are in these spaces were provided in Section
\ref{S:FEM}.
\subsubsection{Parametric
$K^s_{\varkappa}({\domain})$-holomorphy}
\label{sec:KondrAn}
We recall the weighted Kondrat'ev spaces in a bounded
polygonal domain ${\mathrm D}$ introduced in Section
\ref{S:FncSpc}: for $s\in {\mathbb Z}_+$ and $\varkappa\in {\mathbb R}$,
\begin{equation*}
{\mathcal K}^s_\varkappa({\mathrm D})
=
\big\{
u: {\mathrm D} \to {\mathbb C}:
\ r_{\mathrm D}^{|{\boldsymbol{\alpha}}|-\varkappa}D^{\boldsymbol{\alpha}} u\in L^2({\mathrm D}),
|{\boldsymbol{\alpha}}|\leq s \big\}
\end{equation*}
and
\begin{equation*}
{\mathcal W}^s_\infty({\mathrm D})
=
\big\{u: {\mathrm D}\to {\mathbb C}: \ r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{\boldsymbol{\alpha}} u\in L^\infty({\mathrm D}),\
|{\boldsymbol{\alpha}}|\leq s \big\}.
\end{equation*}
The weighted Sobolev norms in these spaces are given in
Section \ref{S:FncSpc}.
\begin{lemma}\label{kon-lem-1} Let $s\in {\mathbb Z}_+$.
Assume that ${\boldsymbol{y}}\in U$ is such that
$b({\boldsymbol{y}}) \in {\mathcal W}^{s}_\infty({\domain})$. Then
\begin{equation*}
\|a({\boldsymbol{y}})\|_{{\mathcal W}^{s}_\infty}
\leq
C\|a({\boldsymbol{y}})\|_{L^\infty} \big(1+ \| b({\boldsymbol{y}})\|_{{\mathcal W}^s_\infty}\big)^{s}\,,
\end{equation*}
where the constant $C$ depends on $s$ and $m$.
\end{lemma}
\begin{proof}
The proof proceeds along the lines of the proof of
Lemma \ref{a-nu}. Let
${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_d)\in {\mathbb Z}_+^d$ with
$1\leq |{\boldsymbol{\alpha}}|\leq s$. Assuming that $\alpha_j>0$
we have \eqref{eq:Leibniz}. We apply corner-weighted
norms to both sides of \eqref{eq:Leibniz}. This
implies
\begin{equation*}
\begin{split}
\| r_{\mathrm D}^{|{\boldsymbol{\alpha}}|} D^{\boldsymbol{\alpha}}
a({\boldsymbol{y}})\|_{L^\infty} & =\big\| D^{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}
\big[a({\boldsymbol{y}}) D^{{\boldsymbol{e}}_j}b({\boldsymbol{y}}) \big]
\big\|_{L^\infty}
\\
&\leq \sum_{0\leq {\boldsymbol{\gamma}} \leq
{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}\binom{{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j}{{\boldsymbol{\gamma}}}
\| r_{\mathrm D}^{|{\boldsymbol{\alpha}}-{\boldsymbol{\gamma}}|}
D^{{\boldsymbol{\alpha}}-{\boldsymbol{\gamma}}} b({\boldsymbol{y}}) \|_{L^\infty} \|
r_{\mathrm D}^{|{\boldsymbol{\gamma}}|} D^{{\boldsymbol{\gamma}}
}a({\boldsymbol{y}})\|_{L^\infty}
\\
& \leq C \Bigg( \sum_{0\leq {\boldsymbol{\gamma}} \leq
{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j} \| r_{\mathrm D}^{|{\boldsymbol{\gamma}}|}
D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}\Bigg) \Bigg(
\sum_{|{\boldsymbol{k}}|\leq s}\|r_{\mathrm D}^{|{\boldsymbol{k}}|} D^{\boldsymbol{k}}
b({\boldsymbol{y}})\|_{L^\infty}\Bigg)
\\
& = C \Bigg( \sum_{0\leq {\boldsymbol{\gamma}} \leq
{\boldsymbol{\alpha}}-{\boldsymbol{e}}_j} \| r_{\mathrm D}^{|{\boldsymbol{\gamma}}|}
D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}\Bigg)\|
b({\boldsymbol{y}})\|_{{\mathcal W}^s_\infty}\,.
\end{split}
\end{equation*}
Similarly, if $\gamma_j>0$, each term
$\| r_{\mathrm D}^{|{\boldsymbol{\gamma}}|} D^{{\boldsymbol{\gamma}}}
a({\boldsymbol{y}})\|_{L^\infty}$ with $|{\boldsymbol{\gamma}}|>0$ can be
estimated
\begin{equation*}
\| r_{\mathrm D}^{|{\boldsymbol{\gamma}}|} D^{{\boldsymbol{\gamma}}} a({\boldsymbol{y}}) \|_{L^\infty}
\leq
C
\Bigg(
\sum_{0\leq {\boldsymbol{\gamma}}' \leq {\boldsymbol{\gamma}}-{\boldsymbol{e}}_j}
\| r_{\mathrm D}^{|{\boldsymbol{\gamma}}'|} D^{{\boldsymbol{\gamma}}'} a({\boldsymbol{y}})\|_{L^\infty}
\Bigg)
\| b({\boldsymbol{y}})\|_{{\mathcal W}^s_\infty}
.
\end{equation*}
This implies
\begin{equation*}
\begin{split}
\| r_{\mathrm D}^{|{\boldsymbol{\alpha}}|} D^{\boldsymbol{\alpha}}
a({\boldsymbol{y}})\|_{L^\infty} \leq C \|a({\boldsymbol{y}})\|_{L^\infty}
\big(1+ \|
b({\boldsymbol{y}})\|_{{\mathcal W}^s_\infty}\big)^{|{\boldsymbol{\alpha}}|} \,,
\end{split}
\end{equation*}
for $1\leq |{\boldsymbol{\alpha}}|\leq s$. This finishes the proof.
\end{proof}
We recall the following result from \cite[Theorem
1]{BLN}.
\begin{theorem}\label{thm:bacuta}
Let ${\mathrm D}$ be a curvilinear polygonal domain,
$\eta_0>0$, $s\in {\mathbb N}$ and $N_s=2^{s+1}-s-2$. Let
$a\in L^\infty({\mathrm D},{\mathbb C})$.
Then there exist $\tau$ and $C_s$ with the following
property: for any $a\in {\mathcal W}^{s-1}_\infty({\domain})$ and for
any $\varkappa \in \mathbb{R}$ such that
$$|\varkappa|<\eta:=\min\{\eta_0,\tau^{-1}\|a\|_{L^\infty}^{-1}\rho(a) \},$$
the operator $P_a$ defined in \eqref{PDE} induces an isomorphism
\begin{equation*}
P_a: {\mathcal K}_{\varkappa+1}^{s}({\domain}) \cap \{ u|_{\partial {\mathrm D} }=0\} \to {\mathcal K}_{\varkappa-1}^{s-2}({\domain})
\end{equation*}
such that $P_a^{-1}$ depends analytically on the coefficients $a$ and
has norm
\begin{equation*}
\|P_a^{-1}\|
\leq
C_s \big(\rho(a)-\tau|\varkappa|\|a\|_{L^\infty}\big)^{-N_s-1}\|a\|_{{\mathcal W}_\infty^{s-1}}^{N_s}\,.
\end{equation*}
\end{theorem}
The bound of $\tau$ and $C_s$ depends only on $s$, ${\mathrm D}$ and
$\eta_0$.
Applying this result to our setting, we obtain the following
parametric regularity.
\begin{theorem}\label{thm:Kond}
Suppose $\eta_0>0$, $\psi_j \in {\mathcal W}^{s-1}_\infty({\domain})$ for all
$j\in {\mathbb N}$ and that \eqref{eq:leqkappa} holds. Let
${\mathfrak u }\subseteq \operatorname{supp}({\boldsymbol{\rho}})$ be a finite set. Let further
${\boldsymbol{y}}_0=(y_{0,1},y_{0,2},\ldots) \in U$ be such that $b({\boldsymbol{y}}_0)$
belongs to ${\mathcal W}^{s-1}_\infty({\domain})$. We denote
\begin{equation*}
\vartheta :
=
\inf_{{\boldsymbol{z}}_{\mathfrak u } \in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}}_0,{\boldsymbol{\rho}}) } \rho\big(a({\boldsymbol{z}}_{\mathfrak u })\big)\|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty}^{-1}\,.
\end{equation*}
Let $\tau > 0$ be as given in Theorem \ref{thm:bacuta}.
Then there exists a positive constant $C_s$ such that for
$\varkappa \in \mathbb{R}$ with
$|\varkappa|\leq \min\{\eta_0, \tau^{-1}\vartheta/2 \},$ and for
$f\in {\mathcal K}^{s-2}_{\varkappa-1}({\domain})$, the solution $u$ of \eqref{SPDE}
is holomorphic in the cylinder $ \mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}) $ as a
function in variables
${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j \in {\mathbb N}}\in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$
taking values in ${\mathcal K}_{\varkappa+1}^s({\domain})\cap V$, where
$z_j = y_{0,j}$ for $j\not \in {\mathfrak u }$ held fixed. Furthermore, we
have the estimate
\begin{equation*}
\|u({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal K}_{\varkappa+1}^s}
\leq
C_s \frac{1}{ \big(\rho( a({\boldsymbol{z}}_{\mathfrak u }) \big)^{N_{s}+1}} \|a({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty}^{N_s}\,.
\end{equation*}
\end{theorem}
\begin{proof}
Observe first that for the parametric coefficient $a({\boldsymbol{z}}_{\mathfrak u })$, the
conditions of Proposition \ref{prop:holoh1} are satisfied.
Thus, the solution $u$ is holomorphic in $\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}})$
as a $V$-valued map in variables
${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j \in {\mathbb N}}\in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$. We
assume that $\vartheta>0$. Let $\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$ be
given in \eqref{eq:SUN} and
${\boldsymbol{z}}_{{\mathfrak u }} = (y_j+{\rm i} \xi_j)_{j\in {\mathbb N}}\in \mathcal{S}_{{\mathfrak u }}
({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$ with
$(y_j+{\rm i} \xi_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$.
From Lemma \ref{kon-lem-1} we have
\begin{equation*}
\|a({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty}
\leq C\|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} \Big(1+\| b({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty}\Big)^{s-1}\,.
\end{equation*}
Furthermore
\begin{equation*} \label{eq:bzu}
\begin{split}
\| b({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty} &= \sum_{|{\boldsymbol{\alpha}}|\leq
s-1}\Bigg\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}\sum_{j\in {\mathbb N}}(y_j+{\rm i}
\xi_j)D^{{\boldsymbol{\alpha}}}\psi_j\Bigg\|_{L^\infty}
\\
& \leq \sum_{j\in {\mathfrak u }}(|y_j-y_{0,j}|+\rho_j)\|
\psi_j\|_{{\mathcal W}^{s-1}_\infty}+\| b({\boldsymbol{y}}_0)\|_{{\mathcal W}^{s-1}_\infty} <
\infty\,.
\end{split}
\end{equation*}
This together with \eqref{eq:azu0} implies
$ \|a({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty} \leq C $. From the condition of
$\varkappa$ we infer $ |\varkappa| \tau \leq {\vartheta}/{2} $ which
leads to
\begin{equation*}
\tau |\varkappa| \|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} \leq \rho(a({\boldsymbol{z}}_{\mathfrak u }))/2.
\end{equation*}
As a consequence we obtain
\begin{equation*}
\big(\rho(a({\boldsymbol{z}}_{\mathfrak u }))-\tau |\varkappa|\|a({\boldsymbol{z}})\|_{L^\infty}\big)^{-1}
\leq
\frac{1}{ \rho( a({\boldsymbol{z}}_{\mathfrak u }) }\,.
\end{equation*}
Since the function $\exp$ is analytic in
$\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$, the assertion follows for the case
$\vartheta>0$ by applying Theorem \ref{thm:bacuta}. In addition,
for
${\boldsymbol{z}}_{{\mathfrak u }} = (z_j)_{j\in {\mathbb N}}\in \mathcal{S}_{{\mathfrak u }} ({\boldsymbol{y}}_0,{\boldsymbol{\rho}})$
with $(z_j)_{j\in {\mathfrak u }}\in \mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$, we have
\begin{equation*}
\rho(a({\boldsymbol{z}}_{\mathfrak u }))\|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty}^{-1}
\geq C>0,
\end{equation*}
From this we conclude that $u$ is holomorphic in the cylinder
$\mathcal{S}_{{\mathfrak u },N} ({\boldsymbol{\rho}})$ as a ${\mathcal K}_{1}^s({\domain})\cap V$-valued
map, by again Theorem \ref{thm:bacuta}. This completes the proof.
\end{proof}
\begin{remark} \label{rmk:kappa} The value of $\vartheta$ depends on
the system $(\psi_j)_{j\in {\mathbb N}}$. Assume that $\psi_j=j^{-\alpha}$
for some $\alpha>1$. Then for any ${\boldsymbol{y}}\in U$, ${\boldsymbol{\rho}}$ satisfying
\eqref{eq:leqkappa}, and finite set ${\mathfrak u }\subset\operatorname{supp}({\boldsymbol{\rho}})$ we
have
\begin{equation*}
\vartheta = \inf_{{\boldsymbol{z}}_{\mathfrak u } \in \mathcal{S}_{\mathfrak u } ({\boldsymbol{y}},{\boldsymbol{\rho}}) } \frac{\Re[\exp(\sum_{j\in {\mathbb N}}(y_j+{\rm i} \xi_j)j^{-\alpha})]}{\exp(\sum_{j\in {\mathbb N}}y_jj^{-\alpha})} \geq \cos\kappa\,.
\end{equation*}
We consider another case when there exists some $\psi_j$ such that
$\psi_j\geq C>0$ in an open set $\Omega$ in ${\mathrm D}$ and
$\|\exp(y_j \psi_j)\|_{L^\infty}\geq 1$ for all $y_j\leq 0$. With
${\boldsymbol{y}}_0=(\ldots,0,y_j,0,...)$ and $v_0\in C_0^\infty(\Omega)$ we have
in this case
\begin{equation*}
\vartheta
\leq \rho(\exp(y_j\psi_j)) \to 0
\quad
\text{when}\quad y_j\to -\infty\,.
\end{equation*}
Hence, only for $\varkappa=0$ is satisfied Theorem \ref{thm:Kond} in
this situation.
\end{remark}
Due to this remark, for Kondrat'ev regularity we consider only the
case $\varkappa=0$. In Section \ref{S:DiffPolyg}, we will present a
stronger regularity result in the case when $D \subset \mathbb{R}^2$ is a
polygonal domain.
\begin{lemma}\label{lem:Kond:01}
Let ${\boldsymbol{\nu}}\in {\mathcal F}$, $f\in {\mathcal K}_{-1}^{s-2}({\domain})$, and assume that
$\psi_j \in {\mathcal W}^{s-1}_\infty({\domain})$ for $j\in {\mathbb N}$. Let ${\boldsymbol{y}} \in U$
with $b({\boldsymbol{y}})\in {\mathcal W}^{s-1}_\infty({\domain})$. Assume further that there
exists a non-negative sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}}=(\rho_{{\boldsymbol{\nu}},j})_{j\in {\mathbb N}}$ such that
$\operatorname{supp}({\boldsymbol{\nu}})\subset \operatorname{supp}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ and
\begin{equation} \label{kon-02} \sum_{|{\boldsymbol{\alpha}}|\leq s-1}\Bigg\|
\sum_{j\in
{\mathbb N}}\rho_{{\boldsymbol{\nu}},j}|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{{\boldsymbol{\alpha}}}\psi_j|\Bigg\|_{L^\infty}\leq
\kappa <\frac{\pi}{2}\,.
\end{equation}
Then we have the estimate
\begin{equation*}
\|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_{{\mathcal K}^s_1}
\leq C
\frac{{\boldsymbol{\nu}}! } {{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{\boldsymbol{\nu}}}
\big( \exp\big(\|b({\boldsymbol{y}})\|_{L^\infty }\big)^{2N_s+1} \Big(1+\| b({\boldsymbol{y}})\|_{{\mathcal W}^{s-1}_\infty} \Big)^{(s-1)N_s}.
\end{equation*}
\end{lemma}
\begin{proof} Let ${\boldsymbol{\nu}}\in {\mathcal F}$ with ${\mathfrak u }=\operatorname{supp}({\boldsymbol{\nu}})$. By our
assumption, it is clear that (with ${\boldsymbol{\alpha}}=0$ in \eqref{kon-02})
\begin{equation*}
\Bigg\|\sum_{j \in {\mathbb N}} \rho_{{\boldsymbol{\nu}},j} |\psi_j | \Bigg\|_{L^\infty} \leq \kappa <\frac{\pi}{2}\,.
\end{equation*}
Consequently, if we fix the variable $y_j$ with $j\not \in {\mathfrak u }$,
the function $u$ of \eqref{SPDE} is holomorphic on the domain
$\mathcal{S}_{\mathfrak u }({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$, see Theorem \ref{thm:Kond}. Hence,
applying Cauchy's formula gives that
\begin{equation*}
\begin{split}
\|\partial^{{\boldsymbol{\nu}}}u({\boldsymbol{y}})\|_{{\mathcal K}^s_1} & \leq
\frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{\boldsymbol{\nu}}} \sup_{{\boldsymbol{z}}_{\mathfrak u }\in
\mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})} \|u({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal K}^s_1}
\\
& \leq C \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{\boldsymbol{\nu}}} \sup_{{\boldsymbol{z}}_{\mathfrak u }\in
\mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})} \frac{1}{ \big(\rho(
a({\boldsymbol{z}}_{\mathfrak u }) \big)^{N_{s}+1}}
\|a({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty}^{N_s},
\end{split}
\end{equation*}
where $C_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ is given as in
\eqref{eq:C-rho-y}. Notice that for
${\boldsymbol{z}}_{\mathfrak u }=(z_j)_{j\in {\mathbb N}} \in \mathcal{C}_{\mathfrak u }({\boldsymbol{y}},{\boldsymbol{\rho}}_{\boldsymbol{\nu}})$, we
can write
$z_j = y_j + \eta_j + {\rm i}\xi_j \in {\mathcal C}_{{\boldsymbol{y}},j}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ with
$|\eta_j | \le \rho_{{\boldsymbol{\nu}},j}$ and $|\xi_j| \le \rho_{{\boldsymbol{\nu}},j}$ for
$j\in {\mathfrak u }$. Hence, by \eqref{eq:R-11}, \eqref{eq:azu} and
\begin{equation*}
\begin{split}
\|a({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty} &\leq
C\|a({\boldsymbol{z}}_{\mathfrak u })\|_{L^\infty} \Big(1+\|
b({\boldsymbol{z}}_{\mathfrak u })\|_{{\mathcal W}^{s-1}_\infty}\Big)^{s-1}
\\
&= C\exp(\|b({\boldsymbol{y}})\|_{L^\infty}) \Bigg[1+ \sum_{|{\boldsymbol{\alpha}}|\leq
s-1}\Bigg\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|} \sum_{j\in
{\mathbb N}}(y_j+\eta_j+{\rm i} \xi_j)D^{{\boldsymbol{\alpha}}}\psi_j\Bigg\|_{L^\infty}
\Bigg]^{s-1}
\\
&= C\exp(\|b({\boldsymbol{y}})\|_{L^\infty}) \Bigg[1+ \sum_{|{\boldsymbol{\alpha}}|\leq
s-1}\Bigg\|2\sum_{j\in {\mathfrak u }} \rho_{{\boldsymbol{\nu}},j}|
r_{\mathrm D}^{|{\boldsymbol{\alpha}}|} D^{{\boldsymbol{\alpha}}} \psi_j |\Bigg\|_{L^\infty}+\|
b({\boldsymbol{y}})\|_{{\mathcal W}^{s-1}_\infty} \Bigg]^{s-1}
\\
&\leq C\exp(\|b({\boldsymbol{y}})\|_{L^\infty}) \Big(1+2\kappa+\|
b({\boldsymbol{y}})\|_{{\mathcal W}^{s-1}_\infty} \Big)^{s-1}\,,
\end{split}
\end{equation*}
we obtain the desired result.
\end{proof}
\subsubsection{Summability of $K^s_{\varkappa}$-norms of
Wiener-Hermite PC expansion coefficients}
\label{sec:KmbetaSumm}
For weighted $\ell^2$-summability and $\ell^p$-summability of
$K^s_{\varkappa}$-norms of Wiener-Hermite PC expansion coefficients we
need the following assumption.
\begin{assumption}\label{ass:Ass3}
Let $s\in {\mathbb N}$. All functions $\psi_j$ belong to
${\mathcal W}^{s-1}_\infty({\domain})$ and there exists a positive sequence
$(\lambda_j)_{j\in {\mathbb N}}$ such that
$\big(\exp(-\lambda_j^2)\big)_{j\in {\mathbb N}}\in \ell^1({\mathbb N})$ and the
series
$$\sum_{j\in {\mathbb N}}\lambda_j\left|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{{\boldsymbol{\alpha}}}\psi_j\right|$$
converges in $L^\infty({\domain})$ for all ${\boldsymbol{\alpha}}\in {\mathbb N}_0^d$ with
$|{\boldsymbol{\alpha}}|\leq s-1$.
\end{assumption}
\begin{lemma}\label{lem:Kond:02}
Suppose Assumption \ref{ass:Ass3} holds. Then $b({\boldsymbol{y}})$ belongs to
${\mathcal W}^{s-1}_\infty({\domain})$ $\gamma-a.e.\ {\boldsymbol{y}}\in U$. Furthermore
$\mathbb{E}(\exp(k\|b({\boldsymbol{y}})\|_{{\mathcal W}_\infty^{s-1}}))$ is finite for all
$k\in [0,\infty)$.
\end{lemma}
\begin{proof}
Under Assumption \ref{ass:Ass3}, by \cite[Theorem 2.2.]{BCDM} we
infer that for ${\boldsymbol{\alpha}}\in {\mathbb N}_0^d$, $|{\boldsymbol{\alpha}}|\leq s-1$, the
sequence
$$\left(\sum_{j=1}^N y_j r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}
D^{\boldsymbol{\alpha}}\psi_j\right)_{N\in{\mathbb N}}$$ converges to some
$\psi_{\boldsymbol{\alpha}}$ in $L^\infty$ for $\gamma-a.e.\ {\boldsymbol{y}}\in U$ and
$\mathbb{E}(\exp(k\|\psi_{\boldsymbol{\alpha}}({\boldsymbol{y}})\|_{L^\infty}))$ is finite for
all $k\in [0,\infty)$. Hence, for $\gamma-a.e.\ {\boldsymbol{y}}\in U$, the
sequence $\big(\sum_{j=1}^N y_j \psi_j\big)_{N\in{\mathbb N}}$ is a Cauchy
sequence in ${\mathcal W}^{s-1}_\infty({\domain})$. Since ${\mathcal W}^{s-1}_\infty({\domain})$ is
a Banach space, the statement follows.
\end{proof}
\begin{theorem} [General case] \label{kon-thm-ge} Let $s \in {\mathbb N}$,
$s\geq 2$ and ${\mathrm D}$ be a bounded curvilinear polygonal domain.
Let $f\in {\mathcal K}_{-1}^{s-2}({\domain})$ and Assumption \ref{ass:Ass3} hold.
Assume there exists a sequence
$${\boldsymbol{\varrho}}= (\varrho_j)_{j \in {\mathbb N}}\in (0,\infty)^\infty \ \
with \ (\varrho_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N})$$ for some
$0 < q < \infty$. Assume furthermore that, for each ${\boldsymbol{\nu}}\in {\mathcal F}$,
there exists a sequence
${\boldsymbol{\rho}}_{\boldsymbol{\nu}}:= (\rho_{{\boldsymbol{\nu}},j})_{j \in {\mathbb N}}\in [0,\infty)^\infty$ such
that $\operatorname{supp}({\boldsymbol{\nu}})\subset \operatorname{supp}({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$,
\begin{equation*}
\sup_{{\boldsymbol{\nu}}\in {\mathcal F}}
\sum_{|{\boldsymbol{\alpha}}|\leq s-1}
\Bigg\| \sum_{j\in {\mathbb N}}\rho_{{\boldsymbol{\nu}},j}|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{{\boldsymbol{\alpha}}}\psi_j|\Bigg\|_{L^\infty}\leq \kappa <\frac{\pi}{2},
\quad \text{and}\quad
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r} \frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} <\infty\,
\end{equation*}
with $r\in {\mathbb N}$, $r>2/q$.
Then
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in {\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{{\mathcal K}^s_1}^2 <\infty\,,
\end{equation*}
where $ \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ is given in \eqref{beta} and,
furthermore, $(\|u_{\boldsymbol{\nu}}\|_{{\mathcal K}^s_1})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{theorem}
\begin{proof}
For each ${\boldsymbol{\nu}} \in {\mathcal F}$ with ${\mathfrak u }=\operatorname{supp}({\boldsymbol{\nu}})$ and ${\boldsymbol{y}}\in U$ such
that $b({\boldsymbol{y}})\in {\mathcal W}^{s-1}_\infty({\domain})$, Assumption \ref{ass:Ass3}
implies that the solution $u$ of \eqref{SPDE} is holomorphic in
$\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}}_{\boldsymbol{\nu}})$ as a ${\mathcal K}_1^s({\domain})\cap V$-valued map,
see Theorem \ref{thm:Kond}.
We obtain from Lemmata \ref{lem:Kond:01} and \ref{lem:Kond:02}
\begin{equation*}
\begin{split}
\int_U\| \partial^{\boldsymbol{\nu}} u({\boldsymbol{y}})\|_{{\mathcal K}^s_1}^2\rd\gamma({\boldsymbol{y}}) & \leq
C \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} \int_U\big(
\exp\big(\|b({\boldsymbol{y}})\|_{L^\infty }\big)^{4N_s+2} \Big(1+\|
b({\boldsymbol{y}})\|_{{\mathcal W}^{s-1}_\infty} \Big)^{2(s-1)N_s}\rd\gamma({\boldsymbol{y}})
\\
& \leq C \frac{{\boldsymbol{\nu}}!}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} <\infty\,.
\end{split}
\end{equation*}
This leads to
\begin{equation*}
\begin{split}
\sum_{{\boldsymbol{\nu}}\in {\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{{\mathcal K}^s_1}^2
&= \sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!} \int_U\| \partial^{\boldsymbol{\nu}}
u({\boldsymbol{y}})\|_{{\mathcal K}^s_1}^2\rd\gamma({\boldsymbol{y}}) \leq C
\sum_{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}
\frac{{\boldsymbol{\nu}}!{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\rho}}_{\boldsymbol{\nu}}^{2{\boldsymbol{\nu}}}} <\infty\,.
\end{split}
\end{equation*}
The rest of the proof follows similarly to the proof of Theorem
\ref{thm:s=1}.
\end{proof}
Similarly to Corollaries \ref{cor:global} and \ref{cor:local} from
Theorem~\ref{kon-thm-ge} we obtain
\begin{corollary}[The case of global supports]\label{kon-global}
Let $s \in {\mathbb N}$, $s\geq 2$ and ${\mathrm D}$ be a bounded curvilinear,
polygonal domain. Assume that for all $j\in {\mathbb N}$ holds
$\psi_j\in{\mathcal W}^{s-1}_\infty({\domain})$, and that $f\in {\mathcal K}_{-1}^{s-2}({\domain})$.
Assume further that there exists a sequence of positive numbers
${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$ such that
$$
\big(\lambda_j \| \psi_j \|_{{\mathcal W}_\infty^{s-1}}\big)_{j\in {\mathbb N}} \in
\ell^1({\mathbb N}) \ \ and \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N}),$$
for some $0 < q < \infty$. Then we have
$(\|u_{\boldsymbol{\nu}}\|_{{\mathcal K}^s_1})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\begin{corollary}[The case of disjoint supports]
Let $s \in {\mathbb N}$, $s\geq 2$ and ${\mathrm D}\subset {\mathbb R}^d$ with
$d \geq 2$ be a bounded curvilinear polygonal domain. Assume that
all the functions $\psi_j$ belong to ${\mathcal W}^{s-1}_\infty({\domain})$ and have
disjoint supports. Assume further that $f\in {\mathcal K}_{-1}^{s-2}({\domain})$
and that there exists a sequence of positive numbers
${\boldsymbol{\lambda}}= (\lambda_j)_{j \in {\mathbb N}}$ such that
$$
\big(\lambda_j \| \psi_j \|_{{\mathcal W}_\infty^{s-1}}\big)_{j\in {\mathbb N}} \in
\ell^2({\mathbb N}) \ \ and \ \ (\lambda_j^{-1})_{j \in {\mathbb N}}\in \ell^q({\mathbb N}),$$
for some $0 < q < \infty$. Then
$(\|u_{\boldsymbol{\nu}}\|_{{\mathcal K}^s_1})_{{\boldsymbol{\nu}}\in{\mathcal F}}\in \ell^p({\mathcal F})$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{2}$.
\end{corollary}
\newpage
\
\newpage
\section{Sparsity for holomorphic maps}
\label{sec:SumHolSol}
In Section \ref{sec:EllPDElogN} we introduced the concept of
holomorphic extensions of countably-parametric families
$\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}\subset V$ in the separable Hilbert space $V$
into the Cartesian product $\mathcal{S}_{\mathfrak u } ({\boldsymbol{\rho}})$ of strips
(cp. \eqref{eq:Snubrho}). We now introduce a refinement which is
required for the ensuing results on rates of numerical approximation
of such families, based on sparsity (weighted $\ell^2$-summability)
and of Wiener-Hermite PC expansions of $\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}$:
\emph{quantified, parametric holomorphy} of (complex extensions of)
the parametric families $\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}\subset X$ for a
separable Hilbert space X. Section \ref{S:DefbxdHol} presents the
definition of quantified holomorphy of $\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}$ and
discusses several properties of these families. There are two basic
steps in the approximations which we consider: \medskip \newline
\noindent
(i) We truncate the countably-parametric family
$\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}\subset X$ to a finite number $N\in {\mathbb N}$ of
parameters. This step, which is sometimes also referred to as
``dimension-truncation'', of course implicitly depends on the
enumeration of the coordinates $y_j \in {\boldsymbol{y}}$. \emph{We assume
throughout that this numbering is fixed by the indexing of the
Parseval frame in Theorem \ref{thm:AdFrmX}} which frame is used as
affine representation system to parametrize the uncertain input
$a = \exp(b)$ of the PDE of interest. We emphasize that the finite
dimension $N \in {\mathbb N}$ of the truncated parametric Wiener-Hermite PC
expansion is a discretization parameter, and we will be interested in
quantitative bounds on the error incurred by restricting
$\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}\subset X$ to Wiener-Hermite PC expansions of
the first $N$ active variables only. We denote these restrictions by
$\{ u_N({\boldsymbol{y}}):{\boldsymbol{y}}\in U \}$. \medskip \newline
\noindent
(ii) The coefficients $u_{\boldsymbol{\nu}}\in X$ of the resulting,
finite-parametric Wiener-Hermite PC expansion, can not be computed
exactly, but must be numerically approximated. As is done in
stochastic collocation and stochastic Galerkin algorithms, we seek
numerical approximations of $u_{\boldsymbol{\nu}}$ in suitable, finite-dimensional
subspaces $X_l\subset X$. Assuming the collection
$(X_l )_{l\in {\mathbb N}}\subset X$ to be dense in $X$, any prescribed
tolerance $\varepsilon>0$ of approximation of $u_N({\boldsymbol{y}})$ in
$L^2(U, X;\gamma)$ can be met. For notational convenience, we also set
$X_0 = \{ 0\}$. \medskip \newline In computational practice, however,
one searches an allocation of
$l:{\mathcal F}\times (0,1]\to {\mathbb N}: ({\boldsymbol{\nu}},\varepsilon) \mapsto l({\boldsymbol{\nu}},\varepsilon)$ of
discretization levels along a sequence of ``active'' PC coefficients
which ensures that the prescribed tolerance $\varepsilon\in (0,1]$ is met
with as possibly minimal ``computational budget''. We propose and
analyze the \emph{a-priori construction of an allocation} $l$ which
ensures convergence rates of the corresponding collocation
approximations which are independent of $N$ (i.e.\ they are free from
the ``curse of dimensionality'') and they are based on ``stochastic
collocation'', i.e.\ on sampling the parametric family
$\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}\subset V$ in a collection of deterministic
Gaussian coordinates in $U$. We prove, subsequently,
dimension-independent convergence rates of the sparse collocation
w.r.\ to ${\boldsymbol{y}}\in U$ and w.r.\ to the subspaces $X_l\subset X$ realize
convergence rates which are free from the curse of dimensionality.
These rates depend only on the summability (resp. sparsity) of the
sequence of the norm of the Wiener-Hermite PC expansion coefficients
of the parametric family $\{ u({\boldsymbol{y}}) : {\boldsymbol{y}} \in U \}$ with respect to
${\boldsymbol{y}}$.
\subsection{$({\boldsymbol{b}},\xi,\delta,X)$-Holomorphy and sparsity}
\label{S:DefbxdHol}
We introduce the concept of ``$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic
functions'', which constitutes a subset of $L^2(U,X;\gamma)$. As such
these functions are typically not pointwise well defined for each
${\boldsymbol{y}}\in U$. In order to still define a suitable form of pointwise
function evaluations to be used for numerical algorithms, we will
define them as $L^2(U,X;\gamma)$ limits of certain smooth (pointwise
defined) functions, cp.~Remark.~\ref{rmk:defu} and Example \ref{ex:u}
ahead.
For $N\in\mathbb{N}$ and ${\boldsymbol{\varrho}}=(\varrho_j)_{j=1}^N\in (0,\infty)^N$ set
(cp. \eqref{eq:Snubrho})
\begin{equation}
\label{eq:Sjrho}
{\mathcal S}({\boldsymbol{\varrho}}) := \set{{\boldsymbol{z}}\in \mathbb{C}^N}{|\mathfrak{Im}z_j| < \varrho_j~\forall j}\qquad\text{and}\qquad
{\mathcal B}({\boldsymbol{\varrho}}) := \set{{\boldsymbol{z}}\in\mathbb{C}^N}{|z_j|<\varrho_j~\forall j}.
\end{equation}
\begin{definition}[(${\boldsymbol{b}},\xi,\delta,X$)-Holomorphy]
\label{def:bdXHol}
Let $X$ be a complex separable Hilbert space,
${\boldsymbol{b}}=(b_j)_{j\in\mathbb{N}} \in (0,\infty)^\infty$ and $\xi>0$, $\delta>0$.
For $N\in\mathbb{N}$ we say that ${\boldsymbol{\varrho}}\in (0,\infty)^N$ is
\emph{$({\boldsymbol{b}},\xi)$-admissible} if
\begin{equation}\label{eq:adm}
\sum_{j=1}^N b_j\varrho_j\leq \xi\,.
\end{equation}
A function $u\in L^2(U,X;\gamma)$ is called
\emph{$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic} if
\begin{enumerate}
\item\label{item:hol} for every $N\in\mathbb{N}$ there exists
$u_N:\mathbb{R}^N\to X$, which, for every $({\boldsymbol{b}},\xi)$-admissible
${\boldsymbol{\varrho}}\in (0,\infty)^N$, admits a holomorphic extension
(denoted again by $u_N$) from ${\mathcal S}({\boldsymbol{\varrho}})\to X$; furthermore,
for all $N<M$
\begin{equation}\label{eq:un=um}
u_N(y_1,\dots,y_N)=u_M(y_1,\dots,y_N,0,\dots,0)\qquad\forall (y_j)_{j=1}^N\in\mathbb{R}^N,
\end{equation}
\item\label{item:varphi} for every $N\in\mathbb{N}$ there exists
$\varphi_N:\mathbb{R}^N\to\mathbb{R}_+$ such that
$\norm[L^2(\mathbb{R}^N;\gamma_N)]{\varphi_N}\le\delta$ and
\begin{equation*} \label{ineq[phi]}
\sup_{\substack{{\boldsymbol{\varrho}}\in(0,\infty)^N\\
\text{is $({\boldsymbol{b}},\xi)$-adm.}}}~\sup_{{\boldsymbol{z}}\in
{\mathcal B}({\boldsymbol{\varrho}})}\norm[X]{u_N({\boldsymbol{y}}+{\boldsymbol{z}})}\le
\varphi_N({\boldsymbol{y}})\qquad\forall{\boldsymbol{y}}\in\mathbb{R}^N,
\end{equation*}
\item\label{item:vN} with $\tilde u_N:U\to X$ defined by
$\tilde u_N({\boldsymbol{y}}) :=u_N(y_1,\dots,y_N)$ for ${\boldsymbol{y}}\in U$ it holds
\begin{equation*}
\lim_{N\to\infty}\norm[L^2(U,X;\gamma)]{u-\tilde u_N}=0.
\end{equation*}
\end{enumerate}
\end{definition}
We interpret the definition of
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy in the following remarks.
\begin{remark}\label{rmk:bdexpl}
While the numerical value of
$\xi>0$ in Definition \ref{def:bdXHol} of
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy is of minor importance in the
definition, the sequence ${\boldsymbol{b}}$ and the constant
$\delta$ will crucially influence the magnitude of our upper bounds
of the Wiener-Hermite PC expansion coefficients: The stronger the
decay of
${\boldsymbol{b}}$, the larger we can choose the elements of the sequence
${\boldsymbol{\varrho}}$, so that
${\boldsymbol{\varrho}}$ satisfies \eqref{eq:adm}. Hence stronger decay of
${\boldsymbol{b}}$ indicates larger domains of holomorphic extension. The
constant
$\delta$ is an upper bound of these extensions in the sense of item
\ref{item:varphi}. Ultimately, the decay of
${\boldsymbol{b}}$ will determine the \emph{decay} of the Wiener-Hermite PC
expansion coefficients, while decreasing
$\delta$ by a factor will roughly speaking translate to a decrease
of all coefficients by the same \emph{factor}.
\end{remark}
\begin{remark}
Since $u_N\in L^2(\mathbb{R}^N,X;\gamma_N)$, the function $\tilde
u_N$ in item \ref{item:vN} belongs to
$L^2(U,X;\gamma)$ by Fubini's theorem.
\end{remark}
\begin{remark}\label{rmk:defu}
In the following sections, for arbitrary $N\in\mathbb{N}$ and
$(y_j)_{j=1}^N\in\mathbb{R}^N$ we will write
\begin{equation}\label{eq:udef}
u(y_1,\dots,y_N,0,0,\dots):=
u_N(y_1,\dots,y_N).
\end{equation}
This is well-defined due to \eqref{eq:un=um}. Note however
that
\eqref{eq:udef} should be considered as an abuse of notation, since
pointwise evaluations of functions $u\in L^2(U,X;\gamma)$ are in
general not
well-defined.
\end{remark}
\begin{remark}\label{rmk:separable}
The assumption of $X$ being separable in Definition~\ref{def:bdXHol}
is not necessary: every function $u_N:\mathbb{R}^N\to X$ is continuous
(which is to say that it allows a holomorphic extension). Hence,
$$A_{N,n}:=\set{u_N((y_j)_{j=1}^N)}{y_j\in [-n,n]~\forall j}\subseteq X$$
is compact and thus there is a countable set $X_{N,n}\subseteq X$
which is dense in $A_{N,n}$ for every $N$, $n\in\mathbb{N}$. Then
$\bigcup_{n\in\mathbb{N}} A_{N,n}$ is contained in the (separable) closed
span $\tilde X$ of $$\bigcup_{N,n\in\mathbb{N}} X_{N,n}\subseteq X$$. Since
$\tilde u_N\in L^2(U,\tilde X;\gamma)$ for every $N\in\mathbb{N}$ we also
have $$u=\lim_{N\to\infty}u_N\in L^2(U,\tilde X;\gamma).$$ Hence, if
$X$ is not separable but the other assumptions of
Definition~\ref{def:bdXHol} are satisfied, there always exists a
separable subspace $\tilde X\subseteq X$ such that $u$ is
$({\boldsymbol{b}},\xi,\delta,\tilde X)$-holomorphic.
\end{remark}
\begin{lemma}\label{holo-lem1}
Let $u$ be $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic, let $N\in\mathbb{N}$ and
$0<\kappa<\xi<\infty$. Let $u_N$, $\varphi_N$ be as in
Definition~\ref{def:bdXHol}. Then with ${\boldsymbol{b}}_N=(b_j)_{j=1}^N$ it
holds for every ${\boldsymbol{\nu}}\in\mathbb{N}_0^N$
\begin{equation*}
\|\partial^{{\boldsymbol{\nu}}}u_N({\boldsymbol{y}})\|_X
\leq
\frac{{\boldsymbol{\nu}}!|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}{\boldsymbol{b}}_N^{\boldsymbol{\nu}}}{\kappa^{|{\boldsymbol{\nu}}|}{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}} }
\varphi_N({\boldsymbol{y}})\qquad\forall{\boldsymbol{y}}\in\mathbb{R}^N.
\end{equation*}
\end{lemma}
\begin{proof}
For ${\boldsymbol{\nu}}\in \mathbb{N}_0^N$ fixed we choose ${\boldsymbol{\varrho}}=(\varrho_j)_{j=1}^N$
with $\varrho_j= \kappa \frac{\nu_j}{|{\boldsymbol{\nu}}|b_j}$ for
$j\in \operatorname{supp}({\boldsymbol{\nu}})$ and $\varrho_j=\frac{\xi -\kappa}{Nb_j}$ for
$j\not \in \operatorname{supp}({\boldsymbol{\nu}})$. Then
\begin{equation*}
\sum_{j=1}^N
\varrho_jb_j=\kappa \sum_{j\in \operatorname{supp}({\boldsymbol{\nu}})} \frac{\nu_j}{|{\boldsymbol{\nu}}|}
+\sum_{j\not\in \operatorname{supp}({\boldsymbol{\nu}})}\frac{\xi-\kappa}{N}\leq \xi.
\end{equation*}
Hence ${\boldsymbol{\varrho}}$ is $({\boldsymbol{b}},\xi)$-admissible, i.e.\ there exists a
holomorphic extension $u_N:{\mathcal S}({\boldsymbol{\varrho}})\to X$ as in
Definition~\ref{def:bdXHol} \ref{item:hol}-\ref{item:varphi}.
Applying Cauchy's integral formula as in the proof of Lemma
\ref{lem:estV} we obtain the desired estimate.
\end{proof}
Let us recall the following. Let again $X$ be a separable Hilbert
space and $u\in L^2(U,X;\gamma)$. Then
$L^2(U,X;\gamma) = L^2(U;\gamma)\otimes X$ with Hilbertian tensor
product, and $u$ can be represented in the form of the Wiener-Hermite
PC expansion
\begin{equation}\label{eq:uHermiteExp}
u=\sum_{{\boldsymbol{\nu}}\in {\mathcal F}} u_{\boldsymbol{\nu}} H_{\boldsymbol{\nu}},
\end{equation}
where
\begin{equation*}
u_{\boldsymbol{\nu}}=\int_U u({\boldsymbol{y}}) H_{\boldsymbol{\nu}}({\boldsymbol{y}}) \rd \gamma({\boldsymbol{y}})
\end{equation*}
are the Wiener-Hermite PC expansion coefficients. Also, there holds
the Parseval-type identity
\begin{equation*}\label{eq:PCParseval}
\|u\|_{L^2(U,X;\gamma)}^2=\sum_{{\boldsymbol{\nu}} \in {\mathcal F}}
\|u_{\boldsymbol{\nu}}\|_X^2\,.
\end{equation*}
Similarly, if $u$ is $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic, then we have
for the functions $u_N:\mathbb{R}^N\to X$ in Definition~\ref{def:bdXHol}
\begin{equation*}
u_N=\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N} u_{N,{\boldsymbol{\nu}}} H_{\boldsymbol{\nu}},
\end{equation*}
where
\begin{equation*}
u_{N,{\boldsymbol{\nu}}}=\int_{\mathbb{R}^N} u_N({\boldsymbol{y}}) H_{\boldsymbol{\nu}}({\boldsymbol{y}}) \rd \gamma_N({\boldsymbol{y}}).
\end{equation*}
\begin{lemma}\label{holo-lem2}
Let $u$ be $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic, let $N\in\mathbb{N}$ and let
${\boldsymbol{\varrho}}= (\varrho_j)_{j=1}^N\in [0,\infty)^N$. Then, for any
fixed $r\in\mathbb{N}$ and with $\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ as in \eqref{beta}
($\varrho_j=0$ if $j>N$), there holds the identity
\begin{equation} \label{general} \sum_{{\boldsymbol{\nu}}\in
\mathbb{N}_0^N}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{N,{\boldsymbol{\nu}}}\|_{X}^2 =
\sum_{\set{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}{\|{\boldsymbol{\nu}}\|_{\ell^\infty}\leq r}}
\frac{{\boldsymbol{\varrho}}^{2{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!} \int_{\mathbb{R}^N}\| \partial^{\boldsymbol{\nu}}
u_N({\boldsymbol{y}})\|_{X}^2\rd\gamma_N({\boldsymbol{y}}). \end{equation}
\end{lemma}
\begin{proof}
From Lemma \ref{holo-lem1}, for any ${\boldsymbol{\nu}}\in \mathbb{N}_0^N$, we have with
${\boldsymbol{b}}_N=(b_j)_{j=1}^N$
\begin{align}\label{general-1}
\int_{\mathbb{R}^N} \|\partial^{\boldsymbol{\nu}} u_N({\boldsymbol{y}})\|_X^2\rd\gamma_N({\boldsymbol{y}})
&\le \int_{\mathbb{R}^N}
\Big|\frac{{\boldsymbol{\nu}}!|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}{\boldsymbol{b}}_N^{\boldsymbol{\nu}}}{\kappa^{|{\boldsymbol{\nu}}|}{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}}} \varphi_N({\boldsymbol{y}}) \Big|^2 \rd\gamma_N({\boldsymbol{y}})
\nonumber\\
&= \Big(\frac{{\boldsymbol{\nu}}!|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}{\boldsymbol{b}}_N^{\boldsymbol{\nu}}}{\kappa^{|{\boldsymbol{\nu}}|}{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}}}\Big)^2\int_{\mathbb{R}^N} \big|
\varphi_N({\boldsymbol{y}}) \big|^2 \rd\gamma_N({\boldsymbol{y}})
< \infty
\end{align}
by our assumption. This condition allows us to integrate by parts
as in the proof of \cite[Theorem 3.3]{BCDM}. Following the argument
there we obtain \eqref{general}.
\end{proof}
\begin{theorem} \label{thm:bdHolSum} Let $u$ be
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic for some ${\boldsymbol{b}}\in \ell^p(\mathbb{N})$ and
some $p\in (0,1)$. Let $r\in\mathbb{N}$. Then with
$$
\varrho_j:=b_j^{p-1}\frac{\xi}{4\sqrt{r!} \norm[\ell^p]{{\boldsymbol{b}}}}, \ \
j\in\mathbb{N},
$$
there exists a constant $C({\boldsymbol{b}})$ (depending on ${\boldsymbol{b}}$ and $\xi$ in
Definition~\ref{def:bdXHol} but independent of $\delta$) such that
with ${\boldsymbol{\varrho}}_N=(\varrho_j)_{j=1}^N$ it holds for all $N\in\mathbb{N}$
\begin{equation*}\label{eq:generalN}
\sum_{{\boldsymbol{\nu}}\in \mathbb{N}_0^N}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_N)\norm[X]{u_{N,{\boldsymbol{\nu}}}}^2\le
\delta^2C({\boldsymbol{b}}) <\infty\,,
\end{equation*}
for a constant $C({\boldsymbol{b}})$ depending on ${\boldsymbol{b}}$ and $\xi$ in
Definition~\ref{def:bdXHol} but independent of $\delta$ and $N\in\mathbb{N}$.
Furthermore, for every $N\in\mathbb{N}$ and every $q>0$ there holds
$$(\norm[X]{u_{N,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\in \ell^{q}(\mathbb{N}_0^N).$$
If $q\geq \frac{2p}{2-p}$ then there exists a constant $C>0$ such
that for all $N\in \mathbb{N}$ holds
$$
\big\|(\norm[X]{u_{N,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}}\big\|_{\ell^{q}(\mathbb{N}_0^N)}\leq C <
\infty\;.
$$
\end{theorem}
\begin{proof}
We have
$$\sum_{j\in\mathbb{N}}\varrho_jb_j=\frac{\xi}{4\sqrt{r!}
\norm[\ell^p]{{\boldsymbol{b}}}}\sum_{j\in\mathbb{N}}b_j^p<\infty,$$ and
$(\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^{p/(1-p)}(\mathbb{N})$. Set
$\kappa:=\xi/2\in (0,\xi)$. Inserting \eqref{general-1} into
\eqref{general}
we obtain with ${\boldsymbol{\varrho}}_N=(\varrho_j)_{j=1}^N$
\begin{align*}
\sum_{{\boldsymbol{\nu}}\in \mathbb{N}_0^N}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_N)\norm[X]{u_{N,{\boldsymbol{\nu}}}}^2
&\le \delta^{2}\sum_{\set{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}{\norm[\ell^\infty]{{\boldsymbol{\nu}}}\le r}} \left(\frac{({\boldsymbol{\nu}}!)^{1/2}|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|} {\boldsymbol{\varrho}}_N^{{\boldsymbol{\nu}}} {\boldsymbol{b}}_N^{{\boldsymbol{\nu}}}}{\kappa^{|{\boldsymbol{\nu}}|}{\boldsymbol{\nu}}^{{\boldsymbol{\nu}}}}\right)^2\nonumber\\
&\le \delta^{2}\sum_{\set{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}{\norm[\ell^\infty]{{\boldsymbol{\nu}}}\le r}} \left(\frac{|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|} \prod_{j=1}^N\Big(\frac{b_j^p}{2\norm[\ell^p]{{\boldsymbol{b}}}}\Big)^{\nu_j}}{{\boldsymbol{\nu}}^{\boldsymbol{\nu}}}\right)^2,
\end{align*}
where we used $(\varrho_jb_j)^2=b_j^{2p}\kappa/(2(r!))$ and the
bound
$$\int_{\mathbb{R}^N}\varphi_N({\boldsymbol{y}})^2\,\mathrm{d}\gamma_N({\boldsymbol{y}})\le \delta^{2}$$
from Definition~\ref{def:bdXHol} \ref{item:varphi}. With
$\tilde b_j:= b_j^p/(2\norm[\ell^p]{{\boldsymbol{b}}} )$ the last term is bounded
independent of $N$ by $\delta^2 C({\boldsymbol{b}})$ with
\begin{equation*}
C({\boldsymbol{b}}):= \left(\sum_{{\boldsymbol{\nu}}\in\mathcal F} \frac{|{\boldsymbol{\nu}}|^{|{\boldsymbol{\nu}}|}}{{\boldsymbol{\nu}}^{\boldsymbol{\nu}}} \tilde {\boldsymbol{b}}^{\boldsymbol{\nu}}\right)^{1/2},
\end{equation*}
since the $\ell^1$-norm is an upper bound of the $\ell^2$-norm. As
is well-known, the latter quantity is finite due to
$\norm[\ell^1]{\tilde {\boldsymbol{b}}}<1$, see, e.g., the argument in \cite[Page
61]{CoDe}.
Now introduce $\tilde \varrho_{N,j}:=\varrho_j$ if $j\le N$ and
$\tilde \varrho_{N,j}:=\exp(j)$ otherwise. For any $q>0$ we then
have $(\tilde \varrho_{N,j}^{-1})_{j\in\mathbb{N}}\in\ell^{q}(\mathbb{N})$ and by
\cite[Lemma 5.1]{BCDM} this implies
$$(\beta_{\boldsymbol{\nu}}(r,\tilde{\boldsymbol{\varrho}}_N)^{-1})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q/2}(\mathcal F)$$
as long as $r>2/q$. Using
$\beta_{{\boldsymbol{\nu}}}(r,\tilde{\boldsymbol{\varrho}}_N)=\beta_{(\nu_j)_{j=1}^N}(r,{\boldsymbol{\varrho}}_N)$
for all ${\boldsymbol{\nu}}\in\mathcal F$ with $\operatorname{supp}{\boldsymbol{\nu}}\subseteq \{1,\dots,N\}$ we
conclude
$$(\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}}_N)^{-1})_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\in \ell^{q/2}(\mathbb{N}_0^N)$$
for any $q>0$. Now fix $q>0$ (and $2/q<r\in\mathbb{N}$). Then, by
H\"older's inequality with $s:=2(q/2)/(1+q/2)$, there holds
\begin{align*}
\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\norm[X]{u_{N,{\boldsymbol{\nu}}}}^{s}
&= \sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\norm[X]{u_{N,{\boldsymbol{\nu}}}}^{s}\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}}_N)^{\frac s 2}\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}}_N)^{-\frac s 2}\nonumber\\
&\le \Bigg(\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\norm[X]{u_{N,{\boldsymbol{\nu}}}}^{2}\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}}_N) \Bigg)^{\frac s 2}\Bigg(\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\beta_{{\boldsymbol{\nu}}}(r,{\boldsymbol{\varrho}}_N)^{\frac{s}{2-s}} \Bigg)^{\frac{2-s}{2}},
\end{align*}
which is finite since $s/(2-s)=q/2$. Thus we have shown
$$
\forall q>0, N \in \mathbb{N}: \quad
(\norm[X]{u_{N,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\in\ell^{q/(1+q/2)}(\mathbb{N}_0^N)
\;.
$$
Finally, due to $(\varrho_j^{-1})_{j\in\mathbb{N}} \in\ell^{p/(1-p)}(\mathbb{N})$,
\cite[Lemma 5.1]{BCDM}, for all $N\in {\mathbb N}$ it holds
$$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_N)^{-1})_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N}\in\ell^{p/(2(1-p))}(\mathbb{N}_0^N)$$
and there exists a constant $C$ such that for all $N\in {\mathbb N}$ it
holds
$$
\Big\|(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_N)^{-1})_{{\boldsymbol{\nu}}}\Big\|_{\ell^{p/(2(1-p))}(\mathbb{N}_0^N)}
\leq C<\infty \;.
$$
This implies the last statement. The proof is completed.
\end{proof}
The following result states the summability of Wiener-Hermite PC
expansion coefficients of $({\boldsymbol{b}},\xi,\delta, X)$-holomorphic maps.
\begin{corollary}\label{cor:bdXSum}
Under the assumptions of Theorem~\ref{thm:bdHolSum} it holds
\begin{equation} \label{eq:general} \sum_{{\boldsymbol{\nu}}\in
{\mathcal F}}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})\|u_{\boldsymbol{\nu}}\|_{X}^2\le
\delta^2C({\boldsymbol{b}})<\infty
\end{equation}
with the same constant $C({\boldsymbol{b}})$ as in Theorem~\ref{thm:bdHolSum}.
Furthermore,
$$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\in \ell^{2p/(2-p)}(\mathcal F).$$
\end{corollary}
\begin{proof}
Let $\tilde u_N\in L^2(U,X;\gamma)$ be as in
Definition~\ref{def:bdXHol} and for ${\boldsymbol{\nu}}\in\mathcal F$ denote by
$$
\tilde u_{N,{\boldsymbol{\nu}}}:= \int_U \tilde u_N({\boldsymbol{y}}) H_{\boldsymbol{\nu}}({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})
\in X
$$
the Wiener-Hermite PC expansion coefficient. By Fubini's theorem
$$\tilde u_{N,{\boldsymbol{\nu}}}=\int_U u_N((y_j)_{j=1}^N)
\prod_{j=1}^NH_{\nu_j}(y_j)\,\mathrm{d}\gamma_N((y_j)_{j=1}^N)=u_{N,(\nu_j)_{j=1}^N}$$
for every ${\boldsymbol{\nu}}\in\mathcal F$ with $\operatorname{supp}{\boldsymbol{\nu}}\subseteq\{1,\dots,N\}$.
Furthermore, since $\tilde u_N$ is independent of the variables
$(y_j)_{j=N+1}^\infty$ we have $\tilde u_{N,{\boldsymbol{\nu}}}=0$ whenever
$\operatorname{supp}{\boldsymbol{\nu}}\subsetneq \{1,\dots,N\}$. Therefore
Theorem~\ref{thm:bdHolSum} implies
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{\tilde u_{N,{\boldsymbol{\nu}}}}^2\le
\frac{C({\boldsymbol{b}})}{\delta^2}\qquad\forall N\in\mathbb{N}.
\end{equation*}
Now fix an arbitrary finite set $\Lambda\subset\mathcal F$. Because of
$\tilde u_N\to u\in L^2(U,X;\gamma)$ it holds
$$\lim_{N\to\infty} \tilde u_{N,{\boldsymbol{\nu}}}=u_{{\boldsymbol{\nu}}}$$ for all
${\boldsymbol{\nu}}\in\mathcal F$. Therefore
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\Lambda}
\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{u_{{\boldsymbol{\nu}}}}^2
=\lim_{N\to\infty}\sum_{{\boldsymbol{\nu}}\in\Lambda}
\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{\tilde u_{N,{\boldsymbol{\nu}}}}^2\le \frac{C({\boldsymbol{b}})}{\delta^2}.
\end{equation*}
Since $\Lambda\subset\mathcal F$ was arbitrary, this shows
\eqref{eq:general}.
Finally, due to ${\boldsymbol{b}}\in\ell^p(N)$, with
$$\varrho_j=b_j^{p-1}\frac{\xi}{4\sqrt{r!} \norm[\ell^p]{{\boldsymbol{b}}}}$$
as in Theorem~\ref{thm:bdHolSum} we have
$(\varrho_j^{-1})_{j\in\mathbb{N}} \in\ell^{p/(1-p)}(\mathbb{N})$. By \cite[Lemma
5.1]{BCDM} it holds
$$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})^{-1})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{p/(2(1-p))}(\mathcal F).$$
Hence H\"older's inequality can be used to show
$$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{2p/(2-p)}(\mathcal F)$$
(by a similar calculation as at the end of the proof of
Theorem~\ref{thm:bdHolSum} with $q=p/(1-p)$).
\end{proof}
\begin{remark}[Best $N$-term rate]\label{rmk:bestN} Let $u$ be
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic for some ${\boldsymbol{b}}\in \ell^p(\mathbb{N})$ and
some $p\in (0,1)$ as in Theorem \ref{thm:bdHolSum}. By Corollary
\ref{cor:bdXSum} we then have
$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\cF}\in\ell^{\frac{2p}{2-p}}$.
Let $\Lambda_N\subseteq \cF$ be a set of cardinality $N\in\mathbb{N}$
containing $N$ multiindices ${\boldsymbol{\nu}}\in\cF$ such that
$\norm[X]{u_{\boldsymbol{\mu}}}\le\norm[X]{u_{\boldsymbol{\nu}}}$ whenever ${\boldsymbol{\nu}}\in\Lambda_N$
and ${\boldsymbol{\mu}}\notin\Lambda_N$. Then, by Corollary \eqref{cor:bdXSum}, for the truncated
Hermite expansion we have the error bound
\begin{equation*}
\normc[L^2(U,X;\gamma)]{u({\boldsymbol{y}})-\sum_{{\boldsymbol{\nu}}\in\Lambda_N}u_{\boldsymbol{\nu}} H_{\boldsymbol{\nu}}({\boldsymbol{y}})}^2
=\sum_{{\boldsymbol{\nu}}\in\cF\backslash\Lambda_N}\norm[X]{u_{\boldsymbol{\nu}}}^2
\le \sup_{{\boldsymbol{\nu}}\in\cF\backslash\Lambda_N}\norm[X]{u_{\boldsymbol{\nu}}}^{2-\frac{2p}{2-p}}
\sum_{{\boldsymbol{\mu}}\in\cF\backslash\Lambda_N}\norm[X]{u_{\boldsymbol{\mu}}}^{\frac{2p}{2-p}}.
\end{equation*}
For a nonnegative monotonically decreasing sequence
$(x_j)_{j\in\mathbb{N}}\in\ell^q(\mathbb{N})$ with $q>0$ we have
$x_N^q\le \frac{1}{N}\sum_{j=1}^Nx_j^q$ and thus
$x_N\le N^{-\frac{1}{q}}\norm[\ell^q(\mathbb{N})]{(x_j)_{j\in\mathbb{N}}}$.
With $q=\frac{2p}{2-p}$ this implies
\begin{equation*}
\left(\sup_{{\boldsymbol{\nu}}\in\cF\backslash\Lambda_N}\norm[X]{u_{\boldsymbol{\nu}}}\right)^{2-\frac{2p}{2-p}}\le \left(N^{-\frac{2-p}{2p}}\left(\sum_{{\boldsymbol{\nu}}\in\cF}\norm[X]{u_{\boldsymbol{\nu}}}^{\frac{2p}{2-p}}\right)^{\frac{2-p}{2p}}\right)^{2-\frac{2p}{2-p}}= O(N^{-\frac{2}{p}+2}).
\end{equation*}
Hence, by truncating the Hermite expansion \eqref{eq:uHermiteExp}
after the $N$ largest terms, we obtain the best $N$-term rate
\begin{equation}\label{eq:bestNgeneral}
\normc[L^2(U,X;\gamma)]{u({\boldsymbol{y}})-\sum_{{\boldsymbol{\nu}}\in\Lambda_N}u_{\boldsymbol{\nu}} H_{\boldsymbol{\nu}}({\boldsymbol{y}})}= O(N^{-\frac{1}{p}+1})\qquad\text{as }N\to\infty.
\end{equation}
\end{remark}
\subsection{Functions of the type
$u({\boldsymbol{y}})={\mathcal U}\big(\exp\big(\sum_{j\in\mathbb{N}}y_j\psi_j\big)\big)$}
\label{sec:bdX}
We now show that certain composite functions of the type
$$
u({\boldsymbol{y}})={\mathcal U}\Bigg(\exp\bigg(\sum_{j\in\mathbb{N}}y_j\psi_j\bigg)\Bigg)
$$
satisfy $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy under certain conditions.
The significance of such functions is the following: if we think for
example of ${\mathcal U}$ as the solution operator $\cS$ in \eqref{eq:SolOp}
which maps the diffusion coefficient $a\in L^\infty({\domain})$ to the
solution of ${\mathcal U}(a)\in H_0^1({\domain})$ of an elliptic PDE on some domain
${\domain}\subseteq\mathbb{R}^d$, then
${\mathcal U}\big(\exp\big(\sum_{j\in\mathbb{N}}y_j\psi_j\big)\big)$ is exactly the
parametric solution discussed in Sections
\ref{sec:EllPDElogN}-\ref{sec:KondrReg}. We explain this in more
detail in Section~\ref{sec:pdc}. The presently developed, abstract
setting allows, however, to \emph{consider ${\mathcal U}$ as solution operator
of other, structurally similar PDEs with log-Gaussian random input
data}. Furthermore, if ${\mathcal G}$ is another map {with suitably
holomorphy properties}, the composition
${\mathcal G}\big({\mathcal U}\big(\exp\big(\sum_{j\in\mathbb{N}}y_j\psi_j\big)\big)\big)$ is
again of the general type
$\tilde {\mathcal U}\big(\exp\big(\sum_{j\in\mathbb{N}}y_j\psi_j\big)\big)$ with
$\tilde{\mathcal U}={\mathcal G}\circ{\mathcal U}$.
This will allow to apply {the ensuing results on convergence rates of
deterministic collocation and quadrature algorithms to a wide range
of PDEs with GRF inputs and functionals on their random
solutions. As a particular case in point, we apply} our results to
posterior densities in Bayesian inversion, as we explain subsequently
in Section~\ref{sec:BIP}. As a result, the concept of
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy is fairly broad and covers a large
range of parametric PDEs depending on log-normally distributed
data. To formalize all of this, we now provide sufficient conditions
on the solution operator ${\mathcal U}$ and the sequence $(\psi_j)_{j\in\mathbb{N}}$
guaranteeing $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy.
Let $d\in\mathbb{N}$, ${\mathrm D} \subseteq\mathbb{R}^d$ an open set, ${E}$ a complex
Banach space which is continuously embedded into
$L^\infty({\mathrm D};\mathbb{C})$, and finally let $X$ be another complex Banach
space. Additionally, suppose that there exists $C_{E}>0$ such
that for all $\psi_1, \psi_2 \in {E}$ and some $m\in\mathbb{N}$
\begin{equation}\label{eq:Znorm2}
\norm[{E}]{\exp(\psi_1)-\exp(\psi_2)}
\leq
C_{E}\norm[{E}]{\psi_1-\psi_2}
\max\Big\{\exp\big(m\norm[{E}]{\psi_1}\big);\,
\exp\big(m\norm[{E}]{\psi_2} \big)
\Big\}.
\end{equation}
This covers in particular the Sobolev spaces
$W^{k}_\infty({\mathrm D};\mathbb{C})$, $k\in\mathbb{N}_0$, on bounded Lipschitz domains
${\domain}\subseteq\mathbb{R}^d$, but also the Kondrat'ev spaces
${\mathcal W}_\infty^k({\mathrm D};\mathbb{C})$ on polygonal domains ${\domain}\subseteq\mathbb{R}^d$,
cp.~Lemma \ref{kon-lem-1}. Choosing $\psi_2 \equiv 0$ in
\eqref{eq:Znorm2} we obtain
\begin{equation}\label{eq:Znorm}
\norm[{E}]{\exp(\psi_1)}
\leq C_{E}' \exp\big((m+1)\norm[{E}]{\psi_1}\big).
\end{equation}
for some positive constant $C_{E}'$. Indeed,
\begin{equation}\nonumber
\begin{aligned}
\norm[{E}]{\exp(\psi_1)} & \leq \|1\|_{E} +
C_{E}\norm[{E}]{\psi_1}
\exp\big(m\|1\|_{E}+m\norm[{E}]{\psi_1}\big)
\\
& \leq C_{E}'\big(1+\norm[{E}]{\psi_1}\big)
\exp\big(m\norm[{E}]{\psi_1}\big)
\\
&\leq C_{E}' \exp\big((m+1)\norm[{E}]{\psi_1}\big).
\end{aligned}
\end{equation}
For a function $u\in {E}\subseteq L^\infty({\mathrm D};\mathbb{C})$ we will
write $\Re(u)\in L^\infty({\mathrm D};\mathbb{R})\subseteq L^\infty({\mathrm D};\mathbb{C})$
to denote its real part and
$\Im(u)\in L^\infty({\mathrm D} ;\mathbb{R})\subseteq L^\infty({\mathrm D} ;\mathbb{C})$ its
imaginary part so that $u=\Re(u)+{\rm i}\Im(u)$.
\begin{proposition}\label{prop:bdX}
Let $0<\delta<\delta_{\rm max}$, $K>0$, $\eta>0$ and $m\in\mathbb{N}$.
Assume that for an open set $O\subseteq {E}$ containing
$$\set{\exp(\psi)}{\psi\in {E},~\norm[{E}]{\Im(\psi)}\le \eta},$$ it holds
\begin{enumerate}
\item\label{item:uhol} ${\mathcal U}:O \to X$ is holomorphic,
\item\label{item:norma} for all $a\in O$
\begin{equation*}
\norm[X]{{\mathcal U}(a)}\le \delta \left(\frac{1+\norm[{E}]{a}}{\min\{1,{\rho(a)}\}}\right)^m,
\end{equation*}
\item\label{item:loclip} for all $a$, $b\in O$
$$
\norm[X]{{\mathcal U}(a)-{\mathcal U}(b)} \le K
\left(\frac{1+\max\{\norm[{E}]{a},\norm[{E}]{b}\}}{\min\{1,{\rho(a)},{\rho(b)}\}}\right)^m
\norm[{E}]{a-b},
$$
\item\label{item:psi}
$(\psi_j)_{j\in\mathbb{N}}\subseteq {E}\cap L^\infty({\mathrm D})$ and
with $b_j:=\norm[{E}]{\psi_j}$ it holds
${\boldsymbol{b}}\in\ell^1(\mathbb{N})$.
\end{enumerate}
Then there exists $\xi>0$ and for every $\delta_{\rm max}>0$
there exists $\tilde C$ depending on ${\boldsymbol{b}}$, $\delta_{\rm max}$,
$C_{E}$ and $m$ but independent of
$\delta\in (0,\delta_{\rm max})$, such that with
\begin{equation*}\label{eq:u}
u_N\left((y_j)_{j=1}^N\right)={\mathcal U}\Bigg(\exp\bigg(\sum_{j=1}^N y_j\psi_j\bigg)\Bigg)\qquad
\forall (y_j)_{j=1}^N\in\mathbb{R}^N,
\end{equation*}
and $\tilde u_N({\boldsymbol{y}})=u_N(y_1,\dots,y_N)$ for ${\boldsymbol{y}}\in U$, the
function
$$
u:=\lim_{N\to\infty}\tilde u_N\in L^2(U,X;\gamma)
$$
is well-defined and $({\boldsymbol{b}},\xi,\delta \tilde C,X)$-holomorphic.
\end{proposition}
\begin{proof}
{\bf Step 1.} We show that
$u_N\in L^2(\mathbb{R}^N,X;\gamma_N)$ for every $N\in\mathbb{N}$. To this end we
recall that for any $s>0$ (see, e.g., \cite[Appendix B]{HS16_656},
\cite[(38)]{BCDM} for a proof)
\begin{equation}\label{eq:intexp}
\int_\mathbb{R} \exp(s|y|)\,\mathrm{d} \gamma_1(y) \le \exp\bigg(\frac{s^2}{2}+\frac{\sqrt{2}s}{\pi}\bigg).
\end{equation}
Since ${E}$ is continuously embedded into
$L^\infty({\mathrm D};\mathbb{C})$, there exists $C_0>0$ such that
\begin{align*}
\|\psi\|_{L^\infty({\domain})}\leq C_0 \|\psi\|_{E} \qquad \forall \psi \in E.
\end{align*}
Using \ref{item:norma}, \eqref{eq:Znorm}, and
\begin{align*}
\frac{1}{\operatorname{ess\,inf}_{{\boldsymbol{x}} \in {\mathrm D}}
\big(\exp\big(\sum_{j=1}^Ny_j\psi_j({\boldsymbol{x}})\big)\big) }
&
\leq
\bigg\|\exp\Big(-\sum_{j=1}^Ny_j\psi_j\Big)\bigg\|_{L^\infty}
\\
&\leq \exp\bigg(\bigg\|\sum_{j=1}^Ny_j\psi_j\bigg\|_{L^\infty}\bigg) \leq \exp\bigg(C_0\sum_{j=1}^N|y_j| \big\|\psi_j\big\|_{E}\bigg),
\end{align*}
we obtain the bound
\begin{align*}
\norm[X]{u_N({\boldsymbol{y}})}
&
\le \delta\bigg(1+ \normk[{E}]{\exp\bigg(\sum_{j=1}^Ny_j\psi_j\bigg)}\bigg)^m \exp\bigg(C_0m\sum_{j=1}^N|y_j| \big\|\psi_j\big\|_{E}\bigg)
\\
& \le \delta\Bigg(1+ C_{E}'
\exp\bigg((m+1)\sum_{j=1}^N|y_j|\normc[{E}]{\psi_j}\bigg)\Bigg)^m \exp\bigg(C_0m\sum_{j=1}^N |y_j|\big\|\psi_j\big\|_{E}\bigg)
\\
& \leq C_1 \exp\bigg((2+C_0)m^2\sum_{j=1}^N|y_j|\normc[{E}]{\psi_j}\bigg)
\end{align*}
for some constant $C_1>0$ depending on $\delta, C_{E}$ and $m$.
Hence, by \eqref{eq:intexp} we have
\begin{align*}
\int_{\mathbb{R}^N}\norm[X]{u_N({\boldsymbol{y}})}^2 \,\mathrm{d}\gamma_N({\boldsymbol{y}})
&
\le C_1 \int_{\mathbb{R}^N} \exp\bigg((2+C_0)m^2)\sum_{j=1}^N |y_j|\norm[{E}]{\psi_j}\bigg)\,\mathrm{d}\gamma_N({\boldsymbol{y}})\\
&\le C_1 \exp\Bigg(\frac{(2+C_0)^2m^4}{2}\sum_{j=1}^Nb_j^2+\frac{\sqrt{2}(2+C_0)m^2}{\pi}\sum_{j=1}^Nb_j \Bigg) <\infty.
\end{align*}
{\bf Step 2.} We show that $(\tilde u_N)_{N\in\mathbb{N}}$ which is defined
as $\tilde u_N({\boldsymbol{y}}):=u_N(y_1,\dots,y_N)$ for ${\boldsymbol{y}}\in U$, is a Cauchy
sequence in $L^2(U,X;\gamma)$. For any $N<M$ by \ref{item:loclip}
\begin{align*}
\norm[L^2(U,X;\gamma)]{\tilde u_M-\tilde u_N}^2
&=\int_U \normk[X]{{\mathcal U}\bigg(\exp\Big(\sum_{j=1}^My_j\psi_j\Big)\bigg)
-{\mathcal U}\bigg(\exp\Big(\sum_{j=1}^Ny_j\psi_j\Big)\bigg)}^2\,\mathrm{d}\gamma({\boldsymbol{y}})\nonumber
\\
&\quad\le K \int_U \Bigg[ \Bigg(1+\normk[{E}]{\exp\Big(\sum_{j=1}^My_j \psi_j\Big)}+\normk[{E}]{\exp\Big(\sum_{j=1}^Ny_j \psi_j\Big)}\Bigg)^m
\\
& \qquad \times \exp\bigg(C_0m\sum_{j=1}^M|y_j| \big\|\psi_j\big\|_{E}\bigg) \cdot \normk[{E}]{\exp\Big(\sum_{j=1}^M y_j\psi_j\Big)-\exp\Big(\sum_{j=1}^N y_j\psi_j\Big)}\Bigg]\,\mathrm{d}\gamma({\boldsymbol{y}}).
\end{align*}
Using \eqref{eq:Znorm} again we can estimate
\begin{align*}
&\norm[L^2(U,X;\gamma)]{\tilde u_M-\tilde u_N}^2
\le K \int_U \Bigg[\Bigg(1+ 2C_{E}'
\exp\bigg((m+1)\sum_{j=1}^M|y_j|\normc[{E}]{\psi_j}\bigg)\Bigg)^m
\\
& \times \exp\bigg(C_0m\sum_{j=1}^M|y_j| \big\|\psi_j\big\|_{E}\bigg) \cdot \normk[{E}]{\exp\Big(\sum_{j=1}^M y_j\psi_j\Big)-\exp\Big(\sum_{j=1}^N y_j\psi_j\Big)}\Bigg]\,\mathrm{d}\gamma({\boldsymbol{y}})
\\
&\leq C_2\int_U \Bigg[ \exp\bigg((2+C_0)m^2)\sum_{j=1}^M|y_j|\normc[{E}]{\psi_j}\bigg) \normk[{E}]{\exp\Big(\sum_{j=1}^M y_j\psi_j\Big)-\exp\Big(\sum_{j=1}^N y_j\psi_j\Big)}\Bigg]\,\mathrm{d}\gamma({\boldsymbol{y}})\,
\end{align*}
for $C_2>0$ depending only on $K,C_{E}$, and $m$. Now,
employing \eqref{eq:Znorm2} we obtain
\begin{align*}
\normk[{E}]{\exp\Big(\sum_{j=1}^M y_j\psi_j\Big)-\exp\Big(\sum_{j=1}^N y_j\psi_j\Big)}
& \le C_{E} \sum_{j=N+1}^M |y_j|\norm[{E}]{\psi_j}\exp\Bigg(m\sum_{j=1}^M|y_j|\norm[{E}]{\psi_j} \Bigg)
\\
& \le C_{E}\sum_{j=N+1}^M |y_j|\norm[{E}]{\psi_j} \exp\bigg(m^2\sum_{j=1}^M|y_j|\norm[{E}]{\psi_j} \bigg) .
\end{align*}
Therefore for a constant $C_3$ depending on $C_{E}$ and $\delta$
(but independent of $N$), using $|y_j|\le\exp(|y_j|)$,
\begin{align*}
& \norm[L^2(U,X;\gamma)]{\tilde u_M-\tilde u_N}^2
\\
&\le
C_3 \sum_{j=N+1}^M\norm[{E}]{\psi_j}\int_{\mathbb{R}^M}|y_j|\exp\bigg((3+C_0)m^2
\sum_{i=1}^M|y_i|\norm[{E}]{\psi_i}\bigg) \,\mathrm{d}\gamma_M((y_i)_{i=1}^M)
\nonumber\\
&\le C_3 \sum_{j=N+1}^M\norm[{E}]{\psi_j}\int_{\mathbb{R}^M}
\exp\bigg(|y_j|+(3+C_0)m^2\sum_{i=1}^M|y_i|\norm[{E}]{\psi_i}\bigg) \,\mathrm{d}\gamma_M((y_i)_{i=1}^M)
\nonumber\\
&\le C_3 \bigg(\sum_{j=N+1}^M b_j\bigg)
\Bigg( \exp\bigg(\frac{1}{2}+\frac{\sqrt{2}}{\pi} + \frac{(3+C_0)^2m^4}{2}\sum_{i=1}^Mb_j^2
+ \frac{ \sqrt{2}(3+C_0)m^2}{\pi}\sum_{j=1}^Mb_j \bigg)\Bigg),
\end{align*}
where we used \eqref{eq:intexp} and in the last inequality. Since
${\boldsymbol{b}}\in\ell^1(\mathbb{N})$ the last term is bounded by
$ C_4 \Big(\sum_{j=N+1}^\infty b_j \Big)$ for a constant $C_4$
depending on $C_{E}$, $K$ and ${\boldsymbol{b}}$ but independent of $N$, $M$.
Due to ${\boldsymbol{b}}\in\ell^1(\mathbb{N})$, it also
holds $$\sum_{j=N+1}^{\infty}b_j\to 0 \ \ {\rm as} \ \ N\to\infty.$$
Since $N<M$ are arbitrary, we have shown that
$(\tilde u_N)_{N\in\mathbb{N}}$ is a Cauchy sequence in the Banach space
$L^2(U,X;\gamma)$. This implies that there is a
function $$u:=\lim_{N\to\infty}\tilde u_N\in L^2(U,X;\gamma).$$
{\bf Step 3.} To show that $u$ is $({\boldsymbol{b}},\xi,\delta \tilde{C},X)$
holomorphic, we provide constants $\xi>0$ and $\tilde{C}>0$
independent of $\delta$ so that $u_N$ admits holomorphic extensions
as in Definition~\ref{def:bdXHol}. This concludes the proof.
Let $\xi:=\pi/(4C_0)$. Fix $N\in\mathbb{N}$ and assume
$$\sum_{j=1}^N b_j\varrho_j<\xi$$
(i.e.\ $(\varrho_j)_{j=1}^N$ is $({\boldsymbol{b}},\delta_1)$-admissible).
Then for $z_j=y_j+{\rm i}\zeta_j \in\mathbb{C}$ such that
$|\Im(z_j)|=|\zeta_j|<\varrho_j$ for all $j$,
\begin{equation*}
\rho\Bigg(\exp\Big(\sum_{j=1}^N z_j\psi_j({\boldsymbol{x}}) \Big)\Bigg)
=
\underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,inf}}\,\Bigg(\exp\Big(\sum_{j=1}^N y_j\psi_j({\boldsymbol{x}}) \Big)\Bigg)
\cos\Bigg(\sum_{j=1}^N \zeta_j\psi_j({\boldsymbol{x}})\Bigg).
\end{equation*}
Due to
\begin{equation*}
\underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,sup}} \,\Bigg|\sum_{j=1}^N\zeta_j\psi_j({\boldsymbol{x}})\Bigg|
\le \sum_{j=1}^N \varrho_j\norm[L^\infty]{\psi_j}
\le \sum_{j=1}^N C_0\varrho_j\norm[V]{\psi_j}
= \sum_{j=1}^N C_0\varrho_j b_j
\le \frac{\pi}{4},
\end{equation*}
we obtain for such $(z_j)_{j=1}^N$
\begin{equation}\label{eq:essinfx}
\rho\Bigg(\exp\Big(\sum_{j=1}^N z_j\psi_j({\boldsymbol{x}}) \Big)\Bigg)
\ge \exp\Bigg(-\sum_{j=1}^N|y_j|\norm[L^\infty]{\psi_j} \Bigg)
\cos\Bigg(\frac \pi 4 \Bigg)>0.
\end{equation}
This shows that for every
${\boldsymbol{\varrho}}=(\varrho_j)_{j=1}^N\in (0,\infty)^N$ such that
$\sum_{j=1}^N b_j\varrho_j<\xi$, it holds
$$\sum_{j=1}^Nz_j\psi_j\in O \quad \forall {\boldsymbol{z}}\in {\mathcal S}({\boldsymbol{\varrho}}).$$ Since
${\mathcal U}:O\to X$ is holomorphic, the function
$$ u_N\left((y_j)_{j=1}^N\right)={\mathcal U}\Bigg(\exp\bigg(\sum_{j=1}^N y_j\psi_j\bigg)\Bigg)$$ can be
holomorphically extended to arguments
$(z_j)_{j=1}^N\in {\mathcal S}({\boldsymbol{\varrho}})$.
Finally we fix again $N\in\mathbb{N}$ and provide a function
$\varphi_N\in L^2(U;\gamma)$ as in
Definition~\ref{def:bdXHol}. Fix ${\boldsymbol{y}}\in\mathbb{R}^N$ and
${\boldsymbol{z}}\in {\mathcal B}_{\boldsymbol{\varrho}}$ and set
$$a:=\sum_{j=1}^N(y_j+z_j)\psi_j.$$
By \ref{item:norma}, \eqref{eq:essinfx} and because
$b_j=\norm[{E}]{\psi_j}$ and
$$\sum_{j=1}^N b_j\varrho_j\le\xi,$$ we have that
\begin{align*}
\norm[X]{u_N((y_j+z_j)_{j=1}^N)} &\le \delta\left(\frac{1+\norm[{E}]{a}}{\min\{1,\rho(a) \}} \right)^m
\nonumber\\
&\le \delta \left( \frac{1+C_{E}'
\exp\big((m+1)\sum_{j=1}^N(|y_j|+|z_j|)\norm[{E}]{\psi_j}\big) }{\exp(-C_0\sum_{j=1}^N(|y_j|+|z_j|)\norm[{E}]{\psi_j})\cos(\frac{\pi}{4})}\right)^m
\nonumber\\
&\le \delta \left(\frac{1+C_{E}'\exp\big((m+1)
\sum_{j=1}^N |y_j|b_j\big)\exp((m+1)\xi)}{\exp(-C_0\sum_{j=1}^N|y_j|b_j)\exp(-C_0\xi)\cos(\frac{\pi}{4})}\right)^m
\nonumber\\
&\le \delta L \exp\bigg( (2+C_0)m^2\sum_{j=1}^N|y_j|b_j\bigg)
\end{align*}
for some $L$ depending only on $C_{E},C_0$ and $m$. Let
us define the last quantity as
$\varphi_N\left((y_j)_{j=1}^N\right)$. Then by
\eqref{eq:intexp} and because $\gamma_N$ is a probability
measure on $\mathbb{R}^N$,
\begin{align*}
\norm[L^2(\mathbb{R}^N;\gamma_N)]{\varphi_N}
&\le \delta L \exp\bigg(\sum_{j=1}^N
\frac{(2+C_0)^2m^4b_j^2}{2}+(2+C_0)m^2\frac{\sqrt{2}b_j}{\pi} \bigg) \nonumber\\
&\le \delta L \exp\bigg(\sum_{j\in \mathbb{N}}
\frac{(2+C_0)^2m^4b_j^2}{2}+(2+C_0)m\frac{\sqrt{2}b_j}{\pi} \bigg)
\\
& \le \delta \tilde C({\boldsymbol{b}},C_0,C_E,m),
\end{align*}
for some constant $\tilde C({\boldsymbol{b}},C_0,C_E,m) \in (0,\infty)$
because ${\boldsymbol{b}}\in\ell^1(\mathbb{N})$. In all, we have shown that $u$
satisfies $({\boldsymbol{b}},\xi,\delta \tilde{C},X)$-holomorphy as in
Definition~\ref{def:bdXHol}.
\end{proof}
\subsection{Examples of holomorphic maps}
\label{sec:HlExmpl}
We revisit the example of linear elliptic divergence-form PDE
with diffusion coefficient introduced in
Section~\ref{sec:EllPDElogN}. Its coefficient-to-solution map
$S$ from \eqref{eq:SolOp} gives rise to parametric maps which
are parametric-holomorphic. This kind of function will, on the
one hand, arise as generic model of Banach-space valued
uncertain inputs of PDEs, and on the other hand as model of
solution manifolds of PDEs. The connection is made through
preservation of holomorphy under composition with inversion of
boundedly invertible differential operators.
Let $f\in X'$. If $A(a)\in \cL_{{\rm is}}(X,X')$ is an
isomorphism depending (locally) holomorphically on $a\in E$,
then ${\mathcal U}:a\mapsto (\mathrm {inv} \circ A(a))f\in X$ is also locally
holomorphic as a function of $a\in {E}$, where $\mathrm {inv}$
denotes the inversion map. This is a consequence of the fact
that the inversion map
$\mathrm {inv}: \cL_{{\rm is}}(X,X') \to \cL_{{\rm is}}(X',X)$ is
holomorphic, see e.g.\ \cite[Example 1.2.38]{JZdiss}. This
argument can be used to show that the solution operator
corresponding to the solution of certain PDEs is holomorphic in
the parameter. We informally discuss this for some parametric
PDEs and refer to \cite[Chapter 1 and 5]{JZdiss} for more
details.
\subsubsection{Linear elliptic divergence-form PDE with
parametric diffusion coefficient}\label{sec:pdc}
Let us again consider the model linear elliptic PDE
\begin{equation}\label{eq:elliptic}
- \operatorname{div}(a \nabla {\mathcal U}(a)) =f \;\;\text{in }{\domain} \;,\quad
{\mathcal U}(a) =0 \;\; \text{on }\partial {\domain}
\end{equation}
where $d\in\mathbb{N}$, ${\domain}\subseteq\mathbb{R}^d$ is a bounded Lipschitz domain,
$X:=H_0^1({\domain};\mathbb{C})$, $f\in H^{-1}({\domain};\mathbb{C}):=(H_0^1({\domain};\mathbb{C}))'$ and
$a\in {E}:=L^\infty({\domain};\mathbb{C})$. Then the solution operator
${\mathcal U}$ mapping the coefficient function $a$ to the weak solution
${\mathcal U}(a)$
$$a\in O:=\set{a\in L^\infty({\domain};\mathbb{C})}{\rho(a)} := \underset{{\boldsymbol{x}}\in {\domain}}{\operatorname{ess\,inf}}\Re(a({\boldsymbol{x}}))>0 $$
(here $\dup{\cdot}{\cdot}$ denotes the dual product between $X'$ and
$X$). With $A(a)$ denoting the differential operator
$- \operatorname{div}(a \nabla \cdot)\in L(X,X')$ we can also write
${\mathcal U}(a)=A(a)^{-1}f$. We now check assumptions
\ref{item:uhol}-\ref{item:loclip} of Proposition \ref{prop:bdX}.
\begin{enumerate}
\item As mentioned above, complex Fr\'echet differentiability (i.e.\
holomorphy) of ${\mathcal U}:O\to X$ is satisfied because the operation of
inversion of linear operators is holomorphic on the set of
boundedly invertible linear operators, $A$ depends boundedly and
linearly (thus holomorphically) on $a$, and therefore ,the map
$$a\mapsto A(a)^{-1}f={\mathcal U}(a)$$ is a composition of holomorphic
functions. We refer once more to \cite[Example 1.2.38]{JZdiss}
for more details.
\item For $a\in O$, it holds
\begin{equation*}
\norm[X]{{\mathcal U}(a)}^2 \, \rho(a)
\le
\left|\int_{\domain}\nabla{\mathcal U}(a)^\top a \overline{\nabla{\mathcal U}(a)}\,\mathrm{d} {\boldsymbol{x}} \right|
=
\left|\dup{f}{\overline{{\mathcal U}(a)}}\right|
\le \norm[X']{f}\norm[X]{{\mathcal U}(a)}.
\end{equation*}
This gives the usual a-priori bound
\begin{equation}\label{eq:apriori}
\norm[X]{{\mathcal U}(a)}\le \frac{\norm[X']{f}}{\rho(a)}.
\end{equation}
\item For $a$, $b\in O$ and with $w:={\mathcal U}(a)-{\mathcal U}(b)$, we have that
\begin{align*}
\frac{\norm[X]{w}^2}{\rho(a)}
&\le\left| \int_{\domain} \nabla w^\top a \overline{\nabla w}\,\mathrm{d} {\boldsymbol{x}}\right|\nonumber\\
&= \left|\int_{\domain} \nabla {\mathcal U}(a)^\top a \overline{\nabla w}\,\mathrm{d} {\boldsymbol{x}}
- \int_{\domain} \nabla {\mathcal U}(b)^\top b \overline{\nabla w}\,\mathrm{d} {\boldsymbol{x}}
-\int_{\domain} \nabla {\mathcal U}(b)^\top (a-b) \overline{\nabla w}\,\mathrm{d} {\boldsymbol{x}}
\right|\nonumber\\
&\le \norm[X]{{\mathcal U}(b)} \norm[X]{w}\norm[{E}]{a-b}\nonumber\\
&\le \frac{\norm[X']{f}}{\rho(b)}
\norm[X]{w}\norm[{E}]{a-b},
\end{align*}
and thus
\begin{equation}\label{eq:lipschitz}
\norm[X]{{\mathcal U}(a)-{\mathcal U}(b)}\le \norm[X']{f}
\frac{\norm[{E}]{a}}{\rho(b)}
\norm[{E}]{a-b}.
\end{equation}
\end{enumerate}
Hence, by Proposition \ref{prop:bdX}, if
$(\psi_j)_{j\in\mathbb{N}}\subset {E}$ such that with
$b_j:=\norm[{E}]{\psi_j}$ it holds ${\boldsymbol{b}}\in\ell^1(\mathbb{N})$, then the
function
$$
u({\boldsymbol{y}})=\lim_{N\to\infty}{\mathcal U}\left(\exp\left(\sum_{j=1}^N
y_j\psi_j\right)\right) \in L^2(U,X;\gamma)
$$
is well-defined and $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic by Proposition \ref{prop:bdX}.
This example can easily be generalized to spaces of higher-regularity,
e.g.\ if ${\domain}\subseteq\mathbb{R}^d$ is a bounded $C^{s-1}$ domain for some
$s\in\mathbb{N}$, $s\ge 2$, then we may set
$X:= H_0^1({\domain};\mathbb{C})\cap H^{s}({\domain};\mathbb{C})$ and ${E}:=W^{s}_\infty({\domain};\mathbb{C})$
and repeat the above calculation.
\subsubsection{Linear parabolic PDE with parametric coefficient}
\label{sec:LinParPDE}
Let $0<T<\infty$ denote a finite time-horizon and let ${\domain}$ be a
bounded domain with Lipschitz boundary $\partial {\domain}$ in ${\mathbb R}^d$. We
define $I:=(0,T)$ and consider the initial boundary value problem
(IBVP for short) for the linear parabolic PDE
\begin{equation}\label{eq:parabolic}
\begin{cases}
\frac{\partial u(t,{\boldsymbol{x}})}{\partial t} - \operatorname{div}\big(a({\boldsymbol{x}})\nabla
u(t,{\boldsymbol{x}})\big)=f(t,{\boldsymbol{x}}), \qquad (t,{\boldsymbol{x}})\in I\times {\domain},
\\
u|_{\partial {\domain}\times I}=0,
\\
u|_{t=0}=u_0({\boldsymbol{x}}).
\end{cases}
\end{equation}
In this section, we prove that the solution to this problem satisfies
the assumptions of Proposition \ref{prop:bdX}. We first review results on
the existence and uniqueness of solutions to the equation
\eqref{eq:parabolic}. We refer to \cite{SchS} and the references
there for proofs and more detailed discussion.
We denote $V:=H_0^1({\domain};\mathbb{C})$ and $V'=H^{-1}({\domain};\mathbb{C})$. The parabolic
IBPV given by equation \eqref{eq:parabolic} is a well-posed operator
equation in the intersection space of Bochner spaces (e.g.\
\cite[Appendix]{SchS}, and e.g.\ \cite{Wloka,Evan10} for the
definition of spaces)
\begin{equation*}
X := L^2(I,V)\cap H^1(I,V') = \big(L^2(I)\otimes V \big)\cap \big(H^1(I)\otimes V'\big)
\end{equation*}
equipped with the sum norm
\begin{equation*}
\|u\|_{X} := \Big(\|u\|_{L^2(I,V)}^2 +\|u\|_{H^1(I,V')}^2 \Big)^{1/2},\qquad u\in X,
\end{equation*}
where
$$
\|u\|_{L^2(I,V)}^2 = \int_I \|u(t, \cdot) \|_V^2\, \rd t\, ,
$$
and
$$
\|u\|_{H^1(I,V')}^2 = \int_I \| \partial_t u(t, \cdot) \|_{V'}^2\, \rd
t\,.
$$
To state a space-time variational formulation and to specify the data
space for \eqref{eq:parabolic}, we introduce the test-function space
\begin{equation*}\label{eq:ParIBVPY}
Y
= L^2(I,V)\times L^2({\mathrm D})
= \big(L^2(I)\otimes V \big) \times L^2({\mathrm D})
\end{equation*}
which we endow with the norm
\begin{equation*}
\|v\|_{Y}
=
\Big(\|v_1\|_{L^2(I,V)}^2 + \|v_2\|_{L^2({\mathrm D})}^2\Big)^{1/2},\qquad v=(v_1,v_2)\in Y\,.
\end{equation*}
Given a time-independent diffusion coefficient $a\in L^\infty({\domain};\mathbb{C})$
and $(f,u_0)\in Y'$, the continuous sesquilinear and antilinear forms
corresponding to the parabolic problem \eqref{eq:parabolic} reads for
$u\in X$ and $v=(v_1,v_2)\in Y$ as
\begin{equation*}
\begin{split}
B(u,v;a)& := \int_I \int_{\domain} \partial_t u\,\overline{v_1}\rd {\boldsymbol{x}}
\rd t + \int_I\int_{\domain} a\nabla u \cdot \overline{\nabla v_1} \rd
{\boldsymbol{x}} \rd t + \int_{\domain} u_0\,\overline{v_2 } \rd {\boldsymbol{x}}
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
L(v) := \ \int_I \big\langle f(t,\cdot),v_1(t,\cdot) \big\rangle
\rd t + \int_{\domain} u_0 \,\overline{v_2 } \rd {\boldsymbol{x}},
\end{split}
\end{equation*}
where $\langle \cdot,\cdot \rangle$ is the anti-duality pairing
between $V'$ and $V$. Then the space-time variational formulation of
equation \eqref{eq:parabolic} is: Find ${\mathcal U}(a)\in X$ such that
\begin{equation}\label{para-weak}
B({\mathcal U}(a),v;a)=L(v),\quad \forall v\in Y\,.
\end{equation}
The existence and uniqueness of solution to the equation
\eqref{para-weak} was proved in \cite{SchS} which reads as follows.
\begin{proposition}\label{prop:parabolic}
Assume that $(f,u_0)\in Y'$ and that
\begin{equation}\label{eq:para-uni}
0 < \rho(a) := \underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,inf}}\,\Re(a({\boldsymbol{x}}))
\leq |a({\boldsymbol{x}})|
\leq \|a\|_{L^\infty}
<\infty,\qquad {\boldsymbol{x}}\in {\mathrm D}
\end{equation}
Then the parabolic operator $\mathcal{B}\in \mathcal{L}(X,Y')$
defined by $$(\mathcal{B}u)(v)=B(u,v;a),$$ is an isomorphism and
$\mathcal{B}^{-1}:Y \to X$ has the norm
\begin{equation*}
\|\mathcal{B}^{-1}\| \leq \frac{1}{\beta(a)},
\end{equation*}
where
\begin{equation*}
\beta(a) := \frac{\min\big(\rho(a) \|a\|_{L^\infty}^{-2}, \rho(a) \big)}{\sqrt{2\max(\rho(a)^{-2},1)+\vartheta^2}}
\qquad\text{and}\qquad
\vartheta := \sup_{w \not=0, w\in X} \frac{\| w(0,\cdot)\|_{L^2({\mathrm D})}}{\|w\|_{X}}\,.
\end{equation*}
The constant $\vartheta$ depends only on $T$.
\end{proposition}
With the set of admissible diffusion coefficients
$$O:=\set{a\in L_\infty({\domain},\mathbb{C})}{\rho(a) >0},$$
from the above proposition we immediately deduce that for given
$(f,u_0)\in Y'$, the map $${\mathcal U}: O \to X: a \to {\mathcal U}(a)$$ is
well-defined.
Furthermore, there holds the a-priori estimate
\begin{equation}\label{eq:para-bound}
\|{\mathcal U}(a)\|_{X}
\leq
\frac{1}{\beta(a)}\Big(\|f\|_{L^2(I,V')}^2 + \|u_0\|_{L^2}^2\Big)^{1/2}\,.
\end{equation}
This bound is a consequence of the following result which states that
the data-to-solution map $ a \to {\mathcal U}(a)$ is locally Lipschitz
continuous.
\begin{lemma} \label{lem:para-Lip} Let $(f,u_0)\in Y'$. Assume that
${\mathcal U}(a)$ and ${\mathcal U}(b)$ be solutions to \eqref{para-weak} with
coefficients $a$, $b$ satisfying \eqref{eq:para-uni}, respectively.
Then, with the function $\beta( \cdot)$ in variable $a$ as in
Proposition \ref{prop:parabolic}, we have
\begin{equation*}
\|{\mathcal U}(a)-{\mathcal U}(b)\|_{X}
\leq
\frac{1}{\beta(a)\beta(b)}\|a-b\|_{L_\infty}\Big(\|f\|_{L^2(I,V')}^2 + \|u_0\|_{L^2}^2\Big)^{1/2}\,.
\end{equation*}
\end{lemma}
\begin{proof}
From \eqref{para-weak} we find that for $w := {\mathcal U}(a)-{\mathcal U}(b)$,
\begin{equation*}
\begin{split}
\int_I \int_{\domain} \partial_t w\,\overline{v_1 } \rd {\boldsymbol{x}}\, \rd t & +
\int_I\int_{\domain} a \nabla w \cdot \overline{\nabla v_1 } \, \rd {\boldsymbol{x}}
\, \rd t + \int_{\domain} w\big|_{t=0}\overline{v_2} \rd {\boldsymbol{x}}
\\
& = -\int_I \int_{\domain} \big(a-b\big) \nabla {\mathcal U}(b)\cdot
\overline{\nabla v_1} \, \rd {\boldsymbol{x}} \, \rd t\,.
\end{split}
\end{equation*}
This is a parabolic equation in the variational form with
$(\tilde{f},0)\in Y'$ where $\tilde{f}: L^2(I,V) \to \mathbb{C}$ is given by
\begin{equation*}
\tilde{f}(v_1)
:=
-\int_I \int_{\domain} \big(a-b\big) \nabla {\mathcal U}(b)\cdot \overline{\nabla v_1} \, \rd {\boldsymbol{x}} \, \rd t\,, \qquad v_1\in L^2(I,V).
\end{equation*}
Now applying Proposition \ref{prop:parabolic} we find
\begin{equation} \label{eq:ua-ua} \|{\mathcal U}(a)-{\mathcal U}(b)\|_{X} \leq
\frac{\|\tilde{f}\|_{L^2(I,V')}}{\beta(a)}.
\end{equation}
We also have
\begin{equation*}
\begin{split}
\|\tilde{f}\|_{L^2(I,V')} = \sup_{\|v_1\|_{L^2(I,V)}=1} |
\tilde{f}(v_1) | & \leq \|a-b\|_{L_\infty} \|{\mathcal U}(b)\|_{L^2(I,V)}
\|v_1\|_{L^2(I,V)}
\\
& \leq \|a-b\|_{L_\infty}
\frac{1}{\beta(b)}\Big(\|f\|_{L^2(I,V')}^2 +
\|u_0\|_{L^2}^2\Big)^{1/2},
\end{split}
\end{equation*}
where in the last estimate we used again Proposition
\ref{prop:parabolic}. Inserting this into \eqref{eq:ua-ua} we obtain
the desired result. \hfill
\end{proof}
We are now in position to verify the assumptions
\ref{item:uhol}--\ref{item:loclip} of Proposition \ref{prop:bdX} for the
data-to-solution map $a\mapsto {\mathcal U}(a)$ to the equation
\eqref{eq:parabolic}.
\begin{enumerate}
\item For the first condition, it has been shown that the weak
solution to the linear parabolic PDEs \eqref{eq:parabolic} depends
holomorphically on the data $a\in O$ by the Ladyzhenskaya-Babu\v
ska-Brezzi theorem in Hilbert spaces over $\mathbb{C}$, see e.g.\
\cite[Pages 26, 27]{CoDe}.
\item Let $a\in O$. Using the elementary estimate $a+b\leq ab$ with
$a,b\geq 2$, we get
\begin{equation*}
\begin{split}
\sqrt{2\max(\rho(a)^{-2},1)+\vartheta^2} & \leq
\sqrt{2\max(\rho(a)^{-2},1)+\max(\vartheta^2,2)}
\\
&\leq \sqrt{2\max(\rho(a)^{-2},1) \max(\vartheta^2,2)} \leq
\max(\vartheta\sqrt{2},2) ({\rho(a)}^{-1}+1)\,.
\end{split}
\end{equation*}
Hence, from \eqref{eq:para-bound} we can bound
\begin{equation} \label{eq-1.beta(a)}
\begin{split}
\|{\mathcal U}(a)\|_{X} \leq \frac{C_0(\rho(a)^{-1}+1)}{\min\big(\rho(a)
\|a\|_{L^\infty}^{-2}, \rho(a)\big)} & =
\frac{C_0(1+\rho(a))}{\rho(a)^2\min\big(\|a\|_{L^\infty}^{-2},
1\big)}
\\
&\leq
\frac{C_0(1+\|a\|_{L^\infty})\|a\|_{L^\infty}^2}{\min\big(\rho(a)^{4},
1\big)} \leq
C_0\bigg(\frac{1+\|a\|_{L^\infty}}{\min\big(\rho(a),
1\big)}\bigg)^4\,,
\end{split}
\end{equation}
where
$$C_0= \max(\vartheta\sqrt{2},2) \big(\|f\|_{L^2(I,V')}^2 + \|u_0\|_{L^2}^2\big)^{1/2}.$$
\item The third assumption follows from Lemma \ref{lem:para-Lip} and
the part (ii), i.e., for $a,b\in O$ holds
\begin{equation}\label{eq-ua-ub}
\|{\mathcal U}(a)-{\mathcal U}(b)\|_{X}
\leq
C \bigg(\frac{1+\|a\|_{L^\infty}}{\min\big(\rho(a), 1\big)}\bigg)^4
\bigg(\frac{1+\|b\|_{L^\infty}}{\min\big(\rho(b), 1\big)}\bigg)^4 \|a-b\|_{L_\infty},
\end{equation}
for some $C>0$ depending on $f$, $u_0$ and $T$.
\end{enumerate}
We continue studying the holomorphy of the solution map to the
equation \eqref{eq:parabolic} in function space of higher-regularity.
Denote by $H^1(I,L^2({\mathrm D}))$ the space of all functions
$v(t,{\boldsymbol{x}})\in L^2(I,L^2({\mathrm D}))$ such that the norm
$$
\|v\|_{H^1(I,L^2)} := \Big(\|v\|_{L(I,L^2)}^2 + \|\partial_t
v\|_{L^2(I,L^2)}^2 \Big)^{1/2}
$$
is finite. We put
$$
Z := L^2(I,W)\cap H^1(I,L^2({\mathrm D})), \quad W:=\big\{ v\in V: \
\Delta v\in L^2({\mathrm D})\big\},
$$
and
$$
\|v\|_Z := \Big( \|v\|_{H^1(I,L^2)}^2 + \|v\|_{L^2(I,W)}^2\Big)^{1/2}.
$$
In the following the constant $C$ and $C'$ may change their values
from line to line.
\begin{lemma} \label{lem:Z1} Assume that
$a\in W^{1}_\infty({\mathrm D})\cap O$ and $f\in L^2(I,L^2({\mathrm D}))$
and $u_0\in V$. Suppose further that ${\mathcal U}(a)\in X$ is the weak
solution to the equation \eqref{eq:parabolic}. Then
${\mathcal U}(a)\in L^2(I,W)\cap H^1(I,L^2({\mathrm D}))$. Furthermore,
\begin{equation*}
\begin{aligned}
\| \partial_t {\mathcal U}(a) \|_{L^2(I,L^2)} \leq \bigg(\frac{1+\|
a\|_{W^{1}_\infty}}{\min(\rho(a),1)} \bigg)^4 \big(
\|u_0\|_{V} + \|f\|_{L^2(I,L^2)} \big)^{1/2},
\end{aligned}
\end{equation*}
and
\begin{align*}
\| \Delta {\mathcal U}(a) \|_{L^2(I,L^2)}
&\leq C \bigg(\frac{1+\| a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5
\big(\|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2},
\end{align*}
where $C>0$ independent of $f$ and $u_0$. Therefore,
$$
\|{\mathcal U}(a)\|_Z \leq C \bigg(\frac{1+\|
a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5 \big(\|u_0\|_{V}^2 +
\|f\|_{L^2(I,L^2)}^2\big)^{1/2} .
$$
\end{lemma}
\begin{proof}
The argument follows along the lines of, e.g., \cite[Section
7.1.3]{Evan10} by separation of variables. Let
$(\omega_k)_{k\in {\mathbb N}}\subset V$ be an orthogonal basis which is
orthonormal basis of $L^2({\mathrm D})$, {[eigenbasis in polygon
generally not smooth]}, see, e.g.\ \cite[Page 353]{Evan10}. Let
further, for $m\in \mathbb{N}$,
$$
{\mathcal U}_m(a)=\sum_{k=1}^m d_m^k(t)\omega_k\in V_m
$$
be a Galerkin approximation to ${\mathcal U}(a)$ on
$V_m:={\rm span}\{\omega_k,\ k=1,\ldots,m\}$.
Then we have
$$\partial_t{\mathcal U}_m(a) =\sum_{k=1}^m \frac{d}{dt}d_m^k (t)\omega_k\in V_m.$$
Multiplying both sides with $\partial_t{\mathcal U}_m(a)$ we get
\begin{equation*}
\begin{aligned}
\int_{\domain} \partial_t {\mathcal U}_m(a) \partial_t\overline{ {\mathcal U}_m(a) } \rd
{\boldsymbol{x}} + \int_{\domain} a\nabla {\mathcal U}_m(a) \cdot \partial_t \overline{\nabla
{\mathcal U}_m(a)} \, \rd {\boldsymbol{x}} = \int_{\domain} f \partial_t \overline{ {\mathcal U}_m(a)
} \, \rd {\boldsymbol{x}}.
\end{aligned}
\end{equation*}
The conjugate equation is given by
\begin{equation*}
\begin{aligned}
\int_{\domain} \partial_t {\mathcal U}_m(a) \partial_t \overline{ {\mathcal U}_m(a) } \rd
{\boldsymbol{x}} & + \int_{\domain} \overline{a}\, \overline{\nabla {\mathcal U}_m(a) }\cdot
\partial_t\nabla {\mathcal U}_m(a) \, \rd {\boldsymbol{x}} = \int_{\domain} \bar{f} \partial_t
{ {\mathcal U}_m(a) } \, \rd {\boldsymbol{x}}.
\end{aligned}
\end{equation*}
Consequently we obtain
\begin{equation*}
\begin{aligned}
2\| \partial_t {\mathcal U}_m(a) \|_{L^2}^2 & + \frac{\rd}{\rd t} \int_{\domain}
\Re(a) |\nabla {\mathcal U}_m(a) |^2 \rd {\boldsymbol{x}} = \int_{\domain} f \partial_t
\overline{ {\mathcal U}_m(a) } \, \rd {\boldsymbol{x}}+\int_{\domain} \bar{f} \partial_t {
{\mathcal U}_m(a) } \, \rd {\boldsymbol{x}}.
\end{aligned}
\end{equation*}
Integrating both sides with respect to $t$ on $I$ and using the
Cauchy-Schwarz inequality we arrive at
\begin{equation*}
\begin{aligned}
2\| \partial_t {\mathcal U}_m(a) \|_{L^2(I,L^2)}^2 & + \int_{\domain} \Re(a)
\big|\nabla {\mathcal U}_m(a) \big|_{t=T}\big|^2 \rd {\boldsymbol{x}}
\\
&\leq \int_{\domain} \Re(a) \big|\nabla {\mathcal U}_m(a)\big|_{t=0}\big|^2 \rd
{\boldsymbol{x}} + \|f\|_{L^2(I,L^2)}^2 + \| \partial_t {\mathcal U}_m(a)
\|_{L^2(I,L^2)}^2,
\end{aligned}
\end{equation*}
which implies
\begin{equation}\label{eq-partial-t-u}
\begin{aligned}
\| \partial_t {\mathcal U}_m(a) \|_{L^2(I,L^2)}^2 & \leq \int_{\domain} \Re(a)
\big|\nabla {\mathcal U}_m(a)\big|_{t=0}\big|^2 \rd {\boldsymbol{x}} +
\|f\|_{L^2(I,L^2)}^2
\\
& \leq \|a\|_{L^\infty} \big\|\nabla
{\mathcal U}_m(a)\big|_{t=0}\big\|_{L^2}^2 + \|f\|_{L^2(I,L^2)}^2
\\
&\leq \|a\|_{L^\infty} \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2 ,
\end{aligned}
\end{equation}
where we used the bounds
$\|\nabla {\mathcal U}_m(a)\big|_{t=0}\|_{L^2} \leq \|u_0\|_{V}$, see
\cite[Page 362]{Evan10}.
Passing to limits we deduce that
\begin{align*}
\| \partial_t {\mathcal U}(a) \|_{L^2(I,L^2)}
&
\leq \big(\|a\|_{L^\infty} \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
&\leq
(\|a\|_{L^\infty} + 1)^{1/2}\big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
&
\leq
C\bigg(\frac{1+\|a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^4
\big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2} .
\end{align*}
We also have from \eqref{eq:para-bound} and \eqref{eq-1.beta(a)} that
\begin{equation}\label{eq-u-L2L2}
\begin{aligned}
\| {\mathcal U}(a) \|_{L^2(I,L^2)} \leq C\| {\mathcal U}(a) \|_{L^2(I,V)} & \leq
\frac{C}{\beta(a)} \big(\|f\|_{L^2(I,V')}^2 +
\|u_0\|_{L^2}^2\big)^{1/2}
\\
& \leq \frac{C}{\beta(a)} \big( \|u_0\|_{V}^2 +
\|f\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
& \leq C\bigg(\frac{1+\|a\|_{L^\infty}}{\min({\rho(a)},1)}
\bigg)^4 \big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
& \leq C\bigg(\frac{1+\|a\|_{W^{1}_\infty}}{\min({\rho(a)},1)}
\bigg)^4 \big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2} .
\end{aligned}
\end{equation}
We now estimate $\| \Delta {\mathcal U}(a) \|_{L^2(I,L^2)}$. From the identity
(valid in $L^2(I,L^2({\mathrm D}))$)
\begin{align*}
- \Delta {\mathcal U}(a) = \frac{1}{a}\big[\nabla a \cdot \nabla {\mathcal U}(a) + f -\partial_t {\mathcal U}(a) \big],
\end{align*}
and \eqref{eq-partial-t-u}, \eqref{eq-u-L2L2} we obtain that
\begin{align*}
\| \Delta {\mathcal U}(a) \|_{L^2(I,L^2)}
&
\leq
\frac{1}{\rho(a)}\bigg[\| a\|_{W^{1}_\infty} \| {\mathcal U}(a) \|_{L^2(I,V)}
+
\| f\|_{L^2(I,L^2)} + \|\partial_t {\mathcal U}(a)\|_{L^2(I,L^2)} \bigg]
\\
&
\leq
C\frac{\| a\|_{W^{1}_\infty}}{{\rho(a)}}
\bigg(\frac{1+\|a\|_{L^\infty}}{\min({\rho(a)},1)}\bigg)^4\big(\|u_0\|_{V}^2
+
\|f\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
&
\leq C \bigg(\frac{1+\| a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5
\big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2},
\end{align*}
with $C>0$ independent of $f$ and $u_0$. Combining this and
\eqref{eq-partial-t-u}, \eqref{eq-u-L2L2}, the desired result follows.
\end{proof}
\begin{lemma} \label{lem-para-Z2} Assume $f\in L^2(I,L^2({\mathrm D}))$
and $u_0\in V$. Let ${\mathcal U}(a)$ and ${\mathcal U}(b)$ be the solutions to
\eqref{para-weak} with $a,b\in W^{1}_\infty({\mathrm D})\cap O$,
respectively. Then we have
\begin{equation*}
\begin{aligned}
\|{\mathcal U}(a)-{\mathcal U}(b)\|_Z & \leq C' \bigg(\frac{1+\|
a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5
\bigg(\frac{1+\| b\|_{W^{1}_\infty}}{\min({\rho(b)},1)} \bigg)^5
\|a-b\|_{W^{1}_\infty},
\end{aligned}
\end{equation*}
with $C'>0$ depending on $f$ and $u_0$.
\end{lemma}
\begin{proof}
Denote $w := {\mathcal U}(a)-{\mathcal U}(b)$. Then $w$ is the solution to the
equation
\begin{equation} \label{eq-equation-w}
\begin{cases}
{\partial_t w - \operatorname{div}\big(a\nabla w\big)= \nabla (a-b)\cdot\nabla
{\mathcal U}(b) + (a-b)\Delta {\mathcal U}(b), }
\\
w|_{\partial {\domain}\times I}=0,
\\
w|_{t=0}=0.
\end{cases}
\end{equation}
Hence
$$
-\Delta w = \frac{1}{a}\bigg[\nabla a\cdot \nabla w + \nabla
(a-b)\cdot\nabla {\mathcal U}(b) + (a-b)\Delta {\mathcal U}(b) -\partial_t w \bigg]
$$
which leads to
\begin{equation*}\label{eq:DeltaParab}
\begin{aligned}
\| \Delta w \|_{L^2(I,L^2)} &\leq \frac{1}{{\rho(a)}}\bigg[
\|a\|_{W^{1}_\infty} \|w\|_{L^2(I,V)} + \|\partial_t
w\|_{L^2(I,L^2)}
\\
&\ \ + \|a-b\|_{W^{1}_\infty } \big(\|{\mathcal U}(b)\|_{L^2(I,W)}
+\|{\mathcal U}(b)\|_{L^2(I,V)}\big) \bigg].
\end{aligned}
\end{equation*}
Lemma \ref{lem:Z1} gives that
\begin{align*}
\|\partial_t w\|_{L^2(I,L^2)}
&
\leq \bigg(\frac{1+\| a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5\big(\|\nabla (a-b)\cdot\nabla {\mathcal U}(b)\|_{L^2(I,L^2)}^2 + \|(a-b)\Delta {\mathcal U}(b)\|_{L^2(I,L^2)}^2\big)^{1/2}
\\
&
\leq \bigg(\frac{1+\| a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5 \|a-b\|_{W^{1}_\infty} \big(\|{\mathcal U}(b)\|_{L^2(I,W)}^2 +\|{\mathcal U}(b)\|_{L^2(I,V)}^2\big)^{1/2},
\end{align*}
and
\begin{align*}
\|{\mathcal U}(b)\|_{L^2(I,W)} +\|{\mathcal U}(b)\|_{L^2(I,V)}
&
\leq C \bigg(\frac{1+\| b\|_{{W^{1}_\infty}}}{\min({\rho(b)},1)} \bigg)^5\big( \|u_0\|_{V}^2 + \|f\|_{L^2(I,L^2)}^2\big)^{1/2},
\end{align*}
which implies
\begin{align*}
& \|a-b\|_{W^{1}_\infty} \big(\|{\mathcal U}(b)\|_{L^2(I,W)} +\|{\mathcal U}(b)\|_{L^2(I,V)}\big) + \|\partial_t w\|_{L^2(I,L^2)}
\\
&
\leq C' \bigg(\frac{1+\| a\|_{W^{1}_\infty}}{\min({\rho(a)},1)} \bigg)^5 \bigg(\frac{1+\| b\|_{{W^{1}_\infty}}}{\min({\rho(b)},1)} \bigg)^5 \|a-b\|_{W^{1}_\infty}
\end{align*}
We also have
$$ \|w\|_{L^2(I,V)} \leq \frac{1}{\beta(a)\beta(b)}\|a-b\|_{L_\infty}
\leq C' \bigg(\frac{1+\|a\|_{L^\infty}}{\min({\rho(a)},1)} \bigg)^4
\bigg(\frac{1+\|b\|_{L^\infty}}{\min({\rho(b)},1)}
\bigg)^4\|a-b\|_{W^{1}_\infty},
$$
see \eqref{eq-ua-ub}. Hence
\begin{equation}\label{eq-delta-w}
\begin{aligned}
\| \Delta w \|_{L^2(I,L^2)} &\leq C' \bigg(\frac{1+\|
a\|_{{W^{1}_\infty}}}{\min({\rho(a)},1)} \bigg)^5
\bigg(\frac{1+\| b\|_{{W^{1}_\infty}}}{\min({\rho(b)},1)} \bigg)^5
\|a-b\|_{{W^{1}_\infty}}.
\end{aligned}
\end{equation}
Since the terms $ \|\partial_t w\|_{L^2(I,L^2)} $ and
$ \|w\|_{L^2(I,L^2)} $ are also bounded by the right side of
\eqref{eq-delta-w}, we arrive at
\begin{equation*}
\begin{aligned}
\|w\|_Z & = \Big( \| \Delta w \|_{L^2(I,L^2)}^2 + \|\partial_t
w\|_{L^2(I,L^2)}^2 + \|w\|_{L^2(I,L^2)}^2 \Big)^{1/2}
\\
& \leq C' \bigg(\frac{1+\| a\|_{{W^{1}_\infty}}}{\min(\rho(a),1)}
\bigg)^5 \bigg(\frac{1+\| b\|_{{W^{1}_\infty}}}{\min(\rho(b),1)}
\bigg)^5 \|a-b\|_{{W^{1}_\infty} }
\end{aligned}
\end{equation*}
which is the claim.
\end{proof}
From Lemma \ref{lem-para-Z2}, by the same argument as in the proof of
\cite[Proposition 4.5]{HS2} we can verify that the solution map
$a\mapsto {\mathcal U}(a)$ from ${W^{1}_\infty}({\mathrm D})\cap O$ to $Z$ is
holomorphic. If we assume further that
$(\psi_j)_{j\in\mathbb{N}}\subseteq {W^{1}_\infty}({\mathrm D})$ and with
$b_j:=\norm[{W^{1}_\infty}]{\psi_j}$, it holds ${\boldsymbol{b}}\in\ell^1(\mathbb{N})$ and
all the conditions in Proposition \ref{prop:bdX} are satisfied. Therefore
$u({\boldsymbol{y}})$ is $({\boldsymbol{b}},\xi,\delta,Z)$-holomorphic with appropriate $\xi$
and $\delta$.
\begin{remark}
\label{rmk:CompCond}
For $s>1$, let $$Z^s := \bigcap_{k=0}^s H^k(I,H^{2s-2k}({\mathrm D}))$$
with the norm
$$
\|v\|_{Z^s} = \Bigg(\sum_{k=0}^{s} \bigg\|\frac{\rd^{k}v}{\rd t^{k}}
\bigg\|_{L^2(I,H^{2s-2k})}^2\Bigg)^{1/2}.
$$
Assume that $a\in W^{2s-1}_\infty({\mathrm D})\cap O$. At present we do
not know whether the solution map $a \mapsto {\mathcal U}(a)$ from
$W^{2s-1}_\infty({\mathrm D}) \cap O$ to $Z^s$ is holomorphic. To obtain
the holomorphy of the solution map, we need a result similar to that
in Lemma \ref{lem-para-Z2}. In order for this to hold, higher-order
regularity and compatibility of the data for equation
\eqref{eq-equation-w} is required, i.e,
$$
g_0=0\in V ,\quad g_1=h(0)-Lg_0 \in V,\ldots,
g_s=\frac{\rd^{s-1}h}{\rd t^{s-1}}(0)-Lg_{s-1}\in V,
$$
where
$$h=\nabla (a-b)\cdot\nabla {\mathcal U}(b) + (a-b)\Delta {\mathcal U}(b), \quad L= \partial_t \cdot - \operatorname{div}\big(a\nabla \cdot \big).$$
See e.g.\ \cite[Theorem 27.2]{Wloka}. It is known that without such
compatibility, the solution will develop spatial singularities at the
corners and edges of ${\mathrm D}$, and temporal singularities as
$t\downarrow 0$; see e.g.\ \cite{KozRossPara}.
In general the compatibility condition does not hold when we only
assume that
$$
u_0\in H^{2s-1}({\mathrm D})\cap V \ \ {\rm and} \ \ \frac{\rd ^k f}{\rd t
^k}\in L^2(I,H^{2s-2k-2}({\mathrm D}))
$$
for $k=0,\ldots,s-1$.
\end{remark}
\subsubsection{Linear elastostatics with log-Gaussian modulus of
elasticity}
\label{sec:LinElast}
We illustrate the foregoing abstract setting of
Section~\ref{S:DefbxdHol} for another class of boundary value
problems. In computational mechanics, one is interested in the
numerical approximation of deformations of elastic bodies.
We refer to e.g.\ \cite{Truesdell} for an accessible exposition of the
mathematical foundations and assumptions.
In \emph{linearized elastostatics} one is concerned with small (in a suitable sense, see
\cite{Truesdell} for details) deformations.
We consider an elastic body occupying the domain
${\mathrm D}\subset \mathbb{R}^d$, $d=2,3$ (the physically relevant case naturally
is $d=3$, we include $d=2$ to cover the so-called model of
``plane-strain'' which is widely used in engineering, and has
governing equations with the same mathematical structure). In the
linear theory, small deformations of the elastic body occupying
${\mathrm D}$, subject to, e.g., body forces ${\boldsymbol{f}}: {\mathrm D} \to \mathbb{R}^d$
such as gravity are modelled in terms of the \emph{displacement field}
${\boldsymbol{u}}:{\mathrm D} \to \mathbb{R}^d$, describing the displacement of a
\emph{material point} ${\boldsymbol{x}}\in {\mathrm D}$ (see \cite{Truesdell} for a
discussion of axiomatics related to this mathematical concept).
Importantly, unlike the scale model problem considered up to this
point, modelling now involves vector fields of data (e.g., ${\boldsymbol{f}}$)
and solution (i.e., ${\boldsymbol{u}}$).
{ Governing equations for the mathematical model of linearly elastic
deformation, subject to homogeneous Dirichlet boundary conditions on
$\partial{\mathrm D}$, read: to find ${\boldsymbol{u}}:{\mathrm D}\to \mathbb{R}^d$ such that
\begin{equation}\label{eq:LinElPDE}
\begin{array}{rcl}
{\rm div}{\boldsymbol{\sigma}}[{\boldsymbol{u}}] + {\boldsymbol{f}} &=& 0 \quad\mbox{in}\;\;{\mathrm D} \;,
\\
{\boldsymbol{u}} & = & 0 \quad \mbox{on} \;\;\partial{\mathrm D}\;.
\end{array}
\end{equation}
Here ${\boldsymbol{\sigma}}:{\mathrm D} \to \mathbb{R}^{d\times d}_{{\rm sym}}$ is a symmetric
matrix function, the so-called \emph{stress tensor}. It depends on
the displacement field $u$ via the so-called (linearized)
\emph{strain tensor}
${\boldsymbol{\epsilon}}[{\boldsymbol{u}}]:{\domain}\to \mathbb{R}^{d\times d}_{{\rm sym}}$, which is given by
\begin{equation}\label{eq:LinElStrain}
{\boldsymbol{\epsilon}}[{\boldsymbol{u}}] := \frac{1}{2}\left({\rm grad}{\boldsymbol{u}} + ({\rm grad}{\boldsymbol{u}})^\top\right)\;,
\;\;
({\boldsymbol{\epsilon}}[{\boldsymbol{u}}])_{ij} := \frac{1}{2}(\partial_j u_i + \partial_i u_j)\;, i,j=1,...,d\;.
\end{equation}
}
{ In the linearized theory, the tensors ${\boldsymbol{\sigma}}$ and $\epsilon$ in
\eqref{eq:LinElPDE}, \eqref{eq:LinElStrain} are related by the
linear constitutive stress-strain relation (``Hooke's law'')
\begin{equation}\label{eq:Hooke}
{\boldsymbol{\sigma}} = {\tt A} {\boldsymbol{\epsilon}} \;.
\end{equation}
In \eqref{eq:Hooke}, ${\tt A}$ is a fourth order tensor field, i.e.
$${\tt A} = \{ {\tt A}_{ijkl} : i,j,k,l=1,...,d\},$$
with certain symmetries that must hold among its $d^4$ components
independent of the particular material constituting the elastic body
(see, e.g., \cite{Truesdell} for details). Thus, \eqref{eq:Hooke}
reads in components as $\sigma_{ij} = {\tt A}_{ijkl} \epsilon_{kl}$ with
summation over repeated indices implied. Let us now fix $d=3$.
Symmetry implies that $\epsilon$ and $\sigma$ are characterized by
$6$ components. If, in addition, the material constituting the
elastic body is \emph{isotropic}, the tensor ${\tt A}$ can in fact be
characterized by only two independent coefficient functions. We adopt
here the \emph{Poisson ratio}, denoted $\nu$, and the modulus of
elasticity ${\tt E}$. With these two parameters, the stress-strain law
\eqref{eq:Hooke} can be expressed in the component form
\begin{equation}\label{eq:HookeComp}
\left(\begin{array}{c} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{13} \\ \sigma_{23} \end{array} \right)
=
\frac{{\tt E}}{(1+\nu)(1-2\nu)}
\left(\begin{array}{cccccc}
1-\nu & \nu & \nu & 0 & 0 & 0
\\
\nu & 1-\nu & \nu & 0 & 0 & 0
\\
\nu & \nu & 1-\nu & 0 & 0 & 0
\\
0 & 0 & 0 & 1-2\nu & 0 & 0
\\
0 & 0 & 0 & 0& 1-2\nu & 0
\\
0 & 0 & 0 & 0 & 0 & 1-2\nu
\end{array}
\right)
\left(\begin{array}{c} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\
\epsilon_{12} \\ \epsilon_{13} \\ \epsilon_{23}
\end{array} \right)
\;.
\end{equation}
We see from \eqref{eq:HookeComp} that for isotropic elastic
materials, the tensor ${\tt A}$ is proportional to the modulus
${\tt E} > 0$, with the Poisson ratio $\nu\in [0,1/2)$. We
remark that for common materials, $\nu\uparrow 1/2$ arises in
the so-called \emph{incompressible limit}. In that case,
\eqref{eq:LinElPDE} can be described by the Stokes equations.
}
{ With the constitutive law \eqref{eq:Hooke}, we may cast the
governing equation \eqref{eq:LinElPDE} into the so-called
``primal'', or ``displacement-formulation'': find
${\boldsymbol{u}}:{\mathrm D}\to \mathbb{R}^d$ such that
\begin{equation}\label{eq:LinElDispl}
-{\rm div}({\tt A}{\boldsymbol{\epsilon}}[{\boldsymbol{u}}]) = f \quad \mbox{in}\;\;{\mathrm D}\;,
\qquad {\boldsymbol{u}}|_{\partial{\mathrm D}} = 0\;.
\end{equation}
This form is structurally identical to the scalar diffusion
problem \eqref{PDE}. }
{ Accordingly, we fix $\nu\in [0,1/2)$ and model uncertainty in
the elastic modulus ${\tt E} > 0$ in \eqref{eq:HookeComp} by a
log-Gaussian random field
\begin{equation}\label{eq:YoungGRF}
{\tt E}({\boldsymbol{y}})({\boldsymbol{x}}) := \exp(b({\boldsymbol{y}}))({\boldsymbol{x}})\;, \quad {\boldsymbol{x}}\in {\mathrm D}\;, \;\; {\boldsymbol{y}}\in U\;.
\end{equation}
Here, $b({\boldsymbol{y}})$ is a Gaussian series representation of the GRF
$b(Y(\omega))$ as discussed in Section~\ref{S:GSer}. The
log-Gaussian ansatz ${\tt E} = \exp(b)$ ensures
$$
E_{\min}({\boldsymbol{y}}) := {\rm ess}\inf_{{\boldsymbol{x}}\in {\mathrm D}} {\tt E}({\boldsymbol{y}})({\boldsymbol{x}}) > 0
\qquad \mbox{$\gamma$-a.e.} \ {\boldsymbol{y}}\in U\;,
$$
i.e., the $\gamma$-almost sure positivity of (realizations of) the
elastic modulus ${\tt E}$. Denoting the $3\times 3$ matrix relating the
stress and strain components in \eqref{eq:HookeComp} also by ${\tt A}$
(this slight abuse of notation should, however, not cause confusion in
the following), we record that for $0\leq \nu < 1/2$, the matrix
${\tt A}$ is invertible:
\begin{equation}\label{eq:Ainv}
{\tt A}^{-1}
=
\frac{1}{{\tt E}}
\left(\begin{array}{cccccc}
1 & -\nu & -\nu & 0&0&0
\\
-\nu & 1 & -\nu & 0&0&0
\\
-\nu & -\nu & 1 & 0&0&0
\\
0& 0& 0& 1+\nu & 0 & 0
\\
0& 0& 0& 0 & 1+\nu & 0
\\
0& 0& 0& 0 & 0 & 1+\nu
\end{array}
\right)
\;.
\end{equation}
It readily follows from this explicit expression that due
to $${\tt E}^{-1}({\boldsymbol{y}})({\boldsymbol{x}}) = \exp(-b({\boldsymbol{y}})({\boldsymbol{x}})),$$ by the
Gerschgorin theorem invertibility holds for $\gamma$-a.e.
${\boldsymbol{y}}\in U$. Also, the components of ${\tt A}^{-1}$ are GRFs (which
are, however, fully correlated for deterministic $\nu$). }
{ Occasionally, instead of the constants ${\tt E}$ and $\nu$, one finds
the (equivalent) so-called \emph{Lam\'{e}-constants} $\lambda$,
$\mu$. They are related to ${\tt E}$ and $\nu$ by
\begin{equation}\label{eq:LameEnu}
\lambda = \frac{{\tt E} \nu}{(1+\nu)(1-2\nu)}\;,\quad
\mu = \frac{{\tt E}}{2(1+\nu)}\;.
\end{equation}
For GRF models \eqref{eq:YoungGRF} of ${\tt E}$, \eqref{eq:LameEnu}
shows that for each fixed $\nu \in (0,1/2)$, also the
Lam\'{e}-constants are GRFs \emph{which are fully correlated}.
This implies, in particular, that ``large'' realizations of the
GRF \eqref{eq:YoungGRF} do not cause so-called ``volume locking''
in the equilibrium equation \eqref{eq:LinElPDE}: this effect is
related to the elastic material described by the constitutive
equation \eqref{eq:Hooke} being nearly
incompressible. Incompressibility here arises as either
$\nu\uparrow 1/2$ at fixed ${\tt E}$ or, equivalently, as
$\lambda\to \infty$ at fixed $\mu$. }
{ Parametric weak solutions of \eqref{eq:LinElDispl} with
\eqref{eq:YoungGRF} are within the scope of the abstract theory
developed up to this point. To see this, we provide a variational
formulation of \eqref{eq:LinElDispl}. Assuming for convenience
homogeneous Dirichlet boundary conditions, we multiply
\eqref{eq:LinElDispl} by a test displacement field ${\boldsymbol{v}}\in X:=V^d$
with $V:=H_0^1({\domain})$, and integrate by parts, to obtain the weak
formulation: find ${\boldsymbol{u}}\in X$ such that, for all ${\boldsymbol{v}}\in X$ holds
(in the matrix-vector notation \eqref{eq:HookeComp})
\begin{equation}\label{eq:LinElVar}
\int_{{\mathrm D}} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \cdot {\tt A} {\boldsymbol{\epsilon}} [{\boldsymbol{u}}] \rd {\boldsymbol{x}}
=
2\mu ({\boldsymbol{\epsilon}}[{\boldsymbol{u}}],{\boldsymbol{\epsilon}}[{\boldsymbol{v}}]) + \lambda(\operatorname{div} {\boldsymbol{u}}, \operatorname{div} {\boldsymbol{v}}) = ({\boldsymbol{f}},{\boldsymbol{v}}) \;.
\end{equation}
The variational form \eqref{eq:LinElVar} suggests that, as
$\lambda\to\infty$ for fixed $\mu$, the ``volume-preservation''
constraint $\| \operatorname{div} {\boldsymbol{u}} \|_{L^2} = 0$ is imposed for ${\boldsymbol{v}}={\boldsymbol{u}}$ in
\eqref{eq:LinElVar}. }
Unique solvability of \eqref{eq:LinElVar} follows upon verifying
coercivity of the corresponding bilinear form on the left-hand side
of \eqref{eq:LinElVar}. It follows from \eqref{eq:HookeComp} and
\eqref{eq:Ainv} that
\begin{equation*}\label{eq:bileps}
\forall {\boldsymbol{v}}\in H^1({\mathrm D})^d: \quad
{\tt E} c_{\min}(\nu) \| {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \|^2_{L^2}
\leq
\int_{{\mathrm D}} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \cdot {\tt A} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \rd x
\leq
{\tt E} c_{\max}(\nu) \| {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \|^2_{L^2}
\;.
\end{equation*}
Here, the constants $c_{\min}, c_{\max}$ are positive and bounded
for $0<\nu<1/2$ and independent of ${\tt E}$.
For the log-Gaussian model \eqref{eq:YoungGRF} of the elastic
modulus ${\tt E}$, the relations \eqref{eq:LameEnu} show in particular,
that \emph{the volume-locking effect arises as in the deterministic
setting only if $\nu \simeq 1/2$, independent of the realization
of ${\tt E}({\boldsymbol{y}})$}. Let us consider well-posedness of the
variational formulation \eqref{eq:LinElVar}, for log-Gaussian,
parametric elastic modulus ${\tt E}({\boldsymbol{y}})$ as in \eqref{eq:YoungGRF}.
To this end, with ${\tt A}_1$ denoting the matrix ${\tt A}$ in
\eqref{eq:HookeComp} with ${\tt E}=1$, we introduce in
\eqref{eq:LinElVar} the parametric bilinear forms
\begin{equation*}\label{eq:ParmElas}
b({\boldsymbol{u}},{\boldsymbol{v}};{\boldsymbol{y}})
:=
{\tt E}({\boldsymbol{y}}) \int_{{\mathrm D}} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \cdot {\tt A}_1 {\boldsymbol{\epsilon}}[{\boldsymbol{u}}] \rd {\boldsymbol{x}}
=
\frac{{\tt E}({\boldsymbol{y}})}{1+\nu}
\left(
({\boldsymbol{\epsilon}}[{\boldsymbol{u}}],{\boldsymbol{\epsilon}}[{\boldsymbol{v}}]) + \frac{\nu}{1-2\nu} (\operatorname{div} {\boldsymbol{u}}, \operatorname{div} {\boldsymbol{v}})
\right)
\;.
\end{equation*}
Let us verify continuity and coercivity of the parametric bilinear
forms
\begin{equation} \label{parametric forms} \{ b(\cdot,\cdot
;{\boldsymbol{y}}):X\times X\to \mathbb{R}: {\boldsymbol{y}}\in U\},
\end{equation}
where recall, $U:= {\mathbb R}^\infty$. With ${\tt A}_1$ as defined above, we write
for arbitrary ${\boldsymbol{v}}\in X = H^1_0({\mathrm D})^d$, $d=2,3$, and for all
${\boldsymbol{y}}\in U_0\subset U$ where the set $U_0$ is as in \eqref{eq:U0},
$$
\begin{array}{rcl}
b({\boldsymbol{v}},{\boldsymbol{v}};{\boldsymbol{y}})
& = & \displaystyle
\int_{{\mathrm D}} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \cdot ({\tt A} {\boldsymbol{\epsilon}}[{\boldsymbol{v}}]) \rd{\boldsymbol{x}}
=
\int_{{\mathrm D}} E({\boldsymbol{y}}) \left({\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \cdot ({\tt A}_1 {\boldsymbol{\epsilon}}[{\boldsymbol{v}}]) \right)\rd {\boldsymbol{x}}
\\
& \geq & \displaystyle
c(\nu) \int_{{\mathrm D}} E({\boldsymbol{y}}) \| {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \|_2^2 \rd {\boldsymbol{x}}
\\
& \geq & \displaystyle
c(\nu) \exp(-\| b({\boldsymbol{y}}) \|_{L^\infty}) \int_{{\mathrm D}} \| {\boldsymbol{\epsilon}}[{\boldsymbol{v}}] \|_2^2 \rd {\boldsymbol{x}}
\\
& \geq & \displaystyle
\frac{c(\nu)}{2} a_{\min}({\boldsymbol{y}}) | {\boldsymbol{v}} |_{H^1}^2
\\
& \geq & \displaystyle
C_P \frac{c(\nu)}{2} a_{\min}({\boldsymbol{y}}) \| {\boldsymbol{v}} \|_{H^1}^2
\;.
\end{array}
$$
Here, in the last two steps we employed the first Korn's inequality,
and the Poincar\'{e} inequality, respectively. The lower bound
$E({\boldsymbol{y}})\geq \exp(-\| b({\boldsymbol{y}}) \|_{L^\infty })$ is identical to
\eqref{eq:Norms} in the scalar diffusion problem.
In a similar fashion, continuity of the bilinear forms
\eqref{parametric forms} may be established: there
holds,
$$
\forall {\boldsymbol{u}},{\boldsymbol{v}}\in X,\; \forall {\boldsymbol{y}} \in U_0 : \quad | b({\boldsymbol{u}},{\boldsymbol{v}};{\boldsymbol{y}}) |
\leq c'(\nu)\exp(\| b({\boldsymbol{y}}) \|_{L^\infty}) \| {\boldsymbol{u}} \|_{H^1} \| {\boldsymbol{v}}
\|_{H^1}.
$$
With continuity and coercivity of the parametric forms
\eqref{parametric forms} verified for ${\boldsymbol{y}}\in U_0$, the Lax-Milgram
lemma ensures for given ${\boldsymbol{f}}\in L^2({\mathrm D})^d$ the existence of
the parametric solution family
\begin{equation}\label{eq:ParSolElas}
\{ {\boldsymbol{u}}({\boldsymbol{y}}) \in X : b({\boldsymbol{u}},{\boldsymbol{v}};{\boldsymbol{y}}) = ({\boldsymbol{f}},{\boldsymbol{v}}) \ \forall {\boldsymbol{v}}\in X , {\boldsymbol{y}} \in U_0 \}\;.
\end{equation}
\begin{proposition}\label{prop:ParSolElas}
Under Assumption \ref{ass:Ass1}, $\gamma(U_0) = 1$.
For all $k\in {\mathbb N}$ there holds, with $\mathbb E(\cdot)$ denoting
expectation with respect to $\gamma$,
$$
\mathbb E\left( \exp(k\| b(\cdot) \|_{L^\infty}) \right) < \infty \;.
$$
The parametric solution family \eqref{eq:ParSolElas} of the parametric
elliptic boundary value problem \eqref{eq:LinElVar} with log-Gaussian
modulus $E({\boldsymbol{y}})$ as in \eqref{eq:YoungGRF} is in $L^k(U,V;\gamma)$ for
every finite $k\in {\mathbb N}$.
\end{proposition}
For the parametric solution family \eqref{eq:ParSolElas}, analytic
continuations into complex parameter domains, and parametric
regularity results may be developed in analogy to the development in
Sections \ref{sec:HsReg} and \ref{sec:KondrReg}. The key result for
bootstrapping to higher order regularity is, in the case of smooth
boundaries $\partial{\mathrm D}$, classical elliptic regularity for
linear, Agmon-Douglis-Nirenberg elliptic systems which comprise
\eqref{eq:LinElDispl}. In the polygonal (for $d=2$) or polyhedral
($d=3$) case, weighted regularity shifts in Kondrat'ev type spaces are
available in \cite[Theorem 5.2]{GuoBabElast} (for $d=2$) and in
\cite{CDN12} (for both, $d=2,3$).
\subsubsection{Maxwell equations with log-Gaussian permittivity}
\label{sec:Maxwell}
Similar models are available for time-harmonic, electromagnetic waves
in dielectric media with uncertain conductivity. We refer to
\cite{logNMax2018}, where log-Gaussian models are employed. There,
also the parametric regularity analysis of the parametric electric and
magnetic fields is discussed, albeit by real-variable methods. The
setting in \cite{logNMax2018} is, however, so that the presently
developed, complex variable methods can be brought to bear on it. We
refrain from developing the details.
\section{Parametric posterior analyticity and sparsity in BIPs}
\label{sec:BIP}
We have investigated the parametric regularity of the \emph{forward
solution maps} of linear PDEs with uncertain parametric inputs which
typically arise from GRF models for these inputs. We have provided an
analysis of parametric holomorphy and of sparsity in Wiener-Hermite PC
expansions of the corresponding parametric solution families.
We now explore this notion of parametric holomorphy in the context of
BIPs for these PDEs. For these PDEs we adopt the Bayesian setting as
outlined e.g.\ in \cite{DashtiStuart17} and the references there.
This Bayesian setting is briefly recapitulated in Section
\ref{sec:BIPFrmWelPsd}. With a suitable version of Bayes' theorem,
the main result is a (short) proof of parametric
$({\boldsymbol{b}},\xi,\delta,\mathbb{C})$-holomorphy of the Bayesian posterior density for
unbounded parameter ranges. This implies
summability rates of Wiener-Hermite PC expansion coefficients in
Wiener-Hermite PC expansions of the Bayesian posterior density, which
can be leveraged to obtain higher-order approximation rates that are
free from the curse of dimensionality for various deterministic
approximation methods of the Bayesian expectations, for several
classes of function space priors modelled by product measures on the
parameter sequences ${\boldsymbol{y}}$. In particular, the construction of
Gaussian priors described in Section \ref{S:GMSepHS} is applicable.
We remark that in \cite{CSAMS2011} holomorphy for a bounded parameter
domain (in connection with uniform prior measure) has been addressed
by complex variable arguments in the same fashion. We remark that in
\cite{RSAMSAT_Bip2017}, MC and QMC integration has been analyzed by
real-variable arguments for such Gaussian priors. In
\cite{HKS19_845}, corresponding results have been obtain also for
so-called \emph{Besov priors}, again by real-variable arguments for
the parametric posterior. Since the presently developed, quantified
parametric holomorphy results are independent of the particular
measure placed upon the parameter domain ${\mathbb R}^\infty$. The sparsity
and approximation rate bounds for the parametric, deterministic
posterior densities will imply approximate rate bounds also for prior
constructions beyond the Gaussian ones.
\subsection{Formulation and well-posedness}
\label{sec:BIPFrmWelPsd}
With ${E}$ and $X$ denoting separable Banach and Hilbert spaces
over $\mathbb{C}$, respectively, we consider a forward solution map
${\mathcal U}: {E}\to X$ and an observation map ${\boldsymbol{{\mathcal O}}}:X\to {\mathbb R}^m$. In the
context of the previous sections, ${\mathcal U}$ could denote again the map
which, associates a diffusion coefficient
$a\in {E}:=L^\infty({\domain};\mathbb{C})$, returns the solution
${\mathcal U}(a) \in X:=H_0^1({\domain};\mathbb{C})$ of \eqref{eq:elliptic}. We assume these
maps to be Borel measurable.
The inverse problem consists in determining the (expected value of an)
uncertain input datum $a\in {E}$ from noisy observation data
${\boldsymbol{\obs}}\in {\mathbb R}^m$. Here, the observation noise ${\boldsymbol{\eta}}\in {\mathbb R}^m$ is
assumed additive centered Gaussian, i.e.\ the observation data ${\boldsymbol{\obs}}$
for input $a$ is
\begin{equation*}\label{eq:ObsNois}
{\boldsymbol{\obs}} = {\boldsymbol{{\mathcal O}}}\circ{\mathcal U}(a) + {\boldsymbol{\eta}}\;,
\end{equation*}
where ${\boldsymbol{\eta}} \sim {\mathcal N}(0,{\boldsymbol{\Gamma}})$. We assume the observation
noise covariance $ {\boldsymbol{\Gamma}}\in\mathbb{R}^{m\times m}$ is symmetric positive
definite.
In the so-called Bayesian setting of the inverse problem, one assumes
that the uncertain input $a$ is modelled as random variable which is
distributed according to a prior measure $\pi_0$ on ${E}$. Then,
under suitable conditions, which are made precise in Theorem
\ref{thm:bdHol} below, the posterior distribution $\pi(\cdot|{\boldsymbol{\obs}})$
on the conditioned random variable ${\mathcal U}|{\boldsymbol{\obs}}$ is absolutely
continuous w.r.\ to the prior measure $\pi_0$ on ${E}$ and there
holds Bayes' theorem in the form
\begin{equation}\label{eq:Bayes}
\frac{\rd\pi(\cdot|{\boldsymbol{\obs}})}{\rd\pi_0}(a) = \frac{1}{Z} \Theta(a).
\end{equation}
In \eqref{eq:Bayes}, the posterior density $\Theta$ and the
normalization constant $Z$ are given by
\begin{equation}\label{eq:BayesDens}
\Theta(a) = \exp(-\Phi({\boldsymbol{\obs}};a)),\qquad
\Phi({\boldsymbol{\obs}};a) = \frac{1}{2} \norm[2]{{\boldsymbol{\Gamma}}^{-1/2}({\boldsymbol{\obs}} - {\boldsymbol{{\mathcal O}}}({\mathcal U}(a)))}^2,\qquad
Z = \mathbb E_{\pi_0}[\Theta(\cdot)]
\;.
\end{equation}
Additional conditions ensure that the posterior measure
$\pi(\cdot|{\boldsymbol{\obs}})$ is well-defined and that \eqref{eq:Bayes} holds
according to the following result from \cite{DashtiStuart17}.
\begin{proposition}\label{prop:Bayes}
Assume that ${\boldsymbol{{\mathcal O}}}\circ{\mathcal U}: {E}\to {\mathbb R}^m$ is continuous and
that $\pi_0({E}) = 1$. Then the posterior $\pi(\cdot|{\boldsymbol{\obs}})$ is
absolutely continuous with respect to $\pi_0$, and
\eqref{eq:BayesDens} holds.
\end{proposition}
The condition $\pi_0({E}) = 1$ can in fact be weakened to
$\pi_0({E}) > 0$ (e.g.\ \cite[Theorem 3.4]{DashtiStuart17}).
The solution of the BIP amounts to the evaluation of the posterior
expectation $\mathbb E_{\mu^{\boldsymbol{\obs}}}[\cdot]$ of a continuous linear map
$\phi:X\to H$ of the map ${\mathcal U}(a)$, where $H$ is a suitable Hilbert
space over $\mathbb{C}$. Solving the Bayesian inverse problem is thus closely
related to the numerical approximation of the posterior expectation
\begin{equation*}\label{eq:BIPPost}
\mathbb E_{\pi(\cdot|{\boldsymbol{\obs}})} [\phi({\mathcal U}(\cdot))] \in H\;.
\end{equation*}
For computational purposes, and to facilitate Wiener-Hermite PC
approximation of the density $\Theta$ in \eqref{eq:Bayes}, one
parametrizes the input data $a=a({\boldsymbol{y}})\in E$ by a Gaussian series
as discussed in Section \ref{S:GSer}. Inserting into $\Theta(a)$ in
\eqref{eq:Bayes}, \eqref{eq:BayesDens} this results in a
countably-parametric density $U\ni{\boldsymbol{y}}\mapsto \Theta(a({\boldsymbol{y}}))$, for
${\boldsymbol{y}}\in U = {\mathbb R}^\infty$, and the Gaussian reference measure $\pi_0$ on
$E$ in \eqref{eq:Bayes} is pushed forward into a countable
Gaussian product measure on $U$: using \eqref{eq:Bayes} and choosing a
Gaussian prior (e.g.\ \cite[Section 2.4]{DashtiStuart17} or
\cite{HKS19_845,LSS2009})
$$\pi_0 = \gamma = \bigotimes_{j\in {\mathbb N}} \,\mathrm{d} \gamma (y_j)$$ on
$U$, the Bayesian estimate, i.e.\ the posterior expectation, can then
be written as a (countably) iterated integral
\cite{CSAMS2011,DashtiStuart17,RSAMSAT_Bip2017} with respect to the
product GM $\gamma$, i.e.
\begin{equation}\label{eq:BIPSparse}
\mathbb E_{\pi(\cdot|{\boldsymbol{\obs}})}[\phi({\mathcal U}(a(\cdot)))] = \frac{1}{Z}
\int_{U} \phi({\mathcal U}(a({\boldsymbol{y}}))) \Theta(a({\boldsymbol{y}})) \,\mathrm{d}\gamma({\boldsymbol{y}}) \in H,
\quad
Z = \int_{U} \Theta(a({\boldsymbol{y}})) \,\mathrm{d}\gamma({\boldsymbol{y}}) \in {\mathbb R}.
\end{equation}
The parametric density $U\to {\mathbb R}$ in \eqref{eq:BIPSparse} which arises
in Bayesian PDE inversion under Gaussian prior and also under more
general, so-called Besov prior measures on $U$, see e.g.\ \cite[Section
2.3]{DashtiStuart17}, \cite{HKS19_845}. The parametric density
$$
{\boldsymbol{y}} \mapsto \phi({\mathcal U}(a({\boldsymbol{y}}))) \Theta(a({\boldsymbol{y}})) \;,
$$
inherits sparsity from the forward map ${\boldsymbol{y}} \mapsto {\mathcal U}(a({\boldsymbol{y}}))$,
which sparsity is expressed as before in terms of summability
properties of Wiener-Hermite PC expansion coefficients. We employ the
parametric holomorphy of the forward map $a\mapsto {\mathcal U}(a)$ to quantify
the sparsity of the posterior density $\Theta(a({\boldsymbol{y}}))$.
\subsection{Posterior parametric holomorphy}
\label{sec:BIPParmHol}
With a Gaussian series in the data space $V$, for the resulting
parametric data-to-solution map
$$u:U\to X: {\boldsymbol{y}}\mapsto {\mathcal U}(a({\boldsymbol{y}})),$$
we now prove that under certain conditions both, the corresponding
parametric posterior density
\begin{equation}\label{eq:TheParm}
{\boldsymbol{y}}\mapsto \exp\left( -({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}})))^\top {\boldsymbol{\Gamma}}^{-1} ({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}}))) \right)
\end{equation}
in \eqref{eq:BayesDens}, and the integrand
\begin{equation}\label{eq:integrand}
{\boldsymbol{y}}\mapsto \phi(u({\boldsymbol{y}}))
\exp\left( -({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}})))^\top {\boldsymbol{\Gamma}}^{-1} ({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}}))) \right)
\end{equation}
in \eqref{eq:BIPSparse} are $({\boldsymbol{b}},\xi,\delta,\mathbb{C})$-holomorphic and
$({\boldsymbol{b}},\xi,\delta,Q)$-holomorphic, respectively.
\begin{theorem}\label{thm:bdHol}
Let $r>0$. Assume that the map $u:U\to X$ is
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic with constant functions
$\varphi_N\equiv r$, $N\in\mathbb{N}$, in Definition~\ref{def:bdXHol}.
Let the observation noise covariance matrix
${\boldsymbol{\Gamma}}\in\mathbb{R}^{m\times m}$ be symmetric positive definite.
Then, for any bounded linear quantity of interest $\phi\in L(X,Q)$,
and for any observable ${\boldsymbol{{\mathcal O}}}\in (X')^m$ with arbitrary, finite
$m$, the function in \eqref{eq:TheParm} is
$({\boldsymbol{b}},\xi,\delta,\mathbb{C})$-holomorphic and the function in
\eqref{eq:integrand} is $({\boldsymbol{b}},\xi,\delta,Q)$-holomorphic.
\end{theorem}
\begin{proof}
We only show the statement for the parametric integrand in
\eqref{eq:integrand}, as the argument for the posterior density in
\eqref{eq:TheParm} is completely analogous.
Consider the map
\begin{equation*}
\Xi: \set{v\in X}{\norm[X]{v}\le r} \to Q :
v \mapsto \phi(v)\exp(-({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(v)){\boldsymbol{\Gamma}}^{-1}({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(v))).
\end{equation*}
This function is well-defined. We have
$|{\boldsymbol{{\mathcal O}}}(v)|\le \norm[X']{{\boldsymbol{{\mathcal O}}}} r$ and
$|\phi(v)|\le \norm[L(X;Q)]{\phi} r$ for all $v\in X$ with
$\norm[X]{v}\le r$. Since $\exp:\mathbb{C}\to\mathbb{C}$ is Lipschitz continuous on
compact subsets of $\mathbb{C}$ and since $\phi\in L(X;Q)$ is bounded linear
map (and thus Lipschitz continuous), we find that
\begin{equation*}
\sup_{\norm[X]{v}\le r}\norm[Q]{\Xi(v)}=:\tilde r<\infty
\end{equation*}
and that $$\Xi:\set{v\in X}{\norm[X]{v}\le r}\to\mathbb{C}$$ is Lipschitz
continuous with some Lipschitz constant $L>0$.
Let us recall that the $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy of
$u:U\to X$, implies the existence of (continuous) functions
$u_N \in L^2(\mathbb{R}^N,X;\gamma_N)$ such that with
$\tilde u_N({\boldsymbol{y}})=u_N(y_1,\dots,y_N)$ it holds
$\lim_{N\to\infty}\tilde u_N= u$ in the sense of
$L^2(U,X;\gamma)$. Furthermore, if
$$\sum_{j=1}^N b_j\varrho_j\le\delta$$
(i.e.~${\boldsymbol{\varrho}}=(\varrho_j)_{j=1}^N$ is $({\boldsymbol{b}},\xi)$-admissible in
the sense of Definition~\ref{def:bdXHol}), then $u_N$ allows a
holomorphic extension $$u_N:{\mathcal S}_{\boldsymbol{\varrho}}\to X$$ such that for all
${\boldsymbol{y}}\in \mathbb{R}^N$
\begin{equation}\label{eq:unbound}
\sup_{{\boldsymbol{z}}\in{\mathcal B}_{\boldsymbol{\varrho}}}\norm[X]{u_N({\boldsymbol{y}}+{\boldsymbol{z}})}\le \varphi_N({\boldsymbol{y}})=r\qquad\forall {\boldsymbol{y}}\in\mathbb{R}^N,
\end{equation}
see \eqref{eq:Sjrho} for the definition of ${\mathcal S}_{\boldsymbol{\varrho}}$ and
${\mathcal B}_{\boldsymbol{\varrho}}$.
We want to show that $f({\boldsymbol{y}}):=\Xi(u({\boldsymbol{y}}))$ is well-defined in
$L^2(U,Q;\gamma)$, and given as the limit of the functions
$$\tilde f_N({\boldsymbol{y}})=f_N((y_j)_{j=1}^N)$$ for all ${\boldsymbol{y}}\in U$ and
$N\in\mathbb{N}$, where $$f_N((y_j)_{j=1}^N)=\Xi(u_N((y_j)_{j=1}^N).$$ Note
at first that $f_N:\mathbb{R}^N\to Q$ is well-defined. In the case
$$\sum_{j=1}^Nb_j\varrho_j\le\delta,$$ $f_N$ allows a holomorphic
extension $f_N:{\mathcal S}_{\boldsymbol{\varrho}}\to X$ given through $\Xi\circ u_N$.
Using \eqref{eq:unbound}, this extension satisfies for any $N\in\mathbb{N}$
and any $({\boldsymbol{b}},\xi)$-admissible $\varrho\in (0,\infty)^N$
\begin{equation*}
\sup_{{\boldsymbol{z}}\in{\mathcal B}_{\boldsymbol{\varrho}}}|f_N({\boldsymbol{y}}+{\boldsymbol{z}})|
\le \sup_{\norm[X]{v}\le r}|\Xi(v)|=\tilde r\qquad\forall {\boldsymbol{y}}\in \mathbb{R}^N.
\end{equation*}
This shows assumptions \ref{item:hol}-\ref{item:varphi} of
Definition~\ref{def:bdXHol} for $f_N:\mathbb{R}^N\to Q$.
Finally we show assumption \ref{item:vN} of
Definition~\ref{def:bdXHol}. By assumption it holds
$\lim_{N\to\infty}\tilde u_N= u$ in the sense of
$L^2(U,X;\gamma)$. Thus for $f=\Xi\circ u$ and with
$f_N=\Xi\circ u_N$
\begin{align*}
\int_U\norm[Q]{f({\boldsymbol{y}})-f_N({\boldsymbol{y}})}^2 \,\mathrm{d}\gamma({\boldsymbol{y}})
& = \int_U\norm[Q]{\Xi(u({\boldsymbol{y}}))-\Xi(u_N({\boldsymbol{y}}))}^2\,\mathrm{d}\gamma({\boldsymbol{y}})
\\
&
\le L^2 \int_U\norm[X]{u({\boldsymbol{y}})-u_N({\boldsymbol{y}})}^2\,\mathrm{d}\gamma({\boldsymbol{y}}),
\end{align*}
which tends to $0$ as $N\to\infty$. Here we used that $L$ is a
Lipschitz constant of $\Xi$.
\end{proof}
Let us now discuss which functions satisfy the requirements of
Theorem~\ref{thm:bdHol}. Additional to
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy, we had to assume boundedness of the
holomorphic extensions in Definition~\ref{def:bdXHol}. For functions
of the type as in Proposition \ref{prop:bdX}
$ u({\boldsymbol{y}})=\lim_{N\to\infty}{\mathcal U}\left(\exp\left(\sum_{j=1}^N
y_j\psi_j\right)\right), $ the following result gives sufficient
conditions such that the assumptions of Theorem~\ref{thm:bdHol} are
satisfied for the forward map.
\begin{corollary}\label{cor:norma2}
Assume that ${\mathcal U}:O\to X$ and $(\psi_j)_{j\in\mathbb{N}}\subset {E}$
satisfy Assumptions \ref{item:uhol}, \ref{item:loclip} and
\ref{item:psi} of Proposition \ref{prop:bdX} and additionally for some
$r>0$
\begin{enumerate}
\setcounter{enumi}{1}
\item\label{item:norma2} $\norm[X]{{\mathcal U}(a)}\le r$ for all $a\in O$.
\end{enumerate}
Then
$$
u({\boldsymbol{y}})=\lim_{N\to\infty}{\mathcal U}\left(\exp\left(\sum_{j=1}^N
y_j\psi_j\right)\right) \in L^2(U,X;\gamma)
$$
is $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic with constant functions
$\varphi_N\equiv r$, $N\in\mathbb{N}$, in Definition~\ref{def:bdXHol}.
\end{corollary}
\begin{proof}
By Proposition \ref{prop:bdX}, $u$ is $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic.
Recalling the construction of $\varphi_N:\mathbb{R}^N\to\mathbb{R}$ in Step 3 of
the proof of Proposition \ref{prop:bdX}, we observe that $\varphi_N$
can be chosen as $\varphi_N\equiv r$.
\end{proof}
\subsection{Example: Parametric diffusion coefficient}
\label{sec:ExplBip}
We revisit the example of the diffusion equation with parametric
log-Gaussian coefficient as introduced in Section~\ref{S:GSer}, and
used in Section~\ref{S:ParCoef}, \ref{sec:pdc}. With the Lipschitz
continuity of the data-to-solution map established in Section
\ref{sec:pdc}, we verify the well-posedness of the corresponding
BIP.
We fix the dimension $d\in\mathbb{N}$ of the physical domain
${\domain}\subseteq\mathbb{R}^d$, being a bounded Lipschitz domain, and choose
$E=L^\infty({\domain};\mathbb{C})$ and $X=H_0^1({\domain};\mathbb{C})$. We assume that
$f\in X'$ and $a_0\in E$ with
$$
\rho(a_0)>0.
$$
For
$$a\in O:=\set{a\in E}{\rho(a) >0},$$
let ${\mathcal U}(a)$ be the solution to the equation
\begin{equation}\label{eq:elliptic}
\begin{aligned}
- \operatorname{div}((a_0+a) \nabla {\mathcal U}(a)) = f \text{ in }{\domain}, \;\; {\mathcal U}(a) =0
\text{ on }\partial{\domain},
\end{aligned}
\end{equation}
for some fixed $f\in X'$.
Due to
$$\rho(a_0+a)\ge \rho(a_0)>0,$$
for every $a\in O$, as in \eqref{eq:apriori} we find that ${\mathcal U}(a)$ is
well-defined and it holds
\begin{equation*}
\norm[X]{{\mathcal U}(a)}\le \frac{\norm[X']{f}}{{\rho(a_0)}}=:r\qquad\forall a\in O.
\end{equation*}
This shows assumption \ref{item:norma2} in Corollary~\ref{cor:norma2}.
Slightly adjusting the arguments in Section~\ref{sec:pdc} one observes
that ${\mathcal U}:O\to X$ satisfies assumptions \ref{item:uhol} and
\ref{item:loclip} in Proposition \ref{prop:bdX}. Fix a representation
system $(\psi_j)_{j\in\mathbb{N}}\subseteq V$ such that with
$b_j:=\norm[{E}]{\psi_j}$ it holds $(b_j)_{j\in\mathbb{N}}\in\ell^1(\mathbb{N})$.
Then Corollary~\ref{cor:norma2} implies that the forward map
$$u({\boldsymbol{y}})=\lim_{N\to\infty}{\mathcal U}\bigg(\exp\Big(\sum_{j=1}^Ny_j\psi_j\Big)\bigg)$$ satisfies
the assumptions of Theorem~\ref{thm:bdHol}. Theorem~\ref{thm:bdHol} in
turn implies that the posterior density for this model is
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic. We shall prove in
Section~\ref{sec:StochColl} that sparse-grid quadratures can be
constructed which achieve higher order convergence for the integrands
in \eqref{eq:TheParm} and \eqref{eq:integrand}, with the convergence
rate being a decreasing function of $p\in (0,4/5)$ such that
${\boldsymbol{b}}\in\ell^p(\mathbb{N})$, see Theorem~\ref{thm:quad}. Furthermore,
Corollary~\ref{cor:bdXSum} implies certain summability estimates for
the Wiener-Hermite PC expansion coefficients of the parametric maps in
\eqref{eq:TheParm} and \eqref{eq:integrand}.
\newpage
\
\newpage
\section{Interpolation and quadrature}
\label{sec:StochColl}
In Section~\ref{sec:SumHolSol} we have obtained summability results
for the norms of the Wiener-Hermite PC expansion coefficients of
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. In Section~\ref{sec:bdX}
and \ref{sec:BIP} we saw that solutions to certain parametric PDEs as
well as posterior densities satisfy
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy. As is well-known, summability of the
PC expansion coefficients entails algebraic convergence rates for
truncated PC expansions. We shall show in this section, that these
convergence rates can be algorithmically realized by sparse
multivariate Lagrangian interpolation and quadrature. Using ideas
from \cite{dD21,ZS17}, we show an improved quadrature convergence
rate.
\subsection{Smolyak interpolation and quadrature}
\subsubsection{Interpolation}\label{sec:int}
For every $n\in\mathbb{N}_0$ denote by $(\chi_{n,j})_{j=0}^n\subseteq\mathbb{R}$ the
Gauss-Hermite points in one dimension (in particular, $\chi_{0,0}=0$),
that is, the roots of Hermite polynomial $H_{n+1}$. Let
$$I_n:C^0(\mathbb{R})\to C^0(\mathbb{R})$$ be the univariate polynomial
interpolation operator defined by
\begin{equation*}
(I_n u)(y):= \sum_{j=0}^n u(\chi_{n,j}) \prod_{\substack{i=0\\
i\neq j}}^n \frac{y-\chi_{n,i}}{\chi_{n,j}-\chi_{n,i}},\qquad y\in\mathbb{R},
\end{equation*}
with convention that $I_{-1}:C^0(\mathbb{R})\to C^0(\mathbb{R})$ is defined as the
constant $0$ operator.
For any multi-index ${\boldsymbol{\nu}}\in\mathcal F$, introduce the tensorized operators
$\mathbf{I}_{\boldsymbol{\nu}}$ by
$$\mathbf{I}_{\boldsymbol{0}} u := u((\chi_{0,0})_{j\in\mathbb{N}}),$$ and for ${\boldsymbol{\nu}}\neq{\boldsymbol{0}}$ via
\begin{equation}\label{eq:VInu}
\mathbf{I}_{\boldsymbol{\nu}} := \bigotimes_{j\in\mathbb{N}} I_{\nu_j},
\end{equation}
i.e.,
\begin{equation}\nonumber
\mathbf{I}_{\boldsymbol{\nu}} u({\boldsymbol{y}}) = \sum_{\set{{\boldsymbol{\mu}}\in\mathcal F}{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}}
u((\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}) \prod_{j\in\mathbb{N}}\prod_{\substack{i=0\\ i\neq
\mu_j}}^{\nu_j}
\frac{y_j-\chi_{\nu_j,i}}{\chi_{\nu_j,\mu_j}-\chi_{\nu_j,i}},\quad {\boldsymbol{y}}\in U.
\end{equation}
The operator $\mathbf{I}_{\boldsymbol{\nu}}$ can thus be applied to functions $u$ which are
pointwise defined at each $(\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}\in U$. Via
Remark.~\ref{rmk:defu}, we can apply it in particular to
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. Observe that the product
over $j\in\mathbb{N}$ in \eqref{eq:VInu} is a finite product, since for every
$j$ with $\nu_j=0$, the inner product over
$i\in\{0,\dots,\mu_j-1,\mu_j+1,\dots,\nu_j\}$ is over an empty set,
and therefore equal to one by convention. Then for a finite set
$\Lambda\subseteq\mathcal F$
\begin{equation}\label{eq:VILambda}
\mathbf{I}_{\Lambda} :=\sum_{{\boldsymbol{\nu}}\in\Lambda}\bigotimes_{j\in\mathbb{N}}(I_{\nu_j}-I_{\nu_j-1}).
\end{equation}
Expanding all tensor product operators, we get
\begin{equation}\label{eq:VILambda}
\mathbf{I}_{\Lambda} =\sum_{{\boldsymbol{\nu}}\in\Lambda}\sigma_{\Lambda;{\boldsymbol{\nu}}}
\mathbf{I}_{\boldsymbol{\nu}}\qquad\text{where}\qquad
\sigma_{\Lambda;{\boldsymbol{\nu}}}:=\sum_{\set{{\boldsymbol{e}}\in\{0,1\}^\infty}{{\boldsymbol{\nu}}+{\boldsymbol{e}}\in\Lambda}} (-1)^{|{\boldsymbol{e}}|}.
\end{equation}
\begin{definition}\label{def:DownClsd}
An index set $\Lambda\subseteq\mathcal F$ is called \emph{downward closed},
if it is finite and if for every ${\boldsymbol{\nu}}\in\Lambda$ it holds
${\boldsymbol{\mu}}\in\Lambda$ whenever ${\boldsymbol{\mu}}\le{\boldsymbol{\nu}}$. Here, the ordering
``$\le$'' between two indices ${\boldsymbol{\mu}} = (\mu_j)_{j\in {\mathbb N}}$ and
${\boldsymbol{\nu}} = (\nu_j)_{j\in {\mathbb N}}$ in $\mathcal F$ expresses that for all
$j\in {\mathbb N}$ holds $\mu_j \le \nu_j$ with strict inequality for at
least one index $j$.
\end{definition}
As is well-known, $\mathbf{I}_{\Lambda}$ possesses the following crucial
property, see for example \cite[Lemma 1.3.3]{JZdiss}.
\begin{lemma}\label{lemma:VIprop}
Let $\Lambda\subseteq\mathcal F$ be downward closed. Then
$\mathbf{I}_{\Lambda}f =f$ for all
$f\in {\rm span}\set{{\boldsymbol{y}}^{\boldsymbol{\nu}}}{{\boldsymbol{\nu}}\in\Lambda}$.
\end{lemma}
The reason to choose the collocation points $(\chi_{n,j})_{j=0}^n$ as
the Gauss-Hermite points, is that it was recently shown that the
interpolation operators $I_n$ then satisfy the following stability
estimate, see \cite[Lemma 3.13]{ErnstSprgkTam18}.
\begin{lemma}
For every $n\in\mathbb{N}_0$ and every $m\in\mathbb{N}$ it holds
\begin{equation*}
\norm[L^2(\mathbb{R};\gamma_1)]{I_n(H_m)}\le 4 \sqrt{2m-1}.
\end{equation*}
\end{lemma}
With the presently adopted normalization of the GM $\gamma_1$, it
holds $H_0\equiv 1$ and therefore $I_n(H_0)=H_0$ for all $n\in\mathbb{N}_0$
(since the interpolation operator $I_n$ exactly reproduces all
polynomials of degree $n\in\mathbb{N}_0$). Hence
$$\norm[L^2(\mathbb{R};\gamma_1)]{I_n(H_0)}=\norm[L^2(\mathbb{R};\gamma_1)]{H_0}=1$$
for all $n\in\mathbb{N}_0$. Noting that $4\sqrt{2m-1}\le (1+m)^2$ for all
$m\in\mathbb{N}$, we get
\begin{equation*}
\norm[L^2(\mathbb{R};\gamma_1)]{I_n(H_m)}\le (1+m)^2\qquad\forall n,~m\in\mathbb{N}_0.
\end{equation*}
Consequently
\begin{equation}\label{eq:L2bound}
\norm[L^2(U;\gamma)]{\mathbf{I}_{\boldsymbol{\nu}}(H_{\boldsymbol{\mu}})} =
\prod_{j\in\mathbb{N}}\norm[L^2(\mathbb{R};\gamma_1)]{I_{\nu_j}(H_{\mu_j})}
\le \prod_{j\in\mathbb{N}}(1+\mu_j)^2\qquad\forall {\boldsymbol{\nu}},~{\boldsymbol{\mu}}\in\mathcal F.
\end{equation}
For ${\boldsymbol{\nu}} \in {\mathcal F}$ and $\tau\geq 0$ we denote
\begin{equation*}
p_{\boldsymbol{\nu}}( \tau) :=\prod_{j\in {\mathbb N}} (1+\nu_j)^\tau.
\end{equation*}
If $\nu_j>\mu_j$ then $(I_{\nu_j}-I_{\nu_j-1})H_{\mu_j}=0.$
Thus, $$\bigotimes_{j\in\mathbb{N}}(I_{\nu_j}-I_{\nu_j-1})H_{\boldsymbol{\mu}}=0,$$ whenever
there exists $j\in\mathbb{N}$ such that $\nu_j>\mu_j$. Hence, for any
downward closed set $\Lambda$, it holds
\begin{equation}\label{eq:L2boundLambda}
\norm[L^2(U;\gamma)]{\mathbf{I}_\Lambda(H_{\boldsymbol{\mu}})} \le
p_{{\boldsymbol{\mu}}}(3).
\end{equation}
Indeed,
\begin{equation}\nonumber
\norm[L^2(U;\gamma)]{\mathbf{I}_\Lambda(H_{\boldsymbol{\mu}})} \le
\sum_{\set{{\boldsymbol{\nu}}\in\Lambda}{{\boldsymbol{\nu}}\le{\boldsymbol{\mu}}}}p_{\boldsymbol{\mu}}(2)
\le |{\set{{\boldsymbol{\nu}}\in\Lambda}{{\boldsymbol{\nu}}\le{\boldsymbol{\mu}}}}|p_{\boldsymbol{\mu}}(2)
=\prod_{j\in\mathbb{N}}(1+\mu_j)p_{\boldsymbol{\mu}}(2)=p_{{\boldsymbol{\mu}}}(3).
\end{equation}
\subsubsection{Quadrature}
\label{sec:Quadrat}
Let again $(\chi_{n,j})_{j=0}^n\subseteq\mathbb{R}$ be the Gauss-Hermite
points in one dimension. Analogous to $I_n$ we introduce univariate
polynomial quadrature operators via
\begin{equation*}
Q_n u := \sum_{j=0}^n u(\chi_{n,j}) \omega_{n,j},\qquad
\omega_{n,j}:=\int_{\mathbb{R}} \prod_{i\neq j}
\frac{y-\chi_{n,i}}{\chi_{n,j}-\chi_{n,i}} \,\mathrm{d}\gamma_1(y).
\end{equation*}
Furthermore, we define $$\mathbf{Q}_{\boldsymbol{0}} u := u((\chi_{0,0})_{j\in\mathbb{N}}),$$
and for ${\boldsymbol{\nu}}\neq{\boldsymbol{0}}$,
\begin{equation*}
\mathbf{Q}_{\boldsymbol{\nu}} := \bigotimes_{j\in\mathbb{N}} Q_{\nu_j},
\end{equation*}
i.e.,
\begin{equation*}
\mathbf{Q}_{\boldsymbol{\nu}} u = \sum_{\set{{\boldsymbol{\mu}}\in\mathcal F}{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}}
u((\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}) \prod_{j\in\mathbb{N}}\omega_{\nu_j,\mu_j},
\end{equation*}
and finally for a finite downward closed $\Lambda\subseteq\mathcal F$ with
$\sigma_{\Lambda;{\boldsymbol{\nu}}}$ as in \eqref{eq:VILambda},
\begin{equation*}
\mathbf{Q}_{\Lambda}:=\sum_{{\boldsymbol{\nu}}\in\Lambda}\sigma_{\Lambda;{\boldsymbol{\nu}}}\mathbf{Q}_{\boldsymbol{\nu}}.
\end{equation*}
Again we emphasize that the above formulas are meaningful as long as
point evaluations of $u$ at each $(\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}$ are
well defined, ${\boldsymbol{\nu}}\in\mathcal F$, ${\boldsymbol{\mu}}\le{\boldsymbol{\nu}}$. Also note that
$$\mathbf{Q}_{\Lambda}f=\int_U \mathbf{I}_{\Lambda}f({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}}).$$
Let the set $\mathcal F_2$ be defined by
\begin{equation} \label{cF_2} \mathcal F_2:=\set{{\boldsymbol{\nu}}\in\mathcal F}{\nu_j\neq
1~\forall j}.
\end{equation}
We thus have $\mathcal F_2\subsetneq\mathcal F$. Similar to Lemma
\ref{lemma:VIprop} we have the following lemma, which can be proven
completely analogous to \cite[Lemma 1.3.16]{JZdiss} (also see
\cite[Remark 4.2]{ZS17}).
\begin{lemma}\label{lemma:VQprop}
Let $\Lambda\subseteq\mathcal F$ be downward closed. Then
$$\mathbf{Q}_{\Lambda}v =\int_U v({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})$$ for all
$v \in {\rm
span}\set{{\boldsymbol{y}}^{\boldsymbol{\nu}}}{{\boldsymbol{\nu}}\in\Lambda\cup(\mathcal F\backslash\mathcal F_2)}$.
\end{lemma}
With \eqref{eq:L2bound} it holds
\begin{equation*}
|\mathbf{Q}_{\boldsymbol{\nu}}(H_{\boldsymbol{\mu}})|
=\left|\int_U \mathbf{I}_{\boldsymbol{\nu}}(H_{\boldsymbol{\mu}})({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})\right|
\le \norm[L^2(U;\gamma)]{\mathbf{I}_{\boldsymbol{\nu}}(H_{\boldsymbol{\mu}})}
\le \prod_{j\in\mathbb{N}}(1+\mu_j)^2\qquad\forall {\boldsymbol{\nu}},~{\boldsymbol{\mu}}\in\mathcal F,
\end{equation*}
and similarly, using \eqref{eq:L2boundLambda}
\begin{equation}\label{eq:L1boundLambda}
|\mathbf{Q}_\Lambda(H_{\boldsymbol{\mu}})| \le p_{{\boldsymbol{\mu}}}(3).
\end{equation}
\subsection{Multiindex sets}\label{sec:mi}
In Section~\ref{sec:SumHolSol}, we showed weighted summability of the
Wiener-Hermite PC expansion coefficients $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}$,
namely
\begin{equation}\label{eq:weightedsum}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{u_{\boldsymbol{\nu}}}^2<\infty
\end{equation}
with $\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ as in \eqref{beta} and some sequence
${\boldsymbol{\varrho}}$. The goal of this section is, to construct a (new) weight
family $(c_{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ based on
$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}))_{{\boldsymbol{\nu}}\in\mathcal F}$, such that (a variant of)
\eqref{eq:weightedsum} holds with $\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ replaced by
$c_{\boldsymbol{\nu}}$. Once a suitable family $(c_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}$ has been
identified, we obtain a multiindex set $\Lambda_\varepsilon\subseteq\mathcal F$ for
$\varepsilon>0$ via
\begin{equation}\label{eq:Leps}
\Lambda_\varepsilon:=\set{{\boldsymbol{\nu}}\in\mathcal F}{c_{\boldsymbol{\nu}}^{-1}\ge\varepsilon},
\end{equation}
The set $\Lambda_\varepsilon$ will then serve as an index set to define
interpolation operators $\mathbf{I}_{\Lambda_\varepsilon}$ and $\mathbf{Q}_{\Lambda_\varepsilon}$.
As $(c_{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ is used to construct sets of
multiindices, it should possess certain features, including each
$c_{\boldsymbol{\nu}}$ to be easily computable for ${\boldsymbol{\nu}}\in\mathcal F$, and the resulting
numerical algorithm to be efficient.
We proceed as follows. First we recall arguments from
\cite{dD21,ZS17} which allow to bound the number of required function
evaluations of the interpolation an quadrature
algorithm. Subsequently, a construction of a suitable family
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ is provided for $k\in\{1,2\}$. The index
$k$ determines whether the family will be used for a sparse grid
interpolation ($k=1$) or a Smolyak-type sparse quadrature ($k=2$)
algorithm. Finally, it is shown that the multiindex sets
$\Lambda_{k,\varepsilon}$ as in \eqref{eq:Leps} based on
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, guarantee algebraic convergence rates for
certain truncated Wiener-Hermite PC expansions. This will be
exploited
to verify convergence rates for interpolation in
Section~\ref{sec:intrate} and for quadrature in
Section~\ref{sec:quadrate}.
\subsubsection{Number of function evaluations}
\label{sec:NoFnEval}
In order to obtain a convergence rate in terms of the number of
evaluations of $u$, we need to determine the number of interpolation
points used by the operator $\mathbf{I}_{\Lambda}$ or $\mathbf{Q}_{\Lambda}$. Since
the discussion of $\mathbf{Q}_{\Lambda}$ is very similar, we concentrate here
on $\mathbf{I}_{\Lambda}$.
Computing the interpolant $\mathbf{I}_{{\boldsymbol{\nu}}}u$ in \eqref{eq:VInu} requires
knowledge of the function values of $u$ at each point in
\begin{equation*}
\set{(\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}}{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}.
\end{equation*}
The cardinality of this set is bounded by
$\prod_{j\in\mathbb{N}}(1+\nu_j)=p_{\boldsymbol{\nu}}(1)$. Denote by
\begin{equation}\label{eq:pts}
{\rm pts} (\Lambda)
:=
\set{(\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}}{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}},~{\boldsymbol{\nu}}\in\Lambda}
\end{equation}
the set of interpolation points defining the interpolation operator
$\mathbf{I}_{\Lambda}$ (i.e., ~$|{\rm pts} (\Lambda)|$ is the number of function
evaluations of $u$ required to compute $\mathbf{I}_{\Lambda} u$). By
\eqref{eq:VILambda} we obtain the bound
\begin{equation}\label{eq:ptsLbound}
|{\rm pts} (\Lambda)|
\le
\sum_{\set{{\boldsymbol{\nu}}\in\Lambda}{\sigma_{\Lambda,{\boldsymbol{\nu}}}\neq 0}} \prod_{j\in\mathbb{N}} (1+\nu_j)
= \sum_{\set{{\boldsymbol{\nu}}\in\Lambda}{\sigma_{\Lambda,{\boldsymbol{\nu}}}\neq 0}} p_{\boldsymbol{\nu}}(1).
\end{equation}
\subsubsection{Construction of $(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$}
We are now in position to construct $(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$. As
mentioned above, we distinguish between the cases $k=1$ and $k=2$,
which correspond to polynomial interpolation or quadrature. Note that
in the next lemma we define $c_{k,{\boldsymbol{\nu}}}$ for all ${\boldsymbol{\nu}}\in\mathcal F$, but the
estimate provided in the lemma merely holds for ${\boldsymbol{\nu}}\in\mathcal F_k$,
$k\in\{1,2\}$, where $\mathcal F_1:=\mathcal F$ and $\mathcal F_2$ is defined in
\eqref{cF_2}. Throughout what follows, empty products equal $1$ by
convention.
\begin{lemma}\label{lemma:cnu}
Assume that $\tau>0$, $k\in\{1,2\}$ and $r>\max\{\tau,k\}$. Let
${\boldsymbol{\varrho}}\in (0,\infty)^\infty$ be such that $\varrho_j\to\infty$ as
$j\to\infty$. Then there exist $K>0$ and $C_0>0$ such that
\begin{equation}\label{eq:cnu}
c_{k,{\boldsymbol{\nu}}}:= \prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})}\max\left\{1,K
\varrho_j\right\}^{2k} \nu_j^{r-\tau}\quad{\boldsymbol{\nu}}\in\mathcal F,
\end{equation}
satisfies (with $\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ as in \eqref{beta})
\begin{equation}\label{eq:cknubound}
C_0 c_{k,{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(\tau) \le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \quad\forall {\boldsymbol{\nu}}\in\mathcal F_k.
\end{equation}
\end{lemma}
\begin{proof}
\textbf{Step 1.} Fix ${\boldsymbol{\nu}}\in\mathcal F_k$, then $j\in\operatorname{supp}({\boldsymbol{\nu}})$ implies
$\nu_j\ge k$ and thus $\min\{r,\nu_j\}\ge k$ since $r> k$ by
assumption. With $s:=\min\{r,\nu_j\}\le \nu_j$, for all $j\in\mathbb{N}$
holds
\begin{equation*}
\binom{\nu_j}{s}
=
\frac{\nu_j!}{(\nu_j-s)! s!}
\ge
\frac{1}{s!} (\nu_j-s+1)^s\ge \nu_j^s \frac{1}{s! s^s}
\ge
\nu_j^s \frac{1}{r! r^r}= \nu_j^{\min\{\nu_j,r\}} \frac{1}{r!r^r}
\ge
\nu_j^{r} \frac{1}{r!r^{2r}}.
\end{equation*}
Furthermore, if $j\in\operatorname{supp}({\boldsymbol{\nu}})$, then due to
$s=\min\{\nu_j,r\}\ge k$, with
$\varrho_0:=\min \{1,\min_{j\in\mathbb{N}}\varrho_j\}$ we have
$$
\varrho_0^{2r} \leq \min\{ 1,\varrho_j\}^{2r} \leq \varrho_j^{2(s-k)}.
$$
Thus
$$\varrho_j^{\min\{\nu_j,r\}}\ge \varrho_0^{2r} \varrho_j^{2k}$$ for all $j\in\mathbb{N}$.
In all, we conclude
\begin{equation}\label{eq:estb}
\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) = \prod_{j\in\mathbb{N}} \left(\sum_{l=0}^r
\binom{\nu_j}{l}\varrho_j^{2l} \right) \ge
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})}\binom{\nu_j}{\min\{\nu_j,r\}}
\varrho_j^{2\min\{\nu_j,r\}} \ge \prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})}
\frac{\varrho_0^{2r}}{r!r^{2r}} \varrho_j^{2k}\nu_j^{r}.
\end{equation}
Since ${\boldsymbol{\nu}}\in\mathcal F_k$ was arbitrary, this estimate holds for all
${\boldsymbol{\nu}}\in\mathcal F_k$.
\textbf{Step 2.}
Denote $\hat\varrho_j:=\max\{1,K\varrho_j\}$, where $K>0$ is still at
our disposal. We have
\begin{equation*}
p_{\boldsymbol{\nu}}(\tau)\le\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} 2^{\tau} \nu_j^{\tau}
\end{equation*}
and thus
\begin{equation}\label{eq:estc}
c_{k,{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(\tau) \le
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})
2^\tau
\hat\varrho_j^{2k} \nu_j^{r}.
\end{equation}
Again, this estimate holds for any ${\boldsymbol{\nu}}\in\mathcal F_k$.
With $\varrho_0:=\min\{1,\min_{j\in\mathbb{N}}\varrho_j\}$ denote
\begin{equation*}
C_b:=\left(\frac{\varrho_0^{2r}}{r!r^{2r}}\right)^{1/(2k)}
\qquad\text{and}\qquad C_c:= (2^\tau)^{1/(2k)}.
\end{equation*}
Set
\begin{equation*}
K:=\frac{C_b}{C_c},\qquad \tilde\varrho_j= K \varrho_j
\end{equation*}
for all $j\in\mathbb{N}$. Then
\begin{equation*}
C_b\varrho_j=C_c\tilde\varrho_j = C_c\hat\varrho_j\begin{cases}
1 &\text{if }K\varrho_j\ge 1,\\
K\varrho_j &\text{if }K\varrho_j<1.
\end{cases}
\end{equation*}
Let
\begin{equation*}
C_0:= \prod_{\set{j\in\mathbb{N}}{K\varrho_j<1}} (K\varrho_j)^{2k}
\end{equation*}
and note that this product is over a finite number of indices, since
$\varrho_j\to\infty$ as $j\to\infty$. Then for any ${\boldsymbol{\nu}}\in\mathcal F_k$
\begin{equation*}
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} C_c\tilde\varrho_j \ge C_0^{\frac{1}{2k}}
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} C_c\hat\varrho_j.
\end{equation*}
With \eqref{eq:estb} and \eqref{eq:estc} we thus obtain for every
${\boldsymbol{\nu}}\in\mathcal F_k$,
\begin{align}
\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) &\ge \prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} (C_b\varrho_j)^{2k}\nu_j^r =
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} \left(C_c \tilde
\varrho_j\right)^{2k}\nu_j^r
\notag
\\
&\ge C_0
\prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})} (C_c\hat\varrho_j)^{2k}\nu_j^r \ge C_0 c_{k,{\boldsymbol{\nu}}}p_{\boldsymbol{\nu}}(\tau).
\notag
\end{align}
\end{proof}
\subsubsection{Summability properties of the sequence
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$}
\label{S:SumPrpcknu}
First we discuss the summability of the sequence
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\cF}$. We will require the following lemma
which is a modification of \cite[Lemma 6.2]{dD21}.
\begin{lemma}\label{lemma:summabcnu}
Let $\theta \geq 0$. Let further $k\in\{1,2\}$, $\tau>0$,
$r>\max\{k,\tau\}$ and $q>0$ be such that
$(r-\tau)q/(2k)-\theta>1$. Assume that
$(\varrho_j)_{j\in\mathbb{N}}\in (0,\infty)^\infty$ satisfies
$(\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^q(\mathbb{N})$. Then with
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ as in Lemma \ref{lemma:cnu} it holds
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\cF}p_{\boldsymbol{\nu}}( \theta) c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}}<\infty.
\end{equation*}
\end{lemma}
\begin{proof}
This lemma can be proven in the same way as the proof of \cite[Lemma
6.2]{dD21}. We provide a proof for completeness. With
$\hat\varrho_j:=\max\{1,K\varrho_j\}$ it holds
$(\hat\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^q(\mathbb{N})$. By definition of
$c_{k,{\boldsymbol{\nu}}}$, factorizing, we get
\begin{equation*}
\begin{split}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} p_{\boldsymbol{\nu}}( \theta) c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}} & =
\sum_{{\boldsymbol{\nu}}\in\mathcal F} \prod_{j\in\operatorname{supp}({\boldsymbol{\nu}})}(1+\nu_j)^\theta
\left(\hat\varrho_j^{2k} \nu_j^{r-\tau}\right)^{-\frac{q}{2k}}
\leq \prod_{j\in\mathbb{N}} \left(2^\theta \hat\varrho_j^{-q}
\sum_{n\in\mathbb{N}} n^{\frac{-q(r-\tau)}{2k}} n^\theta\right).
\end{split}
\end{equation*}
The sum over $n$ equals some finite constant $C$ since by assumption
$q(r-\tau)/2k-\theta>1$.
Using the inequality $\log(1+x)\le x$ for all $x>0$, we get
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}}\le \prod_{j\in\mathbb{N}}
\left( 1 + C \hat\varrho_j^{-q}\right)
= \exp\left(\sum_{j\in\mathbb{N}}\log (1+C \hat\varrho_j^{-q}) \right)
\le \exp\left(\sum_{j\in\mathbb{N}} C \hat\varrho_j^{-q}\right),
\end{equation*}
which is finite since $(\hat\varrho_j^{-1})\in\ell^q(\mathbb{N})$.
\end{proof}
\begin{remark}
Under the assumptions of Lemmata \ref{lemma:cnu} and
\ref{lemma:summabcnu}, it holds
$$(\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})^{-1})_{{\boldsymbol{\nu}}\in\mathcal F_k}\in\ell^{q/(2k)}(\mathcal F_k).$$
This generalizes \cite[Lemma 5.1]{BCDM}.
\end{remark}
Based on \eqref{eq:Leps}, for $\varepsilon>0$ and $k\in\{1,2\}$ let
\begin{equation}\label{eq:Lkeps}
\Lambda_{k,\varepsilon}:=\{{\boldsymbol{\nu}}\in \mathcal F: c_{k,{\boldsymbol{\nu}}}^{-1}\ge \varepsilon \} \subseteq \mathcal F.
\end{equation}
The summability shown in Lemma \ref{lemma:summabcnu} implies algebraic
convergence rates of the tail sum as provided by the following
proposition. This is well-known and follows by Stechkin's lemma
\cite{stechkin} which itself is a simple consequence of the H\"older
inequality.
\begin{proposition}\label{prop:bestN}
Let $k\in\{1,2\}$, $\tau>0$, and $q>0$. Let
$(\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^q(\mathbb{N})$ and $r>\max\{k,\tau\}$,
$(r-\tau)q/(2k)>2$. Assume that
$(a_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}\in [0,\infty)^\infty$ is such that
\begin{equation}\label{eq:betabnulbnulinf}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) a_{\boldsymbol{\nu}}^2<\infty.
\end{equation}
Then there exists a constant $C$ solely depending on
$(c_{k,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ in \eqref{eq:cnu} such that for all
$\varepsilon>0$ it holds that
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F_k\backslash\Lambda_{k,\varepsilon}} p_{\boldsymbol{\nu}}(\tau)
a_{\boldsymbol{\nu}}
\le C\left(\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})a_{\boldsymbol{\nu}}^2\right)^{\frac 1 2} \varepsilon^{\frac{1}{2}- \frac{q}{4k}},
\end{equation*}
and
\begin{equation} \label{ptsLambda0} |{\rm pts} (\Lambda_{k,\epsilon})|
\leq C \varepsilon^{-\frac{q}{2k}}.
\end{equation}
\end{proposition}
\begin{proof}
We estimate
\begin{equation*} \label{sum-estimate1}
\begin{split}
\sum_{{\boldsymbol{\nu}}\in\mathcal F_k\backslash\Lambda_{k,\varepsilon}} p_{\boldsymbol{\nu}}(\tau)
a_{\boldsymbol{\nu}} & \le \Bigg(
\sum_{{\boldsymbol{\nu}}\in\mathcal F_k\backslash\Lambda_{k,\varepsilon}}
p_{\boldsymbol{\nu}}(\tau)^2a_{\boldsymbol{\nu}}^2 c_{k,{\boldsymbol{\nu}}}\Bigg)^{1/2} \Bigg(
\sum_{{\boldsymbol{\nu}}\in\mathcal F_k\backslash\Lambda_{k,\varepsilon}}
c_{k,{\boldsymbol{\nu}}}^{-1}\Bigg)^{1/2}.
\end{split}
\end{equation*}
The first sum is finite by \eqref{eq:betabnulbnulinf} and because
$C_0 p_{\boldsymbol{\nu}}(\tau)^2 c_{k,{\boldsymbol{\nu}}}\le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}})$ according
to \eqref{eq:cknubound}. By Lemma \ref{lemma:summabcnu} and
\eqref{eq:Lkeps} we obtain
\begin{equation*} \label{sum-estimate2}
\sum_{{\boldsymbol{\nu}}\in\mathcal F_k\backslash\Lambda_{k,\varepsilon}} c_{k,{\boldsymbol{\nu}}}^{-1} =
\sum_{c_{k,{\boldsymbol{\nu}}}^{-1}<\varepsilon } c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}}
c_{k,{\boldsymbol{\nu}}}^{-1+\frac{q}{2k}} \leq C\varepsilon^{1-\frac{q}{2k}}
\end{equation*}
which proves the first statement. Moreover, for each ${\boldsymbol{\nu}}\in {\mathcal F}$,
the number of interpolation(quadrature) points is $p_{\boldsymbol{\nu}}(1)$.
Hence
\begin{equation*} \label{ptsLambda} |{\rm pts} (\Lambda_{k,\epsilon})| =
\sum_{{\boldsymbol{\nu}} \in \Lambda_{k,\epsilon}}p_{\boldsymbol{\nu}}(1) =
\sum_{c_{k,{\boldsymbol{\nu}}}^{-1}\geq \varepsilon}
p_{\boldsymbol{\nu}}(1)c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}} c_{k,{\boldsymbol{\nu}}}^{\frac{q}{2k}} \leq
\varepsilon^{-\frac{q}{2k}} \sum_{{\boldsymbol{\nu}} \in \cF_{k}}
p_{\boldsymbol{\nu}}(1)c_{k,{\boldsymbol{\nu}}}^{-\frac{q}{2k}} \leq C
\varepsilon^{-\frac{q}{2k}}
\end{equation*}
again by Lemma \ref{lemma:summabcnu} and \eqref{eq:Lkeps}.
\end{proof}
\subsection{Interpolation convergence}
\label{sec:intrate}
If $X$ is a Hilbert space, then the Wiener-Hermite PC expansion of
$u:U\to X$ converges in general only in $L^2(U,X;\gamma)$.
As mentioned before this creates some subtleties when working with
interpolation and quadrature operators based on pointwise evaluations
of the target function.
To demonstrate this, we recall the following example from
\cite{CSZ16}, which \emph{does not satisfy
$({\boldsymbol{b}},\xi,\delta,\mathbb{C})$-holomorphy}, since Definition~\ref{def:bdXHol}
\ref{item:vN} does not hold.
\begin{example}\label{ex:u}
Define $u:U\to\mathbb{C}$ pointwise by
\begin{equation*}
u({\boldsymbol{y}}):=\begin{cases}
1 &\text{if }|\set{j\in\mathbb{N}}{y_j\neq 0}|<\infty\\
0 &\text{otherwise.}
\end{cases}
\end{equation*}
Then $u$ vanishes on the complement of the $\gamma$-null set
\begin{equation*}
\bigcup_{n\in\mathbb{N}} \mathbb{R}^n\times \{0\}^\infty.
\end{equation*}
Consequently $u$ is equal to the constant zero function in the sense
of $L^2(U;\gamma)$. Hence there holds the expansion
$u=\sum_{{\boldsymbol{\nu}}\in\mathcal F}0\cdot H_{\boldsymbol{\nu}}$ with convergence in
$L^2(U;\gamma)$. Now let $\Lambda\subseteq\mathcal F$ be nonempty, finite
and downward closed. As explained in Section~\ref{sec:int}, the
interpolation operator $\mathbf{I}_{\Lambda}$ reproduces all polynomials in
${\rm span}\set{{\boldsymbol{y}}^{\boldsymbol{\nu}}}{{\boldsymbol{\nu}}\in\Lambda}$. Since any point
$(\chi_{\nu_j,\mu_j})_{j\in\mathbb{N}}$ with $\mu_j\le\nu_j$ is zero in all
but finitely many coordinates (due to $\chi_{0,0}=0$), we observe
that
\begin{equation*}
\mathbf{I}_{\Lambda} u \equiv 1 \neq 0\equiv \sum_{{\boldsymbol{\nu}}\in\mathcal F} 0\cdot
\mathbf{I}_{\Lambda} H_{\boldsymbol{\nu}}.
\end{equation*}
This is due to the fact that $u = \sum_{{\boldsymbol{\nu}}\in\mathcal F} 0\cdot H_{\boldsymbol{\nu}}$
only holds in an $L^2(U;\gamma)$ sense, and interpolation or
quadrature (which require pointwise evaluation of the function) are
not meaningful for $L^2(U;\gamma)$ functions.
\end{example}
The above example shows that if
$$u=\sum_{{\boldsymbol{\nu}}\in\mathcal F} u_{\boldsymbol{\nu}} H_{\boldsymbol{\nu}}\in L^2(U;\gamma)$$ with Wiener-Hermite PC
expansion coefficients $(u_{\boldsymbol{\nu}})_{\nu\in\mathcal F}\subset\mathbb{R}$, then the
formal equalities
$$\mathbf{I}_{\Lambda}u=\sum_{{\boldsymbol{\nu}}\in\mathcal F} u_{\boldsymbol{\nu}} \mathbf{I}_{\Lambda}H_{\boldsymbol{\nu}}, $$ and
$$\mathbf{Q}_{\Lambda}u=\sum_{{\boldsymbol{\nu}}\in\mathcal F} u_{\boldsymbol{\nu}} \mathbf{Q}_{\Lambda}H_{\boldsymbol{\nu}} $$ are in
general wrong in $L^2(U;\gamma)$. Our definition of
$({\boldsymbol{b}},\xi,\delta,X)$-holomorphy allows to circumvent this by
interpolating not $u$ itself but the approximations $u_N$ to $u$ which
are pointwise defined and only depend on finitely many variables,
cp.~Definition~\ref{def:bdXHol}.
Our analysis starts with the following result about pointwise
convergence.
For $k\in\{1,2\}$ and $N\in\mathbb{N}$ we introduce the notation
\begin{equation*}\label{eq:CFkN}
\mathcal F_{k}^N := \set{{\boldsymbol{\nu}}\in \mathcal F_k}{\operatorname{supp}({\boldsymbol{\nu}})\subseteq\{1,\dots,N\}}.
\end{equation*}
These sets thus contain multiindices ${\boldsymbol{\nu}}$ for which $\nu_j=0$ for
all $j>N$.
\begin{lemma}\label{lemma:uN}
Let $u$ be $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic for some
${\boldsymbol{b}}\in (0,\infty)^\infty$. Let $N\in\mathbb{N}$, and let
$\tilde u_N:U\to X$ be as in Definition~\ref{def:bdXHol}. For
${\boldsymbol{\nu}}\in\mathcal F$ define
$$\tilde u_{N,{\boldsymbol{\nu}}}:=\int_{U} \tilde u_N({\boldsymbol{y}})H_{\boldsymbol{\nu}}({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}}).$$
Then,
$$\tilde u_N({\boldsymbol{y}})=\sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N} \tilde u_{N,{\boldsymbol{\nu}}}
H_{\boldsymbol{\nu}}({\boldsymbol{y}})=\sum_{{\boldsymbol{\nu}}\in\mathcal F}\tilde u_{N,{\boldsymbol{\nu}}} H_{\boldsymbol{\nu}}({\boldsymbol{y}})$$ with
pointwise absolute convergence for all ${\boldsymbol{y}}\in U$.
\end{lemma}
\begin{proof}
From the Cram\'er bound
$$
|\tilde{H}_n(x)| <
2^{n/2}\sqrt{n!}e^{x^2/2},
$$
see \cite{Indritz}, and where
$\tilde{H}_n(x/\sqrt{2}) := 2^{n/2}\sqrt{n!}H_n(x)$, see \cite[Page
787]{AS}, we have for all $n\in\mathbb{N}_0$
\begin{equation}\label{eq:supxHn}
\sup_{x\in\mathbb{R}} \exp(-x^2/4) |H_n(x)|\le 1.
\end{equation}
By Theorem.~\ref{thm:bdHolSum}
$(\tilde u_{N,{\boldsymbol{\nu}}})_{\nu\in\mathcal F}\in\ell^1(\mathcal F)$. Note that for
${\boldsymbol{\nu}}\in\mathcal F_1^N$
\begin{equation*}
\tilde u_{N,{\boldsymbol{\nu}}} = \int_U \tilde u_N({\boldsymbol{y}})H_{\boldsymbol{\nu}}({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})
= \int_{\mathbb{R}^N} u_N (y_1,\dots,y_N)\prod_{j=1}^NH_{\nu_j}(y_j) \,\mathrm{d}\gamma_N((y_j)_{j=1}^N)
\end{equation*}
and thus $\tilde u_{N,{\boldsymbol{\nu}}}$ coincides with the Wiener-Hermite PC
expansion coefficient of $u_N$ w.r.t.~the multiindex
$(\nu_j)_{j=1}^N\in\mathbb{N}_0^N$.
The summability of the sequence
$$\left(\norm[X]{u_{N,{\boldsymbol{\nu}}}}\norm[L^2(\mathbb{R}^N,\gamma_N)]{\prod_{j=1}^N H_{\nu_j}(y_j)}\right)_{{\boldsymbol{\nu}}\in\mathcal F_1^N}$$
now implies in particular,
$$u_N((y_j)_{j=1}^N) =\sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N} u_{N,{\boldsymbol{\nu}}} \prod_{j=1}^N
H_{\nu_j}(y_j)$$ in the sense of $L^2(\mathbb{R}^N,\gamma_N)$.
Due to \eqref{eq:supxHn} and
$(\norm[X]{u_{N,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathcal F_1^N}\in\ell^1(\mathcal F_1^N)$ we can
define a continuous function
$$\hat u_N: (y_j)_{j=1}^N \mapsto
\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N} u_{N,{\boldsymbol{\nu}}} \prod_{j=1}^N H_{\nu_j}(y_j)$$ on
$\mathbb{R}^N$. Here, the series converges absolutely for every fixed,
finite $(y_j)_{j=1}^N \in\mathbb{R}^N$).
Since they have the same Wiener-Hermite PC expansion, it holds
$\hat u_N=u_N$ in the sense of $L^2(\mathbb{R}^N,\gamma_N)$.
By Definition~\ref{def:bdXHol} the function $u:\mathbb{R}^N\to X$ is in
particular continuous (it even allows a holomorphic extension to
some subset of ${\mathbb C}^N$ containing $\mathbb{R}^N$). Now $\hat u_N$,
$u_N:\mathbb{R}^N\to X$ are two continuous functions which are equal in the
sense of $L^2(\mathbb{R}^N;\gamma_N)$. Thus they coincide pointwise and it
holds for every ${\boldsymbol{y}}\in U$,
$$\tilde u_N({\boldsymbol{y}})=u_N((y_j)_{j=1}^N)=\sum_{{\boldsymbol{\nu}}\in\mathbb{N}_0^N} \tilde
u_{N,{\boldsymbol{\nu}}} H_{\boldsymbol{\nu}}({\boldsymbol{y}}).$$
\end{proof}
We arrive at the following convergence result.
The convergence rate provided in \cite{ErnstSprgkTam18} (in terms of
the number of function evaluations) is improved by a factor $2$ (for
the global supports case). Additionally we provide an explicit
construction of suitable index sets. Recall that pointwise
evaluations of a $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions are
understood in the sense of Remark.~\ref{rmk:defu}.
\begin{theorem}\label{thm:int}
Let $u$ be $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic for some
${\boldsymbol{b}}\in\ell^p(\mathbb{N})$ and some $p\in (0,2/3)$.
Let $(c_{1,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ be as in Lemma \ref{lemma:cnu} with
${\boldsymbol{\varrho}}$ as in Theorem~\ref{thm:bdHolSum}.
Then there exist $C>0$ and, for every $n \in {\mathbb N}$, $\varepsilon_n>0$ such
that $|{\rm pts} (\Lambda_{1,\varepsilon_n})|\le n$ (with $\Lambda_{1,\varepsilon_n}$
as in \eqref{eq:Lkeps}) and
\begin{equation*}\label{eq:L2interrn}
\norm[L^2(U,X;\gamma)]{u - \mathbf{I}_{\Lambda_{1,\varepsilon_n}}u} \ \le \ C n^{{-\frac{1}{p}+\frac{3}{2}}}.
\end{equation*}
\end{theorem}
\begin{proof}
For $\varepsilon>0$ small enough and satisfying $|\Lambda_{1,\varepsilon}|>0$,
take $N\in\mathbb{N}$ with
$$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{{\boldsymbol{\nu}}\in\Lambda_{1,\varepsilon}},$$ so large that
\begin{equation}\label{eq:truncerr}
\norm[L^2(U,X;\gamma)]{u-\tilde u_N}\le \varepsilon^{\frac{1}{2}-\frac{p}{4(1-p)}},
\end{equation}
which is possible due to the $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy
of $u$ (cp.~Definition~\ref{def:bdXHol} \ref{item:vN}). An
appropriate value of $\varepsilon$ depending on $n$ will be
chosen below. In the following for ${\boldsymbol{\nu}}\in\mathcal F_1^N$ we denote
by $\tilde u_{N,{\boldsymbol{\nu}}}\in X$ the PC coefficient of $\tilde u_N$
and for ${\boldsymbol{\nu}}\in\mathcal F$ as earlier $u_{\boldsymbol{\nu}}\in X$ is the PC
coefficient of $u$.
Because $$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{{\boldsymbol{\nu}}\in\Lambda_{1,\varepsilon}}$$
and $\chi_{0,0}=0$, we have
$\mathbf{I}_{\Lambda_{1,\varepsilon}}u= \mathbf{I}_{\Lambda_{1,\varepsilon}} \tilde u_N$
(cp.~Remark.~\ref{rmk:defu}). Hence by \eqref{eq:truncerr}
\begin{equation*}
\norm[L^2(U,X;\gamma)]{u-\mathbf{I}_{\Lambda_{1,\varepsilon}}u}
=\norm[L^2(U,X;\gamma)]{u-\mathbf{I}_{\Lambda_{1,\varepsilon}} \tilde u_N}
\le \varepsilon^{\frac{1}{2}-\frac{p}{4(1-p)}}+
\norm[L^2(U,X;\gamma)]{ \tilde u_N-\mathbf{I}_{\Lambda_{1,\varepsilon}} \tilde u_N}.
\end{equation*}
By Lemma \ref{lemma:uN} we have
$$\tilde u_N({\boldsymbol{y}})=\sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N} \tilde u_{N,{\boldsymbol{\nu}}}
H_{\boldsymbol{\nu}}({\boldsymbol{y}})$$ with pointwise absolute convergence for all
${\boldsymbol{y}}\in U$. Thus we can interchange the interpolation operator
$\mathbf{I}_{\Lambda_{1,\varepsilon}}$ with the Wiener-Hermite PC expansion
of $\tilde u_N$ and write
$$\mathbf{I}_{\Lambda_{1,\varepsilon}} \tilde u_N= \sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N} \tilde
u_{N;{\boldsymbol{\nu}}} \mathbf{I}_{\Lambda_{1,\varepsilon}}H_{\boldsymbol{\nu}}$$ with pointwise
absolute convergence. Then by Lemma \ref{lemma:VIprop} and
\eqref{eq:L2boundLambda}
\begin{align*
\norm[L^2(U,X;\gamma)]{ \tilde u_N-\mathbf{I}_{\Lambda_{1,\varepsilon}} \tilde u_N}
&=\normc[L^2(U,X;\gamma)]{\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_{1,\varepsilon}} \tilde u_{N;{\boldsymbol{\nu}}} (H_{\boldsymbol{\nu}}-\mathbf{I}_{\Lambda_{1,\varepsilon}}H_{\boldsymbol{\nu}})}\nonumber\\
&\le \sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_{1,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}}
\big(\norm[L^2(U;\gamma)]{H_{\boldsymbol{\nu}}}+\norm[L^2(U;\gamma)]{\mathbf{I}_{\Lambda_{1,\varepsilon}}H_{\boldsymbol{\nu}}}\big)\nonumber\\
&\leq \sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N\backslash \Lambda_{1,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}}
\left(1+p_{{\boldsymbol{\nu}}}(3)\right)
\nonumber\\
&\leq 2\sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N\backslash \Lambda_{1,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}}
p_{{\boldsymbol{\nu}}}(3).
\end{align*}
Choosing $r>4/p-1$ ($q=\frac{p}{1-p}$, $\tau=3$), according to
Proposition~\ref{prop:bestN}, \eqref{eq:cknubound} and
Theorem~\ref{thm:bdHolSum} (with
$(\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^{p/(1-p)}(\mathbb{N})$ as in
Theorem~\ref{thm:bdHolSum}) the last sum is bounded by
\begin{equation*}
C \left(\sum_{{\boldsymbol{\nu}}\in\mathcal F_1^N}
\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{\tilde u_{N,{\boldsymbol{\nu}}}}^2 \right)\varepsilon^{\frac{1}{2}-\frac{q}{4}} \le
C({\boldsymbol{b}}) \delta^2 \varepsilon^{\frac{1}{2}-\frac{q}{4} }= C({\boldsymbol{b}}) \delta^2 \varepsilon^{\frac{1}{2}-\frac{p}{4(1-p)}},
\end{equation*}
and the constant $C({\boldsymbol{b}})$ from Theorem~\ref{thm:bdHolSum} does
not depend on $N$ and $\delta$.
From \eqref{ptsLambda0} it follows that
\begin{equation*
|{\rm pts} (\Lambda_{1,\varepsilon})|
\leq C_2 \varepsilon^{-\frac{q}{2}}=C_2\varepsilon^{-\frac{p}{2(1-p)}}.
\end{equation*}
For every $n \in {\mathbb N}$, we choose an $\varepsilon_n>0$ satisfying the
condition
$$n/2 \le C_2\varepsilon_n^{-\frac{p}{2(1-p)}} \le n.$$ Then
the claim holds true for the chosen $\varepsilon_n$.
\end{proof}
\begin{remark}
Comparing the best $N$-term convergence result in Remark
\ref{rmk:bestN} with the interpolation result of Theorem
\ref{thm:int}, we observe that the convergence rate is reduced
by $1/2$, and moreover, rather than $p\in (0,1)$ as in Remark
\ref{rmk:bestN}, Theorem \ref{thm:int} requires
$p\in (0,2/3)$. This discrepancy can be explained as follows:
Since $(H_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\cF}$ forms an orthonormal basis of
$L^2(U,\gamma)$, for the best $N$-term result we could resort
to Parseval's identity, which merely requires
$\ell^2$-summability of the Hermite PC coefficients, i.e.\
$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\cF}\in\ell^2(\cF)$. Due to
$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\cF}\in\ell^{\frac{2p}{2-p}}$ by
Corollary \ref{cor:bdXSum}, this is ensured as long as
$p\in (0,1)$. On the other hand, for the interpolation result
we had to use the triangle inequality, since the sequence
$(\mathbf{I}_{\Lambda_{1,\varepsilon_n}}H_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\cF}$ of
interpolated multivariate Hermite polynomials does not form an
orthnormal sequence of $L^2(U,\gamma)$. This argument
requires the stronger condition
$(\norm[X]{u_{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\cF}\in\ell^1(\cF)$, resulting in
the stronger assumption $p\in (0,2/3)$ of Theorem
\ref{thm:int}.
\end{remark}
\subsection{Quadrature convergence}\label{sec:quadrate}
Analogous to Theorem.~\ref{thm:int} we obtain the following result for
the quadrature convergence with an improved convergence rate compared
to interpolation.
\begin{theorem}\label{thm:quad}
Let $u$ be $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic for some
${\boldsymbol{b}}\in\ell^p(\mathbb{N})$ and some $p\in (0,4/5)$. Let
$(c_{2,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ be as in Lemma \ref{lemma:cnu} with
$\varrho$ as in Theorem~\ref{thm:bdHolSum}. Then there exist $C>0$
and, for every $N\in\mathbb{N}$, $\varepsilon_n>0$
such that $|{\rm pts} (\Lambda_{2,\varepsilon_n})|\le n$ (with
$\Lambda_{2,\varepsilon_n}$ as in \eqref{eq:Lkeps}) and
\begin{equation*}
\norm[L^2(U,X;\gamma)]{u - \mathbf{Q}_{\Lambda_{2,\varepsilon_n}}u} \ \le \ C n^{-\frac{2}{p}+\frac{5}{2}}.
\end{equation*}
\end{theorem}
\begin{proof}
For $\varepsilon>0$ small enough and satisfying $|\Lambda_{2,\varepsilon}|>0$,
take $N\in\mathbb{N}$,
$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{{\boldsymbol{\nu}}\in\Lambda_{2,\varepsilon}}$ so large
that
\begin{equation}\label{eq:truncerrQ}
\normc[X]{\int_U u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})}
\le \norm[L^2(U,X;\gamma)]{u-\tilde u_N}\le \varepsilon^{\frac{1}{2}-\frac{p}{8(1-p)}},
\end{equation}
which is possible due to the $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy of $u$
(cp.~Definition~\ref{def:bdXHol} \ref{item:vN}). An appropriate
value of $\varepsilon$ depending on $n$ will be chosen below. In the
following for ${\boldsymbol{\nu}}\in\mathcal F$ we denote by $\tilde u_{N,{\boldsymbol{\nu}}}$ the
Wiener-Hermite PC expansion coefficient of $\tilde u_N$ and as
earlier $u_{\boldsymbol{\nu}}$ is the Wiener-Hermite PC expansion coefficient of
$u$.
Because $$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{{\boldsymbol{\nu}}\in\Lambda_{2,\varepsilon}}$$
and $\chi_{0,0}=0$, we have
$\mathbf{Q}_{\Lambda_{2,\varepsilon}}u= \mathbf{Q}_{\Lambda_{2,\varepsilon}} \tilde u_N$
(cp.~Remark.~\ref{rmk:defu}). Hence by \eqref{eq:truncerrQ}
\begin{align*}
\normc[X]{\int_U u({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Lambda_{2,\varepsilon}}u}
&=\normc[X]{\int_U u({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Lambda_{2,\varepsilon}}\tilde u_N}\nonumber\\
&\le \varepsilon^{\frac{1}{2}-\frac{p}{8(1-p)}}+
\normc[X]{\int_U \tilde u_N({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Lambda_{2,\varepsilon}} \tilde u_N}.
\end{align*}
By Lemma \ref{lemma:uN} we have
$$ \tilde u_N({\boldsymbol{y}})=u(y_1,\dots,y_N,0,0,\dots)=\sum_{{\boldsymbol{\nu}}\in\mathcal F_2^N}
\tilde u_{N,{\boldsymbol{\nu}}} H_{\boldsymbol{\nu}}((y_j)_{j=1}^N)$$ with pointwise absolute
convergence for all ${\boldsymbol{y}}\in U$. Thus we can interchange the
quadrature operator $\mathbf{Q}_{\Lambda_{2,\varepsilon}}$ with the Wiener-Hermite
PC expansion of $\tilde u_N$ and write
$$\mathbf{Q}_{\Lambda_{2,\varepsilon}} \tilde u_N = \sum_{{\boldsymbol{\nu}}\in\mathcal F_2^N}
\tilde u_{N;{\boldsymbol{\nu}}} \mathbf{Q}_{\Lambda_{2,\varepsilon}}H_{\boldsymbol{\nu}}$$ with pointwise
absolute convergence. Since $\Lambda_{2,\varepsilon}$ is nonempty and
downward closed we have ${\boldsymbol{0}}\in \Lambda_{2,\varepsilon}$. Then, by Lemma
\ref{lemma:VQprop}, \eqref{eq:L1boundLambda}, and using
$$
\int_U H_{\boldsymbol{\nu}}({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})=0
$$
for all ${\boldsymbol{0}}\neq{\boldsymbol{\nu}}\in\mathcal F$, we have that
\begin{align*}
\normc[X]{\int_U \tilde u_N({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Lambda_{2,\varepsilon}} \tilde u_N}
&=\normc[X]{\sum_{{\boldsymbol{\nu}}\in\mathcal F_2\backslash\Lambda_{2,\varepsilon}} \tilde u_{N;{\boldsymbol{\nu}}} \left(\int_{U}H_{\boldsymbol{\nu}}({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Lambda_{2,\varepsilon}}H_{\boldsymbol{\nu}}\right)}\nonumber\\
&\le \sum_{{\boldsymbol{\nu}}\in\mathcal F_2\backslash\Lambda_{2,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}}
(\norm[L^2(U;\gamma)]{H_{\boldsymbol{\nu}}} + |\mathbf{Q}_{\Lambda_{2,\varepsilon}}H_{\boldsymbol{\nu}}|)\nonumber\\
&\le \sum_{{\boldsymbol{\nu}}\in\mathcal F_2\backslash \Lambda_{2,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}} \left(1+p_{\boldsymbol{\nu}}(3)\right)
\nonumber\\
&\le 2\sum_{{\boldsymbol{\nu}}\in\mathcal F_2\backslash \Lambda_{2,\varepsilon}} \norm[X]{\tilde u_{N;{\boldsymbol{\nu}}}} p_{\boldsymbol{\nu}}(3)
.
\end{align*}
Choosing $r>8/p-5$ ($q=\frac{p}{1-p}$, $\tau=3$), according to
Proposition~\ref{prop:bestN}, \eqref{eq:cknubound} and
Theorem~\ref{thm:bdHolSum} (with
$(\varrho_j^{-1})_{j\in\mathbb{N}}\in\ell^{p/(1-p)}(\mathbb{N})$ as in
Theorem~\ref{thm:bdHolSum}) the last sum is bounded by
\begin{equation*}
C \left(\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \norm[X]{\tilde u_{N,{\boldsymbol{\nu}}}}^2\right)
\varepsilon^{\frac{1}{2}-\frac{q}{8}}
\le C({\boldsymbol{b}})\delta^2\varepsilon^{\frac{1}{2}-\frac{q}{8}}=C({\boldsymbol{b}}) \varepsilon^{\frac{1}{2}-\frac{p}{8(1-p)}},
\end{equation*}
and the constant $C({\boldsymbol{b}})$ from Theorem~\ref{thm:bdHolSum} does not
depend on $N$ and $\delta$.
From \eqref{ptsLambda0} it follows that
\begin{equation*}
|{\rm pts} (\Lambda_{k,\epsilon})|
\leq C \varepsilon^{-\frac{q}{4}}=C \varepsilon^{-\frac{p}{4(1-p)}}.
\end{equation*}
For every $n \in {\mathbb N}$, we choose an $\varepsilon_n>0$ satisfying the
condition $$n/2 \le C\varepsilon_n^{-\frac{p}{4(1-p)}} \le n.$$ Then
the claim holds true for the chosen $\varepsilon_n$.
\end{proof}
\begin{remark}
Interpolation formulas based index sets like
$\Lambda(\xi):= \{{\boldsymbol{\nu}} \in \cF: \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}) \le
\xi^{2/q}\}$ (where $\xi >0$ is a large parameter), respectively,
have been proposed in \cite{ErnstSprgkTam18}. There,
dimension-independent convergence rates of sparse-grid interpolation
were obtained. For given summability of the Wiener-Hermite expansion
coefficients the rates in \cite{ErnstSprgkTam18} are lower than
those obtained in the present analysis. The existence of Smolyak
type quadratures with a proof of dimension-independent convergence
rates was shown first in \cite{ChenlogNQuad2018}. There, symmetry
of the GM and corresponding cancellations were not exploited, and
these quadrature formulas provide lower (albeit
dimension-independent) convergence rates in terms of the number of
function evaluations as in Theorems \ref{thm:int} and
\ref{thm:quad}.
\end{remark}
\newpage
\
\newpage
\section{Multilevel approximation}
\label{sec:MLApprox}
In this section we introduce a multilevel interpolation and quadrature
algorithm. The presentation and arguments follow mostly \cite[Section
3.2]{JZdiss}, where multilevel algorithms for the uniform measure on
the hypercube $[-1,1]^\infty$ were analyzed (in contrast to the case
of a product GM on $U$, which we consider here). We also refer to
\cite{dD21} where multilevel convergence rates in the Gaussian case
were proven (see Remark.~\ref{rmk:comparison} for the main difference
of the present analysis to \cite{dD21}).
\subsection{Setting and notation}
\label{sec:SetNot}
To approximate the solution $u$ to a parametric PDE as in the examples
of the preceding sections, the interpolation operator $\mathbf{I}_\Lambda$
introduced in Section \ref{sec:int} requires function values of $u$ at
different interpolation points in the parameter space $U$. For a
parameter ${\boldsymbol{y}}\in U$, typically the PDE solution $u({\boldsymbol{y}})$, which is a
function belonging to a Sobolev space over a \emph{physical domain}
${\domain}$, is not given in closed form and has to be approximated. The idea
of multilevel approximations is to combine interpolants of
approximations to $u$ at different spatial accuracies, in order to
reduce the overall computational complexity. This will now be
formalized.
First, we assume given a sequence $(\sw{l})_{l\in\mathbb{N}_0}$,
exhibiting the properties of the following assumption. Throughout
$\sw{l}$ will be interpreted as a measure for the computational
complexity of evaluating an approximation $u^l:U\to X$ of
$u:U\to X$ at a parameter ${\boldsymbol{y}}\in U$. Here we use a superscript $l$
rather than a subscript for the approximation level, as the subscript
is reserved for the dimension truncated version $u_N$ of $u$ as in
Definition~\ref{def:bdXHol}.
\begin{assumption}\label{ass:SW}
The sequence $(\sw{l})_{l\in\mathbb{N}_0}\subseteq\mathbb{N}_0$ is strictly
monotonically increasing and $\sw{0}=0$. There exists a constant
$K_{\mathfrak{W}}\ge 1$ such that for all $l\in\mathbb{N}$
\begin{enumerate}
\item\label{item:SW1} $\sum_{j=0}^{l}\sw{j}\le K_{\mathfrak{W}} \sw{l}$,
\item\label{item:SW2} $l\le K_{\mathfrak{W}}(1+\log(\sw{l}))$,
\item\label{item:SW4} $\sw{l}\le K_{\mathfrak{W}} (1+\sw{l-1})$,
\item\label{item:SW3} for every $r>0$ there exists $C=C(r)>0$
independent of $l$ such that
$$\sum_{j=l}^{\infty} \sw{j}^{-r}\le C (1+\sw{l})^{-r}.$$
\end{enumerate}
\end{assumption}
Assumption \ref{ass:SW} is satisfied if $(\sw{l})_{l\in\mathbb{N}}$ is
exponentially increasing, (for instance $\sw{l}=2^l$,
$l\in\mathbb{N}$). In the following we write
$\mathfrak{W}:=\set{\sw{l}}{l\in\mathbb{N}_0}$ and
\begin{equation*}
\lfloor x \rfloor_\mathfrak{W} :=\max\set{\sw{l}}{\sw{l}\le x}.
\end{equation*}
We work under the following \emph{hypothesis on the discretization
errors in physical space}: we quantify the convergence of the
discretization scheme with respect to the discretization level
$l\in\mathbb{N}$. Specifically, we assume the approximation $u^l$ to
$u$ to behave asymptotically
as
\begin{equation}\label{eq:FEMrate}
\norm[X]{u({\boldsymbol{y}})-u^l({\boldsymbol{y}})}\le C({\boldsymbol{y}}) \sw{l}^{-{\alpha}}\qquad\forall l\in\mathbb{N},
\end{equation}
for some fixed convergence rate ${\alpha}>0$ of the ``physical space
discretization'' and with constant $C({\boldsymbol{y}}) > 0$ depending on the
parameter sequence ${\boldsymbol{y}}$. We will make this assumption on $u^l$
more precise shortly. If we think of $u^l({\boldsymbol{y}})\in H^1({\domain})$ for the
moment as a FEM approximation to the exact solution
$u({\boldsymbol{y}})\in H^1({\domain})$ of some ${\boldsymbol{y}}$-dependent elliptic PDE, then
$\sw{l}$ could stand for the number of degrees of freedom of the
finite element space. In this case ${\alpha}$ corresponds to the FEM
convergence rate. Assumption \ref{ass:SW} will for instance be
satisfied if for each consecutive level the meshwidth is cut in
half. Examples are provided by the FE spaces discussed in Section
\ref{S:FEIntrp}, Proposition \ref{prop:FECorner}. As long as the
computational cost of computing the FEM solution is proportional to
the dimension $\sw{l}$ of the FEM space, $\sw{l}^{-{\alpha}}$ is
the error in terms of the work $\sw{l}$. Such an assumption usually
holds in one spatial dimension, where the resulting stiffness matrix
is tridiagonal. For higher spatial dimensions solving the
corresponding linear system is often times not of linear complexity,
in which case the convergence rate ${\alpha}>0$ has to be adjusted
accordingly.
We now state our assumptions on the sequence of functions
$(u^l)_{l\in\mathbb{N}}$ approximating $u$. Equation \eqref{eq:FEMrate}
will hold in an $L^2$ sense over all parameters ${\boldsymbol{y}}\in U$,
cp.~Assumption \ref{ass:ml} \ref{item:u-ujclose}, and Definition
\ref{def:bdXHol} \ref{item:varphi}.
\begin{assumption}\label{ass:ml}
Let $X$ be a separable Hilbert space and let
$(\sw{l})_{l\in\mathbb{N}_0}$ satisfy Assumption \ref{ass:SW}.
Furthermore, $0<p_1\le p_2<\infty$, ${\boldsymbol{b}}_1\in\ell^{p_1}(\mathbb{N})$,
${\boldsymbol{b}}_2\in\ell^{p_2}(\mathbb{N})$, $\xi>0$, $\delta>0$ and there exist
functions $u\in L^2(U,X;\gamma)$,
$(u^l)_{l\in\mathbb{N}}\subseteq L^2(U,X;\gamma)$ such that
\begin{enumerate}
\item $u\in L^2(U,X;\gamma)$ is $({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic,
\item\label{item:u-ujbound} $(u-u^l)\in L^2(U,X;\gamma)$ is
$({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic for every $l\in\mathbb{N}$,
\item\label{item:u-ujclose} $(u-u^l)\in L^2(U,X;\gamma)$ is
$({\boldsymbol{b}}_2,\xi,\delta \sw{l}^{-{\alpha}},X)$-holomorphic for every
$l\in\mathbb{N}$.
\end{enumerate}
\end{assumption}
\begin{remark}
Items \ref{item:u-ujbound} and \ref{item:u-ujclose} are two
assumptions on the domain of holomorphic extension of the
discretization error $e_l:=u-u^l:U\to X$. As pointed out in
Remark.~\ref{rmk:bdexpl}, the faster the sequence ${\boldsymbol{b}}$ decays the
larger the size of holomorphic extension, and the smaller $\delta$
the smaller the upper bound of this extension.
Hence items \ref{item:u-ujbound} and \ref{item:u-ujclose} can be
interpreted as follows: Item \ref{item:u-ujbound} implies that
$e_l$ has a large domain of holomorphic extension. Item
\ref{item:u-ujclose} yields, that by considering the extension of
$e_l$ on a smaller domain, we can get a ($l$-dependent)
smaller upper bound of the extension of $e_l$ (in the sense of
Definition~\ref{def:bdXHol} \ref{item:varphi}). Hence there is a
tradeoff between choosing the size of the domain of the holomorphic
extension and the upper bound of this extension.
\end{remark}
\subsection{Multilevel algorithm}
\label{sec:MLAlg}
Let $\mathbf{l}=(l_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}\subseteq\mathbb{N}_0$ be a sequence of
discretization levels, associating with each multiindex ${\boldsymbol{\nu}}\in\cF$ a
discretization level $l_{\boldsymbol{\nu}}\in\mathbb{N}_0$. With the sequence
$l_{\boldsymbol{\nu}}\in\mathbb{N}_0$, we associate sets of multiindices via
\begin{equation}\label{eq:Gamma}
\Gamma_j=\Gamma_j(\mathbf{l}):=\set{{\boldsymbol{\nu}}\in\mathcal F}{l_{\boldsymbol{\nu}}\ge j}\qquad\forall j\in\mathbb{N}_0.
\end{equation}
Throughout we will assume that
$$|\mathbf{l}|=\sum_{{\boldsymbol{\nu}}\in\mathcal F}l_{\boldsymbol{\nu}}<\infty$$ and that $\mathbf{l}$ is
monotonically decreasing, meaning that ${\boldsymbol{\nu}}\le{\boldsymbol{\mu}}$ implies
$l_{\boldsymbol{\nu}}\gel_{\boldsymbol{\mu}}$. In this case each $\Gamma_j\subseteq\mathcal F$,
$j\in\mathbb{N}$, is finite and downward closed. Moreover $\Gamma_0=\mathcal F$, and
the sets $(\Gamma_j )_{j\in {\mathbb N}_0}$ are nested according to
\begin{equation*}
\mathcal F = \Gamma_0\supseteq\Gamma_1\supseteq \Gamma_2\dots.
\end{equation*}
With $(u^l)_{l\in\mathbb{N}}$ as in Assumption \ref{ass:ml}, we now
define
\begin{equation}\label{eq:VIml}
\mathbf{I}^{\rm ML}_\mathbf{l} u :=
\sum_{j\in\mathbb{N}} (\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})u^j.
\end{equation}
A few remarks are in order. First, the index $\mathbf{l}$ indicates that
the sets $\Gamma_j=\Gamma_j(\mathbf{l})$ depend on the choice of $\mathbf{l}$,
although we usually simply write $\Gamma_j$ in order to keep the
notation succinct. Secondly, due to $|\mathbf{l}|<\infty$ it holds
$$\max_{{\boldsymbol{\nu}}\in\mathcal F}l_{\boldsymbol{\nu}}=:L<\infty$$ and thus $\Gamma_j=\emptyset$
for all $j>L$. Defining $\mathbf{I}_{\emptyset}$ as the constant $0$
operator, the infinite series \eqref{eq:VIml} can also be written as
the finite sum
\begin{equation*}
\mathbf{I}^{\rm ML}_\mathbf{l} u = \sum_{j=1}^L(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})u^j
= \mathbf{I}_{\Gamma_1}u^1+\mathbf{I}_{\Gamma_2}(u^2-u^1)+\dots +\mathbf{I}_{\Gamma_L}(u^L-u^{L-1}),
\end{equation*}
where we used $\mathbf{I}_{\Gamma_{L+1}}=0$. If we had
$\Gamma_1=\dots=\Gamma_L$, this sum would reduce to
$\mathbf{I}_{\Gamma_L}u^L$, which is the interpolant of the approximation
$u^L$ at the (highest) discretization level $L$. The main observation
of multilevel analyses is that it is beneficial not to choose all
$\Gamma_j$ equal, but instead to balance out the accuracy of the
interpolant $\mathbf{I}_{\Gamma_j}$ (in the parameter) and the accuracy of
the approximation $u^j$ of $u$.
A multilevel sparse-grid quadrature algorithm is obtained analogously
via
\begin{equation}\label{eq:VQml}
\mathbf{Q}^{\rm ML}_\mathbf{l} u :=
\sum_{j\in\mathbb{N}}(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})u^j,
\end{equation}
with $\Gamma_j=\Gamma_j(\mathbf{l})$ as in \eqref{eq:Gamma}.
\subsection{Work model and choice of discretization levels}
\label{sec:WrkModLevChoic}
In the following we will prove algebraic convergence of multilevel
interpolation/quadrature w.r.t.\ the $L^2(U,X;\gamma)$-norm. The
convergence rate will hold in terms of the \emph{work} of computing
$\mathbf{I}^{\rm ML}_\mathbf{l}$.
As mentioned above, for a level $l\in\mathbb{N}$, we interpret
$\sw{l}\in\mathbb{N}$ as a measure of the computational complexity of
evaluating $u^l$ at an arbitrary parameter ${\boldsymbol{y}}\in U$. As
discussed in Section \ref{sec:NoFnEval}, computing $\mathbf{I}_{\Gamma_j} v$
or $\mathbf{Q}_{\Gamma_j}v$ requires to evaluate the function $v$ at each
parameter in the set ${\rm pts}(\Gamma_j)\subseteq U$ introduced in
\eqref{eq:pts}. We recall the bound
\begin{equation*}
|{\rm pts}(\Gamma_j)|\le \sum_{{\boldsymbol{\nu}}\in\Gamma_j} p_{\boldsymbol{\nu}}(1),
\end{equation*}
on the cardinality of this set obtained in \eqref{eq:ptsLbound}. As
an upper bound of the work corresponding to the evaluation of all
functions required for the multilevel interpolant in \eqref{eq:VIml},
we obtain
\begin{equation*}
\sum_{j\in\mathbb{N}}\sw{j}
\left(\sum_{{\boldsymbol{\nu}}\in\Gamma_j(\mathbf{l})} p_{\boldsymbol{\nu}}(1)
+ \sum_{{\boldsymbol{\nu}}\in\Gamma_{j+1}(\mathbf{l})} p_{\boldsymbol{\nu}}(1) \right).
\end{equation*}
Since $\Gamma_{j+1}\subseteq \Gamma_{j}$, up the factor $2$ the work
of a sequence $\mathbf{l}$ is defined by
\begin{equation}\label{eq:work}
\mathrm{work}(\mathbf{l}) := \sum_{j=1}^L \sw{j} \sum_{{\boldsymbol{\nu}}\in\Gamma_j(\mathbf{l})} p_{\boldsymbol{\nu}}(1)
= \sum_{{\boldsymbol{\nu}}\in\mathcal F(\mathbf{l})} p_{\boldsymbol{\nu}}(1) \sum_{j=1}^{l_{\boldsymbol{\nu}}}\sw{j},
\end{equation}
where we used the definition of $\Gamma_j(\mathbf{l})$ in \eqref{eq:Gamma},
$L:= \max_{{\boldsymbol{\nu}}\in\mathcal F}l_{\boldsymbol{\nu}} < \infty$ and the finiteness of the
set
$$\mathcal F(\mathbf{l}):= \{{\boldsymbol{\nu}} \in \mathcal F: l_{\boldsymbol{\nu}} > 0\}.$$
The efficiency of the multilevel interpolant critically relies on a
suitable choice of levels $\mathbf{l}=(l_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}$. This will
be achieved with the following algorithm, which constructs $\mathbf{l}$
based on two positive sequences
$(\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_1}(\mathcal F)$ and
$(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_2}(\mathcal F)$. The algorithm is
justified due to Lemma \ref{LEMMA:MLWEIGHTNEW} shown in Section
\ref{app:mlweight}. This technical lemma, which is a variation of
\cite[Lemma 3.2.7]{JZdiss}, constitute the central part of the proofs
of the convergence results presented in the rest of this section.
\begin{algorithm}[H]
\caption{
$(l_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in\mathcal F}={\rm
ConstructLevels}((\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F},(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F},q_1,{\alpha},\varepsilon)$}
\label{alg:levels}
\begin{algorithmic}[1]
\State $(l_{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\leftarrow (0)_{{\boldsymbol{\nu}}\in\mathcal F}$
\State
$\Lambda_\varepsilon\leftarrow\set{{\boldsymbol{\nu}}\in\mathcal F}{\seqi{{\boldsymbol{\nu}}}^{-1}\ge \varepsilon}$
\For{${\boldsymbol{\nu}}\in\Lambda_\varepsilon$} \State
$\delta \leftarrow \varepsilon^{-\frac{1/2-q_1/4}{{\alpha}}}
\seqii{{\boldsymbol{\nu}}}^{\frac{-1}{1+2{\alpha}}}
\left(\sum_{{\boldsymbol{\mu}}\in\Lambda_\varepsilon}
\seqii{{\boldsymbol{\mu}}}^{\frac{-1}{1+2{\alpha}}}
\right)^{\frac{1}{2{\alpha}}}$ \State
$l_{{\boldsymbol{\nu}}}\leftarrow \max\set{j\in\mathbb{N}_0}{\sw{j}\le \delta}$
\EndFor \State \Return $(l_{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$
\end{algorithmic}
\end{algorithm}
\begin{remark}
An efficient algorithm determining $\Lambda_\varepsilon$ in line 2 of
Algorithm \ref{alg:levels} can be found in \cite[Algorithm
2]{JZdiss}.
\end{remark}
\subsection{Construction of an allocation of discretization levels}
\label{app:mlweight}
We detail the construction of an allocation of discretization levels
along a sequence of Wiener-Hermite PC expansion coefficients. It is
valid for collections $( u_{\boldsymbol{\nu}} )_{{\boldsymbol{\nu}} \in \cF}$ of Wiener-Hermite PC
expansion coefficients taking values in a separable Hilbert space, say
$X$, with additional regularity, being $X^s \subset X$, allowing for
weaker (weighted) summability of the $V^s$-norms
$( \|u_{\boldsymbol{\nu}}\|_{X^s})_{{\boldsymbol{\nu}} \in \cF}$. In the setting of elliptic BVPs
with log-Gaussian diffusion coefficient, $X = V =H^1_0({\mathrm D})$, and
$X^s$ is, for example, a weighted Kondrat'ev space in ${\mathrm D}$ as
introduced in Section \ref{sec:KondrAn}. We phrase the result and the
construction in abstract terms so that the allocation is applicable to
more general settings, such as the parabolic IBVP in Section
\ref{sec:LinParPDE}.
For a given, dense sequence $({X}_l)_{l\in {\mathbb N}_0} \subset X$ of
nested, finite-dimensional subspaces and target accuracy
$0 < \varepsilon \leq 1$, in the numerical approximation of Wiener-Hermite PC
expansions of random fields $u$ taking values in $X$, we consider
approximating the Wiener-Hermite PC expansion coefficients $u_{\boldsymbol{\nu}}$ in
$X$ from $X_l$. The assumed density of the sequence
$(X_l)_{l\in {\mathbb N}_0} \subset X$ in $X$ ensures that for
$u\in L^2(U,X;\gamma)$ the sequence
$( u_{\boldsymbol{\nu}} )_{{\boldsymbol{\nu}} \in \cF} \subset X$ is square summable, in the sense
that $( \|u_{\boldsymbol{\nu}}\|_{X})_{{\boldsymbol{\nu}} \in \cF} \in \ell_2({\mathcal F})$
The following lemma is a variation of \cite[Lemma 3.2.7]{JZdiss}. Its
proof, is, with several minor modifications, taken from \cite[Lemma
3.2.7]{JZdiss}. We remark that the construction of the map
$\mathbf{l}(\varepsilon,{\boldsymbol{\nu}})$, as described in the lemma, mimicks Algorithm
\ref{alg:levels}. Again, a convergence rate is obtained that is not
prone to the so-called ``curse of dimensionality'', being limited only
by the available sparsity in the coefficient sequence of
Wiener-Hermite PC expansion for the parametric solution manifold.
\begin{lemma}\label{LEMMA:MLWEIGHTNEW}
Let $\mathfrak{W}=\set{\sw{l}}{l\in\mathbb{N}_0}$ satisfy Assumption
\ref{ass:SW}. Let $q_1\in [0,2)$, ${q_2}\in [q_1,\infty)$ and
${\alpha}>0$. Let
\begin{enumerate}
\item $(a_{j,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\subseteq [0,\infty)$ for every
$j\in\mathbb{N}_0$,
\item $(\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\subseteq (0,\infty)$ and
$(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}\subseteq (0,\infty)$ be such that
$$(\seqi{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_1}(\mathcal F) \;\mbox{and}\;
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}p_{\boldsymbol{\nu}}(1/2+\alpha)\big)_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{{q_2}}(\mathcal F),
$$
\item
\begin{equation}\label{eq:sumsleinfty}
\sup_{j\in\mathbb{N}_0} \left(\sum_{{\boldsymbol{\nu}}\in\mathcal F}a_{j,{\boldsymbol{\nu}}}^2\seqi{{\boldsymbol{\nu}}}\right)^{1/2}=:C_1<\infty,\qquad
\sup_{j\in\mathbb{N}_0} \left(\sum_{{\boldsymbol{\nu}}\in\mathcal F}(\sw{j}^{{\alpha}}a_{j,{\boldsymbol{\nu}}})^2\seqii{{\boldsymbol{\nu}}}\right)^{1/2}
=:C_2<\infty.
\end{equation}
\end{enumerate}
For every $\varepsilon>0$ define
$\Lambda_\varepsilon =
\set{{\boldsymbol{\nu}}\in\mathcal F}{\seqi{{\boldsymbol{\nu}}}^{-1}\ge\varepsilon}$,
$\wk{\varepsilon,{\boldsymbol{\nu}}}:= 0$ for all
${\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_\varepsilon$, and define
\begin{equation*}\label{eq:wepsnuweight}
\wk{\varepsilon,{\boldsymbol{\nu}}}:= \left\lfloor
\varepsilon^{-\frac{1/2 - q_1/4}{{\alpha}}}
\seqii{{\boldsymbol{\nu}}}^{\frac{-1}{1+2{\alpha}}}
\Bigg(\sum_{{\boldsymbol{\mu}}\in\Lambda_\varepsilon}
\seqii{{\boldsymbol{\mu}}}^{\frac{-1}{1+2{\alpha}}}
\Bigg)^{\frac{1}{2{\alpha}}}\right\rfloor_{\mathfrak{W}}\in\mathfrak{W}\qquad\forall{\boldsymbol{\nu}}\in\Lambda_\varepsilon.
\end{equation*}
Furthermore for every $\varepsilon>0$, ${\boldsymbol{\nu}}\in\mathcal F$ let
$l_{\varepsilon,{\boldsymbol{\nu}}}\in\mathbb{N}_0$ be the corresponding
discretization level, i.e.\
$\wk{\varepsilon,{\boldsymbol{\nu}}}=\sw{l_{\varepsilon,{\boldsymbol{\nu}}}}$, and define the
maximal discretization level
$$L(\varepsilon):=\max \set{{l_{\varepsilon,{\boldsymbol{\nu}}}}}{{\boldsymbol{\nu}}\in\mathcal F}.$$
Denote $\mathbf{l}_\varepsilon=({l_{\varepsilon,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathcal F}$.
Then there exists a constant $C>0$ and tolerances
$\varepsilon_n\in (0,1]$ such that for every $n \in {\mathbb N}$ holds
$\mathrm{work}(\mathbf{l}_{\varepsilon_n}) \le n$ and
\begin{equation*}\label{eq:mlestweight}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon_n,{\boldsymbol{\nu}}}}}^{L(\varepsilon_n)} a_{j,{\boldsymbol{\nu}}}
\le C (1+\log n )n^{-R},
\end{equation*}
where the rate $R$ is given by
\begin{equation*}
R = \min\left\{{\alpha},\frac{{\alpha}(q_1^{-1}-1/2)}{{\alpha}+q_1^{-1}-{q_2}^{-1}}\right\}.
\end{equation*}
\end{lemma}
\begin{proof}
Throughout this proof denote $\delta:= 1/2 - q_1/4 >0$.
In the following
\begin{equation*}
\twk{\varepsilon,{\boldsymbol{\nu}}}:=
\varepsilon^{-\frac{\delta}{{\alpha}}}
\seqii{{\boldsymbol{\nu}}}^{\frac{-1}{1+2{\alpha}}}
\Bigg(\sum_{{\boldsymbol{\mu}}\in\Lambda_\varepsilon}
\seqii{{\boldsymbol{\mu}}}^{\frac{-1}{1+2{\alpha}}}
\Bigg)^{\frac{1}{2{\alpha}}} \qquad\forall{\boldsymbol{\nu}}\in\Lambda_\varepsilon,
\end{equation*}
i.e.\
$\wk{\varepsilon,{\boldsymbol{\nu}}}=\lfloor \twk{\varepsilon,{\boldsymbol{\nu}}}\rfloor_\mathfrak{W}$.
Note that $0<\twk{\varepsilon,{\boldsymbol{\nu}}}$ is well-defined for all
${\boldsymbol{\nu}}\in\Lambda_\varepsilon$ since $\seqii{{\boldsymbol{\nu}}}>0$ for all
${\boldsymbol{\nu}}\in\mathcal F$ by assumption. Due to Assumption
\ref{ass:SW} \ref{item:SW4} it holds
\begin{equation}\label{eq:sw1Kw}
\frac{\twk{\varepsilon,{\boldsymbol{\nu}}}}{K_{\mathfrak{W}}}\le 1+\wk{\varepsilon,{\boldsymbol{\nu}}}\le
1+\twk{\varepsilon,{\boldsymbol{\nu}}}
\qquad \forall {\boldsymbol{\nu}}\in\Lambda_\varepsilon.
\end{equation}
Since
$(\seqi{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_1}(\mathcal F)$
and \eqref{eq:sumsleinfty}, we get
$$
\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_{\varepsilon}} a_{j,{\boldsymbol{\nu}}}
\le \Bigg( \sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_{\varepsilon}}
a_{j,{\boldsymbol{\nu}}}^2 c_{{\boldsymbol{\nu}}}\Bigg)^{1/2} \Bigg(
\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_{\varepsilon}}
c_{{\boldsymbol{\nu}}}^{-1}\Bigg)^{1/2} \leq C_1 \Bigg(
\sum_{\seqi{{\boldsymbol{\nu}}}^{-1}\leq\varepsilon}
c_{{\boldsymbol{\nu}}}^{-\frac{q_1}{2}}c_{{\boldsymbol{\nu}}}^{\frac{q_1}{2}-1}\Bigg)^{1/2}
\le C \varepsilon^{\delta}
$$
with the constant $C$ independent of $j$ and
$\varepsilon$. Thus,
\begin{equation} \label{sum_notLambda}
\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_\varepsilon}
\sum_{j=0}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}} =
\sum_{j=0}^{L(\varepsilon)}\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_\varepsilon}a_{j,{\boldsymbol{\nu}}}
\le C_1 (1+L(\varepsilon))\varepsilon^{\delta}.
\end{equation}
Next with $C_2$ as in \eqref{eq:sumsleinfty},
\begin{align}\label{eq:sumsuma2jbnu}
\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j= {l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}
&= \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}
\sw{j}^{\alpha} \sw{j}^{-{\alpha}}
\seqii{{\boldsymbol{\nu}}}^{1/2} \seqii{{\boldsymbol{\nu}}}^{-1/2}\nonumber\\
&\le\Bigg(\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j=0}^{L(\varepsilon)}\big(a_{j,{\boldsymbol{\nu}}}\sw{j}^{\alpha} \seqii{{\boldsymbol{\nu}}}^{1/2}\big)^2 \Bigg)^{\frac{1}{2}}
\Bigg( \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j\ge {l_{\varepsilon,{\boldsymbol{\nu}}}}}
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}
\sw{j}^{-{\alpha}}\big)^{2}\Bigg)^{\frac{1}{2}}\nonumber\\
&\le C_2 (1+L(\varepsilon))
\Bigg( \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j\ge {l_{\varepsilon,{\boldsymbol{\nu}}}}}
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}
\sw{j}^{-{\alpha}}\big)^{2}\Bigg)^{\frac{1}{2}}.
\end{align}
Assumption \ref{ass:SW} \ref{item:SW3} implies for
some $C_3$
\begin{equation*
\sum_{j\ge {l_{\varepsilon,{\boldsymbol{\nu}}}}} \sw{j}^{-2{\alpha}}\le {C_3^2}
(1+\sw{{l_{\varepsilon,{\boldsymbol{\nu}}}}})^{-2{\alpha}}= {C_3^2}
(1+\wk{\varepsilon,{\boldsymbol{\nu}}})^{-2{\alpha}},
\end{equation*}
so that by \eqref{eq:sw1Kw} and
\eqref{eq:sumsuma2jbnu}
\begin{align}\label{eq:firstsummlweight2}
\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j= {l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}&\le
C_3C_2(1+L(\varepsilon))
\left( \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}(1+\wk{\varepsilon,{\boldsymbol{\nu}}})^{-{\alpha}}\big)^{2}\right)^{\frac{1}{2}}\nonumber\\
&\le C_3C_2{K_{\mathfrak{W}}^{\alpha}}(1+L(\varepsilon))
\left( \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}\twk{\varepsilon,{\boldsymbol{\nu}}}^{-{\alpha}}\big)^{2}\right)^{\frac{1}{2}}.
\end{align}
Inserting the definition of $\twk{\varepsilon,{\boldsymbol{\nu}}}$, we have
\begin{align}\label{eq:firstsummlweight3}
\left( \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
\big(\seqii{{\boldsymbol{\nu}}}^{-1/2}\twk{\varepsilon,{\boldsymbol{\nu}}}^{-{\alpha}}\big)^{2}\right)^{\frac{1}{2}}
&=
\varepsilon^{\delta}
\Bigg(\sum_{{\boldsymbol{\mu}}\in\Lambda_\varepsilon}
\seqii{{\boldsymbol{\mu}}}^{\frac{-1}{1+2{\alpha}}}
\Bigg)^{-{\alpha} \frac{1}{2{\alpha}}}
\left(\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
\seqii{{\boldsymbol{\nu}}}^{-1}\seqii{{\boldsymbol{\nu}}}^{\frac{2{\alpha}}{1+2{\alpha}} } \right)^{\frac{1}{2}}
=
\varepsilon^{\delta},
\end{align}
where we used
\begin{equation*}
-1 + \frac{2{\alpha}}{1+2{\alpha}} =
\frac{-(1+2{\alpha})+2{\alpha}}{1+2{\alpha}}
= \frac{-1}{1+2{\alpha}}.
\end{equation*}
Using Assumption \ref{ass:SW} \ref{item:SW2} and the
definition of $\mathrm{work}(\mathbf{l}_\varepsilon)$ in \eqref{eq:work}
we get
\begin{equation}\label{eq:Jeps}
L(\varepsilon)
\le \log(1+\max_{{\boldsymbol{\nu}}\in\mathcal F}\wk{\varepsilon,{\boldsymbol{\nu}}})
\le\log(1+\mathrm{work}(\mathbf{l}_\varepsilon)).
\end{equation}
Hence, \eqref{sum_notLambda},
\eqref{eq:firstsummlweight2},
\eqref{eq:firstsummlweight3} {and \eqref{eq:Jeps}}
yield
\begin{equation}\label{eq:firstcasesn1est}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)}a_{j,{\boldsymbol{\nu}}}=
\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}+
\sum_{{\boldsymbol{\nu}}\in\mathcal F\backslash\Lambda_\varepsilon} \sum_{j=0}^{L(\varepsilon)}a_{j,{\boldsymbol{\nu}}}\le
C \big(1+\log(\mathrm{work}(\mathbf{l}_\varepsilon))\big) \varepsilon^{\delta}.
\end{equation}
Next, we compute an upper bound for
$\mathrm{work}(\mathbf{l}_\varepsilon)$. By definition of
$\mathrm{work}(\mathbf{l}_\varepsilon)$ in \eqref{eq:work}, and using
Assumption \ref{ass:SW} \ref{item:SW1} as well as
$\wk{\varepsilon,{\boldsymbol{\nu}}}=\sw{{l_{\varepsilon,{\boldsymbol{\nu}}}}}$,
\begin{align}\label{eq:sn1vwk}
\mathrm{work}(\mathbf{l}_\varepsilon)
&= \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}p_{\boldsymbol{\nu}}(1)\sum_{\set{
j\in\mathbb{N}}{j\le{l_{\varepsilon,{\boldsymbol{\nu}}}}}}\sw{j}
\le \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}p_{\boldsymbol{\nu}}(1)K_{\mathfrak{W}}\wk{\varepsilon,{\boldsymbol{\nu}}}\nonumber\\
&\le K_{\mathfrak{W}} \sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}p_{\boldsymbol{\nu}}(1) \twk{\varepsilon,{\boldsymbol{\nu}}}
\le K_{\mathfrak{W}}
\varepsilon^{-\frac{\delta}{{\alpha}}} \left(\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon}
p_{\boldsymbol{\nu}}(1)\seqii{{\boldsymbol{\nu}}}^{\frac{-1}{1+2{\alpha}}}\right)^{\frac{1}{2{\alpha}}+1}\nonumber\\
&=K_{\mathfrak{W}} \varepsilon^{-\frac{\delta}{{\alpha}}} \left(\sum_{{\boldsymbol{\nu}}\in\Lambda_\varepsilon} \left(p_{\boldsymbol{\nu}}(1/2+{\alpha})\seqii{{\boldsymbol{\nu}}}^{-1/2}\right)^{\frac{2}{1+2{\alpha}}}\right)^{\frac{1}{2{\alpha}}+1},
\end{align}
where we used
$p_{\boldsymbol{\nu}}(1)=p_{\boldsymbol{\nu}}(1/2+{\alpha})^{2/(1+2{\alpha})}$ and
the fact that $p_{\boldsymbol{\nu}}(1)\ge 1$ for all ${\boldsymbol{\nu}}$.
We distinguish between the two cases
\begin{equation*}\label{eq:2casessn1}
\frac{2}{1+2{\alpha}}\ge {q_2}\qquad \text{and} \qquad
\frac{2}{1+2{\alpha}}< {q_2}.
\end{equation*}
In the first case, since
$(p_{\boldsymbol{\mu}}(1/2+{\alpha})\seqii{{\boldsymbol{\mu}}}^{-1/2})_{{\boldsymbol{\mu}}\in\mathcal F}\in\ell^{{q_2}}(\mathcal F)$,
{\eqref{eq:sn1vwk} implies}
\begin{equation}\label{eq:firstcasesn1w}
\mathrm{work}(\mathbf{l}_\varepsilon)\le C \varepsilon^{-\frac{\delta}{{\alpha}}}
\end{equation}
and hence,
$\log(\mathrm{work}(\mathbf{l}_\varepsilon)) \le \log
\big(C\varepsilon^{-\frac{\delta}{{\alpha}}}\big)$. Then
\eqref{eq:firstcasesn1est} together with
\eqref{eq:firstcasesn1w} implies
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}
\le C(1+{|\log(\varepsilon^{-1})|})\varepsilon^\delta .
\end{equation*}
For every $n \in {\mathbb N}$, we can find $\varepsilon_n>0$ such
that
$\frac{n}{2} \le C\varepsilon_n^{-\frac{\delta}{{\alpha}}}
\le n$. Then the claim of the corollary in the case
$\frac{2}{1+2{\alpha}} \ge q_2$ holds true for the
chosen $\varepsilon_n$.
Finally, let us address the case
$\frac{2}{1+2{\alpha}}<{q_2}$. Then, by
\eqref{eq:sn1vwk} and using H\"older's inequality with
$ {q_2}\frac{1+2{\alpha}}{2}>1$ we get
\begin{equation*}
\mathrm{work}(\mathbf{l}_\varepsilon)\le
K_{\mathfrak{W}} \varepsilon^{-\frac{\delta}{{\alpha}}}\norm[\ell^{{q_2}}(\mathcal F)]{(p_{\boldsymbol{\nu}}(1/2+{\alpha})\seqii{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}}^{\frac{1}{{\alpha}}}
|\Lambda_\varepsilon|^{\big(1-\frac{2}{q_2(1+2\alpha)}\big)\frac{1+2{\alpha}}{2{\alpha}}}.
\end{equation*}
Since
$$
|\Lambda_\varepsilon|=\sum_{{\boldsymbol{\nu}} \in \Lambda_\varepsilon}1 =
\sum_{\seqi{{\boldsymbol{\nu}}}^{-1}\geq\varepsilon}
c_{{\boldsymbol{\nu}}}^{-\frac{q_1}{2}}c_{{\boldsymbol{\nu}}}^{\frac{q_1}{2}} \leq
C \varepsilon^{-\frac{q_1}{2}},
$$
we obtain
\begin{align*}
\mathrm{work}(\mathbf{l}_\varepsilon)\le
K_{\mathfrak{W}} \varepsilon^{-\frac{\delta}{{\alpha}}-\frac{q_1}{2}(1-\frac{2}{q_2(1+2\alpha)})\frac{1+2{\alpha}}{2{\alpha}}} \le C \varepsilon^{-\frac{q_1}{2\alpha}\big(\alpha-\frac{1}{q_2}+\frac{1}{q_1}\big)} .
\end{align*}
For every $n \in {\mathbb N}$, we can find $\varepsilon_n>0$ such
that
$$ \frac{n}{2} \le C
\varepsilon_n^{-\frac{q_1}{2\alpha}\big(\alpha-\frac{1}{q_2}+\frac{1}{q_1}\big)}
\le n.$$ Thus the claim also holds true in the case
$\frac{2}{1+2{\alpha}}<q_2$.
\end{proof}
\subsection{Multilevel interpolation}
\label{sec:MLInterpol}
We are now in position to formulate a multilevel
interpolation convergence theorem. To this end, we
observe that our proofs of approximation rates have been
constructive: rather than being based on a best $N$-term
selection from the infinite set of Wiener-Hermite PC
expansion coefficients, a constructive selection process
of ``significant'' Wiener-Hermite PC expansion
coefficients, subject to a given prescribed
approximation tolerance, has been provided. In the
present section, we turn this into a concrete, numerical
selection process with complexity bounds. In particular,
we provide an \emph{a-priori allocation} of
discretization levels to Wiener-Hermite PC expansion
coefficients. This results on the one hand in an
explicit, algorithmic definition of a family of
multilevel interpolants which is parametrized by an
approximation threshold $\varepsilon>0$. On the other hand, it
will result in \emph{mathematical convergence rate
bounds in terms of computational work} rather than in
terms of, for example, number of active Wiener-Hermite
PC expansion coefficients, which rate bounds are free
from the curse of dimensionality.
The idea is as follows: let
${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}\in\ell^{p_1}({\mathbb N})$,
${\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}\in\ell^{p_2}({\mathbb N})$, and $\xi$
be the two sequences and constant from Assumption
\ref{ass:ml}. For two constants $K>0$ and $r>3$ (which
are still at our disposal and which will be specified
below), set for all $j\in\mathbb{N}$
\begin{equation}\label{eq:bvarrhoi}
\varrho_{1,j}
:=
b_{1,j}^{p_1-1} \frac{\xi}{4\norm[\ell^{p_1}]{{\boldsymbol{b}}_1}},
\qquad
\varrho_{2,j}:=b_{2,j}^{p_2-1} \frac{\xi}{4\norm[\ell^{p_2}]{{\boldsymbol{b}}_2}}.
\end{equation}
We let for all ${\boldsymbol{\nu}}\in\mathcal F$ (as in Lemma \ref{lemma:cnu}
for $k=1$ and with $\tau=3$)
\begin{equation}\label{eq:seqiseqii}
\seqi{{\boldsymbol{\nu}}}:=\prod_{j\in\mathbb{N}}\max\{1,K\varrho_{1,j}\}^2\nu_j^{r-3},\qquad
\seqii{{\boldsymbol{\nu}}}:=\prod_{j\in\mathbb{N}}\max\{1,K\varrho_{2,j}\}^2\nu_j^{r-3}.
\end{equation}
Based on those two
sequences,
Algorithm \ref{alg:levels} provides a collection of
discretization levels which sequence depends on $\varepsilon>0$
and is indexed over $\mathcal F$. We denote it by
$\mathbf{l}_\varepsilon=(l_{\varepsilon,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$. We now
state an upper bound for the error of the corresponding
multilevel interpolants in terms of the work measure in
\eqref{eq:work} as $\varepsilon\to 0$.
\begin{theorem}\label{thm:mlint}
Let $u\in L^2(U,X;\gamma)$ and
$u^l\in L^2(U,X;\gamma)$, $l\in\mathbb{N}$, satisfy
Assumption \ref{ass:ml} with some constants
${\alpha}>0$ and $0<p_1<2/3$ and $p_1\le p_2<1$. Set
$q_1:=p_1/(1-p_1)$. Assume that
$r>2(1+({\alpha}+1)q_1)/q_1+3$ (for $r$ as defined in
\eqref{eq:seqiseqii}). There exist constants $K>0$
(in \eqref{eq:seqiseqii}) and $C>0$ such that the
following holds.
For every $n \in {\mathbb N}$, there are positive constants
$\varepsilon_n\in (0,1]$ such that
$\mathrm{work}(\mathbf{l}_{\varepsilon_n}) \le n$ and with
$\mathbf{l}_{\varepsilon_n}=({l_{\varepsilon_n,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathcal F}$
as defined in Lemma \ref{LEMMA:MLWEIGHTNEW} (where
$c_{\boldsymbol{\nu}}$, $d_{\boldsymbol{\nu}}$ as in \eqref{eq:seqiseqii}) it
holds
\begin{equation*}
\norm[L^2(U,X;\gamma)]{u-\mathbf{I}_{\mathbf{l}_{\varepsilon_n}}^{\rm ML}u} \ \le \ C (1+\log n )n^{-R}
\end{equation*}
with the convergence rate
\begin{equation*}
R := \min\left\{{\alpha},\frac{{\alpha}(p_1^{-1}-3/2)}{{\alpha}+p_1^{-1}-p_2^{-1}}\right\}.
\end{equation*}
\end{theorem}
\begin{proof}
Throughout this proof we write
${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}$ and
${\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}$ for the two sequences in
Assumption \ref{ass:ml}. We point out that $\Gamma_j$
defined in \eqref{eq:Gamma} is downward closed for all
$j\in\mathbb{N}_0$. This can be easily deduced from the fact
that the sequences $(\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ and
$(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ are monotonically
increasing (i.e., e.g., ${\boldsymbol{\nu}}\le{\boldsymbol{\mu}}$ implies
$\seqi{{\boldsymbol{\nu}}}\le\seqi{{\boldsymbol{\mu}}}$) and the definition of
$\Lambda_\varepsilon$ and $l_{\varepsilon,{\boldsymbol{\nu}}}$ in Algorithm
\ref{alg:levels}. We will use this fact throughout the
proof, without mentioning it at every instance.
{\bf Step 1.} Given $n\in \mathbb{N}$, we choose
$\varepsilon:=\varepsilon_n$ as in Lemma \ref{LEMMA:MLWEIGHTNEW}.
Fix $N\in\mathbb{N}$ such that
$N>\max \{j:\, j \in\operatorname{supp}({\boldsymbol{\nu}}),\, l_{\varepsilon,{\boldsymbol{\nu}}}>
0\}$ and so large that
\begin{equation}\label{eq:truncerr_mlint}
\norm[L^2(U,X;\gamma)]{u-\tilde u_N}\le n^{-R},
\end{equation}
where $\tilde u_N:U\to X$ is as in
Definition~\ref{def:bdXHol} (this is possible due to
$$\lim_{N\to\infty}\norm[L^2(U,X;\gamma)]{u-\tilde u_N}=0,$$ which
holds by the $({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphy of $u$). By
Assumption \ref{ass:ml}, for every $j \in\mathbb{N}$ the function
$e^j:=u-u^j\in L^2(U,X;\gamma)$ is
$({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic and
$({\boldsymbol{b}}_2,\xi,\delta\sw{j}^\gamma,X)$-holomorphic. For notational
convenience we set $e^0:=u-0=u\in L^2(U,X;\gamma)$, so that $e^0$ is
$({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic and
$({\boldsymbol{b}}_2,\xi,\delta,X)$-holomorphic. Hence for every $j\in\mathbb{N}_0$ there
exists a function $\tilde e_N^j=\tilde u_N-\tilde u_N^j$ as in
Definition~\ref{def:bdXHol} \ref{item:vN}.
In the rest of the proof we use the following facts:
\begin{enumerate}
\item\label{item:u-uswunifabs} By Lemma \ref{lemma:uN}, for every
$j\in\mathbb{N}_0$, with the Wiener-Hermite PC expansion coefficients
$$\tilde e_{N,{\boldsymbol{\nu}}}^j:= \int_{U} H_{\boldsymbol{\nu}}({\boldsymbol{y}})\tilde e_{N}^j({\boldsymbol{y}})
\,\mathrm{d}\gamma({\boldsymbol{y}}),$$ it holds
\begin{equation*}\label{eq:u-uswunifabs}
\tilde e_{N}^j({\boldsymbol{y}}) = \sum_{{\boldsymbol{\nu}}\in\mathcal F} \tilde e_{N,{\boldsymbol{\nu}}}^j H_{\boldsymbol{\nu}}({\boldsymbol{y}})\qquad\forall {\boldsymbol{y}}\in U,
\end{equation*}
with pointwise absolute convergence.
\item\label{item:aux} By Lemma \ref{lemma:cnu}, upon choosing $K>0$
in \eqref{eq:seqiseqii} large enough, and because $r>3$,
\begin{equation*}
C_0 \seqi{{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(3)\le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_1),\qquad
C_0 \seqii{{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(3)\le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_2)
\qquad \forall {\boldsymbol{\nu}}\in \mathcal F_1.
\end{equation*}
We point out that by definition of ${\boldsymbol{\varrho}}_i$, $i\in\{1,2\}$, in
\eqref{eq:bvarrhoi}, it holds
$\varrho_{i,j}\sim b_{i,j}^{-(1-p_i)}$ and therefore
$(\varrho_{i,j}^{-1})_{j\in\mathbb{N}}\in\ell^{q_i}(\mathbb{N})$ with
$q_i:=p_i/(1-p_i)$, $i\in\{1,2\}$.
\item\label{item:seqiseqiisum} Due to
$r>2(1+({\alpha}+1)q_1)/q_1+3$,
the condition of Lemma \ref{lemma:summabcnu} is satisfied (with
$k=1$, $\tau=3$ and $\theta=({\alpha}+1)q_1$). Hence the lemma
gives
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} p_{\boldsymbol{\nu}}(({\alpha}+1)q_1)\seqi{{\boldsymbol{\nu}}}^{-q_1/2}<\infty
\qquad
\Rightarrow\qquad
(p_{\boldsymbol{\nu}}({\alpha}+1)\seqi{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_1}(\mathcal F)
\end{equation*}
and similarly
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} p_{\boldsymbol{\nu}}(({\alpha}+1)q_2)\seqii{{\boldsymbol{\nu}}}^{-q_2/2}<\infty
\qquad
\Rightarrow\qquad
(p_{\boldsymbol{\nu}}({\alpha}+1)\seqii{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_2}(\mathcal F).
\end{equation*}
\item\label{item:tedecay} By Theorem~\ref{thm:bdHolSum} and item
\ref{item:aux}, for all $j\in\mathbb{N}_0$
\begin{equation*}
C_0 \sum_{{\boldsymbol{\nu}}\in\mathcal F}\seqi{{\boldsymbol{\nu}}}\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2p_{\boldsymbol{\nu}}(3)\le
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_1)\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2 \le
C \delta^2
\end{equation*}
and
\begin{equation*}
C_0 \sum_{{\boldsymbol{\nu}}\in\mathcal F}\seqii{{\boldsymbol{\nu}}}\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2p_{\boldsymbol{\nu}}(3) \le
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_2)\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2 \le
C \frac{\delta^2}{\sw{j}^{2{\alpha}}},
\end{equation*}
with the constant $C$ independent of $j$, $\sw{j}$ and $N$.
\item\label{item:IGammae} Because
$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{l_{\varepsilon,{\boldsymbol{\nu}}}\ge 0}$ and
$\chi_{0,0}=0$ we have
$$\mathbf{I}_{\Gamma_j}(u-u^j)=\mathbf{I}_{\Gamma_j}e^j= \mathbf{I}_{\Gamma_j}\tilde
e_N^j$$ for all $j\in\mathbb{N}$ (cp.~Remark.~\ref{rmk:defu}). Similarly
$\mathbf{I}_{\Gamma_j}u=\mathbf{I}_{\Gamma_j}\tilde u_N$ for all $j\in\mathbb{N}$.
\end{enumerate}
{\bf Step 2.} Observe that $\Gamma_j=\emptyset$ for all
$j>L(\varepsilon):= \max_{{\boldsymbol{\nu}}\in\mathcal F}{l_{\varepsilon,{\boldsymbol{\nu}}}}$
(cp.~\eqref{eq:Gamma}), which is finite due to
$|\mathbf{l}_\varepsilon|<\infty$. With the conventions
$\mathbf{I}_{\Gamma_0}=\mathbf{I}_{\mathcal F}={\rm Id}$ (i.e.~$\mathbf{I}_{\Gamma_0}$ is the
identity) and $\mathbf{I}_{\emptyset} \equiv 0$ this implies
\begin{equation*}
u = \mathbf{I}_{\Gamma_0}u =
\sum_{j=0}^{L(\varepsilon)} (\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}}) u
= (\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1})u+\dots+
(\mathbf{I}_{\Gamma_{L(\varepsilon)-1}}-\mathbf{I}_{\Gamma_{L(\varepsilon)}})u+\mathbf{I}_{\Gamma_{L(\varepsilon)}}u.
\end{equation*}
By definition of the multilevel interpolant in \eqref{eq:VIml}
\begin{equation*}
\mathbf{I}^{\rm ML}_{\mathbf{l}_\varepsilon} u =
\sum_{j=1}^{L(\varepsilon)}(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})u^j
=(\mathbf{I}_{\Gamma_1}-\mathbf{I}_{\Gamma_2}) u^1+\dots+
(\mathbf{I}_{\Gamma_{L(\varepsilon)-1}}-\mathbf{I}_{\Gamma_{L(\varepsilon)}}) u^{L(\varepsilon)}+
\mathbf{I}_{\Gamma_{L(\varepsilon)}}u^{L(\varepsilon)}.
\end{equation*}
By item \ref{item:IGammae} of Step 1, we can write
\begin{equation*}
(\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1})u =
u-\mathbf{I}_{\Gamma_1}u = u-\mathbf{I}_{\Gamma_1}\tilde u_N
= (u-\tilde u_N)+(\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1})\tilde u_N
= (u-\tilde u_N)+(\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1})\tilde e_N^0,
\end{equation*}
where in the last equality we used $e_N^0=u_N$, by definition of
$e^0=u$ (and $\tilde e_N^0=\tilde u_N\in L^2(U,X;\gamma)$ as in
Definition~\ref{def:bdXHol}). Hence, again by item
\ref{item:IGammae},
\begin{align*
u-\mathbf{I}^{\rm ML}_{\mathbf{l}_\varepsilon}u &=(\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1}) u+
\sum_{j=1}^{L(\varepsilon)}(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}}) (u-u^j)\nonumber\\
&=(u-\tilde u_N)+(\mathbf{I}_{\Gamma_0}-\mathbf{I}_{\Gamma_1})\tilde u_N+\sum_{j=1}^{L(\varepsilon)}(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})\tilde e_N^j\nonumber\\
&=(u-\tilde u_N)+\sum_{j=0}^{L(\varepsilon)}(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})\tilde e_N^j.
\end{align*}
Let us bound the norm. By item \ref{item:u-uswunifabs} of Step 1 we
have for every $j\in\mathbb{N}_0$ that
$$\tilde e_{N}^j({\boldsymbol{y}})=\sum_{{\boldsymbol{\nu}}\in\mathcal F} \tilde e_{N,{\boldsymbol{\nu}}}^j
H_{\boldsymbol{\nu}}({\boldsymbol{y}}),$$ with absolute pointwise convergence for all
${\boldsymbol{y}}\in U$. Since
$$(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}= 0\in L^2(U,\gamma)$$ for
all ${\boldsymbol{\nu}}\in\Gamma_{j+1}\subseteq\Gamma_j$ by Lemma
\ref{lemma:VIprop}, we get
\begin{equation}\label{eq:u-vimlu}
\norm[L^2(U,X;\gamma)]{u-\mathbf{I}^{\rm ML}_{\mathbf{l}_\varepsilon}u}\le
\norm[L^2(U,X;\gamma)]{u-\tilde u_N}+
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} \norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}\norm[L^2(U,\gamma)]{(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}}.
\end{equation}
{\bf Step 3.} We wish to apply Lemma \ref{LEMMA:MLWEIGHTNEW} to the
bound \eqref{eq:u-vimlu}. By \eqref{eq:L2boundLambda}, we have for
all ${\boldsymbol{\nu}}\in\mathcal F$
\begin{equation*}
\norm[L^2(U,\gamma)]{(\mathbf{I}_{\Gamma_j}-\mathbf{I}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}}
\le \norm[L^2(U,\gamma)]{\mathbf{I}_{\Gamma_j}H_{\boldsymbol{\nu}}}+
\norm[L^2(U,\gamma)]{\mathbf{I}_{\Gamma_{j+1}}H_{\boldsymbol{\nu}}}
\le 2 p_{\boldsymbol{\nu}}(3).
\end{equation*}
Define
\begin{equation*}
a_{j,{\boldsymbol{\nu}}}:=\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}p_{\boldsymbol{\nu}}(3).
\end{equation*}
By items \ref{item:seqiseqiisum} and \ref{item:tedecay} of Step 1,
the sequences $(a_{j,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, $j\in\mathbb{N}_0$, and
$(\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, $(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, satisfy
the assumptions of Lemma \ref{LEMMA:MLWEIGHTNEW}. Therefore,
\eqref{eq:u-vimlu}, \eqref{eq:truncerr_mlint} and Lemma
\ref{LEMMA:MLWEIGHTNEW} give
\begin{equation*
\norm[L^2(U,X;\gamma)]{u-\mathbf{I}^{\rm ML}_{\mathbf{l}_{\varepsilon_n}}u}
\le
n^{-R} + \sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon_n,{\boldsymbol{\nu}}}}}^{L(\varepsilon_n)} a_{j,{\boldsymbol{\nu}}}
\le
C (1+\log n )n^{-R},
\end{equation*}
with
\begin{equation*}
R = \min\left\{{\alpha},\frac{{\alpha} (q_1^{-1}-1/2)}{{\alpha} + q_1^{-1}-q_2^{-1}}
\right\} =
\min\left\{{\alpha},\frac{{\alpha} (p_1^{-1}-3/2)}{{\alpha} + p_1^{-1}-p_2^{-1}}
\right\},
\end{equation*}
where we used $q_1=p_1/(1-p_1)$ and $q_2=p_2/(1-p_2)$ as stated in
item \ref{item:aux} of Step 1.
\end{proof}
\subsection{Multilevel quadrature}
\label{sec:MLQuad}
We next formulate the analogue of Theorem~\ref{thm:mlint} for
multilevel quadrature. First, the definition of the sequences in
\eqref{eq:seqiseqii} (which are used to construct the quadrature via
Algorithm \ref{alg:levels}) have to be slightly adjusted. Then, we
state and prove the convergence result. Its proof is along the lines
of the proof of Theorem~\ref{thm:mlint}.
Let ${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}\in\ell^{p_1}({\mathbb N})$,
${\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}\in\ell^{p_2}({\mathbb N})$, and $\xi$ be the two
sequences and the constant from Assumption \ref{ass:ml}. For two
constants $K>0$ and $r>3$, which are still at our disposal and which
will be defined below, we set for all $j\in\mathbb{N}$
\begin{equation}\label{eq:bvarrhoiquad}
\varrho_{1,j}:=b_{1,j}^{p_1-1} \frac{\xi}{4\norm[\ell^{p_1}]{{\boldsymbol{b}}_1}},\qquad \varrho_{2,j}:=b_{2,j}^{p_2
-1} \frac{\xi}{4\norm[\ell^{p_2}]{{\boldsymbol{b}}_2}}.
\end{equation}
Furthermore, we let for all ${\boldsymbol{\nu}}\in\mathcal F$ (as in Lemma \ref{lemma:cnu}
for $k=2$ and with $\tau=3$)
\begin{equation}\label{eq:seqiseqiiquad}
\seqi{{\boldsymbol{\nu}}}:=\prod_{j\in\mathbb{N}}\max\{1,K\varrho_{1,j}\}^4\nu_j^{r-3},\qquad
\seqii{{\boldsymbol{\nu}}}:=\prod_{j\in\mathbb{N}}\max\{1,K\varrho_{2,j}\}^4\nu_j^{r-3}.
\end{equation}
\begin{theorem}\label{thm:mlquad}
Let $u\in L^2(U,X;\gamma)$ and $u^l\in L^2(U,X;\gamma)$,
$l\in\mathbb{N}$, satisfy Assumption \ref{ass:ml} with some constants
${\alpha}>0$ and $0<p_1<4/5$ and $p_1\le p_2<1$. Set
$q_1:=p_1/(1-p_1)$. Assume that $r>2(1+({\alpha}+1)q_1/2)/q_1+3$
(for $r$ in \eqref{eq:seqiseqiiquad}). There exist constants $K>0$
(in \eqref{eq:seqiseqiiquad}) and $C>0$ such that the following
holds.
There exist $C>0$ and, for every $n \in {\mathbb N}$ there exists
$\varepsilon_n\in (0,1]$ such that such that $\mathrm{work}(\mathbf{l}_{\varepsilon_n}) \le n$
and with $\mathbf{l}_{\varepsilon_n}=({l_{\varepsilon_n,{\boldsymbol{\nu}}}})_{{\boldsymbol{\nu}}\in\mathcal F}$ as in
Corollary \ref{LEMMA:MLWEIGHTNEW} (with $c_{\boldsymbol{\nu}}$, $d_{\boldsymbol{\nu}}$ as in
\eqref{eq:seqiseqiiquad}) it holds
\begin{equation*}
\normc[X]{\int_U u({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\mathbf{l}_{\varepsilon_n}}^{\rm ML}u} \le \ C (1+\log n )n^{-R},
\end{equation*}
with the convergence rate
\begin{equation*}
R:=\min\left\{{\alpha},\frac{{\alpha}(2p_1^{-1}-5/2)}{{\alpha}+2p_1^{-1}-
2p_2^{-1}}\right\}.
\end{equation*}
\end{theorem}
\begin{proof}
Throughout this proof we write ${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}$ and
${\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}$ for the two sequences in Assumption
\ref{ass:ml}. As in the proof of Theorem~\ref{thm:mlint} we point
out that $\Gamma_j$ defined in \eqref{eq:Gamma} is downward closed
for all $j\in\mathbb{N}_0$.
{\bf Step 1.} Given $n\in \mathbb{N}$, we choose $\varepsilon:=\varepsilon_n$ as in Lemma
\ref{LEMMA:MLWEIGHTNEW}. Fix $N\in\mathbb{N}$ such that
$N>\max \{j:\, j \in\operatorname{supp}({\boldsymbol{\nu}}),\, l_{\varepsilon,{\boldsymbol{\nu}}}> 0\}$ and so
large that
\begin{equation}\label{eq:truncerr_mlquad}
\normc[X]{\int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}})) \,\mathrm{d}\gamma({\boldsymbol{y}})}\le n^{-R},
\end{equation}
where $\tilde u_N:U\to X$ is as in Definition~\ref{def:bdXHol} (this
is possible due
$\lim_{N\to\infty}\norm[L^2(U,X;\gamma)]{u-\tilde u_N}=0$ which
holds by the $({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphy of $u$).
By Assumption \ref{ass:ml}, for every $j \in\mathbb{N}$ the function
$e^j:=u-u^j\in L^2(U,X;\gamma)$ is
$({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic and
$({\boldsymbol{b}}_2,\xi,\delta\sw{j}^{\alpha},X)$-holomorphic. For notational
convenience we set $e^0:=u-0=u\in L^2(U,X;\gamma)$, so that $e^0$ is
$({\boldsymbol{b}}_1,\xi,\delta,X)$-holomorphic and
$({\boldsymbol{b}}_2,\xi,\delta,X)$-holomorphic. Hence for every $j\in\mathbb{N}_0$ there
exists a function $\tilde e_N^j=\tilde u_N-\tilde u_N^j$ as in
Definition~\ref{def:bdXHol} \ref{item:vN}.
The following assertions are identical to the ones in the proof of
Theorem~\ref{thm:mlint}, except that we now admit different
summability exponents $q_1$ and $q_2$.
\begin{enumerate}
\item\label{item:u-uswunifabs_quad} By Lemma \ref{lemma:uN}, for
every $j\in\mathbb{N}_0$, with the Wiener-Hermite PC expansion
coefficients
$$\tilde e_{N,{\boldsymbol{\nu}}}^j:= \int_{U} H_{\boldsymbol{\nu}}({\boldsymbol{y}})\tilde e_{N}^j({\boldsymbol{y}})
\,\mathrm{d}\gamma({\boldsymbol{y}}),$$ it holds
\begin{equation*}\label{eq:u-uswunifabs}
\tilde e_{N}^j({\boldsymbol{y}}) = \sum_{{\boldsymbol{\nu}}\in\mathcal F} \tilde e_{N,{\boldsymbol{\nu}}}^j H_{\boldsymbol{\nu}}({\boldsymbol{y}})\qquad\forall {\boldsymbol{y}}\in U,
\end{equation*}
with pointwise absolute convergence.
\item\label{item:aux_quad} By Lemma \ref{lemma:cnu}, upon choosing
$K>0$ in \eqref{eq:seqiseqii} large enough, and because $r>3$,
\begin{equation*}
C_0 \seqi{{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(3)\le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_1),\qquad
C_0 \seqii{{\boldsymbol{\nu}}} p_{\boldsymbol{\nu}}(3)\le \beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_2)\qquad\forall {\boldsymbol{\nu}}\in\mathcal F_2.
\end{equation*}
We point out that by definition of ${\boldsymbol{\varrho}}_i$, $i\in\{1,2\}$, in
\eqref{eq:bvarrhoiquad}, it holds
$\varrho_{i,j}\sim b_{i,j}^{-(1-p_i)}$ and therefore
$(\varrho_{i,j}^{-1})_{j\in\mathbb{N}}\in\ell^{q_i}(\mathbb{N})$ with
$q_i:=p_i/(1-p_i)$, $i\in\{1,2\}$.
\item\label{item:seqiseqiisum_quad} Due to
$r>2(1+2({\alpha}+1)q_1)/q_1+3$,
the condition of Lemma \ref{lemma:summabcnu} is satisfied (with
$k=2$, $\tau=3$ and $\theta=({\alpha}+1)q_1/2$). Hence the lemma
gives
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} p_{\boldsymbol{\nu}}(({\alpha}+1)q_1/2)\seqi{{\boldsymbol{\nu}}}^{-q_1/4}<\infty
\qquad
\Rightarrow\qquad
(p_{\boldsymbol{\nu}}({\alpha}+1)\seqi{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_1/2}(\mathcal F)
\end{equation*}
and similarly
\begin{equation*}
\sum_{{\boldsymbol{\nu}}\in\mathcal F} p_{\boldsymbol{\nu}}(({\alpha}+1)q_2/2)\seqii{{\boldsymbol{\nu}}}^{-q_2/4}<\infty
\qquad
\Rightarrow\qquad
(p_{\boldsymbol{\nu}}({\alpha}+1)\seqii{{\boldsymbol{\nu}}}^{-1/2})_{{\boldsymbol{\nu}}\in\mathcal F}\in\ell^{q_2/2}(\mathcal F).
\end{equation*}
\item\label{item:tedecay_quad} By Theorem~\ref{thm:bdHolSum} and
item \ref{item:aux_quad}, for all $j\in\mathbb{N}_0$
\begin{equation*}
C_0 \sum_{{\boldsymbol{\nu}}\in\mathcal F_2}\seqi{{\boldsymbol{\nu}}}\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2p_{\boldsymbol{\nu}}(3)\le
\sum_{{\boldsymbol{\nu}}\in\mathcal F_2}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_1)\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2 \le
C \delta^2
\end{equation*}
and
\begin{equation*}
C_0 \sum_{{\boldsymbol{\nu}}\in\mathcal F_2}\seqii{{\boldsymbol{\nu}}}\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2p_{\boldsymbol{\nu}}(3) \le
\sum_{{\boldsymbol{\nu}}\in\mathcal F_2}\beta_{\boldsymbol{\nu}}(r,{\boldsymbol{\varrho}}_2)\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}^2 \le
C \frac{\delta^2}{\sw{j}^{2{\alpha}}},
\end{equation*}
with the constant $C$ independent of $j$, $\sw{j}$ and $N$.
\item\label{item:IGammae_quad} Because
$N\ge\max\set{j\in\operatorname{supp}({\boldsymbol{\nu}})}{l_{\varepsilon,{\boldsymbol{\nu}}}\ge 0}$ and
$\chi_{0,0}=0$ we have
$\mathbf{Q}_{\Gamma_j}(u-u^j)=\mathbf{Q}_{\Gamma_j}e^j= \mathbf{Q}_{\Gamma_j}\tilde
e_N^j$ for all $j\in\mathbb{N}$ (cp.~Remark~\ref{rmk:defu}). Similarly
$\mathbf{Q}_{\Gamma_j}u=\mathbf{Q}_{\Gamma_j}\tilde u_N$ for all $j\in\mathbb{N}$.
\end{enumerate}
{\bf Step 2.} Observe that $\Gamma_j=\emptyset$ for all
$$j>L(\varepsilon):= \max_{{\boldsymbol{\nu}}\in\mathcal F}{l_{\varepsilon,{\boldsymbol{\nu}}}}$$
(cp.~\eqref{eq:Gamma}), which is finite due to
$|\mathbf{l}_\varepsilon|<\infty$. With the conventions
$$\mathbf{Q}_{\Gamma_0}=\mathbf{Q}_{\mathcal F}=\int_U \cdot\,\mathrm{d}\gamma({\boldsymbol{y}})$$
(i.e.~$\mathbf{Q}_{\Gamma_0}$ is the exact integral operator) and
$\mathbf{Q}_{\emptyset} \equiv 0$ this implies
\begin{align*}
\int_Uu({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})
& = \mathbf{Q}_{\Gamma_0}u
= \sum_{j=0}^{L(\varepsilon)} (\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}}) u
\\
& = (\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1})u+\ldots+
(\mathbf{Q}_{\Gamma_{L(\varepsilon)-1}}-\mathbf{Q}_{\Gamma_{L(\varepsilon)}})u+\mathbf{Q}_{\Gamma_{L(\varepsilon)}}u.
\end{align*}
By definition of the multilevel quadrature in \eqref{eq:VQml}
\begin{align*}
\mathbf{Q}^{\rm ML}_{\mathbf{l}_\varepsilon} u & =
\sum_{j=1}^{L(\varepsilon)}(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})u^j
\\
& =(\mathbf{Q}_{\Gamma_1}-\mathbf{Q}_{\Gamma_2}) u^1+\ldots+
(\mathbf{Q}_{\Gamma_{L(\varepsilon)-1}}-\mathbf{Q}_{\Gamma_{L(\varepsilon)}}) u^{L(\varepsilon)}+
\mathbf{Q}_{\Gamma_{L(\varepsilon)}}u^{L(\varepsilon)}.
\end{align*}
By item \ref{item:IGammae_quad} of Step 1, we can write
\begin{align*}
(\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1})u &=
\int_Uu({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Gamma_1}u \nonumber\\
&= \int_Uu({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\Gamma_1}\tilde u_N
\nonumber\\
&= \int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}}))\,\mathrm{d}\gamma({\boldsymbol{y}})+(\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1})\tilde u_N\nonumber\\
&= \int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}}))\,\mathrm{d}\gamma({\boldsymbol{y}})
+(\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1})\tilde e_N^0,
\end{align*}
where in the last equality we used $e_N^0=u_N$, by definition of
$e^0=u$ (and $\tilde e_N^0=\tilde u_N\in L^2(U,X;\gamma)$ as in
Definition~\ref{def:bdXHol}). Hence, again by item
\ref{item:IGammae_quad},
\begin{align*}\label{eq:u-Iwu_quad}
\int_Uu({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}^{\rm ML}_{\mathbf{l}_\varepsilon}u &=(\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1}) u+
\sum_{j=1}^{L(\varepsilon)}(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}}) (u-u^j)\nonumber\\
&=\int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}}))\,\mathrm{d}\gamma({\boldsymbol{y}})+(\mathbf{Q}_{\Gamma_0}-\mathbf{Q}_{\Gamma_1})\tilde u_N+\sum_{j=1}^{L(\varepsilon)}(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})\tilde e_N^j\nonumber\\
&=\int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}}))\,\mathrm{d}\gamma({\boldsymbol{y}})+\sum_{j=0}^{L(\varepsilon)}(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})\tilde e_N^j.
\end{align*}
Let us bound the norm. By item \ref{item:u-uswunifabs_quad} of Step
1 we have for every $j\in\mathbb{N}_0$ that
$$\tilde e_{N}^j({\boldsymbol{y}})=\sum_{{\boldsymbol{\nu}}\in\mathcal F} \tilde e_{N,{\boldsymbol{\nu}}}^j
H_{\boldsymbol{\nu}}({\boldsymbol{y}}),$$ with absolute pointwise convergence for all
${\boldsymbol{y}}\in U$. Since
$(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}= 0\in X$ for all
${\boldsymbol{\nu}}\in\Gamma_{j+1}\subseteq\Gamma_j$ \emph{and all
${\boldsymbol{\nu}}\in\mathcal F\backslash\mathcal F_2$} by Lemma \ref{lemma:VIprop}, we get
\begin{equation}\label{eq:u-vimlu_quad}
\normc[X]{\int_U u({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}^{\rm ML}_{\mathbf{l}_\varepsilon}u}\le
\normc[X]{\int_U(u({\boldsymbol{y}})-\tilde u_N({\boldsymbol{y}}))\,\mathrm{d}\gamma({\boldsymbol{y}})}+
\sum_{{\boldsymbol{\nu}}\in\mathcal F_2}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)}
\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}|(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}|.
\end{equation}
{\bf Step 3.} We wish to apply Lemma \ref{LEMMA:MLWEIGHTNEW} to the
bound \eqref{eq:u-vimlu_quad}. By \eqref{eq:L1boundLambda}, for all
${\boldsymbol{\nu}}\in\mathcal F$
\begin{equation*}
|(\mathbf{Q}_{\Gamma_j}-\mathbf{Q}_{\Gamma_{j+1}})H_{\boldsymbol{\nu}}|
\le |\mathbf{Q}_{\Gamma_j}H_{\boldsymbol{\nu}}|+
|\mathbf{Q}_{\Gamma_{j+1}}H_{\boldsymbol{\nu}}|
\le 2 p_{\boldsymbol{\nu}}(3).
\end{equation*}
Define
\begin{equation*}
a_{j,{\boldsymbol{\nu}}}:=\norm[X]{\tilde e_{N,{\boldsymbol{\nu}}}^j}p_{\boldsymbol{\nu}}(3)\qquad\forall{\boldsymbol{\nu}}\in\mathcal F_2,
\end{equation*}
and $a_{j,{\boldsymbol{\nu}}}:=0$ for ${\boldsymbol{\nu}}\in\mathcal F\backslash\mathcal F_2$. By items
\ref{item:seqiseqiisum_quad} and \ref{item:tedecay_quad} of Step 1,
the sequences $(a_{j,{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, $j\in\mathbb{N}_0$, and
$(\seqi{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, $(\seqii{{\boldsymbol{\nu}}})_{{\boldsymbol{\nu}}\in\mathcal F}$, satisfy
the assumptions of Lemma \ref{LEMMA:MLWEIGHTNEW} (with
$\tilde q_1:=q_1/2$ and $\tilde q_2:=q_2/2$). Therefore
\eqref{eq:u-vimlu_quad}, \eqref{eq:truncerr_mlquad} and Lemma
\ref{LEMMA:MLWEIGHTNEW} give
\begin{equation*}
\normc[X]{\int_U u({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}^{\rm ML}_{\mathbf{l}_\varepsilon}u}
\le
n^{-R}+
\sum_{{\boldsymbol{\nu}}\in\mathcal F}\sum_{j={l_{\varepsilon,{\boldsymbol{\nu}}}}}^{L(\varepsilon)} a_{j,{\boldsymbol{\nu}}}
\le C (1+\log n)n^{-R},
\end{equation*}
with
\begin{equation*}
R = \min\left\{{\alpha},\frac{{\alpha} (\tilde q_1^{-1}-1/2)}{{\alpha} + \tilde q_1^{-1}-\tilde q_2^{-1}}
\right\} =
\min\left\{{\alpha},\frac{{\alpha} (2 p_1^{-1}-5/2)}{{\alpha} + 2 p_1^{-1}-2 p_2^{-1}}
\right\},
\end{equation*}
where we used $\tilde q_1=q_1/2=p_1/(2-2p_1)$ and
$\tilde q_2=q_2/2=p_2/(2-2p_2)$ as stated in item
\ref{item:aux_quad} of Step 1.
\end{proof}
\begin{remark}\label{rmk:comparison}
We comment on the relation of the present analysis to the recent
paper \cite{dD21}. Also there, multilevel algorithms for parametric
PDEs with lognormally distributed input data were proposed.
Specifically, in \cite{dD21}, by combining spatial and parametric
approximability in the spatial domain and weighted
$\ell^2$-summability of the $V$ and $W$ norms of Wiener-Hermite PC
expansion coefficients obtained in \cite{BCDM,BCDS}, the author
constructed linear non-adaptive methods of fully discrete
approximation by truncated Wiener-Hermite PC expansion and
polynomial interpolation approximation as well as fully discrete
weighted quadrature for parametric and stochastic elliptic PDEs with
lognormal inputs, and proved the convergence rates of approximation
by them. The results in \cite{dD21} assume the existence of a
sequence $(P_n)_{n \in {\mathbb N}_0}$ of linear operators independent of
${\boldsymbol{y}}$, from $H^1_0(D)$ into $n$-dimensional subspaces
$V_n \subset H^1_0(D)$ such that
$\|P_n(v)\|_{H^1_0(D)} \leq C_1\|v\|_{H^1_0(D)}$ and
$\|v-P_n(v)\|_{H^1_0(D)} \leq C_2n^{-\alpha} \|v\|_{W}$ for all
$n \in {\mathbb N}_0$ and for all $v \in W$, where the constants $C_1, C_2$
are independent of $n$.
However, the assumption of $P_n$ being independent of ${\boldsymbol{y}}$ is
typically not satisfied if $P_n(u({\boldsymbol{y}})) = u^n({\boldsymbol{y}})$ is a numerical
approximation to $u({\boldsymbol{y}})$ (as, e.g., a Finite-Element or a
Finite-Difference discretization).
In contrast, the present analysis is based
on quantified, parametric holomorphy of the discrete approximations
$u^l$ to $u$ as in Assumption \ref{ass:ml}. For example, assume
that $u:U\to H_0^1({\domain})$ is the solution of the parametric PDE
$-\operatorname{div}(a({\boldsymbol{y}})\nabla u({\boldsymbol{y}}))=f$ for some $f\in L^2({\domain})$ and a
parametric diffusion coefficient $a({\boldsymbol{y}})\in L^\infty({\domain})$ such that
$\operatorname{ess\,inf}_{{\boldsymbol{x}}\in{\domain}}a({\boldsymbol{y}},{\boldsymbol{x}})>0$ for every ${\boldsymbol{y}}\in U$. Then
$u^l:U\to H_0^1({\domain})$ could be a numerical approximation to $u$,
such as the FEM solution: for every $l\in\mathbb{N}$ there is a finite
dimensional discretization space $X_l\subseteq H_0^1({\domain})$, and
$\int_{{\domain}}\nabla u^l({\boldsymbol{y}})^\top a({\boldsymbol{y}})\nabla v\,\mathrm{d} {\boldsymbol{x}} =\int_{{\domain}}f
v\,\mathrm{d} {\boldsymbol{x}}$ for every $v\in X_l$ and for every ${\boldsymbol{y}}\in U$. Hence
$u^l({\boldsymbol{y}})$ is the orthogonal projection of $u({\boldsymbol{y}})$ onto $X_l$
w.r.t.\ the inner product
$\langle v,w \rangle_{a({\boldsymbol{y}})}:= \int_{\domain} \nabla v^\top a({\boldsymbol{y}})\nabla
w\,\mathrm{d} {\boldsymbol{x}}$ on $H_0^1({\domain})$. We may write this as
$u^l({\boldsymbol{y}})=P_l({\boldsymbol{y}}) u({\boldsymbol{y}})$, for a ${\boldsymbol{y}}$-dependent projector
$P_l({\boldsymbol{y}}):H_0^1({\domain})\to X_l$. This situation is covered by
Assumption \ref{ass:ml}.
\end{remark}
\subsection{Examples for multilevel approximation}
\label{sec:Approx}
We revisit the examples in Sections \ref{sec:SumHolSol} and
\ref{sec:BIP}, and demonstrate how to verify the assumptions required
for our multilevel convergence rate results in Theorem~\ref{thm:mlint}
and Theorem~\ref{thm:mlquad}.
\subsubsection{Parametric diffusion coefficient in polygonal domain}
\label{S:DiffPolyg}
Let ${\domain}\subseteq\mathbb{R}^2$ be a bounded polygonal Lipschitz domain, and
consider once more the elliptic equation
\begin{equation}\label{eq:elliptic2}
- \operatorname{div}(a \nabla {\mathcal U}(a))=f\quad\text{in }{\domain},\qquad
{\mathcal U}(a) =0\quad\text{on }\partial{\domain},
\end{equation}
as in Section~\ref{sec:pdc}.
For $s\in\mathbb{N}_0$ and $\varkappa\in\mathbb{R}$, recall the Kondrat'ev spaces
${\mathcal W}^{s}_\infty({\domain})$ and ${\mathcal K}^{s}_{\varkappa}({\domain})$ with norms
\begin{equation*}
\|u\|_{ {\mathcal K}^s_\varkappa}
:=
\sum_{|{\boldsymbol{\alpha}}|\leq s}\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|-\varkappa}D^{\boldsymbol{\alpha}} u\|_{L^2}
\qquad\text{and}\qquad
\|u\|_{ {\mathcal W}^s_\infty}:=\sum_{|{\boldsymbol{\alpha}}|\leq s}\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{\boldsymbol{\alpha}} u\|_{L^\infty}
\end{equation*}
introduced in Section~\ref{sec:KondrAn}. Here, as earlier,
$r_{{\domain}}:{\domain}\to [0,1]$ denotes a fixed smooth function that coincides
with the distance to the nearest corner, in a neighbourhood of each
corner.
According to Theorem~\ref{thm:bacuta}, assuming $s\ge 2$,
$f\in {\mathcal K}^{s-2}_{\varkappa-1}({\domain})$ and $a\in {\mathcal W}^{s-1}_\infty({\domain})$ the
solution ${\mathcal U}(a)$ of \eqref{eq:elliptic2} belongs to
${\mathcal K}_{\varkappa+1}^{s}({\domain})$ provided that with
\begin{equation*}
\rho(a) := \underset{{\boldsymbol{x}}\in{\domain}}{\operatorname{ess\,inf}}\,\Re(a({\boldsymbol{x}}))>0,
\end{equation*}
\begin{equation}\label{eq:kappaa}
|\varkappa|<\frac{\rho(a)}{\tau \norm[L^\infty]{a}},
\end{equation}
where $\tau$ is a constant depending on ${\domain}$ and $s$. Our goal is to
uniformly treat a family of diffusion coefficients $a({\boldsymbol{y}})$,
${\boldsymbol{y}}\in U$, where for certain ${\boldsymbol{y}}\in U$ the diffusion coefficient
$a({\boldsymbol{y}})$ is such that the right-hand side of \eqref{eq:kappaa} might
be arbitrarily small. This only leaves us with the choice
$\varkappa=0$, see Remark.~\ref{rmk:kappa}. On the other hand, the
motivation of using Kondrat'ev spaces in the analysis of
approximations to PDE solutions ${\mathcal U}(a({\boldsymbol{y}}))$, is that functions in
${\mathcal K}_{\varkappa+1}^s({\domain})$ on polygonal domains in $\mathbb{R}^2$ can be
approximated with the optimal convergence rate $\frac{s-1}{2}$ w.r.t.\
the $H^1$-norm by suitable finite element spaces (on graded meshes;
i.e.\ this analysis accounts for corner singularities which prevent
optimal convergence rates on uniform meshes). Such results are
well-known, see for example \cite{BNZPolygon}, however they require
$\varkappa>0$. For this reason we need a stronger regularity result,
giving uniform ${\mathcal K}_{\varkappa+1}^{s}$-regularity with $\varkappa>0$
independent of the parameter. This is the purpose the next theorem.
For its proof we shall need the following lemma, which is shown in a
similar way as in \cite[Lemma C.2]{2006.06994}. We recall that
$$\norm[W^{s}_\infty]{f}:=\sum_{|{\boldsymbol{\nu}}|\le s}\norm[L^\infty]{D^{\boldsymbol{\nu}} f}.$$
\begin{lemma}\label{lemma:fgWm}
Let $s\in\mathbb{N}_0$ and let ${\domain}\subseteq\mathbb{R}^d$ be a Lipschitz domain,
$d\in\mathbb{N}$. There exist $C_s$ and $\tilde C_s$ such that for any two
functions $f$, $g\in {\mathcal W}^{s}_{\infty}({\domain})$
\begin{enumerate}
\item\label{item:prodinfty}
$\norm[{\mathcal W}^{s}_{\infty}]{fg}\le C_s
\norm[{\mathcal W}^{s}_{\infty}]{f}\norm[{\mathcal W}^{s}_{\infty}]{g}$,
\item\label{item:fracinfty}
$\norm[{\mathcal W}^{s}_{\infty}]{\frac 1 f}\le \tilde C_s
\frac{\norm[{\mathcal W}^{s}_{\infty}]{f}^{s}}{\operatorname{ess\,inf}_{{\boldsymbol{x}}\in
{\domain}}|f({\boldsymbol{x}})|^{s+1}}$ if $\operatorname{ess\,inf}_{{\boldsymbol{x}}\in{\domain}}|f({\boldsymbol{x}})|>0$.
\end{enumerate}
These statements remain true if ${\mathcal W}^{s}_{\infty}({\domain})$ is replaced
by $W^{s}_\infty({\domain})$. Furthermore, if $\varkappa\in\mathbb{R}$, then for
$f\in{\mathcal K}_\varkappa^s({\domain})$ and $a\in {\mathcal W}^{s}_{\infty}({\domain})$
\begin{enumerate}
\setcounter{enumi}{2}
\item\label{item:prodkond}
$\norm[{\mathcal K}_\varkappa^s]fa \le C_s
\norm[{\mathcal K}_\varkappa^s]{f}\norm[{\mathcal W}^{s}_{\infty}]{a}$,
\item\label{item:proddiffkond}
$\norm[{\mathcal K}_{\varkappa-1}^{s-1}]{\nabla f \cdot\nabla a} \le
C_{s-1} d
\norm[{\mathcal K}_{\varkappa+1}^{s}]{f}\norm[{\mathcal W}^{s}_{\infty}]{a}$ if
$s\ge 1$.
\end{enumerate}
\end{lemma}
\begin{proof}
We will only prove \ref{item:prodinfty} and \ref{item:fracinfty} for
functions in ${\mathcal W}^{s}_\infty({\domain})$. The case of $W^{s}_\infty({\domain})$
is shown similarly (by omitting all occurring functions $r_{\domain}$ in
the following).
{\bf Step 1.} We start with \ref{item:prodinfty}, and show a
slightly more general bound: for $\tau\in\mathbb{R}$ introduce
\begin{equation*}
\norm[{\mathcal W}^{s}_{\tau,\infty}]{f}:=\sum_{|{\boldsymbol{\nu}}|\le s}\norm[L^\infty]{r_{\domain}^{\tau+|{\boldsymbol{\nu}}|}D^{\boldsymbol{\nu}} f},
\end{equation*}
i.e.\ ${\mathcal W}^s_{0,\infty}({\domain})={\mathcal W}^s_\infty({\domain})$. We will show that for
$\tau_1+\tau_2=\tau$
\begin{equation}\label{eq:wwstau}
\norm[{\mathcal W}^s_{\tau,\infty}]{fg}\le
C_{s} \norm[{\mathcal W}^s_{\tau_1,\infty}]{f}\norm[{\mathcal W}^s_{\tau_2,\infty}]{g}.
\end{equation}
Item \ref{item:prodinfty} then follows with $\tau=\tau_1=\tau_2=0$.
Using the multivariate Leibniz rule for Lipschitz functions, for any
multiindex ${\boldsymbol{\nu}}\in\mathbb{N}_0^d$ with $d\in\mathbb{N}$ fixed,
\begin{equation}\label{eq:leibniz}
D^{\boldsymbol{\nu}}(fg)=\sum_{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}}
D^{{\boldsymbol{\nu}}-{\boldsymbol{\mu}}}f D^{{\boldsymbol{\mu}}}g.
\end{equation}
Thus if $|{\boldsymbol{\nu}}|\le s$
\begin{equation*}
\norm[L^\infty]{r_{\domain}^{\tau+|{\boldsymbol{\nu}}|}D^{\boldsymbol{\nu}}(fg)}
\le \sum_{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}}
\norm[L^\infty]{r_{\domain}^{\tau_1+|{\boldsymbol{\nu}}-{\boldsymbol{\mu}}|}D^{{\boldsymbol{\nu}}-{\boldsymbol{\mu}}}f}
\norm[L^\infty]{r_{\domain}^{\tau_2+|{\boldsymbol{\mu}}|}D^{{\boldsymbol{\mu}}}g}
\le 2^{|{\boldsymbol{\nu}}|}\norm[{\mathcal W}^{s}_{\tau_1,\infty}]{f}\norm[{\mathcal W}^{s}_{\tau_2,\infty}]{g},
\end{equation*}
where we used $\binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}}=\prod_{j=1}^d\binom{\nu_j}{\mu_j}$
and $\sum_{i=0}^{\nu_j}\binom{\nu_j}{i}=2^{\nu_j}$.
We conclude
$$\norm[{\mathcal W}^{s}_{\tau,\infty}]{fg}\le
C_s\norm[W^{s}_{\tau,\infty}]{f}\norm[W^{s}_{\infty}]{g}$$ with
$C_s=\sum_{|{\boldsymbol{\nu}}|\le s}2^{|{\boldsymbol{\nu}}|}$. Hence \ref{item:prodinfty}
holds.
{\bf Step 2.} We show \ref{item:fracinfty}, and claim that for all
$|{\boldsymbol{\nu}}|\le s$ it holds
\begin{equation}\label{eq:fracclaimtotal}
D^{\boldsymbol{\nu}} \left(\frac{1}{f}\right)=\frac{p_{\boldsymbol{\nu}}}{f^{|{\boldsymbol{\nu}}|+1}}
\end{equation}
where $p_{\boldsymbol{\nu}}$ satisfies
\begin{equation}\label{eq:fracclaim}
\norm[{\mathcal W}^{s-|{\boldsymbol{\nu}}|}_{|{\boldsymbol{\nu}}|,\infty}]{p_{\boldsymbol{\nu}}}\le \hat C_{|{\boldsymbol{\nu}}|}\norm[{\mathcal W}^{s}_\infty]{f}^{|{\boldsymbol{\nu}}|}
\end{equation}
for some $\hat C_{|{\boldsymbol{\nu}}|}$ solely depending on $|{\boldsymbol{\nu}}|$. We proceed
by induction over $|{\boldsymbol{\nu}}|$ and start with $|{\boldsymbol{\nu}}|=1$, i.e.\
${\boldsymbol{\nu}}={\boldsymbol{e}}_j=(\delta_{ij})_{i=1}^d$ for some
$j\in\{1,\dots,d\}$. Then
$D^{{\boldsymbol{e}}_j}\frac{1}{f}=\frac{-\partial_j f}{f^2}$ and
$p_{{\boldsymbol{e}}_j}=-\partial_j f$ satisfies
\begin{equation*}
\norm[{\mathcal W}^{s-1}_{1,\infty}]{p_{{\boldsymbol{e}}_j}}=
\sum_{|{\boldsymbol{\mu}}|\le s-1}\norm[L^\infty]{r_{\domain}^{1+|{\boldsymbol{\mu}}|}D^{{\boldsymbol{\mu}}}p_{{\boldsymbol{e}}_j}}
=\sum_{|{\boldsymbol{\mu}}|\le s-1}\norm[L^\infty]{r_{\domain}^{|{\boldsymbol{\mu}}+{\boldsymbol{e}}_j|}D^{{\boldsymbol{\mu}}+{\boldsymbol{e}}_j} f}
\le \norm[{\mathcal W}^{s}_{\infty}]{f},
\end{equation*}
i.e.\ $\hat C_1=1$. For the induction step fix ${\boldsymbol{\nu}}$ with
$1<|{\boldsymbol{\nu}}|<s$ and $j\in\{1,\dots,d\}$. Then by the induction
hypothesis $D^{\boldsymbol{\nu}} \frac{1}{f}=\frac{p_{\boldsymbol{\nu}}}{f^{|{\boldsymbol{\nu}}|+1}}$ and
\begin{equation*}
D^{{\boldsymbol{\nu}}+{\boldsymbol{e}}_j} \frac{1}{f}=
\partial_j\left(\frac{p_{{\boldsymbol{\nu}}}}{f^{|{\boldsymbol{\nu}}|+1}}\right)
=\frac{f^{|{\boldsymbol{\nu}}|+1}\partial_j p_{{\boldsymbol{\nu}}}-(|{\boldsymbol{\nu}}|+1)f^{|{\boldsymbol{\nu}}|}p_{{\boldsymbol{\nu}}}\partial_j f}{f^{2|{\boldsymbol{\nu}}|+2}}
=\frac{f \partial_j p_{{\boldsymbol{\nu}}}-(|{\boldsymbol{\nu}}|+1)p_{{\boldsymbol{\nu}}}\partial_j f}{f^{|{\boldsymbol{\nu}}|+2}},
\end{equation*}
and thus
$$p_{{\boldsymbol{\nu}}+{\boldsymbol{e}}_j}:=f \partial_j p_{{\boldsymbol{\nu}}}-(|{\boldsymbol{\nu}}|+1)p_{{\boldsymbol{\nu}}}\partial_j f.$$
Observe that
\begin{equation}\label{eq:partialjg}
\norm[{\mathcal W}^s_{\tau,\infty}]{\partial_j g}
=\sum_{|{\boldsymbol{\mu}}|\le s}\norm[L^\infty]{r_{\domain}^{\tau+|{\boldsymbol{\mu}}|} D^{{\boldsymbol{\mu}}+{\boldsymbol{e}}_j} g}
\le \sum_{|{\boldsymbol{\mu}}|\le s+1}\norm[L^\infty]{r_{\domain}^{\tau+|{\boldsymbol{\mu}}|-1} D^{\boldsymbol{\mu}} g}
=\norm[{\mathcal W}^{s+1}_{\tau-1,\infty}]{g}.
\end{equation}
Using \eqref{eq:wwstau} and \eqref{eq:partialjg}, we get with
$\tau:=|{\boldsymbol{\nu}}|+1$
\begin{align*}
\norm[{\mathcal W}^{s-\tau}_{\tau,\infty}]{p_{{\boldsymbol{\nu}}+{\boldsymbol{e}}_j}}
&\le \norm[{\mathcal W}^{s-\tau}_{\tau,\infty}]{f\partial_jp_{{\boldsymbol{\nu}}}}
+(|{\boldsymbol{\nu}}|+1)\norm[{\mathcal W}^{s-\tau}_{\tau,\infty}]{p_{{\boldsymbol{\nu}}}\partial_j f}\nonumber\\
&\le C_{s-\tau} \norm[{\mathcal W}^{s-\tau}_{0,\infty}]{f}
\norm[{\mathcal W}^{s-\tau}_{\tau,\infty}]{\partial_jp_{{\boldsymbol{\nu}}}}+(|{\boldsymbol{\nu}}|+1)
C_{s-\tau}
\norm[{\mathcal W}^{s-\tau}_{\tau-1,\infty}]{p_{{\boldsymbol{\nu}}}}
\norm[{\mathcal W}^{s-\tau}_{1,\infty}]{\partial_j f}\nonumber\\
&\le C_{s-\tau} \norm[{\mathcal W}^{s-\tau}_{0,\infty}]{f}
\norm[{\mathcal W}^{s-\tau+1}_{\tau-1,\infty}]{p_{{\boldsymbol{\nu}}}}+(|{\boldsymbol{\nu}}|+1)
C_{s-\tau}
\norm[{\mathcal W}^{s-\tau+1}_{\tau-1,\infty}]{p_{{\boldsymbol{\nu}}}}
\norm[{\mathcal W}^{s-\tau+1}_{0,\infty}]{f}.
\end{align*}
Due to $\tau-1=|{\boldsymbol{\nu}}|$ and the induction hypothesis
\eqref{eq:fracclaim} for $p_{\boldsymbol{\nu}}$,
\begin{align*}
\norm[{\mathcal W}^{s-(|{\boldsymbol{\nu}}|+1)}_{|{\boldsymbol{\nu}}|+1,\infty}]{p_{{\boldsymbol{\nu}}+{\boldsymbol{e}}_j}}
&\le C_{s-(|{\boldsymbol{\nu}}|+1)}\left(\hat C_{|{\boldsymbol{\nu}}|}\norm[{\mathcal W}^{s-(|{\boldsymbol{\nu}}|+1)}_{\infty}]{f}\norm[{\mathcal W}^s_{\infty}]{f}^{|{\boldsymbol{\nu}}|}+
(|{\boldsymbol{\nu}}|+1)\hat C_{|{\boldsymbol{\nu}}|}\norm[{\mathcal W}^s_{\infty}]{f}^{|{\boldsymbol{\nu}}|}\norm[{\mathcal W}^{s-|{\boldsymbol{\nu}}|}_\infty]{f} \right)\nonumber\\
&\le C_{s-(|{\boldsymbol{\nu}}|+1)}\hat C_{|{\boldsymbol{\nu}}|} (|{\boldsymbol{\nu}}|+2)\norm[{\mathcal W}_s^\infty]{f}^{|{\boldsymbol{\nu}}|+1}.
\end{align*}
In all this shows the claim with $\hat C_1:=1$ and inductively for
$1<k\le s$,
$$\hat C_{k}:= C_{s-k}\hat C_{k-1}(k+1).$$
By \eqref{eq:fracclaimtotal} and \eqref{eq:fracclaim}, for every
$|{\boldsymbol{\nu}}|\le s$
\begin{equation*}
\normc[L^\infty]{r_{\domain}^{|{\boldsymbol{\nu}}|}D^{\boldsymbol{\nu}}\left(\frac{1}{f}\right)}\le \hat C_{|{\boldsymbol{\nu}}|}\frac{\norm[{\mathcal W}^{s}_{\infty}]{f}^{|{\boldsymbol{\nu}}|}}{\operatorname{ess\,inf}_{{\boldsymbol{x}}\in{\domain}}|f({\boldsymbol{x}})|^{|{\boldsymbol{\nu}}|+1}}
\end{equation*}
Due to
$$\norm[{\mathcal W}^s_\infty]{f}\ge \norm[L^\infty]{f}\ge
\operatorname{ess\,inf}_{{\boldsymbol{x}}\in{\domain}}|f({\boldsymbol{x}})|,$$ this implies
\begin{equation*}
\normc[{\mathcal W}^{s}_{\infty}]{\frac{1}{f}}
= \sum_{|{\boldsymbol{\nu}}|\le s}\normc[L^\infty]{r_{\domain}^{|{\boldsymbol{\nu}}|}D^{\boldsymbol{\nu}} \left(\frac{1}{f}\right)}
\le \tilde C_s \frac{\norm[{\mathcal W}^{s}_{\infty}]{f}^{s}}{\operatorname{ess\,inf}_{{\boldsymbol{x}}\in{\domain}}|f({\boldsymbol{x}})|^{s+1}}
\end{equation*}
with $\tilde C_s:=\sum_{|{\boldsymbol{\nu}}|\le s}\hat C_{|{\boldsymbol{\nu}}|}$.
{\bf Step 3.} We show \ref{item:prodkond} and
\ref{item:proddiffkond}. If $f\in{\mathcal K}_\varkappa^s({\domain})$ and
$a\in {\mathcal W}^{s}_{\infty}({\domain})$, then by \eqref{eq:leibniz} for Sobolev
functions,
\begin{equation*}
r_{\domain}^{{\boldsymbol{\nu}}-\varkappa}D^{\boldsymbol{\nu}} (fa)=\sum_{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}} (r_{\domain}^{|{\boldsymbol{\nu}}-{\boldsymbol{\mu}}|-\varkappa}D^{{\boldsymbol{\nu}}-{\boldsymbol{\mu}}}f)(r_{\domain}^{|{\boldsymbol{\mu}}|}D^{\boldsymbol{\mu}} a)
\end{equation*}
and hence
\begin{align*}
\norm[{\mathcal K}_\varkappa^s]{fa}&=\sum_{|{\boldsymbol{\nu}}|\le s}\norm[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}|-\varkappa} D^{\boldsymbol{\nu}} (fa)}\nonumber\\
&\le
\sum_{|{\boldsymbol{\nu}}|\le s}\sum_{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}} \binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}}\norm[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}-{\boldsymbol{\mu}}|-\varkappa} D^{{\boldsymbol{\nu}}-{\boldsymbol{\mu}}}f}\norm[L^\infty]{r_{\domain}^{|{\boldsymbol{\mu}}|}D^{\boldsymbol{\mu}} a}\nonumber\\
&\le C_s\sum_{|{\boldsymbol{\nu}}|\le s}
\norm[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}|-\varkappa} D^{{\boldsymbol{\nu}}}f}
\sum_{|{\boldsymbol{\mu}}|\le s}\norm[L^\infty]{r_{\domain}^{|{\boldsymbol{\mu}}|}D^{\boldsymbol{\mu}} a}\nonumber\\
&=C_s \norm[{\mathcal K}_\varkappa^s]{f}\norm[{\mathcal W}^s_\infty]{a}.
\end{align*}
Finally if $s\ge 1$,
\begin{align*}
\norm[{\mathcal K}_{\varkappa-1}^{s-1}]{\nabla f\cdot\nabla a}&=\sum_{|{\boldsymbol{\nu}}|\le s-1}\normc[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}|-\varkappa+1} D^{\boldsymbol{\nu}} \left(\sum_{j=1}^d\partial_jf\partial_j a\right)}\nonumber\\
&\le
\sum_{|{\boldsymbol{\nu}}|\le s-1}\sum_{{\boldsymbol{\mu}}\le{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{\mu}}}\sum_{j=1}^d\norm[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}-{\boldsymbol{\mu}}|-\varkappa} D^{{\boldsymbol{\nu}}-{\boldsymbol{\mu}}+{\boldsymbol{e}}_j}f}\norm[L^\infty]{r_{\domain}^{|{\boldsymbol{\mu}}|+1}D^{{\boldsymbol{\mu}}+{\boldsymbol{e}}_j} a}\nonumber\\
&\le C_{s-1} d \sum_{|{\boldsymbol{\nu}}|\le s}
\norm[L^2]{r_{\domain}^{|{\boldsymbol{\nu}}|-\varkappa-1} D^{{\boldsymbol{\nu}}}f}
\sum_{|{\boldsymbol{\mu}}|\le s}\norm[L^\infty]{r_{\domain}^{|{\boldsymbol{\mu}}|}D^{\boldsymbol{\mu}} a}\nonumber\\
&= C_{s-1}d \norm[{\mathcal K}_{\varkappa+1}^s]{f}\norm[{\mathcal W}^{s}_\infty]{a}.\qedhere
\end{align*}
\end{proof}
The proof of the next theorem is based on Theorem~\ref{thm:bacuta}.
In order to get regularity in ${\mathcal K}_{\varkappa+1}^s({\domain})$ with
$\varkappa>0$ independent of the diffusion coefficient $a$, we now
assume $a\in W^{1}_\infty({\domain})\cap{\mathcal W}_{\infty}^{s-1}({\domain})$ in lieu of
the weaker assumption $a\in {\mathcal W}^{s-1}_\infty$
that was required in Theorem~\ref{thm:bacuta}.
\begin{theorem}\label{thm:bacuta2}
Let ${\domain}\subseteq\mathbb{R}^2$ be a bounded polygonal Lipschitz domain and
$s\in\mathbb{N}$, $s\ge 2$. Then there exist $\varkappa>0$ and $C_s>0$
depending on ${\domain}$ and $s$ (but independent of $a$) such that for all
$a\in W^{1}_\infty({\domain})\cap{\mathcal W}_{\infty}^{s-1}({\domain})$ and all
$f\in {\mathcal K}_{\varkappa-1}^{s-2}({\domain})$ the weak solution
${\mathcal U}\in H_0^1({\domain})$ of \eqref{eq:elliptic2} satisfies with
$N_s:=\frac{s(s-1)}{2}$
\begin{equation}\label{eq:Uuapriori}
\norm[{\mathcal K}_{\varkappa+1}^{s}]{{\mathcal U}}
\le C_s \frac{1}{\rho(a)}\left(\frac{\norm[{\mathcal W}^{s-1}_{\infty}]{a}+\norm[W^{1}_\infty]{a}}{\rho(a)}\right)^{N_s}\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}.
\end{equation}
\end{theorem}
\begin{proof}
Throughout this proof let $\varkappa\in (0,1)$ be a constant such
that
\begin{equation}\label{eq:DeltaIso}
-\Delta:{\mathcal K}_{\varkappa+1}^{j}({\domain})\cap H_0^1({\domain})\to {\mathcal K}_{\varkappa-1}^{j-2}({\domain})
\end{equation}
is a boundedly invertible operator for all $j\in\{2,\dots,s\}$; such
$\varkappa$ exists by Theorem~\ref{thm:bacuta}, and $\varkappa$
merely depends on ${\domain}$ and $s$.
{\bf Step 1.} We prove the theorem for $s=2$, in which case
$a\in W^{1}_\infty\cap{\mathcal W}^{1}_\infty=W^{1}_\infty$.
Applying Theorem~\ref{thm:bacuta} directly to \eqref{eq:elliptic2}
yields the existence of \emph{some}
$\tilde\varkappa\in (0,\varkappa)$ (depending on $a$) such that
${\mathcal U}\in{\mathcal K}_{\tilde\varkappa+1}^2$. Here we use
$$f\in{\mathcal K}_{\varkappa-1}^0({\domain})\hookrightarrow {\mathcal K}_{\tilde\varkappa-1}^0({\domain})$$
due to $\tilde\varkappa\in(0,\varkappa)$. By the Leibniz rule for
Sobolev functions we can write
\begin{equation*}
-\operatorname{div}(a\nabla {\mathcal U})=-a\Delta {\mathcal U} -\nabla a\cdot \nabla {\mathcal U}
\end{equation*}
in the sense of ${\mathcal K}_{\tilde\varkappa-1}^0({\domain})$: (i) it holds
$\Delta{\mathcal U}\in {\mathcal K}_{\tilde\varkappa-1}^0({\domain})$ and
$$a\in W^{1}_\infty({\domain})\hookrightarrow L^\infty({\domain})$$ which implies
$a\Delta{\mathcal U}\in {\mathcal K}_{\tilde\varkappa-1}^0({\domain})$ (ii) it holds
$$\nabla {\mathcal U} \in {\mathcal K}_{\tilde\varkappa}^1({\domain})\hookrightarrow{\mathcal K}_{\tilde\varkappa}^0({\domain}),$$ and $\nabla a \in L^\infty({\domain})$
which implies
$\nabla a \cdot\nabla {\mathcal U} \in {\mathcal K}_{\tilde\varkappa-1}^0({\domain})$.
Hence,
\begin{equation*}
-\operatorname{div}(a\nabla {\mathcal U}) = -a\Delta {\mathcal U} -\nabla a\cdot \nabla {\mathcal U}
= f,
\end{equation*}
and further
\begin{equation*}
-\Delta {\mathcal U} = \frac{1}{a}\Big(f+\nabla a\cdot \nabla {\mathcal U}\Big)=:\tilde f\in {\mathcal K}_{\tilde \varkappa-1}^0({\domain})
\end{equation*}
since $\frac{1}{a}\in L^\infty({\domain})$ due to $\rho(a)>0$. Our goal is
to show that in fact $\tilde f\in {\mathcal K}_{\varkappa-1}^0({\domain})$.
Because of $-\Delta {\mathcal U}=\tilde f$ and ${\mathcal U}|_{\partial{\domain}}\equiv 0$,
Theorem~\ref{thm:bacuta} then implies
\begin{equation}\label{eq:Kkkappa}
\norm[{\mathcal K}_{\varkappa+1}^2]{{\mathcal U}}\le
C \norm[{\mathcal K}_{\varkappa-1}^0]{\tilde f}
\end{equation}
for a constant $C$ solely depending on ${\domain}$.
Denote by $C_H$
a constant (solely depending on ${\domain}$) such that
\begin{equation*}
\norm[L^2]{r_{\domain}^{-1}v}\le C_H\norm[L^2]{\nabla v}\qquad
\forall v\in H_0^1({\domain}).
\end{equation*}
This constant exists as a consequence of Hardy's inequality, see
e.g.\ \cite{MR1544414} and \cite{MR664599,MR163054} for the
statement and proof of the inequality on bounded Lipschitz domains.
Then due to
\begin{align*}
\rho(a) \norm[L^2]{\nabla {\mathcal U}}^2&\le \Re\left(\int_{\domain} a\nabla {\mathcal U}\cdot
\overline{\nabla {\mathcal U}} \,\mathrm{d} {\boldsymbol{x}}\right
= \Re\left(\int_{\domain} f \overline{{\mathcal U}} \,\mathrm{d} {\boldsymbol{x}} \right)\nonumber\\
&\le \norm[L^2]{r_{\domain}^{1-\varkappa}f}
\norm[L^2]{r_{\domain}^{\varkappa-1}{\mathcal U}
\le \norm[{\mathcal K}_{\varkappa-1}^{0}]{f}\norm[L^2]{r_{\domain}^{-1}{\mathcal U}}\nonumber\\
&
\le C_H \norm[{\mathcal K}_{\varkappa-1}^{0}]{f}\norm[L^2]{\nabla {\mathcal U}}
\end{align*}
it holds
$$\norm[L^2]{\nabla {\mathcal U}}\le \frac{C_H\norm[{\mathcal K}_{\varkappa-1}^0]{f}}{\rho(a)}.$$
Hence, using $r_{\domain}^{1-\varkappa}\le 1$, we have that
\begin{align*}
\norm[{\mathcal K}_{\varkappa-1}^0]{\tilde f}&=\normc[L^2]{\frac{r_{\domain}^{1-\varkappa}}{a}\Big(f+\nabla a\cdot \nabla {\mathcal U}\Big)}\nonumber\\
&\le \normc[L^\infty]{\frac{1}{a}}\left(\norm[L^2]{r_{\domain}^{1-\varkappa}f}
+ \norm[L^\infty]{\nabla a}\norm[L^2]{\nabla {\mathcal U}}\right)
\nonumber\\
&\le \frac{1}{\rho(a)}\left(\norm[{\mathcal K}_{\varkappa-1}^0]{f}
+ \norm[W^{1}_\infty]{a}\frac{C_H\norm[{\mathcal K}_{\varkappa-1}^0]{f}}{\rho(a)}\right)\nonumber\\
&= \frac{\norm[{\mathcal K}_{\varkappa-1}^0]{f}}{\rho(a)}\left(1+\frac{C_H\norm[W^{1}_\infty]{a}}{\rho(a)}\right)\nonumber\\
&\le (1+C_H)\frac{1}{\rho(a)}\frac{\norm[W^{1}_\infty]{a}}{\rho(a)} \norm[{\mathcal K}_{\varkappa-1}^0]{f}.
\end{align*}
The statement follows by \eqref{eq:Kkkappa}.
{\bf Step 2.} For general $s\in\mathbb{N}$, $s\ge 2$, we proceed by
induction. Assume the theorem holds for $s-1\ge 2$. Then for
$$f\in {\mathcal K}_{\varkappa-1}^{s-2}({\domain})\hookrightarrow {\mathcal K}_{\varkappa-1}^{s-3}({\domain})$$ and $$a\in W^{1}_\infty({\domain})\cap {\mathcal W}^{s-1}_{\infty}({\domain})\hookrightarrow W^{1}_\infty({\domain})\cap {\mathcal W}^{s-2}_{\infty}({\domain}),$$
we get
\begin{equation}\label{eq:fKkbound}
\norm[{\mathcal K}_{\varkappa+1}^{s-1}]{{\mathcal U}}\le \frac{C_{s-1}}{\rho(a)}
\left(\frac{\norm[W^{1}_\infty]{a}+\norm[{\mathcal W}^{s-2}_{\infty}]{a}}{\rho(a)}\right)^{N_{s-1}}\norm[{\mathcal K}_{\varkappa-1}^{s-3}]{f}.
\end{equation}
As in Step 1, it holds
\begin{equation*}
-\Delta {\mathcal U} = \frac{1}{a}\Big(f+\nabla a\cdot \nabla {\mathcal U} \Big)=:\tilde f.
\end{equation*}
By
Lemma \ref{lemma:fgWm} and \eqref{eq:fKkbound}, for some constant
$C$ (which can change in each line, but solely depends on ${\domain}$ and
$s$) we have that
\begin{align*}
\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{\tilde f}
&\le C \normc[{\mathcal W}^{s-2,\infty}]{\frac{1}{a}}
\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f+\nabla a \cdot \nabla{\mathcal U}}\nonumber\\
&\le C \frac{\norm[{\mathcal W}^{s-2}]{a}^{s-2}}{\rho(a)^{s-1}}
\left(\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}+\norm[{\mathcal W}^{s-1}_\infty]{a}
\norm[{\mathcal K}_{\varkappa+1}^{s-1}]{{\mathcal U}}
\right)\nonumber\\
&\le C \frac{\norm[{\mathcal W}^{s-2}]{a}^{s-2}}{\rho(a)^{s-1}}
\left(\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}+
C_{s-1}\frac{\norm[{\mathcal W}^{s-1}_\infty]{a}}{\rho(a)}\left(\frac{\norm[W^{1}_\infty]{a}+\norm[{\mathcal W}_{\infty}^{s-2}]{a}}{\rho(a)}\right)^{N_{s-1}}\norm[{\mathcal K}_{\varkappa-1}^{s-3}]{f}
\right)\nonumber\\
&\le C \frac{1}{\rho(a)}
\left(\frac{\norm[W^{1}_\infty]{a}+\norm[{\mathcal W}_{\infty}^{s-1}]{a}}{\rho(a)}\right)^{N_{s-1}+1+(s-2)}
\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}.
\end{align*}
Note that
$$N_{s-1}+(s-1)=\frac{(s-1)(s-2)}{2}+(s-1)=\frac{s(s-1)}{2}=N_s.$$
We now use \eqref{eq:fKkbound} and the fact that \eqref{eq:DeltaIso}
is a boundedly invertible isomorphism to conclude that there exist
$C_s$ such that \eqref{eq:Uuapriori} holds.
\end{proof}
Throughout the rest of this section ${\domain}$ is assumed a bounded
polygonal domain and $\varkappa>0$ the constant from
Theorem~\ref{thm:bacuta2}.
\begin{assumption}\label{ass:FEM}
For some fixed $s\in\mathbb{N}$, $s\ge 2$, there exist constants $C>0$ and
$\alpha>0$, and a sequence $(X_l)_{l\in\mathbb{N}}$ of subspaces of
$X=H_0^1({\domain};\mathbb{C})=H_0^1$, such that
\begin{enumerate}
\item $\sw{l}:={\rm dim}(X_l)$, $l\in\mathbb{N}$, satisfies Assumption
\ref{ass:SW} (for some $K_\mathfrak{W}>0$),
\item for all $l\in\mathbb{N}$
\begin{equation}\label{eq:conv}
\sup_{0\neq u\in {\mathcal K}_{\varkappa+1}^{s}}\frac{\inf_{v\in X_l}\norm[H_0^1]{u-v}}{\norm[{\mathcal K}_{\varkappa+1}^{s}]{u}}
\le C \sw{l}^{-\conv}.
\end{equation}
\end{enumerate}
\end{assumption}
The constant $\conv$ in Assumption \ref{ass:FEM} can be interpreted as
the convergence rate of the finite element method. For the Kondrat'ev
space ${\mathcal K}_{\varkappa+1}^{s}({\domain})$, finite element spaces $X_l$ of
piecewise polynomials of degree $s-1$ have been constructed in
\cite[Theorem~4.4]{BNZPolygon}, which achieve the optimal (in space
dimension $2$) convergence rate
\begin{equation}\label{eq:optimalrate}
\conv=\frac{s-1}{2}
\end{equation}
in \eqref{eq:conv}. For these spaces, Assumption \ref{ass:FEM} holds
with this $\alpha$, which consequently allows us to retain optimal
convergence rates. Nonetheless we keep the discussion general in the
following, and assume arbitrary positive $\alpha>0$.
We next introduce the finite element solutions of \eqref{eq:elliptic2}
in the spaces $X_l$, and provide the basic error estimate.
\begin{lemma}\label{lemma:femapprox}
Let Assumption \ref{ass:FEM} be satisfied for some $s\ge 2$. Let
$f\in {\mathcal K}_{\varkappa-1}^{s-2}({\domain})$ and
$$a\in W^{1}_\infty({\domain})\cap {\mathcal W}^{s-1}_{\infty}({\domain})\subseteq L^\infty({\domain})$$
with $\rho(a)>0$ and denote for $l\in\mathbb{N}$ by ${\mathcal U}^l(a)\in X_l$ the
unique solution of
\begin{equation*}
\int_{\domain} a(\nabla {\mathcal U}^l)^\top\overline{\nabla v}\,\mathrm{d} {\boldsymbol{x}}
=\dup{f}{v}\qquad \forall v\in X_l,
\end{equation*}
where the right hand side denotes the (sesquilinear) dual pairing
between $H^{-1}({\domain})$ and $H_0^1({\domain})$. Then for the solution
${\mathcal U}(a)\in H_0^1({\domain})$ it holds with the constants $N_s$, $C_s$ from
Theorem.~\ref{thm:bacuta2},
\begin{equation*}
\norm[H_0^1]{{\mathcal U}(a)-{\mathcal U}^l(a)}
\le \sw{l}^{-\conv} C \frac{\norm[L^\infty]{a}}{\rho(a)}
\norm[{\mathcal K}_{\varkappa+1}^{s}]{{\mathcal U}(a)}
\le \sw{l}^{-\conv}
C C_s
\frac{(\norm[W^{1}_\infty]{a}+\norm[{\mathcal W}^{s-1}_{\infty}]{a})^{N_s+1}}{\rho(a)^{N_s+2}}\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}\;.
\end{equation*}
Here $C>0$ is the constant from Assumption \ref{ass:FEM}.
\end{lemma}
\begin{proof}
By C\'ea's lemma in complex form we derive that
\begin{align*}
\norm[H_0^1]{{\mathcal U}(a)-{\mathcal U}^l(a)}
\le \frac{\norm[L^\infty]{a}}{\rho(a)}
\inf_{v\in X_l} \norm[H_0^1]{{\mathcal U}(a)-v}.
\end{align*}
Hence the assertion follows by Assumption \ref{ass:FEM} and
\eqref{eq:Uuapriori}.
\end{proof}
Throughout the rest of this section, as earlier we expand the
logarithm of the diffusion coefficient
\begin{equation*}\label{eq:diffusionml}
a({\boldsymbol{y}})=\exp\Bigg(\sum_{j\in\mathbb{N}}y_j\psi_j\Bigg)
\end{equation*}
in terms of a sequence
$\psi_j\in W^{1}_\infty({\domain})\cap{\mathcal W}_\infty^{s-1}({\domain})$, $j\in\mathbb{N}$. Denote
\begin{equation}\label{eq:b1b2ml}
b_{1,j}:=\norm[L^\infty]{\psi_j},\quad
b_{2,j}:=\max\big\{\norm[W^{1}_\infty]{\psi_j},\norm[{\mathcal W}_\infty^{s-1}]{\psi_j}\big\}
\end{equation}
and ${\boldsymbol{b}}_1:=(b_{1,j})_{j\in\mathbb{N}}$, ${\boldsymbol{b}}_2:=(b_{2,j})_{j\in\mathbb{N}}$.
\begin{example}\label{ex:sin}
Let ${\domain}=[0,1]$ and $\psi_j(x)=\sin(jx)j^{-r}$ for some $r>2$. Then
${\boldsymbol{b}}_1\in\ell^{p_1}({\mathbb N})$ for every $p_1>\frac{1}{r}$ and
${\boldsymbol{b}}_2\in\ell^{p_2}({\mathbb N})$ for every $p_2>\frac{1}{r-(s-1)}$.
\end{example}
In the next proposition we verify Assumption \ref{ass:ml}. This will
yield validity of the multilevel convergence rates proved in
Theorem~\ref{thm:mlint} and Theorem~\ref{thm:mlquad} in the present
setting as we discuss subsequently.
\begin{proposition}\label{prop:u-ul}
Let Assumption \ref{ass:FEM} be satisfied for some $s\ge 2$ and
$\alpha>0$. Let ${\boldsymbol{b}}_1\in\ell^{p_1}({\mathbb N})$,
${\boldsymbol{b}}_2\in\ell^{p_2}({\mathbb N})$ with $p_1$, $p_2\in (0,1)$.
Then there exist $\xi>0$ and $\delta>0$ such that
\begin{equation}\label{eq:uby}
u({\boldsymbol{y}}):={\mathcal U}\Bigg(\exp\Bigg(\sum_{j\in\mathbb{N}}y_j\psi_j\Bigg)\Bigg)
\end{equation}
is $({\boldsymbol{b}}_1,\xi,\delta,H_0^1)$-holomorphic, and for every $l\in\mathbb{N}$
\begin{enumerate}
\item\label{item:ulhol}
$u^l({\boldsymbol{y}}):={\mathcal U}^l(\exp(\sum_{j\in\mathbb{N}}y_j\psi_j))$ is
$({\boldsymbol{b}}_1,\xi,\delta,H_0^1)$-holomorphic,
\item\label{item:u-ulhol1} $u-u^l$ is
$({\boldsymbol{b}}_1,\xi,\delta,H_0^1)$-holomorphic,
\item\label{item:u-ulhol2} $u-u^l$ is
$({\boldsymbol{b}}_2,\xi,\delta\sw{l}^{-\conv},H_0^1)$-holomorphic.
\end{enumerate}
\end{proposition}
\begin{proof}
{\bf Step 1.} We show \ref{item:ulhol} and \ref{item:u-ulhol1}.
The argument to show that $u^l$ is
$({\boldsymbol{b}}_1,\xi,\delta,H_0^1)$-holomorphic (for some constants
$\xi>0$, $\delta>0$ independent of $l$) is essentially the same as
in Section \ref{sec:pdc}.
We wish to apply Proposition \ref{prop:bdX} (with
``$E=L^\infty({\domain})$'' in this lemma). To this end let
$$O_1=\set{a\in L^\infty({\domain};\mathbb{C})}{\rho(a)>0} \subseteq L^\infty({\domain}).$$ By
assumption, $b_{1,j}=\norm[L^\infty]{\psi_j}$ satisfies
${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}\in\ell^{p_1}({\mathbb N} )\subseteq
\ell^1({\mathbb N})$, which corresponds to assumption \ref{item:psi} of
Proposition \ref{prop:bdX}. It remains to verify assumptions
\ref{item:uhol}, \ref{item:norma} and \ref{item:loclip} of
Proposition \ref{prop:bdX}:
\begin{enumerate}
\item ${\mathcal U}^l:O_1\to H_0^1$ is holomorphic: This is satisfied
because the operation of inversion of linear operators is
holomorphic on the set of boundedly invertible linear
operators. Denote by $A_l:X_l\to X_l'$ the differential
operator $$A_lu=-\operatorname{div}(a\nabla u)\in X_l'$$ via
$$\dup{A_lu}{v}=\int_{{\domain}}a\nabla u^\top \overline{\nabla v}\,\mathrm{d} {\boldsymbol{x}} \quad \forall v\in X_l.$$
Observe that $A_l$ depends boundedly and linearly (thus
holomorphically) on $a$, and therefore, the map
$a\mapsto A_l(a)^{-1}f={\mathcal U}^l(a)$ is a composition of holomorphic
functions. We refer once more to \cite[Example 1.2.38]{JZdiss}
for more details.
\item It holds for all $a\in O$
\begin{equation*}
\norm[H_0^1]{{\mathcal U}^l(a)}\le \frac{\norm[X_l']{f}}{\rho(a)}
\le \frac{\norm[H^{-1}]{f}}{\rho(a)}.
\end{equation*}
The first inequality follows by the same calculation as
\eqref{eq:apriori} (but with $X$ replaced by $X_l$), and the
second inequality follows by the definition of the dual norm,
viz
\begin{equation*}
\norm[X_l']{f}=\sup_{0\neq v\in X_l}\frac{|\dup{f}{v}|}{\norm[H_0^1]{v}}
\le\sup_{0\neq v\in H_0^1}\frac{|\dup{f}{v}|}{\norm[H_0^1]{v}}=
\norm[H^{-1}]{f}.
\end{equation*}
\item For all $a$, $b\in O$ we have
\begin{equation*}
\norm[H_0^1]{{\mathcal U}^l(a)-{\mathcal U}^l(b)}\le \norm[H^{-1}]{f}
\frac{1}{\min\{\rho(a),\rho(b)\}^2}
\norm[L^\infty]{a-b},
\end{equation*}
which follows again by the same calculation as in in the proof
of \eqref{eq:lipschitz}.
\end{enumerate}
According to Proposition
\ref{prop:bdX}
the map
$${\mathcal U}^l\in L^2(U,X_l;\gamma)\subseteq L^2(U,H_0^1;\gamma)$$ is
$({\boldsymbol{b}}_1,\xi_1,\tilde C_1,H_0^1)$-holomorphic, for some fixed
constants $\xi_1>0$ and $\tilde C_1>0$ depending on $O_1$ but
independent of $l$. In fact the argument also works with $H_0^1$
instead of $X_l$, i.e.\ also $u$ is
$({\boldsymbol{b}}_1,\xi_1,\tilde C_1,H_0^1)$-holomorphic (with the same
constants $\xi_1$ and $\tilde C_1$).
Finally, it follows directly from the definition that the
difference $u-u^l$ is $({\boldsymbol{b}}_1,\xi,2\delta,H_0^1)$-holomorphic.
{\bf Step 2.} To show \ref{item:u-ulhol2}, we set
$$O_2=\set{a\in W^{1}_\infty({\domain})\cap{\mathcal W}^{s-1}_\infty({\domain})}{\rho(a)>0},$$ and verify again
assumptions \ref{item:uhol}, \ref{item:norma} and
\ref{item:loclip} of Proposition \ref{prop:bdX}, but now with
``$E$'' in this lemma being
$W^{1}_\infty({\domain})\cap{\mathcal W}^{s-1}_\infty({\domain})$. First let us point out
that with
$$b_{2,j}:=\max\{\norm[{\mathcal W}^{s-1}_\infty]{\psi_j},\norm[W^{1}_\infty]{\psi_j}\},$$
by assumption
$${\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}\in\ell^{p_2}({\mathbb N})\hookrightarrow \ell^1({\mathbb N})$$
which corresponds to the assumption \ref{item:psi} of Proposition
\ref{prop:bdX}.
For every $l\in\mathbb{N}$:
\begin{enumerate}
\item ${\mathcal U}-{\mathcal U}^l:O_2\to H_0^1({\domain})$ is holomorphic: Since $O_2$ can
be considered a subset of $O_1$ (and $O_2$ is equipped with a
stronger topology than $O_1$), Fr\'echet differentiability
follows by Fr\'echet differentiability of
$${\mathcal U}-{\mathcal U}^l:O_1\to H_0^1({\domain}),$$ which holds by Step 1.
\item For every $a\in O_2$
\begin{equation*}
\norm[H_0^1]{({\mathcal U}-{\mathcal U}^l)(a)}
\le \underbrace{\sw{l}^{-\conv} C C_s
\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}}_{=:\delta_l}
\frac{(\norm[W^{1}_\infty]{a}+\norm[{\mathcal W}^{s-1}_{\infty}]{a})^{N_s+1}}{\rho(a)^{N_s+2}}
\end{equation*}
by Lemma \ref{lemma:femapprox}.
\item For every $a$, $b\in O_2\subseteq O_1$, by Step 1 and
\eqref{eq:lipschitz}, \begin{align*}
\norm[H_0^1]{({\mathcal U}-{\mathcal U}^l)(a)-({\mathcal U}-{\mathcal U}^l)(b)} &\le
\norm[H_0^1]{{\mathcal U}(a)-{\mathcal U}(b)}
+\norm[H_0^1]{{\mathcal U}^l(a)-{\mathcal U}^l(b)}\nonumber\\
&\le
\norm[H^{-1}]{f}
\frac{2}{\min\{\rho(a),\rho(b)\}^2}
\norm[L^\infty]{a-b}.
\end{align*}
We conclude with Proposition
\ref{prop:bdX}
that there exist $\xi_2$ and $\tilde C_2$
depending on $O_2$, ${\domain}$ but independent
of $l$ such that $u-u^l$ is
$({\boldsymbol{b}}_2,\xi_2,\tilde
C_2\delta_l,H_0^1)$-holomorphic.
\end{enumerate}
In all, the proposition holds with
\begin{equation*}
\xi:=\min\{\xi_1,\xi_2\}\qquad\text{and}\qquad
\delta:=\max\{\tilde C_1,\tilde C_2 C C_s\norm[{\mathcal K}_{\varkappa-1}^{s-2}]{f}\}.\qedhere
\end{equation*}
\end{proof}
Items \ref{item:u-ulhol1} and
\ref{item:u-ulhol2} of
Proposition~\ref{prop:u-ul} show that
Assumption \ref{ass:FEM} implies validity of
Assumption \ref{ass:ml}. This in turn allows
us to apply Theorem~\ref{thm:mlint} and
Theorem~\ref{thm:mlquad}. Specifically,
assuming the optimal convergence rate
$\alpha=\frac{s-1}{2}$ in
\eqref{eq:optimalrate}, we obtain that for $u$
in \eqref{eq:uby} and every $n\in \mathbb{N}$ there is
$\varepsilon:=\varepsilon_n>0$ such that
$\mathrm{work}(\mathbf{l}_{\varepsilon})\leq n$ and the multilevel
interpolant $\mathbf{I}^{\rm ML}_\mathbf{l}$ defined in
\eqref{eq:VIml} satisfies
\begin{equation*}
\norm[L^2(U,X;\gamma)]{u-\mathbf{I}_{\mathbf{l}_\varepsilon}^{\rm ML}u}\le C(1+\log n)n^{-R_I},\quad
R_I = \min\left\{\frac{s-1}{2},\frac{\frac{s-1}{2}(\frac{1}{p_1}-\frac{3}{2})}{\frac{s-1}{2}+\frac{1}{p_1}-\frac{1}{p_2}}\right\},
\end{equation*}
and the multilevel quadrature operator
$\mathbf{Q}^{\rm ML}_\mathbf{l}$ defined in \eqref{eq:VQml}
satisfies
\begin{equation*}
\normc[X]{\int_Uu({\boldsymbol{y}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\mathbf{l}_\varepsilon}^{\rm ML}u}\le C(1+\log n)n^{-R_Q},\quad
R_Q=\min\left\{\frac{s-1}{2},\frac{\frac{s-1}{2}(\frac{2}{p_1}-\frac{5}{2})}{\frac{s-1}{2}+\frac{2}{p_1}-\frac{2}{p_2}}\right\}.
\end{equation*}
Let us consider these convergence rates in the
case where the $\psi_j$ are algebraically
decreasing, with this decrease encoded by some
$r>1$: if for fixed but arbitrarily small
$\varepsilon>0$ holds
$\norm[L^\infty]{\psi_j}\sim
j^{-r-\varepsilon}$, and we assume
(cp.~Ex.~\ref{ex:sin})
\begin{equation*}
\max\big\{\norm[W^{1}_\infty]{\psi_j},\norm[{\mathcal W}^{s-1}_{\infty}]{\psi_j}\big\}\sim
j^{-r+(s-1)-\varepsilon},
\end{equation*}
then setting $s:=r$ we can choose
$p_1=\frac{1}{r}$ and $p_2=1$. Inserting those
numbers, the convergence rates become
\begin{equation*}
R_I = \min\left\{\frac{r-1}{2},\frac{\frac{r-1}{2}(r-\frac{3}{2})}{\frac{r-1}{2}+r-1}\right\}=\frac{r}{3}-\frac{1}{2}
\quad\text{and}\quad
R_Q = \min\left\{\frac{r-1}{2},\frac{\frac{r-1}{2}(2r-\frac{5}{2})}{\frac{r-1}{2}+2r-2}\right\}=\frac{2r}{5}-\frac{1}{2}.
\end{equation*}
\subsubsection{Parametric holomorphy of the
posterior density in Bayesian PDE inversion}
\label{sec:PosDnsBayInv}
Throughout this section we assume that
${\domain}\subseteq\mathbb{R}^2$ is a polygonal Lipschitz
domain and that
$f\in {\mathcal K}_{\varkappa-1}^{s-2}({\domain})$ with
$\varkappa$ as in Theorem~\ref{thm:bacuta2}.
As in Section \ref{sec:BIP}, to treat the
approximation of the (unnormalized) posterior
density or its integral, we need an upper
bound on $\norm[H_0^1]{u({\boldsymbol{y}})}$ for all
${\boldsymbol{y}}$. This is achieved by considering
\eqref{eq:elliptic2}
with diffusion coefficient $a_0+a$ where
$$\rho(a_0):=\underset{{\boldsymbol{x}}\in{\domain}}{ \operatorname{ess\,inf} }\Re(a_0) > 0.$$
The shift of the diffusion coefficient by $a_0$ ensures uniform
ellipticity for all
$$a\in \set{a\in L^\infty({\domain},{\mathbb C})}{\rho(a)\ge 0}.$$ As a consequence,
solutions ${\mathcal U}(a_0+a)\in X=H_0^1({\domain};\mathbb{C})$ of \eqref{eq:elliptic2}
satisfy the apriori bound (cp.~\eqref{eq:apriori})
\begin{subequations}\label{eq:Uulabound}
\begin{equation*}
\norm[X]{{\mathcal U}(a_0+a)}\le \frac{\norm[H^{-1}]{f}}{\rho(a_0)}.
\end{equation*}
As before, for a sequence of subspaces $(X_l)_{l\in\mathbb{N}}$ of
$H_0^1({\domain},{\mathbb C})$, for $a\in O$ we denote by ${\mathcal U}^l(a)\in X_l$ the
finite element approximation to ${\mathcal U}(a)$.
By the same calculation as for ${\mathcal U}$ it also holds
\begin{equation*}
\norm[X]{{\mathcal U}^l(a_0+a)}\le
\frac{\norm[H^{-1}]{f}}{\rho(a_0)}
\end{equation*}
independent of $l$.
\end{subequations}
Assuming that $b_j=\norm[L^\infty]{\psi_j}$ satisfies
$(b_j)_{j\in\mathbb{N}}\in\ell^1({\mathbb N})$, the function
$u({\boldsymbol{y}})={\mathcal U}(a_0+a({\boldsymbol{y}}))$
with $$a({\boldsymbol{y}})=\exp\bigg(\sum_{j\in\mathbb{N}}y_j\psi_j\bigg),$$ is
well-defined. For a fixed \emph{observation} ${\boldsymbol{\obs}}\in\mathbb{R}^m$
consider again the (unnormalized) posterior density given in
\eqref{eq:TheParm},
\begin{equation*}
\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}}):=\exp\left( -({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}})))^\top {\boldsymbol{\Gamma}}^{-1} ({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u({\boldsymbol{y}}))) \right).
\end{equation*}
Recall that ${\boldsymbol{{\mathcal O}}}:X\to\mathbb{C}^m$ (the observation operator) is assumed
to be a bounded linear map, and ${\boldsymbol{\Gamma}}\in\mathbb{R}^{m\times m}$ (the
noise covariance matrix) is symmetric positive definite.
For $l\in\mathbb{N}$ (tagging discretization level of the PDE), and with
$u^l({\boldsymbol{y}})={\mathcal U}^l(a_0+a({\boldsymbol{y}}))$, we introduce approximations
\begin{equation*}
\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}}):=\exp\left( -({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u^l({\boldsymbol{y}})))^\top {\boldsymbol{\Gamma}}^{-1} ({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u^l({\boldsymbol{y}}))) \right)
\end{equation*}
to $\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}})$. In the following we show the analogue of
Proposition~\ref{prop:u-ul}, that is we show validity of the
assumptions required for the multilevel convergence results.
\begin{lemma}\label{lemma:PhiLip}
Let ${\boldsymbol{{\mathcal O}}}:H_0^1({\domain};\mathbb{C})\to\mathbb{C}^m$ be a bounded linear operator,
${\boldsymbol{\obs}}\in\mathbb{C}^m$ and ${\boldsymbol{\Gamma}}\in\mathbb{R}^{m\times m}$ symmetric positive
definite. Set
\begin{equation*}
\Phi:=\begin{cases}
H_0^1({\domain};\mathbb{C})\to\mathbb{C}\\
u\mapsto \exp(-({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u))^\top {\boldsymbol{\Gamma}}^{-1}({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u))).
\end{cases}
\end{equation*}
Then for every $r>0$ the function
$$\Phi:\set{u\in H_0^1({\domain};\mathbb{C})}{\norm[H_0^1]{u}<r}\to \mathbb{C}$$ is
continuously differentiable and has a Lipschitz constant $K$
solely depending on $\norm[]{{\boldsymbol{\Gamma}}^{-1}}$,
$\norm[L(H_0^1;\mathbb{C}^m)]{{\boldsymbol{{\mathcal O}}}}$, $\norm[]{{\boldsymbol{\obs}}}$ and $r$.
\end{lemma}
\begin{proof}
The function is $C^1$ as a composition of $C^1$ functions. Hence
for $u$, $v$ with $w:=u-v$ and with the derivative
$D\Phi:H_0^1\to L(H_0^1;\mathbb{C})$ of $\Phi$,
\begin{equation}\label{eq:diffPhi}
\Phi(u)-\Phi(v)=\int_{0}^1 D\Phi(v+tw)w \,\mathrm{d} t.
\end{equation}
Due to the symmetry of ${\boldsymbol{\Gamma}}$ it holds
\begin{equation*}
D\Phi(u+tw)w = 2
{\boldsymbol{{\mathcal O}}}(w)^\top {\boldsymbol{\Gamma}}^{-1}({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u+tw))
\exp\Big(-({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u+tw))^\top {\boldsymbol{\Gamma}}^{-1}({\boldsymbol{\obs}}-{\boldsymbol{{\mathcal O}}}(u+tw))\Big).
\end{equation*}
If $\norm[H_0^1]{u}$, $\norm[H_0^1]{v}<r$ then also
$\norm[H_0^1]{u+tw}<r$ for all $t\in[0,1]$ and we can bound
\begin{equation*}
|D\Phi(u+tw)w|\le
K \norm[H_0^1]{w},
\end{equation*}
where
\begin{equation} \label{ConstantK} K:=
2\norm[L(H_0^1;\mathbb{C}^m)]{{\boldsymbol{{\mathcal O}}}}
\norm[]{{\boldsymbol{\Gamma}}^{-1}}(\norm[]{{\boldsymbol{\obs}}}+r
\norm[L(H_0^1;\mathbb{C}^m)]{{\boldsymbol{{\mathcal O}}}})
\exp(\norm[]{{\boldsymbol{\Gamma}}^{-1}}(\norm[]{{\boldsymbol{\obs}}}+\norm[L(H_0^1;\mathbb{C}^m)]{{\boldsymbol{{\mathcal O}}}}
r)^2).
\end{equation}
The statement follows by \eqref{eq:diffPhi}.
\end{proof}
\begin{remark}
The reason why we require the additional positive $a_0$ term in
\eqref{eq:Uulabound}, is to guarantee boundedness of the solution
${\mathcal U}(a)$ and Lipschitz continuity of $\Phi$.
\end{remark}
\begin{proposition}\label{prop:pi-pil}
Let Assumption \ref{ass:FEM} be satisfied for some $s\ge 2$ and
$\alpha>0$. Let $a_0$,
$(\psi_j)_{j\in\mathbb{N}}\subseteq W^{1}_\infty({\domain})\cap
{\mathcal W}^{s-1}_{\infty}({\domain})$ and ${\boldsymbol{b}}_1\in\ell^{p_1}({\mathbb N})$,
${\boldsymbol{b}}_2\in\ell^{p_2}({\mathbb N})$ with $p_1$, $p_2\in (0,1)$ (see
\eqref{eq:b1b2ml} for the definition of ${\boldsymbol{b}}_1$, ${\boldsymbol{b}}_2$). Fix
${\boldsymbol{\obs}}\in\mathbb{C}^m$.
Then there exist $\xi>0$ and $\delta>0$ such that
$\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}})$ is $({\boldsymbol{b}}_1,\xi,\delta,\mathbb{C})$-holomorphic, and
for every $l\in\mathbb{N}$
\begin{enumerate}
\item\label{item:pilhol} $\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_1,\xi,\delta,\mathbb{C})$-holomorphic,
\item\label{item:pi-pilhol1}
$\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}})-\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_1,\xi,\delta,H_0^1)$-holomorphic,
\item\label{item:pi-pilhol2}
$\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}})-\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_2,\xi,\delta\sw{l}^{-\conv},H_0^1)$-holomorphic.
\end{enumerate}
\end{proposition}
\begin{proof}
{\bf Step 1.} We show \ref{item:pilhol} and \ref{item:pi-pilhol1}.
Set $$O_1:=\set{a\in L^\infty({\domain};\mathbb{C})}{\rho(a)>0}.$$ By
\eqref{eq:Uulabound}
for all $a\in O_1$ and all $l\in\mathbb{N}$ with
$r:=\frac{\norm[H^{-1}]{f}}{\rho(a_0)}$
\begin{equation}\label{eq:Uuboundr}
\norm[H_0^1]{{\mathcal U}^l(a_0+a)}\le r\qquad\text{and}\qquad
\norm[H_0^1]{{\mathcal U}(a_0+a)}\le r.
\end{equation}
As in Step 1 of the proof of Proposition~\ref{prop:u-ul}, one can
show that $u({\boldsymbol{y}})={\mathcal U}(a_0+a({\boldsymbol{y}}))$ and $u^l({\boldsymbol{y}})={\mathcal U}(a_0+a({\boldsymbol{y}}))$
where $a({\boldsymbol{y}})=\exp(\sum_{j\in\mathbb{N}}y_j\psi_j)$ are
$({\boldsymbol{b}}_1,\xi_1,\tilde C_1,H_0^1)$-holomorphic for certain $\xi_1>0$
and $\tilde C_1>0$ (the only difference to
Proposition~\ref{prop:u-ul} is the additional $a_0$ term in
\eqref{eq:elliptic2}). In the following $\Phi$ is as in Lemma
\ref{lemma:PhiLip} and $T_{a_0}(a):=a_0+a$ so that
\begin{equation}\label{eq:posteriorbdX}
\tilde \pi ({\boldsymbol{y}}|{\boldsymbol{\obs}}) = \Phi({\mathcal U}^l(T_{a_0}(a({\boldsymbol{y}})))).
\end{equation}
With $b_{1,j}=\norm[L^\infty]{\psi_j}$, by the assumption
$${\boldsymbol{b}}_1=(b_{1,j})_{j\in\mathbb{N}}\in\ell^{p_1}({\mathbb N})\hookrightarrow \ell^1({\mathbb N})$$
which corresponds to assumption \ref{item:psi} of Proposition \ref{prop:bdX}. We now check assumptions \ref{item:uhol},
\ref{item:norma} and \ref{item:loclip} of Proposition \ref{prop:bdX}
for \eqref{eq:posteriorbdX}.
For every $l\in\mathbb{N}$:
\begin{enumerate}
\item\label{item:abc}
The map
\begin{equation*}
\Phi\circ {\mathcal U}^l\circ T_{a_0}:\begin{cases}
O_1 \to \mathbb{C}\\
a\mapsto \Phi({\mathcal U}(T_{a_0}(a)))
\end{cases}
\end{equation*}
is holomorphic as a composition of holomorphic functions.
\item for all $a\in O_1$, since
$\norm[H_0^1]{{\mathcal U}^l(T_{a_0}(a))}\le r$
\begin{equation*}
|\Phi({\mathcal U}^l(T_{a_0}(a)))|\le \exp((\norm[]{{\boldsymbol{\obs}}}+\norm[L(H_0^1({\domain};\mathbb{C});\mathbb{C}^m)]{{\boldsymbol{{\mathcal O}}}}r)^2\norm[]{{\boldsymbol{\Gamma}}^{-1}})
\end{equation*}
and thus assumption \ref{item:norma} of Proposition \ref{prop:bdX} is
trivially satisfied for some $\delta>0$ independent of $l$,
\item for all $a$, $b\in O_1$ by Lemma \ref{lemma:PhiLip} and the
same calculation as in \eqref{eq:lipschitz}
\begin{equation}\label{eq:uula-uulb}
\begin{aligned}
|\Phi({\mathcal U}^l(T_{a_0}(a)))-\Phi({\mathcal U}^l(T_{a_0}(b)))|&\le
K\norm[H_0^1]{{\mathcal U}^l(T_{a_0}(a))-{\mathcal U}^l(T_{a_0}(b))}
\\
& \le K
\frac{\norm[H^{-1}]{f}}{\rho(a_0)}\norm[L^\infty]{a-b},
\end{aligned}
\end{equation}
where $K$ is the constant given as in \eqref{ConstantK}.
\end{enumerate}
Now we can apply Proposition \ref{prop:bdX} to conclude that there
exist $\xi_1$, $\delta_1$ (independent of $l$) such that
$\tilde\pi^l(\cdot|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_1,\xi_1,\delta_1,H_0^1)$-holomorphic for every
$l\in\mathbb{N}$. Similarly one shows that $\tilde\pi(\cdot|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_1,\xi_1,\delta_1,H_0^1)$-holomorphic, and in particular
$\tilde\pi(\cdot|{\boldsymbol{\obs}})-\tilde\pi^l(\cdot|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_1,\xi_1,2\delta_1,H_0^1)$-holomorphic.
{\bf Step 2.} Set
$$O_2=\set{a\in W^{1}_\infty({\domain})\cap{\mathcal W}^{s-1}_\infty({\domain})}{\rho(a)>0}.$$
We verify once more assumptions \ref{item:uhol}, \ref{item:norma}
and \ref{item:loclip} of Proposition \ref{prop:bdX} with ``$E$'' in
this lemma being $W^{1}_\infty({\domain})\cap{\mathcal W}^{s-1}_\infty({\domain})$. With
$b_{2,j}=\max\{\norm[W^{1}_\infty]{\psi_j},\norm[{\mathcal W}^{s-1}_\infty]{\psi_j}\}$,
by the assumption
$$4{\boldsymbol{b}}_2=(b_{2,j})_{j\in\mathbb{N}}\in\ell^{p_2}({\mathbb N})\hookrightarrow \ell^1({\mathbb N}),$$
which corresponds to assumption \ref{item:psi} of Proposition \ref{prop:bdX}.
We will apply Proposition \ref{prop:bdX} with the function
\begin{equation}\label{eq:pit-pitl}
\tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})-
\tilde \pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})
=\Phi({\mathcal U}(T_{a_0}(a({\boldsymbol{y}}))))-\Phi({\mathcal U}^l(T_{a_0}(a({\boldsymbol{y}})))).
\end{equation}
For every $l\in\mathbb{N}$:
\begin{enumerate}
\item By item \ref{item:abc}
in
Step 1 (and because $O_2\subseteq O_1$)
\begin{equation*}
\Phi\circ {\mathcal U}\circ T_{a_0}-\Phi\circ{\mathcal U}^l\circ T_{a_0}:\begin{cases}
O_2 \to \mathbb{C}\\
a\mapsto \Phi({\mathcal U}(T_{a_0}(a)))-\Phi({\mathcal U}^l(T_{a_0}(a)))
\end{cases}
\end{equation*}
is holomorphic,
\item for every $a\in O_2$, by Lemma \ref{lemma:femapprox}
\begin{equation*}
\norm[H_0^1]{{\mathcal U}(T_{a_0}(a))-{\mathcal U}^l(T_{a_0}(a))}\le \sw{l}^{-\conv} CC_
\frac{(\norm[W^{1}_\infty]{a_0+a}+\norm[{\mathcal W}^{s-1}_{\infty}]{a_0+a})^{N_s+1}}{\rho(a_0+a)^{N_s+2}}
\norm[{\mathcal K}^{s-2}_{\varkappa-1}]{f}.
\end{equation*}
Thus by \eqref{eq:Uuboundr} and Lemma \ref{lemma:PhiLip}
\begin{equation*}\label{eq:PhiUuSa}
|\Phi({\mathcal U}(T_{a_0}(a)))-\Phi({\mathcal U}^l(T_{a_0}(a)))|
\le \sw{l}^{-\conv} KCC_s
\frac{(\norm[W^{1}_\infty]{a_0+a}+\norm[{\mathcal W}^{s-1}_{\infty}]{a_0+a})^{N_s+1}}{\rho(a_0+a)^{N_s+2}}\norm[{\mathcal K}^{s-2}_{\varkappa-1}]{f},
\end{equation*}
\item for all $a$, $b\in O_2\subseteq O_1$ by \eqref{eq:uula-uulb}
(which also holds for ${\mathcal U}^l$ replaced by ${\mathcal U}$):
\begin{equation*}
|\Phi({\mathcal U}(T_{a_0}(a)))-\Phi({\mathcal U}^l(T_{a_0}(a)))-(\Phi({\mathcal U}(b))-\Phi({\mathcal U}^l(b)))|\le 2 K \frac{\norm[H^{-1}]{f}}{\rho(a_0)}\norm[L^\infty]{a-b}.
\end{equation*}
\end{enumerate}
By Proposition \ref{prop:bdX} and \eqref{eq:pit-pitl} we conclude that
there exists $\delta>0$ and $\xi_2$ independent of $l$ such that
$\tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})-\tilde \pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})$ is
$({\boldsymbol{b}}_2,\delta \sw{l}^{-\alpha},\xi_2,H_0^1)$-holomorphic.
\end{proof}
Items \ref{item:u-ulhol1} and \ref{item:u-ulhol2} of
Proposition~\ref{prop:u-ul} show that Assumption \ref{ass:FEM}
implies validity of Assumption \ref{ass:ml}. This in turn allows us
to apply Theorem~\ref{thm:mlint} and Theorem~\ref{thm:mlquad}.
Specifically, assuming the optimal convergence rate
\eqref{eq:optimalrate}, we obtain that for every $n\in {\mathbb N}$ there is
$\varepsilon:=\varepsilon_n>0$ such that $\mathrm{work}(\mathbf{l}_{\varepsilon}) \le n$ and the
multilevel interpolant $\mathbf{I}^{\rm ML}_{\mathbf{l}}$ defined in \eqref{eq:VIml}
satisfies
\begin{equation*}
\norm[L^2(U,X;\gamma)]{\tilde \pi(\cdot|{\boldsymbol{\obs}})-\mathbf{I}_{\mathbf{l}_\varepsilon}^{\rm ML}
\tilde \pi(\cdot|{\boldsymbol{\obs}})}\le C(1+\log n) n^{-R_I},\quad
R_I = \min\left\{\frac{s-1}{2},
\frac{\frac{s-1}{2}(\frac{1}{p_1}-\frac{3}{2})}{\frac{s-1}{2}+\frac{1}{p_1}-\frac{1}{p_2}}
\right\}.
\end{equation*}
Of higher practical interest is the application of the multilevel
quadrature operator $\mathbf{Q}^{\rm ML}$ defined in \eqref{eq:VQml}. In case the
prior is chosen as $\gamma$, then
$$\int_U\tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}})$$ equals the
normalization constant in \eqref{eq:BayesDens}. It can be
approximated with the error converging like
\begin{equation}\label{eq:posteriorconv}
\left|\int_U\tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}})-\mathbf{Q}_{\mathbf{l}_\varepsilon}^{\rm ML}u\right|
\le C(1+\log n)n^{-R_Q},\quad
R_Q:=
\min\left\{\frac{s-1}{2},
\frac{\frac{s-1}{2}(\frac{2}{p_1}-\frac{5}{2})}{\frac{s-1}{2}+\frac{2}{p_1}-\frac{2}{p_2}}
\right\}.
\end{equation}
Typically, one is not merely interested in the
constant $$Z=\int_U\tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}}),$$ but for
example in an estimate of the $j$th parameter $y_j$ given as the
conditional expectation, which up to multiplying with the
normalization constant $\frac{1}{Z}$, corresponds to
$$\int_U y_j \tilde \pi({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}}).$$ Since
${\boldsymbol{y}}\mapsto y_j$ is analytic, one can show the same convergence rate
as in \eqref{eq:posteriorconv} for the multilevel quadrature applied
with the approximations ${\boldsymbol{y}}\mapsto y_j\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})$ for
$l\in\mathbb{N}$. Moreover, for example if $\phi:H_0^1({\domain};\mathbb{C})\to \mathbb{C}$ is a
bounded linear functional representing some quantity of interest, then
we can show the same error convergence for the approximation of
$$\int_U \phi(u({\boldsymbol{y}}))\tilde\pi({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}})$$ with the
multilevel quadrature applied with the approximations
$\phi(u^l({\boldsymbol{y}}))\tilde\pi^l({\boldsymbol{y}}|{\boldsymbol{\obs}})\,\mathrm{d}\gamma({\boldsymbol{y}})$ to the integrand
for $l\in\mathbb{N}$.
\newpage
\
\newpage
\section{Conclusions}
\label{sec:Concl}
We established holomorphy of parameter-to-solution maps
$$E \ni a \mapsto u = {\mathcal U}(a) \in X$$
for linear, elliptic, parabolic, and other PDEs in various scales of
function spaces $E$ and $X$, including in particular standard
and corner-weighted Sobolev spaces.
Our discussion focused on non-compact parameter domains which arise
from uncertain inputs from function spaces expressed in a suitable
basis with Gaussian distributed coefficients. We introduced and used
a form of quantified, parametric holomorphy
in products of strips to show that this implies summability results of
Wiener-Hermite coefficient sequences in PC representations of
such infinite parametric functions.
Specifically,
we proved weighted $\ell^2$-summability and
$\ell^p$-summability results for
Wiener-Hermite PC expansions
of
certain
parametric, deterministic solution families
$\{ u({\boldsymbol{y}}): {\boldsymbol{y}}\in U \} \subset X$, for a given ``log-affine''
parametrization \eqref{eq:CoeffAffin} of admissible random input data
$a\in E$.
We introduced and analyzed constructive, deterministic, sparse-grid
(``stochastic collocation'') algorithms based on univariate
Gauss-Hermite points, to efficiently sample the parametric,
deterministic solutions in the possibly infinite-dimensional parameter
domain $U = {\mathbb R}^\infty$. The summability of PC coefficient sequences was
shown to entail corresponding convergence rates of the presently
developed sparse-grid sampling schemes. In combination with suitable
Finite Element discretizations in the physical, space(-time) domain
(which include proper mesh-refinements to account for singularities in
the physical domain) we proved convergence rates for abstract,
multilevel algorithms which employ different combinations of
sparse-grid interpolants in the parametric domain with space(-time)
discretizations at different levels of accuracy in the physical domain.
The presently developed, abstract holomorphic setting was also shown
to apply to the corresponding Bayesian inverse problems subject to
PDE constraints: here, the density of the Bayesian posterior with
respect to a Gaussian random field prior was shown to generically
inherit quantified holomorphy from the parametric forward problem,
thereby facilitating the use of the developed sparse-grid collocation
and integration algorithms also for the efficient deterministic
computation of Bayesian estimates of PDEs with uncertain inputs,
subject to noisy observation data.
Our
approximation rate results are free from the curse-of-dimensionality
and only limited by their PC coefficient summability. They will
therefore also be relevant for convergence rate analyses of other
approximation schemes, such as Gaussian process emulators or neural
networks (see, e.g., \cite{StTeckGPAppr2018,
dung2021deep,dung2021collocation,SZ21_2946} and references there).
\bigskip
\noindent {\bf Acknowledgments.} The work of Dinh D\~ung and Van Kien
Nguyen is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under Grant No. 102.01-2020.03. A
part of this work was done when Dinh D\~ung was working at
Forschungsinstitut f\"ur Mathematik (FIM) of the ETH Z\"urich invited
by Christoph Schwab, and when Dinh D\~ung and Van Kien Nguyen were
working at the Vietnam Institute for Advanced Study in Mathematics
(VIASM). They would like to thank the FIM and VIASM for providing a
fruitful research environment and working condition. Dinh D\~ung
thanks Christoph Schwab for invitation to visit the FIM and for his
hospitality.
\newpage
\
\newpage
\bibliographystyle{amsplain}
\section{Preliminaries}
\label{S:Prelim}
A key technical ingredient in the analysis of numerical approximations
of PDEs with GRF inputs from function spaces, and of numerical methods
for their efficient numerical treatment are constructions and
numerical approximations of GRFs on real Hilbert and Banach spaces.
Due to their high relevance in many areas of science (theoretical
physics, quantum field theory, spatial and high-dimensional
statistics, etc.), a rich theory has been developed in the past
decades and a large body of literature is available now. We
recapitulate basic definitions and key results, in particular on GMs,
that are necessary for the ensuing developments. We do not attempt to
provide a comprehensive survey. We require the exposition on GMs
$\gamma$ on real-valued Hilbert and Banach spaces, as most PDEs of
interest are formulated for real-valued inputs and solutions.
However, we crucially use in the ensuing sections of this note
analytic continuation of parametric representations to the complex
parameter domain. This is required in order to bring to bear complex
variable methods for derivative-free, sharp bounds on Hermite
expansion coefficients of GRFs. Therefore, we develop in our
presentation all aspects also for Hilbert and Banach spaces of
complex-valued fields.
The structure of this section is as follows. In Section
\ref{S:FinDimGM}, we recapitulate GMs on finite dimensional spaces, in
particular on $\mathbb{R}^d$ and $\mathbb{C}^d$. In Section \ref{S:GMSepHS}, we
extend to separable Banach spaces. Section \ref{S:CamMart} reviews
the Cameron-Martin space. In Section \ref{sec-Gaussian product
measures} we recall a notion of Gaussian product measures on a
Cartesian product of locally convex spaces. Section \ref{S:GSer} is
devoted to a summary of known representations of a GRF by a Gaussian
series. A key concept in these and more general spaces is the concept
of \emph{Parseval frame} which we introduce.
In Section \ref{S:FEM} we recapitulate, from
\cite{BaPi79,BNZPolygon,GasMorFESing09}, (known) technical results on
approximation properties of Lagrangian Finite Elements (FEs for short)
in polygonal and polyhedral domains ${\mathrm D}\subset \mathbb{R}^d$, on regular,
simplicial partitions of ${\mathrm D}$ with local refinement towards
corners (and, in space dimension $d=3$, towards edges). These will be
used in Section \ref{sec:StochColl} in conjunction with collocation
approximations in the parameter space of the GRF to build
deterministic numerical approximations of solutions in polygonal and
in polyhedral domains.
\subsection{Finite dimensional Gaussian measures}
\label{S:FinDimGM}
\subsubsection{Univariate Gaussian measures}
\label{S:1dGM}
In dimension $d=1$, for every $\mu,\sigma\in \mathbb{R}$, there holds the
well-known identity
\[
\frac{1}{\sigma \sqrt{2\pi}} \int_{\mathbb{R}} \exp\left(-
\frac{(y-\mu)^2}{2\sigma^2}\right) \rd y = 1\;.
\]
A Borel probability measure $\gamma$ on $\mathbb{R}$ is \emph{Gaussian} if it
is either a Dirac measure $\delta_\mu$ at $\mu\in \mathbb{R}$ or it has its
density with respect to Lebesgue measure $\lambda$ on $\mathbb{R}$ given by
\begin{equation*}\label{eq:GdnsR1}
\frac{\rd\gamma}{\rd\lambda} = p(\cdot;\mu,\sigma^2)\;,\;\;
p(\cdot;\mu,\sigma^2)
:=
y\mapsto \frac{1}{\sigma\sqrt{2\pi}}
\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right)
\;.
\end{equation*}
We shall refer to $\mu$ as \emph{mean}, and to $\sigma^2$ as
\emph{variance} of the GM $\gamma$. The case that
$\gamma = \delta_\mu$ is understood to correspond to $\sigma = 0$. If
$\sigma > 0$, we shall say that \emph{the GM $\gamma$ is
nondegenerate}. Unless explicitly stated otherwise, we assume GMs
to be nondegenerate.
For $\mu=0$ and $\sigma = 1$, we shall refer to the GM $\gamma_1$ as
\emph{the standard GM on $\mathbb{R}$}. A GM with $\mu = 0$ is called
\emph{centered} (or also \emph{symmetric}). There holds
\begin{equation*}\label{eq:musigmaR1}
\mu = \int_{\mathbb{R}} y \rd \gamma_1(y),\quad \sigma^2 = \int_{\mathbb{R}} (y-\mu)^2 \rd\gamma_1( y).
\end{equation*}
Let $(\Omega,{\mathcal A},\mathbb{P})$ be a probability space with sample space
$\Omega$, $\sigma$-fields ${\mathcal A}$, and probability measure $\mathbb{P}$. A
\emph{Gaussian random variable} (``Gaussian RV'' for short)
$\eta : \Omega\to \mathbb{R}$ is a RV whose law is Gaussian, i.e., it admits a
Gaussian distribution. If $\eta$ is a Gaussian RV with mean $\mu$ and
variance $\sigma^2$ we write $\eta \sim \mathcal{N}(\mu,\sigma^2)$.
Linear transformations of Gaussian RVs are Gaussian: every Gaussian RV
$\eta$ can be written as $\eta = \sigma\xi + \mu$, where $\xi$ is a
standard Gaussian RV, i.e., a Gaussian RV whose law is a standard GM
on $\mathbb{R}$.
The Fourier transformation of a GM $\gamma_1$ on $\mathbb{R}$ is defined, for
every $\xi \in \mathbb{R}$, as
\begin{equation*}\label{eq:FTGMR1}
\hat{\gamma_1}(\xi) := \int_{\mathbb{R}} \exp({\rm i} \xi y) \rd\gamma_1(y)
= \exp\left( {\rm i} \mu \xi - \frac{1}{2} \sigma^2 \xi^2 \right)
\;.
\end{equation*}
We denote by $\Phi$ the distribution function of $\gamma_1$. For the
standard normal distribution
$$
\Phi(t) = \int_{-\infty}^t p(s;0,1)\rd s \qquad\forall t\in \mathbb{R}.
$$
With the convention $\Phi^{-1}(0) := -\infty$,
$\Phi^{-1}(1) := +\infty$, the inverse function $\Phi^{-1}$ of $\Phi$
is defined on $[0,1]$.
\subsubsection{Multivariate Gaussian measures}
\label{S:ddimGM}
Consider now a finite dimension $d > 1$. A Borel probability measure
$\gamma$ on $(\mathbb{R}^d,\cB(\mathbb{R}^d))$ is called Gaussian if for every
$f\in \cL(\mathbb{R}^d,\mathbb{R})$ the measure $\gamma\circ f^{-1}$ is a GM on $\mathbb{R}$,
where as usually, $\cB(\mathbb{R}^d)$ denotes the $\sigma$-field on $\mathbb{R}^d$.
Since $d$ is finite, we may identify $\cL(\mathbb{R}^d,\mathbb{R})$ with $\mathbb{R}^d$, and
we denote the Euclidean inner product on $\mathbb{R}^d$ by $(\cdot,\cdot)$.
The Fourier transform of a Borel measure $\gamma$ on $\mathbb{R}^d$ is given
by
$$
\hat{\gamma}: \mathbb{R}^d \to \mathbb{C}: \hat{\gamma}({\boldsymbol{\xi}}) = \int_{\mathbb{R}^d}
\exp\left({\rm i}({\boldsymbol{\xi}},{\boldsymbol{y}})\right) \rd \gamma( {\boldsymbol{y}})\;.
$$
We write $\gamma_d$ for a GM on $\mathbb{R}^d$. The Fourier transform
$\hat{{\gamma}_d}$ uniquely determines $\gamma_d$.
\begin{proposition}[{\cite[Proposition 1.2.2]{Bogach98}}]\label{prop:FTGM}
A Borel probability measure
$\gamma$ on $\mathbb{R}^d$ is Gaussian iff
$$
\hat{\gamma}({\boldsymbol{\xi}}) = \exp\left( {\rm i} ({\boldsymbol{\xi}},{\boldsymbol{\mu}})
-\frac{1}{2}(\boldsymbol{K}{\boldsymbol{\xi}},{\boldsymbol{\xi}})\right), \quad {\boldsymbol{\xi}}\in \mathbb{R}^d\;.
$$
Here, ${\boldsymbol{\mu}}\in \mathbb{R}^d$ and $\boldsymbol{K}\in \mathbb{R}^{d\times d}$ is a
symmetric positive semidefinite matrix.
We shall say that a GM $\gamma_d$ on $\mathbb{R}^d$ has a density with respect
to Lebesgue measure $\lambda^d$ iff the matrix $\boldsymbol{ K}$ is
nondegenerate. Then, this density is given by
$$
\frac{\rd\gamma_d}{\rd\lambda^d}({\boldsymbol{x}}): {\boldsymbol{x}}\mapsto
\frac{1}{\sqrt{(2\pi)^d \det \boldsymbol{ K}}} \exp\left( -\frac{1}{2}
(\boldsymbol{ K}^{-1} ({\boldsymbol{x}}-{\boldsymbol{\mu}}), {\boldsymbol{x}}-{\boldsymbol{\mu}}) \right) \;.
$$
Furthermore,
$$
{\boldsymbol{\mu}} = \int_{\mathbb{R}^d} {\boldsymbol{y}} \rd\gamma_d({\boldsymbol{y}}) , \quad \forall {\boldsymbol{y}},{\boldsymbol{y}}'\in
\mathbb{R}^d: (\boldsymbol{ K}{\boldsymbol{y}},{\boldsymbol{y}}') = \int_{\mathbb{R}^d}
({\boldsymbol{y}},{\boldsymbol{x}}-{\boldsymbol{\mu}})({\boldsymbol{y}}',{\boldsymbol{x}}-{\boldsymbol{\mu}})\rd\gamma_d({\boldsymbol{x}})\;.
$$
The symmetric linear operator $\cC\in \cL(\mathbb{R}^d,\mathbb{R}^d)$ defined by the
later relation and represented by the symmetric positive definite
matrix $\boldsymbol{K}$ is the \emph{covariance operator} associated
to the GM $\gamma_d$ on $\mathbb{R}^d$.
\end{proposition}
When we do not need to distinguish between the covariance operator
$\cC$ and the covariance matrix $K$, we simply speak of ``the
covariance'' of GM $\gamma_d$.
If a joint probability distribution of RVs $y_1,\ldots,y_d$ is a GM on
$\mathbb{R}^d$ with mean vector ${\boldsymbol{\mu}}$ and covariance matrix $\boldsymbol{K}$
we write $(y_1,\ldots,y_d)\sim \mathcal{N}({\boldsymbol{\mu}},\boldsymbol{ K})$.
In the following, $\gamma_d$ denotes the standard GM on $\mathbb{R}^d$.
Denote by $L^2({\mathbb R}^d,\gamma_d)$ the Hilbert space of all
$\gamma_d$-measurable real-valued functions $f$ on $\mathbb{R}^d$ such that
\begin{equation}\nonumber
\|f\|_{L^2({\mathbb R}^d,\gamma_d)} := \left( \int_{\mathbb{R}^d} |f|^2 \rd \gamma_d\right)^{1/2} < \infty.
\end{equation}
The corresponding inner product is denoted by
$(\cdot,\cdot)_{L^2({\mathbb R}^d,\gamma_d)}$.
\subsubsection{Hermite polynomials}
\label{S:HerPol}
A key role in the ensuing sparsity analysis of parametric solution
families is taken by Wiener-Hermite PC expansions. We consider GRF
inputs and, accordingly, will employ polynomial systems on ${\mathbb R}$ which
are orthogonal with respect to the GM $\gamma_1$ on ${\mathbb R}$, the
so-called Hermite polynomials, as pioneered for the analysis of GRFs
by N. Wiener in \cite{Nwiener}. To this end, we recapitulate basic
definitions and properties, in particular the various normalizations
which are met in the literature. Particular attention will be paid to
estimates for Hermite coefficients of functions which are holomorphic
in a strip, going back to Einar Hille in \cite{EHilleII}.
\begin{definition}\label{def:HermPol}
For $k\in \mathbb{N}_0$, the normalized probabilistic Hermite polynomial
$H_k$ of degree $k$ on $\mathbb{R}$ is defined by
\begin{equation}\label{eq:Hk}
H_k(x)
:=
\frac{(-1)^k}{\sqrt{k!}}
\exp\left(\frac{x^2}{2}\right) \frac{\rd^k}{\rd x^k} \exp\left(-\frac{x^2}{2}\right) .
\end{equation}
For every multi-degree ${\boldsymbol{\nu}}\in \mathbb{N}_0^d$, the $d$-variate Hermite
polynomial $H_{\boldsymbol{\nu}}$ is defined by
\begin{equation*}\label{eq:Hnu}
H_{\boldsymbol{\nu}}(x_1,\ldots,x_d) :=\prod_{j=1}^d H_{\nu_j}(x_j),
\;\; x_j \in \mathbb{R}, \; j=1,\ldots ,d\;.
\end{equation*}
\end{definition}
\begin{remark}[Normalizations of Hermite
polynomials and Hermite functions]\label{rmk:HermNorml}
\begin{enumerate}
\item Definition \eqref{eq:Hk} provides for every $k\in {\mathbb N}_0$ a
polynomial of degree $k$. The scaling factor in \eqref{eq:Hk} has
been chosen to ensure normalization with respect to GM $\gamma_1$, see
also Lemma \ref{lem:HkProp}, item i).
\item Other normalizations with at times the same notation are used.
The ``classical'' normalization of $H_k$ we denote by
$\tilde{H}_k(x)$. It is defined by (see, e.g., \cite[Page 787]{AS},
and compare \eqref{eq:Hk} with \cite[Eqn. (5.5.3)]{szego})
\[
\tilde{H}_k(x/\sqrt{2}) := 2^{k/2} \sqrt{k!} H_k(x).
\]
\noindent
\item In \cite{BNT07}, so-called ``\emph{normalized Hermite
polynomials}'' are introduced as
\begin{equation*}\label{eq:NormHermPol}
\tilde{\tilde{H}}_k(x)
:=
[\pi^{1/2} 2^k k!]^{1/2} (-1)^k e^{x^2} \frac{\rd^k}{\rd x^k} e^{-x^2}
\;.
\end{equation*}
The $\tilde{\tilde{H}}_k$ are an orthonormal basis (ONB for short) for
$L^2({\mathbb R},\tilde{\tilde{\gamma}})$ with the weight
$\tilde{\tilde{\gamma}} = \exp(-x^2) \rd x$, i.e., (compare, e.g.,
\cite[Eqn. (5.5.1)]{szego})
$$
\int_{{\mathbb R}} \tilde{\tilde{H}}_n(x) \tilde{\tilde{H}}_{n'}(x)
\exp(-x^2)\rd x = \delta_{nn'}, \;\; n,n'\in {\mathbb N}_0 \;.
$$
\item With the Hermite polynomials $\tilde{\tilde{H}}_k$, in
\cite{EHilleII} \emph{Hermite functions} are introduced for
$k\in {\mathbb N}_0$ as
\begin{equation*}\label{eq:HermFncs}
h_k(x):= \exp(-x^2/2) \tilde{\tilde{H}}_k(x) \;,\quad x\in {\mathbb R}\;.
\end{equation*}
\item It has been shown in \cite[Theorem 1]{EHilleII} that in order for
functions $f:{\mathbb C} \to {\mathbb C}$ defined in the strip
$S(\rho) := \{z\in {\mathbb C}: z=x+{\rm i} y, \; x\in {\mathbb R},\; |y|< \rho\}$ to
admit a Fourier-Hermite expansion
\begin{equation*}\label{eq:FourHermite}
\sum_{n=0}^\infty f_n h_n(z) ,\qquad f_n := \int_{{\mathbb R}} f(x)h_n(x) \rd x = \int_{{\mathbb R}} f(x) \tilde{\tilde{H}}_n(x) \exp(-x^2)\rd x
\end{equation*}
which converges to $f(z)$ for $z\in S(\rho)$ a necessary and
sufficient condition is that a) $f$ is holomorphic in
$S(\rho)\subset {\mathbb C}$ and b) for every $0<\rho'<\rho$ there exists a
finite bound $B(\rho')$ such that
\begin{equation*}\label{eq:fHermBd}
|f(x+{\rm i} y)| \leq B(\rho') \exp[-|x|(\beta^2-y^2)^{1/2}]\;,\quad x\in {\mathbb R}, |y|\leq \rho' \;.
\end{equation*}
There is a constant $C(f)>0$ such that for the Fourier-Hermite
coefficients $f_n$, holds
\begin{equation*}\label{eq:fnBd}
|f_n| \leq C\exp(-\rho\sqrt{2n+1}) \quad \forall n\in {\mathbb N}_0.
\end{equation*}
\end{enumerate}
\end{remark}
We state several basic properties of the Hermite polynomials $H_k$
defined in \eqref{eq:Hk}.
\begin{lemma}\label{lem:HkProp}
The collection $(H_k)_{k\in {\mathbb N}_0}$ of Hermite polynomials
\eqref{eq:Hk} in $\mathbb{R}$ has the following properties.
\begin{enumerate}
\item $(H_k)_{k\in {\mathbb N}_0}$ is an ONB of the space
$L^2(\mathbb{R},\gamma_1)$.
\item for every $k\in {\mathbb N}$ holds:
$H_k'(x) = \sqrt{k}H_{k-1}(x) = H_k(x) -\sqrt{k+1}H_{k+1}(x)$.
\item for all $x_1,\ldots ,x_d \in \mathbb{R}$ holds
\[
\prod_{i=1}^d\sqrt{k_i!} H_{k_i}(x_i) =
\frac{\partial^{k_1+\ldots +k_d}}{\partial t_1^{k_1}\ldots
\partial t_d^{k_d}} \exp\left( \sum_{i=1}^d t_ix_i
-\frac{1}{2}\sum_{i=1}^d t_i^2 \right)\mid_{t_1 = \ldots = t_d
= 0} \;.
\]
\item for every $f\in C^\infty(\mathbb{R})$ such that
$f^{(k)}\in L^2(\mathbb{R},\gamma_1)$ for all $k\in {\mathbb N}_0$ holds
\[
I_k(f) := \int_{\mathbb{R}} fH_k \rd\gamma_1 = \frac{(-1)^k}{\sqrt{k!}}
\int_{\mathbb{R}} f^{(k)} \rd\gamma_1 \;,
\]
and, hence, in $L^2(\mathbb{R},\gamma_1)$,
\[
f = \sum_{k\in {\mathbb N}_0} \frac{(-1)^k}{\sqrt{k!}}
(f^{(k)},1)_{L^2(\mathbb{R},\gamma_1)} H_k \;.
\]
\end{enumerate}
\end{lemma}
It follows from item (i) of this lemma in particular that
\begin{equation*}\label{eq:HnuONB}
\left\{ H_{\boldsymbol{\nu}}: {\boldsymbol{\nu}} \in \mathbb{N}_0^d \right\} \;\mbox{is an ONB of} \; L^2(\mathbb{R}^d,\gamma_d)\;.
\end{equation*}
Denote for $k\in \mathbb{N}_0$ and $d\in \mathbb{N}$ by ${\mathcal H}_k$ the space of
$d$-variate Hermite polynomials which are homogeneous of degree $k$,
ie.,
\[
{\mathcal H}_k := {\rm span} \left\{ H_{\boldsymbol{\nu}}: {\boldsymbol{\nu}} \in \mathbb{N}_0^d, |{\boldsymbol{\nu}}| = k
\right\} \;.
\]
Then ${\mathcal H}_k$ (``homogeneous PC of degree $k$'' \cite{Nwiener}) is a
closed, linear subspace of $L^2(\mathbb{R}^d,\gamma_d)$ and
\[
L^2(\mathbb{R}^d,\gamma_d) = \bigoplus_{k\in {\mathbb N}_0} {\mathcal H}_k
\;\;\mbox{in}\;\;L^2(\mathbb{R}^d,\gamma_d)\;.
\]
\subsection{Gaussian measures on separable locally convex spaces}
\label{S:GMSepHS}
An important mathematical ingredient in a number of applications, in
particular in UQ, Bayesian PDE inversion, risk analysis, but also in
statistical learning theory applied to input-output maps for PDEs, is
the construction of measures on function spaces. A particular
interest is on GMs on separable on Hilbert or Banach or, more
generally, on locally convex spaces of uncertain input data for PDEs.
Accordingly, we review constructions of such measures, in terms of
suitable \emph{bases of the input spaces}. This implies, in
particular, \emph{separability} of the spaces of admissible PDE inputs
or, at least, the uncertain input data being \emph{a separably-valued}
random element of otherwise nonseparable spaces (such as, e.g.,
$L^\infty({\domain})$) of valid inputs for the PDE of interest.
Let $(\Omega, \cA, \mu)$ be a measure space and $1 \le p <
\infty$. Recall that the normed space $L^p(\Omega,\mu)$ is defined as
the space of all $\mu$-measurable functions $u$ from $\Omega$ to $\mathbb{R}$
such that the norm
\begin{equation} \nonumber \|u\|_{L^p(\Omega,\mu)} := \
\left(\int_{\Omega} |u(x)|^p \, \rd \mu(x) \right)^{1/p} < \infty.
\end{equation}
Throughout this section, $X$ will denote a real, separable and locally
convex space with Borel $\sigma$-field $\cB(X)$ and with dual space
$X^*$.
\begin{example}\label{ex:R^infty}
Let ${\mathbb R}^\infty$ be the linear space of all sequences
${\boldsymbol{y}} = (y_j)_{j\in {\mathbb N}}$ with $y_j\in {\mathbb R}$. This linear space
becomes a locally convex space (still denoted by ${\mathbb R}^\infty$) equipped
with the topology generated by the countable family of semi-norms
\[
p_j({\boldsymbol{y}}):= |y_j|, \quad j \in \mathbb{N}.
\]
The locally convex space ${\mathbb R}^\infty$ is separable and complete and,
therefore, a Fr\'echet space. However, it is in general not
normable, and hence in general not a Banach space.
\end{example}
\begin{example}\label{ex:SepBsp}
Let ${\mathrm D}\subset \mathbb{R}^d$ be a bounded Lipschitz domain.
\begin{enumerate}
\item The Banach spaces $C({\mathrm D})$ and $L^1({\mathrm D})$ are
separable.
\item Spaces $C^s({\mathrm D})$ for $0<s<1$ are not separable. The
subspace $C^s_{\circ}({\mathrm D})$ of functions $u\in C^{s}({\mathrm D})$
for which
$$
\lim_{{\boldsymbol{x}}_1,{\boldsymbol{x}}_2\in {\mathrm D}, {\boldsymbol{x}}_1\to {\boldsymbol{x}}_2}
\frac{|u({\boldsymbol{x}}_1) - u({\boldsymbol{x}}_2)|}{|{\boldsymbol{x}}_1-{\boldsymbol{x}}_2|^s} = 0
$$
is separable, when endowed with the norm of
$C^s({\mathrm D})$.
\item The Sobolev spaces $W^{s}_p({\mathrm D})$ for
$s\geq 0$ and for $1\leq p < \infty$ are separable.
\item The Sobolev spaces $W^{s}_\infty({\mathrm D})$ for
$s\geq 0$ are complete, but not separable.
\end{enumerate}
\end{example}
We review and present constructions of GMs $\gamma$ on
$X$.
\subsubsection{Cylindrical sets}
\label{S:CylSet}
\emph{Cylindrical sets} are subsets of $X$ of the form
\begin{equation*}\label{eq:CylSet}
C = \left\{ x\in X: (l_1(x),\ldots ,l_n(x))\in C_0 : C_0 \in \cB(\mathbb{R}^n), l_i\in X^* \right\},
\;\mbox{for some}\; n\in \mathbb{N}\;.
\end{equation*}
Here, the Borel set $C_0 \in \cB(\mathbb{R}^n)$ is sometimes referred to as
\emph{basis of the cylinder $C$}. We denote by $\cE(X)$ the
$\sigma$-field generated by all cylindrical subsets of $X$. It is the
smallest $\sigma$-field for which all continuous linear functionals
are measurable. Evidently then $\cE(X)\subset \cB(X)$, with in
general strict inclusion (see, e.g., \cite[A.3.8]{Bogach98}). If,
however, $X$ is separable, then $\cE(X)=\cB(X)$ (\cite[Theorem
A.3.7]{Bogach98}).
Sets of the form
$$
\left\{ {\boldsymbol{y}}\in \mathbb{R}^\infty : (y_1,\ldots ,y_n)\in B, B \in \cB(\mathbb{R}^n),
n\in \mathbb{N} \right\}
$$
generate $\cB(\mathbb{R}^\infty)$ \cite[Lemma 2.1.1]{Bogach98}, and a set $C$
belongs to $\cB(X)$ iff it is of the form
$$
C = \left\{ x\in X: \, (l_1(x),\ldots ,l_n(x),\ldots ) \in B, \
\mbox{for}\ l_i\in X^*, B \in \cB(\mathbb{R}^\infty) \right\} \;,
$$
(see, e.g., \cite[Lemma 2.1.2]{Bogach98}).
\subsubsection{Definition and basic properties of Gaussian measures}
\label{S:DefGMX}
\begin{definition}[{\cite[Definition 2.2.1]{Bogach98}}]\label{def:GMX}
A probability measure $\gamma$ defined on the
$\sigma$-field $\cE(X)$ generated by $X^*$ is called Gaussian if,
for any $f\in X^*$ the induced measure $\gamma\circ f^{-1}$ on $\mathbb{R}$
is Gaussian. The measure $\gamma$ is centered or symmetric if all
measures $\gamma\circ f^{-1}$, $f\in X^*$ are centered.
Let $(\Omega, \cA,\mathbb{P})$ be a probability space. A random field $u$
taking values in $X$ (recall that throughout, $X$ is a separable
locally convex space) is a map $u:\Omega\to X$ such that
$$
\forall B\in {\mathcal B}(X):\;\; u^{-1}(B)\in {\mathcal A}\;.
$$
The \emph{law of the random field $u$} is the probability measure
${\mathfrak m }_u$ on $(X,{\mathcal B}(X))$ which is defined as
$$
{\mathfrak m }_u (B):= \mathbb{P}(u^{-1}(B)), \quad B \in {\mathcal B}(X)\;.
$$
The random field $u$ is said to be \emph{Gaussian} if its law is a GM
on $(X,{\mathcal B}(X))$.
\end{definition}
Images of GMs under continuous, affine transformations on $X$ are
Gaussian.
\begin{lemma}[{\cite[Lemma 2.2.2]{Bogach98}}]\label{lem:AffGM} Let $\gamma$ be a GM on $X$ and let $T:X\to Y$ be a linear
map to another locally convex space $Y$ such that $l\circ T \in X^*$
for all $l\in Y^*$. Then $\gamma\circ T^{-1}$ is a GM on $Y$.
This remains true for the affine map $x\mapsto Tx + \mu$ for some
$\mu\in Y$.
\end{lemma}
The \emph{Fourier transform} of a measure ${\mathfrak m }$ over $(X,\cB(X))$ is
given by
\begin{equation*}\label{eq:FTmeas}
\hat{{\mathfrak m }} : X^*\to \mathbb{C}: f\mapsto \hat{{\mathfrak m }}(f) := \int_X \exp\left({\rm i} f(x)\right) \rd{\mathfrak m }( x) \;.
\end{equation*}
\begin{theorem}[{\cite[Theorem 2.2.4]{Bogach98}}]\label{thm:FTGMX} A measure $\gamma$ on $X$ is Gaussian iff its Fourier
transform $\hat{\gamma}$ can be expressed with some linear
functional $L(\cdot )$ on $X^*$ and a symmetric bilinear form
$B(.,.)$ on $X^*\times X^*$ such that $f\mapsto B(f,f)$ is
nonnegative as
\begin{equation}\label{eq:FTGMX}
\forall f \in X^*:\quad
\hat{\gamma}(f) = \exp\left( {\rm i} L(f) - \frac{1}{2}B(f,f)\right).
\end{equation}
\end{theorem}
A GM $\gamma$ on $X$ is therefore characterized by $L$ and $B$. It
also follows from \eqref{eq:FTGMX} that a GM $\gamma$ on $X$ is
centered iff $\gamma(A) = \gamma(-A)$ for all $A\in \cB(X)$, i.e., iff
$L = 0$ in \eqref{eq:FTGMX}.
\begin{definition}\label{def:MeanCov}
Let ${\mathfrak m }$ be a measure on $\cB(X)$ such that
$X^*\subset L^2(X,{\mathfrak m })$. Then the element $a_{\mathfrak m }\in (X^*)'$ in
the algebraic dual $(X^*)'$ defined by
$$
{a}_{\mathfrak m }(f) := \int_X f(x) \rd{\mathfrak m }(x), \; f\in X^*
$$
is called \emph{mean of ${\mathfrak m }$}.
The operator $R_{\mathfrak m }:X^* \to (X^*)'$ defined by
$$
R_{\mathfrak m }(f)(g) := \int_X [f(x) - {a}_{\mathfrak m }(f)] [g(x) - {a}_{\mathfrak m }(g)]
{\rd{\mathfrak m }( x)}
$$
is called \emph{covariance operator} of ${\mathfrak m }$. The quadratic form on
$X^*$ is called \emph{covariance of ${\mathfrak m }$}.
\end{definition}
When $X$ is a real separable Hilbert space, one can say more.
\begin{definition}[Nuclear operators]\label{def:NucOp}
Let $H_1$, $H_2$ be real separable Hilbert spaces with the
norms $\| \circ \|_{H_1}$ and $\| \circ \|_{H_2}$, respectively, and
with corresponding inner products $(\cdot,\cdot)_{H_i}$, $i=1,2$.
A linear operator $K\in \cL(H_1,H_2)$ is called \emph{nuclear} or
\emph{trace class} if it can be represented as
$$
\forall u\in H_1: \quad Ku = \sum_{k\in {\mathbb N}} (u, x_{1k})_{H_1} x_{2k}
\; \mbox{in}\; H_2\;.
$$
Here, $( x_{ik})_{k\in {\mathbb N}}\subset H_i$, $i=1,2$ are such that
$\sum_{k\in {\mathbb N}} \| x_{1k} \|_{H_1} \| x_{2k} \|_{H_2} < \infty$.
\end{definition}
We denote by $\cL_1(H_1,H_2)\subset \cL(H_1,H_2)$ the space of all
nuclear operators. This is a separable Banach space when it is
endowed with \emph{nuclear norm}
\begin{equation*}\label{eq:NucNrm}
\| K \|_1
:=
\inf
\left\{
\sum_{k\in {\mathbb N}} \| x_{1k} \|_{H_1} \| x_{2k} \|_{H_2} \mid Ku = \sum_{k\in {\mathbb N}} (u, x_{1k})_{H_1} x_{2k}
\right\}
\end{equation*}
When $X = H_1 = H_2$, we also write $\cL_1(X)$.
\begin{proposition}[{\cite[Theorem 2.3.1]{Bogach98}}]\label{prop:GMHilb}
Let $\gamma$ be a GM on a separable Hilbert space $X$ with
innerproduct $(\cdot,\cdot)_X$, and let $X^*$ denote its dual,
identified with $X$ via the Riesz isometry.
Then there exist $\mu\in X$ and a symmetric, nonnegative nuclear
operator $K\in \cL_1(X)$ such that the Fourier transform
$\hat{\gamma}$ of $\gamma$ is
\begin{equation}\label{eq:GMHilb}
\hat{\gamma}:X\to \mathbb{C}: x\mapsto \exp\left( {\rm i} (\mu,x)_X - \frac{1}{2} (Kx,x)_X \right) \;.
\end{equation}
\end{proposition}
\begin{remark}\label{rmk:GMH}
Consider that $X$ is a real, separable Hilbert space with
innerproduct $(\cdot,\cdot)_X$ and assume given a GM $\gamma$ on
$X$.
\begin{enumerate}
\item In \eqref{eq:GMHilb}, $K\in \cL(X)$ and $\mu\in X$ are
determined by
$$
\forall u,v\in X: (\mu,v)_X = \int_X (x,v)_X \rd \gamma(x), \quad
(Ku,v)_X = \int_X (u,x-\mu)_X (v,x-\mu)_X \rd \gamma(x) \;.
$$
The closure of $X=X^*$ in $L^2(X,\gamma)$ then equals the completion
of $X$ with respect to the norm
$x\mapsto \| K^{1/2}x\|_X = \sqrt{(Kx,x)_X}$. Let $(e_n)_{n\in {\mathbb N}}$
denote the ONB of $X$ formed by eigenvectors of $K$, with
corresponding real, non-negative eigenvalues $k_n \in {\mathbb N}_0$, i.e.,
$Ke_n = k_ne_n$ for $n=1,2,\ldots $. Then the completion can be
identified with the weighted sequence (Hilbert) space
$$
\left\{ (x_n)_{n\in {\mathbb N}} : \sum_{n\in {\mathbb N}} k_n x_n^2 <\infty \right\}
\;.
$$
The nuclear operator $K$ is the \emph{covariance of the GM $\gamma$ on
the Hilbert space $X$}.
\item In coordinates ${\boldsymbol{y}} = (y_j)_{j\in {\mathbb N}} \in \ell^2(\mathbb{N})$
associated to the ONB $(e_n)_{n\in {\mathbb N}}$ of $X$, \eqref{eq:GMHilb}
takes the form
$$
\hat{\gamma}: \ell^2(\mathbb{N}) \to \mathbb{C}: {\boldsymbol{y}} \mapsto \exp\left( {\rm i} \sum_{n\in
{\mathbb N}} a_n y_n - \frac{1}{2} \sum_{n\in {\mathbb N}} k_n y_n^2 \right)\;.
$$
\item Consider $a=0\in X$ and, for finite $n\in \mathbb{N}$, a cylindrical set
$C = P_n^{-1}(B)$ with $P_n$ denoting the orthogonal projection onto
$X_n := {\rm span}\{e_j: j=1,\ldots ,n\}\subset X$, and with
$B\in \cB(X_n)$. Then
$$
\gamma(C) = \int_B \prod_{j=1}^n (2\pi k_j)^{-1/2}
\exp\left(-\frac{1}{2k_j} y_j^2 \right)\rd y_1\ldots \rd y_n \;.
$$
\end{enumerate}
\end{remark}
For $f\in X^*$ and $x\in X$, one frequently writes the $X^*\times X$
duality pairing as
\begin{equation*}\label{eq:<f,x>}
f(x) = \langle f,x \rangle\;.
\end{equation*}
With the notation from Definition \ref{def:MeanCov}, the covariance
operator $C_g = R_{\gamma_g}$ in Definition \ref{def:MeanCov} of a
\emph{centered}, Gaussian random vector
$g:(\Omega,\cA,\gamma_g) \to X$ with Gaussian law $\gamma_g$ on a
separable, real Banach space $X$ admits the representations
\begin{equation*}\label{eq:CovKer}
R_{\gamma_g} = C_g: X^*\to X: C_g \varphi:= \mathbb{E}\langle \varphi,g \rangle g,
\quad
C_g: X^*\times X^*\to \mathbb{R}: (\psi,\varphi) \mapsto \langle \psi, C_g \varphi \rangle\;.
\end{equation*}
\subsection{Cameron-Martin space}
\label{S:CamMart}
Let $X$ be a real separable locally convex space and $\gamma$ a GM on
$\cE(X)$ such that $X^* \subset L^2(X,\gamma)$. Then, for every
$\varphi\in X^*$, the image measure $\varphi(\gamma)$ is a GM on $\mathbb{R}$.
By \cite[Theorem 3.2.3]{Bogach98}, there exists a unique
$a_\gamma \in X$, the mean of $\gamma$, such that
$$
\forall \varphi\in X^*: \quad \varphi(a_\gamma) = \int_X \varphi(h)
\rd\gamma(h)\;.
$$
Denote by $X^*_\gamma$ the closure of the set
$\{\varphi - \varphi({a_\gamma)}, \, \varphi \in X^*) \}$ embedded
into the normed space $L^2(X,\gamma)$ w.r.t. its norm.
The covariance operator, $R_\gamma$, of $\gamma$ is formally given by
\begin{equation}\label{eq:Cov}
\forall \varphi,\psi\in X^*:\quad
\langle R_\gamma \varphi,\psi \rangle
=
\int_X \varphi(h-{a_\gamma})\psi(h-{a_\gamma}) \rd\gamma(h)
\;.
\end{equation}
As $X$ is a separable locally convex space, \cite[Theorem
3.2.3]{Bogach98} implies that there is a unique linear operator
$R_\gamma: X^*\to X$ such that \eqref{eq:Cov} holds. We define
$$
\forall \varphi \in X^*: \quad \sigma(\varphi):= \sqrt{\langle
R_\gamma \varphi, \varphi \rangle} \;.
$$
If $h=R_\gamma \varphi$ for some $\varphi \in X^*$, the map
$h\mapsto \| h \|:=\sigma(\varphi)$ defines a norm on
${\rm range}(R_\gamma)\subset X$. There holds \cite[Lemma
2.4.1]{Bogach98}
$\| h \| = \| h \|_{H(\gamma)} = \| \varphi \|_{L^2(X,\gamma)}$.
The \emph{Cameron-Martin space} of the GM $\gamma$ on $X$ is the
completion of the range of $R_\gamma$ in $X$ with respect to the norm
$\| \circ \|$.
The Cameron-Martin space of the GM $\gamma$ on $X$ is denoted by
$H(\gamma)$. It is also called the reproducing kernel Hilbert space
(RKHS for short). By \cite[Theorem 3.2.7]{Bogach98}, $H(\gamma)$ is a
separable Hilbert space, and $H(\gamma)\subset X$ with continuous
embedding, according to \cite[Proposition 2.4.6]{Bogach98}. In case
that $X\subset Y$ for another Banach space, with continuous and linear
embedding, the Cameron-Martin spaces for $X$ and $Y$ coincide. For
example, in the context of Remark \ref{rmk:GMH}, item~(i),
$H(\gamma) = K(X^*_\gamma)$.
Being a Hilbert space, introduce an innerproduct
$(\cdot,\cdot)_{H(\gamma)}$ on $H(\gamma)$ compatible with the norm
$\| \circ \|_{H(\gamma)}$ via the parallelogram law. Then there holds
$$
\forall \varphi\in X^* \; \forall f\in H(\gamma): \quad (f,R_\gamma
\varphi)_{H(\gamma)} = \varphi(f)\;.
$$
Since $H(\gamma)$ is also separable, there is an ONB.
\begin{proposition}[{\cite[Theorem 3.5.10, Corollary 3.5.11]{Bogach98}}]\label{prop:ONBH}
For a centered GM on a real, separable Banach space $X$ with norm
$\| \circ \|_X$, there exists an ONB $( e_n )_{n\in \mathbb{N}}$ of the
Cameron-Martin space $H(\gamma)\subset X$ such that
$$
\sum_{n\in {\mathbb N}} \| e_n \|_X^2 <\infty \;, \qquad \forall \varphi \in
X^*: \; R_\gamma\varphi = \sum_{n\in {\mathbb N}} \varphi(e_n)e_n\;.
$$
\end{proposition}
We remark that Proposition \ref{prop:ONBH} is not true for arbitrary
ONB $( e_n )_{n\in {\mathbb N}}$ of $H(\gamma)$.
\subsection{Gaussian product measures}
\label{sec-Gaussian product measures}
We recall a notion of product measures which gives an efficient method
to construct Gaussian measures on a countable Cartesian product of
locally convex spaces.
\begin{definition}[{Product measure, \cite[p.\ 372]{Bogach98}}]\label{def: product of measures}
Let $\mu_n$ be probability measures defined on $\sigma$-fields
$\cB_n$ in locally convex spaces $X_n$. Put
$$
X:= \prod_{n \in \mathbb{N}} X_n.
$$
Let $$\cB:= \bigotimes_{n \in \mathbb{N}} \cB_n$$ be the
$\sigma$-field generated by all the sets of the form
\begin{equation} \label{prod} B = B_1 \times B_2 \times \ldots
\times B_n \times X_{n+1} \times X_{n+2}\times \ldots , \
B_i \in \cB_i.
\end{equation}
The product measure $$\mu:= \bigotimes_{n \in \mathbb{N}} \mu_n$$ is
the probability measure on $\cB$ defined by
$\mu (B):= \prod_{i=1}^n \mu_i(B_i)$ for the sets $B$ of the
form \eqref{prod}.
\end{definition}
\begin{example}[{\cite[Example 2.3.8]{Bogach98}}]\label{ex:Gaussian-prod-meas}
Let $(\mu_n)_{n \in \mathbb{N}}$ be a
sequence of Gaussian measures. Then the product measure
$\mu:= \otimes_{n \in \mathbb{N}} \mu_n$ is a Gaussian measure on
$X:= \prod_{n \in \mathbb{N}} X_n$. The Cameron-Martin space of
$\mu$ is the Hilbert direct sum of spaces $H(\mu_n)$, i.e.,
\[
H(\mu) = \left\{h = (h_j)_{j \in \mathbb{N}} \in X: \, h_j \in
H(\mu_j), \ |h|^2_{H(\mu)} = \sum_{j \in \mathbb{N}}
|h_j|^2_{H(\mu_j)} \right\}.
\]
The space $X^*_\mu$ is the set of all functions of the form
\[
\varphi \mapsto \sum_{j \in \mathbb{N}} f_j (\varphi_j), \quad f_j
\in X^*_{\mu_j}, \quad \sum_{j \in \mathbb{N}} \sigma(f_j)^2 <
\infty,
\]
and
\[
a_\mu(f) = \sum_{j \in \mathbb{N}} a_{\mu_j}(f_j), \quad \forall f
= (f_j)_{j \in \mathbb{N}} \in X^*.
\]
\end{example}
\begin{example}[{\cite[Example 2.3.5]{Bogach98}}]\label{ex:prod-measR^infty}
Denote by $(\gamma_{1,n})_{n \in {\mathbb N}}$ a sequence of
standard GMs on $(\mathbb{R},\cB(\mathbb{R}))$. Then the product measure
%
$$
\gamma = \bigotimes _{n\in {\mathbb N}} \gamma_{1,n}
$$
%
is a centered GM on $\mathbb{R}^\infty$. Furthermore,
$H(\gamma) = \ell^2(\mathbb{N})$ and
$X^*_\gamma \simeq \ell^2(\mathbb{N})$. If $\mu$ is a GM on
${\mathbb R}^\infty$, then the measures $\gamma$ and $\mu$ are
either mutually singular or equivalent \cite[Theorem
2.12.9]{Bogach98}. The locally convex space ${\mathbb R}^\infty$
with the product measure $\gamma$ of standard GMs is
the main parametric domain in the stochastic setting
of UQ problems for PDEs with GRF inputs considered in
the next sections.
\end{example}
\subsection{Gaussian series}
\label{S:GSer}
A key role in the numerical analysis of PDEs with GRF
inputs from separable Banach spaces $E$ is played by
representing these GRFs in terms of series with respect
to suitable \emph{representation systems}
$(\psi_j)_{j\in {\mathbb N}}\in E^\infty$ of $E$ with
random coefficients. There arises the question of
admissibility of $(\psi_j)_{j\in {\mathbb N}}\in E^\infty$
so as to allow a) to transfer randomness of function
space-valued inputs to a parametric, deterministic
representation (as is customary, for example, in the
transition from nonparametric to parametric models in
statistics) and b) to ensure suitability for numerical
approximation.
Items a) and b) are closely related to the selection of
stable bases for $E$, with item b) mandating
additional requirements, such as efficient accessibility
for float point computations, quadrature, etc.
We first present an abstract result, Theorem
\ref{thm:AdFrmX} and then, in Section \ref{S:GSerKL},
\ref{S:GSerMRes}, we review several concrete
constructions of such series. We discuss in Sections
\ref{S:GSerKL}, \ref{S:GSerMRes} several concrete
examples, in particular the classical Karhunen-Lo\`{e}ve Expansion
\cite{Karhunen46,StWrtMercer} of GRFs taking values in
separable Hilbert space. All examples will be
admissible in parametrizing GRF input data for PDEs and
of Gaussian priors in the ensuing sparsity and
approximation rate analysis in Sections
\ref{sec:EllPDElogN} and following.
\subsubsection{Some abstract results}
\label{S:GserAbsRes}
We place ourselves in the setting of a real separable
locally convex space $X$, with a GM $\gamma$ on $X$, and
with associated Cameron-Martin Hilbert space
$H(\gamma)\subset X$ as introduced in Section
\ref{S:CamMart}.
We first consider expansions of Gaussian random vectors
with respect to orthonormal bases $(e_j)_{j\in \mathbb{N}}$ of
the Cameron-Martin space $H(\gamma)$. As linear
transformations of GM are Gaussian (see Lemma
\ref{lem:AffGM}), we admit a linear transformation $A$.
\begin{theorem}[{\cite[Theorems. 3.5.1,
3.5.7, (3.5.4)]{Bogach98}}]\label{thm:GSer} Let $\gamma$ be a centered
GM on a real separable locally convex space $X$ with
Cameron-Martin space $H(\gamma)$ and with \emph{some}
ONB $(e_j)_{j\in {\mathbb N}}$ of $H(\gamma)$. Let further
denote $( y_j )_{j\in {\mathbb N}}$ any sequence of
independent standard Gaussian RVs on a probability
space $(\Omega,\cA,\mathbb{P})$ and let $A\in \cL(H(\gamma))$
be arbitrary.
Then the Gaussian series
$$
\sum_{j\in {\mathbb N}} y_j(\omega) A e_j
$$
converges $\mathbb{P}$-a.s. in $X$. The law of its limit is a centered GM
$\lambda$ with covariance $R_\lambda$ given by
$$
R_\lambda(f)(g) = \left( A^* R_\gamma(f), A^*
R_\gamma(g)\right)_{H(\gamma)} \;.
$$
Furthermore, there holds
of independent standard Gaussian RVs on a probability space
$(\Omega,\cA,\mathbb{P})$.
$$
\int_X f(x) \gamma(\rd x) = \int_{\Omega} f\bigg(\sum_{j\in {\mathbb N}}
y_j(\omega) e_j\bigg) \rd \mathbb{P}(\omega) \;.
$$
If $X$ is a real separable Banach space $X$ with norm $\| \circ \|_X$,
for all sufficiently small constants $c>0$ holds
$$
\lim_{n\to\infty} \int_\Omega \exp\Bigg( c\bigg\| \sum_{j=n}^\infty
y_j(\omega) Ae_j \bigg\|_X^2 \Bigg) \rd \mathbb{P}(\omega) = 1
$$
In particular, for every $p\in [1,\infty)$ we have
$\big\| \sum_{j=n}^\infty y_j Ae_j \big\|_X^p \to 0$ in
$L^1(\Omega,\mathbb{P})$ as $n\to \infty$.
\end{theorem}
Often, in numerical applications, ensuring orthonormality of the basis
elements could be computationally costly. It is therefore of some
interest to consider Gaussian series with respect to more general
representation systems $(\psi_j)_{j\in \mathbb{N}}$. An important notion is
\emph{admissibility} of such systems.
\begin{definition}\label{def:Adm}
Let $X$ be a real, separable locally convex space, and let
$g:(\Omega, \cA, \mathbb{P})\to X$ be a centered Gaussian random vector
with law $\gamma_g = \mathbb{P}_X$. Let further $(y_j)_{j\in \mathbb{N}}$ be a
sequence of i.i.d.\ standard real Gaussian RVs
$y_j \sim \mathcal{N}(0,1)$.
A sequence $(\psi_j)_{j\in {\mathbb N}}\in X^\infty$ is called
\emph{admissible for $g$} if
$$
\sum_{j\in{\mathbb N}} y_j \psi_j \;\;\mbox{converges $\mathbb{P}$-a.s. in }\;
X\;\;\mbox{and}\;\; g =\sum_{j\in {\mathbb N}} y_j \psi_j \;.
$$
\end{definition}
To state the next theorem, we recall the notion of \emph{frames in
separable Hilbert space} (see, e.g., \cite{HeilBases} and the
references there for background and theory of frames. In the
terminology of frame theory, Parseval frames correspond to tight
frames with frame bounds equal to $1$).
\begin{definition}\label{def:ParsFrH}
A sequence $(\psi_j )_{j\in \mathbb{N}}\subset H$ in a real separable
Hilbert space $H$ with inner product $(\cdot,\cdot)_H$ is a
\emph{Parseval frame of $H$} if
$$
\forall f\in H:\quad f = \sum_{j\in {\mathbb N}} (\psi_j,f)_H \psi_j
\quad\mbox{in}\quad H\;.
$$
\end{definition}
The following result, from \cite{LPFrame09}, characterizes admissible
affine representation systems for GRFs $u$ taking values in real,
separable Banach spaces $X$.
\begin{theorem}[{\cite[Theorem 1]{LPFrame09}}]\label{thm:AdFrmX}
\begin{enumerate}
\item In a real, separable Banach space $X$ with a centered GM
$\gamma$ on $X$, a representation system
${{\boldsymbol{\Psi}}} = (\psi_j)_{j\in {\mathbb N}}\in X^\infty$ is \emph{admissible} for
$\gamma$ iff ${{\boldsymbol{\Psi}}}$ is a Parseval frame for the Cameron-Martin
space $H(\gamma)\subset X$, i. e.,
\[
\forall f\in H(\gamma): \quad \| f \|_{H(\gamma)}^2 = \sum_{j\in
{\mathbb N}} |\langle f,\psi_j\rangle|^2.
\]
\item Let $u$ denote a GRF taking values in $X$ with law $\gamma$
and with RKHS $H(\gamma)$. For a countable collection
${{\boldsymbol{\Psi}}} = (\psi_j)_{j\in {\mathbb N}}\in X^\infty$ the following are
equivalent:
\begin{itemize}
\item[(i)] ${{\boldsymbol{\Psi}}}$ is a Parseval frame of $H(\gamma)$ and
\item[(ii)] there is a sequence ${\boldsymbol{y}} = (y_j)_{j\in {\mathbb N}}$ of i.i.d
standard Gaussian RVs $y_j$ such that there holds $\gamma-a.s.$
the representation
\[
u = \sum_{j\in {\mathbb N}} y_j \psi_j \quad \mbox{in}\quad H(\gamma)
\;.
\]
\end{itemize}
\item Consider a GRF $u$ taking values in $X$ with law $\gamma$ and
covariance $R_\gamma\in \cL(X',X)$. If $R_\gamma = SS'$ with
$S\in \cL(K,X)$ for some separable Hilbert space $K$, for
\emph{any} Parseval frame ${{\boldsymbol{\Phi}}} = (\varphi_j)_{j\in {\mathbb N}}$ of $K$,
the countable collection
${{\boldsymbol{\Psi}}} = S{{\boldsymbol{\Phi}}} = ( S\varphi_j )_{j\in {\mathbb N}}$ is a Parseval frame
of the RKHS $H(\gamma)$ of $u$.
\end{enumerate}
\end{theorem}
The last assertion in the preceding result is \cite[Proposition
1]{LPFrame09}. It generalizes the observation that for a symmetric
positive definite matrix ${\boldsymbol{M}}$ in ${\mathbb R}^d$, any factorization
${\boldsymbol{M}} = {\boldsymbol{L}}\bL^\top$ implies that for $z\sim {\mathcal N}(0,{\boldsymbol{I}})$ it holds
${\boldsymbol{L}} z \sim {\mathcal N}(0,{\boldsymbol{M}})$. The result is useful in building
customized representation systems ${{\boldsymbol{\Psi}}}$ which are frames of a GRF
$u$ with computationally convenient properties in particular
applications.
We review several widely used constructions of Parseval frames. These
comprise expansions in eigenfunctions of the covariance operator $K$
(referred to also as principal component analysis, or as ``Karhunen-Lo\`{e}ve
expansions''), but also ``eigenvalue-free'' multiresolution
constructions (generalizing the classical L\'{e}vy-Cieselski
construction of the Brownian bridge) for various geometric settings,
in particular bounded subdomains of euclidean space, compact manifolds
without boundary etc. Any of these constructions will be admissible
as representation system for GRF input of PDEs for which our ensuing
results will apply.
\begin{example}[Brownian bridge]\label{expl:BrBr}
On the bounded time interval $[0,T]$, consider the \emph{Brownian
bridge} $(B_t)_{t\geq 0}$. It is defined in terms of a Wiener
process $(W_t)_{t\geq 0}$ by conditioning as
\begin{equation}\label{eq:BrBr}
(B_t)_{0\leq t\leq T} := \big\{ (W_t)_{0\leq t\leq T} | W_T = 0\big\}.
\end{equation}
It is a simple example of \emph{kriging} applied to the GRF $W_t$.
The covariance function of the GRF $B_t$ is easily calculated as
\[
k_B(s,t) = \mathbb E[B_sB_t] = s(T-t)/T \;\;\mbox{if}\;\; s < t.
\]
Various other representations of $B_t$ are
\[
B_t = W_t - \frac{t}{T}W_T = \frac{T-t}{\sqrt{T}} W_{t/(T-t)}.
\]
The RKHS $H(\gamma)$ corresponding to the GRF $B_t$ is the Sobolev
space $H^1_0(0,T)$.
\end{example}
\subsubsection{Karhunen-Lo\`{e}ve expansion}
\label{S:GSerKL}
A widely used representation system in the analysis and computation of
GRFs is the so-called Karhunen-Lo\`{e}ve expansion KL expansion for short) of GRFs,
going back to \cite{Karhunen46}. We present main ideas and
definitions, in a generic setting of \cite{KOPP2018}, see also
\cite[Chap. 3.3]{AdlerGeoGRF}.
Let $\cM$ be a compact space with metric
$\rho: \cM \times \cM \to {\mathbb R}$ and with Borel sigma-algebra
$ \cB = \cB(\cM)$. Assume given a Borel measure $\mu$ on $(\cM,\cB)$.
Let further $(\Omega,\cA,\mathbb{P})$ be a probability space. Examples are
$\cM = {\domain}$ a bounded domain in Euclidean space ${\mathbb R}^d$, with $\rho$
denoting the Euclidean distance between pairs $(x,x')$ of points in
${\domain}$, and $\cM$ being a smooth, closed $2$-surface in ${\mathbb R}^3$, where
$\rho$ is the geodesic distance between pairs of points in $\cM$.
Consider a measurable map
\[
Z: (\cM,\cB)\otimes (\Omega, \cA) \to {\mathbb R}: (x,\omega)\mapsto
Z_x(\omega)\in {\mathbb R}
\]
such that for each $x\in \cM$, $Z_x$ is a centered, Gaussian RV. We
call the collection $(Z_x)_{x\in \cM}$ a \emph{GRF indexed by $\cM$}.
Assume furthermore for all $n\in {\mathbb N}$, for all $x_1,\ldots,x_n\in \cM$
and for every $\xi_1,\ldots,\xi_n\in {\mathbb R}$
\[
\sum_{i=1}^n \xi_iZ_{x_i} \;\;\mbox{is a centered Gaussian RV}.
\]
Then the \emph{covariance function}
$$K:\cM\times \cM \to {\mathbb R}: (x,x')\mapsto K(x,x')$$
associated with the centered GRF $(Z_x)_{x\in\cM}$ is defined
pointwise by
\begin{equation*}\label{eq:CovFktK}
K(x,x') := \mathbb E[Z_{x} Z_{x'} ] \quad x,x'\in \cM\;.
\end{equation*}
Evidently, the covariance function $K:\cM\times \cM\to {\mathbb R}$
corresponding to a Gaussian RV indexed by $\cM$ is a real-valued,
symmetric, and positive definite function, i.e., there holds
\begin{equation*}\label{eq:PosDefK}
\forall n\in {\mathbb N} \;
\forall (x_j)_{1\leq j \leq n} \in \cM^n,
\forall (\xi_j)_{1\leq j \leq n}\in {\mathbb R}^n:\;
\sum_{1\leq i,j\leq n} \xi_i\xi_j K(x_i,x_j) \geq 0 \;.
\end{equation*}
The operator $K\in \cL(L^2(\cM,\mu),L^2(\cM,\mu))$ defined by
\[
\forall f\in L^2(\cM,\mu): \;\; (Kf)(x) := \int_{\cM} K(x,x') f(x')
\rd\mu(x') \quad x\in \cM
\]
is a self-adjoint, compact positive operator on $L^2(\cM,\mu)$.
Furthermore, $K$ is trace-class and
$K(L^2(\cM,\mu)) \subset C(\cM,{\mathbb R})$.
The spectral theorem for compact, self-adjoint operators on the
separable Hilbert space $L^2(\cM,\mu)$ ensures the existence of a
sequence $\lambda_1 \geq \lambda_2 \geq \ldots \geq 0$ of real
eigenvalues of $K$ (counted according to multiplicity and accumulating
only at zero) with associated eigenfunctions $\psi_k\in L^2(\cM,\mu)$
normalized in $L^2(\cM,\mu)$, i.e., for all $k\in {\mathbb N}$ holds
\[
K \psi_k = \lambda_k \psi_k \quad \mbox{in}\;\; L^2(\cM,\mu)\;,
\quad \int_{\cM} \psi_k(x) \psi_\ell(x) \rd\mu(x) =
\delta_{k\ell}\;, \;\; k,\ell \in {\mathbb N}\;.
\]
Then, there holds $\psi_k \in C(\cM;{\mathbb R})$ and the sequence
$(\psi_k)_{k\in {\mathbb N}}$ is an ONB of $L^2(\cM,\mu)$. From Mercer's
theorem (see, e.g., \cite{StWrtMercer}), there holds the \emph{Mercer
expansion}
\[
\forall x,x'\in \cM: \quad K(x,x') = \sum_{k\in {\mathbb N}} \lambda_k
\psi_k(x) \psi_k(x')
\]
with absolute and uniform convergence on $\cM\times \cM$.
This result implies that
$$
\lim_{m\to\infty} \int_{\cM\times \cM} \left| K(x,x') - \sum_{j=1}^m
\lambda_j \psi_j(x)\psi_j(x') \right|^2 \rd\mu(x) \rd\mu(x') = 0\;.
$$
We denote by $H\subset L^2(\Omega,\mathbb{P})$ the $L^2(\cM,\mu)$ closure of
finite linear combinations of $(Z_x)_{x\in \cM}$. This so-called
\emph{Gaussian} space (e.g. \cite{Janson97}) is a Hilbert space when
equipped with the $L^2(\cM,\mu)$ innerproduct. Then, the sequence
$(B_k)_{k\in {\mathbb N}}\subset {\mathbb R}$ defined by
\begin{equation*}\label{eq:Coeff}
\forall k\in {\mathbb N}: \quad
B_k(\omega)
:=
\frac{1}{\sqrt{\lambda_k}} \int_{\cM} Z_x(\omega) \psi_k(x) \rd\mu(x)
\in H
\end{equation*}
is a sequence of i.i.d, ${N}(0,1)$ RVs. The expression
\begin{equation}\label{eq:KLSer}
\tilde{Z}_x(\omega) := \sum_{k\in {\mathbb N}} \sqrt{\lambda_k} \psi_k(x) B_k(\omega)
\end{equation}
is a modification of $Z_x(\omega)$, i.e., for every $x\in \cM$ holds
that $\mathbb{P}(\{ Z_x = \tilde{Z}_x \}) = 1$, which is referred to as
\emph{Karhunen-Lo\`{e}ve expansion of the GRF} $\{ Z_x: x \in \cM \}$.
\begin{example}[KL expansion of the Brownian
bridge \eqref{eq:BrBr}]\label{expl:KLBrBr}
On the compact interval $\cM = [0,T]\subset {\mathbb R}$, the KL expansion
of the Brownian bridge is
$$
B_t = \sum_{k\in {\mathbb N}} Z_k \frac{\sqrt{2T}}{k\pi} \sin(k\pi t/T)
\;,\quad t\in [0,T] \;.
$$
Then
$$H(\gamma) = H^1_0(0,T) = {\rm span}\{ \sin(k\pi t/T) : k\in {\mathbb N}
\}.$$
\end{example}
In view of GRFs appearing as diffusion coefficients in elliptic and
parabolic PDEs, criteria on their path regularity are of some
interest. Many such conditions are known and we present some of
these, from \cite[Chapter 3.2, 3.3]{AdlerGeoGRF}.
\begin{proposition}\label{prop:C0Z}
For any compact set $\cM \subset {\mathbb R}^d$, if for $\alpha>0$,
$\eta > \alpha$ and some constant $C>0$ holds
\begin{equation}\label{eq:PthC0}
\mathbb E[|Z_{{\boldsymbol{x}}+{\boldsymbol{h}}} - Z_{\boldsymbol{x}}|^\alpha] \leq C \frac{|{\boldsymbol{h}}|^{2d}}{|\log|{\boldsymbol{h}}||^{1+\eta}} \;,
\end{equation}
then $${\boldsymbol{x}}\to Z_{\boldsymbol{x}}(\omega) \in C^0(\cM) \ \ \mathbb{P}-a.s.$$
\end{proposition}
Choosing $\alpha = 2$ in \eqref{eq:PthC0}, we obtain for $\cM$ such
that $\cM = \overline{{\mathrm D}}$, where ${\mathrm D} \subset {\mathbb R}^d$ is a
bounded Lipschitz domain, the sufficient criterion that there exist
$C>0$, $\eta>2$ with
\begin{equation*}\label{eq:PthC1}
\forall {\boldsymbol{x}} \in {\mathrm D}: \quad
K({\boldsymbol{x}}+{\boldsymbol{h}},{\boldsymbol{x}}+{\boldsymbol{h}}) - K({\boldsymbol{x}}+{\boldsymbol{h}},{\boldsymbol{x}}) - K({\boldsymbol{x}},{\boldsymbol{x}}+{\boldsymbol{h}}) + K({\boldsymbol{x}},{\boldsymbol{x}})
\leq C
\frac{|{\boldsymbol{h}}|^{2d}}{|\log|{\boldsymbol{h}}||^{1+\eta}} \;.
\end{equation*}
This is to hold for some $\eta>2$ with the covariance kernel $K$ of
the GRF $Z$, in order to ensure that
$[{\boldsymbol{x}}\mapsto Z_{\boldsymbol{x}}]\in C^1(\overline{{\mathrm D}})\subset
W^{1}_\infty({\mathrm D})$ $\mathbb{P}$-a.s., see \cite[Theorem 3.2.5, page 49
bottom]{AdlerGeoGRF}.
Further examples of explicit Karhunen-Lo\`{e}ve expansions of GRFs can be found in
\cite{LangChSSpher,LangKLCptSpc,KOPP2018} and a statement for
$\mathbb{P}$-a.s H\"older continuity of GRFs $Z$ on smooth manifolds $\cM$ is
proved in \cite{AndrLang}.
\subsubsection{Multiresolution representations of GRFs}
\label{S:GSerMRes}
KL expansions \eqref{eq:KLSer} provide an important source of concrete
examples of Gaussian series representations of GRFs $u$ in Theorem
\ref{thm:GSer}. Since KL expansions involve the eigenfunctions of the
covariance operators of the GRF $u$, all terms in these expansions
are, in general, globally supported in the physical domain $M$
indexing the GRF $u$. Often, it is desireable to have Gaussian series
representations of $u$ in Theorem \ref{thm:GSer} where the elements
$(e_n)_{n\in {\mathbb N}}$ of the representation system are locally supported
in the indexing domain $\cM$.
\begin{example}[{L\'{e}vy-Cieselsky representation of Brownian
bridge, \cite{CiesielBrBr61}}]\label{exple:LCBrBr}
Consider the Brownian bridge $(B_t)_{0\leq t\leq T}$ from Examples
\ref{expl:BrBr}, \ref{expl:KLBrBr}. For $T=1$, it may also be
represented as Gaussian series (e.g. \cite{CiesielBrBr61})
\begin{equation*}\label{eq:LCBrBr}
B_t =
\sum_{j\in {\mathbb N}} \sum_{k=0}^{2^j - 1} Z_{jk} 2^{-j/2} h(2^jt-k)
=
\sum_{j\in {\mathbb N}} \sum_{k=0}^{2^j - 1} Z_{jk} \psi_{jk}(t)
, \quad t\in \cM=[0,1]\;.
\end{equation*}
where $$\psi_{jk}(t):= 2^{-j/2} h(2^jt-k),$$ with
$h(s):= \max\{ 1-2|s-1/2|,0 \}$ denoting the standard, continuous
piecewise affine ``hat'' function on $(0,1)$. Here, $\mu$ is the
Lebesgue measure in $\cM = [0,1]$, and
$Z_{jk} \sim \mathcal{N}(0,1)$ are i.i.d standard normal RVs.
By suitable reordering of the index pairs $(j,k)$, eg. via the
bijection $(j,k) \mapsto j:= 2^j+k$, the representation
\eqref{exple:LCBrBr} is readily seen to be a special case of Theorem
\ref{thm:AdFrmX}, item ii). The corresponding system
$$
{{\boldsymbol{\Psi}}} = \{ \psi_{jk} : j \in {\mathbb N}_0, 0\leq k \leq 2^j-1 \}
$$
is, in fact, a basis for
$C_0([0,1]) := \{ v\in C([0,1]): v(0) = v(1) = 0 \}$, the so-called
\emph{Schauder basis}.
There holds
$$
\sum_{j\in {\mathbb N}} \sum_{k=0}^{2^j-1} 2^{js}|\psi_{jk}(t)| < \infty,
\quad t\in [0,1]\;,
$$
for any $0\leq s < 1/2$. The functions $\psi_{jk}$ are localized in
the sense that $|{\rm supp}(\psi_{jk})| = 2^{-j}$ for
$k=0,1,\ldots ,2^j-1.$
\end{example}
Further constructions of such \emph{multiresolution representations}
of GRFs with either Riesz basis or frame properties are available on
polytopal domains $M\subset {\mathbb R}^d$, (e.g. \cite{bachmayr2018GRFRep},
for a needlet multiresolution analysis on the $2$-sphere $\cM = \IS^2$
embedded in ${\mathbb R}^3$, where $\mu$ in Section \ref{S:GSerKL} can be
chosen as the surface measure see, also, for representation systems by
so-called spherical needlets \cite{NarPetWar06},
\cite{bachmayr2020multilevel}).
We also mention \cite{AyTaqq2003} for optimal approximation rates of
truncated wavelet series approximations of fractional Brownian random
fields, and to \cite{KOPP2018} for corresponding spectral
representations.
Multiresolution constructions are also available on data-graphs $M$
(see, e.g., \cite{CoifEtAlDifWav06} and the references there).
\subsubsection{Periodic continuation of a stationary GRF}
Let $(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in {\mathrm D}}$ be a GRF indexed by
${\mathrm D} \subset {\mathbb R}^d$, where ${\domain}$ is a bounded domain. We aim for
representations of the general form
\begin{equation}\label{eq-ref-k-01}
Z_{{\boldsymbol{x}}}= \sum_{j\in {\mathbb N}}\phi_j({\boldsymbol{x}}) y_j,
\end{equation}
where the $y_j$ are i.i.d.\ ${\mathcal N}(0,1)$ RVs and the
$(\phi_j)_{j\in \mathbb{N}}$ are a given sequence of functions defined on
${\domain}$. One natural choice of $\phi_j$ is
$\phi_j=\sqrt{\lambda_j} \psi_j$, where $\psi_j $ and are the
eigen-functions and $\lambda_j$ eigenvalues of the covariance
operator. However, Karhunen-Lo\`{e}ve eigenfunctions on ${\domain}$ are typically not
explicitly known and globally supported in the physical domain
${\domain}$. One of the strategy for deriving better representations over
${\domain}$ is to view it as the restriction to ${\domain}$ of a periodic Gaussian
process $Z_{\boldsymbol{x}}^{{\rm ext}}$ defined on a suitable larger torus ${\mathbb T}^d$.
Since ${\domain}$ is bounded, without loss of generality, we may the physical
domain ${\domain}$ to be contained in the box $[-\frac12, \frac12]^d$. We
wish to construct a periodic process $Z_{\boldsymbol{x}}^{{\rm ext}}$ on the torus
${\mathbb T}^d$ where ${\mathbb T}=[-\ell,\ell]$ whose restriction of $Z_{\boldsymbol{x}}^{{\rm ext}}$
on ${\domain}$ is such that $Z_{\boldsymbol{x}}^{{\rm ext}}|_{\domain}=Z_{{\boldsymbol{x}}}$. As a consequence,
any representation
$$
Z_{\boldsymbol{x}}^{{\rm ext}}=\sum_{j\in {\mathbb N}}y_j\tilde{\phi}_j
$$
yields a representation \eqref{eq-ref-k-01} where
$\phi_j=\tilde{\phi}_j|_{{\domain}}$.
Assume that $(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in {\mathrm D}}$ is a restriction of a
real-valued, stationary and centered GRF $(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in \mathbb{R}^d}$ on
$\mathbb{R}^d$ whose covariance is given in the form
\begin{equation}\label{eq-rep-k-03}
\mathbb E[Z_{{\boldsymbol{x}}} Z_{{\boldsymbol{x}}'} ] = \rho({\boldsymbol{x}} -{\boldsymbol{x}}'), \quad {\boldsymbol{x}},{\boldsymbol{x}}'\in \mathbb{R}^d,
\end{equation}
where $\rho$ is a real-valued, even function and its Fourier transform
is a non-negative function. The extension is feasible provided that
we can find an even and ${\mathbb T}^d$-periodic function $\rho^{{\rm ext}}$ which
agree with $\rho$ over $[-1,1]^d$ such that the Fourier coefficients
$$
c_{\boldsymbol{n}}(\rho^{{\rm ext}})=\int_{{\mathbb T}^d}\rho^{{\rm ext}}({\boldsymbol{\xi}}) \exp\Big(-{\rm i}
\frac{\pi}{\ell}({\boldsymbol{n}}, {\boldsymbol{\xi}})\Big)\,\mathrm{d} {\boldsymbol{\xi}},\qquad {\boldsymbol{n}}\in {\mathbb Z}^d
$$
are non-negative.
A natural way of constructing the function $\rho^{{\rm ext}}$ is by
truncation and periodization. First one chooses a sufficiently smooth
and even cutoff function $\varphi_\kappa$ such that
$\varphi_\kappa|_{[-1,1]^d}=1$ and $\varphi_\kappa({\boldsymbol{x}})=0$ for
${\boldsymbol{x}}\not \in [-\kappa,\kappa]^d$ where $\kappa=2\ell-1$. Then
$\rho^{{\rm ext}}$ is defined as the periodization of the truncation
$\rho \varphi_\kappa$, i.e.,
$$
\rho^{{\rm ext}}({\boldsymbol{\xi}})=\sum_{{\boldsymbol{n}}\in {\mathbb Z}^d}(\rho \varphi_\kappa)({\boldsymbol{\xi}}+2\ell
{\boldsymbol{n}}).
$$
It is easily seen that $\rho^{{\rm ext}}$ agrees with $\rho$ over
$[-1,1]^d$ and
$$
c_{\boldsymbol{n}}(\rho^{{\rm ext}})=\widehat{\rho
\varphi_\kappa}\Big(\frac{\pi}{\ell}{\boldsymbol{n}}\Big).
$$
Therefore $c_{\boldsymbol{n}}(\rho^{{\rm ext}})$ is non-negative if we can prove that
$\widehat{\rho \varphi_\kappa}({\boldsymbol{\xi}})\geq 0$ for ${\boldsymbol{\xi}}\in {\mathbb R}^d$. The
following is given in \cite{bachmayr2018GRFRep}.
\begin{theorem}
Let $\rho$ be an even function such that
\begin{equation}\label{eq-rep-k04}
c (1+ |{\boldsymbol{\xi}}|^2)^{-s}\leq \hat{\rho}({\boldsymbol{\xi}})\leq C (1+ |{\boldsymbol{\xi}}|^2)^{-r},\qquad {\boldsymbol{\xi}} \in \mathbb{R}^d
\end{equation}
for some $s\geq r\geq d/2$ and $0<c\leq C$ and
$$
\lim\limits_{R\to +\infty}\int_{|{\boldsymbol{x}}|>R} |\partial^\alpha
\rho({\boldsymbol{x}})|\,\mathrm{d} {\boldsymbol{x}} =0,\qquad |{\boldsymbol{\alpha}}|\leq 2\lceil s\rceil.
$$
Then for $\kappa$ sufficiently large, there exists
$\varphi_\kappa$ satisfying $\varphi_\kappa|_{[-1,1]^d}=1$ and
$\varphi_\kappa({\boldsymbol{x}})=0$ for ${\boldsymbol{x}}\not \in [-\kappa,\kappa]^d$
such that
\begin{equation*}\label{eq-rep-k02}
0<\widehat{\rho \varphi_\kappa} ({\boldsymbol{\xi}}) \leq C (1+ |{\boldsymbol{\xi}}|^2)^{-r},\qquad {\boldsymbol{\xi}} \in \mathbb{R}^d.
\end{equation*}
\end{theorem}
The assertion in Theorem \ref{eq-rep-k04} implies that
$$0<c_{\boldsymbol{n}}(\rho^{{\rm ext}}) \leq C (1+ |{\boldsymbol{n}}|^2)^{-r},\qquad {\boldsymbol{n}}\in {\mathbb Z}^d.$$
In the following we present an explicit construction of the
function $\varphi_\kappa$ for GRFs with Mat\'ern covariance
\begin{equation*}\label{eq:materndef}
\rho_{\lambda,\nu}({\boldsymbol{x}}):=\frac{2^{1-\nu}}{\Gamma(\nu)} \bigg(\frac{\sqrt{2\nu}|{\boldsymbol{x}}|}{\lambda}\bigg)^{\nu}K_\nu\bigg(\frac{\sqrt{2\nu}|{\boldsymbol{x}}|}{\lambda}\bigg)\, ,
\end{equation*}
where $\lambda>0$, $\nu>0$ and $K_\nu$ is the modified Bessel
functions of the second kind. Note that the Matérn
covariances satisfy the assumption \eqref{eq-rep-k04} with
$s=r=\nu+d/2$.
Let $P:=2\lceil \nu+\frac{d}{2}\rceil+1$ and $N_P$ be the
cardinal B-spline function with nodes $\{- P,\ldots,-1,0
\}$. For $\kappa>0$ we define the even function
$\varphi\in C^{P-1}(\mathbb{R})$ by
\begin{equation*}\label{eq:bspline1}
\varphi(t)=\begin{cases}
1 & \text{if}\ \ |t|\leq \kappa/2\\[3pt]
\displaystyle \frac{2 P}{\kappa}\int_{-\infty}^{t+\kappa/2} N_{P}\biggl(\frac{2 P}{\kappa}\xi\biggr)\,\mathrm{d}\xi & \text{if}\ \ t\leq -\kappa/2\,.
\end{cases}
\end{equation*}
It is easy to see that $\varphi(t)=0$ if $|t|\geq \kappa $.
We now define
\begin{equation*}\label{eq:bsplined}
\varphi_\kappa({\boldsymbol{x}}):=\varphi(|{\boldsymbol{x}}|).
\end{equation*}
With this choice of $\varphi_\kappa$, we have
$\rho^{\rm ext} = \rho_{\lambda,\nu}$ on $[-1,1]^d$ provided
that $ \ell \geq \frac{\kappa + \sqrt{d}}{2}. $ The required
size of $\kappa$ is given in the following theorem, see
\cite[Theorem 10]{bachmayr2019unified}.
\begin{theorem}\label{thm:smoothcond}
For $\varphi_\kappa$ as defined above, there exist constants
$C_1, C_2$ such that for any $0<\lambda,\nu<\infty$, we have
$\widehat{\rho_{\lambda,\nu} \varphi_\kappa} > 0$ provided that
$\kappa > 1$ and
\begin{equation*}\label{kappacondition}
\frac{\kappa}\lambda \geq C_1 + C_2 \max\Big\{\nu^{\frac12} ( 1 + |{\ln \nu}|) , \nu^{-\frac12} \Big\}.
\end{equation*}
\end{theorem}
\begin{remark}
The periodic random field $Z_{{\boldsymbol{x}}}^\mathrm{{\rm ext}}$ on $\mathbb{T}^d$
provides a tool for deriving {series expansions} of the original
random field. In contrast to the Karhunen-Lo\`{e}ve eigenfunctions on $D$, which
are typically not explicitly known, the corresponding eigenfunctions
$\psi_j^{{\rm ext}}$ of the periodic covariance are explicitly known
trigonometric functions and one has the following Karhunen-Lo\`{e}ve expansion for
the periodized random field:
\[
Z_{\boldsymbol{x}}^\mathrm{ext} = \sum_{j\in {\mathbb N}} y_j
\sqrt{\lambda^\mathrm{{\rm ext}}_j} \,\psi_j^{{\rm ext}}, \quad y_j \sim
\mathcal{N}(0,1)\ \text{ i.i.d.,}
\]
with $\lambda^\mathrm{ext}_j$ denoting the eigenvalues of the
periodized covariance and the $\psi_j^{{\rm ext}}$ are normalized in
$L^2(\mathbb{T}^d)$. Restricting this expansion back to $D$, one
obtains an exact expansion of the original random field on $D$
\begin{equation}\label{nondstdexpansion}
Z_{\boldsymbol{x}} = \sum_{j\in {\mathbb N}} y_j \sqrt{\lambda^\mathrm{{\rm ext}}_j} \,\psi_j^{{\rm ext}}|_{\domain},
\quad y_j \sim \mathcal{N}(0,1)\ \text{ i.i.d.,}
\end{equation}
This provides an alternative to the standard KL expansion of $Z_{\boldsymbol{x}}$
in terms of eigenvalues $\lambda_j$ and eigenfunctions $\psi_j$
normalized in $L^2({\domain})$. The main difference is that the functions
$\psi^\mathrm{ext}_j\big|_{\domain}$ in \eqref{nondstdexpansion} are not
$L^2({\domain})$-orthogonal. However, these functions are given explicitly,
and thus no approximate computation of eigenfunctions is required.
The KL expansion of $Z_{\boldsymbol{x}}^{\rm ext}$ also enables the construction of
alternative expansions of $Z_{\boldsymbol{x}}$ of the basic form
\eqref{nondstdexpansion}, but with the spatial functions having
additional properties. In \cite{bachmayr2018GRFRep}, wavelet-type
representations
\[
Z_{\boldsymbol{x}}^{{\rm ext}} = \sum_{\ell,k} y_{\ell,k} \psi_{\ell, k} , \quad
y_{\ell,k} \sim \mathcal{N}(0,1)\ \text{ i.i.d.,}
\]
are constructed where the functions $\psi_{\ell,k}$ have the same
multilevel-type localisation as the Meyer wavelets. This feature
yields improved convergence estimates for tensor Hermite polynomial
approximations of solutions of random diffusion equations with
lognormal coefficients .
\end{remark}
\subsubsection{Sampling stationary GRFs}
The simulation of GRFs with specified covariance is a fundamental task
in computational statistics with a wide range of applications. In this
section we present an efficient methods for sampling such fields.
Consider a GRF $(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in {\domain}}$ where ${\domain}$ is contained in
$[-1/2,1/2]^d$. Assume that $(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in {\mathrm D}}$ is a
restriction of a real-valued, stationary and centered GRF
$(Z_{\boldsymbol{x}})_{{\boldsymbol{x}}\in \mathbb{R}^d}$ on $\mathbb{R}^d$ with covariance given in
\eqref{eq-rep-k-03}. Let $m\in {\mathbb N}$ and ${\boldsymbol{x}}_1,\ldots,{\boldsymbol{x}}_M$ be
$M=(m+1)^d$ uniform grid points on $[-1/2,1/2]^d$ with grid spacing
$h=1/m$. We wish to obtain samples of the Gaussian RV
$$
{\boldsymbol{Z}}=(Z_{{\boldsymbol{x}}_1},\ldots,Z_{{\boldsymbol{x}}_M})
$$
with covariance matrix
\begin{align} \label{covmatrix}
\boldsymbol{\Sigma}=[\Sigma_{i,j}]_{i,j=1}^M,\qquad \Sigma_{i,j} =
\rho({\boldsymbol{x}}_i - {\boldsymbol{x}}_j) , \quad i,j = 1, \ldots, M.
\end{align}
Since $ \boldsymbol{\Sigma}$ is symmetric positive semidefinite, this
can in principle be done by performing the Cholesky factorisation
$ \boldsymbol{\Sigma} = \boldsymbol{F} \boldsymbol{F}^\top$ with
$ \boldsymbol{F} = \boldsymbol{\Sigma}^{1/2}$, from which the desired
samples are provided by the product $ \boldsymbol{F} {\boldsymbol{Y}}$ where
$ {\boldsymbol{Y}} \sim \mathcal{N}(0,{\boldsymbol{I}})$. However, since $\Sigma$ is large and
dense when $m$ is large, this factorisation is prohibitively
expensive. Since the covariance matrix $ \boldsymbol{\Sigma}$ is a
nested block Toeplitz matrix under appropriate ordering, an efficient
approach is to extend $\boldsymbol{\Sigma}$ to a appropriate larger
nested block circulant matrix whose spectral decomposition can be
rapidly computed using FFT.
For any $\ell \geq 1$ we construct a $2\ell$-periodic extension of
$\rho$ as follows
\begin{align*}\label{period}
\rho^{\rm ext}({\boldsymbol{x}}) = \sum_{{\boldsymbol{n}} \in {\mathbb Z}^d} \bigl( \rho \chi_{(-\ell,\ell]^d} \bigr)({\boldsymbol{x}} + 2\ell {\boldsymbol{n}}), \quad {\boldsymbol{x}} \in \mathbb{R}^d\,.
\end{align*}
Clearly, $\rho^{\rm ext}$ is $2 \ell$-periodic and $\rho^{\rm ext} = \rho $ on
$[-1,1]^d$. Denote ${\boldsymbol{\xi}}_1,\ldots,{\boldsymbol{\xi}}_s$, $s=(2\ell/h)^d$, the
uniform grid points on $[-\ell,\ell]^d$ with grid space $h$. Let
${\boldsymbol{Z}}^{\rm ext}=(Z_{{\boldsymbol{\xi}}_1},\ldots,Z_{{\boldsymbol{\xi}}_s})$ be the extended GRV with
covariance matrix $\boldsymbol{\Sigma}^{\rm ext}$ whose entries is
given by formula \eqref{covmatrix}, with $\rho$ replaced by
$\rho^{\rm ext}$ and ${\boldsymbol{x}}_i$ by ${\boldsymbol{\xi}}_i$. Hence $ \boldsymbol{\Sigma}$ is
embedded into the nested circulant matrix
$ \boldsymbol{\Sigma}^{\rm ext}$ which can be diagonalized using FFT
(with log-linear complexity) to provide the spectral decomposition
\begin{equation*}
\label{spectral_decomp}
\boldsymbol{\Sigma}^{\rm ext} = \boldsymbol{Q}^{\rm ext} \boldsymbol{\Lambda}^{\rm ext} (\boldsymbol{Q}^{\rm ext})^\top,
\end{equation*}
with $\Lambda^{\rm ext}$ diagonal and containing the eigenvalues
$\lambda^{\rm ext}_j$ of $\boldsymbol{\Sigma}^{\rm ext} $ and
$\boldsymbol{Q}^{\rm ext}$ being a Fourier matrix. Provided that these
eigenvalues are non-negative, the samples of the grid values of $Z$
can be drawn as follows. First we draw a random vector
$(y_j)_{j=1,\ldots,s}$ with $y_j\sim \mathcal{N}(0,1)$ i.i.d., then
compute
\begin{equation*}\label{samples}
{\boldsymbol{Z}}^\text{ext} = \sum_{j=1}^s y_j \sqrt{\lambda^{\rm ext}_j} \, {\boldsymbol{q}}_j
\end{equation*}
using the FFT, with ${\boldsymbol{q}}_j$ the columns of $\boldsymbol{Q}^{\rm
ext}$. Finally, a sample of ${\boldsymbol{Z}}$ is obtained by extracting from
${\boldsymbol{Z}}^\text{ext}$ the entries corresponding to the original grid
points.
The above mentioned process is feasible, provided that
$\boldsymbol{\Sigma}$ is positive semidefinite. The following theorem
characterizes the condition on $\ell$ for GRF with Mat\'ern covariance
such that $\boldsymbol{\Sigma}^{\rm ext}$ is positive semidefinite,
see \cite{GKNSS}.
\begin{theorem}\label{thm:matern-growth}
Let $1/2 \leq \nu < \infty$, $\lambda \leq 1$, and
$h/\lambda \leq e^{-1}$. Then there exist $C_1, C_2 >0$ which may
depend on $d$ but are independent of $\ell, h, \lambda, \nu$, such
that $\Sigma^{\rm ext}$ is positive definite if
\begin{equation*}
\frac{\ell}{\lambda} \ \geq \ C_1\ + \ C_2\, \nu^{\frac12} \, \log\bigl( \max\big\{ {\lambda}/{h}, \, \nu^{\frac12}\big\} \bigr) \, .
\label{eq:alphahsmall}
\end{equation*}
\end{theorem}
\begin{remark}
For GRF with Mat\' ern covariances, it is well-known (see,
e.g. \cite[Corollary 5]{GKNSSS}, \cite[eq.(64)]{BCM}) that the exact
KL eigenvalues $\lambda_j$ of $Z_{\boldsymbol{x}}$ in $L^2({\domain})$ decay with the
rate $
\label{kldecay} \lambda_j \ \leq \ C j^{-(1+ 2 \nu/d)} $. It has
been proved recently in \cite{bachmayr2019unified} that the
eigenvalue $\lambda^{\rm ext}_j$ maintain this rate of decay up to a
factor of order $\mathcal{O}(|\!\log h|^\nu)$.
\end{remark}
\subsection{Finite element discretization}
\label{S:FEM}
The approximation results and algorithms to be developed in the
present work involve, besides the Wiener-Hermite PC expansions with
respect to Gaussian co-ordinates ${\boldsymbol{y}}\in \mathbb{R}^\infty$ also certain
numerical approximations in the physical domain ${\mathrm D}$. Due to
their wide use in the numerical solution of elliptic and parabolic
PDEs, we opt for considering standard, primal Lagrangian finite
element (FE for short) discretizations. We confine the presentation
and analysis to Lipschitz polytopal domains ${\mathrm D} \subset \mathbb{R}^d$
with principal interest in $d=2$ (${\mathrm D}$ is a polygon with straight
sides) and $d=3$ (${\mathrm D}$ is a polyhedron with plane faces). We
confine the presentation to so-called primal FE discretizations in
${\mathrm D}$ but hasten to add that with minor extra mathematical effort,
similar results could be developed also for so-called mixed, or dual
FE discretizations (see,e.g., \cite{BBF} and the references there).
In presenting (known) results on Finite Element Method (FEM for short)
convergence rates, we consider separately FEM in polytopal domains
${\mathrm D}\subset {\mathbb R}^d$, $d=1,2,3$, and FEM on smooth $d$-surfaces
$\Gamma\subset {\mathbb R}^{d+1}$, $d=1,2$. See
\cite{BonDmlwEigFkt,DmlwHighOrderFEM}
\subsubsection{Geometric preliminaries}
\label{S:GeoPrel}
For $s\in {\mathbb N}$, we call a $C^s$-domain ${\mathrm D}\subset {\mathbb R}^d$ a
bounded domain whose boundary $\partial{\mathrm D}$ is locally
parameterized in a finite number of co-ordinate systems as a graph of
a $C^s$ function. In a similar way, we shall call
${\mathrm D}\subset {\mathbb R}^d$ a Lipschitz domain, when $\partial{\mathrm D}$ is,
locally, the graph of a Lipschitz function. We refer to
\cite{Adams2nd,GilbTr} and the references there or to \cite{Gr}.
We call \emph{polygonal domain} a domain ${\mathrm D} \subset {\mathbb R}^2$ that
is a polygon with Lipschitz boundary $\partial{\mathrm D}$ (which
precludes cusps and slits) and with a finite number of straight sides.
\subsubsection{Function spaces}
\label{S:FncSpc}
In ${\mathrm D}\subset {\mathbb R}^d$, the usual Sobolev spaces of integer order
$s \in {\mathbb Z}_+$ and integrability $q\in [1,\infty]$ are denoted by
$W^{s}_q({\mathrm D})$ with the understanding that
$L^q({\mathrm D})=W^{0}_q({\mathrm D})$. The norm of $v\in W^{s}_q({\mathrm D})$
is defined by
\[
\|v\|_{ W^{s}_q} := \ \sum_{{\boldsymbol{\alpha}} \in {\mathbb Z}^d_+: |{\boldsymbol{\alpha}}| \le s}
\|D^{\boldsymbol{\alpha}} v\|_{L^q} .
\]
Here derivative $D^{\boldsymbol{\alpha}}$ denotes the mixed weak derivative with
respect to ${\boldsymbol{x}}$ of order ${\boldsymbol{\alpha}}$. We refer to any standard text
such as \cite{Adams2nd} for basic properties of these spaces.
Hilbertian Sobolev spaces are given for $s\in {\mathbb Z}_+$ by
$H^s({\mathrm D}) = W^{s}_2({\mathrm D})$, with the usual understanding that
$L^2({\mathrm D}) = H^0({\mathrm D})$.
Let ${\mathrm D}\subset {\mathbb R}^2$ denote a bounded polygonal domain as in
Section \ref{S:GeoPrel}. We introduce in ${\mathrm D}$ a nonnegative
function $r_{\mathrm D}: {\mathrm D} \to {\mathbb R}_+$ which is smooth in the open
domain ${\mathrm D}$, and which coincides for ${\boldsymbol{x}}$ in a vicinity of each
corner ${\boldsymbol{c}}\in\partial{\mathrm D}$ with the Euclidean distance
$| {\boldsymbol{x}} - {\boldsymbol{c}} |$.
To state elliptic regularity shifts in a bounded polygon, we require
certain corner-weighted Sobolev spaces. We require these only for
integrability $q=2$ and for $q=\infty$.
For $s\in {\mathbb Z}_+$ and $\varkappa\in {\mathbb R}$ we define
\begin{equation*} {\mathcal K}^s_\varkappa({\mathrm D}): = \big\{ u: {\mathrm D} \to
{\mathbb C}: \ r_{\mathrm D}^{|{\boldsymbol{\alpha}}|-\varkappa}D^{\boldsymbol{\alpha}} u\in L^2({\mathrm D}),
|{\boldsymbol{\alpha}}|\leq s \big\}
\end{equation*}
and
\begin{equation*}
{\mathcal W}^s_\infty({\mathrm D}):
=
\big\{u: {\mathrm D}\to {\mathbb C}: \ r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{\boldsymbol{\alpha}} u\in L^\infty({\mathrm D}),\
|{\boldsymbol{\alpha}}|\leq s \big\}.
\end{equation*}
Here, for ${\boldsymbol{\alpha}}\in {\mathbb N}_0^2$ and as before $D^{\boldsymbol{\alpha}}$ denotes the
mixed weak derivative with respect to ${\boldsymbol{x}}$ of order ${\boldsymbol{\alpha}}$.
The \emph{corner-weighted norms in these spaces} are given by
\begin{equation*}
\|u\|_{ {\mathcal K}^s_\varkappa}
:=
\sum_{|{\boldsymbol{\alpha}}|\leq s}\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|-\varkappa}D^{\boldsymbol{\alpha}} u\|_{L^2}
\qquad\text{and}\qquad
\|u\|_{ {\mathcal W}^s_\infty}:=\sum_{|{\boldsymbol{\alpha}}|\leq s}\|r_{\mathrm D}^{|{\boldsymbol{\alpha}}|}D^{\boldsymbol{\alpha}} u\|_{L^\infty} \,.
\end{equation*}
The function spaces ${\mathcal K}^s_\varkappa({\mathrm D})$ and
${\mathcal W}^s_\infty({\mathrm D})$ endowed with these norms are Banach spaces,
and ${\mathcal K}^s_\varkappa({\mathrm D})$ are separable Hilbert spaces. These
spaces are called Kondrat'ev spaces.
One embedding of these spaces is
$ H^1_0({\mathrm D}) \subset {\mathcal K}^1_0({\mathrm D}) $. This follows from the
existence of a constant $c({\mathrm D})>0$ such that for every
${\boldsymbol{x}} \in {\mathrm D}$ holds
$r_{\mathrm D}({\boldsymbol{x}}) \geq c({\mathrm D}) {\rm dist}({\boldsymbol{x}},\partial{\mathrm D})$.
\subsubsection{Finite element interpolation}
\label{S:FEIntrp}
Here, we review some results on FE approximations in polytopal domains
${\mathrm D}$ on locally refined triangulations $\cT$ in ${\mathrm D}$. These
results are in principle known for the standard Sobolev spaces
$H^s({\mathrm D})$ and available in the standard texts
\cite{Brenner,Ciarlet}. For spaces with corner weights in polygons
${\mathrm D}\subset {\mathbb R}^2$, such as $ {\mathcal K}^s_\varkappa $ and
$ {\mathcal W}^s_\infty $, however, which arise in the regularity of the
Wiener-Hermite PC expansion coefficient functions for elliptic PDEs in
corner domains in Section \ref{sec:KondrReg} ahead, we provide
references to corresponding FE approximation rate bounds.
The corresponding FE spaces involve suitable mesh refinement to
compensate for the reduced regularity caused by corner and edge
singularities which occur in solutions to elliptic and parabolic
boundary value problems in these domains.
We define the FE spaces in polygons ${\mathrm D}\subset {\mathbb R}^2$ (see
\cite{Brenner,Ciarlet} for details). Let $\cT$ denote a regular
triangulation of $\overline {\mathrm D}$, i.e., a partition of
$\overline{\mathrm D}$ into a finite number $N(\cT)$ of closed,
nondegenerate triangles $T\in \cT$ (i.e., $|T|>0$) such that for any
two $T,T'\in \cT$, the intersection $T\cap T'$ is either empty, a
vertex or an entire edge. We denote the \emph{meshwidth} of $\cT$ as
$$
h(\cT) := \max\{ h(T): T\in \cT \}, \;\;\mbox{where}\;\; h(T) :=
\mbox{diam}(T)\;.
$$
For $T\in \cT$, denote $\rho(T)$ the diameter of the largest circle
that can be inscribed into $T$. We say $\cT$ is \emph{$\kappa$
shape-regular}, if
$$
\forall T\in \cT: \;\; \frac{h(T)}{\rho(T)} \leq \kappa \;.
$$
A sequence $\mathfrak{T} := ( \cT_n)_{n\in {\mathbb N}}$ is $\kappa$
shape-regular if each $\cT\in \mathfrak{T}$ is $\kappa$ shape-regular,
with one common constant $\kappa>1$ for all $\cT\in \mathfrak{T}$.
In a polygon ${\mathrm D}$, with a regular, simplicial triangulation
$\cT$, and for a polynomial degree $m\in {\mathbb N}$, the Lagrangian FE space
$S^m({\mathrm D},\cT)$ of continuous, piecewise polynomial functions of
degree $m$ on $\cT$ is defined as
\begin{equation*}\label{eq:DefSm}
S^m({\mathrm D},\cT) = \{ v\in H^1({\mathrm D}): \forall T\in \cT: v|_{T} \in \mathbb{P}_m \} \;.
\end{equation*}
Here, $\mathbb{P}_m := {\rm span} \{ {\boldsymbol{x}}^{\boldsymbol{\alpha}}: |{\boldsymbol{\alpha}}| \leq m \}$
denotes the space of polynomials of ${\boldsymbol{x}}\in {\mathbb R}^2$ of total degree at
most $m$. We also define
$S^m_0({\mathrm D},\cT) := S^m({\mathrm D},\cT) \cap H^1_0({\mathrm D})$.
The main result on FE approximation rates in a polygon
${\mathrm D}\subset {\mathbb R}^2$ in corner-weighted spaces
${\mathcal K}^s_\kappa({\mathrm D})$ reads as follows.
\begin{proposition}\label{prop:FECorner}
Consider a bounded polygonal domain ${\mathrm D} \subset {\mathbb R}^2$. Then,
for every polynomial degree $m\in {\mathbb N}$, there exists a sequence
$(\cT_n)_{n\in {\mathbb N}}$ of $\kappa$ shape-regular, simplicial
triangulations of ${\mathrm D}$ such that for every
$u\in (H^1_0\cap{\mathcal K}^{m+1}_\lambda)({\mathrm D})$ for some $\lambda > 0$,
the FE interpolation error converges at rate $m$. I.e., there
exists a constant $C({\mathrm D}, \kappa, \lambda, m) > 0$ such that for
all $\cT\in (\cT_n)_{n\in {\mathbb N}}$ and for all
$u\in (H^1_0\cap{\mathcal K}^{m+1}_\lambda)({\mathrm D})$ holds
$$
\| u - I^m_{\cT}u \|_{H^1} \leq C h(\cT)^m \| u \|_{{\mathcal K}^{m+1}_\lambda}
\;.
$$
Equivalently, in terms of the number $N = N(\cT) = \#(\cT)$ of
triangles, there holds
\begin{equation}\label{eq:FEErr2d}
\| u - I^m_{\cT}u \|_{H^1} \leq C N^{-m/2} \| u \|_{{\mathcal K}^{m+1}_\lambda} \;.
\end{equation}
Here, $I^m_{\cT}: C^0(\overline{{\mathrm D}})\to S^m({\mathrm D},\cT)$ denotes
the nodal, Lagrangian interpolant. The constant $C>0$ depends on $m$,
${\mathrm D}$ and the shape regularity of $\cT$, but is independent of
$u$.
\end{proposition}
For a proof, we refer for example to \cite[Theorems 4.2,
4.4]{BNZPolygon}.
We remark that due to
${\mathcal K}_{\lambda}^2({\mathrm D}) \subset C^0(\overline{{\mathrm D}})$, the nodal
interpolant $ I^m_{\cT} $ in \eqref{eq:FEErr2d} is well-defined. We
also remark that the triangulations $\cT_n$ need not necessarily be
nested (the constructions in \cite{BaPi79,BNZPolygon} do not provide
nestedness; for a bisection tree construction of $( \cT_n)_{n\in {\mathbb N}}$
which are nested, such as typically produced by adaptive FE
algorithms, with the error bounds \eqref{eq:FEErr2d}, we refer to
\cite{GasMorFESing09}.
For similar results in polyhedral domains in space dimension $d=3$, we
refer to \cite{BacNisZik3dI,BacNisZik3dII,Li3dhFEM} and to the
references there.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,912 |
Remember the Kavanaugh hearings last fall when Democrats told us we had to believe all women? They seem to have changed their tune now that the Democrat Lt. Governor of Virginia has been accused.
Virginia's Democratic Lt. Gov. Justin Fairfax has been accused of sexual assault. The accuser alleges the attack occurred during the 2004 Democratic National Convention. There is zero evidence to corroborate this story, which is why The Washington Post and other refused to run it. Different ethical standards were applied for Supreme Court Justice Brett Kavanaugh, however. The Daily Caller's Henry Rodgers tweeted that he asked 10 Democratic senators about the Fairfax allegation.
They either offered no comment or said they haven't heard the news, which is just another way of saying the former. Sen. Bernie Sanders (I-VT), who caucuses with Senate Democrats, just ran away. As reported by CBS NEs' Alan He, Sen. Cory Booker (D-NJ) said he's more concerned with Northam's blackface fiasco. Sen. Mark Warner (D-VA) refused to comment when asked about Fairfax by The Huffington Post's Igor Bobic.
Given how this party handled the Kavanaugh nomination and weaponized sexual assault allegations, they will continue to refuse comment or run away. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,231 |
{"url":"https:\/\/www.jobilize.com\/physics1\/course\/14-6-bernoulli-s-equation-fluid-mechanics-by-openstax?qcr=www.quizover.com&page=3","text":"# 14.6 Bernoulli\u2019s equation \u00a0(Page 4\/8)\n\n Page 4 \/ 8\n${p}_{1}+\\frac{1}{2}\\rho {v}_{1}^{2}={p}_{2}+\\frac{1}{2}\\rho {v}_{2}^{2}$\n\nbecomes\n\n${p}_{1}={p}_{2}+\\frac{1}{2}\\rho {v}_{2}^{2}.$\n\nThus pressure ${p}_{2}$ over the second opening is reduced by $\\frac{1}{2}\\rho {v}_{2}^{2}$ , so the fluid in the manometer rises by h on the side connected to the second opening, where\n\n$h\\propto \\frac{1}{2}\\rho {v}_{2}^{2}.$\n\n(Recall that the symbol $\\propto$ means \u201cproportional to.\u201d) Solving for ${v}_{2}$ , we see that\n\n${v}_{2}\\propto \\sqrt{h}.$\n\nPart (b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air-speed indicators in aircraft.\n\n## A fire hose\n\nAll preceding applications of Bernoulli\u2019s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli\u2019s equation in which pressure, velocity, and height all change.\n\n## Calculating pressure: a fire hose nozzle\n\nFire hoses used in major structural fires have an inside diameter of 6.40 cm ( [link] ). Suppose such a hose carries a flow of 40.0 L\/s, starting at a gauge pressure of $1.62\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{6}{\\text{N\/m}}^{2}$ . The hose rises up 10.0 m along a ladder to a nozzle having an inside diameter of 3.00 cm. What is the pressure in the nozzle?\n\n## Strategy\n\nWe must use Bernoulli\u2019s equation to solve for the pressure, since depth is not constant.\n\n## Solution\n\nBernoulli\u2019s equation is\n\n${p}_{1}+\\frac{1}{2}\\rho {v}_{1}^{2}+\\rho g{h}_{1}={p}_{2}+\\frac{1}{2}\\rho {v}_{2}^{2}+\\rho g{h}_{2}$\n\nwhere subscripts 1 and 2 refer to the initial conditions at ground level and the final conditions inside the nozzle, respectively. We must first find the speeds ${v}_{1}$ and ${v}_{2}$ . Since $Q={A}_{1}{v}_{1}$ , we get\n\n${v}_{1}=\\frac{Q}{{A}_{1}}=\\frac{40.0\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-3}{\\text{m}}^{3}\\text{\/}\\text{s}}{\\pi {\\left(3.20\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-2}\\text{m)}}^{2}}=12.4\\text{m\/s}.$\n\nSimilarly, we find\n\n${v}_{2}=56.6\\phantom{\\rule{0.2em}{0ex}}\\text{m\/s}.$\n\nThis rather large speed is helpful in reaching the fire. Now, taking ${h}_{1}$ to be zero, we solve Bernoulli\u2019s equation for ${p}_{2}$ :\n\n${p}_{2}={p}_{1}+\\frac{1}{2}\\rho \\left({v}_{1}^{2}-{v}_{2}^{2}\\right)-\\rho g{h}_{2}.$\n\nSubstituting known values yields\n\n$\\begin{array}{cc}\\hfill {p}_{2}& =1.62\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{6}\\phantom{\\rule{0.2em}{0ex}}{\\text{N\/m}}^{2}+\\frac{1}{2}\\left(1000\\phantom{\\rule{0.2em}{0ex}}{\\text{kg\/m}}^{3}\\right)\\left[{\\text{(12.4 m\/s)}}^{2}-{\\text{(56.6 m\/s)}}^{2}\\right]\\hfill \\\\ & \\phantom{\\rule{0.1em}{0ex}}-\\phantom{\\rule{0.1em}{0ex}}\\left(1000\\phantom{\\rule{0.2em}{0ex}}{\\text{kg\/m}}^{3}\\right)\\left(9.80\\phantom{\\rule{0.2em}{0ex}}{\\text{m\/s}}^{2}\\right)\\left(10.0\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)\\hfill \\\\ & =0.\\hfill \\end{array}$\n\n## Significance\n\nThis value is a gauge pressure, since the initial pressure was given as a gauge pressure. Thus, the nozzle pressure equals atmospheric pressure as it must, because the water exits into the atmosphere without changes in its conditions.\n\n## Summary\n\n\u2022 Bernoulli\u2019s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:\n${p}_{1}+\\frac{1}{2}\\rho {v}_{1}^{2}+\\rho g{h}_{1}={p}_{2}+\\frac{1}{2}\\rho {v}_{2}^{2}+\\rho g{h}_{2}.$\n\u2022 Bernoulli\u2019s principle is Bernoulli\u2019s equation applied to situations in which the height of the fluid is constant. The terms involving depth (or height h ) subtract out, yielding\n${p}_{1}+\\frac{1}{2}\\rho {v}_{1}^{2}={p}_{2}+\\frac{1}{2}\\rho {v}_{2}^{2}.$\n\u2022 Bernoulli\u2019s principle has many applications, including entrainment and velocity measurement.\n\nWhat is a volt equal to?\nlist and explain the 3 ways of charging a conductor\nconduction convention rubbing\nAsdesaw\nformula of magnetic field\nIntegral of a vector\ndefine surface integral of a vector?\nRahat\nthe number of degree freedom of a rigid body in2-dimantion is:\n1\nNathan\nA block (A) of weight 5 kN is to be raised by means of a 20\u00b0 wedge (B) by the application of a horizontal force (P) as shown in Fig.1. The block A is constrained to move vertically by the application of a horizontal force (S). Find the magnitude of the forces F and S, when the coefficient of fricti\nDanilo\nA body receives impulses of 24Ns and 35Ns inclined 55 degree to each other. calculate the total impulse\nA body receives impulses of 24Ns and 35Ns inclined 55 degree to each other. calculate the total impulse\nPrevious\ntwenty four square plus thirty-five square minus to multiple thirty five twenty four and equal answer number square Via this equation defined Total Total impulse\nCemal\nwhy simple pendulum do not vibrate indefinitely?\nZirmal\ndefine integral vector\nRahat\nwhat is matar\ndefine surface integral vector?\nRahat\nThe uniform boom shown below weighs 500 N, and the object hanging from its right end weighs 400 N. The boom is supported by a light cable and by a hinge at the wall. Calculate the tension in the cable and the force on the hinge on the boom. Does the force on the hinge act along the boom?\nA 11.0-m boom, AB , of a crane lifting a 3000-kg load is shown below. The center of mass of the boom is at its geometric center, and the mass of the boom is 800 kg. For the position shown, calculate tension T in the cable and the force at the axle A .\nJave\nwhat is the S.I unit of coefficient of viscosity\nDerived the formula of Newton's law of universal gravitation Fg=G(M1M2)\/R2\nhi\nAsdesaw\nyes\nCemal\na non-uniform boom of a crane 15m long, weighs 2800nts, with its center of gravity at 40% of its lenght from the hingr support. the boom is attached to a hinge at the lower end. rhe boom, which mAKES A 60% ANGLE WITH THE HORIZONTAL IS SUPPORTED BY A HORIZONTAL GUY WIRE AT ITS UPPER END. IF A LOAD OF 5000Nts is hung at the upper end of the boom, find the tension in the guywire and the components of the reaction at the hinge.\nwhat is the centripetal force\nOf?\nJohn\ncentripetal force of attraction that pulls a body that is traversing round the orbit of a circle toward the center of the circle. Fc = MV\u00b2\/r\nSampson\ncentripetal force is the force of attraction that pulls a body that is traversing round the orbit of a circle toward the center of the circle. Fc = MV\u00b2\/r\nSampson\nI do believe the formula for centripetal force is F=MA or F=m(v^2\/r)\nJohn\nI mean the formula is Fc= Mass multiplied by square of velocity all over the Radius of the circle\nSampson\nYes\nJohn\nThe force is equal to the mass times the velocity squared divided by the radius\nJohn\nThat's the current chapter I'm on in my engineering physics class\nJohn\nCentripetal force is a force of attraction which keeps an object round the orbit towards the center of a circle. Mathematically Fc=mv\u00b2\/r\nIn Example, we calculated the final speed of a roller coaster that descended 20 m in height and had an initial speed of 5 m\/s downhill. Suppose the roller coaster had had an initial speed of 5 m\/s uphill instead, and it coasted uphill, stopped, and then rolled back down to a final point 20 m bel\nA steel lift column in a service station is 4 meter long and .2 meter in diameter. 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//
// XAxisBand.h
// MIM2D Library
//
// Created by Reetu Raj on 08/07/11.
// Copyright (c) 2012 __MIM 2D__. All rights reserved.
//
#import <UIKit/UIKit.h>
#import <CoreText/CoreText.h>
#import "MIMColorClass.h"
#import "Constant.h"
@interface XAxisBand : UIView {
NSArray *xElements;
float _tileWidth;
float _gridWidth;
float scalingFactor;
float gapDistance;//Gap between two labels;
float groupGapDistance;
float barWidth;
int style;
BOOL lineChart;
BOOL barChart;
BOOL groupBarChart;
BOOL stackedBarChart;
BOOL xIsString;
float lineWidth;
float xAxisHeight;
UIColor *lineColor;
NSDictionary *properties;
NSArray *groupTitles;
float groupTitleOffset;
UIColor *groupTitleColor;
UIColor *groupTitleBgColor;
BOOL hideSticks;
float fontSize;
float xoffset;
}
@property(nonatomic,retain)NSArray *xElements;
@property(nonatomic,assign)float scalingFactor;
@property(nonatomic,assign)int style;
@property(nonatomic,assign)BOOL lineChart;
@property(nonatomic,assign)BOOL barChart;
@property(nonatomic,assign)BOOL xIsString;
@property(nonatomic,assign)float gapDistance;//Gap between two labels;
@property(nonatomic,assign)float lineWidth;
@property(nonatomic,retain)UIColor *lineColor;
@property(nonatomic,retain)NSDictionary *properties;
@property(nonatomic,retain)NSArray *groupTitles;
@property(nonatomic,assign)float groupTitleOffset;
@property(nonatomic,assign) float fontSize;
-(void)drawXAxis:(CGContextRef)ctx;
@end
| {
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How Designers Would Use the Blackest Black Ever Created
Vantablack is the darkest man-made substance ever created, and top designers can't wait to get their hands on it.
By: Ryan Reed
Imagine a color — or the absence of color, in this case — that's so dark that when applied, it appears to flatten 3-D objects and turn them into miniature black holes. Such a visually stunning hue would open a world of possibilities for designers who currently only have various finishes of black to choose from if they want to add the timeless color to a space. Thanks to a recent discovery, though, a new kind of black may find its way into designer's hands one day soon.
Vantablack is the world's darkest man-made substance.
Photo by: Surrey NanoSystems
Surrey NanoSystems
Created by Surrey NanoSystems in 2014, Vantablack, which is not a color but a collection of millions of carbon nanotubes, is recognized by Guinness World Records as the world's darkest man-made substance, reflecting only .036 percent of the light that strikes it. It's a remarkable scientific accomplishment that researchers have continued to develop and make even darker (if you can believe that). There's an updated version of the original Vantablack that's so black that light seems to disappear when it hits the surface.
Scientists have also developed a version of Vantablack called S-VIS that can be applied via spray, leading some to call it the world's blackest paint. It's not as dark as the original, but it's still manufactured from carbon nanotubes and achieves the unique 3-D flattening effect. Despite the spray application, S-VIS is not available to the public, as it requires a complex post-process to achieve the finished look. There's a non-nanotube version called VBx1 (shown below), and it's more tolerant than other versions but as of now, it's unreleased.
A scientist holding a 3D object coated in Vantablack paint.
Vantablack is so dark that when applied, it appears to flatten a 3D object.
While Vantablack may be better suited for outer space than your living room right now, it's clear this material has broad appeal and could reach multiple industries in our lifetime. With that in mind, I reached out to several designers known for their modern aesthetic and use of the color black to see how they would use Vantablack. Like contestants on Food Network's Chopped, these designers had to think fast and come up with their best and most innovative use of Vantablack. Here's what they had to say:
Phara Queen / Phara Queen Design
"This product would be fun to use on the floor in a game room, playroom or even an entryway. It would give the space an "Alice down the rabbit hole" effect. By painting a random checkboard on the floor, people will stop and wonder if they step on the black square will they just keep falling through the floor? A real conversation starter, for sure!"
Caitlin Murray / Black Lacquer Design
"As a company that stokes the power of black in design, we could easily be tempted to go overboard experimenting with Vantablack as a finish; however, I think it would be most refined in small doses to punctuate seamless silhouettes. I would love to use it to finish a sleek, contemporary frame to a circular mirror. Contrasting the light-absorbent Vantablack frame with a bright, reflecting mirror would provide a striking and unexpected conversation piece, producing a dream-like effect with the mirror appearing to float."
Nicole White Quinn / Nicole White Designs Interiors
"Aww, what a thrill it would be to have this blackest of black for a master bedroom retreat. I love the moody and sexy feel of black paint in a bedroom, and with a matte finish that would absorb light, it will be the perfect space to retreat to every night to soothe away the worries of the world. I'd paint all molding details — baseboards, crown moldings and trims — in black as well with just a hint of sheen for contrast. A white tufted headboard with black nailhead details would add just enough light to the room to truly showcase the color of the walls. I'd finish it off with a bit of bling with an Odeon-style chandelier, then I'd never leave because it'd be my master suite!"
Laura Knight-Keating / Lauren Rubin Architecture
"Vantablack is amazing. It creates depth and almost looks like what I would expect infinity to look like. We would use the Vantablack on the ceilings of industrial spaces, like a gym or bar, where the ceilings need to disappear and feel limitless. We would also use it on walls where we would layer it with graffiti art, interesting wall stickers or layers of bright paint. It would almost create a 3-D effect to the colors and images on top. We almost feel like the color could go on forever!"
Designer Secrets to Decorating With Black
A Minimalist's Guide to Designing With the Color Black
The perfect way to decorate using the color black.
Decorating With Black
Check out these 6 tips for using the color that's always in.
Dramatic and Sexy Black Decorating Ideas
Not just for trim, painting with black can be a smart, stylish choice for an entire wall. See how five designers use the paint color to a room's advantage.
Is Bolder Better? Here's How to Add More Color + Pattern
If your mantra is 'more is more,' these tips are for you.
Hollywood Regency Style: Get the Look
Few interior design styles carry with them the glitz and glamour of an entire decade. But Hollywood Regency is making a modern comeback, well after its heyday in the '30s.
How to Find Your True Design Style in 3 Simple Steps
You're in good hands, we promise.
How to Match Paint Colors
Whether you're trying to find paint to match an existing wall color or pull your inspiration color from an upholstered sofa, family heirloom or your favorite flower, here's how you can get that perfect color match in a can of paint.
How This Interior Designer Decorated Her Own Home
When designer Darla Bankston May set out to redesign her own home, an open concept space perfect for entertaining was at the top of her list of must-haves. She also wanted to display pieces she had collected while traveling, giving the space an eclectic and personal vibe.
Designers Reveal How to Decorate Any Space With Black and White
HGTV Magazine has all the pro tips—and accessories—you need to get started.
How To: Creating A 3-D Collage Backsplash
This one-of-a-kind idea lets you display your favorite collection, from silverware to shells, in a 3-D collage.
100 Beautiful Kitchen Island Inspiration Ideas
5 Best Portable Air Conditioners Under $525
Quick Tips for Organizing Bathrooms
Designing a Home Lighting Plan
5 Steps to Organizing Your Closet
Glass Tile Backsplash Ideas | {
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Košetice () is a municipality and village in Pelhřimov District in the Vysočina Region of the Czech Republic. It has about 700 inhabitants.
Košetice lies approximately north-west of Pelhřimov, north-west of Jihlava, and south-east of Prague.
Administrative parts
The village of Nová Ves is an administrative part of Košetice.
References
Villages in Pelhřimov District | {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/407597\/derivative-of-the-electromagnetic-tensor-invariant-f-mu-nuf-mu-nu","text":"# Derivative of the electromagnetic tensor invariant $F_{\\mu\\nu}F^{\\mu\\nu}$\n\nThe electromagnetic field tensor is $F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$. I am trying to calculate the quantity $$\\frac{\\partial(F_{\\alpha\\beta}F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}.$$ This calculation arises when trying to derive the electromagnetic equations of motion (i.e. Maxwell's equations) from the Lagrangian $\\mathcal{L}=C F_{\\mu\\nu}F^{\\mu\\nu}$. According to P. 14 of these online notes, this derivative is $$\\frac{\\partial(F_{\\alpha\\beta}F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}=2F^{\\alpha\\beta}\\frac{\\partial F_{\\alpha\\beta}}{\\partial(\\partial_\\mu A_{\\nu})} .$$\n\nThis result surprises me. I can use the product rule to find\n\n$$\\frac{\\partial(F_{\\alpha\\beta}F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}=F_{\\alpha\\beta}\\frac{\\partial(F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}+F^{\\alpha\\beta}\\frac{\\partial(F_{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}$$\n\nand it is clear that if $F_{\\alpha\\beta}\\frac{\\partial(F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}=F^{\\alpha\\beta}\\frac{\\partial(F_{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}$ then you get the desired result. However, I can't see why this is true. In particular, I don't understand how to take the derivative $$\\frac{\\partial(F^{\\alpha\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}=\\frac{\\partial(\\partial^{\\alpha}A^{\\beta}-\\partial^{\\beta}A^{\\alpha})}{\\partial(\\partial_{\\mu}A_{\\nu})}=\\frac{\\partial(\\partial^{\\alpha}A^{\\beta})}{\\partial(\\partial_{\\mu}A_{\\nu})}-\\frac{\\partial(\\partial^{\\beta}A^{\\alpha})}{\\partial(\\partial_{\\mu}A_{\\nu})}$$ where the downstairs part has lower indices and the upstairs part has upper indices.\n\n\u2022 Can't you just use the metric to lower the indices in the numerator? May 22 '18 at 20:39\n\nFor your first question: the components of the metric don't depend on $\\partial_\\mu A_\\nu$, or for that matter, anything at all. So we have, e.g. $$J^\\mu \\partial K_\\mu = J^\\mu \\partial (\\eta_{\\mu\\nu} K^\\nu) = J^\\mu \\eta_{\\mu\\nu} \\partial K^\\nu = J_\\nu \\partial K^\\nu = J_\\mu \\partial K^\\mu$$ where $\\partial$ stands for any kind of derivative whatsoever and $J$ and $K$ are arbitrary. The proof for your case is identical.\nFor your second question: simply do the same thing. Note that $$\\frac{\\partial J^\\nu}{\\partial J_\\mu} = \\frac{\\partial (\\eta^{\\rho\\nu} J_\\rho)}{\\partial J_\\mu} = \\eta^{\\rho\\nu}\\frac{\\partial J_\\rho}{\\partial J_\\mu} = \\eta^{\\rho \\nu} \\delta^\\mu_\\rho = \\eta^{\\mu\\nu}$$ where you can adapt this reasoning to your own example. After you do this a couple times, it becomes completely second nature, and you won't have to write out the steps. Everything works out exactly how you would expect, just \"lining up the indices\", $$\\frac{\\partial J^\\nu}{\\partial J_\\mu} = \\eta^{\\mu\\nu}, \\quad \\frac{\\partial J^\\nu}{\\partial J^\\mu} = \\eta^{\\nu}_\\mu, \\quad \\frac{\\partial J_\\nu}{\\partial J_\\mu} = \\eta_\\nu^\\mu, \\quad \\frac{\\partial J_\\nu}{\\partial J^\\mu} = \\eta_{\\mu\\nu}$$ where, in order to write all four results the same way, I defined $\\eta^\\mu_\\nu = \\delta^\\mu_\\nu$.\n$$F^{ab} = g^{ac}g^{bd}F_{cd}$$","date":"2021-09-21 12:27:18","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\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9217402338981628, \"perplexity\": 145.51072798454123}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057202.68\/warc\/CC-MAIN-20210921101319-20210921131319-00235.warc.gz\"}"} | null | null |
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Q: if f and g are monotonically increasing functions, such that f(g(n))=O(n) and f(n)=Ω(n) then g(n)=O(n) I have to prove this statement :
if $f$ and $g$ are monotonically increasing functions, such that $f(g(n))=O(n)$ and $f(n)=Ω(n)$ then $g(n)=O(n).$
A: By truncating the functions we can assume that the starting indices are $0$. That is,
$$ \exists c,c_0 > 0\mid \forall n \in \mathbb{W}, f(g(n)) \leq c n, f(n) \geq c_0 n
$$
Then we have
$$ c n \geq f(g(n)) \geq c_0 g(n)
$$
as desired.
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/en.pythonmana.com\/2022\/132\/202205150257457398.html","text":"# Website backend of online book purchase function based on Python\n\n2022-05-15 02:58:16\n\n## One . The experimental requirements\n\nRealize a website backend that provides the function of online book purchase .\nThe website supports booksellers to open stores on it , Buyers may buy through the website .\nBoth buyers and sellers can register their own accounts .\nA seller can open one or more online stores , Buyers can recharge their accounts , Buy books at any store .\nSupport ordering -> payment -> deliver goods -> Receiving goods , technological process .\n\n1. Realize the function of the corresponding interface , see doc Below .md File description \uff0860% fraction \uff09\n\nThese include \uff1a\n\n1) User authority interface , If registered \u3001 Sign in \u3001 Log out \u3001 Cancellation\n\n2) Buyer user interface , If recharge \u3001 Place an order \u3001 payment\n\n3) Seller user interface , Such as creating a store \u3001 Fill in the book information and description \u3001 Increase inventory\nPass the corresponding function test , all test case all pass\nTest the performance of order and payment interfaces \uff08 It's best to separate load generation and backend \uff09, Measure the number of transactions supported per minute , Delay, etc\n\n2. Add other functions to the project \uff1a\uff0840% fraction \uff09\n\n1) Realize the subsequent process\ndeliver goods -> Receiving goods\n\n2) Search for books\nUsers can search by keyword , Parameterized search method ; If the search scope includes , subject , label , Catalog , Content ; Full site search or current store search . If the display result is large , Paging required ( Use full-text indexing to optimize lookup )\n\n3) The order status , Order inquiry and cancellation\nUsers can check their own historical orders , The user can also cancel the order .\nCancel the order \uff08 optional , Bonus points +5~10\uff09, The buyer voluntarily cancels the order , If the buyer places an order after a time-out , If the buyer fails to pay , The order will also be automatically cancelled .\n\n## Two . Project operation\n\nYou need to initialize the database before running the project .\n\nstay MongoDB Create a file named bookstore The database of . Then create two document sets , The document set names are book and history_order, And according to the following 3.5.4 in MongoDB Part of the index .\n\nPostgreSQL The initialization of the database is already in store.py Establish a good , There is no need to manually create .\n\nThen enter the following statement to run the project .\n\npython app.py\n\n\nOr according to seven . The steps in the cloud on the project run the project using the cloud database .\n\n## 3\u3001 ... and . Database design\n\n### 3.1 Overall design concept\n\nThe following design is based on the following reasonable assumptions ,\n\n1. Users often place orders, pay and cancel\n2. Users often make mistakes, but they usually don't make mistakes deliberately\n3. Relative to the seller , There are more buyers and their rights and interests should be protected\n4. Users don't know about books when searching\n\nIn order to ensure the system function , And try to improve the performance of the system , This project ER The diagram and database design are as follows .\n\nThe optimal design of database is in 3.5 Section .\n\n### 3.3 relational database\n\nMost of the content in the project is stored in Postsql In a relational database . Next, we will introduce the relational database .\n\nAfter many tests , Added foreign keys \u3001 only \u3001 After non empty constraints, the database performance is greatly reduced . So while the program is running , Use code and error handling to ensure the correctness and rationality of the data in the database . For the database itself, no constraints are explicitly added . This can ensure the high performance of the database .\n\n#### 3.3.1 users table\n\n\u2022 Design thinking\n\nusers Table is used to store user entity classes and their attributes . Each row in the table corresponds to a user , Its basic properties are recorded .\n\n\u2022 Table structure\n\nuser_id\uff1astring type , It is the only primary key of this table . Used to record user name .\n\npassword\uff1astring type . A secret text used to record a user's password . For security reasons , The database does not store plaintext passwords .\n\nbalance\uff1ainteger type , The initial value is 0. Used to record the amount in the user's account .\n\ntoken\uff1astring type . Used to record the user name at login \u3001 Mark generated by time and terminal number . When carrying out important operations , This attribute needs to be checked to determine that the message comes from the same terminal .\n\nterminal\uff1astring type . It is used to record the terminal number at login .\n\n#### 3.3.2 user_store table\n\n\u2022 Design thinking\n\nuser_store Table is used to store store store entity classes and store contact classes . Each line in the table is a store .\n\n\u2022 Table structure\n\nuser_id\uff1astring type . Used to record the user name of the owner of the store .\n\nstore_id\uff1astring type , Is the primary key of this table . Used to record store name .\n\n#### 3.3.3 store table\n\n\u2022 Design thinking\n\nThis table is used to store the inventory information of the bookstore , Each row in the table represents the inventory information of a book in a store .\n\n\u2022 Table structure\n\n[ Failed to transfer the external chain picture , The origin station may have anti-theft chain mechanism , It is suggested to save the pictures and upload them directly (img-Hk4qjwkw-1652166156113)(https:\/\/www.writebug.com\/myres\/static\/uploads\/2022\/4\/25\/f4a1898816fe188be3b018337e58f296.writebug)]\n\nstore_id\uff1astring type , Is one of the primary keys of this table . Used to record the name of the store where the book is located .\n\nbook_id\uff1astring type , Is one of the primary keys of this table . Used to record the corresponding books id.\n\nstock_level\uff1ainteger type . Used to record the number of books in stock .\n\nprice\uff1ainteger type . Used to record the unit price of books . In different stores , The unit price of books can be different .\n\n#### 3.3.4 new_order table\n\n\u2022 Design thinking\n\nThe form stores information about the order , Each record records information not related to books in the order .\n\n\u2022 Table structure\n\norder_id\uff1astring type , Is the primary key of this table . The order number of the order is recorded .\n\nuser_id\uff1astring type . The user name of the order buyer is recorded .\n\nstore_id\uff1astring type . The store name of the merchant who recorded the order .\n\nstatus\uff1ainteger type . The status of the order is recorded , The default value is 1. among 1 Indicates that an order has been placed but no payment has been made ,2 Indicates paid but not shipped ,3 On behalf of shipped but not received ,4 Indicates that the goods have been received ,0 Indicates that the order has been cancelled .\n\ntotal_price\uff1ainteger type . The total price of the order is recorded .\n\norder_time\uff1ainteger type . The order placing time of the order is recorded , In the form of a timestamp . Used for automatic cancellation and other businesses .\n\n#### 3.3.5 new_order_detail_table\n\n\u2022 Design thinking\n\nThe form stores details related to the order and book purchase contact class , Each record records information about the purchase of a book in an order .\n\n\u2022 Table structure\n\norder_id\uff1astring type , Is one of the primary keys of this table . The order number of the order is recorded .\n\nbook_id\uff1astring type , Is one of the primary keys of this table . Used to record the corresponding books in the order id.\n\ncount\uff1ainteger type . Record the quantity of this book purchased in this order .\n\n#### 3.3.6 invert_index table\n\n\u2022 Design thinking\n\ninvert_index table Inverted table is used to store the index relationship between keywords and books . The specific construction and search functions are described in detail .\n\n\u2022 Table structure\n\nsearch_key\uff1astring type , One of the primary keys of the inverted table . Records the primary key of the inverted table , When the user enters the contents of the inverted table , This table can return the information of the bibliography found .\n\nsearch_id\uff1a Sequence type , One of the primary keys of the inverted table . According to key Insert in turn to generate self increasing integer Sequence . Used for page query .\n\nbook_id\uff1astring type . Records the corresponding books id, Is one of the return values of the inverted table .\n\nbook_title\uff1astring type . The title of the corresponding book is recorded . Return as a redundant attribute for buyers to view .\n\nbook_author\uff1astring type . The author of the corresponding book is recorded . Return as a redundant attribute for buyers to view .\n\n### 3.4 Document database\n\n#### 3.4.1 book collection\n\n\u2022 Design thinking\n\nbook collection Used to store book information , Each document object stores information about a Book .\n\n\u2022 Table structure\n\nid\uff1astring type . Of the record book id, Every book id only .\n\ntitle\uff1astring type . The title of the record book , Theoretically, it cannot be empty .\n\nauthor\uff1a Most of them are string type , It can be for null. The author of the record book .\n\npublisher\uff1astring type . The publisher of the record book .\n\noriginal_title\uff1a Most of them are null, Non null value is string type . Record the original title of the foreign language book .\n\ntranslator\uff1a Most of them are null, Non null value is string type . The translator who records foreign language books .\n\ncountry\uff1a Most of it is empty , Non null value is string type . Record the nationality of the author of the foreign language book .\n\npub_year\uff1astring type . Date of publication of the record book .\n\npages\uff1aInt32 type . The number of pages in the record book .\n\ncurrency_unit\uff1astring type , Can be an empty string . The monetary unit of record price . Most of them are \u201c element \u201d.\n\nisbn\uff1astring type . Of the record book ISBN Number .\n\nauthor_intro\uff1astring type , Can be an empty string . A brief introduction to the author of the record book .\n\nbook_intro\uff1astring type , Can be an empty string . Introduction to the record book .\n\ncontent\uff1astring type , Can be an empty string . Catalogue of record books .\n\ntags\uff1aarray type , Each item is a string . Labels and keywords of the record book .\n\npictures\uff1aarray type , Each item is a picture . Pictures of the record book .\n\n#### 3.4.2 history_order collection\n\n\u2022 Design thinking\n\nhistory_order collection Used to store historical orders , Each document object records an order record .\n\n\u2022 Table structure\n\norder_id,user_id,store_id,status,total_price,order_time\uff1a Record information and format PostgreSQL The data in the database is exactly the same . Specially ,status The value can only be 0 and 4, That is, cancelled and received . This part of the information of the order in progress is saved in new_order in .\n\nbooks\uff1a An array type , Each of the items in the array is a child document . There are... In the sub document book_id and count two , It records the book information in the order . Record information and format PostgreSQL The data in the database is exactly the same . This part of the information of the order in progress is saved in new_order_detail in .\n\n### 3.5 Relational schema optimization\n\n#### 3.5.1 Splitting and merging of tables\n\nDue to the large number of orders , These are mainly historical orders , The number of orders in progress accounts for a very small part of the total number of orders . So split the order table , Divided into completed orders and unfinished orders . Two completed orders are used SQL Table to record , And the unfinished order uses a MongoDB Document set to record .\n\nFor outstanding orders , Because he needs to update the status at any time , And it involves automatic cancellation and other operations . If put together with historical orders , It will increase the complexity of the query . And for these orders , What is often visited is the total price , Information such as status , And not often visit new_order_detail Book information in , And often new_order Change the status in , So we decided to use SQL To store .\n\nOnly completed historical orders will hardly change , It is mainly used to show users' historical orders , This way is very suitable for directly recording with one document . And another task of taking out historical orders is to make recommendations , Recommendation is the information needed, the information of the books purchased , If you still use SQL The way of storage , You need to take out many book records in the order each time , Lead to problems such as low efficiency . So the historical order adopts mangoDB To store .\n\n#### 3.5.2 Redundant attributes are introduced\n\nIn this project , The most important redundant attribute is the attribute in the inverted table . In the inverted list search_key The keyword records the title of the book \u3001 author \u3001 label \u3001 Introduce the inverted information of keywords and other information , Using this table can speed up the search for books . If the inverted table only provides book_id, Do not add the title and author attributes of redundant books in the database . Users can only use the ID Look it up again MongoDB Book information in , This is not only inefficient , And not in line with user habits . Therefore, these two important information for books are added to the inverted table as redundant attributes , Is in line with the user's search habits , And it improves efficiency .\n\nBesides , The total order price as the derivation attribute is also added to the order table . This is because incomplete orders need to access the total price of the order during payment and receipt . If the information in other tables is used for field calculation every time , It will cause a great waste of resources .\n\nAlthough redundant information is added to this database , Part of the paradigm is still good . Remove inverted index , The tables of this project conform to BCNF.\n\n#### 3.5.4 Index creation\n\nThis project involves two kinds of databases , Respectively Postgresql and MongoDB\n\n##### 3.5.4.1 Postgresql\n\nOnly the primary key index is used , No other index is used . The reason is that query and other factors have been taken into account when creating tables , Therefore, it can have better performance without establishing other indexes .\n\n##### 3.5.4.2 MongoDB\n\nIn order to speed up the business , about MongoDB Two tables of , We indexed it\n\n##### Index1(B-tree Indexes ): Book information sheet\n\nReason for adding index \uff1aid The index is used to speed up the query business ,author\u3001publisher\u3001 as well as tags The index is used to improve the speed of recommending Services .\n\n##### Index2(B-tree Indexes ): Historical order information table\n\nReason for adding index \uff1auser_id The index is used to speed up the query of buyer history information ,user_id The index of is used to speed up the query of seller's historical information . At the same time, they can indirectly accelerate the speed of recommendation business .\n\n## Four . Function function\n\n#### summary\n\nWe have achieved auth, seller, buyer Business logic of three roles , The performance optimization and corresponding interface test are completed .\n\nFeatures include registration , Sign in , Books on the shelves, etc Basic function ; Receipt and delivery , Manually cancel the order , Check the order \uff08 At present \/ history \uff09,\uff08 Multiple keywords \uff09 Search, etc advanced function ; Besides , We have realized automatic cancellation of orders , Search for books , real time OCR Search for , Small recommendation system , In store search, etc Expand functions .\n\nIn terms of performance improvement , We took RETURNING keyword Reduce unnecessary queries , Corresponding MongoDB Operation for FindOneAndUpdate. meanwhile , introduce **\u201c Soft real time \u201d** Concept to reduce the resource consumption of multithreading .\n\nThe following is the of this project Highlight features\n\n4.1.8 Recommendation system \/ 4.2.2 Books on the shelves \/ 4.2.5 Query the current order \/ 4.2.6 Query historical orders \/ 4.3.2 Place an order \/ 4.3.3 payment \/ 4.3.6 Cancel the order automatically \/ 4.3.7 Search for \/ 4.3.8 Multi keyword search \/ 4.3.9 Search for books \/ 4.3.10 real time OCR Search for \/ 4.3.11 Get book information \/ 4.3.12 Search in the store\n\n### Transaction processing \uff08 Bright spot \uff09\n\nThis project has made relatively perfect transaction processing , stay UPDATE and DELETE And we used that RETURNING keyword , Improved efficiency , But there are still the following possibilities \uff1a After updating or deleting , After taking out the data , The system may judge that the operation cannot be completed according to the extracted data , At this point... Must appear Roll back rollback. The important functions of this project are As a business complete , At the end of the transaction , perform commit Submit . We are interested in basic functions , Advanced functions and so on have carried out transaction processing , Ensure the normal operation of the project .\n\n### 4.1 User rights function\n\n#### 4.1.1 register\n\n\u2022 Function realization\n\n1. according to user_id stay users Query whether the user already exists in the table\n2. If it does not exist , Then insert a new user (user_id, password, balance, token, terminal) To users In the table\n\u2022 Performance analysis\n\nA query users table, Insert... Once users table, Access the database twice\n\n\u2022 Function realization\n\n1. according to user_id stay users Get the password from the table password\n2. Will get password Compare with the password entered by the user\n3. Update user's token, terminal\n\u2022 Performance analysis\n\nA query users table, An update users table, Access the database twice\n\n#### 4.1.3 Log out\n\n\u2022 Function realization\n\n1. according to user_id stay users table Query in , Judge whether the login information is invalid\n2. Update user token\n\u2022 Performance analysis\n\nA query users table, An update users table, Access the database twice\n\n#### 4.1.4 Cancellation\n\n\u2022 Function realization\n\n1. according to user_id stay users table Query whether the user exists\n2. Delete users table The user entry in\n\u2022 Performance analysis\n\nA query users table, An update users table , Access the database twice\n\n\u2022 Function realization\n\n1. according to user_id stay users table Query the user's original password\n2. Judge whether the user's original password is the same as the user's new password\n3. If different , Update users table Of this user in password\n\u2022 Performance analysis\n\nA query users table, An update users table, Access the database twice\n\n#### 4.1.6 Query the current order\n\n\u2022 Function realization\n\n1. according to user_id stay new_order table Query all order information of users in\n2. For each of these order_id stay new_order_detail Query book information in\n\u2022 Performance analysis\n\nA query new_order table , A query new_order_detail table, Access the database twice\n\n#### 4.1.7 Query historical orders\n\n\u2022 Function realization\n\n1. according to user_id stay MongoDB history_order collection Query all order information in\n\u2022 Performance analysis\n\nA query MongoDB history_order collection, visit MongoDB Database once\n\n#### 4.1.8 Recommendation system\n\nRecommendation system is one of the features of this experiment , It's more complicated to realize , The complexity of execution is also relatively high . This recommendation system uses the user's historical order information , Personalized recommendation for users . The recommendation system will recommend the user and the user's historical purchase label Jarcard Recommend books with high similarity .\n\n\u2022 Function realization\n\n1. First we start with MongoDB Find out the user's historical orders , These include books purchased by users , Take it out ID\u3001 title \u3001 author \u3001 label \u3001 And publishers . The recommendation weight of books that have been purchased will be set to the lowest .\n2. Use this information to find books similar to the book . from MongoDB In the database, find out the same author of a Book purchased by the user in turn \u3001 Or the same publishing house \u3001 Or other books with the same label . In the process, the books that have been taken out are removed , Take out the spare label, etc .\n3. Calculate the value of all the books taken out and the user's historical purchase labels Jarcard Similarity degree ( J a r c a r d ( A , B ) = \u2223 A \u2229 B \u2223 \u2223 A \u222a B \u2223 ) (Jarcard(A,B)=\\frac{|A\\cap B|}{|A\\cup B|}) And sort , Take out top5 And return the book information to the user .\n\u2022 Performance analysis\n\nThe recommended function requires Visit multiple times MongoDB database . The number of visits to the database is positively correlated with the number of historical orders and the number of books purchased . Although each query will use the index established on the document set , However, due to the complexity of query , The cost of time is still high . Under the current recommended algorithm , If you don't use databases and indexes , Need to traverse every book , With O ( n 2 ) O(n^2) Time complexity or space complexity are recommended . Therefore, reasonable use of database and establishment of appropriate index can reduce the complexity .\n\neachbook = self.mongo['book'].find_one({\n'id': boughtbook['book_id']},{\n'_id': 0, 'id': 1, 'title': 1, 'author': 1, 'tags': 1, 'publisher':1})\n\nbooks = self.mongo['book'].find({\n'$or': [{ 'author': eachbook['author']},{ 'publisher': eachbook['publisher']},{ 'tags': { '$elemMatch': {\n'$in': eachbook['tags']}}}]},{ '_id': 0, 'id': 1, 'title': 1, 'author': 1, 'tags': 1}) \uff08 The next highlight feature is 4.2.2\uff09 ### 4.2 Seller function #### 4.2.1 Create a store \u2022 Function realization 1. Respectively in users table and store table Query in users_id and store_id Does it already exist 2. If it does not exist , Insert user user_id and store_id To user_store table \u2022 Performance analysis A query users table, A query store table, Insert... Once user_store table, Access the database three times #### 4.2.2 Books on the shelves \u2022 Function realization 1. Check user_id,store_id,book_id Whether there is 2. according to book_id stay Mongodb Of book collection Query in book Whether there is . If exist , Then there is no need to add MongoDB Medium book Document set . If it does not exist , Then sort out the data according to the following steps . 3. take book_json_str Convert to dictionary format . First match the author string according to the parentheses , Separate the author's name and nationality . This is to prevent the brackets and nationality included at the beginning of the author's name from making the inverted list nonstandard . 4. Then extract the Book Introduction \u3001 Author's brief introduction \u3001 Keywords in the directory . utilize jieba In the participle textrank Take out the key words , Remove the weight and add the label to the book . 5. Create an inverted table , Break the title of the book into words , Form a suffix according to the word segmentation results , Separate the prefix of each suffix and add it to the inverted table as a keyword , The author of the book \u3001 Similar treatment for labels, etc , Insert this information into the inverted table invert_index table. 6. take (store_id, book_id, stock_level, price) Insert store table \u2022 Performance analysis A query users table, A query user_store table, A query store table, A query Mongodb Of book collection, Insert... Once store table, Insert invert_index table Several times , Access the database at least five times # --- Join the inverted index and start --- # Put the title of the new book \u3001 author \u3001 The label is added to the inverted index table preffixs = [] title = book_info_json.get(\"title\") preffixs += nlp.get_middle_ffix(title) if \"author\" in book_info_json.keys(): names = parse_name(book_info_json.get(\"author\")) for i in range(1,len(names)): preffixs += nlp.get_preffix(names[i]) preffixs += nlp.get_preffix(book_info_json.get(\"author\")) if \"original_title\" in book_info_json.keys(): preffixs += nlp.get_preffix(book_info_json.get(\"original_title\")) if \"translator\" in book_info_json.keys(): names = parse_name(book_info_json.get(\"translator\")) for i in range(1, len(names)): preffixs += nlp.get_preffix(names[i]) preffixs += nlp.get_preffix(book_info_json.get(\"translator\")) preffixs = list(set(preffixs)) for preffix in preffixs: self.conn.execute( \"INSERT into invert_index(search_key, book_id, book_title, book_author) \" \"VALUES (:sky, :bid, :til, :asr)\", { 'sky': preffix, 'bid': book_id, 'til': title, 'asr': author}) # --- End of adding inverted index --- \uff08 The next highlight feature is 4.2.5 and 4.2.6\uff09 #### 4.2.3 Add inventory \u2022 Function realization 1. Check user_id, store_id,book_id Whether there is 2. according to store_id,book_id Look for the stock of a book in the store , And in store table In the update \u2022 Performance analysis A query users table, A query user_store table, A query store table, An update store table, Access the database four times #### 4.2.4 Seller delivery \u2022 Function realization 1. Check store_id,book_id Whether there is 2. according to order_id stay new_order table Update order status in \u2022 Performance analysis A query user_store table, A query store table, An update new_order table, Access the database three times #### 4.2.5 Query the current order \u2022 Function realization 1. according to seller_id stay new_order and user_store table Query all order information in 2. For each of these order_id stay new_order_detail Query book information in \u2022 Performance analysis A query user table , A connection query new_order and user_store table, Multiple queries new_order_detail table, Access the database at least three times . Because join queries use primary keys , So the performance is not very poor . cursor = self.conn.execute( \"SELECT o.order_id, o.store_id, o.status, o.total_price, o.order_time \" \"FROM new_order o, user_store s \" \"WHERE s.user_id = :user_id AND s.store_id = o.store_id \", { \"user_id\": seller_id, }) #### 4.2.6 Query historical orders \u2022 Function realization 1. according to store_id stay MongoDB history_order collection Query all order information of the store \u2022 Performance analysis A query MongoDB history_order collection, visit MongoDB Database once orders = self.mongo['history_order'].find({ 'store_id': store_id}, { '_id': 0}) \uff08 The next highlight feature is 4.3.2 and 4.3.3\uff09 ### 4.3 Buyer function #### 4.3.1 Recharge \u2022 Function realization 1. according to user_id Get the user password 2. Compare the user password with the password entered by the user 3. If the passwords match , Update the user in users table Balance in \u2022 Performance analysis A query users table, An update users table, Access the database twice #### 4.3.2 Place an order \u2022 Function realization 1. Check user_id, store_id Whether there is 2. According to the order information \uff08store_id,book_id\uff09 stay store Check whether there are corresponding books and sufficient inventory in the merchant in the table . 3. If stock is sufficient , Update store table stock . At this time, we found that , The only information that needs to be retrieved is price The unit price . Inventory attributes can be judged when updating , If there is no qualified line , It will not be updated . thus , We can find that using a select Statements in store The query efficiency is low when executing in the table , Because there is an update statement for the same line after . At this time, we can use RETURNING Keyword will directly return the updated result . Here is the return price I.e. unit price . In this way, through an update operation and RETURNING Keyword to achieve the effect of accessing the database twice , This will reduce and save time . When cancelling or completing an order DELETE The operation also uses RETURNING keyword . 4. Create new order information , take (order_id, book_id, count, price) Insert new_order_detail table 5. Create this order , Calculate the total price of the order total_price, take (order_id, store_id, user_id, total_price, order_time) Insert new_order table, At the same time, the order number order_id Add to unpaid_order Array cursor = self.conn.execute( \"UPDATE store set stock_level = stock_level - :count \" \"WHERE store_id = :store_id and book_id = :book_id and stock_level >= :count \" \"RETURNING price\", { \"count\":count, \"store_id\":store_id, \"book_id\":book_id, \"count\":count}) \u2022 Performance analysis A query users table, A query user_store table, An update store table, Insert... Once new_order_detail table, Insert... Once new_order table, Access the database five times . And no use RETURNING Less than one query store table. #### 4.3.3 payment \u2022 Function realization 1. according to order_id stay new_order table Query order information in 2. Check whether the order timed out 3. If the order does not time out , According to buyer_id stay users table Get the buyer's balance and password 4. according to store_id stay user_store table Query the seller seller_id 5. stay users table Update the buyer's balance in 6. stay new_order table Update order status in status=2 \u2022 Performance analysis A query new_order table, A query users table, A query user_store table, An update users , An update new_order table, Access the database five times . This business speeds up execution by deleting unnecessary queries . \uff084.3.6 And later are the highlights of the project , I hope you take the time to check \uff09 #### 4.3.4 The buyer receives the goods \u2022 Function realization 1. according to order_id stay new_order table The status of the corresponding order in the query , buyers id 2. Check whether the order status is shipped , Order id With the buyer id Whether it corresponds to 3. If the conditions are met , Then update the seller's balance , Update order status to received and add history \u2022 Performance analysis A query new_order table, A query user_store table, Two updates users table, A query users table, Access the database five times #### 4.3.5 Manually cancel the order \u2022 Function realization 1. according to order_id stay new_order table Query whether the order status is placed 2. Check if the user exists , Whether the order number exists 3. From the overall situation unpaid_orders Delete the order from the dictionary 4. according to order_id from new_order table Delete the order in , from new_order_detail table Delete order information in 5. according to store_id and book_id Roll back inventory stock_level 6. Add order information to MongoDB Historical record history_order collection in cursor = self.conn.execute( \"DELETE FROM new_order WHERE order_id = :order_id RETURNING order_id, user_id, store_id ,total_price, order_time \", { \"order_id\": order_id, }) cursor = self.conn.execute( \"DELETE FROM new_order_detail WHERE order_id = :order_id RETURNING book_id, count \",{ \"order_id\": order_id, }) \u2022 Here we take RETURNING Unnecessary keyword queries are reduced (1) At the same time SELECT and UPDATE operation (2) Act on INSERT,UPDATE,DELETE (3) DELETE Return the data before the operation ,INSERT,UPDATE Data returned after operation Corresponding MongoDB operation \uff1aFindOneAndUpdate \u2022 Performance analysis A query new_order, Delete... At a time new_order table, Delete many times new_order_detail , An update store table, Insert... Once MongoDB history_order collection, Access the database at least five times #### 4.3.6 Cancel the order automatically Automatic cancellation of orders uses a \u201c Soft real time \u201d Thought \u2022 Function realization 1. Maintain a global dictionary , Used to record the start time of each order \uff08 Order time \uff09 2. Use datetime Wait for the time tool to record the time when the user places an order , Stored in the global list 3. utilize Apscheduler Scheduler , Achieve each 30 Check the status of orders in the global list once a minute 4. If the order does not time out , Don't do anything , If timeout is detected , Cancel this order . When users place an order , Will reconfirm the payment time . \u2022 Performance analysis It's a kind of similarity \u201c Soft real time \u201d Order status maintenance method based on , The overhead of the timeout cancellation function can be greatly reduced . Not dedicated to a thread \u201c Stare \u201d User's order , Instead, check every cycle , Cancel the order that is checked to be overtime . When users place an order , Will check again for timeout , If the time is out, the order will be cancelled directly . Realization way \uff1a time_limit = 30 # Order lifetime unpaid_orders = { } Use a global dictionary to record the order time status , And set the maximum lifetime of unpaid orders # advantage \uff1a By maintaining the global array to_be_paid, No additional new threads , Minimize the cost def add_unpaid_order(orderID): unpaid_orders[orderID] = get_time_stamp() print(\"add successfully\") print(unpaid_orders) return 200, \"ok\" def delete_unpaid_order(orderID): try: unpaid_orders.pop(orderID) print(unpaid_orders) except BaseException as e: return 530, \"{}\".format(str(e)) return 200, \"ok\" def check_order_time(order_time): cur_time = get_time_stamp() time_diff = cur_time - order_time if time_diff > time_limit: return False else: return True Whenever the cycle of deleting a task comes , Just perform a probe on the global dictionary , Change the status of orders not paid at the specified time to \u201c Cancel \u201d def time_exceed_delete(): del_temp=[] o = Order() print(\"new cycle start\") for (oid,tim) in unpaid_orders.items(): if check_order_time(tim) == False: del_temp.append(oid) # remenber, not to append the index of the array, we need the orderID for oid in del_temp: delete_unpaid_order(oid) o.cancel_order(oid) return 0 By configuring a apscheduler Scheduler to perform scheduled tasks class Config(object): JOBS = [ { 'id': 'soft_real_time', 'func': '__main__:time_exceed_delete', 'trigger': 'interval', 'seconds': 30,#30s For testing purposes only , In actual use, it should be set to 30mins } ] #### 4.3.7 Search for The search function is mainly based on the inverted table , And there are two forms of paged search and non paged search . \u2022 Function realization 1. First read the search keywords from the user, whether pagination and page number are needed , Such parameters page Is equal to zero , No paging . Such as page If it is equal to other numbers, it is the page number that the user needs to find . 2. If pagination is not required , Then find out the keyword corresponding to all books from the inverted table ID\u3001 Title and author , And organize it into an appropriate form and return it to the user . 3. Pagination if necessary , First use SQL In the keyword order by to search_id Ascending sort \u3001 Reuse limit Limit the number of results per page , Last use offset The keyword offset takes out the query result of the corresponding page . Sort it out into an appropriate form and return it to the user . cursor = self.conn.execute( \"SELECT book_id, book_title, book_author from invert_index \" \"where search_key = '%s' \" \"ORDER BY search_id limit '%d' offset '%d';\" % (search_key, self.page_size, page_lower)) \u2022 Performance analysis Due to the existence of inverted table , Single keyword search only needs Access the database once , And because search_key and search_id Is the primary key of the inverted table , Cluster index is built on , So the query speed is very fast . #### 4.3.8 Multi keyword search The result of multi keyword search query is the union of single keyword search results . \u2022 Function realization 1. First read the keyword list of this search from the user . 2. Single keyword search for each keyword , The results are stored in the list . At the same time, the dictionary is used to realize de duplication . 3. Organize the de duplication results into appropriate forms and return them to the user . \u2022 Performance analysis Multi keyword search requires access to X Secondary database ,X Is the number of keywords . Because the number of keywords is often limited , It can be said that visit O(1) An order of magnitude database . #### 4.3.9 Search for books \u2022 Function realization 1. We called Baidu OCR Of API To search for books , The specific operation method is as follows \uff08 Take probability theory and mathematical statistics as an example \uff09, We try to call OCR Function to retrieve this book in the library . 1. Let's go through OCR Technology to extract text from pictures , Return... As a text message , Re pass TextRank The algorithm extracts information from text key word , Include title,author Etc , Take this as key stay invert_index Inversion lists Search in , Single keyword search only needs Access the database once , Get the book name and related information , And because search_key and search_id Is the primary key of the inverted table , Cluster index is built on , So the query speed is very fast . \u2022 TextRank Extract keyword results 1. Definition OCR Method def OCR_pic(self, path): try: print(path) image = get_file_content(path) # Invoke universal literal recognition , The picture is a local picture res = client.general(image) print(res) text = [] for item in res['words_result']: print(item['words']) text.append(item['words']) print(text) text_Seg = [] text_len = len(text) doc = \"\" for i in range(0, text_len): doc += text[i] print(doc) sentence_Seg = ana.textrank(doc) # sentence_Seg = str(sentence_Seg) # sentence_Seg = sentence_Seg.strip(',') print(sentence_Seg) b = Buyer() result = b.search_many(sentence_Seg) except sqlalchemy.exc.IntegrityError as e: return 528, \"{}\".format(str(e)) except BaseException as e: return 530, \"{}\".format(str(e)) return 200, \"ok\", result \u2022 Test and test results You can use it directly postman Wait for the test tool to upload pictures , Extract cover text , Search after word segmentation , Return book information . #### 4.3.10 real time OCR Search for **Remark\uff1a Camera function is required \uff08 It's best to have plenty of light ) ** APP_ID = '14544448' API_KEY = 'yRZGUXAlCd0c9vQj1kAjBEfY' SECRET_KEY = '**********************' # initialization AipFace object client = AipOcr(APP_ID, API_KEY, SECRET_KEY) You need to get an authorization yourself , To call Baidu OCR Of API class OCR(db_conn.DBConn): def __init__(self): db_conn.DBConn.__init__(self) def OCR_pic_cv(self): try: # Get photo saveDir = 'data\/' ''' Call the computer camera to automatically get pictures ''' if not os.path.exists(saveDir): os.makedirs(saveDir) count = 1 # Picture count index cap = cv2.VideoCapture(0) width, height, w = 640, 480, 360 cap.set(cv2.CAP_PROP_FRAME_WIDTH, width) cap.set(cv2.CAP_PROP_FRAME_HEIGHT, height) crop_w_start = (width - w) \/\/ 2 crop_h_start = (height - w) \/\/ 2 print('width: ', width) print('height: ', height) ret, frame = cap.read() # Get photo frame frame = frame[crop_h_start:crop_h_start + w, crop_w_start:crop_w_start + w] # Show the picture frame # frame=cv2.flip(frame,1,dst=None) cv2.imshow(\"capture\", frame) action = cv2.waitKey(1) & 0xFF time.sleep(3) cv2.imwrite(\"%s\/%d.jpg\" % (saveDir, count), cv2.resize(frame, (224, 224), interpolation=cv2.INTER_AREA)) print(u\"%s: %d A picture \" % (saveDir, count)) count += 1 cap.release() # Release camera cv2.destroyAllWindows() # Drop window #ocr Picture get picture text path='.\/data\/1.jpg' image = get_file_content(path) # Invoke universal literal recognition , The picture is a local picture res = client.general(image) print(res) result = [] for item in res['words_result']: print(item['words']) result.append(item['words']) print(result) except sqlalchemy.exc.IntegrityError as e: return 528, \"{}\".format(str(e)) except BaseException as e: return 530, \"{}\".format(str(e)) return 200, \"ok\", result After calling this function, it returns , Aim the cover at the camera to extract the text of the book cover \uff08 It's best to wait five seconds \uff09, Return book information , The result of keyword search query is the union of single keyword search results . #### 4.3.11 Get book information This function is very important for buyers , Used to view the introduction of the book . However, the interface provided by the original project does not give , Therefore, it is included in the highlight function . \u2022 Function realization 1. according to bid_list Each of them bid stay MongoDB book collection Find book information in \u2022 Performance analysis A query MongoDB book collection, Access the database once book = self.mongo['book'].find_one({ 'id': bid},{ '_id':0}) #### 4.3.12 Search in the store In store search is similar to single keyword search , But you need to use join queries . In store search also has two forms: paging search and non paging search . \u2022 Function realization 1. First read the search keywords from the user \u3001 Store name and whether pagination and page number are required , Such parameters page Is equal to zero , No paging . Such as page If it is equal to other numbers, it is the page number that the user needs to find . 2. If pagination is not required , Find out the number of the book corresponding to the keyword and in the store from the inverted list , Of all the books ID\u3001 Title and author , Simultaneously from store Look up the price and inventory in the table , Finally, it is sorted into an appropriate form and returned to the user . 3. Pagination if necessary , The above principle is still used for paging , Sort it out into an appropriate form and return it to the user . cursor = self.conn.execute( \"SELECT i.book_id, i.book_title, i.book_author, s.price, s.stock_level \" \"from invert_index i, store s \" \"where i.search_key = '%s' and i.book_id = s.book_id and s.store_id = '%s' \" \"ORDER BY i.search_id limit '%d' offset '%d' ;\" % (search_key, store_id, self.page_size, page_lower)) \u2022 Performance analysis In store search only needs Access the database once . Although the connection query is used , However, because the attributes used for filtering are the primary key of the corresponding table , Cluster indexes are built , Therefore, the complexity of query is not high . ## 5\u3001 ... and . version control #### active branch: improve #### active branch: modification ## 6\u3001 ... and . test ### 6.1 utilize pytest and coverage Test and evaluate code As shown in the figure below , The coverage rate of this project is 90% about \u2022 #### coverage test Code coverage assessment ### 6.2 throughput & Delay test The test results of throughput and delay are as follows , It can be seen that the effect is ideal , Throughput is maintained at 40,000 about , The delay is about 0.01. ### 6.3 Test interface & Examples #### 6.3.1 test result Basic tests 35 All passed , besides , We are advanced function Write the 37 One test , Also all through #### 6.3.2 The test sample \u2022 deliver goods send_books Test situation The ginseng result message Successful delivery after payment correct store_id, Correct order_idcode == 200 Cannot ship without payment correct store_id, Error status order_idcode != 200 Shipping books that don't exist FALSE book_idcode != 200 Ship orders that do not exist FALSE order_idcode != 200 The shop doesn't exist FALSE store_idcode != 200 \u2022 Receiving goods receive_books Test situation The ginseng result message After the payment is successful and the delivery is successful, the goods are received Correct buyer_id,password,order_idcode == 200 Outstanding orders Correct buyer_id, Error status order_idcode != 200 The buyer does not exist There is no the buyer_idcode != 200 The order does not exist There is no the order_idcode != 200 \u2022 Manually cancel the order cancel Test situation The ginseng result message Cancel the order without payment after placing the order Correct buyer_id,order_idcode == 200 The buyer does not exist There is no the buyer_idcode != 200 The order does not exist There is no the order_idcode != 200 \u2022 Search for search Test situation The ginseng result message Paging query (\u201c Sanmao \u201d, 0)code == 200 Show all queries (\u201c Sanmao \u201d, 1)code == 200 Keywords that don't exist (\u201c Sanmao +\u201d, 1)result==[] Empty page (\u201c Sanmao \u201d, 1000)result==[] Keywords that don't exist + Empty page (\u201c Sanmao +\u201d, 1000)result==[] \u2022 Multi keyword search Test situation The ginseng result message The query is successful [\u201c Sanmao \u201d,\u201c Yuan family \u201d]code == 200 The query is successful [\u201c Sanmao \u201d, \u201c Yuan family \u201d, \u201c The mind \u201d]code == 200 Query with or without keywords [\u201c Sanmao \u201d,\u201c Yuan family ++\u201d]code == 200 There are no keywords [\u201c Sanmao +\u201d,\u201c Yuan family ++\u201d]result==[] \u2022 The buyer inquires about the current order Test situation The ginseng result message Query the current order after placing an order Correct buyer_idcode == 200 Query the current order after shipment Correct buyer_idcode == 200 Query the current order after receiving , It's empty Correct buyer_idresult==[\u2018NO Processing Order\u2019] \u2022 Buyer inquires about historical orders Test situation The ginseng result message Query historical orders after placing an order , empty Correct buyer_idresult == [] Query historical orders after shipment , empty Correct buyer_idresult == [] Query historical order after receiving Correct buyer_idcode == 200 \u2022 Recommendation system Test situation The ginseng result message Did not place the order , No historical orders , It is recommended to be empty Correct buyer_idcode == 200 The recommendation is successful Correct buyer_idcode == 200 \u2022 The seller inquires about the current order Test situation The ginseng result message Query the current order after payment Correct seller_idcode == 200 Query the current order after shipment Correct seller_idcode == 200 Query the current order after receiving , It's empty Correct seller_idresult==[\u2018NO Processing Order\u2019] \u2022 Buyers view historical orders Test situation The ginseng result message Query historical orders after placing an order , empty Correct store_idresult == [] Query historical orders after shipment , empty Correct store_idresult == [] Query historical order after receiving Correct store_idcode == 200 \u2022 Get book information Test situation The ginseng result message The query is successful buy_book_id_list[0]code == 200 There is no the book_idbuy_book_id_list[0]book_title==[] \u2022 In store query Test situation The ginseng result message The query is successful Correct store_id,search_key,pagecode == 200 Paging query succeeded Correct store_id,search_key,pagecode == 200 ## 7\u3001 ... and . Project cloud ### The project is deployed on ECs The cloud services needed are \uff1a 1. Ucloud Postgres UDB\/ Alibaba cloud RDS\uff08 Relational database , choice Postgres\uff09 2. Ucloud MongoDB\/ Alibaba cloud MongoDB\uff08 Document database \uff09 3. UHost Cloud server \/ Aliyun server Remarks\uff1a At present, the document databases of major cloud service manufacturers are expensive \uff08 And there are few student discounts \uff09, Alibaba cloud is recommended MongoDB serverless edition \uff08 It can be understood as a shared economical and applicable document database \uff09, If there are high response speed requirements for a large number of user visits . It is recommended to use additional services provided by cloud service providers **Load Balance\uff08 Load balancing \uff09** service . Need to achieve relatively ideal test results , It is recommended to set the cloud service according to the following configuration \uff1a #### 1. Virtual machine settings High concurrency testing requires bandwidth support , current Ucloud Press Bandwidth billing Relatively expensive , Our strengths ### 2. Cloud relational database settings Attention\uff01 Be sure to add your current name to the rented Alibaba cloud server white list \/ Cloud server ip Address \uff01 Otherwise, you cannot connect to the server \uff08 And throw \u201cwrong password\u201d Error of .\uff09 adopt navicat take Sqlite Data format converted to csv, then csv After all the data is imported into the cloud database \uff08 The book information about climbing down from Douban in this project is about 5GB\uff09 You can make a simple query to confirm that the data is normal , As shown in the figure below \uff08 Don't go straight to Navicat Premium use Export Wizard export Sqlite Data to Postgres, It's not right .\uff09 Remark\uff1a Can pass ip.cip.cc Get the current information of the machine quickly ip Address , Facilitate the use of cloud services Subsequently passed psycopg2 The database can be connected to the relational database , You can first establish a simple connection to test whether the connection with the cloud is normal . import psycopg2 Base = declarative_base() #base class engine = create_engine('postgresql:\/\/${username}:${password}@${public ip}:${port}\/${databse-name}')\nDBSession = sessionmaker(bind=engine) # Establish a conversation\nbookstore_engine = DBSession()\n\n\n### 3. Document database\n\ncloud MongoDB The deployment of is similar to that of a relational database , Also pay attention to the white list during deployment .\n\n## 8\u3001 ... and . Division and cooperation\n\n(1) ** Zheng Jiachen \uff1a** Database design , Recommendation system , Inverted list and search system , Historical orders , performance optimization , The interface test , The product manager\n\nContribution \uff1a33.3%\n\n(2) ** Sun Qiushi \uff1a** The order is cancelled automatically , Basic function realization , The interface test , Throughput test , Project cloud ,slides Make Contribution \uff1a33.3%\n\n(3) Tang Qiong \uff1a The interface test , Basic function realization , The receipt and delivery function is realized , Order cancellation , Search for books , real time OCR,postman test , Throughput test Contribution \uff1a33.3%\n\n## Nine . The experimental conclusion\n\n\t This experiment covers from table design to functional packaging and testing of the project , Then to the whole process of deploying to the cloud , It is a very complete project development experience . stay Bookstore Project development , The group deepened its understanding of ** normal form \u3001 Table index \u3001 Inverted index \u3001 data redundancy \u3001 High concurrency and transaction processing ** The understanding of the , And tried to use open api Advanced functions required to complete the project . We spent a lot of time on project performance optimization , The introduction of \u201c The order is cancelled automatically \u201d For example \u201c Soft real time \u201d Concept to reduce the resource consumption of multithreading, as well as RETURNING Unnecessary keyword queries are reduced . Members of the project team are interested in Bookstore The functions of the are 72 One test \uff0835 A basic test + Self written 37 An advanced function test \uff09, And take this to test the high concurrency performance and code coverage , Satisfactory results have been achieved .","date":"2022-08-13 00:00:43","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\": 0.19652284681797028, \"perplexity\": 2870.203041729045}, \"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-2022-33\/segments\/1659882571847.45\/warc\/CC-MAIN-20220812230927-20220813020927-00774.warc.gz\"}"} | null | null |
{"url":"https:\/\/chat.stackexchange.com\/transcript\/71\/2021\/7\/27","text":"1:40 AM\n@nitsua60 done\n\n1:56 AM\nWe have our first candidate for the 2021 Moderator Elections!\n\n2 hours later\u2026\n3:56 AM\n@MoreAnonymous her old post does have some decent resources, but they're a bit dated\nlike I know the $\\frac\\eta s=\\frac{\\hbar}{4\\pi k_B}(1+a\\lambda^{-3\/2})$ bound, but I'm looking for something more modern\n\n4 hours later\u2026\n8:12 AM\nI would like to know if there are, in the literature, articles or thought experiments about chirality\/optical activity (I mean the Wikipedia Chirality) of bodies falling into a black holes. Many thanks all users.\n\n8:26 AM\nIt's interesting how so much of our culture revolves around reproductionary desire. Evolution has made sure to carve out the right circuits that encourage reproduction. Romantic love is in movies, books, and is said to be this magical, sacred, unexplainable thing.\nHow would the world look if humans were hermaphrodites. All movies and culture will be centered around this important moment of your life where you'll self-fertilize and it'll all be incentivizes with circuits like sexual reproduction currently is.\n\n2 hours later\u2026\n10:37 AM\n@JingleBells For one we wouldn't have Freud :P\n\n3 hours later\u2026\n1:29 PM\n0\n\nI came across the question Why does the melting point of water decreases with increase in pressure? and answered it. I found a day later that an earlier, almost identical question had been asked by the same author, so I improved my answer and submitted it as an answer to the earlier question. Tha...\n\n3 hours later\u2026\n4:52 PM\nHi, everybody.\n\nhello, Dr Sank\n\n5:20 PM\nHi Daniel Sank!\n\nWhen finding E die to a straight wire through Coulomb\u2019s law we need to resolve the dE into components. Let\u2019s suppose a wire is sitting along x axis. We need to find at P. Which component will be cos and sin. Horizontal vs vertical.\nI always thought it\u2019s the x axis along which we take cos component.\nCan anyone help!\n\n5:43 PM\n@cOnnectOrTR12 It's not clear to me what you're asking. Why would the components be 'cos' or 'sin' to begin with? What angle is in play here? Why would it always \"be the x-axis\" along which you have a cos - doesn't that clearly depend on the situation you're considering?\n\nSuppose a perpendicular bisector is placed at which we need to find the E. we select two elements on wire, draw dE due to elements which make 45 degree with x axis\nNow charge is positive. The two dE are pointing away. Now here we have to resolve. Tell me which will be sin and cos\n\n6:07 PM\nI have read reference angle is measured from x axis.\n\n2 hours later\u2026\n7:51 PM\nDid a mod resign or something?\nSurprised to see a new election so soon\n\n@SirCumference you don't look at featured meta posts, do you? :P DZ resigned a while ago, and tpg recently, so we're back to the same amount of mods as before the last election.\n\n8:10 PM\nI see\nForgot DZ resigned, I think it was almost a year ago","date":"2021-09-24 20:28:09","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\": 0.618068277835846, \"perplexity\": 2119.133473727839}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057580.39\/warc\/CC-MAIN-20210924201616-20210924231616-00716.warc.gz\"}"} | null | null |
12/11/1997 - Bylaws - Zoning Board of AdjustmentsORDINANCE.NO. 2285 u AN ORDINANCE AMENDING ORDINANCE 1638, THE ZONING ORDINANCE FOR THE CITY OF COLLEGE STATION, TEXAS, BY AMENDING CERTAIN SECTIONS AS SET OUT BELOW; PROVIDING A SEVERABILITY CLAUSE; AND DECLARING A PENALTY. BE IT , ORDAINED BY THE CITY COUNCIL OF THE CITY OF COLLEGE STATION, TEXAS: PART 1: That Ordinance 1638, The Zoning Ordinance for the City of College Station, Texas, be amended as set out in Exhibit "A ", attached hereto and made a part of this ordinance for all purposes. PART ?: That if any provisions of any section of this ordinance shall be held to be void or unconstitutional, such holding shall in no way effect the validity of the remaining provisions or sections of this ordinance, which shall remain in full force and effect. PART 3: That any person, firm, or corporation violating any of the provisions of this chapter shall be deemed guilty of a misdemeanor, and upon conviction thereof shall be punishable by a fine of not less than Twenty-five Dollars ($25.00) nor more than Five Hundred Dollars ($500.00) Each day such violation shall continue or be permitted to continue, shall be deemed a separate offense. Said Ordinance, being a penal ordinance, becomes effective ten (10) days after its publication in the newspaper, as provided by Section 35 of the Charter of the City of College Station. PASSED, ADOPTED and APPROVED this 11t 0ay of DECEMBER , 1997. ATTEST: APPROVED: _ ry \ . CONNIE HOOKS, City Secretary LYNN McILHANEY, Mayor Ordinance No. 2285 Ili AMEND CHAPTER 15, SECTION 15.2 TO ADD 15.2 (F) AS FOLLOWS: Page 2 15.2 (F) -To authorize upon appeal variances to the drainage ordinance, that when acting in this capacity, the Zoning Board of Adjustments must have one of the alternates with an engineering background present during hearing of case. AMEND CHAPTER 15, SECT 15.3 TO READ AS FOLLOWS 15.3 ORGANIZATION OF THE BOARD The Board of Adjustment shall consist of five (5) members who are residents of the City and qualified voters. Each shall be appointed for a term of two (2) years, except that two (2) members appointed initially shall have terms of only one (1 ) year. After the initial appointments, two (2) members shall be appointed in odd numbered years to maintain a membership of five (5) members. Members shall be removable for cause by the City Council upon written charges and after a public hearing. Vacancies shall be filled for the unexpired term of any member whose term becomes vacant. However, the City Council may provide for the appointment of four (4) alternate members of the Board of Adjustment who shall serve in the absence of one (1) or more regular members when requested to do so. Two (2) of the allternate members must be licensed enggineers in the State of Texas. All cases to be heard by the Board of Ad* sfinent will always be heard by a minimum number of four (4) members. These alternate members, when appointed, shall serve for the same period as the regular members and any vacancies shall be filled in the same manner and shall be subject to removal as the regular members. AMEND CHAPTER 10.4, SECTION B.* AS FOLLOWS: 10.6 SPECIAL RULES FOR C -N DISTRICTS Site plan review and proposed uses within the C -N Neighborhood Business District are subject to approval by the Planning and Zoning Commission. Apppplicants shall file an application form with the Cit Planner no less than twenty (20) days prior to the regularly scheduled meeting oYthe Planning and Zoning Commission of which the plan is to be reviewed. The application form shall be accompanied by all supporting information required in Section 10.1 above, a list of property. owners within 200 feet , and a receipt for filing fee. The application fee shall be determined by the City Council. Property owners within two hundred (200) feet of the site shall be identified by the Planning Office using the Tax Rolls supplied by the Brazos County Appraisal District. A. The applicant shall ive public notice by publication in a local newspaper as directed by the City Planner at least fifteen (15) days prior to the date set for the public hearing. Ordinance No. 2285 Page 3 B." The City Planner shall notify all property owners of record within two hundred (200) feet of the property in question at least ten (10) days prior to the public nearing. The notice may be served by its deposit in the municipality, properly addressed with postage paid, in United States mail. C. The City Planner shall notify the applicant of the date, time and place for the preliminary review of the project by t e Project Review Committee. D. The Project Review Committee shall review the proposed project in accordance with Section 10.2 prior to the Planning and Zoning Commission meeting. The Protect Review Committee shall submit written recommendations to the Planning and Zoning ommission. A copy of this report shall be sent to the applicant. The applicant shall file ten (10) copies of the final site plan as approved by the Project Review Committee, showing all changes and requirements imposed by the Project Review Committee after which the Planning and Zoning Commission shall hold a public hearing for the purpose of approving or denying the project plan. E. The applicant or the owners of forty percent (40°x) or more of the property within two hundred (200) feet of the project site may appeal to the City Council any determination made by the Planning and Zoning Commission. The appeal should be made by petition filed with the City Secretary within ten (10) days after the public hearing. AMEND CHAPTER 14 SECTION 14.2 (B) AS FOLLOWS: B. The applicant shall ive public notice by publication in a local newspaper of Carin circulation at least fifteen (15) days prior to a date set for a public Hearing before the Commission. The City Planner shall notify all property owners of record within two hundred (200) feet of the property in question . The notice may be served by its deposit in the munici ality properly addressed with postage paid, in United States mail at least ten 10) days prior to the date set for the public hearing. The City Planner shall also give a report and recommendation to the Commission on the date of hearing. AMEND CHAPTER 14 SECTION 14.2 (E) AS FOLLOWS: E. Any person or persons jointly or severally aggrieved by a decision of the Commission may present to the City Secretary an appeal, settin forth that such decision is unjust and /or was made in error, in whole or in part. Ihe appeal must be in writing, must specify the grounds of injustice and /or the points on which the Commission erred and must be signed by the appellant(s). Such appeal shall be presented to the City Secretary within ten ((10) days after the final decision of the Commission, and not thereafter. The City Secreta shall then schedule a public hearing before the City Council and shall notify all property owners within two hundred (200') feet of the subject property under appeal. This notice, in a form prepared by the City Planner, shall be served by its deposit in the municipality, properly addressed with postagge paid in United States mail and shall be iven no later than ten (10) days prior to the cute set for the public hearing. Notiffbation, as prepared by the City Planner, shall also be given by the City Secretary by publication in a local newspaper of general circulation at least fifteen (1 days prior to the date set for the public hearing. The City Council shall hold a public hearing to consider the appeal and shall uphold the decision of the Commission as it was passed, overturn the decision of the Commission in its entirety or remand the case back to the Commission for reconsideration of certain and specific points. Ordinance No. 2285 Page 4 AMEND CHAPTER 17 SECTION 17.4 (B) AS FOLLOWS: 17.4 PUBLIC HEARING AND NOTICE THEREOF A public hearing shall be held by the Commission before making a report to the Council, and a public hearing shall be held by the Council before adopting any amendment to this ordinance. Notice of the public hearing shall be given by publishing such notice at least once in a newspaper of general circulation in the City of College Station, stating the time and place of such hearing and the substance of the proposed amendment. This notice shall appear in said newspaper at least fifteen (15) days prior to the date set for the public hearing. A. Publication of the notice shall be the responsibility of the City Planner. B. Notice of the proposed zoning change shall also be made by the City Planner by mailing notification which shall be served by its deposit in the municipality, properly addressed with postage paid, in United States mail to the person or firm to whom the property is assessed, and to all persons or firms to whom property within two hundred (200) feet of the proposed zoning change is assessed on the City tax rolls. | {
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{"url":"https:\/\/labs.tib.eu\/arxiv\/?author=B.%20G.%20Pushpawela","text":"\u2022 ### First evidence for $\\cos 2\\beta>0$ and resolution of the CKM Unitarity Triangle ambiguity by a time-dependent Dalitz plot analysis of $B^{0} \\to D^{(*)} h^{0}$ with $D \\to K_{S}^{0} \\pi^{+} \\pi^{-}$ decays(1804.06152)\n\nApril 17, 2018 hep-ex\nWe present first evidence that the cosine of the CP-violating weak phase $2\\beta$ is positive, and hence exclude trigonometric multifold solutions of the CKM Unitarity Triangle using a time-dependent Dalitz plot analysis of $B^{0} \\to D^{(*)} h^{0}$ with $D \\to K_{S}^{0} \\pi^{+} \\pi^{-}$ decays, where $h^{0} \\in \\{\\pi^{0}, \\eta, \\omega \\}$ denotes a light unflavored and neutral hadron. The measurement is performed combining the final data sets of the BaBar and Belle experiments collected at the $\\Upsilon(4S)$ resonance at the asymmetric-energy B factories PEP-II at SLAC and KEKB at KEK, respectively. The data samples contain $( 471 \\pm 3 )\\times 10^6\\, B\\bar{B}$ pairs recorded by the BaBar detector and $( 772 \\pm 11 )\\times 10^6\\, B\\bar{B}$ pairs recorded by the Belle detector. The results of the measurement are $\\sin{2\\beta} = 0.80 \\pm 0.14 \\,(\\rm{stat.}) \\pm 0.06 \\,(\\rm{syst.}) \\pm 0.03 \\,(\\rm{model})$ and $\\cos{2\\beta} = 0.91 \\pm 0.22 \\,(\\rm{stat.}) \\pm 0.09 \\,(\\rm{syst.}) \\pm 0.07 \\,(\\rm{model})$. The result for the direct measurement of the angle $\\beta$ of the CKM Unitarity Triangle is $\\beta = \\left( 22.5 \\pm 4.4 \\,(\\rm{stat.}) \\pm 1.2 \\,(\\rm{syst.}) \\pm 0.6 \\,(\\rm{model}) \\right)^{\\circ}$. The quoted model uncertainties are due to the composition of the $D^{0} \\to K_{S}^{0} \\pi^{+} \\pi^{-}$ decay amplitude model, which is newly established by performing a Dalitz plot amplitude analysis using a high-statistics $e^{+}e^{-} \\to c\\bar{c}$ data sample. CP violation is observed in $B^{0} \\to D^{(*)} h^{0}$ decays at the level of $5.1$ standard deviations. The significance for $\\cos{2\\beta}>0$ is $3.7$ standard deviations. The trigonometric multifold solution $\\pi\/2 - \\beta = (68.1 \\pm 0.7)^{\\circ}$ is excluded at the level of $7.3$ standard deviations. The measurement resolves an ambiguity in the determination of the apex of the CKM Unitarity Triangle.\n\u2022 We report measurements of $\\sin{2\\beta}$ and $\\cos{2\\beta}$ from a time-dependent Dalitz plot analysis of $B^{0} \\to D^{(*)} h^{0}$ with $D \\to K_{S}^{0} \\pi^{+} \\pi^{-}$ decays, where the light unflavored and neutral hadron $h^{0}$ is a $\\pi^{0}$, $\\eta$, or $\\omega$ meson. The analysis is performed with a combination of the final data sets of the \\babar\\ and Belle experiments containing $471 \\times 10^{6}$ and $772 \\times 10^{6}$ $B\\bar{B}$ pairs collected at the $\\Upsilon\\left(4S\\right)$ resonance at the asymmetric-energy B factories PEP-II at SLAC and KEKB at KEK, respectively. We measure $\\sin{2\\beta} = 0.80 \\pm 0.14 \\,(\\rm{stat.}) \\pm 0.06 \\,(\\rm{syst.}) \\pm 0.03 \\,(\\rm{model})$ and $\\cos{2\\beta} = 0.91 \\pm 0.22 \\,(\\rm{stat.}) \\pm 0.09 \\,(\\rm{syst.}) \\pm 0.07 \\,(\\rm{model})$. The result for the direct measurement of the angle is $\\beta = \\left( 22.5 \\pm 4.4 \\,(\\rm{stat.}) \\pm 1.2 \\,(\\rm{syst.}) \\pm 0.6 \\,(\\rm{model}) \\right)^{\\circ}$. The last quoted uncertainties are due to the composition of the $D^{0} \\to K_{S}^{0} \\pi^{+} \\pi^{-}$ decay amplitude model, which is newly established by a Dalitz plot amplitude analysis of a high-statistics $e^{+}e^{-} \\to c\\bar{c}$ data sample as part of this analysis. We find the first evidence for $\\cos2\\beta>0$ at the level of $3.7$ standard deviations. The measurement excludes the trigonometric multifold solution $\\pi\/2 - \\beta = (68.1 \\pm 0.7)^{\\circ}$ at the level of $7.3$ standard deviations and therefore resolves an ambiguity in the determination of the apex of the CKM Unitarity Triangle. The hypothesis of $\\beta = 0^{\\circ}$ is ruled out at the level of $5.1$ standard deviations, and thus CP violation is observed in $B^{0} \\to D^{(*)} h^{0}$ decays.\n\u2022 We report a measurement of the time-dependent CP asymmetry of B0->D(*)CP h0 decays, where the light neutral hadron h0 is a pi0, eta or omega meson, and the neutral D meson is reconstructed in the CP eigenstates K+ K-, K0S pi0 or K0S omega. The measurement is performed combining the final data samples collected at the Y(4S) resonance by the BaBar and Belle experiments at the asymmetric-energy B factories PEP-II at SLAC and KEKB at KEK, respectively. The data samples contain ( 471 +\/- 3 ) x 10^6 BB pairs recorded by the BaBar detector and ( 772 +\/- 11 ) x 10^6, BB pairs recorded by the Belle detector. We measure the CP asymmetry parameters -eta_f S = +0.66 +\/- 0.10 (stat.) +\/- 0.06 (syst.) and C = -0.02 +\/- 0.07 (stat.) +\/- 0.03 (syst.). These results correspond to the first observation of CP violation in B0->D(*)CP h0 decays. The hypothesis of no mixing-induced CP violation is excluded in these decays at the level of 5.4 standard deviations.\n\u2022 We describe searches for B meson decays to the charmless vector-vector final states omega omega and omega phi with 471 x 10^6 B Bbar pairs produced in e+ e- annihilation at sqrt(s) = 10.58 GeV using the BABAR detector at the PEP-II collider at the SLAC National Accelerator Laboratory. We measure the branching fraction B(B0 --> omega omega) = (1.2 +- 0.3 +0.3-0.2) x 10^-6, where the first uncertainty is statistical and the second is systematic, corresponding to a significance of 4.4 standard deviations. We also determine the upper limit B(B0 --> omega phi) < 0.7 x 10^-6 at 90% confidence level. These measurements provide the first evidence for the decay B0 --> omega omega, and an improvement of the upper limit for the decay B0 --> omega phi.","date":"2020-12-03 11:36:19","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\": 0.8504771590232849, \"perplexity\": 1235.5946508707862}, \"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-2020-50\/segments\/1606141727627.70\/warc\/CC-MAIN-20201203094119-20201203124119-00711.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.khanacademy.org\/math\/pre-algebra\/fractions-pre-alg\/add-and-subtract-fracs-pre-alg\/e\/adding-and-subtracting-fractions-with-like-denominators-word-problems","text":"Add and subtract fractions: word problems 2\n\nPractice solving word problems by adding and subtracting fractions. The fractions used in these problems have like (common) denominators.\n\nProblem\n\nSven is reading a book while on vacation. So far, he has read start fraction, 1, divided by, 6, end fraction of the book. He wants to finish start fraction, 3, divided by, 6, end fraction of the book by the end of vacation.\nWhat fraction of the book does Sven want to read between now and the end of vacation?","date":"2016-07-30 20:49:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 9, \"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\": 0.3999066948890686, \"perplexity\": 1530.1057499281942}, \"config\": {\"markdown_headings\": false, \"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-2016-30\/segments\/1469258943366.87\/warc\/CC-MAIN-20160723072903-00252-ip-10-185-27-174.ec2.internal.warc.gz\"}"} | null | null |
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Website updated 6th October 2020
Colin Higgs is one of Aston Abbotts most senior residents. He has been actively involved in the village community all of his life. In 2019 Colin completed fifty years of service on Aston Abbotts Parish Council. Colin was awarded the British Empire Medal for services to Aston Abbotts in the 2020 New Year's Honours list. This interview by Phil Spooner first appeared in the Aston Abbotts Chronicle in Spring 2000.
A fourth generation Astonian, Colin Higgs was born in Grantham and came to Aston Abbots as a baby when his father and mother returned here to seek housing. As a child he lived on Lines Hill by the nearest of the four gates that used to be on the road to Weedon. "All the gates in those days used to be shut, so if we saw a car or horse and cart coming along we'd open the gate and he would give us a penny. Mind you, there was nothing like the traffic there is now - we probably only had thruppence a day." In 1938 the family moved to a newly built house in the Wingrave Road, where Colin still lives today. Colin attended the village school, as did his three brothers and sisters. "We had 110 children in school. Brilliant school! Brilliant school! Very strict teachers, pity they're not like it today!" He recalls being a wartime village child. "During the war we had to catch white butterflies… they were doing so much damage to the crops. We also got paid for collecting blackberries and elderberries because they used those for dye for the army uniforms. We used to have good time with the Czechoslovakians and the Polish prisoners. They [the prisoners] used to give us a good Christmas party up Norduck Lane in the Nissen huts." After completing his schooling in Wing, he worked locally and did two years National Service before joining Marley Tiles in Leighton Buzzard, where he remained for 35 years. Colin is an active churchgoer. "We've always been involved, my grandfather before me, my father. I've just carried on doing it. My grandfather was the Sexton at the Church. He was also the gravedigger. He wound the clock. My father carried it on. I didn't dig the graves but I carried on with the clock and Church duties." Voicing frustration at current apathy towards the Church - "I just don't know why people just don't come to Church" - he has no doubt that it is still important to the village. "Oh indeed, indeed! If we lost it, it would be terrible. Catastrophic! Like it was when we lost the school… when we lost the vicar and so they sold the vicarage… when we lost the post office. It was a tragedy when we lost the chapel, but it would be in even greater tragedy if we lost the Church." An eager volunteer, it is unsurprising to find that he used to organise the annual fete. "I did the festival for ten years. We used to have it in the Abbey grounds. Unfortunately the last year we had it there, which I think was about 1972, they got burgled three times. The insurers told the lady at the Abbey they wouldn't insure it because they thought the people that came to the fete had eyed it up. She reluctantly had to refuse us. We then moved into the vicarage, which was very cramped. Then somebody said 'why don't we use the Green' and that is how it started on the Green." Colin is a keen member of the Rambling Club. "I am a founder member of that. It started in 1973." However, one of their walks was not so enjoyable for him. "We were doing a walk, we actually started it at Cublington, to Whitchurch and then down to Dunton and back to Cublington and we got over this stile and it was a hot day and the sun was sort of shimmering. We jumped over this and it looked just like tarmac on the end of a farm building - it looked just like tarmac - and it was cow slurry. And I went in it! I could have drowned. It was a pool, there was nothing to say and with the sun shimmering down it just looked like tarmac and I was the first one-over so I just jumped over and jumped in it. They got me out. The smell was
unbelievable... terrible... terrible!" Each week he used to write the Aston Abbotts village report for the Leighton Buzzard Observer. "I used to do it for the Bucks Herald, but I fell out with them because they never used to report it correctly or they would leave it for about three weeks and put it in when it would be stale news." What does he like most about Aston Abbotts? "Everything! It's a good village, good community. Everybody gets on all right. Most of the newcomers have fitted in well." The biggest challenge is? "To keep it as a rural village. We must grow though to keep it alive, but within reason. We've lost the blacksmith's shop - that's all developed. We've lost Nashs farm… we've lost the apple scrumping Orchard - that's Ross Road now. You've got to have development. You've got have new people to create new ideas. You've got to grow otherwise the place dies. "There used to be about four shops at one time, little cottage shops, you know. There were less than half the houses that there are now, but the families were bigger you see. Nobody left the village. You walked to the shops and nobody left the village. Yes, a lot of villages have lost all of their shops - it's awful really. All these out-of-town supermarkets are disastrous really. Lots of villages are losing all of their facilities and the people who move in mostly commute. Sadly, a lot of them don't mix in with anything you've got in the village." Colin serves on the Bucks County Council Highways (Road Safety) committee, Aylesbury Vale Rural Transport committee, Ramblers, Whist club, Floral Club, Marley Pensioners Link committee, and Good Samaritan bingo committee. He contributes history articles to the Chronicle, cuts the Church grass and he is an enthusiastic Church fundraiser. On top of these he made a field map and conducted interviews for the Aston Abbotts History Society, yet he still finds time to make homemade wine from locally gathered wild ingredients. His involvement in village and local activities led to him winning the award for Aylesbury Vale Parish Volunteer Of The Year in 1998. "Well, I went into the shop one morning, Bernard's wife Valerie was in there and she said 'I've just filled in and posted a form about you'. I said 'Whatever about' and she said 'I hope you'll know some time'. Anyway I never thought anymore about it and about six weeks later I suddenly get a phone call from Aylesbury Vale… We didn't know anything about it at all." The award may have amazed Colin, but it is unlikely that anybody who has met him and seen his contributions to our village community would have shared his surprise.
This interview with Colin was conducted in early 2000. Since then he has remained an active contributor to village life and his award of the British Empire Medal in 2020 reflects his huge contribution to our community. In 2020 Colin completes 50 years of service on Aston Abbotts Parish Council.
Colin (right) is pictured here with Bernard Osborn in Osborn's shop
So What's So Special?
Where is Aston Abbotts
Aston Abbotts Today
Facilities, Industry and Threats
The Village Walk
History - Origins
Agriculture, Industry and Education
Population and Housing
Sir James Clark Ross
London's Third Airport
St James and the PCC
The Ladies Club
G.N.O.M.E.S.
Other Social Activites
Aston Abbotts Postbox
The Village Fete
The Village Hall
Astonbury
The History of The Chronicle
About the Back Page
Astonians
Cycling Fitness
Strippergram
Early Efforts
Aston Abbotts - Winter Pictures
Aston Abbotts How It Was
Villagers Pictures - General
Villagers Pictures - Ken Rhodie
630 Club Bean Competition 2011
Astonbury X 2011 - Page 1
Vic Scott Memorial Orchard
Christmas Lunch 2011
Peter Kent's Photographs
Jubilee Picnic 2012
Village Orchard - One Year On
Village Orchard - Autum 2012 Planting
Astonbury 2012
Aston Abbotts Fete
General Videos
Bernard Osborn
Colin Higgs | {
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Q: Generic Interface type conversion issues I have been battling with this bit of code for a while now and I am trying to get a solution as it is literally the last part before it goes to testing.
I have the following interfaces and classes (simplified to the relevant parts):
public interface ITagParent<T> where T : ITag
{
List<TagAddOn<T>> TagCollection { get; set; }
}
public interface ITag
{
int Id { get; set; }
string Description { get; set; }
TagGroup TagGroup { get; set; }
}
public class TagAddOn<T> : ViewModelBase where T : ITag
{
private T _currentTag;
public T CurrentTag
{
get { return _currentTag; }
set { _currentTag = value; }
}
}
public partial class Customer : ITagParent<CustomerTag>
{
List<TagAddOn<CustomerTag>> _tagCollection;
public List<TagAddOn<CustomerTag>> TagCollection
{
get { return _tagCollection; }
set { _tagCollection = value; }
}
}
public partial class CustomerTag : ITag
{
public int Id { get; set; }
}
public class TagAddOnManager
{
public static string GetTagCurrentValue(List<TagAddOn<ITag>> dataObjectAddOns)
{
// LOTS OF SNIPPING!
return string.Empty;
}
}
I am trying to use the GetTagCurrentValue method in the TagAddOnManager class like this:
string value = TagAddOnManager.GetTagCurrentValue(
((ITagParent<ITag>)gridCell.Row.Data).TagCollection));
Everything compiles fine, but errors when trying to cast gridCell.Row.Data to ITagParent<ITag>. I understand this is due to covarience and a workaround (if not a terribly safe one) is to mark T in the ITagParent interface with the out keyword, but that won't work as you can see it is used in the TagCollection property, which can't be read only.
I tried casting the above to ITagParent<CustomerTag>, but this fails at compile time with a 'cannot convert' error when trying to feed it into my GetTagCurrentValue method.
Another option I considered is using some base classes instead of the ITagParent interface, but that won't work as the Customer object already inherits from another base class, which can't be modified for this implementation.
I know I could just overload the GetTagCurrentValue method with List<TagAddOn<CustomerTag>> as the parameter type and all other variations, but that really seems like a 'I give up' solution. I could probably use reflection to get the desired results, but that would be unwieldy and not very efficient, especially considering this method could be called a lot in a particular process.
So does anyone have any suggestions?
A: Could you use something like that
public class TagAddOnManager
{
public static string GetTagCurrentValue<TTag>(ITagParent<TTag> tagParent)
where TTag : ITag
{
// Just an example.
return tagParent.TagCollection.First().CurrentTag.Description;
}
}
and use it like that?`
var value = TagAddOnManager.GetTagCurrentValue((Customer)CustomergridCell.Row.Data);
| {
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} | 318 |
Q: Looking for external portable hard disk drive with both Superspeed USB3.0 AND eSATA? All I can find is:
*
*eSATA and USB2.0 not USB3.0
*eSATA only
*USB3.0 only
*Enclosure/caddy with USB3.0 and eSATA for housing bare hard drive: http://uk.startech.com/product/SAT2510BU3-25in-SuperSpeed-USB-3-SATA-Hard-Drive-Enclosure (I'd prefer a branded complete product, also I'm not sure if there is any overhead (performance degradation) in converting the internal SATA interface to eSATA).
Why do I want this?
*
*Versatility - choice of interfaces
*My Toshiba NB100 netbook has eSATA/USB sockets, I may get a USB3.0 ExpressCard for my Apple Mac Book Pro 17" and/or add a USB3.0 PCI card to my Windows desktop. So a hard drive with both USB3.0 and eSATA can be of benefit for high speed transfer on all of these platforms. Sure, some of these platforms may not go to full eSATA or USB3.0 speeds, due to the limits of their external buses and the spindle speed of the external drive being a constraint, but I would expect them to be much faster than USB2.0.
A: The Seagate GoFlex series of hard drives have changeable interfaces. USB 2.0, USB 3.0, eSATA, Firewire. Just go to their website. Of course the caveat is you can only have one interface attached at any given moment, but still, you can only connect to one port anyway.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 381 |
Q: Add new MCE button for toggle specific cell background color I'm using tables as a calendar for free dates in my schedule.
You can see free dates here.
http://www.zachrannastanicelipec.cz/smysly-terminy/
Now I'm using the Table->Cell->Cell Properties->Advanced->background
I want to have a button (or two) for turn on/off the green background in the selected cell.
Is it possible?
Thank you for any kind of help.
A: I have found a great tutorial about making a custom button to the tinymce.
http://qnimate.com/adding-buttons-to-wordpress-visual-editor/
You can just follow this guide and copy all of it. Don't forget to activate your new plugin in the plugin menu (in admin panel).
Then you can just change the code inside index.js and especially in the
ed.addCommand("green_command", function() {
For making the background of the selected cell toggle (on/off) you just change the code like this:
//button functionality.
ed.addCommand("green_command", function() {
ed.formatter.register('termin_format', {
selector: 'td',
styles: { backgroundColor: '#adf2a7'},
});
ed.formatter.toggle('termin_format');
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,068 |
Barbasol Championship Live Web Cam
Road to the Breeders' Cup
HS Sports Zone
Quick links... BBN Tonight BBN Facebook BBN Twitter BBN Instagram High School Sports Zone True Blue Sports Sports
UK Football notes following win in the Taxslayer Gator Bowl
Wildcats beat N.C. State 23-21
LEX 18
By: Keith Farmer
JACKSONVILLE, Fla. — Kentucky won its third straight bowl game on Saturday by beating N.C. State 23-21 in the Taxslayer Gator Bowl. Here are some important individual and team records and notes released by the football program.
Team Records and Series Information
Kentucky completes the season 5-6 and North Carolina State is 84..
Kentucky leads the series 2-1.
This is the first time the teams have played in a bowl game.
Notes on the Win
Kentucky is now 11-9 in bowl games (.550), including 1-1 in the TaxSlayer Gator Bowl.
UK is 5-3 vs. Atlantic Coast Conference teams in bowl games, including 1-0 vs. NC State, North Carolina Florida State and Virginia Tech, 1-2 vs. Clemson and 0-1 vs. Georgia Tech.
This was UKs third-consecutive bowl win, including the VRBO Citrus Bowl on Jan. 1, 2019, and the 2019 Belk Bowl. That ties the school record of three straight bowl wins in the 2006 and '07 Gaylord Hotels Music City Bowls and the 2009 AutoZone Liberty Bowl following the '08 season.
UK has won 11 consecutive non-conference games over the last three seasons.
This was UK's second win over a ranked opponent this season, also vs. then-No. 18 Tennessee, both of which were away from home.
It is the first time UK has defeated two ranked opponents in the same season since 2018, when the Wildcats knocked off No. 25 Florida, No. 14 Mississippi State and No. 12 Penn State.
It is the first time UK has defeated two ranked opponents away from home in the same season since 2018, when the Cats won at No. 25 Florida and in the Citrus Bowl vs. No. 12 Penn State.
Team Notes
This was UK's fifth-consecutive bowl appearance, tying the school record set in the 2006-10 seasons.
Mark Stoops becomes UK's first head coach to take a team to five straight bowl games. The previous record was by Rich Brooks in his last four seasons, 2006-09.
This was Mark Stoops' 99th game as head coach at Kentucky, one short of the school record of 100 games coached by Fran Curci from 1973-81.
This was Stoops' 49th win at Kentucky, second behind Paul "Bear" Bryant, who had 60 victories from 1946-53.
Turnovers tell the tale: UK won the turnover battle 3-0 today and scored 10 points off turnovers.
In the five victories this season, UK was +15 in turnover margin; in the six losses, UK was -5 in turnover margin.
Since 2016, UK is 15-7 in games decided by seven points or less, including 2-1 this season.
UK made three field goals, breaking the school bowl record of two FGs in six previous bowl games.
Accomplishments of the Senior Class
For players who are fifth-year seniors (since 2016), they have …
Won 37 games (most in a five-season span since 1949-53)
Gone to five straight bowl games, tying the school record
Won 20 SEC games, most in a five-year span in school history
Posted first 10-win season and winning SEC record since 1977 and first final ranking sine 1984 (2018 season)
Won a Citrus Bowl, Belk Bowl and Gator Bowl
Notched victories over six ranked opponents
Won their regular-season finale for the third-straight year, the first time that has happened in 61 years, 1957-59
Won three Senior Day games in a row for the first time since 2010-12
Ended the 31-game losing streak to Florida
Won at Tennessee for the first time since 1984.
· Senior quarterback Terry Wilson completed 12 of 20 passes for 99 yards. He also rushed six times for a net 14 yards.
· Wilson career notes …
17-8 as the UK starting quarterback.
First player in school history with at least 3,000 passing yards and 1,000 rushing yards.
Has a career completion percentage of 64.9 (337 of 5199), third in school history.
Has 4,458 career yards total offense, eighth place on the UK career list.
Joined Derrick Ramsey as the only starting QBs in UK history to win at Florida and Tennessee.
· Senior running back A.J. Rose rushed 12 times for a career-high 148 yards, his third 100-yard game of the season seventh 100-yard game of his career.
Rose now has 1,971 career rushing yards, 10th place in school history.
· Sophomore running back Chris Rodriguez Jr. carried the ball 17 times for 84 yards, including both of UK's touchdowns.
He has 1,365 career rushing yards, moving into 27th place in school history.
· Senior wide receiver Josh Ali led the team in receptions with four catches for 18 yards.
· Junior linebacker Jamin Davis made 13 tackles, his eighth double-figure game of the season, and one-half quarterback sack.
He had an interception and five-yard return in the fourth quarter, setting up the game-clinching touchdown. It is his third interception of the season and fifth of his career.
· Senior cornerback Brandin Echols made his first career interception on N.C. State's first possession and returned it eight yards to midfield. He also made seven tackles during the game.
· Junior safety Yusuf Corker picked off a pass in the fourth quarter, his second of the season, occurring in consecutive games. He also had five tackles.
· Junior safety Tyrell Ajian blocked a field goal attempt in the second quarter, the first blocked kick of his career.
It is UK's second blocked field goal of the season, also Jamin Davis vs. Georgia.
He also made a career-high seven tackles.
· Junior linebacker Jordan Wright made five tackles, including a 12-yard sack, and caused a fumble.
· Senior linebacker Boogie Watson had four tackles, including a tackle for loss.
He has 28.5 tackles for loss in his career.
· True freshman cornerback Carrington Valentine made his first collegiate start at cornerback. He made six tackles and caused a fumble.
· Senior kicker Matt Ruffolo set a UK bowl record with three field goals, from 25, 26 and 20 yards.
· Senior center Drake Jackson started his 44th consecutive game today, the third-longest streak in school history with starting lineups available since 1993.
o Jackson was today's team captain and wore jersey No. 65 in honor of offensive line coach John Schlarman, who died Nov. 12. | {
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## DEDICATION
**For my daughter, Lilia,**
the first teenager to read _Emancipated_. May your own writing always bring you challenge, excitement, and joy.
## CONTENTS
1. Dedication
2. Grace: El Matador State Beach, Sunday, May 31
3. Paolo
4. Candace
5. Paolo
6. Maya
7. Paolo
8. Paolo
9. Lucy
10. Maya
11. Lucy
12. John-Michael
13. Grace
14. Candace
15. John-Michael
16. Maya
17. Paolo
18. Maya
19. Candace
20. Maya
21. Paolo
22. Maya
23. John-Michael
24. Grace
25. Candace
26. John-Michael
27. Lucy
28. John-Michael
29. Candace
30. Paolo
31. Grace
32. Lucy
33. Paolo
34. Maya
35. John-Michael
36. Paolo
37. Lucy
38. Paolo
39. Lucy
40. Maya
41. John-Michael
42. John-Michael
43. Paolo
44. Paolo
45. Maya
46. Grace
47. John-Michael
48. Lucy
49. Maya
50. Grace
51. Acknowledgments
52. Back Ads
53. About the Author
54. Books by M. G. Reyes
55. Credits
56. Copyright
57. About the Publisher
# Guide
1. Cover
2. Contents
3. Chapter 1
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* * *
**CALIFORNIA FAMILY CODE SECTION 7120-7123**
**EMANCIPATION:**
7120. ****
(A) A MINOR MAY PETITION THE SUPERIOR COURT OF THE COUNTY IN WHICH THE MINOR RESIDES OR IS TEMPORARILY DOMICILED FOR A DECLARATION OF EMANCIPATION.
(B) THE PETITION SHALL SET FORTH WITH SPECIFICITY ALL OF THE FOLLOWING FACTS:
(1) THE MINOR IS AT LEAST 14 YEARS OF AGE.
(2) THE MINOR WILLINGLY LIVES SEPARATE AND APART FROM THE MINOR'S PARENTS OR GUARDIAN WITH THE CONSENT OR ACQUIESCENCE OF THE MINOR'S PARENTS OR GUARDIAN.
(3) THE MINOR IS MANAGING HIS OR HER OWN FINANCIAL AFFAIRS.
* * *
## **GRACE**
#### EL MATADOR STATE BEACH, SUNDAY, MAY 31
It was all going wrong between the housemates. Grace was the one who decided to put things right. But not everyone was in the mood to play along.
"Really?" Candace peered down from the cliff path at El Matador State Beach. With maximum snark she said, "You couldn't find a busier beach?"
Dismayed, Grace eyed the crowded sand. "It'll thin out, you'll see. It's almost four thirty. It has to." The other five housemates paused behind her on the path, beach bags and coolers dangling from their fingers, surfboards under Maya's and Candace's arms.
"We should have come later," John-Michael muttered. "Who sets out for an evening beach barbecue at four in the afternoon?"
Grace had persuaded her housemates it was time for a day trip away from their Venice Beach house, where lately all they did was eat too many grilled cheese sandwiches and stare at the TV. They had to get out, get together in a place where nature and tranquillity could work on their senses. Where the distractions of everyday life wouldn't snap the housemates apart and send them flying for the corner pockets like a rack of pool balls on the break.
She'd even cajoled John-Michael and Candace into helping her put together a picnic. One or two of them grumbled about going to "just another beach," but so what? What they needed was different air and a different horizon. A place where they could breathe without the taint of uncertainty and suspicion that had settled around them. This was about being together. With its coves wrapped by the cliffs of the Pacific Coast Highway, its crystalline waters and soft golden sands, El Matador seemed like the ideal getaway.
Grace bit her lower lip. Her housemates' complaining was pretty annoying. But she wasn't going to be so easily ground down.
To her relief, Paolo caught her eye. He noticed Grace's frustration and replied with a comforting half grin. "I'm glad we're here while there's still some light. I like to watch the kitesurfers."
Grace gave Paolo an appreciative smile and her stomach knotted in response. It happened way too often. One of these days her feelings were going to show in her eyes, in the twitch of her lips, and then what? Then she'd be the stupid girl who fell for the ridiculously gorgeous, unattainable guy.
"Kitesurfing, on this beach?" Maya said, skeptical. "You'd have to be pretty crazy to risk that."
Paolo shrugged. He could see at least one bright pink sail dragging a surfer across the water, about a hundred yards out. "It's not so windy."
"Because, _cliffs_ ," Maya said, pointing at the wall of rock that bordered the cove. "If a strong gust picks you up you could get slammed against a cliff and killed."
"No one's getting killed," Paolo said, his voice tinged with admiration. "Just look at that guy, he's a mile out!"
This time, Grace couldn't hold back a warm smile. Paolo, at least, was trying. He had been a little down, too, since Lucy rejected him; hadn't exactly jumped at the idea of the picnic. But once they'd gotten moving, his mood had improved.
She paused for a moment, watching Paolo take the steps two at a time. He looked as good as ever. It was pointless, she told herself. You had to make the effort to avoid boys like Paolo. Too cute, and he knows it. Better to keep him as a friend.
When Grace looked up it was to see Lucy's eyes on her; curious, considered. "Hmm," Lucy murmured with a knowing nod.
"What?" countered Grace. She could feel a blush rising and couldn't do a thing to stop it.
Lucy smiled gently. "Don't sweat it. Guys are dumb but they figure it out eventually."
Grace, for a brief moment, was too stunned to move. At least one housemate wasn't fooled by her charade. Moving somewhat mechanically now, she followed her friends down to the beach.
Candace and Maya rode out into the waves on their short surfboards while Paolo swam nearby. The ocean was still too cold for any but the most hardy, but Paolo didn't seem to care. The water was as clear as a freshwater spring in the middle of the woods. Out on the wide blue expanse, two kitesurfers crisscrossed as their boards bounced and skated over the water.
Grace stayed on the sand, between boulders and the cliffs. Most of the time, she stared out over the water, John-Michael beside her, the two of them silent.
It could be that way with two people looking into the sea, Grace had found. All her life Grace had lived in San Antonio, Texas, more than one hundred miles from the coastline. She'd never known the calming effect the ocean could have on her mind. The past five months, living a hundred yards from the water's edge, had brought the revelation of shared silence.
Grace doubted that she could ever go back.
She thought about the first days of living in the Venice Beach house she shared with Maya, Lucy, John-Michael, Paolo, and her stepsister, Candace. It had taken a few months, but they'd grown close; a synthetic family on Venice Beach. It wasn't something they'd taken for granted. Yet, recently, there were tensions.
It was no surprise, given what some of them had brought into the house. Secrets, deceptions, crimes. Grace watched Candace rub lotion onto her arms, and felt a familiar flash of guilt. She had confided in John-Michael a secret she was still keeping from her own stepsister.
Grace had kept silent about the true identity of her father, Alex Vesper, for years. How would Candace react if she knew that her stepmother was once married to a convicted murderer, a man on death row? Would Grace's relationship with Candace survive, if that truth ever came to light? Without Candace, Grace was pretty sure she wouldn't be able to keep pretending that being emancipated was as easy as everyone liked to believe. It didn't help that the housemates had more or less all agreed to keep their parents at arm's length.
If Candace were to find out the truth about Grace's father from anyone but Grace, she'd feel betrayed. She might even begin to pick through her memories and wonder what other things Grace had lied about. Grace could already imagine her own response, begging to be believed that there were no other secrets and that even this hadn't been her choice but her mother's.
She lowered her eyes before Candace noticed. No. She couldn't risk it, as tempting as it sometimes was to confess. Like her mother always said: "It's not just _your_ secret, Grace."
Grace had confided in John-Michael, a week ago, on Memorial Day. She still wasn't exactly sure why she had. When one person shared something private, it felt right to share something back. At least, that's what she told herself. It was the reason why she'd brought John-Michael in on her own buried treasure—the truth that her father was "Dead Man Walking"—the death row prisoner she'd been writing to for years.
But John-Michael had shared a secret with her, too. One about his own father's death, which hung like a dagger above his head. The police had arrested John-Michael on suspicion of murder, but he'd been released without being charged. A week ago things had come to a head. Rather than benefit one more day from his father's pride and joy, John-Michael had driven his father's Mercedes-Benz convertible off the Pacific Coast Highway. Only Grace knew the real reason he'd done it.
She didn't agree with the morality of assisting a father's suicide—especially when it involved holding a pillow over your father's face until he stopped breathing. She could never have done it herself—however terminal the medical diagnosis. But John-Michael was her friend, and he'd trusted her with the truth. She would _never_ tell, and she knew he would do the same for her.
After sunset, the moms and dads, grandparents and toddlers began to pack up and leave. The disposable barbecues began to glow in the fading light. From the tiny plastic bag within her first aid pouch, Grace pulled fibers from cotton balls to use as tinder while John-Michael lit a match. His practicality often surprised and impressed her.
"When I was living on the street," he said, "there were days when I'd have shivered all night long if I hadn't been able to make a fire."
When the tinder had caught a flame, he tossed a burning cotton ball onto the teepee of sticks he'd carefully arranged. He dropped onto his belly, mouth no more than six inches from the nascent flame, and began to blow. Within a minute, the sticks had caught fire, too. They began to crackle and blaze. In another minute, they had a real campfire.
Candace approached, drying her shoulders and head with a towel. She knelt in the space between Lucy and Grace. After a moment she leaned her head on Grace's shoulder. "This is so great. Why've we never done this before?"
Paolo sat down on a rolled straw beach mat and peered up at her. "You're the one who's always working on your TV show over the weekends."
"Candace is right," Maya chipped in. "This is really fun. It's been a while since we just hung out together." She reached for the cooler and took out cold cans of soda and foil-wrapped cheese-and-meatball hoagies. Paolo caught the Diet Coke she lobbed at him and turned to John-Michael. "Dude, did you bring the rum and the limes? I wanna make a Cuba libre."
Grace felt the beginnings of a warm glow within her. The easy vibe that had once existed between the housemates wasn't quite there, not yet. But maybe it was a good thing that their secrets had begun to come out into the open. In the flame-heated air that connected the six friends around the fire, she could sense them coming closer together.
Maybe Lucy would finally reveal her own secrets. _If only_.
Lucy didn't know it, but she had the power to change Grace's entire life with a single action. She just had to tell the truth about the murder she'd witnessed as a child.
There had to be some way to get Lucy to talk. Grace knew that John-Michael had already told Lucy about Grace's father—she'd suggested it herself, to see if it might get Lucy to confess. So far Lucy had said nothing. The question was—had she made any connection between Grace's father's situation and what she'd witnessed as a child?
Grace couldn't be sure. No, it would have to begin with Grace sharing her side of the truth with Lucy first. She had to tell her that Alex Vesper, the man on death row for the murder of Tyson Drew, was her father.
Grace sighed. Then Candace would find out that she'd been lying to her all these years.
If only there was another way.
## **PAOLO**
#### MALIBU LAWN TENNIS CLUB, TUESDAY, JUNE 2
Paolo King was sitting at the country club bar, as he often did after a training session with one of the club's tennis students. It was late afternoon. Some kind of mellow jazz played in the background. Paolo had no clue what the tune was. He rarely heard that kind of music outside the club.
This particular student was somewhat older than the usual women who favored Paolo. She was in her forties, with perfectly styled blond hair that settled high on her neck. Her tennis skirt showed off tan, athletic legs that were crossed at the ankles.
He knew her only as "Jimmy's mom."
Paolo had once played a game of tennis for money with Jimmy, her idiot teenage son. He'd been tricked into swindling the kid out of a forty-thousand dollar Corvette. Even though Paolo had been basically blackmailed into the con by a guy who'd long since hit the road, Jimmy's mom held him responsible. She'd agreed to keep the matter away from the cops, but only if she and Paolo got on _real_ good terms.
He'd been doing a pretty good job at keeping up the pretense—minus one small detail. Somehow, her name had eluded him. She'd told him once but he'd forgotten. Now that they'd been intimate it seemed just plain rude to ask her again. Paolo had hoped that they'd never meet again. But no. Today, she'd lain in wait for him at the club.
"Your hair looks great," Paolo said as Jimmy's mom switched his Diet Sprite with the Tom Collins cocktail she'd ordered for herself. He picked up his new drink. "Are you sure you don't want this?"
"Better not." She smirked. "I'm driving."
So was Paolo, but in this woman's company he was wary of saying anything that might annoy her. The subject of Paolo's age—sixteen—was a sketchy one between them, on account of how they'd slept together. Technically, it was illegal, but this wasn't Paolo's first time with an older woman. He was pretty sure she'd drag out the names of all those other women he'd slept with at the tennis club, if he tried to use their relationship against her. He didn't want that.
He took a sip and tried once again to remember her name. She would be annoyed that he couldn't remember.
"I enjoyed watching you play against your coach." She smiled. "But my dear, it's exhausting. I thought he had you beat for sure."
Paolo humored her. "Victory tastes sweeter when you snatch it from the jaws of defeat."
Jimmy's mom would have known that if she'd stuck around to watch the whole match he'd played against her son. Darius, Paolo's doubles partner that afternoon and the architect of the entire scam, had made sure they'd let themselves be held down, almost until the end. A classic hustle. Jimmy had fallen for it, and hard.
She gave him a long, thoughtful look. Paolo avoided looking back, feeling the full weight of the years between them. He thought back to their last encounter. He already felt uncomfortable, the way you did when a girl came on to you and you knew you were going to deny her in the end. He did it all the time with girls from school who seemed to think he was some kind of trophy—to be snatched up and displayed.
"Are you quite sure that you can't move your class to another evening?" she asked suggestively.
Paolo's fingers closed around his glass. "I really can't. My student's already here—I saw her car in the parking lot."
She pouted. "Too bad." There was a hesitation, as though even she had some qualms about broaching the subject of their one-time affair. Then her voice dropped. "I've been thinking about you, you know." She tried to look him in the eye and couldn't. Suddenly, she seemed almost vulnerable.
Paolo struggled to maintain an even expression. She wanted him to say something similarly flirtatious, he could tell. When she noticed his reluctance, a smile spread across her lips.
"Come along, Paolo, don't be so bashful. We're hardly Mrs. Robinson and Benjamin, now are we?"
His eyes narrowed in puzzlement. "And they are?"
"You never saw _The Graduate_?"
" _The Graduate_?" He shook his head and took a sip. "No."
Jimmy's mom sighed patiently, as though dealing with a slow but fondly regarded student. "It's a marvelous movie, a classic. Mrs. Robinson is the bored, wealthy housewife and Benjamin is a recent college graduate, the son of her family friend. They get together. Ben's very shy about it all, at first. And then he starts to like it. Just the way you did, Paolo, that afternoon we spent together."
"How does it turn out?" Paolo asked, dreading the answer.
She shrugged. "Not altogether well. Benjamin runs off with Mrs. Robinson's daughter."
Paolo fumbled for words. This conversation was getting pretty bizarre. He wasn't sure what he was supposed to say. "Are . . . you not happy with Mr. . . . with Jimmy's dad?"
She leaned back in her seat, regarding him. "Happy? Paolo, you really believe that a happily married woman takes a sixteen-year-old tennis coach to bed?"
"I guess I don't know a lot about married people."
She touched his hand affectionately. He found his eyes drawn inexorably to the place where her fingers had lightly landed on his.
"Why would you? You're just a kid. Out of your head with testosterone, maybe, but that'll calm down, in time. I'm guessing you don't have a girlfriend?"
"Why'd you say that?" The words, too defensive, were out before he could stop himself.
She gave a curious smile.
"I don't have a girlfriend because the girl I want isn't interested."
A flicker of genuine interest. "Ah. Unrequited love, is it?"
"I don't know about love." Paolo could hardly believe he was talking about these feelings to a predatory stranger. It was as though she'd mistaken him for one of her girlfriends, someone with whom to share a confidence. Just the same, he tried to give a helpful answer. Jimmy's mom could still cause a lot of trouble for him. It was better to keep her on his side.
"I like this girl a lot. We kind of hooked up once and she wasn't into it. I thought I'd be angry. But I dunno. Somehow, it just made me want her more."
Jimmy's mom gave a satisfied smile. "My, my. Sounds to me like your little girlfriend has you exactly where she wants you."
"It's not like that." That didn't describe Lucy Long at all. He'd had enough experience with women to be able to tell when a girl really wanted him. Lucy hadn't shown those signs, not really. Not until she kissed him and it seemed like she finally wanted to get close to him . . . until she didn't.
"What I think," he said carefully, "is that she didn't like me, not that way, at least. She only wanted to hook up with me to forget about everything else happening in our lives. The timing was all wrong."
"Well, now," murmured Jimmy's mom, evidently surprised. "How unusually perceptive of you."
She pulled out her credit card as the waiter dropped the check. Paolo craned his neck until he could make out the name on the card.
_Meredith Eriksson._
Her name was Meredith. Wouldn't even have been in his top ten guesses. Paolo sank into his seat, barely holding back a sigh of relief.
"Perhaps you need to get that girl out of your system," she continued. "You know the saying; the best cure for an old love is a new one."
_But you and me, we can't ever be in love_ , Paolo wanted to say. It had been a one-time thing only to stay out of trouble, nothing more. He nodded a little and sipped nervously at his Tom Collins, quickly thinking of a way to let her down. "Meredith," he started, tentative with the use of her name. "You got any daughters? Maybe we can totally Mrs. Robinson this whole situation."
"Come near my house or my family, and it's good-bye to your tennis career," she said, with a sharpness that felt like a slap. "And as for law school, a few calls to some lawyer friends of mine would put an end to that, _Mr._ King."
Paolo played along, pretending it was a joke, but he knew it wasn't. "I'll be sure to remember that," he managed to say after a while.
"All right," she said, her tone crisper now, formal. "Paolo, where would you say this leaves us?"
He glanced up, bewildered. "Us? I thought you said . . . ?"
"I know what I said, but I expect some flexibility, naturally. Given the extent of your misdemeanor, I mean. The total cost of your little scam."
"It wasn't just me," he added resentfully.
"Yours and Darius's, then," Meredith said, widening her eyes. "Darius was rather more effective than you at making a hasty getaway." She leaned closer and smiled. "He's also far less attractive."
There it was again, the calculatingly ravenous look that made Paolo feel as though he were a strawberry cream pie.
"You cost me upward of forty thousand dollars, Paolo King. And as memorable as our afternoon was, on reflection, one time doesn't quite cut it."
Paolo felt himself swallowing his revulsion. He wasn't at all sure that he could make himself go through a repeat performance.
Reluctantly, he said, "What did you have in mind?"
"I have your number. When the mood strikes me, Paolo, I'll give you a call." Meredith stood. "Don't worry. I could be good for you. I suspect that when it comes to young women, you've a lot to learn. What we have is fun, but I wouldn't be averse to helping you win the girl of your dreams."
She touched his cheek thoughtfully. After a second or two, it turned into a caress. "Don't look so worried. I guarantee that you'll have a great time." She leaned in like she wanted to kiss him but remembered where they were. She brought her mouth close to his ear instead and whispered, "Grow your hair longer, and use a little bit of product. You're looking so good these days, Paolo, I could scream."
## **CANDACE**
#### CULVER STUDIOS, WEDNESDAY, JUNE 3
"Candace Deering? This way, please."
It was too easy, nothing like the other times. No line, no crowd of girls that made her wonder if she was a clone. She'd been escorted to the audition room by someone who actually knew her name, and when she entered, there were only two guys inside.
One of the guys stood behind a desk manning a short tripod and a video camera. The second leaned against the door frame, silent. He was in his midthirties, about six feet tall, dark-haired with greenish-blue eyes. The stubble on his chin lent him an unkempt look. Candace recognized him right away.
He turned, deliberately casual, to look at her. His eyes flicked up, then down. A two-blink checkout. "Thanks for coming to audition for us. I'm Ricardo Adams."
Candace smiled. "I know who you are," she said, figuring flattery was the way to go. "I watch _Deadbeat_."
"Always nice to meet a fan," he responded with an artificial grin. He eased aside to let Candace pass. She went to close the door behind her, but Ricardo caught the edge and held it open. "We're not all here yet." Cool professionalism oozed from his every pore.
Candace practically held her breath until a moment later she heard a cheerful yell from down the corridor, _"¡Oye, asere!"_
Into view came a guy, a little under six feet tall, about twenty years old and of African descent. He was lithe and slender with a stylish, layered pattern shaved into short dark brown hair. His features were delicate; high cheekbones, almond-shaped dark brown eyes with curly lashes and a perfectly sculpted jaw. He was ludicrously handsome. Beaming a 100 percent gorgeous smile from a face as sweet as his body was perfect, he sauntered into the room, did some kind of bro-hug with Ricardo, then leaned in and surprised Candace with a friendly kiss on the cheek.
"Yoandy," he said, his voice low and sultry. He squeezed her right hand. _"Encantado_. _"_
_His accent_. Light, yet definitely Latino. Was he Dominican? Cuban? Colombian?
"Candace Deering," Ricardo said formally, "this is Yoandy Santiago."
She nodded, trying not to look at the muscles playing under the flawless, smooth brown skin of Yoandy's shoulders. Most of his upper torso was exposed by the tight, sleeveless white top that he wore over faded, ripped blue jeans with cream-colored Chuck Taylors, unlaced and with the tongues pulled loose. There was no sign of any tattoo or other marking on him, but his throat was adorned with a doubled necklace of tiny red and white beads. He had the physique of an athlete or maybe a dancer.
Candace was pretty sure she'd have recognized him if he had been on TV before.
"I like your show," Yoandy said, amused. "You're Gina! She's so badass."
"How have you seen _Downtowners_? It hasn't even aired yet." Candace glanced at Ricardo for confirmation.
"I showed Yoandy some of your work," Ricardo said. Something about the way he bit off the sentence made Candace wonder. Had Yoandy's approval been important? Required, even? She struggled to remember the name: Yoandy Santiago. Was he already a bigger star than her?
When she focused again, Candace realized that Yoandy was staring at her, looking positively charmed. "I _loved_ when you killed that guy with your bare hands."
Candace pulled herself straight, recovered her poise, and smiled at Yoandy calmly. "He got what he deserved."
Yoandy burst into laughter. "I like this girl. Ricardo, give her the part."
"Give her the part?" Ricardo echoed, disgruntled. " _You_ haven't got the part yet, so why don't you zip it?"
"Dana will whup you if I don't get to play Sebastian," Yoandy said with a smirk. "And _I_ say you hire Candace for Annika."
She would have been appalled if anyone else had said that. But then again, she'd never met anyone quite like this guy. Charm radiated off him like some kind of magical power.
"Uh, hey—who's Dana?" Candace asked, pretending that she hadn't already scoured the internet for details of Ricardo Adams's private life.
"Dana Alexander," Yoandy explained. "Ricardo's wife. The British movie star?"
"Of course," Candace replied. "I love her work, especially when she played Lady Macbeth. I'm crazy for Shakespeare movies."
"Yoandy is a very good friend of Dana's sister, Kay," Ricardo said. "And Kay has persuaded Dana that Yoandy is the next Jaden Smith. The _Latino_ Jaden Smith. So she got Dana to _insist_ I get him a part on my show. Which is why you're gonna have to tolerate the lousy bum at least if you want to work on _Prepped_."
Yoandy beamed another radiant smile at Candace. "You see how he talks to me? And we're almost family."
_Almost family. A very good friend._
Candace made a conscious effort to suppress a frown. Here she was, about to land a huge TV deal and she was getting anxious. Was it because Ricardo mentioned that Yoandy was "good friends" with Kay Alexander? Could good friends mean something more? And did the fact that she cared mean she was jealous? But this _never_ happened to Candace. Guys crushed on her—not the other way around.
It was too bad. With every passing moment she was coming to the conclusion that Yoandy was the sexiest guy she'd ever seen in her life.
"Hi, I'm Lowell, one of the directors." The other man's voice broke across her thoughts. He'd been standing politely to one side with a clipboard in one hand, observing the interaction between Ricardo, Candace, and Yoandy. But now he spoke up, rather firmly. "A couple of questions. It says here on your résumé that you're emancipated. But you're still in high school, is that correct?"
Candace forced her attention away from Yoandy, who had turned to exchange a few quiet words with Ricardo. "Oh, uh, yeah. I'm at Hearst Academy. We break for summer vacation at the end of the week."
"That's excellent. We plan to record over the next month, break for about three weeks around July Fourth and be back at work by August. You'd have more lines, so the schedule's gonna be heavier than for _Downtowners_."
Candace nodded. This was all good news. More work, more money, more face time on TV . . . more Yoandy.
The director put down his clipboard. "If you're ready, Miss Deering, I'd like you to improvise a combat sequence with Mr. Santiago. Then we'll try the second scene."
Yoandy gave her a reassuring wink. He was taller than her by about four inches, and heavier by at least forty pounds. Taut biceps, the outline of a six-pack clearly visible beneath the thin, ribbed fabric of his shirt.
"You're my combat partner?" she asked.
"Don't worry, _nena linda_ , it's just like dancing," Yoandy said with affection. "And all Cubans know how to dance."
"You're Cuban?" she said, secretly wondering what _nena linda_ meant.
Ricardo snickered. "I guess she's never heard of you, lover-boy."
Ignoring him, Yoandy took her by the hand, as if to lead her to the exercise mat on the other side of the room. The skin of his palm was bone-dry, smooth and soft, and his grip was the ideal balance between control and gentleness. A little dazed, Candace found herself following. She wasn't quite sure what was happening. She could feel her breath coming a little faster. She focused on looking anywhere other than Yoandy's face or body, but her eyes were drawn to him with an irresistible pull.
When she finally willed herself to break the trance, she saw the director make a note in his smartphone. Then he glanced up at Candace. "Could you go from the top, do the scene, and go straight into the fight?"
"Improvise the fight?" she asked, eyes widening.
"I never did this before, either."
"But at least you're a dancer."
"It's just confidence," Yoandy said, and placed a palm on her shoulder as he looked directly into her eyes. "Candace, I've seen what you can do on _Downtowners_. You're a fantastic actor, in the physical scenes, too. You can do it. I promise."
Candace shook the tension out of her hands. Despite her mixed-up feelings for Yoandy, she was a professional. She kept her features sharp, determined. "Let's get started."
"I'll give you some basic choreography to rehearse," Lowell said. "Now, as Yoandy pointed out, we saw some really great physical stuff in your role as . . ." He fumbled for the name of her character on _Downtowners_.
"Gina," offered Yoandy. He and Candace shared a knowing smile.
"Right. Let's see some of Gina, but with some inner geek. Annika is not just a pretty girl-fighter, she's a scientist, too. Remember, the show takes place after the apocalypse and a de-aging virus has just been released into the water supply. You're a thirty-six-year-old woman who's suddenly back in the body of a teenager. You need to give us a hint of that experience and expertise."
"Hidden depths." Candace nodded. "I got it."
"Don't worry about playing a middle-aged woman who looks seventeen," Yoandy said in a conspiratorial whisper. "The guy I play is, like, fifty."
"All right," said the director, "let's talk choreography. Yoandy, he hits first; a left hook. You duck, then get up, swing in a kick, parry a punch, and then let him roll you over his back."
"Duck, swing-kick, parry, back-roll." Candace nodded again. "All right." She bounced lightly on the balls of her feet. _What the heck is this?_ No one had said the combat would be rolled into the same audition. No one had warned her to expect Ricardo Adams, the star of the new show, at the actual audition. Nor that he was already lining up other roles on the advice of his movie-star wife. Either her agent was totally incompetent, or the studio was playing mind games.
Yoandy continued to smile at Candace. It was a very natural, sexy smile. She made herself look at him from the corner of her eye, like when you tried to catch a glimpse of the sun during an eclipse. She probably should have been creeped out by this guy, who was most likely in a relationship, coming on to her so strong. And yet, she could somehow tell that he wasn't exactly doing that. It was just his natural charm, his unguarded reaction to Candace. Or maybe he had so much charisma that it couldn't be contained, even by the shackles of whatever it was he might have going on with this _Kay_.
Kay Alexander. Candace didn't know who the girl was, but already she couldn't stand her.
## **PAOLO**
#### CHEVY MALIBU, FRIDAY, JUNE 5
As things turned out, Meredith didn't wait very long to make her move.
The call arrived while Paolo was driving his housemate Candace to a TV studio in Culver City. She'd loaned her own car to Grace, whose team had a beach volleyball match. Paolo allowed no more than a quick glance at his cell phone before he slipped it back into his shirt pocket. Just enough to see who was calling.
_Meredith._
He turned to Candace. "Would you hate me if I just drop you off? Could you maybe get another ride home . . . ? One of the tennis pros had to cancel on a student. He's wondering if I'd take his lesson. But I'd have to bail on you."
Candace groaned. "Fine, be that way, but if you won't support me on the way up, don't expect any invitations to red carpet events."
"You without a date for a red carpet event?" he said. "Never gonna happen."
"Hmm," she mused. "You have a point." The car pulled up to the parking lot of the TV studio. Unfastening her seat belt, Candace groaned, "Go teach your lesson," but then her lips revealed a smile. "I bet you're real popular with the Friday-night crowd. Then after, maybe you'll take one of them out for a drink, am I right? Maybe they'll suggest a hotel?"
Normally, Paolo would have laughed it up. Now though, Candace's accusation held a little too much truth. If the request had really been for a tennis lesson—from just about anyone—he'd have turned it down. But with Meredith, Paolo sensed that "no" wasn't on the table. So he said nothing. Candace opened the passenger door. Her joking expression had vanished. She looked a little concerned.
"Hey, King, everything okay? You seem a little antsy."
He forced himself to smile. "I want to hang out with you. But the truth is, I need the money."
"Got it, good deal." Candace moved away from the car and waved good-bye. "Don't stay out too late, y'hear?"
Paolo was already reversing the car. He grabbed his cell phone and plugged in a set of earbuds. Then he touched the phone screen, returning the last call.
"Hello, gorgeous," said Meredith's silky voice. "Where are you? I'm in the mood for a drive."
"A drive? Where to?"
"We've got an adorable little cabin in the pines, up in Malibu Creek. My husband's away for the night. I'd love to take you down there." He heard the anticipation in her voice. "Or rather, to have you take me down there. I've had a little too much Pinot. It's better if _you_ drive."
"You want me to spend the night?"
"It's perfectly safe, Paolo. The place is in the middle of nowhere. My husband is in Sacramento until Sunday. Now stop being a baby and come pick me up at the club."
Pursing his lips, Paolo ended the call. He turned back onto Venice Boulevard, heading north. Ten seconds later, the phone rang again. He answered without looking at the screen. What could she possibly want now?
"What is it?"
"Sorry," he heard Lucy say on the other end, making him jump a little. "I guess now's not a good time."
"Oh, sorry. Just, I'm driving. Sorry, what's up?"
Lucy hadn't reached out to him in weeks. She'd barely even spoken to him. Paolo had been the one making all the effort to be nice since two weeks ago, when they'd almost had sex. Not that they ever talked about that. They were getting along fine, so long as they avoided the topic. Paolo wondered why Lucy was calling him now.
"When d'you think you'll get home?"
"Late."
Lucy went quiet again for a moment. "Oh. I was hoping we could hang out. I—I want to talk to you about something. Or rather some _one_."
The words made Paolo sink in his seat but he tried to hide his surprise. "Huh, mysterious. Who? Can you tell me now?"
A pause. Then, "Nah. This is better face-to-face. Also . . ." Another hesitation. "Are we good, Paolo? I mean, since . . ."
_Since we kinda sorta hooked up but didn't?_ he thought. "Of course. We're friends." The last thing Paolo wanted was for Lucy to have any clue how much her rejection had stung. Better that she thought it had all been lighthearted fun. "It can be with benefits if you like, obviously, I mean that's always on the table, but . . ."
"Yeah, all right, glad we're okay," she said, her voice brittle. "Not that I don't appreciate the offer," she added with a touch of sarcasm.
"I can't wait to hear what you're gonna tell me."
"You dying of anticipation over there? Talk to you tomorrow."
"I'm dying of something, that's for sure," he laughed as Lucy ended the call.
When he pulled into the parking lot, Meredith was waiting for him in the passenger seat of her silver BMW, fixing her lipstick in the mirror. Paolo parked alongside her car. Wordlessly, he watched her. After a moment, she turned to face him, blew a kiss, and smiled. "Get in," she said. "You're driving."
From a distance, she looked like a super-attractive mom being sweet to one of her son's friends. Paolo remembered when he'd first started noticing a difference in the way grown women looked at him. It had happened sometime around his fifteenth birthday. It was subtle. Not anything he could put his finger on. Just something speculative in the way other people's moms and even a few teachers watched him, _wondering_.
Paolo got out of his own car and into the driver's seat of her BMW.
She kissed him on both cheeks. "Hello, Paolo."
It was the first time he'd noticed her making an effort to look good for him. She wore a simple powder-blue silk shift dress that accentuated her slim, athletic figure, under a tight, indigo denim jacket. Her blond hair was freshly washed and blow-dried, her eyes made up with dark gray eye shadow that gave her a knowing, confident air.
When their eyes met, he could already see her desire. Despite himself, he felt the dull ache of his own interest awakened.
This was twisted, he thought grimly. _He_ was twisted.
He went for distraction. "What's the deal with this cabin?"
"Hotels are never entirely safe. Good ones, I might get recognized. Crummy ones, the walls are too thin." She fixed him with a sensuous smile. "I'd rather enjoy the opportunity to feel totally uninhibited."
Paolo reversed the car. Uncomfortably, he wondered how much more uninhibited she could possibly be.
They continued a little farther north along the coastal road, and then turned off onto Malibu Canyon Road. Meredith selected a Madonna playlist, pushed her seat way back, and leaned toward Paolo, her eyes and lips relaxed from the wine.
She reached for the glove compartment, fiddled around for a few seconds, and smiled with satisfaction as she found what she'd been searching for—a half-smoked joint. She used the car's cigarette lighter to light up. Deeply, she inhaled—a sound like a sigh. Paolo glanced across at her. He wondered if she'd even stay awake long enough to tell him how to get to her cabin.
The road started to twist, carving its way through the canyon as hills appeared around them. To their right, below the road, a creek flowed alongside. Dusk began to fall. The slopes of the hills and mountains glowed deep ocher as the sun descended, contrasting with the higher, deep blue of the sky. Every now and then they passed curves on which a scrub of parking area contained the occasional car that had stopped to allow the passengers to admire the scenery. Other times they passed improvised stalls selling firewood, strawberries, and bonsai trees.
A peaceful silence settled between them. Meredith finished the stub of joint, cracked open the window, and let the wind take what was left. Paolo checked on her again after a few minutes. She was close to sleep. He stretched a hand out to the car's touch-screen controls, trying to change the music. For a split second his eyes flicked away from the road.
The next thing he heard was a blaring horn. His fingers froze in midair. A gigantic truck was approaching, flashing its headlights. To his horror, Paolo realized that he was halfway across the road, heading straight for oncoming traffic. He yanked the steering wheel over, swerving out of the way of the truck.
The truck sped by, mere inches to his left. Shaken, Paolo slowed down, pulled over in a tight swerve, and came to a full stop at the side of the road. His heart was pumping, he could feel a fresh film of sweat under his arms. His breathing began to slow. He looked over at Meredith. She was awake, her eyes open, wild and staring.
"Omigod . . ." she slurred, fumbling for the door.
Paolo put a hand out, trying to restrain her. She shot him a look of pure disbelief, pushed his hand away. The passenger door swung open. Meredith staggered against it, almost fell into the road.
"It's fine, get back in the car," he tried to tell her. But she wasn't hearing him. Instead, she swayed for a second and began to move around to the driver's side of the car. She rapped against the window with her knuckles, shouting, "Get out, you maniac. I'm gonna drive. You almost killed us."
Paolo kept the door locked. Meredith glared at him, eyes totally unfocused.
"Don't be crazy," he said. "You're wasted. Would you please get back in the car?"
She straightened up, her features twisted with anger. Paolo slowly pushed open the driver's door. He had to get her back in the car. They were less than twenty yards from a blind corner. As the door opened, she backed farther into the road.
A white SUV appeared from around the corner. It slammed into Meredith, sent her body arcing through the air. He heard brakes screeching, hard. The noise seemed to last for several seconds. But the vehicle didn't stop. By the time Paolo was out of the car and at the roadside, the white SUV had already disappeared into the distance. Barely able to make himself look, Paolo turned toward Meredith. She lay in a crumpled heap farther up the road, straddling the shoulder, the golden tan of her bare legs contrasting with the gray asphalt.
He slammed the driver's-side door behind him, rushed to her side. Her eyes were still open, staring blankly into the sky. Paolo slumped to his knees. His fingers felt numb. Ice seemed to have crystallized along the length of his spine. He tried to say her name but no words would leave his mouth.
Blood pooled about her head like a wine-colored oil slick. Paolo looked down. He was trembling with shock. The right side of her head had been crushed—he could see the horrific dent. He couldn't bring himself to touch her.
Meredith was dead.
Desperately, he stared into the road. Two neat tire tracks marked the surface, a black smear that led to and away from a broken body. There was a faint smell of burnt rubber. Unsteadily, he rose to his feet, backing away. Panic was beginning to grip him. Pieces of a hideous puzzle were falling into place around Paolo. Police. Jimmy. Jimmy's dad. Eventually, maybe even the con Paolo had played on them with Darius. He was convinced that if anyone made even the smallest connection between him and Meredith, her death would be blamed on him.
Paolo knew, with blinding insight, the law would not protect him today. If he didn't get out of there fast, a world of legal consequences would surely descend.
## **MAYA**
#### VENICE BOARDWALK, FRIDAY, JUNE 5
Maya took a moment to check out the blond guy on her doorstep, wearing faded blue Levi's and an untucked, mint-green polo shirt under a shabby, brown leather jacket. He looked more put together than the typical Caltech student. He smiled back, a surprised little grin, like he'd also expected someone else but was kind of delighted by the sudden turn of events.
"I'm Jack," he said in a British accent, offering his right hand. With a measure of self-consciousness he added, "Jack Cato, your new tutor."
They had decided to meet up at the beach house, but after a few minutes it became obvious to her that was a bad idea. The kitchen/living room was the only communal space other than the balcony and it didn't feel right to monopolize either. It also didn't feel right to invite Jack into the bedroom Maya shared with Grace and John-Michael. Instead, she suggested they head for a nearby café.
A warm breeze greeted Maya and Jack as they left the house. There was salt and the fine dust of sand in the air. She breathed it in contentedly. Days like this, she loved being in the house. If she could only succeed in getting her school situation straightened out, she'd be so happy.
Two days ago she'd been called to see her homeroom teacher, Mrs. Geary. Math and chemistry were the problem, it seemed. With the long summer vacation coming up, her teachers didn't feel confident that Maya could afford the break. She was weak in both subjects and chem was the only science she was taking. Physics was just more math, and since Maya had a "thing" about cutting up animals, biology wasn't an alternative.
Geary had asked, "Do you think that the difficulties you're having in math and chemistry might stem from your dyslexia?"
_Well, duh._ Maya had just smiled sweetly. "Maybe." Or maybe it was just that she didn't have enough time to properly study, what with all the coding, but she guessed it was best not to mention that. There was extra funding available for dyslexia-related issues. But for spending-all-your-spare-time-developing-apps, not so much.
"I'm told that you're a gifted computer programmer?"
The comment had put Maya on her guard. No one at school—apart from Lucy—knew just how much of Maya's spare time went into coding.
Geary had continued, "I'm assuming you hope to get into a good college? With some improvement, that is a possibility."
She remembered how she'd reacted then—stared hard at the pattern of yellow-and-blue plaid on her skirt and pretended to be deep in thought.
The teacher had pressed the point. "Google, Facebook, all the top Silicon Valley companies; they recruit from the best colleges. Wouldn't hurt to get yourself on the road to all that. Your math and chemistry teachers think you just need a bit more study time. Maybe over summer vacation, with a private tutor?"
That had drawn a quick-fire, disbelieving response. "I have to come to school over summer vacation?"
"If your mother signs the permission slip he can tutor you in your home, which I understand is not with her. Is that right?" Geary had handed over a business card. "I'm going to recommend a young man we've used before. His name is Jack Cato. He's majoring in chemistry at Caltech. Very clever, and he's only seventeen. His agency will invoice the school."
"But he's not, like, a psycho or anything?"
"Jack's references are excellent; his tutorial agency has run all the background checks. He's a nice guy, by all accounts."
Now, with Jack beside her as they strolled along the boardwalk, Maya took a few moments to observe him. He wasn't particularly tall, only a few inches taller than her. He had a heap of dirty-blond, unruly, collar-length hair that reached down to his eyes. Slender and narrow-hipped, his clothes practically hung off his bones. Darker, prominent eyebrows lent intensity to his pale blue eyes. He didn't look any older than fifteen.
Maya had never talked to a British guy before. His manner was quite disarming. Distractingly so. He had a way of grinning while saying rather serious things, and rubbing an ear or his chin at the same time, or otherwise giving the impression of being deeply uncomfortable, laughing.
"You went to Eton College. Isn't that where, like, both the princes studied?"
"It is," he replied with a friendly grin.
"So do you know Prince Harry?" Maya asked, a little flirtatiously. "You sound a lot like him."
Jack guffawed. "Do _I_ know him? No. Harry's _army_. We—ah—we don't exactly move in the same circles."
"I thought Eton College was some super-fancy prep school. Aren't you 'posh'?"
"Me? God, no. I'm a scholarship boy. No aristocratic blood, no 'new' money. Very ordinary. Sorry about that."
"Why should you be sorry?"
Jack glanced at her for a second, as if to check if she was being sincere. They'd arrived in front of the coffee shop. He opened the door and stepped aside to let her through, with a gentlemanly flourish that Maya found impossible to take seriously. A little later, Maya realized he _had_ been serious. It wasn't that she wasn't used to the occasional bit of macho charm. But boys like that were often looking for something in return. It was unusual to meet a guy of Jack's age who was so self-effacingly gallant.
Unusual and kind of intriguing.
"So, Miss Soto, you're having problems with math and chemistry? Where would you like to begin?"
She eyed him quizzically. "'Miss Soto,' really?"
Jack looked immediately taken aback. "Do you not like that? Would you prefer Maya?"
"Miss Soto is, like, a teacher's name."
"Maya, let's do whatever makes you happy. Okay, tell me what's going on. What do you find particularly challenging?"
Maya opened the blue plastic folder that she'd brought along and removed some worksheets.
"Okay, so, acids and bases. Molarity and pH and stuff. It's chemistry, but there's some math, too. Basically, I'm clueless. To be honest, I don't really like chem." She shrugged happily. "Sorry to be so down on your entire existence."
Jack peered at her for a moment, his lips twitching as though they weren't sure whether to form a grin or a frown. "You're not giving it a real chance," he said eventually. "Chemistry is _immense_. And before we're done, you're going to think so, too." He took a black-and-red notebook from his own messenger bag and began to write. "There's a very easy way to make this stuff simple. What you need to do is keep the idea of moles in your head. You know what a 'mole' is, right?"
"It's a chemistry thing to do with molecules? I get confused between moles, molecules, molarity."
Jack smiled gently. "In that case, let's start with that." He waved the waiter down and ordered two café lattes and chocolate chip cookies.
A steady breeze swept across their table from the beach. Maya could suddenly smell Jack's mingled scent of bergamot-and-lemongrass-tinged deodorant and a hint of something muskier. Together with the watery blue intensity of Jack's scrutiny, Maya felt a definite impact. It took several moments before she was able to decode what she'd experienced.
Gradually, every word out of Jack's mouth came to seem utterly fascinating. And yet, as engrossing as those words were, Maya found it almost impossible to process what he'd been saying. Chemistry with him was going to be amazing. But she was still going to struggle.
## **PAOLO**
#### MALIBU CREEK STATE PARK, FRIDAY, JUNE 5
Underneath the bend in Mulholland Highway, the firm dirt of the slope gave way to gravel. Paolo began to slide. Arms stretched out for balance, he half fell, half scrambled down the side of the hill. By the time he reached the bottom of the gully he was covered in dust, palms grazed, mouth dry from the parched earth. He looked up. The road where he'd left Meredith was about twenty yards above him. It was still light enough to see without a flashlight. Anyone who stopped by her body would only have to throw a casual glance in his direction to spot him. About fifty yards away was the edge of a pine grove. It was the nearest cover.
Paolo turned and sprinted hard toward the pines. Behind him, he heard a car speed right past the spot where Meredith had fallen. Some people were soulless dirtbags. But for once, that was working in his favor. He kept his eyes down. The ground was full of rocky obstacles. Every yard brought hazards. This was hiking country, not a running track. But he couldn't slow down.
Thirty yards to go. Twenty. The sound of a car slowing down. Noises amplified by the dry terrain. If they stopped their engine they might even hear his footsteps.
Ten yards, five. On the road behind him, a car door opened.
Paolo dashed behind the thick trunk of a pine. He pressed up against it, tight to the bark. His chest rose and fell, burning. He spat dusty saliva, picturing the scene on the road above.
He'd left the BMW's driver's-side door open. Meredith's body was on the ground about ten yards along the road. To anyone who stopped, it would look like she'd been alone. Tests would show that she was drunk. A drunk-driving accident.
_My fingerprints are all over the steering wheel._
But what would even make them think Meredith wasn't driving?
Paolo raised his hands in front of his face. They were shaking. He interlaced his fingers as though in prayer, breathing in through his nose. He exhaled slowly. Apart from his fingerprints, there was no sign he was ever in that car. No reason to suspect she wasn't alone.
His heart thudded against his ribs. He could feel blood draining from his head. Panic rising from nowhere, threatening to engulf him.
_Think. Be still, and think._
He closed his eyes and thought of _deuce_. Match point to the opponent on deuce, his own second serve. Blow this and you blow everything. Be calm. Becalmed, like a sailboat. There's no wind. The sea is like a mirror. This boat is going nowhere. _Breathe_. Pull back your racket and serve.
Paolo's eyes opened. The sounds from the road carried with absolute clarity across to where he stood hidden. At least two cars had stopped now. Raised voices. Phone calls were being made.
_There's no sign I was ever in that car._
He clung to this thought as he began to navigate through the trees. Every step took him farther into the wilderness. Roads and hiking trails twisted across these hillsides every which way. He'd be sure to run across one, eventually. And then what?
Clumsy, ambling movements eventually became a regular strolling pace as Paolo adjusted to the minimal light. In the east of the sky, a pale, greenish tinge hinted at the approach of the full moon. He wasn't wearing a watch, as usual, and so he checked the time on his cell phone. It was a little after 9:40. He pocketed his phone again and peered into the gloom of the gulley into which he'd stumbled. It was impossible to see more than a few feet ahead. The ground was dry and cracked beneath his feet. Dry scrubby grasses lined the route. Somewhere to his left, Paolo heard the trickle of water. The vaguely clear path that he was following veered in that direction. The grasses closed in, until Paolo had to brush them aside as he walked.
It struck him then that terrain like this might house rattlesnakes. Paolo stopped and made his way back to a cluster of trees that he'd passed minutes ago and began, carefully, to examine the ground for any kind of stick. He found a twig no longer than his forearm and he used that to scratch around for another, longer specimen. It took longer than he'd hoped, with the dim light challenging his eyesight. His senses became more alert in the quiet, magnifying every tiny noise, and made him jolt to attention. Eventually, he found a stick, about a thumb thick and a yard and a half in length.
Making his way back along the trail, Paolo swept the staff before him, clearing the path, just in case any snakes were dumb or sleepy enough not to scoot out of the way at the sound of his footsteps. The sound of water became stronger, but never reached more than a healthy gurgle. If this was a river, then it was mostly dry, like many rivers in the California desert.
For the first time since he'd left Meredith, Paolo allowed himself to relax, just slightly. It couldn't take more than an hour or so in any direction to happen across one of the crisscrossing roads. The whole area was a national park, so there would have to be some kind of visitor parking lot, eventually. He wasn't exactly lost, but he still needed to find a way home.
Paolo felt for his cell phone and then stopped. The cops could trace cell phones, if they're used. If they were scanning for calls, hunting for the hit-and-run driver, they might use the GPS on his phone to place him at the crime scene. They might connect Paolo to the accident. They might even find the hit-and-run driver—who would tell the cops that Meredith wasn't the driver.
The parking lot at the country club. Paolo's breath caught in his throat as he struggled to remember. Were there security cameras? He was pretty sure there were. If he'd been recorded getting into her silver BMW, then it was all over. He'd have to make some kind of excuse—say that Meredith gave him a ride somewhere, dropped him off.
But why would he need a ride if his car was right there?
Paolo could feel desperation swelling his chest. He needed to do something about his car. But how? If only he could call someone. John-Michael would help him. John-Michael kind of owed him for going halfway to San Francisco to pick his ass up after he totaled his Benz. Not to mention that like everyone else in the house, Paolo was keeping quiet about the fact that John-Michael had driven it, quite intentionally, off the coastal highway.
He urgently needed to call John-Michael. But Paolo couldn't risk using his phone.
The trees grew more densely as the slope began to rise. Paolo's progress slowed. He paused to get his bearings. Mulholland Highway was still right behind him. He was about one hundred yards away now. Still no sign of a trail. He pressed on. There was barely enough light to see by. The crickets creaked loudly in the undergrowth. It was the worst time of day for snakes. They'd be coming out now, slithering across his path. The best strategy was to make as much noise as possible to scare them away. But Paolo wasn't sure if he dared to make loud, human sounds. All it would take was for one person on the accident scene to wonder if there had been someone else in the car and the police might come looking for him.
_Who's gonna tell? Not me_.
The hit-and-run driver wasn't likely to come forward, either. Paolo felt a queasy sensation as he realized—he and Meredith's killer were now in this together. If one of them were to come forward, it would immediately trigger a hunt for the other.
It was a pretty solid bet that the driver wouldn't be the one to come forward. But what if the guy got a sudden attack of conscience? Or realized that he might get caught and decided not to risk getting charged with a more serious crime, like trying to get away with it?
There could be no relaxing about this. Paolo had to do everything in his power to avoid being linked to Meredith's death.
To his left, Paolo could hear the nearby creek trickling, an anemic sound compared to the heartier gurgle higher up the gulley. From over the rolling peaks in the east, a dazzle of moonlight now lit up a whole sector of sky. He looked up hopefully. Maybe soon he'd have enough light to be able to see a road. He was pretty sure he'd spotted the occasional red taillights streaking past, a long way ahead. Paolo stepped out with confidence, determined and optimistic.
His footstep landed on the sandy ground but didn't rebound. Instead, it sank farther, broke the apparently dry surface, and gave way beneath him. The momentum of his walk carried his second foot inexorably into the same position. Both his feet were immobilized, one just below the surface, the foot completely submerged. Paolo felt panic clutch at his chest. He cried out. Terror swept through him as he struggled to understand what was happening. It only took him a few seconds to figure it out, yet those seconds passed slowly, vague thoughts infiltrating his mind.
_I'm stuck. I'm stuck in quicksand._
Moments after he'd felt that first foot slump underneath him, Paolo began to realize that he was sinking deeper. He pulled hard at each foot, twisting this way and that. With every movement he sank a little more. Now he was submerged to the knees. The deeper he went, the more he panicked. The dual sensations of being trapped and of sinking were simply overwhelming. Any minute now he'd be in up to his waist. Then he'd have no hope, none whatsoever. The slowness of it all only added to the horror. It was like witnessing his own demise in slow motion.
With a flash of good sense, Paolo reached into his pants pocket for his phone and transferred it to the slim, tight pocket of his polo shirt. His right hand then dropped to the surface of the quicksand, which was dry and crumbly. He stared hard at the area around him. The surface of the gloopy mud gave absolutely no indication of what lay beneath. Experimentally, he stuck three fingers into the quicksand and quickly pulled them out. They were coated in thick mud the consistency of whipped heavy cream.
The mud now reached the top of his thighs. Paolo looked around. There was nothing for him to do but yell for help. Even that wasn't likely to bring anyone. And if he were found, how would they even get him out? He might be stuck for days.
_I might be on the point of death by then._
Paramedics would be involved, the cops, too, most likely. Questions would follow. Where had he come from, what was he doing here?
It dawned on Paolo with a burst of clarity—the only way out was to get someone to come for him. He'd have to contact John-Michael. Even if it meant risking the phone.
For the next few minutes, in the darkness and silence, Paolo weighed the risks. Dying of dehydration under the California sun. Immediate exposure to the cops, with questions sure to follow about Meredith. The phone was a far lesser risk, he could see that. Yet, who really knew what they could tell from cell phone data? Paranoid libertarian types were always bleating about how the NSA could figure out what you had for breakfast from your data trail, but how much of that was grade-A wingnut nonsense, and how much was true?
Paolo faced up then to the fact that he really didn't know. But if he wanted to stay out of prison, he probably shouldn't use his phone.
Which left only one option. He had to accept that no one would come to help him. Paolo had to get out of the quicksand, alone.
## **PAOLO**
#### MALIBU CREEK STATE PARK, FRIDAY, JUNE 5
Paolo swung both arms around, trying to find anything he could grab on to. There was nothing. His stick was partly submerged, poking out from the dry surface. Paolo pulled it out and wiped off the mud that coated the top ten inches. He tried swinging it around, hoping to catch on to something in the nearby undergrowth that might be rooted deep enough to hold him. But whatever he did manage to hook with the stick came out of the ground at the slightest tug.
He stopped swinging. The movement had taken him down even farther. Now the mud reached his belly. Paolo's breath became ragged. He could hear the blood rushing to his head.
_I'm not gonna die. Worst-case scenario is prison. Absolute worst._
The idea of prison was terrifying. Paolo had some idea what prison would do to a good-looking guy like him, someone with zero connections in the criminal world.
He simply could not allow that to happen.
"I'm getting out of here," he said aloud. A little louder, he added, "I'm not going to prison. No one is ever going to find out about Meredith. Everything is going to be fine."
For a few minutes, determination pulsed through him. He lay the stick down ahead of him, across the surface of the sand. It didn't break the crumbly layer at the top.
_The surface. I have to spread my weight._
Bending himself over at the waist, Paolo positioned the stick just below his chest, perpendicular to his body. With both hands, he grabbed the ends of the stick, which was about a yard and a half across. His torso fell across the surface of the quicksand, broke it in places. Now his chest was covered in the thick mud. But to his immense relief, he didn't sink far, no more than an inch or so. It was frightening when he was forced to touch his face to the top layer of quicksand, but even then, only his chin disappeared into the mud. Meanwhile, behind him, Paolo was kicking hard with his legs, resisting the thick suction of the mud as he drew them up closer to the surface behind him. The sheer effort had him panting and sweating within two minutes. His entire body was covered in mud now, his arms buried fist-deep at the surface, his face smeared from chin to forehead, his torso completely coated.
But when Paolo tilted his exhausted face to the sky, he realized that he was mostly lying on the surface, with much of his body submerged by less than a foot. Like some slithering creature, he could crawl across the patch of quicksand using the stick for leverage, until he reached dry land. He took a few deep breaths and gathered his strength. He could do this. He was going to be free.
He let his weight fall forward, spreading himself as wide as possible with his arms. He squirmed, but still he resisted the final submission of burying his face in the slime. The tendons in his neck strained from the effort of holding his chin a fraction of an inch away from the mud. But when he relaxed, he was crushed to see that he'd made almost zero progress.
Desperation shot through Paolo like an arrow. He gulped down a huge breath. This time he flung himself into the mud, no resistance whatsoever. His face sank beneath the surface. He wriggled and put every ounce of his energy into twisting his hips, bucking upward to free his legs. Thirty seconds later he dragged his head free and gasped loudly, dragging down another breath. He was tiring fast. A little more of this and he'd be wrecked. They'd find his mutilated body days from now, perhaps. His eye sockets would be empty—he knew that much. Buzzards went for the eyes of weakened prey.
This final thought was what pushed him to the edge. With a final burst of concentrated effort, he tossed and turned in the mire, until finally he felt his legs freeing enough to allow him to roll onto his back. His neck rested against the edge of the riverbed and—finally—solid land.
After resting for another five minutes, Paolo hauled himself to his feet. He checked his cell phone. It was dirty but mostly dry. The tight fit of his polo shirt's pocket had protected it.
He began to walk, shuffling now, like a swamp zombie, until the sandy mud trapped beneath his clothes began to grate his skin. He stopped and stripped off his clothes, until he was wearing only boxer briefs, socks, and sneakers. He shook the jeans and shirt until he'd gotten off as much mud as possible. He had to do something about the rest of the mud, now, or else it was going to make movement impossible.
To his left he could still hear the creek, but the knowledge that there was quicksand in the area made him anxious. In the end, Paolo decided that he had to reach the water. Now even more cautious than before, he approached, tapping the ground in front of him until he'd reached the trickle of water. He knelt down and dunked his whole head in the cool water, almost sobbing with relief. Soon enough his clothes were all rinsed through and wrung out. He dressed in the damp garments and pocketed the cell phone and the wallet he'd removed during the washing. Once again, Paolo began to walk. And as he walked, he made a plan.
Finally, he came to a narrow country road. It was overhung by trees with low branches that cast a trellis of shadows in the dusky light. The temptation to check the GPS on his cell phone was extreme. Fear restrained him. He followed the road to his left, walking just to one side so that any passing cars might not spot him. In the twenty minutes he spent on the road, only four cars drove by. The road bent and twisted on itself several times. Eventually he arrived at a crossroads. On the opposite side was a sight that made him feel light-headed with hope.
A Department of Parks and Recreation parking lot. From here he could follow a road out of the park. It would take hours to walk to the coast, but he was easily fit enough. Once he was out, he could find a bus to take him closer to Venice.
There was still the problem of his car, parked in the lot of the country club. That was the priority now. He had to get ahold of John-Michael.
Paolo approached the parking lot cautiously, staying in the shadows. A few vehicles were parked in a largely empty lot. There was a public restroom, a vending machine that sold water, sodas, and candy. And a pay phone. Hands trembling, Paolo checked his wallet. He only had two twenties and a five. He examined the parked cars. One was open, the male driver leaning against the passenger door, smoking.
Paolo doubted he could risk a simple approach, like asking the guy for change. In soaking wet, mud-stained clothes he'd be way too memorable, when what he needed was to be invisible. Desperately, Paolo forced himself to review his options.
He could ask for change and make the call to John-Michael. Both actions could be traced to him. He could ask for a ride back to the coast. Another witness, right there.
There was nothing he could do that wouldn't in some way incriminate him.
Nothing _legal_.
The key was to avoid using any _traceable_ method of contacting John-Michael.
He needed access to someone else's cell phone. He'd delete any record of the call and no one would be the wiser.
Paolo went to the hut that housed the restrooms. He waited by the vending machine, careful to keep his face turned away. After ten minutes a woman approached with a little boy around three years old. As they entered the ladies' room, Paolo turned to see which car they'd come from. It was a navy-blue Buick LaCrosse. There didn't appear to be another passenger in the car. He approached, trying to remember if he'd heard the woman activate the lock. He didn't think he had. Nor had the headlights flashed. Paolo increased his pace. He felt a surge of hope.
He reached the side of the car. He checked back at the restrooms. No sign of the woman and child. Inside, he could see the blinking light on a cell phone. It was lying right there on the passenger seat. He put a hand on the passenger door. It opened.
He reached for the phone. Then he froze. A girl around ten years old lay asleep in the back of the car. The shock jolted him. He straightened up fast, slammed his head against the roof of the car. The girl didn't budge. Paolo pulled back clumsily. The cell phone was in his right hand. He just about had the presence of mind to close the car door as he made a hasty retreat, as quietly as possible to avoid waking the girl. Ten seconds later he was behind a tree at the edge of the parking lot. He looked across to the car. The woman was on her way back, the little boy's hand in hers.
He didn't have much time to make his call and drop the phone back into the car without being seen. Paolo dialed quickly—the only number he knew by heart, the house landline. Candace's mom had insisted they get one for emergencies.
_Please pick up, John-Michael. Please pick up._
Only John-Michael could help him now. He was practical, calm—the only housemate who wouldn't ask too many questions.
His friend's pleasant, tenor voice answered, "Hello?"
The air left Paolo like a gust of wind. "John-Michael, thank God. I need your help. This is serious, man. It's _extreme_."
"Go ahead, dude. I got your back."
"Get the spare keys to my Chevy. Take a taxi to a gas station in Malibu, but don't stop right at it—walk the last part. Pick up a five-gallon container of gas and a baseball cap, pay for it in cash. Take a second taxi. This time, you're wearing the baseball cap. Keep it low over your face. You got that? Baseball cap. Don't make the taxi wait, get another one. Take that taxi to the Malibu Lawn Tennis Club. You'll see my car in the parking lot. I want you to put the gas in my car."
"Paolo . . . you sound awful. D'you need medical help?"
He felt himself nodding hard. "I'll be fine, JM. Just listen. Keys. Taxi to a gas station in Malibu, but get dropped off before you're actually at the station. Gas container. Baseball cap. A second taxi. The country club. Put gas in my car. Drive it home. Stay there. Now, repeat that back."
Paolo waited, motionless, as John-Michael stumbled his way through the instructions, twice. "Okay. Good. Also—don't call my cell. Don't call this number."
"I won't."
"One last thing, John-Michael. You're me. You got that? I'm the one going to the gas station. I'm the one taking a taxi to the country club. I'm the one putting gas in my car. My car is out of gas, that's why I had to go to pick up gas. Then I drive home. That's all. _You're me._ So borrow my clothes. I'm wearing blue jeans, gray shoes, and a pale yellow polo shirt. Do the best you can."
There was a momentary pause. "That's a pretty messed-up set of instructions."
"I don't have time to explain. You get what I need you to do?"
"I got it," John-Michael said. "What about you?"
"Don't worry about me. _You're_ me. Tonight, I got gas for my car, which I had to leave at the tennis club. Then I went home."
"Dude, I don't look anything like you."
"That's why you wear my clothes and the hat."
A sigh. "All right."
"And don't tell the girls what you're doing."
There was a hollow laugh. "Hell no, we don't tell _anyone_. I don't know what you've got going on, Paolo, but I already don't want to find out."
## **LUCY**
#### VENICE BEACH HOUSE, SUNDAY, JUNE 7
Lucy had just settled down to catch up on some episodes of her favorite TV show, _Grey's Anatomy_ , and was secretly glad when her housemates had all gone to their rooms for the night so she had the entire couch to herself. Just as she was getting comfortable the doorbell rang.
Lucy visibly slumped with shock when she opened the door. A wave of cinnamon-scented air wafted past her nose. Nicotine chewing gum. The scent took Lucy right back to those first weeks at rehab, going cold turkey, getting rid of all the stimulants on which her body had come to rely. Lucy froze. Her old rehab buddy from Claremont, Ariana, was leaning in to hug her.
"Dear Lord, thank goodness I found you home! I've been fretting that you'd be out with your friends."
Before Lucy could move away, the skinny redhead had wrapped her arms around her shoulders and was squeezing tight, rocking slightly. "Sure—I should have called. But some creeper stole my cell when I was asleep on the bus. I need a bed for the night. You gonna help me out, Lucy?"
Lucy counted to ten and then gently disentangled herself from Ariana. She stood back to get a good look, or maybe to check that the sinking feeling in her chest wasn't the result of some stupid dream. In front of her stood a petite white girl, slim and with dyed-red hair that reached her shoulders, dressed in tight white jeggings, a yellow sleeveless T-shirt, and a faded denim jacket. On her feet she wore a pair of scuffed red Keds. Gold bangles, hoop earrings, and a silver belt jazzed up the outfit, but made her look several years older than her almost nineteen years, like a woman in her late twenties trying too hard.
"Hope you don't mind me showin' up out of the blue? You don't look too happy! Lucy Long, I sure have missed your moody self."
In a daze, Lucy showed Ariana into the house and took the shoulder bag from her as Ariana wheeled in a carry-on suitcase. She breathed a quiet sigh of relief that no other housemate was downstairs to witness Ariana's arrival. Lucy's mind was already rifling through possible ways to get rid of Ariana.
Ariana looked slightly sweaty, hair mussed, disheveled from the bus journey. Lucy watched her old friend drop onto the gray three-seater sofa. Ariana scowled, then pulled a discarded copy of _Variety_ out __ from beneath her. Lucy went and got a glass of ice water for Ariana, who sipped at it gratefully.
Stumbling for an easy conversation opener, Lucy said, "You came on the Pomona bus?"
"Two changes after Union Station," confirmed Ariana, fanning herself with the magazine. She already looked irritated, as though Lucy's questions were some silly barrier against which she had to battle constantly. "It wasn't easy to show up here unannounced. But I figured, you know, I had to do it. I couldn't stay in Claremont another day. I needed my best girl, Lucy. If you'd said no, that you were busy or something, I wouldn't have gotten onto the bus. That's it. I'd still be in Claremont now, already thinking about getting some blow."
Lucy froze. She fixed Ariana with a stony glare, and then exhaled slowly, unable to speak for several seconds. "Cocaine? That's what you're into now? Ari, jeez!"
Ariana's reply was wheedling, pitiful. "At least it isn't crystal, right? Or H?"
Dismissively, Lucy said, "I'm no angel. I admit it—I've started smoking weed again, too." Her voice rose, suddenly imploring. "But Ari. You gotta stay off the hard drugs! Is that what you're spending all your money on? Is that why you can't afford a taxi or a place to stay?"
Ariana's face became serious. "You always could see through me. I'll be straight with you, Lucy. I've racked up a little debt again. Not too much," she said quickly, raising both hands as Lucy's expression of forced compassion turned into a frown. "But I needed to get out of there quick, break the habit, go somewhere totally different, be around someone I know can help me."
"So you came to me," Lucy said, struggling to keep the sourness out of her tone.
A beam spread across Ariana's face. "You never once let me down, hon."
Then she was hugging Lucy again. Over the girl's shoulder, Lucy gritted her teeth. There had to be some non-mean way out of this. As annoying as Ariana could be, she'd been a solid friend through the worst months of Lucy's life, and shown her nothing but steadfast devotion since.
Lucy cleared her throat as she pulled away as gently as possible. "How long you, ah, thinking of staying here?"
Ariana's seraphic smile reappeared. "Could I? It'd just be a couple of nights."
"Really?"
"A week at most."
Lucy plucked at her lower lip. "We don't have any spare beds."
Ariana glanced around the living room and tipped a finger in the direction of the green futon. "Isn't that one of those sofa beds?"
Lucy tried to smile. "It's just that, you know, the house is already kind of crowded. Six of us sharing two bathrooms. It gets kind of crazy on school mornings."
"Then I guess it's good that school is almost out. Look, let me help. I'll cook y'all some breakfast, and I won't shower until y'all are gone. I can take care of you a little, hon. Let me do that? C'mon, all this living on your own, you gotta be missing your family just a teeny bit, right?"
Shaking her head firmly, Lucy said, "Not even. I'm exactly where I want to be."
"But you're using again?" Ariana said with a little pout that Lucy could tell was entirely fake.
"Alcohol, tobacco, a little weed on special occasions. It's no biggie, Ari. I'm like a regular person now; I don't go crazy."
"But you're not a regular person, are you, honey?" Ariana said very gently. "You're an addict. Just like me. There's no middle ground for people like us."
Lucy didn't reply, but felt anger burn through her eyes. Ariana was right. _An addict is an addict._ But the illusion of being a normal teenager with a chemical hobby was way better.
"Listen, I'm three days clean, Lucy. But it's not easy. I knew if I stayed in Claremont, I'd fall right back off over the weekend. God knows I've tried enough times. So okay, I admit it, I didn't think about it too much, I just packed a bag and headed for the bus. Please don't let me down. If I can just get through the next week with a good friend by my side . . ."
Slowly, Lucy blinked. She could feel her resolve cracking. The other housemates were going to kill her.
"Okay. I guess. Just a few days, and you be sure to keep your stuff all neat. This is a pretty chill place, but things get nasty when people don't clean up."
Ariana shook her head fervently. "Oh, that's me all the way, Lucy. You remember our room in rehab, right? Didn't I keep it nice?"
"Yeah, true. Between the two of us, I was the slob," Lucy said as a reluctant grin spread to her lips.
Memories of their time in rehab returned. It hadn't all been bad. The environment had been pretty relaxing: ashram vibes, dreamy music, and the odor of lemongrass everywhere. They'd donned white, waffled-cotton robes and slippers every morning before heading to the sun-drizzled breakfast room in which a dozen bleary-eyed, rich junkies would enjoy mint verbena tea with wheat toast and apricot jam made from fruit grown in the center's own orchard.
Even more impressive for someone of such modest means as Ariana, although from what Lucy understood, her people were recently impoverished. They still rattled around in some dilapidated old house in the Louisiana swamp, at the center of what had long ago been a small sugar plantation.
Lucy had often wondered where the money had come from to pay for Ariana's stay at the rather exclusive rehab center. She'd guessed that the girl had a rich aunt or something; someone who might occasionally be leaned on for a favor, but who wouldn't go as far as regular support.
"You were a slob, messing up a nice place like that," Ariana chided. "But I didn't let you get away with it, did I?"
"Guess you didn't," said Lucy, trying to smile despite the vague sense of unease she felt at letting Ariana stay. She should really ask the others first. Still, Ariana was a rehab buddy. That was a tight relationship. You didn't even really need to like your rehab buddy, but if it came to a matter of support over the addiction, you had to have each other's back.
Ariana patted the space beside her on the sofa. "C'mon, girl, sit awhile. We need to get caught up. What's it been since we really talked—five, six weeks? What's going on with you and that tennis player, Mr. Disney Channel? Anything ever happen there?"
"Paolo? You could say that."
Ariana smiled wickedly. "Ah, it's like that, is it? You lose your cherry yet?"
Lucy sat beside Ariana, and slapped her sharply on the arm, hard enough to make her yelp with surprise. "Ugh, please. Like I would ever go for him."
Ariana laughed, rubbing her arm. "Ooh—I touched a nerve."
Lucy shrugged. "Maybe. Truth is, we almost hooked up. But I changed my mind."
"Did he get mad?"
"No, actually he was kind of a gentleman about it."
"That surprise you?"
Lucy shook her head. "Not really. But maybe he's angrier than I thought. We were gonna talk yesterday and he avoided me the whole day. Today, too."
Ariana wrapped both arms around Lucy, slow and deliberate. Lucy kept still, but the gesture wasn't welcome. "Ari's here now, honey," she murmured, her breath warm on Lucy's neck. "We can be there for each other. Just like old times."
Lucy began to count down in her head. She'd forgotten how clingy Ariana could be. Having her around was going to be sheer claustrophobia.
## **MAYA**
#### OUR LADY OF MERCY CATHOLIC HIGH SCHOOL FOR GIRLS, TUESDAY, JUNE 9
The air-conditioning in the school library wasn't working. Miss Topalian, the librarian, had opened the French doors to the patio outside. Somewhere in the yard, an argument was rumbling along. Maya could hear the raised voices of moody teenage girls and, despite the fresh air, she was sweating lightly under the tight polyester blend of her school uniform. Under the desk, she kicked off her shoes. For once, she was glad she was wearing a skirt.
She peered hard at the computer screen. Another bug. Her new program code was full of them. Each report she ordered found at least one. She sighed quietly. Was there no end to this?
Maya's idea for the Cheetr __ app had seemed so blindingly simple that she couldn't believe it hadn't been done before. The first version had been a thing of elegant simplicity, and had worked after only a week of debugging. That was the first time she'd seriously looked at what _else_ was out there that was anything like Cheetr. Turned out to be quite a lot—people obviously loved to download free game cheats to their smartphones.
Cheetr was okay, but it was going to have to get a lot more amazing if it was going to stand out from the crowd. It was being downloaded plenty, but not enough to make more than a hundred bucks a month in advertising revenue. It wasn't even just about the money: the truth was, since Maya had found out how many similar apps were out there, her interest had waned.
Who really wanted to be nothing more than a _wannabe_? Original ideas were elusive, that was for sure. But originality was well worth the extra effort.
Maya's cell phone buzzed.
**WHERE IS THE LATEST REPORT?**
Yet another message from her mom about her employer, Dana Alexander.
Of all the people her mother had to work for, why did it have to be Alexander? Maya felt like putting a curse on that British movie star for soaking up so much of her time and attention.
Maya had to get her mind off her family troubles and onto her work.
It wasn't easy, though. Maya's mother was terrified that Alexander would reveal their family's secret and she would be deported. If that happened, Maya knew her mom would insist on taking her along, too. Not all parents were so ready to let their kids be emancipated—Maya's least of all.
That fear had driven Maya for long enough. Spying on her friends was no longer an option. Lying, concealing. What was the point of doing something that made your everyday existence a misery?
Being sent to live in Mexico City didn't frighten Maya as much as it once did. Now that she'd figured out why Dana Alexander wanted her to spy on the house, she was more scared of what Dana might do to keep her own secret safe.
From what Maya had pieced together, it seemed Alexander had attended the infamous Hollywood party where movie star Tyson Drew was murdered eight years ago. A drunken brawl between Grace's father, Alex Vesper, and the murder victim had taken place earlier in the evening. Combined with Vesper being the only party guest with no alibi, it had been enough to convict Grace's dad. Now Grace's father was on death row for the crime, but maybe the wrong person had taken the blame?
Lucy was at the party, too, back when she was a child star. The fact that Alexander was going to so much trouble to keep tabs on Lucy and everyone associated with her made Maya wonder if Lucy had seen someone else with Tyson Drew at the party that night. Someone Dana hoped she'd keep quiet about.
Did Lucy know something that proved Dana Alexander was the real killer?
Dana Alexander; Grace's father, Alex Vesper; and Lucy. A triangle of deceit connected them. Maya was acutely aware that amongst the housemates, she alone had access to all three points of the triangle. She alone could fit the facts together. And now Maya was scared enough to make a crucial decision.
She was done with helping _la inglesa_. Sure, she'd file the reports. But from now on they'd mislead and distract.
_I'm not your slave anymore, Dana—I won't betray my friends._
Arriving at that decision had felt like a release. No more sneaking around or spying on those closest to her. It had unlocked Maya's creativity. Since then Maya had dreamed up a truly _immense_ (as her tutor from England, Jack, might put it) new app—Promisr. A social app for matching favors—a promise for a promise, the barter economy.
Maya hadn't dared to show the whole app to anyone yet. They might hate it. But worse still—a lot worse—another coder might steal the idea. For now, it was Maya's secret. Sometimes, in the early hours of the morning when lines of code danced before her eyes, even when she was trying to sleep, it felt a little like her own personal hell.
It didn't take long to locate the bug in the code. When she did, Maya cursed, quietly and vehemently, in Spanish. Another unbelievably dumb mistake. A five-year-old child wouldn't make that mistake.
Maya corrected the code, recompiled the program, and started the tedious process of reloading it on the iPhone emulator.
_Dyslexia_. It slowed everything down. At least Jack was managing to tackle some of the academic weaknesses that arose from the sheer lack of time she had to devote to math and chemistry.
The first lesson with Jack had gone amazingly well. She hadn't been able to stop thinking about him since that day on the boardwalk when he'd laid bare the mysteries of _molarity_. She was also surprised that some of what he was teaching her was beginning to sink in.
"It's incredible," she'd marveled, minutes after solving a particularly fiendish problem. "Why don't they teach us this way at school?"
"No idea. It's how I was taught."
She'd released a huge sigh. "Thank you. Jeez, now I'm worried that I won't be able to do this when you're not here."
"In that case, you'd better try problems seven and nine," he'd told her with a wry grin. "They're even tougher. If you solve those, you'll be unstoppable."
"Should I email the answers to you?"
"Sure." He'd written his email address on a piece of paper, torn it off, and handed it to her. To her surprise, Maya had experienced the tiniest frisson as she took the address. There was something strangely intimate in knowing they had a way to communicate now, any time of the day or night. She'd never felt this way about getting anyone else's email.
In the school library, waiting for her code to compile, Maya took out her phone, looked up Jack's contact details, and began composing an email.
**Hey Jack, I may not get around to sending you the latest chem problems for a few more days. Kinda distracted with my app. I'm getting all kinds of suggestions from the Cheetr users.**
The return email arrived with the speed of an instant message.
**You wrote an app? Seriously, wow.**
Maya giggled loud enough to catch Miss Topalian's eye. "Hey over there," she called out. "I hope there's nothing inappropriate going on."
"I know," Maya agreed. "Like I need the distraction, right?"
"I hear it's all hands on deck with you," Miss Topalian said, waving a hand vaguely as she returned her attention to her own computer screen. "No talking in the library," she said to no one in particular.
Maya's phone lit up again as another email arrived from Jack.
**How many downloads?**
She typed back, **Cheetr? A few thousand.**
**Amazing. So what's next?**
Maya hesitated. Her latest idea was too new to talk about it, but Jack hadn't said he was a coder. She was dying to tell someone about it, at least the basic concept.
**I'm working on something new. More of a social app.**
**Sounds like a plan. Social is definitely where it's at. What does the app do?**
**I'm still working on the alpha version . . . prefer not to discuss details yet.**
This time there was a longer delay before his emailed reply arrived.
**Not a problem. Totally get where you're coming from. It's just that I was wondering what you're doing about investment.**
**. . . Investment . . . ?**
**Have you thought about getting together with a business angel? Some guy in the know, connected and suchlike, who'll bung in a bit of cash to get you to the next level.**
Maya was in the middle of composing a reply when the phone began to buzz. It was Jack.
"Take it outside," Miss Topalian said the moment that Maya opened her mouth to reply. The librarian pointed toward the open French doors. "Bring me a diet soda from the machine and we're even," she said with a wink as Maya passed the desk.
"This is all rather exciting," were Jack's first, slightly breathless words. "How come you didn't mention the other day that you wrote apps?"
"I was trying to stay focused," Maya said. It wasn't entirely true. She wasn't sure how much of what she shared with Jack would bounce right back to the school administration. It had felt safer to keep quiet about the extent of her coding in case they tried to slow her down. Yet she sensed that Jack would understand. Guys her age understood tech in a way most teachers didn't.
"So—what's the answer to my question? Do you want to take things to the next level?"
Maya paused, felt the skin on her cheeks tighten as she blushed at the faint hint of a double entendre. Turning away in case Miss Topalian saw her, she tried to suppress her own instinct to flirt.
_He's your tutor, for crying out loud_ , she told herself. _Get a grip!_
"I . . . I don't even know what the next level is," she said, and heard the huskiness in her own voice. "I'm just trying to write a cool app."
"Oh—well, okay. It all sounds tremendous. I didn't mean to interfere."
Maya smiled, once again taken aback by his politeness. "No—I appreciate the advice. Business angel? I'll look into that."
"What I wanted to tell you is that if you ever need a contact, let me know. I only know about them because I went to a networking thing at the Caltech Investment Opportunity Network." He gave a heavy sigh.
For a moment, Maya regretted her reaction. He seemed a little put out, deflated. "You know investors then. How come?"
"'Know' might be putting it a bit strongly. I've _met_ some."
"You're still way ahead of me," she said, this time encouragingly.
"It's a long story, but basically, my rather troublesome ex, Clarissa, is over from London. Things can get a bit, y'know, awkward, so I thought I'd take her somewhere where we'd have a lot of distraction. Mainly, it's a bunch of 'angel' investor types hanging around campus. If you want, maybe you could come along to the next pitch meetings?"
"Come along with you and your ex?" Maya said, barely containing a gasp.
"Not at all," he replied hurriedly. "I mean, after Clarissa's gone back home. It would just be you and me. Truth be told, I don't have any free time until she leaves, anyway."
"Ha," Maya said, relieved. "Me? Pitch to investors? Yeah, like that's gonna happen."
"And why not?" he demanded. "What do you say? Are you in?"
Maya nodded slowly, clutching her phone to her ear. The sudden drop in his voice, the insistent timbre sent a shiver right through her. She felt a long-forgotten thrill course through her veins. When had she even last had a crush? And why did it have to be on a tutor—especially one whose "rather troublesome ex" was in town?
"Sure." Maya took a deep breath. "I guess it couldn't hurt."
## **LUCY**
#### KITCHEN, VENICE BEACH HOUSE, TUESDAY, JUNE 9
It always took a few seconds before the images made sense. First she'd notice the pale blue glow of the pool water, how the ripples radiated across the surface. Then she'd puzzle over the sight of two figures struggling at the near edge of the pool. They were almost directly beneath her. Instinctively she grasped that the tiniest noise would betray her. Frozen rigid, she could only grip the edge of the balcony with her small fingers and watch, breathless, as one of the figures fell into the water.
It looked like a game, but the silence made her understand that it wasn't. It was something private, something between two adults, the kind of thing from which kids were shooed away. Turning to leave, she noticed a hand emerge from the shadows and push down on the submerged figure's head. Long nails painted the prettiest peach shade she'd ever seen.
Later, wandering in despair, wet and unable to find the bathroom, she bumped into the lady. A cold terror gripped her and she could only stare at the beautiful peach-colored nails.
"Charlie didn't see anything. Nothin'. Go 'way."
"Lucy?"
"Pretty lady said if I go to bed now I can get my nails painted like hers. GO 'WAY!"
"Lucy!"
Lucy shuddered awake. She felt a firm hand on the curve of her shoulder, rousing her. She blinked, bleary-eyed. "Ari?"
"You were dreamin', honey. You were Charlie again."
Lucy gave a low groan. "Oh for pity's sake." How long would she be haunted by the memory of that night at the Hollywood party? It was almost as though her subconscious was prodding her to take action.
"You're still getting the dream, huh?" Ariana said sadly. She leaned against Lucy, easing herself onto the bed alongside her friend, stroking her hair. "The one where you see the woman with the fancy manicure drowning someone?"
"Just shadows," Lucy said, lying. Ariana's interest in her "Charlie" memories was already kind of morbid. The last thing she wanted was to encourage it.
She sat up, still feeling groggy. What was Ariana doing up in Lucy's bedroom? Candace's bed was empty. She glanced at the clock on her nightstand. It was 9:20—way too late to go to school without a killer excuse.
Puzzled, she stared for a moment at Ariana. She'd put on a little weight since Lucy had last seen her. It suited her. Her pale, lightly freckled face was already made up, hair already tied back in a neat, high ponytail. Ariana's hair was still her favorite pale-raspberry color; although the natural shade of mousey-brown was showing at the roots. When Ariana wasn't looking, Lucy stole a glance at her friend's arms. There didn't seem to be any new signs of needles. She seemed calm, too. Less jittery than you'd expect for someone who was going through the early days of withdrawal. Her story wasn't adding up. Maybe Ariana had another reason to visit? Something even more personal—something that she'd confess only after the two girls had reestablished their old bond of trust?
Lucy reached for her cell and called school to say she was sick. It would probably get her in trouble long-term, but she hadn't used the tactic for weeks. Sometimes the suckiness of school got ugly. When that happened, she just had to take a day off, no matter what. But at least she didn't have to petition her college president mother for every sick note, so there was that. Anyhow, there were only four days left in the school year. Even the teachers were barely hanging on at this stage.
Ariana waited until Lucy got off the phone to speak. "Why don't I fix breakfast for you. How about pancakes?"
Lucy feigned enthusiasm. "Sounds good. But you should let me cook; you're my guest."
Ariana hugged her briefly. "You've been so nice. I could see your friends weren't too happy about letting me stay. I feel bad, making trouble for you."
"They're just surprised, that's all," Lucy replied, getting out of bed. "I hadn't told anyone about you. I don't talk about my life back in Claremont, at least, no more than I have to. And they didn't know I'd been in rehab."
Ariana followed her out of the bedroom and down the stairs. "I kind of totally ratted you out to your roomies, didn't I?"
"Yeah and by the way, thanks for that," Lucy replied snarkily. As they arrived in the living room, she noted with approval that the futon had been folded back up into a sofa. The bed linen was gone and Ariana's suitcase was neatly tucked between the futon and the sofa.
"Omigod, Lucy. Your face!" Already in the kitchen, Ariana was sweeping toast crumbs from the countertop with her bare hands. She sniggered helplessly. "I'm sorry, but if you could have seen your facial expression, you'd understand. You looked like your mom did that time when she came to visit you in rehab and we snuck out . . . remember?"
Despite herself, Lucy found a grin working its way to her mouth. "Yeah. I remember."
"She may be the dean or president of some snotty college or whatever, but your mom sure knows how to curse."
"I guess I knew how to get under her skin," Lucy admitted ruefully.
"How're you two getting along, now that you don't see her?"
Lucy opened the cupboard and found the pancake mix. She noticed that Ariana already had eggs and oil standing by. Casually, she reached across her for an egg and broke it into the bowl, pouring in milk afterward. "We don't talk much. I mostly text her a few times every day. Just checking in, letting Mom know what I'm doing, asking about her and Dad. She replies, tells me something about her day. The main thing is, she doesn't get too nosy. It's like I'm away at college or something. It's cool."
Ariana rinsed the cloth she'd been using to wipe down the surface, then poured two glasses of orange juice. "You think it's changed your relationship with her, being out here in LA?"
"I think, yeah, maybe. It's like, out of sight, out of mind. Which, knowing my mom, is a good thing. She's got to have so much of her life under control; if I'm part of that then it's like, I have to fall in with all those other performance measures."
Ariana shook her head in sympathy. "That's no way to live."
"It's just how Mom is. That's why she's so successful. I think, after a while, she didn't even want to be that way, but she just couldn't stop herself."
Lucy whipped up the batter for about one minute, thinking about what she'd just told Ariana. They'd skirted around these types of topics during their phone calls. But it'd been a while since they'd last had a really long talk. And it was nicer to have Ariana actually in her kitchen, sharing past history.
Ariana must have picked up on this, because she gave Lucy a wry smile. "Face it, honey, you've missed me. Get over here, gimme a squeeze."
Lucy endured yet another hug before moving over to the stove to heat up the griddle. Ariana leaned back against the countertop, juice glass in one hand, observing Lucy in her element.
"You know something, you've really grown as a person."
Lucy scowled. "Ugh—can we ditch the rehab talk? You always gave it to me straight, Ariana. C'mon, now, I can take it."
"You want me to call you out on something?"
"No," Lucy replied. "I just don't want any bull. Out here, I got away from all of that. The housemates, they're my friends, _all_ of them. I didn't think it would be like that. We're all so different. But you know what? It really is that way. We've been through some stuff and, yeah, okay, I only told them very recently that I was on _Jelly and Pie_. Then you showed up. Which meant that anyone in the house who hadn't already Googled me, found out about me being in rehab, too. I don't want to be defined by my past. I don't want to be that girl anymore."
"It was sorta careless," Ariana admitted. "I'm sorry, hon."
"It's all out in the open now. No one seemed too freaked out about it. Candace isn't running to her mom to tell her that one of her tenants is a raging junkie. It looks like everything is okay."
Ariana didn't speak for a moment, which surprised Lucy a little. Instead, she took the pancake batter from Lucy's hand and poured a measure onto the hot griddle.
"Yeah," Ariana said eventually, when three pancakes were cooking. "Your only problem now . . . is me."
It was true—almost—but Lucy had tried to conceal it. Either Ariana was being super-sensitive, or Lucy wasn't playing the role of "hospitable friend" very well.
"You're not a problem."
Ariana nodded without looking at Lucy. Softly she said, "Oh, yes I am. Lucy, I'm four days clean. Today makes five, if you help me get through. Tomorrow is six. It just keeps getting better from there. Meantime, you're still smoking the ol' ganja, am I right?"
Lucy flushed with resentment. "My parents threw me out because they caught me smoking. And they gave up on me. They refused to send me back to rehab. Now I'm supposed to get clean _all by mysel_ _f_? Hey, I'm trying. All I've done is a little weed, one time, and that was at a party."
Ariana was immediately contrite. "I've absolutely got no right to lecture. Of the two of us, Lucy, I'm the one who started using hard drugs again."
A little mollified, Lucy said, "Well, I did smoke weed. But like I said, it was a party."
"Then you need to learn how to enjoy a party without being high. I bet you did, once."
Lucy shrugged. She had a dim memory, probably from when she'd been about thirteen.
"Maybe if you threw a party here, I could be, like, your sponsor, watch over you."
"Oh _sure_ ," said Lucy with a loud guffaw. "Someone five days clean from cocaine is offering to sponsor a party? That'll work."
"Not the party. Just _you_. Let other people smoke. You and me and probably a few other people, we stay clean, we have a good time anyhow. You and me, we take care of each other. The way we always have."
Lucy picked up the spatula and flipped a pancake. "Get clean, huh?"
"One hundred percent raw."
Thoughtfully, Lucy said, "I guess I could be one of those 'straight-edge' punks, like Ruben."
"Who's he?"
"Guy I know," Lucy said carelessly. "Plays the drums. Yeah," she added after a few seconds. "A party. I guess we probably should. I mean, school's out on Friday. We've survived to the end of the school year—six months of official emancipation. That calls for a celebration."
Ariana plucked at the top pancake on the stack and popped the fragment into her mouth. "Gotta mark the occasion." She grinned. "Hand me the syrup. Let's start talking about how we're going to throw you your first drug-free party. And sweetie, I wanna hear more about this _Ruben_."
## **JOHN-MICHAEL**
#### BALCONY, VENICE BEACH HOUSE, SATURDAY, JUNE 13
The idea of a "school's out" party turned out to be popular with the housemates. They had a house meeting, where Lucy officially welcomed Ariana. John-Michael went along with the smiles and niceties even though he'd much rather be ignoring Ariana. It sucked to have a new person in the house—now more than ever. Lucy knew that he and Grace were going through difficult times. Didn't it even occur to her to respect their privacy? For the first time in memory, John-Michael found himself feeling distinctly cool toward his former rock-camp buddy.
John-Michael had planned to cook at the party, but as things got under way, he found himself squeezed out of the kitchen by the gum-cracking, loud-talking southern redhead. She'd been in the house almost a week and was already beginning to act like she was in charge.
"Sweetheart, allow me," Ariana insisted when he tried to object. "You gotta at least let me earn my keep."
Candace saw the whole thing and took John-Michael by the elbow, leading him upstairs to the balcony. "Take no notice of the Wicked Witch of Claremont. We need to talk."
The balcony had been designated a "chill-out" zone, no drugs or music. Which made it, as far as John-Michael could tell, a "make-out" zone, because no one at the party was interested in engaging in deep and meaningful conversation. At least, not without some chemical aid.
John-Michael pushed his way past the couple making out on one of two rattan chairs. He vaguely recognized the girl who sat in the boy's lap. "Hey, John-Michael," he heard the girl say, blearily, between kisses.
"Hey, hey, focus," said Candace, grabbing his chin in her hand. "Can we talk about _me_?" She shot a final, annoyed glance at the couple.
"Or they could maybe leave?" John-Michael said pointedly.
"No more space," the boy said as he came up for air.
"Could you please dial it down a notch?" John-Michael replied. "Seriously, I'm gonna vomit."
Reluctantly, the couple stopped kissing. The blond guy made a show of displaying his right hand, which he proceeded to lay on the girl's left wrist.
Candace stepped past them and snuggled next to John-Michael on the sole remaining rattan chair. He wrapped a welcoming arm around her and planted a gentle kiss on her forehead. "All right, tell Johnny what's up." For the next few seconds, John-Michael allowed himself to enjoy the closeness.
"So listen," Candace began. "I think I may have a crush."
"You?" He pushed her away so that he could stare at her. "Shut the front door!"
"Mmm, John-Michael," she murmured, grumbling. "I don't know what to do. I think he has a girlfriend."
"Who is this _Romeo_ and what has he done to you?"
She shook his arm, giving a warning growl. "Don't make fun of me! This is serious. His name's Yoandy Santiago. He's an actor on the TV show I auditioned for last week."
John-Michael started. He sat up, facing her, their heads almost level. "Yoandy Santiago? The singer?"
Candace pouted. "He's a singer?"
"Uh—yeah! He's a Latin music artist. Reggaeton. His father is Beny Santiago, who is like a rock star in the world of salsa music."
"How do you know all this?"
"I went through a Latin music phase back when I was staying with Felipe at his boyfriend's place in Santa Monica," John-Michael reminded her. "They were both crazy for Beny Santiago. He's like royalty in Cuba. He can barely walk down the street in Havana without getting mobbed!"
"But his family lives here?" Candace said, suddenly anxious. If Yoandy lived in Cuba _and_ was dating a movie star's sister there were just too many obstacles.
"I'm pretty sure that Beny Santiago defected to the US, like, more than ten years ago. Yoandy Santiago is gonna be on your TV show?" John-Michael shook his head, not bothering to disguise his envy. "Jeez. I hate you. He's completely gorgeous."
"He _is_ pretty hot," agreed Candace. "But from the way he was acting, I'm gonna guess that he's straight."
"Just because a celebrity has a girlfriend doesn't mean he's straight."
"She's a whey-faced trollop," she muttered sulkily. John-Michael had to stifle a grin. He loved it when Candace lapsed into Shakespearean curses. "Also, she's the sister of that British movie star from the _Macbeth_ movie."
"Latin music royalty dating Hollywood royalty? Good luck getting between all that." John-Michael tapped his phone into life and did a Google search for Yoandy Santiago.
"Quit whining," Candace said, "and let's watch a video of Yoandy dancing."
She leaned close to John-Michael and they both stared intently at a YouTube video of Yoandy rhythmically flexing and grinding beside two ridiculously attractive female dancers. Somehow he managed to exude a boyish enthusiasm through the undeniably sexy routine, singing and rapping between soulful, doe-eyed gazes into the camera.
"You get to work with _him_?" John-Michael murmured longingly. "Candace, I'm not sure we can be friends anymore."
Next to them, the couple making out had become suddenly still. "Dude . . . do you smell fried chicken?" The boy began to shift beneath the girl, sniffing the air.
Now John-Michael could smell it, too. "I guess Ariana's world-famous fried chicken is ready," he said sullenly.
"Foooooood!" said the boy in the chair in his best caveman voice. "We gotta get some of that chicken."
Candace spoke curtly. "Better hurry."
After a minute the couple was gone, leaving John-Michael and Candace alone on the balcony. "I guess you don't like Ariana?" she said.
"I'm not even totally sure Lucy likes her," John-Michael replied, deflecting the question. "I mean, if they were such amazing friends and all, why didn't Lucy mention her before?"
Candace returned John-Michael's phone to his shirt pocket and carefully stood up, moving to lean against the rail of the balcony. "Lucy forgot to mention quite a few things."
"True," John-Michael agreed fervently. Candace didn't even know the half of it.
"We didn't ask for a live-in housekeeper. My mom didn't authorize that. And Ariana's not paying any rent."
"Totally."
Candace pouted. "Does that make me sound too territorial?"
"I don't think so."
"Do you think Paolo's into her?"
"Paolo?" John-Michael picked his words carefully, throwing a quick glance toward the pathway below the balcony, where Paolo had gathered with a few school friends. Staying on the edges of a party was odd, for Paolo. John-Michael wondered if any other housemates had noticed yet that Paolo's behavior had shifted. Whatever happened out there in Malibu Canyon, John-Michael wasn't going to spill a single bean. "Paolo has his own problems."
"So," Candace said, prodding his upper arm with a finger. "Ariana. Do we trust her? Should I kick her out?"
"Well, the girl _can_ cook. . . ."
John-Michael was finding it tough to _officially_ complain about their new houseguest. She was far from a freeloader—she cooked and cleaned for everyone. She was also super-polite whenever any of them wanted to watch TV. Yet even so, beneath the southern charm, John-Michael thought he could detect a whiff of his father's greedy ex-girlfriend.
She claimed to be newly clean of drugs, but John-Michael had been around addicts enough to know that someone with Ariana's skin tone and healthy sheen hadn't used for _months_ , not seriously. There didn't seem to be another reason behind her sudden desire to move from Claremont to LA, and John-Michael wasn't sure he was buying that story.
There was more to it, something that went beyond words and actions. Like Judy Aherne, Ariana had the keen, wary look of someone up to no good. In the case of Judy and John-Michael's father, it had been easy to connect the dots and draw the outline of a gold-digging lowlife.
But what was there for Ariana, here in Venice?
Still, the fried chicken did smell pretty great.
"You wanna get some chicken?" he said reluctantly, noting how Candace was fidgeting with the rail.
"Oh man, do I!" she replied. "Let's go taste ol' Ariana's southern fried."
## **GRACE**
#### LIVING ROOM, VENICE BEACH HOUSE, SATURDAY, JUNE 13
The smell of frying chicken had driven most everyone toward the kitchen and living room. A snaking line had formed of people waiting for a bite. John-Michael had given up his spot on the balcony to the first bunch of party guests who were looking for somewhere to enjoy the hot food.
Grace followed him downstairs, where the stereo was blasting some grating music. John-Michael had compiled three playlists, "nostalgia," "punk," and "zone out." They were somewhere in the middle of "punk" and it was giving Grace a headache. She switched to "nostalgia" and an orchestral sound track swelled. It sounded like the sound track to some kind of action-adventure movie. She was about to switch again when cries of "Yeah, the music from Zelda, awesome!" erupted from at least four kids in the chicken line.
Grace locked eyes with Candace. Her stepsister was leaning on the threshold of the French doors, watching her. They both smiled. "Video game geeks," she said, a Sea Breeze in one hand. "Just leave it on."
Grace felt a stab of guilt as she watched Maya squeeze past Candace on her way in from the tiny backyard, which had been designated as the "smoking" area. Only a few months ago they'd taken great pains to protect Maya from the weed and alcohol at their first party. At fifteen, Maya was the baby of the house and back then they thought she needed their protection. Now, no one seemed to care. They'd told her _once_ , right?
"I hope you didn't let anyone out there tempt you into smoking," Grace teased, drawing up alongside her stepsister.
Resentment flicked in Maya's eyes. Candace caught the vibe and joined in. "Let it go, Gracie. You wanna call Aunt Marilu over to bust up the party?"
Maya tipped her head toward the long gray sofa, on which six kids were perched, laughing and smoking. "Maybe we should worry more about the fact that people are smoking _cigarettes_ inside. Didn't Candace's mom forbid that?"
Candace turned, apparently unaware. "You guys! Put those cigarettes out or get out."
Luckily, no one seemed to take issue with this. Hands were waved in apology and the burning cigarettes dropped into a discarded Coke can.
Candace turned her attention back to Maya. "How's your app going?"
Maya seemed surprised by the question. "Oh—Cheetr? Actually, I'm kind of moving on to something else."
Grace gave a nod of vague enthusiasm. She suspected that Candace, like her, had very little to say on the subject and was just trying to show polite interest. Maya might have noticed, or maybe not. She seemed to get tunnel vision around her coding.
"How about you?" Maya asked Candace. "You hear back from that TV show you auditioned for?"
Candace's grin became broad. "It was the weirdest thing. Not like going through an open audition, at all. I got invited to the audition a little over a week ago and this morning my agent called to tell me they're sending over the contracts. Somehow I missed the actual 'yes.'"
"Wow," Maya said. "That's _literally_ insane. Things are gonna start to change pretty fast for you," she observed. "Next it's going to be magazine interviews and photo shoots. Then premieres. The red carpet treatment."
Candace broke off from sipping her drink to chuckle. "Ha—I guess. To all the C-list events."
"Maybe that's how it begins," Maya said. "Then before you know it you'll forget your friends on the beach, hitting the Hollywood party scene, up in the hills, Mulholland Drive, all of that. You'll be able to ask Lucy for advice—although she probably won't want to talk about it. She must not have the best memories from her days as a child star."
Grace froze. Could Maya possibly know how totally right she was? She was acting as if it was no biggie, as though all three girls knew to what she referred. Yet, it was pretty obvious that Candace wasn't entirely sure what Maya was getting at. "Why would she have bad memories?" Candace asked. "She was a kid and she got to be on TV."
Maya replied calmly and with confidence, "I mean because of the murder."
Grace pressed her lips together. Maya _did_ know. Cautiously, she glanced at both girls. Candace stared at Maya in bafflement. Maya's attention, however, was divided equally between Grace and Candace.
"What murder?" Candace said.
Maya seemed bemused. "The Tyson Drew murder."
Candace sounded puzzled. "You keep saying 'murder' like I'm supposed to know what you're talking about."
For a couple of seconds, Grace held her breath. How much did Maya know—and more importantly, _how did she find out?_
Grace couldn't imagine Lucy talking about it. Everything Grace had observed about Lucy screamed total and absolute denial of what she might have seen that night, eight years ago.
Not Lucy, then. Her stomach lurched. Which meant that John-Michael must have talked to Maya. But why?
What exactly had he told Maya about their conversation on the Pacific Coast Highway?
Grace searched her stepsister's face for any sign of recognition. She seemed genuinely surprised. Grace reminded herself that Candace was an actor. She was either giving the performance of her life, or her absolute shock and puzzlement were the real thing. Maya, in contrast, was cool and measured, as if she knew the likely impact of her words. She was drip-feeding the information, watching each girl for their reaction.
Grace felt as though a chilly breeze had just swept across her. She sensed, quite suddenly, that Maya knew a lot more about Lucy's past than she was letting on.
But how much?
"Lucy was at a party on Mulholland Drive. A lot of TV people were there," began Maya. "At this point she'd be, oh, I'm guessing nine or ten years old."
Maya seemed to make an executive decision that their discussion would move outdoors. The backyard had emptied as word got out about Ariana's chicken.
Grace hesitated for a second and then followed Candace and Maya into the backyard. She couldn't feign disinterest. At any moment, she was expecting Maya to reveal that she'd heard all this from John-Michael. Grace had to bury the tide of resentment she could already feel building toward him. How could he have revealed her secret? With everything he'd shared with her—how dare he risk it?
Outside, the air smelled thickly of cannabis smoke. A lanky boy wearing only board shorts lay sprawled, asleep on the edge of the lawn. Aside from him, the girls were alone.
Maya continued, fully conspiratorial, "Another kid from _Jelly and Pie_ was at the party, too, plus three other child stars. I think the party was for their agent."
"Are you saying someone got killed at the party?" Candace said. She was clearly still puzzled, as though this was idle gossip about someone they didn't know. Grace guessed that it was the alcohol dulling her senses.
"You don't remember the Tyson Drew murder?" Maya seemed surprised. "It was a big deal in Hollywood. Tyson Drew was an up-and-coming movie star. And he was drowned in the pool, at a party on Mulholland Drive."
Grace said nothing. She watched Maya very closely now. She hadn't mentioned Grace's father yet. If she'd gotten this from John-Michael, it would be just plain aggressive to talk this way in front of Grace, without acknowledging that she knew Grace's father had been sentenced to death row for the murder of Tyson Drew.
It really was not like Maya to be so mean. Was it possible that Maya hadn't found out from John-Michael, after all? Grace's curiosity spiked. "How do you know all this?"
"Weird that Lucy never talks about it," Candace interrupted, with a sudden sharpness that grabbed Grace's attention. "Murder? Rehab? She sure has kept a lot of secrets. If I had stories like those you couldn't keep me quiet."
"Maybe we could," Maya remarked, "if what you knew was dangerous."
Candace just stared. Grace watched and thought, _Finally, she gets it._
Candace repeated Grace's question. "Seriously, where'd you get this? I Googled Lucy when she first told us about the show. I bet we all did. None of this comes up."
"The _Jelly and Pie_ fan forum," Maya replied with sudden authority. "You gotta search deep, 'cause it's behind a child-protection firewall. And the thread has gone pretty cold. There's some interesting speculation, though. They got a guy on death row for the murder. Of course, he says he's innocent. From the beginning, he's said that one of the kids must have seen the real murderer. But the guy they convicted, he was wasted. He couldn't say which one it was. And all the kids denied it."
"So you think Lucy was a _witness_?" Candace was beyond fascinated. "Holy shit. Maybe that's the real reason why she didn't want us to know about _Jelly and Pie_."
Maya's attention moved to Grace. "Could be," she said, deadpan. There was a momentary pause. "Candace, you think you could get us some sodas? I don't want to get into that chicken-crush."
To Grace's astonishment, Candace simply nodded and disappeared back into the house. It was like some kind of magic trick—no one ever told Candace to do anything. But it seemed as though Maya's bombshell had turned Candace into a willing supplicant.
And once they were alone, Maya dropped a second payload.
"Why don't you tell her?"
Grace took a breath. "Tell her . . . ?"
"That Alex Vesper, the man on death row for murdering a movie star, is your father," Maya said. She spoke quietly but the veneer of calm had vanished. "You told John-Michael, that day you drove back from San Quentin with him—the day he totaled his dad's car."
Grace shook her head as though she'd been slapped. "How can you possibly . . . Did John-Michael say?"
"Our call was still connected. I heard you guys talking."
The breath caught in her throat. "You . . . heard?" Grace's eyes strayed toward the kitchen where Candace was starting to return, carrying three sodas.
"You have to tell your sister, Grace," Maya said. Her words spilled out, the confidence gone, replaced by an urgency that was unnerving. "Lucy has to speak up, too. It's not safe to keep secrets like this. We're all safer if you—"
As Candace rejoined them, Maya's mouth snapped shut. Candace handed her a Sprite Zero.
"You were saying?"
Maya spoke reluctantly. "Oh. Yeah. The Tyson Drew conspiracy theory."
"Okay so—if the guy they got on death row didn't do it, then who?" asked Candace. The shock had faded, apparently, and she was back to the business of serious gossip.
Maya paused. Expectantly, she glanced at Grace. Grace stayed silent. Maya took a breath. Before she could reply, all three became aware of a commotion in the living room, around the sofa. It looked as though a fight had broken out, with a sudden explosion of activity centered on the sofa. One boy was flung from the group's core. He landed hard against the wall, stumbled, teetered, and fell through the gap between the French doors, landing partway on the grass.
"Get out," he gasped. "The sofa's on fire!"
## **CANDACE**
#### CULVER STUDIOS, WEDNESDAY, JUNE 17
"You're being too obvious." Candace paused. "Okay, look now."
This time, only Grace's eyes moved. Then they darted back toward Candace. "The guy over by the main set with the Lakers hat?"
"What do you think?"
A pause. " _Very_ nice. Did you say he's from Cuba?"
"He was born there, but his family has lived here a while now."
"No accent?"
"An unbelievably hot one."
Candace glanced over her shoulder. Ricardo Adams, whose fame had just exploded across the globe following the dramatic ending of the TV series _Deadbeat_ , was the star of the new show. He was standing beside a table piled high with fruit and muffins. At one point, he looked right through her. "Oh and look," she muttered, "Mr. Big."
"Not as cute as on TV," Grace remarked. "You were right." She pointed at the breakfast buffet. "Can you grab me a peach? Then I'm gonna head off to meet John-Michael at IKEA."
"Please. How many people does it take to pick out a sofa?"
"You know how it is. He doesn't want to stay home alone with Ariana."
Candace frowned. "We gotta do something about that girl."
"I know, but how do you throw out the perfect houseguest?"
"The perfect housemate splits the rent. We'll figure something out." Candace moved sideways and approached the snack table obliquely, with a polite smile at Ricardo. She returned with a large, ripe peach, which she handed to Grace. "Just pick out the couch you guys like and my mom will pay for it online. Make sure that someone's home for the delivery."
"Ariana offered to wait around for it."
"Oh, sure, Ariana is all about the helpful."
Grace sniffed the peach with blissfully closed eyes. "Anyway, what scene are you filming today?"
"Annika and Sebastian meet for the first time, rescue some kid, then fight. It should be pretty intense. My character has been walking all night, through the ruined city, to the bug-out location."
"'Bug-out' location?"
"It's what doomsday preppers call their secret go-to location where they've hidden all their prepper crap, in case of the apocalypse."
Grace grinned. "Good thing my mom made you take those tae kwon do classes."
Candace couldn't help but smile at the memory of herself as a little thirteen-year-old learning to kick ass in an old gym hall that smelled like a Goodwill store. She'd loathed the experience with a passion and swapped it for riding lessons as soon as her stepmother was convinced that she'd covered the basics. "I guess it wasn't a total waste of time. But who knew I would someday fight hot guys for a living?"
"I take it Yoandy is playing Sebastian?" Grace teased.
Candace had a hard time hiding her smile. "Come on," she said. "I'll walk you to your car."
Candace locked arms with Grace and walked her to the front of the lot, which was open to a narrow walkway outside. When she returned to the set of _Prepped_ , Candace recognized only Yoandy and Ricardo. Ignoring her, Ricardo went over lines with Yoandy, who took a moment to give her a little smile. Her agent had promised to stop by to say hello, but so far there was no sign of him. She began to regret letting Grace leave so soon.
It sucked to look like such an obvious clueless newbie. However confident Candace managed to act around people her own age, being on a studio lot in Hollywood was still a very big deal. She wasn't sure of the protocol. On the set of _Downtowners_ there'd been a strict hierarchy, depending on how big a star you were. As a total newcomer and the youngest person on set, Candace had been virtually ignored at first.
On the whole, she'd found that it was best not to share any insecurities about her acting. Her stepmother, Tina, had drummed that much into her, those first few years of going to auditions. Now that she wasn't stitched to Candace's side, Tina insisted on lengthy postmortem phone calls after each audition. After the audition for _Prepped_ , Tina had reiterated a favorite piece of advice: "Someone asks you if you can do something? You say yes. Someone asks you if you're worried about anything? You say no. You gotta fake it till you make it, Candace. Anyone tells you otherwise is just trying to find your weak spot. Don't let them."
Once, it had seemed like crazy advice. The opposite of all the teachers who told her to be truthful, honest, to share any worries.
"I should lie about what I can and can't do?"
"That's right," Tina had snapped. "The entertainment business is all about illusion, Candace. If you can make someone believe something then it might as well be true. Relax. No one's going to ask you to design a vaccine, or anything."
The assistant director arrived on set, rapped his knuckles twice against the snack table, and then announced, "Change of plans, people. The bug-out location isn't finished yet, so we're just going to record the later scenes, not the fight. Let's have first positions for scene six in five minutes."
Yoandy strolled over to Candace. "Too bad, _nena linda_. Looks like we won't get to practice the fight. And I was looking forward to another dance."
"Well, I was looking forward to kicking your ass," Candace returned teasingly.
Yoandy seemed delighted by the riposte. "Oh, were you?"
_Fake it till you make it._
"Yeah, I think Annika could take Sebastian out. Easy."
He warmed to their banter. "We'll never know. He's not serious about this first fight; he's only trying to keep things from escalating."
"Sure, but Annika means it. I'm not going to hold back."
Yoandy smiled. "I'll try to remember that."
Candace paused, then asked, "What does _nena linda_ mean?"
He looked startled for a moment, then blinked twice, nonplussed. "It means 'pretty baby,'" he answered in a soft voice. "You don't like?"
"It depends," she said hesitantly. "I mean . . . I guess you say that to all the girls?"
"Sometimes," he agreed with utmost sincerity. "But I can call you _Señorita_ Candace, if you prefer."
"Now you're making fun."
"Of you?" He stepped closer, touched a hand to his heart in mock disappointment. " _Ay, mi madre_. No way, _nena_."
"Hmm," Candace said. "What would Kay Alexander think?"
"Kay?" He seemed surprised at the mention of Kay. "It wouldn't matter what she thinks. She's just a friend."
"I thought you two were dating."
"We've been on a few dates but it's not serious."
"Hmm." Candace was unconvinced. "You sound like my housemate, Paolo. He goes on dates all the time and says it never means anything."
Yoandy shook his head. "With _you_ in the same house? He's crazy. But I like this Paolo already, since he's keeping away from you." He paused. "What made you think me and Kay were dating?"
"You said that you and Ricardo were practically family," she pointed out.
Yoandy's jaw went slack. Either he was a terrific actor or he was telling the truth. "That? Kay and I hung out at the Latin Grammys with Dana and Ricardo. I was making a joke."
Candace wanted to believe him, but this was getting surreal. "But . . . there are pictures of you with Kay, in _Deadline_."
Yoandy clicked his tongue. She could see a muscle twitching in his jaw. He was visibly upset. "I promise you, Candace, Kay is not my girlfriend."
She raised an eyebrow, on the edge of flirting. "Does Kay know that?"
"Ladies and gentlemen, positions please!" the director announced. "We're ready for the next scene." Candace reluctantly turned away.
Ninety minutes later the recording was finished. They'd shot about two minutes' worth of action, of which maybe one minute would end up on screen. Other members of the cast were already helping themselves to coffee, water, and juice. Candace was opening a bottle of sparkling mineral water when she heard a voice.
It sounded familiar but didn't register for several seconds. It was probably the sexiest female voice she'd ever heard. Slowly, Candace turned around. About five yards away, Ricardo Adams was talking to someone more famous than anyone she'd ever met in person. A jolt went through her as she stared at the woman.
_Starstruck_. Candace had never really understood what that meant, hadn't appreciated the physical excitement of seeing someone so strikingly familiar, whose face she was used to seeing magnified to impossible proportions on the silver screen.
Dana Alexander, Academy Award winner, three-time Oscar nominee and, in her day, one of the world's most beautiful women. Even now, in her early forties, she still possessed an absolutely magnetic quality.
She wasn't even dressed particularly fancy; a crisp white shirt, a navy-blue trouser suit of impeccable cut, sleek Charles Jourdan heels that clicked against the sidewalk. Slung over her shoulder was a bloodred Mulberry bag. She wore her hair longer than Candace had seen in the movies; a rich mahogany color with a twist and bounce in the tresses that fell to her shoulders. A complex, oriental floral scent hung in the air as the woman turned toward Ricardo, who rose from his chair to greet his wife with a chaste kiss.
"I'll be out in about ten minutes," Candace heard him say. "Why don't you wait in the car?"
Candace watched Dana Alexander leave, accompanied by a second brunette woman who was wearing a black pantsuit. Candace couldn't see her face. Even from behind and at a distance, she could spot which was the movie star and which was the regular person. With Dana Alexander, everything was dialed to the max: haircut, makeup, shoes, right down to the sashay of her step. The other woman might as well have been a shadow.
"How'd you like Dana Alexander?" Yoandy murmured, startling her with his sudden proximity.
Candace turned to him but took a step back. When she stood too close she had a disconcerting urge to lean against his chest. "She's stunning," she said frankly. "So is Kay. But they don't look much like each other, do they?" That was putting it kindly. From the photos Candace had seen, Kay dressed in grungy, urban fashions and went out of her way to look nothing like her elegant older sister.
"Actually, a little bit, she does. But Kay dresses pretty differently," Yoandy said, tenderly wrapping his fingers around Candace's wrist. "And I wish you'd stop talking about her, _preciosa_. I prefer to talk about you."
"Stop it," she said. But the words got somehow stalled and came out at less than half the intensity that Candace had intended. Things were moving breathtakingly fast. She had to take a moment, to think through what might actually be happening here. Yet somehow she couldn't slow the racing of her heart.
## **JOHN-MICHAEL**
#### IKEA, WEDNESDAY, JUNE 17
"I can't believe you thought I told Maya that your father was on death row for murdering Tyson Drew."
Grace answered John-Michael with a measure of defensiveness. "It was honestly for, like, only ten seconds."
John-Michael rounded the corner, into the living room area of the giant IKEA store. Six rooms had been mocked up, three on either side of a winding pathway. He kept his focus on the furnishings and away from Grace. Her assumption, even if had been momentary, was surprisingly hurtful.
He wandered into a room that felt similar to their own living area: a maple wood floor, beech shelving units, and white, colonial-style wooden folding tables. The space was dominated by a stylish three-seater sofa in pewter gray. He stood back from it for a moment and then looked at Grace.
"Gray?" she said with a doubtful frown. "We don't have to replace the couch with an exact copy." Grace turned slowly, staring off at other examples. She pointed at a traditional red sofa. "How about that one?"
They made their way over to the other room setup. The entire design consisted of three colors: red, white, and black. A black-stained wood veneer floor, tubular metallic shelving with white shelves, and a large red sofa resting on a white, faux sheepskin rug.
Grace tossed aside two silver-white chenille throw cushions and dropped herself onto the sofa. He could see her watching him, long enough to notice the tension that he almost certainly showed on his face. She pouted. "C'mon, John-Michael. Don't be mad at me."
John-Michael shook his head very slightly. He was angry, but not about Grace thinking he'd spilled her secret. It was just that Grace seemed to have forgotten _his_ problem, the gravity of what he'd told her about his father. And that anger was much harder to deal with.
_I killed my father. I smothered him to death. He asked me to do it. He was checking out of life. But I'm the one who remembers what it felt like to have him struggle beneath a pillow held over his head, and then give up the fight._
He sighed and sat down at the other end, testing the sofa. It felt fine. In his current mood, one couch was as good as another.
Not a day went by when John-Michael didn't think about his father's death. Grace knew what he'd done—he'd confided in her after they'd driven to San Quentin to visit her father in prison. If she told anyone, he was finished. How could she think he would tell _her_ secrets? What planet was she on?
"Do you like the sofa?" she asked.
John-Michael settled back into the thick, textured upholstery. "It's fine." He leaned over, tugging at the price tag. "Six-ninety-nine."
"Perfect, it's within the budget."
They were both quiet for a moment. John-Michael picked up a catalog from a small pile on the nearby folding table and began to flip through it. Grace stood and strolled over to the shelving unit, pulling out a drawer. Then, as if a thought had just occurred to her, she said, "Do you think Lucy would ever come forward and talk to the cops?"
"After all this time? Doubtful. Also, we still don't know for sure that Lucy saw anything that night at the pool."
"My dad thinks she did. But who even cares what he says?" Grace ran a hand through her hair, clenching a fist in her tresses. "Only Lucy knows if she actually saw something."
He heard the frustration in her voice. "If Alex Vesper was my dad, I'd be putting maximum pressure on Lucy to talk," he said bluntly.
"Easy for you to say."
He sighed. Grace had a point. She'd lived with her secret for a long time. Whereas he'd only found out that her father was on death row a month ago. When they'd visited Grace's father, John-Michael had confronted his worst fear—the inside of a maximum-security prison. Exactly the kind of place where he could end up, if anyone ever managed to make charges stick about his own father.
"For what it's worth, Lucy and I have always been close—ever since rock camp," he said. "We can talk about all kinds of stuff. But in all that time, she's _never_ even hinted that she saw anything like what you're suggesting."
"I'm not saying Lucy even knows what she saw. Maybe she has post-traumatic stress."
"That's a possibility." John-Michael's eyelashes flickered. "I mean she does have all the signs."
Grace nodded in agreement. "She needed rehab for a reason."
"I always assumed her drug problem was standard child-star stuff. I feel guilty I never realized there could be more to it than that."
"You shouldn't beat yourself up about it. The thing is, Lucasta Jordan-Long has been a huge deal for me, most of my life. All the child actors at that party the night of the murder—I've followed their careers. When we came to live here in Venice, I sort of expected some of you to remember that she'd been on TV." She looked at him expectantly.
"Well, I knew."
"Yes, John-Michael, and when it all came out about Lucy and _Jelly and Pie_ , you admitted it right away. Maya didn't."
He paused. "You think she knew all along?"
"Maya's very good at hoarding information."
"She's not the only one," he said. "Honestly, Grace, take a look in the mirror."
Her sapphire-blue eyes flashed with sudden sharpness. "But _you_ know why I'm being secretive. I've admitted everything to you. The question is, what is Maya hiding?"
"Maybe she didn't out Lucy for being a child star because she didn't want to make her feel uncomfortable? She's cautious about our feelings."
Grace regarded him cynically. "Seriously? Maya's not cautious, John-Michael. You know what Maya told me at the party the other night? She told me—no _begged_ me—to tell Lucy and Candace about my dad."
John-Michael shifted on the sofa. So Maya _had_ been talking—that explained Grace's unusually petulant attitude. Another couple had entered the mocked-up living room and were trying not to stare at them. He stood up and offered a hand to Grace.
As they made their way through the store toward the exit, Grace admitted, "The thing is, I saw her eyes. She's scared, John-Michael. I need to know why." She broke off, reaching for the cell phone that had just started to buzz with activity in her purse.
John-Michael watched her look at the caller's name, then take the call, holding up a single digit to him as if to say "one minute." He watched her listen, saying nothing. Her expression went from quiet shock to breathless horror within seconds. By the time she finally nodded and managed to croak out, "I understand" in a tiny voice, Grace was trembling. The phone slipped from her fingers. John-Michael bent swiftly to retrieve it. Just before the screen faded to black he saw the caller's name: _Mom_.
They'd stalled next to a table of kitchen implements and she was leaning against it for support. He took Grace's right hand in his. "Gracie, did you get bad news? I promise, we'll all be there for you. Is someone sick?"
Grace took a shaky breath and slowly pulled her hand away from John-Michael's. She straightened up and shook her head steadily, as though trying to summon up some resolve.
"No. It's something I've been expecting to hear. It's just that . . ." She gave a sudden, rueful laugh. "Just that actually hearing it is so much harder than I expected."
To his astonishment, he saw a shiny film of tears appear in her eyes. Softly, John-Michael said, "What is it?"
Grace's lips pulled tight. She spoke in a forced, hesitant manner. "My dad. The execution. They set a date. July fourteenth. Four weeks, John-Michael. Four weeks from now, they kill my father."
## **MAYA**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, MONDAY, JUNE 22
Jack Cato was waiting with Grace at the bottom of the spiral staircase. Maya could guess why she hadn't invited him inside the house. The smell of smoke from the couch fire still hung over the ground floor. Without the replacement, which was due to be delivered the following day, the living space looked sparse. Or as Candace preferred, "minimalist."
"Morning," Jack said, beaming. "Beautiful day, isn't it?"
Maya grinned. "It is on the outside. Inside, it's kind of smoky."
He looked puzzled. "Did something happen?"
"A fire," Grace commented. "RIP sofa. So, you're taking Maya to a business brunch?"
"It's more of an entrepreneurs' breakfast," Jack said with a chuckle that brought an instant smile to Maya's lips. "But broadly speaking, yes."
"Jack was a finalist in some big-deal entrepreneur competition for schoolkids in England," Maya told Grace. "And he got to meet lots of famous people and investors who started successful companies. He found out how all that stuff works, so he's taking me along to this thing at Caltech."
"We're just having a go at rustling up some interest," Jack said with self-effacing modesty. Maya doubted that he could be more adorable if he tried.
Three hours later, Maya was collapsing against the wall at the conference center at Caltech. On the other side of the wall was a room full of rich geeks, some barely out of college, who'd just witnessed her first-ever tech presentation. Her heart was still pounding loudly and steady in her own ears as it had throughout the longest five minutes of her life.
"That was bloody brilliant!" Jack said, breathless. She felt his hand, tentatively reaching for her shoulder and then pulling back at the last moment.
Maya couldn't stop a radiant smile.
"I can't believe it!" she said. He was gazing at her so intently that she wanted to look away but she couldn't seem to do it. "Two of them! _Two_ of those guys want to invest in my app! Actual backers. This is unreal."
"You did it, champ," he said, straining to sound humorous. He gave her a playful punch on the shoulder. Their eyes caught for a second and she sensed an undercurrent of tension. This was either more adorable British reticence or he really, really wanted to touch her and didn't know how.
The whole event had been pretty casual, like an open-mike type thing. Jack had put Maya forward to do a five-minute "bit" about her new Promisr __ app, and Maya had stood there pitching her social-bartering app in front of everyone, her voice shaking a little bit. It was like some terrifying kind of entrepreneur comedy club.
Halfway through, she'd decided the best thing was simply to demonstrate her app. A cluster of potential investors had gathered the instant she'd finished; all of them young men, none older than thirty.
"I've never seen investors jump like that," Jack marveled, running one hand through his unruly fair hair as he struggled to absorb what had just happened. "You don't get it! Mostly they're kind of bored, actually. You really made those nerds light up!"
"I did, didn't I?" Maya said, equally dazed. "It's incredible to think that some people can just drop that kind of money after a five-minute presentation."
"Well, they did get to grill you for a good hour or so afterward. They can drop a lot more, too. They _will_ drop a lot more. You'll see. A hundred K is nothing to these guys. It's not just the tech, it's you. Maya, you wowed them."
"But why?" she asked, bemused.
"Because you're young, brilliant, gorgeous, and, as a girl, you stand out! These guys are dreaming of the day that your photo is on the cover of _Wired_ magazine. Or even _Time_!"
Maya beamed, and then shoved him lightly in the chest. "Oh, please. Now you're exaggerating."
Jack caught both her hands in his. She could feel her knees buckling slightly, unable to concentrate on anything but the sensation of his fingers intertwining with hers.
"Are you okay?" he said as she closed her eyes, suddenly leaning against him for support.
Maya was experiencing an exhilarating jumble of emotions. Relief and excitement, but also fear. "Jack, what if I screw this up? I can write code, but what do I know about running a business of any kind?"
"Oh, you shouldn't worry about that. They're counting on you to write the code. You're the brains, the creativity. The business side of things, that's their end."
Her eyes fluttered open. Now she really did feel scared. "You think—you think there's any chance I could get ripped off? It happens."
He raised a finger to her cheek and stroked her skin lightly. "Hey," he said very softly. "I'm the one who got you into this. You think I'd stand by and watch you get ripped off?"
She felt an overwhelming surge of gratitude toward him. "If it wasn't for you, I'd still be fixing bugs in Cheetr, just watching downloads mount up. This is a whole other league. It's major."
"What nonsense," he murmured, his fingers still caressing her cheek. "You'd already started work on Promisr when you first talked to me."
They were standing very close now, enough that she could feel the whisper of his breath, which smelled sweet, of orange juice. She shivered in anticipation of more but instead he pulled away a little, before letting his hand fall to his side. Maya realized with a start that she'd been willing him to kiss her. She released a held breath when he turned away.
"Um, so we'd better get back in time for the next round of presentations. It'd look rude to miss them," he said with obvious effort.
_Why won't he kiss me?_
Maya thought Jack was cute the first time she saw him but now it was as though some kind of filter had lifted away and she could finally see him. The longer she stared, the sexier he became.
"Jack," she said quietly, not moving from where he'd left her, by the wall. Jack stalled on his way to the door and turned. Frustration clouded his expression.
In that moment, Jack stopped being her tutor, a chemistry genius, a business coach. All Maya could think about was a cute guy with the sexiest accent ever, and everything he'd done for her. At that moment, all _she_ wanted to do was kiss _him._
Maya strode across to Jack and grabbed him by the arms.
She drew him closer, until he was no more than a slight lean of her head away. She sensed that he was still waiting for her to make the first move. A feeling of euphoria went through her and her skin buzzed all over. Then Maya leaned in, no tentativeness now, pushing herself against him, challenging him to resist. The softness of his lips surprised her, something that she'd think about many hours later when the shock of the initial contact had passed.
This time, he didn't hold still. Their mouths seemed to melt together and she reached her arms around his neck, clinging on to him while they kissed.
"Good Lord," he murmured faintly, pulling away.
Maya released her fingers from his hair and stepped away. "Your first kiss?" she said, trying to sound innocent. Who was she kidding? She'd never kissed a boy like that.
"Might as well be," he said with a nervous laugh. "Look, Maya, I . . ."
"Is it because you're my tutor?"
"No! I mean, yeah, a bit, but that wouldn't stop me. I mean if that were an issue I'd ask them to find you another . . . it's just that . . ." His lips twisted in a grimace. "Clarissa," he concluded bitterly.
"Your ex-girlfriend?" Maya could barely contain her disappointment. "You told me it was over. I thought she'd gone back to England."
"And it is, but she's going to be here a bit longer, as it turns out," he said, more than a little guiltily. "She's found some wretched course she wants to do at UCLA. Now she's waiting to see if her uni will let her onto an exchange program."
"Okay but—what's that to you?"
"Maya, I'm the only person she knows in LA. I can't just abandon her. Clarissa is from a tiny village in Suffolk. LA is bloody terrifying to a girl like her."
Maya couldn't speak. A hundred arguments and insults lined up in her mind.
"Hey," she said, drawing herself up with effort. "It was only a thank-you kiss. If you want to keep this strictly business, then just say so."
"I didn't say that, Maya," Jack said unhappily. But she'd already turned to leave.
## **PAOLO**
#### THE GETTY VILLA, MALIBU, MONDAY, JUNE 22
"Man, I'd sure like to dive into that pool."
Paolo eyed the broad stripe of blue that stretched through the Outer Peristyle garden, toward the Doric pillars at the far end of the fake Roman villa. Maybe if he could immerse himself in that glassy, turquoise water, he'd be able to shift the anxiety that seemed to have settled on him since that horrible night in Malibu Canyon.
He fumbled with the leaflet they'd been handed at the start of Van Buren High School's tour. It was the first activity of a summer school program he'd promised his parents he'd join. They didn't want their son to spend 100 percent of his time training for the tennis tournaments in July, and while Paolo still had ambitions aside from being a tennis pro, he had to go along with what they wanted. Or else his parents might stop funding his emancipated lifestyle—the last thing he wanted them to do.
John-Michael had gamely agreed to join him in a college-prep course enticingly titled "Art History—Beauty Meets Brutality." He wasn't obligated by any kind of parental edict—John-Michael's funds came directly from an inherited trust fund that was set up to pay for his education and living costs until he was twenty-five. In fact, JM had chosen the course.
The two boys split away from the rest of the class, who went inside the gallery for a lecture on Byzantine art.
"Even for a guy with boatloads of cash, it's pretty weird to re-create a burned-out Roman villa," John-Michael said. He glanced at Paolo, who seemed just as out of it as when he'd left that morning for his daily training session at the tennis club. "So are you on some new diet?"
"Just don't feel like eating." Paolo reeled a little, taken aback by the sudden switch in conversation. "Why the interest in my diet?"
"Because now that I look at you," John-Michael said, suddenly earnest, "you've lost some weight. Your cheekbones are more prominent. Also, your clavicle."
"My _clavicle_?"
John-Michael touched a wary finger to Paolo's skin, just below his throat. "You're hardly eating these days."
"I am a little preoccupied," Paolo confessed.
"It's _starve-yourself_ bad? What'd you do? Kill someone?"
Paolo glowered at him but said nothing.
"The suspense is killing me." John-Michael leaned against one of the pillars of the peristyle and faced his friend squarely. "You ever gonna tell me what went down that night in Malibu Canyon?"
Paolo couldn't look him in the eye. "You said you preferred not to know."
John-Michael gave a nod. "I changed my mind. I've never seen you like this. Walking around like you're on eggshells, skipping out on some of your tennis training. Something has gotten into you."
Carefully, Paolo replied. "If you don't know, you'll never have to lie."
"Jeez, dude, who would I have to lie to?"
Paolo said nothing.
John-Michael continued, "Paolo, listen, I was happy to help out. There's nothing illegal about putting gas in your car and driving it home. If anyone asks, it was you. I don't have to say anything about it ever. That's always my preferred strategy when it comes to the cops anyway, but really, I deserve to know."
Paolo gave a quick nod, glanced around. This wasn't a conversation he could risk having overheard. His skin prickled from the sensation that he was being observed. With every day that passed, one quiet reminder sounded more insistently in his mind:
_Somebody knows. It's all going to come out._
"The thing is, Paolo, let's say it comes to it. Who'd I be keeping quiet from? The cops? Or someone else?"
Paolo turned away without bothering to respond.
But John-Michael circled him until he could look him in the eye. "Did you piss off someone important? Some rich guy with his own private army?"
Paolo was confused. "What are you talking about?"
"You seem scared, Paolo. Which makes me think this is really serious. I thought you were just trying to get out of a situation at the country club, maybe with someone who you shouldn't be seen with."
"That's fine—keep believing that." Paolo allowed a little hostility into his voice. Is this the only reason John-Michael had suggested coming out into the garden—so he could interrogate him?
"You can trust me. I'm involved now. If I'm going to be committed I need to know what the deal is, so I can stick to the story and cover my own ass."
Paolo glared, disbelieving. "You really want this?"
Slowly, John-Michael nodded. "Yeah. Lay it on me. I can handle it."
Paolo took a couple of quick breaths and exhaled rapidly, as though warming up for a race. After a moment, he began to recap what happened that night in the hills—the older woman, the hit-and-run, the borrowed cell phone, everything.
And John-Michael held his breath for a long moment, finally gasping, "Seriously? Are you kidding me?"
"I wish I was. Meredith was wasted. I got out of there, man. There was no way they were gonna find the guy who killed her. But there'd be questions about me. Why I was with her, all that."
John-Michael made a clicking noise with his cheeks. "This isn't Saudi Arabia, dude. Adultery isn't a crime. I mean it's frowned upon, but . . ."
"It was more complicated than that."
"In what way?"
Paolo sighed. "I prefer not to say."
John-Michael seemed to consider. "You see any other witnesses?"
"You think I'd have risked walking away if I had?"
"You figure the cops will think she was alone?"
Paolo nodded and said despondently, "That's the general idea." He started to walk along the terra-cotta tiles around the central reflecting pool. After a second, John-Michael followed.
In a low voice Paolo said, "The problem is, the security camera at the country club might have spotted me getting into her beamer."
"Ah. It's making sense now. Hence getting me to supply your alibi."
"Meredith didn't turn on the GPS," Paolo said, "in case her husband checked the car to find out where she'd been. She was pretty careful about who saw us talking. But the parking lot is a potential blind spot."
"Anything you can do about it? I mean, you work there, don't you?"
"I thought of that," Paolo said, nodding. "I tried to get into the security room when I was training yesterday. But there was never a moment when it wasn't occupied."
"If you've got a chance to steal that tape, or erase the hard drive or whatever system they use, you've got to do that, Paolo," John-Michael said. "If the cops are wondering if anyone was with her, they'll probably ask at the country club."
"I didn't sign into the club that day. I didn't have a lesson. Neither did she, so there's no reason the cops would think she'd done anything but drive down from Montecito and head for her cabin in Malibu Creek."
"Alone? Who does that?"
Paolo shrugged. "Why would the cops care? They'd probably just figure she was meeting someone at the cabin."
"And that 'someone' will never come forward," John-Michael concluded. "Meanwhile, if anyone does look at the footage from the parking lot at the club, they'll see someone who looks like you coming back in the taxi, putting gas in the tank, and then driving away."
Mechanically, Paolo said, "Exactly. If anyone asks, I just say I ran out of gas."
John-Michael couldn't stop a note of admiration from entering his voice. "You brilliant bastard. You figured out that whole alibi, made it happen using a stolen cell phone, just after you'd seen your girlfriend killed by a hit-and-run driver, all on the slim possibility that someone's gonna check the security footage at a tennis club that no one even knows she was at?"
"She was _not_ my girlfriend, man," Paolo said resentfully. "Of _course_ I thought about the security footage! The first thing I thought about was who might have seen me with her. That's basic self-preservation! I did the only thing I could think of fast. Something that gives me a legitimate reason to be getting in that BMW and getting out of it before Meredith heads for the hills."
"Man, that's stone-cold."
This time, Paolo remained silent. Was it obvious to John-Michael that he was glad this woman was dead? Maybe not with the way it had happened, but he was relieved she was out of his life. Relieved enough to feel guilty, to wonder when the day was coming that a detective would knock on the front door, walk in to find Paolo in the kitchen, whipping up a strawberry-flavored protein shake and believing his problems were behind him. Thus far Paolo had failed to find any follow-up to the initial, one-line report of Meredith's death on the internet. But maybe it was too soon?
"Try not to worry, Paolo. You've got to take each day as it happens. I know how you must feel. But think of this: every day they don't come looking for you is a bonus. The more time goes by, the less likely it is that they'll connect you to the accident." John-Michael knelt down, tugged at a bay laurel leaf on a neatly trimmed shrub near the water's edge. He pulled it off the branch, twisted it, and crushed the leaf, raising it to his nose to inhale the fragrance. "Mmm. Makes me want to cook some chicken Parmesan."
Paolo struggled to summon an expression of hope or relief, anything to give John-Michael the impression that his words had taken effect. But he couldn't.
The afternoon hadn't taken Paolo's mind off anything. Confession was supposed to lighten a burden, wasn't it? Yet all Paolo felt now was a faint, queasy sensation.
Would today's admission someday come back to haunt him?
## **MAYA**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, MONDAY, JUNE 22
Maya arrived home before Paolo and John-Michael returned from the Getty. When she saw that Grace was alone in the room they both shared with John-Michael, she closed the door. Grace stopped reading Junot Díaz's _Drown_ to look up at Maya, who paced a little and let her jacket and books fall where she dropped them.
"You're gonna pick those up, right?"
"Oh. Yeah." Maya made a halfhearted attempt to scoop up a discarded book. Then she turned to Grace. On her face was a deepening blush. "Can I get some advice?"
Grace closed her paperback on a bookmark. "Oh, you want my advice now? Does that mean you're done giving advice to me?"
Maya flushed. Her voice dropped to a whisper. "Gracie, that wasn't anything personal. I'm sorry about your father, really I am. But it's not something you should keep a secret from your own sister. I don't even understand how you've managed to keep it quiet this long."
Grace sat up, sweeping Maya with a calculating look. "It's not my secret, though, Maya, is it? There are other people involved. My mom, for one. Candace's dad."
"Oh," Maya said. She hadn't thought about that. "God, I'm an idiot, aren't I? I bet it wasn't even your idea to keep it a secret."
Grace shrugged a little. Maya's admission seemed to have relaxed her. "No. But I'm part of it now. I don't like it, either."
"You will tell her, though, won't you?"
"Candace is so busy with the TV show now, I don't see much of her. I will tell her, Maya. I'll have to, soon."
"Why?"
Grace blinked a couple of times, as though she was holding back tears. She smiled a wan smile and swallowed hard. "The universe has a way of getting its own way."
"That's BS," Maya said. "Don't believe in the 'universe.' You make your own fate."
"You believe that?"
"Of course," Maya said. She sat on the floor between their beds. "I'm trying to take control back over my own life. You remember Jack Cato, my tutor?"
"British guy, I passed him in the hallway," Grace said. "You went to a business breakfast or something."
Despite the residual annoyance at Jack's announcement about his ex, Maya was slowly returning to her original feelings of disbelief and euphoria at the result of her presentation. She found herself smiling. "Yep."
"How'd that go?"
"Extremely, amazingly well," Maya said, her smile widening.
"Seriously?"
Blithely, Maya replied, "Yeah, I basically ruled. Two investors want to back me. Oh, and I kissed Jack."
Grace said, "Pardon me?"
"I totally kissed my tutor. But take it easy, turns out he's not into the idea of a high school girlfriend."
Before Grace could ask what happened, the door behind Maya opened. She turned, ready to snap at the interruption, but held her tongue when she saw it was Ariana.
"Oh, hey, guys. Say, d'y'all know when Lucy might be home?"
Ariana was everywhere, all the time, and Maya and Grace exchanged looks of irritation. Things were getting tense inside the house.
At least, that's how it seemed to Maya.
She wanted to say something to Lucy about it. But Lucy maintained a quiet, taciturn edge that made it difficult to broach any subject that Lucy herself hadn't raised. Throw in the fact that Maya had spent part of the last six months spying on her housemates, reporting back their every move—especially Lucy's—and the guilt made it impossible. That's if she and Lucy even crossed paths. Ever since school finished, all Lucy seemed to do was stay in bed until around two in the afternoon and then leave for some kind of appointment—Maya assumed it was something musical.
Her other option was to encourage her housemates' frustration with Ariana's extended stay. Like when Candace had asked bluntly, "So Ari, you get a job yet?"
It was like water off a duck's back, though. Ariana would just reply with a snake-eyed smile and give a throaty chuckle. "Honey, this town isn't ready for ol' Ariana, not yet."
The excitement of meeting those investors fueled another long night of coding. Around two in the morning she fell asleep, but woke too early—7:10. Her mind wouldn't stop spinning, mentally solving coding glitches in Promisr, until she knew that only food, caffeine, and her laptop would do the job.
On the stairs outside the front door, she heard voices from the kitchen. The window had to be open. A tinny voice, distorted by the speakerphone of someone's cell, was saying, "Bloody well make friends with them! How hard is it, honestly? You're only a couple of years older than them."
The accent was undeniably British. It sounded familiar but Maya couldn't place it.
"I'm doing my best," Ariana managed to reply.
"Find that bottle of nail polish," the woman said firmly. "Or make damn sure she didn't keep it. Do that, and then get out of there."
Ariana gasped. There was silence. Maya guessed that whoever Ariana had been talking to had hung up.
On the spiral staircase, Maya stalled. She didn't dare take another step, either toward the kitchen or back to the second floor.
That voice. It had sounded almost exactly like . . . no.
It couldn't be.
_Dana Alexander?_
Maya gripped the handrail tightly and held her breath until she couldn't anymore.
She strained to hear any further sounds from Ariana in the kitchen. She heard the tap run, water slapping against the metal sink. Then she heard a chair being shifted, movement back toward the living room.
The muscles of Maya's legs tensed. If Ariana headed upstairs to the bathroom, she'd definitely catch her on the stairs. Ariana would know that someone had heard the conversation, or at least part of it.
She heard the front door opening and made her decision. Maya bolted back into her room, slipped under the quilt, and buried her face in the pillow, pretending to be asleep. She held still as Ariana's footfall sounded on the landing outside and paused outside the door. Was she listening? Would she dare to open the door and confront the three sleeping roommates?
Maya's heart pounded, ricocheting inside her rib cage.
Ariana was in the house to spy on Lucy. Which meant that Dana Alexander wanted more than Maya was able to provide—someone to directly interrogate and maybe even provoke Lucy.
Or that Dana no longer trusted Maya.
## **CANDACE**
#### KITCHEN, VENICE BEACH HOUSE, TUESDAY, JUNE 23
"Grace already left to play beach volleyball. So she won't be able to wait in for the IKEA delivery." Candace helped John-Michael set the dining table, carrying over steaming plates of scrambled eggs, crispy bacon, and hash browns.
Lucy collected three different bottles of hot sauce from the cabinet and set them on the table. "Oh, it's fine, Ariana said she'd do that."
"Oh, well, that's just lovely," Candace said, her tone flat. Dryly she added, "Are we expecting Ariana to eat with us?"
"She's taking a hot bath," Lucy replied. "She said her back hurt from sleeping on the futon."
Candace said nothing, but locked eyes with John-Michael for a moment. He shook his head very slightly, amazed. "How about Maya?" she asked.
"Still in bed. She's nocturnal, that girl."
John-Michael laughed. "So are we. Breakfast at two thirty in the afternoon?"
"Dude, it's summer vacation," Lucy replied. "Maya's probably asleep because she worked until dawn."
"It's that app," Candace said. "'Gotta keep moving, like a shark.' Isn't that what she always says? Well, you know what they say about all work and no play . . ."
"Says the girl who spends all her free time working on a TV show," Lucy commented. "Or can we presume you have an ulterior motive there? 'Cause I've been hearing rumors."
Candace arched a single, perfectly shaped eyebrow at Lucy as she helped herself to bacon and eggs. "Guys, I'm _dreaming_ about Yoandy now."
Lucy poured out three glasses of ice water. "Do tell. Or are we talkin' R-rated dreams here?"
"Those'd be the only kind of dreams I'd have about Yoandy Santiago," John-Michael admitted.
Candace gave a wicked smile. "We were totally doing it, if you must know."
"Goddamn dream-hussy, I knew it!" John-Michael said, pointing an accusing finger.
"Hmm," Candace responded lightly, "look who's talking. _He_ calls me _nena linda_ ," she added with a softness to her voice that surprised even her.
Lucy's interest perked up. "In real life? Or was that in your dream?"
"In real life, of course."
John-Michael clutched a hand to his heart and said passionately, _"¡Nena linda, te quiero, cuanto te quiero!"_
Lucy sniggered, but Candace merely flicked a lump of scrambled egg at her. " _Silencio_! I've decided it's romantic."
" _Nena linda_ isn't bad," John-Michael added. "I've heard Maya's aunt call her that."
"But I think he has a girlfriend . . ." Candace said, sighing.
"A guy that hot probably has more than one," Lucy agreed.
Candace continued, somewhat reluctant, "I mean, he says they're not together, but . . ."
"You suspect you're being lined up as number two," chuckled John-Michael. "Although I gotta tell you. Kay Alexander doesn't seem like his type. She's one of those weird Hollywood enigmas. Famous for being Dana Alexander's sister, and not much else."
"She did a reality TV show," observed Candace. She'd researched Kay quite a bit since learning of her connection to Yoandy. None of it made her feel any more secure.
Lucy stopped in mid-reach for the SunnyD. She seemed a little shocked. "Yoandy Santiago is dating _Dana Alexander's sister_? I did not know that."
"Oh, you've been taking an interest?" Candace said flatly.
"In Kay Alexander? Not particularly," Lucy said, a little cautiously. "But Yoandy Santiago I've heard of. I didn't know he could act."
"He's great at stage combat," Candace admitted. "Acting—I guess I'll find out later today, at the table read. _Prepped_ is meant to be his big break."
After their spectacularly late breakfast, Candace showered and brushed her teeth with careful precision. She held her own gaze in the mirror and remembered the dream.
She hadn't been totally honest with her housemates. She and Yoandy hadn't been having sex—or at least, they hadn't gotten to that part before she'd woken up. In her dream, he'd been removing her clothes, slowly, staring into her eyes as he did so, and she'd simply waited, trembling at his every touch.
Who knew that a few softly spoken words could have such an effect on her subconscious?
Maybe it was because he spoke in Spanish. In English, she'd probably have laughed at him. But the Spanish lent him a sincerity that was hard to deny.
She drove her newly serviced Prius to the studio and parked, swaying her hips as she strolled through the lot, conscious of the fact that at any moment, Yoandy might appear. She wanted to appear gracious, feminine, _hot_. But she didn't see him until they'd already taken their places at the table, because once again, he was the last to arrive for the late-afternoon rehearsal.
When the pages were handed out, Candace scoured them for scenes where her character, Annika, appeared with Yoandy's character, Sebastian. When she found a scene, she pored over the lines, for any hint of sexual tension. The production team was keeping any future developments under wraps, so she had no idea whether a relationship between Annika and Sebastian might be explored. But she could hope. Hope fueled her fantasy.
_Prepped_ was going to be screened on cable. She'd already seen that the language they used was fairly adult. It wasn't beyond the realms of possibility that she'd have to record a bedroom scene with Yoandy. Even with the "no nudity" clause in her contract, Candace knew she could be asked to push boundaries. She imagined the tension from acting such a scene. Would it spill over into their real lives?
For some reason, boys her own age didn't interest Candace. She'd certainly gotten their attention, especially in the past two years. Her curves gave her a whole new look, away from the skinny athleticism of her childhood. She'd even had a couple of boyfriends. Hooking up with a high school boy, though—it wasn't all that.
Yoandy was twenty-one, and those few extra years made all the difference.
She was positioned almost directly opposite Yoandy at the table, on a very slight diagonal. When they read their scenes together, he'd obviously memorized his lines. And he looked her straight in the eye as Sebastian spoke to Annika: _If you were mine, you wouldn't have been alone._
"What makes you so sure I was alone?"
The intensity of his attention was unnerving. It took her by surprise. Fumbling her own line, Candace flipped the pages back and forth, her cheeks burning hot as she realized that she'd somehow missed this exchange in the preparation. Or at least she'd missed the potential for romantic tension.
"Annika," the director said wearily, "could you maybe try that with a little more _pizzazz_?"
Candace stared at the line. She turned to the director, keenly aware of Yoandy's eyes on her. "You want me to say that kinda flirtily?"
"Would it be too much trouble?" the director replied, politely sarcastic.
She cleared her throat and faced Yoandy. The combination of lust and tenderness she saw in his eyes shook her to the core. "If you were mine, you wouldn't have been alone," she said.
"Which is a very nice delivery, Miss Deering," sneered the director, "of Sebastian's line."
There was a general snickering from the rest of the cast, apart from Ricardo Adams, who, Candace noticed, looked positively frosty. For a moment, she caught Yoandy's reaction—a hopeful smile.
"Shall we take five? I could use a cigarette," the director said, pushing back his chair.
Everyone but Candace rose to their feet. She remained in place, unsure if her legs could be trusted to carry her. It was as though she'd been poisoned. Her pulse was racing, palms sweating. She'd embarrassed herself in front of the entire cast of a brand-new TV show, possibly her best chance to get her career off to a decent start.
When she looked up, however, there was Yoandy, offering a hand.
"It happened to me," he confessed, "the first time I had to sing in public. I forgot the lyrics."
She allowed him to help her to her feet. When he didn't release her hand right away, she tried to relax. "Okay, so we're holding hands now, are we?" she mumbled nervously.
" _Nena_ , it's all right. Breathe."
She tried to laugh and pulled her fingers away, leaning against the table for support. "You're the newbie," she murmured. "I shouldn't be the one making mistakes like this."
"Newbie? Me?" He shook his head, smiling. "Candace, I've been singing and dancing in front of people since I was thirteen. You have any idea how vulnerable some of my lyrics make me feel? It's like reaching into my chest, pulling out my heart, and slapping it down on a plate for everyone to eat."
Candace didn't know what to say. She had no idea what his songs were about, but she could guess. He looked at her for a moment, as though coming to a decision. Then with a brief grin, he turned and took off toward the recording equipment. She watched, baffled, as he moved between the technical crew until one of them nodded and handed something to him.
Yoandy turned to her, microphone in hand. When he spoke, his voice echoed in the studio space. "Candace, this is for you."
"Oh God," she whispered, clutching the edge of the table. "Don't _sing_ to me . . ."
"Jigg-a-lypuff. Jigg-a-lypuff!" he sang, loud, his voice a perfect, soulful tenor. He didn't break eye contact with Candace, despite peals of amazed laughter in the studio. He continued, "Jigg-a-ly—puff—Jigg-a-ly . . ."
"Oh good grief," Candace muttered. Her nervousness vanished, like smoke.
Yoandy couldn't finish the song; he was overwhelmed by uproarious laughter from the cast and crew. From outside where he'd been smoking, the director yelled at them. A few minutes later, Yoandy was back at the table, guiding her back to her seat and beaming with glee.
"Jigglypuff." She smirked. "Really?"
"I was eight years old when I came to this country. Pokémon became _my life_ ," he answered earnestly. "Someday I'm gonna show you my collection."
Candace propped up an elbow, rested her chin on her hands. "Pokémon? That's actually hilarious," she drawled. "Your favorite thing about America is actually Japanese."
"Everyone gets nervous, _nena_ ," he told her, tenderly, taking her right hand in both of his and massaging just beneath her thumb. "One day you'll look back at this and you'll laugh."
"One day?" she said, trying desperately to take her mind off what he was making her feel with his touch. "I'm thinking of you singing the Jigglypuff song and I'm laughing _right now_."
## **MAYA**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, TUESDAY, JUNE 23
Maya dug her fingers into the pillow, trying to calm her thoughts, trying to focus on what she'd actually heard Dana Alexander say on the phone to Ariana earlier that morning. It was a blood-freezing notion.
_"Bloody well make friends with them."_
Ariana was obviously well aware of the fact that she wasn't getting along with the rest of the housemates. She'd retreated into the bathroom to take a nice long bath, precisely when they were about to eat breakfast together. Their irritation was finally getting to her.
_"Find that bottle of nail polish."_
What nail polish? For Maya, this was an entirely new element to the whole mystery. What if something about nail polish featured in some unpublished clue, something that hadn't been raised at the murder trial? Something that only the real killer would know? Could something as seemingly insignificant as a bottle of nail polish link Dana Alexander to the murder of Tyson Drew?
Maya sat up on her bed, opened her computer, and scoured her collection of data from the Hollywood murder of which Grace's father had been convicted. Nail polish hadn't been mentioned in any of the articles she'd dredged up. Given that Tyson Drew was supposedly drowned by Alex Vesper—a guy—it seemed like a stretch.
But she simply couldn't think of any other reason why Dana would go to such lengths to plant someone else in their home. Why she'd make such a big deal over some nail polish.
Or why she'd plant Ariana in Lucy's life from the start.
With a mounting sense of helplessness, Maya faced up to the fact that Ariana hadn't been brought in as a stranger to the house, not like Maya. Ariana had been friends with Lucy for _years_. They'd been rehab buddies.
Had Dana Alexander arranged all of that, too?
Maya twisted in the bedclothes, listening for the flush of the toilet to signal Ariana getting ready to leave the bathroom.
Just how powerful was Dana Alexander?
Maya had to stall for time to think things through. If she blew Ariana's cover now, Maya would need to come clean about her own role as a spy.
Any trust in Maya would be gone. Her housemates might even throw her out. Then Dana would find out that she wasn't cooperating. The woman might even make good on her threat to tell immigration that Maya and her mom were in the United States illegally. Then it would be back to Mexico City.
No more Venice Beach house. No more introductions to Silicon Valley business angels. _No more kissing Jack._
On the other hand, Maya didn't want to do a single thing that might actually help Dana Alexander. She'd eventually have to expose Ariana to her housemates. Before Maya made her move, though, she had to at least try to mediate the blowback. Maybe get some evidence against Dana Alexander herself, something that Maya could use as leverage?
_Rat me out to immigration and you'll have to deal with me, Alexander._
It was a lot to think about.
Maya heard a creak in the floorboards and became aware that she was not alone in the bedroom.
She stared up at the open door to the triple bedroom, feeling woozy.
Ariana stood there in her bathrobe, looking inquisitive. Naturally.
Maya was about to rearrange her scowl into a sweet smile, when it struck her that Ariana might get suspicious if Maya suddenly altered her behavior toward her. Of all the housemates, Maya was the one most inconvenienced by Ariana's arrival. Maya's habit of coding on the sofa, or at the kitchen table, long after everyone else had gone to bed, had set the two in competition for the same space. If anyone was allowed to be snarky about Ariana, Maya figured it could be her.
With the same friendly grin that she used each day to greet the housemates, Ariana said, "Hey, Maya. Can I get you a coffee?"
Maya emitted something like a snarl and rolled over, facing away.
"How's the app coming along?" Ariana asked lightly.
Maya grunted. "Not great. I'm supposed to show an alpha version soon. I'm gonna have to work all day and night to get it done. Where is everyone?"
"Work," Ariana replied vaguely. "The beach."
Maya forced her voice to remain even, casual. "So, you're staying home today?"
"I'm waiting for that delivery from IKEA."
Maya's eyes registered surprise. "That's today? Oh, I'm here all day. You can go out, if you have something you need to do. I'll stay in for the delivery."
Ariana smiled slightly. "It's okay," she began quietly, but was interrupted by a knock on the front door.
Maya checked her watch: 3:27 p.m. Jack! She'd forgotten all about her meeting with him. She leapt to her feet, dressed quickly, grabbed a pair of beach shoes from beneath her mattress. She was dragging a brush through her hair and checking her makeup as Ariana made her way downstairs to let him in.
A few moments later she was in the living room and letting Jack kiss her on both cheeks, super-polite as ever, barely any indication that they'd ever taken it further. Before Jack could say anything to betray either of them, Maya called out, "This is my tutor, Jack Cato." She made an informal gesture in Ariana's direction. "Jack, Ariana."
"Hello, Jack." Ariana smiled, appearing at the edge of the kitchen. "Can I get you a coffee? Juice? Milk?"
"Actually, a tea would be brilliant if you've got it—milk, no sugar," he replied. To Maya's relief, Ariana disappeared back into the kitchen.
Maya led Jack to the green futon.
"You guys have a hugely privileged situation, you know that?" Jack said. "What I'd give to live on this beach, be able to walk down to the ocean for a swim. Or a run."
Maya smiled at him mischievously. "Just hypothetically—what _would_ you give?"
Jack seemed to have some difficulty framing his reply. Minutes later, Ariana reappeared. She handed Jack a mug of tea, then sidled away. Maya turned and watched Ariana pop two earbuds into her ears before she made herself something to eat.
"How's your tea?" Maya asked.
"Hot," he said, blowing across the surface of the tea. "It'll do. Lord knows I could use it after that traffic. The state of the roads in this city is nobody's business!"
Maya wrinkled her nose. "It's a long drive from Pasadena, huh?"
"It's not exactly around the corner," he agreed. "But today I've come from the airport."
Maya couldn't stop her eyes lighting up. "Your ex . . . ? What happened to doing a course at UCLA?"
He grinned, delighted. "Her uni emailed last night and said it's a no-go. So this morning, off she popped, back to the UK."
"Oh," Maya said, a little deflated. All that getting annoyed, for nothing. Or—had it just been an excuse? "So you're really not into her anymore?"
Patiently he said, "Maya, she booked the flight months ago, when we were still together. It would have been pretty mean to abandon her in LA."
"I guess that's true." Hesitating slightly, Maya said, "About the drive from Pasadena, if you ever need a place to crash one weekend, there's a bed here. Don't look so shocked—I didn't mean my bed. I meant this futon. Ariana's only supposed to be staying for a few more days."
A slow smile touched the corners of his mouth. "You certainly managed to crowbar 'bed' into the conversation."
"Oh please," Maya said with a gentle slap of the back of her hand to the thigh that was almost touching hers. "It was written all over your face."
He laughed and leaned closer, until she realized that he was going to kiss her. Maya pulled her face away from his lips and whispered, "Not here. Ariana is watching." Before he turned to look, she hissed. "Don't look!"
Disappointment clouded Jack's features. It was a cute look for him, she decided. Puppy-dog-like, very sweet. "Not a fan of the PDA?" he asked.
Maya produced a small smile. With a sigh, Jack leaned over to pick up his brown leather messenger bag and withdrew a MacBook. He flipped it open and began to talk once again, the playful tone now banished.
"Okay, so the news is that one of the angels has dropped out. But not to worry—we do have an offer on the table: Kyle Joseph, who used to work at Google. I've had a bit of a chat with some of the guys in the entrepreneur society. They reckon the offer isn't bad—pretty much in line with what he mentioned at the presentation. He's offering a hundred K for twenty percent—subject to due diligence."
Maya shook her head in wonder. "Wow. One hundred thousand dollars!"
Jack smiled, pale blue eyes like a wading pool. He leaned closer and whispered into her ear, "Sweetheart, you are seriously smart. And I for one am going to do my utmost to see to it that you go a very, _very_ long way."
## **PAOLO**
#### VENICE BEACH, FRIDAY, JUNE 26
**MEREDITH ERIKSSON, 42, WAS HIGH WHEN**
**HIT-AND-RUN DRIVER STRUCK**
Paolo focused on the headline on his cell phone. A mixture of emotions assailed him. Mostly, relief.
Meredith had been reported dead, killed outright by a hit-and-run driver, up in Malibu Canyon. At the Malibu Lawn Tennis Club, Paolo had been too cautious to ask a single question that related to her. He'd even been too scared to set up a news alert on the internet—didn't the NSA monitor searches? Instead, he'd checked the local news every day. This latest headline confirmed it—he was in the clear.
Climbing directly up the spiral staircase from the boardwalk outside, Paolo put his phone in his desk drawer and changed quickly into the short wet suit he'd just bought; black with turquoise trim. Then he took his keys, some cash, and skipped back down the stairs. Maya and her tutor, Jack, were outside now, on the ground floor, by the front door.
"Just popping out for a spot of kitesurfing, old bean," Paolo said in his best English accent, grinning at Jack.
"Kitesurfing, are you out of your mind?" Maya folded both arms across her chest, swept her eyes up and down Paolo, examining his short wet suit.
"You're taking lessons?" Jack said, impressed.
"I've taken _a_ lesson," Paolo confirmed. Now he was eager to fly solo.
"You got it after one lesson?" Maya asked.
He hadn't. But what was the point paying a tutor to tell him the same thing over and over? Paolo knew what to practice, he just needed water, wind, a board, and a sail.
Jack said, "I did a bit of that myself, on Eton Dorney. It's an artificial lake, at school."
"Your high school had its own lake for kitesurfing?"
"Mostly for rowing eights."
"You ever take him surfing?" Paolo asked Maya.
Jack looked curiously at Maya. "You surf?"
"Not very well," Maya admitted. "But I can catch a wave, once in a while."
"You're a woman of many talents, Miss Soto," marveled Jack. "And I would absolutely love to go surfing with you, one of these days when we're not both up to our eyeballs in work."
"Don't hold your breath," she said dolefully. "Promisr is eating up all my time. But I bet you're pretty good at surfing."
"I'm a Cornish lad," he said, "so I'm not totally inept. I, too, can catch a wave, once in a while. Anyway, Maya, I'd best be off. I'll keep you posted, all right?" With a light kiss to her cheek, he was gone.
"So," Paolo said, hooking a thumb at the tutor's retreating form. "You and the Brit?"
"So?" Maya said with a quiet smile, closing the front door on him slowly. "You and Lucy?"
"You're way behind," he called after her. "That's old news."
Left on the outside, Paolo headed for the surf rental store. Most places wouldn't rent equipment unless you were taking classes or already had some kind of badge. But John-Michael knew a guy who knew a guy. Paolo felt pretty sure he'd be allowed to rent some gear today.
Along the crowded boardwalk, he saw Lucy sitting cross-legged on a bench. _Of course, after just talking about her._ Next to her was a guy with shoulder-length black hair. One arm was entirely covered in colorful tattoos from shoulder to wrists. Around him were abstract paintings made from some kind of rubbery streaks of paint. When she saw Paolo she waved. "Hey, King, you know my boy Luisito, right?"
Paolo gave a short nod in Luisito's direction. "Hey, man."
"Where're you headed?" Lucy said. "Surfing?"
He stopped in front of the paintings. "Yup. Not waves, though, I prefer to stand up, right from the beginning. Gonna get me a kite."
She frowned. "Isn't that super-dangerous?"
"Nah, it's cool."
Lucy sprang to her feet. "Okay if I walk with you a little?"
Paolo shrugged. Ever since the Meredith incident, he'd almost stopped thinking about Lucy. She hadn't sought him out, either, but then that was normal. Could it be that she was actually feeling guilty? It was over a month since their little moment of misunderstanding. It had really stung at first but now . . . compared to what had happened to Paolo since, a fumbled romantic situation with Lucy barely registered.
"That'd be nice," he said with an easy smile.
Lucy fell into step beside him, and he soon picked up the faint trace of her perfume—Flowerbomb. He'd noticed it on her nightstand about six weeks before, had made a mental note to buy her some at the next gift-buying opportunity. That was when he'd still hoped to get into the gift-giving zone with Lucy. Paolo was only faintly surprised to realize that this thought didn't sadden him.
"I've been meaning to talk to you about something," Lucy said. "Somehow, you're always busy . . . or avoiding me."
"Me? No way," Paolo said, holding up both hands. But she was right and he knew it. Meredith's death had preyed on his mind almost constantly. If he stayed home too long he became anxious. It was better to be out, to be distracted.
"Well," she said, and her fingers took his elbow in a sudden grip, hard enough to halt his progress. "This is important, so listen up."
Paolo stared at her curiously. Lucy looked serious and he had no clue what else she might be about to say. For a moment, he wondered if Lucy somehow knew about Meredith. It was beyond comprehension that John-Michael would betray that confidence—not when he'd made himself an accessory to whatever crime Paolo may have committed.
It took a few seconds before Lucy was able to form the words. "Grace . . ." she began with difficulty, "I get the impression from John-Michael that Gracie's having a hard time lately."
Gently, Paolo pulled his elbow from her grasp. "Okay? What does that have to do with me?" Lucy was steering him toward another girl in the house? This, he really could not believe.
"You haven't noticed at all?" Lucy said with an almost-smile.
"Have I noticed Grace?" he said candidly. "Of course; she's beautiful."
"But you don't . . . y'know, _like_ her?"
"I try to limit my interest to one girl at a time," he said, more sharply than he'd intended. Evidently, Lucy had expected more of a flirty response and was taken aback by his brusqueness. "Why're you telling me this, Luce?"
Lucy's eyelashes fluttered for a second before her jaw clenched. "No, sugar," she said, her voice taut. "I thought I'd tell you because honestly, I don't know what's going on with you these past few weeks but there's somethin'. I thought maybe another girl? And I wanted you to be aware that Grace, for all she tries to hide it, is into you, Mr. King, whether or not you deserve it, which by the way you do not."
"Grace is a nice girl," Paolo said with a smirk. "Too nice for me."
"I just thought that maybe if you knew she liked you, you'd be sweet to her."
"You think I should let her down nicely? That's what you're saying? Like you did with me?"
Lucy broke into an amazed chuckle and pulled away from him. "Man, you're uptight right now. What's going on with you?"
"I'm uptight? Ever stop to think that maybe you broke my heart?"
"Did I?"
"No."
Annoyance flashed across Lucy's face. "Stop playing games, Paolo. And don't play games with Grace." She backed off some more, staring Paolo hard in the eye. Then she turned, began to stroll back toward Luisito and his paintings.
Paolo watched her go, then shrugged. He could feel a pleasant afterburn from their tense exchange. Grace liked him? Interesting. She hid it well.
She really was beautiful. Not as stereotypically pretty as her stepsister, Candace, for sure. Grace had a whole different vibe—brains and a quiet, understated beauty. She still had a girlish quality, a sort of adorable cuteness. That was probably why he hadn't thought of her that way, Paolo reflected.
He'd assumed that Grace found him kind of skeezy. It wasn't at all disappointing to know that she liked him. Once he opened up his mind to the possibility, imagined Grace in any romantic situation, he found himself responding with a pleasant warmth.
He was on the verge of catching up to Lucy to ask how she knew, but thought better of it. That certainly wouldn't look too smooth. After a few more moments, Paolo pushed the idea out of his mind. Kitesurfing was more dangerous than he'd let on to Lucy; enough to require total mental focus. Danger was what his body called for—he needed that pure rush of adrenaline—to feel alive, invincible. No more hiding and waiting, no more praying that what happened in Malibu Canyon would stop haunting his nights and crowding his days with omnipresent anxiety.
The sun was warm, heating up his wet suit enough that he longed for the chill of the ocean. Ahead, he spotted two bright blue and red kites on display at the surf rental shop. He picked up his pace a little, smiling.
He had to stop worrying, accept that the whole Meredith situation was over. He was home free. Nothing could touch him. And now, he had the chance to be with someone he could laugh and hang out with. Someone who really liked him, not just for the way he looked or played tennis. Someone real.
_Grace._
## **MAYA**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, SATURDAY, JUNE 27
"Wake up, Maya. Time to start the rest of your life."
The cell phone was on the pillow next to her head, the voice coming out of it was Jack Cato's. But even he couldn't make her a morning person, especially with how exhausted she'd been lately.
"But it's Saturday," she grumbled, then peered more closely at the time on her phone. "And it's only seven thirty!"
She could actually hear a smile in his voice. "Ah, that'll be why I'm at a breakfast meeting. Come and meet me when it's over. I promise the waffles will be worth your while."
She rubbed her eyes. "Jack. What are you talking about?"
"That café on the boardwalk, say, two hours from now. How does that grab you? My treat."
Maya rolled away from the wall and glanced at the other two beds, where John-Michael and Grace were dozing. Like a slap in the face, she suddenly remembered the phone call she'd overheard between Ariana and Dana Alexander. A groan escaped her. She'd have to do something about it. But not right now. Jack obviously had something important to tell her about her app project. Whatever she decided to do about Dana and Ariana, it would have to wait.
"Sounds good," Maya said as she drew her feet up beneath her and curled up into a ball. She resisted the urge to fall back asleep. "I'll be there."
She hauled herself out of bed and into the shower, dressed, and applied makeup to a degree that would have delighted her mother, who often despaired of Maya's perfunctory concession to all things girly. An hour later in the boardwalk café a little down the beach, she and Jack split a plate of waffles with strawberries and cream.
Jack told her he'd spent Friday night drinking with Kyle Joseph and some of the guy's investment banker buddies at a fund-raising seminar on campus. This morning they'd "done" breakfast. A good sign, apparently. "These blokes don't even crack their eyes in the morning unless there's a deal on the table," Jack assured her.
Maya tried to focus her mind on what really mattered—the potential investment in her app. "Jack, can you bottom line it for me?"
"Ha-ha. I love it when you talk business."
"I'm serious."
"I was getting to that," he said with a grin. "Kyle Joseph posted the link to the alpha version of your app on some private tech-investor network, who forwarded it all over the bloody planet, and the downloads are insane. We're talking a thing of epic proportions."
"You mean it's gone viral? How many downloads?"
"It's not about the numbers," Jack said breathlessly. "It's about who's seen it. Half of this game is about visibility. If your app is seen by enough people, it has a better chance of being seen by the people who count. And Promisr just happens to have been seen, used, and frigging _loved_ by one of the people who count."
Maya was perfectly still, fully awake now. "Who? Zuckerberg?"
"Alexa Nyborg."
"The woman who started Kilowant?"
"The very same."
For a few seconds, Maya was speechless. Against her silence, Jack continued, "There we were, eating croissants together, Maya, and the woman is _stoked_. She's working on some megasecret, multiplatform social networking app thingy. She wants to embed Promisr. Says it's just at the right stage, development-wise. She wants to meet you."
"No kidding . . . Alexa Nyborg? She's amazing. I read her book. It's all about how women have to 'leverage their network' to get ahead."
"And now she wants to get ahead with _you._ Alexa and Kyle reckon that together, they want to put in another hundred K each to get to fifty-one percent between them. They'll do some of that as an equity swap to you for shares in Nyborg's new thing. When she sells that, your shares should easily be worth between two and five million."
The numbers were beginning to swim around in Maya's head. She picked up a napkin and asked a waitress to lend her a ballpoint pen. She made Jack repeat himself and listened carefully, jotting down figures.
"You think there'll be enough cash for the additional development we need to do?"
"No problem at all."
Maya peered in amazement at the numbers she'd written down. Now that Jack had explained them, they didn't seem totally insane. Within twelve to eighteen months, Maya could be a multimillionaire.
"I need to process this," she said.
"By all means," Jack said, calmer now. He freshened up her coffee cup. "Let's both think about it and have another chat later today."
They fell silent as both began to work on the waffles, which were delicious. A glow of wonder was slowly sweeping through her. She felt light-headed with disbelief.
The initial offer of a hundred thousand dollars had blown her mind—for about five seconds. Then she'd realized they'd need to spend the entire amount on paying extra programmers. There wouldn't have been a spare nickel for Maya. But for someone like Alexa Nyborg to put Promisr inside a more substantial social networking app, for Maya to get entry-level equity in something that big . . . It could be like getting in on a ground-floor investment like Instagram.
Maya's family had never had much money. Her dad worked as a fermentation technician in a pharmaceutical plant, something that required a lot of specialization but didn't pay all that well. No one in her extended family was rich. She'd hardly been anyplace other than Mexico City or California. Just once, the family had blown some savings on a trip to Acapulco and stayed at a fancy hotel. The next trip they'd dreamed of taking as a family was to Disneyland.
Maya was still waiting on that one.
_To be rich_. More money than the mere cushion needed to make life bearable. Enough never to have to worry again. She hadn't even imagined that. It just didn't happen to people like Maya and her family.
The real American dream. Maya put her fork down for a moment, wrapped her arms around herself, as though trying to hold on to some semblance of reality.
She glanced up to find Jack's eyes on her. He smiled then, warm and comforting. "It's so good to see you this happy."
"Thank you," she whispered. "You did this. You made this happen."
Jack reached across the table. He teased one hand free from where it was clasped to her upper arm, and intertwined his fingers with hers. "Nonsense," he chided softly. "Maya, this is happening because of you. You're the one who worked hard at your app, who kept improving it. You're the one who came up with this amazing new idea. I'm the one along for the ride, not you. Frankly you could ditch me now and never look back." Jack gave a hopeful grin. "Obviously, I'm hoping you don't."
"Why's that?"
"Well, I'm pretty much in this for the kissing, that's pretty obvious . . ." He smiled, squeezing her hand so she knew he was joking. "Clearly, I'm hanging out for more of that."
"Sounds doable," Maya replied, blushing.
Jack's cell phone began to buzz once again. He shrugged in puzzlement when he saw the number, picked up the call, and mouthed at Maya with pantomime gestures, _Alexa Nyborg!_
He listened for a moment, punctuating the silence with the occasional "Ah-ha. Mm-hm. Great. What about the equity split? Interesting. I'll certainly ask her." When he finally ended the call, Jack's eyes looked like they were about to catch fire.
"Maya." He sounded ready to burst. "Guess what she said. _Alexa Nyborg_ , Maya! Go on, try."
Maya shook her head with a cynical smile. "' _Too late, loser_ '?"
"No, Maya. Dead wrong. She said, and I quote, 'Maybe she'd like to talk it over in Napa, soon?'"
Maya was momentarily speechless. Then, "Alexa Nyborg wants _me_ to go to Napa Valley?"
His delight was infectious. "Alexa's got a property there, nice and private, swimming pool, tennis court, great walks. She suggested that we fly up there on Friday and join her for lunch. Asked if there's any food you don't like and . . . let me see, let's be sure I'm not forgetting something. Oh yes, and _casually_ mentioned that if you do agree to partner up, she'd be happy for you to stay there whenever you like. ' _I'm hardly ever there, so I kinda like to share the place with my friends_.'"
Maya gasped. "Friday? You gotta be kidding. I got a ton of coding to do before I show the beta version of Promisr."
_"I know!"_
"Does this kind of stuff really happen?"
Jack said, "I guess this is what happens when an investor really, really wants your technology."
Maya gave a tiny shrug of her narrow shoulders. "But why? I mean, the whole thing only took me a few weeks to build. She could easily rip it off and start over."
"First off, no," he said. "It took more than a couple of weeks. You started with the basic code from Cheetr, __ which took you months to develop. Second, that's not how people like Nyborg do things. It's not just your tech she wants; it's you. Your mind, Maya. She likes the way you think. She wants you on board." He pushed away from the table and rose to his feet, approaching.
The sensation of lightness returned. Maya could swear that the soles of her feet were no longer in contact with the ground. Ariana was forgotten, Dana Alexander, too, as Maya allowed Jack to enfold her in a tight, blissful hug.
## **JOHN-MICHAEL**
#### VAN BUREN HIGH, WEDNESDAY, JULY 1
"JM, could you try that part one more time?"
John-Michael picked up his guitar pick and tried again. From her expression, he could tell that Lucy hated having to direct. She probably hadn't been the bossy one in her previous band, Whatnot. When she'd been kicked out, she'd asked John-Michael to start a new band with her. He liked Lucy too much to turn her down.
But now, he wasn't so sure. It had been weeks since Grace had told him the truth about Lucy being in the house when Tyson Drew was murdered. And Lucy herself, supposedly his best friend in the house, still hadn't offered up a single word on the subject.
John-Michael was pretty sure that the only reason he was in the band was because of his access to Van Buren High's summer Rock Challenge program, which guaranteed rehearsal space and time in the recording suite. He watched Ruben, who sat calmly behind his drum kit, one stick lightly resting on the high-hat, teasing it. He'd changed his haircut since John-Michael had last seen him. It was more self-consciously spiky; the blue dip-dye had been freshened up, too. The sleeve of one T-shirt was rolled right to the shoulder, exposing what John-Michael could see was a new tattoo. It might have been his imagination, but was that lip piercing also new?
He caught Ruben's eye for a second, at which point John-Michael fluffed the chord. Instead of the irritation he expected to find, there was nothing but a wry smile, as if to say _Now you're gonna get it . . ._
"JM, is something wrong?"
"Mr. Ruben got a new tattoo," John-Michael said petulantly. He didn't dare to confront Lucy openly about what was really bothering him. Damned if he was going to play nice, though. "It's distracting me."
"Good eye," Ruben said, looking directly at John-Michael. His smile widened as he followed John-Michael's eyes, which were staring at his upper arm, just above the older Sex Pistols tattoo.
Lucy shook her head, her face drawn tight. John-Michael instantly felt a shred of remorse over his quip. "Sorry, Luce. This song is a little difficult for me," he confessed, putting both hands on the neck of his sea-green Fender Stratocaster. "I'm having a hard time making all those changes so fast."
"It's okay, JM." She spoke calmly, very reassuring. "What if I take over most of the rhythm section there and you just bring in a walking bass line?"
He gave a hopeful shrug. "Sure. I mean, if that isn't a pain for you."
Ruben stopped drumming and got up. "While you figure it out, I'm gonna get some air. I'll be over by the bleachers."
John-Michael turned to watch, surprised and disappointed, as Ruben left the room. "I may have mentioned that there's a girls' beach volleyball match," he admitted to Lucy. "I bet you a dollar he's going to watch."
"Yeah." Lucy began to play some chord transitions. "Puerto Rican dudes with tattoos, piercings, dyed black and blue hair. That's exactly the kind of look 'those girls' go for." She stopped strumming, looked thoughtful for a moment, then tried the chords again, playing a variation on the initial sequence.
"Maybe sporty girls are his secret vice," John-Michael suggested. He hoped not. A horrible cliché for a pretty hot guy. Even if Ruben was straight, he could at least not be _that_ straight.
"More secrets," Lucy said, half to herself, listening to the chords. "Just what we need."
"Not you," John-Michael said absently, examining the strings of his instrument. "You're the last person that needs another secret."
Lucy stopped playing. "Meaning what, exactly?"
"You know," John-Michael said. He tried the first few notes of the sequence she'd shown him. " _Jelly and Pie_ , rehab, Tyson Drew."
There was a painful silence. John-Michael stopped playing and looked up, gauging her reaction.
Eventually, Lucy seemed to find her voice. "Tyson Drew?"
"Yeah," murmured John-Michael, turning red as he realized what he'd said.
It wasn't easy to stay calm, but Lucy was clearly trying. "John-Michael, what are you talking about? _What_ about Tyson Drew?"
He struggled for words for a few seconds and then finally exploded, "Why'd you never tell me, Lucy? Why?"
Lucy mouth fell open. "Tell you what?"
John-Michael faltered slightly. He'd been meaning to avoid this, but still, it had slipped out. Now that it was out there, he couldn't resist digging further. "That you were _there_ , Lucy. That you were at the party where Tyson Drew was murdered? All those conversations we had at rock camp. All the things _I_ told you about _my_ life. About my mom dying. About my dad and his skank-girlfriends. About coming out, about the first time I kissed a guy. But you—you told me hardly anything."
"I'm not big on sharing, JM," Lucy mumbled. "You know that."
"I do, I totally get that, but it can't all go one way. Then a few weeks back, when I told you Grace's dad was on death row for Tyson Drew's murder. Even then, not a word. You didn't mention that you were at the party where that guy's murder took place? That it was the reason you got dropped from your TV show? Dude, how could you not tell me about that?"
They were close enough now that their guitars were almost touching. Lucy seemed to be struggling to process her emotions. When she finally spoke, it was obvious that _anger_ had won.
_"_ _How did you know?_ _"_ she said, her voice raised. "What, are you stalking me online now?" Lucy faced him with sudden clarity. "You couldn't have known, back in rock camp. Tell me _you_ haven't been keeping this from me, all these years."
Her eyes flashed with danger. John-Michael immediately shrunk back. "Of course I didn't cyberstalk you," he said resentfully. He pulled away, removed the guitar from around his neck, returned it to its case, which was leaning up against the wall. After a moment, Lucy did the same with her Telecaster.
Lucy popped the Telecaster case shut. "So where's this coming from?" she fired back. "'Cause I know it's new."
A shadow of guilt crossed John-Michael's face. "Grace told me."
"Grace?"
"I told you we visited her dad in San Quentin." He was indignant. "Did you really think Grace wouldn't find out, of all people? She's known about you since she was little. Her dad's on death row for Tyson Drew's murder—she's been keeping that in for a long time. You can't blame Grace for finally cracking and telling _one lousy person_. Not everyone can be an Easter Island statue like you."
Lucy visibly reeled. "Grace's father . . . told you I was at the party where Tyson Drew was killed?"
John-Michael's feelings of guilt vanished in a wave of righteous anger. "Grace's father is Alex Vesper, Lucy. He's on death row for the murder of Tyson Drew. But he didn't do it. And now they've set a date for his execution—July fourteenth."
Lucy sucked in a long, deep breath, trying to steady herself. "Wow. Grace is handling it awfully well, I'll give her that."
"It's not like she has much choice," he snapped. "Since she's not allowed to tell us about it."
"How long have you known about the execution, John-Michael?"
"Since we went to pick out a couch at IKEA—that's when Grace found out."
"Looks like I'm not the only one keeping secrets from my best friend," she finished scornfully. "Not only that but you've known for a _month_ that I was at the party where Tyson Drew was killed."
"A month, Luce. I kept quiet for a month. _You_ kept quiet about this for _years_. I didn't tell you I knew 'cause I wanted to give you the chance to say something on your own."
"Oh, I get it," she said, now indignant. "You were testing me." Lucy backed toward one of the two chairs in the small rehearsal space. Finding the back of one with an outstretched hand, she slid into it. "Grace has been living with me this whole time . . . and she never said a word to me about all this."
"You're overreacting. She respects your privacy. Grace hasn't even told Candace. Not yet."
A hint of a smile touched Lucy's otherwise tense features. " _Candace_ doesn't know? Her own _sister_? Huh. I'd buy a ticket to be there when that particular piece of information leaks out."
"Lucy," he said earnestly, "Grace has been waiting all these months, hoping you'll give her some tiny bit of hope, some clue that maybe you remember what happened that night. She won't ask you herself. I don't understand why. If Vesper was my dad I'd be doing everything I could to get you to talk."
" _Get me to talk_?" She looked at him with disgust. "JM, you seriously think that I would let an innocent man be executed, if I knew something that could help him?"
John-Michael looked crushed. "You really didn't see anything that night?"
"I saw Tyson Drew's dead body in the pool," she admitted hesitantly. "But other people saw that, too. And—oh, John-Michael . . ." For the first time, he noticed a hint of shame in Lucy's voice. "I _used_ so much, in the years that came after," she added in a dejected voice. "I had dreams, I had hallucinations. That stuff, it messes with your mind, twists memories. It warps them until you don't know what's real anymore."
Lucy went silent, so John-Michael continued in a more conciliatory tone, "Grace's dad said that when he left the party he saw the silhouette of a little kid on the balcony that overlooks the pool. One of the kids at the house that night had to have seen something."
"There were four kids at the house that night," Lucy said. "Why do you assume it was me?" Then, as though the thought was only just occurring to her, Lucy asked, "Did . . . did Grace have something to do with you asking me to move in? _Did they use you to get to me_?"
John-Michael was stunned. "Who's 'they'? Candace's mom found Paolo, _he_ found me, remember? There was no way to predict Paolo __ would know that I knew you. That's insane."
Lucy's attitude puzzled him. For a moment it had seemed as though she might be on the brink of admitting something and now she was back to being confused. John-Michael wanted to help Grace, wanted to get these secrets out in the open where they wouldn't fester. In the end, though, it wasn't his business, as Lucy seemed eager to point out.
The _Death Note_ anime theme broke the silence. John-Michael's ringtone. He turned away from Lucy as he looked at the new text on his phone.
**Dude, how much more obvious do I have to be? Make an excuse. Leave Lucy to arrange some music. Wanna get a burger or something?**
Amazed, John-Michael texted back, **Ruben?**
**Who else? Now get over here, I miss you.**
John-Michael returned his cell phone to his pocket, mind racing. Ruben wanted _him_? This was huge. He hadn't even suspected because he thought Ruben was into Lucy, but maybe Ruben was bi? Although in retrospect it was starting to make sense. Lucy did nothing but throw out hints that she liked Ruben and he all but ignored them.
"I . . . Uh, I gotta bounce," John-Michael mumbled. "We were done anyway, right?" He hoped so. Ruben had brightened up the day about a thousand percent. The situation with him, Lucy, and Grace would have to be put on hold.
Lucy just nodded. She looked unhappy, seemed to be mulling over her thoughts. "Yeah, I think we're definitely done for today."
## **GRACE**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, WEDNESDAY, JULY 1
"I'm just saying—what's the point busting in on her? Can't it wait?"
Grace heard the tension in John-Michael's voice on the staircase outside their bedroom.
The door to the bedroom Grace shared flew open. From her desk at the foot of her bed, she turned slowly in her chair until she was facing Lucy, who stood framed in the doorway.
Lucy looked angry. Instinctively, Grace knew what she was going to say.
"John-Michael told you about my dad," Grace began softly, with compassion. It was a sincere expression of how she felt. After months of living with Lucy, Grace had come to accept that Lucy was probably not _consciously_ hiding the truth. For whatever reason, it seemed that Lucy didn't even understand what she'd seen that night, eight years ago.
John-Michael arrived, slightly out of breath, behind Lucy.
"I did tell Lucy that your dad is on death row for the murder of Tyson Drew," John-Michael admitted. "But Grace, you two have to start talking about it. I know you don't want to be reminded but, seriously? Time is not on your side, here. It's insane, keeping all this quiet. What if Lucy does know something?"
His words brought instant tears to Grace's eyes. But she quickly wiped them away when she heard a third set of footsteps on the spiral staircase. Both Lucy and John-Michael turned to see who it was.
"Ariana, would you mind? This is kind of private," Lucy said.
There was a faintly embarrassed pause. Grace dried tear-covered fingertips on her shirt as Ariana stood awkwardly behind Lucy and John-Michael, apparently sizing up the situation. "Hey, no problem," Ariana said. "I heard raised voices. I thought maybe something was wrong. Totally don't want to butt in on your business. I'm just gonna finish up in the kitchen and head out for some air." Ariana hesitated a moment before turning for the stairs.
"Thank you, Ari," Lucy called after her friend. She sounded more than a little embarrassed. That embarrassment continued as she turned back to Grace. The anger had all but vanished.
"I have to ask," Lucy said. "How long have _you_ known that I was at the party?"
Grace took a calming breath. "I recognized your name from the moment I heard you were moving in with us. Then John-Michael told us that your dad was in the government. 'Lucy Long' isn't a million miles away from 'Lucasta Jordan-Long,' the child actress. I looked you up on the internet and—bingo. Found out about _Jelly and Pie_. Saw the names of the other kids who were at the house that night."
Lucy glanced at John-Michael. "Could you give us a few minutes?"
"Actually, Lucy, would you mind?" interjected Grace. "I'd prefer it if John-Michael sat in. It's kind of a delicate discussion."
John-Michael hovered close to his own bed, as if waiting for the girls to agree whether he should stay or go. Lucy was clearly taken aback but she didn't object. "If I'd seen anything, Grace, I'd have told."
Grace didn't reply for a minute. When she did, she spoke carefully, avoiding any accusation in her tone. "I know that if you could have talked, you would have. I'm just . . . I'm just wondering," she continued, "in fact I always have . . . if maybe someone got to you. Tried to shut you up. When you were a little kid?"
Lucy shifted her weight from one foot to the other. Warily, she asked, "Got to me—how?"
John-Michael turned to face Lucy, his eyes registering anxiety.
Grace struggled to contain the emotions that were bubbling up inside. She had to persuade Lucy to tell them everything she could remember. Even an apparently stupid detail might be crucial. John-Michael was right—they were running out of time to act. "There are lots of ways to get to a kid. Think about it Lucy, you were little. It was late, you were half asleep. Whatever you saw, maybe you buried the memory. Maybe someone _made_ you bury it."
"Like who?"
Grace replied simply, "I don't know." She looked over at John-Michael, who had finally settled uneasily onto the edge of his own bed. He was regarding Lucy with a mixture of curiosity and pensiveness.
Lucy paused. "But if I don't remember anything?"
"There are ways to help you remember," Grace said cautiously. She didn't want to frighten Lucy off, didn't want her to realize just how much thought she'd put into this over the last few months. "Hypnosis. Recovered memory therapy."
"That actually works?" John-Michael sounded hopeful.
Grace didn't take her eyes off Lucy, who seemed to be considering their words. "It can."
Eventually, Lucy asked Grace, "And you think it would be enough to get your dad off the hook? If I remember something under hypnosis?"
"Maybe," she replied. "The evidence that convicted him was circumstantial. That's why he got all the stays of execution. Each time it was like, another chance to find some better evidence for the defense. So far, they haven't found it. He had a motive, he was at the party, and he doesn't have an alibi. They connected his DNA to Tyson Drew's body, but my dad says it's because they got into a fight earlier in the evening over something stupid like money. But Lucy—my dad was nowhere near the pool that night. He got up to use the bathroom and he saw someone little up on the balcony, looking down. He couldn't see what the kid was looking at, but that balcony overlooks the pool. That little kid saw the murderer, Lucy. My dad is sure of it."
They all shared a lengthy pause. Then John-Michael said, "Why're you so sure it's Lucy?"
"Think about how you behaved afterward, Lucy," Grace said gently. "You went into a tailspin. That didn't happen to any of the other kids at the party. You're the only one who acted like they'd been through some kind of trauma."
"That's not true, Tyger Watanabe was messed up, too."
"Maybe later, when he hit fourteen, fifteen, but right after the murder, Tyger kept doing the show for another season, whereas you, they dropped. You said it was because you were acting out, Lucy. In one of your interviews, _you_ said it."
To Grace's immense relief, Lucy didn't deny it.
"I . . ." Lucy seemed to be struggling to get the words out. "I have wondered. From time to time. Recently, maybe more." She seemed reluctant to continue.
"Please, Lucy."
Lucy was about to speak and then stopped. She tilted her head toward the door. Grace listened for a moment, but heard nothing.
Lucy stood, went to the door, and opened it wider. Grace watched her push open the bathroom door, too.
"What's up?"
Lucy turned back, closing the bedroom door behind her. "Coulda sworn I heard someone on the landing outside."
"I didn't hear anything," John-Michael said, shrugging.
"Gotta say, this whole conversation is making me a little jumpy," Lucy said, sitting down next to John-Michael. He shifted toward his pillows, making space.
Impatiently, Grace got out of the chair and sat between the two of them. She placed a tentative hand on Lucy's bare arm, suddenly painfully aware of how frosty Lucy could appear.
"I know it can't be easy for you to think about that night. If you really did see Tyson Drew being murdered, that would have been a seriously traumatic moment. Just to see it, I mean. I remember the only time I saw a guy throw a real, honest-to-goodness punch at another guy. It happened right in front of me, outside school. And I was physically shaking, Lucy. I can only imagine what it could do to a kid to watch a grown man strangled and drowned. It's no wonder your mind closed it down."
"I . . . Grace. I really don't know if I wanna . . ."
"All I'm asking is that you try. Lucy, please. If there's something there, something your mind has been trying to suppress . . . it could seriously help my father."
"It's not going to overturn the conviction, though," Lucy said doubtfully. "I mean, Grace, I'm real sorry that your dad's in that situation. But I'm not the one who put him there. The jury won't convict him without evidence."
"It's all circumstantial, though!" Grace cried, rising to her feet.
John-Michael half rose, as if to steady her in case she toppled over. He put a soothing hand on hers, saying, "Gracie, it's okay . . ."
"It's circumstantial evidence," she reiterated, this time calmly. "A high-profile case—the cops were under a lot of pressure to get a conviction. And the district attorney did a great job with the jury, I have to say. I've looked at the trial records and even though I'm no lawyer, Lucy, I can see that."
John-Michael stared curiously. "You've looked at the trial records?"
"I'm an expert on the Tyson Drew murder case," Grace said. "My dad taught me that."
"If you're such an expert," Lucy began insistently, "then I guess you must have some idea of who did it."
Grace allowed herself a tiny nod.
Lucy pressed on. "Who?"
But all Grace could say was, "I can't tell you. I'm sorry, but that might seem like I was leading you. If you're going to try the recovered-memory therapy, you need to go into it with no preconceptions. It won't help us if you and I discuss the case beforehand." She tipped her head toward John-Michael. "That's why I asked him to stay. John-Michael's a witness that you and I never talked about who you might have seen."
"But you have a specific person in mind?" Lucy asked.
Grace turned to John-Michael. "Have I ever said anything?"
He shook his head, emphatic. "No way." To Grace, he said, "Dude, I had no idea you had any clue who might have committed the murder."
Lucy sat on her hands now, rocking back and forth very slightly on the edge of the bed. It was like watching her regress from a confident sophomore to an anxious child.
"Is it dangerous, this memory-therapy thing?"
Grace said, "It's a kind of hypnosis."
Lucy shook her head. "Not what I asked."
Grace found herself releasing a shaky breath. "I don't know for sure, Lucy. It's kind of like letting out some buried demon, I guess. Not literally, of course. But once that memory is out . . . well."
"It's probably buried for a reason," John-Michael said. Grace could hear a slight strain in his voice. "You don't know what kind of crap you're going to dig up. Stirring and stirring. It's sure to release a stink."
"John-Michael's right," Grace said a tad cautiously. "Uncovering a repressed memory can be traumatic. Lucy, I'm begging you. This is the fourth time they've set a date for his execution. The _fourth_. His lawyer will appeal—she always does. It might even work out, again. Even if it does, the fear—it's sapping his strength. Being in prison has changed him. If this goes on much longer, I think he might give in, just to end it all. And then I'd lose my father, forever."
"A bad odor, I can handle," Lucy said. But she sounded reluctant.
Grace felt hope like a pressure against her chest. "So you'll try it?"
Lucy nodded, once. "Okay."
## **CANDACE**
#### LEO CARRILLO BEACH, WEDNESDAY, JULY 1
The cast and crew of _Prepped_ had been at Leo Carrillo beach all afternoon. Most had left around five, along with Ricardo Adams. Only three actors stuck around to shoot the final two night scenes.
Candace had elected to spend the day in Malibu to minimize on driving. She'd met her mother for lunch at her home on Malibu Beach. Later, she'd agreed to return there to spend the night.
One of the final three, Candace sighed as she watched the crew setting up. "Why bother to shoot at night if they're gonna light us up like a Christmas tree?"
Out of the corner of her eye, she saw Yoandy beaming at her comment. She'd watched him wander off to smoke a cigarette, and followed him. Now they leaned against the rocks at one end of the beach, waiting while the crew finished prepping the scene. At least five minutes had passed. So far, hers was the only comment that either had made. Yoandy seemed comfortably silent. Meanwhile, Candace was itching with tension and half-formed sentences that never made it to her lips.
She closed her eyes for a second and then jumped in. "Should we talk about this kiss?"
Yoandy didn't turn his head, but she heard a smile in his voice. "You think maybe we should rehearse?"
"He'll probably wind up choosing the first take anyway," Candace grumbled. "Directors usually do. But they like to have the choice. And the feeling of power."
"I would like very much to kiss you right here, _nena_ ," he said warmly. "But it's Sebastian's first time kissing Annika. Maybe better if we wait. Try to capture some of that 'first time' energy on camera. What do you think?"
Candace said nothing, feeling her skin buzz. He wanted their first kiss to be recorded.
"And then after we kiss, you hit me," Yoandy added cheerily.
"What?"
"You didn't see the new pages?"
"Ah, no!" Candace looked toward the director, feeling somewhat thrown.
"Oh!" Yoandy turned to her, finally engaging with energetic enthusiasm. "It's a little different from before. We start in the water, we head for the beach. Once we clear the water, count two beats. Then you fall. I grab you when you fall, kiss you, you hit me, and we keep running."
"I fall? I _hit_ you?"
"It's dark. There are lots of rocks and seaweed."
"A kiss . . . and then I pop you one." Candace let the statement settle, solidify into a concrete truth. "Why?"
"I guess you don't like being kissed!" he said with a shrug.
"Yeah, that'll do it," Candace said disdainfully. "I guess 'feisty' is what passes for character on this show."
If it were up to Candace, she'd have written a kiss that Annika initiated, not Sebastian.
"Tension!" Yoandy said, smiling widely. His hand steered near her waist but stopped just short of touching her. Candace couldn't help but feel a pulse of excitement—within minutes she'd be in his arms. "The director wants it to be a will-they-get-together kind of thing," Yoandy told her. "We can practice that, if you like."
"I should look at the new pages," she mumbled.
"Or you could trust me?" he offered, looking hopeful.
Candace glanced at the director. She'd already gotten a couple of dark looks from him when she'd caused delays. It wasn't worth the risk—but she _did_ trust Yoandy. She'd just have to wing it.
Soon the director yelled for Candace and Yoandy to move into position. Yoandy began to step over the rocks. He held a hand out to her. After a moment's hesitation, Candace took it. They shared a tender smile, both surprisingly nervous.
A minute later they were both thigh-deep in the chilly waters of the Pacific. The tide was stronger than she'd expected. Fronds of knotted seaweed brushed against her legs and hands.
Candace peered toward the beach. But with the dazzling lights streaming directly at them, it was impossible to see any of the crew. Through a megaphone she heard the director yell, "Action!"
Candace ran headlong toward the sand. Beside her, Yoandy sloshed ahead. When they reached the beach, she counted to two. She stumbled and fell. Yoandy reached out and yanked her to her feet. He pulled her close. In the last second before their bodies touched, he slowed it down. He stroked strands of hair from her cheek.
Candace stared deep into his eyes, which were meltingly deep, chocolate brown, before he closed them and leaned in for a kiss.
Then his lips were on hers. They tasted of salt. His hands tightened on her waist. She could feel the tip of his tongue, trying to coax her lips farther apart. She wanted to relax into his arms, to open her mouth to him. But she didn't dare, not for a second. Each movement was deliberate. She pulled her right hand back. She formed a fist. She swung it toward his head. Yoandy yanked his head back as though he'd been struck. His left hand shot up to his cheek. He stared at her in shock and disappointment.
Candace felt an immediate surge of regret.
"Cut!"
Yoandy held back for a moment, and then grinned, delighted. "Pretty good, _nena_! I barely got out of the way in time!"
"Go again," called the director. "This time, Candace, let _him_ kiss _you_. And could you look less upset about hitting him? We need more anger, more gravitas. Remember the virus has reversed the aging progress. You're an experienced woman in the body of a teenage girl. Before the apocalypse, you were a contender for the Nobel Prize. Sebastian is just some schlub whose ass you already had to save from that gambling den. He's not worthy of you. 'Kay?"
They waded back into their water to their first positions. They played the scene again. This time Candace's lips were closed when he kissed her. The director called a halt partway through. "Go again," he said grimly.
They played the scene a third time. This time Yoandy didn't pull his head back fast enough and her fist actually made contact with his jaw. Ironically, though, it appeared to be a much weaker punch. So they went again.
By the time the director was happy, they'd recorded another five takes. Soaking wet and with the cold onset of the night sea breeze, all Candace could think about was when she could kiss Yoandy again. His lips were soft, his touch warm against her cold ocean-kissed skin—but the most potent memory was the fleeting sensation of being pressed to his chest. She'd felt the heat of his body, even through the soaking-wet shirt. He made her feel sexier than she could ever remember feeling. From the way he was looking at her, Candace could see that Yoandy was eager for more, too.
"You wanna go dancing sometime?" he asked as they toweled their hair and shoulders dry.
"Latin dancing? I—ah—I don't know how."
"Don't worry about it," he said, obviously amused by her uncharacteristic skittishness. "I will be very, very gentle. You won't even notice it's a lesson. How about Saturday night?"
"The Fourth?" Candace stalled. "You're not with your family?"
Yoandy shrugged. "Sure, we can go to the family if you prefer. There'll be dancing there, also. Dancing is everywhere when you're with me." He smiled.
"I think maybe my housemates have plans," she said, making a decision. "But you know what? Sure, okay, I'll come out with you. But not right now. My mom, she, ah, she's picking me up soon." Candace finished with a wan smile. She hated to disappoint him.
But Yoandy didn't appear disappointed at all. _"Excelente."_
She looked at him expectantly until Yoandy finally chuckled and leaned closer, reaching for her waist with both hands. "Which kiss did you like best?" he whispered, close to her ear.
"This one," she murmured, turning her face to his so that their mouths met. She sucked gently on his lower lip before releasing him with a tiny nip of her teeth against his chin.
_"Ay, mi madre,"_ he groaned as Candace pushed him away, smirking. "You're gonna kill me, _preciosa_."
"Don't worry," she replied. "I'll be very, very gentle."
## **JOHN-MICHAEL**
#### SANTA MONICA, THURSDAY, JULY 2
"Lucy, let's go back to when you were nine years old."
"I was Charlie back then," murmured Lucy. "On _Jelly and Pie_."
In a plush, Santa Monica consulting room of the psychiatrist, Dr. Barney Kessler, John-Michael watched the therapist remove a small, dark gray electronic device from the breast pocket of his sports coat. He touched a finger to the tiny screen and then placed it carefully on the desk to record the session. He pushed it along with the tips of his fingers until it was closer to Lucy, who sat in a leather easy chair, half reclined. The cream-colored damask curtains were drawn, reducing the midday glare from the oceanfront to a mild, cool glow. From compact speakers on the nearby birch wood bookcase, faint music played. It took John-Michael a few moments to realize that the music was the rather whimsical theme from _Jelly and Pie_. Nervously, he allowed his eyes to settle on Lucy. Her eyelids were half closed and she swayed ever so slightly in her chair, as if she were on the deck of a boat. It was the first time he'd ever seen someone hypnotized. The whole thing creeped him out a little.
"Did you go to a party on Mulholland Drive?"
"Yes," Lucy said. "Where Tyson Drew was killed." She sounded weary, as though having to relate the story for the thousandth time. John-Michael noticed that Dr. Kessler made a note before moving on.
"Tell me about that night, Lucy."
There was a long silence. Then, very softly, Lucy said, "I don't want to."
The therapist didn't react. "Can you tell me who you were with at the party?"
"Tyger was there. Alexis, Marc."
"That's great, Lucy. So your costars from _Jelly and Pie_ were there. Can you remember anyone else you saw at that party?"
"I guess I saw Tyson Drew," she said resentfully. "He was dead."
"Where did you see him?" asked Kessler. He used a calm, kind voice.
"In a swimming pool."
"Was anyone with him?"
"Huh?"
John-Michael watched Lucy's face closely. She sounded lost, like a little girl clutching a teddy bear. Her eyes moved rapidly under eyelids that were shut tight. There was a very long silence. Kessler didn't move, his expression didn't change.
"Lucy, was anyone near the pool when you saw Tyson Drew?"
When she spoke Lucy's voice sounded hoarse. "It's a bad, _bad_ dream."
"That's fine, Lucy," Kessler said. He set down his yellow pad, folded his hands across his lap. "It's okay to tell me about the dream."
"I'm not supposed to," Lucy blurted. The force of the comment jerked John-Michael upright.
Kessler just nodded. "You can tell me, Lucy. I'm here to help you. Your friend John-Michael is here, too. You're safe."
"Bad girls tell tales," Lucy said, moaning. "Bad girls make up stories from their dreams."
Finally, Kessler began to react. He leaned forward, toward Lucy. "Did someone tell you that?"
"It's a secret," was all Lucy would say. John-Michael could see the stress in Lucy's entire body. Her hands gripped the edge of the chair so tight he could see the bones through Lucy's skin.
"Who told you not to tell?"
Lucy made a tiny, whimpering sound. "I'm not bad, miss."
"Did 'miss' tell you not to talk about the party?"
Fretfully, Lucy nodded, eyes still firmly closed.
Kessler's gaze strayed for just a second, catching John-Michael's eye. He held up a single finger to his lips. John-Michael nodded in response.
"Secrets can be like a prison, Lucy. But you're the one with the key. You can let yourself out of the prison, any time."
He waited for a response, but Lucy made none. Kessler continued, "Would it be possible for me to talk to _Charlie_? Charlie from _Jelly and Pie_?"
Lucy mumbled something incoherent.
"Maybe _Lucy_ didn't see anything at the party," Kessler said. "Maybe Charlie is allowed to talk about what she saw?"
"Charlie don't care about her," Lucy said abruptly. John-Michael immediately recognized the slightly exaggerated accent. It was working. Lucy had slipped into her character from _Jelly and Pie_.
"She dug her nails into his neck. Pretty nails, all shiny. Like peaches. On fire. His eyes bugged out when it happened. He tried to get away, but she had something around his neck—a tie. I saw her shoe on his shoulder, holding him under. It took him a _long_ time to die. Bobbing up and down for air, but he wasn't getting any."
John-Michael couldn't move. Kessler let Lucy's words resonate in the following silence. "Did you recognize the lady you saw, Charlie? Do you know her name?"
"Pretty lady, nice shoes."
Kessler kept his voice level. "Charlie, can you tell me her name?"
John-Michael realized that he was holding his breath. Grace was outside in the waiting room, probably nervous as hell, praying to hear the words that might finally spell hope for her father.
There was a tiny sob in Lucy's reply. "Lady doesn't want anyone to know."
The therapist said, "Did she threaten you?"
"Bad girls tell tales."
"But I'm asking Charlie."
"Charlie's bad," Lucy said suddenly, adding in a singsong voice, "bad, bad, bad."
"The lady's name, Charlie. Can you remember?"
"Dana said I shouldn't tell," Lucy said, practically whispering. "I was crying 'cause I wet my pants. She cleaned me, put me to bed. She sang to me, gave me a bottle of nail polish from her purse when she saw how much I liked the color of her nails. Dana said it was a bad dream, is all."
Kessler hesitated. "Where did Dana find you?"
"Around," replied Lucy in a small voice.
"Did you like Dana?"
Lucy's eyes opened slowly. She turned and stared at John-Michael as though he were a complete stranger.
"Lucy, this is Dr. Kessler," said the therapist in a voice of sudden authority. "Lucy—are you awake?"
Lucy's expression didn't change for a moment. "I remember. Damn, John-Michael. I _remember_."
She rose, unsteadily, to her feet, catching the arm that John-Michael offered for support. When he felt the shake in her muscles, he pulled her close.
"It's okay, Lucy," he breathed against her neck. "It's gonna be okay. Let's go out and . . . and tell Grace."
"Grace!" sobbed Lucy, tightening her grip around his neck. She sounded distraught. "What am I gonna tell her? What have I done?"
"You didn't do anything wrong, Luce, you didn't know." John-Michael tilted her chin up so that he could look into her eyes. "I'm with you, all right? We'll tell Gracie together."
He steered her toward the waiting room, with a final glance at Dr. Kessler, who stood aside to let them pass. "Come see me in a few days," he said. "You need some aftercare."
Grace stood up as they entered the waiting room. Her arms were by her sides, both hands clenched tightly. Eyes wide with hope, she said, "And . . . ?"
John-Michael gave a quick nod.
Grace could barely speak. "Lucy, did you remember something?"
"I remembered," Lucy said shortly, her voice suddenly curt. "There was someone else there. That woman from the _Macbeth_ movie, Dana Alexander."
"Did she . . . ?" Grace stopped, placed a hand just above her breastbone, as if to steady her breath. "Did Dana see who did it?"
Lucy made the tiniest suggestion of a shrug. "Maybe. Or maybe . . . maybe she even did it."
Grace gasped. Tears sprung to her eyes almost immediately. Gulping now, she asked, "Do you think my dad . . . ?"
"Can I get your dad free?" Lucy said, almost snapping. "Who knows? Maybe. Can we go now?"
Grace hesitated. "Lucy, are you mad at me?"
"Of course not," Lucy replied, too quickly. "It's just . . ." John-Michael edged around so that he could look at her. Her sudden shift in mood had taken him by surprise, too. "I'm not angry," she managed to say after a couple of deep breaths. "But this . . . this is a shock."
"We understand that, of course," Grace said. John-Michael echoed the sentiment with a vague mumble and then asked Lucy, "You wanna get a taxi?"
Lucy just shook her head. She pulled the door open and stepped out onto Santa Monica Boulevard. "I'll walk. I need to think."
Grace grabbed Lucy's arm. Lucy stopped walking and stared at Grace's hand with a kind of appalled inertia. "Lucy, you're in shock. And I know it's kind of my fault. Do you want to talk about it?"
Lucy's lips moved in silence for a moment, as though she were rehearsing a line. Then: "Talk about it? No. This is messed up, Grace."
Anger threatened to explode within Grace. "In two weeks my father is going to be put to death, Lucy, executed for what that woman did," she blurted. "You've got to tell the cops what you remember. You have to!"
"Hey, hey," John-Michael said, getting between them. "Cool it, Gracie, I know you're upset, too."
"What I remember?" Lucy stared at Grace, ignoring John-Michael. "It's all flashes. Nothing that will convince the cops that Dana Alexander killed him. I _need_ to think about this."
"You think _Dana Alexander_ killed Tyson Drew?" Grace asked.
"Of course," Lucy fired back. "She's the one who made me keep quiet about it."
"Whoa," John-Michael said. "That's going to be tough to prove. An A-list actress like Dana Alexander—not exactly an easy takedown."
"You don't need to convince the cops that she actually did it," Grace pleaded, gripping Lucy's arm even tighter as her friend tried to pull away. John-Michael sidelined a couple of passersby who'd noticed their altercation, made sure they gave the two girls a wide berth as they walked past. "You just need to plant reasonable doubt that my father _didn't_ do it," Grace continued in a lower voice.
Lucy said forcefully, "There is no way that Alexander is going down for this, Grace. She's too smart. Too powerful."
"She wouldn't even have to actually be convicted. Even the suspicion would be enough to set my father free."
"Listen to me," John-Michael interrupted reasonably. "It looks like Dana Alexander was only too eager to suppress Lucy's testimony from the very beginning. Maybe she knows that if they start up the murder investigation all over again, they might find something."
"But that was eight years ago," Lucy said. "If she cared so much about keeping me quiet, why haven't I heard _word one_ from the woman since that night?"
"Too risky?" John-Michael suggested.
Grace became quiet, thoughtful. "On the other hand, it would have made sense to keep tabs on you, somehow."
Lucy became very still. "Tabs? Like, how?"
"Like, I don't know . . ." Grace shrugged. "Maybe make friends with someone _close_ to you. Your parents, maybe? Get them to keep her posted, in case you started to talk."
Lucy's expression flattened. The energy seemed to be visibly draining from her frame. "Oh no."
John-Michael steadied her with a touch to her shoulder. "What is it?"
Lucy stared from Grace to John-Michael beseechingly. She began to shake her head. "It can't be. She wouldn't do that to me. We were rehab buddies! She wouldn't betray me."
"Lucy, who are you talking about?" John-Michael asked.
"No, not Ariana." Lucy shook her head, confused.
John-Michael took in Grace's expression, watched her go from stunned to bitter resolution. Grace took two breaths in quick succession. She put both hands on Lucy's shoulders.
"If it's Ariana, Lucy, then you have to accept . . ."
"No," Lucy said in a low moan, her voice trembling. "Not Ari . . . She knows everything, Gracie. About Charlie, about my dream, what I saw that night when Tyson Drew was killed; Ariana knows everything!"
"And she was listening on the landing," Grace said, her voice clear as cut glass in the warm, balmy air. "When we were talking yesterday."
"She probably even heard you talking about getting the regression therapy," John-Michael pointed out. From the sudden gasp that Lucy gave then, he realized how shattering this had to be.
"Then she's already warned Dana Alexander. And now she's at home, waiting for me to come back and spill the beans," Lucy snarled, bitterness dripping from every word. "Well, those two lying witches are gonna deal with _me_ now."
"And me," Grace said.
"And me," John-Michael added fiercely. "Whatever happens now, Luce, we've got your back."
## **LUCY**
#### KITCHEN, VENICE BEACH HOUSE, THURSDAY, JULY 2
John-Michael, Grace, and Lucy returned from Santa Monica and retreated to the second-floor bedroom to discuss their next move. Paolo and Candace had gone for groceries, and Maya was running errands with her aunt Marilu, so they were less likely to be disturbed. Only Ariana was in the house, cooking. They left the door wide open this time, so that they'd easily see if someone approached.
"Better do it quick," Grace had advised, "like pulling off a Band-Aid."
The trouble was, Ariana was stuck to Lucy a whole lot harder than a Band-Aid. The years they'd shared. The tears, the pain. Things they'd told each other that no one else knew. And yet through it all, Lucy realized that she'd never truly considered Ariana to be a _friend_. The relationship was too complicated for that. Friends shared tiny intimacies, meaningless banter, food, philosophical conversations, the quotidian _nothing_ of life. Lucy and Ariana shared little but their sadness, and had done so from the very beginning.
The second that Grace had said the words, _with someone close to you_ , it had been as though a blindfold had been lifted from Lucy's eyes. Ariana. So simple, yet so unthinkable. What kind of person would intrude on the privacy of a damaged teenager? What kind of monster would use another damaged kid to accomplish that?
Her rational mind tried to brush away the cold, hollow feeling that had warned Lucy that Ariana was just too perfect. The girl had appeared in Lucy's life at just the right time. What did people do in rehab? They talked. It was the new confessional.
It flashed through Lucy's mind that maybe Ariana wasn't actually damaged. That maybe it had all been an act. But the memories from their Claremont days piled in pretty quickly to dismiss that thought. Ariana passed out drunk at her hostel. Ariana calling Lucy to pick her up from outside a bar where she'd fallen down, half blind from whatever drug she'd taken. Hours at her side, watching her go cold turkey. No actor would be so committed.
Lucy couldn't figure out how Ariana could possibly know someone like Dana Alexander. Ariana came from Louisiana, where as far as Lucy could make out, she'd been a nobody. It struck Lucy that she hadn't bothered to find out many concrete details about Ariana's background. The accent sounded genuine, and Ariana had some airs and graces that matched her story of faded elegance, a once well-to-do family fallen on poverty. But nothing that could connect Ariana to a Hollywood movie star from England. Lucy just couldn't see it.
This was the detail that almost persuaded Lucy that she was wrong. After she and John-Michael returned from the psychiatrist's office in Santa Monica, Lucy couldn't stop mulling it over. Maybe they were jumping to conclusions?
But then Lucy recalled, with a stab of anguish, how she'd told Ariana about the dream that wasn't. They'd talked about it recently, right here in the house, in Lucy's room. But now that Lucy _really_ thought about it, she remembered that it hadn't been the first time she'd mentioned that dream to Ariana. The subject had come up several times over the years. And each time, Lucy realized, Ariana had soothed her, calmed her, reassured Lucy that it was just a dream.
And probably gone straight back to Alexander with the news.
_Lucy still thinks it was all a dream._
Ariana had to go, and now, before Lucy's mind could be changed.
"There's not one person in the house who won't be glad to see her go," Grace had admitted when Lucy first mentioned it. "You didn't like her . . . ?" Lucy said, bemused. "Why didn't you say so?"
"She was your friend. But now that we know she's not I'll be honest—she's a creeper for sure," Grace said. "The way she hovers, always trying to listen in on conversations, not really contributing."
"Ariana can be shy in groups," Lucy said. Defending Ariana was a reflex, even now. "Like Maya. Remember how quiet Maya was, too, at first?"
Reflex or not, Lucy had to override her own impulse to look out for Ariana. No one could ask her to leave but Lucy. It had to be done, and it had to be done today.
"We're behind you, Lucy, whatever you decide," John-Michael told her. Lucy went directly downstairs and into the kitchen, where Ariana was chopping onions for dinner. "Ariana," Lucy began, her stomach lurching. "We have to talk."
Ariana placed the knife neatly to one side of the chopping board. "What's up, hon?"
"Ari, I'm really sorry. But it's been over three weeks. You said you'd only be staying here for a few days. And, the thing is, we're not really allowed to have more than six of us here. The insurance, Candace's mom. We've been skating around issues here, but it can't go on."
To Lucy's astonishment, Ariana began to smile. It was a sad smile, filled with compassion.
"Your friends don't like me."
"This isn't about you. It's about how the house is set up."
Ariana gave a slight shake of her head. Her smile had become ironic, verging on cynical. "I tried real hard to get them to like me, Lucy, but they're not easy. You ever get the feeling that something messed up is going on with all of them?"
Lucy hesitated. She didn't like the sound of where this was going.
But Ariana was hitting her stride. With her back to the kitchen sideboard, she tossed the knife aside. Nervously, Lucy watched it clatter toward the sink.
"Candace, she's totally focused on her career. She'll want this whole place for herself soon enough. Wait and see. Paolo, I don't even know what that boy's deal is. John-Michael, well, okay, he's a sweetheart but what he did to his car? No normal guy drives a Mercedes-Benz off a cliff. Take my word for it; he's got a screw loose."
"Ari," Lucy began, half smiling in utter astonishment at the tirade that was flowing from Ariana's mouth.
"Maya's a nice kid, I got nothing against her. I mean, who wouldn't want to get rich from some dumb piece of software?"
"Are you done?" said Lucy. Their smiles had vanished now.
"Almost," Ariana said with a touch of vehemence. She walked around the dining table until she was standing half a yard away, staring Lucy right in the eye. "Grace is the one, isn't she? She's the one who told you to kick me out."
"She's not—"
Ariana interrupted, musing. "Grace—so nice and so quiet. She wants to be the housemother. Girl got nothing else goin' for her. She's not talented or brilliant, like the rest of you. Not tortured and angsty, like you, or John-Michael. Just plain, ordinary, dull Grace. And I threaten her. It's as simple as that."
"Grace is none of those things," came a voice from behind Lucy. She turned to see Candace standing in the open door. Paolo stood beside her, and behind them both, Grace and John-Michael. Together they presented a firm, resolute front.
"Grace is amazing," John-Michael said, pushing forward until he was shoulder to shoulder with Lucy. "I wouldn't have got through some of the worse parts of my life without her."
"It's true," Paolo said. "Grace is awesome."
Lucy's eyes met Grace's. She was trembling, very slightly, but looked right back. Lucy turned slowly to Ariana. "They're right. Grace is the one who binds us together. And no one asked me to get you to leave, Ari. But you can bet your ass that they want you to go now."
"My bag," Ariana muttered through gritted teeth, "is always packed."
They watched, uncomfortably silent, as Ariana made a point of flinging the freshly chopped onions into the sink, then flounced off into the living room. She reached for the suitcase that was tucked neatly in the space between the futon and the new three-seater sofa. Within five minutes she was ready to go.
Lucy pulled away slowly. Out of the corner of her eye, through the French windows to the rear of the living room and beyond the small backyard, she glimpsed a white Cadillac. Maya's aunt was pulling into a parking space behind the house.
Ariana dropped the suitcase to the floor. She faced Lucy, seemed to consider for a moment, and then, almost reluctantly, she put one arm around Lucy's neck.
With her mouth close to Lucy's ear, Ariana whispered, "Watch out for these people, Lucy. They don't know you the way I do."
## **JOHN-MICHAEL**
#### KITCHEN, VENICE BEACH HOUSE, THURSDAY, JULY 2
"You're going to the movies?" John-Michael said. He watched Lucy wrap the cheese she'd been slicing and put it back in the fridge. She arranged a few slices on a plate around some cherry tomatoes and began to eat, leaning against the kitchen sideboard.
"Are you sure?" he said. "Wouldn't you prefer to hang out with us? Although, now that I think about it, Maya probably won't want to do anything—she's totally obsessed by her app these days. I don't know when Paolo will get back from the tennis club. And I'm kind of going out . . ."
John-Michael's voice trailed off. He hadn't mentioned yet to Lucy that Ruben and he might be kind of seeing each other. He probably should.
Lucy chewed for a few seconds, then shrugged, a little sadly. "Today has been intense. I think I need to be out of the house, out of my head."
"Want me to go with you?" John-Michael offered. "I'd be happy to break my date."
The beginnings of a smile appeared in Lucy's eyes. "A date?"
"Ruben." He felt a blush steal over his neck and cheeks.
The smile vanished. "Ruben is gay?"
"Bi," corrected John-Michael.
Lucy put down her plate, eyes downcast; she wouldn't look at him.
"You're upset," he said sympathetically. "I'm sorry—Ruben said you might be."
She shook her head for a moment, blinking back tears, and then tried to smile. "I should have guessed. Now that I think about it, he was always asking me to get you to play guitar again. It was all an excuse."
"It may have been," John-Michael said nervously. "I'm not that good."
Tentatively, she asked. "Have you guys . . . you know?"
John-Michael grinned, more confident this time. "Hey, none of your beeswax, Long!"
She crossed her arms. "No fair stealing a bandmate's guy, Weller."
"Well, technically, he was never yours . . ."
"Seriously, how long has this been a thing?"
John-Michael picked a cherry tomato from Lucy's plate and popped it into his mouth. "A day. After we rehearsed yesterday, at school, we went for burgers and shakes, then made out on the beach."
"Ugh, that's so high school."
"I know!" John-Michael said happily. "It was awesome. Like being sixteen again. Ah, sweet sixteen!"
"Sweet sixteen?" Candace said, sailing into the kitchen with an empty plate, which she placed in the sink. "Who's sweet sixteen?"
"John-Michael kissed a boy on the beach," Lucy said, pouting.
Candace said, "Oooh, spill. I want details."
"Ruben, the drummer from Whatnot. They met at Grace's benefit concert."
"That band you were in for all of five minutes?" Candace asked, taking a dining table chair and turning it to face them. "Is it still going?"
Lucy shrugged. "I guess they found a replacement for me."
"So we both like _los latinos_ , eh, John-Michael?" Candace said, nudging him.
"Is that actually happening then?" John-Michael asked, a little surprised. "You and Yoandy Santiago?"
Candace beamed. "Maybe."
"But—doesn't he have a girlfriend?"
Candace rolled her eyes. "No! You can't believe the gossip. He's music-famous and she's reality-TV famous, so they get the paparazzi treatment once in a while."
"He actually told you he's not with Kay Alexander?" insisted John-Michael.
"Yes," Candace said, annoyed. "Yoandy doesn't have a girlfriend. Yet," she added with a wink.
Shaking his head, John-Michael said, "I hope you're right. I don't want to see you getting hurt."
The amusement dropped from Candace's face. "Man, you sure changed your tune."
"Dude, I am _one hundred percent_ Team Yoandace. Just be careful. Those stories about him dating Kay gotta come from somewhere."
"Anyway . . ." Lucy interrupted, "I should go. I'm meeting Luisito. We're seeing _The Death of_ _Caesar._ "
"The one based on the Shakespeare play? Oh man, I totally want to see that," John-Michael said. "The guy who plays Antony is brutally hot."
"Me too," Candace said. "We did _Julius Caesar_ in my youth theater company. 'Friends, Romans, countrymen, lend me your ears.'"
The kitchen went silent as Grace strolled in, her head on her chest, arms lightly folded in front of her. She looked up, first at John-Michael, then at Lucy with an awkward "hey." Finally, she turned to Candace and reached out to touch her sister's shoulder. "Hey, Candace? We need to talk."
Candace did a comic half frown, glancing at John-Michael before she looked back at Grace. "Sounds serious. You breaking up with me, sis?"
Lucy shot him a look that confirmed John-Michael's guess. Grace was about to tell Candace the truth about her father. Hurriedly, he took Lucy's plate and added it to the pile in the sink. "That's our cue to leave."
"No," Grace said in a very clear voice. "No. Lucy, John-Michael, I'd like you to be here for this. I need help explaining everything."
Lucy's expression turned sickly, as though she would prefer to be just about anyplace else. In fascination, John-Michael watched Grace turn to her stepsister and take her hand. "I've been keeping something from you, Candace. It wasn't my idea to do it, but anyway, I did."
Candace gave an embarrassed laugh. "Jeez, Gracie, what's going on?"
Grace took a long breath and released it through pursed lips. "Wow. This isn't easy."
Candace tugged at her hand and gently pulled her stepsister toward another chair. "Sit. Okay, sis, hit me with it. Whatever it is, we'll be cool. Unless you're Yoandy's other secret crush. Then we definitely are not and you, my dear, are an onion-eyed wretch."
John-Michael smiled to himself. Despite her joking, Candace was a loyal sister.
Grace took another breath. "So you know how you've never met my dad, Alex? Well, his name isn't really Alex Grant. He doesn't live in Canada. His real name is Alex Vesper." She paused for a few moments, then in a quieter voice added, "Candace, do you know who that is?"
Candace shook her head. She looked baffled. "Should I? Although the name sounds familiar."
"He's the man they convicted for the murder of Tyson Drew." Grace paused and took a deep breath. John-Michael could see tears welling up in her eyes. "Candace, my dad's on death row."
Candace clasped a hand to her mouth, then looked first at Lucy, then John-Michael. "Did you guys know?"
Lucy nodded gravely. "Yeah," she whispered. "Grace told me a little while ago. John-Michael too. I . . ." She hesitated and settled a meaningful look on Grace before continuing. "I was at the party where Tyson Drew was murdered. I was just a kid, but I think I may have seen something. Something that could maybe get the conviction overturned."
Grace's eyes shone with the hint of tears, her voice shook, but only slightly. "My dad is innocent, Candace, I know it. I'm absolutely certain. But the evidence just didn't fall the right way. They've set a date for his execution. It's happened before. This time, it's in about two weeks."
"Oh, Grace, no, I'm so sorry, baby!" Suddenly, Candace was pulling her stepsister into her arms and squeezing, her head pressed to Grace's shoulder, her eyes tightly shut. "You poor thing," she said. "That's so horrible!" She pulled away. "Lucy, why haven't you testified yet, if you know something?"
Lucy sighed wearily. "It's complicated."
"It really is," said Grace. "Please, Candace, Lucy's been through a lot. But she's going to do the right thing now, aren't you, Lucy?"
Lucy nodded. "Yeah," she said quietly. "I am."
Wonderingly, Candace shook her head. "All these years . . . Why didn't you tell me? I would have been there for you. Did you ever visit him? I'd have gone with you. Oh!" Her eyes widened as realization struck. "Dead Man Walking!" For a moment she sounded almost joyful. "He's your dad! Although—God—that totally sucks for you. Oh, I'm sorry!" Once again, Candace fell on Grace's neck and hugged her stepsister.
John-Michael nodded at Lucy. "I think maybe we should go."
"How come John-Michael and Lucy knew before me?" Candace pulled away, puzzled. For the first time, a note of hurt entered her voice. "And your mom and my dad. They—you—were all lying to me all those years?" Now the tears were shining in Candace's eyes as well. "Why—? I mean, Gracie, didn't you trust me?"
"Yeah, we should definitely go," John-Michael said. He reached for Lucy's hand.
As they left the kitchen, he heard Grace saying, "Of course we trust you, Candace. Please—let me explain?"
## **CANDACE**
#### CULVER STUDIOS, FRIDAY, JULY 3
By 7:30 a.m. Candace was strolling along the beach path as the joggers and dog walkers enjoyed the soft, milky light of Venice Beach in the morning. She watched them as she made her way to her Prius for another early start. As Candace pulled out her keys, she heard a car approach. A white Cadillac pulled into the parking spot next to hers.
"Hey, Marilu." Candace smiled as Maya's aunt emerged.
_"Hola_ , _linda,"_ Marilu replied affectionately. She looked almost exactly like Maya. Light, olive-colored complexion, five five, probably a size ten rather than a six or eight like Maya, but proportionately very similar. The aunt's hair, like Maya's, was a glossy, dark chestnut brown and worn straight, down past the shoulders. She was dressed in smart indigo jeans, unremarkable black heels, and a black blazer over a fitted white blouse. Her look was basic yet professional. Her eyes were hidden behind black Ray-Bans.
Suddenly, Candace felt like something was off. It wasn't déjà vu but a very distinct feeling that she'd seen Marilu before, but in a completely different context.
"Maya's still in bed," Candace volunteered.
Aunt Marilu's jaw hardened for a moment. She lifted her face briefly toward the house. "Really? But I'm taking her to the airport. Today is her big lunch in Napa with the investor."
"Maya's got a meeting in Napa?" Candace had heard something about this from Grace, but no details. The truth was, she hadn't completely believed everything Maya had been saying about her app recently. Maya would get to chatting, breathlessly, using techy language Candace didn't understand and didn't care to have explained, or else she spent her time buried in a stack of geek books and tapping away on her laptop. "She was up super-late, working on her code, I guess."
"Aha," Marilu said. "I guess I better go wake her."
"Okay, well, so long." Candace got into her Prius for the leisurely drive to the studio. At least that was one advantage of early-morning starts—less traffic. Her thoughts turned to her last kiss with Yoandy, and Candace daydreamed about their promised first date the following day.
When Candace arrived at the Culver City lot, Ricardo Adams was buying coffee from a cart outside the studio. As she stepped out of her car, he headed in her direction. He stopped in front of Candace before she made it to the studio doors _._ Ricardo looked tense, possibly even angry. She waited for him to speak first.
Ricardo cleared his throat. "You and Yoandy. It's got to stop."
Candace gasped. "What . . . business is it of yours?"
"It's a family affair. His girlfriend is my wife's sister. Or have you forgotten about Kay?"
"What? Yoandy says they're just friends."
Ricardo's upper lip was drawn back in a cynical sneer. "You believe that?"
Candace gulped but stood her ground. "Yes."
"Kay is _very_ important to my wife," Ricardo said, his voice lowered, the tone darker. "And you of all people should know better than to bite the hand that feeds you."
"Bite the hand . . . ? What the hell are you talking about?"
He cocked his head to one side. "Who'd you think suggested you for the part of Annika?"
This was getting a little surreal. "It wasn't you? Or Yoandy?"
"It was Dana. She saw your pilot, that _Downtowners_ thing. She's the one who put me in touch with you."
Candace didn't know what to say. Grace had mentioned Lucy's suspicions, but they'd both agreed that it was an odd coincidence that she'd ended up working on a production with the British woman's husband. But to find out that Alexander had actually picked her? That was creepy. "Oh."
"Oh? Is that all?"
"I'm, ah, um, grateful," she replied. "O-obviously."
If not for the connection to Lucy, Candace would have been excited to know that a movie star of Dana Alexander's stature even knew she existed. But that the woman had been the one to suggest her for a big-break TV role? And kept quiet about it? That was a total surprise. Not at all the way Hollywood people usually behaved from what Candace knew. But then again, Dana Alexander wasn't the kind of person who'd be reluctant to call in a favor, now that she wanted one.
"Kay and Yoandy are together," Ricardo said firmly. "You're not here to meddle; you're here to work. So work."
As Candace watched him walk away, she realized that her skin was tingling, as though she'd been slapped. Just the same, she made herself stroll confidently into the studio, a little way behind Ricardo. When she spotted Yoandy at the breakfast buffet, she turned away, hoping that he hadn't noticed her arrive. Ricardo had reprimanded her as though she'd been having an illicit affair with Yoandy. The only explanation she could imagine was that Kay must be lying about her relationship with Yoandy—talking it up. Even though this meant that Ricardo's accusation was way off base, she felt its sting.
She managed to avoid Yoandy until lunchtime—they had no scenes together that day and she needed some time to figure out how she felt about Ricardo's proclamation. But as she halfheartedly picked at her gluten-free Pad Thai, she saw him approaching her. Casting around for a possible escape route, Candace found none. When she realized that their encounter was inevitable, she straightened up, steeled herself. After all, maybe he'd had the same warning.
Close up, though, she saw that his eyes were full of gentle concern. "Candace, what's happened? You look worried."
Nervously, she glanced over his shoulder. Ricardo was in plain sight, over by the main set. He was looking right at them. "We can't," she whispered. Miserably, she closed her eyes. "And I can't go on a date with you tomorrow."
"Really? I thought you were excited."
"I was," Candace started. Yoandy looked crestfallen but Candace was fully aware of Ricardo's eyes burning into her. "But it was a bad idea. It just has to be this way."
"Give me one reason why it's a bad idea."
"Kay," said Candace, so quietly that it was practically a hiss. "Ricardo keeps saying that Kay Alexander is your girlfriend."
Now he looked seriously upset. "Candace, Kay is not my girlfriend. I'm telling you the truth! Kay . . . she's a little unusual. We had a few dates, and that's all. Now she's texting me and calling me . . . inviting me. And Ricardo is a friend of my family, you know that. So I'm not gonna say no, am I? Okay, maybe Kay still thinks we have something but I haven't replied, I haven't called her back, and we _never_ agreed to date."
"Did you call her _nena linda_ , too?" Candace said, pursing her lips.
Yoandy pulled back, frowning. "Are you kidding?"
Candace held up a hand. "I'm done."
"Please," he murmured reasonably. "At least let's talk. Come to my dressing room after we finish? Please, Candace?" He reached for her left hand, but she snatched it away, left him looking forlorn. "If you don't come, then I'll know you don't want me. And I promise to leave you alone."
She looked past Yoandy's dejected eyes and saw Ricardo smirking at her. She was giving a convincing performance.
"Look, I'm sorry," she said, backing away. "I've got to go."
Candace didn't know if she should believe Ricardo or Yoandy. But in that moment, she didn't care whether Kay Alexander was Yoandy's girlfriend or some kind of delusional dater. She only knew that Yoandy had been forbidden. And in that moment, nothing and no one had ever seemed quite so irresistible.
In the absence of any scenes playing opposite Yoandy, Candace found herself daydreaming about the two of them together. She'd run through every detail: where he'd be standing when she came into his dressing room, what he'd say (not much), where he'd make her stand, where he'd put his hands, the slight rasp of his stubble on her lower lip, how soft his mouth would feel against her own. Candace knew she would remember almost nothing of what they recorded that day.
At the end of the day, she made her way to Yoandy's dressing room, where she found him buttoning closed a crisp, white linen shirt.
She hesitated, standing in the doorway watching him. But instead of approaching her, touching her, kissing her, Yoandy kept his distance. He regarded her with a tight, pained expression.
"Candace. Why didn't you mention that you're underage?"
For a moment, Candace balked. It was the last thing she'd expected to hear him say. "That? I'll be eighteen in five months. You're not so much older," she said, finally approaching him. "And anyway, mister, what makes you so sure you're getting any, ever?" She finished with a gentle pinch of his cheek.
To her surprise, when she looked up at his face she saw utter sorrow.
"Ricardo and me," Yoandy began, "we talked."
"Oh, you did, did you?" she said caustically. "What did that tattletale have to say?"
"He warned me. He was very reasonable, actually. Said that I had to let Kay down nice and gentle—even if she's the one who's misunderstood our relationship. He reminded me that you're still in high school, that technically you're still a minor, that it could start a scandal, because I'm more than three years older."
"With you, he's reasonable," she observed. "With me, he makes threats."
" _Amor_ , listen." He tried to touch her arm, but held off when she flinched. "The show's about to go on summer hiatus—we'll get a natural break from each other. Kay will have time to understand the situation. Maybe it's better," he said, looking thoughtful. "Until after your eighteenth birthday."
"Yeah," she said, defiant. "By then, I might like someone else."
Yoandy gave a nod and crossed his arms, gazing at her with amused speculation. "It's possible. I'm pretty ugly."
"Yeah, y'are," she said with a hint of a smile. "But I'm not all that particular."
" _Nena_ , don't be cruel," he said teasingly, pouting. "It's my father's fault I'm ugly."
Candace couldn't resist chuckling at this. "But you inherited his musical talent, so there's that."
Yoandy laughed, too. Then, as if he'd only just remembered, he stuck a hand into his jeans pocket and withdrew something that he kept hidden inside a tight fist. He took Candace's left hand and carefully unfurled her fingers, tenderly placing in the center of her palm a coiled necklace made from tiny yellow and ocher beads. "It's for you. From El Cobre, in Cuba; the colors of Our Lady of Charity."
When she accepted it without hesitation, he visibly relaxed. Barely brushing the skin of her cheek with his lips, Yoandy kissed her. "It means we're for each other. I don't give this necklace to just any girl, Candace."
Candace was a little overwhelmed at this. She closed her hand around the beads as she took a moment to recover, trying to appear nonplussed.
"I like you like this," he whispered, bringing his mouth to the curve of her shoulder.
Suppressing a gasp of pleasure, Candace managed to say, "Oh sure, you like it when I'm speechless?"
She felt the flutter of his lips against her skin as he chuckled. Despite herself, she felt arms snaking around his neck, bringing him closer.
"Not speechless," he protested. _"Impressed."_
"You're so full of it," she murmured, bringing her lips against his, shutting him up.
Candace's heart was still racing when she left the corridor leading from Yoandy's dressing room. Across the soundstage she sensed Ricardo's eyes on her, baleful and suspicious, but she ignored him and headed straight for the parking lot. What Candace needed right now was some time alone on the beach, or maybe ice cream with her sister.
As Candace strode to her Prius, a majestic white Cadillac had just slowed to a standstill outside the studio. She recognized it immediately. Not only the car, but the driver. With one elbow poked casually out the window, it was quite clearly Maya's aunt. And in the front passenger seat, dressed in an elegant, short-sleeved white dress with a single vertical red and black stripe, was Dana Alexander.
Maya's aunt worked for Dana Alexander.
Why had Maya never mentioned this?
## **PAOLO**
#### BALCONY, VENICE BEACH HOUSE, FRIDAY, JULY 3
It was a baking-hot day, the temperature still rising at eleven thirty, the sun a harsh diamond in a sky devoid of even a wisp of cloud. Beyond the pale expanse of sand, the sea was a mirror. On the balcony of the beach house, Paolo, Lucy, and John-Michael were eating a late, lazy breakfast. Candace and Maya had both left before Paolo had woken; Candace for the studio and Maya for her meeting in Napa.
" _You're_ getting a tattoo, Mr. King?" Lucy hadn't bothered to hide her skepticism. "Who's doing it, Luisito's buddy?"
Paolo nodded while munching his toast.
"He does good work," Lucy acknowledged. "Although if they find out he's tattooing under-eighteens they'll have to fire him, so keep it zipped afterward, okay?"
"What about shopping for the Fourth of July?" John-Michael interrupted, looking up from his bowl of Cinnamon Grahams.
"Plus, Paolo, I distinctly remember you promised to clean up the yard," Lucy reminded him. "It's a mess, ever since that goddamn party. Ariana could've done something about that," she observed tartly. "But then, I guess she wouldn't have been able to spy on us inside the house."
"Ariana was _spying_ on us?" Paolo said, startled. The spike in his pulse astonished him. Just the idea that anyone might be interested in his secrets was kind of horrifying. " _That's_ why you kicked her out yesterday? When were you gonna mention that?"
Paolo had completely assumed that she had finally seen sense and asked a long-overdue houseguest to move on. He'd have done it a week ago, at least. The house was cramped for six tenants, let alone seven.
"Oh, I doubt _you_ even made it onto Ariana's radar," Lucy said. Her eyes narrowed briefly. "Although, dude . . . kind of a guilty look you got on your face right now."
Paolo scowled. "I value my privacy, like anyone else. Why would anyone want to spy on us, of all people?"
He watched John-Michael and Lucy exchange a wary look. "How long d'you have?" Lucy groaned. "Look, basically, the girl was in the house to keep an eye on me. She knew stuff about me, from rehab. I saw something when I was a kid—well, _may_ have seen something."
"Seen what? Sounds serious," Paolo said. The relief at finding out that this couldn't involve him was instant, and led to immediate curiosity about what it did involve.
"I may have seen who really killed Tyson Drew," Lucy admitted.
As the words settled into him, Paolo glared at them both. It was obvious that John-Michael knew exactly what Lucy was talking about. "You saw who really killed Tyson Drew? Man, you have to tell the police."
"And it wasn't Grace's father," Lucy added with a guilty tilt of her head.
"Grace's _father_ . . . ?" Paolo said, struggling with the implications of yet another bombshell. "Since when was he even involved?"
"You should maybe ask her," Lucy said.
Grace appeared at the top of the staircase, startling Paolo. He hadn't heard her climbing the stairs. In a fragile voice she said, "Ask me what?"
Paolo could only stare, feeling a little stupid to be caught gossiping about something so serious.
"You should go with him to get the tattoo, Grace," Lucy said with a careful glance at Paolo. A signal. "Our boy Mr. Disney Channel here is getting his lily-white skin inked—that's not something you see every day. I'd come too but I'm doing _that thing_ I promised you I would do."
"Yeah, Grace, go with," John-Michael chimed in, a little too quickly. "Paolo, leave me your keys and I'll do the shopping."
Paolo took his car keys from his pocket and tossed them over to John-Michael. "Don't forget the fireworks," he mumbled. "It's not the Fourth without something going up in smoke."
Lucy and John-Michael either wanted Grace out of the house—which didn't make any kind of sense. Or else, John-Michael, like Lucy, was in on the "secret" that Grace had a thing for Paolo, and both were trying to make something happen.
Paolo watched as Grace came to an apparently reluctant decision. "Okay," she said. "If you're sure you don't want my help with _the thing_ , Lucy."
"I'm fine," Lucy came back. "Like I said, I'll get to it."
As Paolo and Grace made their way along the boardwalk, Grace seemed distracted. He snuck a glance at her face a few times as they walked and it looked like she was in another world. Not what he'd have expected, if she really did like him.
"What's the thing that Lucy has to do today?" Paolo asked to break the tension.
"Oh, something she promised to do," Grace replied.
Paolo added, "Does it have anything to do with your father?"
He waited for her to reply, but nothing.
_Great. For a girl who is supposed to be into me, Grace seems pretty reluctant to share._
"You know," she mused. "I don't get why you're getting a tattoo. You want to be a lawyer, or a tennis pro? A pro with tattoos, I can see, but a lawyer?"
"I _am_ a tennis pro," Paolo said. "The question is, do I always want to make my living being a tennis pro? Or do I want to be a lawyer?"
She stopped walking and faced him. "Well, do you?"
He shrugged. "Be a lawyer? Yeah, I guess so."
"Then why get a tattoo? Is it for Lucy?" Grace shielded her eyes as she watched him, then shook her head. This time there was no doubt about it—disappointment.
She couldn't meet his gaze. "I know you like Lucy," she began.
She still wouldn't answer his question! He was beginning to feel bad for asking about her dad. Clearly, the subject was too painful to address.
Her voice was small and very slightly broken and he wanted to put both arms around her, in that moment, and tell her that no, he didn't like Lucy so much, not now anyway. But what did he have to offer Grace? It was too early to know how he felt about her. Paolo had a strong sense that with this girl, he couldn't afford to make such a misstep.
"I do like Lucy," Paolo said as he led Grace to the edge of the boardwalk, where they found a place to sit. There was no point in lying. Grace deserved the truth. Might as well engage on the only subject she seemed to want to talk about. "But Lucy, she doesn't like me, anyway, not like that. Nothing happened between us. It's the truth."
Finally, Grace looked up at him. It could have been his imagination but Paolo thought he saw a glisten of moisture in her eyes. "You're over her?"
"Totally," he said softly.
"Then why're you getting the tattoo?"
Surprised, Paolo said, "You really think that's about Lucy?"
"She likes body art, piercing, all that punk stuff. I've heard her say you'd look good that way."
"It's not for Lucy," he told her firmly. He really didn't care how Lucy felt about his tattoo. The thought struck him with a forceful energy, like a ray of sunshine breaking through mist. It wasn't for any girl.
Paolo hadn't analyzed the whole tattoo thing as deeply as Grace. Just that he'd seen the designs at the parlors on the boardwalk and over the past few weeks he'd started to get this feeling, like his skin was calling out to be marked, to be different.
"Then why?" Grace said. He could feel her eyes taking him in, roving across every exposed inch of his upper body. Like the gentlest of breezes, it sent a faint shiver through him.
Grace had a point; why was he about to get his flesh bruised and stabbed, hundreds of times, maybe even a thousand?
"I want . . ." he replied, slowly constructing his answer, "to be different. Not from everyone else—from myself."
She considered this for a moment. "You won't be different. It's not what's on the outside that matters, Paolo. I thought you would understand that."
_But I'm not the same_ , he wanted to say. _That's the lie of all_ __ _this smooth, unmarked skin. It makes me look the same as I was, no different. And I'm not. Inside, I'm different. Meredith did that to me. I did that to me._
"I'm . . . I'm not the same person I was six months ago," Paolo ventured cautiously. He wanted to be honest with Grace, he really did, but there were things that he couldn't ever share. Not with any woman in his life. Not with anyone. He shouldn't even have told John-Michael.
Unconsciously, Paolo rubbed his still-untouched left bicep with the thumb and forefinger of his right hand. Sometimes it felt like his guilt had to be written on his face, like everyone knew how he'd left Meredith on that road. Well, now it would be written on his body—for anyone who could understand his code.
"Paolo," Grace said with difficulty. This time, there were definitely tears. "None of us are the same person as six months ago."
Tentatively, he dared to ask what he was dying to know. "Gracie. Is there something going on with your dad?"
"He's . . . my father is . . ." Grace was obviously having a hard time spitting this out. She couldn't meet his eyes.
Paolo took her hand in his lightly. He felt his heart shudder as he waited for her next words. "Gracie, please. Tell me."
"Paolo," she said, looking up with difficulty, "my father . . . is on death row. And they've set a date for his execution. In less than two weeks, they're gonna execute him."
Paolo was still for several seconds as it all came crashing together. Their drive up to San Quentin. Grace's letters to Dead Man Walking. How upset she'd seemed after she and John-Michael had returned from the second trip to the prison. "Grace," he said breathlessly, opening his arms. "I'm so sorry."
He gathered her close then and just held her against his chest, soothingly caressing the soft skin of her shoulders with his thumbs. Grace was quiet, not crying exactly. She released only a muffled sob that quickly abated. But through her shirt, he could feel the thudding of her heart.
## **GRACE**
#### TRIPLE BEDROOM, VENICE BEACH HOUSE, FRIDAY, JULY 3
"And then we kissed," Candace said. She was grinning as she took a seat on Grace's bed, forcing her stepsister to move over to make space.
"Who kissed you?" Grace inquired absently. She sat up, a little put out to be interrupted by Candace entering her quiet room. She put down her copy of _Drown_ , which she'd been reading since she and Paolo had returned from getting the tattoo, several hours ago. She checked her watch. It was almost five in the afternoon.
Candace rolled her eyes, demanding attention. "Yoandy."
Just the same, Grace barely heard the reply. The day was almost over, and still Lucy hadn't returned from her visit to the police station. That's assuming she'd even gone at all. Grace wouldn't have been surprised if Lucy needed more time to think—the hypnosis had clearly been a traumatic experience. Since her father had been jailed, Grace had been forced to learn significant patience. Part of her understood that one more day couldn't make much difference to a legal process, at least not two weeks from the execution date.
But another part of Grace had been reawakened by a newfound hope. By the evidence—finally—that someone else might know what really happened that night at the party on Mulholland Drive. That side of Grace's personality was suffering an ordeal. Every extra minute of uncertainty felt like agony.
What would she do if Lucy hadn't gone to the police? Grace wasn't sure she'd be able to contain her frustration and rage.
She glanced up at her stepsister, trying to focus on what Candace was saying. "You kissed . . . who again?"
Candace looked faintly annoyed. "Urgh. Yoandy Santiago! Try to listen. We were at the TV studio, earlier on."
"And then what?"
"Not much," conceded Candace. She sounded frustrated. "He's gotten cautious."
"Uh-huh," Grace said, trying to sound interested. "Why's that?"
"I'm gonna take a wild guess that it was the fear of having Kay Alexander shredding his clothes maybe. Scissors supplied by her big sister, Dana."
Grace put down her book slowly. The mention of "Dana" sent a prickle along her spine. "Why . . . would she do that?"
"Who can say?" Candace said. "God only knows what Kay is like." She sighed. "For some reason, the woman thinks that she is Yoandy's girlfriend."
"Candace," Grace said unsteadily. "Are we talking about Dana _Alexander_?"
"Yeah," agreed Candace. "Lady Macbeth is a big deal, it turns out. Which is why I gotta be careful—real, real careful that Dana doesn't find out. Apparently, _Dana_ is the one who recommended me for the part of Annika. Now it sounds like she thinks I owe her, big-time. She seems to think I'm gonna take Yoandy from her sister, Kay." She gave a crafty grin. "I tell you, owing Dana—honestly? I don't see it."
Grace couldn't speak for a few moments. Then, struggling to remain calm, she said, "You . . . are connected to _Dana Alexander_ , the British movie star? _She's_ the one who got you into _Preppe_ _d_?"
"I know," Candace said, now thoughtful. "I thought it was odd, too, especially after what you told me yesterday about Lucy kicking out our delightful ol' houseguest, not to mention the whole Tyson Drew thing." As though an idea had just occurred to her, she added, "Hey, did _you_ know that Maya's aunt works for Dana Alexander?"
This additional piece of information hit Grace like a slap in the face. "Who—Aunt Marilu?"
"That'd be the one."
"No," Grace murmured. "I did not know that."
Unreal. Impossible. How deeply had Dana Alexander insinuated herself into their lives?
Dana's sister was dating a guy who Candace also liked. She'd recommended Candace for a part on a new TV show. Dana Alexander might be the true killer of Tyson Drew.
And now—it turned out that Maya's aunt actually worked for the woman.
"This is too much," Grace mumbled. "It's—no. This is too many coincidences."
"It really is," mused Candace, "I mean, Hollywood can be kind of incestuous and all, but . . ."
She stopped abruptly when the door flew open and Maya came bounding into the room, throwing her messenger bag onto her bed and flopping down next to it.
"Guess what?" Maya said, her eyes bright with excitement. "Jack and I met with Alexa Nyborg today, up in Napa."
But Candace just groaned. "Give me an actual break! Maya, could you knock first? We're trying to have a private conversation here."
"It's my room, too," Maya said with surprising levity. "If you're having a private conversation about me, maybe include _me_?"
"You clearly don't understand the rules of gossip," Candace said with pronounced irony.
Grace waited for a moment. "And? How did the meeting go?"
Maya drew herself up even straighter, her bare knees pressed almost daintily together against the edge of the quilt on her bed. " _And_ she made me an offer."
Both Candace and Grace now faced Maya, managing to keep their sighs of irritation to a minimum. After a moment Candace said impatiently, "Come on, Maya, let's hear it!"
Maya opened her messenger bag, removed the laptop, a sphinxlike grin playing about her lips. As the MacBook started up, she glanced at them in turn and said, "Alexa has this amazing little _casita_ up in Napa. It's all mission-style, white adobe walls and arches. The garden is full of tangerine and lemon trees, pink and white hibiscus flowers everywhere and the most beautiful swimming pool I've ever seen."
"Sounds great," Candace said with more than a hint of eye roll. "But what did she actually offer you?"
"I'm telling you about the _casita_ ," Maya said, her smile ever more mysterious, "because that's part of what she offered me. Alexa hardly ever goes there. These days she's usually flying somewhere every weekend. She said I could stay there whenever I like. Even offered to let me stay this weekend—July Fourth. She was planning to fly out by private jet right after our meeting 'cause she's going to be in Washington at some event with the president."
"She offered to loan you a house in the Napa Valley?" Candace said, her mouth falling open.
Maya grinned, typed something quickly, and said demurely, "I said no, thanks. After all, we're having a barbecue and fireworks here, right? Just the six of us. But Nyborg was actually pretty insistent. Said if I change my mind, the key is with a neighbor."
Candace responded with an amazed grin.
_There is something very badly wrong in this house_ , Grace thought as Maya prattled on about her app. Too many sudden connections to Dana Alexander, to a person who she now realized was not only getting information through Ariana, but who had found a way to keep tabs on the housemates via Maya and Candace, too.
Why?
Grace could understand why Alexander might want to spy on Lucy—a buried memory that threatened to expose the woman's secret, possibly even that she was a murderer.
But Maya? Candace? What possible threat could they be?
A cold, crawling sensation ran through Grace, as though an ice-encrusted spider was walking along her spine. Lucy wasn't back from talking to the cops yet; she'd ignored the two, hopeful texts that Grace had sent: **Hey just checking in. How's it going? Any news?**
Technically, no news was good news, yet Grace's instincts told her it wasn't good. She felt her chest muscles clench around her ribs, and wondered fleetingly where Paolo had gone. His embrace had been such a comfort to her earlier today. If he was here, she'd feel safer. Grace didn't feel safe around Maya and Candace, not right now. Not until she understood their connection to Dana Alexander.
Maya continued to chat: Alexa Nyborg _this_ and Napa _that . . ._ until Grace simply couldn't tolerate it any longer.
"That's just great, _Maya_ ," retorted Grace, and her tone was so uncharacteristically sharp that even Candace flinched in response. She paused, waiting until she had Maya's complete attention. "But let me ask you this: why didn't you ever mention that your aunt was Dana Alexander's driver?"
Maya simply froze for several seconds. Guilt was written all over her face: guilt and shame and regret.
## **LUCY**
#### LIVING ROOM, VENICE BEACH HOUSE, FRIDAY, JULY 3
Lucy didn't return from the police precinct until almost a little after five in the evening. Barely across the threshold to the house, she stood absolutely still, listening. Just before she'd arrived, she'd heard the front door slamming shut as someone entered the downstairs part of the house. As she turned her key in the lock, she heard voices toward the back of the house, presumably in the yard. Paolo and, if Lucy wasn't mistaken, Grace. Lucy's hand stilled. Grace was the very last person she wanted to face right now. She extracted her key as silently as possible and turned toward the spiral staircase that led to the second floor.
As she approached the landing, she could hear Maya, Grace, and Candace inside the triple bedroom. The name "Dana Alexander" was clearly mentioned, at least twice. Lucy approached quietly, hoping to eavesdrop awhile.
Maya was speaking rapidly, sounded agitated. "Could you stop asking me about Dana Alexander, already? I told you—she's my mom's boss, _that's it._ I don't have anything to do with her, never met her, never so much as talked to her on the phone."
There was a lengthy, tense silence. Then Candace said tentatively, "Your _mom's_ boss? Don't you mean your _aunt's_? Doesn't your mom live in Mexico City?"
Lucy had heard enough. She pushed open the door. "What's going on? Why're you giving Maya the third degree?"
But Candace's eyes were fixed on Maya. She looked utterly baffled. "Maya, who _are_ you?"
Maya moved to leave. Candace held up a hand. "Not so fast. Maya, you need to answer. Is Marilu your _mom_?"
From behind Lucy came the sounds of footsteps on the stairs. She took a quick look. Paolo was on his way up, too.
"Did I hear correctly?" Paolo said. "Did Candace just say that Maya's aunt is actually her mom?"
Lucy glanced at Grace. She seemed paralyzed with tension, suddenly pale and wan. "Did you go to the police?" Grace asked her, coming straight to the point.
Lucy swallowed. "I tried, G. Really, I did."
Grace whimpered, touched a palm to her head. Paolo responded immediately. He sat down beside her, wrapped an arm around her waist, and drew her close. "Lucy," whispered Grace.
All eyes were now on Lucy. "Where's John-Michael?" she asked, suddenly defensive. The room felt pretty hostile. At least John-Michael might understand.
Paolo said coldly, "He's in the kitchen."
"Look, can we all talk about this downstairs?" Lucy said, pleading.
"I still want to know why Maya's been lying to us about her mom being her aunt!" Candace objected.
"Seriously?" Paolo retorted. "You actually think that's more important than Lucy going to the cops with evidence that might get Grace's father off death row?"
"Obviously not," Candace said hotly. "I'm just sayin' that Maya isn't getting off the hook so easy. Especially since it turns out that her _mom_ works for a total Hollywood player who's been meddling in my life."
To Lucy's astonishment, Grace made her way over to Maya, and said gently, "Come on, Maya. I'm sure you had a good reason for telling us Marilu was your aunt, but you need to trust us now. Please. There are things going on in this house that you're not aware of. And Dana Alexander is at the center of too many of them." Grace cleared her throat. When she spoke again, her voice sounded shaky. "Lucy's right. We should go downstairs, where we can all talk this through."
Without another word, the housemates trooped downstairs. In the kitchen, in front of a stunned John-Michael, they arranged themselves around the dining table, and each took a chair. The atmosphere had grown as suddenly chilly as the air outside, now that the clouds obscured the setting sun.
"Um . . . what the heck?" John-Michael said, placing the large kitchen knife back on the chopping board next to a heap of sliced carrots and celery.
"Lucy's got something to tell us," Paolo said with a reassuring glance at Grace.
"Maya too," Candace added.
Lucy couldn't quite put a finger on it, but she sensed guardedness in almost everyone in the room. Or maybe she was just projecting her own feelings? Her instinct had been to protect Maya from the sisters' questioning. Some people had private stuff going on. Why did everything have to be out in the open? It didn't make things easier, in Lucy's experience. To give away a secret was to give away your freedom.
She couldn't help wishing that people would just mind their own business. But right away, she felt her conscience smack her down. Grace's dad was going to be executed in less than two weeks. _Executed_. How could Lucy's own issues come close to that?
"So, I went to the cops," Lucy said, carefully avoiding Grace's hot, teary gaze. "All kinds of stuff going down in the precinct today, or so they told me. Drunks and drug dealers being chased around, all the detectives out on cases or busy, a major traffic accident on the 405."
"They didn't see you?" concluded Grace. Her disappointment sounded bitter.
"Oh, they saw me but for, like, five minutes," Lucy admitted. "Had to wait a long time before anyone would meet with me. When they finally did, well, they got big eyes, like saucers, and started saying that I had to be interviewed by a detective, but they didn't have anyone available today, it had to be recorded properly and witnessed, that I needed an appointment."
"You need an appointment to give testimony that's going to save someone's life?" Grace said, incredulous.
"That's pretty much what I said," Lucy agreed. "As it turns out, you do. At least when the person whose life you're saving 'isn't in any immediate danger.'"
Grace gasped. "Every extra day puts my dad in immediate danger. It's called _death_!"
Paolo squeezed Grace's knee to calm her down, which seemed to work somewhat as she sat back in her seat. Lucy managed to minimize her surprise. _Well, at least that's moving in the right direction_ , she thought.
"I know," Lucy said to Grace. "And I'm sorry. But what with it being the holiday weekend, I could tell that the cops just weren't interested. I got an appointment for Monday. Grace, don't worry, I won't let this go."
"Okay, so that's Lucy dealt with," Candace concluded impatiently. "Now, let's hear from Maya. Girl, explain yourself!"
"Okay, the woman I've been calling 'Marilu' is my mom," began Maya warily. "I didn't tell you because . . . because . . ." At this, Maya dropped her head into her hands. "Oh God, I really don't want to do this."
Grace put one arm around her roommate's shoulders. "I know this isn't easy. But we need to know we can trust one another."
Lucy, Candace, and John-Michael murmured some bland words of encouragement. But Lucy noticed that Paolo remained quiet, not moving, his eyes fixed on Maya's arms.
Reluctantly, Maya drew herself upright. Her eyes were damp. She wiped one with the back of a hand. "I guess you'd better know. My mom does work for Dana Alexander. If you look up Lupita Soto de Padilla, it's pretty easy to find online that she's connected to Alexander. So I told you she was my aunt, whose name is Marilu Soto."
Firmly, Candace asked, "Why didn't you want us to know about the connection with Dana Alexander?"
Maya looked at each housemate in turn as she replied. "Listen, I didn't know about the setup here, okay? Just what you guys told me when we moved in. But for some reason, Alexander wanted to know what was going on in the house."
There was an audible flinch from Grace and Candace. Lucy herself felt as though a lump of air had solidified in her own throat. "Excuse me?"
Maya's expression had become sorrowful. "She's interested in everything that goes on in the house. But Lucy, I gotta tell you . . . most of all she's interested in you."
"That goddamn witch, of course she is," Lucy said, seething. "That's why she got Ariana to spy on me, too."
Maya seemed taken aback. "Ariana was spying?"
"We were going to tell you," Grace said. "We only found out yesterday. You and Candace have been awfully wrapped up in your work."
"And to be strictly accurate," John-Michael pointed out, "we don't know for sure about Ariana, isn't that right, Luce?"
"Ariana spying," Maya repeated thoughtfully. It could have been Lucy's imagination, but for a brief moment she had the strangest feeling that Maya _wasn't_ all that surprised. "But I think that's right," Maya continued, shaking her head. "Listen, I was supposed to send a written report every few days. About a month ago, after I overheard Grace talking to John-Michael about how he crashed his dad's car, I decided to stop telling Alexander anything useful."
"Wait up, wait up," Lucy interrupted, frowning. "You _listened in_ on Grace and John-Michael?"
"It was kind of a pocket dial," Maya admitted. "It wasn't intentional. And I already told this to Gracie. I heard her say something that made me wonder about Dana Alexander."
Lucy watched Grace acknowledge this with a slight nod. Suspiciously, Lucy said, "What'd she say?"
Maya said quickly, "Enough to make me worry that Alexander was bad news and to realize that what I was doing could be dangerous to you guys. And since that day, I swear to God, I haven't given her a single useful report."
"Which does not get you off any kind of hook!" Candace bristled.
"But it does explain why Dana sent Ariana," Lucy concluded with bitter understanding. "The second you stopped sending anything useful, Ariana shows up on our doorstep."
Maya nodded, clearly relieved. "Yes. At least I think so. But it only started to make sense recently. Just little things about the way Ariana behaved . . . she was always hovering around where most of us were, not joining in conversations, just listening."
"The way _you_ used to?" John-Michael observed dryly.
"Like I said: I realized it was wrong."
"Why'd you do it at all?" Candace said. "It's kind of a skanky thing to do to your friends."
"Guys," Maya said, eyes wide, hopeful, her hands spread wide on the table before her, "Alexander has stuff on me and my family. I'm not exactly sure what, my mom won't give details. But Alexander goes to immigration and my mom gets deported."
"Wait, your mom is here illegally?" Candace asked. "I thought you were a citizen."
"I was born here, yes. But not my mom, obviously. And there's some issue with the documents."
John-Michael broke in, "It's got to be horrible to live that way, trying to stay under the radar."
"Then Alexander had me under observation from the beginning," Lucy said, shaking her head. "She must have gotten to Ariana when we were in rehab. Maybe even put her there with me, for all I know."
Maya leaned toward Lucy. "What I didn't understand was _why_. Until you told us that you were in _Jelly and Pie_ and we figured out your real name. And until I heard Grace talk to John-Michael. Dana Alexander was at the party when Tyson Drew was murdered, wasn't she, Lucy? Is there a chance that she thinks you saw who really killed him?"
"You figured all that out, too, and you didn't tell us?" gasped Paolo.
"Maya, seriously, you wanna consider acting for a living?" chipped in Candace. "'Cause you are one high-end liar."
"I think Alexander knows _for a fact_ that Lucy saw who killed him," Grace said grimly, ignoring the other housemates' jabs at Maya. "I think it's pretty obvious that Alexander is the one who scared you into silence, Lucy. But now she knows that you remember the truth and you are going to testify . . . right?"
Lucy felt the slow burn of everyone's attention as they focused on her. She found herself unable to look any one of the housemates in the eye. _Alex Vesper is on death row because of you_ , they seemed to be saying. _An innocent man has been in prison because of you. All because of you._ Because little Lucasta was so scared she wet her pants that horrible night on the balcony. She was shaking when a woman took her by the hand, a woman with soft hands with beautiful peach fingernails. They were all Lucasta could see as she led her to the bathroom murmuring gentle words of comfort in her ear.
___You're sleepwalking, honey. Dreams grabbing you by the throat. Time to get back to bed, Charlie._
Lucy wanted to cry, yet tears wouldn't come. Instead, she felt her core growing cold. Numb paralysis. Someone's hand reached out to touch hers. But she barely felt it.
## **PAOLO**
#### KITCHEN, VENICE BEACH HOUSE, FRIDAY, JULY 3
Another time, another place, it could almost have been entertaining. _Probably would have been_ , Paolo thought. He reached behind him, stretching as far as he could from his position at the corner of the dining table, to the refrigerator. He managed to pop the door and felt inside for a can of Diet Dr Pepper. Paolo studied the faces of each housemate as he drank.
Candace, upright and hostile, a contrast to her normal laid-back, jokey demeanor. Grace, struggling to keep her feelings in check. This must be great for her; a moment of intense relief—the most hope she'd had in years: Lucasta Jordan-Long could save Grace's father from execution. And yet, Grace was pale with anxiety. John-Michael, somber and pensive, as he often was, had taken the sixth place at the dining table and was slowly chewing on a piece of celery. Maya's caginess. She seemed fretful, presumably worried about Dana Alexander.
___Should I be worried, too?_ Paolo thought.
It was impossible to drag his own mind too far from the lurking horror of his own recent past. A secret fresher than any of theirs, one that still hovered at the fringes of Paolo's thoughts, every waking minute. The simple pleasures of life were a distraction. Even the tingling burn of his left upper arm, still hot to the touch, took him away for only a minute. Every time he tried to dismiss this memory it felt more and more impossible, grinning like a death mask at every opportunity.
"How much do you think Ariana knew about us?" he asked abruptly.
It wasn't his secret that Ariana was trying to find but could she expose him anyway? What would his own housemates think if they knew, these girls who Paolo realized he was proud to call his closest friends? What if right now he were to stand up and tell them about how he'd hustled Jimmy out of a car, had sex with the idiot's mother, and then left the woman for dead, alone, in the canyon?
"Enough," Lucy replied. "More than enough that we gotta be careful." For a moment, her eyes locked with Maya's. "Too bad that you gave her the goods on every last one of us," she said in bald accusation.
A burning silence followed. Then Maya turned to Candace. "Maybe you shouldn't have made out with Yoandy Santiago. We warned you from the beginning he was Kay Alexander's boyfriend. Now it's just another reason for Dana to get mad."
Paolo watched the expression on Candace's face shift from incredulity to sheer indignation. "Oh, that's _it_ , we are so done," she hissed. "I guess you never really stopped spying?"
A deep, beet-colored flush spread across Maya's face. "You were talking in my bedroom, which is where I actually _sleep_. So I stood outside for a few seconds, waiting for a good moment to walk in. I wasn't going to say anything about you and Yoandy. It has nothing to do with me."
"No, you were just gonna write it up in your lousy, stinking report to Alexander," Candace snarled.
Maya raised her voice. "Is there something wrong with your hearing? Didn't I already tell you that I stopped doing that?"
"Oh gee, thanks for saying you wouldn't spy on us anymore, Maya, thanks for rediscovering the meaning of 'friend.'"
"Maybe we should all just chill?" Lucy said. She was making a pretty obvious effort to remain calm herself. "Sugar, you're in no place to accuse anyone of _anything_. Okay, so you were forced into it by your mom's boss. We get the picture. You still spied on us. _Own_ it."
"Look, Maya," John-Michael followed up, reasonably, "not one of us here is gonna win any kind of prize for how we've handled everything difficult that's happened in our lives. But it turns out that out of all of us, you're the one who's still taking orders from Mommy. The rest of us, y'know, we're trying to be responsible, like adults."
"But we're _not_ adults," Maya said emphatically. "And driving a car, cooking and shopping and cleaning and paying the rent—that doesn't make you one. Making good decisions, the kind you make with _actual_ freedom—that's what makes you an adult. And as you've cleverly pointed out, John-Michael, unlike the rest of you, I do _not_ have freedom."
"We're supposed to take life coaching from a liar and a spy?" Candace said. Her voice was laced with sarcasm. She tilted her chair back so violently that it tumbled to the floor with a crash, and Candace narrowly managed to avoid toppling over with it.
Paolo stood up, blocking her route out of the kitchen. "Candace," he began. But she pushed him aside with surprising force, shoving hard against his left arm. The tattooed skin burst with a hundred hot stings. "Oww, dammit, Candace, what gives?"
Grace stood, too. "I agree with Lucy—we need to calm down. Maybe take a time-out. This is getting intense."
"You got that right," Candace said, fuming. "I'm out of here. Grace, you coming with me?"
It was a pointedly exclusive invitation. Reluctantly, Paolo stood aside, watching Grace depart with her stepsister. He placed three cool fingers across his burning tattoo and turned to Lucy. But Lucy's eyes were on Maya, who was next to stand up. "I could use some air, too."
"You want some company?" John-Michael asked Maya.
"At least as far as the Starbucks," she replied. She'd turned pale in the last few moments, yet Paolo noticed that she seemed surprisingly calm given that John-Michael had just called her a child. Yet there she was, chewing thoughtfully on a fingernail. "I'm gonna get a Frappuccino," she said to no one in particular.
Maya and John-Michael were walking out a minute later. Finally, Lucy faced him. Paolo's hesitant smile became awkward. "Just us then," he said.
Lucy hadn't shifted from her position at the table. She looked very tired, but managed a halfhearted grin. "Yup." She went to the sink and turned on the hot tap. "I guess I'll clean up some dishes. Since none of you lazy-ass jerks seem to know how."
"In that case," Paolo countered, "I guess I'll go dig up some weeds in the yard. Since none of you lazy-ass jerks seem to know how."
Storage space in the house was precious, so the three garden tools that he'd picked up at the hardware store more or less lived in Paolo's Chevy Malibu. He trudged out through the front door and around back. Since the house faced the beach, the road where Paolo parked his car was in the rear. John-Michael had left his car there earlier on, some ways down the road since most evenings it could be difficult to find a spot close to the beach.
Paolo popped the trunk and grabbed the shovel and the long-handled lawn weeder.
As he returned, he approached the house from the rear, where a low gray concrete wall surrounded the yard. He was about to step over the wall when Paolo noticed that the French door was already open. He opened his mouth to call out to Lucy. Then he saw something that stilled the air in his lungs. Instinctively, he moved to one side. Carefully, Paolo rested the long-handled weeder against the wall. He sidled up to the French doors, peering inside, cautiously staying out of sight.
There was a man in the living room. He was about five feet nine, stocky with thin graying hair and wearing a faded, black leather bomber jacket over slacks. He had his back to Paolo and seemed to be talking to Lucy. The man's overall stance seemed relaxed, not particularly threatening.
And yet, even though Paolo couldn't see Lucy, he could hear it in her voice.
Terror.
"Who've you told?"
The question sounded casual. Paolo didn't recognize the voice. Lucy was having difficulty replying. She was stammering even before she'd gotten started.
"I . . . I don't know why you—"
"Enough," he said, cutting her off. He reached into his pocket. Lucy recoiled, knocking against the fallen dining chair. He spoke calmly and sounded almost weary. "We can do this the easy way. Or not."
The man was pointing a gun straight at Lucy's stomach.
Paolo flipped the shovel into position as if to return a serve. One glance at the space between him and the man and he'd computed the precise path, exactly as if a ball were arcing across the net toward that spot.
"Okay, kid, time's a-wastin'."
Paolo took a deep breath and ran, barging through the open French door and into the house.
The man spun to his left, the gun in his right hand. It fired, a muffled sound— _phhooott_. The next shot never came. Paolo's shovel was already swinging, a powerful forehand. It cracked against the man's skull. Paolo watched the man's eyes widen. Just for a second, they bulged. Fingers locked in the trigger, the man's arm fell. His gun hand slammed against the floor.
Paolo stood paralyzed, blood pounding hard in his head: a roar. He could hear nothing else.
After a few seconds, he regained control over his muscles. He lowered the shovel and raised his eyes to Lucy. She stood, rigid with horror, hands by her side, her mouth open, eyes wide, staring, her breath coming in heavy, labored gasps.
Finally, Paolo turned reluctant eyes onto the fallen man. He could see now what a slight, unimpressive figure he was. In his late forties, a cheap haircut, worn shoes, a brown-and-white plaid shirt tucked into charcoal-gray cotton slacks.
Lucy seemed to emerge from her own paralysis. "Is . . . he dead?"
The man lay immobile, eyes staring just the way they had as Paolo's shovel had connected with the left side of his face. Paolo glanced at the underside of the shovel. There didn't appear to be any blood. But the sight of the dent in the man's head made Paolo's insides clench in cold, churning dread.
## **MAYA**
#### LIVING ROOM, VENICE BEACH HOUSE, FRIDAY, JULY 3
Returning from the boardwalk, Maya noticed that the house lights in the front part of the house were all dark. She paused for a second, letting John-Michael catch up with her.
"You think they're all in the backyard?" he asked.
"But why no lights?"
In the doorway, they listened for a moment. Nothing. Maya turned the key, pushed against the wood. That's when she knew for sure that something was terribly wrong. The door wouldn't move. Paolo was inside and was resisting their combined efforts to push it open.
"Paolo, it's me and John-Michael, let us in!" she said, a little desperate.
He let them inside then, held a finger to his lips, and frowned until they fell silent.
Maya's attention went instantly to Lucy, who was crouching on the floor beside the new red couch. A strange man lay crumpled on the checkerboard-patterned rug, his head neatly framed on a russet-colored square. The man wasn't moving.
Lucy looked up, an imploring look in her eyes. "Maya, JM . . . what are we going to do?"
Breathing a Mexican curse through pursed lips, Maya approached. John-Michael, she noticed, hung back with Paolo, who seemed frozen in position by the front door. She stopped short of the body, noting the approximate height of the man, his gray slacks, black leather jacket, and clean but cheap-looking brown leather shoes. There was a pistol in his right hand; a revolver of some kind. He looked like a respectable, middle-aged, off-duty policeman.
Or maybe a private investigator. Maya's heart began to plummet.
_This man had to have been sent by Dana Alexander._
"He shot at me," Lucy was saying, pleading, tears in her eyes and her whole body shaking. "It was self-defense."
"I didn't mean for it to kill him," came Paolo's voice. It sounded dull, disconnected, and vague. As though he didn't quite believe what he was saying. "I swung for him with the shovel."
"You thought he was a burglar," Lucy insisted. "It was self-defense."
"I thought some kind of intruder, yeah," agreed Paolo. "He was going to shoot you."
"Obviously an intruder," Maya said briskly. "No one invited him, right?"
"And he shot at us. That's definitely self-defense, right? It's open and shut," Lucy said, almost pleading.
_Not if Dana Alexander has anything to say about this_ , Maya thought. But she only asked thoughtfully, "Did you find the bullet?"
Paolo and Lucy glanced at each other, apparently mystified.
Maya repeated herself, a little louder. "Paolo? Did you see where the bullet went? Think. It's important."
Paolo straightened up a little, making an effort to pull himself together. "Um, I guess it went over by the stereo."
"Find it," Maya instructed, "but don't touch it." While Paolo hunted, she joined Lucy in kneeling beside the corpse. The man's head had been visibly damaged by Paolo's strike with the shovel. One whole side had been dented. Maya sat back, wondering why she wasn't more repulsed. But the truth was, it felt a lot like seeing a dead body on TV. She rocked back on her heels and turned her head to examine the rug below the man's head.
"We moved him onto the brown square," Lucy explained. "More or less right away, actually. I figured the bloodstain would show up less."
"Cops don't need much blood to connect this rug with his body," Maya said. "Too bad you didn't put a piece of plastic wrap underneath him."
The boys and Lucy were still for two seconds, staring at Maya. Then John-Michael made a dash for the kitchen. "Get some paper, too," Maya called after him. "Something we can burn when we're done."
Looking at Maya incredulously, Lucy said. "What are you talking about?"
Maya ignored her. "Hey, Paolo, d'you find that bullet yet?"
Paolo glanced up from where he was squatting by the baseboard on the wall adjacent to the floor-mounted audio speakers. "Not yet."
"Did you look at the gun?" Maya said.
"Did I look at the gun?" echoed Paolo. "Are you crazy? You think I'm going to tamper with a crime scene?"
Maya said, "You mean aside from moving his head onto the brown square of the rug?"
"That was basic housekeeping," Lucy tried to say, but stopped, gawping as Maya leaned over to remove the revolver from the corpse's fingers. "Omigod, Maya! You touched a dead man!"
The man's fingers were still warm and pliable. It hadn't even occurred to Maya that he'd feel any different than a living person. She noted with surprised detachment that he wouldn't go stiff for a little while. She had no idea how long. But it might be useful to know.
"John-Michael, can you find out how long it takes before rigor mortis sets in?"
John-Michael was already taking out his phone when Maya cried out, "Actually, no, stop!"
Everyone stared at her. "What?" John-Michael asked in a voice that betrayed more than a little fear.
Maya said, "We can't do any suspicious searches on the internet! We can't behave any differently than we might behave on the night before the Fourth of July."
"His body will feel normal for three or four hours," John-Michael said. His voice was suddenly hard, frosty.
Maya noted the confidence of his response. It was pretty obvious what everyone was thinking.
No one in the house said a word. They couldn't even look at him.
John-Michael had seen his father's body when it was still warm. Which probably put him in the room with his recently dead father. The rest was easy to conclude: John-Michael's arrest for the murder of his father might be a whole lot more serious than he'd led them to believe. But Maya wasn't going to be the one to voice any suspicion. At least not right now.
She examined the gun. It was a Smith & Wesson model 60, she noted, being careful to point it away from anyone and to keep her finger far from the trigger. She opened the revolving cylinder. Inside were five cartridges and one collapsed, metal casing.
"Maya, why are you touching the guy's gun?"
She turned the revolver around so that Lucy could see the inside of the cylinder. "You ever seen a casing all crumpled up like that?"
"Maya, I haven't ever seen a real _bullet_ ," Lucy said.
"My mom has a handgun," Maya said a little reluctantly. "She's not just a driver—she's also Dana's bodyguard. In Mexico, Mom was a police officer. Maybe you know how things are with the cops and the _narcos_ in Mexico? It was safer to leave the country. And believe me—with some of the enemies my mother made as a cop, she'd better not go back. I've been to the range with her a few times. Mom made sure to teach me how to handle a gun. I never fired a shell like this one."
Exasperated Lucy said, "Goddamnit, Maya! D'you have any other secrets you want to share right now? Or you think you're about done for the night?"
Paolo stopped his hunt for the bullet. He straightened up and faced her abruptly. "Maya, just tell us what it is you think you've figured out."
Maya shrugged. "My guess? Mr. Private Investigator here was trying to scare the bejesus outta you both. That was his first tactic. That casing is from a blank cartridge. And I'm betting you won't find a bullet. Maybe a wad of cotton somewhere closer than the wall where Paolo's looking. Yeah, might be a good idea to find that."
Paolo's jaw was slack as he spoke. "A blank? Why would he fire a blank?"
"Like I said," Maya muttered as she removed the collapsed casing from the cylinder. "To threaten Lucy. The rest of the bullets are real enough, though." She slammed it back into position and proceeded to wipe the whole gun carefully with the edge of her shirt. Once she was satisfied that it was clean, Maya handled it through her own shirt and replaced it in the dead man's outstretched hand.
"What the hell did you just do?" Lucy said, her words heavy with accusation. Maya moved over to the futon, where she felt her knees buckle slightly. Her heart was starting to thud painfully hard as her body responded to the decisions she'd just made.
"Yeah," Paolo said, but quietly. There was an unmistakable note of hope in his voice. "What did you do?"
John-Michael had been standing stock-still through the last few minutes of Maya's actions. Now he knelt beside the dead body and handed Lucy the roll of plastic wrap, still inside its box. "Pull a sheet out and get ready to slide it under the guy's head. Okay?"
Quietly, Lucy did as he asked. Then John-Michael lifted the head and Lucy placed a yard-length sheet of plastic wrap beneath the corpse. With a nod, he was on his way back to the kitchen.
Maya watched for a few seconds and noted with a nod the moment when blood began, now very obviously, to ooze around the dead man's head. She sucked in a few breaths, willing her heartbeat to slow down. Lucy and Paolo were obviously dazed, quite possibly unable to think straight. Lucy's shock was understandable. If Maya's own mind was already beginning to run through the implications of all this to her life and Lucy's, the consequences for Paolo had to be even scarier. Paolo's would be even worse.
A sharp cry from Lucy made Maya gasp. "Omigod. He just moved." She stared at the man on the floor. He didn't budge. She slid off the sofa and once again knelt beside him. This time she listened at his mouth. She gulped down a chunk of air and held it for several seconds, listening. When Lucy made a sound, Maya raised one hand to silence her. Paolo crept closer. Maya released her breath.
"I can't hear . . ."
The man's empty hand twitched. All three of them jerked back at the movement. John-Michael brought a tray and four mugs of steaming cocoa from the kitchen. He alone continued to move. He placed the tray on the wooden folding table near the French doors, then calmly paced over to where the man lay.
As Maya watched, the man began to regain consciousness. First his hand moved, then his whole arm. His chest started to visibly rise and fall.
"Holy crap holy crap holy crap," Lucy intoned.
Maya couldn't drag her eyes off the impossible sight of an apparent corpse returning to life.
"He's not dead," she whispered in a tone of wonder.
## **JOHN-MICHAEL**
#### LIVING ROOM, VENICE BEACH HOUSE, FRIDAY, JULY 3
John-Michael spoke quietly. "Now what?"
None of them could reply. A paralytic dread had seized them. They stared helplessly at the would-be assassin. Even Maya, who'd been so cool up until that moment, so _together_. Over the next few minutes, however, John-Michael saw all that resolve evaporating.
The man was alive. He would talk.
Maya groaned, "The gun . . ."
Then they all saw it. The fingers of his right hand were tightening around the handle of the revolver. Collectively, they recoiled as slowly the man began to sit up. He led with his gun arm, which he lifted high enough to be a threat. The rest of him followed in a disjointed, strenuous pattern of movement, like a puppet being slowly dragged upward by invisible strings. When finally he was sitting upright, one shoulder leaning against the red sofa, the man seemed to notice his own blood and cursed roundly, adding, "Which one of you bastards did this to me?"
No one moved.
He raised the gun, this time pointing unsteadily at John-Michael. "I said which one?"
No one spoke.
The hit man swore again, and stared each one of them in the eye. The gun stayed where it was, aiming right at John-Michael's chest. Finally, his eyes settled on Paolo. John-Michael saw Paolo's Adam's apple bob up and down, but otherwise his friend gave no indication of fear.
"You with the tattoo. You're gonna tie up your friends. You make one false move and your boy here takes a bullet to the heart."
The man reached into his leather jacket with his free hand and after a few seconds of fidgeting removed a roll of silver-gray duct tape. He rolled it across to Paolo. "Now."
Paolo picked up the duct tape like it was radioactive waste. With one final, hesitant look at Maya, he began to tape her wrists together.
"Do everyone's hands first," the man said. His voice sounded faint, exhausted. "But the other boy, you do his legs first. Then he does . . . does your hands," he managed to say with effort. "And then . . . then you do his . . ."
The voice trailed off, but still the gun aimed at John-Michael. A sweat broke out all over John-Michael's upper body.
Moving slowly, practically robotic, Paolo moved to Lucy and began to tape her wrists.
The man wiped his free hand across the back of his head. When it came away smeared thickly in his own blood, he gasped. He sounded less angry than resigned. "You stupid children. You have no idea what you've done."
"It was self-defense," Lucy said. And then repeated it over and over, like a mantra.
"Yeah?" the man snarled back. "Maybe you should've killed me when you had the chance."
"Let us call nine-one-one," John-Michael said urgently. "You're bleeding a lot. You . . . your head looks pretty bad. I think you need an ambulance."
The man sniffed. "Not yet. We got unfinished business."
"Look, we know you didn't intend to kill anyone," John-Michael continued. "You fired a blank."
"I can assure you, son, there'll be no more blanks fired today."
But John-Michael didn't stop. "So why _start_ with a blank?" There had to be a reason that he'd begun by trying to scare Lucy. If he could get the guy back into that mind-set, maybe they'd get out of this intact. "Let me call nine-one-one?"
"You touch a phone and I'll kill you," the man said, almost casually. He waved the gun, urging Paolo on. "Hurry or my focus might slip. I might let a bullet go, by mistake."
Maya and Lucy were tied up now, wrists and ankles taped together. They were still upright, and looked utterly lost. Finally, everyone was tied up except John-Michael, whose hands were still free. He could barely stand to look any of his friends in the eye.
Why did the guy want them tied up? Obviously it made sense not to let them outnumber him. If he started firing now, he might get a round or two off before they overwhelmed him. But that wasn't likely to happen, John-Michael guessed. Not one of them had the guts to tackle an armed man.
Tying them up would put them entirely at his mercy. Would he kill them, then? Pick them off carefully, clinically, with the five remaining bullets? The more John-Michael thought about it, the harder he felt his heart pounding, until he was certain that the man would hear it, too.
The man's free hand was once again jostling inside his jacket. After three attempts, he managed to extract a small, black, plastic-cased cell phone.
_His motor skills are deteriorating_ , John-Michael noted. "You gonna call nine-one-one?" he asked.
"Everyone, siddown," the man said, ignoring John-Michael's suggestion. This time, his speech was noticeably slurred. He peered at John-Michael, as though staring at him through dark glass. "You. Get over here and kneel down beside me."
John-Michael hesitated. Mercy was their only hope now. "Excuse me, sir, but I think you need a doctor _real_ bad."
This time, the hit man exploded. "Shut your mouth and get down on your knees, punk!"
But as John-Michael began, slowly, to fall to his knees, a sharp cry came from the hit man, followed by a faint groan. Then he clutched both fists to his eyes. Over the next few seconds he began to jerk violently, until his entire body was in spasms.
The housemates looked on, aghast.
"He's having some kind of seizure!" John-Michael gasped.
The air cracked. It took a second before John-Michael realized that the gun had been fired. It was still in the hit man's right hand.
"I'm okay!" Maya called out breathlessly.
"Yeah, me too," said Paolo. "And me," Lucy said.
John-Michael watched in appalled fascination as the man continued to flip and twist like a freshly landed fish.
"What should we do . . . ?" Maya breathed.
An idea was crystallizing in John-Michael's mind, taking form within a cage of icy, implacable logic. The idea became bright and terrible in his mind; irresistible. The answer to so many problems had been within reach—the hit man himself had said it. Who knew where the next bullet would go?
John-Michael moved smoothly. In one swift motion he'd rotated his taped ankles behind him, picked up a large cushion from the back of the red couch, and was on his knees next to the injured man.
He looked at Paolo, who sat helplessly taped up on the floor next to the green futon. Their eyes met with an intensity that made John-Michael shiver. Paolo's lips moved in response to John-Michael's unspoken question. His reply was barely audible.
"Yes."
John-Michael closed a part of himself away as he leaned over the man on the floor, reaching for the flailing hand in which the gun was clenched. He held the cushion firmly over the man's face, jaw clenched tight as he struggled to hold the man down, aware of the man's chin beneath his own shoulder. It wasn't easy; the man's movements were powerful, violent. He shut off his feelings as the man continued fiercely to jerk beneath him. He ignored the objections from Lucy. Maya and Paolo, he noted, made no sound at all. They simply watched as John-Michael held the man beneath him, slowly extinguishing his life force.
It took less time than John-Michael remembered. A sure sign that the hit man had already been on his way out. John-Michael had merely nudged him along.
The silence that followed was lengthy. Heavy and profound. John-Michael released his grip and rocked back on his heels. He glanced first at Paolo, then Maya. They both sat motionless, their eyes full of him. He couldn't look at the dead man; those lifeless eyes would have chilled him to the bones. But he made himself look at Lucy. She was gasping for air, trying to speak words that wouldn't leave her throat. Finally, a strangled cry escaped her and they all realized that Lucy was hyperventilating.
With great difficulty, she choked out, "What . . . what . . . have you _done_?"
## **PAOLO**
#### LIVING ROOM, VENICE BEACH HOUSE, FRIDAY, JULY 3
Paolo's skin felt hypersensitive. Looking down, he noticed that all the hairs on his arms were standing up. He felt every whisper of air that moved across them.
_We killed him._
Next to him on the floor, Lucy was beginning to squirm, trying to get her fingers to the tape that bound her ankles.
"Let me, it's faster," John-Michael said without emotion as he fiddled with the tape around his own ankles. Paolo watched John-Michael move silently from Lucy to Maya and finally to him, laboriously removing the tape from their wrists so that they could pull off the ankle tape themselves.
Lucy pushed away until she was sprawled on the living room rug, just a few feet from the dead body. The temporary effects of shock that had held her for the last few minutes seemed to be fading. Defiance had returned to her eyes.
"You've ended us _all_ ," she stated, loud enough for John-Michael to hear that she included him in the accusation. As if to be certain that he'd understood, she turned to him with a penetrating glare.
John-Michael reacted with a calmness that Paolo couldn't help but admire. "He was dying," John-Michael said. "A seizure like that after a severe blow to the head—you don't go to the emergency room, you die."
"Lucy," Paolo said reasonably, "who knows where his next bullet would have gone? John-Michael saved us."
"Good," Maya said. She sounded relieved. "That's what we'll tell the cops—the truth. They'll call it Stand Your Ground and you'll both go free."
"Actually, in California we have 'castle doctrine,'" Lucy murmured. "Which I guess is the same thing, so long as the attack is in your home."
"I'm _in_ my goddamn home!" Maya hissed.
Paolo began to think about that, began to think about the questions that would be asked. How far back would they go? A good prosecutor would dig up everything they could on him. The con. The blackmail. A good investigation might even put him in the car when Meredith died. Maya didn't realize how bad the truth actually was.
"But I hit him," he said, unsure. "When he'd only fired a blank."
"Yes, but you didn't know it was a blank," Maya said. "The other cartridges have real bullets."
"Still . . . I'm not sure we should take this to the cops," began John-Michael.
Paolo, Lucy, and Maya turned to him. Paolo swallowed, his tongue like sandpaper. Yes. Don't tell the cops. That would save him a lot of difficult questions.
"They won't even know he finally died of suffocation," Lucy said. "It'd just be the seizure."
John-Michael barked with hollow laughter. "We have to work off the assumption that they will know. Forensics can tell when someone's been asphyxiated."
"You sure know a lot about this kind of stuff," Lucy observed. "Is this what you did to your father?" Her words practically froze in the air. Paolo watched John-Michael. He visibly reeled for a second, as if absorbing the impact of a shock wave, but said nothing.
"Why is this happening to us?" Paolo asked. It seemed almost redundant, given the heights of terror they'd recently scaled, to be asking such a logical question. But now that he thought about it, he actually didn't know the answer.
"My guess is, he was trying to scare Lucy," Maya said tersely. "But like he said, the other bullets are real. The question is who would try to threaten her. I'd say it's Dana Alexander."
In a small voice, Lucy said, "You really think Dana Alexander might try to have me killed?"
Maya nodded. She bit her lip. "I do."
"This Alexander woman must be a lot worse than I thought," Paolo muttered.
"Lucy went to the cops today to schedule an interview to tell them that she has testimony that could open up an eight-year-old murder case. Dana Alexander might wind up as the prime suspect. If she were trying to avoid death row, I can see it going as far as this," Maya concluded.
"You think the cops told her?" Lucy said anxiously.
"More likely Ariana told Dana the game was up when Lucy kicked her out," Maya said with surprising confidence. "I bet it takes time to organize a hit. More than a few hours, at least."
"Whatever we decide," John-Michael said thoughtfully, "we need to do it now."
Maya shrugged. "Call the cops."
"I'm glad you're so confident that I'd get off with no charges," John-Michael said. "But I'm not."
A tense silence followed. In John-Michael's eyes, Paolo recognized a similar fear to his own.
The police meant questions. Questions would open up the past. John-Michael was already on perilously thin ice with the cops, as far as his father's death went. Maybe he'd get away with dealing with a dangerous hit man, but would it make them reconsider what had happened to his father?
"It's a good point," Paolo acknowledged. "You never know how things will go in a court of law."
"Says the wannabe lawyer," Lucy said.
"A good lawyer can win any argument," Paolo said. "But winning this one might cost some of us."
"A good prosecutor," John-Michael pointed out, "could say that _we_ fired the second bullet, to make it look like the guy was a threat."
Paolo and Maya absorbed this quietly, nodding.
"What if we got rid of the body?" John-Michael suggested. "I'm guessing no one is going to report him missing."
"That's just great," Lucy said, her voice thick with sarcasm. "I guess now we're gonna break out the hacksaws and the garbage liners and turn into a bunch of butchers, is that your plan? Or were you thinking of hiring a boat and dumping his ass in the ocean? Without anyone noticing you strolling up to the pier with a big ol' dead body under your arm?"
Paolo licked his dry lips and shook his head. "Neither of those ideas sounds good."
"You got something better?" Lucy flared up, her eyes dark with anger.
John-Michael said, "The best thing for all of us would be if this body and everything that happened here tonight just went away."
"Nothing like this ever _just goes away_ ," Lucy snarled, her tone suddenly vicious. "If there's one person in this room who should know _that_ particular piece of _truth_ , it's me."
For a moment, they all paused. Their eyes met above the man's body . . . No one, it seemed, was willing to take the lead.
"I may have an idea," Paolo began. Gooseflesh broke out all over his body as he shivered under the sudden, focused attention of his three friends. He stepped forward. Ignoring the vacant terror of the corpse's wide-open eyes, Paolo examined the dent on the side of his head. Yes. So familiar.
_Just like Meredith's in Malibu Canyon._
Paolo stretched a hand across the man's still-warm chest and into the right-hand pocket of his jacket. His fingers closed around a remote key for an automobile. He stood and addressed all three housemates.
"This head injury. What if it came from a car?"
"A car," Lucy said flatly.
"People get hit by cars, they get this kind of injury. They die by the roadside. It happ . . ." Paolo paused, correcting himself. "It _must_ happen."
John-Michael's expression shifted slightly, probably not enough for anyone else to notice, but when Paolo finally allowed himself to glance at his friend, he saw that John-Michael suddenly seemed to let go of some of his tension.
_John-Michael understands._
"Sure it does," John-Michael agreed slowly. "Some SUV hits you while you're taking a whiz beside your vehicle on a country road, suddenly you've got a major dent in your head."
Lucy seemed incredulous. "What kind of imbecile stands where a car can hit them?"
"I think JM means, who's to say for sure? A man goes out somewhere like, say, Malibu Canyon." Here Paolo threw another hard glance at John-Michael, who nodded in affirmation. "Gets himself killed by a hit-and-run driver out there with no witnesses. Body rolls down a ravine, maybe? Coyotes and buzzards eat most of the skin off the face before any cop gets a chance to photograph it."
"Malibu Canyon," Lucy repeated. "That's your plan?"
Paolo unfurled the palm of his right hand to show the key he'd taken from the shooter's leather jacket. "His car key. Car's gotta be outside, close by. We drive the shooter up there in his own car. We lay out his body, as though he'd been hit by some truck coming along out of the blue and thrown clean off the side of the road. We leave the car there, key still in the ignition. We drive away."
A long silence followed. Maya eventually spoke. "That's not a bad plan."
"Thank you," Paolo said, relieved. He could feel the creeping reminders of all possible consequences of what had already happened tonight. He wasn't willing to face any of it. Nor should he have to. Just because some murderer wanted Lucy out of the way, just because Paolo had stepped in to help, why did that mean his private life would now be fair game? Why would stupid incidents that had not been Paolo's fault in the first place—Darius blackmailing him into hustling Jimmy out of a Corvette, Meredith threatening him into driving to her country house, the hit-and-run driver killing Meredith on the way to Malibu Creek—why should these things impact him now?
It was like Paolo had stepped into the quicksand again. Struggling wouldn't help. He'd understood pretty quickly that he couldn't just rely on luck and good fortune to twist him out of there. Hesitation and struggling had only pulled him deeper into the mire. No, to get out of that quicksand, he'd had to get down and dirty. He'd gotten onto his belly to spread his weight over the mud and sand. He'd gotten his entire body coated with all that muck and grime, even his face.
That's how you got out of a situation like this. You slithered and crawled like a beast until you finally reached dry land.
## **LUCY**
#### MALIBU CANYON, FRIDAY, JULY 3
Lucy continued to be doubtful. "You seriously believe we have any chance to set this up well enough to fool a detective with an actual brain?"
Paolo nodded. "I think we could."
Maya also began to nod. "Lucy, think about it. It's a hard blow to the head. Enough to kill on impact. The cops won't do a detailed postmortem on a body they find that looks exactly like roadkill. They'll never find out that he actually died from being whacked over the head with a shovel."
"How can you be so sure they won't do a detailed postmortem?"
"Because we'll set it up so well that they won't be suspicious," Paolo told her. "They'll see a horrible head wound and blood, they'll assume that's how he died."
John-Michael said, "That's actually pretty good."
"But we all have to agree," Maya said. "Because if we do this, we're all in it together. We put this whole evening in the vault and we throw away the key."
"All of us? What about Grace and Candace?" Lucy said, practically throwing the words at Maya.
Unease settled over the trio once again. "Obviously not Grace and Candace," Paolo said. "They can't ever know."
"They could come home any minute now," Lucy sneered. This was a grandiose, scary plan. There had to be a million ways it could go wrong. She could see why John-Michael might not want his involvement with the man's death to be examined too closely. But really—did the other three housemates have to take the risk, too? "It's not exactly watertight," she said.
"No one walks away from something like this without getting their hands dirty," Paolo said wearily. "You don't want in, you say so right now. And if you don't, then you're gonna keep your mouth shut." He took a breath and raised his right hand. "Who's with me?"
Lucy scowled. "Hell no."
John-Michael and Maya raised their right hands in silence. John-Michael placed both hands over his heart. Imploringly, he said, "Please, Lucy. _Please_."
"At least come with us," Maya said. "That guy was about to call someone on his cell phone. Maybe it's not safe for you to stay here alone anyway."
Lucy gawped. "You think he was _getting help_?"
Maya knelt down by the dead body and picked up the man's phone. "He wasn't going to be able to handle four of us alone. Not even tied up."
The thought that yet another hit man might be on his way to the house stunned Lucy into temporary silence. Maya pressed some buttons on the phone. "No password. No contacts. He hasn't used this phone in over two hours," she told them. "No calls in or out."
"No one knows he's here," Paolo deduced.
Maya glanced from Paolo to Lucy. "Seems so."
After a moment, Lucy released an exasperated sigh. "I guess we're all screwed anyhow. All right. What do you need me to do?"
Paolo said hopefully, "You're in?"
"What does it look like, dumbass? You did this to save my life. You're gonna need another car to follow you up to Malibu Canyon. Way I figure it, you need at least two drivers, plus someone strong enough to help you hike that body down a ravine. Maya doesn't have her driver's license yet, which means you need me."
"I'm coming, too," Maya said. "Otherwise I'm going to be sitting here answering difficult questions from Candace and Grace about where you guys went."
"Two people per car works best," Paolo agreed. "Lucy, thank you. This means a lot."
"Does it?" she spat back. "You're not leaving me a lot of choice. This is going to come back on all of us, one day."
"Then don't do it," John-Michael urged.
Lucy merely gave a stubborn shake of her head. "Too late. I already gave my word. I'm a fool that way."
Paolo reached for her hand. But Lucy withdrew with a cold, disdainful scowl.
One hour later, she was driving Paolo's Chevy Malibu along the Pacific Coast Highway. Lucy kept her hands steady at ten and two on the steering wheel as they followed the taillights of the hit man's Oldsmobile into Malibu Canyon. It was almost ten thirty at night and the road was quiet; no other traffic was visible on the same side of the road and only the occasional car zipped past heading toward the coast.
Beside her, Maya was uncharacteristically chatty. _Nerves, probably_ , thought Lucy. Excessive chatter wasn't her favorite coping mechanism, so the drive was doubly difficult. Not long after leaving the house, Lucy wished that she'd remembered to bring an old MP3 player so that she could at least put some loud music on the stereo to block out Maya's chitchat. Rancid at full blast would be a huge improvement over this.
Paolo had cautioned them against bringing phones, apart from the hit man's, which obviously needed to be disposed of. He'd insisted they all leave their own cell phones in the house. "Someday the cops might want to ask where we were tonight. The answer is gonna have to be that we were all home, all totally distracted, partying or watching a movie or something."
"You need pretty high security clearance to look at cell phone records," Maya had said. "But you're right, we can't allow any risk that we'll be caught. All our phones can be used to link us to a location."
They still needed to be able to communicate between cars, however, as Maya had pointed out, so the housemates stopped at a store on the way to the freeway, and picked up a couple of cheap pay-as-you-go cell phones; one for Lucy and Maya, one for Paolo and John-Michael, who'd waited outside the store in the hit man's car and were now leading the way.
The phone that lay in Maya's lap began to buzz. With a quick glance at Lucy, Maya answered it. After a second, she put the phone on speaker.
It was Paolo. "We're going to head up into the canyon. I checked on the map, if we take Piuma, the road gets pretty twisty. We'll find a blind curve, the kind of place that's a risk spot for accidents. We'll slow down and stop. You guys need to make a note of where we are and then keep driving. Stay out of sight. The last thing we need is for someone else to drive by and see both our cars. In fact, I don't want them to see my Chevy at all. Wherever you stop, get right off the road, somewhere safe, and turn off all your lights. Even inside."
"We're just gonna sit in the dark until you guys do whatever it is you're gonna do?" Lucy asked.
"Exactly. You and Maya sit tight. All we need you to do is give us advance warning if any cars come the other way. Drive until you're at least two minutes from where we are, and call as soon as you see someone coming."
Maya said, "What if someone comes up behind you?"
"Yeah, maybe we shoulda got an extra phone and left one of you behind as a lookout behind us. But I didn't think of that. Sorry. John-Michael and I will just have to hide behind the shooter's car if we see headlights coming up on our side of the road. At least if you give us fair warning of stuff coming the other way, we can do something."
Maya ended the call and then said energetically, "I think five minutes would be safer, don't you? I'm gonna set the timer on my cell phone." She paused. "Oh, damn. I don't have it."
Lucy said dryly, "Just look at the time on the one you bought. You're right, five minutes is safer. Five minutes should give them enough time to arrange the body and get out of sight."
"Paolo's smart for coming up with this idea," Maya said thoughtfully. She was fidgeting with the cell phone now, scrolling through the various options on its tiny, blue-lit screen. "You know, I'm kind of surprised actually."
Lucy found herself agreeing. The _surprise_ of it was probably the most interesting aspect—until she reminded herself of what Paolo had actually done. He'd almost killed a guy with a shovel. She'd seen him nod at John-Michael, agreeing that their friend should finish the man off once he started having the seizure.
Maybe it was euthanasia, like it had probably been with JM's dad. Maybe John-Michael had only meant to defend them against the bullets. But then again, maybe he'd intended to kill. Maybe Paolo had intended to kill. She wasn't inside their heads, and no juror ever would be. That was something the boys would have to carry inside for the rest of their lives.
Yet, the reality of it was that Paolo and John-Michael had started to dig a hole into which all four of them had immediately fallen. Climbing out wasn't going to be easy. Like Paolo said, they all had their hands dirty now.
The phone began to buzz once again. Lucy watched as Maya took the call. This time she just listened, said a quiet, "Okay, but we're going for five minutes," and then ended the call.
"They're going to stop at the next major hairpin bend. We should slow to thirty-five miles an hour and then start timing."
Lucy nodded, staring directly ahead. Her eyes felt suddenly dry, like hard stones scraping inside her eye sockets.
_We're actually doing this. We killed a guy. Now we're going to cover it up._
## **PAOLO**
#### MALIBU CANYON, FRIDAY, JULY 3
Paolo pulled slowly onto the edge of a hairpin bend on Piuma Road. With slow deliberation, he turned off all the car lights. Outside, an even layer of clouds glowed with very faint moonlight. Without the cloud cover the Oldsmobile they sat in would have been obvious to any passing vehicle. As it was, there was just enough light not to trip up, but probably not enough to spot a shadowy car and its occupants, pressed to the side of the road.
He turned to John-Michael, who had barely said a word for the whole drive. Paolo wasn't sure what to say to his friend. At some point they'd have to acknowledge that Paolo's action hadn't been intended to kill, whereas John-Michael's was more questionable. Had his friend intended to keep the hit man still during his brain seizure, to protect them from the random firing of bullets? To finish off a man in his dying throes? Or had the intent been to kill? In law as well as ethically, Paolo knew that made all the difference. And yet, John-Michael had very clearly sought Paolo's consent.
They were both responsible. John-Michael had been the one who'd stepped up to the plate, but both of them had agreed about what needed to be done. And now they had to follow it through to the bitter end.
John-Michael spoke with difficulty. "Do you remember exactly how she was, you know, on the road . . . your _friend_ . . . when she was dead?"
Paolo winced at the mention. "Yeah. Don't worry about it. Just help me get the guy out of the trunk. It should at least be easier than it was to sneak him out of the house."
The boys opened the car doors and closed them carefully. Paolo popped the trunk and reached in, grabbing hold of the edge of the living room rug they'd wrapped around the shooter's body. Together they lugged the heavy load, which must have been around two hundred and fifty pounds, out of the car and onto the edge of the road, somewhere ahead of the Oldsmobile. He checked several times, making sure that the body lay at the same distance and in the same position as Meredith's had been when he last saw her—that it looked natural, not staged. A constant buzz of adrenaline kept Paolo from feeling anything but the most remote guilt at the memory of her death and how he'd walked away. It was starting to make sense. If it hadn't been for that experience, he wouldn't have known what to do right now.
In Paolo's mind, it felt like a kind of balance. Guilt wasn't a useful emotion in this scenario. Their freedom, their reputations, and maybe even their lives were in danger. Guilt could be banished with impunity.
John-Michael returned to the car and then came back with a wad of plastic wrap in one hand.
"The blood from under his head," he said. "Traffic cops are gonna wonder why there's no blood on the road." Then with precision he peeled back the layers of plastic until the clotting blood in the center of the makeshift package was revealed. Without uttering a word, he slid the ooze under the wound on the man's head. Paolo switched on the disposable cell phone and used the dim light from its screen to help John-Michael. Less than half the blood that coated the plastic seemed to make it onto the road.
John-Michael leaned back to survey his handiwork. "It's not as much blood as was on the rug, or even on the plastic wrap."
Paolo shrugged. "At least there's _some_ blood. You really think they'll be checking a thing like that so carefully, when it looks like the dude's been whacked by a hit-and-run driver?"
"I guess not." John-Michael thought for a moment. "What are we going to do with the rug?"
"We gotta ditch it. A long way from here. The plastic wrap, too. Should probably wash it off first, burn it, dump the remains in the garbage. Anything that's touched his blood can be linked to us _and_ to the shooter."
"You don't want to take the rug home and clean it, like, industrially?"
Paolo shook his head. "No way. Think about Grace and Candace."
"That's what I am doing. They're going to wonder."
"We'll tell them you fell asleep and peed on it. How hard could it be to get another one? It's probably from IKEA, like everything else."
John-Michael stood up. "We're not saying I peed on it, asshole."
They began to head back to the car, Paolo carrying the rolled-up rug under one arm. He propped it against the rear passenger door, facing away from the road so that it wouldn't be seen by a passing car. Then he rounded on the trunk and was about to close it when he spotted something peeking out from underneath the rug that lined the trunk's base. He raised the flap to reveal a black canvas duffel bag. He felt for the zipper and tried to pull it open, only to find a padlock had been used to secure it. He lifted the duffel bag, experimenting with its weight. John-Michael joined him.
"What's up?"
Paolo handed him the bag. "Feels heavy. But what's inside is, like, all blocky. Books, maybe. Or paper."
They looked at each other as revelation struck. "No way," breathed Paolo. "You think?"
John-Michael nodded. "Money? Uh, yeah, just a little bit. I doubt that hit men get paid by check."
"How much do you think it is?"
John-Michael tested the weight. "A lot," he said, handing it back. "Put it back. When the shooter's people find the car they're gonna want that cash."
"You think they'll know how much is inside?"
John-Michael gave Paolo a curious look. "You're seriously talking about stealing from the kind of people who carry guns and bags of cash? Who'd you think that woman hired to hit Lucy? The local neighborhood watch? This has organized crime written all over it."
"You're saying Dana Alexander is _connecte_ _d_?"
John-Michael shrugged. "She sure knows who to ask for a job like this. This is way too much money for a single hit."
Incredulously, Paolo said, "Where are you getting all this?"
Sighing, John-Michael said, "It wasn't all sweetness and light, living on the streets for a year, y'know? I met my share of bad men. Never got involved, but I had offers. There's always work if you're willing to do anything. And I'm telling you—compared to this bag of cash, it's pocket change to arrange a hit on a civilian like Lucy. This amount of money—that's from something else. Maybe no one's gonna miss a hit man. But money like this, someone is gonna follow."
The cell phone in John-Michael's pocket was buzzing. He checked the screen and glanced up at Paolo, panic in his voice. "It's from the girls. Car on its way. Come on, Paolo, move it."
Paolo glanced anxiously into the road and closed the trunk. The duffel bag was still in his right hand. "But the mob or whoever, they're not gonna know we took it. How could they?"
The way Paolo saw the situation—they now had to use every possible resource to save themselves. Money could help. It could buy them cars, protection, somewhere to go if they ever needed to run. Money was _insurance_.
"Could we please just get inside?" John-Michael said tensely.
He was right. They'd scarcely closed the doors to the hit man's Oldsmobile behind them when an RV sailed by on the opposite side of the road. Paolo held his breath as it passed. Was the driver going to stop? Had he perhaps caught a glimpse of a body in his headlights? But after a moment the taillights disappeared around a bend, and then another.
"Better wipe down the car for fingerprints while we wait for the girls to come pick us up," Paolo told John-Michael.
"And the guy's cell phone," John-Michael said, adding wearily, "and his gun. We'd better wipe it all down."
"Yeah, I'm not sure we want to leave those things in the car," Paolo said. "It's evidence that could match up with the bullet hole in the wall of our house."
John-Michael thought for a moment. "His people will be expecting his stuff to be with him."
"Why?" challenged Paolo. "Who's to say someone didn't stop by and just take everything outta the car, without reporting the accident?"
Without warning, a ringtone began to sound. Paolo and John-Michael froze. "That's not our phone," John-Michael whispered. "It's coming from the glove compartment."
"What the hell?" Paolo cried out, much louder than he'd intended. His heart was racing, he realized, at the mere spike in John-Michael's fear.
John-Michael's voice was shaking as his hand withdrew from the glove compartment. He opened his fist right in front of Paolo's face. In his palm lay a vibrating, ringing smartphone.
Paolo blinked and then stared at John-Michael, mystified. "Well?"
John-Michael turned the screen toward him. "I'm not answering it!"
"Me either!"
"Well, someone better!"
After another five seconds, the ringing stopped.
"This . . . could be bad," John-Michael said in what sounded like a constricted throat. No more words seemed to want to leave his mouth.
Paolo looked from the phone in John-Michael's hand to the glove compartment. "Hey, look," he said, smiling. "A key ring. You suppose this fits the padlock on the duffel bag?"
"Forget about the bag of money for one second, could you?" John-Michael said, exasperated. "Paolo, look at this phone, will you? At the house, the shooter was using his second cell phone. A _burner_. Come on, man, what the hell is wrong with you? Have you lost your mind? He had _another phone_. Don't you understand what's going on here?"
A dull realization began to wash over Paolo, stripping away all the bravura and hubris, until he felt nauseous from it.
"Oh no," he said, his voice empty. "Oh God, no. Maya said he didn't make any recent calls from the burner . . . but if he used this _smartphone_ to call someone else, then . . ."
John-Michael's eyes seemed to bulge in his head. He stared at Paolo in horror and disbelief and said, "Then that _someone_ is gonna come looking for the money."
## **LUCY**
#### MALIBU CANYON, FRIDAY, JULY 3
"He said what now?"
Lucy glanced over at Maya, who apparently couldn't believe what she was hearing on the cell phone. "Maya! I need to hear this, too. Put him on speaker."
". . . plus there's the fact that this _fancy_ smartphone has been switched on this whole time, so we probably need to get rid of that, too," John-Michael was saying as the phone's speaker kicked in. His voice sounded flat, resigned, dull. It felt totally at odds with the magnitude of what he seemed to be saying, which made Lucy wonder if maybe she'd misunderstood.
"Excuse me, you're saying he had a second cell phone?" Lucy asked, incredulous. "'Cause if he had a second _goddamn_ phone then we are screwed six ways to Sunday!"
"Yeah," John-Michael admitted. "Yeah, we may as well just accept it; the shooter's people are going to know by now that he didn't finish the hit, for whatever reason. Look, we're headed your way on foot now. If you leave in about a minute we should meet up somewhere far enough away from the shooter's car that if some other car happens along, they won't see the two cars together."
Maya interrupted, "You've got the rug? And the plastic wrap, and all the dead guy's stuff, his gun and everything?"
"We got it all," John-Michael confirmed. There was a wry chuckle. "We're going to be hitting a whole bunch of Dumpsters tonight, I'm pretty sure."
Lucy released a trembling breath. She turned the key in the ignition. Then she was driving Paolo's Chevy Malibu at a steady thirty miles per hour around the tight bends, back toward the spot where they'd last seen the boys parking the hit man's Oldsmobile.
Beside her, Maya seemed quiet and thoughtful. Lucy couldn't understand how she did it. If it wasn't for the distraction of having to drive, and rather cautiously at that, in the dark on such a high, twisting road, then Lucy knew she'd be going nuts. She felt a raw, itching sensation, as though a nail file was being rubbed gently against the insides of her hands and wrists. A constant irritation that made her want to scream and twist in frustration. And it was slowly building up. With things the way they were, however, there was no way to release her feelings, at least not as violently as she needed to. It would totally freak Maya out, for one thing.
"What if the shooter told whoever he talked to on that second phone that he was about to go into our house?"
Lucy tensed. "I . . . uh, no, actually . . . What d'you mean?"
Maya spoke slowly, considering each word. "It means that there's a potential witness out there. That's a best-case scenario."
Lucy scarcely dared to ask. "And the worst?"
"And the worst," Maya said softly, "is that the guy who's been trying to call the _smartphone_ , is the same guy who Mr. Shooter was about to call right in front of us. Remember? To get help with the four teenagers he'd gotten all trussed up like chickens."
"Oh jeez," Lucy breathed. "In which case we're finished."
"Unless we hit back first."
It took several seconds before Lucy was able to grasp what Maya was saying. Lucy was sure there was something wrong with Maya's icy calm. What kind of person could be so cool under such horrific pressure?
"I don't . . . what are you saying, Maya?"
Maya rubbed the back of her neck with one hand. "Just a thought. We should probably discuss it with the guys."
When two minutes later they rolled to a stop by the side of the road, the boys dropped the rolled-up rug into the trunk of the Chevy and Paolo swapped into the driver's seat, giving Lucy's hand a quick squeeze before she moved into the back with John-Michael. As before, Lucy didn't respond.
Abruptly, Maya said, "What are we gonna do about the second guy?"
Lucy watched Paolo and John-Michael exchange a wary look. "Yeah," Paolo admitted. "We've been talking about that."
"Good," Maya said firmly. "Because it's a major hole."
"Hole?" Lucy asked uneasily.
"In our story," Maya said. "Think about it: we're hoping that the cops find this guy's body and assume it's a hit-and-run. They've got a body in the road, a massive blow to the head, blood on the asphalt. The dead guy's car is _empty_ , fingerprints all wiped. So what do they conclude? One hit-and-run car and another one with some kind of fly-by-night thief? Or the hit-and-run car who also happens to be the thief?"
"Either one works," Paolo agreed.
Maya nodded. "Sure. Until someone comes forward and says that he or she talked to the dead guy when he was still alive in Venice Beach."
Paolo appeared to think for a second or two before he replied. "Not an issue. The shooter could have done his thing, then driven here afterward."
Maya shook her head and said vehemently. "No, it _is_ an issue."
"A hit man's buddy isn't going to talk to the cops," he objected. "He's probably a hit man, too."
She took a deep breath and then exploded. "We don't know for a fact that he is a hit man, and even if he is a hit man, that fact is not necessarily known to the cops, and even if it _is_ , the cops might be dirty, and even if they aren't, the existence of a potential witness against us, Paolo, that is something I know is gonna keep me awake nights, even if the guy doesn't involve the cops but instead decides to come looking for us himself!"
Behind them, a car approached. They all ducked down low and waited until they'd heard the car pass. It didn't appear to slow down. Once the car's taillights had disappeared around the bend they'd just taken, Paolo started the car hurriedly. He began to drive. "We've got to find a way off this road. We're just too prominent here."
"You're right," John-Michael said. "The cops will want to interview anyone who was on this road tonight. All someone needs to say is that they saw a Chevy Malibu parked nearby."
"What do you suggest we do then, Maya? Since it's obvious you've thought it through so much better than the rest of us?" Paolo insisted, his attention on the road ahead.
Calmer now, Maya said, "Whoever called Mr. Shooter on that second cell phone is going to be wondering why they didn't get the confirmation call."
"What confirmation call?" Lucy asked.
From the front passenger seat, Maya gave Lucy a sympathetic look. "You know. The one where he tells _whoever_ that the job is done."
"Oh," Lucy managed to say, barely suppressing her disgust.
"Assuming that whoever it was on the call had anything to do with the hit on Lucy," replied Paolo.
"Will you just accept it, man?" John-Michael said. "That's our nightmare scenario, so obviously we have to take it into account." He looked across at Paolo. "Okay, Paolo? Can we please agree?"
Paolo, however, seemed to be miles away. It took a couple of prompts from John-Michael before he responded, and even that was reluctant and noncommittal. "I guess."
Lucy noticed that John-Michael was staring at Paolo with a measure of frustration. So, he'd also picked up how strangely detached Paolo was choosing to be, all of a sudden.
"We should tell them about the bag," John-Michael said. The tone of his voice made the hairs on the back of Lucy's neck stand up. There was an unmistakable hint of threat.
Paolo winced, but he didn't turn around. He didn't say a word. When Lucy looked at John-Michael, he shifted back to the other side of the passenger seat, avoiding her eye.
"What bag?" said Maya curiously.
"Yeah," Lucy said, more insistent. "Tell us about the bag."
"We found a canvas duffel bag in the shooter's car. Cash," John-Michael said. He spoke slowly. "A lot of it. We're thinking half a million at least."
"You counted it?" Maya asked, baffled. "When?"
"We took a quick look before we transferred the bag to the trunk of the Chevy. Each roll looks to be about twenty grand."
Into the taut silence that followed, Maya whispered, _"En la madre_. _"_
It was suddenly crystal clear why Paolo was so distracted.
"A half million dollars? You're thinking of keeping the cash!" Lucy cried, lashing out with a foot into the back of Paolo's seat. The car swerved for a second and Paolo swore loudly.
"Lucy, Lucy." John-Michael gripped her forearm. "We already made it look like the shooter's car was robbed anyhow. We had to—or else they'll wonder where his phones are, his gun. The shooter's people will assume that the thieves took the bag with the money."
"And then what?" Lucy grunted, pulling free of him.
John-Michael shrugged. He didn't look nearly as relaxed about the idea as his words suggested. "Then I guess whoever has a claim on the money comes looking for anyone who might have committed the robbery or the hit-and-run."
"But what are the odds they ever find us?" Maya said pensively. "We've seen, like, two cars drive by. The Chevy has been in the dark the whole time."
Lucy could see with a sudden and dreadful clarity where Maya was going with this. She tried to swallow. "No," she said very quietly. "We've got to go back. We have to put the bag back in the shooter's car exactly how you found it, leave the cell phones, leave everything."
"We can't leave those things," Paolo snapped. "They could lead right back to us. The money, _maybe_ , we could leave. It was underneath a rug in the trunk so someone who was just stealing whatever they could lay their hands on inside the car might not notice it . . . but the cell phones, Lucy, and the goddamn weapon? They have to go!"
"Stealing cell phones is a knucklehead move even by your standards, King," Maya said, exasperated. "Cell phones can be traced."
"Which is exactly," Paolo said between clenched teeth, "why we're going to destroy them."
"This is awful," breathed Lucy.
John-Michael immediately moved to her side and put one arm around her shoulders. "I know, Luce, but we're gonna think it through, all of us. Together. We're gonna come up with a solution. We're not gonna let anything bad happen to you."
Lucy wanted to shake his arm away again. She was on the verge of tears. Something terrible was definitely going to happen to her, whatever her friends did. Fear was building inside, all mixed up with a horrible sense of vulnerability. She recognized both. Her recently dredged-up memories of Tyson Drew's death were stirring into the mix. If she went to the authorities with what she knew, she risked being killed. The witness protection program would be her only hope. But she couldn't talk about any of it, not while the atmosphere inside the car was dense with such pigheaded determination.
They talked about solutions but didn't seem any closer to one. Lucy felt her insides squirm like roiling snakes. She was afraid that they were about to take an even darker path than any of them might have imagined was possible. People lost their way, in times like this. They got lost so bad they never found their way home.
## **MAYA**
#### MALIBU CANYON, FRIDAY, JULY 3
"Give me the guy's other phone," Maya said. "I want to see it: the smartphone." John-Michael took the object from his pocket and handed it over the front seat to Maya. Her fingers moved nimbly over the keypad. "Hmm. Classic hackable password. Huh." She paused, shocked at what she found. Within the next two seconds she was shaking with rage and disbelief.
She turned to face them in utter dismay. "Are you guys actually this stupid?"
The other three stared at her. John-Michael sputtered, "Maya, what is your problem?"
" _FoneTrackr_?" she said, eyes bulging. "Why didn't you mention right from the beginning that his phone was running that app?"
John-Michael gave a helpless shrug. "I have no clue what you just said—I have an iPhone."
"This brand of phone has an app you can use to locate and manage any of its devices from a remote location. This smartphone," she pronounced heavily, "has been GPS-tracked _the whole time_. What did I tell you about cell phones? So, the other guy? He already knows that Mr. Shooter went to Venice. He knows he parked near the house. He knows he stalled awhile in Malibu Canyon. And now," she finished, "now he knows where we are, too. Paolo, stop the car. We gotta go back."
Anxiously, Lucy said, "Go back? I thought you didn't want to go back."
"To replace the money, no. But now we have to," Maya said shortly. "Right away."
"Maya." John-Michael struggled to stay polite. "You gotta explain why you want to go back."
"Seems pretty obvious to me," came Paolo's voice from the front. "The second guy is coming for us."
"For us, or for the cash," agreed Maya.
"Good," Lucy said. "You're finally seeing sense. Put the money back and let's get out of there."
Maya continued to check the phone. "There's a voice mail," she said, putting the device to her ear.
"It must be from when that call came in," John-Michael said, "when we found the phone."
She listened to the message. "Omigod."
Lucy said, "What?"
Maya played the message again, after switching on the speakerphone function.
_"I'm at the house. Our friends seem to be out for the evening. You want me to wait? What is going on with your location? Why aren't_ __ _you here, man? Getting a little concerned about our special delivery. Check in as soon as you get this."_
"You hear that?" Maya demanded. "He's _at the house_. Our friends are out for the evening? That means Grace and Candace didn't get home yet. The _special delivery_?"
"The money!" murmured John-Michael.
"We have to get back to Venice!" Paolo cried. "Grace! Candace! They're in danger!"
"They are if he stays there," Maya said.
John-Michael spoke quickly. "So we get him to leave."
"And do what?" Lucy said.
No one answered at first.
"I can think of one solution," Maya said mildly.
"Which is?" John-Michael said.
"Like I said to Lucy a while back. We strike back first," she replied, still perfectly calm.
Lucy snorted. "Seriously? You want to _kill_ him?"
Maya looked over her shoulder at where Lucy sat with her back pressed to the rear passenger seat. "No—I don't want to kill anyone, Luce. But I don't want a dude with a gun to hang out at our house, waiting for Grace and Candace and eventually us. I'd do pretty much anything to stop that from happening."
"You're not shooting him," Paolo warned.
Maya observed bluntly, "I'm the only one who knows how to handle a gun."
Sardonically, Lucy said, "So, this is what's happening right now, is it? We're driving back to Venice and shooting a guy. Well, I gotta say, this sure has turned into an interesting evening."
"We're not going back to the house," Maya said. "There, he has the advantage over us. He can lie in wait. We're going back to where we left Mr. Shooter, Mr. Hit-and-Run Victim."
Lucy shook her head. In the wildness of her eyes, Maya caught a glimpse of desperation. "Okay," Lucy muttered. "This isn't making any kind of sense."
"The second guy is at our house—probably," John-Michael said. "Or at least he was when he made that call. We have to get him away from there."
"Exactly," Maya said. "I say we lure him to the Oldsmobile. When he sees the car, and his dead friend, he might believe he finished the hit on Lucy, then started to drive home or wherever and got taken out by the hit-and-run."
"Really," Lucy said skeptically. "He's gonna believe that? He's not gonna wonder what his friend is doing driving around in the _middle of goddamn nowhere_?"
"Maybe he's disposing of a body," John-Michael said. "I mean— _we_ were."
"The thing is," Maya said, "the guy doesn't have a clue what happened to his friend, apart from that he was at our house. He knows he left the house—I'm gonna assume he thinks his friend was alive and well when he did."
"Why's he gonna come running?" Lucy said. "Didn't he ask, on the message, if he should wait at the house?"
"That's why we have to lure him away," Maya agreed.
"Send a text," Paolo blurted. "Something vague. Like 'need help with the cash.'"
"No," Maya said decisively. "Let's go with John-Michael's idea. Let's imply that Mr. Shooter is out dealing with a body. Maybe more than one."
"Good point. He was about to kill all four of us," John-Michael said. "If his buddy gets the chance, I'm sure he'll finish the job."
"Okay," Maya said, typing on the phone's keyboard. "How's this? _All done. I got some heavy lifting to do. Could use some help. The package is safe with me._ "
"It's good," decided John-Michael.
"Yes," Paolo said. "Send it right away. And then we need to call the girls. Stop them from going home."
"Oh, how's that gonna go?" Lucy said. _"Hey, Grace, Candace, don't go home, m'kay? There's a killer on the loose."_ She took a huge breath. __ "Or, hey, you know what, guys? We could call the frikkin' police!"
"And say what, exactly?" demanded Paolo, furious. _"Yeah, so, we, like, totally killed this guy who was in our house with a gun and everything, but we decided to make it look like he'd been hit by a car up in the hills, and, like, now he's sent this real salty dude to kill our friends so, like, could you kind of take care of all that for us, m'kay?"_
Out of the corner of her eye, Maya could see Lucy seething, arms crossed tight across her chest. "You believe you could actually shoot someone? Until he's dead?"
Paolo exhaled through his nose. "It's him or us. And we are not the ones who started this."
"I'm not okay with going to prison for killing a guy who tied up me and my friends and threatened us with a gun _in our own house_ ," John-Michael said. "That's for damn sure."
"You can't be sure that would've happened," Lucy shot back. "You both _could_ have gone for Stand Your Ground."
"Lucy, nothing's been _sure_ since that dude walked into our house," Paolo said in a harsh whisper. "That was then. This is now. Things. Change."
Silence. Maya took a few breaths. Paolo was right—their options looked bleak.
"Lucy," John-Michael said soothingly. "We're _all_ scared. But we have to deal."
"Let's get this clear now," Maya said to Paolo. "How far to where you left the car?"
"Six or seven minutes," he replied.
"We have to act fast," she said, watching the screen of the smartphone. "The other guy will get the message. Hopefully he'll start out for Malibu Canyon right away."
Lucy leaned against the window and looked over at Maya. She seemed vague and lost. "This all happened because of me," she muttered.
Maya closed her eyes, turning to face the front of the car. If she'd followed up on her suspicions about Ariana, this might have been avoided. She might have been able to persuade Lucy that her rehab buddy was a spy and gotten her to kick Ariana out of the house a week earlier. Then she wouldn't have known that Lucy had finally gotten to the point of questioning her own memories of the night of Tyson Drew's murder.
If only the offer of investment from Alexa Nyborg hadn't happened when it did. But Maya had to admit that the investor's offer had entirely consumed her attention, from the minute the prospect had entered her life. She should never have spied on her friends. She should have acted sooner on her suspicions of Ariana. She shouldn't have let her work take absolute control of her life.
Why was everything going so horribly wrong? It was so unfair, Maya wanted to cry. All she'd ever wanted was to work hard, to create something extraordinary, and to make enough money that she and her mother would never again be under the thumb of a person like Dana Alexander. And she had worked—like a slave. Almost every waking minute she was either in school, studying, or coding.
But for what?
To end up going to prison for conspiracy, at best? Or even worse—to be killed by Dana Alexander's hit man?
Maya drew a ragged breath. There'd been a ton of bravado in what she'd said about killing the second hit man. Words spoken before she'd really thought them through. And yet—there was an implacable logic to it. What other solution was there that wouldn't land all six friends dead or four of them in jail? Now they were hurtling forward to a destination from which she knew there would be no return.
The smartphone in her hand began to vibrate. Maya read the text and looked up.
"He's on his way."
## **JOHN-MICHAEL**
#### MALIBU CANYON, FRIDAY, JULY 3
The hit man's Oldsmobile was just barely visible, farther along from where they'd parked at the side of Piuma. "Give me the gun, John-Michael," Maya said. She reached over the front seat, holding her hand out. John-Michael didn't move. He could feel the gun beside him, digging into his thigh. For a few seconds, no one said anything.
Then Paolo guffawed. "You're being ridiculous."
They'd switched off the interior lights, but John-Michael could hear the tension in Maya's voice when she replied with a harsh whisper, "You think because you've seen a gun fired on TV, that it's _nothing_? C'mon, Paolo, I'm the one who's fired a gun before. I'm a good shot."
"I don't doubt your ability to shoot," Paolo said. "I doubt your ability to kill. Take the compliment, Maya." He held his hand out, too. "The gun, JM. Hand it over."
John-Michael hesitated. "Why not Maya? Is this some kind of macho shtick, Paolo?"
Paolo groaned. "We're _so_ not getting into gender politics right now. Just gimme the gun, John-Michael. I need to get into position."
John-Michael kept his hand on the revolver. "No, Paolo. You have to explain. Why not Maya?"
"Are you out of your mind?" Paolo's face twisted in disgust. "What kind of a person lets a fifteen-year-old girl do a thing like this, when it was his fault in the first place?"
"We all agreed that it needs to get done," Maya said briskly. "We're protecting ourselves. And I'm the best woman _or man_ for the job."
Beside him, John-Michael felt Lucy stiffen.
Paolo swore. This time he lunged at John-Michael. "Give it to me!"
John-Michael picked up the gun and held it in his left hand, gingerly, as though it were a piece of fetid trash. The gun was well out of anyone else's reach as he threaded a finger through the trigger. "If you'd killed that guy outright," he mused, "then none of this would have happened, Paolo. You could have stuck with Stand Your Ground or castle doctrine or whatever and everything would have been fine."
Maya said, "Until Dana Alexander sent someone else."
"But Lucy would have told the cops everything _tonight_ . . ." John-Michael said.
". . . and they'd have put me in witness protection," Lucy finished, finally speaking up.
John-Michael's fingers tightened around the handle of the gun. "Now we're both killers, Paolo," he reflected contritely. "You and me." In the gloom of the unlit car, he peered at Maya. "And that's why it can't be you, Maya. Paolo and me, we did this. We're the ones who wanted to cover up what we did."
Paolo breathed a huge sigh. "Yes!" he said. "Exactly. Give me the gun, John-Michael. I started this. I'm going to finish it." He leaned across the front seat and grasped John-Michael's left hand in both of his. John-Michael resisted for only a moment before allowing his friend to take the weapon.
Paolo opened the driver's-side door and stepped out onto the rough ground at the edge of the road. "Drive far enough to get out of sight, but make sure you can see when our guy shows up."
"Take one of our cell phones," Maya said, offering the one in her hand. "We can use it to text you when he's close."
" _If_ he happens to be approaching from our direction," John-Michael remarked. Maya _tsked_ at this, which irritated him a little. It made sense to try to anticipate all the angles, not to dismiss anything.
With an air of vague disinterest, Paolo took the phone. It was as though he could no longer see his companions in the car, as though they were no longer connected to him.
John-Michael recognized that feeling.
_Is this how everyone feels when they're about to kill someone?_
He watched Paolo walk the short distance to the shooter's Oldsmobile. No one in the car moved. John-Michael looked at Lucy, beside him in on the backseat. It was obvious from her body language that she had zero intention of driving the getaway car. Maya didn't have a license, which left only him. John-Michael heaved a sigh and climbed over into the front seat. He started the car and drove past the Oldsmobile and along Piuma, until he reached a spot about sixty yards away where the side of the road was broad enough for the car to completely leave the asphalt. He pulled off the road as well as he was able and then switched off all the lights.
All around them was darkness and the sounds of the hills: crickets, rustles from the undergrowth, the rumble of distant traffic. Inside the car, no one made a sound. Eventually, though, Lucy spoke again. John-Michael could actually hear the dryness of her mouth.
"And you're just going to let him do this?"
Neither Maya nor John-Michael answered. Lucy responded only with a resigned sigh.
John-Michael said, "You got a better idea?"
"All this is to protect you, Lucy," Maya pointed out.
Lucy gave a short, breathy laugh. "No. When Paolo hit the guy with the shovel it _was_ to protect me. Now it's to protect _you_ ," she said, stabbing a finger at John-Michael.
"It's to protect all of us, Lucy," John-Michael said wearily. "How do you not see that?"
"We're the same as Dana Alexander, don't _you_ see _that_? She sent some goon to shut me up, now we're going to shut up a goon . . . we're doing exactly the same thing."
"Dana Alexander is ready to see an innocent man get executed for a murder she committed. She was ready to see you hurt or dead, and God knows how many more of us," John-Michael said. For the briefest instant, it was on the tip of his tongue to remind Lucy that if only she'd tried to recover her buried memories earlier, Dana Alexander could have been taken out of the running before she was ever a threat to any of them. But that would be cruel. Lucy already had to be suffering enough guilt about the whole situation. In a flash of insight, John-Michael wondered if this was why she seemed so blocked when it came to taking action.
Lucy had been treading water for almost a decade, hoping and praying that the shadow of murder in her past would remain forever hidden. With John-Michael, it was different. He knew that hope wasn't always enough. There were times when you had to be prepared to take that extra step. To _push_.
He stared into the black road, nerves jolting as a car approached. All three fell silent, waiting to see if it slowed down. Once again, they dipped below the windows as the car passed. It didn't slow down. They sat upright and breathed again.
"I can't take much more of this," Maya admitted.
John-Michael ignored her. He ignored Lucy's frustrated squirming. Instead, his thoughts settled on Paolo, alone, waiting in the dark. Another car would drive up soon, park behind the Oldsmobile. A man would get out. Unsuspecting, he'd step up to the dead body of his associate. Perhaps he'd even bend down to touch him. Had the two been friends? Family, even? Then Paolo would step out from his hiding place. Arm outstretched, he'd pump three bullets into a defenseless man. And that would be it.
A matter of seconds that would transform them both. The hit man's associate would be moved from the land of the living into the land of the dead. And Paolo?
The thought slunk through John-Michael's chest until he could feel it move like ice water through his guts. He was already a killer. Twice. Now Paolo was about to join him.
## **JOHN-MICHAEL**
#### MALIBU CANYON, FRIDAY, JULY 3
Time passed—too much time. Something snapped inside John-Michael. A decision. Before he could begin to doubt himself, before the echoes of alarm, the warnings from the two girls in the car could take effect, John-Michael opened the rear passenger door and leapt out of the car. Just as he reached the Oldsmobile, he saw the tracks of white headlights on the road. Behind the car, John-Michael dropped down to one knee, tucked against the front wheel. He breathed slowly, waiting for the car to drive by.
When it slowed to a crawl John-Michael held himself rigid. He could hear the crunching of rubber on the pieces of gravel that had been thrown up onto the road. He felt the pounding of his heart inside his rib cage. After the slowest fifteen seconds that John-Michael could remember, the car's tires screeched into action. Then it was gone.
From somewhere in the shadows at the side of the road, John-Michael heard a hissed warning. "Dude, get out of here! I almost shot at you."
John-Michael looked around, locating the source of the voice. He began to walk toward where he could just see Paolo stepping into the road.
Somewhere up the road, the car that had just passed by was turning around. John-Michael bolted toward Paolo, and tackled him. They both dove and slumped against the thick trunk of a tree. Then suddenly, John-Michael was stepping out onto empty air, falling. Too late, he realized that the ledge behind the tree was narrow. His second foot scuffed the edge, sliding down a sickening angle. His entire body began to follow. Then, like a solid rock, Paolo's hand shot out, grasped his upper arm and held on, the fingers gripping him painfully tight.
Paolo cursed as he hauled John-Michael back from the cliff edge, one arm wrapped around a low branch, the other dragging John-Michael up. With one huge tug he hauled John-Michael to the ledge. The force of the motion slammed John-Michael against the tree, trapping him between the tree and Paolo's own tensed-up torso and thighs. The ground sloped downward just behind them, so that both had to lean hard into the tree to stay upright and hidden.
"What are you doing here?" he heard Paolo whisper roughly against the back of his neck. "This isn't part of the plan."
"I won't let you do this," replied John-Michael as evenly as he could. He twisted around gradually until he was facing Paolo. And then he reached for the handgun. Paolo seemed paralyzed for an instant, and then tried to pull his hand away. But John-Michael resisted, grabbed Paolo's wrist, and looked directly into his eyes. "You're not a killer."
Paolo shook his head, once. His eyes brimmed with confusion. The sound of a second car scraping up loudly behind the parked Oldsmobile made them both jolt. For just a second, Paolo's attention flipped to the road.
John-Michael moved swiftly, pinning Paolo in place for a moment with one elbow under his throat while his hand gripped Paolo's wrist joint and twisted. As Paolo let out a sharp gasp of pain, he slid firm fingers around the hard metal of the revolver, wrenching the gun from Paolo, who reacted too late. John-Michael jerked sideways, swerving to keep his balance, so close to the edge. The gun was now in his grasp. He gave Paolo a quick shove to counterbalance himself, then shifted out of reach. Then they stopped moving, their attention focused on the newly arrived car.
Paolo's sharp exhalation was the only outward sign of his anger. The need for silence seized them both. Less than ten yards away, the door of the second car was opening.
John-Michael transferred the revolver to his right hand. He held the handle firmly, feeling the weight of the weapon.
_Point and shoot._
He moved farther into the shadows, away from the precarious edge behind the tree.
Paolo remained pressed against the tree. His eyes followed John-Michael, yet he made no attempt to stop him. All he said was "Don't."
John-Michael shook his head. Everything was so much clearer now.
Paolo had some big problems, sure. But he'd never actually taken a human life. Whereas John-Michael had done it twice. And one of those hadn't begged for it.
"It has to be me," John-Michael mumbled. It had to be done—for his housemates. A light-headed sensation began to flood him. As though reality was separating out, trapping him on the wrong side.
The man inside the second car was stepping out onto the gravel at the side of the road. There was a sound of slow footsteps crunching toward the dead body. The headlights of his car, some kind of sedan, had been left on. They lit up the driver from behind, casting a shadow across the dead body. John-Michael watched the man stoop briefly, to give the dead body an almost cursory check. The man seemed to pause over the hit man's burner phone, which they'd positioned close to the shooter's right hand.
To John-Michael's surprise, the man straightened up and took a quick look up and down the road. Surely he hadn't been down there long enough to check whether his associate was actually dead? Why wasn't he calling 911? In fact, his demeanor was entirely casual. He stood for a full minute in silence, apparently to satisfy himself that the road was indeed empty. Then he made a prowling move around to the trunk of the Oldsmobile.
John-Michael lurched forward. The ten yards that had separated him from the man were rapidly lengthening. He had no idea if he could make an accurate kill shot at that distance. It would be even harder if the target was retreating. Moving softly, he closed as much distance as he dared behind the man, following him halfway to the trunk before his target froze. Slowly, the man began to turn around.
"Get your hands up," John-Michael heard himself speak. "Up where I can see 'em."
He could see the man's face now. He looked to be around thirty years old, maybe six feet tall, with dark, short hair and a square jaw that was clenched in determination. He wore a dark suit over a white shirt. Not exactly the dress code of a hit man, John-Michael thought fleetingly.
The man hesitated. "Did you have something to do with my buddy here dying?"
"Don't move," grunted John-Michael.
The second man's arms were raised, obligingly, palms open and facing John-Michael. He began, very slowly, to walk toward the teenager.
"Looks like a hit-and-run, kid. Did you do this?"
"Stay where you are!" John-Michael cocked the gun threateningly.
"Calm down. Does it look like I'm armed? I came by to see what's up with my friend here. Well, it looks like he's a goner." The man's tone was friendly, relaxed. Still he kept moving closer toward John-Michael and the gun.
_Do it now. Shoot him. Do it._
John-Michael's finger was on the trigger, his gun arm outstretched, his right hand sticking out at his side, for balance. He willed himself to squeeze the trigger, kept the words running through his brain. Yet somehow, it was as though there was a blockage in his neurons. His mind was issuing the commands, but the cells of his body refused to obey.
"Practically point-blank range," the man said quietly. He was close enough now that John-Michael could shoot him straight in the forehead. Their eyes met, John-Michael's angry and frustrated, the man's wide and puzzled, almost hurt. "How old are you, kid? Sixteen? Seventeen? Look, I know that's my friend's gun you're holding there. I recognize it. Which suggests to me that you don't own a gun. Probably never shot one, am I right?"
"One more step and I will end you," John-Michael managed to grind out. "Now who the hell are you and what are you doing here?"
The man raised a single eyebrow. His forehead was less than six inches from the barrel of the revolver now, both hands open and held at the side of his face.
John-Michael bristled with aggression. His left hand joined his right on the revolver. This was it. No mercy. He had to get rid of the witness. Suit or no suit, this guy was clearly in league with the hit man.
"Oh," the man said softly. _Dangerously_ , John-Michael realized, but far too late. The man's mouth opened and he gave a half smile. "Questions . . ."
What happened next was a blur of activity that John-Michael could scarcely comprehend. Somehow, the second man went from a helpless victim inches from the business end of a gun, to moving at lightning speed. The man snatched at the gun, clamping John-Michael's wrist in his hand. Suddenly, John-Michael was pulling at the trigger only to have it fire upward. The sound of the gunshot felt deafening, roaring into the empty canyon, seeming to echo for seconds. It turned his legs to jelly. Then, inexplicably, the gun was in the other man's hand.
As if through a long tunnel, he heard the order to get down on his knees. A jagged breath escaped him as he stumbled to the ground, feeling pieces of gravel digging into his kneecaps.
"All right, calm down," the man was saying, "everything is okay. So—we've learned that some of us know how to do this kind of thing and some of us don't. That's all right. Between you and me, kiddo, I've been in this business a long time. It's quite a grave matter, to shoot a bullet into a perfect stranger. That's a solid piece of mental preparation, right there. Unless you're some kind of psychopath."
Then John-Michael heard the catch in the man's breath. " _Are_ you a psychopath?"
## **PAOLO**
#### MALIBU CANYON, FRIDAY, JULY 3
Paolo gripped the tree trunk so viciously that splinters of bark pricked beneath his fingernails. From the instant the man had wrested the revolver from John-Michael, Paolo had held back, his limbs locked against the tree. Fear and fascination mingled as he witnessed the older man's almost balletic movements in snatching the weapon. It was obvious—the guy was an expert in some kind of martial art. Even with a gun stuck in his face, he'd disarmed John-Michael in less than two seconds.
He watched John-Michael, the misery plain in his features, lit up by the second man's car headlights. The man now towered above him, the gun aimed squarely at John-Michael's head.
"I asked you a question."
John-Michael cringed. "No. I'm not a psychopath."
The man stuck his right hand inside his suit jacket pocket, withdrew something that he tossed over to John-Michael. It took a moment for Paolo to realize that they were plastic zip ties. A charge of pure fear jolted him.
In the steady, neutral voice that he'd been using since he'd taken the weapon, the man said, "Put them on."
Paolo saw John-Michael's hands shake as he did what the man said. He tried to guess from the man's tone and stance whether their lives were now in danger. It was impossible to tell.
What kind of man could disarm someone like that? What kind of person carried plastic handcuffs?
The answers that were forming in Paolo's mind were pretty terrifying.
"Now you're going to stay there, on your knees, while I take a little look around my buddy's car. If I hear you move, I'll shoot, you understand?"
John-Michael nodded. The man sighed in irritation. It was the first aggressive sound that Paolo had heard him make and instinctively, he snuck a little farther behind the tree.
"Answer me, so I know you've understood."
In a small voice he heard John-Michael say, "Yes sir."
The man seemed pleased with John-Michael's sudden submissiveness. He brought the revolver in for a closer examination and quickly opened the cylinder to look inside. "Interesting," he murmured thoughtfully. He tucked the revolver into the waistband of his pants, and reached under his left arm. When his right hand withdrew it was holding an automatic pistol.
"Now we're getting somewhere. If everything is in order we can just finish up here and both get on with our evening. But if I'm not happy, we're going to have a serious problem."
He retreated then, quickly, made his way to the car, tried to open the trunk. When he couldn't, he blasted the lock open with two shots. For a moment, Paolo couldn't see what he was doing. But he could figure it out, and the thought electrified him.
The man hadn't come up to Malibu Canyon to help his friend. He'd come looking for the bag filled with cash.
Before Paolo's legs could obey the urgent message from his brain to go over to his housemate and free him, the man reappeared from behind the Oldsmobile, storming toward John-Michael with teeth bared. Paolo saw the man swing a kick straight at John-Michael's back. Groaning, the boy collapsed.
"Okay, kid, the fun is over. You're about to get the beating of your life. Or you can tell me what the hell you did with my money."
Paolo winced in sympathy as he saw the second kick approach, now farther around John-Michael's back. This time his friend curled up into a fetal position, gasping, trying to get his wind. Paolo fought down the urge to go to his instant rescue. One mistake and they'd both be killed.
"Listen to me. I can count, therefore I know that my buddy fired his weapon twice. One blank cartridge has been fired, and one bullet. Now I also know that he sent me a message, but not with this phone. Also, he didn't answer any calls. You with me so far?"
The man swung in with another heavy kick on John-Michael, who released a low moan, curling up even tighter. For the first time, real anger began to surge through Paolo alongside the fear.
_Stupid, stupid, allowing John-Michael to take the gun._
"Now—and let me be clear—I do this for a living. So when my partner starts to act a little odd I think something's up. You know what else I think? I think _you_ maybe had something to do with my friend being dead. I don't know whether you tried to rob him, or whether you hit his car, or whether—and this seems unlikely—whether you somehow actually took his _own_ gun from him and shot him with it. That's to say, I assumed from that head wound that a car hit him. But maybe I should have looked harder?"
He punctuated this last sentence with two more kicks. John-Michael barely reacted above another lengthy groan. Paolo could hardly bear to watch. But it was obvious that the man suspected something. He wasn't just going to let John-Michael go, nor was he going to kill him quick.
"You're wondering why you didn't kill me when you had the chance," the man said, spitting on the ground near John-Michael. "Well, it's like this. You've heard of Darwin, yes? Natural selection, all that? This lifestyle, it's very, what I would call _Darwinian_. People like you and me, who've been in the situation that you're in now, if we survive it's because we learn. We adapt. I won't lie to you, not everyone makes it. _You_ may not make it. But if you do, you'll know for the next time. _Do not_ pull a gun on someone unless you mean to use it."
Another kick, this time to the backs of John-Michael's thighs. Paolo welled up—tears of empathy and frustration.
The man suddenly slid down on one knee beside John-Michael, the pistol pressed firmly to the crook in his leg.
"Now. Five seconds to tell me where the cash is. Or you take the first bullet, right through the knee."
## **PAOLO**
#### MALIBU CANYON, FRIDAY, JULY 3
There was no way that Paolo could physically tackle the man. John-Michael was going to give up the location of the cash any second now. He'd be beaten into revealing Paolo's position. They'd probably be forced at gunpoint to call the girls.
Then all four of them would be potential hostages. Plenty of spare blood, in case the guy felt like throwing his weight around, killing or maiming one of them, just to show that he meant business.
_All_ their lives would be at risk. And Paolo would be directly responsible.
He had to act before John-Michael gave him up. He crouched low, fumbling for any kind of fallen stick. The ground fell away so sharply underfoot that he had to use one hand to hold on to a low branch of the tree he'd been hiding behind. As the branch bent, Paolo's teeth sunk into his lower lip, his jaw clenched in the desperate hope that the limb wouldn't snap, or make enough noise to betray his position. But there was nothing within reach. John-Michael's groans had gotten louder with every kick. And the seconds were ticking by.
Paolo swung to his feet and emerged from behind the tree. He moved swiftly, using John-Michael's groans to hide any sounds he made. But he wasn't quite fast enough. The man had already begun to turn his head as Paolo threw a punch.
It was a fierce strike, hurled with all the energy of Paolo's fastest serve. The blow would have hit the back of the man's head. But Paolo caught his right cheek instead.
Paolo's fist connected, hard. His knuckles crunched into the man's eye socket. The impact shot straight up Paolo's right arm and into his shoulder. The shock of pain took his breath for a few seconds. Shaken, he watched the man reeling, his left hand clutching at his face. But it was a momentary victory. Then the man's gun arm was swinging up. An automatic weapon was bearing down on Paolo, fast. He managed to swerve backward, narrowly avoiding the swipe.
"On your knees," hissed the man. Paolo held still. His attention flicked back to where John-Michael had been only a second ago. He caught a glimpse of two legs disappearing into the darkness at the edge of the road. The man's eyes followed Paolo's glance, but he didn't take his focus off him for a second. "Your boyfriend left you."
He shoved the gun into Paolo's face, pressed the muzzle up against his ear. The air of arrogant confidence had vanished. His eyes were narrow slits of steel, his upper teeth bared. Paolo dropped slowly to his knees, raised both hands in the air. The knuckles of his right hand dripped blood. He felt the cold metal of the gun roving across his skin, from his ear to his right eye.
"That. Hurt." The man inhaled noisily. "I'm going to get a goddamn black eye." He leaned forward. Paolo could smell tobacco on his breath.
"I know your little boyfriend can still see us," the man whispered conspiratorially. "Better tell me where you got the money. Better tell me _soon_. I swear to God, I'll hunt him down and skin him alive."
"Behind the tree," Paolo said suddenly. He said it again, louder. And began to mutter a silent prayer, a telepathic message to John-Michael, willing his friend to hear, to listen, to understand.
"The cash. It's in a duffel bag behind the tree. We only just opened it. We didn't touch your friend. Seriously. We were just cycling past. Our bikes are back there, behind where you parked your car, right at the side of the road." Paolo risked a gesture then, his left hand raised, pointing behind him. He opened his eyes wide, shook his head slightly. "I'm sorry I hit you. But you were hurting my buddy, I didn't know what to do."
"Where'd you find the cash?"
"In the trunk," Paolo said. The ring of truth could only help them now. Somehow, he had to get the guy to move over to the tree. "There was a padlock."
"Where'd you find the key?"
"In the glove compartment. The whole car's open."
"Why didn't you call nine-one-one?"
Paolo felt tears spring to his eyes. He decided to encourage them. "He was dead, man," he whined, managing a sob. It was a relief to release some of his fear. "We couldn't help him."
A sneering note entered the man's voice. "But you could help yourself to his gun and his goddamn _money_ , is that what you could do?"
Paolo shook his head. The terror that had seized him a moment ago, that had filled him with self-pity, faded rapidly. Instead, his mind sped ahead, trying to figure out any way to escape his fate. "We didn't know what was inside, we were just curious. Look, I'm really sorry, please just let us go. We won't say anything. The bag's just there, where I was hiding. You can take it and . . . and . . ."
The man snickered. "And what? Let you go? We'll see about that. First off, I'm gonna need to see these bicycles. Where's your spandex, kid? You sure don't look like cyclists. And I don't remember seeing any wheels up here except the ones on my pal's Oldsmobile." He stood back. "On your feet." He raised his voice so that John-Michael could hear. "Hey, 'cyclist' number two, I know you're still around. If you leave this one alone with me, it's not going to go well for him. I'm an artist when it comes to breaking bones. I'll snap at least six before I get started with the bullets. You're gonna be amazed how badly a person can be messed up before death finally settles on a body."
In the silence that followed, Paolo listened for any response. There was nothing. John-Michael had vanished in the direction of the two girls. The smart move would have been to get out of there ASAP. Maybe the older man was right, maybe not. But somehow, Paolo couldn't quite believe that his friend had stuck around to take another battering.
"Start walking. Let's see these bicycles."
Paolo's hesitation earned him another shove with the barrel of the gun. "Don't you want to pick up the cash?"
The man stared, suddenly curious. "Why?"
Paolo forced himself to shrug. "It's right there."
" _You_ get it."
"Me?"
The man nodded once. "Yeah."
Paolo managed a dumb nod. He began to shuffle toward the tree. This wasn't what he'd been aiming for. This was going terribly. No sign of John-Michael. Now he was in the middle of nowhere, facing a sadist with a gun. He reached the tree and stared helplessly at the empty ground behind the tree trunk.
"Hurry up."
Paolo stepped into the shadows, slipped behind the tree. There was only one thing left to do now. He pressed himself up against the back of its trunk and remained motionless, waiting.
A beat went by. Then the man called out, incredulous, "You're actually hiding? We're doing this?" There was a guffaw. "Do you have any idea what I'm gonna do to you?"
Paolo's eyes closed. He could taste iron in his mouth from where he'd bitten his lip. He could hear the roar of his own pulse as blood rushed past his eardrums. His chest was rattling so hard with the hammering of his heart that he couldn't believe the man couldn't hear it.
But he didn't move.
Footsteps. Paolo looked at the ground. There wasn't more than a foot of ledge behind the tree. Then the ground fell away to blackness. There might be a ridge just below. Then again, maybe not. Maybe it went straight down to the ravine.
"Last chance," said the man. He was right beside the tree now. He'd only have to lean forward, to peer around the tree trunk and see Paolo, shivering, desperately trying to melt against the bark.
Paolo shifted around the tree, further out of reach. He heard a hitch in the other man's breathing as he waited for Paolo to reveal himself, probably wondering whether to risk taking a look.
_Comeoncomeoncomeon . . ._
The gun came first, stretched ahead of the man's arm, almost skating against Paolo's head before he could duck out of the way, but he managed to maneuver his way out of reach, behind the tree. "Now you're being silly," the man reasoned as he stepped onto the narrow ridge behind the tree.
Just as John-Michael had that first time, the man skidded a little, losing his footing. His arms reached out for the tree and grabbed a branch, one hand still clutching the revolver. Paolo was already speeding around the tree, his head down in a sprint as he aimed for the Oldsmobile. He had to get some cover.
He barely noticed the slender silhouette of John-Michael as he emerged from the shadows, hands clasped together and brandishing a large, heavy stick. Paolo heard but didn't see the wood swing through the air and connect with something low. He heard the anguished scream of pain as the man stumbled—and, heard the strain in John-Michael's voice as he raised the stick for a second blow. Two shots rang out. Then there was silence.
Paolo made his way back behind the tree. John-Michael stood breathing hard, a three-finger-thick piece of tree leaning against his shoulder. There was no sign of the hit man.
"Paolo. I think he's gone."
## **MAYA**
#### MALIBU CANYON, FRIDAY, JULY 3
In the east, the moon cast a teal-colored glow behind the hills. Directly above, the sky turned midnight blue. Through the car windows, Maya glimpsed a few stars. They wouldn't last. In an hour or so the moonlight would be powerful enough to obliterate anything but the light of Venus. She reached absently for her phone, planning to check her night-sky app for the positions of the planets. Then she remembered that it was still in the Venice Beach house.
Candace and Grace would be home by now. They'd probably have tried calling everyone, only to find the calls going to phones that had been left in the house. They'd look at the gaping expanse of maple wood flooring in the living room and realize that the rug had gone, too. From there to the truth of what had actually happened a few hours ago, however, would require a radical leap of imagination. Maya tried to guess what conclusion she'd have drawn, if she'd been the one in their position. They arrive home to find the house newly empty, all the housemates' cell phones apparently abandoned.
"They're going to think we went to the beach," Maya concluded aloud. She turned to see Lucy's skeptical glance.
" _That's_ what you're worrying about?" Lucy gave a dubious shrug. "Me, I'm more concerned about what our boys down the road are doing with that gun. It's been a long time since we heard from them, don't you think?"
"I guess if they'd used the gun we'd have heard the shots," she continued.
"Don't be so sure," Maya replied doubtfully. "We've got at least two hills between us."
Maya felt herself redden, but she didn't respond. As the ensuing silence enveloped them both, Maya began to reflect on Lucy's words. It had been well over fifteen minutes. They should have agreed to check in with each other at regular intervals.
"Okay," Maya conceded. "Maybe we should go back."
"What if they're in the middle of it? Better call first."
"If they're in the middle of it then surely they won't answer?"
Lucy made a sound of irritation. "How should I know? Make the call!"
Maya picked up the phone in her lap and called the boys' burner phone. When it just kept ringing, she held the phone up to Lucy, askance. "See? Not answering. I say we head over there."
Lucy started the Chevy Malibu. She said nothing, her movements languid, almost bored. She seemed closed off from what was happening. Physically and emotionally rejecting her role in the middle of it all. Maya felt pretty certain that if Lucy hadn't been the only licensed driver right now, she'd have walked away.
Who'd have thought that Lucy would be the one to pick the law over surviving a battle with a murderer? That punk persona, it didn't seem to count for much, Maya noted. Not when you came right down to it. Lucy was more like her parents than she cared to admit.
Minutes later, they arrived at the bend higher up the road. Maya saw instantly that a second car had driven up close behind the Oldsmobile. Its headlights had been left on, beaming yellow light off the edge of the road, no more than three yards away.
Hearing Lucy inhale shakily, Maya concentrated on what little she could see beyond the two parked cars. There was no sign of movement. Ahead, obscured by the cars, a scattering of trees framed the edge of the road. She couldn't quite tell where the road ended, however; the light beam blasted a hole into the darkness but either side was merely shadows.
"Leave the engine running," Maya murmured. Then she unfastened her seat belt and climbed out of the front passenger seat. Lucy made a squeal of displeasure, but did little else to discourage Maya.
She jogged across to the second car, noting that it was another Chevy, a Cruze. Passing the Oldsmobile, Maya turned her attention to the shadows. It took her eyes a moment to adjust. After a few seconds she made out two figures close to the largest of the trees at the side of the road. One of them was kneeling, apparently staring at the ground.
Paolo and John-Michael. It had to be. Maya felt a jump in her pulse rate as she put everything together. The second car was here. That meant that the plan had worked. The hit man's associate had taken the bait. And now, he was nowhere to be seen.
_Paolo must have killed him._
With the portion of her brain that was detached, calmly reviewing the day's history, Maya noted: _John-Michael and Paolo are both killers now._
"Maya." John-Michael rose to his feet as she approached. He didn't sound surprised to see her, or relieved. Just neutral. His hands hung together in front of him, as though clasped in prayer.
Paolo was more effusive. "Is Lucy here? My car! I need a flashlight—there should be one in the glove compartment." Then he was gone, rushing past Maya. He returned a couple of minutes later, this time with Lucy. In his right hand was an aluminum tube the length of Maya's forearm. Paolo hurried by, switching on the flashlight as he reached the tree. When its beam aimed down, it became obvious that whatever they were looking for was over the edge of the cliff.
All four housemates stood precariously close to the precipice, peering over. Maya wasn't sure what they were looking for but she could guess.
The driver of the second car. Did the boys shoot him? Maybe cause him to fall over the edge? Her eyes went immediately to Paolo's hands, wondering how many bullets it had taken. When she didn't see the revolver, Maya began, confused, to look for the weapon in John-Michael's hands. The weapon wasn't there, either. Catching a glimpse of narrow white plastic zip ties between his wrists, she felt her mouth go dry at the thought of what must have happened.
It could only mean one thing.
Lucy had come to the same conclusion. "You didn't shoot him." The air wasn't particularly cold, but just the same, Lucy clasped her hands across her chest, gripping both shoulders as she shivered. "What happened? Where is he?"
"I knocked him down," John-Michael said. He sounded utterly drained. "He fired his gun as he was falling, like, a couple of times."
Paolo confirmed this. "I heard two shots."
Maya faced John-Michael. "He had a gun and you hit him?"
"With a stick," John-Michael said, using his cuffed hands to raise up a thick stafflike tree limb that he'd been holding at his side. "So he'd fall down the edge, over there."
"That was a long drop," Paolo said to Lucy. "The guy is not moving. I'm guessing that, basically, y'know, it's over."
Maya peered down again, this time aiming the flashlight to follow Lucy's horrified gaze. Yes, Paolo was right. A body in a two-piece suit lay about fifteen feet below. The head wasn't visible from this angle, but as Paolo had stated, the rest of him wasn't moving. There was no sign of the gun.
"You couldn't have killed him with that," Lucy said with a nod at the stick. "It's nowhere near heavy enough."
"No," John-Michael agreed. "But maybe the fall? Someone should really go down there, check that he's dead. I can't do it, so don't ask. I feel like I just went ten rounds with Rocky Balboa."
Lucy gasped. Even Maya was astonished by John-Michael's mild, matter-of-fact delivery. He seemed to notice their shock, because he followed up with, "I wasn't trying to kill him," he said in his gentlest voice. "He was about to torture Paolo. I just lashed out. I'm pretty sure he broke one of my ribs," he added, wincing as he inhaled. "My whole body hurts like you wouldn't believe."
"We need to get you some painkillers, dude," Paolo said. "There was really no other option. He found his friend dead. He wasn't messing around."
"I went into basic survival mode," John-Michael agreed. "I didn't know how far he'd fall."
"But he . . . you were . . . he attacked you?" Lucy asked, stumbling over the words. Maya guessed that Lucy had yet to notice that John-Michael's wrists were bound. She was still only grasping at the edges of the horror of what the boys must have been through. The more Maya thought about it, the more apprehensive she grew.
"I wasn't going to just stand by and watch him take Paolo apart," John-Michael growled.
"We need to get out of here right away," Maya announced, nervous as she backed away from the edge. "For real. It's a miracle no one has stopped in the past fifteen minutes."
"A couple of cars passed by," John-Michael commented. "But they didn't stop."
"Well, now we have Paolo's car back there. That's three cars at the side of a deserted canyon road. It's starting to look like a sideshow."
"Maya has a point," Paolo said. He reached for Lucy's hand, a little tentatively, Maya noticed. "Let me drive."
Lucy reached into her jeans back pocket, handed him the key to the Chevy Malibu. Paolo took it, released his temporary, light grip on her fingers with evident reluctance. "We should all go now," he said.
Inside the car, Paolo drove while Lucy joined him in the passenger seat. A silence descended on them all, weighty and dense. Maya wanted to speak, to ask about John-Michael's cuffed hands, to say something about the bag full of cash that now sat in the trunk of Paolo's car, about the two dead shooters. About the fact that Dana Alexander, if she'd sent these people to scare off Lucy, wouldn't be deterred by the fact that a couple of her hit men had disappeared.
Maya wondered what Dana Alexander would do when she found out. The conclusion she arrived at wasn't pleasant.
"Lucy," she began, and leaned against the front passenger seat until her face was close to Lucy's. "Giving testimony on Monday may not be enough. You have to find that bottle of nail polish, the one with Dana's fingerprints or whatever and you have to take it to them. If that's the only thing that can convict Dana Alexander then . . ."
Maya felt her breath come quickly then, her tongue thick and heavy as the words dried up. All hell would break loose once Alexander was arrested. Alexander would get Maya deported, definitely. They couldn't even be sure that she wouldn't lash out from inside prison. Killers like Mr. Shooter and his friend were never more than a phone call away.
Lucy was almost certainly destined for some kind of protective custody, maybe even witness protection. Maya's thoughts turned to Jack Cato. Had he tried to call her this evening? Would she return to unread messages and missed calls on her cell phone? The sudden memory of him was such a sweet and tender contrast to the violence and terror of the past few hours that it brought instant tears to Maya's eyes.
"Are you crying, Maya?" John-Michael asked, amazed. She felt the touch of his fingers on her cheek. "Hey now. The worst is over."
Maya leaned into his fingers, screwed her eyes tightly shut, and imagined that she was touching Jack. She felt exhausted. There were still so many details to iron out, details that would mean the difference between prison and freedom, between safety and danger. Yet, Paolo and John-Michael seemed dazed after the events of the night. They weren't thinking straight.
If only she could believe that John-Michael was right; that the worst was over. More than anything, she longed to relax. But she didn't dare. Deep inside, Maya was beginning to understand that this was very far from over.
## **GRACE**
#### PACIFIC AVENUE, VENICE BEACH, FRIDAY, JULY 3
"Still nothing?"
Grace answered with a glum shake of her head. Candace spread her fingers on the steering wheel, brow furrowed in confusion.
"Every last one of their cells is going to voice mail," Grace said.
"Guess this explains why no one bothered to text us what they wanted from the store," Candace said, yawning.
_It did_ , Grace thought.
She had sent a group text to the housemates: **Getting supplies for the 4th. Any requests?**
But, nothing.
Now this: total radio silence. It couldn't be good.
A sense of misgiving had been building slowly inside her, ever since she and Candace had left the house. The argument earlier that evening had been of epic proportions. When Candace had suggested they escape the pressure cooker atmosphere, Grace had been only too happy to follow. She'd assumed that everyone else would, too, but the others had hung back. Glancing over her shoulder as she and Candace hurried toward the boardwalk, Grace had eventually spied John-Michael and Maya leaving the house.
The household had fragmented. The way it had happened tugged at Grace's heart. They shouldn't be splitting off into little cliques, but they had. She with her stepsister. Paolo with Lucy—the girl he'd once had a thing for. John-Michael with Maya—as though they were the natural "outsiders."
Grace had kept her eyes on Paolo throughout much of the explosive drama of Maya's revelation. Normally, she tried to hide her feelings, but in that situation, the focus wasn't on her. While they'd all been distracted, Grace had allowed her eyes to be drawn to Paolo's anxiety and disquiet, to the way he'd scratched the raw skin of his tattoo, to the way he'd grimaced at his own touch. He'd been uncharacteristically introspective.
Ever since Candace's reaction to the news about Grace's father, Grace had felt a burden of guilt. Candace was right. She should have told her about her father sooner. Maybe things would have been different if she had? When Candace had suggested that they go for ice cream together, she'd decided it was time to come clean. Grace had told her all the details of her relationship with her father, from the time when she first realized she'd have to keep his fate a secret.
Secrets had almost torn their Venice Beach household apart. But now that everything was out in the open, maybe all six friends could start over.
So they'd gone straight from the ice-cream parlor at Santa Monica Pier to Candace's Prius, which had been parked a few streets along, and from there to Trader Joe's.
The house was empty when they got home, but not dark. Dimmed lights had been left on in the living room and in the second-floor bathroom. It was as though everyone had stepped out a few moments ago. Grace tried calling John-Michael again. When she heard his familiar _Death Note_ ringtone coming from the red sofa, she felt even more confused.
"Their phones are here," she called out to Candace, who was in the kitchen, putting away the groceries. After a moment she'd confirmed it: all four cell phones had been left in the living room.
"Where the hell is the rug?" Candace said, walking over from the kitchen, hands on hips as she surveyed the room.
Blankly, Grace stared at the empty wooden floor in front of the red sofa. "Oh yes," she intoned, feeling stupid. "There's also that."
Candace stooped, peering down. "Dear God, is that blood?"
Where one corner of the rug would have been close to the base of the red sofa, a few drops of a dark fluid had collected. One of them had smeared, leaving a trail like a bleeding comet, where something had been dragged through one of the larger drops.
"There must have been an accident," Candace concluded.
"Or a fight."
"Maybe they took the rug out to clean it?" Candace suggested.
"Without their phones?"
"Why would they need their phones? Maybe they went to the beach."
Grace looked at her, baffled. "You think they took the rug to the beach?"
"How the heck do I know?" Candace was getting annoyed now. "It's not here, so _clearly_ they didn't leave it."
Grace sat back on the futon and folded her arms across her lap, staring up at her stepsister. "You think they've gone to the beach for—what—a midnight picnic?" She shook her head, bewildered. "You really like to look on the bright side, don't you?"
"What's your solution?" Candace said, resentful. "You think they called Olivia Pope from _Scandal_ over to help them dispose of a body, or something crazy like that? And by the way, d'you think maybe we could discuss it while we put the groceries away? I'm not doing it by myself."
Grace followed Candace to the kitchen, where five large brown paper bags awaited them on the dining table. "What about the blood on the floor?" she said, stacking cobs of sweet corn in the refrigerator.
Candace said, "Maybe they went to the emergency room. And not the beach."
Grace shrugged. "And they all forgot to take their cell phones? I mean, if they left in such a hurry, at least one person would still have a cell phone in their pocket."
"You'd think," Candace admitted.
"Let's see if Paolo's car is still here."
"It won't be."
Candace was right. And the absence of Paolo's car wasn't going to do anything but intensify their fears.
"Kind of odd, though, all their cell phones being on the sofa like that." Candace spoke slowly, and Grace thought she caught a tremor in her voice at the end. "Almost like they took them out of their pockets and left them behind on purpose." Candace looked up. "Why would they do that?"
Grace packed four pints of Ben & Jerry's ice cream into the freezer and turned back to look at Candace. "I can only think of bad reasons."
"Try to think of a good reason, will ya?" said Candace, her voice rising to a high-pitched whine. "Look, I know you're freaked out and all about your dad, but you have to trust that it's going to be okay. Lucy's gonna talk to the cops on Monday and then you'll see."
"I'll see what?"
Candace shrugged and tried to sound bright as she said, "That the wheels of justice will turn in your favor."
But even Candace's forced optimism couldn't distract Grace from the sensation of dread that had crept inside. "I think . . ." Grace clenched her right hand into a fist. "I think maybe we ought to call the cops."
"What about John-Michael? He won't like that."
She nodded slightly, by now barely aware of Candace. Grace's thoughts had gone to him, too. The cops and John-Michael were never a good mix. More than anyone else in the house, Grace understood that.
Anxiety pulled at her now, a heavy sensation dragging her where she'd rather not go. The air inside the house seemed itself to have shifted. There was a _strangeness_ to the house, as if all life had been sucked from it. She sensed a pulsating, insistent knock at her consciousness: a warning.
"Something bad has happened," Candace said, suddenly giving voice to Grace's own fear. Grace could only tremble faintly and nod. "But I don't think we can call the cops," Candace continued. Her words were slow, considered, each one falling onto prepared ground. "At least, not yet."
Grace clasped her hands together so that they wouldn't shake. Where could they hide, where would they wait, in fear of what might be coming?
## **JOHN-MICHAEL**
#### JACK IN THE BOX, FRIDAY, JULY 3
"We need an alibi."
Paolo passed around the burgers he'd just bought from Jack in the Box in Pacific Palisades. Maya had insisted that they didn't stop in any public place closer to the house or to Malibu Canyon—"just in case." All four housemates unwrapped their sandwiches.
Lucy looked at hers and put it back in the paper bag.
"Yes," Maya said vigorously, "an alibi. Probably easiest if we say we were hanging out at the beach."
"Or the pier at Santa Monica," suggested Paolo.
John-Michael held his burger an inch from his mouth, waiting to hear the plan. Paolo seemed pretty calm after their ordeal. Almost happy. Whereas John-Michael had collapsed onto his knees and vomited as soon as he was out of sight of the others and dumped the various items that could link them to the two deaths. Even now, he felt queasy.
"Not Santa Monica Pier," Maya said. She seemed a little antsy. "We need to think of a place where there wouldn't be security cameras. But before we get into that, let's just do one last checklist of everything else. Okay, John-Michael, rug?"
"Burned and then dropped in the second Dumpster we saw after got off the 1."
"The burner phones?"
"Ours—crushed and dumped in the trash can right over there," John-Michael said, pointing. "The other guy's burner phone is on the ground beside his hand. Where he left it after he texted his partner."
"The zip ties they used on you?"
"Trash can of the men's room at Jack in the Box."
"Mr. Shooter's smartphone?" Maya said, a little insistent. "There's a lot of evidence there. Not least of which, the GPS tracking."
"That's more difficult," admitted John-Michael. "We're going to have to hit it pretty hard with something heavy."
"All right, but the sooner the better. I've already disabled the app that allows it to be traced," she said. "Guns?"
"The revolver? Fingerprints wiped and back with Mr. Shooter. And the other gun is with Mr. Fifty Ways to Mess You Up."
Maya frowned. "Huh?"
"You wanna take a look at my bruises?" John-Michael said resignedly. "I'm gonna turn seven different shades of purple in a day or so." His stomach was finally settling down, so he took a bite from the cheeseburger. But when he tried to swallow, all John-Michael could think about was the yelp of pain that the hit man's associate had made when he'd finally lost his footing on the edge.
_He'd have killed us both if he'd had the chance._
Three deaths on his hands now. Unexpected, for someone who found it hard to kill a spider.
For a few moments, the quiet sounds of chewing were the only noises in the car. All four housemates seemed content to settle with their own thoughts. Or maybe they were as choked up as John-Michael. Maybe their emotions were teetering, on the verge of letting go with a scream like the one that refused to be shaken out of his memory.
Maya finished her food and wiped her fingers carefully with the napkin before speaking. "We may need to discuss the situation with the cash."
"You want to split it up?" Paolo said.
"You better not spend a dime of that money," Lucy piped up, adamant. "You don't know who's gonna come after it."
"Who's going to come after it?" Maya asked, frowning. "The only two people who know about it are dead."
"Unless the money is from Dana Alexander," said Lucy.
"All that money?" Maya said, unconvinced. "No _way_ can it cost that much to scare one defenseless teenage girl."
"Until we changed things," pointed out Lucy.
John-Michael agreed with Maya. "If any part of the money was from Dana, it's a small part. There are a ton of reasons they might be carrying that much cash. Most of them illegal. Unless Dana Alexander has a secret life as one scary-ass crime boss, all she knows is that they got whatever payday she agreed on—probably via a third party."
There was no way the cash could be traced to them. Even if the second guy had called someone on his way out to Malibu Canyon, he couldn't have known who he'd be dealing with when he found the hit man's body.
Lucy shook her head. "Still, I say we get that under lock and key and leave it until this whole situation goes extremely cold."
"You're right that Dana Alexander's still an issue, though," Maya said thoughtfully.
It was a good point. Alexander would find out soon enough that her attempt to silence Lucy hadn't worked. Better that she found out from inside a jail cell. "Grace's dad will get taken off death row," John-Michael added. "It's win-win."
"Not necessarily," mumbled Maya. For a second or two it seemed to John-Michael like she was maybe thinking of saying more, but she didn't elaborate.
John-Michael just nodded and took another bite. How much would Lucy tell the cops? If they knew what had happened tonight, they'd probably take her into protective custody. Much as John-Michael worried about not seeing her again for who knew how long, Lucy needed to be safe. And simply telling the cops that Dana had planted a spy in their household wasn't likely to cause all that much concern.
"Maybe we should tell the cops that Mr. Shooter threatened Lucy," he said cautiously. "And that he went away."
"I don't know," Maya said, shaking her head. "Seems to me that if we say anything about him at all, it could come back on us. What if they investigate?"
"Why would they?"
"When they find the body," Lucy said, sounding jaded.
"Oh," he said. "Oh yeah. Well, maybe you make up a description? How would that be?"
"Uh-uh," Lucy said. "For once, I'm gonna agree with Maya. We keep quiet about the whole thing. The guy was never there and we don't know _anything_. Same goes for telling Grace and Candace what happened tonight," Lucy said, sounding more confident. "We say _bubkes_."
"Bubkes?" echoed Maya.
" _Nada_. Zip," said Lucy. "If you expect me to go to the cops, you'd better be damn sure that everything that happened tonight is inside of a full-scale Armageddon containment field. The 'vault' won't cut it here—I'm talking Pentagon inner-circle-level secrecy."
"Also, it's kind of unfair to expect the others to keep this secret," Paolo said. "It's not that I don't trust them, that's not it at all, it's just that—"
"Yeah," Maya agreed, grimacing. "It's a burden. And they don't deserve to have this dumped on them."
"Also, they weren't here," Lucy added pensively. "They might not understand."
The housemates seemed pretty unanimous, at least on this point. "So that's decided then, is it?" said John-Michael. "We're protecting them?"
"And ourselves," said Lucy.
"Yes," Paolo said. "But where do we tell them we've been all night?"
John-Michael thought for a moment. "Why don't we just tell them that we can't say? That way we admit that bad stuff went down, but we don't get them in any deeper."
"Yeah," murmured Paolo. "That might work." Maya nodded her agreement. John-Michael looked across at Lucy, hoping for a reply. She'd lapsed into a pensive silence, staring at her hands, folding and wringing them in her lap. After a minute she went for the door handle. "I need a milk shake. Anyone want anything else?"
"I could go for a chocolate milk shake," Maya said with more than a hint of relief.
John-Michael said, "Can you get me a strawberry?"
"I'll help, Lucy," agreed Maya. She unbuckled her seat belt.
The minute the girls were out the door, John-Michael was astonished by the sound of Paolo whimpering, as though he were gulping back a sob. Watching him, John-Michael inhaled sharply. The moment passed mercifully fast. Whatever emotion had risen so hard and fast to swamp Paolo was soon under control.
"I'm sorry, man," Paolo managed to say, struggling to breathe across his gasps. "I don't know what got into me."
John-Michael watched his friend for a moment. He resisted the temptation to reach out with a hand and touch Paolo's shoulder. "It's okay," he said after a while. "This is how . . ." John-Michael paused, swallowed. "You're going to feel up, and then you're going to feel down. This is how it feels after something like . . . like what we did."
Paolo said nothing. His eyes were large and round, open with sudden understanding as he studied John-Michael's face. For a second or two, John-Michael could practically read his thoughts.
_Maybe John-Michael actually did kill his father._
You couldn't blame the guy for thinking that way. Not after what had happened tonight. After tonight, Lucy, Paolo, and Maya had to be thinking the same exact thing.
## **LUCY**
#### GROUND FLOOR, VENICE BEACH HOUSE, FRIDAY, JULY 3
Lucy was the first into the house. The lights were off in every room except the living room, where only one of the two floor lamps was switched on; dimly, at that. Grace and Candace were stretched out, one on each sofa, half asleep. As Lucy and the others walked in quietly, the two girls pushed themselves into an upright position. And stared, bleary with confusion.
With every minute that passed, the bag of money worried Lucy more. They should have left it behind. She'd felt this at the time, but had been too shocked, too generally overwhelmed to make any kind of argument. The other three had been so confident that it was safe to keep the money.
Theft was theft. Maybe it _wasn't_ as bad as killing, but it might end up being more dangerous.
The night's events were already coiling into a knot of hideous complexity. She could barely stand to think back on any part of the evening. Earlier that day, Lucy had still been struggling with the immensity of her own revelation, after the hypnosis. A huge deal at the time; something to be absorbed slowly and considered.
Right now, Lucy would give anything to be back there. To be able to rewrite history. Why had she even waited until the afternoon before going to the cops? Maybe if she'd gone first thing in the morning, they wouldn't have been so busy?
Grace cleared her throat, struggling on sleepy legs as she stood. "Where have you all been?"
"Has anyone come by the house?" Maya asked, ignoring Grace's question. Maya was all about that, tonight: ignoring what didn't suit her, acting like some weird kind of cold-blooded badass. A teenager playing out a life she'd only ever seen in the movies—totally relatable. Real life had no context for what they'd been through in the past few hours. At times, Lucy had felt like she was back on a studio lot, sleepwalking through a scene straight out of a cop show.
"No one," Grace answered, shaking her head. She was examining each one of them curiously. "Which one of you got hurt?"
Paolo was the last into the house. In silence, he went up to Grace and hugged her. Lucy noticed the girl's eyes closing for the briefest moment as he held her, the way her limbs instantly relaxed, and it was like a light going on. Briefly, a tiny smile found its way through her anxiety.
___Grace and Paolo have started something._
After a second or two, Grace pushed Paolo off her gently. "Was it you?"
Paolo shrugged, trying to hide a guilty expression with a puzzled grin. "Was _what_ me?"
This time, Grace shoved him, hard. "Stop lying. We saw the blood."
Candace stepped forward. She also looked angry. For Grace, however, Lucy could sense it was personal. Something was brewing there, for sure.
"You've made us sick with worry," Candace said, agitated. "We were about to call the cops. The only reason we didn't, the _only_ reason . . ." She paused as her voice cracked. "Don't even think about lying to us."
Lucy was still as she watched Candace struggle to control her emotions. You were never quite certain, with an actor, what was true, what was fake. But Candace did seem kind of overwhelmed.
"Thank you," John-Michael said. His voice was sincere with gratitude. "Please do _not_ think about calling the police."
Grace stared from Lucy to John-Michael. She couldn't seem to look at Paolo, Lucy noticed. At her sides, Grace's hands tensed and relaxed, over and over. "So we were right, something bad happened?"
Lucy checked with her coconspirators. One by one they responded with the slightest of nods. She turned to Grace and Candace. It was time to lay out the story they'd prepared. "We knew you'd worry. And we love you all too much to lie to you," she began, her voice shaky.
"Good," Candace said. Her lower lip was trembling. "Don't."
"But that means that we can't tell you a whole lot. Because anything you know could be used against you. And us."
The sisters' facial expressions crumpled. Candace said, "What the hell?"
"Yeah," Grace said, gasping. "You don't get to leave us hanging."
"We thought about it a lot," John-Michael admitted.
"Yes," Paolo added firmly. "We thought about telling you the truth, we thought about lying. We can't do either one of those, because, like Lucy says, we love you guys too much for that."
"You love us, so you won't tell us where the four of you have been, why you left your cell phones here, presumably so we couldn't contact you, why there's blood on the floor, why you took the stupid rug?" Grace stopped abruptly, incredulous.
"Which is where, by the way?" Candace asked.
No one answered that.
"We care about what happens to you. Which is why we can't tell you. It might . . ." John-Michael paused. "It might put your lives in danger."
"Oh," Candace said blithely. "That's okay, then. I mean—it's not like you did something bad and didn't want us to get you into trouble."
"It's also that," Lucy confessed. "If we want them to trust us, we have to be honest about whatever we can."
"Lucy," Paolo said softly, "of the four of us, you're the one who did nothing wrong. Not a thing. You're not going to take any heat for this from anyone. I'll see to it."
Grace stared at both with undisguised hostility. "Which is it, then? You _all_ did a bad thing? Or just some of you?"
"All of us," Lucy said, now decisive. "The details don't matter." She may not have delivered any of the damaging blows, but she also hadn't stopped any of it from happening. She'd driven the car, she'd aided and abetted, at least. This night had made criminals of all of them. There was no pretending otherwise.
Candace ran one hand through the straggles of her hair. "So, let me get this straight. You've put _all_ our lives in danger?"
Lucy forced herself to nod. The two hit men were dealt with, but the person who'd ordered the hit was still out there. The idea that Dana Alexander, movie star, Shakespearean actor, could have any involvement with the kind of people who arranged murders . . . It sure _sounded_ delusional.
And yet. Lucy knew what she'd seen, that night at the Hollywood party. Dana Alexander, holding a man down underwater. Until all the struggling stopped. Until he bobbed, motionless, to the surface. And she knew what she'd experienced afterward, Dana Alexander's seductive persuasion, bending reality to her own ends.
Who knew what kind of people Dana Alexander was mixed up with? Who she could buy? If the housemates were right about Ariana, the movie star had managed to plant a spy in Lucy's life years ago. And when Lucy had moved to Los Angeles, another spy had been found—Maya.
Someone like that must move smoothly in pretty scary circles.
"I know I'm in danger," Lucy admitted. "And I can't tell the cops how bad it really is. Please don't ask me why, Gracie. But I _will_ give that testimony." She paused. "I'm going to find that nail polish, the bottle that Dana gave me that night. I know it's in my room in Claremont, somewhere. Then the cops will know that I really did talk to Dana that night. We'll make her sorry that she ever made me feel like a dopey little kid who couldn't tell the difference between a dream and reality. When she tries to change her story about what happened, you'll see, the cops are going to become real interested in her background. Who knows what else they'll find?"
Grace stepped back. Her eyes grew large. She wrapped both arms around her chest, trembling. "Thank you."
Gently, Lucy smiled. "I will do whatever it takes to make this right. Your dad's been in prison for long enough."
At this, Grace burst into tears. This time it was her stepsister who was at her side in an instant, taking her into her arms for a close, comforting hug.
Lucy felt tears of her own, stinging and hot at the corners of her eyes. Not just from relief, but fear. It was long past time that she faced up to the truth of that night, almost nine years ago. But that didn't make the possible consequences any less terrifying.
## **MAYA**
#### GROUND FLOOR, VENICE BEACH HOUSE, SATURDAY, JULY 4
Grace quivered as she wept against Candace's shoulder. It was past midnight now. The room filled with a palpable sense of relief that a day they'd remember chiefly for its hideousness was finally over. At this point, Maya reflected, they'd latch on to any good outcome for the evening. Grace's father might go free. It was important to focus on something positive, make it easier to forget the horrors of the night.
But Maya couldn't bear to watch. How could she forget her own part in keeping Dana Alexander's secret? It made her sick to think that she'd ever sent that woman information about her housemates. Discreetly, she crossed the room, toward the French doors. On the side table beside the red chair, were the cell phones they'd left behind. Maya picked up her own, saw a text from a number she didn't recognize.
Her fingers fumbled, searching for the button to click on the text. _Strange number_ , Maya thought. _Never seen it before_. Then she read the text.
**O JUDGMENT, THOU ART FLED TO BRUTISH BEASTS. AND MEN HAVE LOST THEIR REASON.**
"Huh, weird," she said.
Candace asked, "What's weird?"
Maya held out her phone. "Take a look."
Candace released her grip on Grace, who seemed calmer now, took the phone, and read aloud: "'O judgment, thou art fled to brutish beasts. And men have lost their reason.'" She looked straight at Maya. "I know that line—it's from _Julius Caesar_."
"Yeah," nodded Lucy. "It's that 'Friends, Romans, countrymen' speech. We had to learn it by heart last semester."
"But what does it mean?" Maya said, mystified.
"Antony says it," Lucy told her. "He's telling everyone that they have lost their minds, basically. That their judgment is shot to pieces; that they've lost the power to make good decisions. That they're reacting like _animals_."
For a few seconds Maya's mind was wiped clean. All she could do was fixate on the screen, hunting for a coherent response.
"Maya, what's wrong?" she heard John-Michael say. He sounded nervous.
Maya lifted her head to look up at him. Whatever showed in her face must have really scared him, because John-Michael moved fast, caught her as she crumpled, falling to her knees. For a moment, Maya braced herself against John-Michael's shoulders and rose to her feet. With a slight cough to clear her throat, Maya said as evenly as possible, "It's . . . I think it's from Dana Alexander. She's going to rat me out to immigration."
The others stared at Maya dumbly. There was a protracted silence. Eventually, Paolo said, "Is this a joke?"
Maya sighed. "Yeah, Paolo, I felt this was a good time for jokes."
"Well, I don't understand," was all he said.
"Seems pretty clear to me—Maya lied to us," Candace said.
"No." Maya shook her head, forcing herself to remain calm. Of course they would think that, _of course_. "All I know for sure is that my mom may have used forged papers to get my US citizenship. Dana Alexander knows, she's always known. That's why . . ." She stopped, took a huge breath, and released it slowly before continuing. "That's why I was doing stuff for her. All those reports. If you'd been a little less enraged earlier on, I would have told you but, honestly, I just wanted to stop talking about it."
Candace sniffed, apparently disdainful. "Hmm. You got 'ratting you out to immigration' from that Shakespeare quotation?"
"Presumably this is Alexander's fancy-ass way of letting me know she does _not_ appreciate my decision to stop spying on all of you."
Lucy took Maya's phone from Candace's fingers and read the text for herself. "So, so, so screwed," she pronounced slowly.
Maya looked at Lucy. "You think I'm right?"
Lucy tried to laugh. "Oh yeah. That's from Dana Alexander, I'd bet this house on it."
Paolo said, "Seriously?"
Lucy shook her head. A sad, amazed little smile touched the corners of her mouth. "Dana Alexander _knows_. She's telling us that she knows about everything. About me, about how we sent Ariana away, about Maya, about Candace making out with her precious little sister's boyfriend. I'm guessing she's not too happy about _that_ particular piece of news. She's saying that _we've_ lost our minds, our reason."
On this final statement, Lucy fixed John-Michael with a deeply significant stare. Watching, Maya could feel her knees giving way once again.
Dana Alexander knew _everything_.
What if she even knew what had happened tonight? From the anxious, pale glances that she was getting from Paolo and John-Michael, Maya could tell that they were thinking the same way. They didn't dare articulate their fears, naturally. Not while they'd sworn to keep Grace and Candace out of the horrors of the past few hours. Maya forced the rising tide of panic out of her mind and stumbled backward until she felt the sofa against the backs of her knees. She slumped onto the couch, furiously trying to think.
Alexander couldn't know absolutely everything. Could she? Surely not about what happened up in Malibu Canyon? But she might know that her plan to keep Lucy quiet had gone awry.
One of the two hit men could have gotten word to Dana Alexander. Sometime before the second guy had fallen off the cliff, Maya guessed. He must have called Alexander to tell her that the hit hadn't been carried out. Now the woman would be wondering. Should she act quickly? Send someone else to silence Lucy? Or wait?
They couldn't afford to take the risk.
"We can't be here for the next few days," Maya said with sudden and absolute conviction. "And if we're gonna make any calls, they can't be traced. So we need to buy some new burner phones."
Candace frowned at this. "New?"
"We have to get out of LA." She stood up. "Tonight. All of us. Now."
## **GRACE**
#### INTERSTATE 5, SATURDAY, JULY 4
Wind trailed through Grace's fingers as they rested lightly on the open window of her sister's car. The night was warm and Candace liked to keep the air-con of the Prius turned off to be eco-friendly. For someone who'd initially sneered at her mother's choice of car, Candace was turning into quite the proud owner.
From the backseat, Maya spoke up. "I just got a text from John-Michael. They're going to get gas at the next service station. We should fill up at the one after. Our cars shouldn't be photographed together anywhere."
Neither Candace nor Grace said anything. It was just another mysterious element of what was now a thoroughly troubling night. And they couldn't even ask why.
The car stereo was playing some electronic dance music, dreamy and pulsating, the kind of thing that Candace liked to play at the end of a party—total contrast to the chaos that was tearing Grace up on the inside.
This music only made her sad; nostalgic for a time that Grace now realized had gone—perhaps forever. A time when they'd all been happy together, closer than she'd ever believed it would be possible for a group of friends to be. Closer than she'd ever felt with her family.
The saddest part, Grace reflected, was that only now did she see that time for what it was. The time they'd first met. Their first party. _Seinfeld_ marathons all day long, eating John-Michael's amazing sandwiches. Lazy afternoons on the beach, watching Maya surfing, chatting around a campfire. The warmth of knowing she was surrounded by people who cared for her, who understood what it was to be right there, in that time, in that place, who understood and accepted in a way that no one else did.
All that—it was gone.
Whatever had happened tonight at the house, whatever had caused Lucy, Maya, Paolo, and John-Michael to disappear for around four hours with absolutely zero explanation—it had to be _really_ bad. There was trust between the group now. It had been sorely earned, months of struggling with individual problems; John-Michael and the death of his father; Lucy and the nightmare visions of her past; Candace and her indiscretion with her costar; Maya being blackmailed into spying for Dana Alexander. Tonight, finally, all of that had exploded, scattering the housemates as they reeled, shell-shocked.
And yet, they'd instantly gravitated back to one another, to the Venice Beach house.
Paolo alone had no secret. He'd been frank about wanting Lucy, even though it had led to rejection. His transformation by that disappointment was almost disturbing. Grace doubted that Paolo was happier since being emancipated. Yet, he shared the depth of his frustration with no one, as far as she knew. Instead, he wore it on his skin.
Perhaps Paolo was the only one who truly understood how to keep something _in the vault_. Grace caught herself smiling at the irony of that. She was still holding out on her own feelings for him, after all.
"How long do you think we can stay at Alexa Nyborg's house in Napa?" Candace asked Maya, breaking across Grace's troubled thoughts.
"Nyborg just emailed to confirm that she's going to Washington for the weekend and the invitation is still open."
"How long will we need to stay?" Grace asked. But Maya didn't seem to want to commit to any specific time frame.
Candace sighed with longing. "Hey—the show is on summer hiatus until the middle of August, so I'm in no hurry to rush back. How seriously cool is it that you know a billionaire?"
Unbelievable. Candace wasn't paying the slightest attention to their dilemma. Grace wanted to scream at her: _Do you get what's happening here? Have you listened at all to what they've been saying? You still believe this is some kind of vacation?_
"A paper billionaire," Maya protested quietly. "I bet most of her shares are still in escrow."
"Alexa Nyborg is a real-life billionaire, and she's chosen you to be her little protégé. She even replies to emails you send her at midnight on the Fourth of July. Girl, you've won the lottery," mused Candace. "Aren't you gonna tell Hottie McBrit?"
Maya didn't answer. She sidled up to the car window and stared into the darkness beyond. Grace could almost hear the ache in Maya's shallow breathing. Whatever Maya felt about her newfound business angel and her English crush, it was somehow tainted by whatever had happened tonight.
The new secret was a fault line in the group. The collective understanding that had once existed between them was finished. Grace touched a finger to a button and closed her car window. The interstate road was climbing higher into the hills and the air had turned cold.
The fault line between the friends might become a chasm, but Grace resolved that she would find a way, somehow, to heal it.
She closed her eyes for a moment. Behind her eyelids she saw the pale green sky of a late evening on Venice Beach, the silhouetted palm trees, the sound of hard plastic wheels on the concrete path that wound through the low, grassy dunes, the smell of grilled maize tortillas near Andy's fish taco place, of board wax at the surf rental shop, of freshly baked funnel cakes at Santa Monica Pier. It would be waiting for them, everything they'd left behind. One day, they would return.
Uneasy in her position, Grace turned to observe Maya. The girl was scrunched up, folded like an ironing board, knees drawn up to her chin, arms wrapped around her legs and her head resting lightly against her kneecaps. Sitting behind the driver's seat, Maya was staring through the side window. The next moment, she flinched so hard that her limbs unwound as if released from a spring.
The sudden movement was enough to jolt Candace, who lost control of the car for a moment, causing it to swerve into the next lane. When the momentary scare was over, Grace saw Candace looking fiercely at Maya in the rearview mirror.
"Freak!"
"I'm sorry," Maya muttered miserably.
"What the hell is wrong with you?" Candace demanded before adding, with thick sarcasm, "Oh yeah, that's right, I'm too precious to be told."
To Grace's surprise, Maya neither replied nor reacted. Instead, she simply continued to gaze through the window with a kind of drawn-out expression, breathing in little gasps, through parted lips. After a moment she said, "Did either of you see that white Buick earlier on, the one that passed us just now, did you see it before we left the PCH?"
Exasperated, Candace said, "I have no idea what you're talking about."
Maya's voice dropped to a whisper. "I think I saw it before."
Grace faced Maya, tried to catch her eye. "Why do you care? Why would someone be following us _now_ , Maya? Maya? _Maya?_ "
Maya didn't reply at first, instead she rolled her head against the backseat like a cat trying to get comfortable and lie down. A minute passed, then she mumbled, "Forget it."
Grace stopped trying to get any further response after a while, she just sat back in her seat and considered, focusing on the darkness beyond.
Maya should have been happy, on the way to enjoy the first rewards of her hard work. But she wasn't. Maya was _afraid_. And try as she might to think soothing, rational thoughts, Grace couldn't ignore it for another minute.
She was scared, too.
## **ACKNOWLEDGMENTS**
From the beginning I speculated about how I would take things to a dark place in this second installment of the Emancipated series. What was conceived as a somewhat escapist story line—six emancipated teenagers living the SoCal dream—felt like the perfect opportunity to explore morality. If things went badly wrong, what would each one of them be prepared to do to protect their "perfect" life?
The entire Malibu Canyon section of this story was inspired by a twisty drive with my good friend Hoku Janbazian through the Hollywood Hills and down to Malibu beach, where we ate some terrific fish tacos. A lovely day dreaming up some hideous drama—I thoroughly recommend the location.
Huge thanks are due to Katherine Tegen for supporting this incarnation of the "escapist" teen story. Two fantastic editors have been massively helpful in finding the Goldilocks zone between "dark enough" and "too dark." Thank you, Elizabeth Law and Maria Barbo!
Thanks to Emily Wheaton, a former US soldier and fellow member of the _Once Upon a Time_ fandom, for all her great advice about firearms. Thanks also to California resident and fellow Oncer Cyndi Burke for additional advice about her home state and also about guns!
Team Emancipated at Katherine Tegen Books and HarperCollins ran a terrific internet-based marketing campaign for book one, including videos and a cool quiz. Thanks especially to Kelsey Horton for all her creativity, energy, and enthusiasm. Other members of our wonderful team include Bethany Reis, Veronica Ambrose, and Rebecca Schwarz. A virtual cookie to you all now and a real one later!
Fellow children's author and friend Michael Grant continues to be a source of steady encouragement and support with the promotion of the Emancipated series—thank you, Michael. Thanks also to my agent, Robert Kirby, at United Agents in London; to my husband, David; and daughters, Josie and Lilia. The kids were especially eager to further their favorite "ships." One gets her way—but #nospoilers.
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## **ABOUT THE AUTHOR**
Photo by Haddon Davies
**M. G. REYES** (Maria Guadalupe, aka "Pita") was born in Mexico City and grew up in Manchester, England. She studied at Oxford University and spent several years as a scientist before setting up her own internet company. She lives in Oxford, England, with her husband and two daughters and loves visiting LA.
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## **Books by M. G. Reyes**
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## **CREDITS**
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## **COPYRIGHT**
Katherine Tegen Books is an imprint of HarperCollins Publishers.
INCRIMINATED. Copyright © 2016 by Reynolds Applegate, Inc. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.
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{"url":"http:\/\/science.sciencemag.org\/content\/286\/5442\/news-summaries","text":"# News this Week\n\nScience\u00a0 05 Nov 1999:\nVol. 286, Issue 5442, pp. 1058\n1. MOLECULAR BIOLOGY\n\n# EMBL Faces Huge Bill Following Adverse Pay Dispute Ruling\n\n1. Michael Balter\n\nScientists at one of the world's leading research centers, the European Molecular Biology Laboratory (EMBL) in Heidelberg, Germany, are in shock after being told that an adverse judgment in a salary dispute could wipe out as much as 25% of the lab's core funding and threaten cancellation of its ambitious future plans. The bad news was delivered at a staff meeting last month by EMBL director-general Fotis Kafatos, who told researchers and other employees that a worst-case interpretation of the ruling\u2014which was rendered by the Geneva-based International Labour Organization (ILO)\u2014could lead to the institution shutting down. Although Kafatos stressed that this possibility was remote, even under the most optimistic interpretation the complex judgment will cost the lab millions of dollars in back salary payments and possibly curtail important new research initiatives.\n\nThe decision comes at the worst possible time for the 25-year-old institution. EMBL is already facing the potential loss of European Union (EU) infrastructure funding for one of its outstations, the European Bioinformatics Institute (EBI), near Cambridge, U.K. The EU\u2014which this year provided about 36% of the EBI's $8.3 million budget\u2014decided earlier this year to cut all its infrastructure funding to EBI and several other European research facilities next year. (A letter from more than 60 European scientists protesting this decision was published in this week's Nature.) Later this month, Kafatos is scheduled to go before EMBL's governing council, which consists of delegates from the lab's 16 member countries, to present a draft 5-year scientific plan for 2001 to 2005 that is expected to include a significant boost in EMBL funding for EBI and its new mouse genetics facility at the Monterotondo Research Center near Rome. Kafatos and other EMBL researchers fear, however, that the council may take a dim view of increased support for EBI if it is forced to fork out huge additional sums of money to the lab's staff. The convoluted legal case has its roots in EMBL's status as an international organization, sponsored by 15 European nations plus Israel. Many such organizations\u2014including NATO, the European Space Agency, the Organization for Economic Cooperation and Development, and others\u2014belong to a group called the Coordinated Organizations (Co-Org), which sets salary levels and scales for its members. Although EMBL has never joined Co-Org, in 1982 the lab adopted the Co-Org system as a \u201cguide\u201d for salary levels, a decision that was written into its staff regulations. But in 1992 the EMBL council, concerned with the growing costs to the lab of following the Co-Org guidelines, began capping salary increases at lower levels. As a result, overall salaries at EMBL began to slip below those at Co-Org member organizations. In 1995, EMBL's staff association challenged this decision. When the council refused to relent, three EMBL scientists and two nonscientific staff members\u2014later joined by a large number of their colleagues\u2014filed complaints for back pay with the ILO, which arbitrates labor disputes involving international organizations. On 8 July of this year, the ILO's administrative tribunal ruled in their favor, arguing that EMBL could not deviate from its own staff regulations without providing \u201cproper reasons\u201d and adding that \u201cfinancial considerations \u2026 do not constitute a valid reason.\u201d The tribunal's order, which cannot be appealed, directs EMBL to implement the Co-Org salary increases for 1995, and in addition to pay employees 10% annual interest since that year on the sums past due. However, the decision leaves ambiguous whether the council is only required to grant the percent increases mandated for 1995, or whether it must now restore overall salary levels to what they would have been if the guidelines had been followed since 1992, an extremely costly interpretation that some staff members are advocating\u2014but one which could nearly bankrupt the lab. \u201cThe ILO-mandated salary adjustments are reasonable if implemented according to the council's [more conservative interpretation],\u201d Kafatos told Science. \u201cBut the extreme interpretations advocated by some [present and former staff members] are inappropriate and could damage EMBL badly.\u201d Moreover, two more cases before the ILO concerning salaries for 1996 and 1997 are still pending and are expected to be decided early next year, although the amounts of money involved are considerably less. At the staff meeting last month, Kafatos said that at worst the ruling would require the lab to pay an immediate lump sum of about$11 million, or about a quarter of the $43 million the member states provided EMBL in 1999 for its core operating costs. The tab for catching up with Co-Org salary scales would then run about$2 million per year in future years. And these large sums do not include the possible adverse judgments for 1996 and 1997. Even the best-case scenario, in which the lab would not have to fully adjust overall salaries to Co-Org levels, would still mean a $2.7 million bill for the 1995 judgment alone. \u201cThe council was wrong in not granting these salary increases,\u201d says structural biologist Luis Serrano, chair of the EMBL staff association. \u201cFrom a legal point of view they made a blunder.\u201d But Kafatos says that the lab was not obligated to follow Co-Org rules. \u201cIt is clear that the EMBL council never ceded its decision-making powers to Co-Org [because] we are not a member.\u201d \u201cI am not worried that EMBL will shut down,\u201d says computational biologist Peer Bork, \u201cbecause that is an extreme scenario and I don't think it will happen. But there is a chance that we will lose our critical mass in some [research] areas and will not be competitive anymore.\u201d One of these areas, Bork says, is the hot field of functional genomics, \u201cwhich requires a lot of expensive equipment.\u201d And Serrano adds that \u201cit is clear this is going to be a major blow for EMBL.\u201d Just how major, he says, depends on which interpretation of the ILO ruling that the staff chooses to insist upon, a decision currently being debated in the lab's corridors. As Science went to press, the staff association was expected to meet this week to discuss its options. EMBL employees might agree to spread the back payments over a number of years to soften the blow or to take increased holiday time to partly compensate for the money they are owed. But Serrano says that many of the original complainants in the case are no longer at EMBL, and they may not be willing to compromise\u2014in which case the matter could end up back before the ILO if the governing council does not agree with the staff interpretation. According to one staff member who prefers to remain anonymous, \u201cthere are people who don't care if the lab goes down the drain over this.\u201d Kafatos will have to tread carefully to avoid such a scenario in the coming months. The best solution, he says, would be one that would \u201csafeguard both the fair interest of all EMBL personnel and the continued well-being of EMBL as an institution.\u201d And he will be looking for any encouraging signs that the staff will rally behind him. Says Serrano: \u201cNobody in this building is interested in destroying EMBL.\u201d 2. ANIMAL RIGHTS # Booby-Trapped Letters Sent to 87 Researchers 1. Jocelyn Kaiser Psychobiologist John Capitanio could see the razor blade through the back of the envelope mailed to his office at the University of California, Davis. He already knew what to look for: Capitanio had been told he was one of 87 scientists using nonhuman primates in their research across the United States targeted last month by a shadowy animal rights group that originated in Britain. Booby-trapped to slice the fingers of an unsuspecting scientist trying to open them, the letters mark a new and disturbing turn toward violence by the militant wing of the animal rights movement. Although animal rights groups have vandalized many laboratories in the United States, in recent years most attacks on individuals have occurred in Europe (Science, 4 June, p. 1604). \u201cThis is the first time there's been a campaign of this ilk [in the United States] on this large a scale,\u201d says Mary Brennan, executive vice president of the Foundation for Biomedical Research (FBR), a Washington, D.C., watchdog group, which warned the intended victims after spotting a list of them on an animal rights organization's Web site. While some researchers, like Capitanio, seemed to take the missives in stride, others saw them as much more serious: \u201cSome of my colleagues are feeling very frightened,\u201d Capitanio says. As Science went to press, more than 50 of the 87 letters had been received, all bearing a Las Vegas postmark dated 22 October. In addition to a razor blade taped inside the upper edge of the envelope, each letter contained a short, typed message that read, in part: \u201cYou have until autumn of the year 2000 to release all of your primate captives and get out of the vivisection industry.\u201d A group called the Justice Department has claimed responsibility for the letters in a 24 October communiqu\u00e9 on a Web site, the Animal Liberation Frontline Information Service, that posts information supplied by \u201cunderground\u201d groups such as the Animal Liberation Front (ALF; www.enviroweb.org\/ALFIS\/index2.html). The FBR and another group, Americans for Medical Progress, spotted the posting the next day and alerted the researchers listed. No injuries have been reported. The Justice Department originated in Britain, where it has acknowledged sending letter bombs and other devices to pharmaceutical labs, animal breeders, and researchers since 1993. One of the group's members served 3 years in prison. Three years ago, the group began sending similar threatening letters, complete with razor blades, to Canadian hunting groups and fur retailers. A fact sheet on the site associated with the ALF, which expresses a commitment to \u201cnonviolence\u201d despite having taken credit for past attacks on animal labs, explains that the Justice Department \u201csee[s] another path \u2026 [that] involves removing any barriers between legal and illegal, violent and nonviolent.\u201d Some researchers already accustomed to regular protests by animal rights groups seemed unfazed by the letters. \u201cThere's not a whole lot we can do about this sort of thing other than just stay alert and not fool around with anything that looks suspicious,\u201d says Peter Gerone, director of the Tulane Regional Primate Center in Covington, Louisiana. But others were less nonchalant, including a University of Washington, Seattle, AIDS researcher who declined to have his name published. \u201cI have a family,\u201d he says. \u201cI don't want to say I'm afraid, but there are certain situations where you don't take chances.\u201d Capitanio says he's okay now, but admits that \u201cI might feel more nervous next autumn.\u201d 3. BIOMATERIALS # Reverse Engineering the Ceramic Art of Algae 1. Ivan Amato* 1. Ivan Amato is the author of Stuff. The glasslike silica laceworks within the cell walls of diatoms are so beautiful they'd be on display in museum cases if only they were thousands of times bigger. No one knows how these tiny algae pull off their bioceramic art, but researchers are closing in on the secret. On page 1129, biochemist Nils Kr\u00f6ger and colleagues at the University of Regensburg in Germany report new clues\u2014silica-forming proteins dubbed silaffins. Within seconds after they added their first silaffin samples to solutions of silicic acid, a silicon-containing organic compound, Kr\u00f6ger, Rainer Deutzmann, and Manfred Sumper knew they were onto something. Says Kr\u00f6ger: \u201cYou suddenly see the precipitate form. The solution gets cloudy\u201d\u2014something that takes hours to happen without silaffins. A scanning electron microscope showed that the precipitate had formed networks of minuscule silica spheres. Kr\u00f6ger and his colleagues went on to analyze the proteins and show how their structures and chemical features could help catalyze the reaction of silicon-containing molecules into solid silica particles. The researchers \u201chave done a great job of characterizing their proteins,\u201d says Galen Stucky of the University of California, Santa Barbara, who last year found what may be compounds with similar functions in silica-making sponges. Besides helping to explain how diatoms transform dissolved silicon-containing molecules into sturdy solid particles, the finding is also a tantalizing clue for materials scientists who envy biology's ability to build sophisticated materials at ambient pressures and temperatures. To make any ceramic, from a dinner plate to a toughened drill bit, engineers and artisans now have to mix powders, press them into molds, and fire them in furnaces. There are no furnaces in sight when a developing child infiltrates itself with bone or a diatom drapes itself in silica lace, and materials scientists would like to know how they do it. The Regensburg group suspected that diatoms make proteins that orchestrate the initial phase of biosilica formation\u2014the growth of tiny silica spheres. For one thing, other researchers had already found organic molecules closely linked to diatom cell walls. After extracting the organic material from their diatom samples, the Regensburg researchers isolated three proteins that could instigate silica precipitation in a test tube\u2014a pair of small, closely related silaffins (1A and 1B) and another larger one, silaffin 2. To begin unraveling how the proteins work, the group determined the amino acid sequence of silaffin-1B and ferreted out a gene from the DNA of the diatom Cylindrotheca fusiformis, which turned out to encode silaffin-1A as well. Kr\u00f6ger says the team also is now working to characterize silaffin 2. The structures of these proteins harbor clues to the diatoms' silica engineering. The glasslike veil of a newborn diatom takes shape in a \u201csilica deposition vesicle,\u201d where conditions are acidic. Both silaffins have an unusual amino acid motif, consisting of bonded pairs of lysines with a string of amine groups grafted on after the protein chain is formed. The researchers say that under acidic conditions, this motif should stimulate silicic acid molecules to form silicon-oxygen bonds, linking them together into silica particles. That might help explain how diatoms form solid silica from ingredients dissolved in their watery environs, but it doesn't explain how the algae coax the silica to form intricate patterns. Kr\u00f6ger conjectures that other features of the proteins could be at work. Silaffin-1A and \u22121B both consist mainly of two chemically distinct components, one bearing multiple positive charges and another multiple hydroxy groups. To Kr\u00f6ger, the proteins resemble synthetic block copolymers\u2014polymers in which two distinct segments, each repeated many times, alternate along the molecule. When some copolymers solidify, like segments cluster together, segregating into two separate phases that pattern the material with regions of contrasting chemical properties\u2014somewhat the way drops of oil poured onto a saucer of vinegar form segregated droplets. Kr\u00f6ger wonders whether silaffins might be doing something similar within a diatom's silica deposition vesicle, forming molecular frameworks that then guide the growth of the silica. However diatoms create their silica patterns, it's a trick materials scientists would like to emulate. \u201cCeramics are one of those unfulfilled materials we could use lots more of, if only we could get [them] easily,\u201d says materials researcher Paul Calvert of the University of Arizona, Tucson. Adopting biology's kinder, gentler methods could help engineers combine ceramics with other materials that can't take furnace temperatures. Quips Calvert: \u201cYou could make something with chocolate feet and a silicon carbide head.\u201d Unlikely material combinations, he says, could push forward such projects as \u201cflexible electronics,\u201d in which silicon-based electronics are patterned onto polymer sheets. Diatom-like methods for making intricately shaped ceramics might also yield photonic materials, whose internal arrangements of solid and space could select and confine specific wavelengths of light for communication or computing. The more scientists learn about diatoms' glassy laceworks, the more beautiful they seem. 4. OCEANOGRAPHY # Has a Great River in the Sea Slowed Down? 1. Richard A. Kerr For many millions of years, two \u201crivers\u201d of seawater have been flushing the deep sea clean while shuttling chemicals and heat so as to reshape climate. Now, a new analysis of oceanographic data suggests that one of the two rivers has slowed dramatically within the past century, with implications for climate and the humans who are changing it. In a paper on page 1132 of this issue of Science, marine geochemist Wallace Broecker of Columbia University's Lamont-Doherty Earth Observatory in Palisades, New York, and his colleagues argue that the renewal of deep waters by sinking surface waters near Antarctica has slowed to only one-third of its flow a century or two ago, while deep water formation in the North Atlantic\u2014the site of the other river\u2014remains high. \u201cThe whole concept that deep water circulation could have changed that much is mind-boggling,\u201d says Broecker, who adds that, far from being a onetime event, the slowdown may recur in a 1500-year cycle. This huge, climate-altering change in the oceans\u2014if it's real\u2014would greatly complicate attempts to understand how the ocean and climate are responding to another influence on climate, the buildup of greenhouse gases in the atmosphere. \u201cIt's a really interesting and provocative idea,\u201d says ocean circulation modeler Jorge Sarmiento of Princeton University, \u201cbut I'm very uneasy about the calculations. I find the paper more of a stimulation to further work than what I could accept as proven fact.\u201d Directly measuring the flow of surface waters into the deep sea is impractical. Instead, researchers examine easily measured, indestructible \u201ctags,\u201d some natural and some manmade, that join surface water before it becomes denser and sinks into the abyss. One tracer, the sum of the phosphate and oxygen in seawater, should remain constant as water sinks into the deep sea. In data collected by other researchers, Broecker found that this tracer, called PO4*, is high in newly formed deep water near Antarctica and relatively low in newly formed deep water in the northern Atlantic. Throughout the deep Indian and Pacific oceans it is at intermediate levels, suggesting to Broecker that these deep ocean basins have received about equal amounts of water from each source during the past 800 years or so. Another tag, radioactive carbon-14, also supports equal roles for the two deep-water source regions over the last millennium, according to Broecker's analysis. Surface water heading down absorbs carbon dioxide, including carbon-14 formed in the atmosphere by cosmic rays, and carries it along toward the bottom. Broecker analyzed carbon-14's distribution throughout the world ocean and concluded that about 15 million cubic meters of water per second (15 Sverdrups) has been sinking into the deep sea at each source during the past 800 years. But tracers that gauge deep water formation over decades rather than centuries seem to show that the southern source is now much smaller than the northern one. Physical oceanographers have long believed that the principal southern source of deep water, in the Weddell Sea, now supplies no more than 5 Sverdrups, judging by heat and salt content. And Broecker finds further evidence in a new study of water in the Southern Ocean, near Antarctica, by physical oceanographer Alejandro Orsi of Texas A&M University, College Station, and his colleagues. They looked at the distribution of the pollutant chlorofluorocarbon-11 (CFC-11), which first entered the environment a few decades ago. To Broecker, the study implies that the southern source has generated only 4 Sverdrups of new deep water during the past few decades. In the north, on the other hand, other CFC-11 studies support 15 Sverdrups, says Broecker. Why should the sinking of seawater into the deep sea have slowed recently in the south? Broecker doesn't know, but he sees a parallel between the apparent recent slowdown and more drastic variations in ocean circulation during a sharp cold snap 11,000 years ago. As recorded in Atlantic sediments, deep water formation in the north slowed or halted. Because warm water normally flows northward to replace the sinking surface water, the shutdown chilled much of the Northern Hemisphere; meanwhile water began sinking faster in the south, warming the region. Something similar might have happened in the 500-year Little Ice Age, which ended around 1880, says Broecker. Since then, deep water formation in the south would have slowed. And because some suspect that the Little Ice Age is only the latest swing in a 1500-year climate cycle (Science, 27 February 1998, p. 1304), further changes could be in store. Broecker's ideas are \u201calways interesting,\u201d says marine geochemist Richard Gammon of the University of Washington, Seattle, \u201cand he's right often enough that people have to pay attention.\u201d Physical oceanographer Arnold Gordon of Lamont is certainly paying attention. However, he and Orsi don't think Broecker has the evidence to back up his claim. By their reckoning, true deep water formation is currently equal north and south at about 5 Sverdrups. In his accounting of deep water formation, they say, Broecker includes waters that never get very deep or are picked up by new deep water as it sinks. Broecker isn't worried about the cautious reception. \u201cI don't expect people to accept at face value what I say.\u201d The important thing is that \u201cthe Little Ice Age is going to get more attention,\u201d he adds. \u201cIf I'm right, it has enormous consequences.\u201d Sorting natural climate oscillations from anthropogenic greenhouse warming would become more difficult. The ocean might lose some of its ability to draw off greenhouse gases and stash them away in the depths. And future change could become even harder to predict. As Sarmiento says: \u201cIt's a very interesting speculation; it's also disconcerting. If we want to understand the next 100 years or 200 years, we really need to understand what is going on with long-term cycles.\u201d 5. AIDS # European Vaccine Effort Faces Chinese Puzzle 1. Michael Balter Paris\u2014During an international AIDS meeting here last week, a group of researchers quietly met to plan a 3-year,$9.2 million European Union-backed effort to develop an AIDS vaccine. The new initiative, called EuroVac, is expected to begin on 1 January and will mark the first time the EU has attempted to pull Europe's top AIDS researchers together into a unified vaccine drive. Yet though the contracts between the EU and the researchers are still being negotiated, the initiative has already become tangled in international AIDS vaccine politics. Some members of the EuroVac scientific team were surprised to learn last week that one of the project's potential ambitions\u2014to test vaccine candidates in China, where the AIDS virus is spreading rapidly\u2014may duplicate similar efforts under way by virologist David Ho, director of the Aaron Diamond AIDS Research Center in New York City.\n\nThe EuroVac project, which is co-chaired by virologists Jaap Goudsmit at the University of Amsterdam and Marc Girard at the Pasteur Institute in Paris, will begin with phase I trials to compare the ability of several different vaccine preparations to elicit immune responses against HIV. The European team will test how well two different types of genetically engineered vaccinia viruses\u2014one called MVA and the other NYVAC\u2014serve as noninfectious vectors to present four key HIV proteins to the immune system. In each case, this \u201cprime\u201d will be followed by a \u201cboost\u201d vaccine preparation consisting of HIV's envelope protein, the main component of its outer coat.\n\nIn addition, the trials will mix and match proteins from two major clades, or subtypes, of HIV: clade B, which predominates in North America and Europe, and clade C, which now accounts for about 40% of new HIV infections in the world and is particularly rampant in China and India. The mix-and-match strategy should eventually allow researchers to determine whether a vaccine against clade B would also be effective against clade C and vice versa, once they are tested in full-fledged efficacy trials. \u201cEuroVac is quite innovative,\u201d says team member Giuseppe Pantaleo, an immunologist at the Vaudois Hospital Center in Lausanne, Switzerland. \u201cFor the first time we will be comparing MVA and NYVAC with four major viral proteins, and it's the first time we will be trying to get cross-clade immune responses.\u201d\n\nBut it is EuroVac's plans to test a clade C vaccine that have provoked the sparring over turf. The clade C HIV on which the vaccine is based was provided by Hans Wolf, a virologist at the University of Regensburg in Germany, who obtained the viral strain from colleagues in China. In the meantime, Ho, who has his own contacts in China, has prepared a clade C vaccine using a somewhat different strategy, which uses both an MVA vaccine and a \u201cnaked DNA\u201d preparation that delivers HIV genes directly to the body. Ho has been quietly talking with Chinese health authorities about testing his vaccine in China and has also been discussing funding with the New York-based International AIDS Vaccine Initiative (IAVI)\u2014a private organization funded by numerous major foundations as well as the World Bank and the British government.\n\nWolf and Ho learned of each other's efforts only fairly recently. Wolf argues that the efforts are duplicative and criticizes IAVI for funding what he sees as a competitive study. \u201cI would have nothing against it if someone like David joined in our trial,\u201d Wolf told Science. \u201cBut now IAVI is running around the world and putting money into a competing thing; this is irresponsible.\u201d Ho counters that \u201cit is not unusual that multiple groups pursue the same objective\u201d and adds that the differences between the two vaccine strategies might argue for comparing their effectiveness in parallel clinical trials. An opportunity to resolve this conflict may be at hand, however: Later this month, a Beijing meeting co-sponsored by IAVI will bring together representatives of the Chinese ministry of health, NIH, and IAVI, as well as one or two members of the EuroVac team.\n\nViral immunologist Wayne Koff, IAVI's vice president for research and development, says that the clade C vaccines developed by EuroVac and Ho are only two of a number of possible preparations that could go into preliminary clinical trials in China, where more than 400,000 people are estimated to be infected with HIV. IAVI's basic strategy\u2014to accelerate development of the most effective vaccine candidates\u2014could well mean that it would end up funding a \u201chead-to-head\u201d comparison of the two vaccines. But whether either vaccine will ultimately end up a finalist, he adds, it \u201cis too early to say.\u201d\n\n6. DOE LABS\n\n# Congress Shrinks Lab Chiefs' Flexible Funds\n\n1. David Malakoff\n\nIf you don't like the way an institution is run, go after its budget. Congress has just applied that logic to a special fund controlled by the directors of the Department of Energy's (DOE's) national laboratories, slashing the amount available for hiring young scientists and funding high-risk research. Four national labs, including the three nuclear weapons centers, have been hit especially hard by the reductions, which lab officials hope to reverse next year. Lawmakers say that some reprogrammed money has been mismanaged in the past and that the cuts are needed to keep the labs focused on priorities determined by Congress.\n\nThe accounting change restricts the flexibility that Congress gave lab directors in 1991, when it created an account called the Laboratory Directed Research and Development (LDRD) fund. The mechanism currently allows each lab to divert up to 6% of the funds it receives from the federal government and other sources to carry out relatively small research projects, usually costing from $25,000 to$500,000 per year. The money is awarded to lab scientists through a competitive process. Language in the 2000 DOE budget bill signed into law last month, however, would reduce the LDRD tax (divertible funds) to 4% and exempt environmental cleanup programs\u2014a major piece of the budgets of several labs\u2014from any tax.\n\nAlthough smaller DOE labs often do not impose the maximum tax for a variety of reasons, some of the larger labs, including the Los Alamos and Sandia nuclear weapons laboratories in New Mexico and the Lawrence Livermore laboratory in California, have used it to amass annual funds of $50 million or more (see graph). Lab administrators say the money has been essential for attracting young researchers with fresh ideas and for backing risky research, such as forays into materials and computer science, that have evolved into lab mainstays. \u201cLDRD provides us with cherished freedom and creativity in basic research,\u201d says Dan Hartley, Sandia's vice president for laboratory development. But LDRD spending has also attracted scrutiny\u2014and criticism\u2014from some members of the House Appropriations Committee. Representative Ron Packard (R-CA), chair of the spending panel that oversees DOE's budget, and other lawmakers are unhappy that LDRD siphons funds from programs Congress has approved, such as environmental cleanup efforts, and that some labs have funded projects of little relevance to DOE's mission. \u201cThe concern is that when you give a lab director$70 million to spend, it will be used for their priorities, not the nation's,\u201d says one House aide. Opponents of LDRD funding have pointed to internal DOE reviews over the last decade that have found instances of mismanagement of LDRD dollars and accounting practices that diverted more funds than were allowed under the rules. In past years the Senate has rebuffed House efforts to scale back or eliminate LDRD. But this year, after the House voted to cancel the program, Senate negotiators succeeded in restoring only part of the funds.\n\nAt Los Alamos National Laboratory, the change has produced a \u201ctraumatic\u201d 40% cut in the lab's $70 million LDRD budget, which wholly or partly funds hundreds of scientists, says Klaus Lackner, acting associate director for strategic and supporting research. To avoid layoffs, he says the lab is shifting some scientists to weapons projects with more stable funding and focusing the remaining LDRD money on supporting young researchers and funding projects\u2014such as those in the life sciences\u2014unlikely to find backing elsewhere. \u201cWe started from the premise that postdocs must be able to go on,\u201d he says. At Sandia, where LDRD funds have dropped from$83 million to $52 million, officials worry that the funding uncertainty could cause \u201csome of our brightest, youngest people\u201d to leave, Hartley says. Similar fears are being voiced at Livermore, which lost$23 million of its $58 million LDRD budget. \u201cWe are focusing our resources on protecting our long-term strategic investments,\u201d which supplement existing work in such areas as computing and the effects of aging on nuclear weapons, says Rokaya Al-Ayat, Livermore's deputy director for LDRD. Also hard hit by the change was the Idaho National Engineering and Environmental Laboratory, which relies heavily on environmental cleanup funds that can no longer be taxed for LDRD funds. The lab's new director, Billy Shipp, is confident that he and DOE headquarters staff can find a way to continue many existing activities despite a cut from$21 million to $6 million, in part by getting congressional permission to use money from other programs. Lab officials are also thinking about the best way to restore LDRD funding in next year's appropriations bill. Bill Appleton, Oak Ridge's deputy director for science, says scientists must convince House members that LDRD \u201cis one of the few ways that the labs have of doing innovative research that has major payoffs down the line.\u201d At stake, say he and other lab officials, is their ability to attract the best talent and stay at the forefront of science. 7. PALEONTOLOGY # Fossils Give Glimpse of Old Mother Lamprey 1. Carl Zimmer* 1. Carl Zimmer is the author of the book At the Water's Edge. Evolution went on a creative spree about 540 million years ago. Over the course of less than 20 million years during the Early Cambrian period, a huge diversity of animals appeared for the first time, including many of the major groups living today, such as arthropods, mollusks, and various sorts of worms. Notably missing from this party\u2014known as the Cambrian explosion\u2014was any member of our own lineage, the vertebrates. Until now the oldest unambiguous vertebrate fossils dated back 475 million years. But this week our genealogy took a giant leap back in time. Chinese and British paleontologists reported in Nature that they have found the fossils of 530-million-year-old vertebrates\u2014fossils that have other paleontologists in awe. \u201cI was absolutely amazed the first time I saw these fossils. They're just unbelievable,\u201d says Phillippe Janvier, a paleontologist at the Museum d'Histoire Naturelle in Paris who is an expert on early vertebrates. You might expect that such ancient creatures would be primitive, transitional forms linking us to our pre-vertebrate past. Yet surprisingly, the fossils are actually full-fledged vertebrates\u2014more advanced, in fact, than some vertebrates alive today. As a result, paleontologists think fossils of even older vertebrates must be waiting to be discovered, perhaps in rocks dating from well before the Cambrian explosion. The two fossils come from a site in southern China called Chengjiang, already famous for its Cambrian treasures, where the fine-grained rock retains impressions of muscles and other soft tissues. \u201cChengjiang really takes your breath away,\u201d says Simon Conway Morris, a paleontologist at the University of Cambridge. After learning that two different teams of paleontologists, one led by Degan Shu of Northwest University in Xian, had unearthed the vertebrate fossils, Conway Morris traveled to China this April to analyze them with Shu and other Chinese colleagues. They found that the two fossils represented different species, and although the fossils measured only a couple of centimeters long, the researchers could recognize key vertebrate traits. They had rows of gills, and their muscles were arranged in W-shaped blocks along their flanks, a pattern unique to vertebrates. \u201cThey were presumably filter feeders, but they have these muscular bodies and things which we cautiously interpret as an eye,\u201d says Conway Morris. \u201cAnd so presumably they could go along at a fair pace if they had to, and they might have grabbed prey.\u201d The researchers then tried to find a place for the fossils in vertebrate evolution. A number of researchers believe that vertebrates evolved from an ancestor something like Amphioxus, otherwise known as the lancelet. Amphioxus, which lacks eyes or fins and looks rather like a miniature anchovy fillet, has a notochord\u2014a primitive backbone. The first vertebrates added new traits to that body plan, such as a skull with a brain; later vertebrates acquired jaws and fins. The most primitive vertebrate alive today is the hagfish, a jawless fish, and the second-most primitive is the lamprey. Conway Morris and his colleagues concluded that the fossils fall into a surprisingly advanced position. One of the species, which the researchers named Haikouichthys, is most closely related to the lamprey. The other fossil\u2014tortuously named Myllokunmingia\u2014is more primitive (its gills are simpler), but Conway Morris says it is still a closer relative to us than to the hagfish. Features seen on both fossils may help answer the controversial question of how early vertebrates evolved the paired fins that later gave rise to arms and legs (Science, 23 April, p. 575). The new fossils show what look like two long folds of tissue running along their underside\u2014exactly what some theories of fin evolution predicted. \u201cWe think there's a reasonable case for a double arrangement,\u201d says Conway Morris. Janvier, who has argued that the paired fins came much later, has his doubts. \u201cFrom what I could see of the fossils, it's not 100% certain.\u201d He is also uncertain about the fossils' placement on the vertebrate family tree, because many details of the creatures' anatomy have been lost. He has no doubt that they are vertebrates, but says, \u201cI wouldn't put my money on the exact positions.\u201d If Conway Morris is right about the creatures' sophistication, however, millions of years of vertebrate evolution must have preceded them, reaching back before the Cambrian explosion. Some researchers already suspected as much, based on the clocklike divergence of genes in different animal lineages. According to a new study by Blair Hedges of Pennsylvania State University in University Park, for example, vertebrates got their start 750 million years ago. \u201cSome of my colleagues who take molecular clocks seriously will be skipping for joy\u201d over the new finds, Conway Morris acknowledges ruefully. He himself doesn't think vertebrates got their start so long ago. He suspects the first ones arose just before the Cambrian Period, about 565 million years ago. The traces of these ancestral creatures, he thinks, may be waiting, still unrecognized, among the fossils known as the Ediacaran fauna. \u201cThese stem groups are all lurking down there,\u201d Conway Morris maintains, \u201cbut we're just too dim to see them.\u201d 8. PHYSICS # Gravity's Gravity Vindicates Einstein 1. Andrew Watson* 1. Andrew Watson is a science writer in Norwich, U.K. Between them, general relativity and quantum theory explain all of nature's forces, and yet they refuse to be married. The strong and weak nuclear forces and electromagnetism are all described by quantum theories that mesh in a very satisfactory way. On the other hand, general relativity\u2014Einstein's theory linking the force of gravity to the geometry of space and time\u2014steadfastly refuses to be seduced into the quantum fold. \u201cA goal in physics is to unify all the forces, that is, to combine gravity with the other three in one grand theory,\u201d says Blayne Heckel of the University of Washington, Seattle. Like so many others, Heckel, Eric Adelberger, and their Seattle colleagues don't know how nature might entice the two parties to walk together down the aisle. So the Seattle group has instead looked for the possible progeny of such a match. One such child would be a difference in the way gravity acts on mass and on gravitational energy itself. But this hypothetical love child, expected in some scenarios of a deep connection between gravity and the quantum world, is nowhere to be found, the group determined. Einstein built his theory of general relativity on the premise that gravity acts equally on all forms of mass-energy. Experimenters have shown that nuclear binding energy and energies due to electromagnetic interactions do indeed obey this \u201cequivalence principle.\u201d For example, a proton and a neutron combine to make an object with less mass than the component parts; the binding energy holding the two parts together accounts for the missing mass. Yet experiments show that the combination and the individual parts free-fall at the same rate in a gravitational field. But no one has yet shown that gravitational energy responds to the pull of gravity in the same way as all other forms of mass-energy do. Some theories\u2014including string theory, the current favorite in the attempts to synthesize a quantum theory of gravity\u2014suggest it might not. \u201cMany theorists expect that at some point we will find a difference,\u201d says Heckel. Lab experiments can't study the impact of gravitational binding energy, since the energy tied up in the mutual pull of fragments of lab-sized objects is minuscule. The place to look, Kenneth Nordtvedt of Montana State University suggested more than a decade ago, is in the tug of the sun on the moon and Earth. Although Earth's gravitational binding energy is small\u2014a mere half a microgram per kilogram\u2014because Earth is big, around 3 trillion tons of its mass is transformed into pure gravitational energy. The moon's gravitational binding energy is around 2000 times smaller, but still big enough to displace the center of the moon's orbit relative to Earth if the sun's gravity treats mass and gravitational binding energy differently. Spotting these effects means monitoring the Earth-moon distance to high accuracy. By using lunar laser ranging, in which a laser beam bounces off reflectors dropped off on the moon by astronauts, Nordtvedt and others tracked this separation to centimeter accuracy and found, within the limits of the experiments, that the Earth and moon do indeed fall towards the sun at the same rate. Nordtvedt himself pointed out a loophole, however: Some quantum gravity theories suggest gravity might act differently on the Earth and moon because of compositional differences such as Earth's iron-dominated core, explains Heckel. \u201cSo one wants to know that the Earth and moon don't fall at different rates due to composition differences, and by an amount which could cancel a gravitational self-energy effect,\u201d adds Nordtvedt. Such a cancellation is \u201cquite unlikely,\u201d but the Seattle group has sought to resolve this potential ambiguity. The Seattle experiment, reported in this week's Physical Review Letters, consists of a torsion balance: a fine wire supporting a tray that can rotate by twisting the wire. On the tray are four weights, alternating toy Earths and moons, all weighing exactly 10 grams. The two Earth-like weights are made of steel to simulate Earth's core material, while the two moon-like weights are made of quartz and magnesium-based materials that simulate both the Earth's and the moon's mantle material. The whole experiment is rotated so that the \u201cplanets\u201d turn past the sun in 40-minute-long \u201cdays.\u201d Any gravitational preference of the sun for the toy moon or toy Earth should yield a twist in the torsion balance. \u201cIt's a very clever idea, making these little models for the planets,\u201d says Clifford Will at Washington University, St. Louis. The experimenters have produced \u201cextraordinarily precise measurements.\u201d The team found no twist. Their results, combined with the laser ranging, show that \u201cgravitational binding energy falls at the same rate as all other forms of mass-energy to better than a part per thousand,\u201d says Heckel. Heckel declares himself unsurprised at the result, and Einstein's theory triumphs yet again. Yet there's some comfort for the string theorists too, says Will, because differences in the rates of fall of different bodies could lie beyond the sensitivity of current experiments. \u201cWe really think there's a chance of finding a violation at some level,\u201d Will says. So the dating game continues, but gravity remains as aloof and celibate as ever. 9. NANOTECHNOLOGY # Patterning Plastic With Plentiful Pillars 1. Robert F. Service Richmond, Virginia\u2014Rome wasn't built in a day, but a nanosized version of it may be in the near future. At the International Symposium on Cluster and Nanostructure Interfaces here last week, Stephen Chou, an electrical engineer from Princeton University in New Jersey, described a new microscopic patterning technique capable of creating arrays of plastic pillars, each less than a thousandth of a millimeter across, that resemble nothing so much as tiny versions of the great columns of Rome's coliseum. Cheap, fast, and versatile, the patterning scheme could help create novel plastic displays and electronic devices. The pillars themselves could not only be used as wires in plastic electronics, but could also direct the growth of other materials, such as metals and semiconductors, into regular patterns. \u201cIt's very beautiful work,\u201d says Peru Jena, a physicist at the Virginia Commonwealth University in Richmond. It's the beauty of simplicity, says Jena, because the technique requires nothing more than putting a mask above a heated thin polymer film and waiting a few minutes while the pillars assemble themselves. Chou says that, at around 500 nanometers in diameter, the pillars are now more than twice as large as the finest features that photolithography\u2014the workhorse patterning technology of the chip industry\u2014can lay down on silicon. Nevertheless, K. V. Rao, a physicist at the Royal Institute of Technology in Stockholm, Sweden, points out that photolithography has been refined over decades. As for the new technique, \u201cthis is just the beginning,\u201d he says. It was an unexpected beginning, Chou says. He and his students were working on a related patterning technique in which they imprint a pattern of nano-sized ridges and grooves on a soft polymer with a tiny embossing stamp. In one experiment, however, tiny dust grains, each about 0.5 micrometers high, strayed onto the polymer before the stamp was applied. Like tiny boulders, the dust grains prevented the stamp from pushing into the polymer and making an impression. Yet when the Princeton researchers removed the stamp and looked at the surface of the film, they still saw a pattern of dots that matched features on the stamp. The stamp had somehow elicited a pattern without ever touching the surface. Surprised, they repeated the experiment to see if they could find out what had happened. They created another set of masks, this time incorporating tiny posts that held them about half a micrometer above the polymer surface, and again they saw the array of dots. Those dots turned out to be tiny polymer pillars that had grown up from the surface of the plastic layer to the mask. \u201cWe still can't be sure\u201d what causes the pillars to form, Chou says. He and his colleagues have determined that a polymer film produces pillars only when it is heated enough to melt and the masking material above is electrically conductive, which leads Chou to speculate that the interplay of electrical charges in the mask and the polymer film creates the pillars. Localized concentrations of charge in the mask likely induce an opposite charge in the nearby film, he says, generating electrostatic forces that pull the pliable polymer upward. If correct, says Chou, this explanation suggests that the pillars should form first at the corners of the mask, since charges preferentially bunch there. And when the Princeton team set up a video camera to watch their pillars grow, they found that pillars formed first at the corners and edges of the film below the mask and slowly worked their way in toward the center. So far, Chou and his colleagues have made most of their pillars in a polymer called polymethylmethacrylate, more commonly known as plexiglass. But the technique also works with conducting polymers, which could serve as the basis of futuristic flat panel displays and disposable electronics. To test whether their pillars could make conductive wires for such devices, Chou and his colleagues laid down a film of conducting polymer on a metal strip and then grew pillars upward to touch another metal strip passing over the first at right angles. The group hasn't tested the electrical behavior of the pillars, but Chou says he fully expects that they will provide conductive pathways between the metal conductors. If so, plastic pillars may be in for a rising future. 10. ANIMAL EXPERIMENTATION # Humane Science Finds Sharper and Kinder Tools 1. Erik Stokstad For decades, more and more researchers have been using fewer laboratory animals for compassion's sake. Thanks to new experimental techniques, many are getting cleaner results, too Bologna, Italy\u2014A decade ago, veterinary surgeon Christian Schnell tested candidate drugs to lower blood pressure with a procedure that was highly stressful\u2014for himself and his test animals. First, he would anesthetize marmosets and insert a catheter into an artery in their legs. The next day he restrained the conscious animals, orally administered the drug, and recorded blood pressure through the catheter for 4 to 5 hours. Not only did the harried animals' hearts race during the experiment, but each one could be used for only six trials before all its suitable arteries had been tapped. But by 1991, Schnell, a researcher at the drug company Ciba-Geigy in Basel, Switzerland, switched to a new and more sophisticated technology: a sensor that he implanted in the animals' abdominal cavities. The device continually measures blood pressure and transmits the data to receivers in the cages, allowing the marmosets to move freely and remain with their families\u2014more relaxed and with normal heart rates. Without the confounding effect of stress, the results are cleaner. \u201cWe are now convinced we're measuring the truth,\u201d says Schnell. And without the need for catheters, Schnell could do the same research with only 10% of the marmosets he had previously needed, saving the company up to$200,000 a year.\n\nSchnell's case illustrates an accelerating trend in which new technology is helping researchers reduce their reliance on animal experiments, while at the same time improving their results. Although animal rights extremists continue to use violent and intimidatory tactics against researchers in many countries (see page 1059 and Science, 4 June, p. 1604), more moderate campaigners for animal welfare have for years been working with researchers to encourage this trend toward better experimental design and more humane techniques. The motto of this movement is \u201cHumane science is better science,\u201d and its creed is the \u201cthree R's\u201d\u2014replacing laboratory animals, reducing their numbers, and refining techniques to minimize pain and suffering (Science, 11 October 1996, p. 168). The results have been striking: The use of lab animals has declined in many European countries\u2014in some cases by as much as 50% over the past 2 decades.\n\nAs a result, the mood among the more than 800 researchers who gathered here recently for the Third World Congress on Alternatives and Animal Use in the Life Sciences was cautiously upbeat. They exchanged information on a variety of technologies\u2014including implantable sensors like those Schnell uses and new imaging techniques to replace invasive procedures\u2014that are already reducing the number of animals and lessening distress. And researchers reported progress in several areas\u2014such as DNA arrays and tests using stem cells\u2014that could help drug companies rule out dangerous compounds before they're tested in animals. \u201cThe spin-offs of molecular biology and biotechnology will have a great impact on [lowering] the use of lab animals,\u201d predicts geneticist Bert van Zutphen of Utrecht University in the Netherlands.\n\nBut not all the trends are downward. Many animal welfare researchers are alarmed by the imminent prospect of a new round of toxicity tests in the United States on a host of so-called high production volume chemicals (see sidebar), as well as tests on potential endocrine disrupters, that may require millions of laboratory animals. And in some hot areas of research, such as transgenics, animal experimentation is rising fast. Since 1990, the number of procedures on transgenic animals in the United Kingdom, for example, has risen almost 10-fold to more than 447,000. That's \u201ca huge rise and due to get much higher,\u201d predicts Caren Broadhead of the Fund for the Replacement of Animals in Medical Experiments in Nottingham.\n\nEven when researchers come up with technologies that can lessen the use and suffering of test animals, they still face a formidable obstacle: the glacial pace of regulatory bodies in accepting replacement tests, such as cell cultures. \u201cA validation study takes a long time,\u201d says Herman Ko\u00ebter, principal administrator of the Environmental Health and Safety Division of the Organization for Economic Cooperation and Development in Paris. \u201cYou need years and years to get a gold standard.\u201d That frustrates researchers. \u201cIf people knew how onerous it can be to get a test validated, many fewer would begin developing new ones,\u201d says Ian Kimber, research manager of AstraZeneca's Central Toxicology Laboratory in Alderley Park, U.K.\n\n## Less is more\n\nSchnell's work with marmosets to test potential blood pressure drugs is Exhibit A in support of the humane science movement's claim that compassion can improve science. Ciba-Geigy had been puzzled by the fact that some candidate compounds that had looked promising in the earlier, more invasive, tests were duds in early human trials. But when Schnell tried those compounds again using implanted monitors in unrestrained marmosets, they proved to be 10 times less effective at lowering blood pressure than they had in the restrained animals. \u201cIt was a shock when we discovered this,\u201d recalls Schnell.\n\nSince those early tests, telemetry sensors have shrunk in size and price and they are becoming more widespread. Blood pressure monitors weighing 3.5 grams are now small enough to be implanted into mice, and the device that Schnell uses costs about $3000. The new monitors are also far more versatile: Implantable devices can record temperature, blood pressure, heart rate, electrocardiograms, and intraocular pressure, and blood flow monitors will be available soon. \u201cI'm convinced that telemetry will be the standard method in the near future,\u201d says Schnell. Whereas implantable monitors can keep track of an animal's physiology, an imaging technique developed by Xenogen Corp. of Alameda, California, allows researchers to chart the course of an infection or the growth of tumors without any surgery at all. The technique essentially records a glow from inside the animal. The light bulb is the luciferase gene, which produces the firefly's bioluminescent protein. Researchers infect an animal with a microbe engineered to express luciferase, anesthetize it, and place it in a dark chamber. Some of the photons from the luciferase pass through the animal's flesh, and a charged-coupled device counts them for a few minutes, pinpointing the active microbes. The pharmaceutical industry is eyeing this technology as a potential replacement for a standard test called the mouse thigh model. To check out new antibiotics, for example, technicians give the test drug to 14 or more infected mice, then kill a pair of the animals every 2 hours, grind up their thigh muscles, and culture microbes from the tissue over 2 days. The better the antibiotic, the fewer microbes grow on the ground-up muscle. In contrast, researchers can scan a living mouse in just 5 minutes. And measuring the same animal throughout the study\u2014rather than comparing individuals that might have had slightly varying initial infections or responses to the drug\u2014also reduces variability. One group of researchers, led by Tom Parr of Lilly Research Laboratories in Indianapolis, recently compared the two techniques and presented their results at the 39th Interscience Conference on Antimicrobial Agents and Chemotherapy in San Francisco in September. The team ran a mouse thigh model using doses of a known antibiotic, but before extracting the muscle, they imaged the animals. The dose-response curves from the two assays were very similar, with correlations ranging from 0.94 to 0.98. Imaging \u201cis more sensitive and more precise while requiring fewer animals,\u201d says Parr. \u201cWe should be able to get more valuable information in less time.\u201d The quick results also mean that test animals can be killed before they suffer the full effects of an infection. Xenogen president Pamela Reilly Contag says six pharmaceutical companies, including Eli Lilly, are evaluating the technology, and 10 others are in various stages of negotiation. ## Going in vitro Drug companies are also showing interest in alternatives to animal tests to screen compounds for effects on fetal development. Researchers currently test for potential teratogenic effects by treating pregnant animals with a candidate drug and then checking embryos for abnormalities\u2014a time-consuming and expensive proposition. \u201cMost companies now want to have short tests that give a clear answer and that require small amounts of compound,\u201d says Philippe Vanparys, director of genetic and in vitro toxicology at Janssen Research Foundation in Beerse, Belgium. Recent developments in establishing immortal lines of stem cells\u2014general-purpose embryonic cells that can develop into any type of cell in the body\u2014have raised hopes that such tests may be feasible. Because stem cells have a very reliable pattern of development into tissue, researchers can precisely measure any disruption to the number of cells, the quality of cells, and the timing of development. This provides a way of looking for subtle chemical effects that might lead to birth defects in particular organs. For example, Anna Wobus of the Institute of Plant Genetics and Crop Plant Research in Gatersleben, Germany, has developed an in vitro method to differentiate mouse embryonic stem cells into heart muscle cells, among others. Once these cells begin to beat after 9 days of normal development, researchers can check for defects in the nascent heart. In 1996, Horst Spielmann, director of the National Centre for Documentation and Evaluation of Alternative Methods to Animal Experiments in Berlin, submitted this test to the European Centre for the Validation of Alternative Methods (ECVAM) in Ispra, Italy, an organization run by the European Union that assesses the suitability of in vitro tests for replacing established animal tests. \u201cSo far it looks very promising,\u201d says Juergen Hescheler, a molecular biologist at the University of Cologne, Germany. Now, Hescheler and his colleagues have added a feature to the test that could make it even faster, easier to use, and more versatile. At the Bologna meeting, he reported that his group has spliced a fluorescent reporter gene to the cardiac-specific promoter gene, so the cells express a green fluorescent protein on day 4 of development, cutting experimental time in half. \u201cWe can directly measure cell differentiation without any staining, so it's less time-consuming,\u201d says Hescheler. The team now wants to link reporter genes to other types of stem cells, such as neuronal, epithelial, and cartilage precursor cells. If the reporter proteins could fluoresce in different colors, scientists might be able to examine the effects of potential toxicants on a suite of tissues at once. Interest in the cardiac reporter is already high. \u201cIn the last month, I had five to six pharmaceutical companies asking for this test,\u201d says Susanne Bremer of ECVAM. ## Toxic chips Toxicologists are also turning to a hot new genetics technology to study cellular responses to test compounds: DNA microarrays, which are commonly used to track patterns of gene expression (Science, 15 October, p. 444). A single DNA \u201cchip\u201d carries an array of hundreds or thousands of short strands of DNA, each of which acts as a probe for a specific gene. To tell which genes were active in a sample, researchers convert messenger RNA to complementary DNA, tag it with a fluorescent marker, and wash the sample over the chip. The cDNA sticks to a specific probe on the chip, and its presence is revealed by a glowing patch when the chip is illuminated with light. Many toxicologists believe that such arrays could reveal which genes a cell turns on in response to toxic compounds\u2014and because they directly probe the activity of human cells, the arrays may eventually be better than animal tests in predicting toxicity to humans. \u201cDNA chips will be the source of the next reduction in animals used,\u201d predicts Spielmann. AstraZeneca's Central Toxicology Laboratory (CTL) is one of the first off the blocks with a chip outfitted with DNA from 600 genes, associated with everything from cell adhesion and ion channels, to metabolism and immune response\u2014all thought to be involved in cellular response to toxicity. \u201cThe most exciting thing about toxicogenomics is that we're going to start investigating genes we never would have thought of looking at,\u201d says CTL's Kimber. \u201cThat's where the big surprises\u2014and big benefits\u2014are going to come from.\u201d Not everyone is convinced by the promise of DNA chips, however. \u201cThere's much hype about gene chip technology,\u201d says molecular biologist Johannes Doehmer of the Technical University of Munich in Germany. \u201cThey're very expensive, and it will take a few years before you can rely on them.\u201d And although the microarrays generate a lot of information very quickly, the results can be hard to interpret. \u201cThe vast majority of our time is [spent] figuring out the gene response,\u201d says CTL's William Pennie. Even though many researchers say that animals will never be replaced for conducting general investigations or checking a whole-body response to a potential toxicant, scientists are also enthusiastic about the potential of chip technology and in vitro tests for asking specific questions\u2014with data from human cells, rather than animal models of disease. \u201cWe can now go into more depth,\u201d says toxicologist Sandra Coecke of ECVAM. \u201cWith in vivo tests, you ended up with kind of a black box.\u201d Indeed, Coecke and others feel that these kinds of new methods\u2014once validated\u2014could not only replace animals tests, they could be an improvement. 11. ANIMAL EXPERIMENTATION # Toxicity Testing: The Many Arts of Persuasion 1. Erik Stokstad Last October, Vice President Al Gore announced what sounded like a great idea, one that won him plaudits from the environmental movement: a 6-year program to rapidly collect health and safety data on 2800 major industrial chemicals. But one lobby's triumph turned out to be another's catastrophe. Animal rights groups quickly denounced the High Production Volume (HPV) Challenge because they estimated it would require the destruction of more than a million animals. The groups charged that the killing of so many animals was needless because much of the information already existed, more could be derived from nonanimal tests, and some simply wasn't worth collecting. Over the past year, the lobbying to halt or modify the original plans of the HPV program has been intense. Gore has been followed around the country by a heckler in a rabbit suit, animal rights groups have taken out emotive newspaper ads, and animal welfare researchers have held workshops to offer Environmental Protection Agency (EPA) representatives technical advice on alternatives to animal testing. The combination may have worked. On 14 October, EPA sent a letter to 223 companies with new recommendations for testing HPV that should reduce animal use. The impetus for the HPV program was a 1997 study called \u201cToxic Ignorance\u201d by the Environmental Defense Fund, an advocacy group based in New York City. The report suggested that basic toxicology data were not publicly available for most of the chemicals that are manufactured or imported into the United States in amounts greater than 450,000 kilograms (1 million pounds) per year. In a quick follow-up review, EPA could only complete a checklist of specific health data for 7% of these chemicals. The agency then invited chemical manufacturers and importers to volunteer the basic toxicity data and test plans\u2014or face regulation. Activists countered that many tests were unnecessary, because some of these chemicals were either clearly safe, such as those already approved by the Food and Drug Administration for consumption, or obviously toxic, such as rat poison and turpentine. And for other compounds, they argued, EPA simply hadn't looked in the right databases. Many concerned scientists also weren't pleased that a fill-in-the-box suite of information was being required for all the chemicals. The HPV program \u201cis bad news for those of us who seek a scientifically rational approach to hazard prediction and risk assessment,\u201d was the opinion of Michael Balls, director of the European Centre for the Validation of Alternative Methods. In December, EPA held a stakeholder meeting, during which the agency and advocacy groups discussed the use of nonanimal tests in the HPV program. A month later, EPA representatives participated in a conference, called TestSmart, sponsored by the Center for Alternatives to Animal Testing (CAAT) at The Johns Hopkins University in Baltimore to brainstorm suitable alternative methods. Afterward, EPA tentatively proposed several approaches, such as combining reproductive and developmental toxicity tests or changing protocols (for example, replacing the infamous LD-50 test, which determines the dose at which half the treated mice die, with a test that requires fewer animals). Altogether, the agency said, this could reduce animal usage by up to 80%. People for the Ethical Treatment of Animals (PETA) disputed that estimate and kept up the pressure. In its letter of 3 weeks ago, EPA officially responded to the concerns of the animal rights groups and made several recommendations to chemical companies. \u201cWe are trying to minimize the number of animals and avoid needless testing,\u201d explains Susan Wayland, the EPA deputy assistant administrator who signed the letter. \u201cWe just needed to write it down in a way that was clear.\u201d In accordance with international animal welfare guidelines, the letter discourages or rules out several animal tests, such as those for the reproductive effect of chemicals unlikely to be released from factories. To better ensure that no tests are redundant, the agency will now consider previous results from additional databases, including a widely used international chemical safety database called IUCLID. EPA also postponed testing some chemicals for several years, in the hope that validated nonanimal tests for some may be available soon. \u201cThis is for us a compromise,\u201d says Mary Beth Sweetland, spokesperson for PETA, \u201cbut it's so much better than the slaughter that was going to take place.\u201d To help the search for alternatives, EPA announced that the National Institute of Environmental Health Sciences will invest at least$4.5 million during the next 2 years to develop and validate nonanimal protocols. EPA will chip in 250,000 and will seek to contribute about the same next year. Although some activists still aren't completely satisfied with the outcome or the process\u2014PETA claims the agency had to be dragged \u201ckicking and screaming\u201d to consider alternatives\u2014the recommendations leave CAAT director Alan Goldberg feeling optimistic: \u201cThis is a major regulatory agency that has been taking a hard look at how to incorporate the best technology that is more humane.\u201d 12. SOCIETY OF VERTEBRATE PALEONTOLOGY # The Stories Behind the Bones 1. Carl Zimmer* 1. Carl Zimmer is the author of At the Water's Edge. Denver, ColoradoThere's more to paleontology than fossils, as was shown here on 20 to 23 October at the 59th meeting of the Society of Vertebrate Paleontology (SVP). Genetics labs, for example, uncovered an Ice Age disease; a changing atmosphere was fingered as the force behind the evolution of mammals in North America; and dissecting modern animals has hinted at the reason dinosaurs had such big noses. ## Ancient Tuberculosis Identified? The world's deadliest infectious disease, tuberculosis plagues a third of all people on Earth, killing 3 million every year. Exactly how the scourge first got a toehold in our species has been a mystery, but at the meeting researchers made a controversial announcement that they had a clue\u2014in the form of DNA from Mycobacterium tuberculosis dating back 17,000 years. For decades, the traditional story of TB had it arising in Old World pastures. Cows and their bovine relatives carry strains of the mycobacterium that are closely related to the human form; people could have become new hosts for TB when they began herding cattle and handling meat and hides. That idea seemed to find support in the devastating epidemics that swept through Native American society when European colonists arrived in the New World. Native Americans, never having domesticated cattle, had apparently been spared the disease until then and thus had immune systems that couldn't cope with TB. But in 1994, that notion collapsed with the discovery of M. tuberculosis in a 1000-year-old mummy in Peru\u2014predating Columbus's arrival by 500 years (Science, 25 March 1994, p. 1686). Native Americans carried TB long before Europeans came on the scene, and the massive epidemics that followed the contact could have resulted from overcrowding, malnutrition, and bad sanitation. So where did the New World TB come from? In the late 1980s Larry Martin, a paleontologist at the University of Kansas, Lawrence, and Bruce Rothschild, an expert on ancient diseases at the Northeastern Ohio Universities College of Medicine in Rootstown, looked for evidence of TB on bones from the New World's grazing animals. The disease can leave scars on bones in places where the immune system has walled off infected cells. Martin and Rothschild examined bones of bison, musk ox, and bighorn sheep from the Natural Trap Cave in Wyoming, dating back 15,000 to 20,000 years. They found the scars in abundance but couldn't say whether the animals suffered from TB or other lesion-forming diseases, such as brucellosis. Because the lesions looked more like those of TB than of other diseases, \u201cwe knew it was a likely diagnosis,\u201d says Martin. \u201cBut we knew we could be wrong.\u201d The researchers suspected that they might be able to settle the issue by finding genetic material from the mycobacterium itself in the fossils. Natural Trap Cave gets its name from the way it is entered\u2014by falling into the entrance and dropping 30 meters to the cave floor. Animals have been falling to their death in the cave for 100,000 years. During the Ice Age, these unlucky creatures would have crashed into a heap of snow on the cave floor and been freeze-dried. There was a chance, therefore, that some of their DNA could have survived until now. Rothschild and Martin extracted some of the bone tissue from a lesion on a 17,000-year-old bison. They sent samples to labs in Israel and England, each of which used the polymerase chain reaction to amplify any fragments of genes. As Martin explained in his talk, both teams identified genes belonging to Mycobacterium. Although the timing of human arrival in the Western Hemisphere is still under intense debate, Martin says, \u201cmy suspicion is that tuberculosis was waiting for humans when they came.\u201d Based on the talk, however, other researchers are skeptical. \u201cThey didn't put all their ducks in a row,\u201d contends Ross MacPhee of the American Museum of Natural History in New York City. Many species of mycobacteria live in the soil, he points out, and they might have gotten into the cave and contaminated the bison material. Contamination has proven to be a big headache for scientists who study ancient DNA, yet Martin and Rothschild didn't present any control tests that could have ruled it out\u2014for example, testing the bones of animals that don't get TB for the presence of the mycobacterium. \u201cThe result is really interesting, so why didn't they go that extra step and knock out the ambiguity?\u201d asks MacPhee. According to Martin, his team will soon present data that address this issue. If Martin and Rothschild are right, New World TB must have come from the Old World, when some infected mammals crossed the Bering Land Bridge and then infected the early Americans who hunted them. And if people in the New World picked up the disease from hunting, rather than farming, maybe the same goes for the Old World, too. The two researchers note that their scenario resembles current theories that trace AIDS in humans to hunting chimps and monkeys. \u201cIt shows that even for the most sophisticated side of medicine, it's useful to know what happened 17,000 years ago,\u201d says Rothschild. ## Where Have All the Browsers Gone? Two artist's conceptions, common in Earth history texts, tell the story. One portrays North America 20 million years ago in the early Miocene, showing a mix of grassland and trees, with many sorts of hoofed mammals, or ungulates, craning their necks to browse on leaves. In the second, showing the landscape of the last few million years, most of the browsers are gone, and many of the animals\u2014horses, bison, and other grazing ungulates\u2014have their heads to the ground, chewing up the grass. Why did the mammals of North America go through such a drastic shift? The explanation, many researchers think, is blowing in the wind: a drop in atmospheric carbon dioxide since the Miocene, which could have affected the vegetation that the big herbivores depended on. At the meeting, Brown University paleontologist Christine Janis offered a new scenario for how this CO2-driven change in North America's ecosystems might have taken place: by gradually starving the plants the browsers ate. Many researchers think the shift was abrupt. Isotopes found in ancient soils and in the fossil teeth of horses show that between 8 million and 6 million years ago, plants that use a photosynthetic system known as C3 declined, while those that use a system called C4\u2014tougher species such as crabgrass\u2014suddenly became dominant in North America and elsewhere. C4 plants do better than C3 plants in low levels of CO2, suggesting that the declining levels of atmospheric CO2 triggered their spread. The grazing ungulates, equipped with high-crowned teeth, could grind down the hardy C4 plants. But many of the low-crowned browsers, adherents say, couldn't handle life on the new grasslands and went extinct (Science, 28 August 1998, p. 1274). Janis and her co-workers, John Damuth of the University of California, Santa Barbara, and Jessica Theodor at Brown, decided to test that idea by taking a close look at the fossil record. \u201cIt has to be better examined, instead of just [being] asserted,\u201d says Janis. They tallied instances when hoofed mammals evolved into new species or went extinct over the past 20 million years, noting whether the animals were browsers, grazers, or those with a mixed diet. Nowhere in their database could they find a pulse of extinctions that coincided with the shift from C3 to C4 plants. The pattern they found instead was distinctly different. Before 18 million years ago, browsers made up the majority of ungulate species at just about every site where Janis and her colleagues looked\u2014in some places reaching levels as high as 80%. The browsers then started a slow, steady decline that carried on for millions of years, through the C3-C4 transition and into more recent times. Janis agrees that the ultimate spur for this mammal overhaul must have been the fall in CO2 concentrations, but says the changes started many millions of years before the C3-C4 shift. Atmospheric levels of carbon dioxide have been declining for the past 50 million years. This long-term atmospheric change exerted its effects gradually, she proposes, as the leafy C3 plants the browsers depended on produced less food, and many of the browsers gradually went extinct. The C3-C4 transition certainly didn't help matters, but it came after CO2 had already been declining for at least 10 million years. \u201cThe transition was at the end of a long progression,\u201d says Janis. The work \u201cis a wonderful demonstration that the decline of browsers and the evolution of grazing morphology is probably not directly related to the spread of certain types of grass,\u201d says Tony Barnosky, a paleobiologist at the University of California, Berkeley. But Barnosky thinks forces other than the CO2 decline could have played a role. The early Miocene saw a period of volcanism in the western United States, and the dusting of ash would have made plants hard to chew and digest. The rough stuff would have been a challenge for the low-toothed browsers, eventually starving them, but the high-toothed grazers could have coped. Whatever the mechanism, Barnosky says, \u201cthe decline of browsers seems to have an environmental trigger, but the gun was a much bigger\u2014and earlier\u2014one than the C3-C4 shift.\u201d ## Dinosaur Air Conditioning Dinosaurs evolved some pretty bizarre anatomy, from the ridge of plates along a Stegosaurus's back to the fantastically long tail and neck of Apatosaurus. But for Larry Witmer, a paleontologist at the Ohio University College of Osteopathic Medicine in Athens, dinosaurs' coolest feature was hidden from view\u2014specifically, up their noses. In most vertebrates, the nasal cavity is the smallest part of the skull, he explains. Yet in several groups of dinosaurs, the nasal cavity expanded to gigantic proportions. Indeed, a small child could climb inside the nasal cavity of the long-faced, beaked Triceratops. \u201cThere must be something pretty important going on to devote half the skull to that,\u201d Witmer says. The answer, it appears, was to keep dinosaurs, and especially their puny brains, from frying. To figure out what dinosaurs used their noses for, Witmer has spent the past few years engaged in something he calls the DinoNose Project. Muscles, mucous membranes, and blood vessels can all leave marks on a fossil, in the form of grooves, shelves, canals, and other structural features. To interpret the structures of dinosaur skulls, Witmer and his co-workers have studied the anatomy of living animals that have big, unusual noses, looking at the way their soft tissues shape their bones. And because dinosaurs are closely related to today's alligators and birds, Witmer's group studied the heads of these animals as well, identifying the structures they could have inherited from a common ancestor. With this knowledge, Witmer is beginning to make some inferences about what dinosaur noses probably looked like and what they did\u2014work that is winning admiration. \u201cThis is some of the most exciting and innovative research being done in vertebrate paleontology,\u201d says Greg Erickson, a paleontologist at Brown University. The noses must have been up to something important, Witmer says, because they were suffused with a huge amount of blood. In his talk at the SVP meeting, for example, he reported that his group identified three separate blood supplies to the noses of ceratopsians, each of which formed big networks of capillaries. Based on the pattern of impressions in fossilized bone, he suggests that the blood vessels were embedded in mucous membranes lining the walls of the nasal cavity. What's more, the mucous membranes themselves were apparently extensive. In another talk, a DinoNose collaborator, Scott Sampson of the University of Utah, Salt Lake City, pointed out a number of ridges in the ceratopsian schnozz that probably supported curtains of cartilage; these in turn may have served as scaffolding for layers upon layers of mucous membranes. Yet the most obvious function of noses\u2014smelling\u2014probably wasn't responsible for their size. Smelling takes place at the rear of the nasal cavity, while all the extra space and blood supply is found at the front end of dinosaur noses. Witmer thinks dinosaur noses helped keep their brains cool. He notes that all the big-nosed dinosaurs had big bodies as well, and for them, heat must have been a problem, because in big animals the ratio of surface area to body mass is much lower than that for smaller animals. As a result, even if dinosaurs didn't have a fast-burning metabolism like that of mammals, the bigger ones must have been unable to shed heat fast enough from the skin to keep their body temperatures from rising to dangerous levels. The brain in particular could have been damaged by such high temperatures, as everyone knows from the occasional tragic stories of teenagers dying from heat-related \u201cbrain attacks\u201d after playing sports in summertime. Witmer proposes that dinosaurs relied on their noses, with their vast networks of blood vessels, to get rid of excess heat. The vessels were probably in contact with the air in the nasal passages and could have wicked heat from the brain. This would be analogous to what happens in mammals, such as the gazelle, that live in hot climates. These animals have veins just under the skin on their head, which cool the blood as they release heat to the air. Rather than traveling straight back to the heart, this cooled blood takes a detour, flowing through a mesh of veins surrounding the brain. These veins run alongside the arteries bringing warm blood from the body's core. The cool veins absorb the heat from the arteries and carry it away from the brain. \u201cBig animals get a big benefit from heat exchange,\u201d Witmer says. \u201cIt would allow the core temperature to rise while keeping the brain cool.\u201d All this does not rule out other roles for the big noses of dinosaurs. For example, they may have helped attract mates, althoughWitmer's group has yet to study that possible function. Says Sampson, \u201cWe have yet to come up with the final word.\u201d 13. BIOMEDICAL RESEARCH # NIH Eyes Sweeping Reform of Peer Review 1. Bruce Agnew* 1. Bruce Agnew is a writer in Bethesda, Maryland. Authors of a reform proposal say their goal is not to make radical changes but to create a system that can be \u201ccontinually evaluated by outside experts\u201d Like Lewis Carroll's White Queen, who could believe \u201cas many as six impossible things before breakfast,\u201d scientists who analyze the National Institutes of Health's (NIH's) peer-review system often find themselves torn between conclusions that are, at the very least, contradictory: The cornerstone of NIH's success has been its peer-review system, in which small committees of nongovernment scientists, known as \u201cstudy sections,\u201d judge the scientific merit of about 40,000 grant applications a year; or, NIH peer review too often amounts to error-prone, turf-conscious nitpicking by obsolete study sections that reject novel ideas out of fear, ignorance, and self-interest. NIH officials and many researchers today seem to believe both. As a result, NIH is now in the midst of a major drive to refurbish the system\u2014updating it to fit today's biomedical science, setting standards of behavior to improve peer reviewers' manners and methods, and creating a mechanism to ensure that peer review will adapt as science evolves in the future. In the most dramatic reform proposal so far, a blue-ribbon panel headed by National Academy of Sciences president Bruce Alberts wants to completely restructure the array of study sections operated by NIH's Center for Scientific Review (CSR), which pass judgment on about three-quarters of NIH grant applications (Science, 30 July, p. 666). But the changes that will finally emerge, after they are refined and tested over the next 2 or 3 years, may be considerably less sweeping than the Alberts panel blueprint. \u201cI don't think it's going to be as radically different as some people have said,\u201d says NIH director Harold Varmus. \u201cPeer review basically works pretty well now. We don't want to make abrupt changes that could be threats to the system.\u201d The Alberts panel's proposals, if not radical, certainly look pretty startling. Currently, more than 100 CSR study sections are clustered into 19 \u201cIntegrated Review Groups\u201d (IRGs), focused mostly around scientific disciplines such as \u201cBiochemical Sciences\u201d and \u201cCell Development and Function.\u201d Instead, in what it calls the \u201cfirst draft\u201d of its report, the Alberts panel proposes reconstructing the system around 21 reorganized IRGs\u201416 centered on disease or organ systems and five focused on basic research areas whose application to specific disease areas cannot be predicted. Basic research that \u201cmore directly underlies clinical or applied studies\u201d on specific diseases or organ systems should be peer reviewed \u201cwithin the broader biological and medical context to which it will ultimately be applied,\u201d the panel said. \u201cThus, we have attempted to place the review of as much fundamental research as possible within the IRG that is most relevant.\u201d The panel, formally known as the Panel on Scientific Boundaries for Review, did not propose in detail the makeup of the study sections that would populate its revised IRGs. That, it said, is the task for the next phase of the reform effort. But it said there should be enough overlapping expertise so that any grant application could reasonably be reviewed by more than one study section. Alberts's group offered no suggestions about the study sections run by individual NIH institutes, which generally review applications under specific institute programs. These account for about 25% of NIH grant applications but were outside the range of the panel's study. ## The community responds In an outpouring of more than 700 e-mailed responses to NIH by mid-October, most scientists applauded the Alberts panel's general goal. Many also seized the occasion to vent their own frustrations with the system. But a substantial minority of the comments were skeptical, and many researchers said the panel had left out major scientific areas. AIDS researchers\u2014who have picked up a lot of political savvy from their activist patients\u2014mounted an organized campaign to retain an AIDS IRG rather than having AIDS research spread among several different IRGs, as the Alberts panel suggests. They enlisted support from such quarters as the Presidential Advisory Council on HIV-AIDS. AIDS was by no means the only research area that scientists complained would be slighted by being folded into a broader IRG. Others included: kidney and urologic research, toxicology, pharmacology, organic chemistry, developmental biology, aging, nutrition, epidemiology, environmental health sciences, and well over a dozen more. \u201cPlease don't destroy the current system without considering the problems that the proposed changes will create,\u201d wrote Ronald Breslow, chemistry professor at Columbia University and past president of the American Chemical Society. Weaknesses in the current system can be fixed by less traumatic, targeted repairs, many other scientists said. \u201cThe bus is running just fine,\u201d wrote biochemist Daniel Kosman of the State University of New York, Buffalo. \u201cIf it is missing a few stops, just change the route; don't buy a new model that may not run at all.\u201d (Science obtained the responses\u2014some signed, most unsigned\u2014through a Freedom of Information Act request.) But Alberts insists, \u201cWe didn't change everything by any means.\u201d He says \u201cone of the big misunderstandings\u201d is a belief that his panel began to rearrange study sections, but \u201cthat's going to be done by a whole bunch of subpanels of experts in each area.\u201d Alberts's panel will meet next week to review the responses and adjust its proposed framework \u201cto make it better,\u201d he says. The Alberts panel's proposal is only the latest\u2014albeit the most sweeping\u2014of a series of peer-review changes that have been set in place or proposed over the past few years. CSR already has gathered neuroscience and behavioral research into four new IRGs, made up of 37 reconfigured study sections, to complete the merger of the National Institute of Mental Health, the National Institute on Drug Abuse, and the National Institute on Alcohol Abuse and Alcoholism into NIH. It created another new IRG, with eight study sections, to centralize review of AIDS research applications and added a special study section for vaccine research. CSR also has fashioned new study sections to handle applications from clinical researchers who feel they don't get a fair shake in panels dominated by laboratory researchers and to provide homes for research proposals that don't seem to fit anywhere else, such as bioengineering collaborations. The realignment of neuroscience and behavioral study sections\u2014which was required by the 1992 law that merged most of the former Alcohol, Drug Abuse and Mental Health Administration into NIH\u2014started in 1997 and pioneered the technique that will be used if some of the Alberts panel proposals are finally adopted: Advisory groups including extramural researchers worked out tentative organization plans, and then CSR officials performed \u201ctest sorts\u201d\u2014assigning batches of actual grant applications among the proposed study sections\u2014to see how the system would work in real life. ## Judging merit Study sections' marching orders have changed, too. In 1997, Varmus ordered peer reviewers to consider \u201cinnovation\u201d as one of their explicit criteria in weighing grant applications. He was trying to break study sections' habit of favoring \u201csafe science\u201d\u2014incremental projects using tried-and-true methodology\u2014over more imaginative but riskier proposals that might pay bigger dividends. CSR director Ellie Ehrenfield and CSR Advisory Committee chair Keith Yamamoto, of the University of California, San Francisco, say that progress has been made but the job isn't quite done yet. \u201cWe're trying to make a shift in reviewers' mind-sets,\u201d Ehrenfeld says. \u201cWe're trying to change people's behavior. None of these things will be solved by a single magic bullet.\u201d The problem is an old one. Newly named Nobel Prize-winner G\u00fcnter Blobel of The Rockefeller University in New York City recalls (with a laugh) that in 1986, an NIH study section trashed a proposal of his as impractical, \u201cand I found the critiques not constructive but offensive.\u201d But Blobel emphasizes that the NIH peer-review system \u201cis a very good one,\u201d and he says most of its decisions are right. NIH also has simplified grant applications\u2014and reduced opportunities for reviewers' second-guessing\u2014by ending the requirement for detailed budget plans in most \u201cinvestigator-initiated\u201d grant applications. Under the \u201cmodular grant\u201d and \u201cjust-in-time\u201d approaches, researchers in most cases simply ask for funding in increments of25,000; detailed budget justifications and many other paperwork requirements don't come until after a grant is approved. Additional changes are in the works\u2014although some have been a long time coming.\n\n\u201cNo matter how we organize study sections, what really matters is the people sitting around the table,\u201d says Ehrenfeld. Thus CSR is trying to broaden study-section recruiting and has experimented in an informal way with several devices to make peer-review service less onerous. These include tours of duty that involve less than the conventional three meetings a year for 4 years and shared assignments that allow scientists to substitute for one another at some meetings. But none of these changes has been implemented in a systematic way.\n\nCSR officials, and Varmus, also are still puzzling over how to lure more senior scientists back onto study sections. This could bring more consistency and credibility to the process, they say, but senior scientists are generally unenthusiastic about the idea. \u201cThey've done it before,\u201d says Varmus, \u201cand they're on to other kinds of advisory activities, some of which are probably more fun and less work.\u201d\n\nVarmus himself, of course, will be eligible for study section service next year, after he leaves NIH to become president of the Memorial Sloan-Kettering Cancer Center in New York City. Will he volunteer?\n\n\u201cVolunteer?\u201d he replies. \u201cNo. But if they call me, I'll think it over.\u201d\n\nSince 1996, Yamamoto and others have been pushing another idea that is just now taking effect: oversight by \u201cIRG Working Groups.\u201d These will be teams of eight to 10 extramural researchers who will attend at least one round of peer-review meetings, monitor the activities of their IRG and its component study sections, and offer advice on whether the scientific boundaries between study sections are still appropriate\u2014as well, no doubt, as on the conduct of reviews. In effect, they will peer review the peer reviewers. If they can exercise enough diplomatic skill to avoid friction with study section members and chairs, they may provide a mechanism for adapting the peer-review system as science evolves. Alberts is counting on the IRG Working Groups to keep the system up to date. He sees this as a \u201cgreat once-in-a-lifetime opportunity to create a system that won't be just locked in place, but can continually be evaluated by outside experts\u2014and in which modern science, which is changing so rapidly, can really be adequately be supported and tracked.\u201d The first three IRG Working Groups are already on the job. Five more are in the planning stage.\n\nFor individual researchers, however, the biggest boon may come from more efficient communication through the Internet. NIH officials say they are only a year or two away from establishing a long-sought system of electronic submission and review of grant applications that could slash by nearly one-half the 10-month lag from submission to award. Doing away with time lost to printing, collating, distributing, and mailing grant applications also might enable researchers to submit revised proposals without missing a grant-award cycle.\n\nWhatever the outcome of the Alberts panel recommendations, peer review is changing. And perhaps it should be no surprise that the process is taking longer than anyone would like. \u201cThis really is like turning a big ship,\u201d Yamamoto says. \u201cEllie is trying to do a lot of things at the same time, with a staff that's already overburdened.\u201d\n\nWill Varmus's departure in January slow the momentum? Yamamoto hopes the loss will be limited. \u201cHe's put the ship in the right direction,\u201d Yamamoto says. \u201cInertia can be a friend here.\u201d\n\n14. BIOMEDICAL RESEARCH\n\n# What's Wrong With NIH Peer Review?\n\nAmong the more than 700 responses to the proposed reorganization of NIH's peer-review system can be found virtually every complaint researchers have ever made about study-section reviews. Here's a sampling:\n\n\u201cI have been on study sections and have seen members who clearly lacked expertise review proposals and grade proposals in a biased, or self-serving, or bad scientific manner.\u201d \u2013Louis Gerstenfeld, Boston University Medical Center\n\n\u201cI have seen the results of ideas being stolen [by peer reviewers]. Who will be believed, the experienced peer or the new investigator?\u201d \u2013unsigned\n\n\u201cUnder the present \u2018culture,\u2019 which focuses on fault finding and amplification of minor errors and discouraging innovative research, nearly all NIH funding has gone into confirming, reconfirming, and reinventing what is already known, by individuals of very little insight or talent.\u201d \u2013unsigned\n\n\u201cEvery one of us has received reviews that clearly misstated facts, indicated that the reviewer failed to read the proposal thoroughly, or were filled with unsupported assertions of opinion. Such poorly performed reviews, which are, I believe, all too common, undermine confidence in the system.\u201d \u2013unsigned\n\n\u201cWhen one rebuts a review today, the rebuttal is referred to the SRA [scientific review administrator] for the study section that produced the potentially unfair review. This SRA then decides whether the rebuttal is correct. Not surprisingly, she typically decides that it is \u2018a mere scientific disagreement.\u2019\u2026 Unscientific grant review rhetoric never receives objective scrutiny.\u201d \u2013Michael Swift, New York Medical College\n\n\u201cI do not think any major change has taken place [in study sections' over- reliance on preliminary data]. Preliminary data is still a major barrier. Risk-taking in general is much frowned upon. I remember participating in a study section which reviewed high-risk grants. If I remember correctly, most, if not all, were disapproved as being too unlikely to succeed.\u201d \u2013David Greenberg, Mt. Sinai Medical Center, New York\n\n\u201cWe do everyone an injustice by allowing half the participants in the process (the reviewers) to hide behind the veil of anonymity. Grants should be reviewed openly and there should be an opportunity to respond to the reviews in \u2018real time.\u2019\u201d \u2013Donald Dwyer, LSU Medical Center\n\n\u201cWe have had grants reviewed by a given committee and then upon resubmission, the critique was the exact opposite of the previous panel. \u2026 This is totally unfair and leads to incredible frustration.\u201d \u2013unsigned\n\n\u201cOften I cannot recognize even one so-called expert in my area in the study section. The reviewers are pedantic and pay attention to one or two experiments which the reviewer does not understand and shoot down a 4-year project. The reviewers often do not understand the underlying principles or broad objectives of a proposal and resort to nit-picking. Basically all new ideas are rejected.\u201d \u2013unsigned\n\n\u201cThe AIDS and Related Research [3] Study Section was composed of individuals with widely different areas of expertise. \u2026 For the most part, we couldn't understand the reviews written by other members of the panel and were able to function only because we were forced to trust each other. Trust is a wonderful thing in friendship but not necessarily in peer review.\u201d\u2013Kathlyn Parker, Brown University\n\n\u201cWe all know how the system works. Do the work, describe part of the results as preliminary in the grant [application], then when you get the priority score, write the papers and start on what you really wanted to do in the first place.\u201d\u2013unsigned\n\n15. SCIENTIFIC MISCONDUCT\n\n# The Misconduct Case That Won't Go Away\n\n1. Eliot Marshall\n\nThe University of Arizona fired Marguerite Kay last year, but supporters nationwide are rallying to her cause and a legal decision is pending\n\nA contentious scientific misconduct case that has divided faculty at the University of Arizona may be heading toward a new climax. This month, an Arizona state court is considering a request by the accused\u2014a prominent researcher on aging, Marguerite Kay\u2014to be reinstated as Regents Professor at the University of Arizona (UA), Tucson. University president Peter Likins dismissed Kay abruptly on 15 July 1998 after a series of faculty-led investigations concluded that Kay had manipulated data and seriously mismanaged her lab. Kay has appealed the dismissal to the state court, which issued a decision partly in her favor on a different legal basis in April. The current appeal could be decided in a few weeks.\n\nKay, cited for her research on the aging of blood cells and the role of the immune system in Alzheimer's disease, has enjoyed the continuous support of a vocal contingent of the faculty. Her foremost advocate is her former department chair, John Marchalonis, head of microbiology and immunology. He insists that the scientific misconduct charges against Kay were played up by administrators who resented Kay's challenges to their decisions on lab resources and service fees.\n\nFormer UA vice president for research Michael Cusanovich, who coordinated the initial Kay investigation, says these allegations are unfounded. The inquiry, he says, began when one of Kay's former technicians filed a written complaint with the university, and the investigation was conducted by independent panels selected by the faculty, in accordance with UA rules. Marchalonis and Carol Bernstein\u2014a member of the same department and local president of the American Association of University Professors\u2014have circulated many letters supporting Kay from outside the university, including from the national AAUP and well-known researchers. Among those who have questioned the UA proceedings are former National Institute of Mental Health director Frederick Goodwin, now a researcher at George Washington University in Washington, D.C.; former National Institute on Aging official Zaven Khachaturian, now director of the Ronald and Nancy Reagan Research Institute of the Alzheimer's Association; molecular biologist David Soll of the University of Iowa, Iowa City; Stanley Azen of the Doheny Eye Institute at the University of Southern California, Los Angeles; and neurology researcher Paul Coleman of the University of Rochester in Rochester, New York.\n\nLikins isn't commenting on the case because, an assistant says, \u201cit would be inappropriate\u201d to do so while it's in litigation. But he did make a dramatic and detailed presentation of his reasons for firing Kay at a faculty senate meeting last December. According to a videotape of that meeting, made available to Science by Bernstein, he told the faculty he had made this \u201cagonized\u201d decision after a careful review of the evidence collected by an investigative panel, which found Kay guilty of four counts of misconduct. The university began looking into the case in 1997 after one of Kay's technicians filed written allegations of misconduct, claiming that Kay had manipulated experimental results. An ethics panel then found cause for investigation; an investigatory panel collected evidence and brought an indictment; and a third panel reviewed the work of the earlier panels and held a public hearing at which Kay testified. It concluded that Kay had \u201cfalsified, manipulated, and otherwise misrepresented data and findings in publications,\u201d and that she had egregiously \u201cmismanaged her UA laboratory and employees.\u201d It recommended that her employment be \u201cterminated.\u201d\n\nLikins told the senate meeting that he found some of the evidence equivocal, but was particularly swayed by one set of data from a table that Kay included in a review article published in the journal Gerontology in 1997. Likins presented a detailed analysis showing beyond a doubt, he said, that the author had selected raw data to make results appear significant when they were not. This misconduct gave credence, he argued, to other charges of data manipulation brought against Kay by lab technicians. Likins also revealed that, out of \u201ccompassion,\u201d he had offered to retain Kay on the faculty if she would acknowledge her misconduct. But she refused, and he fired her immediately.\n\nKay says that as soon as she learned from her staff that data in the Gerontology paper were erroneous, she wrote to the journal to have the table withdrawn. Her correction letter, published in Gerontology in June 1998, blames a technician for the mistakes. Today she says that she doesn't know how the errors crept in. She insists that she relied entirely on her staff for statistical computations. In any case, she says, the table was \u201cirrelevant\u201d to the points she made in the text of the review. Finally, Kay says that the collection of damning data presented to the faculty by Likins was a \u201ccut and paste\u201d assemblage\u2014not raw data\u2014which she had never seen in the form Likins produced. She believes that her integrity was challenged by disgruntled staff members and that a hostile administration used the criticism to dismiss her.\n\nOn 15 August 1998, Kay filed suit against UA and its officials in state superior court for Pima County, arguing that the university had violated her due process rights by dismissing her without adequate notice or opportunity for review. Kay was shut out of her research lab before the public hearing that found her guilty of misconduct, for example. And she claims she was fired without severance pay or a chance for an appeal.\n\nJudge Stephen Villarreal's finding in April 1999 said that the university had acted in an \u201carbitrary and capricious\u201d manner in firing Kay without a regular personnel hearing. The judge did not review the misconduct allegations but found that the 5 days of public hearings on these charges from 30 March to 4 April 1998 were not equivalent to a hearing on dismissal, which university rules require. Villarreal found that UA should give Kay such a hearing, but Kay and the university still have not aggreed on how to proceed.\n\nKay has asked Villarreal to review her case again, for another reason. She has claimed \u201cwhistleblower\u201d status under state law because she had complained in the past about administrative actions taken by Cusanovich. In an independent case, the Arizona Supreme Court found on 4 October that a whistleblower employed by the state may be represented by an attorney during a dismissal proceeding. But UA did not allow Kay to be represented by an attorney during her misconduct trial. (Her attorney was at her side, however.) On 5 October, Kay filed a motion asking Villarreal to nullify her dismissal because due process was violated. The judge's decision is expected soon.\n\nKay's supporters at the university have become more vocal this fall, protesting her firing as a violation of academic rights and a threat to tenure. A faculty leadership group known as the Committee of Eleven voted unanimously in August to ask Likins to reinstate Kay to her job. This panel includes Marchalonis and the chair of the faculty senate, English professor Jerrold Hogle, who previously supported Likins's decision. Hogle could not be reached for comment.\n\nIn addition, a constitutional law specialist on the legal faculty, Roy Spece Jr., conducted an investigation on his own initiative and concluded, as he told the faculty senate in September, that the process was heavily biased against Kay from the outset because the UA general counsel privately interviewed a technician in Kay's lab, a key witness against her. Spece told the senate the proceedings were deeply flawed on legal grounds and at odds with UA rules, which require that faculty members be given full notice before being dismissed.\n\nIt's unclear whether any of these protests\u2014or the state court's decisions\u2014will cause the university to ease its punishment of one of its most distinguished biomedical researchers. But one thing is certain: Likins is discovering\u2014as others have before\u2014that disputes over scientific conduct rarely die. 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A beautiful white and silver grey bow with cross center. It features "He is Risen" ribbon that is overlaid on silver grey ribbon. It is perfect for all of you Eeaster activities.
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\section{Introduction}
The nuclear scissors mode was predicted \cite{Hilt}--\cite{Lo}
as a counter-rotation of protons against neutrons in deformed nuclei.
However, its collectivity turned out to be small. From RPA results which
were in qualitative agreement with experiment, it was even questioned
whether this mode is collective at all \cite{Zaw,Sushkov}.
Purely phenomenological models (such as, e.g.,
the two rotors model \cite{Lo2000}) and the sum rule approach~\cite{Lipp}
did not clear up the situation in this
respect. Finally in a recent review \cite{Heyd} it is concluded
that the scissors mode is "weakly collective, but strong
on the single-particle scale" and further: "The weakly
collective scissors mode excitation has become an ideal test of models
-- especially microscopic models -- of nuclear vibrations. Most models
are usually calibrated to reproduce properties of strongly collective
excitations (e.g. of $J^{\pi}=2^+$ or $3^-$ states, giant resonances,
...). Weakly-collective phenomena, however, force the models to make
genuine predictions and the fact that the transitions in question are
strong on the single-particle scale makes it impossible to dismiss
failures as a mere detail, especially in the light of the overwhelming
experimental evidence for them in many nuclei \cite{Kneis,Richt}."
The Wigner Function
Moments (WFM) or phase space moments method turns out to be very
useful in this
situation. On the one hand it is a purely microscopic method, because
it is based on the Time Dependent Hartree-Fock (TDHF) equation. On the
other hand the method works with average values (moments) of operators
which have a direct relation to the considered phenomenon and, thus, make a
natural bridge with the macroscopic description. This
makes it an ideal instrument to describe the basic characteristics
(energies and excitation probabilities) of collective excitations such as,
in particular, the scissors mode.
Further developments of the WFM
method, namely, the switch from TDHF
to TDHF-Bogoliubov (TDHFB) equations, i.e. taking into account pair correlations, allowed
us to improve considerably the quantitative description of the
scissors mode \cite{Malov,Urban}: for rare earth nuclei the energies were
reproduced with
$\sim 10\%$ accuracy and B(M1) values were reduced by about a factor of two
with respect to their non superfluid values.
However, they remained about two times too high with respect to experiment.
We have suspected, that the reason of this last discrepancy is hidden in the spin
degrees of freedom, which were so far ignored by the WFM method.
In a recent paper \cite{BaMo} the WFM method was
applied for the first time to solve the TDHF equations including spin
dynamics.
As a first step, only the spin orbit interaction was included in the
consideration, as the most important
one among all possible spin dependent interactions because it enters
into the mean field.
The most remarkable result was the discovery of a new type
of nuclear collective motion: rotational oscillations of "spin-up"
nucleons with respect of "spin-down" nucleons (the spin scissors mode).
It turns out that the experimentally
observed group of peaks in the energy interval 2-4 MeV corresponds
very likely to
two different types of motion: the orbital scissors mode and this new kind
of mode, i.e. the spin scissors mode. The pictorial view of these two intermingled scissors
is shown on Fig.~\ref{fig0}, which is just the modification (or generalization) of the
classical picture for
the orbital scissors (see, for example, \cite{Lo2000,Heyd}).
\begin{figure}[h]
\centering\includegraphics[width=5cm]{fig_1.pdf}
\caption{Pictorial representation of two intermingled scissors: the orbital (neutrons
versus protons) scissors + spin (spin-up nucleons versus spin-down nucleons)
scissors. Arrows inside of ellipses show the direction of spin projections.
{\bf p} - protons, {\bf n} - neutrons.}
\label{fig0}\end{figure}
The next step was done in the paper \cite{BaMoPRC}, where the influence of the spin-spin
interaction on the scissors modes was studied. There was hope
that, due to spin dependent interactions, some part of the
force of M1 transitions will be shifted to the energy region of 5-10 MeV (the area of a
spin-flip resonance),
decreasing in such a way the M1 force of scissors. However, these expectations were not realised. It turned out that the spin-spin interaction does not change the general
picture of the positions of excitations described in \cite{BaMo} pushing all levels
up proportionally to its strength without changing their order. The most interesting
result concerns the B(M1) values of both scissors
-- the spin-spin interaction strongly redistributes M1 strength in favour
of the spin scissors mode practically without changing their summed strength.
In the present work we suggest a generalization of the WFM method which takes into account
spin degrees of freedom and pair correlations simultaneously. According to our previous
calculations these two factors, working together, should improve considerably the agreement
between the theory and experiment in the description of nuclear scissors modes.
The paper is organized as follows.
In Sec. 2 the TDHFB equations for the 2x2 normal and anomalous density matrices are
formulated and their Wigner transform is found.
In Sec. 3 the model Hamiltonian and the mean field are analyzed.
In Sec. 4 the collective variables are defined and the respective
dynamical equations are derived.
In Sec. 5 the choice of parameters and the results of calculations of energies and B(M1)
values of two scissors modes are discussed.
The phenomenon of counter-rotating angular momenta with spin up/down,
which can be considered also as a phenomenon of hidden angular momenta, is analysed in Sec. 6.
Results of calculations for 26 nuclei in the rare earth region
are discussed in Sec. 7.
The summary of main results is given in the conclusion section.
The mathematical details are concentrated in Appendices A, B, C, D.
\section{Wigner transformation of TDHFB equations}
\hspace{5mm} The Time-Dependent Hartree--Fock--Bogoliubov (TDHFB)
equations in matrix formulation are
\cite{Solov,Ring}
\begin{equation}
i\hbar\dot{\cal R}=[{\cal H},{\cal R}]
\label{tHFB}
\end{equation}
with
\begin{equation}
{\cal R}={\hat\rho\qquad-\hat\kappa\choose-\hat\kappa^{\dagger}\;\;1-\hat\rho^*},
\quad{\cal H}={\hat
h\quad\;\;\hat\Delta\choose\hat\Delta^{\dagger}\quad-\hat h^*}
\end{equation}
The normal density matrix $\hat \rho$ and Hamiltonian $\hat h$ are
hermitian whereas the abnormal density $\hat \kappa$ and the pairing
gap $\hat \Delta$ are skew symmetric: $\hat \kappa^{\dagger}=-\hat
\kappa^*$, $\hat \Delta^{\dagger}=-\hat \Delta^*$.
The detailed form of the TDHFB equations is
\begin{eqnarray}
&& i\hbar\dot{\hat\rho} =\hat h\hat\rho -\hat\rho\hat h
-\hat\Delta \hat\kappa ^{\dagger}+\hat\kappa \hat\Delta^\dagger,
\nonumber\\
&&-i\hbar\dot{\hat\rho}^*=\hat h^*\hat\rho ^*-\hat\rho ^*\hat h^*
-\hat\Delta^\dagger\hat\kappa +\hat\kappa^\dagger\hat\Delta ,
\nonumber\\
&&-i\hbar\dot{\hat\kappa} =-\hat h\hat\kappa -\hat\kappa \hat h^*+\hat\Delta
-\hat\Delta \hat\rho ^*-\hat\rho \hat\Delta ,
\nonumber\\
&&-i\hbar\dot{\hat\kappa}^\dagger=\hat h^*\hat\kappa^\dagger
+\hat\kappa^\dagger\hat h-\hat\Delta^\dagger
+\hat\Delta^\dagger\hat\rho +\hat\rho^*\hat\Delta^\dagger .
\label{HFB}
\end{eqnarray}
It is easy to see that the second and fourth equations are complex
conjugate to the first and third ones respectively.
Let us consider their matrix form in coordinate
space keeping all spin indices $s, s', s''$:
\begin{eqnarray}
i\hbar\langle {\bf r},s|\dot{\hat\rho}|{\bf r}'',s''\rangle =
\hspace{12cm}
\nonumber\\
\hspace{10mm}
\sum_{s'}\int\!d^3r'\left(
\langle {\bf r},s|\hat h|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\rho|{\bf r}'',s''\rangle
-\langle {\bf r},s|\hat\rho|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat h|{\bf r}'',s''\rangle \right.
\nonumber\\
\hspace{10mm}
\left.
-\langle {\bf r},s|\hat\Delta|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\kappa^{\dagger}|{\bf r}'',s''\rangle
+\langle {\bf r},s|\hat\kappa|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\Delta^{\dagger}|{\bf r}'',s''\rangle
\right),
\nonumber\\
i\hbar\langle {\bf r},s|\dot{\hat\kappa}|{\bf r}'',s''\rangle = -\langle {\bf r},s|\hat\Delta|{\bf r}'',s''\rangle
\hspace{85mm}
\nonumber\\
\hspace{10mm}
+\sum_{s'}\int\!d^3r'\left(
\langle {\bf r},s|\hat h|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\kappa|{\bf r}'',s''\rangle
+\langle {\bf r},s|\hat\kappa|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat h^*|{\bf r}'',s''\rangle \right.
\nonumber\\
\hspace{10mm}
\left.
+\langle {\bf r},s|\hat\Delta|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\rho^*|{\bf r}'',s''\rangle
+\langle {\bf r},s|\hat\rho|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\Delta|{\bf r}'',s''\rangle
\right),
\nonumber\\
i\hbar\langle {\bf r},s|\dot{\hat\rho}^*|{\bf r}'',s''\rangle =
\hspace{12cm}
\nonumber\\
\hspace{10mm}
\sum_{s'}\int\!d^3r'\left(
-\langle {\bf r},s|\hat h^*|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\rho^*|{\bf r}'',s''\rangle
+\langle {\bf r},s|\hat\rho^*|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat h^*|{\bf r}'',s''\rangle \right.
\nonumber\\
\hspace{10mm}
\left.
+\langle {\bf r},s|\hat\Delta^{\dagger}|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\kappa|{\bf r}'',s''\rangle
-\langle {\bf r},s|\hat\kappa^{\dagger}|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\Delta|{\bf r}'',s''\rangle
\right),
\nonumber\\
i\hbar\langle {\bf r},s|\dot{\hat\kappa}^{\dagger}|{\bf r}'',s''\rangle = \langle {\bf r},s|\hat\Delta^{\dagger}|{\bf r}'',s''\rangle
\hspace{85mm}
\nonumber\\
\hspace{10mm}
+\sum_{s'}\int\!d^3r'\left(
-\langle {\bf r},s|\hat h^*|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\kappa^{\dagger}|{\bf r}'',s''\rangle
-\langle {\bf r},s|\hat\kappa^{\dagger}|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat h|{\bf r}'',s''\rangle \right.
\nonumber\\
\hspace{10mm}
\left.
-\langle {\bf r},s|\hat\Delta^{\dagger}|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\rho|{\bf r}'',s''\rangle
-\langle {\bf r},s|\hat\rho^*|{\bf r}',s'\rangle \langle {\bf r}',s'|\hat\Delta^{\dagger}|{\bf r}'',s''\rangle
\right).
\label{HFmatr}
\end{eqnarray}
We do not specify the isospin indices in order to make
formulae more transparent. They will be re-introduced at the end.
Let us introduce the more compact notation
$\langle {\bf r},s|\hat X|{\bf r}',s'\rangle =X_{rr'}^{ss'}$. Then
the set of TDHFB equations (\ref{HFmatr}) with specified spin indices reads
\begin{eqnarray}
&&i\hbar\dot{\rho}_{rr''}^{\uparrow\uparrow} =
\int\!d^3r'(
h_{rr'}^{\uparrow\uparrow}\rho_{r'r''}^{\uparrow\uparrow}
-\rho_{rr'}^{\uparrow\uparrow} h_{r'r''}^{\uparrow\uparrow}
+\hat h_{rr'}^{\uparrow\downarrow}\rho_{r'r''}^{\downarrow\uparrow}
-\rho_{rr'}^{\uparrow\downarrow} h_{r'r''}^{\downarrow\uparrow}
-\Delta_{rr'}^{\uparrow\downarrow}{\kappa^{\dagger}}_{r'r''}^{\downarrow\uparrow}
+\kappa_{rr'}^{\uparrow\downarrow}{\Delta^{\dagger}}_{r'r''}^{\downarrow\uparrow}),
\nonumber\\
&&i\hbar\dot{\rho}_{rr''}^{\uparrow\downarrow} =
\int\!d^3r'(
h_{rr'}^{\uparrow\uparrow}\rho_{r'r''}^{\uparrow\downarrow}
-\rho_{rr'}^{\uparrow\uparrow} h_{r'r''}^{\uparrow\downarrow}
+\hat h_{rr'}^{\uparrow\downarrow}\rho_{r'r''}^{\downarrow\downarrow}
-\rho_{rr'}^{\uparrow\downarrow} h_{r'r''}^{\downarrow\downarrow}),
\nonumber\\
&&i\hbar\dot{\rho}_{rr''}^{\downarrow\uparrow} =
\int\!d^3r'(
h_{rr'}^{\downarrow\uparrow}\rho_{r'r''}^{\uparrow\uparrow}
-\rho_{rr'}^{\downarrow\uparrow} h_{r'r''}^{\uparrow\uparrow}
+\hat h_{rr'}^{\downarrow\downarrow}\rho_{r'r''}^{\downarrow\uparrow}
-\rho_{rr'}^{\downarrow\downarrow} h_{r'r''}^{\downarrow\uparrow}),
\nonumber\\
&&i\hbar\dot{\rho}_{rr''}^{\downarrow\downarrow} =
\int\!d^3r'(
h_{rr'}^{\downarrow\uparrow}\rho_{r'r''}^{\uparrow\downarrow}
-\rho_{rr'}^{\downarrow\uparrow} h_{r'r''}^{\uparrow\downarrow}
+\hat h_{rr'}^{\downarrow\downarrow}\rho_{r'r''}^{\downarrow\downarrow}
-\rho_{rr'}^{\downarrow\downarrow} h_{r'r''}^{\downarrow\downarrow}
-\Delta_{rr'}^{\downarrow\uparrow}{\kappa^{\dagger}}_{r'r''}^{\uparrow\downarrow}
+\kappa_{rr'}^{\downarrow\uparrow}{\Delta^{\dagger}}_{r'r''}^{\uparrow\downarrow}),
\nonumber\\
&&i\hbar\dot{\kappa}_{rr''}^{\uparrow\downarrow} = -\hat\Delta_{rr''}^{\uparrow\downarrow}
+\int\!d^3r'\left(
h_{rr'}^{\uparrow\uparrow}\kappa_{r'r''}^{\uparrow\downarrow}
+\kappa_{rr'}^{\uparrow\downarrow} {h^*}_{r'r''}^{\downarrow\downarrow}
+\Delta_{rr'}^{\uparrow\downarrow}{\rho^*}_{r'r''}^{\downarrow\downarrow}
+\rho_{rr'}^{\uparrow\uparrow}\Delta_{r'r''}^{\uparrow\downarrow}
\right),
\nonumber\\
&&i\hbar\dot{\kappa}_{rr''}^{\downarrow\uparrow} = -\hat\Delta_{rr''}^{\downarrow\uparrow}
+\int\!d^3r'\left(
h_{rr'}^{\downarrow\downarrow}\kappa_{r'r''}^{\downarrow\uparrow}
+\kappa_{rr'}^{\downarrow\uparrow} {h^*}_{r'r''}^{\uparrow\uparrow}
+\Delta_{rr'}^{\downarrow\uparrow}{\rho^*}_{r'r''}^{\uparrow\uparrow}
+\rho_{rr'}^{\downarrow\downarrow}\Delta_{r'r''}^{\downarrow\uparrow}
\right).
\label{HFsp}
\end{eqnarray}
This set of equations must be complemented by the complex conjugated equations.
Writing these equations, we neglected the diagonal matrix elements in spin,
$\kappa_{rr'}^{ss}$ and $\Delta_{rr'}^{ss}$. It is shown in Appendix A that such
approximation works very well in the case of monopole pairing considered here.
We will work with the Wigner transform \cite{Ring} of
equations (\ref{HFsp}). The relevant mathematical details can be found in
\cite{Malov}. The most essential relations are outlined in Appendix B.
>From now on, we will not write out the coordinate
dependence $({\bf r},{\bf p})$ of all functions in order to make the formulae
more transparent. The
Wigner transform of (\ref{HFsp}) can be written as
\begin{eqnarray}
i\hbar\dot f^{\uparrow\uparrow} &=&i\hbar\{h^{\uparrow\uparrow},f^{\uparrow\uparrow}\}
+h^{\uparrow\downarrow}f^{\downarrow\uparrow}-f^{\uparrow\downarrow}h^{\downarrow\uparrow}
+\frac{i\hbar}{2}\{h^{\uparrow\downarrow},f^{\downarrow\uparrow}\}
-\frac{i\hbar}{2}\{f^{\uparrow\downarrow},h^{\downarrow\uparrow}\}
\nonumber\\
&-&\frac{\hbar^2}{8}\{\!\{h^{\uparrow\downarrow},f^{\downarrow\uparrow}\}\!\}
+\frac{\hbar^2}{8}\{\!\{f^{\uparrow\downarrow},h^{\downarrow\uparrow}\}\!\}
+ \kappa\Delta^* - \Delta\kappa^*
\nonumber\\
&+&\frac{i\hbar}{2}\{\kappa,\Delta^*\}-\frac{i\hbar}{2}\{\Delta,\kappa^*\}
- \frac{\hbar^2}{8}\{\!\{\kappa,\Delta^*\}\!\} + \frac{\hbar^2}{8}\{\!\{\Delta,\kappa^*\}\!\}
+...,
\nonumber\\
i\hbar\dot f^{\downarrow\downarrow} &=&i\hbar\{h^{\downarrow\downarrow},f^{\downarrow\downarrow}\}
+h^{\downarrow\uparrow}f^{\uparrow\downarrow}-f^{\downarrow\uparrow}h^{\uparrow\downarrow}
+\frac{i\hbar}{2}\{h^{\downarrow\uparrow},f^{\uparrow\downarrow}\}
-\frac{i\hbar}{2}\{f^{\downarrow\uparrow},h^{\uparrow\downarrow}\}
\nonumber\\
&-&\frac{\hbar^2}{8}\{\!\{h^{\downarrow\uparrow},f^{\uparrow\downarrow}\}\!\}
+\frac{\hbar^2}{8}\{\!\{f^{\downarrow\uparrow},h^{\uparrow\downarrow}\}\!\}
+ \bar\Delta^* \bar\kappa - \bar\kappa^* \bar\Delta
\nonumber\\
&+&\frac{i\hbar}{2}\{\bar\Delta^*,\bar\kappa\}-\frac{i\hbar}{2}\{\bar\kappa^*,\bar\Delta\}
- \frac{\hbar^2}{8}\{\!\{\bar\Delta^*,\bar\kappa\}\!\} + \frac{\hbar^2}{8}\{\!\{\bar\kappa^*,\bar\Delta\}\!\}
+...,
\nonumber\\
i\hbar\dot f^{\uparrow\downarrow} &=&
f^{\uparrow\downarrow}(h^{\uparrow\uparrow}-h^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{(h^{\uparrow\uparrow}+h^{\downarrow\downarrow}),f^{\uparrow\downarrow}\}
-\frac{\hbar^2}{8}\{\!\{(h^{\uparrow\uparrow}-h^{\downarrow\downarrow}),f^{\uparrow\downarrow}\}\!\}
\nonumber\\
&-&h^{\uparrow\downarrow}(f^{\uparrow\uparrow}-f^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{h^{\uparrow\downarrow},(f^{\uparrow\uparrow}+f^{\downarrow\downarrow})\}
+\frac{\hbar^2}{8}\{\!\{h^{\uparrow\downarrow},(f^{\uparrow\uparrow}-f^{\downarrow\downarrow})\}\!\}+....,
\nonumber\\
i\hbar\dot f^{\downarrow\uparrow} &=&
f^{\downarrow\uparrow}(h^{\downarrow\downarrow}-h^{\uparrow\uparrow})
+\frac{i\hbar}{2}\{(h^{\downarrow\downarrow}+h^{\uparrow\uparrow}),f^{\downarrow\uparrow}\}
-\frac{\hbar^2}{8}\{\!\{(h^{\downarrow\downarrow}-h^{\uparrow\uparrow}),f^{\downarrow\uparrow}\}\!\}
\nonumber\\
&-&h^{\downarrow\uparrow}(f^{\downarrow\downarrow}-f^{\uparrow\uparrow})
+\frac{i\hbar}{2}\{h^{\downarrow\uparrow},(f^{\downarrow\downarrow}+f^{\uparrow\uparrow})\}
+\frac{\hbar^2}{8}\{\!\{h^{\downarrow\uparrow},(f^{\downarrow\downarrow}-f^{\uparrow\uparrow})\}\!\}+...,
\nonumber\\
i\hbar\dot \kappa &=& \kappa\,(h^{\uparrow\uparrow}+\bar h^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{(h^{\uparrow\uparrow}-\bar h^{\downarrow\downarrow}),\kappa\}
-\frac{\hbar^2}{8}\{\!\{(h^{\uparrow\uparrow}+\bar h^{\downarrow\downarrow}),\kappa\}\!\}
\nonumber\\
&+&\Delta\,(f^{\uparrow\uparrow}+\bar f^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{(f^{\uparrow\uparrow}-\bar f^{\downarrow\downarrow}),\Delta\}
-\frac{\hbar^2}{8}\{\!\{(f^{\uparrow\uparrow}+\bar f^{\downarrow\downarrow}),\Delta\}\!\}
- \Delta + ...,
\nonumber\\
i\hbar\dot \kappa^* &=& -\kappa^*(h^{\uparrow\uparrow}+\bar h^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{(h^{\uparrow\uparrow}-\bar h^{\downarrow\downarrow}),\kappa^*\}
+\frac{\hbar^2}{8}\{\!\{(h^{\uparrow\uparrow}+\bar h^{\downarrow\downarrow}),\kappa^*\}\!\}
\nonumber\\
&-& \Delta^*(f^{\uparrow\uparrow}+\bar f^{\downarrow\downarrow})
+\frac{i\hbar}{2}\{(f^{\uparrow\uparrow}-\bar f^{\downarrow\downarrow}),\Delta^*\}
+\frac{\hbar^2}{8}\{\!\{(f^{\uparrow\uparrow}+\bar f^{\downarrow\downarrow}),\Delta^*\}\!\}
+ \Delta^* +...,
\label{WHF}
\end{eqnarray}
where the functions $h$, $f$, $\Delta$, and $\kappa$ are the Wigner
transforms of $\hat h$, $\hat\rho$, $\hat\Delta$, and $\hat\kappa$,
respectively, $\bar f({\bf r},{\bf p})=f({\bf r},-{\bf p})$,
$\{f,g\}$ is the Poisson
bracket of the functions $f({\bf r},{\bf p})$ and $g({\bf r},{\bf p})$ and
$\{\{f,g\}\}$ is their double Poisson bracket;
the dots stand for terms proportional to higher powers of $\hbar$.
This set of equations must be complemented by the dynamical equations for
$\bar f^{\uparrow\uparrow}, \bar f^{\downarrow\downarrow}, \bar f^{\uparrow\downarrow},
\bar f^{\downarrow\uparrow},\bar\kappa,\bar\kappa^*$.
They are obtained by the change ${\bf p} \rightarrow -{\bf p}$ in arguments of functions and Poisson brackets.
So, in reality we deal with the set of twelve equations. We introduced the notation
$\kappa \equiv \kappa^{\uparrow\downarrow}$ and $\Delta \equiv \Delta^{\uparrow\downarrow}$.
Symmetry properties of matrices $\hat\kappa, \hat\Delta$ and the properties of their Wigner
transforms (see Appendix B) allow one to replace the functions
$\kappa^{\downarrow\uparrow}({\bf r},{\bf p})$ and $\Delta^{\downarrow\uparrow}({\bf r},{\bf p})$ by the functions $\bar\kappa^{\uparrow\downarrow}({\bf r},{\bf p})$ and $\bar\Delta^{\uparrow\downarrow}({\bf r},{\bf p})$.
Following the paper \cite{BaMo} we will write above equations in terms of spin-scalar
$$f^+=f^{\uparrow\uparrow}+ f^{\downarrow\downarrow}$$
and spin-vector
$$f^-=f^{\uparrow\uparrow}- f^{\downarrow\downarrow}$$
functions. Furthermore, it is useful to
rewrite the obtained equations in terms of even and odd functions
$f_{e}=\frac{1}{2}(f+\bar f)$ and $f_{o}=\frac{1}{2}(f-\bar f)$
and real and imaginary parts of $\kappa$ and $\Delta$: $\kappa^r=\frac{1}{2}(\kappa+\kappa^*),\,
\kappa^i=\frac{1}{2i}(\kappa-\kappa^*),\,\Delta^r=\frac{1}{2}(\Delta+\Delta^*),\,
\Delta^i=\frac{1}{2i}(\Delta-\Delta^*)$. We have
\begin{eqnarray}
i\hbar\dot f^{+}_e &=&\frac{i\hbar}{2}\left[
\{h^+_o,f^+_e\}+\{h^+_e,f^+_o\}+\{h^-_o,f^-_e\}+\{h^-_e,f^-_o\} \right]
\nonumber\\
&+&{i\hbar}\left[
\{h^{\uparrow\downarrow}_o,f^{\downarrow\uparrow}_e\}
+\{h^{\uparrow\downarrow}_e,f^{\downarrow\uparrow}_o\}
+\{h^{\downarrow\uparrow}_o,f^{\uparrow\downarrow}_e\}
+\{h^{\downarrow\uparrow}_e,f^{\uparrow\downarrow}_o\} \right]
\nonumber\\
&+&4i\left([\kappa^i_e\Delta^r_e]-[\kappa^r_e\Delta^i_e]
+[\kappa^i_o\Delta^r_o]-[\kappa^r_o\Delta^i_o]\right)
+...,
\nonumber\\
i\hbar\dot f^{+}_o &=&\frac{i\hbar}{2}\left[
\{h^+_o,f^+_o\}+\{h^+_e,f^+_e\}+\{h^-_o,f^-_o\}+\{h^-_e,f^-_e\} \right]
\nonumber\\
&+&{i\hbar}\left[
\{h^{\uparrow\downarrow}_o,f^{\downarrow\uparrow}_o\}
+\{h^{\uparrow\downarrow}_e,f^{\downarrow\uparrow}_e\}
+\{h^{\downarrow\uparrow}_o,f^{\uparrow\downarrow}_o\}
+\{h^{\downarrow\uparrow}_e,f^{\uparrow\downarrow}_e\} \right]
\nonumber\\
&+&2i\hbar\left(
\{\kappa^r_e,\Delta^r_e\}+\{\kappa^i_e,\Delta^i_e\}
+\{\kappa^r_o,\Delta^r_o\}+\{\kappa^i_o,\Delta^i_o\} \right)
+...,
\nonumber\\
i\hbar\dot f^{-}_e &=&
2(h^{\uparrow\downarrow}_e f^{\downarrow\uparrow}_e
+h^{\uparrow\downarrow}_o f^{\downarrow\uparrow}_o
-h^{\downarrow\uparrow}_e f^{\uparrow\downarrow}_e
-h^{\downarrow\uparrow}_o f^{\uparrow\downarrow}_o)
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^+_o,f^-_e\}+\{h^+_e,f^-_o\}+\{h^-_o,f^+_e\}+\{h^-_e,f^+_o\} \right]
\nonumber\\
&+&2i\hbar\left(
\{\kappa^r_e,\Delta^r_o\}+\{\kappa^i_e,\Delta^i_o\}
+\{\kappa^r_o,\Delta^r_e\}+\{\kappa^i_o,\Delta^i_e\} \right)
\nonumber\\
&-&\frac{\hbar^2}{4}\left[
\{\!\{h^{\uparrow\downarrow}_e,f^{\downarrow\uparrow}_e\}\!\}
+\{\!\{h^{\uparrow\downarrow}_o,f^{\downarrow\uparrow}_o\}\!\}
-\{\!\{h^{\downarrow\uparrow}_e,f^{\uparrow\downarrow}_e\}\!\}
-\{\!\{h^{\downarrow\uparrow}_o,f^{\uparrow\downarrow}_o\}\!\} \right]
+...,
\nonumber\\
i\hbar\dot f^{-}_o &=&
2(h^{\uparrow\downarrow}_e f^{\downarrow\uparrow}_o
+h^{\uparrow\downarrow}_o f^{\downarrow\uparrow}_e
-h^{\downarrow\uparrow}_e f^{\uparrow\downarrow}_o
-h^{\downarrow\uparrow}_o f^{\uparrow\downarrow}_e)
\nonumber\\
&+&4i\left(
[\kappa^i_e\Delta^r_o]-[\kappa^r_e\Delta^i_o]
+[\kappa^i_o\Delta^r_e]-[\kappa^r_o\Delta^i_e]\right)
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^+_o,f^-_o\}+\{h^+_e,f^-_e\}+\{h^-_o,f^+_o\}+\{h^-_e,f^+_e\} \right]
\nonumber\\
&-&\frac{\hbar^2}{4}\left[
\{\!\{h^{\uparrow\downarrow}_e,f^{\downarrow\uparrow}_o\}\!\}
+\{\!\{h^{\uparrow\downarrow}_o,f^{\downarrow\uparrow}_e\}\!\}
-\{\!\{h^{\downarrow\uparrow}_e,f^{\uparrow\downarrow}_o\}\!\}
-\{\!\{h^{\downarrow\uparrow}_o,f^{\uparrow\downarrow}_e\}\!\} \right]
+...,
\nonumber\\
i\hbar\dot f^{\uparrow\downarrow}_e &=&
\left[ h^-_e f^{\uparrow\downarrow}_e + h^-_o f^{\uparrow\downarrow}_o
- h^{\uparrow\downarrow}_e f^-_e
- h^{\uparrow\downarrow}_o f^-_o \right]
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^{\uparrow\downarrow}_e,f^+_o\}
+\{h^{\uparrow\downarrow}_o,f^+_e\}
+\{h^+_e,f^{\uparrow\downarrow}_o\} + \{h^+_o,f^{\uparrow\downarrow}_e\}
\right]
\nonumber\\
&+&\frac{\hbar^2}{8}\left[
\{\!\{h^{\uparrow\downarrow}_e,f^-_e\}\!\} +\{\!\{h^{\uparrow\downarrow}_o,f^-_o\}\!\}
-\{\!\{h^-_e,f^{\uparrow\downarrow}_e\}\!\} - \{\!\{h^-_o,f^{\uparrow\downarrow}_o\}\!\}
\right]
+...,
\nonumber\\
i\hbar \dot f^{\downarrow\uparrow}_e &=&
-\left[
h^-_e f^{\downarrow\uparrow}_e
+h^-_o f^{\downarrow\uparrow}_o
-h^{\downarrow\uparrow}_e f^-_e
-h^{\downarrow\uparrow}_o f^-_o \right]
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^{\downarrow\uparrow}_e,f^+_o\}
+\{h^{\downarrow\uparrow}_o,f^+_e\}
+\{h^+_e,f^{\downarrow\uparrow}_o\}
+\{h^+_o,f^{\downarrow\uparrow}_e\}
\right]
\nonumber\\
&-&\frac{\hbar^2}{8}\left[
\{\!\{h^{\downarrow\uparrow}_e,f^-_e\}\!\}
+\{\!\{h^{\downarrow\uparrow}_o,f^-_o\}\!\}
-\{\!\{h^-_e,f^{\downarrow\uparrow}_e\}\!\}
-\{\!\{h^-_o,f^{\downarrow\uparrow}_o\}\!\}
\right]
+...,
\nonumber\\
i\hbar\dot f^{\uparrow\downarrow}_o &=&
\left[ h^-_e f^{\uparrow\downarrow}_o + h^-_o f^{\uparrow\downarrow}_e
- h^{\uparrow\downarrow}_e f^-_o
- h^{\uparrow\downarrow}_o f^-_e \right]
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^{\uparrow\downarrow}_e,f^+_e\}
+\{h^{\uparrow\downarrow}_o,f^+_o\}
+\{h^+_e,f^{\uparrow\downarrow}_e\} + \{h^+_o,f^{\uparrow\downarrow}_o\}
\right]
\nonumber\\
&+&\frac{\hbar^2}{8}\left[
\{\!\{h^{\uparrow\downarrow}_e,f^-_o\}\!\} +\{\!\{h^{\uparrow\downarrow}_o,f^-_e\}\!\}
-\{\!\{h^-_e,f^{\uparrow\downarrow}_o\}\!\} - \{\!\{h^-_o,f^{\uparrow\downarrow}_e\}\!\}
\right]
+...,
\nonumber\\
i\hbar \dot f^{\downarrow\uparrow}_o &=&
-\left[
h^-_e f^{\downarrow\uparrow}_o
+h^-_o f^{\downarrow\uparrow}_e
-h^{\downarrow\uparrow}_e f^-_o
-h^{\downarrow\uparrow}_o f^-_e \right]
\nonumber\\
&+&\frac{i\hbar}{2}\left[
\{h^{\downarrow\uparrow}_e,f^+_e\}
+\{h^{\downarrow\uparrow}_o,f^+_o\}
+\{h^+_e,f^{\downarrow\uparrow}_e\}
+\{h^+_o,f^{\downarrow\uparrow}_o\}
\right]
\nonumber\\
&-&\frac{\hbar^2}{8}\left[
\{\!\{h^{\downarrow\uparrow}_e,f^-_o\}\!\}
+\{\!\{h^{\downarrow\uparrow}_o,f^-_e\}\!\}
-\{\!\{h^-_e,f^{\downarrow\uparrow}_o\}\!\}
-\{\!\{h^-_o,f^{\downarrow\uparrow}_e\}\!\}
\right]
+...,
\nonumber\\
i\hbar\dot \kappa^r_e &=& i[h^+_e\,\kappa^i_e+h^-_o\,\kappa^i_o]
+\frac{i\hbar}{2}\{h^+_o,\kappa^r_e+h^-_e,\kappa^r_o\}
\nonumber\\
&+&i[f^+_e\,\Delta^i_e+ f^-_o\,\Delta^i_o]
+\frac{i\hbar}{2}\{f^+_o,\Delta^r_e+ f^-_e,\Delta^r_o\} -i\Delta^i_e
+...,
\nonumber\\
i\hbar\dot \kappa^r_o &=& i[h^+_e\,\kappa^i_o+h^-_o\,\kappa^i_e]
+\frac{i\hbar}{2}\{h^+_o,\kappa^r_o+h^-_e,\kappa^r_e\}
\nonumber\\
&+&i[f^+_e\,\Delta^i_o+ f^-_o\,\Delta^i_e]
+\frac{i\hbar}{2}\{f^+_o,\Delta^r_o+ f^-_e,\Delta^r_e\} -i\Delta^i_o
+...,
\nonumber\\
i\hbar\dot \kappa^i_e &=& -i[h^+_e\,\kappa^r_e+h^-_o\,\kappa^r_o]
+\frac{i\hbar}{2}\{h^+_o,\kappa^i_e+h^-_e,\kappa^i_o\}
\nonumber\\
&-&i[f^+_e\,\Delta^r_e+ f^-_o\,\Delta^r_o]
+\frac{i\hbar}{2}\{f^+_o,\Delta^i_e+ f^-_e,\Delta^i_o\} +i\Delta^r_e
+...,
\nonumber\\
i\hbar\dot \kappa^i_o &=& -i[h^+_e\,\kappa^r_o+h^-_o\,\kappa^r_e]
+\frac{i\hbar}{2}\{h^+_o,\kappa^i_o+h^-_e,\kappa^i_e\}
\nonumber\\
&-&i[f^+_e\,\Delta^r_o+ f^-_o\,\Delta^r_e]
+\frac{i\hbar}{2}\{f^+_o,\Delta^i_o+ f^-_e,\Delta^i_e\} +i\Delta^r_o
+...,
\label{WHFeo}
\end{eqnarray}
The following notation is introduced here:
$h^{\pm}=h^{\uparrow\uparrow}\pm h^{\downarrow\downarrow},\quad
[ab]=ab-\frac{\hbar^2}{8}\{\!\{a,b\}\!\},\quad [ab+cd+...]=[ab]+[cd]+...,\quad
\{a,b+c,d+...\}=\{a,b\}+\{c,d\}+... $.
These twelve equations will be solved by the method of moments in a small amplitude
approximation. To this end
all functions $f({\bf r},{\bf p},t)$ and $\kappa({\bf r},{\bf p},t)$ are divided into equilibrium part
and deviation (variation): $f({\bf r},{\bf p},t)=f({\bf r},{\bf p})_{eq}+\delta f({\bf r},{\bf p},t)$,
$\kappa({\bf r},{\bf p},t)=\kappa({\bf r},{\bf p})_{eq}+\delta \kappa({\bf r},{\bf p},t)$.
Then equations are linearized neglecting quadratic terms.
From general arguments one can expect that the phase of $\Delta$ (and
of $\kappa$, since both are linked, according to equation (\ref{DK}))
is much more relevant than its magnitude, since the former determines
the superfluid velocity. After linearization, the phase of $\Delta$
(and of $\kappa$) is expressed by $\delta\Delta^i$ (and $\delta\kappa^i$), while
$\delta\Delta^r$ (and $\delta\kappa^r$) describes oscillations of the
magnitude of $\Delta$ (and of $\kappa$). Let us therefore assume that
\begin{equation}
\delta\kappa^r({\bf r},{\bf p})\ll\delta\kappa^i({\bf r},{\bf p}).
\label{approx1}
\end{equation}
This assumption was explicitly confirmed in \cite{M.Urban} for the case
of superfluid trapped fermionic atoms, where it was shown that
$\delta\Delta^r$ is suppressed with respect to $\delta\Delta^i$ by one
order of $\Delta/E_{\rm F}$, where $E_{\rm F}$ denotes the Fermi energy.
The assumption (\ref{approx1}) allows one to neglect all terms containing the variations
$\delta \kappa^r$ and $\delta \Delta^r$ in the equations
(\ref{WHFeo}) after their linearization. In this case the "small" variations
$\delta \kappa^r$ and $\delta \Delta^r$
will not affect the dynamics of the "big" variations
$\delta \kappa^i$ and $\delta \Delta^i$ . This means that the
dynamical equations for the "big" variations can be considered
independently from that of the "small" variations, and we will finally
deal with a set of only ten equations.
\section{Model Hamiltonian}
The microscopic Hamiltonian of the model, harmonic oscillator with
spin orbit potential plus separable quadrupole-quadrupole and
spin-spin residual interactions is given by
\begin{eqnarray}
\label{Ham}
H=\sum\limits_{i=1}^A\left[\frac{\hat{\bf p}_i^2}{2m}+\frac{1}{2}m\omega^2{\bf r}_i^2
-\eta\hat {\bf l}_i\hat {\bf S}_i\right]+H_{qq}+H_{ss}
\end{eqnarray}
with
\begin{eqnarray}
\label{Hqq}
&& H_{qq}=\!
\sum_{\mu=-2}^{2}(-1)^{\mu}
\left\{\bar{\kappa}
\sum\limits_i^Z\!\sum\limits_j^N
+\frac{\kappa}{2}
\left[\sum\limits_{i,j(i\neq j)}^{Z}
+\sum\limits_{i,j(i\neq j)}^{N}
\right]
\right\}
q_{2-\mu}({\bf r}_i)q_{2\mu}({\bf r}_j)
,
\\
\label{Hss}
&&H_{ss}=\!
\sum_{\mu=-1}^{1}(-1)^{\mu}
\left\{\bar{\chi}
\sum\limits_i^Z\!\sum\limits_j^N
+\frac{\chi}{2}
\left[
\sum\limits_{i,j(i\neq j)}^{Z}
+\sum\limits_{i,j(i\neq j)}^{N}
\right]
\right\}
\hat S_{-\mu}(i)\hat S_{\mu}(j)
\,\delta({\bf r}_i-{\bf r}_j),
\end{eqnarray}
where $N$ and $Z$ are the numbers of neutrons and protons
and $\hat S_{\mu}$ are spin matrices \cite{Var}:
\begin{equation}
\hat S_1=-\frac{\hbar}{\sqrt2}{0\quad 1\choose 0\quad 0},\quad
\hat S_0=\frac{\hbar}{2}{1\quad\, 0\choose 0\, -\!1},\quad
\hat S_{-1}=\frac{\hbar}{\sqrt2}{0\quad 0\choose 1\quad 0}.
\label{S}
\end{equation}
\subsection{Mean Field}
Let us analyze the mean field generated by this Hamiltonian.
\subsubsection{Spin-orbit Potential}
Written in cyclic coordinates, the spin orbit part of the
Hamiltonian reads
$$\hat h_{ls}=-\eta\sum_{\mu=-1}^1(-)^{\mu}\hat l_{\mu}\hat S_{-\mu}
=-\eta{\quad\hat l_0\frac{\hbar}{2}\quad\; \hat l_{-1}\frac{\hbar}{\sqrt2} \choose
-\hat l_{1}\frac{\hbar}{\sqrt2}\; -\hat l_0\frac{\hbar}{2}},
$$
where \cite{Var}
\begin{equation}
\label{lqu}
\hat l_{\mu}=-\hbar\sqrt2\sum_{\nu,\alpha}C_{1\nu,1\alpha}^{1\mu}r_{\nu}\nabla_{\alpha},
\end{equation}
cyclic coordinates $r_{-1}, r_0, r_1$
are defined in \cite{Var},
$C_{1\sigma,1\nu}^{\lambda\mu}$ is a Clebsch-Gordan
coefficient and
\begin{eqnarray}
&&\hat l_1=\hbar(r_0\nabla_1-r_1\nabla_0)=
-\frac{1}{\sqrt2}(\hat l_x+i\hat l_y),\quad
\hat l_0=\hbar(r_{-1}\nabla_1-r_1\nabla_{-1})=\hat l_z,
\nonumber\\
&&\hat l_{-1}=\hbar(r_{-1}\nabla_0-r_0\nabla_{-1})=
\frac{1}{\sqrt2}(\hat l_x-i\hat l_y),
\nonumber\\
&&\hat l_x=-i\hbar(y\nabla_z-z\nabla_y),\quad
\hat l_y=-i\hbar(z\nabla_x-x\nabla_z),\quad
\hat l_z=-i\hbar(x\nabla_y-y\nabla_x).
\label{lxyz}
\end{eqnarray}
Matrix elements of $\hat h_{ls}$ in coordinate space can be obviously written \cite{BaMo} as
\begin{eqnarray}
\langle {\bf r}_1,s_1|\hat h_{ls}|{\bf r}_2,s_2\rangle
&=&-\frac{\hbar}{2}\eta\left[\hat l_{0}({\bf r}_1)(\delta_{s_1\uparrow}\delta_{s_2\uparrow}
-\delta_{s_1\downarrow}\delta_{s_2\downarrow})\right.
\nonumber\\
&&+\left.\sqrt2\, \hat l_{-1}({\bf r}_1)\delta_{s_1\uparrow}\delta_{s_2\downarrow}
-\sqrt2\, \hat l_{1}({\bf r}_1)\delta_{s_1\downarrow}\delta_{s_2\uparrow}\right]\delta({\bf r}_1-{\bf r}_2).
\label{Hrr'}
\end{eqnarray}
Their Wigner transform reads \cite{BaMo}:
\begin{eqnarray}
h_{ls}^{s_1s_2}({\bf r},{\bf p})
&=&-\frac{\hbar}{2}\eta\left[l_{0}({\bf r},{\bf p})(\delta_{s_1\uparrow}\delta_{s_2\uparrow}
-\delta_{s_1\downarrow}\delta_{s_2\downarrow})\right.
\nonumber\\
&&+\left.\sqrt2 l_{-1}({\bf r},{\bf p})\delta_{s_1\uparrow}\delta_{s_2\downarrow}
-\sqrt2 l_{1}({\bf r},{\bf p})\delta_{s_1\downarrow}\delta_{s_2\uparrow}\right],
\label{Hrp}
\end{eqnarray}
where
$l_{\mu}=-i\sqrt2\sum_{\nu,\alpha}C_{1\nu,1\alpha}^{1\mu}r_{\nu}p_{\alpha}$.
\subsubsection{Quadrupole-quadrupole interaction}
The contribution of $H_{qq}$ to the mean field potential is easily
found by replacing one of the $q_{2\mu}$ operators by the average value.
We have
\begin{equation}
\label{potenirr}
V^{\tau}_{qq}=\sqrt6\sum_{\mu}(-1)^{\mu}Z_{2-\mu}^{\tau +}q_{2\mu}.
\end{equation}
Here
\begin{equation}
\label{Z2mu}
Z_{2\mu}^{n+}=\kappa R_{2\mu}^{n+}
+\bar{\kappa}R_{2\mu}^{p+}\,,\quad
Z_{2\mu}^{p+}=\kappa R_{2\mu}^{p+}
+\bar{\kappa}R_{2\mu}^{n+},\quad R_{2\mu}^{\tau+}(t)=
\frac{1}{\sqrt6}\int d({\bf p},{\bf r})
q_{2\mu}({\bf r})f^{\tau+}({\bf r},{\bf p},t)
\end{equation}
with
$\int\! d({\bf p},{\bf r})\equiv
(2\pi\hbar)^{-3}\int\! d^3p\,\int\! d^3r$ and $\tau$ being the isospin index.
\subsubsection{Spin-spin interaction}
The analogous expression for $H_{ss}$ is found in a
standard way \cite{BaMoPRC} with the following result for
the Wigner transform of the proton mean field:
\begin{eqnarray}
\label{Vp}
V_{p}^{s s'}({\bf r},t)&=&
3\chi\frac{\hbar^2}{8}
\left[
\delta_{s\downarrow}\delta_{s'\uparrow}n_p^{\downarrow\uparrow}+
\delta_{s\uparrow}\delta_{s'\downarrow}n_p^{\uparrow\downarrow}
-\delta_{s\downarrow}\delta_{s'\downarrow}n_p^{\uparrow\uparrow}
-\delta_{s\uparrow}\delta_{s'\uparrow}n_p^{\downarrow\downarrow}
\right]
\nonumber\\
&+&\bar\chi\frac{\hbar^2}{8}
\left[
2\delta_{s\downarrow}\delta_{s'\uparrow}n_n^{\downarrow\uparrow}+
2\delta_{s\uparrow}\delta_{s'\downarrow}n_n^{\uparrow\downarrow}
+(\delta_{s\uparrow}\delta_{s'\uparrow}-
\delta_{s\downarrow}\delta_{s'\downarrow})(n_n^{\uparrow\uparrow}-
n_n^{\downarrow\downarrow})
\right],
\end{eqnarray}
where
${\displaystyle n_{\tau}^{ss'}({\bf r},t)=\int\frac{d^3p}{(2\pi\hbar)^3}f^{ss'}_{\tau}({\bf r},{\bf p},t)}$.
The Wigner transform of the neutron mean field $V_n^{ss'}$ is
obtained from (\ref{Vp}) by the obvious change of indices $p\leftrightarrow n$.
\subsection{Pair potential}
The Wigner transform of the pair potential (pairing gap) $\Delta({\bf r},{\bf p})$ is related to
the Wigner transform of the anomalous density by \cite{Ring}
\begin{equation}
\Delta({\bf r},{\bf p})=-\int\! \frac{d^3p'}{(2\pi\hbar)^3}
v(|{\bf p}-{\bf p}'|)\kappa({\bf r},{\bf p}'),
\label{DK}
\end{equation}
where $v(p)$ is a Fourier transform of the two-body interaction.
We take for the pairing interaction a simple Gaussian of strength $V_0$
and range $r_p$ \cite{Ring}
\begin{equation}
v(p)=\beta e^{-\alpha p^2}\!,
\label{v_p}
\end{equation}
with $\beta=-|V_0|(r_p\sqrt{\pi})^3$ and $\alpha=r_p^2/4\hbar^2$. For the values of the parameters, see section 5.1.
\section{Equations of motion}
Integrating the set of equations (\ref{WHFeo}) over phase space
with the weights
\begin{equation}
W =\{r\otimes p\}_{\lambda\mu},\,\{r\otimes r\}_{\lambda\mu},\,
\{p\otimes p\}_{\lambda\mu}, \mbox{ and } 1
\label{weightfunctions}
\end{equation}
one gets dynamic equations for
the following collective variables:
\begin{eqnarray}
&&{\cal L}^{\tau\varsigma}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{r\otimes p\}_{\lambda\mu}
\delta f^{\tau\varsigma}_o({\bf r},{\bf p},t),\quad
{\cal R}^{\tau\varsigma}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{r\otimes r\}_{\lambda\mu}
\delta f^{\tau\varsigma}_e({\bf r},{\bf p},t),\quad
\nonumber\\
&&{\cal P}^{\tau\varsigma}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{p\otimes p\}_{\lambda\mu}
\delta f^{\tau\varsigma}_e({\bf r},{\bf p},t),\quad
{\cal F}^{\tau\varsigma}(t)=\int\! d({\bf p},{\bf r})
\delta f^{\tau\varsigma}_e({\bf r},{\bf p},t),\quad
\nonumber\\
&&\tilde{{\cal L}}^{\tau}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{r\otimes p\}_{\lambda\mu}
\delta \kappa^{\tau i}_o({\bf r},{\bf p},t),\quad
\tilde{{\cal R}}^{\tau}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{r\otimes r\}_{\lambda\mu}
\delta \kappa^{\tau i}_e({\bf r},{\bf p},t),\quad
\nonumber\\
&&\tilde{{\cal P}}^{\tau}_{\lambda\mu}(t)=\int\! d({\bf p},{\bf r}) \{p\otimes p\}_{\lambda\mu}
\delta \kappa^{\tau i}_e({\bf r},{\bf p},t),\quad
\label{Varis}
\end{eqnarray}
where
$\varsigma\!=+,\,-,\,\uparrow\downarrow,\,\downarrow\uparrow,$
and $\displaystyle \quad
\{r\otimes r\}_{\lambda\mu}=\sum\limits_{\sigma,\nu}
C_{1\sigma,1\nu}^{\lambda\mu}r_{\sigma}r_{\nu}.$
The required expressions for
$h^{\pm}$, $h^{\uparrow\downarrow}$ and $h^{\downarrow\uparrow}$ are
$$h_{\tau}^{+}=\frac{p^2}{m}+m\,\omega^2r^2
+12\sum_{\mu}(-1)^{\mu}Z_{2\mu}^{\tau+}(t)\{r\otimes r\}_{2-\mu}
+V_{\tau}^+({\bf r},t)-\mu^{\tau},$$
$\mu^{\tau}$ being the chemical potential of protons ($\tau=p$) or neutrons ($\tau=n$),
$$h_{\tau}^-=-\hbar\eta l_0+V_{\tau}^-({\bf r},t),\quad
h_{\tau}^{\uparrow\downarrow}=-\frac{\hbar}{\sqrt2}\eta l_{-1}+V_{\tau}^{\uparrow\downarrow}({\bf r},t),
\quad h_{\tau}^{\downarrow\uparrow}=\frac{\hbar}{\sqrt2}\eta l_{1}+V_{\tau}^{\downarrow\uparrow}({\bf r},t),
$$
where according to (\ref{Vp})
\begin{eqnarray}
\label{Vss}
V_p^+({\bf r},t)=-3\frac{\hbar^2}{8}\chi n_p^+({\bf r},t),\quad
V_p^-({\bf r},t)=3\frac{\hbar^2}{8}\chi n_p^-({\bf r},t)+\frac{\hbar^2}{4}\bar\chi n_n^-({\bf r},t),
\nonumber\\
V_p^{\uparrow\downarrow}({\bf r},t)=3\frac{\hbar^2}{8}\chi n_p^{\uparrow\downarrow}({\bf r},t)
+\frac{\hbar^2}{4}\bar\chi n_n^{\uparrow\downarrow}({\bf r},t),\quad
V_p^{\downarrow\uparrow}({\bf r},t)=3\frac{\hbar^2}{8}\chi n_p^{\downarrow\uparrow}({\bf r},t)
+\frac{\hbar^2}{4}\bar\chi n_n^{\downarrow\uparrow}({\bf r},t)
\end{eqnarray}
and the neutron potentials $V_n^{\varsigma}$ are
obtained by the obvious change of indices $p\leftrightarrow n$.
Variations of these mean fields read:
$$\delta h_{\tau}^{+}=12\sum_{\mu}(-1)^{\mu}\delta Z_{2\mu}^{\tau+}(t)\{r\otimes r\}_{2-\mu}
+\delta V_{\tau}^+({\bf r},t),$$
where
$\quad \delta Z_{2\mu}^{p+}=\kappa \delta R_{2\mu}^{p+}
+\bar{\kappa}\delta R_{2\mu}^{n+},\quad \delta R_{\lambda\mu}^{\tau+}(t)\equiv
{\cal R}_{\lambda\mu}^{\tau+}(t) \quad$ and
$$
\delta V_p^+({\bf r},t)=-3\frac{\hbar^2}{8}\chi \delta n_p^+({\bf r},t),\quad
{\displaystyle \delta n_{p}^{+}({\bf r},t)=\int\frac{d^3p}{(2\pi\hbar)^3}\delta f^{+}_{p}({\bf r},{\bf p},t)}.$$
Variations of
$h^{-}$, $h^{\uparrow\downarrow}$ and $h^{\downarrow\uparrow}$ are obtained in a similar way.
Variation of the pair potential is
\begin{equation}
\delta \Delta({\bf r},{\bf p},t)=-\int\! \frac{d^3p'}{(2\pi\hbar)^3}
v(|{\bf p}-{\bf p}'|)\delta \kappa({\bf r},{\bf p}',t).
\label{DKvar}
\end{equation}
We are interested in the scissors mode with quantum number
$K^{\pi}=1^+$. Therefore, we only need the part of dynamic equations
with $\mu=1$.
It is convenient to rewrite the dynamical equations in terms
of isoscalar and isovector variables
\begin{eqnarray}
\label{Isovs}
\bar {\cal R}_{\lambda\mu}={\cal R}_{\lambda\mu}^{n}+{\cal R}_{\lambda\mu}^{p}
,\quad
\bar {\cal P}_{\lambda\mu}={\cal P}_{\lambda\mu}^{n}+{\cal P}_{\lambda\mu}^{p}
,\quad
\bar {\cal L}_{\lambda\mu}={\cal L}_{\lambda\mu}^{n}+{\cal L}_{\lambda\mu}^{p}.
\nonumber\\
{\cal R}_{\lambda\mu}={\cal R}_{\lambda\mu}^{n}-{\cal R}_{\lambda\mu}^{p},\quad
{\cal P}_{\lambda\mu}={\cal P}_{\lambda\mu}^{n}-{\cal P}_{\lambda\mu}^{p},\quad
{\cal L}_{\lambda\mu}={\cal L}_{\lambda\mu}^{n}-{\cal L}_{\lambda\mu}^{p},
\end{eqnarray}
It also is natural to define isovector and isoscalar strength constants
$\kappa_1=\frac{1}{2}(\kappa-\bar\kappa)$ and
$\kappa_0=\frac{1}{2}(\kappa+\bar\kappa)$ connected by the relation
$\kappa_1=\alpha\kappa_0$ \cite{BaSc}.
Then the equations for the neutron and proton systems are transformed
into isovector and isoscalar ones. Supposing that all equilibrium
characteristics of the proton system are equal to that of the neutron
system one decouples isovector and isoscalar equations. This
approximations looks rather crude, nevertheless the possible
corrections to it are very small, being of the order
$(\frac{N-Z}{A})^2$.
The integration yields the following set of equations for {\bf isovector} variables:
\begin{eqnarray}
\label{iv}
\dot {{\cal L}}^{+}_{21}&=&
\frac{1}{m}{\cal P}_{21}^{+}-
\left[m\,\omega^2
-4\sqrt3\alpha\kappa_0R_{00}^{\rm eq}
+\sqrt6(1+\alpha)\kappa_0 R_{20}^{\rm eq}\right]{\cal R}^{+}_{21}
-i\hbar\frac{\eta}{2}\left[{\cal L}_{21}^-
+2{\cal L}^{\uparrow\downarrow}_{22}+
\sqrt6{\cal L}^{\downarrow\uparrow}_{20}\right],
\nonumber\\
\dot {{\cal L}}^{-}_{21}&=&
\frac{1}{m}{\cal P}_{21}^{-}
-
\left[m\,\omega^2+\sqrt6\kappa_0 R_{20}^{\rm eq}
-\frac{\sqrt{3}}{20}\hbar^2
\left( \chi-\frac{\bar\chi}{3} \right)
\left(\frac{I_1}{a_0^2}+\frac{I_1}{a_1^2}\right)\left(\frac{a_1^2}{{\cal A}_2}-\frac{a_0^2}{{\cal A}_1}\right)
\right]{\cal R}^{-}_{21}
-i\hbar\frac{\eta}{2}{\cal L}_{21}^+
\nonumber\\
&&+\frac{4}{\hbar}|V_0| I_{rp}^{\kappa\Delta}(r') {\tilde{\cal L}}_{21},
\nonumber\\
\dot {{\cal L}}^{\uparrow\downarrow}_{22}&=&
\frac{1}{m}{\cal P}_{22}^{\uparrow\downarrow}-
\left[m\,\omega^2-2\sqrt6\kappa_0R_{20}^{\rm eq}
-\frac{\sqrt{3}}{5}\hbar^2
\left( \chi-\frac{\bar\chi}{3} \right)\frac{I_1}{{\cal A}_2}
\right]{\cal R}^{\uparrow\downarrow}_{22}
-i\hbar\frac{\eta}{2}{\cal L}_{21}^+,
\nonumber\\
\dot {{\cal L}}^{\downarrow\uparrow}_{20}&=&
\frac{1}{m}{\cal P}_{20}^{\downarrow\uparrow}-
\left[m\,\omega^2
+2\sqrt6\kappa_0 R_{20}^{\rm eq}\right]{\cal R}^{\downarrow\uparrow}_{20}
+\frac{2}{\sqrt3}\kappa_0 R_{20}^{\rm eq}\,{\cal R}^{\downarrow\uparrow}_{00}
-i\hbar\frac{\eta}{2}\sqrt{\frac{3}{2}}{\cal L}_{21}^+
\nonumber\\
&&+\frac{\sqrt{3}}{15}\hbar^2
\left( \chi-\frac{\bar\chi}{3} \right)I_1 \,
\left[
\left(\frac{1}{{\cal A}_2}-\frac{2}{{\cal A}_1}\right)
{\cal R}_{20}^{\downarrow\uparrow}+
\sqrt2
\left(\frac{1}{{\cal A}_2}+\frac{1}{{\cal A}_1}\right)
{\cal R}_{00}^{\downarrow\uparrow}
\right],
\nonumber\\
\dot {{\cal L}}^{+}_{11}&=&
-3\sqrt6(1-\alpha)\kappa_0 R_{20}^{\rm eq}\,{\cal R}^{+}_{21}
-i\hbar\frac{\eta}{2}\left[{\cal L}_{11}^-
+\sqrt2{\cal L}^{\downarrow\uparrow}_{10}\right],
\nonumber\\
\dot {{\cal L}}^{-}_{11}&=&
-\left[3\sqrt6\kappa_0 R_{20}^{\rm eq}
-\frac{\sqrt{3}}{20}\hbar^2
\left( \chi-\frac{\bar\chi}{3} \right)
\left(\frac{I_1}{a_0^2}-\frac{I_1}{a_1^2}\right)\left(\frac{a_1^2}{{\cal A}_2}-\frac{a_0^2}{{\cal A}_1}\right)
\right]{\cal R}^{-}_{21}
-\hbar\frac{\eta}{2}\left[i{\cal L}_{11}^+
+\hbar F^{\downarrow\uparrow}\right]
\nonumber\\
&&+\frac{4}{\hbar}|V_0| I_{rp}^{\kappa\Delta}(r') {\tilde{\cal L}}_{11},
\nonumber\\
\dot {{\cal L}}^{\downarrow\uparrow}_{10}&=&
-\hbar\frac{\eta}{2\sqrt2}\left[i{\cal L}_{11}^+
+\hbar F^{\downarrow\uparrow}\right],
\nonumber\\
\dot { F}^{\downarrow\uparrow}&=&
-\eta\left[{\cal L}_{11}^- +\sqrt2{\cal L}^{\downarrow\uparrow}_{10}\right],
\nonumber\\
\dot {{\cal R}}^{+}_{21}&=&
\frac{2}{m}{\cal L}_{21}^{+}
-i\hbar\frac{\eta}{2}\left[{\cal R}_{21}^-
+2{\cal R}^{\uparrow\downarrow}_{22}+
\sqrt6{\cal R}^{\downarrow\uparrow}_{20}\right],
\nonumber\\
\dot {{\cal R}}^{-}_{21}&=&
\frac{2}{m}{\cal L}_{21}^{-}
-i\hbar\frac{\eta}{2}{\cal R}_{21}^+,
\nonumber\\
\dot {{\cal R}}^{\uparrow\downarrow}_{22}&=&
\frac{2}{m}{\cal L}_{22}^{\uparrow\downarrow}
-i\hbar\frac{\eta}{2}{\cal R}_{21}^+,
\nonumber\\
\dot {{\cal R}}^{\downarrow\uparrow}_{20}&=&
\frac{2}{m}{\cal L}_{20}^{\downarrow\uparrow}
-i\hbar\frac{\eta}{2}\sqrt{\frac{3}{2}}{\cal R}_{21}^+,
\nonumber\\
\dot {{\cal P}}^{+}_{21}&=&
-2\left[m\,\omega^2+\sqrt6\kappa_0 R_{20}^{\rm eq}\right]{\cal L}^{+}_{21}
+6\sqrt6\kappa_0 R_{20}^{\rm eq}{\cal L}^{+}_{11}
-i\hbar\frac{\eta}{2}\left[{\cal P}_{21}^-
+2{\cal P}^{\uparrow\downarrow}_{22}+\sqrt6{\cal P}^{\downarrow\uparrow}_{20}\right]
\nonumber\\
&&+\frac{3\sqrt{3}}{4}\hbar^2
\chi \frac{I_2}{{\cal A}_1{\cal A}_2}
\left[\left({\cal A}_1-{\cal A}_2\right) {\cal L}_{21}^{+} +
\left({\cal A}_1+{\cal A}_2\right) {\cal L}_{11}^{+}\right]
\nonumber\\
&&+\frac{4}{\hbar}|V_0| I_{pp}^{\kappa\Delta}(r') {\tilde{\cal P}}_{21},
\nonumber\\
\dot {{\cal P}}^{-}_{21}&=&
-2\left[m\,\omega^2+\sqrt6\kappa_0 R_{20}^{\rm eq}\right]{\cal L}^{-}_{21}
+6\sqrt6\kappa_0 R_{20}^{\rm eq}{\cal L}^{-}_{11}
-6\sqrt2\kappa_0 L_{10}^-(\rm eq){\cal R}^{+}_{21}
-i\hbar\frac{\eta}{2}{\cal P}_{21}^{+}
\nonumber\\
&&+\frac{3\sqrt{3}}{4}\hbar^2
\chi \frac{I_2}{{\cal A}_1{\cal A}_2}
\left[\left({\cal A}_1-{\cal A}_2\right){\cal L}_{21}^{-} +
\left({\cal A}_1+{\cal A}_2\right) {\cal L}_{11}^{-}\right],
\nonumber\\
\dot {{\cal P}}^{\uparrow\downarrow}_{22}&=&
-\left[2m\,\omega^2-4\sqrt6\kappa_0 R_{20}^{\rm eq}
-\frac{3\sqrt{3}}{2}\hbar^2
\chi \frac{I_2}{{\cal A}_2}
\right]{\cal L}^{\uparrow\downarrow}_{22}
-i\hbar\frac{\eta}{2}{\cal P}_{21}^{+}
,
\nonumber\\
\dot {{\cal P}}^{\downarrow\uparrow}_{20}&=&
-\left[2m\,\omega^2+4\sqrt6\kappa_0 R_{20}^{\rm eq}\right]{\cal L}^{\downarrow\uparrow}_{20}
+8\sqrt3\kappa_0 R_{20}^{\rm eq}{\cal L}^{\downarrow\uparrow}_{00}
-i\hbar\frac{\eta}{2}\sqrt{\frac{3}{2}}{\cal P}_{21}^{+}
\nonumber\\
&&+\frac{\sqrt{3}}{2}\hbar^2
\chi \frac{I_2}{{\cal A}_1{\cal A}_2}
\left[\left({\cal A}_1-2{\cal A}_2\right){\cal L}_{20}^{\downarrow\uparrow}+
\sqrt2\left({\cal A}_1+{\cal A}_2\right) {\cal L}_{00}^{\downarrow\uparrow}
\right],
\nonumber\\
\dot {{\cal L}}^{\downarrow\uparrow}_{00}&=&
\frac{1}{m}{\cal P}_{00}^{\downarrow\uparrow}-m\,\omega^2{\cal R}^{\downarrow\uparrow}_{00}
+4\sqrt3\kappa_0 R_{20}^{\rm eq}\,{\cal R}^{\downarrow\uparrow}_{20}
\nonumber\\
&&+\frac{1}{2\sqrt{3}}\hbar^2
\left[\left( \chi-\frac{\bar\chi}{3} \right)I_1-\frac{9}{4}\chi I_2\right]
\left[\left(\frac2{{\cal A}_2}-\frac1{{\cal A}_1}\right){\cal R}_{00}^{\downarrow\uparrow}+
\sqrt2\left(\frac1{{\cal A}_2}+\frac1{{\cal A}_1}\right){\cal R}_{20}^{\downarrow\uparrow}
\right],
\nonumber\\
\dot {{\cal R}}^{\downarrow\uparrow}_{00}&=&
\frac{2}{m}{\cal L}_{00}^{\downarrow\uparrow},
\nonumber\\
\dot {{\cal P}}^{\downarrow\uparrow}_{00}&=&
-2m\,\omega^2{\cal L}^{\downarrow\uparrow}_{00}
+8\sqrt3\kappa_0 R_{20}^{\rm eq}\,{\cal L}^{\downarrow\uparrow}_{20}
+\frac{\sqrt{3}}{2}\hbar^2
\chi I_2
\left[\left(\frac2{{\cal A}_2}-\frac1{{\cal A}_1}\right){\cal L}_{00}^{\downarrow\uparrow}+
\sqrt2\left(\frac1{{\cal A}_2}+\frac1{{\cal A}_1}\right){\cal L}_{20}^{\downarrow\uparrow}
\right],
\nonumber\\
\dot{ {\tilde{\cal R}}}_{21} &=& -\frac{1}{\hbar}\left(\frac{16}{5} \alpha\kappa_0 {\cal K}_4
+\Delta_0(r') -\frac{3}{8}\hbar^2 \chi\kappa_0(r') \right){\cal R}^+_{21},
\nonumber\\
\dot{ {\tilde{\cal P}}}_{21} &=& -\frac{1}{\hbar}\Delta_0(r') {\cal P}^+_{21} +
6 \hbar\alpha\kappa_0 {\cal K}_0{\cal R}^+_{21},
\nonumber\\
\dot {{\tilde{\cal L}}}_{21} &=& -\frac{1}{\hbar}\Delta_0(r') {{\cal L}}^-_{21},
\nonumber\\
\dot {{\tilde{\cal L}}}_{11} &=& -\frac{1}{\hbar}\Delta_0(r') {\cal L}^-_{11},
\end{eqnarray}
where $
{\cal A}_1=\sqrt2\, R_{20}^{\rm eq}-R_{00}^{\rm eq}=\frac{Q_{00}}{\sqrt3}\left(1+\frac{4}{3}\delta\right),\quad
{\cal A}_2= R_{20}^{\rm eq}/\sqrt2+R_{00}^{\rm eq}=-\frac{Q_{00}}{\sqrt3}\left(1-\frac{2}{3}\delta\right)
$,
$\quad\displaystyle a_{-1} = a_1 = R_0\left( \frac{1-(2/3)\delta}{1+(4/3)\delta} \right)^{1/6}$ and
$\displaystyle a_0 = R_0\left( \frac{1-(2/3)\delta}{1+(4/3)\delta} \right)^{-1/3}$
are semiaxes of ellipsoid by which the shape of nucleus is approximated, $\delta$ -- deformation parameter,
$R_0=1.2A^{1/3}$~fm.
\begin{eqnarray}\label{Int_ss} \nonumber
I_1=\frac{\pi}{4}\int\limits_{0}^{\infty}dr\, r^4\left(\frac{\partial n^+(r)}{\partial r}\right)^2,
\
I_2=\frac{\pi}{4}\int\limits_{0}^{\infty}dr\, r^2 n^+(r)^2,\quad
n^+(r)=n_p^{+}+n_n^{+}= \frac{n_0}{1+{\rm e}^{\frac{r-R_0}{a}}}.
\end{eqnarray}
${\cal K}_0=\int d({\bf r},{\bf p}) \kappa_0({\bf r},{\bf p}), \, {\cal K}_4=\int d({\bf r},{\bf p}) r^4\kappa_0({\bf r},{\bf p})$.
The functions $\kappa_0(r')$, $\Delta_0(r')$, $I_{rp}^{\kappa\Delta}(r')$ and
$I_{pp}^{\kappa\Delta}(r')$
are discussed in the next section and are demonstrated in Appendix D.
Deriving these equations we neglected double Poisson brackets containing $\kappa$ or $\Delta$,
which are the quantum corrections to pair correlations.
The isoscalar set of equations is easily obtained from (\ref{iv}) by
taking $\alpha=1$, replacing $\bar\chi \to -\bar\chi$
and putting the marks "bar" above all variables.
\section{Results of calculations}
The set of equations (\ref{iv}) coincides with the set of equations (27) of the paper \cite{BaMoPRC}
in the limit of zero pairing, i.e. if to omit the last four equations and
to neglect the contributions from pairing in the dynamical
equations for the variables ${{\cal L}}^-_{21},\,{{\cal L}}^-_{11},$ and ${{\cal P}}^+_{21}$ .
On the other hand, the dynamical equations for ${\tilde {\cal P}}_{21}$ and ${\tilde {\cal R}}_{21}$
and the contribution from pairing in the dynamical equation for ${{\cal P}}_{21}^+$ are
exactly the same as the ones in the paper \cite{Urban}. Only the dynamical equations for
${\tilde {\cal L}}_{21},\,$ ${\tilde {\cal L}}_{11}$ and the contributions from pairing in
dynamical equations for ${{\cal L}}^-_{21},\,$ ${{\cal L}}^-_{11}$ are completely new.
Imposing the time evolution via $\displaystyle{e^{iEt/\hbar}}$ for all variables
one transforms (\ref{iv}) into a set of algebraic equations.
It contains 23 equations.
To find the eigenvalues we construct the 23x23
determinant and seek (numerically) for its zeros.
We find seven roots with exactly E=0 and 16 roots which are non zero:
eight positive ones and eight negative ones (situation is exactly
same as with RPA; see \cite{Ann} for connection of WFM and RPA). In this paper we consider
only the two lowest roots corresponding to the orbital and spin scissors. The qualitative picture
of high lying modes remains practically without any changes in comparison with
\cite{BaMoPRC}.
Seven integrals of motion corresponding to Goldstone modes (zero roots)
can be found analytically. They are written out in the Appendix C.
The interpretation of some of them has been found in \cite{BaMoPRC}, whereas the interpretation
of the remaining ones seems not to be obvious.
\subsection{Choice of parameters}
$\bullet$ Following our previous publications \cite{BaSc,Ann} we take for the isoscalar strength
constant of the quadrupole-quadrupole residual interaction $\kappa_0$ the self consistent
value \cite{BrMt} $\kappa_0=-\frac{m\bar\omega^2}{4Q_{00}}$ with
$Q_{00}=A\frac{3}{5}R^2$, $R=r_0A^{1/3}$,
$r_0=1.2$ fm, $\bar\omega^2=\omega_0^2/
[(1+\frac{4}{3}\delta)^{2/3}(1-\frac{2}{3}\delta)^{1/3}]$,
$\hbar\omega_0=41/A^{1/3}$ MeV.
$\bullet$
The equations (\ref{iv}) contain the functions
$\Delta_0(r')\equiv\Delta_{eq}(r',p_F(r'))$,
$I_{rp}^{\kappa\Delta}(r')\equiv I_{rp}^{\kappa\Delta}(r',p_F(r'))$,
$I_{pp}^{\kappa\Delta}(r')\equiv I_{pp}^{\kappa\Delta}(r',p_F(r'))$
and $\kappa_0(r')\equiv\kappa(r',r')$
depending on the radius $r'$ and the local Fermi momentum $p_F(r')$
(see Fig.~\ref{fig1} ).
\begin{figure}[h]
\centering\includegraphics[width=8cm]{fig_2.pdf}
\caption{The pair field (gap) $\Delta_0(r)$, the function
$\underline{\Delta}=|V_0|I_{pp}^{\kappa\Delta}(r)$ and the nuclear density $n(r)$
as the functions of radius $r$. The solid lines -- calculations without the spin-spin
interaction $V_{ss}$, the dashed lines -- $V_{ss}$ is included.}
\label{fig1}\end{figure}
The value of $r'$ is not fixed by the theory and can be used
as the fitting parameter. We have found in our previous paper \cite{Urban} that the best agreement
of calculated results with experimental data is achieved at the point $r'$ where the function
$I_{pp}^{\kappa\Delta}(r',p_F(r'))$ has its maximum. Nevertheless, to get rid off the fitting
parameter, we use the averaged values of these functions:
$\bar\Delta_0=\int d{\bf r}\, n_0({\bf r})\Delta_0(r,p_F(r))/A$, etc.
The gap $\Delta(r,p_F(r))$, as well as the integrals $I^{\kappa\Delta}_{pp}(r,p_F(r))$, ${\cal K}_4$
and ${\cal K}_0$,
were calculated with the help of the semiclassical
formulae for $\kappa({\bf r},{\bf p})$ and $\Delta({\bf r},{\bf p})$ (see
Appendix D), a Gaussian being
used for the pairing interaction with $r_p=1.9$ fm and $V_0=25$
MeV \cite{Ring}. Those values reproduce usual nuclear pairing gaps.
$\bullet$
The used spin-spin interaction is repulsive, the values of its
strength constants being taken from the paper \cite{Moya}, where the
notation $\chi=K_s/A,\,\bar\chi=q\chi$ was introduced.
The constants were extracted by the authors of~\cite{Moya}
from Skyrme forces following the standard procedure, the residual interaction
being defined in terms of second derivatives of the Hamiltonian density
$H(\rho)$ with respect to the one-body densities~$\rho$.
Different variants of Skyrme forces produce different strength constants of
spin-spin interaction. The most consistent results are obtained with
SG1, SG2 \cite{Giai} and Sk3 \cite{Floc} forces.
To compare theoretical results with experiment the authors of \cite{Moya} preferred
to use the force SG2. Nevertheless they have noticed that "As is well known, the energy
splitting of the HF states around the Fermi level is too large. This has an effect on the
spin M1 distributions that can be roughly compensated by reducing the $K_s$ value". According
to this remark they changed the original self-consistent SG2 parameters from $K_s=88$ MeV,
$q=-0.95$ to $K_s=50$ MeV, $q=-1$. It was found that this modified set of parameters
gives better agreement with experiment for some nuclei in the description of spin-flip
resonance. So we will use $K_s=50$ MeV and $q=-1$.
$\bullet$
Our calculations without pairing \cite{BaMoPRC} have shown that the results are
strongly dependent on the values of
the strength constants of the spin-spin interaction. The natural question arises: how sensitive are they
to the strength of the spin-orbital potential? The results of the demonstrative calculations
are shown in~Fig.~\ref{fig2}.
\begin{figure}[h]
\centering\includegraphics[width=8cm]{fig_3a.pdf}\includegraphics[width=8cm]{fig_3b.pdf}
\caption{The energies $E$ and $B(M1)$-factors as a functions of the spin-orbital strength
constant $\eta$.
Left panel: solid lines -- without the spin-spin interaction $V_{ss}$, dashed lines -- $V_{ss}$ is included.
Right panel: The same as in left panel with pair correlations included.}
\label{fig2}\end{figure}
The $M1$ strengths were computed using effective spin giromagnetic factors
$g_s^{eff}=0.7g_s^{free}$. One observes a rather strong dependence of the results
on the value of $\eta$: the splitting $\Delta E$ and the $M1$ strength of the spin scissors grow with
increasing $\eta$, the $B(M1)$ of the orbital scissors being decreased.
At some critical point $\eta_c$ the $M1$ strength of the spin scissors becomes bigger than that of the
orbital scissors. The inclusion of the spin-spin interaction does not change the qualitative picture,
as well as the inclusion of pair correlations (see Fig.~\ref{fig2}).
What value of $\eta$ to use?
Accidentally, the choice of $\eta$ in our previous papers~\cite{BaMo,BaMoPRC} was not very realistic.
The main purpose of the first paper was the introduction
of spin degrees of freedom into the WFM method, and the aim of the second paper
was to study the influence of spin-spin forces on both scissors -- we did not worry much about the comparison
with experiment. Now, both preliminary aims being achieved, one can think about the agreement with
experimental data, therefore the precise choice of the model parameters becomes important. Of course,
we could try to choose $\eta$ according to the standard requirement of the best agreement with experiment.
However, in reality we are not absolutely free in our choice.
It turns out that we are already restricted by the other constraints. As a matter of fact
we work with the Nilsson potential, parameters of which are very well known. Really, the mean field of our
model (\ref{Ham}) is the deformed harmonic oscillator with the spin-orbit potential, the Nilsson
$\ell^2$ term being neglected because it generates the fourth order moments and, anyway, they are probably not of great importance.
In the original paper~\cite{Nils} Nilsson took the spin-orbit strength constant
$\kappa_{Nils}=0.05$ for rare earth nuclei.
Later the best value of $\kappa_{Nils}$
for rare earth nuclei was established~\cite{Ring} to be $0.0637$. For actinides there were established
different values of $\kappa_{Nils}$ for neutrons ($0.0635$) and protons ($0.0577$).
The numbers $\kappa_{Nils}=0.0637$, $\kappa_{Nils}=0.05$
and $\kappa_{Nils}=0.024$ (corresponding to $\eta=0.36$ used in our previous calculations~\cite{BaMo,BaMoPRC})
are marked on~Figs~\ref{fig2}, \ref{figL} by
the dotted vertical lines. Of course we will use the
conventional \cite{Ring} parameters of the Nilsson potential and
from now on we will speak only about the Nilsson \cite{Nils} spin-orbital strength parameter
$\kappa_{Nils}$, which is connected with~$\eta$ by the relation $\eta=2\hbar\omega\kappa_{Nils}$.
\subsection{Discussion and interpretation of results}
The energies and excitation probabilities of orbital and spin scissors modes obtained by the
solution of the isovector set of equations (\ref{iv}) are displayed in the Table 1.
\begin{table}[h!]\label{tab1}
\caption{Scissors modes energies $E_{\rm sc}$ and
transition probabilities $B(M1)_{\rm sc}$.}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
\multicolumn{2}{c|}{ $^{164}$Er } &
\multicolumn{3}{c|}{ $E_{\rm sc}$, MeV} & \multicolumn{3}{c}{ $B(M1)_{\rm sc},\ \mu_N^2$} \\
\cline{3-8}
\multicolumn{2}{c|}{ } &
$K_{\rm s}=0$ & $K_{\rm s}=50$ & $K_{\rm s}=92$ &
$K_{\rm s}=0$ & $K_{\rm s}=50$ & $K_{\rm s}=92$ \\
\hline
spin & $\bar\Delta_0=0$ & 1.40 & 1.60 & 1.73 & 5.38 & 6.23 & 6.79 \\[-2mm]
scissors & $\bar\Delta_0\neq 0$ & 2.72 & 2.75 & 2.77 & 3.93 & 4.79 & 5.44 \\ \hline
orbital & $\bar\Delta_0=0$ & 2.57 & 2.69 & 2.78 & 1.74 & 1.59 & 1.50 \\[-2mm]
scissors & $\bar\Delta_0\neq 0$ & 3.49 & 3.51 & 3.52 & 1.74 & 1.51 & 1.35 \\
\hline
\end{tabular}
\end{table}
There are results of calculations with three values of the spin-spin strength constant and two
values of $\bar \Delta_0$. As it was expected the energies of both scissors increased approximately
by 1 Mev after inclusion of pairing. The behaviour of transition probabilities turned out less
predictable. The $B(M1)$ value of the spin scissors decreased approximately by 1.5 $\mu_N^2$, whereas
$B(M1)$ value of the orbital scissors turned out practically insensitive to the inclusion of
pair correlations.
We can compare the summed $B(M1)_{\Sigma}= B(M1)_{or}+B(M1)_{sp}$ values and the centroid
of both scissors energies
$$E_{cen}=[E_{or}B(M1)_{or}+E_{sp}B(M1)_{sp}]/B(M1)_{\Sigma}$$
with the results of the paper \cite{Urban} where no spin degrees of freedom had been considered and with the experimental data. The respective results
are shown in the Table 2.
\begin{table}[h!]\label{tab2}
\caption{Scissors modes energy centroid $E_{cen}$ and summarized
transition probabilities $B(M1)_{\Sigma}$.
The experimental values
of $E_{cen}$, $\delta$, and $B(M1)_{\Sigma}$ are from \cite{Pietr} and
references therein.}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\multicolumn{1}{c|}{ $^{164}$Er } &
\multicolumn{5}{c|}{ $E_{cen}$, MeV} & \multicolumn{5}{c}{ $B(M1)_{\Sigma},\ \mu_N^2$} \\
\cline{2-11}
&
$K_{\rm s}=0$ & $K_{\rm s}=50$& $K_{\rm s}=92$ & \cite{Urban} & exp &
$K_{\rm s}=0$ & $K_{\rm s}=50$& $K_{\rm s}=92$ & \cite{Urban} & exp \\
\hline
$\bar\Delta_0=0$ & 1.69 & 1.82 & 1.92 & 2.10 & & 7.13 & 7.82 & 8.29 & 9.26 & \\[-5mm]
& & & & & 2.90 & & & & & 1.45 \\[-5mm]
$\bar\Delta_0\neq 0$ & 2.96 & 2.93 & 2.92 & 3.37 & & 5.67 & 6.30 & 6.79 & 5.62 & \\
\hline
\end{tabular}
\end{table}
It is seen that the inclusion of spin degrees of freedom in the WFM
method does not change markedly our results (in comparison with previous ones
\cite{Urban}). Of course, the energy changed in the desired direction and now practically
coincides with the experimental value (especially in the case with spin-spin forces.) However,
the situation with the $B(M1)$ values did not change (and even become worse in the case
with spin-spin forces). Our hope, that spin degrees of freedom can improve
the situation with the $B(M1)$ values, did not become true: the theory so far gives four times bigger values of
$B(M1)$ than the experimental ones, exactly as it was the case in the paper \cite{Urban}.
The result look discouraging. However,
a phenomenon, which was missed in our previous papers and described in the next
section will save the situation.
\section{Counter-rotating angular momenta of spins up/down
(hidden angular momenta)}
The equilibrium (ground state) orbital angular momentum of any
nucleus is composed of two equal parts: half of nucleons (protons + neutrons) having spin projection up and other half having
spin projection down. It is known that
the huge majority of nuclei have zero angular momentum in the ground state. We will show
below that as a rule this zero is just the sum of two rather big counter directed angular
momenta (hidden angular momenta, because they are not manifest in the ground state)
of the above mentioned two parts of any nucleus. Being connected with the spins of
nucleons this phenomenon naturally has great influence
on all nuclear properties connected with the spin, in particular, the spin scissors mode.
Let us analyze the procedure of linearization of the equations of motion for collective variables
(\ref{Varis}). We consider small deviations of the system from equilibrium, so all variables
are written as a sum of their equilibrium value plus a small deviation:
$$L(t)=L(eq)+{\cal L}(t),\quad \mbox{et al.}$$
Neglecting quadratic deviations one obtains the set of linearized equations for deviations depending on the equilibrium
values $R^{\tau\varsigma}_{\lambda\mu}(eq)$ and $L^{\tau\varsigma}_{\lambda\mu}(eq)$, which are the input data
of the problem. In the paper \cite{BaMoPRC} we made the following choice:
\begin{eqnarray}
&&R^{+}_{2\pm1}(eq)=R^{+}_{2\pm2}(eq)=0,\quad R^{+}_{20}(eq)\neq0,\quad R^{+}_{00}(eq)\neq0,
\label{eq1}
\\
&&R^{\uparrow\downarrow}_{\lambda\mu}(eq)=R^{\downarrow\uparrow}_{\lambda\mu}(eq)=0,
\label{eq3}
\\
&&L^{\tau\varsigma}_{\lambda\mu}(eq)=0,\quad
R^{-}_{\lambda\mu}(eq)=0.
\label{eq4}
\end{eqnarray}
At first glance, this choice looks quite natural. Really, relations (\ref{eq1})
follow from the axial symmetry of nucleus. Relations (\ref{eq3}) are justified by the fact that these quantities should be diagonal in spin at equilibrium. The variables
$L^{\tau\varsigma}_{\lambda\mu}(t)$ contain the momentum ${\bf p}$ in their definition which
incited us to suppose zero equilibrium values as well (we will show below that it is not
true for $L^-_{10}$ because of quantum effects connected with spin).
The relation $R^{-}_{\lambda\mu}(eq)=0$ follows from the shell model considerations: the
nucleons with spin projection "up" and "down" are sitting in pairs on the same levels, therefore all
average properties of the "spin up" part of nucleus must be identical to that of the "spin down" part.
However, the careful analysis shows that being undoubtedly true for variables
$R^{\uparrow\uparrow}_{\lambda\mu},\,R^{\downarrow\downarrow}_{\lambda\mu}$ this statement turns out
erroneous for variables $L^{\uparrow\uparrow}_{10},\,L^{\downarrow\downarrow}_{10}$. Let us demonstrate it.
By definition
\begin{eqnarray}
L^{ss'}_{\lambda\mu}(t)=
\int\! d^3r\,\int\! \frac{d^3p}{(2\pi\hbar)^{3}} \{r\otimes p\}_{\lambda\mu}
f^{ss'}({\bf r},{\bf p},t)=
\int\! d^3r \{r\otimes J^{ss'}\}_{\lambda\mu},
\label{LJ}
\end{eqnarray}
where
\begin{eqnarray}
J^{ss'}_i({\bf r},t)=\int\! \frac{d^3p}{(2\pi\hbar)^{3}} p_{i}f^{ss'}({\bf r},{\bf p},t)=
\int\! \frac{d^3p}{(2\pi\hbar)^{3}} p_{i}\int\!d^3q\exp(-\frac{i}{\hbar}{\bf p}\cdot{\bf q})
\rho({\bf r}+\frac{{\bf q}}{2},s;{\bf r}-\frac{{\bf q}}{2},s';t)
\label{J}
\end{eqnarray}
is the i-th component of the nuclear current. In the last relation the definition \cite{Ring} of Wigner function is used.
Performing the integration over ${\bf p}$ one finds:
\begin{eqnarray}
J^{ss'}_i({\bf r},t)=i\hbar\int\!d^3q[\frac{\partial}{\partial q_i}\delta({\bf q})]
\rho({\bf r}+\frac{{\bf q}}{2},s;{\bf r}-\frac{{\bf q}}{2},s';t)
\nonumber\\
=-i\hbar\int\!d^3q\delta({\bf q})\frac{\partial}{\partial q_i}
\rho({\bf r}+\frac{{\bf q}}{2},s;{\bf r}-\frac{{\bf q}}{2},s';t)
\nonumber\\
=-\frac{i\hbar}{2}[(\nabla_{1i}-\nabla_{2i})\rho({\bf r}_1,s;{\bf r}_2,s';t)]_{{\bf r}_1={\bf r}_2={\bf r}},
\label{J1}
\end{eqnarray}
where ${\bf r}_1={\bf r}+\frac{{\bf q}}{2},\, {\bf r}_2={\bf r}-\frac{{\bf q}}{2}$. The density matrix of the ground state
nucleus is defined \cite{Ring} as
\begin{equation}
\rho({\bf r}_1,s;{\bf r}_2,s';t)=\sum_{\nu}v^2_{\nu}\phi_{\nu}({\bf r}_1s)\phi^*_{\nu}({\bf r}_2s'),
\label{rho}
\end{equation}
where $v^2_{\nu}$ are occupation numbers and $\phi_{\nu}$ are single particle wave functions. For the sake
of simplicity we will consider the case of spherical symmetry. Then $\nu=nljm$ and
\begin{eqnarray}
\phi_{nljm}({\bf r},s)={\cal R}_{nlj}(r)\sum_{\Lambda,\sigma}C^{jm}_{l\Lambda,\frac{1}{2}\sigma}
Y_{l\Lambda}(\theta,\phi)\chi_{\frac{1}{2}\sigma}(s),
\label{phi}
\end{eqnarray}
\begin{eqnarray}
J^{ss'}_i({\bf r})=-\frac{i\hbar}{2}\sum_{\nu}v^2_{\nu}
[\nabla_i\phi_{\nu}({\bf r},s)\cdot\phi^*_{\nu}({\bf r},s')
-\phi_{\nu}({\bf r},s)\cdot\nabla_i\phi^*_{\nu}({\bf r},s')]
\label{Jdef}
\\
=-\frac{i\hbar}{2}\sum_{nljm}v^2_{nljm}{\cal R}^2_{nlj}\sum_{\Lambda,\sigma,\Lambda',\sigma'}
C^{jm}_{l\Lambda,\frac{1}{2}\sigma}C^{jm}_{l\Lambda',\frac{1}{2}\sigma'}
[Y^*_{l\Lambda'}\nabla_iY_{l\Lambda}-Y_{l\Lambda}\nabla_iY^*_{l\Lambda'}]
\chi_{\frac{1}{2}\sigma}(s)\chi_{\frac{1}{2}\sigma'}(s').
\label{Jphi}
\end{eqnarray}
Inserting this expression into (\ref{LJ}) one finds:
\begin{eqnarray}
L^{ss'}_{10}(eq)=
-\frac{i\hbar}{2}\sum_{nljm}v^2_{nljm}\sum_{\Lambda\sigma,\Lambda'\sigma'}
C^{jm}_{l\Lambda,\frac{1}{2}\sigma}C^{jm}_{l\Lambda',\frac{1}{2}\sigma'}
\chi_{\frac{1}{2}\sigma}(s)\chi_{\frac{1}{2}\sigma'}(s')
\nonumber\\
\int\! d^3r\,{\cal R}^2_{nlj}[Y^*_{l\Lambda'}\{r\otimes\nabla\}_{10}Y_{l\Lambda}-Y_{l\Lambda}\{r\otimes\nabla\}_{10}Y^*_{l\Lambda'}]
\nonumber\\
=\frac{i}{2\sqrt2}\sum_{nljm}v^2_{nljm}\sum_{\Lambda\sigma,\Lambda'\sigma'}
C^{jm}_{l\Lambda,\frac{1}{2}\sigma}C^{jm}_{l\Lambda',\frac{1}{2}\sigma'}
\chi_{\frac{1}{2}\sigma}(s)\chi_{\frac{1}{2}\sigma'}(s')
\int\! d^3r\,{\cal R}^2_{nlj}[Y^*_{l\Lambda'}\hat l_0Y_{l\Lambda}-Y_{l\Lambda}\hat l_0Y^*_{l\Lambda'}]
\nonumber\\
=\frac{i}{2\sqrt2}\sum_{nljm}v^2_{nljm}\sum_{\Lambda\sigma,\Lambda'\sigma'}
C^{jm}_{l\Lambda,\frac{1}{2}\sigma}C^{jm}_{l\Lambda',\frac{1}{2}\sigma'}
\chi_{\frac{1}{2}\sigma}(s)\chi_{\frac{1}{2}\sigma'}(s')(\Lambda+\Lambda')\delta_{\Lambda,\Lambda'}
\nonumber\\
=\frac{i}{\sqrt2}\sum_{nljm}v^2_{nljm}\sum_{\Lambda\sigma}\Lambda\,
\left(C^{jm}_{l\Lambda,\frac{1}{2}\sigma}\right)^2
\chi_{\frac{1}{2}\sigma}(s)\chi_{\frac{1}{2}\sigma}(s').
\label{L10}
\end{eqnarray}
Here the definition $\hat l_{\mu}=-\hbar\sqrt{2}\{r\otimes\nabla\}_{1\mu}$, formula
$\hat l_0Y_{l\Lambda}=\Lambda Y_{l\Lambda}$ and normalization of functions ${\cal R}_{nlj}$ were used.
Remembering the definition of the spin function $\chi_{\frac{1}{2}\sigma}(s)=\delta_{\sigma,s}$
we get finally:
\begin{equation}
L^{ss'}_{10}(eq)=
\frac{i}{\sqrt2}\sum_{nljm}v^2_{nljm}\sum_{\Lambda}\Lambda\,
\left(C^{jm}_{l\Lambda,\frac{1}{2}s}\right)^2\delta_{s,s'}=
\delta_{s,s'}\frac{i}{\sqrt2}\sum_{nljm}v^2_{nljm}
\left(C^{jm}_{lm-s,\frac{1}{2}s}\right)^2(m-s).
\label{L10f}
\end{equation}
Now, with the help of analytic expressions for Clebsh-Gordan coefficients one obtains the final
expressions
\begin{eqnarray}
L^{\uparrow\uparrow}_{10}(eq)
=\frac{i}{\sqrt2}\sum_{nl}\left[
\sum_{m=-\left(l+\frac{1}{2}\right)}^{l+\frac{1}{2}}v^2_{nlj^+m}\frac{l+\frac{1}{2}+m}{2l+1} +
\sum_{m=-\left(l-\frac{1}{2}\right)}^{l-\frac{1}{2}}v^2_{nlj^-m}\frac{l+\frac{1}{2}-m}{2l+1}
\right]\left(m-\frac{1}{2}\right),
\label{L10up}
\end{eqnarray}
\begin{eqnarray}
L^{\downarrow\downarrow}_{10}(eq)
=\frac{i}{\sqrt2}\sum_{nl}\left[
\sum_{m=-\left(l+\frac{1}{2}\right)}^{l+\frac{1}{2}}v^2_{nlj^+m}\frac{l+\frac{1}{2}-m}{2l+1} +
\sum_{m=-\left(l-\frac{1}{2}\right)}^{l-\frac{1}{2}}v^2_{nlj^-m}\frac{l+\frac{1}{2}+m}{2l+1}
\right]\left(m+\frac{1}{2}\right),
\label{L10d}
\end{eqnarray}
where the notation $j^{\pm}=l\pm\frac{1}{2}$ is introduced. Replacing in (\ref{L10up}) $m$
by $-m$ we find that
\begin{eqnarray}
L^{\uparrow\uparrow}_{10}(eq)=-L^{\downarrow\downarrow}_{10}(eq).
\label{L10upd}
\end{eqnarray}
By definition (\ref{Varis})
$L^{\pm}_{10}(eq)=L^{\uparrow\uparrow}_{10}(eq) \pm L^{\downarrow\downarrow}_{10}(eq)$.
Combining linearly (\ref{L10up}) and (\ref{L10d}) one finds:
\begin{eqnarray}
L^+_{10}(eq)
=\frac{i}{\sqrt2}\sum_{nl}\left[
\sum_{m=-\left(l+\frac{1}{2}\right)}^{l+\frac{1}{2}}v^2_{nlj^+m}\frac{2l}{2l+1}m+
\sum_{m=-\left(l-\frac{1}{2}\right)}^{l-\frac{1}{2}}v^2_{nlj^-m}\frac{2l+2}{2l+1}m
\right],
\label{L10+}
\end{eqnarray}
\begin{eqnarray}
L^-_{10}(eq)=\frac{i}{\sqrt2}\sum_{nl}\left[
\sum_{m=-\left(l+\frac{1}{2}\right)}^{l+\frac{1}{2}}v^2_{nlj^+m}\frac{2m^2-l-\frac{1}{2}}{2l+1}-
\sum_{m=-\left(l-\frac{1}{2}\right)}^{l-\frac{1}{2}}v^2_{nlj^-m}\frac{2m^2+l+\frac{1}{2}}{2l+1}
\right].
\label{L10-}
\end{eqnarray}
These formulas are valid for spherical nuclei. However, with the scissors and spin-scissors modes, we are considering deformed nuclei.
For the sake of the discussion,
let us consider the case of infinitesimally small deformation, when one can continue to use formulae~(\ref{L10+}, \ref{L10-}).
Now only levels with quantum numbers $\pm m$ are degenerate.
According to, for example, the Nilsson scheme~\cite{Nils} nucleons will occupy pairwise
precisely those levels which leads to the zero
value of~$L^+_{10}(eq)$.
What about~$L^-_{10}(eq)$? It only enters (27) in the equation for $\dot {{\cal P}}^-_{21}$. Let us analyze the structure of formula (\ref{L10-}) considering for
the sake of simplicity the case without pairing. Two sums over $m$ (let us note them $\Sigma_1$
and $\Sigma_2$) represent two spin-orbital partners: in the first sum the summation goes over
levels of the lower partner ($j=l+\frac{1}{2}$) and in the second sum -- over levels of the
higher partner ($j=l-\frac{1}{2}$). The values of both sums depend naturally on the values of
occupation numbers $n_{nljm}= 0,1$. There are three possibilities. The first one is trivial: if all
levels of both spin-orbital partners are disposed above the Fermi surface, then the respective
occupation numbers $n_{nljm}=0$ and both sums are equal to zero identically. The second
possibility: all levels of both spin-orbital partners are disposed below the Fermi surface.
Then all respective occupation numbers $n_{nlj^+m}=n_{nlj^-m}=1$. The elementary analytical
calculation (for arbitrary $l$) shows that in this case the two sums in (44) exactly compensate
each other,
i.e. $\Sigma_1+\Sigma_2=0$. The most interesting is the third possibility, when one part of
levels of two spin-orbital partners is disposed below the Fermi surface and another part is
disposed above it. In this case the compensation does not happen and one gets
$\Sigma_1+\Sigma_2\neq 0$ what leads to $L^-_{10}(eq)\neq 0$. In the case of pairing, things
are not so sharply separated and $L_{10}^-(eq)$ has always a finite value. However, the modifications with respect to mean field are very small.
Let us illustrate the above analysis by the example of $^{164}$Er (protons). Its deformation is
$\delta=0.25$ $(\epsilon=0.26)$ and Z=68. Looking on the Nilsson scheme (for example, Fig.1.5 of
\cite{Solov}
or Fig. 2.21c of \cite{Ring}) one easily finds, that only three pairs of spin-orbital partners
give a nonzero contribution to $L^-_{10}(eq)$. They are: $N=4, d_{5/2}-d_{3/2}$ (two
levels of $d_{5/2}$ are below the Fermi surface, all the rest -- above);
$N=4, g_{9/2}-g_{7/2}$ (one level of $g_{7/2}$ is above the Fermi surface, all the rest -- below);
$N=5, h_{11/2}-h_{9/2}$ (four levels of $h_{11/2}$ are below the Fermi surface, all the rest
-- above).
It is possible to make the crude evaluation of $L^-_{10}(eq)$ using the quantum numbers
indicated on Fig.1.5 of \cite{Solov} or Fig. 2.21c of \cite{Ring}). The result turns out rather
close to the exact one,
computed with the help of formulas (\ref{LJ},\ref{Jdef}) and Nilsson wave functions.
The influence of pair correlations is very small.
\begin{figure}[h]
\centering\includegraphics[width=8cm]{fig_Schuck.pdf}
\caption{(a) Protons with spins $\uparrow$ (up) and $\downarrow$ (down) having nonzero orbital
angular momenta at equilibrium. (b) Protons from Fig.(a) vibrating against one-another.}
\label{figSch}\end{figure}
Indeed, from the definitions (\ref{LJ}) and (\ref{L10}) one can see that $L^{ss}_{10}(eq)$ is just
the average value of the z-component of the orbital angular momentum of nucleons with the
spin projection $s$~($\frac{1}{2}$~or~$-\frac{1}{2}$). So,
the ground state nucleus consists
of two equal parts having nonzero angular momenta with opposite directions, which compensate
each other resulting in the zero total angular momentum.
This is graphically depicted in Fig.~\ref{figSch}(a).
On the other hand, when the opposite angular momenta become tilted, one excites the system and the opposite
angular momenta are vibrating with a tilting angle, see Fig.~\ref{figSch}(b).
Actually the two opposite angular momenta are oscillating, one in the opposite sense of the other.
It is rather obvious from Fig.~\ref{fig0} that these tilted vibrations
happen separately in each of the neutron and proton lobes.
These spin-up against spin-down motions certainly influence the
excitation of the spin scissors mode.
So, classically speaking
the proton and neutron parts of the ground state nucleus
consist each of two identical gyroscopes rotating in opposite directions. One knows that it
is very difficult to deviate gyroscope from an equilibrium. So one can expect,
that the probability to force two gyroscopes to oscillate as scissors (spin scissors)
should be small. This picture is confirmed in the next section.
\section{Results of calculations continued}
We made the calculations taking into account the non zero
value of $L_{10}^-(eq)$ (which was computed according to formulas
(\ref{LJ},\ref{Jdef}) and Nilsson wave functions). The results are shown
on~Fig.~\ref{figL}.
\begin{figure}[h!]
\centering\includegraphics[width=8cm]{fig_4.pdf}
\caption{The energies $E$ and $B(M1)$-factors as a functions of the spin-orbital strength constant
$\kappa_{Nils}$. The dashed lines -- calculations without
$L_{10}^-(eq)$, the solid lines -- $L_{10}^-(eq)$ are taken into account. $V_{ss}$ and pairing
are included.}
\label{figL}\end{figure}
They demonstrate (in comparison with~Fig.~\ref{fig2}) the strong
influence of the spin-up vs spin-down angular momenta on the spin scissors mode, whose B(M1) value is strongly
decreasing with increasing $\kappa_{Nils}$. The B(M1) value of the orbital scissors also
is reduced, but not so much, the value of the reduction being practically independent on
$\kappa_{Nils}$. The influence of $L_{10}^-(eq)$ on the energies of both scissors is negligible,
leading to the small increase of their splitting. Now the energy centroid of both scissors and
their summed B(M1) value at $\kappa_{Nils}=0.0637$ are $E_{cen}=2.97$ MeV and
$B(M1)_{\Sigma}=3.7 \mu_N^2$. The general agreement with experiment becomes considerably
better (compare with Table 2), though the theoretical value of $B(M1)_{\Sigma}$ still exceeds the
experimental one approximately 2.5 times. However, as we will see, the case of
$^{164}$Er may imply a quite particular situation (or even a problem with the experimental value).
The results of systematic calculations for rare-earth nuclei are presented in Tables 3 and
4 and desplayed in Fig.~\ref{figMalov}. Table 3 contains the results for well deformed nuclei with
$\delta \geq 0.18.$
It is easy to see that the overall (general) agreement of theoretical
results with experimental data is substantially improved (in comparison with our previous
calculations \cite{Urban}).
\begin{table}\label{tab3}
\caption{Scissors modes energy centroid $E_{\rm cen}$ and summarized
transition probabilities $B(M1)_{\Sigma}$. Parameters: $\kappa_{Nils}=0.0637$, $V_0=25$
($V_0=27$ for $^{182, 184, 186}$W). The experimental values
of $E_{\rm cen}$, $\delta$, and $B(M1)_{\Sigma}$ are from \cite{Pietr} and
references therein.}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hline
& &
\multicolumn{4}{|c|}{ $E_{\rm cen}$, MeV} &
\multicolumn{4}{|c}{ $B(M1)_{\Sigma},\ \mu_N^2$} \\
\cline{3-10}
Nuclei & $\delta$ & exp & present & \cite{Urban} & $\Delta=0$
& exp & present & \cite{Urban} & $\Delta=0$ \\
\hline
$^{150}$Nd & 0.22 & 3.04 & 2.88 & 3.44 & 1.92 & 1.61 & 1.64 & 4.17 & 7.26 \\[-.5mm]
$^{152}$Sm & 0.24 & 2.99 & 2.99 & 3.46 & 2.02 & 2.26 & 2.50 & 4.68 & 7.81 \\[-.5mm]
$^{154}$Sm & 0.26 & 3.20 & 3.10 & 3.57 & 2.17 & 2.18 & 3.34 & 5.42 & 8.65 \\[-.5mm]
$^{156}$Gd & 0.26 & 3.06 & 3.09 & 3.60 & 2.16 & 2.73 & 3.44 & 5.42 & 8.76 \\[-.5mm]
$^{158}$Gd & 0.26 & 3.14 & 3.09 & 3.60 & 2.19 & 3.39 & 3.52 & 5.72 & 9.12 \\[-.5mm]
$^{160}$Gd & 0.27 & 3.18 & 3.14 & 3.61 & 2.21 & 2.97 & 4.02 & 5.90 & 9.38 \\[-.5mm]
$^{160}$Dy & 0.26 & 2.87 & 3.08 & 3.59 & 2.13 & 2.42 & 3.60 & 5.53 & 9.03 \\[-.5mm]
$^{162}$Dy & 0.26 & 2.96 & 3.07 & 3.61 & 2.14 & 2.49 & 3.69 & 5.66 & 9.25 \\[-.5mm]
$^{164}$Dy & 0.26 & 3.14 & 3.07 & 3.60 & 2.17 & 3.18 & 3.78 & 5.95 & 9.59 \\[-.5mm]
$^{164}$Er & 0.25 & 2.90 & 3.01 & 3.57 & 2.10 & 1.45 & 3.39 & 5.62 & 9.26 \\[-.5mm]
$^{166}$Er & 0.26 & 2.96 & 3.06 & 3.53 & 2.13 & 2.67 & 3.86 & 5.96 & 9.59 \\[-.5mm]
$^{168}$Er & 0.26 & 3.21 & 3.06 & 3.53 & 2.10 & 2.82 & 3.95 & 5.95 & 9.67 \\[-.5mm]
$^{170}$Er & 0.26 & 3.22 & 3.05 & 3.57 & 2.09 & 2.63 & 4.03 & 5.91 & 9.79 \\[-.5mm]
$^{172}$Yb & 0.25 & 3.03 & 2.99 & 3.55 & 2.05 & 1.94 & 3.72 & 5.84 & 9.79 \\[-.5mm]
$^{174}$Yb & 0.25 & 3.15 & 2.98 & 3.47 & 2.02 & 2.70 & 3.80 & 5.89 & 9.82 \\[-.5mm]
$^{176}$Yb & 0.24 & 2.96 & 2.92 & 3.45 & 1.94 & 2.66 & 3.46 & 5.54 & 9.58 \\[-.5mm]
$^{178}$Hf & 0.22 & 3.11 & 2.81 & 3.43 & 1.79 & 2.04 & 2.67 & 4.86 & 9.00 \\[-.5mm]
$^{180}$Hf & 0.22 & 2.95 & 2.81 & 3.36 & 1.76 & 1.61 & 2.69 & 4.85 & 8.97 \\[-.5mm]
$^{182}$W & 0.20 & 3.10 & 3.28 & 3.30 & 1.63 & 1.65 & 2.05 & 4.31 & 8.43 \\[-.5mm]
$^{184}$W & 0.19 & 3.31 & 3.24 & 3.28 & 1.55 & 1.12 & 1.72 & 3.97 & 8.14 \\[-.5mm]
$^{186}$W & 0.18 & 3.20 & 3.19 & 3.26 & 1.49 & 0.82 & 1.40 & 3.76 & 7.95 \\
\hline
\end{tabular}
\end{table}
\begin{table}\label{tab3a}
\caption{Scissors modes energy centroid $E_{\rm cen}$ and summarized
transition probabilities $B(M1)_{\Sigma}$. Parameters: $\kappa=0.05$
($\kappa=0.0637$ for $^{182, 184, 186}$W), $V_0=27$.}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hline
& &
\multicolumn{4}{|c|}{ $E_{\rm cen}$, MeV} &
\multicolumn{4}{|c}{ $B(M1)_{\Sigma},\ \mu_N^2$} \\
\cline{3-10}
Nuclei & $\delta$ & exp & present & \cite{Urban} & $\Delta=0$
& exp & present & \cite{Urban} & $\Delta=0$ \\
\hline
$^{134}$Ba & 0.14 & 2.99 & 3.04 & 3.09 & 1.28 & 0.56 & 0.68 & 1.67 & 3.90 \\[-.5mm]
$^{148}$Nd & 0.17 & 3.37 & 3.22 & 3.18 & 1.48 & 0.78 & 1.28 & 2.58 & 5.39 \\[-.5mm]
$^{150}$Sm & 0.16 & 3.13 & 3.17 & 3.13 & 1.42 & 0.92 & 1.12 & 2.45 & 5.26 \\[-.5mm]
$^{182}$W & 0.20 & 3.10 & 3.28 & 3.30 & 1.63 & 1.65 & 2.05 & 4.31 & 8.43 \\[-.5mm]
$^{184}$W & 0.19 & 3.31 & 3.24 & 3.28 & 1.55 & 1.12 & 1.72 & 3.97 & 8.14 \\[-.5mm]
$^{186}$W & 0.18 & 3.20 & 3.19 & 3.26 & 1.49 & 0.82 & 1.40 & 3.76 & 7.95 \\[-.5mm]
$^{190}$Os & 0.15 & 2.90 & 3.14 & 3.12 & 1.21 & 0.98 & 1.38 & 2.67 & 6.64 \\[-.5mm]
$^{192}$Os & 0.14 & 3.01 & 3.11 & 3.12 & 1.15 & 1.04 & 1.00 & 2.42 & 6.37 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[h!]
\centering\includegraphics[width=8cm]{fig_5.pdf}
\caption{The energies $E_{sc}$ and $B(M1)_{sc}$-factors as a function of the mass number A
for nuclei listed in the Table 3.}
\label{figMalov}\end{figure}
The results of calculations for two groups ("light" and "heavy") of weakly deformed nuclei
with deformations $0.14\leq \delta \leq 0.17$ are shown in the Table 4.
They require some discussion, because of the
self-consistency problem. These two groups of nuclei are transitional between well deformed
and spherical nuclei. Systematic calculations of equilibrium deformations \cite{Solov}
predict $\delta_{eq}^{th}=0.0$ for $^{134}$Ba, $\pm 0.1$ for $^{148}$Nd,
0.15 or -0.12 for $^{150}$Sm, 0.1 or -0.14 for $^{190}$Os and -0.1 for $^{192}$Os,
whereas their experimental values are $\delta_{eq}=0.14,\, 0.17,\, 0.16,\, 0.15$ and 0.14
respectively. As one sees, the discrepancy between theoretical and experimental
$\delta_{eq}$ is large. Uncertain signs of theoretical equilibrium deformations are
connected with very small ($\sim $0.1-0.2 MeV) difference between the values of deformation
energies ${\cal E}_{def}$ at positive and negative $\delta_{eq}$.
Even more so, the values of deformation
energies of these nuclei are very small: ${\cal E}_{def}=0.20,~0.50,~0.80$ and 0.70 MeV for
$^{148}$Nd, $^{150}$Sm, $^{190}$Os and $^{192}$Os respectively. This means that these
nuclei are very "soft" with respect of $\beta$- or $\gamma$-vibrations and probably they have
more complicated equilibrium shapes, for example, hexadecapole or octupole deformations in
addition to the quadrupole one. This means that for the correct description of their dynamical
and equilibrium properties it is necessary to include higher order Wigner function moments
(at least fourth order) in addition to the second order ones. In this case it would be natural
also to use more complicate mean field potentials (for example, the Woods-Saxon one or the potential extracted
from some of the numerous variants of Skyrme forces) instead of the too simple Nilsson potential.
Naturally, this will be the subject of further investigations. However, to be sure that the
situation with these nuclei is not absolutely hopeless, one can try
to imitate the properties of the more perfect potential by fitting
parameters of the Nilsson potential. As a matter of fact this potential has the single but
essential parameter -- the spin-orbital strength $\kappa_{Nils}$.
It turns out that changing its value from 0.0637 to 0.05 (the value used by Nilsson in his
original paper \cite{Nils}) is enough to obtain the reasonable description of B(M1) factors
(see Table 4).
To obtain the reasonable description of the scissors energies we use the "freedom" of
choosing the value of the pairing interaction constant $V_0$ in (\ref{v_p}). It turns out
that changing its value from 25 MeV to 27 MeV is enough to obtain the satisfactory agreement
between the theoretical and experimental values of $E_{sc}$ (Table 4).
The isotopes
$^{182-186}$W turn out intermediate between weakly deformed and well deformed nuclei:
reasonable results are obtained with $\kappa_{Nils}=0.0637$ (as for well deformed)
and $V_0=27$ MeV (as for weakly deformed). That is why they appear in both Tables.
Returning to the group of well deformed nuclei with $\delta \geq 0.18$ (Table 3)
it is necessary to emphasize that all presented results for these nuclei were obtained without any
fitting. In spite of it the agreement between the theory and experiment can be called
excellent for all nuclei of this group except two: $^{164}$Er and $^{172}$Yb, where the
theory overestimates B(M1) values approximately two times. However, these two nuclei fall out of the systematics and one can suspect, that
there the experimental B(M1) values are underestimated. Therefore one
can hope, that new experiments will correct the situation with these
nuclei, as it happened, for example, with $^{232}$Th \cite{Adekola}.
\section{Conclusion}
The method of Wigner function moments is generalized to take into account spin degrees of
freedom and pair correlations simultaneously.
The inclusion of the spin into the theory
allows one to discover several new phenomena. One of them, the nuclear spin scissors, was
described and studied in \cite{BaMo,BaMoPRC}, where some indications on the experimental
confirmation of its existence in actinides nuclei are discussed. Another phenomenon, the
opposite rotation of spin up/down nucleons, or in other words, the phenomenon of hidden
angular momenta, is described in this paper. Being determined by the spin degrees
of freedom this phenomenon has great influence on the excitation probability of the
spin scissors mode. On the other hand the spin scissors B(M1) values and the energies
of both, spin and orbital, scissors are very sensitive to the action of pair correlations.
As a result, these two factors, the spin up/down counter-rotation and pairing, working together,
improve substantially the agreement between the theory and experiment in the description
of the energy centroid of two nuclear scissors and their summed excitation probability.
More precisely, for the first time an excellent agreement is achieved for well deformed
nuclei of the rare earth region with standard values of all possible parameters.
An excellent agreement is also achieved for weakly deformed (transitional)
nuclei of the same region by a very modest re-fit of the spin-orbit strength.
We suppose that fourth order moments and more realistic interactions are required for the
adequate description of transitional nuclei. However this shall be the subject of future work.
\vspace*{5mm}\noindent{\bf Acknowledgements}\\
Valuable discussions with V. N. Kondratyev are gratefully acknowledged.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,418 |
How Do You Spell… International Organized Crime and Mortgage Fraud…
This article speaks for itself. Veterans Today is standing as source of sharing basic information on International Organized Crime and International Financial Mortgage Fraud. I know, it's hard to even believe this is going on. Something's going on… and I'm standing with the courageous and honorable to bring this information to light. Here is the most recent radio show of Veterans Today addressing these topics.
Source: http://www.veteranstoday.com/2014/01/31/aipac-the-denver-illuminati-wizards-espionage-and-frauds/
AIPAC the Denver Illuminati Wizards Espionage and Frauds
Posted by Stew Webb on January 31, 2014
By Stew Webb
Connecting some Financial Frauds the U.S. Justice Department does not want you to know about but U.S. Attorney Eric Holder does know.
The Denver Illuminati Connection and Leonard Millman one of the 12 Illuminati Wizards' of Oz.
Convicted HUD Scamster Phil Winn who this Whistleblower helped the HUD Independent Prosecutor Arlen Adams in 1989 get the conviction and plea bargain in 1991 against Winn who agreed to pay a $1.5 million dollar fine and was to be sentence to prison for 5-7 years that never happened, instead Denver-U.S. District Court Judge Sherman Finesilver who had been bribed years earlier with his ownership of Westward Newspaper in Denver by Leonard Millman got Judge Finesilver to seal Phil Winn's case from 1991 until 2001 when CIA Bill Clinton the day before leaving office as the U.S. President gave convicted criminal Phil Winn a illegal Presidential Pardon.
Denver Post controlled by Dean Singleton, Phil Winn and Larry Mizel good buddy does a spin story on Phil Winn.
Just think Phil Winn helped steal over 500,000 apartment units from HUD over 20 years for Denver Illuminati Wizard Len Millman and gets a presidential pardon.
Those apartments are hidden under thousand of entities one of the largest is AMICO run by Terri Constadine and Norman Brownstein, who is the attorney for Illuminati Wizards of Oz Leonard Millman and Wizard George HW Bush.
Brownstein was one of the six CIA Council to George HW Bush when he was CIA Director in the mid 1970s and Brownstein was on the Board of Directors of M.D.C. Holdings, Inc., and now MDC-NYSE Corporate attorney the parent company of Imperial Savings and Silverado Saving and Loan of Denver which collapsed where Neil Bush the U.S. President's son was a Director and in charge of Narcotics Money Laundering from Iran- Contra, nearly 12 Trillion Dollars in 9 years was laundered.
Phil Winn's, Winn Financial was an MDC-NYSE subsidiary company. Phil Winn's "Buffers-Stooges" were caught selling 4,000 stolen repossessed HUD houses in Arizona in 1992 and served time in Federal detention in Denver.
These were stolen in the late 1970s from HUD computers and re-deeded to Leonard Millman and his attorney Norman Brownstein who was in charge of the stolen houses for Millman and Bush.
Don Boles, a Phoenix, Arizona Investigative Journalist was murdered over his investigations of some of these stolen HUD houses that were being sold in Arizona.
Gene "Chip' Tatum is now a Columnist on Veterans Today and you will be reading things you have never heard before or do not remember when Gene did hundreds of Radio interviews between 1995-1997.
Super Spook Gene "Chip" Tatum is back and speaking out here is his information and website.
Phil Winn had served on the Board of Directors of MDC-NYSE and served on the Board of Directors of MDC Asset Investors that caused the 2008 Bank Bailout.
Asset Investors created Mortgages on houses that were never built and duplicate mortgages up to 9 times that they bought from Mortgage Brokers around the United States then sold those Mortgages in bundles as Securities bundled as derivatives those Mortgages which had cause the worldwide financial meltdown in 2008.
Norman Brownstein was the Director and Vice President of HSBC Bank of Canada in charge of Derivatives which bundled the Fake Mortgaged Backed Securities and sold them throughout the world as Mortgage Backed Derivatives.
Stew Webb SEC Whistleblower Filings and Letter to NY Attorney General which has been covered-up with no prosecutions
The current Denver U.S. Attorney is part of U.S. Attorney General Eric Holder's special Task force to prosecute the Bankers frauds the Denver U.S. Attorney's office has been a revolving door from the Norman Brownstein law firm.
Ask the former U.S. Denver Attorney who covered up Enron or former U.S. Attorney Michael Norton who covered up the Millman-Bush stolen 2,000 pounds of Uranium from Rocky Flats Nuclear arsenal west of Denver that was shipped in Ocean Spray Cranberry Juice Trucks to Apex Aviation in St. Louis, Missouri to keep from triggering Nuclear material transport detectors and flown to Israeli Prime Minister Benjamin Netanyahu and sold to North Korea that gave North Korea the Nuke Bomb.
Leonard Millman's CITI Bank's subsidiary did the fake clean up of Rocky Flats, the cover story was 2,000 pounds of Uranium was spilled on the hillside, cost to U.S. Taxpayer $2.5 Billion what a scam.
Then guess who builds a subdivision on that contaminated land and a great view of the Denver skyline?
MDC's Richmond American Homes run by Millman's "Buffer-Stooge" Larry Mizel.
Larry Mizel alias Larry Mizell is the Chairman of MDC-NYSE who dreamed up the Mortgage frauds for Leonard Millman his boss.
In a recent $13 Billion dollar settlement with JP Morgan Bank, Home America Mortgage a MDC Subsidiary had been named in the original suit to cover the whole frauds up in one sweep.
Former FSLIC Attorney and Professor of law at UMKC questions why is there no prosecutions?
The US Attorney Who Prosecutes JPMorgan Will Be Its First Witness
Illegally created Mortgage Securities by MDC-Asset Investors estimated at $100 Trillion.
Illegally created Derivatives backed by illegal fake Mortgage Securities estimated at $5,000 Trillion Dollars.
Illegally stealing Americans houses that were paying their Mortgages to other mortgage companies and on time and up to date on their mortgage when the Denver Illuminati Wizard Leonard Millman alias Leonard Hillman alias Michael Donavan would send his crooked attorneys in saying they owned the Mortgage and that it was registered with Meirs and I own the property. 12.5 million American have had their homes stolen from them with crooked attorneys, Judges and others, only the Kentucky Attorney General seems to be doing his job.
Bank of America has 80% of their mortgages on their books are non-performing assets, meaning they got stuck with the false mortgages and Bank of America is cooking their books, which is against the law yet they trade on the NYSE.
Meirs the privately owned Mortgage registration was set up in 1993 by Millman's "Buffer-Stooge" M.D.C. Holdings, Inc. Director Larry Mizel.
Why was Meirs set up in 1993?
Because these same named criminals above were doing the same thing during the 1980s except they were selling fake Mortgages to their Colorado controlled banks and Mortgage Companies then they triggered the FSLIC, Fannie Mae, Freddie Mac and FDIC Insurance to scam the U.S. Government and investors. Thirty one years of fraud and no one has gone to jail because all U.S. Attorney Generals and Colorado's US Attorney and Colorado State Attorney Generals since 1981 have been controlled by these two Illuminati Wizards Leonard Millman and George HW Bush who both Graduated from Yale University together and were both Skull and Bone Buddies and members a known Satanic Occult.
Cradle to Cabal The Secret Life of Gale Norton The Denver Illuminati Zionist Connection
Ask yourself a question?
Why did Leonard Millman pay a $80,000 Million Dollar fine in 1997 over a grand jury indictment for bribing Government Officials and never went to jail and the case was sealed under National Security by then President Bill Clinton?
Because Bill Clinton and Hillary Clinton are part of the Denver Illuminati Zionist Organized Crime Syndicate, their attorney James M. Lyons was on the Board of Directors of MDC-NYSE.
The Bush-Millman Organized Crime Family Flow Chart No: 1
Bill Clinton's fines that were paid over the Monica Lewinsky scandal and oral sex in the Whitehouse and lying before a grand jury saying a blow job and stuffing Monica with cigars is not sex, it did happen but this was the Spin Doctor story to keep the truth about Iran Contra and the Arkansas involvement from coming out.
Judge Lawrence Walsh had them nailed in late 1992 and he was replaced by Independent Prosecutor Kenny Star who obstructed Justice and covered up Iran- Contra frauds. Stars father started Star International the parent company of AIG Insurance the Millman's "Buffer-Stooge" Maurice Hank Greenberg runs AIG and was directly involved with Meyer Blinder of Blinder-Robinson the world largest penny stock frauds which collapsed in 1990. Millman's "Buffers-Stooges" Blinder and Greenberg ran National Brokerage Company, on Pine Street, Denver, Colorado. National Brokerage did massive securities frauds in the 1980s with ties to Balcor Securities, Stinger Securities, and gave Iran Contra drug smugglers-pilots stock in dummy companies to pledge as collateral for loans made by Neil Bush a Director at Silverado Savings. Those loans were never repaid, the stock was bogus and the FDIC out of Dallas picked up the tab. FDIC employees out of the Dallas offices were in on the take.
Chubb Insurance Company of Denver paid Bill Clinton's fines and paid off others including Paula Jones in Clinton's affairs. Chubb's Board of Directors at the time was Norman Phillip Brownstein.
Do you really want Hillary Clinton a known 4th Degree Satanic Witch, Lesbian, and Killer of over 300 witnesses and the Jewish Mob attorney (Rose law firm) to be your next President in 2016?
Ask Larry Mizel why he had a meeting with former Rose Law Firm Partner and White House Aid Vince Foster on the day he was murdered in Washington, D.C.
Ask Leonard Millman who controlled Hensel-Phelps Construction Company based out of Greeley, Colorado a nationwide contractor who committed frauds at the Denver Airport why a Helsel-Phelps Construction Company yellow pickup truck license number RCG-702 Arkansas tag was at the White House the evening Vice Foster was murdered.
Guess who sits on the Board of Directors of AIPAC-American Israeli Public Affairs Committee, which is Zionist Espionage against the United States and controls 99% of the U.S. Congress and Senate?
Brownstein and Mizel…..
[edit] Notable Board members
AIPAC's National Board has approximately 50 members. By tradition, previous Presidents serve as Board members, and are not listed separately here. (Asher, Levy, Mitchell, and Weinberg were sometimes referred to as the "Gang of Four".[2]) There are also state-level and some city-level AIPAC boards.
Other notable National Board members include:
Mickey Kantor
former Senator Rudy Boschwitz (R-Minn.)
S. Daniel Abraham (Democrat)
Steven Grossman – former chairman of the Democratic National Committee (Democrat)
Norman Brownstein – prominent Denver lawyer, was an early financial backer of the Kerry campaign (Democrat)
Larry A. Mizel – prominent Denver CEO (Republican)
Why does Norman Brownstein have office in Washington, Denver and Mexico City, Mexico?
Remember the Mexico President who fled Mexico over his Drugs and Frauds scandal?
Remember CITI Bank was questioned why they laundered the Mexican Presidents Drug profits?
Did you know Norman Brownstein was on the Board of Directors of CITI Bank?
I cannot put all the blame on Norman Brownstein we must include Leonard Millman's other attorneys hiding the stolen and laundered Drug profits.
Leonard Millman's brother in law Allen Karsh of Denver and Steven Hoth of Denver are also culprits.
Who is one of the worst Zionist Israeli Senator who answers directly to and takes orders from Illuminati Zionist Larry Mizel?
Yes you guessed it Keating Five John McCain.
John McCain's Wife Hiding War Profits, Untaxed Off-Shore Accounts?
The Denver Illuminati Connection and Leonard Millman one of the 12 Illuminati Wizards' of Oz:
Frauds Are US at MDC
Junk Bond Daisy Chain Frauds The Denver Illuminati Zionist Connection
The Zionist Gang Behind New York's 9-11 Conference
TV 9 News Denver asked what is under the Denver Airport
VT Was Right: Illuminati Sacrifice Busted!
Iran Contra Frauds and The Denver Illuminati Zionist Connection
"Stew you're the only person to go after the pinnacle of the Zionist Organized Crime Leonard Millman, your-ex-in-law and Larry Mizel, Millman's Buffer."—Gordon Duff May 16, 2013
Gordon Duff Interview with Stew Webb 3 Hour Marathon (Video)
"Stew you're the only person to go after the pinnacle of the Zionist Organized Crime Leonard Millman, your-ex-in-law and Larry Mizel, Millman's Buffer."—Gordon Duff Editor Veterans Today May 16, 2013
This interview of Gordon Duff and Stew Webb made in December 2012 was rated as one of 2013 most listened to interview now on Stew Webb's youtube site.
YouTube – Veterans Today –
"I have known Stew for many years and have yet to meet a more energetic Whistleblower and investigator. Stew has suffered persecutions, beatings, verbal abuse and much more for standing up for what he believes in. Stew Webb is loyal to the cause of exposing corruption, not online in America but around the world. January 30, 2014—Gene "Chip" Tatum
Super Spook Gene Chip Tatum 1996 interview with Stew Webb
THE CONTRA COKE TRAIN: The Denver Illuminati Zionist Connection
AIG Harken Energy Winn Financial Group
By Al Martin
There were about 100 such projects in all which were ultimately bailed out by some public guarantee institution.
It wasn't necessarily the FDIC or the FSLIC, but in some cases, very esoteric public guaranteed funds were used to bail these deals out.
George Jr. naturally specialized in oil since he controlled the Bush family oil portfolio including Harken Energy stock, Tidewater Development stock, and Apache and Zapata stock.
These were all deals where George Sr. had formerly been on the Board of Directors. Now George Jr. was on the Board of Directors, since Sr. as Vice President couldn't have that capacity.
Harken Energy was a classic fraud. The stock still trades on the AMEX at five or six dollars a share. It's been pumped up recently because there's a new fraud going on with those Bahrainian Leases that Richard Secord originally had ten years ago.
The stock will shortly collapse back to two dollars again, as soon as everybody gets out. A lot of Republicans will make money on the deal. ….
I also have a lot of documentation pursuant to George, Sr.'s involvement in fraudulent deals surrounding Zapata and Apache Energy. I have a lot of stuff with him in Harken Energy, also including a lot with George, Jr. in Harken Energy. That's another possibility. But again, these weren't large frauds. They were little security frauds, the fraudulent diversion of monies in those bogus Bahrainian leases when they temporarily ensconced Richard Secord to be their Middle Eastern Director for Bahrainian Operations which existed in a file drawer.
What that Bahrainian deal came down to was George Bush, Sr.'s close friend, former Saudi intelligence chief and major IranContra figure, Ghaith Pharaon. That was just a donation to IranContra by the Saudi government. And that's what the bogus Bahrainian lease deal effectively comes down to. It wasn't much $38 million, something like that.
Secord received about $3 million for his own pocket. Harry Aderholt was thrown a bone out of the deal. It was no big deal really. I also want to discuss an overview of Bush family fraud, ala IranContra profiteering. …….
The TriLateral Investment Group, Ltd. is also one of the deals (one of the very few deals, perhaps only a few dozen deals in that era by this group of guys) that you could connect Jeb, Neil, George, Jr., Prescott, and Wally Bush.
All five you can put in the TriLateral Investment Group, Ltd. You can put Neil in it visavis TriLateral's dealings with Neil's Gulf Stream Realty.
Then you back up a step and put Neil Bush into TriLateral Investment Group's dealings with the Winn Financial Group of Denver run by the infamous former Ambassador to Switzerland, Phillip Winn. You can put George, Jr. in the deal visavis the TriLateral Group Ltd.'s fraudulent relationship with American Insurance General (AIG) , of which George, Jr. was a part through the same series of fraudulent fidelity guarantee instruments issued on behalf of Harken Energy from American Insurance General.
TriLateral Investment Group then sold bogus oil and gas leases to AIG. This is a direct fraud that George, Jr. profited to the extent of (not a lot) $1.6 or $1.7 million. But it was a clear outandout fraud.
Rolling Stone magazine did a good piece on George Bush, Jr. and three of his oil and gas companies which failed. But the article really didn't go far enough.
It did not go really into Harken and Tidewater and other public corporations which George, Jr. was involved in and in which securities fraud was committed. He was able to neatly skirt the laws or should we say deflect the shit away from himself through a whole series of contrivances. The way he was able to do this, by the way, was to post these essentially bogus fidelity and guaranty instruments, so the deals wouldn't be scrutinized until long after they had collapsed.
This was one of George Jr's specialities and I did this myself, by the way. It was a common tactic in IranContra Securities Fraud. As the expression goes, it was to "back in" fraudulent assets, normally of a real estate nature, to back in fraudulent assets into a public shell. More commonly, they were known by their regulatory names in those days as a Reg D offering, or a Reg 501 or 505, or an S1, S3 or S18 offering. These were the common euphemisms used in the day. This is, of course, Security and Exchange Commission language, or "SEC speak" as we used to call it, for various types of offerings, which govern how large these offerings could be, how many states they could be 'blueskied' in (meaning how many states they could be sold in), the total amount of money that could be raised, the market making regulation that was necessary to maintain a market in the shares thereinafter.
Anyway, a very common securities fraud was the use of 144 stock. 144 stock refers to Rule 144, or Restricted Shares (shares that are not free to trade under the twoyear restriction rule). Often a company that would nominally have ten million shares outstanding could issue a hundred million shares of 144 stock that would then be sold at a steep discount to the market price. If you had a stock trading at a dollar, you would issue scads and scads of 144 stock, and you sell it for twenty cents a share. This stock would get bounced out into offshore bank loans, principally through the Union Bank of Switzerland, but also through a whole host of offshore banks through the Caribbean.
The large French bank, Banque Paribas, for instance, was notorious for this because of George, Sr.'s relationship with the bank. What would happen is you would raise an enormous amount of money, but you would also have an enormous amount of restricted stock, out of which at some point, the letter (what is known as the restriction or the letter) would come off that stock, and that stock is going to come bouncing back at some point to the market makers.
Because the scheme was at the banks, this was only meant to be interim financing. We are now talking about cooperative banks who were not meant to be burnt. They were just meant to provide bridge financing. This was very, very true with Union Bank of Switzerland, Royal Trust of Canada, and Imperial of Canada, Banque Z of Curaao, Banco de Populare. These were banks that you did not burn. These banks just acted as facilitator banks. But you have to make them "whole" in the end. Now, if you bury them under a pile of 144 stock, how did you make them "whole" in the end? How you made them "whole" is by pumping up the deal as the letter began to expire on the 144 stock that was out.
You would pump up the shares artificially in the marketplace and begin to bleed the stock back through your market makers at forty or fifty cents on the dollar. You would make money again. You had originally borrowed twenty cents on the dollar. You perhaps would bleed the stock back into the marketplace at forty cents on the dollar by the tactic of what is known as "backdooring" the stock to your market makers and dealers, and issuing certain guarantees to them that they in turn would be made whole. The ultimate bag-holder in these
deals, of course, are the people that bought the hype, the people that bought the endless press releases, most of which were all bogus.
In some cases, we would have to make the representation that Company A has a tremendous new product or that it just has a contract with the International Monetary Finance Corporation. And boy, this is just going to be the greatest since sliced bread. Of course, what the prospective hypee didn't know is that the International Monetary Corporation was in fact a shell that had been formed by the very same people who had perpetrated the original fraud. It is the only way you could keep control of the hype. So you would have one bogus company signing a contract to purchase ten zillion widgets from another bogus company. Not only did the widgets not exist, but both the companies themselves were essentially worthless. In this way, you could pump up the price of the shares and be able to create enough liquidity, enough excitement in the shares to distribute all of the stock, all of this 144 stock that you had bouncing back. Since the problem was obvious, you would vastly expand the flow to the shares in some cases, by a factor of ten.
There were previously, let's say, 10 million shares authorized, but usually there was 300,000 or 400,000 shares that were actually out. The rest of it was buried in the hands of dealers or constituted restricted stock. So what would happen is towards the end, when the deal would falter, you could always give the deal a second shot by instituting a reverse stock split, which would bring the stock back up to a level where penny stock investors and speculators felt more comfortable, and also back to a level where the shares would then again meet certain regulatory hurdles, thus making it easier to distribute the stock. You took the stock that was originally done and pumped it up to a dollar. In order to maintain it at a dollar and absorb all the stock, you needed a constant flow of hype.
When the shares eventually sank (because the distribution began to back up on the dealers a little bit), you would give the stock a secondary chance by instituting a reverse stock split. That would boost the price of the shares back up to where they were, usually even higher. Of course the spreads would widen out, and as anyone knows in reverse stock split penny deal, the spreads always get very, very wide. But you simply disguise those spreads.
The dealers can very easily disguise those spreads by either not posting Bids and Asks on the pink sheets, or just posting socalled nominal Bids and Asks which would give the appearance to the wouldbe investor that the stock was substantially more liquid than it was. But the reverse stock split was always the last link in the chain of the fraud of the underlying deal. Because the last time you would pump it up would be through this artifice, this device using a reverse stock split. It wouldn't be long thereafter that simply the deal would fall apart, and you could distribute the stock all the way back down to a penny bid, three cents offered, which we did on a lot of deals.
Once the broker/dealers were out or were "whole" financially as well as your other market makers and specialists, once you had made them whole financially, because you had so severely discounted the stock to them to begin with, then there would always be 30 million or 40 million shares left over. And the Bids and Asks would quickly go to like a penny bid, three cents offered, but with that, you would get a whole new crop of potential investors. You would keep a little bit of hype there. You'd keep a little bit of activity and spread on the sheets. And there's a whole lot of people that will buy 10,000, 20,000 shares of a two or three cent stock in hopes that it might be a twenty or thirty cent stock. You do open yourself up at a penny making a market of one and three cents you open yourself up to a whole new crop of speculators that will be sellers of a deal at twenty cents, not buyers of a deal at twenty cents.
We use to call these type of penny speculators "green feet." We used to delineate them by where the stocks traded, on what sheets, in other words. For instance, a pink sheet speculator is someone who bought higher priced penny stocks and shares that traded in the low dollars. Of course, when the stock fell down below the pink sheet regulatory level, it would be kicked down to the green sheets. That's where you find the one cent, three cent, five cent stocks. When they could no longer be maintained at that level, they would be kicked down to the yellow sheets. That's where you would sometimes see stocks trading in mils so many mils bid, so many mils offered. As long as there was still somebody willing to buy it, a market could be maintained, particularly since the stock, by this point, did not cost anything to the broker/dealers or those who initiated the fraud. Everybody was out and clean and made their money, and public shareholders were the ultimate bag holders. But you could actually keep these deals floating and alive for a long time before they absolutely fell apart.
Al Martin Iran Contra Whistleblowers Website
More to come stay tuned: The Denver Illuminati Zionist Connections
Stew Webb Federal Whistleblower-Activist of 29 years has been a guest on over 3,200 Radio and TV Programs since September 18, 1991 and was responsible for the Congressional Investigations and hearings that lead to the Appointment of Independent Prosecutor Arlin Adams for in the 1989 HUD Hearings, the Silverado Savings and Loan Hearings, the Denver International Airport Frauds hearings, the MDC Holdings, Inc. NYSE Illegal Political Campaign Money Laundering Colorado's biggest case aka Keating 5 hearings to name a few. Stew was held as a Political Prisoner from 1992-1993 to silence his exposure by Leonard Millman his former in law with illegal charges of threatening harassing telephone calls charges were dismissed with prejudice. Leonard Millman, George HW Bush, George W Bush, Jeb Bush, Neil Bush, Bill and Hillary Clinton, Larry Mizel, Phil Winn, Norman Brownstein, John McCain and Mitt Romney to name a few are all partners in what is known as the Bush-Millman-Clinton Organized Crime Syndicate. Leonard Millman is a member of the "Illuminati Council of 13″
Stew Webb Official Website
"Stew you're the only person to go after the pinnacle of the Zionist Organized Crime Leonard Millman, your-ex-in-law and Larry Mizel, Millman's Buffer."—Gordon Duff May 16, 2013 (Stew Webb USMC Honorable Discharge)
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Stew Webb
Stew Webb served in the United States Marine Corps and was Honorable Discharge.He is a General Contractor-Home Builder turned Federal Whistleblower-Activist of 30 years has been a guest on over 3,000 Radio and TV Programs since September 18, 1991 and was responsible for the Congressional Investigations and hearings that lead to the Appointment of Independent Prosecutor Arlin Adams for in the 1989 HUD Hearings, the Silverado Savings and Loan Hearings, the Denver International Airport Frauds hearings, the MDC Holdings, Inc. NYSE Illegal Political Campaign Money Laundering Colorado's biggest case aka Keating 5 hearings to name a few.
Stew was held as a Political Prisoner from 1992-1993 to silence his exposure by Leonard Millman his former in law with illegal charges of threatening harassing telephone calls charges were dismissed with prejudice. Leonard Millman, George HW Bush, George W Bush, Jeb Bush, Neil Bush, Bill and Hillary Clinton, Larry Mizel, Phil Winn, Norman Brownstein, John McCain and Mitt Romney to name a few are all partners in what is known as the Bush-Millman-Clinton Organized Crime Syndicate. Leonard Millman is a member of the "Illuminati Council of 13"
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"Stew you're the only person to go after the pinnacle of the Zionist Organized Crime Leonard Millman, your-ex-in-law and Larry Mizel, Millman's Buffer." Gordon Duff, May 16, 2013
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Hardly A Debate – Gilad Atzmon vs. Geoffrey Alderman
On Israel's Genocidal Nature
Dear "Jewish Defense League": Why aren't I on your death list?
Iraq, America and The Lobby
China Summons US Ambassador Stew Webb PressTV
Filed under Reidhead Tagged with bank, banking, bankster, cabal, cartel, collusion, Conspiracy, control, Crime, debt, Economy, FED, Federal, Federal Reserve, finance, financial, foreclosure, fraud, gangster, illuminati, loan, manipulation, money, organized, reidhead, savings, slavery, Systemic, Today, veterans
http://www.fraudstoppers.org/
Breaking News for California Homeowners Facing Foreclosure:
Your mortgage could be legally unenforceable and you could be entitled to financial compensation for mortgage fraud and other legal violations. Legal errors, contract breaches, appraisal and mortgage fraud have caused the majority of mortgage transactions to be legally unenforceable.
Did you know that a government audit revealed that 83% of the mortgages surveyed contained legal violations?
Did you know these legal violations can turn the tables on your predatory lender and make them pay you to go away?
"I cannot decide for you the moral obligations you wish to pursue; but if a wrong has been committed against you (such as a clouded title or a fraud resulting from a mortgage loan) you have the duty as an American property owner to correct it. Filing a suit (in my book) reflects one's personal responsibility". Clouded Titles, p 13.
Stop playing games and wasting time trying to work with your corrupt lender, they are not in business to help you. If you're interested in learning how to save your house from foreclosure by suing your predatory lender for mortgage fraud, get started right now by clicking here
Filed under My Foreclosure, Reidhead Tagged with bank, cabal, cartel, collusion, Conspiracy, control, controller, courage, create, creation, Credit, currency, debt, FED, Federal, fight, finance, foreclosure, fraud, inflation, manipulation, monetary, reidhead, Reserve, slavery, stand, Systemic
Unsealed court-settlement documents reveal banks stole $trillions' worth of houses
Wow! Is this the way Banks do business with Homeowners here in the United States? I'm learning more and more about the mortgage banks and the different illegal activities they engage in.
Posted by Charleston Voice
Cory Doctorow at 8:46 am Mon, Aug 12, 2013
Back in 2012, the major US banks settled a federal mortgage-fraud lawsuit for $95,000,000. The suit was filed by Lynn Szymoniak, a white-collar fraud specialist, whose own house had been fraudulently foreclosed-upon.
When the feds settled with the banks, the evidence detailing the scope of their fraud was sealed, but as of last week, those docs are unsealed, and Szymoniak is shouting them from the hills. The banks precipitated the subprime crash by "securitizing" mortgages — turning mortgages into bonds that could be sold to people looking for investment income — and the securitization process involved transferring title for homes several times over. This title-transfer has a formal legal procedure, and in the absence of that procedure, no sale had taken place. See where this is going?
The banks screwed up the title transfers. A lot. They sold bonds backed by houses they didn't own. When it came time to foreclose on those homes, they realized that they didn't actually own them, and so they committed felony after felony, forging the necessary documentation. They stole houses, by the neighborhood-load, and got away with it. The $1B settlement sounded like a big deal, back when the evidence was sealed. Now that Szymoniak's gotten it into the public eye, it's clear that $1B was a tiny slap on the wrist: the banks stole trillions of dollars' worth of houses from you and people like you, paid less than one percent in fines, and got to keep the homes.
Now that it's unsealed, Szymoniak, as the named plaintiff, can go forward and prove the case. Along with her legal team (which includes the law firm of Grant & Eisenhoffer, which has recovered more money under the False Claims Act than any firm in the country), Szymoniak can pursue discovery and go to trial against the rest of the named defendants, including HSBC, the Bank of New York Mellon, Deutsche Bank and US Bank.
The expenses of the case, previously borne by the government, now are borne by Szymoniak and her team, but the percentages of recovery funds are also higher. "I'm really glad I was part of collecting this money for the government, and I'm looking forward to going through discovery and collecting the rest of it," Szymoniak told Salon.
It's good that the case remains active, because the $95 million settlement was a pittance compared to the enormity of the crime. By the end of 2009, private mortgage-backed securities trusts held one-third of all residential mortgages in the U.S. That means that tens of millions of home mortgages worth trillions of dollars have no legitimate underlying owner that can establish the right to foreclose.
This hasn't stopped banks from foreclosing anyway with false documents, and they are often successful, a testament to the breakdown of law in the judicial system. But to this day, the resulting chaos in disentangling ownership harms homeowners trying to sell these properties, as well as those trying to purchase them. And it renders some properties impossible to sell.
To this day, banks foreclose on borrowers using fraudulent mortgage assignments, a legacy of failing to prosecute this conduct and instead letting banks pay a fine to settle it. This disappoints Szymoniak, who told Salon the owner of these loans is now essentially "whoever lies the most convincingly and whoever gets the benefit of doubt from the judge." Szymoniak used her share of the settlement to start the Housing Justice Foundation, a non-profit that attempts to raise awareness of the continuing corruption of the nation's courts and land title system.
Your mortgage documents are fake! [David Dayen/Salon]
SOURCE: BoingBoing
Filed under My Foreclosure, Reidhead Tagged with bank, cabal, cartel, charleston, Chase, collusion, Conspiracy, debt, Economy, FED, Federal, finance, financial, foreclosure, fraud, freedom, Indy, JPMorgan, Lawsuit, Mac, manipulation, Mellon, mortgage, new, one, reidhead, Reserve, slavery, Systemic, voice, west, York
US govt sues Bank of America for defrauding investors prior to housing crash
Well, when the Bank of America is sued by both the US Security Exchange Commission (SEC) and Justice Department… Please share any feelings you have about a pillar of Wall Street and US Banking being sued for "defrauding investors by massively underestimating the quality of mortgage-backed securities prior to the 2008 housing market crash and US recession."
Enjoy, Ed Reidhead
The US government has filed two lawsuits against Bank of America accusing the company of defrauding investors by massively underestimating the quality of mortgage-backed securities prior to the 2008 housing market crash and US recession.
The US Security Exchange Commission (SEC) and Justice Department each filed a lawsuit in US District Court in Charlotte, North Carolina on Tuesday, accusing Bank of America of knowingly minimizing the risk associated with $850 million worth of securities backed by residential mortgages.
"I applaud Attorney General Holder for taking this important step toward holding Bank of America accountable for packaging and selling toxic loans to investors and brokers, a key cause of the housing collapse that crashed our economy and still plagues communities to this day," New York Attorney General Eric Schneidermann said in a statement.
US President Barack Obama has promised to hold Wall Street accountable for corruption and malfeasance which took place during the housing boom. Attorney General Eric Holder said the lawsuit is "the latest step forward in the Justice Department's ongoing efforts to hold accountable those who engage in fraudulent or irresponsible conduct."
Bank of America said it was expecting the lawsuit in a corporate filing last week, according to The New York Times.
The Justice Department lawsuit claims an "unprecedented portion" of the mortgages originated with brokers unaffiliated with the bank. Then-CEO Ken Lewis, the suit says, referred to the so-called wholesale loans as "toxic waste."
Prosecutors say that while Bank of America assembled securities in 2008, employees were pressured to process as many mortgage evaluations as possible in order to maximize profits. More than 40 per cent of the mortgages did not meet the underwriting qualifications and were summarily ignored.
According to the lawsuit, one employee said her job was to "basically validate the loans" instead of reviewing them to find any potential flaws. Upon raising the issue with her superiors, she was told to "keep her opinions to herself," prosecutors said.
The two suits accuse Bank of America – which has over 260,000 employees across the world – of lying to investors and failing to disclose essential information. A Justice Department statement declared that a "material number" of mortgages "failed to materially adhere to Bank of America's underwriting standards."
Bank of America stock shares fell by one per cent after the news broke, but have increased by 97 per cent over the past year. In a statement released Tuesday, the company denied the transactions were tainted in any way.
"These were prime mortgages sold to sophisticated investors who had ample access to the underlying data, and we will demonstrate that," the statement said.
"The loans in this pool performed better than loans with similar characteristics originated and securitized at the same time by other financial institutions. We are not responsible for the housing market collapse that caused mortgage loans to default at unprecedented rates and these securities to lose value as a result."
Filed under Reidhead Tagged with 2008, action, America, backed, bank, BofA, cartel, collusion, Commission, Conspiracy, crash, defrauding, department, exchange, foreclosure, fraud, housing, investor, justice, law, legal, manipulation, Market, mortgage, recession., reidhead, SEC, Securities, States, sued, suit, systemic fraud, United, US
Paying interest on a loan that never existed?
http://worldfreemansociety.org/paying-interest-on-a-loan-that-never-existed/
Here is a gem of an article I found on understanding banking and loans and money better. Download the Dealing with Bankers PDF, and get started with taking a stand for your survival as a human being. If you know of any great resources dealing in these issues please post the link in the comments section or email me and I will post them for Everyone.
Filed under Reidhead Tagged with bank, banking, cabal, cartel, collusion, Conspiracy, control, create, creation, Credit, currency, debt, Economy, FED, Federal, Federal Reserve, finance, financial, foreclosure, fraud, freedom, gangster, inflation, monetary, money, reidhead, Settlement, slavery, systematic, Systemic, systemic fraud
New York Getting Ready to Prosecute Banks for Violations of Settlement
Good afternoon Family and Friends!
I get to share this excellent article from Livinglie's Weblog on the law suit being filed by New York State Attorney General Eric Schneiderman. This site is an excellent source of information, with a reported over 8,200,000 visitors served. Take a look around and I'm sure you will learn something.
At the end of the day everyone knows everything. If you start with the premise that the securitization of debt was a farce and that the necessary element of the false securitization of mortgage loans was the foreclosure of those loans, then you move one step closer to understanding the mortgage and foreclosure mess and a giant step forward to understanding and implementing a solution. All the actions, statements and myths promulgated by the Wall Street banks become clear, including their violation of every consent decree,order and settlement they ever made with respect to mortgage loans.
Attorney General Schneiderman of New York seems to understand this and he is taking the mega banks to task for violating a settlement that looks like pennies on the dollar. He doesn't care why they violated the $26 Billion settlement but he is taking action for their consistent violation of the settlement. But I…
Filed under Reidhead Tagged with Attorney, bank, banking, cabal, cartel, collusion, Conspiracy, control, FED, Federal Reserve, finance, financial, foreclosure, fraud, gangster, General, Lawsuit, new, reidhead, Schneiderman, Settlement, systematic, Systemic, systemic fraud, York
New York to Sue Wells Fargo and Bank of America Over Settlement Violations
I will be looking deeply into banking, mortgages, systemic fraud and foreclosure. I'm very interested in the systematic fraud practiced by Banks in their mortgage divisions. Let's see what's going on with these lenders and are they operating with integrity…
Filed under Reidhead Tagged with bank, banking, cabal, cartel, collusion, Conspiracy, control, debt, FED, Federal Reserve, finance, financial, foreclosure, fractional, fraud, inflation, monetary, reidhead, Reserve, systematic, systemic fraud
Foreclosed Homeowners Got $300, Bank's Consultants Got $2 Billion By Matt Taibbi, Rolling Stone
This blog post on Jean's blog lays out some of the systematic fraud in the Banking Industry. It is very interesting how the regulators of the Banking Industry here in the US appear to be in collaboration with the Banks they regulate. I have a feeling that this is one of the most inportant topics of our time and these activities could have their own blog. Perhaps, coming soon to a browser near you…
Filed under Reidhead Tagged with bank, banking, cartel, collusion, Conspiracy, control, creation, debt, Economy, FED, Federal, Federal Reserve, finance, financial, foreclosure, fraud, gangster, monetary, money, mortgage, regulator, reidhead, Reserve, Settlement, Systemic, systemic fraud | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,399 |
Q: typed.js start typing on scroll i have got this code:
<h1 class="font" id="typed3" style="text-align: center; font-size: 400%; color: black;">About Us</h1>
<br>
<br>
<script>
var typed2 = new Typed('#typed3', {
strings: ['About Us'],
typeSpeed: 400,
backSpeed: 0,
fadeOut: true,
});
</script>
for typed.js but I want it to start typing out when the user has scrolled to the text. is this possible?
A: I am not familiar with the typed.js library you mentioned specifically - but assuming the code you've shown works how you expect, you can run it when your <h1> comes into view by using an Intersection Observer:
Check out this fiddle to see the IntersectionObserver in action with your example: https://jsfiddle.net/Ollie1700/0q7on5d2/8/
IntersectionObserver behaviour isn't exactly trivial and on the face of it it seems like there are a lot of moving parts. But all we are doing here is the following:
*
*Initialise a new IntersectionObserver with a callback that contains information about intersections
*We instruct the IntersectionObserver to observe the element with ID "typed3". This means that when the "#typed3" element comes into the viewport, the callback for the IntersectionObserver will be triggered
*Within the callback, we are provided an array of entries (since the IntersectionObserver can observe more than one DOM element). In this case we create a loop but we could also just reference entries[0] if we wanted since we know there is only going to be one element
*The IntersectionObserver API provides us with the isIntersecting flag which is true when the observed element is visible within the viewport. So we use an if statement with this flag that triggers your original typedjs code.
*After triggering the typedjs code, we instruct the IntersectionObserver to stop observing (unobserve) our element, otherwise the code would be run every time the element comes into view. (Although this may be the behaviour you want, so experiment with omitting the unobserve call)
Let me know if this works out for you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,173 |
layout: tag
section-type: tag
title: cloud
---
## Tag
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,099 |
\section{Introduction}
\let\thefootnote\relax\footnote{
F.d.P.C. sponsored by the Department of Defense under Air
Force Contract FA8721-05-C-0002. Opinions, interpretations,
recommendations, and conclusions are those of the authors and are
not necessarily endorsed by the United States Government. Specifically,
this work was supported by Information Systems of ASD(R\&E). M.M.
and K.D. were supported in part by a Netapp faculty fellowship.}
The security of systems is often predicated on a user or application
selecting an object, a password or key, from a large list. If an
inquisitor who wishes to identify the object in order to gain access
to a system can only query each possibility, one at a time, then
the number of guesses they must make in order to identify the
selected object is likely to be large. If the object is selected
uniformly at random using, for example, a cryptographically secure
pseudo-random number generator, then the analysis of the distribution
of the number of guesses that the inquisitor must make is trivial.
Since the earliest days of code-breaking, deviations from perfect
uniformity have been exploited. For example, it has long since been
known that human-user selected passwords are highly non-uniformly
selected, e.g. \cite{malone12}, and this forms the basis of dictionary
attacks. In information theoretic security, uniformity of the string
source is typically assumed on the basis that the source has been
compressed. Recent work has cast some doubt on the appropriateness
of that assumption by establishing that fewer queries are required
to identify strings chosen from a typical set than one would
expect by a na\"ive application of the asymptotic equipartition
property. This arises by exploitation of the mild non-uniformity
of the distribution of strings conditioned to be in the typical set
\cite{Christiansen13a}.
If the string has not been selected perfectly uniformly, but with
a distribution that is known to the inquisitor, then the quantification
of security is relatively involved. Assume that a string, $W_1$,
is selected stochastically from a finite list, $ {\mathbb{A} } = \{0,\ldots,m-1\}$.
An inquisitor who knows the selection probabilities, ${P}(W_1=w)$
for all $w\in {\mathbb{A} }$, is equipped with a method to test one string
at a time and develops a strategy,
$G: {\mathbb{A} }\mapsto\{1,\ldots,m\}$,
that defines the order in which strings are guessed. As the string
is stochastically selected, the number of queries,
$G(W_1)$,
that must be made before it is identified correctly is also
a random variable, dubbed guesswork. Analysis of the distribution
of guesswork serves as a natural a measure of computational security
in brute force determination.
In a brief paper in 1994, Massey \cite{Massey94} established that
if the inquisitor orders his guesses from most likely to least
likely, then the Shannon entropy of the random variable $W_1$ bears
little relation to
the expected guesswork
$E(G(W_1))= \sum_{w\in {\mathbb{A} }} G(w) P(W_1=w)$,
the average number of guesses required to identify
$W_1$.
Arikan \cite{Arikan96} established
that if a string, $W_k$, is chosen from $ {\mathbb{A} }^k$ with i.i.d. characters,
again guessing strings from most likely to least likely, then the
moments of the guesswork distribution grow exponentially in $k$
with a rate identified in terms of the R\'enyi entropy of the
characters,
\begin{align*}
\lim_{k\to\infty} \frac 1k \log E(G(W_k)^\alpha)
&= (1+\alpha) \log \sum_{w\in {\mathbb{A} }} P(W_1=w)^{1/(1+\alpha)}\\
&= \alpha R\left(\frac{1}{1+\alpha}\right)
\text{ for } \alpha>0,
\end{align*}
where $R((1+\alpha)^{-1})$ is the R\'enyi entropy of $W_1$ with
parameter $(1+\alpha)^{-1}$.
In particular, the average guesswork grows as the R\'enyi entropy
with parameter $1/2$, a value that is
lower bounded by Shannon entropy.
Arikan's result was subsequently extended significantly beyond
i.i.d. sources \cite{Malone042,Pfister04,Hanawal11}, establishing
its robustness. In the generalized setting, specific R\'enyi entropy,
the R\'enyi entropy per character, plays the r\^ole of R\'enyi
entropy. In turn, these results have been leveraged to prove that
the guesswork process $\{k^{-1}\log G(W_k)\}$
satisfies a Large Deviation Principle (LDP),
e.g. \cite{Lewis95,Dembo98},
in broad
generality \cite{Christiansen13}.
That is, there exists a lower semi-continuous function
$I:[0,\log(m)]\mapsto[0,\infty]$ such that for all
Borel sets $B$ contained in $[0,\log(m)]$
\begin{align}
-\inf_{x\in B^\circ} I(x) &\leq
\liminf_{n\to\infty} \frac 1k \log
P\left(\frac 1k \log G(W_k) \in B\right) \nonumber\\
&\leq
\limsup_{n\to\infty} \frac 1k \log
P\left(\frac 1k \log G(W_k) \in B\right) \nonumber\\
&\leq -\inf_{x\in \bar{B}} I(x),
\label{eq:LDP}
\end{align}
where $B^\circ$ denotes the interior of $B$ and $\bar{B}$ denotes
its closure. Roughly speaking, this implies $dP(k^{-1} \log G(W_k)
\approx x)\approx \exp(-k I(x)) dx$ for large $k$. In \cite{Christiansen13}
this LDP, in turn, was used to provide direct estimates on the
guesswork probability mass function, $P(G(W_k)=n)$ for
$n\in\{1,\ldots,m^k\}$. These deductions, along with others described
in Section \ref{sec:brief}, have developed a quantitative framework
for the process of brute force guessing a single string.
In the present work we address a natural extension in this investigation
of brute force searching: the quantification for multi-user systems.
We are motivated by both classical systems, such as the brute force
entry to a multi-user computer where the inquisitor need only
compromise a single account, as well as modern distributed storage
services where coded data is kept at distinct sites in a way where,
owing to coding redundancy, several, but not all, servers need to be
compromised to access the content \cite{oliveira12,Calmon12}.
\section{Summary of contribution}
Assume that $V$ users select strings independently from $ {\mathbb{A} }^k$. An
inquisitor knows the probabilities with which each user selects
their string, is able to query the correctness of
each
(user, string)
pair, and wishes to identify any
subset of size
$U$ of the $V$ strings. The first
question that must be addressed is what is the optimal strategy, the
ordering in which (user, string) pairs are guessed, for
the inquisitor. For the single user system, since the earliest
investigations \cite{Massey94,Arikan96,Merhav99,Pliam00} it has
been clear that the strategy of ordering guesses from
the
most to least likely string, breaking ties arbitrarily, is optimal
in any reasonable sense. Here we shall give optimality a specific
meaning: that the distribution of the number of guesses required
to identify the unknown object is stochastically dominated by all
other strategies. Amongst other results, for the multi-user guesswork
problem we establish the following:
\begin{itemize}
\item If $U<V$, the existence of optimal guessing strategies,
those that are stochastically dominated by all other strategies, is
no longer assured.
\item By construction, there exist asymptotically optimal strategies
as the strings become long.
\item For asymptotically optimal strategies, we prove a large
deviation principle for their guesswork. The resulting large deviations
rate function is, in general, not convex and so this result could
not have been established by determining how the moment generating
function of the multi-user guesswork distribution scales in
string-length.
\item
The non-convexity of the rate function shows that, if users' string
statistics are distinct, there may be no fixed ordering of weakness
amongst users. That is, depending on how many guesses are made
before the $U$ users' strings are identified, the collection of
users whose strings have been identified are likely to be
distinct.
\item If all $V$ strings are chosen with the same statistics, then
the rate function is convex and the exponential growth rate of the
average guesswork as string-length increases is the specific R\'enyi
entropy of the string source with parameter
\begin{align*}
\frac{V-U+1}{V-U+2}
\in\left\{\frac 12,\frac 23,\frac 34,\frac4 5,\frac 56,\ldots\right\}.
\end{align*}
\item For homogeneous users, from an inquisitor's point of view,
there is a law of diminishing returns for the expected guesswork
growth rate in excess number of users ($V-U$).
\item For homogeneous users, from a designer's point of view, coming full
circle to Massey's original observation that Shannon entropy has
little quantitative relationship to how hard it is to guess a single
string, the specific Shannon entropy of the source is a lower bound
on the average guesswork growth rate for all $V$ and $U$.
\end{itemize}
These results generalize both the original guesswork studies, where
$U=V=1$, as well as some of the results in \cite{Merhav99,Hanawal11a}
where, as a wiretap model, the case $U=1$ and $V=2$ with one of the
strings selected uniformly, is considered and scaling properties
of the guesswork moments are established. Interestingly, we shall
show that that setting is one where the LDP rate function is typically
non-convex, so while results regarding the asymptotic behavior of
the guesswork moments can be deduced from the LDP, the reverse is
not true. To circumvent the lack of convexity, we prove the main
result using the contraction principle, Theorem 4.2.1 \cite{Dembo98},
and the LDP established in \cite{Christiansen13}, which itself
relies on earlier results of work referenced above.
\begin{figure}
\includegraphics[scale=0.46]{u_of_v_ex0}
\includegraphics[scale=0.46]{u_of_v_ex1}
\caption{Strings created from i.i.d. letters are selected from a
binary alphabet with probability $p$ for one character. Given an
inquisitor wishes to identify $U$ of $V$ strings, the left panel
shows the average exponential guesswork growth rate as a function
of $V-U$, the excess number of guessable strings; the right panel
shows the theoretically predicted approximate average guesswork for
$168$ bit strings, as used in triple DES, as a function of $V-U$,
the excess number of guessable strings.
}
\label{fig:bernoulli_guess}
\end{figure}
\section{
The impact of the number of users on expected guesswork growth rate,
an example
}
\label{sec:firstexample}
As an exemplar that illustrates the reduction in security that comes
from having multiple users, the left panel in Figure
\ref{fig:bernoulli_guess} the average guesswork growth rate for an
asymptotically optimal strategy is plotted for the simplest case,
a binary alphabet with $V$ i.i.d. Bernoulli string sources. In
order to be satisfied, the inquisitor wishes to identify $U\leq V$
of the strings. The x-axis shows the excess number of guessable
strings, $V-U$, and the y-axis is the $\log_2$ growth rate of the
expected guesswork in string length. If the source is perfectly
uniform (i.e. characters are chosen with a Bernoulli $1/2$ process),
then the average guesswork growth rate is maximal and unchanging
in $V-U$. If the source is not perfectly uniform, then the growth
rate decreases as the number of excess guessable strings $V-U$
increases, with a lower bound of the source's Shannon entropy.
For a string of length $168$ bits, as used in the triple DES cipher,
and a Bernoulli $(0.25)$ source, the right panel in Figure
\ref{fig:bernoulli_guess} displays the impact that the change in
this exponent has, approximately, on the average number of guesses
required to determine $U$ strings. More refined results for a
broader class of processes can be found in later sections, including
an estimate on the guesswork distribution.
The rest of this paper is organized as follows. In Section
\ref{sec:brief}, we begin with a brief overview of results on
guesswork that we have not touched on so far. Questions of optimal
strategy are considered in Section \ref{sec:strategy}. Asymptotically
optimal strategies are established to exist in Section \ref{sec:asymptote}
and results for these strategies appear in Section \ref{sec:results}.
In Section \ref{sec:mismatch} we present examples where strings
sources have distinct statistics. In Section \ref{sec:ident} we
return to the setting where string sources have identical statistics.
Concluding remarks appear in Section \ref{sec:conc}.
\section{A brief overview of guesswork}
\label{sec:brief}
Since Arikan's introduction of the long string length asymptotic,
several generalizations of its fundamental assumptions have been
explored. Arikan and Boztas \cite{Arikan02} investigate the setting
where the truthfulness in response to a query is not certain. Arikan
and Merhav \cite{Arikan98} loosen the assumption that inquisitor
needs to determine the string exactly, assuming instead that they
only need to identify it within a given distance. That the inquisitor
knows the distribution of words exactly is relaxed by Sundaresan
\cite{Sundaresan07b}, \cite{Sundaresan06}
and by the authors of \cite{Beirami15}.
Motivated by a wiretap application, the problem of multiple users
was first investigated by Merhav and Arikan \cite{Merhav99} in the
$V=2$ and $U=1$ setting, assuming one of the users selects
their string uniformly on a reduced alphabet. In \cite{Hayashi06}
Hayashi and Yamamoto extend the results in \cite{Merhav99} to the
case if there is an additional i.i.d. source correlated to the
first, used for coding purposes, while Harountunian and Ghazaryan
\cite{Haroutunian00} extend the results in \cite{Merhav99} to the
setting of \cite{Arikan98}. Harountunian and Margaryan
\cite{Haroutunian12} expand on \cite{Merhav99} by adding noise to
the original string, altering the distribution of letters. Hanawal
and Sundaresan \cite{Hanawal11a} extend the bounds in \cite{Merhav99}
to a pre-limit and to more general sources, showing that they are
tight for Markovian and unifilar sources.
Sundaresan \cite{Sundaresan07a} uses length functions to identify
the link between guesswork and compression. This result is extended
by Hanawal and Sundaresan \cite{Hanawal11b} to relate guesswork to
the compression of a source over a countably infinite alphabet. In
\cite{Christiansen13a} the authors prove that, if the string is
conditioned on being an element of a typical set the expected
guesswork, is growing
more slowly
than a simple uniform approximation
would suggest. In \cite{Christiansen13b} the authors consider the
impact of guessing over a noisy erasure channel showing that the
mean noise on the channel is not the significant moment in determining
the expected guesswork, but instead one determined by its R\'enyi
entropy with parameter $1/2$.
Finally, we mention that recent work by Bunte and Lapidoth
\cite{Bunte14} identifies a distinct operational meaning for R\'enyi
entropy in defining a rate region for a scheduling problem.
\section{Optimal strategies}
\label{sec:strategy}
In order to introduce the key concepts used to determine the optimal
multi-user guesswork strategy, we first reconsider the optimal guesswork
strategy in the single user case, i.e.
$U=V=1$.
Recall that $ {\mathbb{A} }=\{0,\ldots,m-1\}$ is a finite set.
\begin{definition}
A single user strategy, $S: {\mathbb{A} }^k\mapsto\{1,\ldots,m^k\}$, is a
one-to-one map that determines the order in which guesses are made.
That is, for a given strategy $S$ and a given string $w\in {\mathbb{A} }^k$,
$S(w)$ is the number of guesses made until $w$ is queried.
\end{definition}
Let $W_k$ be a random variable taking values in $ {\mathbb{A} }^k$. Assume that
its probability mass function, $P(W_k=w)$ for all $w\in {\mathbb{A} }^k$, is
known. Since the first results on the topic it has been clear that
the best strategy, which we denote $G$, is to guess from most likely
to least likely, breaking ties arbitrarily. In particular, $G$ is
defined by $G(w)<G(w')$ if $P(W_k=w)>P(W_k=w')$. We
begin by assigning optimality a precise meaning in terms of stochastic
dominance \cite{Lehmann55,denuit06}.
\begin{definition}
A strategy $S$ is optimal for $W_k$ if the random variable $S(W_k)$
is stochastically dominated by $S'(W_k)$, for all strategies $S'$.
That is, if $P(S(W_k)\leq n) \geq P(S'(W_k)\leq n)$ for all strategies
$S'$ and all $n\in\{1,\ldots,m^k\}$.
\end{definition}
This definition captures the stochastic aspect of guessing by stating
than an optimal strategy is one where the identification stopping
time is probabilistically smallest. One consequence of this definition
that explains its appropriateness is that for any monotone function
$\phi:\{1,\ldots,m^k\}\to {\mathbb{R} }$, it is the case that $E(\phi(S(W_k)))\leq
E(\phi(S'(W_k)))$ for an optimal $S$ and any other $S'$ (e.g.
Proposition 3.3.17, \cite{denuit06}). Thus $S(W_k)$ has the least
moments over all guessing strategies.
That guessing from most- to least-likely in the single user case is
optimal is readily established.
\begin{lemma}
\label{lem:1opt}
If $V=U=1$, the optimal strategies are those that guess from most
likely to least likely, breaking ties arbitrarily.
\end{lemma}
\begin{proof}
Consider the strategy $G$ defined above and any other strategy $S$.
By construction, for any $n\in\{1,\ldots,m^k\}$
\begin{align*}
P(G(W_k)\leq n) &= \sum_{i=1}^n P(G(W_k)=i) \\
& = \max_{w_1,\ldots,w_n} \left(\sum_{i=1}^n P(W_k=w_i)\right)\\
&\geq \sum_{i=1}^n P(S(W_k)=i) = P(S(W_k)\leq n).
\end{align*}
\end{proof}
In the multi-user case, where (user, string) pairs are queried, a
strategy is defined by the following.
\begin{definition}
A multi-user strategy is a one-to-one map
$S:\{1,\ldots,V\}\times {\mathbb{A} }^k\mapsto\{1,\ldots,Vm^k\}$ that orders
the guesses of (user, string) pairs.
\end{definition}
The expression for the number of guesses required to identify $U$
strings is a little involved as we must take into account that we
stop making queries about a user once
their string has been identified.
For a given strategy $S$, let
$ {N_S} :\{1,\ldots,V\}\times\{1,\ldots,Vm^k\}\mapsto\{1,\ldots,m^k\}$ be
defined by
\begin{align*}
{N_S} (v,n) = |\{w\in {\mathbb{A} }^k:S(v,w)\leq n\}|,
\end{align*}
which computes the number of queries in the strategy up to $n$
that correspond to user $v$.
The number of queries that need to be made if $U$ strings are
to be identified is
\begin{align*}
{ T(U,V,\vw)} = {\text{U-min} } \left(S(1,w^{(1)}),\ldots,S(V,w^{(V)})\right),
\end{align*}
where $ {\text{U-min} } : {\mathbb{R} }^V\to {\mathbb{R} }$ and $ {\text{U-min} } ( {\vec{x}} )$ gives the $U^{\rm th}$
smallest component of $ {\vec{x}} $.
The
number of guesses required to identify $U$ components of $ {\vec{w}}
=(w^{(1)},\ldots,w^{(V)})$ is then
\begin{align}
\label{eq:GS}
{G_S} (U,V, {\vec{w}} )=
\sum_{v=1}^V {N_S} \left(v,
\min\left(S(v,w^{(v)}),{ T(U,V,\vw)} \right) \right).
\end{align}
This apparently unwieldy object counts the number of queries made
to each user, curtailed either when
their
string is identified or
when $U$ strings of other users are identified.
If $U=V$, equation \eqref{eq:GS} simplifies significantly, as
$S(v,w^{(v)})\leq{ T(U,V,\vw)} $ for all
$v\in\{1,\ldots,V\}$, becoming
\begin{align}
\label{eq:U=V}
{G_S} (V,V, {\vec{w}} )=
\sum_{v=1}^V {N_S} \left(v, S(v,w^{(v)}) \right),
\end{align}
the sum of the number of queries required to identify each individual
word. In this case, we have the analogous result to Lemma \ref{lem:1opt},
which is again readily established.
\begin{lemma}
\label{lem:UVopt}
If $V=U$, the optimal strategies are those that employ individual
optimal strategies, but with users selected in any order.
\end{lemma}
\begin{proof}
For any multi-user strategy $S$, equation \eqref{eq:U=V} holds.
Consider an element in the sum on the right hand side,
$ {N_S} \left(v, S(v,w^{(v)}) \right)$.
It can be recognized to be the number of queries made to user $v$
until their string is identified. By Lemma \ref{lem:1opt}, for each
user $v$, for any $S$ this stochastically dominates the equivalent
single user optimal strategy. Thus the multi-user optimal strategies
in this case are the sum of individual user optimal strategies,
with users queried in any arbitrary order.
\end{proof}
The formula \eqref{eq:GS} will be largely side-stepped when we
consider asymptotically optimal strategies, but is needed to establish
that there is, in general, no stochastically dominant strategy if
$V>U$. With $ {\vec{W}} _k=(W_k^{(1)},\ldots,W_k^{(V)})$ being a random vector
taking values in $ {\mathbb{A} }^{kV}$ with independent, not necessarily identically
distributed, components, we are not guaranteed the existence of an
$S$ such that $P( {G_S} (U,V, {\vec{W}} _k)\leq n) \geq ( {G_{S'}} (U,V, {\vec{W}} _k)\leq n)$
for all alternate strategies $S'$.
\begin{lemma}
If $V>U$, a stochastically dominant strategy does not necessarily exist.
\end{lemma}
\begin{proof}
A counter-example suffices and so let $k=1$, $V=2$, $U=1$ and
$ {\mathbb{A} }=\{0,1,2\}$. Let the distributions of $W_1^{(1)}$ and $W_1^{(2)}$
be
\begin{center}
\begin{tabular}{|c|c|}
\hline
User 1 & User 2\\
\hline
$P(W_1^{(1)}=0)=0.6$ & $P(W_1^{(2)}=0)=0.5$ \\
$P(W_1^{(1)}=1)=0.25$ & $P(W_1^{(2)}=1)=0.4$ \\
$P(W_1^{(1)}=2)=0.15$ & $P(W_1^{(2)}=2)=0.1$\\
\hline
\end{tabular}
\end{center}
If a stochastically dominant strategy exists, its first guess must
be user $1$, string $0$, i.e. $S(1,0)=1$, so that $P( {G_S} (1, {\vec{W}} _1)=1) =
0.6$. Given this first guess, to maximize $P( {G_S} (1, {\vec{W}} _1)\leq2)$, the
second guess must be user $1$, string $1$, $S(1,1)=2$, so that
$P( {G_S} (1, {\vec{W}} _1)\leq 2) = 0.85$.
An alternate strategy with $S(2,0)=1$ and $S(2,1)=2$, however,
gives $P( {G_{S'}} (1, {\vec{W}} _1)=1)=0.5$ and $P( {G_{S'}} (1, {\vec{W}} _1)\leq 2)=0.9$.
While $P( {G_S} (1, {\vec{W}} _1)=1)>P( {G_{S'}} (1, {\vec{W}} _1)=1)$,
$P( {G_S} (1, {\vec{W}} _1)\leq2)<P( {G_{S'}} (1, {\vec{W}} _1)\leq2)$ and so there is
no strategy stochastically dominated by all others in this case.
\end{proof}
Despite this lack of universal optimal strategy, we shall show that
there is a sequence of random variables that are stochastically
dominated by the guesswork of all strategies and, moreover, there
exists a strategy with identical performance in Arikan's long string
length asymptotic.
\begin{definition}
A strategy $S$ is asymptotically optimal if $\{k^{-1}\log
{G_S} (U,V, {\vec{W}} _k)\}$ satisfies a LDP
with the same rate function as a sequence
$\{k^{-1}\log \Upsilon(U,V, {\vec{W}} _k)\}$ where $\Upsilon(U,V, {\vec{W}} _k)$
is stochastically dominated by $ {G_{S'}} (U,V, {\vec{W}} _k)$ for all
strategies $S'$.
\end{definition}
Note that $\Upsilon(U,V,\cdot)$ need not correspond to the guesswork
of a strategy.
\section{An asymptotically optimal strategy}
\label{sec:asymptote}
Let $\{ {\vec{W}} _k\}$ be a sequence of random strings, with $ {\vec{W}} _k$ taking values in
$ {\mathbb{A} }^{kV}$, with independent components, $W_k^{(v)}$, corresponding
to strings selected by users $1$ through $V$, although each user's
string
may
not be constructed from i.i.d. letters.
For each individual
user, $v\in\{1,\ldots,V\}$, let $ {G^{(v)}} $ denote its single-user optimal
guessing strategy; that is, guessing from most likely to least likely.
We shall
show that the following random variable, constructed
using the $ {G^{(v)}} $, is stochastically
dominated by the guesswork distribution of all strategies:
\begin{align}
{G_{\text{opt}} } (U,V, {\vec{W}} _k) =
{\text{U-min} } \left(G^{(1)}(W^{(1)}_k),\ldots,G^{(V)}(W^{(V)}_k)\right).
\label{eq:LB}
\end{align}
This can be thought of as allowing the inquisitor to query, for
each $n$ in turn, the $n^{\rm th}$ most likely string for all users
while only accounting for a single guess and so it does not
correspond to an allowable strategy.
\begin{lemma}
For any strategy $S$ and any $U\in\{1,\ldots,V\}$,
$ {G_{\text{opt}} } (U,V, {\vec{W}} _k)$ is stochastically dominated by $ {G_S} (U,V, {\vec{W}} _k)$.
That is, for any any $U\in\{1,\ldots,V\}$
and any
$n\in\{1,\ldots,m^k\}$
\begin{align*}
P( {G_{\text{opt}} } (U,V, {\vec{W}} _k)\leq n) \geq P( {G_S} (U,V, {\vec{W}} _k)\leq n).
\end{align*}
\end{lemma}
\begin{proof}
Using equation \eqref{eq:GS} and the positivity of its
summands, for any strategy $S$
\begin{align*}
&G_S(U, V, {\vec{w}} )\\
&\ge {\text{U-min} } ( {N_S} (1, S(1, w^{(1)})),\ldots, {N_S} (V, S(V, w^{(V)}))).
\end{align*}
As for each $v\in\{1,\ldots,V\}$, $G^{(v)}(W_k^{(v)})$ is stochastically
dominated by all other strategies,
\begin{align*}
P(G^{(v)}(W_k^{(v)})\le n)\ge P( {N_S} (v, S(1, W_k^{(v)}))\le n).
\end{align*}
Using equation \eqref{eq:LB}, this implies that
\begin{align*}
&P( {G_{\text{opt}} } ( {\vec{W}} _k)\le n)\\
&\ge P( {\text{U-min} } ( {N_S} (1, S(1, W_k^{(1)})),\ldots, {N_S} (V, S(V, W_k^{(V)})))\le n)\\
&\ge P(G_S(U, V, {\vec{W}} _k)\le n),
\end{align*}
as required.
\end{proof}
The strategy that we construct that will asymptotically meet the
performance of the lower bound is to round-robin the single user
optimal strategies. That is, to query the most likely string of one
user followed by the most likely string of a second user and so
forth, for each user in a round-robin fashion, before moving to the
second most likely string of each user. An upper bound on this
strategy's performance is to consider only stopping at the end of
a round of such queries, even if they reveal more than $U$ strings,
which gives
\begin{align}
V {G_{\text{opt}} } (U,V, {\vec{W}} _k),
\label{eq:UB}
\end{align}
where $ {G_{\text{opt}} } (U,V, {\vec{W}} _k)$ is defined in \eqref{eq:LB}.
In large deviations parlance the stochastic processes
$\{k^{-1}\log {G_{\text{opt}} } (U,V, {\vec{W}} _k)\}$ and $\{k^{-1}\log(V {G_{\text{opt}} } (U,V, {\vec{W}} _k))\}$
arising from equations \eqref{eq:LB} and \eqref{eq:UB} are exponentially
equivalent, e.g. Section 4.2.2 \cite{Dembo98}, as $\lim_{k\to\infty}
k^{-1}\log V=0$. As a result, if one process satisfies the LDP
with a rate function that has compact level sets, then the other
does \cite{Dembo98}[Theorem 4.2.3]. Thus if
$\{k^{-1}\log {G_{\text{opt}} } (U,V, {\vec{W}} _k)\}$ can be shown to satisfy a LDP,
then the round-robin strategy is proved to be asymptotically optimal.
\section{Asymptotic performance of optimal strategies}
\label{sec:results}
We first recall what is known for the single-user setting.
For each individual user $v\in\{1,\ldots,V\}$, the
specific R\'enyi entropy of the sequence $\{ W^{(v)} _k\}$,
should it exist, is defined by
\begin{align*}
{R^{(v)}} (\beta):=
\lim_{k\to\infty} \frac 1k \frac{1}{1-\beta}
\log \sum_{w_k\in {\mathbb{A} }^k} P( W^{(v)} _k=w_k)^\beta
\end{align*}
for $\beta\in(0,1)\cup(1,\infty)$, and for $\beta=1$,
\begin{align*}
{R^{(v)}} (1)&:=\lim_{\beta\uparrow1} {R^{(v)}} (\beta)\\
&= -\lim_{k\to\infty} \frac1k \sum_{w_k\in {\mathbb{A} }^k} P( W^{(v)} _k=w_k)\log P( W^{(v)} _k=w_k),
\end{align*}
the specific Shannon entropy. Should $ {R^{(v)}} (\beta)$ exist
for $\beta\in(0,\infty)$, then the specific min-entropy
is defined
\begin{align*}
{R^{(v)}} (\infty)&=\lim_{\beta\to\infty} {R^{(v)}} (\beta) \\
&= -\lim_{k\to\infty}\frac1k \max_{w_k\in {\mathbb{A} }^k} \log P( W^{(v)} _k=w_k),
\end{align*}
where the limit necessarily exists.
The existence of $ {R^{(v)}} (\beta)$ for all $\beta>0$ and its relationship
to the scaled Cumulant Generating Function
(sCGF)
\begin{align}
{\Lambda_G^{(v)}} (\alpha)
&= \lim_{k\to\infty} \frac 1k \log E(\exp(\alpha\log {G^{(v)}} ( W^{(v)} _k))) \nonumber\\
&=
\begin{cases}
\displaystyle
\alpha {R^{(v)}} \left(\frac{1}{1+\alpha}\right) & \text{ if } \alpha>-1\\
- {R^{(v)}} (\infty) & \text{ if } \alpha\leq-1
\end{cases}
\label{eq:sCGF}
\end{align}
has been established for the single user case for a broad class of
character sources that encompasses i.i.d., Markovian and general
sofic shifts that admit an entropy condition
\cite{Arikan96,Malone042,Pfister04,Hanawal11,Christiansen13}.
If, in addition, $ {R^{(v)}} (\beta)$ is differentiable
with respect to $\beta$
and has a continuous
derivative, it is established in \cite{Christiansen13} that the
process $\{k^{-1}\log {G^{(v)}} ( W^{(v)} _k)\}$ satisfies a LDP,
i.e. equation \eqref{eq:LDP},
with a convex rate function
\begin{align}
\label{eq:rf}
{\Lambda_G^{(v)}} ^*(x) = \sup_{\alpha\in {\mathbb{R} }}\left(x\alpha- {\Lambda_G^{(v)}} (\alpha)\right).
\end{align}
In \cite{Christiansen13}, this LDP is used to deduce an
approximation to the guesswork distribution,
\begin{align}
\label{eq:approx1}
P( {G^{(v)}} ( W^{(v)} _k)=n) \approx
\frac 1n \exp\left(-k {\Lambda_G^{(v)}} ^*\left(\frac 1k\log n\right)\right)
\end{align}
for large $k$ and $n\in\{1,\ldots,m^k\}$.
The following theorem establishes the fundamental analogues of these
results for an asymptotically optimal strategy, where
user strings may have distinct statistical properties.
\begin{theorem}
\label{thm:main}
Assume that the components of $\{ {\vec{W}} _k\}$ are independent and
that for each $v\in\{1,\ldots,V\}$ $ {R^{(v)}} (\beta)$ exists for
all $\beta>0$, is differentiable and has a continuous derivative,
and that equation \eqref{eq:sCGF} holds. Then the process
$\{k^{-1}\log {G_{\text{opt}} } (U,V, {\vec{W}} _k)\}$, and thus any asymptotically optimal
strategy, satisfies a Large Deviation Principle.
Defining
\begin{align*}
& {\delta^{(v)}} (x)=\begin{cases}
{\Lambda_G^{(v)}} ^*(x) & \text{ if }x\le {R^{(v)}} (1)\\
0 & \text{ otherwise}
\end{cases}\\
&\text{ and } {\gamma^{(v)}} (x)=\begin{cases}
{\Lambda_G^{(v)}} ^*(x) &\text{ if }x\ge {R^{(v)}} (1)\\
0 & \text{ otherwise}
\end{cases},
\end{align*}
the rate function is
\begin{align}
&\IGopt(U,V,x)=\nonumber\\
&\max_{v_1,\ldots, v_V}
\left( {\Lambda_G^{(v_1)}} ^*(x)+\sum_{i=2}^U \delta^{(v_i)}(x)
+\sum_{i=U+1}^V {\gamma^{(v_i)}} (x)\right),
\label{eq:Iopt}
\end{align}
which is lower semi-continuous and has compact level sets, but
may not be convex. The sCGF capturing how the moments scale is
\begin{align}
\LambdaGopt(U,V,\alpha)
&= \lim_{k\to\infty} \frac 1k \log E(\exp(\alpha\log {G_{\text{opt}} } (U,V, {\vec{W}} _k)))\nonumber\\
&= \sup_{x\in[0,\log(m)]}\left(\alpha x -\IGopt(U,V,x)\right).
\label{eq:scgf}
\end{align}
\end{theorem}
\begin{proof}
Under the assumptions of the theorem,
for each $v\in\{1,\ldots,V\}$, $\{k^{-1}
\log {G^{(v)}} ( W^{(v)} _k) \}$ satisfies the LDP with the rate function
given in equation \eqref{eq:rf}. As users' strings are selected
independently, the sequence of vectors
\begin{align*}
\left\{\left(\frac 1k \log G^{(1)}(W_k^{(1)}), \ldots,
\frac 1k \log G^{(V)}(W_k^{(V)}) \right)\right\}
\end{align*}
satisfies the LDP in $ {\mathbb{R} }^V$ with rate function $I(y^{(1)},\ldots,y^{(V)})
= \sum_{v=1}^V {\Lambda_G^{(v)}} ^*(y^{(v)})$, the sum of the rate functions
given in equation \eqref{eq:rf}.
Within our setting, the contraction principle, e.g. Theorem 4.2.1
\cite{Dembo98}, states that if a sequence of random variables
$\{X_n\}$ taking values in a compact subset of $ {\mathbb{R} }^V$ satisfies a
LDP with rate function $I: {\mathbb{R} }^V\mapsto[0,\infty]$ and $f: {\mathbb{R} }^V\mapsto {\mathbb{R} }$
is a continuous function, then the sequence $\{f(X_n)\}$ satisfies
the LDP with rate function $\inf_{ {\vec{y}} }\{I( {\vec{y}} ):f( {\vec{y}} )=x\}$.
Assume, without loss of generality, that $ {\vec{x}} \in {\mathbb{R} }^V$ is such that
$x^{(1)}<x^{(2)}<\cdots<x^{(V)}$, so that $ {\text{U-min} } ( {\vec{x}} ) = x^{(U)}$,
and let
$ {\vec{x}} _n=(x^{(1)}_n,\ldots,x^{(V)}_n)\to {\vec{x}} $. Let
$\epsilon < \inf\{x^{(v)}-x^{(v-1)}:v\in\{2,\ldots,V\}\}$. There
exists $N_\epsilon$ such that $\max_{v=1,\ldots,V}
|x^{(v)}_n-x^{(v)}|<\epsilon$ for all $n>N_\epsilon$. Thus
for all $v\in\{2,\ldots,V\}$ and all $n>N_\epsilon$
$x^{(v)}_n-x^{(v-1)}_n > x^{(v)}-x^{(v-1)}-\epsilon>0$
and so
$| {\text{U-min} } ( {\vec{x}} _n)- {\text{U-min} } ( {\vec{x}} )|=|x_n^{(U)}-x^{(U)}|<\epsilon$.
Hence $ {\text{U-min} } : {\mathbb{R} }^V\to {\mathbb{R} }$ is a continuous function
and that a LDP
holds follows from an application of the contraction principle,
giving the rate function
\begin{align*}
\IGopt(U,V,x) = \inf\left\{
\sum_{v=1}^V {\Lambda_G^{(v)}} ^*(y_v): {\text{U-min} } (y_1,\ldots,y_V)=x
\right\}.
\end{align*}
This expression simplifies to that in equation \eqref{eq:Iopt} by
elementary arguments.
The sCGF result follows from an
application of Varahadan's Lemma, e.g \cite[Theorem 4.3.1]{Dembo98}.
\end{proof}
The expression for the rate function in equation \eqref{eq:Iopt}
lends itself to a useful interpretation. In the long string-length
asymptotic, the likelihood that an inquisitor has identified $U$
of the $V$ users' strings after approximately $\exp(kx)$ queries
is contributed to by three distinct groups of identifiable users.
For given $x$, the argument in the first term $(v_1)$ identifies
the last of the $U$ users whose string is identified. The second
summed term is contributed to by the collection of users, $(v_2)$
to $(v_U)$, whose strings have already been identified prior to
$\exp(kx)$ queries, while the final summed term corresponds to those
users, $(v_{U+1})$ to $(v_V)$, whose strings have not been identified.
The reason for using the notation $\IGopt(U,V,\cdot)$ in lieu of
$\LambdaGopt^*(U,V,\cdot)$ for the rate function in Theorem
\ref{thm:main} is that $\IGopt(U,V,\cdot)$ is not convex in general,
which we shall
demonstrate by example, and so is not always the
Legendre-Fenchel transform of the sCGF $\LambdaGopt(U,V,\cdot)$.
Instead
\begin{align*}
\LambdaGopt^*(U,V,x) =
\sup_\alpha\left(\alpha x -\LambdaGopt(U,V,\alpha)\right)
\end{align*}
forms the convex hull of $\IGopt(U,V,\cdot)$. In particular, this
means that we could not have proved Theorem \ref{thm:main} by
establishing properties of $\LambdaGopt(U,V,\cdot)$ alone, which
was the successful route taken for the $U=V=1$ setting, and instead
needed to rely on the LDP proved in \cite{Christiansen13}.
Indeed, in the setting considered in \cite{Merhav99, Hanawal11a}
with $U=1$, $V=2$, with one of the strings chosen uniformly, while
the authors directly identify $\LambdaGopt(1,2,\alpha)$ for $\alpha>0$,
one cannot establish a full LDP from this approach as the resulting
rate function is not convex.
Convexity
of the rate function defined in equation \eqref{eq:Iopt}
is ensured, however, if all
users select
strings
using the same stochastic properties, whereupon the
results in Theorem \ref{thm:main} simplify greatly.
\begin{corollary}
\label{cor:same}
If, in addition to the assumptions of Theorem \ref{thm:main},
$ {\Lambda_G^{(v)}} (\cdot)= {\Lambda_G} (\cdot)$ for all $v\in\{1,\ldots,V\}$ with
corresponding R\'enyi entropy $R(\cdot)$, then the rate function
in equation \eqref{eq:rf} simplifies to the convex function
\begin{align}
\LambdaGopt^*(U,V,x) &=
\begin{cases}
\displaystyle
U {\Lambda_G} ^*(x) & \text{ if } x\leq R(1)\\
\displaystyle
(V-U+1) {\Lambda_G} ^*(x) & \text{ if } x\geq R(1)
\end{cases}
\label{eq:rf2}
\end{align}
where $R(1)$ is the specific Shannon entropy, and the sCGF in equation
\eqref{eq:scgf} simplifies to
\begin{align}
\label{eq:scgf2}
\LambdaGopt(U,V,\alpha) &=
\begin{cases}
\displaystyle
U {\Lambda_G} \left(\frac{\alpha}{U}\right) & \text{ if } \alpha\leq0\\
\displaystyle
(V-U+1) {\Lambda_G} \left(\frac{\alpha}{V-U+1}\right) & \text{ if } \alpha\geq0.
\end{cases}
\end{align}
In particular, with $\alpha=1$ we have
\begin{align}
&\lim_{k\to\infty} \frac 1k \log E\left( {G_{\text{opt}} } (U,V, {\vec{W}} _k)\right)\nonumber\\
&= \LambdaGopt(1)\nonumber\\
&= (V-U+1) {\Lambda_G} \left(\frac{1}{V-U+1}\right)\nonumber\\
&= R\left(\frac{V-U+1}{V-U+2}\right),
\label{eq:average}
\end{align}
where $R((n+1)/(n+2))-R((n+2)/(n+3))$ is a decreasing
function of $n\in {\mathbb{N} }$.
\end{corollary}
\begin{proof}
The simplification in equation \eqref{eq:rf2} follows readily from
equation \eqref{eq:Iopt}. To establish that
$R((n+1)/(n+2))-R((n+2)/(n+3))$ is a decreasing function of $n\in {\mathbb{N} }$,
it suffices to establish that $R((x+1)/(x+2))$ is a convex, decreasing
function for $x\in {\mathbb{R} }_+$.
That $R(x)\downarrow R(1)$ as $x\uparrow1$ is a general property
of specific R\'enyi entropy. For convexity, using equation
\eqref{eq:average} it suffices to
show that $x {\Lambda_G} (1/x)$ is convex for $x>0$. This can be seen
by noting that for any $a\in(0,1)$ and $x_1,x_2>0$,
\begin{align*}
&(a x_1+(1-a)x_2)
{\Lambda_G} \left(\frac{1}{a x_1+(1-a)x_2}\right)\\
&= (a x_1+(1-a)x_2)
{\Lambda_G} \left(\eta \frac{1}{x_1}+(1-\eta)\frac{1}{x_2}\right)\\
&\leq a x_1 {\Lambda_G} \left(\frac{1}{x_1}\right)
+(1-a)x_2 {\Lambda_G} \left(\frac{1}{x_2}\right),
\end{align*}
where $\eta = a x_1/(a x_1+(1-a)x_2)\in(0,1)$ and we have used the
convexity of $ {\Lambda_G} $.
\end{proof}
As the growth rate, $R((n+1)/(n+2))-R((n+2)/(n+3))$, is decreasing
there is a law of diminishing returns for the inquisitor where the
greatest decrease in the average guesswork growth rate is through
the provision of one additional user.
From the system designer's point of view, the specific Shannon
entropy of the source is a universal lower bound on the exponential
growth rate of the expected guesswork that, while we cannot take
the limit to infinity, is tight for large $V-U$.
Regardless
of whether the rate function $\IGopt(U,V,\cdot)$ is
convex, Theorem \ref{lem:approx}, which follows, justifies the approximation
\begin{align*}
P( {G_{\text{opt}} } (U,V, {\vec{W}} _k)=n) \approx
\frac 1n \exp\left(-k\IGopt\left(U,V,\frac 1k\log n\right)\right)
\end{align*}
for large $k$ and $n\in\{1,\ldots,m^k\}$. It is analogous to that
in equation \eqref{eq:approx1}, first developed in \cite{Christiansen13},
but there are additional difficulties that must be overcome to
establish it. In particular, if $U=V=1$, the likelihood that the
string is identified at each query is a decreasing function of guess
number, but this is not true in the more general case.
As a simple example, consider $U=V=2$, $ {\mathbb{A} }=\{0,1\}$, strings of
length $1$ and strings chosen uniformly. Here the probability of
guessing both strings in one guess is $1/4$, but at the second guess
it is $3/4$. Despite this lack of monotonicity, the approximation
still holds in the following sense.
\begin{theorem}
\label{lem:approx}
Under the assumptions of Theorem \ref{thm:main}, for any $x \in [0,
\log m)$ we have
\begin{align*}
&\lim_{\epsilon \downarrow 0}\liminf_{k\rightarrow \infty}
\frac 1k \log \inf_{n \in K_k(x, \epsilon)}P( {G_{\text{opt}} } (U,V, {\vec{W}} _k)=n)\\
&=\lim_{\epsilon \downarrow 0}\limsup_{k\rightarrow \infty}
\frac 1k \log \sup_{n \in K_k(x, \epsilon)}P( {G_{\text{opt}} } (U,V, {\vec{W}} _k)=n)\\
&=-I_{ {G_{\text{opt}} } }(U, V, x)-x,
\end{align*}
where
\begin{align*}
K_k(x, \epsilon)=\{n:n \in (\exp(k(x-\epsilon)), \exp((k(x+\epsilon)))\}
\end{align*}
is the collection of guesses made in a log-neighborhood of $x$.
\end{theorem}
\begin{proof}
The proof follows the ideas in \cite{Christiansen13} Corollary 4,
but with the added difficulties resolved by isolating the last word
that is likely to be guessed and leveraging the monotonicity it its
individual likelihood of being identified.
Noting the definition of $K_k(x, \epsilon)$ in the statement
of the theorem, consider for $x\in(0,\log(m))$
\begin{align*}
&\sup_{n \in K_k(x, \epsilon)}P( {G_{\text{opt}} } (U,V, {\vec{W}} _k)=n)\nonumber\\
=& \sup_{n \in K_k(x, \epsilon)}\sum_{(v_1, \ldots, v_V)}
P( {G^{(v_1)}} (W_k^{(v_1)})=n)\\
&\prod_{i=2}^{U}P( {G^{(v_i)}} (W_k^{(v_i)})\le n)
\prod_{i=U+1}^{V}P( {G^{(v_i)}} (W_k^{(v_i)})\ge n)\\
\le& \sup_{n \in K_k(x, \epsilon)}
\max_{(v_1, \ldots, v_V)}(V!)P( {G^{(v_1)}} (W_k^{(v_1)})=n)\\
&\prod_{i=2}^{U}P( {G^{(v_i)}} (W_k^{(v_i)})\le n)
\prod_{i=U+1}^{V}P( {G^{(v_i)}} (W_k^{(v_i)})\ge n)\\
\le& \sup_{n \in K_k(x, \epsilon)}
\max_{(v_1, \ldots, v_V)}(V!)P( {G^{(v_1)}} (W_k^{(v_1)})=n)\\
&\prod_{i=2}^{U}
P\left(\frac 1k \log {G^{(v_i)}} (W_k^{(v_i)})\le x-\epsilon\right)\\
&\prod_{i=
U+1
}^{V}P\left(\frac 1k \log {G^{(v_i)}} (W_k^{(v_i)})\ge x+\epsilon\right)\nonumber\\
&\le \inf_{n \in K_k(x-2\epsilon, \epsilon)}\max_{(v_1, \ldots, v_V)}(V!)
P\left(\frac1k \log {G^{(v_1)}} (W_k^{(v_1)})=n\right)\\
&\prod_{i=2}^{U}
P\left(\frac 1k \log {G^{(v_i)}} (W_k^{(v_i)})\le x+\epsilon\right)\\
&\prod_{i=U+1}^{V}P\left(\frac 1k \log {G^{(v_i)}} (W_k^{(v_i)})\ge x-\epsilon\right).\nonumber
\end{align*}
The first equality holds by definition of $ {G_{\text{opt}} } (U,V,\cdot)$. The
first inequality follows from the union bound over all possible
permutations of $\{1, \ldots, V\}$. The second inequality utilizes
$k^{-1}\log n \in (x-\epsilon, x+\epsilon)$ if $n\in K_k(x, \epsilon)$,
while the third inequality uses the monotonic decreasing probabilities
in guessing a single user's string.
Taking $\lim_{\epsilon \downarrow 0}\limsup_{k\rightarrow
\infty}k^{-1}\log$ on both sides of the inequality, interchanging
the order of the max and the supremum, using the continuity
of $ {\Lambda_G^{(v)}} (\cdot)$ for each $v\in\{1,\cdots,V\}$, and the
representation of the rate function $\IGopt(U,V,\cdot)$ in equation
\eqref{eq:Iopt}, gives the upper bound
\begin{align*}
&\lim_{\epsilon \downarrow 0}\limsup_{k\rightarrow \infty}\frac 1k \log \sup_{n \in K_k(x, \epsilon)}
P( {G_{\text{opt}} } ( {\vec{W}} _k)=n) \\
&\leq-\IGopt(U, V, x)-x.
\end{align*}
Considering the least likely guesswork in the ball leads to a
matching lower bound. The other case, $x=0$, follows similar logic,
leading to the result.
\end{proof}
We next provide some illustrative examples of what these results
imply, returning to using $\log_2$ in figures.
\section{Mismatched Statistics Example}
\label{sec:mismatch}
The potential lack of convexity in the rate function of Theorem
\ref{thm:main}, equation \eqref{eq:Iopt}, only arises if users'
string statistics are asymptotically distinct. The significance of
this lack of convexity on the phenomenology of guesswork can be
understood in terms of the asymptotically optimal round-robin
strategy: if the rate function is not convex, there is no single
set of users whose strings are most vulnerable.
That is, if $U$ strings are recovered after a small number of
guesses, they will be from one set of users, but after a number of
guesses corresponding to a transition from the initial convexity
they will be from another set of users.
This is made explicit in the following corollary to Theorem
\ref{thm:main}.
\begin{corollary}
If $\IGopt(U,V,x)$ is not convex in $x$, then there is there is no
single set of users whose strings will be identified in the long
string length asymptotic.
\end{corollary}
\begin{proof}
We prove the result by establishing the contraposition: if a single set
of users is always most vulnerable, then $\IGopt(U,V,x)$ is convex.
Recall the expression for $\IGopt(U,V,x)$ given in equation
\eqref{eq:Iopt}
\begin{align*}
&\IGopt(U,V,x)=\\
&\max_{v_1,\ldots, v_V}
\left( {\Lambda_G^{(v_1)}} ^*(x)+\sum_{i=2}^U \delta^{(v_i)}(x)
+\sum_{i=U+1}^V {\gamma^{(v_i)}} (x)\right),
\end{align*}
As explained after Theorem \ref{thm:main}, for given $x$ the set
of users $\{(v_1),\ldots,(v_U)\}$ corresponds to those users whose
strings, on the scale of large deviations, will be identified by
the inquisitor after approximately $\exp(kx)$ queries. If this set
is unchanging in
$x$, i.e. the same set of users is identified
irrespective of $x$, then both of the functions
\begin{align*}
\left( {\Lambda_G^{(v_1)}} ^*(x)+\sum_{i=2}^U \delta^{(v_i)}(x)\right)
\text{ and }
\sum_{i=U+1}^V {\gamma^{(v_i)}} (x)
\end{align*}
are sums of functions that are convex in $x$, and so are convex
themselves. Thus the sum of them, $\IGopt(U,V,x)$, is convex.
\end{proof}
This is most readily
illustrated by an example that falls within the two-user setting
of \cite{Merhav99}, where one string is constructed from uniformly
from i.i.d. bits and the other string from non-uniformly selected
i.i.d. bytes.
\begin{figure}
\begin{center}
\includegraphics[scale=0.46]{nonconvex.pdf}
\end{center}
\caption{User 1 picks a uniform bit string. User 2 picks a non-uniform
i.i.d. byte string. The straight line starting at $(0,1)$ displays
${ {\Lambda_G} ^{(1)}}^*(x)$, the large deviations rate function for
guessing the uniform bit string. The convex function starting
below it is ${ {\Lambda_G} ^{(2)}}^*(x)$, the rate function for guessing
the non-uniform byte string. The highlighted line, which is the
minimum of the two rate functions until $x=1$ and then $+\infty$
afterwards, displays $I_{ {G_{\text{opt}} } }(1, 2, x)$, as determined by
\eqref{eq:Iopt}, the rate function for an inquisitor to guess one
of the two strings. Its non-convexity demonstrates that initially
it is the bytes that are most likely to be revealed by brute force
searching, but eventually it is the uniform bits that are more
likely to be identified. The Legendre-Fenchel transform of the
scaled cumulant generating function of the guesswork distributions
would form the convex hull of the highlighted line and so this
could not be deduced by analysis of the asymptotic moments.
}
\label{fig:nonconvex}
\end{figure}
Let $ {\mathbb{A} }=\{0,\ldots,7\}$, $U=1$ and $V=2$. Let one character source
correspond to the output of a cryptographically secure pseudo-random
number generator. That is, despite having a byte alphabet, the source
produces perfectly uniform i.i.d. bits,
\begin{align*}
P(W^{(1)}_1=i) &=
\begin{cases}
1/2 & \text{ if } i\in\{0,1\}\\
0 & \text{ otherwise}.
\end{cases}
\end{align*}
The other source can be thought of as i.i.d. bytes generated by a
non-uniform source,
\begin{align*}
P(W^{(2)}_1=i) &=
\begin{cases}
0.55 & \text{ if } i=0\\
0.1 & \text{ if } i\in\{1,2\}\\
0.05 & \text{ if } i\in\{3,\ldots,7\}.
\end{cases}
\end{align*}
This models the situation of a piece of data, a string from the
second source, being encrypted with a shorter, perfectly uniform
key. The inquisitor can reveal the hidden string by guessing either
the key or the string. One might suspect that either the key or
the string is necessarily more susceptible to being guessed, but
the result is more subtle.
Figure \ref{fig:nonconvex} plots the rate functions for guessing
each of the user's strings individually as well as the rate function
for guessing one out of two, determined by equation \eqref{eq:Iopt},
which in this case is the minimum of the two rate function where
they are finite. The y-axis is the exponential decay-rate in string
length $k$ of the likelihood of identification given approximately
$\exp(k x)$ guesses, where $x$ is on the x-axis, have been made.
The rate function reveals that if the inquisitor identifies one of
the strings quickly, it will be the non-uniform byte string, but
after a certain number of guesses it is the key, the uniform bit
string, that is identified.
Attempting to obtain this result by taking the Legendre Fenchel
transform of the sCGF identified in \cite{Merhav99} results in the
convex hull of this non-convex function, which has no real meaning.
This explains the necessity for the distinct proof approach taken
here if one wishes to develop estimates on the guesswork distribution
rather than its moments.
\section{Identical Statistics Examples}
\label{sec:ident}
When the string statistics of users are asymptotically the same,
the resulting multi-user guesswork rate functions are convex by
Corollary \ref{cor:same}, and the r\^ole of specific Shannon entropy
in analyzing expected multi-user guesswork appears. This is the
setting that leads to the results in Section \ref{sec:firstexample}
where it is assumed that character statistics are i.i.d., but not
necessarily uniform.
An alternate means of departure from string-selection uniformity
is that the appearance of characters within the string may be
correlated. The simplest model of this is where string symbols are
governed by a Markov chain with arbitrary starting distribution and
transition matrix
\begin{align*}
\left(
\begin{array}{cc}
1-a & a \\
b & 1-b
\end{array}
\right),
\end{align*}
where $a,b\in(0,1)$. The specific R\'enyi entropy of this character
source can be evaluated, e.g. \cite{Malone042}, for $\beta\neq1$ to be
\begin{align*}
R(\beta) =& \frac{1}{1-\beta}\log
\left(
(1-a)^\beta+(1-b)^\beta \right. \\
& \left.+
\sqrt{((1-a)^\beta-(1-b)^\beta)^2 + 4(ab)^\beta }
\right)
-\frac{1}{1-\beta}
\end{align*}
and $R(1)$ is the Shannon entropy
\begin{align*}
R(1) &=
\frac{b}{a+b}H(a)+\frac{a}{a+b}H(b),
\end{align*}
where $H(a) = -a\log(a)-(1-a)\log(1-a)$.
Figure \ref{fig:gap_mark} shows $R(1/2)-R(1)$ the difference between
the average guesswork growth rate for a single user system versus
one for an arbitrarily large number of users as $a$ and $b$ are
varied. Heavily correlated sources or those with unlikely characters
give the greatest discrepancy in security.
\begin{figure}
\begin{center}
\includegraphics[scale=0.46]{sec_gap_markov}
\end{center}
\caption{Markovian string source over a binary alphabet $ {\mathbb{A} }=\{0,1\}$
with $a$ being the probability of a 1 after a 0 and $b$ being the
probability of a 0 after a 1. The plots shows the difference in
average guesswork exponent for a single user system and a system
with an arbitrarily large number of users, a measure of computational
security reduction.
}
\label{fig:gap_mark}
\end{figure}
If $a=b$, then the stationary likelihood a symbol is a $0$ or $1$
is equal, but symbol occurrence is correlated. In that setting, the
string source's specific R\'enyi entropy gives for $\beta\neq1$
\begin{align*}
R(\beta) =& \frac{1}{1-\beta}\log
\left(
(1-a)^\beta + a^\beta
\right),
\end{align*}
which is the same as a Bernoulli source with probability $a$ of one
character. Thus the results in Section \ref{sec:firstexample} can
be re-read with the Bernoulli string source with parameter $p=a$
substituted for a Markovian string source whose stationary distribution
gives equal weight to both alphabet letters, but for which character
appearance is correlated.
\section{Discussion}
\label{sec:conc}
Since Massey \cite{Massey94} posed the original guesswork problem
and Arikan \cite{Arikan96} introduced its long string asymptotic,
generalizations have been used to quantify the computational security
of several systems, including being related to questions of loss-less
compression. Here we have considered what appears to be one of the
most natural extensions of that theory, that of multi-user computational
security. As a consequence of the inherent non-convex nature of
the guesswork rate function unless string source statistics are
equal for all users, this development wasn't possible prior to the
Large Deviation Principle proved in \cite{Christiansen13}. The
results therein themselves relied on the earlier work that determined
the scaled cumulant generating function for the guesswork for a
broad class of process \cite{Arikan96,Malone042,Pfister04,Hanawal11}.
The fact that rate functions can be non-convex encapsulates that
distinct subsets of users are likely to be identified depending on
how many unsuccessful guesses have been made. As a result, a simple
ordering of string guessing difficulty is inappropriate in multi-user
systems and suggests that quantification of multi-user computational
security is inevitably nuanced.
The original analysis of the asymptotic behavior of single user
guesswork identified an operational meaning to specific R\'enyi
entropy. In particular, the average guesswork grows exponentially
in string length with an exponent that is the specific R\'enyi
entropy of the character source with parameter $1/2$. When users'
string statistics are the same, the generalization to multi-user
guesswork identifies a surprising operational r\^ole for specific
R\'enyi entropy with parameter $n/(n+1)$ for each $n\in {\mathbb{N} }$ when
$n$ is the excess number of strings that can be guessed. Moreover,
while the specific Shannon entropy of the string source was found
in the single user problem to have an unnatural meaning as the
growth rate of the expected logarithm of the guesswork, in the
multi-user system it arises as the universal lower bound on the
average guesswork growth rate.
For the asymptote at hand, the key message is that there is a law
of diminishing returns for an inquisitor as the number of users
increases. For a multi-user system
designer,
in contrast to the single character, single user system
introduced in \cite{Massey94},
Shannon entropy is the appropriate measure of
expected guesswork for systems with many users.
Future work might consider the case where the $V$ strings are not
selected independently, as was assumed here, but are instead linear
functions of $U$ independent strings. A potential application of
such a case, suggested by Erdal Arikan (Bilkent University) in a
personal communication, envisages the use of multi-user guesswork
to characterize the behavior of parallel concatenated decoders
operating on blocks of convolutionally encoded symbols passed though
a preliminary algebraic block Maximum Distance Separable (MDS) code,
e.g. \cite{shu2004}. The connection between guessing and convolutional
codes was first established by Arikan \cite{Arikan96}.
Decoding over a channel may, in general, be viewed as guessing a
codeword that has been chosen from a list of possible channel input
sequences, given the observation of an output sequence formed by
corrupting the input sequence according to some probability law
used to characterize the channel, e.g \cite{Christiansen13b}.
Considering sequential decoding of convolutional codes, first
proposed by Wozencraft \cite{Wozencraft57}, that guessing may
constitute an exploration along a decision tree of the possible
input sequences that could have led to the observed output sequence,
as modeled by Fano \cite{Fano63}. If the transmitted rate, given
by the logarithm of the cardinality of possible codewords, falls
below the cut-off rate, then results in \cite{Arikan96} prove that
the guesswork remains in expectation less than exponential in the
length of the code. Beyond the cut-off rate, it becomes exponentially
large. One may view such a result as justifying the frequent use
of cut-off rate as as a practical, engineering characterization of
the limitations of block and convolutional codes.
Consider now the following construction of a type of concatenated
code \cite{shu2004}, which is a slight variant of that proposed by
Falconer \cite{Falconer67}. The original data, a stream of i.i.d.
symbols, is first encoded using an algebraic block MDS code. For a
block MDS code, such as a Reed-Solomon code \cite{shu2004}, over a
codeword constituted by a sequence of $V$ symbols, correct reception
at the output of any $U$ symbols from the $V$ allows for correct
decoding, where the feasibility of a pair of $V$ and $U$ depends
on the family of codes. For every $U$ input symbols in the data
stream, $V$ symbols are generated by the algebraic block MDS code.
Note that these symbols may be selected over a set of large
cardinality, for instance by taking each symbol to be a string of
bits. As successive input blocks of length $U$ are processed by the
block MDS code, these symbols form $V$ separate streams of symbols.
Each of these $V$ streams emanating from the algebraic block MDS
code is coded using a separate but identical convolutional encoder.
The $V$ convolutional codewords thus obtained are dependent, even
though any $U$ of them are mutually independent. This dependence
is imputed by the fact that the $V$ convolutional codewords are
created by $U$ original streams that form the input of the block
MDS encoder. The $V$ convolutional codewords constitute then the
inputs to $V$ mutually independent, Discrete Memoryless Channels
(DMCs), all governed by the same probability law. In Falconer's
construct, such parallel DMCs are embodied by time-sharing equally
a single DMC. While Falconer envisages independent DMCs governed
by a single probability law, as is suitable in the setting of
interleaving over a single DMC, we may readily extend the scheme
to the case where the parallel DMCs have different behaviors. Such
a model is natural in wireless settings where several channels are
used in parallel, say over different frequencies. While the behavior
of such channels is often well modeled as being mutually independent,
and the channels individually are well approximated as being DMCs,
the characteristics of the channels, which may vary slowly in time,
generally differ considerably from each other at any time.
Decoding
uses the outputs of the $V$ DMCs as follows. For each DMC,
the output is initially individually decoded using sequential
decoding so that, in the words of Falconer, "controlled by the Fano
algorithm, all $[V]$ sequential decoders simultaneously and
independently attempt to proceed along the correct path in their
own trees". The dependence among the streams produced by the original
application of the block MDS code entails that, when $U$ sequential
decoders each correctly guesses a symbol, the correct guesses
determine a block of $U$ original data symbols. The latter are
communicated to all remaining $V-U$ sequential decoders, eliminating
the need for them to continue producing guesses regarding that block
of $U$ original data symbols. The sequential decoders then proceed
to continue attempting to decode the next block of $U$ original
data symbols. This scheme allows the $U$ most fortunate guesses out
of $V$ to dominate the performance of the overall decoder. A
sequential decoder that was a laggard for one block of the original
$U$ symbols may prove to be a leader for another block of $U$
symbols.
\section*{Acknowledgments:}
The authors thank Erdal Arikan (Bilkent University) for informative
feedback and for pointing out the relationship between multi-user
guesswork and sequential decoding.
They also thank the anonymous reviewers for their feedback
on the paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,337 |
\section{Introduction}
Solitons arise in many model field theories\cite{Cole} where
nontrivial time-independent solutions to the classical field equations
exist. The fluctuations of quantum fields about classical
configurations are difficult to study and can qualitatively affect,
even destabilize, solitons. Topological arguments support the
stability of some particularly interesting solitons. However, much less is
known about the fate of ``non-topological solitons'' that appear as
minima in the classical action, but have no deeper claim to stability
when quantum effects are taken into account.
The quantum corrections to the energies of classical field configurations
are
typically highly divergent in $3+1$ dimensions. In some cases, like
the Skyrme model\cite{Sk}, the underlying theory is
non-renormalizable, so the quantum
contribution to the soliton's energy is unavoidably cut-off dependent
and ambiguous, like any other radiative correction in
a non-renormalizable theory.
In this Letter we describe a systematic and efficient procedure for
calculating the quantum fluctuations about time-independent field
configurations in renormalizable field theories.
We show that all divergences can be removed by the
same renormalization procedure that renders the perturbative sector of
the theory finite. The only ``ambiguities'' are the well known scheme
and scale dependences of the renormalization prescription that are
resolved completely in the perturbative sector. The result of our
program is a renormalized quantum ``effective energy'',
whose
non-trivial minima (if they exist) describe solitons in the
quantum theory.
First we show how to regulate and renormalize the divergences in the
sum over quantum fluctuations. Then we develop calculational methods
that are efficient and practical enough that quantum effects can be
included in a search for stable field configurations. Our results
take the form of an effective energy, ${\cal E}(\phi(\vec x),m,\{g\})$,
depending on the ``profile function'' of the renormalized field,
$\phi(\vec x)$, the renormalized mass, $m$, and various renormalized
couplings,
$\{g\}$, defined in the usual perturbative sector of the model. One may
then search over the parameter space characterizing $\phi(\vec x)$ for
minima of ${\cal E}$ while holding $m$ and $\{g\}$ fixed. Here we
treat the simple case of a charged scalar coupled to the field $\phi$.
Our methods can be generalized straightforwardly to models including
fermions, gauge fields and self-coupled scalars. However, our
approach is limited in that we work only to order $\hbar$, and we
only consider spherically symmetric profile functions.
The possibility that the top quark in the standard model might be
described as a non-topological soliton \cite{Schechter}
provided the original
motivation for our work. The top quark Yukawa coupling, $g_t$,
leads to its mass, $g_tv$, where $v$ is the
Higgs vacuum expectation value.
For large $g_t$ it
would therefore appear favorable to suppress the Higgs condensate in
the vicinity of the $t$-quark, and the top quark would be a sort of
``bag''. However, gradient and potential energy terms in the Higgs
sector of the classical action oppose the creation of a such a bag in the
Higgs condensate. In order to study the problem at the quantum level
it is necessary to regulate and renormalize the divergences in the
$t$-quark fluctuations about a deformation of the Higgs condensate.
It is crucial to hold the renormalized parameters of the standard
model fixed while varying the possible profiles of the Higgs field.
Bagger and Naculich studied this problem by making a derivative
expansion \cite{BN}. They also worked in a large-$N$ approximation,
in which there are no quantum corrections above order $\hbar$.
We would expect that if the $t$-quark is a bag, then the
Higgs field will vary significantly --- $\Delta\phi\sim v$ --- over
distance scales of order the Compton wavelength of the top quark ---
$\lambda \sim 1/g_tv$. However, the $t$-quark mass, $g_tv$,
also sets the scale for the derivative expansion.
Thus all derivatives are of
the same size, making the expansion unreliable. Our method
is designed for such situations.
We can trace elements of our approach back to Schwinger's work on QED
in strong fields \cite{Sch}. Schwinger studied the energy of the
electron's quantum fluctuations --- the ``Casimir energy'' --- in the
presence of a prescribed, static configuration of electromagnetic
fields. He isolated the divergences in low orders of perturbation
theory. Our work can be viewed as an extension
of Schwinger's to situations where the field is
determined self-consistently by minimizing the total energy of the
system including the Casimir energy. In addition we complete the
renormalization program and develop practical computational methods in
three dimensions. Dashen, Hasslacher and Neveu renormalized the
divergent contributions to the energy of the $\phi^4$ kink and
sine-Gordon soliton in 1+1
dimensions using a simple version of the method we propose
here \cite{DHN}. Ambiguities in these models recently pointed out
and studied by Rebhan and van Nieuwenhuizen \cite{vN} can also be
resolved with our methods.
Studies of solitons in renormalizable
models often note that the divergences in the quantum contribution to
the soliton energy can be cancelled by the available
counterterms \cite{other}. However, we are not aware of any
work in 3+1 dimensions
in which renormalization of the field configuration
energy is done in a manner
consistent with on-shell mass and coupling constant renormalization in
the perturbative sector.
\section{Formalism}
We consider a renormalizable field theory with a real scalar field $\phi$
coupled to a charged scalar $\psi$. We take the classical potential
$V(\phi)\propto (\phi^2-v^2)^2$, and
$\psi$ acquires a mass through spontaneous symmetry breaking.
At the quantum level we put aside the
$\phi$ self-couplings and consider only the effects of the $\phi-\psi$
interactions. We further restrict ourselves to ${\cal O}(\hbar)$
effects in the quantum theory, which correspond to one-loop diagrams.
Our model is defined by the classical action
\begin{eqnarray}
S[\phi,\psi]&=&\int d^4x\left\{ \frac{1}{2}
(\partial_\mu\phi)^2-\frac{\lambda}{4!}(\phi^2-v^2)^2 +
\partial_\mu\psi^\ast\partial^\mu\psi - g\psi^\ast\phi^2\psi\right.
\\ \nonumber
&&\ \ \left. + a (\partial_\mu\phi)^2 - b (\phi^2-v^2) - c (\phi^2-v^2)^2
\phantom{\frac{1}{2}}\right\}\ ,
\label{I.1}
\end{eqnarray}
where we have separated out the three counterterms necessary for
renormalization and written them in a convenient form.
At one-loop order in $\psi$, these are the only
counterterms required.
We quantize around the classical vacuum $\phi=v$ and define
$h=\phi-v$, so that
\begin{eqnarray}
\label{I.1a}
S[h,\psi] &=& \int d^4x\left\{\frac{1}{2}(\partial_\mu h)^2 - \frac{m^2}{8v^2}
(h^2 + 2vh)^2 + \partial_\mu\psi^\ast\partial^\mu\psi -
M^2\psi^\ast\psi - g(h^2+2vh)\psi^\ast \psi\right. \\ \nonumber
&&\ \ \left. + a(\partial_\mu h)^2 - b(h^2+2hv) - c(h^2+2hv)^2
\phantom{\frac{1}{2}}\right\}
\end{eqnarray}
where $M=\sqrt{g}v$ is the $\psi$ mass and $m^2 = \lambda v^2/3$
is the $h$ mass.
The one-loop quantum effective action for $h$ is obtained by
integrating out $\psi$ to leading order in $\hbar$.
We are interested in time-independent field configurations
$h=h(\vec x)$, for which the effective action yields an
effective energy ${\cal E}[h]$ that has three parts:
\begin{equation}
{\cal E}[h] = {\cal E}_{\rm cl}[h] + {\cal E}_{\rm ct}[h] + {\cal
E}_{\psi}[h]\ ,
\end{equation}
where ${\cal E}_{\rm cl}[h]$ is the classical energy of $h$,
\begin{equation}
{\cal E}_{\rm cl}[h] = \int d^3x\ \left\{ \frac{1}{2} |\vec\nabla h|^2 +
\frac{m^2}{8v^2} (h^2 + 2vh)^2 \right\}\ ,
\label{I.1b}
\end{equation}
${\cal E}_{\rm ct}[h]$ is the counterterm contribution,
\begin{equation}
{\cal E}_{\rm ct}[h] = \int d^3x\left\{ a |\vec\nabla h|^2 +
b(h^2+2hv) + c(h^2+2hv)^2 \right\}\ ,
\label{I.1b1}
\end{equation}
and ${\cal E}_{\psi}[h]$ is the one-loop quantum contribution from
$\psi$. ${\cal E}_{\rm ct}[h]$ and ${\cal E}_{\psi}[h]$
are divergent, but we will see explicitly that these divergences
cancel for any configuration $h(\vec x)$.
We fix the counterterms by applying renormalization conditions
in the perturbative sector of the theory. Having done so, we have
defined the theory for all $h(\vec x)$. We choose the on-shell
renormalization conditions
\begin{equation}
\Sigma_1=0,\quad \Sigma_2(m^2) = 0,\quad {\rm and}\quad
\left.\frac{d\Sigma_2}{dp^2}\right|_{m^2} = 0,
\label{I.2a}
\end{equation}
where $\Sigma_1$ and $\Sigma_2(p^2)$ are the one- and two-point functions
arising only from the loop and counterterms as seen in
Fig.~\ref{figure1}.
\begin{figure}
$$
\BoxedEPSF{fig1.ps scaled 500}
$$
\caption{One-loop diagrams.}
\label{figure1}
\end{figure}
We denote the one-loop diagrams with one insertion by
$\Omega$ and with two insertions by $\Pi(p^2)$, and find
\begin{eqnarray}
\Sigma_1 &= &2vg\Omega + 2vb\ ,\nonumber\\
\Sigma_2(p^2) & = &(2vg)^2 \Pi(p^2) +g\Omega + b + (2v)^2c + a p^2\ .
\label{I.2b}
\end{eqnarray}
Defining
\begin{equation}
\Pi'(p^2) \equiv\frac{d\Pi(p^2)}{dp^2}\ ,
\end{equation}
the renormalization conditions eq.(\ref{I.2a}) then yield
\begin{equation}
a = -(2vg)^2 \Pi'(m^2)\ ,\ b = -g\Omega\ ,\ c = g^2(m^2 \Pi'(m^2)-
\Pi(m^2))\ ,
\end{equation}
which we then substitute into the counterterm energy, eq.~(\ref{I.1b1}).
Now we consider the calculation of ${\cal E}_{\psi}[ h]$. This energy
is the sum over zero point energies, $\frac{1}{2}\hbar\omega$, of the modes of
$\psi$ in the presence of $ h(\vec x)$,
\begin{equation}
{\cal E}_\psi[ h] = \sum_\alpha E_\alpha[ h]
\label{I.3}
\end{equation}
where $E_\alpha$ are the positive square roots of the eigenvalues of a
single particle Hamiltonian, $\hat H$, given by
\begin{equation}
\hat H = -\vec\nabla^2 + M^2 + g(h^2 + 2vh).
\label{I.4}
\end{equation}
The fact that $\psi$ is complex accounts for the absence of $\frac{1}{2}$
in eq.~(\ref{I.3}).
${\cal E}_\psi$ is highly divergent. However our model is renormalizable and
therefore the counter\-terms fixed in the presence of the trivial $ h$ {\it must\/}
cancel all divergences in ${\cal
E}_\psi$. Rather than attempt to regulate the divergent sum in
eq.~(\ref{I.3}) directly, we study the density of states that defines the
sum. We can
isolate the terms that lead to
divergences in ${\cal E}_\psi$ and renormalize them using conventional
methods.
For fixed $h(\vec x)$ the spectrum of
$\hat H$ given in eq.~(\ref{I.4}) consists of a finite
number (possibly zero) of
normalizable bound states and a continuum beginning at $M^2$,
parameterized by
$k$, with $E(k)=+\sqrt{k^2+M^2}$. Furthermore, $\hat H$ depends on $h$ only
through the combination
\begin{equation}
\chi = h^2+2hv\ ,
\end{equation}
so we can consider ${\cal E}_\psi$ to be a functional
of $\chi$. We restrict ourselves to spherically symmetric $h$. Then
\begin{equation}
{\cal E}_\psi[\chi] = \sum_j E_j + \sum_\ell (2\ell +1)
\int dk \rho_\ell(k) E(k)
\end{equation}
where $\rho_\ell(k)$ is the density of states in $k$ in the $\ell^{\rm th}$
partial wave and the $E_j$ are the bound state energies. $\rho_\ell(k)$
is finite, but the sum over $\ell$ and the integral over $k$ are
divergent. Furthermore
\begin{equation}
\rho_\ell(k) = \rho_\ell^{\rm free}(k)
+ \frac{1}{\pi}\frac{d\delta_\ell(k)}{dk}\ ,
\label{I.6}
\end{equation}
where $\delta_\ell(k)$ is the usual scattering phase shift for the
$\ell^{\rm th}$ partial wave, and $\rho_\ell^{\rm free}(k)$
is the free ($g=0$) density of states. This
relationship between the density of states and the derivative of the
phase shift is shown for example in \cite{Sch}.
At the outset, we subtract $\rho^{\rm free}(k)$ from the density
of states since we wish to compare ${\cal E}_\psi[\chi]$ to ${\cal E}_\psi[0]$.
Viewing ${\cal E}_{\psi}[\chi]$ as the sum of one loop
diagrams, we see that only the diagrams with one or two insertions
of $g\chi$ are divergent. A diagram with $n$ insertions corresponds to
the $n^{\rm th}$ term in the Born expansion, so all possible
divergences can be eliminated by subtracting the first and second Born
approximations from the phase shifts that determine the
density of states. Standard methods allow us to construct the
Born approximation for the phase shifts \cite{Schiff}, which is a
power series in the ``potential'' $g\chi$.
We define the combination
\begin{equation}
\bar\delta_\ell(k) \equiv \delta_\ell(k) - \delta^{(1)}_\ell(k) -
\delta^{(2)}_\ell(k)\ ,
\label{I.7}
\end{equation}
where $\delta^{(1)}_\ell(k)$ and $\delta^{(2)}_\ell(k)$ are the first
and second Born approximations to $\delta_l(k)$. We then have
\begin{eqnarray}
{\cal E}_\psi [\chi]&=& \sum_j E_j + \sum_\ell (2\ell+1)
\int_0^\infty dk
\frac{1}{\pi} \frac{d\bar\delta(k)}{dk}E(k) + g\Omega
\int \frac{d^3\! p}{(2\pi)^3} \tilde \chi(\vec p) \\ \nonumber
&&\ + g^2 \int \frac{d^3\! p}{(2\pi)^3} \Pi(-\vec p\;\mbox{}^2)
|\tilde\chi(\vec p)|^2
\label{I.7a}
\end{eqnarray}
where
\begin{equation}
\tilde \chi(\vec p) =\int d^3x \chi(\vec x) e^{-i\vec p \cdot \vec x}\ ,
\end{equation}
and likewise for $\tilde h(\vec p)$. Both $\tilde h$ and
$\tilde \chi$ are real and
depend only on $q\equiv|\vec p|$ for spherically symmetric $h$.
We have subtracted out the order $g$ and $g^2$ contributions by using
$\bar \delta_\ell(k)$ instead of $\delta_\ell(k)$, and added them back
in by using their explicit diagrammatic representation in terms of the
divergent constant $\Omega$ and the divergent function $\Pi(p^2)$.
We can now combine ${\cal E_\psi}$ and ${\cal E_{\rm ct}}$ and obtain a
finite result:
\begin{eqnarray}
{\cal E_\psi} + {\cal E_{\rm ct}} &=& \sum_j E_j + \sum_\ell (2\ell+1)
\int_0^\infty dk \frac{1}{\pi} \frac{d\bar\delta_{\ell}(k)}{dk}E(k)
+\Gamma_2[h]
\label{I.8}
\end{eqnarray}
where
\begin{equation}
\Gamma_2[h] =
g^2 \int \frac{q^2dq}{2\pi^2} \left[\left(\Pi(-q^2)
- \Pi(m^2) + m^2 \Pi'(m^2)\right) \tilde \chi(q)^2
+4v^2q^2 \Pi'(-q^2) \tilde h(q)^2 \right] \ .
\label{I.8a}
\end{equation}
$\Pi$ is log divergent, but both
$\{\Pi(-q^2)
- \Pi(m^2)\}$ and $\Pi'$ are finite, so $\Gamma_2[h]$ is finite as well.
Each term in the Born approximation to the phase shift goes to zero at
$k=0$, so by Levinson's theorem $\bar\delta_\ell(0) = \delta_\ell(0) =
\pi n_\ell$ where $n_\ell$ is the number of bound states with
angular momentum $\ell$. As $k\to\infty$, $\delta_\ell(k)$ falls off
like $\frac{1}{k}$, $\delta^{(1)}_\ell(k)$ falls off
like $\frac{1}{k}$, and $\delta^{(2)}_\ell(k)$ falls off
like $\frac{1}{k^2}$. Since the Born approximation becomes exact
at large $k$, $\bar\delta_\ell(k)$ falls like $\frac{1}{k^3}$.
Thus we see that the first subtraction renders each
integral over $k$ convergent. The second subtraction makes the
$\ell$-sum convergent. We are then free to integrate by parts in
(\ref{I.8}), obtaining
\begin{equation}
{\cal E}[ h]= {\cal E}_{\rm cl}[h]
+\Gamma_2[ h]
- \frac{1}{\pi}\sum_\ell (2\ell+1)\int_0^\infty dk\:
\bar\delta_\ell(k)\frac{k}{E(k)} +\sum_j (E_j-M) \ .
\label{I.9}
\end{equation}
In this expression
we see that each bound state contributes its binding energy, $E_j-M$,
so that the energy varies smoothly as we strengthen $h$ and bind more states.
As noted in the Introduction, the representation of the Casimir energy
as a regulated sum/integral over phase shifts plus a limited number of
Feynman graphs was derived by Schwinger for the case of a prescribed
background field. Our aim is to develop it into a practical tool to
study the stability of non-trivial field configurations $h(r)$.
\section{Calculational Methods}
In this Section we describe the method that allows us to construct
${\cal E}[ h]$ as a functional of $ h$ and search for stationary
points. We now consider the calculation of each of the terms in
eq.~(\ref{I.9})
in turn. The classical contribution to the action is evaluated directly by
substitution into eq.~(\ref{I.1b}).
$\Gamma_2[h]$ of eq.~(\ref{I.8a}) is obtained from a Feynman diagram
calculation,
\begin{eqnarray}
\Gamma_2 [h]= \frac{g^2}{(4\pi)^2}\int\frac {q^2dq}{2\pi^2}
\int_0^1 &dx& \left\{ \left[
\log\frac{M^2+q^2x(1-x)}{M^2-m^2x(1-x)}
-\frac{m^2 x(1-x)}{M^2-m^2x(1-x)} \right] \tilde \chi (q)^2\right.\\
\nonumber &&\left. - \frac{x(1-x)}{M^2-m^2x(1-x)}
\ 4v^2q^2\tilde{h}(q)^2
\right\}\ . \label{II.1}
\end{eqnarray}
The partial wave phase shifts and Born approximations are calculated
as follows. The radial wave equation is
\begin{equation}
-u_\ell^{\prime\prime} +\left[\frac{\ell(\ell+1)}{r^2}
+g\chi(r)\right]u_\ell=k^2 u_\ell, \
\label{II.2}
\end{equation}
where $k^2>0$, and $\chi(r)\to 0$ as $r\to \infty$.
We introduce two linearly independent solutions to
eq.~(\ref{II.2}), $u_\ell^{(1)}(r)$ and $u_\ell^{(2)}(r)$, defined by
\begin{eqnarray}
u_\ell^{(1)}(r) &=& e^{i\beta_\ell(k,r)}r h_\ell^{(1)}(kr) \\
\nonumber
u_\ell^{(2)}(r) &=& e^{-i\beta_\ell^\ast(k,r)}r h_\ell^{(2)}(kr)
\label{II.3}
\end{eqnarray}
where $h_\ell^{(1)}$ is the spherical
H\"ankel function asymptotic to $e^{ikr}/r$ as $r\rightarrow\infty$,
$h_\ell^{(2)}(kr) = h_\ell^{(1)\ast}(kr)$, and
$\beta_\ell(k,r) \to 0$ as $r \to \infty$, so that
$u_\ell^{(1)}(r)\to e^{ikr}$ and
$u_\ell^{(2)}(r)\to e^{-ikr}$ as $r\to\infty$. The scattering
solution is then
\begin{equation}
u_{\ell}(r) = u_\ell^{(2)}(r) + e^{2i\delta_\ell(k)} u_\ell^{(1)}(r)\ ,
\end{equation}
and obeys $u_{\ell}(0) = 0$. Thus we obtain
\begin{equation}
\delta_\ell(k)=2\:{\rm Re}\:\beta_\ell(k,0).
\label{II.4}
\end{equation}
Furthermore, $\beta_\ell$ obeys a
simple, non-linear differential equation obtained by substituting
$u^{(1)}_\ell$ into eq.~(\ref{II.2}),
\begin{equation}
-i\beta_\ell^{\prime \prime}
-2ikp_\ell(kr)\beta_\ell^\prime
+2(\beta_\ell^\prime)^2+\frac{1}{2} g\chi(r) = 0\ ,
\label{II.5}
\end{equation}
where primes denote differentiation with respect to $r$, and
\begin{equation}
p_\ell(x)=\frac{d}{dx} \ln \left[ xh_\ell^{(1)}(x)\right]
\label{II.6}
\end{equation}
is a simple rational function of $x$.
We solve eq.~(\ref{II.5}) numerically, integrating from $r=\infty$
to $r=0$
with $\beta_\ell^\prime(k,\infty)=\beta_\ell(k,\infty)=0$, to get the
exact phase
shifts. To get the Born approximation to $\beta_\ell$, we solve the
equation
iteratively, writing $\beta_\ell = g\beta_{\ell 1} + g^2\beta_{\ell 2} +
\dots$, where $\beta_{\ell 1}$ satisfies
\begin{equation}
-i\beta_{\ell 1}^{\prime \prime}
-2ikp_\ell(kr)\beta_{\ell 1}^\prime+\frac{1}{2} \chi(r) = 0
\label{II.7}
\end{equation}
and $\beta_{\ell 2}$ satisfies
\begin{equation}
-i\beta_{\ell 2}^{\prime \prime}
-2ikp_\ell(kr)\beta_{\ell 2}^\prime
+2(\beta_{\ell 1}^\prime)^2 = 0..
\label{II.8}
\end{equation}
We can solve efficiently for $\beta_{\ell 1}$ and $\beta_{\ell 2}$
simultaneously
by combining these two equations into a coupled differential equation
for the
vector $(\beta_{\ell 1},~\beta_{\ell 2})$. This method is much faster than
calculating the Born terms directly as iterated integrals in $r$ and will
generalize easily to a theory requiring higher-order counterterms.
Having found the phase shifts, we then use Levinson's theorem to
count bound states. We then find
the energies of these bound states by using a shooting method to
solve the corresponding eigenvalue equation. We use the effective
range approximation \cite{Schiff} to calculate the phase shift and
bound state energy near the threshold for forming an s-wave bound state.
\section{results}
For the model at hand, we calculated the energy ${\cal E}[ h]$ for a
two-parameter ($d$ and $w$) family of gaussian backgrounds
\begin{equation}
h(r)=-dve^{-r^2 v^2/2w^2}\ .
\label{IV.1}
\end{equation}
In Fig.~\ref{figure2}, we show results which are representative of our
findings in general. We plot the energy of configurations with
fixed $d=1$ as a function of $w$, for $g=1,2,4,8$ (from top to bottom).
We note that to this order, for $g=8$ the vacuum is unstable to the
formation of large $\phi=0$ regions.
\begin{figure}
$$
\BoxedEPSF{fig2.ps scaled 500}
$$
\caption{${\cal E}[h]$ in units of $v$, for $d=1$ and
$g=1,2,4,8$, as a function of $w$.}
\label{figure2}
\end{figure}
To explore whether the charged scalar forms a non-topological soliton
in a given $\phi$ background, we add to ${\cal E}[ h]$
the energy of a ``valence'' $\psi$ particle in the lowest bound state.
We then compare this total energy to $M$, the energy of the $\psi$ particle
in a flat background, to see if the soliton is
favored. This is the scalar model analogue of $t$-quark bag
formation.
For fixed $g$ and $m$, we varied $h$ looking for
bound states with energy $E$ such that $E+{\cal E}[ h]<M$. However,
for those values of $g$ and $m$ where we did find such solutions, we
always found that by increasing $w$, we could make ${\cal E}[ h]<0$,
so that the vacuum is unstable, as we pointed out above in the case
of $g=8$ in
Fig.~\ref{figure2}. Thus we find that if
we stay in the $g,m$ parameter region where the vacuum is stable, the
minimum is at $h=0$, so there are no nontopological solitons.
Although we did not find a non-trivial solution at one-loop order
in this simple model, our calculation demonstrates the practicality
of our method. We can effectively characterize and search the space
of field configurations, $h(r)$, while holding the renormalized
parameters of the theory fixed. The same methods can be used
to study solitons in theories with richer structure.
\section{derivative expansion}
Our results are exact to one-loop order. The derivative expansion, which is
often applied to problems of this sort, should be accurate for slowly varying
$h(r)$. We found it useful to compare our results with the derivative
expansion
for two reasons: first, we can determine the range of validity (in $d$
and $w$)
of the derivative expansion; and second, where the derivative expansion is
expected to be valid, it provides a check on the accuracy of our
numerical work
and C++ programming. Where expected, the two calculations did agree to the
precision we specified (1 \%).
In our model, the first two terms in the derivative expansion of
the one-loop effective Lagrangian can be calculated to be
\begin{equation}\label{V.1}
{\cal L}_{\rm 1} = {\cal L}_{\rm ct} + \alpha z+\beta z^2
+ {g^2v^4\over 32\pi^2}\left[
(1+z)^2\ln (1+z)-z-{3\over2}z^2\right]+
{g\over48\pi^2v^2}{1\over 1+z} (\partial_\mu z)^2 \ ,
\end{equation}
where $z=g\chi/M^2=(h^2+2hv)/v^2$, $\alpha$ and $\beta$ are
cutoff-dependent constants, and ${\cal L}_{\rm ct}$ is the same
counterterm Lagrangian as we used in Sec. 2. For $\phi^4$
scalar field theory a similar result was first derived in
\cite{Coleman}. The last term above is proportional to $(\partial
h)^2$, and is completely cancelled by a finite counterterm that
implements the renormalization prescription of Sec. 2. In this
prescription, counterterms also cancel the $\alpha z$ and $\beta z^2$
terms above. Thus the ${\cal O}(p^2)$ derivative expansion for the effective
energy, to be compared with the phase shift expression for ${\cal E}[h]$, is
\begin{equation}\label{V.2}
{\cal E}_{\rm DE}[h]=\int d^3\! x\ \left\{
\frac{1}{2} (\vec{\nabla}h)^2 +{m^2\over8v^2}(h^2+2hv)^2+
{g^2v^4\over32\pi^2}\left[ (1+z)^2\ln (1+z )-
z-{3\over2}z^2\right]\right\}\ .
\end{equation}
The results of the comparison with the phase shift method can
be seen in Fig.~\ref{figure3} for $d = 0.25$,
and $g=4$. A similar pattern holds in general for other values of both
$d$ and $g$. As the width becomes larger, the two results merge. This is
as expected, since it is for large widths, and thus small gradients,
that we expect the derivative expansion to yield accurate results. As the
width tends to zero, both results tend to zero, and the fact that the plot
tends to 1 simply indicates that the derivative expansion result goes to
zero faster than the phase shift result.
\begin{figure}
$$
\BoxedEPSF{fig3.ps scaled 500}
$$
\caption{$( {\cal E} -{\cal E}_{\rm DE})/{\cal E}$
for $d=0.25$, $g=4$, as a function of $w$.}
\label{figure3}
\end{figure}
\section{conclusions}
We have presented a numerically tractable method for evaluating the one-loop
effective energy of a static background field configuration in a
renormalizable quantum field theory. Since we rely on calculating
phase shifts, the method is only suitable for rotationally invariant
(or generalized rotationally invariant) backgrounds. The model explored
in the present work is particularly simple and does not support any
solitons. However, our methods could just as well be applied to any
renormalizable field theory. They can be used to study the one-loop quantum
stability of field configurations in the standard electroweak model as well as
various unified models that support monopoles, strings and the like.
Topologically non-trivial field configurations with maximal symmetry, like the
``hedgehog solutions'' in chiral models can be studied in this fashion.
Ultimately we hope to be able to reliably determine whether large Yukawa
couplings may yield solitons in the standard electroweak model.
\vspace*{1cm}
We would like to thank J.~Goldstone, R.~Jackiw, K.~Johnson, K.~Kiers,
S.~Naculich, B.~Scarlet, R.~Schrock, M.~Tytgat, and P.~van Nieuwenhuizen for
helpful conversations, suggestions and references. This work is
supported in part by funds provided by the U.S. Department of Energy
(D.O.E.) under cooperative research agreement \#DF-FC02-94ER40818.
and by the RIKEN BNL Research Center. N.~G. is supported in part by an NSF
Fellowship.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,360 |
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object --+
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<strong class="uidshort">DryCreek</strong>
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<hr />
<p>Subclass for SpeedLoader created for the purpose of organization
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<table border="0" cellpadding="0" cellspacing="0" width="100%">
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<td align="left"><span class="table-header">Instance Methods</span></td>
<td align="right" valign="top"
><span class="options">[<a href="#section-InstanceMethods"
class="privatelink" onclick="toggle_private();"
>hide private</a>]</span></td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.DryCreek-class.html#__init__" class="summary-sig-name">__init__</a>(<span class="summary-sig-arg">self</span>)</span><br />
Initializes subclass DryCreek</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#DryCreek.__init__">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.DryCreek-class.html#weatherStationData_csv" class="summary-sig-name">weatherStationData_csv</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">filename</span>)</span><br />
Returns:
returns a list of the data and their headers</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#DryCreek.weatherStationData_csv">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.DryCreek-class.html#getColumnIndexAndName" class="summary-sig-name">getColumnIndexAndName</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">string</span>,
<span class="summary-sig-arg">tableHeaders</span>)</span><br />
Returns the index and name of the first header that is found with a
name like the string given.</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#DryCreek.getColumnIndexAndName">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.DryCreek-class.html#getColumn" class="summary-sig-name">getColumn</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">index</span>,
<span class="summary-sig-arg">dataRows</span>)</span><br />
Takes the csv data table and parses out a particular _column based on
the index of that _column (found using getColumnIndexAndName)</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#DryCreek.getColumn">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a name="convertDate_mdY_HM"></a><span class="summary-sig-name">convertDate_mdY_HM</span>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">dates</span>)</span></td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#DryCreek.convertDate_mdY_HM">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td colspan="2" class="summary">
<p class="indent-wrapped-lines"><b>Inherited from <code>object</code></b>:
<code>__delattr__</code>,
<code>__format__</code>,
<code>__getattribute__</code>,
<code>__hash__</code>,
<code>__new__</code>,
<code>__reduce__</code>,
<code>__reduce_ex__</code>,
<code>__repr__</code>,
<code>__setattr__</code>,
<code>__sizeof__</code>,
<code>__str__</code>,
<code>__subclasshook__</code>
</p>
</td>
</tr>
<tr bgcolor="#e8f0f8" >
<th colspan="2" class="group-header"
> Inherited from <a href="speedloader.SpeedLoader-class.html">SpeedLoader</a></th></tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a name="argumentParse_cmd"></a><span class="summary-sig-name">argumentParse_cmd</span>(<span class="summary-sig-arg">self</span>)</span><br />
Stubbed method</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.argumentParse_cmd">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.SpeedLoader-class.html#fetch_gui" class="summary-sig-name">fetch_gui</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">entries</span>,
<span class="summary-sig-arg">lastCall</span>=<span class="summary-sig-default"><code class="variable-quote">'</code><code class="variable-string">false</code><code class="variable-quote">'</code></span>)</span><br />
Fetches the data from the gui frame; closes out window if it is the
last call</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.fetch_gui">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.SpeedLoader-class.html#getEntries_gui" class="summary-sig-name">getEntries_gui</a>(<span class="summary-sig-arg">self</span>)</span><br />
Returns the gui form _entries</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.getEntries_gui">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.SpeedLoader-class.html#makeForm_gui" class="summary-sig-name">makeForm_gui</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">root</span>,
<span class="summary-sig-arg">fields</span>)</span><br />
Creates a gui frame</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.makeForm_gui">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.SpeedLoader-class.html#openFileDialog_gui" class="summary-sig-name">openFileDialog_gui</a>(<span class="summary-sig-arg">self</span>)</span><br />
Opens a basic file dialog that browses to a file</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.openFileDialog_gui">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td width="15%" align="right" valign="top" class="summary">
<span class="summary-type"> </span>
</td><td class="summary">
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr>
<td><span class="summary-sig"><a href="speedloader.SpeedLoader-class.html#substring" class="summary-sig-name">substring</a>(<span class="summary-sig-arg">self</span>,
<span class="summary-sig-arg">myString</span>,
<span class="summary-sig-arg">myList</span>)</span><br />
Find substring finds a particular string within a list</td>
<td align="right" valign="top">
<span class="codelink"><a href="speedloader-pysrc.html#SpeedLoader.substring">source code</a></span>
</td>
</tr>
</table>
</td>
</tr>
</table>
<!-- ==================== PROPERTIES ==================== -->
<a name="section-Properties"></a>
<table class="summary" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr bgcolor="#70b0f0" class="table-header">
<td colspan="2" class="table-header">
<table border="0" cellpadding="0" cellspacing="0" width="100%">
<tr valign="top">
<td align="left"><span class="table-header">Properties</span></td>
<td align="right" valign="top"
><span class="options">[<a href="#section-Properties"
class="privatelink" onclick="toggle_private();"
>hide private</a>]</span></td>
</tr>
</table>
</td>
</tr>
<tr>
<td colspan="2" class="summary">
<p class="indent-wrapped-lines"><b>Inherited from <code>object</code></b>:
<code>__class__</code>
</p>
</td>
</tr>
</table>
<!-- ==================== METHOD DETAILS ==================== -->
<a name="section-MethodDetails"></a>
<table class="details" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr bgcolor="#70b0f0" class="table-header">
<td colspan="2" class="table-header">
<table border="0" cellpadding="0" cellspacing="0" width="100%">
<tr valign="top">
<td align="left"><span class="table-header">Method Details</span></td>
<td align="right" valign="top"
><span class="options">[<a href="#section-MethodDetails"
class="privatelink" onclick="toggle_private();"
>hide private</a>]</span></td>
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<a name="__init__"></a>
<div>
<table class="details" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr><td>
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr valign="top"><td>
<h3 class="epydoc"><span class="sig"><span class="sig-name">__init__</span>(<span class="sig-arg">self</span>)</span>
<br /><em class="fname">(Constructor)</em>
</h3>
</td><td align="right" valign="top"
><span class="codelink"><a href="speedloader-pysrc.html#DryCreek.__init__">source code</a></span>
</td>
</tr></table>
<p>Initializes subclass DryCreek</p>
<dl class="fields">
<dt>Overrides:
object.__init__
</dt>
</dl>
</td></tr></table>
</div>
<a name="weatherStationData_csv"></a>
<div>
<table class="details" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr><td>
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr valign="top"><td>
<h3 class="epydoc"><span class="sig"><span class="sig-name">weatherStationData_csv</span>(<span class="sig-arg">self</span>,
<span class="sig-arg">filename</span>)</span>
</h3>
</td><td align="right" valign="top"
><span class="codelink"><a href="speedloader-pysrc.html#DryCreek.weatherStationData_csv">source code</a></span>
</td>
</tr></table>
<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>filename</code></strong> - the full path plus the file name</li>
</ul></dd>
<dt>Returns:</dt>
<dd>returns a list of the data and their headers</dd>
</dl>
</td></tr></table>
</div>
<a name="getColumnIndexAndName"></a>
<div>
<table class="details" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr><td>
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr valign="top"><td>
<h3 class="epydoc"><span class="sig"><span class="sig-name">getColumnIndexAndName</span>(<span class="sig-arg">self</span>,
<span class="sig-arg">string</span>,
<span class="sig-arg">tableHeaders</span>)</span>
</h3>
</td><td align="right" valign="top"
><span class="codelink"><a href="speedloader-pysrc.html#DryCreek.getColumnIndexAndName">source code</a></span>
</td>
</tr></table>
<p>Returns the index and name of the first header that is found with a
name like the string given. With this information, one can grab the
column from the csv that they desire.</p>
<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>string</code></strong> - The header name that is being searched within the column titles</li>
<li><strong class="pname"><code>tableHeaders</code></strong> - The list of column titles (from the parsed csv)</li>
</ul></dd>
<dt>Returns:</dt>
<dd>the index and name of the first header that is found with the
given string name</dd>
</dl>
</td></tr></table>
</div>
<a name="getColumn"></a>
<div>
<table class="details" border="1" cellpadding="3"
cellspacing="0" width="100%" bgcolor="white">
<tr><td>
<table width="100%" cellpadding="0" cellspacing="0" border="0">
<tr valign="top"><td>
<h3 class="epydoc"><span class="sig"><span class="sig-name">getColumn</span>(<span class="sig-arg">self</span>,
<span class="sig-arg">index</span>,
<span class="sig-arg">dataRows</span>)</span>
</h3>
</td><td align="right" valign="top"
><span class="codelink"><a href="speedloader-pysrc.html#DryCreek.getColumn">source code</a></span>
</td>
</tr></table>
<p>Takes the csv data table and parses out a particular _column based on
the index of that _column (found using getColumnIndexAndName)</p>
<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>index</code></strong> - index of the _column</li>
<li><strong class="pname"><code>dataRows</code></strong> - the list of rows of the data that includes all columns</li>
</ul></dd>
<dt>Returns:</dt>
<dd>the _column of data values as a parsed list (removing other
_column data)</dd>
</dl>
</td></tr></table>
</div>
<br />
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,960 |
{"url":"https:\/\/support.bioconductor.org\/p\/9138042\/","text":"DESeq2 with two continuous variable in a design\n2\n0\nEntering edit mode\nsofia \u2022 0\n@725046bd\nLast seen 17 months ago\n\nHi everyone, Hi Michael,\n\nI am using DESeq2 to analyze small RNA sequencing results, and I am very uncertain about the results I am getting for a particular design, so I would really appreciate some help.\n\nThe following is my coldata and my design\n\nhead(colDesign)\nAge BMI\n\nsample_01 17.0 23.03\nsample_02 16.1 23.66\nsample_03 10.4 18.56\nsample_04 15.3 39.57\nsample_05 14.5 25.19\nsample_06 18.1 18.88\n\ndds=DESeqDataSetFromMatrix(countData = cts_bmi,\ncolData = colDesign,\ndesign = ~ Age+BMI+Age:BMI)\nDESeq(dds)\nresultsNames(dds)\n[1] \"Intercept\" \"Age\" \"BMI\" \"Age.BMI\"\n\n\nAs you can see, I want to study the interaction between 2 continuous variables, age and BMI. I read the interaction examples on the help page and I also read Michael's answer on a thread regarding the interaction between a discrete and a continuous variable, from what I learned is that, there's no reference point for the continuous variable, it's all embedded in the intercept? So for my design, results(dds, name=\"Age.BMI\") will give the effect of BMI across all ages? Is that correct? And also if I want to use LRT test, should the reduced model be reduced=~Age+BMI?\n\nThank you in advance, any help is greatly appreciated!\n\nDESeq2 RNASeq R design \u2022 1.6k views\n2\nEntering edit mode\n@mikelove\nLast seen 7 hours ago\nUnited States\n\nBasically, correct. I'd recommend however to center and scale the numeric covariates for model fitting improvement. Then the coefficients represent changes with one SD of the variable. You can back-convert back by multiplying the coefficient by SD(Age) or SD(BMI) if you want the coefficients on original scale.\n\n0\nEntering edit mode\n\nThank you very much Mike! I have another similar question, so I have a design design= Sex+Sex:Age, and the resultsNames are \"Intercept\" \"Sex_M_vs_F\" \"SexF.Gestational.Age..wks.\" \"SexM.Gestational.Age..wks.\" If my understanding is right, results(dds, name=\"Sex_M_vs_F) is the main effect (the gender effect across all ages), and if I do results(dds, contrast=list(\"SexF.Gestational.Age..wks.\" \"SexM.Gestational.Age..wks.\")), it will test if the age effect is different between males and females? Isn't that the same as the main effect?\n\n0\nEntering edit mode\n\nSee the interactions section of the vignette, and then for further questions I'd recommend consulting with a local statistician. In particular, the main effect is not averaging across all ages.\n\n0\nEntering edit mode\nLaia \u25b4 10\nLast seen 3 days ago\nBelgium\n\nHi,\n\nI have a similar question. In my design I have two continuous variables that are the conditions at which I cultured my cells: different substrate stiffnesses (3 stiffnesses), and different flow rates applied (5 flow rates). I wrote down the units in colData because otherwise it gave error when fitting the model.\n\n> print(colData)\nstiffness shear\nAA_2_val_sorted.bam 100_kpa 25_dyn\nAC_2_val_sorted.bam 10_kpa 25_dyn\nAE_2_val_sorted.bam 1_kpa 15_dyn\nAF_2_val_sorted.bam 1_kpa 15_dyn\nAH_2_val_sorted.bam 1_kpa 25_dyn\nAI_2_val_sorted.bam 10_kpa 5_dyn\nAJ_2_val_sorted.bam 100_kpa 5_dyn\nAL_2_val_sorted.bam 100_kpa 15_dyn\nAM_2_val_sorted.bam 100_kpa 25_dyn\nAO_2_val_sorted.bam 10_kpa 40_dyn\nAR_2_val_sorted.bam 1_kpa 15_dyn\nAS_2_val_sorted.bam 1_kpa 25_dyn\n(...)\n\n\nSamples are independent (only 1 measure per sample = only 1 timepoint), and I have 3 or 4 replicates per condition. For 1_kpa, there is no 40_dyn flow condition, only static, 5_dyn, 15_dyn, 25_dyn (4 flow rates instead of 5).\n\nI want to see if there is an interaction of shear and stiffness (hope I have enough power), but when I fit the design:\n\ndds <- DESeqDataSetFromMatrix(countData = counts_data,\ncolData = colData,\ndesign = ~ stiffness + shear + stiffness:shear)\n\n\nI get:\n\nError in checkFullRank(modelMatrix) :\nthe model matrix is not full rank, so the model cannot be fit as specified.\nOne or more variables or interaction terms in the design formula are linear\ncombinations of the others and must be removed.\n\nvignette('DESeq2')\nIn DESeqDataSet(se, design = design, ignoreRank) :\nsome variables in design formula are characters, converting to factors\n\n\nI checked this, and I read several answers in Bioconductor, but still don't know how should I fit my model. Do I need to nest some factor to avoid 1kpa having only 4 flow conditions while 10kpa and 100kpa have both 5 flow conditions?\n\nAny help with understanding this would be very much appreciated.\n\n0\nEntering edit mode\n\nI'd recommend consulting with a local statistician or someone familiar with linear models in R, for help designing your statistical analysis.","date":"2023-02-04 11:51:46","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\": 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\": 0.5667136311531067, \"perplexity\": 4411.7245356885705}, \"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-2023-06\/segments\/1674764500126.0\/warc\/CC-MAIN-20230204110651-20230204140651-00376.warc.gz\"}"} | null | null |
import { Dialog, DialogContent, DialogTitle } from '@rmwc/dialog';
import React from 'react';
import { useIsEmulatorDisabled } from '../common/EmulatorConfigProvider';
export const AppDisconnected: React.FC<
React.PropsWithChildren<unknown>
> = () => {
const disabled = useIsEmulatorDisabled();
if (!disabled) {
return null;
}
return (
<Dialog open preventOutsideDismiss>
<DialogTitle>Emulator Suite Disconnected</DialogTitle>
<DialogContent>
All emulators for this project have stopped running. To restart all
emulators, at the command line, run{' '}
<code>firebase emulators:start</code>.
</DialogContent>
</Dialog>
);
};
export default AppDisconnected;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,598 |
Q: Is it true that if $\phi :G \rightarrow G $ defined by $ \phi(x)=x^{4}$ is a homomorphism, then $G$ is abelian?
Is it true that if $\phi :G \rightarrow G $ defined by $ \phi(x)=x^{4}$ is a homomorphism, then $G$ is abelian?
By the property of homomorphisms we see that $\forall x,y \in G $ we have
$$\begin{align}
\phi(xy) = \phi(x)\phi(y) &\Rightarrow (xy)^4 = x^4 y^4 \\
&\Rightarrow (yx)^3=x^3y^3.
\end{align}$$
I couldn't find of a way to get $xy=yx$ from this.
I tried to find a counter-example by considering small non-abelian groups such as $D_3$ and $S_4$ however for those, there were always elements such that $\phi(xy) \neq \phi(x)\phi(y)$
A: With the dihedral group $D_4$ acting on four elements, the fourth power of any element is the identity. And sending each element to the identity is always a homomorphism, but $D_4$ is not abelian.
| {
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Saturday Morning Soundtrack: "Grave Digging" by The Classic Crime (2008) (Driving Mix Edition!)
Well, it's another week putting together the Saturday Morning Soundtrack, and we'll be taking a break from our rock and roll count down (count up?) to celebrate a milestone. Yes, it's the fiftieth post in the series, but that's not the only milestone I reached this week. As some of you know, one of my favorite hobbies is to get my car out on the open road. It's something I do as often as I can, and when I do, I like to have a great soundtrack to accompany me. For me, nothing beats 5-6 hours of continuous driving, while the radio churns out a steady stream of my favorite songs.
Of course, I also believe that variety is the spice of life, and so I hate hearing the same tracks every time I hit the highway. This has led me to put together a massive playlist of my favorite driving songs--high-tempo rockers that keep the pedal to the metal, moody atmospheric pieces that serve as navigator when you're cruising on an empty highway at night, and everything in between. The list includes hard rock, heartland rock, alternative hits, as you'd expect. But it also features some genres that you might not typically associate with driving, including instrumental numbers like Zoe Keating's "Exurgency," and even a few a a cappella numbers. (I've never been a genre snob.) Some of the songs, I've loved since I was a teen. Some were given to me by friends, here and gone. Others I've only recently discovered. Together, I like to think they do as good a job as anything of telling the story of my travels, capturing the large and various, complex, idiosyncratic, and often contradictory aspects of my persona.
This week, I reached song #1000 on the driving mix. For those wondering, that means I could listen to my playlist continuously for 2 days, 22 hours, 20 minutes, and 31 seconds, before ever hearing a repeat song. Not bad!
I can't take all the credit though--this song was recently recommended to me by my friend (and partner in tailgating crime) Dominique Bruno, and it's a good thing, too, because I've got a road trip coming up in a few weeks, and this song melts blacktop.
Saturday Morning Soundtrack @ Youtube
"Grave Digging" by The Classic Crime (2008)
I suppose it could be bad juju to include a song called "Grave Digging" on your driving CD, but the whole James-Dean-drive-fast-and-die-young theme certainly finds its roots in American car culture. And (knock on wood), if I were going to go before my time, I supposed I'd want it to happen while doing something I love--like driving.
That said, despite the fact that the singer of this song is "sitting in the suicide lane on highway 99," this isn't really a song about traveling recklessly in the traditional sense. At least, it doesn't have to be. In most travel narratives, we're presented with a character who is either moving toward someone (the coming home tale) or running away. In this song, our leading man probably falls into that second category. He's at war with himself. He's looking for signs. And though he's playing the martyr card early and clearly here, there's also a defiant sound to the music itself that cuts against what would otherwise be unfettered self-pity and melodrama.
The song follows a simple verse-chorus pattern, and like many contemporary rock tracks, it borrows its sound from a number of hard rock influences--the chugging electric guitar, reminiscent of Def Leppard's "Turbo Lover" or Golden Earring's "Radar Love"; the rhythmic chant straight out of AC/DC; the screaming chorus Metallica-esque; and the guitar solo played at what could be described as "Yngwie Malmsteen speed." It's not dramatically different from what many other post-grunge and nu-metal bands are doing, but that's okay--this song doesn't have to reinvent the wheel to be a successful road trippin' song (no pun intended).
For some reason, whenever I judge a night driving song (especially one with a fast tempo), I always think of an interview I read once, in which Bob Seger described the experience of listening to "Hollywood Nights" for the first time. He'd just finished the recording and popped the tape in on the way home, and when the song finished, he looked down to see that his speedometer was pinned above 100 miles-per-hour. "Grave Digging" has that quality to it. It's the perfect song for when you're flying down the highway on a dark, moonless night, and you've got a lot of ground to cover before you reach your final destination.
The only thing that would make the song better, of course, is if it had more cowbell.
So what do you think? Was this song worthy of #1000? Share your thoughts in the comments below. And, if you would like to write a Guest Entry for the "Saturday Morning Soundtrack" series where you creatively respond to one of your favorite rock songs, don't hesitate to contact me with queries. | {
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Q: How to call javascript function from code-behind I wrote a javascript with a asp.net page.
In Asp.net Page
<HTML> <HEAD>
<script type="text/javascript">
function Myfunction(){
document.getElementId('MyText').value="hi";
}
</script>
</HEAD> <BODY>
<input type="text" id="MyText" runat="server" /> </BODY>
In Code-behind
Private Sub Page_Load(ByVal sender As Object, ByVal e As System.EventArgs)
Handles Me.Load
If Session("My")= "Hi" Then
I want to call "Myfunction" javascript function
End If
End Sub
How can I do?
A: One way of doing it is to use the ClientScriptManager:
Page.ClientScript.RegisterStartupScript(
GetType(),
"MyKey",
"Myfunction();",
true);
A: This is a way to invoke one or more JavaScript methods from the code behind.
By using Script Manager we can call the methods in sequence. Consider the below code for example.
ScriptManager.RegisterStartupScript(this, typeof(Page), "UpdateMsg",
"$(document).ready(function(){EnableControls();
alert('Overrides successfully Updated.');
DisableControls();});",
true);
In this first method EnableControls() is invoked.
Next the alert will be displayed.
Next the DisableControls() method will be invoked.
A: There is a very simple way in which you can do this. It involves injecting a javascript code to a label control from code behind. here is sample code:
<head runat="server">
<title>Calling javascript function from code behind example</title>
<script type="text/javascript">
function showDialogue() {
alert("this dialogue has been invoked through codebehind.");
}
</script>
</head>
..........
lblJavaScript.Text = "<script type='text/javascript'>showDialogue();</script>";
Check out the full code here: http://softmate-technologies.com/javascript-from-CodeBehind.htm (dead)
Link from Internet Archive: https://web.archive.org/web/20120608053720/http://softmate-technologies.com/javascript-from-CodeBehind.htm
A: If the order of the execution is not important and you need both some javascript AND some codebehind to be fired on an asp element, heres what you can do.
What you can take away from my example:
I have a div covering the ASP control that I want both javascript and codebehind to be ran from. The div's onClick method AND the calendar's OnSelectionChanged event both get fired this way.
In this example, i am using an ASP Calendar control, and im controlling it from both javascript and codebehind:
Front end code:
<div onclick="showHideModal();">
<asp:Calendar
OnSelectionChanged="DatepickerDateChange" ID="DatepickerCalendar" runat="server"
BorderWidth="1px" DayNameFormat="Shortest" Font-Names="Verdana"
Font-Size="8pt" ShowGridLines="true" BackColor="#B8C9E1" BorderColor="#003E51" Width="100%">
<OtherMonthDayStyle ForeColor="#6C5D34"> </OtherMonthDayStyle>
<DayHeaderStyle ForeColor="black" BackColor="#D19000"> </DayHeaderStyle>
<TitleStyle BackColor="#B8C9E1" ForeColor="Black"> </TitleStyle>
<DayStyle BackColor="White"> </DayStyle>
<SelectedDayStyle BackColor="#003E51" Font-Bold="True"> </SelectedDayStyle>
</asp:Calendar>
</div>
Codebehind:
protected void DatepickerDateChange(object sender, EventArgs e)
{
if (toFromPicked.Value == "MainContent_fromDate")
{
fromDate.Text = DatepickerCalendar.SelectedDate.ToShortDateString();
}
else
{
toDate.Text = DatepickerCalendar.SelectedDate.ToShortDateString();
}
}
A: asp:run javascript method
Add this line to the bottom of the page before </form> tag, at least under the js function you wrote.
the reason of doning this is avoid calling your method before your
browse knowing what is the funcion and finally do nothing.
<% Response.Write($"<script>yourfunction('{Config.id}');</script>"); %>
ps: I've tried all methods up there but nothing worked fine for me. And I figure out this easy and wonder way of calling js method on my own!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,181 |
<?php
use SilverStripe\Dev\BuildTask;
/**
* Updates the available SilverStripe versions.
*
* @package mysite
*/
class UpdateSilverStripeVersionsTask extends BuildTask
{
/**
* {@inheritDoc}
* @var string
*/
protected $title = 'Update SilverStripe Versions';
/**
* {@inheritDoc}
* @var string
*/
protected $description = 'Updates the available SilverStripe versions';
/**
* @var SilverStripeVersionUpdater
*/
private $updater;
/**
* @param SilverStripeVersionUpdater $updater
*/
public function __construct(SilverStripeVersionUpdater $updater)
{
$this->updater = $updater;
}
/**
* {@inheritDoc}
* @param SS_HTTPRequest $request
*/
public function run($request)
{
$this->updater->update();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,360 |
{"url":"http:\/\/www.gulfcoastcampingresort.com\/of-medicine-vnft\/c514f8-cucl-oxidation-number","text":"Covid Vs Asthma Cough, Importance Of Computer Literacy, Nick Anderson Band, Data Center Certification, Mike Bloomberg Facebook, Jtx 6000 Instruction Manual, \" \/>\n\n# Gulf Coast Camping Resort\n\n## cucl oxidation number\n\nCopper(II) chloride is light brown when anhydrous ... CuO + 2HCl \u2192 CuCl 2 + H 2 O Uses. So letting the oxidation number of Cu to be x (purely for calculation sake) x + 4(-1) = -2. x= +2. The oxidation state of Cu in CuCl2 (cupricchloride) is +2and that of Cl- is -1. Answer Save. The oxidation number of simple ions is equal to the charge on the ion. Since is in column of the periodic table, it will share electrons and use an oxidation state of . The term covers a large and diverse body of processes. For example, the oxidation number of Na + is +1; the oxidation number of N 3-is -3. Find the Oxidation Numbers CCl_4. The oxidation number of hydrogen is +1 when it is combined with a nonmetal as in CH 4, NH 3, H 2 O, and HCl. Oxidation Reduction ( Redox Reactions ) Objective A Given the formula for a neutral or charged molecule, determine the oxidation number for each atom in the molecule.. Oxidation Number: a charge assigned to an atom according to a set of rules. Cupric Chloride - CuCl 2 Copper(II) Chloride Copper Chloride Copper(2+) Chloride Cupric Chloride Anhydrous Coclor Copper(II) Chloride (1:2) Copper Chloride (CuCl2) Cupric Dichloride Molar Mass of Cl2Cu Oxidation State of Cl2Cu You learnt that, ... Chlorine belongs to halogen group series and their oxidation number is -1 when they form compounds with metals such as NaCl, MgCl 2, CaBr 2. Oxygen is not in a peroxide. Anonymous. For example, the oxidation number of Na + is +1; the oxidation number of N 3-is -3. This is the full list of oxidation states for this molecule. Anonymous. 11. 1 decade ago. They allow chemists to do things such as balance redox (reduction\/oxidation) equations. It also contains chloride ions. Hope this helps! Oxygen typically has an oxidation number of -II, and it does here. Certain elements have the same oxidation number in almost all their compounds. e.g. For nitric acid, HNO 3, based on rule number 3 hydrogen has a +1 Oxidation Number and each oxygen atom has a -2 Oxidation Number. for elements that we know and use these to figure out oxidation number for Cu.----------GENERAL RULESFree elements have an oxidation state of zero (e.g. Gauri. To find the correct oxidation state of Cu in CuCl (Copper (I) chloride), and each element in the compound, we use a few rules and some simple math. oxidation number +2 Fluorine always has an oxidation number of \u20131. The most common oxidation numbers are -1 (as in HCl and NaCl) and 0 (as in Cl2). Oxygen almost always has an oxidation number of -2, except in peroxides (H 2 O 2) where it is -1 and in compounds with fluorine (OF 2) where it is +2. What is the oxidation number for each element in CuCO3 (the three is a subscript) and how did you find it? Using this information we can figure out the oxidation number for the element C in CO. Assign an oxidation number of -2 to oxygen (with exceptions). Fluorine in compounds is always assigned an oxidation number of -1. Oxidation numbers are assigned to elements using these rules: Rule 1: The oxidation number of an element in [\u2026] This means that you can write ON_\"chlorine\" + ON_\"oxygen\" = -1 Now, oxidation numbers are all about electronegativity. We know Oxygen generally shows a oxidation number of -2. The oxidation number of an element in any elementary substance is zero. Literally, the oxidation states for any covalent compounds, e.g (CO) and ionic compounds, e.g(NaCl) is Zero, because the arbitary charge (oxidation states) of its individual ions or elements will balance the total charge of the compound to Zero. Anhydrous CuCl 2 adopts a distorted cadmium iodide structure. Answer Save. The usual oxidation number of hydrogen is +1. So 3 xx -2 + C_(ON) = -2. The elements with oxidation number of 1+ are those is group 1 of the periodic table (H, Li, Na, K, Rb, Cs, and Fr). -2 B. So, chlorine's only negative oxidation number \u2026 2. To find the correct oxidation state of Cu in CuCl2 (Copper (II) chloride), and each element in the compound, we use a few rules and some simple math.First, since the CuCl2 doesn\u2019t have an overall charge (like NO3- or H3O+) we could say that the total of the oxidation numbers for CuCl2 will be zero since it is a neutral compound.We write the oxidation number (O.N.) Since there are three oxygen atoms that is a total Oxidation Number of 3 x -2 = -6 from the oxygen atoms. Since is in column of the periodic table, it will share electrons and use an oxidation state of . Oxidation numbers are bookkeeping numbers. It contains a copper(1+). For example, the oxidation number of chlorine in Cl2, phosphorus in P4, and sulfur in S8 is 0. The sum of the formal oxidation numbers in an a complex or compound ion must sum to the charge of the ion. In oxidation-reduction processes, the driving force for chemical change is in the exchange of electrons between chemical species. Assigning Oxidation Numbers. As the oxidation number of Cu went from +2 to 0, it means Cu was reduced (gained electrons). Oxidation numbers are positive or negative numbers, but don\u2019t confuse them with positive or negative charges on ions or valences. *Response times vary by subject and question complexity. Its purpose is to help you keep track of electrons as they move from one atom or molecule to the next. D. 0. check_circle Expert Answer. To find the correct oxidation state of Cu in CuCl2 (Copper (II) chloride), and each element in the compound, we use a few rules and some simple math. 0 0. 2. It has a role as a molluscicide and an agrochemical. oxidation number +1 Group 2 always forms +2 ions: Mg2+, Ca2+, etc. To find the correct oxidation state of Cu in CuCl (Copper (I) chloride), and each element in the compound, we use a few rules and some simple math.First, since the CuCl doesn\u2019t have an overall charge (like NO3- or H3O+) we could say that the total of the oxidation numbers for CuCl will be zero since it is a neutral compound.We write the oxidation number (O.N.) Many oxidation-reduction reactions are as common and familiar as fire, the rusting and dissolution of metals, the Click hereto get an answer to your question \ufe0f The oxidation number of C in CH4, CH3Cl, CH2Cl2, CHCl3 and CCl4 are respectively : assign oxidation numbers to all the atoms in each of the following. 1 Answer. +4 C. +2 D. 0 Application Catalyst for oxidation of alcohols under aerobic conditions. 3. With reaction of chlorine, copper is oxidized to copper +2 ion. Anonymous. for elements that we know and use these to figure out oxidation number for Cu.----------GENERAL RULESFree elements have an oxidation state of zero (e.g. Na, Fe, H2, O2, S8).In an ion the all Oxidation numbers must add up to the charge on the ion.In a neutral compound all Oxidation Numbers must add up to zero.Group 1 = +1Group 2 = +2Hydrogen with Non-Metals = +1Hydrogen with Metals (or Boron) = -1Fluorine = -1Oxygen = -2 (except in H2O2 or with Fluorine)Group 17(7A) = -1 except with Oxygen and other halogens lower in the group---------- Oxygen almost always has an oxidation number of 2 except in peroxides h 2 o 2 where it is \u2026 Compound Ions Oxidation No. Chlorine can show oxidation numbers of -1, 0, +1, +3, +5 and +7. A. SrCr2O7 B. CuCl2 C. O2 D. H2O2 E. MgCO3 F. Ag G. PbSO3 H. PbO2 I. Na2C2O4 J. CO2 K. (NH4)2Ce(SO4)3 L. Cr2O3 I'm only having trouble with a few of them but I didn't want to throw the numbering off, any help is appreciated. This makes {eq}CuCl_2 {\/eq} the oxidizing agent as it caused Zn to be oxidized. The sum of the oxidation numbers in a polyatomic ion is equal to the charge of the ion. The oxidation number of oxygen (O) in compounds is usually -2, but it is -1 in peroxides. is it still +2 or is it +1 ? 6.1 An Introduction to Oxidation-Reduction Reactions 211 Oxidation-Reduction and Molecular Compounds The oxidation of nitrogen to form nitrogen monoxide is very similar to the oxidation of zinc to form zinc oxide. The oxidation number is a positive or negative number that is assigned to an atom to indicate its degree of oxidation or reduction. Since one chloride ion has a charge of -1, then that would be its oxidation state. Na, Fe, H2, O2, S8). 2- x 3 = 6- total. 9 years ago. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. assign oxidation numbers to all the atoms in each of the following. Properties. The oxidation number of a free element is always 0. Na, Fe, H2, O2, S8).In an ion the all Oxidation numbers must add up to the charge on the ion.In a neutral compound all Oxidation Numbers must add up to zero.Group 1 = +1Group 2 = +2Hydrogen with Non-Metals = +1Hydrogen with Metals (or Boron) = -1Fluorine = -1Oxygen = -2 (except in H2O2 or with Fluorine)Group 17(7A) = -1 except with Oxygen and other halogens lower in the group------- Let's try an example. Copper(I) chloride is an inorganic chloride of copper in which the metal is in the +1 oxidation state. What is the oxidation number of carbon in oxalate ion, C_2O_4^(2-)? Since is in column of the periodic table, it will share electrons and use an oxidation \u2026 Ryan. The oxidation number of Cu in CuO is II, i.e. Also Cl has an oxidation number of -1. Rule 2: The oxidation number of a monatomic (one-atom) ion is the same as the charge on the ion, for example: Rule 3: The sum of all oxidation numbers in a neutral compound is zero. Oxidation state of cu in cucl2. Copper(II) chloride (CuCl 2) reacts with several metals to produce copper metal or copper(I) chloride (CuCl) with oxidation of the other metal. Select one: A. To convert copper(II) chloride to copper(I) chloride, it can be convenient to reduce an aqueous solution with sulfur dioxide as the reductant: 2 CuCl 2 + SO 2 + 2 H 2 O \u2192 2 CuCl + 2 HCl + H 2 SO 4 A clean and highly efficient alcohol oxidizing system, using hydrogen peroxide in the presence of CuCl 2 \/Bifunctional ionic liquid (CuCl 2 \/BIL), under solvent free condition has been developed. 1 decade ago. In oxidation-reduction processes, the driving force for chemical change is in the exchange of electrons between chemical species. Therefore it donated 4 \u2026 N 2(g) + O 2(g) \u2192 2NO(g) 2Zn(s) + O 2(g) \u2192 2ZnO(s) Relevance. C. +2. The as-prepared For an ion, the oxidation state is the charge. 0 0. Favorite Answer. i think u didnt check out the charge of the entire molecule which is -2. so Cl has an oxidation state of -1, and if there are 4 Cl, this would be -4,and Cu will hav +2 . -2. The carbon is fully oxidized. The element is in the seventh element on the Periodic Table, which means that it has seven outer shell electrons.. The oxidation number of the sulfur atom in the SO 4 2-ion must be +6, for example, because the sum of the oxidation numbers of the atoms in this ion must equal -2. Copper (Cu) by itself has an oxidation state of zero since free elements have an oxidation state of zero (e.g. Favorite Answer. It reacts with sodium hydroxide to make copper(II) hydroxide. :) 0 0. ion exchange in nitrogen is controlled by the interdiffusion process of Cu + and K + in the glass, whereas ion exchange in air seems to be controlled by the Cu + \u2192Cu 2+ oxidation reaction. The metal is used as an alloying agent for steel. Oxidation-reduction reaction, any chemical reaction in which the oxidation number of a participating chemical species changes. You're dealing with the hypochlorite anion, \"ClO\"^(-), so right from the start you know that the oxidation numbers of the atoms that make up the anion must be equal to -1, the overall charge of the ion. For copper this is +2 and for iron this is +3. I know that oxygen is -2 in this but how can I find out the oxidation numbers of the other two elements? ; When oxygen is part of a peroxide, its oxidation number is -1. 2. However the oxidation state of cu in cucl cuprous chloride is 1 and that of cl is 1. The oxidation proceeds smoothly with 0.01 mmol of CuCl 2 \/BIL and 30% of H 2 O 2 to give the carboxylic acids with excellent yield. There are 3 oxygens in the ion. But chlorine can have other oxidation numbers, for example: +1 in NaClO Repeating the above steps (2) and (3) can obtain ultrathin Pt on Au\/GCE (Pt(CuCl)x\/Au\/GCE, here x denotes the number of CuCl-electrodeposition\/GRR cycles). General description CuCl 2 was used in studies investigating pleural fibrosis resulting from chronic inhalation of indium phosphide (InP) particles in mice.. The alkaline earth metals (group II) are always assigned an oxidation number of +2. The alkali metals (group I) always have an oxidation number of +1. Oxidation state is equal to the number of valence electrons that carbon is supposed to have, minus the number of valence electrons around carbon in our drawings, so let's count them up after we've accounted for electronegativity. * Corresponding authors a Department of Applied Chemistry, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan E-mail: kyama@appchem.t.u-tokyo.ac.jp b Department of Chemistry and Biotechnology, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Since is in column of the periodic table, it will share electrons and use an oxidation state of . The reaction rate expression was first order with respect to Hg(0). Anonymous. A. SrCr2O7 B. CuCl2 C. O2 D. H2O2 E. MgCO3 F. Ag G. PbSO3 H. PbO2 I. Na2C2O4 J. CO2 K. (NH4)2Ce(SO4)3 L. Cr2O3 I'm only having trouble with a few of them but I didn't want to throw the numbering off, any help is appreciated. For an ion, the oxidation state is the charge.Since one chloride ion has a charge of -1, then that would be its oxidation state. Therefore oxidation number of Cl2 in SO2Cl2 is -1*2=-2. Here the charge is -2. 4. The given coordination compound is, Let the oxidation state of P be, 'x' Hence, the oxidation state of P is +5. The atoms in He and N 2, for example, have oxidation numbers of 0. The oxidation number of a free element is always 0. n is the highest oxidation number M metal can show. SO 4 2-: (+6) + 4(-2) = -2. The oxidation number of an element in a monatomic ion is equal to the charge on that ion. Cu(II)O (Copper(I) oxide (cuprous oxide, Cu2O), a red powder. Vanadium(V) oxide as a Catalyst. Properties. In this study, the reaction kinetics for a heterogeneous oxidation reaction of elemental mercury (Hg(0)) vapor with CuCl 2 was studied in a fixed-bed reactor using 2%(wt) CuCl 2 \/\u03b1-Al 2 O 3 between 100 and 180 \u00b0C for Hg(0) oxidation after air preheater at a typical coal-fired power plant. It releases chlorine and turns into copper(I) chloride when heated very hot. M is the metal which has variable oxidation numbers and X 2 is the halogen. Formula Metal Oxidation Number Name (use roman numerals) CuCl +1 Copper (I) Chloride CuCl 2 FeO Fe 2O 3 SnS SnS 2 PbO PbO 2 SbF 3 SbF 5 BiBr 3 BiBr 5 Hg 2O HgO Multiple Oxidation Number Metal Lower Oxidation Number Higher Oxidation Number Copper (Cu) \u2026 When the oxidation number of an atom increases (when going from reactants to products), the atom was oxidized (it lost electrons). The usual oxidation number of hydrogen is +1. Oxidation number, also called oxidation state, the total number of electrons that an atom either gains or loses in order to form a chemical bond with another atom.. Each atom that participates in an oxidation-reduction reaction is assigned an oxidation number that reflects its ability to acquire, donate, or share electrons. The oxidation number of a monatomic ion equals the charge of the ion. Median response time is 34 minutes and may be longer for new subjects. Here it is bonded to element symbol so the oxidation number on Oxygen is -2. I know that oxygen is -2 in this but how can I find out the oxidation numbers of the other two elements? Oxidation state is equal to the number of valence electrons that carbon is supposed to have, minus the number of valence electrons around carbon in our drawings, so let's count them up after we've accounted for electronegativity. See Answer. Therefore, it is C, IV^+. In CuCl, copper is at its +1 oxidation state. The oxidation number of chlorine can be -1, 0, +1, +3, +4, +5, or +7, depending on the substance containing the chlorine. Posted in Chemistry You need the oxidation number for Cl, so we will use only the ClO3 1-. Example: Assign oxidation numbers to each atom $\\mathrm{H_2SO_4}$ Start with what you know: H and O. H: +1; O: \u20132; To assign the oxidation number to sulfur, take note of Rule 4. The oxidation number of sodium in the Na + ion is +1, for example, and the oxidation number of chlorine in the Cl-ion is -1. It is in \u201cchlorate\u201d so the oxidation number of oxygen is 2-. Select one: A. What is the oxidation number for each element in CuCO3 (the three is a subscript) and how did you find it? Can figure out the oxidation numbers ( +1 ) from the hydrogen atom give an number. Group 2 always forms +2 ions: Li+, Na+, K+, etc +4 C. +2 D. 0 is! So2Cl2 is -1 * 2=-2 2 adopts a distorted cadmium iodide structure * 2=-4 0... The other two elements an oxidation number is a positive or negative charges on ions or valences numbers ( )... Hydrogen atom give an oxidation number for Cl, or chlorine, is Cl-1 of Cl- is -1 *.! Term covers a large and diverse body of processes, and it does here { eq } CuCl_2 \/eq. To do things such as balance redox ( reduction\/oxidation ) equations atoms that is a )! Degree of oxidation states for this molecule... CuO + 2HCl \u2192 2! Electrons between chemical species and how did you find it all their compounds longer for new subjects e.g. Cooper is a positive or negative number that is a subscript ) and how did you find it, (! Cu in CuCl, copper is At its +1 oxidation state Li+, Na+, K+, etc of.! With sodium hydroxide to make copper ( II ) oxide ( cupric oxide, CuO,! Sulfur in S8 is 0 1- are in group 17, the state. Know oxygen generally shows a oxidation number of Na + is +1 ; the oxidation numbers of.. 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\section{Introduction}
The evolutionary status of classical Be stars is a frequently raised and
yet unsolved question. The main issue is to determine whether the Be
phenomenon appears at a given stage in the evolutionary track of every B
star, or it originates in the conditions of formation of some stars, which
include fast rotation and probably other facts. A fundamental element in
this discussion is the study of Be stars in open clusters, in two
different ways: i./ the determination of the Be star positions in the
cluster photometric diagrams; and, ii./ the study of the abundance of Be
stars as a function of the cluster age.
It is well known that Be stars usually occupy anomalous positions in the
colour-magnitude diagrams, lying above the main sequence.
Early attempts to explain the Be phenomenon suggested that Be stars occur
during the core contraction phase following the exhaustion of hydrogen
(Schmidt-Kaler 1964). Later, however, it was observed a significant
fraction of Be stars close to the ZAMS (Schild \& Romanishin 1976), and
today it is generaly accepted that they occupy the whole main sequence
band and different evolutionary states (Mermilliod 1982; Slettebak 1985)
and therefore they are not confined to any particular evolutionary phase.
It is well established that the anomalous positions in the
photometric diagrams can be explained in terms of the contribution of the
circumstellar continuum emission to the photometric indices (Fabregat et
al. 1996; Fabregat \& Torrej\'on 1998; and references therein).
Extensive studies of the abundances of Be stars in open clusters have been
done by Mermilliod (1982) and recently by Grebel (1997). Both authors
obtained similar results, finding Be stars in clusters of all ages, with a
peak frequency in clusters with turn-off at spectral types B1-B2, and
a regular decrease with increasing age afterwards.
Nevertheless, these kind of studies face some difficulties which make
their conclusions somewhat uncertain. The purpose of this paper is to
critically review the previous work in this field, and to present a new
study of the abundance of Be stars in clusters of different ages taking
into account new considerations and observational results.
\section{Critical review of previous work}
In this section we will address the main drawbacks which affect the
determination of the Be star abundace as a function of the cluster age.
Whenever possible we will propose solutions to these problems.
\subsection{The ages of the open clusters}
The usual way to determine the age of a star cluster is by means of
isochrone fitting. The most extended technique is to transform the
theoretical isochrone from the $L$ - $T_{\rm eff}$ plane to the
observational colour-magnitude plane, and then directly compare it with
the observational photometric data.
In young open clusters the isochrone fitting is difficult by two main
problems affecting the observational data. The usual presence of
differential reddening across the cluster face widens the observed main
sequence. For the clusters we are dealing with in this paper, the presence
of Be stars which generally occupy anomalous positions in the
colour-magnitude diagrams, also contributes to a further main-sequence
widening. Hence, the fit of a particular isochrone can be a very uncertain
process. For instance, recent age determinations for the cluster with the
higest Be star abundance in the Galaxy, NGC 663, are the following: 21 Myr
(Leisawitz 1988), 9 Myr (Tapia et al. 1991), 12-15 Myr (Phelp \& Janes
1994) and 23 Myr (this work, Sect. 3). The difference in the age
determinations amounts to a factor of three.
As an attempt to solve this problem, we have investigated the different
photometric indices which are commonly used as horizontal axis in the
observational HR diagrams, with regard to the B star region of the main
sequence. In Table 1 we present for each index its variation along the B
star sequence, the photometric accuracy which is usually reached in the
photometric data, the sampling of the main sequence -- the ratio between
the index variation and the accuracy -- and how the interstellar reddening
affects the index.
\begin{table}
\begin{center}
\caption{Photometric indices commonly used as $T_{\rm eff}$ indicators in
the observational HR diagrams.}
\begin{tabular}{ccccc}
\hline
&&&&\\
Index & B range & accuracy & sampling & $E(i)/E(B-V)$ \\
&&&&\\
\hline
&&&&\\
$(B-V)$ & 0.30 & 0.010 & 30 & 1.0 \\
$(U-B)$ & 1.00 & 0.020 & 50 & 0.7 \\
$(V-R)$ & 0.15 & 0.010 & 15 & 0.8 \\
$(V-I)$ & 0.40 & 0.010 & 40 & 1.6 \\
$(b-y)$ & 0.10 & 0.005 & 20 & 0.7 \\
$c_{1}$ & 1.10 & 0.020 & 55 & 0.2 \\
&&&&\\
\hline
\end{tabular}
\end{center}
\end{table}
In view of this table, the best sampling is obtained with the $(U-B)$
colour in the Johnson system and the $c_{1}$ index in the Str\"omgren
system. $c_{1}$ has the additional advantage of being much less affected by
reddening. Furthermore, the $M_{V}$ - $c_{1}$ diagram allows an efficient
segregation of the Be stars from the absorption line B stars (Fabregat et
al. 1996). Therefore we propose that the most efficient way to determine
reliable ages for very young clusters is the isochrone fitting to the
observational $M_{V}$ - $c_{1}$ HR diagram. In Section 3 we will base our
discussion of the Be stars abundances as a function of the cluster age on
ages determined in this way.
\subsection{Determination of Be star frecuencies}
Spectroscopic surveys devoted to the detection of Be stars in open
clusters are scanty in the literature. The only clusters for which the
abundance of Be stars have been exhaustively studied are h and $\chi$
Persei (NGC 869 and NGC 884) and NGC 663. The last systematic, although
not exhaustive, survey dates from 1976 with the work of Schild \&
Romanishin.
In the recent years new detection techniques based on CCD
imaging photometry are being applied to study the Be star abundances in
clusters in the Galaxy (Capilla \& Fabregat 1999) and in the Magellanic
Clouds (Grebel et al. 1992; Grebel 1997; Keller et
al. 1999). These studies, however, only provide lower limits of the
abundance for no more than 10 clusters. We are still far from having a
statistically significant sample of open clusters with well determined Be
star abundances.
\subsection{Classical versus Herbig Be stars}
In this paper we deal only with the abundances of classical Be stars.
There exists other classes of early-type emission line stars. Among them
the most conspicuous are the so called Herbig Ae/Be stars. These objects
are pre-main sequence stars in which the line emission originates in
circumstellar material remaining from the proto-stellar cloud from which
the star was formed (for a recent review of Herbig Ae/Be stars, see
Waters \& Waelkens 1998). The observational
characteristics of classical and Herbig Be stars, at least in the optical
region, are very similar, making a very difficult task to differenciate
between the two types. An efficient segregation can be made in the
far-infrared region, where the Herbig Ae/Be stars show an
important excess caused by the presence of dust, which is lacking in
classical Be stars.
Grebel (1997) includes in her study the clusters NGC 2244, NGC 6611 and IC
2944. All of them are very young open clusters (age $<$ 6 Myr)
associated to bright emission nebulosities, and are still
undergoing stellar formation (Hillenbrand et al. 1993; P\'erez et
al. 1987; Reipurt et al. 1997; de Winter et al. 1997). Hillenbrand et al.
(1993) found a large number (27) of H$\alpha$ emission line stars in NGC 6611.
They pay special atention to the study of the strong emiters W235 and
W503, and show evidence that these stars are Herbig Ae/Be stars instead of
classical Be stars. For the rest of the stars the situation is much more
uncertain, but they concluded that all emission line stars in NGC 6611 are
pre-main sequence objects instead of classical Be stars. De Winter et al.
(1997), in a smaller sample in the same cluster, found 11 emission line
stars. They classified three of them as Herbig Ae/Be stars, and agreed
with Hillenbrand et al. that most probably the rest of the emission line
objects are also pre-main sequence stars.
In the same way, Reipurth et al. (1997) analysed 7 emission line stars in
the \ion{H}{ii} region IC 2944. They classified all of them as young
pre-main sequence objects, probably Herbig Ae/Be stars. Van den Anker et
al. (1997) found 5 emission line stars in the star-forming cluster NGC
6530. They classified 3 as Herbig Ae/Be, and concluded that the remaining
two most probably are of the same type.
We can conclude that, despite the difficulty of differenciating
between classical and Herbig Be stars, there are in the literature several
positive identifications of Herbig Ae/Be stars in the youngest,
star-forming open clusters. Conversely, no bona fide classical Be star
has yet been reported among them.
\section{The Be star abundance as a function of the cluster age}
There are only four galactic open clusters with a known distinctly high
abundance of Be stars, namely more than 15 Be stars, or more than 25\% of
Be stars among their observed B stars. They are NGC 663, NGC 869, NGC 884
and NGC 3760. As we argued in Sect. 2.1, we will assume for these
clusters the ages determined through isochrone fitting to the
observational $M_{V}$ - $c_{1}$ HR diagram. These ages are 14 Myr for NGC 869
and NGC 884 (Fabregat et al. 1996) and 24 Myr for
NGC 3766 (Shobbrook 1985, 1987). For NGC 663, in Fabregat et al. (1996) we
assumed the age of 21 Myr, because it was the age determination found in
the literature which shown the best agreement with our {\em uvby} data.
However, in Fig. 2 of this reference it can be seen that the 21 Myr
isochrone does not make the best fit to the data in the $M_{V}$ - $c_{1}$
diagram.
A much better fit is obtained with an age of 23 Myr, and we will assume
this value as the cluster age. We conclude that, when using the isochrone
fitting to the $M_{V}$ - $c_{1}$ diagram as age estimator, the galactic
clusters with the highest frecuency of Be stars occupy the very narrow age
interval of 14-24 Myr.
Grebel (1997) report on three more clusters in the Magellanic Clouds,
with Be stars abundances comparable or even higher than the
clusters referred to in the above paragraph. The ages of these clusters
fall in the same range than those of the above galactic clusters. They are
NGC 330 (19 Myr), NGC 2004 (20 Myr) and NGC 1818 (25 Myr). Notice that
the age reported by Grebel for NGC 1818 is between 25 and 30 Myr, but
looking at her Fig. 1 we find that the 25 Myr fits better the
data. The same conclusion is reached by Van Veber and Vanbeveren (1997).
Dieball and Grebel (1998) studied three more clusters in the LMC, namely
SL 538 (18$\pm$2 Myr), NGC 2006 (22.5$\pm$2.5 Myr) and KMHK 19 (16
Myr). They found between 5\% to 12\% of Be stars among the observed
clusters B stars.
Keller at al. (1999) also searched for Be stars in the clusters NGC 330
and NGC 346 in the SMC, and NGC 1818, NGC 1948, NGC 2004 and NGC 2100 in
the LMC. They found a large amount of Be stars in all these clusters, with
frecuencies ranging from 13\% to 34\% of the clusters main sequence B
stars. For the three clusters not in common with Grebel (1997), the age
determinations in the literature are 15 Myr for NGC 1948
(Vallenari et al. 1993) and 18 Myr for NGC 2100 (Cassatella et al.
1996). We will exclude NGC 346 from this discussion because it is a much
younger cluster, embedded in N66, the largest and brightest \ion{H}{ii}
region in the SMC (Kudritzki et al. 1989; Massey et al. 1989).
All the age determinations for the Magellanic Clouds clusters are derived
from $BVRI$ photometry, and so we consider it less reliable that the ages
obtained from {\em uvby} photometry, for the reasons explained in Sect. 2.3.
Keller et al. consider these ages uncertain by factors of 2 to 3. Despite
of this fact, the coincidence between these ages and the age interval we
determined from the {\em uvby} photometry of the galactic clusters is
overwhelming. Only one Magellanic Clouds cluster fall very marginally
outside of the age interval of 14-24 Myr, namely NGC 1818 (25 Myr).
In Table 2 we resume the ages and Be star abundances of
all discussed "Be star rich" clusters. When comparing the abundances of
the Galactic and Magellanic Clouds clusters it should be noted that
abundances in the latter are derived from single-epoch surveys, while in
the former they came from more than 50 years of study. In fact, it turns
out that the Be star abundance of the Magellanic Clouds clusters is
significantly higher, and this can be attributed to the
different metalicity (Maeder et al. 1999).
\begin{table}
\begin{center}
\caption{The clusters with the highest Be star abundance in the Galaxy and
Magellanic Clouds.}
\begin{tabular}{lrrl}
\hline
& & & \\
Cluster & Age & N$_{\rm Be}$/N$_{\rm OB}$ & Ref. \\
& & & \\
\hline
& & & \\
Milky Way &&& \\
& & & \\
NGC 663 & 23 & 40\% & Sanduleak (1990) \\
NGC 869 & 14 & 25--50\% & Waelkens et al. (1990) \\
NGC 884 & 14 & 25--50\% & Waelkens et al. (1990) \\
NGC 3766 & 24 & 36\% & Shobbrook (1985, 1987) \\
& & & \\
SMC &&& \\
& & & \\
NGC 330 & 19 & 34\% & Keller et al. (1999) \\
& & & \\
LMC &&& \\
& & & \\
NGC 1818 & 25 & 21\% & Keller et al. (1999) \\
NGC 1948 & 15 & 14\% & Keller et al. (1999) \\
NGC 2004 & 20 & 13\% & Keller et al. (1999) \\
NGC 2006 & 22 & 12\% & Dieball \& Grebel (1998) \\
NGC 2100 & 18 & 28\% & Keller et al. (1999) \\
SLC 538 & 18 & 12\% & Dieball \& Grebel (1998) \\
KMHK 1019 & 16 & 5\% & Dieball \& Grebel (1998) \\
& & & \\
\hline
\end{tabular}
\end{center}
\end{table}
From the above, we conclude that the clusters with a high abundance of Be
stars occupy a very narrow range of ages, namely between 14 and 25 Myr.
For older clusters the percentage of Be stars decrease, as shown by
Mermilliod (1982) and Grebel (1997). We have to study now the abundances
in the younger clusters. There are not in the literature enough
observational data to make a complete analysis, but we have been able to
collect several pieces of evidence pointing towards the same conclusion:
the very paucity of Be stars in clusters younger than 10 Myr. In the
following paragraphs we will review some of them.
We have performed a first search in the WEBDA database of open cluster
data (Mermilliod 1999). We found 64 clusters younger than 10 Myr. Among
them, 9 clusters contain 3 or more Be stars. They include NGC 2244 (3 Be
stars), NGC 6530 (18), NGC 6611 (20) and IC 2944 (8). All these clusters
have been discussed in Sect. 2.3, where we show that their emission line
objects are Herbig Be stars instead of classical Be stars. NGC 6823 (5)
lies in a bright \ion{H}{ii} region with associated dark clouds (Stone
1988). NGC 7380 (3) is associated to the molecular cloud regions Sh2-142
and NGC 7380 E, and contains pre-main sequece stars among which several
Herbig Ae/Be stars are identified (Chavarr\'\i a-K. et al. 1994). IC
1590 (4) is embedded in the nebulosity of NGC 281, also identified as the
bright \ion{H}{ii} emission region Sharpless 184 (Guetter \& Turner 1997).
One more cluster, IC 1805 (2) is located in an \ion{H}{ii} region, and
associated to a large molecular cloud (Ninkov et al. 1995; Heyer et al.
1996). For the same arguments exposed in Sect. 2.3, we consider that the
emission line objects in the last four clusters are likely to be pre-main
sequence objects. The two remaining Be star rich clusters are NGC 884 (17)
and NGC 957 (5). NGC 884 has been discussed in the above paragraphs, where
we assumed and justified an age of 14 Myr. Meynet et al.
(1993) also derived an age of 14 Myr from isochrone fitting to $UBV$
photometric data. The age of 8 Myr in the WEBDA
database is clearly wrong. The same can be said for NGC 957, whose age in
the database is 6 Myr. This cluster has an age of 15 Myr in the last
edition of the Lyng{\aa } catalogue (Lyng{\aa } 1987), which would place
it in the age range we have established for the maximum Be star abundance.
Maeder, Grebel \& Mermilliod (1999) assumed ages of 13 and 16 Myr
respectively for NGC 884 and NGC 957.
Among the remaining 54 clusters younger than 10 Myr in the WEBDA database,
one contain two Be stars (NGC 6871) and three one Be star (NGC 6383, NGC
7235 and Hogg 16). Even in these very few cases doubts still remain
on the cluster ages and the nature of the emission line objects.
The emission line star in NGC 6383 has been studied by Th\'e et al.
(1985), who cannot decide whether it is a classical Be star or a pre-main
sequence object. A recent determination of the age of Hogg 16 gives
a result of 25 Myr (V\'azquez \& Feinstein 1991). 50 more clusters
younger than 10 Myr in the WEBDA database have no Be stars detected so
far.
Balona and co-workers (Balona 1994; Balona \& Koen 1994; Balona \& Laney
1995, 1996) obtained CCD {\em uvby}$\beta$ photometry for several young open
clusters. They did not make any particular investigation of the Be star
content of the clusters they observed, but we have searched their
photometric lists for objects with emission in the H$\beta$ line.
A $\beta$ index equal to 2.55 correspond to an equivalent width of the H$\beta$
line equal to 0. i.e., a photospheric absorption line completely filled-in
by emission (Fabregat \& Torrej\'on 1999). Hence, $\beta < 2.55$
indicates that the H$\beta$ line is in emission. We have searched the above
referred to photometric lists for OB stars (\byo $<$ 0.05) with $\beta <
2.55$.
This search will also detect, as well as classical Be stars, other kinds
of early-type emission line objects, like Of and OBIa stars. To exclude
these stars we have introduced the additional restriction of $M_V > -4.5$.
A proof of the reliability of this last restriction
can be found in the following data: the two emission line stars in NGC
3293 brighter than $-$4.5 have spectral classification in the literature;
they are star 3, with type B0.5Ib (Feast 1958), and star 4, type B0Ib
(Morgan et al. 1955). In NGC 6231 there are 8 emission line stars brighter
than $-$4.5, and 7 among them have spectral types given by Levato \&
Malaroda (1980). Two are OBI supergiants and three more Of. Even if the
remaining three are Be stars -- Oe in this case -- this would not affect
the main conclussions of this work, as we will comment on later.
60\% of the Be stars in NGC 663, NGC 869 and NGC 884 observed by Fabregat
et al. (1996) would have been detected by applying the above criteria.
This percentage can be considered as the typical detection capability of a
photometric survey. Photometric surveys never detect the whole content of
Be stars by the two following reasons: i./ surveys through photometric
filters only detect stars with high level of line emission, loosing the
mild emitters which only can be identified by spectroscopic means; ii./
the Be phenomenon is variable, and at a given time only a fraction of Be
stars are in a phase of line emission.
After all these considerations, the final results of our survey in the
photometric data published by Balona and co-workers are given in Table 4.
Except in the case of NGC 2362, the ages reported have been derived from
the pulsational properties of the $\beta$ Cephei stars present in each
cluster by Balona et al. (1997).
As it can be seen the clusters in the age interval 4-10 Myr are almost
lacking of Be stars. No Be stars have been found in the two younger
clusters, with ages of 4-5 Myr. A few have been detected in the two
older, with ages of 9-12 Myr.
\begin{table}
\begin{center}
\caption{Clusters with CCD {\em uvby}$\beta$ photometry obtained by Balona and
co-workers. For each cluster we indicate the age, number of OB stars
observed and number of Be stars.}
\begin{tabular}{cccc}
\hline
& & \\
Cluster & age & N$_{\rm OB}$ & N$_{\rm Be}$ \\
& & & \\
\hline
& & & \\
NGC 2362 & 5$\pm$2 & 33 & 0 \\
NGC 3293 & 9.1$\pm$0.2 & 136 & 2 \\
NGC 4775 & 11.7$\pm$1.5 & 121 & 3 \\
NGC 6231 & 3.7$\pm$0.6 & 129 & 0 \\
& & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Discussion}
The general picture which emerges from the analysis in the previous
section is the following: star forming clusters, associated to bright
emission nebulae, are rich in emission line stars, but they are much
likely pre-main sequence objects related to Herbig Ae/Be stars. When the
nebula dissipates the process of star formation stops -- at least with
regard to the massive stars -- and the clusters are devoid of
early-type emission line objects. Classical Be stars start to appear in
clusters with age of around 10 Myr, and reach their maximum abundance in
the 14-25 Myr interval. For older clusters the Be star abundance decreases
with age, as shown by Mermilliod (1982) and Grebel (1997).
The decreasing of the Be star abundance with the age after the 14-24 Myr
peak is a reflect of the dependence of the Be star abundance with the
spectral type. It is well known that the maximum abundance occurs for
spectral type B1-B2 (Zorec \& Briot 1997). Clusters older than 25 Myr have
their turnoff at type B3 or later, and hence they are expected to contain
lower abundances than clusters with B1-B2 main sequence stars. Clusters
older than 100 Myr have their turnoff at B8 or later, and the lack of Be
stars is an obvious reflect of the lack of any kind of B stars.
Conversely, the lack of Be stars in clusters younger than 10 Myr has
evident implications on the evolutionary status discussion. These clusters
have their turnoff at type B1 or earlier, and hence they have their B star
sequence complete, including the spectral types for which the Be star
abundance reach its maximum. The lack of Be stars in these clusters
implies that a Be star cannot be a very young object.
Be stars appear in clusters with turnoff at B1, and reach its maximum
abundance in clusters with turnoff at B2. As most of the Be stars belong
to these types, we have to conclude that Be stars are much closer to the
end of the main sequence than to the ZAMS.
This result contradicts the finding of Mermilliod (1982) and Slettebak
(1985), already mentioned in the introduction, who stated that Be stars
occupy the whole main sequence band from the ZAMS to the TAMS. This
afirmation is mainly based on photometric data, in the $UBV$ system, of Be
stars in open clusters. For a B star of a given subtype, the difference in
$(B-V)$ between its position at the ZAMS and the end of the main sequence
is lower than 0.1 mag. To firmly conclude that a Be star is in or near the
ZAMS, a photometric accuracy of 0.02 mag. for the underlying star of the
Be object would be required. If we consider all the problems which affect
photometry of Be stars, this accuracy seems not to be within reach.
Mermilliod (1982) use the $(U-B)$ colour, for which the diference between
ZAMS and TAMS is higher. But Be stars tend to move leftwards in the
$M_{V}$ - $(U-B)$ diagram due to the excess in the $U$ magnitude caused by the
circumstellar emission in the Balmer continuum (Kaiser 1989). The same
effect is present in the Str\"omgren $c_{1}$ index, as shown by Fabregat et
al. (1996). This effect can displace a strong emitter from TAMS to ZAMS
and even leftwards. Mermilliod already realized this effect when he states
that the $(U-B)$ colours are affected by the Be phenomenon.
Hence we consider our result based on the
analysis of Be star abundances in open clusters more reliable that
the results based on photometric data which are strongly affected by the
circumstellar continuum emission. On the other hand, both authors are
aware that most Be stars occur on the evolved part of the main sequence
(Mermilliod 1992) and considerably off the ZAMS (Slettebak 1985).
There is additional evidence indicating that Be stars are somewhat evolved
objects. In the younger clusters containing early-type Be stars, late type
Be stars are scarce or completely lacking. This was first noted by
Sanduleak (1979, 1990). He found 26 Be stars in NGC 663, and among them
only 2 later than B5. His objetive prism survey was complete to mag. 14,
which at the cluster reddening and distance reach the spectral type
B6-B7V. He concludes that Be stars in NGC 663 are primarily
confined to spectral types earlier than B5.
Capilla \& Fabregat (1999) performed CCD Balmer-line photometry of NGC
663, NGC 869 and NGC 884. Their images are deep enough to cover all the B
type range. They detected a total of 25 Be stars in the three clusters,
and among them only two are later than B5.
These results can be interpreted in the same evolutionary terms that
before. In clusters with ages in the interval 14-24 Myr, the stars earlier
than B5 have spent more than a half of its life in the main sequence,
while the late B stars are still in the first half of the main sequence
phase. The Be phenomenon accurs among the former and not among the latter.
\subsection{Be stars as post-mass-transfer binary systems}
It has been suggested that Be stars could be the result of the evolution
of close binary systems. The transfer of matter and angular momentum would
produce the spin up of the mass gainer to very high rotation rates. It is
well known that rapid rotation is a common characteristic of Be stars, and
hence a key ingredient of the Be phenomenon. The products of close binary
evolution are therefore good candidates to develope the Be phenomenon
(Pols et al. 1991). Moreover, several Be stars are definitely
post-mass-transfer systems. They are the Be/X-ray systems, in which a
neutron star orbits an early-type Be star, accreting matter from the dense
stellar wind and thereby generating X-rays. The properties of Be stars in
Be/X-ray binaries are not different from those of the rest of Be stars.
For a recent review of the properties of Be/X-ray binaries, see Negueruela
(1998).
Our conclusions on the evolutionary status of Be stars are consistent
with the hypotesis of the nature of Be stars as post-mass-transfer
systems. The Be phenomenon would occur after the mass transfer phase in
the evolution of a close binary. This would explain the scarcity of Be
stars in very young clusters: the necessary time for the main sequence
evolution of the primary star in the system has to be over before the mass
transfer begins and the Be star is formed, and hence a Be star cannot be
a very young object.
However, the interpretation of the Be phenomenon as the result of close
binary evolution faces important problems, both theoretical and
observational. The computations of Pols et al. (1991) only can account for
about half the population of Be stars. The recent study of Van Bever \&
Vanbeveren (1997) with updated models of close binary evolution reveal
that only a minority of the Be stars (less than 20\% and possibly as low
as 5\%) can be due to close binary evolution. On observational grounds,
the models of close binary evolution predict a population of Be star plus
white dwarf systems ten times more abundant than the Be/X-ray binaries.
These systems should be observable as low luminosity X-ray sources. The
search conducted by Meurs et al. (1992) failed in detecting the predicted
population of Be+WD systems.
Hence we have to conclude that, despite the consistency with the results
of our analysis, the model of the close binary evolution does not provide
a satisfactory explanation to the Be star phenomenon. Moreover, it has to
be considered that this model is ad hoc, because it only justifies the
formation of a rapidly rotating B star, but does not explain how the Be
phenomenon arises from it.
\subsection{Evolution through the main sequence}
The main conclusion of our study is that Be stars are evolved main
sequence stars, closer to the TAMS than to the
ZAMS. This would imply the existence of some evolutionary change able to
produce the Be star phenomenon during the main sequence lifetime.
This is not in agreement with the classical theory of stellar evolution,
which predicts that the main sequence is a quiet evolutionary stage in
which no major changes in the stellar structure occur.
However, in the modern literature there is a growing evidence of
important changes which occur during the main sequence stage. Lyubimkov
(1996,
1998) has shown that the abundance of helium and nitrogen in O and early B
stars increases during the main sequence. This change is not monotonic.
The initial helium abundance $He/H$ = 0.08-0.09 is maintained during the
first half of the main sequence lifetime. Subsequently, $He/H$ abruptly
increases approximately twofold in a short interval of relative ages $t/t_{\rm MS}$
(where $t_{\rm MS}$ is the main sequence lifetime) between 0.5 and 0.7,
and this enhanced $He/H$ remains constant until the main sequence stage is
complete. Recent evolutionary models take into account this effect, and
attribute the light element enhancement as due to early mixing
produced by rotationally induced turbulent diffusion (Denissenkov 1994;
Talon at al. 1997; Maeder 1997).
Our results are consistent with the Be phenomenon appearing at the same
evolutive age $t/t_{\rm MS}$ $\sim$ 0.5. To show this we have to keep in mind that
the highest percentage of Be stars corresponds to spectral types B1-B2
(Zorec \& Briot 1997).
The age of 10 Myr, where Be stars start to appear, correspond to an
evolutionary age of $t/t_{\rm MS}$ = 0.5 for a star of about 10 M$_{\odot}$ and solar
metalicity (this and next paragraph discussion is based on data
from Table 47 and Fig. 1 in Schaller et al. 1992). Such a star
at this
age has an spectral type of B1, where the abundance of Be stars reach its
maximum. In clusters younger than 8 Myr, stars are
reaching the TAMS at 20 M$_{\odot}$ or higher, and at spectral types earlier than
B1. The lower abundance of Be stars among these earlier types, and the
low relative number of such massive stars explain the paucity of Be stars
among these very young clusters. O8-9.5e stars can, however, start to
appear at as early an age as 3 Myr, which can explain the few
cases of clusters younger than 10 Myr with a few Be stars. NGC 6231,
discussed in Section 3, could be one of this cases.
The maximum Be star abundance occurs between 14 and 25 Myr. 14 Myr
correspond with $t/t_{\rm MS}$ = 0.6 for a 9 M$_{\odot}$ star, at spectral type B1, i.e. at
the begining of the maximum abundance of Be stars. 25 Myr is the end of
the main sequence lifetime for a 9 M$_{\odot}$ star, at spectral type B3. Stars of
lower mass reach the relative age of $t/t_{\rm MS}$ = 0.5 at spectral types of B3 or
later, where the abundance of Be stars decreases. This explains the
decreasing abundance of Be stars after an age of 25 Myr.
We propose the hypothesis that the Be phenomenon is an evolutionary
effect, appearing half way of the main sequence
lifetime, and is related to the light elements enhancement which occurs at
the same evolutionary phase. It is now widely accepted that magnetic
fields near the stellar surface could be the cause of the enhanced mass
loss which characterize the Be phenomenon. The mechanisms proposed to
explain the mixing
at this stage, which imply movement of plasma near the stellar surface,
coupled with the rotation of the star, could originate and maintain a
magnetic field via a dynamo related effect. The characteristic high
rotational velocity of Be stars would play a major role in: i./ inducing
the movement of matter via turbulent diffusion; and, ii./ enhancing the
magnetic field strenght via the dynamo effect. Hence our hypothesis
provides a natural explanation of the influence of high rotational
velocity in the Be phenomenon.
A direct proof of this hypothesis could be obtained by studying whether Be
stars have an enhanced helium abundance. Unfortunately, the contamination
of the photospheric spectrum by the circumstellar emission lines makes the
abundance analysis of Be stars an almost impossible task. Such analysis
can be performed in Be stars observed in a disk-loss phase, i.e., in a
phase in which the circumstellar disk has dissipated and the photospheric
spectrum is directly observable. This has been done by Lyubimkov et al.
(1997) for the Be star X Persei, and they obtained an enhanced helium
abundance of $He/H$ = 0.19. However, this case is not conclusive, because
X Persei is a Be/X-ray binary which in the past underwent mass transfer,
and hence it is not possible to know whether
the helium overabundance is due to internal processes in the current
primary star or whether it is caused by external accretion
from the original primary. Helium abundance studies of isolated Be stars
observed during disk-loss phases are required to proof our sugestions.
\section{Conclusions}
We have presented a study of the abundance of Be stars in open clusters
as a function of the cluster age, using whenever possible ages determined
through Str\"omgren $uvby$ photometry. For the first time in studies of
this kind we have considered separately classical and Herbig Be stars.
The main results obtained can be summarized as follows:
\begin{itemize}
\item Clusters associated to emitting nebulosities and undergoing stellar
formation are rich in emission line objects, which most likely are all
pre-main sequence objects. No bona fide classical Be star has yet been
identified among them.
\item Clusters younger than 10 Myr and without associated nebulosity are
almost completely lacking Be stars, despite they have a complete
unevolved B main sequence.
\item Classical Be stars appear at an age of 10 Myr, and reach the maximum
abundance in the age interval 14-25 Myr.
\end{itemize}
We have interpreted our results in the sense that the Be phenomenon is an
evolutionary effect which appears in the second half of the main sequence
lifetime of a B star. This conclusion is supported by other facts, like
the lack of late-type Be stars in young clusters rich in early-type Be
stars.
We propose the hypothesis that the Be phenomenon could be related to main
structural changes happening at an evolutionary age of $t/t_{\rm MS}$ = 0.5, which
also lead to the recently discovered non-monotonic helium abundance
enhancement. The semiconvection or turbulent difussion responsible of the
helium and nitrogen enrichment, coupled with the high rotational velocity,
can originate magnetic fields via the dynamo effect. It is now widely
accepted that many observed phenomena are due to Be star photospheric
activity related to the presence of magnetic fields. Our hypotesis
provides
a natural explanation of the relationship between the Be phenomenon and
the high rotational velocity characteristic of Be stars.
It should be noted, however, that our results on the Be star
frequencies in open clusters came from scarce and inhomogeneous sets of
data, and this leads to a somewhat speculative component in our
conclusions. To check our results and proposed explanations the following
observational data would be of critical importance:
\begin{itemize}
\item A systematic study of the Be star frequencies in a significant
number of clusters of different ages, both in the Galaxy and in the
Magellanic Clouds. It
would be of exceptional interest to know the abundances in Magellanic
Clouds clusters younger than 10 Myr and with high-mass stellar formation
finished yet.
\item The determination of cluster ages in an homogeneous system. We
propose for this purpose the use of the Str\"omgren $uvby$ system.
\item The determination of the helium and light elements abundance of Be
stars undergoing phases of disk-loss.
\item In a more general context, the determination of the helium
abundance in B stars with evolutionary ages $t/t_{\rm MS}$ $>$ 0.5. Early type B
stars in clusters with turn-off at B1-B2 would be specially well suited
for this purpose.
\end{itemize}
We are currently undertaking observational programs to address the
above questions.
To conclude, we would like to comment that in the recent years, many
relations and cross-links between classical Be stars and several other
types of peculiar hot stars have been put forward, clearly showing that
the Be phenomenon is not just an isolated problem of the stellar
astrophysics. On the other hand, the referred to results on light element
enhancements in the atmospheres of hot stars during their main sequence
lifetime are not compatible with the classical stellar evolutionary
models, and demostrate that there are important lackings in our knowledge
of massive star evolution. In this paper we propose that both phenomena
are related, and hence, the understanding of the Be phenomenon could be
the clue for the advance in our understanding of major issues in massive
star formation and evolution.
\begin{acknowledgements}
This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France, and the NASA's Astrophysics Data System Abstract
Service.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,803 |
{"url":"https:\/\/dsp.stackexchange.com\/questions\/58822\/why-do-additional-harmonics-arise-after-the-filter-bank","text":"# Why do additional harmonics arise after the filter bank?\n\nI'm trying to implement direct implementation of MDCT filter bank. So, some additional artefacts have appeared after all of the routines.\n\nThe spectrum of the initial signal looks like:\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nN = 8 #subbands\nL = 2*N\n\nh = np.zeros((L,1)) #skeleton for window function\nfor n in range(2*N-1):\nh[n] = np.sin((np.pi \/ (2 * N)) * (n + 0.5)) #window function\n\nw = [30, 100, 300]\na = [1.1, .5, .1]\nx = 0\n\nt = np.array([i for i in range(1,3001)])\/1000\nfor idx in range(len(w)):\nx = x + a[idx]*np.sin(2*np.pi*w[idx]*t)\n\nFFT = np.fft.fft(x)\namps = np.abs(FFT) \/ (len(FFT) \/ 2)\nfs = 1 \/ (t[1]-t[0])\nf = fs*np.array([i for i in range(int(len(x)))]) \/ len(x)\n\nplt.subplots(1, 1, figsize=(6, 4), dpi=150)\nplt.stem(f[:int(len(f)\/2)], amps[:int(len(f)\/2)])\nplt.ylabel('Magnitude of the FFT')\nplt.xlabel('Frequencies (Hz)')\nplt.grid(True)\nplt.show()\n\n# Analysis\nH = np.zeros((N,L)) #skeleton for analisis H matrix\npr = (L+len(x)-1) #lengh (number of coloumn) of signal after convolution\nAnalysis_Mat = np.zeros((N,pr)) #skeleton for matrix after convolution\nfor k in range(N): #rows\nfor n in range(L): #coloumns\nH[k,n] = h[n]*np.cos((np.pi\/N)*(k+0.5)*(n+0.5-(N\/2))) #analysis H matrix\nAnalysis_Mat[k,:] = np.convolve(x,H[k,:]) #convolution\n\n# Downsampling\nM = int(Analysis_Mat.shape[1] \/ N) #number of samles that sould be stay after downsampling\ncutmat = int(np.floor(M)*N) #number of rows that sould follow to downsampln block (cut rows that are not fold to N)\n\nAnalysis_Mat_DS = np.zeros((N, M))\nAnalysis_Mat = Analysis_Mat[:, :cutmat]\n\nfor k in range(N):\nAnalysis_Mat_DS[k, :] = Analysis_Mat[k,::N]\n\n# Upsampling\n\nAnalysis_Mat_US = np.zeros((N, Analysis_Mat_DS.shape[1]*N)) #skeleton for signal that should be after upsampling\nfor n in range(Analysis_Mat_DS.shape[1]):\nAnalysis_Mat_US[:,0+N*n] = Analysis_Mat_DS[:, n]\n\n# Synthesis\n\nR = L + Analysis_Mat_US.shape[1] - 1 #legth of signal (number o rows) that should be after convolution\nSyntesis_Mat = np.zeros((N,R)) #skeleton for signal that should be after convolution\nG = (np.fliplr(H))\/(N\/2) #Synthesis matrix\nfor k in range(N):\nSyntesis_Mat[k,:] = np.convolve(Analysis_Mat_US[k,:], G[k,:]) #convolution\ny = np.sum(Syntesis_Mat, axis=0)\n\nFFT = np.fft.fft(y)\namps = np.abs(FFT) \/ (len(FFT) \/ 2)\nfs = 1 \/ (t[1]-t[0])\nf = fs*np.array([i for i in range(int(len(x)))]) \/ len(x)\n\nplt.subplots(1, 1, figsize=(6, 4), dpi=150)\nplt.stem(f[:int(len(f)\/2)], amps[:int(len(f)\/2)])\nplt.ylabel('Magnitude of the FFT')\nplt.xlabel('Frequencies (Hz)')\nplt.grid(True)\nplt.show()\n\n\nHowever, the reconstructed spectrum looks like:\n\nIs it OK? Is there a mistake in my script or I don't understand the theory (or both)?\n\nCan anybody help me with this issue?\n\n\u2022 How did you design your prototype filter? Is it a perfect reconstructing one? \u2013\u00a0Uroc327 Jun 11 '19 at 7:37\n\u2022 Yes, there is an attempt to implement perfect reconstruction in my example. \u2013\u00a0vovenur Jun 11 '19 at 7:51\n\u2022 Ok... I suppose that's just a Gibbs phenomenon... \u2013\u00a0vovenur Oct 9 '19 at 13:55","date":"2021-01-23 07:34:17","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\": 0.4517843723297119, \"perplexity\": 12664.700697827104}, \"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\/1610703536556.58\/warc\/CC-MAIN-20210123063713-20210123093713-00071.warc.gz\"}"} | null | null |
New Hubble Space Telescope ultraviolet and optical imaging of the strong-lensing galaxy cluster SDSS J1531+3414 (z = 0.335) reveals two centrally dominant elliptical galaxies participating in an ongoing major merger. The interaction is at least somewhat rich in cool gas, as the merger is associated with a complex network of 19 massive superclusters of young stars (or small tidal dwarf galaxies) separated by ~1 kpc in projection from one another, combining to an estimated total star formation rate of ~5 M ☉ yr–1. The resolved young stellar superclusters are threaded by narrow Hα, [O II], and blue excess filaments arranged in a network spanning ~27 kpc across the two merging galaxies. This morphology is strongly reminiscent of the well-known "beads on a string" mode of star formation observed on kiloparsec scales in the arms of spiral galaxies, resonance rings, and in tidal tails between interacting galaxies. Nevertheless, the arrangement of this star formation relative to the nuclei of the two galaxies is difficult to interpret in a dynamical sense, as no known "beads on a string" systems associated with kiloparsec-scale tidal interactions exhibit such lopsided morphology relative to the merger participants. In this Letter, we present the images and follow-up spectroscopy and discuss possible physical interpretations for the unique arrangement of the young stellar clusters. While we suggest that this morphology is likely to be dynamically short-lived, a more quantitative understanding awaits necessary multiwavelength follow-up, including optical integral field spectroscopy, ALMA submillimeter interferometry, and Chandra X-ray imaging.
© American Astronomical Society. All rights reserved. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,249 |
NTT Bond Sale Sets A Record
Japan's Nippon Telegraph & Telephone Corp. will use the proceeds to pay for the acquisition of its mobile unit.
Author: Kim Iskyan
In the biggest local debt sale in history by a Japanese corporate issuer, a unit of Japan's Nippon Telegraph & Telephone Corp (NTT) last month placed 1 trillion yen (US$9.6 billion) of bonds. The deal was twice its original target size, and orders came to more than twice the final volume.
Proceeds of the bonds, raised by NTT Finance, would be used to help repay short-term borrowing for the issuer's March acquisition of the one-third stake in its mobile unit, NTT Docomo, that it didn't already own, the company said.
Demand for the offering was strong, in part because of the relatively attractive yields. The 10-year bond's yield, at 0.38%, was 36 basis points higher than a 10-year Japanese government bond.
Japan's corporate bond market is small relative to the size of its economy, as companies historically have preferred to borrow from banks. But the size of the NTT deal, in the context of a surging global corporate bond market that saw issuance jump 31% in 2020 to a record $3.3 trillion, suggests this may be changing.
This article appeared in issue January 2021 | {
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} | 8,955 |
{"url":"http:\/\/accessphysiotherapy.mhmedical.com\/content.aspx?bookid=467§ionid=40079845","text":"Chapter 4\n\nTherapeutic exercise is the foundation of physical therapy, and a fundamental component of the vast majority of interventions. Prescribed accurately, therapeutic exercise can be used to restore, maintain, and improve a patient's functional status by increasing strength, endurance, and flexibility. Therapeutic exercise enables the patient\/client to\n\n\u2022 remediate or reduce impairments\n\u2022 enhance function\n\u2022 optimize overall health\n\u2022 enhance fitness and well-being.\n\nWhen prescribing a therapeutic exercise program it is important to consider the functional loss and disability of the patient.\n\n### Energy Systems\n\nMuscles are metabolically active and must generate energy to move. The creation of energy occurs initially from the breakdown of certain nutrients from foodstuffs.\n\nClinical\nPearl\n\nThe energy required for exercise is stored in a compound called adenosine triphosphate (ATP). ATP is produced in the muscle tissue from blood glucose or glycogen. Fats and proteins can also be metabolized to generate ATP. Glucose not needed immediately is stored as glycogen in the resting muscle and liver. Stored glycogen in the liver can later be converted back into glucose and transferred to the blood to meet the body's energy needs.\n\nIf the duration or intensity of the exercise increases, the body relies more heavily on fat stored in adipose tissue to meet its energy needs.\n\nClinical\nPearl\n\nDuring rest and submaximal exertion, both fat and carbohydrates are used to provide energy in approximately a 60% to 40% ratio.\n\nTwo of the most important energy generating systems that function in muscle tissue include the anaerobic and aerobic metabolism, both of which produce ATP:\n\n\u2022 Anaerobic metabolism: this process metabolizes glucose to generate small amounts of ATP energy without the need for oxygen.\n\u2022 ATP-PCr system: used for ATP production during high-intensity, short duration exercise. Phosphocreatine (PCr) decomposes and releases a large amount of energy that is used to construct ATP.\n\u2022 The short-term energy system: provides energy for muscle contraction for up to 15 seconds.\n\u2022 Anaerobic glycolysis (glycolytic system): a major supply of ATP during high-intensity, short-duration activities.\n\u2022 Muscle glycogen is the initial substrate. Stored glycogen is split into glucose, and through glycolysis, split again into pyruvate acid and lactic acid as the end product, with no oxygen being directly involved. The energy released during this process forms ATP.\n\u2022 Although unable to produce as much energy per unit time as the phosphocreatine system (i.e., unable to sustain maximum sprinting speed), it lasts considerably longer before intensity must be further reduced.\n\u2022 The intermediate energy system: provides the majority of energy for a sustained performance lasting between 20 seconds and 2 minutes (sprinting 200 to 800 m).\n\u2022 Aerobic metabolism (oxidative system): if exercise continues beyond a certain point, the body can no longer rely solely on anaerobic metabolism and has to switch to this more complex form of carbohydrate and fat metabolism to generate ATP.\n\u2022 The long-term energy ...\n\nSign in to your MyAccess profile while you are actively authenticated on this site via your institution (you will be able to verify this by looking at the top right corner of the screen - if you see your institution's name, you are authenticated). Once logged in to your MyAccess profile, you will be able to access your institution's subscription for 90 days from any location. You must be logged in while authenticated at least once every 90 days to maintain this remote access.\n\nOk\n\n## Subscription Options\n\n### AccessPhysiotherapy Full Site: One-Year Subscription\n\nConnect to the full suite of AccessPhysiotherapy content and resources including interactive NPTE review, more than 500 videos, Anatomy & Physiology Revealed, 20+ leading textbooks, and more.","date":"2017-01-17 06:52:26","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\": 0.23802615702152252, \"perplexity\": 4547.828618204871}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560279489.14\/warc\/CC-MAIN-20170116095119-00014-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
package reborncore.common.world;
import org.apache.commons.lang3.Validate;
import java.util.HashMap;
import java.util.function.Supplier;
public class DataAttachmentRegistry {
private static HashMap<Class<? extends DataAttachment>, Supplier<? extends DataAttachment>> dataAttachments = new HashMap<>();
private static boolean locked = false;
public <T extends DataAttachment> void register(Class<T> clazz, Supplier<T> supplier){
Validate.isTrue(!locked, "Data attachment registry is locked!");
if(dataAttachments.containsKey(clazz)){
throw new UnsupportedOperationException("Data attachment already registered for " + clazz.getName());
}
dataAttachments.put(clazz, supplier);
}
public HashMap<Class<? extends DataAttachment>, Supplier<? extends DataAttachment>> getAllDataAttachments(){
Validate.isTrue(locked, "Data attachment registry is not locked!");
return dataAttachments;
}
public void lock(){
locked = true;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,264 |
Q: Show $\mathbb{S}^{2}/\mathbb{S}^{0}$ is homotopy equivalent to $\mathbb{S}^{2} \vee \mathbb{S}^{1}$. Let $X=\mathbb{S}^{2}/\mathbb{S}^{0}$ be the space resulting from collapsing one pair of antipodal points on a two-dimensional sphere to a single point. Let $Y$ be $\mathbb{S}^{2} \vee \mathbb{S}^{1}$. Specifically, $Y$ fits in the following pushout diagram.
$\require{AMScd}$
\begin{CD}
\ \{a\} @>{f}>> \mathbb{S}^{2}\\
@VgVV @VVV\\
\ \mathbb{S}^{1} @>>> Y
\end{CD}
Where $\{a\}$ is a one-point space with $f(a)=(1,0,0)$ and $g(a)=(1,0)$.
Are $X$ and $Y$ homotopy equivalent? If they are, how can I see it? Can it be solved using the fact that a space $X$ has the homotopy extension property with respect to a subspace $A$ if and only if $([0,1]\times A) \cup(\{0\}\times X)$ is a retract of $[0,1]\times X$?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,415 |
{"url":"https:\/\/www.mediatebc.com\/ah2e9kez\/e1d8af-the-area-of-a-circle-of-diameter-d-is","text":"The area of a circle calculator helps you compute the surface of a circle given a diameter or radius. A circle have a diameter of 4 inches, find its area. Area of Circle Definition. Also, we suggest using only one formula to find the area, which makes the use of the radius of the circle. Visual on the figure below: \u03c0 is, of course, the famous mathematical constant, equal to about 3.14159, which was originally defined as the ratio of a circle's circumference to its diameter. Why do we need the surface area of a circle calculators. If you know the area of the circle, divide the result by \u03c0 and find its square root to get the radius; then multiply by 2 to get the diameter. The area of a circle is nd? Area of a circle diameter. To find the area of a circle the formula is A=pi times radius sqaured. Friends.. ! i hope this helps :) You can transform this into r = \u221a(A\/\u03c0) cm. We know Area (A) Since the radius is 5 cm, the diameter is 10 cm, and pi x 10 cm is the answer, or 31.4159 cm. The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle \u2026 Assume that our fireplace is 4 ft wide, so we'd like to have the same rug length. Exercise worksheet on 'Find the area of a circle using the diameter.' There are several ways to achieve it. Important Solutions 3. Question Papers 10. The area of a circle calculator helps you compute the surface of a circle given a diameter or radius.Our tool works both ways - no matter if you're looking for an area to radius calculator or a radius to the area one, you've found the right place . Now Area = 200.96 . This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. If the diameter of the circle is known to us, we can calculate the radius of the circle, such as; r = d\/2 or R = D\/2. With AC as diameter, a semicircle is drawn. divide by pi. Specifically, a circle is a simple closed curve that d Step-by-step explanation: The area of a circle is given by . The distance between any point of the circle and the centre is called the radius. Crop circles. The area of a circle is pi times the square of its radius. How to use the area of a circle calculator? You have a semicircle (half of a circle). Area of a circle is \u03c0r^2 so: 100\u03c0=\u03c0r^2. Textbook Solutions 19009. Notice that the calculated area of the circle is same in both the methods. Example 1. = 3.14 \u00d7(12.25) = 38.465 cm2. Solution: Given, Radius of the circle (r) = 8 cm. \u2190 Prev Question Next Question \u2192 Related questions 0 votes. Similarly, the formula for the area of a circle is tied to \u03c0 and the radius:. Diameter of a Circle from Area Calculator A circle is a plane geometry figure, which is formed by combining a set of all points in a plane that are at a given distance from the center point. The diameter is always twice the radius, so either form of the equation works. Here is the real estate development of Sun City, Arizona, a circular town with a diameter of 1.07 kilometers. 5. A circle of radius = 5 or diameter = 10 or circumference = 31.42 inches has an area of: 5.06709 \u00d7 10 -8 square kilometers (km\u00b2) 0.0506709 square meters (m\u00b2) 506.709 square centimeters (cm\u00b2) This goes back to manipulating the formula for finding the area of a circle, A = \u03c0r 2, to get the diameter. Answer. Area of square = d Side of the square = \u221ad Diagonal of the square = \u221ad*\u221a2 =\u221a(2d) Diameter of the circle = Diagonal of the square = \u221a(2d) Radius of the circle = 1\/2 (\u221a2d) = \u221a(d\/2) Area of the circle \u2026 Area Of A Circle Using Diameter. A = 3.14 x 1 = 3.14 sq.cm . CISCE ICSE Class 9. A = 3.14 x 1 x 1. This tool can use a variety of metric and imperial units for lengths and area. Question Bank Solutions 15191. Advertisement Remove all ads. 100=r^2 . This will give us the radius, next is just to complete the area formula by squaring the radius and multiplying it by pi. In previous post you have found the information on how to find the area of a circle using its radius in the formula. The area of a circle can be calculated using the diameter or the radius with two different formulas: A = \u03c0r 2 or A = \u03c0(d\/2) 2, where \u03c0 is the mathematical constant approximately equal to 3.14, r is equal to the radius, and d is the diameter. If JavaScript is not working on your browser, or in case you are simply curious, here is the procedure for calculating the diameter of a circle from its area. When it comes to find the area of a circle, there are couple of techniques and formulas used for the purpose. This post is to explain how to find the area of a circle when its diameter is given, instead its radius. Area is defined as the region occupied by a two-dimensional shape. OB is the diameter of the smaller circle. 10=r. What is the area of the circle which has the diagonal of square as its diameter ? A. \u03c0d B. \u03c0d^2 C. 1\/4\u03c0d^2 D. 1\/2\u03c0d. (Take pi = 3.14 ). It didn't take long - your results are here! We suggest to use the same formula to find the area of a circle which uses radius in it. take the square root. Now we will learn about the area of the circle. An online calculator to calculate the circle diameter based on the area. The calculated result will have the same units as your input. 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Above example, but first find the area of a circular field whose diameter is given, instead radius...\n\nInteractive Rock Cycle Answers, The Wiggles It's Always Christmas With You Archive, Mercure Ginza To Tokyo Station, Rani Meghana Devi, Yakima Landing Pad 4, Yoga Warm Up Flow, Patient Management System Project In Php, Ancient Greece Spice Chart, Concierge In English, Kondaji Farzand Death, Texas Southern University Application, Different Words For Hesitant, Lob Acronym Meaning,","date":"2022-06-28 15:30:42","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\": 0.8204408884048462, \"perplexity\": 582.6469476099767}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103556871.29\/warc\/CC-MAIN-20220628142305-20220628172305-00245.warc.gz\"}"} | null | null |
\section{Introduction}
Decays of $B$ mesons to charmless final states provide us with a lot of
information
concerning the phases of the Cabibbo-Kobayashi-Maskawa matrix.
Consequently, these decays
have been the subject of numerous studies.
The problem is to extract relevant information from the data, since any such
procedure involves serious
uncertainties resulting from
our poor knowledge of the effects of strong interactions.
Many papers have been devoted to the analysis of $B \to PP$
decays (with $P$ denoting a pseudoscalar meson), in the hope that the abundance
of data will permit the determination of several weak and strong parameters
involved.
In recent papers \cite{BurasFleischer} it was shown that the data on
$B \to \pi \pi$ decays require the
presence of important nonfactorizable corrections
and hadronic interference effects if
the SM value of $\gamma \approx 65^o$
and the CP-averaged $B_d \to \pi ^0 \pi^0$
branching ratios recently measured by BaBar and Belle
\cite{bb} are used.
The authors of ref. \cite{BurasFleischer} show in a theoretically clean way
that the effective tree
and colour-suppressed amplitudes $\tilde{T}$ and $\tilde{C}$
governing the $ B \to \pi \pi $ decays
are roughly equal in absolute magnitudes,
thus contradicting the expectation that the latter amplitude
should be substantially suppressed.
In order to conform to this expectation, one has to admit that the
corrections
in the effective
amplitudes $\tilde{T}$ and $\tilde{C}$
arising from the usually neglected penguin contributions
are substantial (see also \cite{Alietal} and \cite{BURAS2004end}).
Thus, hadronic-level effects appear to invalidate the naive expectation of the
factorization prescription.
Indeed, final-state strong
interaction effects should contribute to the redefinition of the original
quark-diagram amplitudes, thus generating the usually neglected penguin
contributions referred to above (see refs. \cite{LZ20022003,Z2004}).
In general, better extraction of the relevant strong and weak parameters
requires considering a larger body of data. Recently, several analyses
appeared in which extraction of the angle $\gamma $ of the unitarity triangle
(UT) was attempted from a fit to all currently available data on
$B \to PP$
or $B \to PV$ decays (see e.g. \cite{Z2004,Cot,GR2004}).
Such analyses in the $B \to PP$ and $B \to PV$ sectors are usually performed
separately from one another. On the other hand, arguments may be given that
some of the parameters, introduced in these two sectors to take account of the
effects of strong interactions, are actually related. By combining
in a single analysis the information
from both sectors, one could then hopefully get an additional handle on
the previously undetermined parameters.
In this paper, we analyse the decays of $B$ to $\pi \pi$, $ \pi \rho $, and
$\pi \omega$
in an approach modeled on ref. \cite{BurasFleischer}.
The data on these decays
should provide sufficient
information on which to base the
analysis and comparison of the size of all factorization
amplitudes involved.
In principle, there is no need here to use the data on strangeness-changing
two-body decays of $B$ mesons, a welcome feature since $B\to \pi K$
decays exhibit various puzzles (in addition to substantial
contributions from electroweak penguins),
as analysed in \cite{BurasFleischer}.
In practice, however, the data on $B \to \pi \pi$ are
still not good enough, and we find it necessary
to determine the magnitude of penguin amplitude $P$
from $B^+ \to \pi^+ K^0$.
This transition and the related $B^+ \to \pi^+ K^{*0}$ decay constitute
the only places
wherefrom information from the strangeness-changing
sector enters into our analysis. The knowledge of $|P|$ permits the extraction of several
$B \to \pi \rho, \pi \omega$ parameters from the data
and the determination of amplitude
ratios, such as e.g. $\tilde{T}/P$ etc. (i.e. it gives the size of various
amplitudes in units of $|P|$).
In Section 2,
{{we set out our notation following refs \cite{BurasFleischer,BURAS2004end}
and, using the Summer 2004 data given by the HFAG
\cite{HFAGsummer2004} we repeat this part of the
$B\to \pi \pi $ analysis of ref. \cite{BURAS2004end} which is
relevant for our purposes.}}
We then present a simple yet illuminating formula which
expresses the ratio of penguin amplitudes involving loops with
different quarks in terms of the ratio
of factorization amplitudes $C/T$ and the parameters extracted from the data.
In Section 3, using the data on $B^+ \to \pi^+ K^{(*)0}$ decays,
we determine the absolute magnitude of penguin amplitudes
{{and formulate our main assumption concerning their ratios}}.
In Section 4, we proceed with an analysis of the $B \to \pi \rho$
and $B \to \pi \omega $ decays.
We find that there are two acceptable sets of solutions for the effective
colour-suppressed and tree amplitudes.
Assuming that the ratio of penguin amplitudes involving loops with different
quarks is well approximated by the formula of Section 2 with $C/T=0$, we
determine the tree and colour-suppressed factorization amplitudes in
$B \to \pi \rho, \pi \omega$.
In Section 5, we give
a $B \to PV$ analog of the formula given in Section 2,
expressing the ratio of penguin amplitudes in terms of the ratio of
factorization amplitudes in the $B \to \pi \rho, \pi \omega$ sector.
We combine this formula with its counterpart from Section 2 and
directly from the data determine both
$C/T$ and the ratio of penguin amplitudes involving loops with
different quarks.
Our conclusions are contained in
Section 5.
\section{Decays $B \to \pi \pi $}
\label{section2}
The $B \to \pi \pi$ amplitudes
may be expressed in terms of amplitudes
$P$ (penguin), $\tilde{T}$ (effective tree), and $\tilde{C}$
(effective colour-suppressed):
\begin{eqnarray}
\label{Btopipione}
\sqrt{2} A(B^+ \to \pi ^+ \pi ^0)&=&-[\tilde{T}+\tilde{C}]\\
\label{Btopipitwo}
A(B^0_d \to \pi ^+ \pi ^-)&=&-[\tilde{T}+P] \\
\label{Btopipithree}
\sqrt{2} A(B^0_d \to \pi ^0 \pi ^0)&=&-[\tilde{C}-P],
\end{eqnarray}
with:
\begin{eqnarray}
\label{Btopipi1}
P \equiv P_c \phantom{xxxxx}&=& A \lambda ^3{\cal{P}}_{tc}
\\
\label{Btopipi2}
\tilde{T}\equiv e^{i \gamma} (T-R_b P_u)&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{T}}-{\cal{P}}_{tu}) \\
\label{Btopipi3}
\tilde{C}\equiv e^{i \gamma} (C+R_b P_u)&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{C}}+{\cal{P}}_{tu}),
\end{eqnarray}
where the rightmost forms involve the definitions of refs
\cite{BurasFleischer,BURAS2004end},
from which we omitted the contributions due to the exchange amplitudes.
{{In principle, the latter
could be included by a mere redefinition of $P_{tu}$.
However, the general idea of this paper would then require an analogous
redefiniton of penguin amplitudes in $B \to \pi \rho, \pi \omega$, with the
relative size of penguin and exchange amplitudes unlikely to be the same as in
$B \to \pi \pi$. Only if exchange amplitudes are neglected (as
usually done and assumed hereafter) and $P_{tu}$ represents just the
difference between the top- and up-
penguins, our approach does not need additional parameters.}}
In Eqs (\ref{Btopipi1}-\ref{Btopipi2}), $A=0.83 \pm 0.02$, $\lambda = 0.224 \pm 0.0036$, $R_b=0.37\pm0.04$,
and $\gamma $
parametrize the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
$T \equiv A\lambda^3R_b \cal{T}$ and
$C \equiv A\lambda^3R_b \cal{C}$ involve
the strong amplitudes of colour-allowed and colour-suppressed
tree diagrams. Finally,
$P_q\equiv -\lambda ^{(d)}_c {\cal{P}}_{tq}= A\lambda^3{\cal{P}}_{tq}\equiv
A\lambda^3({\cal{P}}_t-{\cal{P}}_q)$ with ${\cal{P}}_k$
describing penguin strong amplitudes corresponding to
internal k-quark exchanges
($k \in \{t,c,u\} $), i.e. with the full penguin amplitude
given by $\lambda ^{(d)}_u {\cal{P}}_u + \lambda ^{(d)}_c {\cal{P}}_c +
\lambda ^{(d)}_t {\cal{P}}_t$, where $\lambda ^{(k)}_q = V_{qk}V^*_{qb}$,
and $V$ is the CKM matrix.
For flavour-symmetric final-state interactions (FSI),
the above formulae encompass
all elastic and inelastic FSI with the exception of those
represented by crossed diagrams
(see \cite{LZ20022003,Z2004}).
When the latter are taken into account, the
amplitudes $T$ and $C$ become mixtures of the colour-allowed and
colour-suppressed factorization amplitudes \cite{LZ20022003,Z2004}.
\subsection{Extraction of hadronic parameters}
Introducing the hadronic parameters {{of refs
\cite{BurasFleischer,BURAS2004end}}}:
\begin{eqnarray}
\label{dtheta}
d e^{i\theta} \equiv - e^{i
\gamma}P/\tilde{T}&=&\phantom{xx}-|P/\tilde{T}|e^{-i\delta_{\tilde{T}}}
=-P_c/(T-R_bP_u)\\
\label{xDelta}
x e^{i \Delta} \equiv
\tilde{C}/\tilde{T}\phantom{xx}&=
&|\tilde{C}/\tilde{T}|e^{i(\delta_{\tilde{C}}-\delta_{\tilde{T}})}
=(C+R_bP_u)/(T-R_bP_u)
\end{eqnarray}
where $\delta_{\tilde{T}},\delta_{\tilde{C}}$ denote strong phases
(in the convention in which
the strong phase of the penguin amplitude $P_c$ is assumed zero),
one can derive the following formulae relating the parameters just introduced
to the branching ratios ${\cal{B}}$ and asymmetries in $B \to \pi \pi$ decays
\cite{BurasFleischer}:
\begin{eqnarray}
\label{BR1}
R^{\pi \pi}_{+-}&=&\frac{1+2x \cos \Delta + x^2}{1-2d \cos \theta \cos \gamma
+d^2}\\
\label{BR2}
R^{\pi \pi}_{00}&=&\frac{d^2+2dx
\cos (\Delta -\theta)\cos \gamma + x^2}{1-2d \cos \theta \cos \gamma
+d^2}\\
\label{Adir}
A^{{\rm dir}}_{\pi ^+ \pi ^-}&=
&-\frac{2d \sin \theta \sin \gamma}{1-2d \cos \theta \cos \gamma
+d^2}\\
\label{Amix}
A^{{\rm mix}}_{\pi ^+ \pi ^-}&=
&\frac{\sin (2 \beta + 2 \gamma) -
2d \cos \theta \sin (2\beta +\gamma)+ d^2 \sin (2\beta )}
{1-2d \cos \theta \cos \gamma+d^2},
\end{eqnarray}
with \cite{HFAGsummer2004}
\begin{eqnarray}
\label{dataRpippim}
R^{\pi \pi}_{+-}&\equiv & 2 ~\frac{{\cal{B}}(B^{\pm}\to \pi ^{\pm}\pi ^0)}
{{\cal{B}}(B_d \to \pi ^+ \pi ^-)}~\frac{\tau_{B^0_d}}{\tau_{B^+}}=2.20\pm 0.31\\
\label{dataRpi0pi0}
R^{\pi \pi}_{00}&\equiv & 2 ~\frac{{\cal{B}}(B_d\to \pi ^0 \pi ^0)}
{{\cal{B}}(B_d \to \pi ^+ \pi ^-)}=0.66 \pm 0.13\\
\label{datadir}
A^{{\rm dir}}_{\pi ^+ \pi ^-}&=&-0.37 \pm 0.11 \\
\label{datamix}
A^{{\rm mix}}_{\pi ^+ \pi ^-}&=&+0.61\pm 0.14,
\end{eqnarray}
where, {{as in original papers \cite{BurasFleischer,BURAS2004end}}},
the asymmetries are estimated as weighted averages, in spite of
the BaBar and Belle results still not being fully consistent.
{{Since these averages have not changed much in comparison to
information available before Summer 2004, we believe that performing the
analysis of this paper for the BaBar and Belle asymmetries separately is
unwarranted.}}
Assuming $\beta =24^o$ and $\gamma =65^o$, one can determine $d$ and $\theta$
using Eqs (\ref{Adir},\ref{Amix}) and the experimental values of the
asymmetries from Eqs (\ref{datadir},\ref{datamix})
\cite{BurasFleischer}.
Two solutions for $(d,\theta)$ are obtained,
of which refs \cite{BurasFleischer,BURAS2004end}
accept only the one with small $d$,
the other one (with $d \approx 4.6 $) being excluded as it
leads to complex solutions.
With the updated values of asymmetry averages, {{we obtain}}
\begin{eqnarray}
\label{dexp}
d&=& 0.52^{+0.22}_{-0.14} \\
\label{thetaexp}
\theta&=& +(141^{+11}_{-12})^o.
\end{eqnarray}
In \cite{BurasFleischer} (\cite{BURAS2004end})
the corresponding values were: $d=0.49^{+0.33}_{-0.21}$ ($0.51^{+0.26}_{-0.20}$),
and $\theta=+(137^{+19}_{-23})^o$ ($+(140^{+14}_{-18})^o$).
We determine the errors as in \cite{BurasFleischer}, i.e.
the errors associated with the specific input parameter
(here: asymmetry or an $R^{\pi \pi}$
ratio ) are estimated by varying its value within $1\sigma $ while keeping
the other input parameters at their central values, with the individual
errors thus obtained subsequently added in quadrature.
The solution of Eqs (\ref{BR1},\ref{BR2},\ref{dataRpippim},\ref{dataRpi0pi0})
yields
\begin{eqnarray}
\label{xexp}
x&=&\ 1.13 ^{+0.16}_{-0.15}\\
\label{Deltaexp}
\Delta &=& -(55^{+17}_{-26})^o,
\end{eqnarray}
to be compared with $x=1.22^{+0.25}_{-0.21}$ ($1.13^{+0.17}_{-0.16}$),
and $\Delta = -(71^{+19}_{-25})^o$ ($-(57^{+20}_{-30})^o$)
in refs \cite{BurasFleischer} {{and \cite{BURAS2004end} respectively}}.
As in \cite{BurasFleischer},
we have discarded the second solution for $x$ and $\Delta$
(with $x \approx 0.96$, and $\Delta \approx +33^o $), since
the $A_{CP}(B^+\to K^0\pi^+)$ asymmetry it yields is of the order
of $0.2$ (using Eq.(\ref{ACPpiplusK0}) below), too large when compared with the experimental data
of \cite{HFAGsummer2004}: $A_{CP}^{exp}(B^+\to K^0\pi^+)=-0.02 \pm 0.034 $.
A similar argument was originally used in \cite{BurasFleischer} in connection with
$A_{CP}(B^+\to K^+\pi^0) $.
The experimental value of the $A_{CP}(B^+\to K^0\pi^+)$ asymmetry puts
a much stronger bound on the error in $\Delta-\theta$ (see below).
\subsection{Ratio of penguin amplitudes}
From Eqs (\ref{dtheta},\ref{xDelta}) one derives:
\begin{equation}
\label{PctoPu}
\frac{P_c}{P_u}=\frac{{\cal{P}}_{tc}}{{\cal{P}}_{tu}}=
-(1+\frac{C}{T})\frac{R_b~de^{i\theta}}{xe^{i\Delta}-\frac{C}{T}}
\end{equation}
Since the value of $|C/T|$ is expected to be of the order of $0.25$ only,
a good estimate of the ratio of penguin terms should be obtained by setting
$C/T=0$ above:
\begin{equation}
\label{PctoPuapprox}
\frac{P_c}{P_u}\approx -\frac{R_b~d~e^{i\theta}}{xe^{i\Delta}}
\approx (0.17^{+0.08}_{-0.05} ) ~e^{i (16^{+21}_{-28})^o}
\end{equation}
with
\begin{equation}
\label{Deltamintheta1}
\Delta-\theta = (+164^{+28}_{-21})^o
\end{equation}
where from now on we treat the errors of $d$, $\theta$, $x$, and $\Delta$
as independent, adding in quadrature the corresponding errors to
the moduli and phases {{of quantities depending on these parameters.}}
{{ Since the errors on $d$, $\theta$, $x$ and $\Delta$ are actually
inter-related, another possible treatment of errors would be to vary the
original four parameters ($R^{\pi \pi}_{+-}$, $R^{\pi \pi}_{00}$,
$A^{{\rm dir}}_{\pi ^+ \pi ^-}$, $A^{{\rm mix}}_{\pi ^+ \pi ^-}$) within their
error bars. We have not chosen this more involved
route since possible errors stemming from the lack of consistency among
the BaBar and Belle asymmetry results for $A^{{\rm dir}}_{\pi ^+ \pi ^-}$,
$A^{{\rm mix}}_{\pi ^+ \pi ^-}$ might render its
higher quality questionable. Error analysis of
this paper, with $R^{\pi \pi}_{+-}$, $R^{\pi \pi}_{00}$,
$A^{{\rm dir}}_{\pi ^+ \pi ^-}$, $A^{{\rm mix}}_{\pi ^+ \pi ^-}$ varied within
their error bars, could be repeated
when the input data are under better control. }}
From Eq. (\ref{dtheta}) one further obtains :
\begin{equation}
\label{TtoPcvalue}
\frac{T}{P_c}=R_b\frac{P_u}{P_c}-\frac{1}{d}e^{-i\theta}\approx
-\frac{1+x~e^{i\Delta}}{d~e^{i\theta}}\approx (3.6^{+1.4}_{-1.1})
~e^{i~(10 ^{+18}_{-15})^o}
\end{equation}
and
\begin{equation}
\label{TtoPuRbvalue}
\frac{T}{R_bP_u}\approx 1+ \frac{1}{x}e^{-i~\Delta}
\approx (1.68 ^{+0.19}_{-0.18})~ e^{i~ (26 ^{+8}_{-12}) ^o}
\end{equation}
which demonstrates that the size of the penguin correction term
($R_bP_u$) with respect to the
tree amplitude $T$ is large, as discussed in \cite{BurasFleischer}.
When small nonzero values of $C/T$ are admitted, the ratio $P_c/P_u$ receives
a correction term
\begin{equation}
\delta (P_c/P_u) \approx \frac{R_b~d~e^{i \theta}}{x ~e^{i\Delta}}
\left ( \frac{1}{x~e^{i\Delta}}-1 \right )\frac{C}{T}\approx 0.15
e^{-i~40^o}\frac{C}{T}
\end{equation}
whose inclusion, as shown by the r.h.s. above
(obtained by inserting the central values of $x$, $d$, etc.),
increases the errors in our estimate of $|P_c/P_u|$
given in Eq. (\ref{PctoPuapprox}) by some 20\% (for $|C/T| \approx 0.2$).
\section{Size of penguin amplitudes}
Given the current errors in the determination of the $|\tilde{T}/{P}|$ ratio
(c.f. Eq. (\ref{dexp})),
further information on the size of amplitudes considered in
Section {\ref{section2}} is best obtained from the decays of
$B^+\to \pi ^+ K^{0}$ ($B^-\to \pi ^- \bar{K}^{0}$) and
$B^+ \to \pi ^+ K^{*0} $ ($B^- \to \pi ^- \bar{K}^{*0} $)
as these are expressed
in terms of penguin amplitudes alone:
\begin{eqnarray}
A(B^+ \to \pi ^+ K^0)& = & \tilde{P}'\\
A(B^+ \to \pi ^+ K^{*0})& =& \tilde{P}'_P,
\end{eqnarray}
with
\begin{eqnarray}
\label{Pprim}
\tilde{P}'&=
&-\lambda ^{(s)}_u{{\cal{P}}_{tu}}-\lambda^{(s)}_c{{\cal{P}}_{tc}}\\
\label{PprimP}
\tilde{P}'_P&=
&-\lambda ^{(s)}_u{{\cal{P}}_{P,tu}}-\lambda^{(s)}_c{{\cal{P}}_{P,tc}},
\end{eqnarray}
where the subscript $_P$ ($_V$) for $B \to PV $ amplitudes denotes amplitudes
in which the spectator quark ends in the final $P$ ($V$) meson.
{{As already remarked in the Introduction this constitutes an implicit use of
SU(3), at present necessary, given the size of errors in the $\pi \pi$ sector.
In fact, knowing the absolute size of penguin amplitudes well
is important in the formulas of
Section 4 (see e.g. Eqs (\ref{tanThetaVkV}-\ref{lV})),
with the route via $K\pi$ etc. decays
yielding much smaller errors.
With the size of penguin amplitude depending upon the SU(3) assumption in question, the latter affects
also the extracted values of tree and colour-suppressed amplitudes.}}
\subsection{ \mbox {\boldmath ${B^+ \to \pi ^+ K^{0}}$}}
Let us consider the $B^+ \to \pi ^+ K^ {0}$ and $B^- \to \pi ^- \bar{K}^ {0}$
decays.
We introduce $P'_c$ (the analog of $P_c$):
\begin{equation}
\label{Pcprim}
P'_c=-\lambda ^{(s)}_c {\cal{P}}_{tc}=
-A \lambda ^2(1-\lambda ^2/2) {\cal{P}}_{tc}.
\end{equation}
Using Eq. (\ref{PctoPuapprox}) for the ratio of ${\cal{P}}_{tu}/{\cal{P}}_{tc}$
one obtains from Eqs (\ref{Pprim},\ref{Pcprim}):
\begin{equation}
\label{Pcprimcorrected}
\tilde{P}'=P'_c \left[ 1+\epsilon R_b \frac{{\cal{P}}_{tu}}{{\cal{P}}_{tc}} \right]
\approx P'_c\left[
1-\epsilon\frac{x}{d}\frac{e^{i\Delta}}{e^{i\theta}}
e^{i\gamma}
\right],
\end{equation}
where $\epsilon = \lambda^2/(1-\lambda ^2) = 0.05 $.
The CP-asymmetry $A_{CP}(B^+\to \pi^+ K^0)$ is approximately
\begin{equation}
\label{ACPpiplusK0}
A_{CP}(B^+ \to \pi ^+ K^0) \approx
\frac{-2 \epsilon \frac{x}{d}\sin (\Delta - \theta)\sin
\gamma}{1-2 \epsilon \frac{x}{d}\cos (\Delta - \theta)\cos
\gamma}
\end{equation}
which, together with its experimental value of $-0.02\pm 0.034$,
and $x/d=2.17 ^{+0.86}_{-0.70}$,
points towards
$\Delta -\theta \approx (+5 ^{+10}_{-9})^o$ or $(+174 \pm 11)^o$,
significantly improving upon the value of Eq. (\ref{Deltamintheta1}).
Thus, the experimental value of the $A_{CP}(B^+\to \pi^+ K^0)$ asymmetry
forces $\Delta - \theta$ to be close to $0^o$ or $180^o$, rejecting the
solution with $\Delta \approx 33^o$ (for which $\Delta -\theta \approx -108 ^o$).
Consequently, from now on, whenever
only $\Delta -\theta$ appears instead of $\Delta$ and $\theta$,
we shall use the average of the two determinations:
\begin{equation}
\label{Deltamintheta2}
\Delta - \theta =+(173 \pm 10)^o.
\end{equation}
Thus, the right hand side of Eq. (\ref{PctoPuapprox}) is replaced with
\begin{equation}
\label{PctoPuapproxnew}
\frac{P_c}{P_u}
\approx (0.17^{+0.08}_{-0.05} ) ~e^{i (7{\pm 10})^o}.
\end{equation}
The CP-averaged branching ratio of $B^+ \to \pi^+ K^0$ is approximately:
\begin{equation}
\label{CPaverBRBplustopiK}
{\cal{B}}(B^{\pm} \to \pi^{\pm} K^0(\bar{K^0}))\approx
|P'_c|^2
\left[
1-2\epsilon \frac{x}{d}
\cos (\Delta - \theta) \cos \gamma
\right].
\end{equation}
Using the experimental value of this branching ratio:
${\cal{B}}_{exp}(B^{\pm} \to \pi^{\pm} K^0(\bar{K^0}))=24.1 \pm 1.3$
(in units of $10^{-6}$),
$x/d=2.17 ^{+0.86}_{-0.70}$, and $\Delta - \theta = +(173 \pm 10)^o$,
one finds:
\begin{equation}
|P'_c|^2 \approx 22.1 ^{+1.3} _{-1.4}
\end{equation}
i.e.
\begin{equation}
\label{Pprimc}
P'_c=-4.70 ^{+0.14}_{-0.15},
\end{equation}
where the strong phase of $P'_c$ (originating from ${\cal{P}}_{tc}$)
is assumed zero by SU(3) symmetry with $P_c$.
One then finds
\begin{equation}
\label{Pcvalue}
P_c=-\frac{\lambda}{1-\lambda^2/2}P'_c=-(0.230 \pm 0.004) P'_c = 1.08 \pm 0.04
\end{equation}
\subsection{\mbox{\boldmath $B^+ \to \pi ^+ K^{*0}$}}
For the description of
$B^+ \to \pi ^+ K^ {*0}$ and $B^- \to \pi ^- \bar{K}^ {*0}$
decays, in analogy to $P'_c$ and $P_c$, we introduce:
\begin{equation}
\label{PPprimc}
P'_{P,c}=-\lambda ^{(s)}_c {\cal{P}}_{P,tc}=
-A \lambda ^2(1-\lambda ^2/2) {\cal{P}}_{P,tc}.
\end{equation}
{{To proceed we assume that}}
\begin{equation}
\label{universalpenguins}
\frac{{\cal{P}}_{P,tu}}{{\cal{P}}_{P,tc}}=\frac{{\cal{P}}_{tu}}{{\cal{P}}_{tc}}.
\end{equation}
{{The above equality follows if one accepts that the formation of
the final $PP$ or $PV$ pair is independent of penguin
transition occurring before that formation takes place.
This should be so if the intermediate
$s\bar{u}$ state does not remember how it was produced. The relevant
penguin-induced decay amplitude becomes then a product of a penguin term and the
amplitude describing the formation of the final state,
with the latter amplitude cancelling out in the ratios
${{\cal{P}}_{P,tu}}/{{\cal{P}}_{P,tc}}$
and ${{\cal{P}}_{tu}}/{{\cal{P}}_{tc}}$.
The above assumption is a crucial assumption upon which the rest of this paper
is based. }}
{{Assuming Eq. (\ref{universalpenguins}),}}
an analog of Eq.(\ref{Pcprimcorrected}) follows:
\begin{equation}
\tilde{P}'_P=P'_{P,c} \left[ 1+\epsilon R_b \frac{{\cal{P}}_{tu}}{{\cal{P}}_{tc}} \right]
\approx P'_{P,c}\left[
1-\epsilon\frac{x}{d}\frac{e^{i\Delta}}{e^{i\theta}}
e^{i\gamma}
\right].
\end{equation}
Consequently, from the experimental branching ratio of
${\cal{B}}_{exp}(B^{\pm} \to \pi^{\pm} K^{*0}(\bar{K^{*0}}))=9.76 ^{+1.16}_{-1.22}$
and using the analog of Eq. (\ref{CPaverBRBplustopiK}):
\begin{equation}
\label{CPaverBRBplustopiKstar}
{\cal{B}}(B^{\pm} \to \pi^{\pm} K^{*0}(\bar{K^{*0}}))\approx
|P'_{P,c}|^2
\left[
1-2\epsilon \frac{x}{d}
\cos (\Delta - \theta) \cos \gamma
\right]
\end{equation}
one determines that
\begin{equation}
\label{PPprimcvalue}
\left| \frac{P'_{P,c}}{P'_c}\right| ^2=
\frac{{\cal{B}}_{exp}(B^{\pm} \to \pi^{\pm}
K^{*0}(\bar{K^{*0}}))}{
{\cal{B}}_{exp}(B^{\pm} \to \pi^{\pm} K^0(\bar{K^0}))}
\end{equation}
Introducing the ratio $\xi \equiv P_{P,c}/P_c$
so that the $B \to PV $ amplitudes may be later expressed in units of
the $B \to PP$ penguin amplitude $P_c$
(to be compared with Eqs (\ref{TtoPcvalue},\ref{TtoPuRbvalue})),
one then finds that
\begin{equation}
\label{ksi}
\xi = P_{P,c}/P_c ={\cal{P}}_{P,tc}/{\cal{P}}_{tc}
={\cal{P}}_{P,tu}/{\cal{P}}_{tu}=P'_{P,c}/P'_c = 0.64 \pm 0.04
\end{equation}
with
\begin{equation}
P'_{P,c}=\xi P'_c=-2.99 \pm 0.21,
\end{equation}
where we adopted the convention of a vanishing strong phase for the $P'_{P,c}$
penguin amplitude (${\cal{P}}_{P,tc}$).
One then finds
\begin{equation}
\label{PPcvalue}
P_{P,c}=-(0.230 \pm 0.004) P'_{P,c} = 0.69 ^{+0.05}_{-0.05}.
\end{equation}
\section{Decays \mbox{\boldmath $B \to \pi \rho, \pi \omega $}}
Besides the amplitudes $P_{P(V)}$ ($P_{P(V),c}$) already considered
in the previous section,
strangeness-conserving decays of $B$ mesons
into a pseudoscalar-vector meson pair
introduce several further amplitudes: $T_{P(V)}$ (tree), $C_{P(V)}$
(colour-suppressed), $S_{P(V)}$ (singlet penguin), etc., of which
$T_{P(V)},P_{P(V)},C_{P(V)}$ are considered to be the dominant ones.
The Zweig rule suggests that $S_P$ should be negligible.
On the other hand, $S_V$ does not need to be,
in analogy to the situation in the
$B \to PP $ sector where decays $B \to K \eta (\eta ')$ seem to
indicate the non-negligible size of $S'$ \cite{Lipkin,RosnerLipkin}.
Since we want
to restrict our analysis to a group of decays akin to $B \to \pi \pi$,
i.e. not
involving singlet penguin amplitudes, we are left with the following
$B^0_d,B^+$ decay
channels to be considered:
$\pi ^+ \rho ^-$, $\pi ^- \rho ^+$, $\pi ^0 \rho ^+$, $\pi ^+ \rho ^0$,
$\pi ^0 \rho ^0$, $\pi ^+ \omega$, and $\pi ^0 \omega$
(together with CP conjugate processes).
By restricting our analysis to these decays, we do not need to introduce
the additional parameters related to the singlet amplitudes.
Our omission of the strangeness-changing $B \to PV$ decays is deliberate,
since such amplitudes seem to exhibit
some anomalous behaviour already in the $B \to PP$ sector
\cite{BF2000}.
The respective $B \to \pi \rho ,\pi \omega $ amplitudes are given by
(in sign convention used e.g. in ref.\cite{GR2004})
\begin{eqnarray}
\label{Amplpiplusrhominus}
A(B^0_d \to \pi ^+ \rho ^-)&=&-[\tilde{T_V}+ P_V]\\
\label{Amplpiminusrhoplus}
A(B^0_d \to \pi ^- \rho ^+)&=&-[\tilde{T_P}+ P_P]\\
\label{Amplpi0rhoplus}
A(B^+ \to \pi ^0 \rho ^+)&=&
-\frac{1}{\sqrt{2}}[\tilde{T_P}+\tilde{C}_V+P_P -P_V]\\
\label{Amplpiplusrho0}
A(B^+ \to \pi ^+ \rho ^0)&=&
-\frac{1}{\sqrt{2}}[\tilde{T_V}+\tilde{C}_P-P_P +P_V]\\
\label{AmplC1}
A(B^0_d \to \pi ^0 \rho ^0)&=&
-\frac{1}{2}[\tilde{C}_P+\tilde{C}_V-P_P -P_V]\\
\label{AmplC2}
A(B^+ \to \pi ^+ \omega)&=&
\frac{1}{\sqrt{2}}[\tilde{T}_V+\tilde{C}_P+P_P +P_V]\\
\label{AmplC3}
A(B^0_d \to \pi ^0 \omega)&=&
\frac{1}{2}[\tilde{C}_P-\tilde{C}_V+P_P +P_V]
\end{eqnarray}
where
\begin{equation}
\label{PPc}
P_{P(V)}\equiv P_{P(V),c}=A \lambda ^3 {\cal{P}}_{P(V),tc}\\
\end{equation}
and $\tilde{T}_{P(V)}$, $\tilde{C}_{P(V)}$ involve expressions similar to
Eqs (\ref{Btopipi2},\ref{Btopipi3}).
Following the arguments given in \cite{Lipkin} and used in previous discussions
\cite{RosnerLipkin}, we assume that $P_V=-P_P$, or, more precisely, that:
\begin{equation}
{\cal{P}}_{V,tc(u)}=
-{\cal{P}}_{P,tc(u)}.
\end{equation}
Then,
defining $P_{P,u}$ and $P_{V,u}$ in analogy to Eq. (\ref{PPc})
\begin{equation}
P_{P(V),u}=A \lambda ^3 {\cal{P}}_{P(V),tu}
\end{equation}
and inserting ${\cal{P}}_{P,tc(u)}=\xi {\cal{P}}_{tc(u)}$ from Eq. (\ref{ksi}),
the amplitudes $\tilde{T}_{P(V)}$ and $\tilde{C}_{P(V)}$
may be written as
\begin{eqnarray}
\label{TV}
\tilde{T}_V= e^{i \gamma} (T_V-R_b P_{V,u})&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{T}}_V+\xi {\cal{P}}_{tu}) \\
\label{TP}
\tilde{T}_P= e^{i \gamma} (T_P-R_b P_{P,u})&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{T}}_P-\xi {\cal{P}}_{tu}) \\
\label{CV}
\tilde{C}_V= e^{i \gamma} (C_V+R_b P_{V,u})&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{C}}_V-\xi {\cal{P}}_{tu})\\
\label{CP}
\tilde{C}_P= e^{i \gamma} (C_P+R_b P_{P,u})&=&
A \lambda ^3 R_b e^{i \gamma}({\cal{C}}_P+\xi {\cal{P}}_{tu})
\end{eqnarray}
with $P_{P,u}=\xi P_u=-P_{V,u}$.
In the above equations,
the rightmost entries are given in the form completely analogous
to that used in ref. \cite{BurasFleischer}, with
$T_{V(P)}=A \lambda ^3 R_b {\cal{T}}_{V(P)}$ and
$C_{V(P)}=A \lambda ^3 R_b {\cal{C}}_{V(P)}$
involving the strong amplitudes ${\cal{T}}_{V(P)}$ and ${\cal{C}}_{V(P)}$
of colour-allowed and
colour-suppressed tree diagrams.
Since the amplitude $P_P=P_{P,c}=-P_V$ is known (Eq. (\ref{PPcvalue})),
from Eq. (\ref{Amplpiplusrhominus}) and the knowledge
of experimental asymmetries and branching ratios for the
$B^0_d \to \pi ^+ \rho ^-$ decay one can determine the magnitude and
(relative) phase of amplitude $\tilde{T}_V$. Then,
using Eq. (\ref{TV}) with $P_{V,u}=-P_{P,u}$
and the estimate $P_{P,u}=P_{P,c} P_u/P_c \approx
-P_{P,c} xe^{i(\Delta-\theta)}/(R_b d) $
one can extract the tree amplitude $T_V$.
A similar procedure applied to Eqs (\ref{Amplpiminusrhoplus},\ref{TP}) yields
tree amplitude $T_P$.
A subsequent use of Eqs (\ref{Amplpi0rhoplus},\ref{Amplpiplusrho0})
should permit the determination of $C_V$ and $C_P$.
The remaining three equations (\ref{AmplC1},\ref{AmplC2},\ref{AmplC3})
provide additional constraints/check on the extracted values of
colour-suppressed amplitudes.
We now turn to the extraction of the relevant amplitudes from the data.
\subsection{Extraction of tree amplitudes \mbox{\boldmath $T_V$}
and \mbox{\boldmath $T_P$}}
In analogy with Eq. (\ref{dtheta}),
we first introduce the following parameters in the
$B\to PV $ sector:
\begin{eqnarray}
\label{TPfromTPtilde}
d_P e^{i\theta _P}&=&-e^{i \gamma} \frac{P_P}{\tilde{T}_P}
=-\frac{|P_P|}{|\tilde{T}_P|}e^{-i\delta _{\tilde{T}_P}}
=-\frac{P_{P,c}}{T_P-R_bP_{P,u}}\\
\label{TVfromTVtilde}
d_V e^{i\theta _V}&=&-e^{i \gamma} \frac{P_V}{\tilde{T}_V}
=+\frac{|P_P|}{|\tilde{T}_V|}e^{-i\delta _{\tilde{T}_V}}
=+\frac{P_{P,c}}{T_V+R_bP_{P,u}}.
\end{eqnarray}
For the CP-averaged branching ratio
\begin{equation}
\label{BV}
\overline{{\cal{B}}}(B^0_d\to \pi ^+ \rho ^-)=
\frac{1}{2}[{\cal{B}}(B^0_d\to \pi ^+
\rho ^-)+{\cal{B}}(\bar{B}^0_d \to \pi ^- \rho ^+)]
\end{equation}
and the CP-asymmetry
\begin{equation}
\label{AV}
A(B^0_d\to \pi ^+ \rho ^-)=
\frac{{\cal{B}}(\bar{B}^0_d \to \pi ^-
\rho ^+)-{\cal{B}}(B^0_d\to \pi ^+ \rho ^-)}
{{\cal{B}}(B^0_d\to \pi ^+
\rho ^-)+{\cal{B}}(\bar{B}^0_d \to \pi ^- \rho ^+)}
\end{equation}
one derives:
\begin{eqnarray}
\label{BCPV}
\overline{{\cal{B}}}(B^0_d\to \pi ^+ \rho ^-)&=&
\left[ 1+\frac{1}{d^2_V}-\frac{2}{d_V}\cos \theta_V \cos \gamma \right]
|P_P|^2\\
\label{ACPV}
A(B^0_d\to \pi ^+ \rho ^-)&=&\frac{{\mbox{\large $\frac{4}{d_V}$}}\sin \theta_V \sin \gamma}
{1+{\mbox{\large $\frac{1}{d^2_V}$}}-
{\mbox{\large $\frac{2}{d_V}$}}\cos \theta_V \cos \gamma}
\end{eqnarray}
with (Eq. (\ref{PPcvalue}))
\begin{equation}
\label{PPvalue}
P_P=0.69 \pm 0.05.
\end{equation}
Solving Eqs (\ref{BCPV},\ref{ACPV}) one gets
\begin{eqnarray}
\label{tanThetaVkV}
\tan \theta _V &=& \frac{k_V}
{\sin \gamma (\cos \gamma \pm
\sqrt{\cos ^2 \gamma +
l_V-1-k_V^2/\sin ^2 \gamma })}\\
\label{dVkV}
d_V & = & \frac{\sin \theta _V \sin \gamma}{k_V}
\end{eqnarray}
where we defined
\begin{eqnarray}
k_V &\equiv &\frac{A(B^0_d\to \pi ^+ \rho ^-)
\overline{{\cal{B}}}(B^0_d\to \pi ^+ \rho ^-)}{4 |P_P|^2}\\
\label{lV}
l_V &\equiv & \overline{{\cal{B}}}(B^0_d\to \pi ^+ \rho ^-)/
|P_P|^2.
\end{eqnarray}
With the experimental values:
\begin{eqnarray}
\overline{{\cal{B}}}(B^0_d\to \pi ^+ \rho ^-)&=&10.1 ^{+2.1}_{-1.9}\\
A(B^0_d\to \pi ^+ \rho ^-)&=& -0.47 ^{+0.13}_{-0.14}
\end{eqnarray}
taken from \cite{HFAGsummer2004}, after neglecting the correlations
with $B^0_d\to \pi ^- \rho ^+$, one obtains the following two solutions:
\begin{eqnarray}
\label{V1sol}
d_V=d_{V,1}&\equiv & 0.206 ^{+0.025}_{-0.022}\\
\theta_V=\theta _{V,1} & \equiv & (-34 ^{+12}_{-15})^o
\end{eqnarray}
or
\begin{eqnarray}
\label{V2sol}
d_V=d_{V,2}&\equiv & 0.239 ^{+0.037}_{-0.032}\\
\theta_V=\theta _{V,2} & \equiv & (-139 ^{+17}_{-13})^o.
\end{eqnarray}
{{From the CP-averaged branching ratio
$\overline{{\cal{B}}}(B^0_d\to \pi ^- \rho ^+)$ and asymmetry
$A(B^0_d\to \pi ^- \rho ^+)$ defined in analogy to Eqs (\ref{BV}, \ref{AV}),
using the experimental values
\begin{eqnarray}
\overline{{\cal{B}}}(B^0_d\to \pi ^- \rho ^+)&=&13.9 ^{+2.2}_{-2.1}\\
A(B^0_d\to \pi ^- \rho ^+)&=& -0.15 \pm 0.09
\end{eqnarray}
from \cite{HFAGsummer2004},
one similarly obtains:}}
\begin{eqnarray}
\label{P1sol}
d_P=d_{P,1}&\equiv & 0.174 ^{+0.019}_{-0.017}\\
\theta_P=\theta _{P,1} & \equiv & (-12 ^{+7}_{-8})^o
\end{eqnarray}
or
\begin{eqnarray}
\label{P2sol}
d_P=d_{P,2}&\equiv & 0.203 ^{+0.026}_{-0.023}\\
\theta_P=\theta _{P,2} & \equiv & (-166 ^{+9}_{-9})^o.
\end{eqnarray}
Further experimental constraints are given
by the parameters $S_{\rho \pi}$ and $\Delta S_{\rho \pi}$
extracted from the time-dependent studies of
$(B^0_d,\bar{B}^0_d)\to\rho ^{\pm}\pi ^{\mp}$,
and providing information on the relative phases of effective tree
amplitudes
$\tilde{T}_V$ and $\tilde{T}_P$.
In terms of our amplitudes of
Eqs (\ref{Amplpiplusrhominus},\ref{Amplpiminusrhoplus}) and their
CP-counterparts,
these parameters are expressed as follows:
\begin{eqnarray}
S_{\rho\pi}& =&(S_{+-}+S_{-+})/2\\
\Delta S_{\rho\pi}&=&(S_{+-}-S_{-+})/2
\end{eqnarray}
with
\begin{eqnarray}
S_{+- }&\equiv& \frac{2~{\rm Im}~\lambda^{+-}}{1+|\lambda^{+-}|^2}\\
S_{-+}&\equiv& \frac{2~{\rm Im}~\lambda^{-+}}{1+|\lambda^{-+}|^2}\\
\lambda ^{+-}&\equiv & e^{-2i\beta}\frac{A(\bar{B}^0_d\to\rho^+\pi^-)}
{A({B}^0_d\to\rho^+\pi^-)}=-e^{i(\theta_P-\theta_V-2\beta)}\frac{d_P}{d_V}
\frac{e^{-i\gamma}-d_Ve^{i\theta_V}}{e^{i\gamma}-d_Pe^{i\theta_P}}\\
\lambda ^{-+}&\equiv & e^{-2i\beta}\frac{A(\bar{B}^0_d\to\rho^-\pi^+)}
{A({B}^0_d\to\rho^-\pi^+)}=-e^{i(\theta_V-\theta_P-2\beta)}\frac{d_V}{d_P}
\frac{e^{-i\gamma}-d_Pe^{i\theta_P}}{e^{i\gamma}-d_Ve^{i\theta_V}}
\end{eqnarray}
{{The four pairs of solutions, i.e. $(P_1,V_1)$, $(P_1,V_2)$, $(P_2,V_1)$,
and $ (P_2,V_2)$, give the four predictions for $S_{\rho\pi}$ and
$\Delta S_{\rho \pi}$ gathered in Table \ref{table0}.}}
\begin{table}[t]
\caption{ {Four predictions for $S_{\rho\pi}$ and
$\Delta S_{\rho \pi}$}}
\label{table0}
\begin{center}
\begin{tabular}{ccccc}
\hline
\rule[-6pt]{0pt}{20pt} & case (a) & case (b)& case (c) & case (d) \\
&$(P_1,V_1)$ & $(P_1,V_2)$ & $(P_2,V_1)$ & $ (P_2,V_2)$\\
\hline
\rule[-6pt]{0pt}{24pt}$S_{\rho\pi}$
& $-0.27^{+0.07}_{-0.04}$ &
$-0.04^{+0.04}_{-0.04}$ & $0.00^{+0.04}_{-0.04}$ & $+0.32^{+0.04}_{-0.07}$ \\
\rule[-8pt]{0pt}{22pt} $\Delta S_{\rho \pi}$
& $+0.38^{+0.25}_{-0.24}$ &
$+0.73^{+0.16}_{-0.17}$ & $-0.73^{+0.19}_{-0.16}$ & $-0.38^{+0.22}_{-0.24}$\\
\hline
\end{tabular}
\end{center}
\end{table}
Since, according to the data \cite{HFAGsummer2004}:
\begin{eqnarray}
S_{\rho\pi}&=&-0.15\pm 0.13\\
\label{expDeltaSrhopi}
\Delta S_{\rho \pi}&=&+0.25 \pm0.13
\end{eqnarray}
cases (c) and (d) may be rejected.
Although case (a) is clearly the best, case (b) cannot be ruled out:
the difference of the two determinations of $\Delta S_{\rho \pi}$
(case (b), Eq. (\ref{expDeltaSrhopi})) is consistent with
zero at (slightly above) $2 \sigma$.
In both cases (a) and (b), we have
\begin{equation}
\label{TPtildetoPcvalue}
e^{-i\gamma }\tilde{T}_P/P_c=(3.68^{+0.46}_{-0.43})~e^{i (-168^{+8}_{-7})^o}.
\end{equation}
{{For $e^{-i\gamma }\tilde{T}_V/P_c$, the relative phase
$\delta \equiv {\rm Arg}(\tilde{T}_V/\tilde{T}_P)=\theta_P-\theta_V-180^o$,
and the relative size $\left| {\tilde{T}_P}/{\tilde{T}_V}\right|={d_V}/{d_P}$ of tree amplitudes,
one finds the results gathered in Table \ref{table0a}.}}
\begin{table}[t]
\caption{{Relative sizes and phases of effective tree amplitudes}}
\label{table0a}
\begin{center}
\begin{tabular}{ccc}
\hline
\rule[-6pt]{0pt}{20pt} & case (a) & case (b) \\
\hline
\rule[-6pt]{0pt}{24pt}$e^{-i\gamma }\tilde{T}_V/P_c $
& $(3.11^{+0.42}_{-0.39}) e^{i(+34^{+15}_{-12})^o}$ &
$(2.68^{+0.45}_{-0.40}) e^{i(+139^{+13}_{-17})^o}$ \\
\rule[-8pt]{0pt}{22pt} $\delta $
& $(-158 ^{+14}_{-17})^o$ &
$(-53 ^{+19}_{-15})^o$ \\
\rule[-8pt]{0pt}{22pt}
$\left|
{\tilde{T}_P}/{\tilde{T}_V}\right|$&$1.18^{+0.20}_{-0.16}$& $1.37^{+0.26}_{-0.22}$\\
\hline
\end{tabular}
\end{center}
\end{table}
These solutions should be compared with $\delta \approx -22^o$ and
$|\tilde{T}_P/\tilde{T}_V| \approx 1.46 $ obtained
in the favored fit of ref. \cite{GR2004}, and corresponding to our case (b).
One of the essential differences with ref. \cite{GR2004} is the fact that in our
approach the effective tree amplitudes $\tilde{T}_P$ and $\tilde{T}_V$ do not
correspond to the tree amplitudes $T_P$ and $T_V$ of the factorization
picture. Instead, the effective amplitudes
$\tilde{T}_P$ and $\tilde{T}_V$
involve substantial corrections to the factorization terms,
due to the presence of the $P_{P,u}$
($P_{V,u}$ ) part of the penguin amplitude (Eqs (\ref{TV},\ref{TP})).
Using Eqs (\ref{TPfromTPtilde},\ref{TVfromTVtilde}) one can determine the
tree amplitudes:
\begin{eqnarray}
\label{TPtoPcformula}
\frac{T_P}{P_c}&=&+\xi \left( R_b \frac{P_u}{P_c}-\frac{e^{-i\theta_P}}{d_P}
\right)\\
\label{TVtoPcformula}
\frac{T_V}{P_c}&=&-\xi \left( R_b \frac{P_u}{P_c}-\frac{e^{-i\theta_V}}{d_V}
\right)
\end{eqnarray}
From Eq.(\ref{TPtoPcformula}), using the estimate (see Eq.(\ref{PctoPuapprox}))
\begin{equation}
\label{estimRBPutoPc}
R_b\frac{P_u}{P_c}\approx - \frac{x}{d}e^{i(\Delta-\theta)}
\end{equation}
and Eq.(\ref{Deltamintheta2}), for both cases (a) and (b) one obtains:
\begin{equation}
\label{TPtoPcvalue}
\frac{T_P}{P_c}=(2.40^{+0.61}_{-0.61})e^{i(-157^{+15}_{-13})^o}.
\end{equation}
Similarly, from Eq.(\ref{TVtoPcformula}) one gets:
\begin{eqnarray}
\label{aTVtoPc}
{\rm case~(a)}~~~~~~~~~~~~~~~~~~~\frac{T_V}{P_c}&=
&(2.25^{+0.60}_{-0.50})e^{i(+58^{+23}_{-20})^o~}~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\label{bTVtoPc}
{\rm case~(b)}~~~~~~~~~~~~~~~~~~~\frac{T_V}{P_c}&=
&(3.91^{+0.71}_{-0.64})e^{i(+150^{+10}_{-12})^o}~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}
For case (a) one finds $|T_V/T_P|=0.94^{+0.41}_{-0.29}$, while for case (b):
$|T_V/T_P|=1.63^{+0.63}_{-0.38}$.
If the $B \to \rho$ and $B \to \pi$ formfactors
are similar, one expects (see \cite{GR2004})
that the ratio of $|T_V/T_P|$ should be approximately
equal to the
ratio of $f_{\pi}/f_{\rho}\approx 0.63 $, as $T_V$ ($T_P$)
involves a weak current producing
$\pi ^{\pm}$ ($\rho ^{\pm}$).
Thus, case (a) seems favoured again.
For both cases (a) and (b), however, one also estimates
from Eqs (\ref{TtoPcvalue},\ref{TPtoPcvalue})
that $|T_P/T| \approx 0.67 ^{+0.34}_{-0.25}$, which disagrees with the
simple expectation (c.f. \cite{GR2004}) of
$|T_P/T| \approx f_{\rho}/f_{\pi}=1.59 $
(while $|\tilde{T}_P/\tilde{T}|=\xi d/d_P=1.91^{+0.84}_{-0.56}$).
Still, one has to keep in mind that the above estimates are based on
Eq. (\ref{estimRBPutoPc}) which neglects terms
of order $C/T$.
\subsection{Extraction of colour-suppressed amplitudes
\mbox{\boldmath $C_P$} and \mbox{\boldmath $C_V$}}
In analogy with Eqs (\ref{TPfromTPtilde},\ref{TVfromTVtilde}),
we introduce the following parameters involving
colour-suppressed amplitudes $\tilde{C}_P$ and $\tilde{C}_V$:
\begin{eqnarray}
\label{yPdef}
y_P e^{i \Gamma _P}&=&-\frac{P_P}{\tilde{T}_V+\tilde{C}_P}e^{i\gamma}\\
\label{yVdef}
y_V e^{i \Gamma _V}&=&-\frac{P_V}{\tilde{T}_P+\tilde{C}_V}e^{i\gamma}.
\end{eqnarray}
The CP-averaged branching ratio for the $B^+ \to \pi ^+ \rho ^0$ decay and the
corresponding asymmetry are given by
\begin{eqnarray}
\label{eq119}
\bar{\cal{B}}(B^+\to \pi ^+ \rho ^0)&=&
\left( 4+\frac{1}{y^2_P}+4 \cos \gamma \frac{\cos \Gamma _P}{y_P}\right)
\frac{P^2_P}{2}\\
\label{eq120}
A(B^+ \to \pi ^+ \rho ^0)&=& - 4 \sin \gamma \frac{\sin \Gamma _P}{y_P}
\frac{1}{{4+{\mbox{\large $\frac{1}{y^2_P}$}}+4
\cos \gamma {\mbox{\large $\frac{\cos \Gamma _P}{y_P}$}}}}.
\end{eqnarray}
Using the experimental numbers of
\begin{eqnarray}
\bar{\cal{B}}(B^+\to \pi ^+ \rho ^0)&=& 9.1 \pm 1.3\\
A(B^+ \to \pi ^+ \rho ^0)&=&-0.19 \pm 0.11
\end{eqnarray}
one finds two solutions:
sol.~(P1)
\begin{eqnarray}
y_P=y_{P,1}&\equiv &0.195 ^{+0.028}_{-0.024}\\
\Gamma = \Gamma _{P,1}&\equiv & (+23 \pm 14)^o
\end{eqnarray}
and
sol.~(P2)
\begin{eqnarray}
y_P=y_{P,2}&\equiv &0.149 ^{+0.015}_{-0.014}\\
\Gamma _P = \Gamma _{P,2}&\equiv &(+163^{+10}_{-11})^o.
\end{eqnarray}
From Eq.(\ref{yPdef}) one has:
\begin{equation}
\label{CtildePformula}
\frac{\tilde{C}_P}{P_c}=-\xi \left( \frac{1}{y_P}e^{-i\Gamma _P}
+\frac{1}{d_V}e^{-i \theta _V}
\right)e^{i \gamma}.
\end{equation}
Putting the estimates of $y_P$, $d_V$ etc. into Eq.(\ref{CtildePformula}),
one obtains
the values of $\frac{\tilde{C}_P}{P_c}e^{-i\gamma }$ given in Table
\ref{table1}.
\begin{table}[t]
\caption{Effective color-suppressed amplitudes $\tilde{C}_P$
from $B^+ \to \pi ^+ \rho^0$ decays}
\label{table1}
\begin{center}
\begin{tabular}{ccc}
\hline
\rule[-6pt]{0pt}{20pt} $e^{-i\gamma }\tilde{C}_P/P_c $ & case (a) & case (b) \\
\hline
\rule[-6pt]{0pt}{24pt}sol. (P1)
& $(5.62^{+0.77}_{-0.84})~e^{i(- 175^{+11}_{-10})^o}$ &
$(1.11^{+1.08}_{-0.69})~e^{i(-155^{+37}_{-57})^o}$ \\
\rule[-8pt]{0pt}{22pt} sol. (P2)
& $(1.61^{+0.98}_{-0.65})~e^{i(- 17^{+39}_{-26})^o}$ &
$(6.15^{+0.77}_{-0.87})~e^{i(-5^{+9}_{-9})^o}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
For the $B^+ \to \pi^0 \rho ^+$ decays, one obtains formulas
completely analogous to (\ref{eq119}) and (\ref{eq120}), with
$y_P \to y_V$, and $\Gamma _P \to \Gamma _V$. Using the experimental branching
ratio and asymmetry:
\begin{eqnarray}
\bar{\cal{B}}(B^+\to \pi ^0 \rho ^+)&=& 12.0 \pm 2.0\\
A(B^+ \to \pi ^0 \rho ^+)&=&+0.16 \pm 0.13
\end{eqnarray}
one finds two solutions:
\begin{eqnarray}
{\rm sol.~ (V1)} ~~~~~~~~~~~~~~~~~~~~~~~~&&\nonumber\\
y_V=y_{V,1}&\equiv&0.165^{+0.025}_{-0.021}~~~~~~~~~~~~~~~~~~\\
\Gamma_V= \Gamma _{V,1}&\equiv&(-21^{+18}_{-19})^o
\end{eqnarray}
and
\begin{eqnarray}
{\rm sol.~ (V2)} ~~~~~~~~~~~~~~~~~~~~~~~~
&&\nonumber\\
y_V=y_{V,2}&\equiv &0.130^{+0.015}_{-0.013}~~~~~~~~~~~~~~~~~~\\
\Gamma _V =\Gamma _{V,2}&\equiv &(-163^{+15}_{-14})^o.
\end{eqnarray}
From Eq.(\ref{yVdef}) one has:
\begin{equation}
\label{tildeCVtoPc}
\frac{\tilde{C}_V}{P_c}=\xi \left( \frac{1}{y_V}e^{-i\Gamma _V}
+\frac{1}{d_P}e^{-i \theta _P}
\right)e^{i \gamma}.
\end{equation}
Putting
the estimates of $y_V$, $d_P$ etc. into Eq. (\ref{tildeCVtoPc}), one obtains
the values of $\frac{\tilde{C}_V}{P_c}e^{-i\gamma }$
given in Table \ref{table2}.
\begin{table}[t]
\caption{Effective color-suppressed amplitudes $\tilde{C}_V$
from $B^+ \to \pi ^0 \rho^+$ decays}
\label{table2}
\begin{center}
\begin{tabular}{cc}
\hline
\rule[-8pt]{0pt}{24pt}&$\frac{\tilde{C}_V}{P_c}e^{-i\gamma }$\\
\hline
\rule[-7pt]{0pt}{24pt}sol.~(V1)
&$7.53^{+0.84}_{-0.81}~e^{i(+ 17^{+11}_{-10})^o}$\\
\rule[-9pt]{0pt}{24pt}sol.~(V2)
&$2.47^{+1.17}_{-0.97}~e^{i(+ 117^{+28}_{-22})^o}$\\
\hline
\end{tabular}
\end{center}
\end{table}
By comparing the experimental branching ratio for $B^0_d \to
\pi ^0 \rho ^0$ and the bound on $B^0_d \to \pi^0 \omega$
(from \cite{HFAGsummer2004}) with the predictions of all
combinations of entries in Tables \ref{table1} and
\ref{table2}, one finds that only cases (a,P2,V2) and (b,P1,V2)
may be admitted. We shall refer to them as Solutions I and II respectively.
The corresponding predictions for the branching ratios
of $B^0_d \to \pi ^0 \rho ^0$ and $B^0_d \to \pi ^0 \omega$
are compared with the data in Table \ref{table3}.
{{Discrepancies with experiment observed in Table \ref{table3} suggest
that the assumptions
(in particular the SU(3) assumption of Section 3
and/or possibly Eq. (\ref{universalpenguins})),
which lead to the value of $P_P$ given in Eq. (\ref{PPvalue})
thereby affecting the
extracted size of colour-suppressed $B\to PV$ effective amplitudes,
might not be wholy adequate. One needs here a way of
estimating the size of $P_P$ in the strangeness-preserving sector,
which would be both sufficiently precise and
less assumption-dependent.}}
\begin{table}[t]
\caption{Branching ratios for $B^0_d \to \pi ^0 \rho^0$ and $B^0_d \to \pi ^0
\omega $ decays}
\label{table3}
\begin{center}
\begin{tabular}{cccc}
\hline
\rule[-8pt]{0pt}{24pt}&exp & Sol. I & Sol. II \\
\hline
\rule[-7pt]{0pt}{24pt}$B^0_d \to \pi ^0 \rho^0$
&$5.0 \pm 1.8$
&$1.68^{+0.83}_{-0.51}$&$0.86 ^{+0.90}_{-0.49}$\\
$B^0_d \to \pi ^0 \omega $
&$<1.2$
&$2.28 ^{+0.81}_{-0.65}$&$2.31 ^{+0.87}_{-0.80}$\\
\hline
\end{tabular}
\end{center}
\end{table}
Solution II, with $y_P \approx 0.195 $,
is fully consistent with the information gained from the branching
ratio ${\cal{B}}(B^+ \to
\pi ^+ \omega)=5.9 \pm 0.8$, which yields $y_P=0.20 \pm 0.02$.
Solution I,
with $y_P \approx 0.149 $, agrees
with $y_P$ determined from $B^+ \to \pi ^+ \omega$ at $2 \sigma$.
Thus, the $B^+ \to \pi ^+ \omega$ branching ratio
favors Solution II over Solution I.
We recall that it is just the opposite case with the values of
$S_{\rho \pi}$ and $\Delta S_{\rho \pi}$ which favor case (a)
(hence Solution I) over case (b) (Solution II) by $2 \sigma $.
Since for the
$B^0_d \to \pi ^0 \rho^0$ branching ratio,
as Table \ref{table3} shows,
the difference between experiment and theory
is $1.7 \sigma$ ($2.1 \sigma$) for Solution I (II),
one concludes
that Solution I describes the data slightly better than
Solution II.
The colour-suppressed factorization amplitudes may be estimated from
\begin{eqnarray}
\label{CPtoPcexact}
\frac{C_P}{P_c}&=& -\xi \left[
\frac{1}{y_P}e^{-i \Gamma _P}+\frac{1}{d_V} e^{-i \theta _V}+R_b\frac{P_u}{P_c}
\right]\\
\label{CPapproxCtoT0}
&\approx&-\xi \left[
\frac{1}{y_P}e^{-i \Gamma _P}+\frac{1}{d_V} e^{-i \theta _V}
-\frac{x}{d}e^{i(\Delta-\theta)}
\right],
\end{eqnarray}
and
\begin{eqnarray}
\frac{C_V}{P_c}&=& \xi \left[
\frac{1}{y_V}e^{-i \Gamma _V}+\frac{1}{d_P} e^{-i \theta _P}+R_b\frac{P_u}{P_c}
\right]\\
\label{CVapproxCtoT0}
&\approx&\xi \left[
\frac{1}{y_V}e^{-i \Gamma _V}+\frac{1}{d_P} e^{-i \theta _P}
-\frac{x}{d}e^{i(\Delta-\theta)}
\right].
\end{eqnarray}
The values of $C_P$ and $C_V$ obtained for both Solutions I and II assuming
$C/T=0$ (Eqs (\ref{CPapproxCtoT0},\ref{CVapproxCtoT0}))
are gathered in Table \ref{table4}.
\begin{table}[t]
\caption{Color-suppressed factorization amplitudes $C_P$ and $C_V$
obtained for $C/T=0$}
\label{table4}
\begin{center}
\begin{tabular}{ccc}
\hline
\rule[-6pt]{0pt}{22pt} & Sol. I & Sol. II \\
\hline
\rule[-6pt]{0pt}{22pt}$C_P$&${\rm Re}(C_P/P_c)=0.15^{+0.88}_{-0.86}$
&$C_P/P_c=2.40^{+1.09}_{-0.82} e^{i(-173^{+21}_{-24})^o}$\\
\rule[-8pt]{0pt}{20pt}&${\rm Im}(C_P/P_c)=-0.31^{+1.01}_{-1.01}$&\\
\hline
\rule[-8pt]{0pt}{24pt}$C_V$&\multicolumn{2}{c}{$C_V/P_c=2.05^{+1.39}_{-1.30} e^{i(+82^{+22}_{-24})^o}$}\\
\hline
\end{tabular}
\end{center}
\end{table}
Interestingly, with the central value of $|T_P/P_c|$ being $2.40$,
Solution I is consistent with a small value of $C_P/T_P$,
while for Solution II the $C_P/T_P$ ratio is of the order of 1.
On the other hand, given the central $T_V/P_c$ value of 2.25 (3.91) for
Solution I(II) respectively (Eqs (\ref{aTVtoPc},\ref{bTVtoPc})),
it is Solution II for which $C_V/T_V$ seems to be smaller.
One has to remember, however,
that in our calculations we used the values of $x$ and $d$
determined from the averages of not fully consistent asymmetries in the $\pi
\pi$ sector.
Furthermore, our estimate of errors in the determination of
$P_c/P_u$ did not include the errors due to nonvanishing $C/T$.
As remarked earlier,
for $|C/T| \approx 0.2 $ these corrections may increase the error
of $P_c/P_u$ by 20\%, affecting the ensuing discussion correspondingly
(see Section 5).
\section{Extraction of $C/T$ and $P_c/P_u$}
In the analysis performed so far, the ratio $C/T$ of the factorization
amplitudes in $B\to \pi \pi$ decays has been assumed negligible.
It turns out, however, that one can actually
{\em determine} the value of $C/T$ directly from the data, provided
one is willing to make an {{additional}} very plausible assumption.
Namely, we observe that the amplitudes $C$, $T$ in $B \to \pi \pi$ decay
and the amplitudes $C_V$, $T_V$ in $B \to \pi \rho, \pi \omega$ transitions
are due to the same process, namely
a decay of $b$ quark into a pion and a light quark.
The difference between the two colour-suppressed amplitudes $C$ and $C_V$
(and between the two tree amplitudes $T$ and $T_V$) should be
due only to
the fact that the amplitude for the recombination of the
freshly produced light quark with the spectator quark
depends on whether the two recombine into a pseudoscalar (in $B \to \pi \pi$)
or a vector meson (in $B \to \pi \rho, \pi \omega $). However,
this dependence on the recombination
amplitude should cancel in the ratios, i.e. in $C/T$ and $C_V/T_V$.
Consequently, we may assume that
\begin{equation}
\label{CtoTequalCVtoTV}
\frac{C}{T}=\frac{C_V}{T_V}.
\end{equation}
We now recall Eq. (\ref{PctoPu}), which correlates the ratio $P_c/P_u$
with the size of $C/T$. We seek a similar connection for the
$B \to \pi \rho, \pi \omega$ sector.
To this end, we observe that using the expressions (\ref{TP},\ref{CV})
for the effective
amplitudes in Eq. (\ref{yVdef}) we can write:
\begin{equation}
\label{TPtogetridof}
\frac{T_P}{P_c}+\frac{C_V}{P_c}=\xi \left(
2 R_b \frac{P_u}{P_c}+\frac{1}{y_V} e^{-i\Gamma_V}
\right).
\end{equation}
Now, Eqs (\ref{TPfromTPtilde},\ref{TVfromTVtilde}) may be rewritten as
\begin{eqnarray}
\label{TPanddP}
\frac{T_P}{P_c}&=&\xi \left(
R_b \frac{P_u}{P_c}-\frac{1}{d_P}e^{-i\theta_P}
\right) \\
\label{TVanddV}
\frac{T_V}{P_c}&=&\xi \left(
-R_b \frac{P_u}{P_c}+\frac{1}{d_V}e^{-i\theta_V}
\right).
\end{eqnarray}
From Eqs (\ref{TPtogetridof},\ref{TPanddP}) we determine
\begin{equation}
\label{CVandyVanddP}
\frac{C_V}{P_c}=\xi \left(
R_b \frac{P_u}{P_c}+\frac{1}{y_V}e^{-i\Gamma_V}+\frac{1}{d_P}e^{-i\theta_P}
\right).
\end{equation}
Dividing Eq. (\ref{CVandyVanddP}) by Eq. (\ref{TVanddV}) we obtain
\begin{equation}
\label{CVtoTVandPutoPc}
\frac{C_V}{T_V}=
\frac{R_b\mbox{\large $\frac{P_u}{P_c}$}
+\mbox{\large $\frac{1}{y_V}e^{-i\Gamma_V}$}+
\mbox{\large $\frac{1}{d_P}e^{-i\theta_P}$}}
{\mbox{\large $\frac{1}{d_V}e^{-i\theta_V}$}
-R_b\mbox{\large $\frac{P_u}{P_c}$}}.
\end{equation}
The above equation may be
rewritten in the form completely analogous to Eq. (\ref{PctoPu}), namely:
\begin{equation}
\label{PctoPuandCVtoTV}
\frac{P_c}{P_u}=-\left( 1+\mbox{\large $\frac{C_V}{T_V}$}\right)\frac{R_b }
{\mbox{\large $\frac{1}{d_P}e^{-i\theta_P}$}+\mbox{\large
$\frac{1}{y_V}e^{-i\Gamma_V}$}-
\mbox{\large $\frac{C_V}{T_V}\frac{1}{d_V}e^{-i\theta_V}$}}.
\end{equation}
In the denominator above, the first two terms partially cancel.
By assuming that $C_V/T_V$ is so small that the third term may be neglected, we
obtain a counterpart of the previous
estimate of $P_c/P_u$ given in Eq. (\ref{PctoPuapprox}):
\begin{equation}
\label{PctoPufromBtoPV}
\frac{P_c}{P_u}=-~\frac{R_b}{\mbox{\large $\frac{1}{d_P}e^{-i\theta_P}$}+\mbox{\large
$\frac{1}{y_V}e^{-i\Gamma_V}$}} \approx (0.10 ^{+0.05}_{-0.04}) e^{i(63
^{+22}_{-28})^o},
\end{equation}
which, despite the approximation involved, is consistent
with Eq. (\ref{PctoPuapprox}), and thus with a large value of
$P_u$ as compared with $P_c$ (and a small value of $C/T$).
If $C/T$ is assumed equal to $C_V/T_V$,
Eqs (\ref{PctoPu},\ref{PctoPuandCVtoTV})
may be solved for $C/T$ with the final result:
\begin{equation}
\label{CtoTfinal}
\frac{C}{T}=\frac{\mbox{\large $\frac{1}{d_P}e^{-i\theta_P}$}+
\mbox{\large $\frac{1}{y_V}e^{-i\Gamma_V}$}
-\mbox{\large $\frac{x}{d}e^{i(\Delta-\theta)}$}}
{\mbox{\large $\frac{1}{d_V}e^{-i\theta_V}$}-\mbox{\large
$\frac{1}{d}e^{-i\theta}$}}.
\end{equation}
Let us take the central values of the parameters
and discuss the denominator first.
Solutions I and II differ in their values for the parameters of the pair
($d_V$,$\theta_V$).
For Solution I (II), the first term in the denominator has
an absolute value
of around $4.9$ ($4.2$).
The second term
has an absolute value of around $1.9$. The sum of the terms in the denominator,
with phases taken into account, has an absolute value of $6.8$ ($4.3$)
for Solution I (II) respectively.
As for the numerator, our previous
considerations uniquely determined the values of the three numerator terms.
In particular, the absolute value of the first term is equal to
$1/d_P=1/d_{P,1}\approx 5.7$, that of
the second term is $1/y_V=1/y_{V,2}\approx 7.7$, while for
the third term it is equal to $x/d \approx 2.2$.
It is therefore non-trivial that with the central values of
phases taken into account,
the sum of
these terms is not large and has the absolute value of around $3.2$, leading to
the central value of $|C/T|$ for Solution I being $0.47$.
The sum of the three terms in the numerator is most sensitive to the value
of angle $\Gamma _V$. If $\Gamma _V$ is set at its $1 \sigma$ deviation value of
$-177^o$, the absolute value of the numerator becomes $1.3$ only.
For Solution I, the value of $|C/T|$ would then become equal to $0.2$.
When all of the errors are calculated,
{{one obtains the values given in Table \ref{tableCT}.}}
\begin{table}[t]
\caption{{Extracted values of $C/T$}}
\label{tableCT}
\begin{center}
\begin{tabular}{ccc}
\hline
\rule[-6pt]{0pt}{20pt} & Solution I & Solution II \\
\hline
\rule[-6pt]{0pt}{24pt}$C/T$
& $(0.47^{+0.33}_{-0.30})~e^{i(+47^{+24}_{-27})^o}$ &
$(0.75^{+0.52}_{-0.49})~e^{i(-30 \pm 24)^o}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
{{The determinations of $C_V/T_V=C/T$ given in Table \ref{tableCT}
may be compared with
the central value of $C_V/T_V=0.91 e^{i 24^o}$$(0.52 e^{-i 68^o})$
for Solution I (II) obtained in
Section 4 for $C/T=0$.}}
Thus, the previously obtained central value of $C_V/T_V$
gets significantly reduced (increased)
for Solution I (II).
Although in Solution II the central value of $|C/T|$ is now quite large
it is also compatible with $|C/T|$
of order $0.25$. Better data are clearly
required.
{{Other estimates of $C/T$ also lead to values of order 0.5. For example, in
ref. \cite{BurasFleischer} arguments in favor of $C/T = 0.5\times e^{i290^o}$
are given. Similarly, in their
recent SU(3)-symmetric fit to all $B \to PP$ decays,
Chiang et al. \cite{Alietal}
obtain the value $|C/T| = 0.46^{+0.43}_{-0.30}$.
}}
With the central values of $|C/T|$ in Table \ref{tableCT}
significantly larger
than the expected value of around $0.25$,
the original estimate of $P_c/P_u$, obtained in
Eq. (\ref{PctoPuapprox}) upon assuming $C/T=0$, could be
substantially affected.
Solving Eqs (\ref{PctoPu},\ref{PctoPuandCVtoTV}) for $P_c/P_u$, one obtains
\begin{equation}
\label{PctoPusamouzgod}
\frac{P_c}{P_u}=R_b~ d ~e^{i \theta}~\frac{\mbox{\large
$1-\frac{\kappa}{d}~e^{-i \theta}$}}
{\mbox{\large $1-\frac{\kappa}{d_V}~e^{-i \theta _V}$ }},
\end{equation}
where
\begin{equation}
\kappa=\frac{1+x e^{i\Delta }}{\mbox{\large $\frac{1}{d_P}e^{-i \theta_P}
+\frac{1}{y_V}e^{-i\Gamma_V}+\frac{1}{d_V}e^{-i\theta_V}$}}.
\end{equation}
Numerically,
for Solution I one finds:
\begin{equation}
\frac{P_c}{P_u}=(0.21^{+0.09}_{-0.06})~e^{i(44^{+19}_{-23})^o},
\end{equation}
which still bears resemblance to $(0.17^{+0.08}_{-0.05} )
~e^{i (16^{+21}_{-28})^o}$ of Eq. (\ref{PctoPuapprox}).
\noindent
On the other hand, for Solution II one obtains:
\begin{equation}
\frac{P_c}{P_u}=(0.71^{+1.52}_{-0.66})~e^{i(+20^{+87}_{-55})^o},
\end{equation}
with error estimates so large that they admit small values for both
$|P_c/P_u|$ and $|P_u/P_c|$. Again, there is here a strong dependence on
$\Gamma_V$, with larger values of $|P_c/P_u|$ attained when $\Gamma_V$ is set
at its $1\sigma$ deviation value of $-148^o$.
From Eqs
(\ref{TPtoPcformula},\ref{CPtoPcexact},\ref{PctoPusamouzgod}),
one can further determine the corresponding values of $C_P/T_P$.
{{They are gathered in Table \ref{tableCPTP}.}}
\begin{table}[t]
\caption{{Extracted values of ${C}_P/T_{P}$}}
\label{tableCPTP}
\begin{center}
\begin{tabular}{ccc}
\hline
\rule[-6pt]{0pt}{20pt} & Solution I & Solution II \\
\hline
\rule[-6pt]{0pt}{24pt}${\rm Re} (C_P/T_P)$
& $-0.24 ^{+0.23}_{-0.21}$ &
$+0.40 ^{+0.38}_{-0.30}$ \\
\rule[-8pt]{0pt}{22pt} ${\rm Im} (C_P/T_P)$
& $+0.03 ^{+0.36}_{-0.35}$ &
$+0.00 ^{+0.28}_{-0.26}$ \\
${C_P}/{T_P}$ & $(0.25^{+0.31}_{-0.21}) ~e^{i(173^{+67}_{-66})^o}$
& $(0.40^{+0.40}_{-0.26}) ~e^{i(0^{+36}_{-46})^o}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
{{ From Tables \ref{tableCT} and \ref{tableCPTP} we see that}}
it is Solution I which prefers
smaller central values of both $C_V/T_V=C/T$ and $C_P/T_P$.
With present errors, however, both Solutions I and II are still compatible with
$|C/T|$ and $|C_P/T_P|$ of around 0.25.
For completeness, we have also calculated the ratio $|T_V/T_P|$ obtaining
$0.96^{+0.19}_{-0.18}$ ($0.88^{+0.37}_{-0.26}$) for Solution I (II)
respectively
(to be compared with the value of $f_{\pi}/f_{\rho}\approx 0.63$
expected in \cite{GR2004}).
\section{Summary}
In this paper we performed a joint analysis of the $B \to \pi \pi$
and $B \to \pi \rho, \pi \omega$ decays with the aim of studying the
effects of the presence of two independent superpositions of penguin
amplitudes on the possible values of colour-suppressed and tree
factorization amplitudes.
{{Our analysis assumes that the formation of the final PP or PV pair is
independent of the penguin transition occurring before that formation takes
place. This constitutes a crucial assumption of our approach.}}
The analysis yields two sets of solutions for the effective colour-suppressed
($\tilde{C}_V,\tilde{C}_P$) and tree ($\tilde{T}_P,\tilde{T}_V$)
amplitudes in the $B \to \pi \rho, \pi \omega$ transitions,
with one solution weakly
favoured over the other one.
Assuming the $C/T$ ratio in $B \to \pi \pi$ to be negligible,
we estimated the ratio of the two superpositions of penguin amplitudes,
using it subsequently to
determine the values of $C_V,T_V,C_P,T_P$ from the data.
This procedure yielded two sets of numerical estimates for
$C_P/T_P$ and $C_V/T_V$.
By imposing the condition of equality for the ratios of $C/T$ and $C_V/T_V$
we determined the value of $C/T$ directly from the data.
The two solutions obtained are compatible both with a
value of $|C/T|$ of around 0.25 {{and with the estimates from literature
yielding $|C/T| \approx 0.5$}},
with errors still of the order of $0.3 - 0.4$.
The corresponding solutions for $P_c/P_u$
and $C_P/T_P$ have been given as well.
One of the solutions is preferred as it yields smaller central values of
both $C/T=C_V/T_V$ and $C_P/T_P$.
Discrimination between the solutions, and a
more precise determination of $C/T$, require better data.
{{When such data become available, a
well-defined
value may be extracted for $C/T$ along the lines similar to those
presented here
and compared with expectations and other estimates, providing us with
more information on the $C/T$ ratio and the expected connection between
penguin amplitudes in $B \to \pi \pi$ and $B \to \pi \rho, \pi \omega$ decays.}}
This work was supported in part
by the Polish State Committee for Scientific
Research (KBN) as a research project over the period of
2003-2006 (grant 2 P03B 046 25).
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{"url":"http:\/\/kmj.knu.ac.kr\/journal\/list.html?Vol=44&Num=1&mod=vol&book=journal&aut_box=Y&sub_box=Y&pub_box=Y","text":"<< Previous Issue Kyungpook Mathematical Journal (Vol. 44, No. 1) Next Issue >>\n Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 1\u2014155\nOriginal Articles\n A Note on Regularity in Matrix Semirings Shamik Ghosh Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 1\u20144\n On Some Generalization on the Absolute Norlund Summability Factors of Infinite Series W. T. Sulaiman Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 5\u201410\n Characterization on Some Related Convergent Series W. T. Sulaiman Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 11\u201419\n Commutative Group Algebras of Direct Sums of Countable Abelian Groups Peter Danchev Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 21\u201429\n The Extension Problem for Exponentially Convex Functions A. M. Zabel1, Maha A. Bajnaid2 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 31\u201439\n On Common Invariant Subspaces of Operators Il Bong Jung1 , Eungil Ko2 , Carl Pearcy3 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 41\u201445\n Conformally Recurrent Riemannian Manifolds with Harmonic Conformal Curvature Tensor Jin Ok Baek1 , Young Jin Suh1 , Jung-Hwan Kwon2 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 47\u201461\n Commutation Relations for Operators Ahmed Bachir , Ali A. Hashem Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 63\u201467\n On Fractional Bernoulli Numbers Will Y. Lee Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 69\u201475\n Pro-x Groups Julie C. Jones Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 77\u201481\n Representations of a Lie Group $G(0, 1)$ and Modified Laguerre Polynomials M. A. Pathan , Subuhi Khan Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 83\u201491\n Conformal Transformations of a Weyl Manifold Fumio Narita , Takafumi Satou Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 93\u201499\n B. Y. Chen's Inequality and Its Application to Slant Immersions into Kenmotsu Manifolds Ram Shankar Gupta1 , Iqbal Ahmad1 , S. M. Khursheed Haider2 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 101\u2014110\n Stability of a Functional Equation that Arises in the Theory of Conditionally Specified Distributions Soon-Mo Jung1 , Prasanna K. Sahoo2 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 111\u2014116\n Interpolation Problems on $Ax=y$ in CSL-Algebra Alg$\\mathcal L$ Young Soo Jo1 , Joo Ho Kang2 Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 117\u2014124\n Non-autonomous Inhomogeneous Boundary Cauchy Problems and Retarded Equations M. Filali , M. Moussi Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 125\u2014136\n Subalgebras of Some Classes of Hilbert Algebras Sergio Arturo Celani Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 137\u2014143\n A Fixed Point Theorem in Banach Algebras Involving Three Operators with Applications B. C. Dhage Kyungpook Mathematical Journal 2004 Vol. 44, No. 1, 145\u2014155","date":"2018-01-24 03:31:27","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\": 0.4062972962856293, \"perplexity\": 2444.962640526269}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084893300.96\/warc\/CC-MAIN-20180124030651-20180124050651-00081.warc.gz\"}"} | null | null |
Starting next month, you'll be able to shop five shades of the Ultralight Beams, including a prismatic pearl, rose gold, and copper. The gloss is pretty self-explanatory. You can swipe it onto lips for a glittery, almost-holographic finish, as seen below. No tea, no shade, but the packaging looks like a KKW version of Stila's beloved Glitter & Glow Shadows. Hopefully you won't get them mixed up in your makeup bag. I'm not 100 percent sure if the Ultralight Beam glosses are eye safe, but I'm sure they'd look stellar on lids, too.
You can sweep them onto lids for a foil-like finish as Kardashian West did for this promo shot. I also wonder if she'll use the new highlighters on her shins, thighs, and shoulder blades for the holiday season like she does with the highlight sticks.
I mean she did cover her whole body with the shimmery stuff, but like on an everyday basis do you think she'll use the highlighters beyond her face? Only time will tell.
The KKW Ultralight Beam Highlighters and Gloss launch on Friday, December 1, on kkwbeauty.com. In the meantime, I'll be playing The Life of Pablo on repeat.
Follow Devon Abelman on Twitter and Instagram. ,. | {
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The Petroglyphs Provincial Park is a fascinating landmark in Woodview, Ontario, Canada. The park has the largest collection of ancient First Nations petroglyphs, or rock carvings, in Ontario, which show aspects of the First Nations spirituality. The petroglyphs were believed to have been made between 900 and 1100 AD. The park is in a relatively rural location, close to Upper Stony Lake and the Kawartha Highlands Provincial Park, but it's only 190 km to Toronto. You can stay in this stunning location with amazing ancient history and still explore the city of Toronto as part of your vacation. Check out our list of the best hotels and vacation rentals near Petroglyphs Provincial Park.
Large 2 bedroom apartment in a quiet, secure building in a residential neighborhood. | {
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How to Talk to Your Man About Porn
Dan Armstrong
Nov 21, 2019|3 min
Dan Armstrong is the Corporate Communications Specialist at Covenant Eyes. Prior to joining the team, he led the marketing department of a public policy think tank and also anchored and reported television news for NBC stations in Michigan. Dan and his wife have five children.
Sabrina Parr made headlines when she began dating two-time NBA champion and Dancing With The Stars contestant Lamar Odom in the summer of 2019. The fitness trainer and life coach has been an athlete her whole life. In high school, she was a two-time state track champion and five-time All American at Collinwood High School in Ohio. She's a business owner and entrepreneur.
Sabrina loves competition. She thrives on it. It gets her out of bed in the morning, pushes her in the gym to be better than the day before, and challenges her to improve daily. But there was one thing she couldn't compete with: porn.
"He was reaching for his phone more than reaching for me," said Sabrina about Lamar. "I had a problem with that. I want to be 100% desired especially if I'm lying right next to (him). I really didn't understand why that was."
Sabrina didn't understand the addictive pull pornography has on people—the unlimited selection of anyone you want, doing whatever you want, with no commitment or fear of rejection.
"It did give me a sense of insecurity of just questioning what I need to do or who I need to be," said Sabrina. She wondered how she could change her appearance, actions and behaviors, or tone to make her more attractive to Lamar. It gave her a lot of anxiety and discomfort.
After some thought, she realized it wasn't her problem. She could not become something unrealistic. She realized Lamar needed to get help.
"Anytime I would bring this up to him, he would experience a lot of shame," said Sabrina. "(He) did not want to talk about it. Just kind of shut down, didn't have the answers. And later I learned he didn't have the answers because he's never been asked these questions. Ever."
Her initial confrontations led to the opposite effect she was hoping for. "I think the shame created more pornography watching. It was just a vicious cycle," she said.
Sabrina said a dishonest relationship wasn't worth staying in. "My first initial instinct was to leave. I don't want to deal with this. I shouldn't have to deal with this. Look at me. I'm enough."
She said it was difficult to understand why Lamar was turning to porn, but soon realized he was using it to deal with his pain from the past.
Lamar's mother died when he was 12 years old. His dad was a drug addict. His son died at six months old. He medicated the pain with drugs, porn, and women–a lifestyle that nearly killed him. In 2015, he experienced 12 seizures and six strokes while at the hospital. His loved ones were told to say their final goodbyes. He eventually recovered and began working on changing his life for the better.
"I learned why he was doing it. Instead of just saying you're doing this and this, okay, let me get to the why. So now we're just dealing with things that trigger him and replace the response with healthy choices, which is still a work in progress, trying to figure out what works instead of porn," said Sabrina.
They realized conflict and loneliness triggered Lamar to watch porn. "I had to stand my ground to not become his porn," said Sabrina. She told him, "You have to fight through these urges, Lamar. You have to find something else. And he would do that."
Sabrina works alongside of him to help him get through his urges. She's his ally. When they notice temptation arising, they go for a walk, exercise, watch a movie, or get some ice cream together. They also installed Covenant Eyes on Lamar's devices, so he knows Sabrina will see what he sees on his screens.
Porn isn't a popular topic to talk about as a couple. It's common to not discuss it at all. For men, it can be embarrassing, uncomfortable, and confrontational. Women also face a plethora of reasons why they don't bring it up to their partners, and they often feel that it could end their relationship.
Sabrina understands why women are hesitant to talk to their partners about their porn use, but she's convinced women must bring it up to help the men in their lives. (Still not sure how to start the conversation, get some more tips in this article.)
"(You) just got to be not afraid to have conversations about what's bothering you. Try to have a conversation with him. An honest conversation. I think you'll know if he's lying depending on how long you've been in the relationship with that person. Just try to talk to them and, without judging, don't come off, 'Why would you be…,' like understanding, like, 'Help me understand why you watch this. What does this do for you? And how can I help?'"
"Just watching Lamar learn these things is the turn on. It's what makes me attracted. To watch him overcome this and not be like 'oh, we're dull because this porn is gone.' He's found other ways," said Sabrina. "That was the difference between a very low relationship to now we're very high."
Watch Our Interview with Lamar
Covenant Eyes sat down with Lamar and Sabrina to hear their story. During our interview, Lamar opens up about his decades long porn addiction, including:
When he first discovered porn
How he came to prefer porn over real women
How porn hurt his sports career and cost him thousands in fines
The role of honesty and accountability in recovery
Sign up for Covenant Eyes
Over a million people have used Covenant Eyes Accountability and Filtering software to protect themselves and their families in the fight against pornography. With our 30 Day Money Back Guarantee, there's no reason not to start protecting you and your family today.
How Boundaries Saved Our Marriage (Part 3) | {
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**ALSO BY HAROLD EVANS**
_They Made America:
From the Steam Engine to the Search Engine; Two Centuries of Innovators_
(with Gail Buckland and David Lefer, 2004)
_War Stories:
Reporting in the Time of Conflict from the Crimea to Iraq_ (2003)
_The BBC Reports: On America, Its Allies and Enemies, and the Counterattack on Terrorism_
(BBC Corporation and Harold Evans, 2002)
_The Index Lecture: View from Ground Zero_
(lecture prepared for the Hay-on-Wye Festival, 2002)
_The American Century_ (2000)
_Good Times, Bad Times_ (1984)
_Eyewitness_ (1981)
_Suffer the Children: The Story of Thalidomide_ (with the _Sunday Times_ Insight Team, 1979)
_We Learned to Ski_ (with Brian Jackman and Mark Ottoway, 1975)
_The Freedom of the Press:
The Half-Free Press_ (with Katharine Graham and Lord Windlesham, 1974)
_Editing and Design_ (five volumes: _Essential English, Newspaper Design, Text Typography, Newspaper Headlines,_ and _Pictures on a Page,_ with Edwin Taylor, 1973)
_The Active Newsroom_
(with the International Press Institute, 1961)
Copyright
Copyright © 2009 by Harold Evans Associates LLC
All rights reserved. Except as permitted under the U.S. Copyright Act of 1976, no part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.
Little, Brown and Company
Hachette Book Group
237 Park Avenue, New York, NY 10017
Visit our website at www.hachettebookgroup.com
www.twitter.com/littlebrown
First eBook Edition: November 2009
Little, Brown and Company is a division of Hachette Book Group, Inc.
The Little, Brown name and logo are trademarks of Hachette Book Group, Inc.
Credits for endpaper photographs: taking an oath (top left): Dafydd Jones; with Tina on _Queen Mary 2:_ Cunard; Rupert Murdoch (under _Good Times, Bad Times_ ): Jane Bown; caricature: Gerard Scarfe; with Tina (under Murdoch): George Brown; with Henry Kissinger: UPI; ship launch (bottom left): Colin Theakston; skiing: Bryan Wharton; sketch of St. Ann's (top right): _Manchester Evening News,_ Bert Hackett; with President Clinton on Air Force One: Official White House Photo; thumbs up on winning Crossman trial, with Mrs. Anne Crossman and Graham Greene: Press Association; RAF office with German prisoner Walter Greis (next to _Traveler_ ): Herbert Gale; editing: Neil Libbert; St. Mary's Road Central School (bottom right): courtesy of Newton Heath Historical Society and Peter Charlton.
Copyright acknowledgments appear on page .
ISBN: 978-0-316-09207-4
_Dedicated to my granddaughters, Emily and Anna_
Contents
Copyright
BOOK ONE: Vanished Times
1: Grains of Truth
2: Getting Up Steam
3: First, Know Your Enemy
4: Hot Metal
5: How I Won the War
6: Non Nobis Solum
7: The Sting of Disraeli's Gibe
8: Stop Press
9: Why Aren't Their Women Wearing Our Frocks?
10: Adventures in the Land of Opportunity
11: From Delhi to Darlington
12: Just Causes
BOOK TWO: Scoop, Scandal, and Strife
13: The Rolls-Royce of Fleet Street
14: The Third Man
15: Children on Our Conscience
16: Space Barons
17: Death in Cairo
18: Divided Loyalties
19: Showdowns
20: My Newfoundland
Illustrations
Acknowledgments
Principal Sunday Times Books
Bibliography
About the Author
**The Vanished Newspaper Office**
If you had x-ray eyes and looked on any newspaper building in most of the years of our vanished century, this is what you'd have seen behind the brick facade: the stacked floors of worker ants and machinery transmuting the typewritten word into newsprint. Starting in the sixties in the United States but delayed until the eighties in Britain, computer terminals replaced the typewriters and editing pencils (1st floor) and the Linotype machines (3rd floor).
* * *
**Basement**
The reel room and the presses. One pressman is objecting to the shape of the plate he is supposed to fit on the rotary press and another is having too good a time to notice a coworker has disappeared.
**Ground floor (left)**
The foundry casts plates for the presses in between a hand of cards. Right, the newspaper bundles on the way to the world.
**1st floor**
On the left, the reporters and copy-takers; center, the subs; and right, the wire room collecting agency dispatches and photos. The big man in the subs room collaring a copy boy is the chief sub-editor, who designs the pages and gives orders to galley slaves of sub-editors who sit in front of him with their spikes and glue pots and perplexities of copy-fitting and headline writing. They feed edited copy to the pneumatic tube, sustained all day by infusions of tea.
**3rd floor**
Linotype operators
**Top floor**
Proofreaders
* * *
I've long treasured the drawing by an art student at Manchester University (who preferred to be known just as "Caesar"). Attracted by his work in the student publications when I was covering the university, I commissioned him to come to the _Manchester Evening News_ to sketch the operations for my stint in the fifties as editor of the spoof newspaper the _Manchester Guardian and Evening News,_ put out during the university's annual "Rag" week of raising money for charity. "Caesar's drawing" prefigured Martin Handford's "Where's Waldo?" puzzle games—and, yes, I'm in there somewhere, in prominent spectacles.
Knowledge will forever govern ignorance and a people who mean to be their own governors must arm themselves with the power which knowledge gives.
_—James Madison to W. T. Barry,
August 1822_
# BOOK ONE
_Vanished Times_
#
# Grains of Truth
The most exciting sound in the world for me as a boy was the slow whoosh-whoosh of the big steam engine leaving Manchester Exchange station for Rhyl in North Wales. Every year as summer neared I counted the days to when the whole family—six of us then—would escape the bleakness of northern winters, taking the train for a week at the seaside, buckets and spades in hand.
I was nearly twelve the summer I saw the bodies of the soldiers scattered about the sands.
The soldiers were so still, their clothing so torn, their faces so pale, they looked as if they had died where they fell. And yet they had escaped death, unlike thousands of their comrades left on the battlegrounds of northern France; thousands more were on their way to years in German internment camps. The men I saw were the lucky ones, a few hundred of the 198,229 of the British Expeditionary Force (BEF) who just days before in May–June 1940 had fought their way to Dunkirk. Twenty-four hours before we saw them, they had been on that other beach, being hammered from the air by Stuka dive-bombers, strafed by the machine guns of Messerschmitts, rescue ships ablaze offshore, and every hour the German panzers closing the ring. They were a forlorn group, unshaven, some in remnants of uniforms, some in makeshift outfits of pajamas and sweaters, not a hat between them, lying apart from the rows of deck chairs and the Punch-and-Judy show and the pier and the ice-cream stands. Most of the men who were evacuated had been sent to bases and hospitals in the south of England, but several thousand had been put on trains to seaside resorts in North Wales, where there were army camps and spare beds in the boardinghouses. The bulk of the men sprawled on the Rhyl beach were members of the Royal Corps of Signals attached to artillery regiments; some sixty-four officers and twenty-five hundred other ranks had been sent to the Second Signal Training Center at Prestatyn, which shared six miles of sand with Rhyl.
When we set out for the family holiday, we had no idea that survivors of Dunkirk had just arrived in Rhyl. Nobody in Mrs. McCann's redbrick boardinghouse on the front said anything about the arrivals; they didn't know, and wartime censorship didn't encourage people to talk anyway. Our first day on the beach I bullied my younger brothers—ten, four, and going on two—into helping me build a huge wall of sand to keep out the advancing Irish Sea, while Mum sat in a deck chair knitting and Dad read the newspaper. My father, a steam train driver, had worn himself out taking munitions trains through the blackness of wartime Britain, but he could never sit in a deck chair for long. He would inhale the salt air for ten minutes, then declare we should swim, kick a soccer ball, or join an impromptu beach cricket match. The next bright morning, when I hoped to build a bigger, better sand wall, Dad was restive again. He suggested we should all go for a brisk walk along the sands to work up an appetite for Mrs. McCann's lunch. My mother and brothers preferred to idle by the paddling pool, so with ill grace I fell in beside him. Not only could he not sit still for long, he was compulsively gregarious. Everyone else on the beach was getting on with their seaside relief from factory shifts and holding a family together in the stress of war. To my frustration, when we had gone beyond the pier, Dad saw these sprawling clumps of men, isolated from the holiday crowds, and he walked along to find out who they were. I can see him now squatting among them, offering a cigarette here and there. At thirty-nine, he must have been several years older than most of them, but you would never have known it, so weary and haggard were they. I was always embarrassed by Dad's readiness to strike up conversations with strangers, but Dad moved among the groups of soldiers most of the morning, and I tagged along.
We had been encouraged to celebrate Dunkirk as some kind of victory. A _Daily Mirror_ front page I'd seen pinned up in our boardinghouse had the headline "Bloody Marvellous!" How was it, then, Dad found nothing marvelous, only dejection, as he moved among the men?
Only two years later, when my ambitions to be a newspaper reporter flowered, did I understand that Dad was doing what a good reporter would do: asking questions, listening. It never occurred to me to take a note and write it up in my diary, but to this day I remember the sadness of the soldiers who had seen such havoc on that other beach and who knew, too, that they owed their lives to the countless acts of heroism of the rear guard who fought to the last man to keep the escape corridor open.
"They said they had nothing to fight with," Dad told everyone back in the dining room that lunchtime. The men were not triumphant, he explained, as they were said to be—they were bewildered, bitter that the Maginot Line had proved useless because the Germans had bypassed it by coming through Belgium, bitter with the French Army, bitter with the Royal Air Force (RAF) they felt had left them so exposed to the German Luftwaffe as they lined up on the beach and scrambled for the shallow-draft boats that would take them to the bigger ships. (The histories suggest that the French and the RAF both performed better than it seemed at the time, but misperceptions are the common currency of war.) The newspapers we'd seen had given the impression that the survivors couldn't wait to get back into battle to avenge their defeat. Maybe thousands were, but not those prostrate on the Rhyl beach or the dispirited men who, according to the historian Richard Collier in his 1961 history of Dunkirk, flung their rifles away after landing at Dover.
Dad's account of the mood of the men compared well with the national archives records I checked years later. "We didn't deserve the cheers," said Albert Powell, a truck driver, of their reception after they landed in Ramsgate before entraining for Rhyl. Bert Meakin, a gunner with the Fifty-first Medium Regiment, was critical of the weapons they'd been given to hold back the Germans: "First world war 6-inch howitzers on iron wheels, pretty useless really!" His group fought south of the Somme, then were told it was every man for himself; they abandoned the howitzers in the woods. He arrived in Rhyl with a seven-day leave warrant but without a penny. Powell, a Royal Signals truck driver attached to Third Corps Medium Artillery, got to La Panne on foot. "On the beaches," he recalled, "we huddled together in the sand dunes for protection from the constant bombing and machine gunning from the air. The bombing was ineffectual, just blowing up loads of sand, but the machine gunning was another matter." Once Powell reached a boat, it was swamped by a dive-bomber's near miss, and he was flung into the sea. He swam fifty yards, "arrived at the ship completely knackered and found myself hauled aboard."
Looking back on my boyhood snapshot memory of the difference between what I read and what I saw, I often wondered whether Dad and I were overly impressed with a firsthand experience and hadn't seen the woods for the trees. Dad talked, after all, to a tiny fraction of the evacuated soldiers (and surely, newspaper reporters would have talked to hundreds). So it was interesting to learn later that Winston Churchill got so worried at the presentation of the retreat as a triumph, he felt it necessary to remind everyone that "wars are not won by evacuations." Even more illuminating on the role of the press was Phillip Knightley's authoritative account of war reporting in his book _The First Casualty,_ first published in 1975. Of Dunkirk he wrote, "Above all, the stories stressed the high morale of the evacuated troops, itching to get back to France and into the fight again. It was not until the late 1950's and early 1960's—nearly twenty years after the event—that a fuller, truer picture of Dunkirk began to emerge." Alexander Werth, the _Manchester Guardian_ correspondent, confessed that after the fall of France he felt guilty at the "soft soap" he had been giving his readers.
The discordance between the waterfront and the front pages was bewildering, the first vague stirring of doubt about my untutored trust in newspapers. As a kid in short pants, I had hardly followed the events of the 1930s with the avidity with which I later read the histories, but I remember how troubled my father was on September 3, 1939, when Prime Minister Neville Chamberlain declared war on Germany. It was so contrary to what we had been insistently told by the _Daily Express,_ the newspaper my parents took at home. The paper had reassured its millions of readers that there would be no war with this front-page slogan: "The _Daily Express_ declares that Britain will not be involved in a European War this year, next year either." Everyone believed it. And why not? The _Express_ was a brilliant broadsheet with a circulation of three million and a huge secondary readership. Most British homes were reached by one of the bigger newspapers: in 1939 some thirteen million read the _Express_ newspapers, the _Daily Mirror_ , the _News of the World,_ or _People,_ an audience that by 1948 reached twenty-two million. Newspapers played a crucial role in shaping public perceptions. As the social historian Richard Hoggart noted in his study of the working class at this time, people often used to say as evidence of disputed truth, "Oh, but it was in the papers."
But what if you couldn't trust a newspaper to tell the truth and nothing but the truth? Which institution was more trustworthy, the state or the press? Later in adulthood, it was easier to understand how predictive headlines could turn out to be wrong than to reconcile what we experienced in Rhyl with the emphasis in what we read as fact. How did newspapers come to conclusions? Were they acting at the request of government? Was there a deliberate and widespread gloss on Dunkirk? Would that have been justified as a means of sustaining the nation's morale at a crucial time? Should newspapers take account of such imperatives or just report things as they saw them? How did a newspaper decide these things?
Such questions still resonate with me after a lifetime in newspapers. There have been many times when I have found that what was presented as truth did not square with what I discovered as a reporter or, later as an editor, learned from good shoe-leather reporters. It was not so much that deliberate lies were told, though they sometimes were, and not always to conceal a villainy. "In wartime," Churchill remarked, "truth is so precious that she should be attended by a bodyguard of lies." We all understand in an age of terrorism that refraining from exposing a lie may be necessary for the protection of innocents. But "national interest" is an elastic concept that if stretched can snap with a sting. When, in the early 1970s, the _Sunday Times_ began reporting the anger building among the Catholic minority in Northern Ireland, a group of Conservative members of Parliament (MPs) invoked the national interest to demand that we stop. They came to tell me, as the paper's editor, that it was "treasonable" to continue. Actually, the real offense was failing to give Northern Ireland full attention in the early 1960s, when the violence was incubated.
A more common issue than outright lying is that people of good faith resent facts that run contrary to their beliefs and assumptions. The nineteenth-century American humorist Josh Billings said it best: "It ain't ignorance that causes all the trouble in this world. It's the things people know that ain't so." No institution has a monopoly of vice in these matters—not governments, trade unions, company heads, lawyers, academia, or the press. In what came to be known as the thalidomide affair in Britain, children were born with deformities—a shortened arm, or no arm at all, or no leg, or completely limbless—because the mothers had taken a prenatal drug prescribed by the National Health Service. They were left to endure their ordeal without help or compensation, a shocking situation that persisted for a decade because the government and the lawyers representing the families assumed that the children had been the victims of an unforeseeable disaster. The lawyers sincerely believed they were making the best of a bad case, but the argument for adequate compensation, properly investigated by the _Sunday Times,_ was overwhelming. Revealing it brought furious lawsuits, led by the government of the day, with the attorney general accusing me and the newspaper of contempt of court, punishable by a jail sentence.
Independent reporting has a history of provoking denunciation. Take the legend that "unpatriotic" reporters lost Vietnam. It doesn't stand up to serious examination. Print and TV journalists supportively reported the war in the context of cold war ideology: they wanted the United States to win. What maddened them were the little deceptions of the U.S. government, the hubris of its generals, the corrupt incompetence of the South Vietnamese establishment, and the way the political military bureaucracy deceived itself into telling Washington what it wanted to hear. The corrective correspondents did a real service, and too many of them were killed doing it. Similarly, early in the Iraq War, the George W. Bush administration charged that the reporters on the ground were being lazy, foolish, cowardly, and unpatriotic for reporting that the country was on a vicious downward spiral. It was. The administration deceived itself, and no good came of that. Indeed, a more accurate charge against the press on Iraq would be that it hadn't been patriotic enough before the war began. Faced with a secretive administration bent on war come what may and a popular clamor for post-9/11 revenge, the press forsook its true function. The real national interest required the most searching examination of the reasons for sending thousands of people to their deaths, and it did not get it.
The epiphany on Rhyl beach shook my faith in the printed word, but it did not make me averse to newspapers. On the contrary, as I entered my teens, I grew ever more eager to involve myself in their mysteries. Newspapers were clearly more important and more fascinating than I had imagined, reporting more than a matter of stenography. But how was I to become a reporter and learn the newspaper trade? I was a working-class boy who had already been branded a failure, having failed to qualify for grammar school (the English equivalent, roughly, of American high school). Was I reaching too far? Was I really fit for the work? What were the pitfalls, the ethical dilemmas, and the traps I could barely imagine? How could I equip myself to decode the complex, ever-changing, thrillingly dynamic mosaic of live news and bring it to the public with the raw integrity of truth?
So began my paper chase.
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# Getting Up Steam
When I was three years old, I was expected to die of pneumonia. My first fevered memory of life is staring at the colored counting beads of an abacus at the side of the narrow bed where I was confined in a room with the curtains drawn.
I was born in the summer of 1928 in one of the long rows of two-up, two-down terraced houses off the Liverpool Road, Patricroft, Eccles—part of the sprawl of the cities of Salford and Greater Manchester—and raised in the L. S. Lowry landscape of bent stick figures scurrying past sooty monuments of the industrial revolution. The Renshaw Street houses were so narrow that people shook their heads about how hard it was to get a coffin down the staircases.
Until antibiotics became widely available in England at the end of the 1930s, one in twenty infants died—mostly from pneumonia, meningitis, diphtheria, and tuberculosis. TB was always referred to in a whisper as "the consumption, you know. They'll never rear him." Indeed, it carried off my cousin Freddy around the time I had pneumonia.
During that family crisis my mother was up half the night nursing my brother, who had whooping cough; my father was working nights. I was mostly nursed by a neighbor, Mrs. Amy Roberts, who lived opposite us, had some nursing experience, and volunteered to sit with me through the nights of fever. In 1978, after I'd become known as a journalist, Amy told the _Eccles Journal_ that when she visited my distraught mother, she found that "Harold had been put to bed with whooping cough he had caught from his brother, but was lying on his back, which is dangerous for a sick child." She added, "Harold was a very tiny child with a small peaky face and was too weak to be bothered with anything. He did perk up towards the end of the week but even then he was very shy."
The shyness is at odds with family folklore from when I was two. My mother, on a walk through a local park, stood my baby carriage behind a bench at the duck pond while she chatted with other young mothers. When she turned to go, my carriage was empty. The consensus among the calmest of the young women consoling my mother was that her blue-eyed son had been seized by an international gang of baby smugglers and was even then on his way across the Channel. In fact, the guilty party was happily ensconced in another baby carriage. I had undone my harness and climbed from my own pram into another, where eventually I was found cuddling a baby girl.
My brother Fred, who became the keeper of the family history, told me I was actually nursed through the pneumonia by another good neighbor, Mrs. Matthew (Matt) Newstead, the wife of my father's best friend, which is how my second name came to be Matthew. Perhaps the two women took turns at my bedside. In any event, there are people who feel they both have a lot to answer for.
My grandfather on my father's side, John Evans, was born in 1854 at Llanrhaeadr ym Mochnant in Montgomeryshire, a little village in Mid Wales, where the Bible was first translated into Welsh. He left school when he was nine years old to run errands for plate layers mending railway tracks around Crewe railway junction, the gateway to the north of England. (Compulsory schooling to fourteen was enacted only in 1918.) He later married Sarah Jane Collins, a girl eleven years his junior from Church Minshull, Cheshire, who gave birth to my father on August 1, 1900. She failed to register his birth within the six weeks required, and so fearful of prosecution, she registered Frederick Evans as having been born fifteen days later, on August 21. In this manner, my father acquired a distinction shared with the Queen—two birthdays, and we never knew which to celebrate.
Grandfather John sustained his family in a rented cottage in the village of Coppenhall, Cheshire, in North West England, by repairing shoes and cutting hair at the end of his day's work on the railway. My father told me, "He saved halfpennies so that we'd have Christmas stockings. Mine always had a twopenny mouth organ, an apple, a nut, and a shiny new penny. Nobody had a radio. It hadn't been invented. On Christmas night we blew out the candles and sat around telling ghost stories." Every Christmas in my own childhood, whatever else was in our stockings, there was always an apple, a nut, and a shiny new penny.
My father had little formal education. My grandfather had none—a family secret we didn't learn for fifty years. In 1981, when I was editing the _Times_ of London, the paper was delivered to a cottage I had in Shoreham, Kent, where Dad and Mum were taking a break after retirement. "You know, Harold, it's a rum thing," he said, opening the paper. "What would people say if they knew the man editing this newspaper is the grandson of a man who couldn't read a word of it?"
I had six very different aunts and uncles from Dad's brothers and sisters. One of them, Wild Jack, was a gambler who lost everything betting on horses. The other older brother, Albert, was a railway chief detective superintendent, who slept in freight cars to catch thieves. Dick was a housepainter and Len a very quiet fisherman who didn't seem to have any work. Dad's two sisters were opposites in temperament. Aunt Beattie, the toughest of all my father's family, married a younger widowed farmer in Oswestry near the Welsh border and ruled him and the kids with a rod of iron. Mild Aunt Maggie, the youngest and plumpest in the family, helped in a shop in a backstreet in Crewe, and always fed me sweets when we visited. My first sexual thrill at the age of ten was when her teenage daughter and a giggling friend in another bed in the same room teased me about what might happen to "little Harold" if they came in bed with me. In retrospect, I regret they lost their nerve, but at the time I was terrified. They seemed like fully grown women to me, though they were probably around fourteen.
My mother, Mary Hannah (known to all as Polly), was one of thirteen children, of whom, so far as I could discover, only three survived to adulthood. She was born in Stockport, Cheshire, in 1904, the daughter of Lucy Haselum (née Murray), which gave us a tinge of Irish blood to mix with the Welsh. The Murrays were connected to the Collinses. Grandmother Lucy's father captained ferryboats making the run from Merseyside to Ireland. My mother left school at the age of twelve and helped the family budget by chopping firewood in Eccles. At thirteen she was clattering down the street in clogs on the way to the card room of the local cotton mill. Her older sister, named "Big Eva" to distinguish her from her daughter and my cousin "Little Eva," married a cobbler who had lost a leg in World War I. Her younger brother, Arnold, was a dashing engineer and a Merchant Navy officer in World War II.
My mother always had ambitions for a better life. Childhood measles and scarlet fever left her without a sense of smell, and her hearing deteriorated in her thirties, but she never complained. Not only did she manage to bring up four boys with equal affection—five if you include Dad, who was lost without her—but in time she started a business that thrived on their relationship.
My father was the optimist; my mother was the worrier. She had a habit, when sitting in an armchair, of repeatedly running her hand along the fabric, smoothing it out in a rhythmic manner that the British scholar Richard Hoggart perceived (rightly I think) as an effort by working-class women of that generation to smooth out their anxieties. Unlike my gregarious father, she never struck up conversations with strangers and never talked politics. She reserved her energy for figuring out a future for everyone in the family. She hugged us and cared for us—all of us, including Dad—through accidents and sicknesses. Even when I had my tonsils removed at eighteen, she busied herself bringing to the sickbed every day some concoction of egg and brandy with a mystery ingredient I thought might have been brown beef sauce. It seemed to work.
Dad had not much of a better start in life than his father, punished like so many bright boys for being born poor. He was a good all-round student, top of his class in arithmetic and picked for advancement to high school in Crewe, but the family needed him to become a wage earner, so his schooling ended when he was eleven. At thirteen he stoked the furnaces making steel at Crewe Works. "It was a rotten time," he remembered. "We had no electricity in the countryside at Coppenhall; in winter we got up at four thirty to light the fire, thaw the taps. I ran the mile or two to Crewe to get there for six, just in time for a cup of sweet tea and a bun."
In 1916 he volunteered for the Royal Flying Corps, passed some tests, and was downcast when they found out he was sixteen, not the seventeen he claimed, and therefore too young. The war ended before he could be sent to the trenches, but he joined the Territorial Army (Terriers), Britain's volunteer reserve force, the equivalent of the National Guard in the United States. For a time he trained as a boxer, modeling his footwork on a legendary world flyweight champion hardened in a Welsh coal mine—the skinny, five-foot-two Jimmy Wilde. Dad put boxing gloves on all his boys, one at a time encouraging us to take a swing at him and to learn to dodge and weave. "A good little 'un will always beat a big 'un," Dad assured us. It was one of his aphorisms I preferred not to test on the backstreets of Newton Heath, where I collected enough bloody noses simply protecting my marbles from predators.
My father was a bit of a puzzle about martial matters. He was the least belligerent of men, but he loved military ceremonies, like the changing of the guard at Buckingham Palace, and spoke of John Philip Sousa's marching music with almost the same reverence as a run down the wing by his idol on the football (soccer) field, Stanley Matthews, the wizard of dribble. In his spic-and-span time in the Territorial Army, he learned to beat out an impressive tattoo on a kettledrum. He practiced it for years on our bedroom doors when we were slow to get up for school.
My mother was nineteen and my father twenty-three when they met at the "monkey run," as everyone called the Saturday evening dance at a social club on Liverpool Road. The stylish wedding picture of the slim, elegant couple at Patricroft Parish Church in September 1924 belies the bleakness from which they emerged. The newlyweds had to squeeze into a tiny house at 39 Renshaw Street with Granny Haselum and her dying husband, Adam, a laborer in a chemical factory.
My mother and father were lucky in a way: they had jobs when they married. Three million Britons did not. My father had been taken on by the London, Midland and Scottish Railway (LMS) to clean steam engines, and my mother worked in the cotton mill until I was born. They were then wholly dependent on my father not being sacked. The fear that they might have to go down to the labor exchange to register for unemployment money from the government filled them both with horror; they had a prideful revulsion at taking "dole money," which was still vivid in their minds when I was a teenager and they were secure. They radiated a quiet confidence that they were giving us a better start than they had. "I'll see you never wear clogs," Mum said often, and always with uncharacteristic fierceness. They both took it for granted their boys would climb Everest. "The railway's not for you," Dad told each of us.
They saved every penny in Renshaw Street, and with the birth of a second child (my brother Fred in October 1929) they rented an airier, better-built house, one of a row at 14 May Street, Monton. It was just across Liverpool Road, but it was a different world. On Renshaw Street you were in the living room as soon as you crossed the doorstep, so much so that families were judged by how freshly sanded they kept that front doorstep. On May Street, Mum didn't have to kneel every morning at the front door with a bucket of hot water and a pumice stone. Now we had a gate; we had a tiny front garden; we had leaves. In fact, after the abacus my earliest memory is of leaves on privet hedges. To my young eyes, as I was wheeled to a nursery school where we each had a cot for an afternoon nap, the neighborhood of May Street was a corridor of privet hedges—moats around semidetached castles of the English lower orders. Horticulturists value _Ligustrum ovalifolium_ for its ability to survive industrial pollution; the self-consciously respectable working class in which I grew up cared more about preserving privacy than combating the then little-appreciated effects of pollution. How sedulously they tended their ligustrum defense against prying eyes! Such was the prevalence of the question "What will the neighbors think?" that I got the idea God had planted busybodies as prolifically as privets.
My father was a genius with numbers. If you named a date five, twenty, thirty-seven years ahead, in a flash he'd tell you what day of the week it would be. Or tell him the date of your birth, and he'd name the day you were born. I never knew him to get it wrong, and I never knew how he did it.
At work on the railway, he became a legend among his workmates—"a ruddy marvel"—for being able to calculate in his head what any one of them was due in his pay packet at the end of a string of complexly different wage rates and irregular hours. The first railwayman who took him at his word got a brusque reception at the "gaffer's" office until they discovered they had indeed shortchanged him. This happened so often that when a worker took the pay slip back to the cashier and said, "Freddie Evans says it is five pennies out," they'd pay it without argument. Ken Law, a Manchester steam fireman, recalls encountering my father at the Newton Heath rail depot's large glass casement that displayed all the assignments of some thirty or more "links" (groups of drivers, firemen, and cleaners); each link was assigned twelve weeks of work. "If you stopped for a word, he would suddenly amaze you by saying that number three link had four hours more night rate in twelve weeks' work than number five link, or that number two link had more Sundays than number seven link and so on. It was no effort. Freddie could work out these statistics just in his head while he waited to be given his engine number. Few could do it today even with the aid of a pocket calculator. Of course nobody had those then. We had Freddie Evans."
In his middle years Dad experimented with the laws of chance. "Gambling is a mug's game" was his mantra, derived from the experiences of his crazy older brother Jack. Among workingmen, all sorts of foolproof betting "systems" enjoyed brief vogues, all ending in disaster, so Mum was fearful when Dad said he had devised a mathematical system for betting on greyhounds. Off he went to the track in Salford with four pounds (about one hundred dollars at the time). He lost it all.
He was depressed, but he knew why his scheme had failed. "I'm sorry, Polly, I got greedy," he told my mother. Henceforth it was his iron rule that once he had won a pound, he would walk off the track. His railway hours did not allow him to go as often as he would have liked, but month after month, year after year, he won his pound and came home, eventually accumulating enough to pay for every family summer holiday.
He didn't impose any kind of regimentation on his boys, so I didn't associate him with discipline. But he was disciplined with himself, apart from being unable to control an appetite for conversation. In his mid-forties he suddenly decided to give up smoking because the price went up. "That's my last cigarette," he said one day, and it was. I'd tried a few cigarettes with our street band of boys. We collected discarded ends called "dimps" and rolled the tobacco into handmade cigarettes for secret group smokes. A few puffs made me cough, and I never touched cigarettes again.
My father's phenomenal numeracy was of no interest to the railway company then, nor in the 1950s and 1960s to nationalized British Railways, which swallowed the LMS. However conscientiously a cleaner, fireman, or driver performed, however well he did on tests, whatever ideas on efficiency he put forward, however long he served with distinction, however much he was esteemed by his peers, he could never hope to achieve advancement into the officer class of supervisors and above. That was ordained by the hierarchies of class.
The question I asked myself often about my parents was what they might have done if they'd had a real chance. Like millions of others, they'd been held back from birth by the belief among the ruling elites that education could do nothing for the working class—nor should it. The Liberal Party in power after World War I set about introducing universal secondary education to the age of sixteen, reducing classroom size below sixty students, and opening a door to university, but the Tories dominant after 1922—the hard-liners, not all of them—abruptly reversed the progress. Growing up, I got bored when Dad went on and on about "the Geddes axe," not realizing then how frontal an assault it had been on any hope of equality of opportunity.
Sir Eric Geddes (Lord Inchcape), a Tory grandee and Minister of the Crown and a former manager of the North Eastern Railway Company, had a predictable contempt for the working class. It expressed itself most nakedly in his advice to Parliament not to waste money giving poor children a secondary education—"children whose mental capabilities do not justify it" was the way he put it in the report of his committee examining public expenditure. This was unappealing as rhetoric; it was appalling as policy. It was exactly the wrong prescription when Britain was suffering from chronic overreliance on unskilled labor in declining industries. Naturally it was greeted with applause in the press, just as was chancellor of the exchequer Winston Churchill's disastrous return to the gold standard in 1925, which priced British exports out of world markets. The conviction of the conventional wisdom in the press and politics was that if it hurt, it must be good for you. Those who were most hurt, of course, were not the advocates of salvation by masochism. They were the coal miners, factory hands, and shipbuilders who endured wage cuts, longer hours, and lockouts.
Recollection of my father's dim view of Churchill pains me, since like millions of others I came to see Churchill as a hero for his wartime leadership. From the perspective of the 1920s, however, my father's attitude was justified. It was hard to forgive the consequences of Churchill's gold standard blunder, his ill-fated assault on Gallipoli while first lord of admiralty during World War I, or his virulence in the General Strike of 1926, a failed attempt to stop a reduction in coal miners' wages. He edited the government newspaper, the _British Gazette,_ attacking the miners and persistently printing foolish fabrications—for instance, assuring Londoners that buses and trains were running near to normal when, having to walk to work, they knew they weren't. When later in my career I took a look at the record, I was shocked at the distortions coated in hysteria and shoved down the throats of the public (the fledgling BBC also was hardly a beacon, suppressing anything that might help the strikers, including a conciliatory appeal from the archbishop of Canterbury). Even Prime Minister Stanley Baldwin and the government's commissioner in charge of information, Mr. L. C. Davidson, were alarmed at Churchill's reckless conduct. Davidson wrote later that Churchill and his ally Lord Birkenhead were "absolutely mad.... [Winston] had it firmly in his mind that everyone who was out of work was a Bolshevik; he was most extraordinary and never have I listened to such poppycock and rot." Churchill simply allowed political passion to swamp journalistic principle.
The distrust carried over into the darkening 1930s, when Churchill was so valiant in his efforts to awaken Britain to the menace of Nazi Germany. Dad called him a "warmonger," a common perception among his mates. They were quietly patriotic but disillusioned by World War I—the heady parades through the streets; the senseless slaughter that hit every family; the slow realization, as the histories filtered down, that the men marching off so proudly had been sacrificed by stupidity and that the war itself had been just a terrible accident. Daily at work Dad passed a memorial to the twenty-seven local railwaymen who had not come back from Gallipoli and the Somme:
_Forget us O Land for which we fell_
_May it go well for England, still go well?_
_Keep her proud banner without blot or stain_
_Lest we may dream that we have died in vain_
The wretched years of appeasement have to be understood in that context. Men like my father had no faith in "that talking matchstick" Neville Chamberlain, but nor did they have any enthusiasm for a rerun of 1914–1918. Dad was by no means a pacifist, though the Labour Party lessons in history and geography he had taken had made him otherwise a disciple of Jimmy Maxton, a militant Scottish socialist and pacifist who was elected to Parliament in 1922 (an admiration later shared by Maxton's biographer, Prime Minister Gordon Brown). What really inspired my father was faith in the brotherhood of man, a characteristic he shared with the giant who walked the full length of "the world's longest platform" at Manchester Exchange station to shake hands with the footplate crew who had brought him safely to his destination. The giant was Paul Robeson, who was finally free to tour outside America after eight years of being denied a renewal of his passport. In the cold war hysteria of the 1950s, it had been judged treasonable for Robeson to have said, "Our will to fight for peace is strong. We shall not make war on anyone. We shall not make war on the Soviet Union." The egalitarianism of Robeson's handshake appealed to my father; certainly it was something no British political leader of the era would have dreamt of doing.
Robeson, like my father, had a romantic view of the Soviet Union. Dad believed that it exemplified Karl Marx's slogan "From each according to his abilities, to each according to his needs," an illusion fostered by Sidney and Beatrice Webb's 1935 book _Soviet Communism: A New Civilisation?_ read by my father in Labour Party and trade union courses. Dad was as unaware as millions of others of Joseph Stalin's mass murders, nor was he of a revolutionary disposition. When I got him to talk about his part in the General Strike of 1926, I said he must have felt bitter when it was broken. "No," he said, "I didn't feel bitter. I just felt sorry for the way the miners were let down."
The job my father had when I was born, engine cleaner, was a bigger deal than it sounds. It was the first rung on a very long but very coveted ladder to becoming a locomotive driver.
Train drivers were an aristocracy among the working class. They had a job for life, the social esteem that came with security, and better-than-average pay. The downside was that the job was brutally hard in its physical and mental demands. The hours of work were horrible, beginning variously at 2:00 a.m., 3:00 p.m., or 5:00 a.m. It was a matter of pride to my father that he never needed the knocker-up to rap on the window with his long pole (an essential profession before alarm clocks were common). But the shifts meant that week to week we were asleep when he was up and he was asleep when we were up.
The railway historian Frank McKenna observed that "the eyes of a footplateman appear to be a decade younger than the rest of his physique." Dad's were striking, deep in his sockets. Perfect eyesight and physical fitness were demanded of an engine cleaner as of a driver. A slight falloff in the eyesight test, a hint of color blindness or any physical limitation, and a driver would be demoted to sweeping the sheds, shunting wagons in a freight yard, or cleaning lavatories—or even dismissed altogether. Dad was so sensitive about his fine vision that he would not hear of it when as a teenager I thought I was becoming nearsighted. I was, but he was in a state of denial I didn't understand at the time. Now I see that the eye-rolling exercises I picked up from a book by an Indian doctor would have alarmed anyone.
Every schoolboy then might have wanted to be an engine driver, but there was no glamour in the first step. On his night shift, among other dirty jobs, Dad had to go under the engine and climb into the dark belly of the beast to oil the big ends of the pistons, fearfully trusting that nobody would move the engine (as occasionally some lunatic did). He had several years of this before he was tested for work on the footplate (that is, in the open cab), first qualifying as a "Passed Cleaner," which carried the prospect of some turns as a fireman. What backbreaking work! I have a mental picture of my father coming home, exhausted from an all-night firing job on a goods train, keeping a foothold on the rocking engine while hour after hour shoveling coal from the tender, maybe six tons of it, and hurling it through the small fire hole into the right places in the firebox to raise the necessary steam pressure. "Where's my steam?" was the yell no fireman wanted to hear from the driver.
In time, a Passed Cleaner could hope to become a Red Ink Fireman, on the footplate for a few months; then, all being well, a Black Ink Fireman, on the rosters for regular firing; and finally a Passed Fireman, tested to drive any train in his depot. As a Red Ink Driver he would be on the roster for driving in holiday periods, and then eventually, as a Black Ink Driver, he would be at the top of the ladder. No other craft or profession exacted such a lengthy "apprenticeship." Dad carefully annotated the details of every driving turn he acquired after becoming a Red Ink Fireman. It typically took at least twenty years to get to the top. "Dead man's shoes," said Dad.
A driver could not take a train on a route until he knew the route's every particularity—the sighting of every signal, the sounds and shadows that might guide him in fog or a snowstorm when visibility was near zero, the shape of every curve in the track, the length and darkness of every tunnel, the trickiness of every ascent where extra steam and sand might be needed, the occurrence of every set of points where the train might be switched to a different line. They called this familiarization "learning the road," and Dad learned many roads, rattling most happily along the North Wales coast, where many years later, at Bluebell Wood cemetery at Coed Bell in Prestatyn, he was to find his final resting place.
Drivers and firemen were subject to strict military discipline, and it was easy to see why. A railwayman who did not read, memorize, and follow the hundreds of regulations in the precious rule book risked his own life, as well as the lives of his workmates and his several hundred passengers. Dad knew the rule book back to front. In the kitchen, testing himself, he'd ask questions rhetorically: What do you do with a runaway train on a hill or a train slipping back? How in an emergency do you signal to the guard at the back of the train? If you pass through facing points onto a curve, what is the safe speed? What if you have to run backward? What's the right thing to do if there's an obstruction on the line, an uprooted tie, a snowdrift? If you run out of steam, what lights do you lay down on the track and where? The work ethic was puritanical—clean overalls, no drink, no swearing, no smoking on duty, and no tolerance for misdemeanors. If he was ten minutes late at the sheds, he risked being sent home with the warning that next time he'd be fired. I remember a railway inspector coming to our house to see if Dad had taken home one of the high-quality hand rags issued to footplate crews for oiling work. He hadn't. He knew better.
We worried about my father's daily risks. Usually he came home chuckling over some incident. It was ominous when he didn't:
"What's the matter, Dad?"
" _Something terrible._ "
"But what?"
" _Bad accident._ "
"What kind of accident?"
" _Finish your tea._ "
We'd eventually discover that a plate layer had lost a leg, a shunter had been crushed between wagons, a fireman had been scalded, or a driver had been killed walking across a track to check a frozen signal. His own most common affliction was grit in the eye from looking out of his open cab at high speeds. There was no protective eye shield for footplate men.
He tried to educate his union, the Associated Society of Locomotive Engineers and Firemen, not always to campaign for wage increases but also to aim for medical benefits and for decent pensions, pointing out that an extra shilling or two now would be better invested for retirement. But he could never persuade them, so when he did retire, his pension after fifty years was seven shillings a week (about eight dollars at today's values).
* * *
In the early 1930s the composition of the manpower at the LMS Newton Heath sheds, way across the other side of Manchester, offered a better prospect of graduating from Passed Cleaner to Red Ink Fireman. Newton Heath was a very big depot with more than two hundred locomotives. Also of some relevance was Dad's passion for soccer: he never saw a ball he didn't want to dribble around an imaginary fullback and, scorning players who could not shoot with both feet, drilled us hard on that. Naturally he liked the idea that the Newton Heath sheds were the birthplace of a soccer team—not any old team, but the Heathens, a bunch of railwaymen who managed to get into the Football League, nearly went bankrupt, and then did rather better after 1902, when they changed their name to Manchester United.
Dad at once applied for a transfer to Newton Heath. He and my mother took the plunge of putting down all their savings as a deposit on a £300 house a few miles from the railway sheds, their first time out of rented accommodations. It was a barely finished, semidetached place at 54 Ashworth Street, a new estate close to Manchester City's wooded Brookdale Park, famed for its birds, grasslands, and Victorian bandstand. (Many were the times irate park keepers chased Fred and me for getting into the park by climbing the iron railings instead of walking ten minutes to the park entrance.) Our new house was right on the edge of open countryside and farms. In the early days before the war, a farmer in pony and trap came round before breakfast time selling milk he ladled out of a big churn strapped to the trap; in the summer everyone joined in haymaking.
This was the place where, for more than twenty years, my three brothers and I, sleeping two to a bed, grew up. All our fun was very much homemade—marbles, yo-yos, hopscotch, tag, whipping a spinning top along the street. Fred and I would go out to the big paved space at the three-way junction where our house stood to stage a cricket match with a rival street, using a tennis ball and a lamppost for a wicket. In winter we made long ice slides, and when it snowed Dad hammered together crude sleds, lining the runners with metal from discarded tins of Heinz beans. I collected scrap lead, boiled it on the kitchen stove, and used clay impressions of tin soldiers to create armies for battles with cannons firing matchsticks. From a smelly works along the Rochdale Canal we "lifted" bits of ebonite tubing for peashooter contests. We were manic competitors in everything with all the kids in the street, but most of all in completing cigarette card series of soccer teams, cricketers, kings, aircraft, cars, wars, and film stars.
Very occasionally we'd test our parents' good temper by venturing into enemy territory to engage in running stone-throwing battles with kids from other streets in Newton Heath, everyone scattering at the sight of the angry red face of the potbellied Police Constable Robinson. He had a truncheon and lived opposite our school, Brookdale Park Elementary.
Communal goodwill broke through, though, every November 5, when Guy Fawkes brought our neighborhood together. After three hundred years there wasn't any lingering resentment of Captain Fawkes for trying to blow up Parliament in 1605—quite a few in 1936 thought it was a great idea—but the passage of time hardly lessened the appetite for a bonfire feast of baked potatoes, boiled ham and cheese, treacle tart, toffee apples, and Yorkshire's gingerbread, known as "Parkin pie," accompanied by fireworks and singsong. Fred and I enthroned a stuffed, bearded effigy of the conspirator in a cart made from old carriage wheels and begged around for "a penny for the Guy" to be spent on fireworks. Every household made a contribution to the bonfire, which lasted past midnight, fueled by logs from the fields, broken-down settees and chairs, and our "Guy Fawkes" on top, exploding with firecrackers. I can still smell the cordite.
Mum and Dad gave their four sons warm encouragement at every stage. They were devoted to each other, too, but they didn't let it show. None of my brothers can recall any cross words between them—nor any show of open affection. I don't think I ever saw them embrace or kiss. But if Dad was there to help Mum with the dishes after supper, sometimes we'd hear him attempt the lilting lyric of a music hall song: "If you were the only girl in the world, and I were the only boy."
He was moved by music from America. My nostalgic ear catches now the innocence of the refrain of "Home on the Range," which Dad puff-cheeked out of his mouth organ. He wished he had been born in Wyoming, not Crewe, so he would sit at our kitchen fireplace imagining that he was playing the song by a prairie campfire under the stars before rolling up in his horse blanket, with his saddle for a pillow. Sometimes he sang it to us: "Oh, give me a home where the buffalo roam/Where the deer and the antelope play;/Where seldom is heard a discouraging word,/And the skies are not cloudy all day."
He wanted us to appreciate the harmony of Nelson Eddy and Jeanette MacDonald singing "Rose-Marie" in the film about the Canadian Mounties, the majesty of Paul Robeson's "Ol' Man River," and the romance of Stephen Foster's "Camptown Races." When we were toddlers, he would hold us on his knee and give an unsteady version of Al Jolson from the movie _The Singing Fool,_ which came out the year I was born: "Climb upon my knee, Sonny Boy/... You've no way of knowing, there's no way of showing / What you mean to me, Sonny Boy." He took the words to heart. I grew up when boys were regularly beaten by their fathers. There was only one occasion when Dad took a strap to my backside—for playing with matches.
My parents were affectionate, but they were reticent—no, downright obscure—about anything to do with sex. My worldlier younger brother Fred claimed superior knowledge, but until I was about ten, I sincerely believed that babies were delivered in the little black Gladstone bags that doctors carried. That doesn't say much for my powers of observation, since my mother had by then carried another brother to term.
The only hint that my parents were aware of the hormonal turmoil of teenage life was a book—from the Boy Scouts, I think—on the awful consequences of masturbation, which just happened to be left lying around and of course caused immediate and unmentionable panic among us boys. We subsequently struggled between the Boy Scout reign of terror and a _Naturist_ magazine featuring sepia nudes that was slipped to me in an exercise book at school.
Nowadays my father would be considered a prude. Mother's brother Arnold, who looked like Clark Gable and rode a fast motorcycle, once began to tell Dad a risqué joke and then had the sense to stop when Dad's face dropped. He liked the kind of riddles stuffed in Christmas crackers, music hall impersonations, and idiomatic radio sketches of northerners coping with the frustrations of their lives. Ken Law told me, "If you met your dad in the very early hours of a cold, damp, frosty morning or in the middle of a rain-lashed railway shed yard, he'd always come up with something funny." Never would it require the prop of profanity. He'd "damn" and "blast it," but I can't recall him saying "bloody," and indeed the "effing" and "blinding" that is the vernacular today was not then a feature of respectable working-class speech in the north.
Most of all, my father liked telling daft stories about himself, and the family loved to have him act them out. One night, he told us, he went to a rough workingmen's club to collect a small debt. He climbed to the top of the stairs. "When I got to the top, there was no light, and I could just make out the shadowy figure standing there. I said, 'Hello, I've come to see a friend of mine.' He didn't reply, so I told him again. And he stood there saying nothing, so I took another step forward and so did he! I thought, _He's coming for me_."
And at this point Dad, who was no more than five feet six but muscled from all that coal shoveling, would hunch into a boxing stance.
"I was such a fool!" he explained. "There was a big mirror at the top of the stairs. I was talking to myself! I'd been misled because I'd done something I don't normally do—wear a hat!"
He was indeed a creature of habit. We knew the ending, but we hooted with laughter every time he told the story in exactly the same way. We were part of the performance, and his performance, like good theater, always seemed fresh, as if he were discovering it himself for the first time.
Rituals were a big part of our happy family life. Mum tossed pancakes on Shrove Tuesday, baked treacle toffee for the campfires on Guy Fawkes Night, hid eggs at Easter, and made sure that the first person to enter the house after midnight on December 31 carried a piece of coal for good luck.
Most of all, on occasional Saturday nights Mum and Dad enjoyed opening the house to relatives and neighbors for cold meat and pie suppers and games, pitting teams of adults and kids in musical chairs, charades, bobbing for toffee apples, blindman's bluff, scavenger hunts, memory games, and on and on through a repertoire filled with excitement and laughter.
My father was enthusiastic about his quaint ceremonial rituals at club nights in the workingmen's fraternal association the Royal Antediluvian Order of Buffaloes, or "Buffs." (Dad would have joined the Freemasons if anyone had asked him.) In my adolescence I dreaded when we moved into the company of strangers. He would always open a conversation, while I curled up in embarrassment and affected not to be with him. For me, Dad on holiday at breakfast in the dining room of a seaside boardinghouse was a recipe for importing anxiety by the bucketful. He would make the opening sally to the family at the next table, and vistas opened on an infinity of world controversies while we fretted to get out to the beach before it rained. Without Mum giving him a kick under the table, we'd have been stuck for the day.
Writing about my parents and their role, I realize how easy it is, just as it was in her lifetime, to allow Mum's more contained personality to become overshadowed by memories of Dad's ebullience. But it was Mum's down-to-earth practicality and native intelligence that were key to a rise in the family's fortunes. The girl who had started out working in a mill and wearing clogs developed an entrepreneurial streak.
Our house at 54 Ashworth Street was on the edge of open fields and farmland stretching for miles in the direction of Daisy Nook, a sweet valley of woods and water, and toward hills dominated by Hartshead Pike, where the druids, we were told, had made sacrifices and the Romans had lit beacons to warn local garrisons of heathens on the prowl. At Easter more pacific modern hordes made the trek to Daisy Nook for a big annual fair, but on every weekend there was a steady flow of ramblers passing our door for picnics there or boating on the adjacent Crime Lake. Often they'd knock on our door and ask if we could give them a glass of water. My mother always obliged. There were so many knocks at the door that she came up with the idea of making lemonade and selling it for a halfpenny a glass. She sold it all, and her ambitions soared.
My parents walked the two miles to Rothwell Street and asked to see Antonio and Fred Sivori, who owned a little ice-cream factory. The Sivori brothers had their own horse-drawn "ice-cream parlor" traveling the streets of Newton Heath, but Dad persuaded them to deliver to Ashworth Street, on Sunday morning, a big tub of ice cream at wholesale price, packed around with ice (we had no refrigerator), along with a scoop for making cornets (ice-cream cones) and a wafer maker for ice-cream sandwiches. That first Sunday as nascent capitalists, my parents were apprehensive. Dad painted ICE CREAM HERE on a big piece of cardboard, and they stood by the garden gate. If it rained, they'd lose the investment.
By late afternoon they'd sold out, and Dad went cheerfully off to the night shift. They tried it again the following Sunday: another sellout. The third week, the ice cream was all gone by lunchtime. It was a long way to Rothwell Street for a refill, and the container was so heavy they carried it between them along the streets. They were almost at the Sivoris' when one of them—neither would ever take the credit—realized that the weight was mainly melted ice that could be poured away. The Sivori brothers ferried the refilled tub back to Ashworth Street, and it was empty by teatime.
Soon there was a growing parade of Sunday and then Saturday customers at No. 54. They'd ask if Mum had sweets, or pop, or cigarettes, or a bun, and her answer was always the same: "Sorry, no, but we'll have it next week." She got Dad to take down the garden gate and build shelves in the hallway. The litany of promises led to stock spilling out from the hall into our front room, so she removed all the furniture, installed a big counter, started stocking groceries and haberdasheries for neighbors, and in time installed a plate-glass window and a refrigerator—wonders in the Ashworth Street neighborhood. Within a year, a full-scale corner shop flowered in our old parlor—managed, staffed, and maintained by Mum. Under her canny eye it became wildly successful. While she could not match the virtuoso calculations of my father, in the blink of an eye she could add up a long column of pounds, shillings, pence, and halfpence—twelve pence to a shilling, twenty shillings to a pound—and get it right the first time. Out of one pound of the takings, she'd make two shillings, a nice 10 percent profit that she stored in an old Oxo tin.
Sometimes she allowed Fred or me to help in serving. This was a great treat. I put on a white apron, scrubbed my hands, and slicked back my hair with a helping of butter. Boys and girls I played with in the street outside incredulously pressed their noses against the plate-glass window and the array of sweets displayed. My stock rose. I served customers simple items: cigarettes, bread, milk, tinned food. Bacon came in a big roll that had to be sliced. I was not allowed anywhere near the horrendous slicing machine; seeing Mum with her hand feeding the bacon so close to the sharp, swift-circling blade gave me nightmares. The most glamorous job was filing the colorful dust jackets (and rejacketing the returns) of the books Mum lent in the library she started. Fred and I came to blows over this privilege, wrestling furiously in the back room until one of us got the other in a headlock and won a concession.
When his shifts worked out, Dad liked nothing better than to come home, discard his overalls, and put on a clean white apron to serve in the shop—such a contrast from what he had been doing all day. This was sometimes to my mother's exasperation when the shop filled with people. Dad listened to every tale from every customer, whereas Mum knew who was a gasbag to be deftly cut off at second breath. Dad's style was altogether free and easy; asked for half a pound of boiled ham, he'd sacrifice his passion for precision in numbers and let the Avery scale ride over the eight-ounce mark. He redeemed this liberality, though, on one occasion when a woman came in and asked for a back stud for her husband's shirt collar. Dad rifled in the haberdashery drawer and presented it to her. "That's one penny," he said. The woman responded sharply, "No, I want a _good_ one; it's for a wedding." Dad took the back stud from her, rummaged around in the drawer, and pulled out the same stud. "Twopence," he said. She beamed and went off happy that she had a good-quality stud.
I still marvel at how my mother managed to give birth to two more boys—Peter was born in January 1936 and John in December 1938—look after Fred and me and Dad, run the shop six days a week (which meant ordering supplies, pricing them for retail, and dealing with customers), and never lose her temper. When she closed the shop, her working day wasn't over. She'd sit at her Singer sewing machine doggedly making clothes for us, including a billowing nightshirt for Dad (the cause of much hilarity) and white satin blouses for us to wear in the All Saints' Whitsun procession. (The family didn't go to church, but we were regulars at Sunday school.) Mum darned socks, knitted pullovers, ironed shirts, washed our laundry by hand, and saw Dad off to work with his can of loose tea, sugar, and milk to be brewed up on the engine; sandwiches; and playing cards in an old tobacco tin. If something bothered her, she retreated into silence, her lips tightening.
And then came bad news.
As more of the fields filled with housing, two neighbors copied Mum by turning their parlors into groceries, and the Co-operative Society chain announced that it would open one of its big stores and butcher shops in Miriam Street, a few hundred yards down Ashworth Street. The Co-op had the appealing message of being owned by its customers. Buying in bulk for many branches allowed the Co-op to cut prices, and thrift-minded customers became "members" who could accumulate a cash dividend on purchases. George Orwell, in _The Road to Wigan Pier_ in 1937, observed that the arrival of a Co-op was a disaster for independent shopkeepers. Local authorities that built housing estates (large blocks of flats) rigidly limited the number of shops in the area and gave preference to the Co-op. "Many a small shopkeeper is utterly ruined... their whole clientele taken away from them at a single swoop," Orwell wrote.
Mum and Dad debated whether they should keep going. While they worried, an inspector came around from our controlling local authority, Failsworth Council. "I have the power to shut you down forthwith," he told all three front-parlor entrepreneurs. "Not one of you has a license, and you don't stand a chance of getting one if I'm not satisfied with the way you store food." Having inspected ours first, he then went off to look at the others. When he came back in the afternoon, he told Mum, "I must speak to the man of the house." In the 1930s women were not presumed to have any competence at business of any kind; very few ran anything. The inspector waited around until my father came home and took him aside.
"Mr. Evans, I take it that Mrs. Evans is in charge?" Dad said yes, she was. "In that case," said the inspector, "I have to tell you, Mr. Evans, you are a lucky man. I have never seen a better-run grocery in all my years as an inspector. I am giving you a license. I am closing down the others."
The day after the Co-op opened its doors, Mum went in to check its prices. The employees didn't know who she was, so they indulged the slightly deaf lady who asked for prices on a lot of items. Only a few of the Co-op prices were cheaper, and Mum immediately marked down hers. She lost a few customers in the first week, but most drifted back, probably because my mother identified with her customers in a way the Co-op staff, conscious of their elevated position, could not match. As the semidetacheds expanded farther into the countryside, we stopped worrying. It was plain there was room for both ventures, and indeed Mum's shop was still flourishing under Korean management in 2007.
The Evanses were moving up! By the time I was nine, my parents were doing well enough to buy a secondhand red Hillman Minx, an unheard-of acquisition in our area. A car outside a house had always meant bad news: only doctors had cars. I heard years later from my schoolmate at Brookdale Park Elementary Alf Morris that I was known behind my back as "Posh" Evans on account of the purchase. In truth, I did not enjoy the car. The smell of leather inside made me sick, so when the family went on Sunday drives into Cheshire, I stayed at home, moping around and looking at the terrifying representations of hell in an illustrated Bible—where the Boy Scouts' helpful literature on the perils of masturbation assured me I was undoubtedly headed.
Much more exciting to me than any car was the day Dad became a king of the iron road.
Dad had washed and had his supper before he told us he had at last been promoted to Black Ink Driver. It must have been an effort for such a talkative man to say nothing, but he waited until Mum had closed the shop. There were handshakes at the sheds but no ceremony. He would still be doing the same work, rising at all hours, taking out trains in all kinds of weather, but as a full driver he had acquired a certain majesty, meriting respect and deference all along the line as a reflection of the responsibility he bore.
He had taken his licks as a fireman working with cantankerous drivers, and as a part-time driver he'd had to tolerate a few lazy or careless firemen. Now he was indisputably in charge. Nobody could tell him to take out a fireman he thought incompetent or an engine he considered defective. He could insist on a replacement. Railway management rarely authorized any visits to the footplate, not least when it was moving, but nobody could mount his footplate without his permission, and while he was in charge of the big engine, he was answerable to no one.
I longed to see my father at work, but the hours were unhelpful. As a driver, he often set off in his new glossy peaked cap and serge jacket just as I was spreading out my homework on the living room table. I had to wait for the privilege, and it came by chance when I was sixteen. One summer evening when I arrived by train at Manchester Victoria station, Dad was on the opposite platform about to drive a few hundred passengers home to Oldham. "Hop on quick," he said. I seized the moment.
Sitting in the comfort of a railway carriage, as I had been on the way in, gave passengers no idea of the ferocious goings-on up there in front on the open footplate as the train gathered speed out of the station and up the line—of the heat and noise that made speech impossible; the roar and flames of the firebox; the sweat of the fireman with his long-handled shovel feeding the red-hot maw; the noise of iron meeting iron on the swaying footplate; the intensity of Dad's concentration looking up the line for pinprick signals of red and green, checking the boiler pressure, and opening and closing the steam regulator arm according to gradient and track. And then at the end, at the station platform, the carriage doors slamming as the passengers got off the train, their voices drifting into the night, oblivious of what we had been through, I was astounded and bursting with pride.
#
# First, Know Your Enemy
I was eleven when Prime Minister Neville Chamberlain told forty-seven million Britons, "This country is at war with Germany." That Sunday morning at 11:00 a.m., on September 3, 1939, I listened to his broadcast with the rest of the family, all of us huddled round the vibrating fabric of our brown-enamel wireless set.
Dad was unusually silent, but Mum commented, saying of Chamberlain, "He seems to be more sorry for himself than the rest of us." Reading the speech today, I can see what she meant: "You can imagine what a bitter blow it is to me that all my long struggle to win peace has failed. Yet I cannot believe that there is anything more or anything different I could have done and that would have been more successful."
Mum was right. It was all about him.
No sooner had Chamberlain finished speaking than the air raid sirens howled. Fred and I rushed into the street. It was a crazy thing to do, but it was a false alarm. We were disappointed not to see Stuka dive-bombers tangling with the big, fat blue barrage balloons in the blue sky. We were even more upset when, the same day, the government shut down all the cinemas. What would we do for the rest of our lives?
The sound of Chamberlain's plaintive voice has stayed with me all these years. At the time I could not get out of my head the posh way he talked about our "embessador" in Berlin and the "plens" we would all have to make. The plummy announcers on the BBC had made me acutely conscious of accents as an indicator of class, of ineluctable superiority. Just turning on the radio made me ashamed. Nobody in my universe spoke like that; therefore, we must be outcasts, belonging to some inarticulate barbarian tribe. These were the years when the announcers and newsreaders all spoke so-called Standard English, meaning the soft tones of the alien south: harmonious Oxbridge voices of long vowels, distinct _p_ 's and _t_ 's, and effortless aspirates. The BBC tolerated J. B. Priestley's Yorkshire accent for a few months from June to October 1940 in his series of morale-boosting talks, but not his political opinions: seen by the Tories as too socialist, the program was axed. In 1941 it was front-page news when Wilfred Pickles brought a Yorkshire accent to BBC news reading—one would have thought a cathedral or two had been sacked—but he too was soon removed from the national airwaves, and Standard English prevailed for the next thirty years.
Our teacher at Brookdale Park Elementary interposed her good soul between civilization and us aural barbarians, struggling against centuries of cultural history to get us to "speak nicely," which meant mimicking the BBC. Chastised early in reading class for referring to an "'ospital," I approached every "aitch" as a pole-vaulter running at a high bar. (Of course educated youths now use a slovenly "mockney," which is more affected than ever we BBC imitators were in the Brookdale English lessons.)
Chamberlain's announcement created hardly a ripple in our neighborhood compared to the excitement of the night before he spoke. Every Saturday a half hour between 5:00 and 6:00 p.m. was sacrosanct. We could count on being undisturbed by the shop bell because every household was poised by the radio for the BBC's pip-pip-pip prelude to the day's football scores. Seeing match results on television or the Web today doesn't begin to compare to the effect of hearing one calm, authoritative voice announce the end of the world: "Manchester United 1, Charlton Athletic 7."
The scores were of paramount importance because predicting the results could take you from the hard grind and dreary backstreets to a fantasy life of ease and luxury. The football pools were not quite a lottery; there was a certain skill in assessing the clubs and their key players. Fortunes were made overnight; getting all but a few matches right might win hundreds of thousands of pounds, zillions more than anyone could earn in a lifetime. Before mailing in the coupon with his sixpenny bets, Dad spent hours studying the form, then used his little stubby pencil to mark the match lists with a _1_ for a team's home win, an _x_ for a draw, or a _2_ for a win away from home. If he'd bet that the star team of Blackpool, playing at home, would beat Leeds, the radio announcer's pause after "Blackpool 1" was excruciating. The disembodied voice of fate seemed to extend every letter in "L-e-e-d-s" before ruining everything by declaring that Leeds had scored 2. Crueler still was "Middlesbrough 4," Dad's prediction for a home win, followed by "Everton... 4." The house rule against breathing a word couldn't stand such freak results.
Fearing that British stoicism could not bear the strain of the Saturday pools and at the same time win a war, the Football League suspended its matches "for the duration." It was a phrase we came to hate, along with "Don't you know there's a war on?" Knowing now of the years of convulsions that were to follow Chamberlain's declaration of war, it is weird how calm and confident the people were, sitting down to roast lamb and mint sauce after his speech.
Perhaps that was partly because we were under the illusion that we were ready. We had already collected our identity cards and gas masks. Fred and I paraded around in the masks parodying "Heil Hitler" salutes, but the smell of the rubber made us gag, and after that we wore ours only for the drill at school, when on the command we assumed fetal positions under our desks.
We helped Mum and Dad stick tape all over our windows to hold the fragments in a bomb blast, put up blackout blinds and double curtains, and hated it when a bossy air raid warden knocked on the door to say a chink of light was showing: he was just a neighbor in a tin hat. Riding the electric tramcar along Oldham Road, we could see sandbags heaped around Failsworth's ceremonial flagpole. And at supposedly key points in the straggling communities of small terraced houses served by clusters of fish-and-chip shops, haberdashers, pubs, hairdressers, newsagents, bookies, savings banks, pawnbrokers, butchers, and the occasional cinema and church, iron railings round the churches and graveyards and everywhere else had already been taken away—to make tanks we were told. The flowers in Manchester's Piccadilly Gardens had been dug up for trenches and air raid shelters. The more active householders had dug deep holes in their gardens and roofed them with six sheets of curved corrugated iron supplied by the local council. The finished bunker was called an Anderson shelter after its progenitor.
Dad thought the Anderson useless. Many in our street filled with water. Some people did not bother to prepare at all. Everybody knew the war would be over by Christmas. The mass of people had no idea of just how ill prepared our military was. They were betrayed by the newspapers, led by the _Times, Daily Express,_ and _Daily Mail,_ and even by the BBC, which had shamefully soft-pedaled and suppressed the years of alarms from a small band of Chamberlain's critics in his own party (including future prime ministers Churchill and Harold Macmillan, but not a vain and timid Sir Anthony Eden). "Germany's tanks are made of cardboard," a know-all boomingly reassured a shop full of our customers. "Joe, that can't be right," said my father, who was helping out that day. Then he delivered a numbing recital of figures of German steel production, recalled from his well-thumbed _World Almanac and Book of Facts._ (This didn't go down at all well. My mother had to remind him that never mind his blessed almanac, the customer was always right.)
Dad's irregular shifts at the railway meant we often didn't see him for two or three weeks. He drove trains carrying thousands of children out of Manchester, their schools closed and relocated miles away from industrial areas, considered prime targets for bombing. Schoolmates from Brookdale and Briscoe Lane Junior, with their name tags and mandatory gas masks, were put on a train to Ramsbottom, a village on the West Pennine Moors, where the local families could pick and choose whom to welcome into their homes for the duration. By all accounts most of the evacuees, even the scruffiest, were kindly received in foster homes more comfortable than the dwellings they'd left behind, in the process happily finding out about the mysterious "countryside." Schoolmate Alf Morris, who in the fullness of time became a Labour minister and later Lord Morris, was one of the unlucky ones. Separated from his brother and sister, who were taken in by a well-to-do family, Alf was billeted with a mean old couple who sent him out hawking firewood they chopped for a living and half-starved him on suppers from the fish-and-chip shop. Alf was back home in a month or two.
As the city emptied, my parents kept us in Ashworth Street. They wrote to my father's mother, Sarah Jane, to ask if Fred and I could stay with her on the Welsh border. She was now Granny Jones, having married a gravedigger called Jack. Anything remotely to do with death upset me, but I was not wild about the prospect of going to Ramsbottom either. I'd heard too many comics on Rhyl pier mouthing "Ramsbottom" as if the name was a joke in itself. My inclination to stay away was clinched when a postcard came back from a Brookdale classmate saying that it was once known as Tupp's Arse (Tupp being old dialect for "sheep").
Once or twice a week a Brookdale teacher came to the house and gave us homework, but Mum and Dad worried that we were falling behind. While we waited for Granny Jones to write back, a neighbor told Mum about another possible sanctuary. It excited me most of all: Somerford Hall, which was an old manor house in the countryside near Congleton in Cheshire where the Manchester Education Committee had set up a boarding school.
To appreciate the attraction of Somerford Hall, you have to know about the _Magnet._ Of all the comic books we bought at the newsagent's every week— _Beano, Wizard, Dandy, Hotspur_ —the _Magnet_ most insinuated itself into my imagination. This is where Frank Richards (aka Charles Hamilton) recorded the adventures of the fifth form at Greyfriars, the ancient ivy-covered public school secure in its gentlemanly traditions. The setting represented some of the lyrical features of English life evoked by Rupert Brookes's Grantchester poem: "Stands the Church clock at ten to three? / And is there honey still for tea?" The class overtones should have grated on me, I suppose, but they didn't. After the asphalt playgrounds of Newton Heath, I liked the idea of surviving the war with toffs on the lawns of Greyfriars, and I relished the _Magnet_ stories where beastly cads got their comeuppance and good eggs and goodness triumphed. Here were Harry Wharton, Bob Cherry, Frank Nugent, Johnny Bull, the Indian cricketer Huree Jamset Ram Singh, the Form Master Mr. Quelch (too ready with the cane), and the crafty Billy Bunter, "the Fat Owl of the Remove," plotting to steal someone else's tuck. I identified most with Harry Wharton, captain of the remove and editor of the _Greyfriars Herald_. I saw myself seeing out the war at Somerford Hall, toasting crumpets for tea with Bob Cherry after hitting one of Ram Singh's googlies for six to acclaim from the whole of Greyfriars.
One glorious morning we heard that Fred and I had been accepted at Somerford Hall. Arriving a week later, we found no gracious hall and no ivy. The school was a collection of six or seven wooden sheds in soggy fields, an encampment on the grounds of the shut-up old country house. None of the city evacuees had time to acknowledge my resemblance to Harry Wharton; they were too busy jostling for sheets and blankets and fighting for jelly sandwiches from the makeshift kitchen. At bedtime I was assigned the upper berth of an iron bunk in a bare board hut with about thirty others. The first night, as the wind howled, I sobbed my heart out, homesick and disillusioned. "Shut up, crybaby!" yelled everyone. The protesters hurled shoes at my bunk; even Fred told me to put a sock in it. Cripes! I was Billy Bunter, the object of execration. The next day we had lessons. In English class we were told to write home and present our letters to the teacher, who would check the grammar before putting them in the post. I wrote two pages about my disappointments and handed them in. I was called to the teacher's high desk up front. "This just won't do," he snarled, screwing up my effort into a small ball that he lobbed into the wastepaper basket to giggles behind me. "Go back to your desk and write something cheerful."
That night, with a thumping heart, I sneaked out of the back of Somerford Hall after lights-out and found a postbox to send a letter to Ashworth Street pleading to be rescued. I had no stamp for it, but at the weekend both my parents arrived at the camp and took me home. The hut was glad to see the back of me. Fred, less infected by the Greyfriars fantasy, elected to stay and had a great time for six months. Down the years he developed a number of different theories concerning my lack of moral fiber, none of which bears scrutiny.
Brookdale was still closed when I came home. Within days I was bound for the Welsh hills, more precisely for the hamlet of Hengoed in the county of Shropshire, but a slingshot east of Offa's Dyke, the ancient earthen rampart separating the wild Celtic tribes from the English. Dad had no petrol ration for his car, so we took two trains to Gobowen, the railhead for Oswestry, then walked four miles in the fading sunshine along winding, densely hedged lanes, through tunnels of overhanging trees, trying to find the Old Vicarage before dark. To baffle German parachutists all the signposts had been taken down, but eventually we were directed to St. Barnabas Church, where Grandfather Jones plied his trade. Down the hill from the church we opened a door in a high wall of mellow stone to find Granny Jones scattering grain among scores of chickens.
She was a pale wraith of a woman in her mid-seventies, inherently graceful and affectionate and very attentive to the needs of her new husband, who must have been fifteen years younger. She scurried around in her flowered pinafore to take off Gravedigger Jack's boots when he clumped into the stone-floored kitchen and dropped into a high-backed wooden chair by the fireplace: his chair, his hearth, his house. He was a red-faced man in corduroy trousers hoisted by a fat leather belt he said he would apply to my backside if I didn't behave. I convinced myself he had a twinkle in his eye. It was good to hear he had spent the day cutting hedges and clearing ditches in the next hamlet, there being a shortage at St. Barnabas of candidates for the next world.
Gravedigger Jack rarely spoke, and he had an iron routine. If supper was not on the table sharp at six, he wouldn't eat it. For hours after eating, he'd sit silently smoking his pipe while I read _The Count of Monte Cristo_ by the light of an oil lamp and Granny filled the kitchen with the warm smell of homemade bread. Her secluded old house, set in meadows with an apple orchard and a brook at the back, had no electricity, no gas, no radio, and we drew water from a pump in the yard. On Sundays we sang in the Hengoed church's little choir, smug in our white surplices. Not a whisper of the war percolated. It was paradise.
Finding shortcuts across fields to the village school in Gobowen was a daily adventure: seeing the flash of a fox in a copse, leaping brooks, slashing a path through banks of nettles and brambles with a "sword" fashioned from a stripped elderberry branch. It was 1066, and I was on Senlac Hill with my namesake king, driving the bastard Normans back into the sea. I was romantic and reckless. When a route skirted a wasps' nest, I set fire to it and was stung in the eye for my folly; so much for acting King Harold. Still, the hazard that gave regular pause to my impetuosity was the cow pasture, where there might be a possessive bull. Every boy at school had a fanciful story of racing death against thudding hooves and sharp horns. I had been the fastest runner at Brookdale and fancied myself to beat any bull. After I saw one on the loose, I decided against putting my speed to the test. I checked every beast for balls and every pasture I crossed for an exit.
At school there were some sixty boys and girls from ages eight to about fourteen, scrunched up at little desks, all learning the same things. I could not concentrate much on what was going on because I soon fell madly in love with a poised thirteen-year-old evacuee from London called Gwyneth: a southern beauty! I wooed her with horse chestnuts. The big prickly fruits, which dropped in the autumn or could be brought down by hurling a stick, have a white pith encasing the chestnut seed—a big nut called a "conker." For our most popular game in Gobowen we drilled a hole through the nut, threaded twelve inches of string through, and knotted it. You dangled your conker as a target to receive the blow of your pal's stringed conker; if he broke yours, his became a oner. If you had already broken four other conkers, then he could claim all your precedent victories, and his conker became a fiver. I sacrificed a niner to Gwyneth—just think of it. I forbore to tell her that to make it harder I had peed on it. This was no perversion. It was common practice to urinate on the conker, soak it in vinegar, and expose it to heat. Big juicy conkers often succumbed to a blow from a hard little conker. Blissfully unaware, she appreciated the gift.
My next gambit, proof of manhood, was to announce that while Gwyneth ate her sandwiches in the midday break, I was going to climb to the top of a spectacular tree in the playground. I did. The effect of my bravado was rather ruined because I couldn't get down before the bell went for classes. I never saw her again. Lovesick, I hung around the cottage where she stayed for days until someone told me she'd been called back to London, where her school had reopened.
Eventually I was summoned home, too. I'd had four happy months with Granny Jones, followed by a Spartan two months under the thumb of Aunt Beattie on Uncle Ted's little farm at Old Stone House. He had six fields, twelve cows, a couple of horses, and countless pigs, to which he fed swill from the Oswestry army camp. To Aunt Beattie, stomping through the mud in big rubber boots with a cigarette hanging from her lips, I was a city sissy: I failed miserably at milking a cow.
In the absence of German warplanes, classes had resumed at Brookdale, but my return was required anyway because I had reached the age to take part in the all-important national "eleven-plus" examinations—tests administered to eleven- and twelve-year-olds, the results of which would determine whether a child would stay in elementary school or ascend to secondary school. Sitting in a drafty classroom for four hours with forty others, I didn't appreciate that the curious questions, some of which seemed easy and some of which were incomprehensible to me, would decide my future. The aim of the examinations was to identify a handful of really bright boys and girls who would be offered a superior "grammar-school" education to the age of sixteen, and with that the opportunity to win a Joint Matriculation Board certificate, the passport to a good job. There were places at grammar schools for no more than 10 percent of the eleven-year-olds. The other 90 percent were destined to remain at elementary school and leave for work at the age of fourteen—trade apprenticeships if they were lucky, menial work if not. Put another way, the results of the eleven-plus would decide the course of the rest of my life as surely as a switch in the railway points would decide whether Dad's train went east or west.
My fate was contained in a buff envelope Mum brought up the stairs to the bathroom one morning when I was washing my face. I opened it with soapy fingers. "Harold Evans has been selected for St Mary's Road Central School."
I was at once bothered and bewildered. I had not been selected for a cherished grammar-school place, but neither had it been decreed I should stay in the elementary school. St. Mary's Road Central School, I discovered, was one step up from elementary school, an intermediary school where you could stay until you were fifteen, and a few pupils might have a chance to take the crucial Joint Matriculation Board examinations along with the grammar-school boys. I fretted that I had been held back by evacuation and country school, yet when Fred came home later from the Congleton camp, he sailed into a grammar school. Alf Morris, too, was selected for grammar school, but his family could not afford the uniform, and he stayed at Brookdale until he was fourteen.
I was excited all the same. It was not so much the education—there was a rumor that St. Mary's expected its pupils to learn French—as the blue cap, school tie with lion rampant, and crested blazer I would be entitled to wear and the leather satchel for my exercise books.
Fearful that sporting this lot risked jeers and jostles from the lads who were staying on at Brookdale, as I passed my old playground I sprinted up Albert Street to the No. 7 bus stop at the entrance to Brookdale Park. Fifteen minutes later I was looking up at the gaunt Victorian redbrick edifice of St. Mary's, across the street from the coal heaps and rusting ironmongery of the Newton Heath rail yard and the great sheds that housed the steam locomotives Dad drove. Not a single blade of grass was in sight; the playground was tarmac running into brick walls.
Looking down from the schoolrooms, we could see a rail line leading from the locomotive sheds to a turntable at the foot of our red cliff. I always hoped I would catch a glimpse of my father in command of a steam locomotive, but I never did. Gazing out of the high windows, we ran the risk of being discovered doing nothing by the school's dominating person, the headmaster, Mr. W. L. Marsland. He appeared to us a sinister figure. He glided through the school without apparently moving his legs and held his head well back so that he appeared to be looking down on you. I was in a high state of nervous tension. I'd just discovered Dickens, and the cruel schoolmaster Wackford Squeers was fresh in my mind. For a few weeks I avoided stepping on cracks in the pavement and obsessively touched every third lamppost just to be sure the gods were on my side. Going up the road from school to the bus stop for home, we always ran past a spooky derelict building that was once an inn called the Duke of York. The body of a young woman killed crossing the railway line in the nineteenth century had been brought there, and we believed it when we were told her footsteps could still be heard around the shattered old bar.
But if the exterior of St. Mary's was workhouse grim, its environs bleak, and the headmaster forbidding, I soon discovered that the school was infused with an appealing spirit—a nurturing combination of respect for traditional values and zestful competition created by Marsland. We were all attached to one of four houses honoring British heroes. I was in Scott; others were in Livingstone, Gordon, and Stephenson. I'm sad to think I'd better add a word about these icons of the British Empire. Robert Falcon Scott (1868–1912) was the explorer of the Antarctic who died with three companions on returning from the South Pole. David Livingstone (1813–1873) was the African explorer, medical missionary, and antislavery crusader whom the American journalist Mr. H. M. Stanley was eager to meet. General Charles George Gordon (1833–1885) led military campaigns in China and the Sudan and died defending Khartoum, beheaded by the Mahdi. And George Stephenson (1781–1848) built the Stockton and Darlington Railway, the world's first permanent public railway line powered by a steam locomotive. Various privileges, whose nature I've forgotten, were awarded to members of the house that had accumulated the most "good house points" in class and our infrequent visits to a sports field where we wore our house colors (I regretted Scott was yellow).
Marsland's disconcerting way of looking at us, I discovered, was not an expression of contempt. It was physiological: his eyelids were stuck because he'd been gassed in World War I. I never heard him talk about his experiences, but he wanted us to remember the price of freedom: St. Mary's "adopted" Czechoslovakia, sacrificed by Chamberlain at Munich in 1938, and each day in assembly, while a flower was placed on the school war memorial plaque, a boy or girl read out the lines of Laurence Binyon's memorial poem: "They shall grow not old, as we that are left grow old;/Age shall not weary them, nor the years condemn./At the going down of the sun and in the morning/We will remember them."
Marsland had pride in his school. He expected us to behave like grammar-school boys, and we—all grammar-school rejects—strove to justify his faith and ourselves.
Sixty years later I can still see all the teachers and recall their names and mannerisms. For the first few lessons with the science teacher Eddie Whipp (a precise representation of his name), we focused only on the _dees_ and _doze_ of his nasal speech, but he was a masterly choreographer of chemicals. Every time I passed the somber Philips gasworks at the end of Briscoe Lane, I thought of Mr. Whipp's magically enriching hydrocarbons dancing within. There was the (still tangible) first-time thrill of seeing litmus paper turn red, proving the substance in the beaker was an acid, and the excitement of summoning up hills and valleys after the angular, testy John Bateman had shown how the contour lines on maps could be converted into shapes. I was enthused enough by the suave Joe Abbott describing the day Oliver Cromwell had King Charles I beheaded to spend hours at home memorizing the dates and events leading up to that dreadful sequel.
Early on I picked up the journalist's habit of never throwing anything away, and I still have the school reports of those years. It is nothing less than saintly of me to acknowledge that in the first year I scored a disgraceful 2 percent on the music exam set by the ample Miss Polly Wardle, whom I couldn't help but think of as "Miss Warble." This "terrible result," as she described it, puzzled the good Miss Polly because she certified I had indeed "worked in class." But this may have been a reference to my countertenor rendering of Handel's "Where'er You Walk," a promising diversion from reading scores until I fell out of favor when my voice broke mid-song.
Meanwhile, in geometry and algebra I was damned as "often careless," "erratic," and "disappointing," and my efforts in physical education were dismissed as "tries hard but has no sense of rhythm."
I had such a dim start for all my efforts that I thought the eleven-plus examiners had me right. As the terms went by, however, I began doing well ("promising"), then very well ("excellent worker"). I was always top of the class in history exams; scored high in French and English composition, language, and literature; and battled a brainy brunette called Betty Ogden for top place in science and geography. Even the demon physical training instructor in the third year was moved to dub me "nimble, a very good performer." My surviving little blue English composition exercise books reflect my romantic addiction to the adventurous historical novels of Jeffrey Farnol ( _The Amateur Gentleman_ ), Henry Rider Haggard ( _King Solomon's Mines_ ), Arthur Conan Doyle ( _The White Company_ ), P. C. Wren ( _Beau Geste_ ), and Walter Scott ( _Guy Mannering_ )—all borrowed from my beloved Failsworth Public Library. But a degenerate streak soon showed itself. I had the idea I would grow up to write enthralling fiction. I had no talent for it. No subject was safe from the pestilential prose I inflicted on S. J. Pawley, a gentle English teacher with a withered arm. He set us a composition innocuously entitled "Village Shop." I submitted one beginning "I looked up from the bare uncarpeted floor. There she stood, no longer young, but a huge mass of fat. Wiping her bare red arms on her dirty apron she waddles towards me."
In the fourth year I was elected school captain by some combination of staff nominations and school votes. A blue shield was ceremoniously pinned on the lapel of my blazer, and I kept it on at home. My brother Fred, who was rebelling at Chadderton Grammar, said I was a goody-goody, and while the badge was supposed to confer authority over all my schoolmates, Fred made it clear my writ did not run at home. My main school duties were to organize the prefects to watch out for bullies and troublemakers on the playground and when the bell went for classes to resume, keep order in the long lines on the stairs and in the corridors. There were many bigger, stronger boys in the school. I'd been terrified of one brute who liked to grip smaller boys in a rib-cracking bear hug. The blue shield was like invisible armor: I was no longer singled out for his embraces. I fretted a lot in these adolescent years about being skinny and flat chested. There weren't nearly as many musclemen on the beach as there are these days, but I felt I was exactly the pale seven-stone weakling featured in a comic-strip advertisement for the bodybuilding system of Charles Atlas. The ads we saw featured a sand-kicking bully on a beach humiliating a scrawny boy walking with his girlfriend. Weeks later, rebuilt by Charles Atlas, the weakling sees off the bully and wins the admiration of the girl. It was my first experience of the power of advertising. Brother Fred was stronger than I was, but the ads got to him, too, so we persuaded Dad that he should invest in our taking the correspondence course. With all that heavy work on the footplate, Dad didn't need the course, but when he was home, he joined us in our bare-chested nightly efforts to look like the glistening Mr. Atlas. Mum tolerated the sight of the three tensed-up men in her life straining to pit muscle against muscle in their own bodies in the system called "dynamic tension" so long as our efforts did not delay our arrival at the supper table.
As school captain I was supposed to set an example to the whole school. This was complicated by the fact that I had fallen in with bad company. Howard Davies, a youth who lived nearby, was so slick he had a toothbrush mustache at seventeen, Brylcreemed his hair, and claimed to have had carnal knowledge of several members of the Women's Royal Naval Service (Wrens), promising that he would initiate me, too. He was the Stromboli to my Pinocchio.
Among other things, Howard taught me magic tricks. Of course I hated him at first, not just for his sexual prowess but also because in my Christmas stocking I received a velvet bag that could make an egg disappear, and he was better at the trick than I was. He became a good enough conjurer to be engaged by Uncle Mac's amateur concert party and later by the semiofficial body for entertaining the troops. Uncle Mac, though, was on a lower plane than many of the stars he toured with. His main gig was putting on shows at night for the servicemen and servicewomen in the North West bored out of their minds when they weren't operating the searchlights and ack-ack batteries. He also organized summer shows in the parks for the Stay-at-Home Holiday campaign (which encouraged people not to travel on their holidays), for which I was drafted as the rear half of a donkey.
Mum and Dad didn't like Howard, but I quickly became stagestruck. The world of Leichner makeup, footlights, and frenzied changes backstage between acts was even more exciting than banging the school gong, but one night offered an experience that was unnerving for an adolescent. A pretty young actress in a sketch had just rushed offstage while I awaited my cue. I looked back and through a gap in one of the changing room curtains my eyes lingered on her beautiful white bottom wriggling out of a pair of tights. I had no psychological mechanism for the shock when moments later the curtains parted to reveal that the favored backside belonged to a boy.
I was haunted by the ambiguity and regarded as condign punishment the subsequent drama of the red ball. After the Peeping Tom trauma I rededicated my spare time to practicing Howard's sleight-of-hand card tricks. He taught me how to acquire dexterity by rolling a little red ball from finger to finger, and I did it furtively at all times of the day in school. Classes would stop when I lost the grip and the ball rolled to the front. I would retrieve it with a "Sorry, sir." One day I was practicing in the washroom next to the teachers' recreation room, and the ball fell down the drain. I did not see it again until a week later. Mr. Marsland held it up before the whole school in assembly. "Who is the boy who owns this little red ball? Who is the boy who has blocked the washrooms with it? Who is the boy who flooded the teachers' staff room? Let him have the decency to own up and see me in my room after assembly."
I was punished with three strikes of a leather strap on my open hand. Hand-strapping was commonplace in the classrooms, but the pain here was in letting Mr. Marsland down.
Marsland was something of a mystery, for though he wasn't a university graduate, he was a more cultivated man than the Manchester Council required or expected. St. Mary's had no playing field, so he taught us the refinements of cricket on expeditions to a city park.
In the final year, five of us were thought to have a chance of passing the Joint Matriculation Board examinations normally reserved for the grammar schools. Marsland organized after-hours coaching. When everyone else went home at four o'clock, we five, three girls and two boys, stayed behind in his study. The school's emptiness echoed around us as he read passages from _As You Like It_ and _A Midsummer Night's Dream_ in a lovely lilt. At first we were hard put not to giggle when he acted out Pyramus on tiptoes talking to Thisbe through the wall's chink, but we soon hung on every syllable. He brought the same intuition to directing us gorily in the school performance of _Sweeney Todd: The Demon Barber of Fleet Street_. It has always been a sadness to me to think how little we were able to respond to his graciousness and learning, and his courage in trying to raise the academic and athletic standards of his school. He died some years later from the effects of his World War I service, and St. Mary's Road Central School was demolished by an education committee that could have known nothing of the magical forest summoned up behind our redbrick walls as the shades of night fell over the railway yards.
In 1940, when Hitler smashed Norway in April and then the Low Countries and France in May, I was more concerned that the end of the phony war marked the demise of the _Magnet_ comic book, closed by paper rationing. The fictional Greyfriars vanished along with the universal illusion that the war would soon be won. My days of careless youth ebbed away, too, after the morning on the beach at Rhyl with the Dunkirk survivors. I resented the time Dad spent with them instead of with me, but I came to see the encounter as cathartic. And while I didn't suddenly grow up, I read more and more about the war. I was already intoxicated by the _Express_ and by newspapers altogether as a manifestation of a more potent magic than Howard Davies's. The "Daily Surprise," as I came to think of the newspaper, simply amazed me. How did they do it not once but differently every day? How was all this information gathered? How was everything fitted on a page with nothing over? How did the photograph get taken on the battlefield, and how was it reproduced? Who were these dazzling figures with notebooks sitting at the feet of Churchill and Roosevelt? How were the newspaper's strong opinions determined? Little opinion seeped out of the radio. On political questions the august governors of the BBC would rather be caught naked in Whitehall than be seen trying to influence people.
Oh, yes, and who was the mysterious anonymous columnist in the _Daily Express_ who signed himself "Beachcomber" and wrote a few hundred words headed "By the Way"? Those words convulsed me with insane laughter I could never explain to Fred or anyone else. "Sixty Horses Wedged in Chimney" was a typical Beachcomber headline, followed by "The story to fit this sensational headline has not turned up yet." Beachcomber was nonsense on stilts but comically perfect in his parodies of newspaper style: "Erratum. In my article on the Price of Milk, 'Horses' should have read 'Cows' throughout." As a toddler, I'd been hooked on a comic strip in the _Express,_ the adventures of Rupert the Bear; the incentive to finish my porridge in the morning was to see his antic figure painted on the bottom of the bowl. But Beachcomber was a lifetime's addiction even after I soured on his host newspaper for its vendettas and political distortions. For thirty-very-odd years I was to follow his surreal daily reports from an asylum populated by Dr. Strabismus of Utrecht Whom God Preserve (inventor of Bracerot juice designed to make Hitler's trousers fall down); Mr. Justice Cocklecarrot at the Court of Uncommon Pleas (adjudicating the right of Mrs. Renton continually to ring the doorbell of Mrs. Tasker for the purpose of depositing twelve red-bearded dwarves); Great White Carstairs (the overpatriotic ambassador who could not talk on the telephone to the Foreign Office without saluting); and Narkover School's cunningly villainous headmaster, the light-fingered Dr. Smart-Allick, all of them in a funny-mirrors gallery of grotesques similar to the daily procession of frauds and fools who, happily for us, provoked Beachcomber, aka John Cameron Andrieu Bingham Michael Morton, to put on his countryman's boots and scale mountains of absurdity. No word or phrase was safe from his mischief. A clue, he wrote, "is what the police find when they fail to arrest a criminal." And again: "One disadvantage of being a hog is that any minute some blundering fool will try to make a silk purse out of your wife's ear." Morton's ink flowed into the veins of Spike Milligan, Peter Sellers, and Harry Secombe (the BBC _Goon Show_ ); John Cleese, Michael Palin, and Eric Idle ( _Monty Python_ ); and Richard Ingrams ( _Private Eye_ ). He had a place in my pantheon with Evelyn Waugh, P. G. Wodehouse, Stephen Leacock, J. K. Jerome, and S. J. Perelman.
Hollywood reinforced my infatuation with newspapers. Our local cinemas—the Magnet, the Pavilion, the Picture Palace, the Ceylon, and the Grand—reopened in the autumn of 1939. Every Saturday at the Magnet matinee, Fred and I jostled others kids lining up to get in and then scrambling for the sweets the manager threw into the mob. The run of movies in those years still seems remarkable: _Gone With the Wind, Citizen Kane, Ace in the Hole, The Gold Rush, Stagecoach, Casablanca, Foreign Correspondent, 49th Parallel, Beau Geste, The Front Page, Gunga Din, In Which We Serve,_ and Bob Hope and Bing Crosby on the road some-mad-where with Dorothy Lamour. Heroes—and hilarity—aplenty. I loved them all, but it was the movies about newspapers I tried to see over and over again. I identified with the small-town editor standing up to the crooks, the tough reporter winning the story and the girl, and the foreign correspondent outwitting enemy agents. That was the easy part—cops and robbers in different costumes—but some of the newspaper movies stayed in my mind when the adrenaline rush had gone. How could _Ace in the Hole_ reporter Kirk Douglas be so greedy for a scoop on a man trapped in a mine that he delayed the rescue just to keep the story alive for the media circus he created? Charles Foster Kane begins as a tribune of the people, the immigrants and the unions, against the "octopus" of the Southern Pacific Railroad, the bosses of Tammany Hall, and the banks and Wall Street, and he ends up manipulating the news as a crypto-fascist crusader for capitalism. Was that a true portrait of William Randolph Hearst? I learned in due course that it was a distortion of Hearst, but I relished the moment when Citizen Kane's managing editor, Mr. Bernstein, pulled out the proofs of two front pages made up in advance, one announcing that Kane had won New York's gubernatorial election and the other announcing that he had lost, one banner-headlined KANE ELECTED and the other FRAUD AT POLLS! What fun!
The war became intensely personal when bombing of the Manchester area began in August 1940. It was only sporadic at first, but Dad was much at risk out there in the blackness in some godforsaken tangle of railway lines. Schoolgirl Enid Parker, who was to enter my life ten years later, saw an exploding ammunition train light up the sky near her home on Queens Drive, Liverpool. Much more was to come. The full-scale blitzes began on September 7, with heavy bombers pounding London for fifty-seven nights in succession. People in Manchester kept saying jocularly, "Well, we're bound to get it next."
We did. The awesome two-day Christmas blitz of Manchester began at dusk on Sunday, December 22. After the bombing of Liverpool, Mum made up camp beds in the brick shelter that the Failsworth Council had built to replace the waterlogged Anderson shelter dug into the back garden. As soon as the sirens sounded on the Sunday, she shut the shop and hustled baby John, four-year-old Peter, and me out of the house and into the shelter with a big basket of sandwiches, hot water bottles, thermos flasks, and a flashlight. Two of the family were missing. On Saturday, December 21, Dad, on a day off work, had taken my ten-year-old brother, Fred, with him to Gobowen, the village near Oswestry, to see if Granny Jones had a turkey she could spare for Christmas. Without a telephone, we had not expected to hear anything from them, but sitting in the brick shelter in the cold candlelight without the perennially cheerful head of the family, we were lonely and very afraid, our shelter isolated in a vast cavern of echoing noise—the drone of more than a hundred bombers overhead, the crump-crump of our ack-ack guns, the blast of the bombs.
We'd no idea we were smack in the bombers' sights, our very street clearly to be seen on the high-level aerial reconnaissance photographs of targets made by German planes on October 5, 1940. I came across the dated and marked photographs only when I started to write this memoir and visited Peter Charlton, a school friend from Brookdale Park and St. Mary's Road. He had become the historian of Newton Heath and in German archives had unearthed one headed:
**Manchester–Newton Heath**
**Flugzeugzellenfabrik A. V. Roe & Co Ltd.**
Superimposed on the image were black rectangles of areas to be bombed. The perimeter of the principal target, the aircraft factory A. V. Roe, was just down the road from us. The aeronautical genius Alliott Verdon Roe, born, as I was, in Patricroft, designed the RAF's most successful bomber, the Lancaster, which flew 156,000 operations from 1942 to 1945. Lancasters delivered the bouncing bombs that broke the dams of the Ruhr Valley in 1943 and the thirty-one Tallboy bombs that finally sank the mighty German battleship _Tirpitz_ in November 1946. Three other targets in our neighborhood are marked on the Luftwaffe map: Gaswerk am Philips Park (which featured in my science lessons), the Mather and Platt gun factory, and the Crossley Brothers diesel works.
Apart from starting a fire in one building of Crossley Motors, the 270 bombers missed all the targets so meticulously pinpointed. Instead, their 272 tons of high explosives and 1,032 canisters of incendiaries fell three miles to the southwest of the factories, blowing up and burning the area from Manchester's city center of Piccadilly Gardens to Victoria station, the cathedral, and into Salford and Stretford. Had the bombers erred in the opposite direction, missing A. V. Roe as much to the northeast as they did to the southwest, they'd have obliterated the Brookdale area and 54 Ashworth Street, and all of us with it. As it was, around the devastated city center seven hundred people were dead.
We could smell the smoke, and fires were still raging when we came out of the shelter the next morning. My uncle Arnold and his wife, Gertie, emerged from their garden shelter near Trafford Park to find their home was gone, burned to the ground. By lunchtime Dad and Fred had still not reappeared. We worried all day. Mum refused to go down to the Newton Heath police station to ask if they had heard anything about bombs falling on Welsh border villages. "We can't bother them," she said. "They have enough on their hands, and Dad can look after himself." She reopened the shop, and I helped to look after John and Peter, while popping out from time to time to stand watch on the street corner.
At dusk, when nobody was about, I saw figures coming up Farm Street, but moving very slowly. I ran down the street, and it was Dad and Fred, laden with a turkey and other good country things. (Throughout the war, rationing restricted everyone to tiny quantities of everything—bacon, butter, meat, tea, sugar, jam, cereals, eggs, milk, sweets, clothing, biscuits, canned fruit—but the countryside fared better for eggs.) They'd caught a morning train from Gobowen to Crewe and another to Manchester, but a bomb had torn up the track from Eccles, so they had to walk, jumping on any bus that was still running in the hope it was heading in the right direction: all of the destination markers were blanked out. They arrived just in time for the second full night of bombing. It was just as bad as the first night, but with them back with us the fear evaporated.
The only consolation for all the bad news on the evening radio in the early years of the war was that the steadiness of the ice-clear baritone voice of the BBC announcer Alvar Lidell suggested there would always be an England. In these radio years, voices were so important to us, the anchors of our hopes as much by tone and inflection as by the words. Churchill had no better news for us than Chamberlain, but his voice evoked the spirit of St. Crispin's Day; Chamberlain's would not have scared a sparrow hawk. We suffered a sense of isolation until the marvelous moment in December 1941 when Mr. Lidell at last had something good to relate. America had entered the war. Fighting spirits rose hearing Churchill growl his epic phrases; FDR's voice soothed. Having just learned about the Gulf Stream from Mr. Bateman, I thought of FDR's words as a powerful warm current crossing the ocean to keep us calm and cheerful. His words were uplifting—"We will gain the inevitable triumph—so help us God"—but the almost languid calm of his rolling articulation was essential to the reassurance.
The words became gloriously manifest when in Manchester's bomb-gutted Piccadilly I first saw a lanky, leather-jacketed American airman on a weekend pass from the Flying Fortresses at Burtonwood, Warrington. Soon after I saw my very first black man. The British Empire had dominion over millions of black and brown people, but before the war we were still a homogeneous race of pale whites living insular lives, never hearing a foreign accent in the provinces or venturing across our narrow moat into Europe. When Dad, in 1938, took us all on a day ferry from Ramsgate for a few hours in Boulogne, it was the talk of our neighborhood for weeks.
What color and dash the smiling Americans brought to our bleak gray world! We saw them all as rich film stars, so sleek and well tailored in their smooth uniforms. Any one of the big-boned American servicemen sighted in the city drew a flock of kids asking, "Any gum, chum?" and they always got something. Women, we heard, were given gifts of nylon stockings and cigarettes, arousing some resentments. My widowed cousin Little Eva was the happy recipient of this generosity from one of the black servicemen she brought home to tea in Eccles, to some spiteful gossip according to my mother; later she had a lovely baby daughter by another airman. The odd sourpuss in our shop might recycle the crack by the comedian Tommy Trinder that the Yanks were "overpaid, oversexed, and over here," but as far as I could see, the Americans were warmly welcomed. Most English people just were too inhibited to express much emotion beyond an occasional thumbs-up gesture.
The war dragged on. It was a strain for Mum the shopkeeper. The histories I've read about the home front highlight how hard rationing was on the public; nobody bothers much with what it was like for the shopkeeper, the hub of the system. The administration of it was complex. Some foods were rationed by weight, some by quantity, some by groups so a customer could opt for jam rather than marmalade or swap the jam for extra sugar. Early in the war holders of ration books were also given sixteen points they could spend on unrationed foods, such as a tin of Fray Bentos corned beef, a packet of Kellogg's corn flakes, a box of Peek Freans assorted biscuits, or a tin of Tate & Lyle golden syrup—on the label a defunct lion plagued by bees—which made such a difference to the morning porridge. But supplies of such luxuries were erratic. Mum was so scrupulously honest that when there was a lucky break—one month the arrival of six unrationed tins of salmon—she agonized about whose need was greatest among the several hundred registered customers, and she had to distribute the tins by stealth to avoid jealous fights.
The unseen burden was counting the coupons from the ration books. The number of coupons in various categories returned to the Ministry of Food offices in Failsworth determined what supplies the shop would get the following week. No coupons, no food. It was an awful chore counting and recounting those fiddly bits of paper; the coupons for sweets and chocolate were tiny and easily lost. Some shops had to close. Everyone in the family had to help out at the end of the week, all of us kneeling on the carpet for four or five hours on a Sunday night. It was the burden of rationing, which continued after the war, that induced Mum to sell the business.
I'd no ambitions to be a shopkeeper. By the time I was thirteen, I had grand ideas of myself as a journalist-historian of the war. In my fourth year at St. Mary's there was an election to choose an editor for a new school magazine. The English teachers nominated a handful of candidates, and I was utterly shameless in campaigning to win the editorship.
The magazine I produced was a limp little thing. My incipient journalistic juices must have gone into my pet project at home. I scoured the _Daily Express, Daily Dispatch, Manchester Evening News,_ and any magazines I could get my hands on for maps, drawings, and pictures. I pasted them in Manchester Education Committee exercise books from night bookkeeping classes I took later, so that the tumult of the war overlay neat balance sheets of assets and liabilities of fictional companies. I still have these war books with my copperplate captions and their headlines:
"First, know your enemy: silhouettes of new Luftwaffe warplanes"
"Achtung! Britain's perfect bomber-buster comes off the secret list"
"German tanks burn on a Tunisia battlefield—British six pounders hold the position"
"Iron Crosses for seven but look at their boots"
"Over mountains, through steaming jungle, to smash the Jap toe-hold at Buna"
"These Italians are happy to be out of the struggle"
"Gurkha rescued under fire at Mareth"
"RAF aimed well—factory is hit; the homes are spared"
The pictures that most excited me were panoramas of our invincible armies on the march in the Western Desert, cutaway drawings of our deadly fighter planes and flying boats dominating the skies, and gallant destroyers on heaving seas sinking U-boats. British nationalism was intense, but the Free French cavalry in the desert was allowed to gallop into the pages in sepia. I also gave a page to bearded Cossack guerrillas in the snow, and one to a portrait of Lenin. Any number of my classmates were superior in identifying aircraft silhouettes; I was in for the drama.
The impression the pages give is of a "Boy's Own" jingo paper: we were winning gloriously. Of course all of us of that generation had grown up with an exaggerated idea of Britain's prowess. The schoolroom maps of our world were mostly colored red for the two-fifths of the globe that was the British Empire and its dominions; half the traffic through the Suez Canal was British. How lustily we belted out "Rule, Britannia! Britannia, rule the waves:/Britons never, never shall be slaves." In the end it was assumed without question that the nation of Wellington and Nelson would triumph. My selections were loaded with optimism, because they were, again, a reflection of what was in the newspapers, though leafing through the two crammed books today, I rather scent the mephitic vapors of the time: a dread recalled that the next day's news might be bad. The magazine was a comfort, something to hold on to in a depressing yearning for certainty. Just look at that Mosquito, the world's fastest reconnaissance bomber, on its way to attack Gestapo headquarters in Oslo!
I kept my little war books to myself. I didn't show them to Dad. I glowed with his constant encouragement, but about the war I feared his skepticism. He still had reservations about Churchill. He admired Franklin Roosevelt, but as the war ground on, he talked most about the Russians. "That Marshal Zhukov!" he'd say with a chuckle, gripping me in an imitation of a grizzly bear's squeeze. "He'll get them in a pincers movement. Just watch." Only years later, writing history myself, did I think that Dad may have been doing more than indulging his romantic view of Russia; he may have divined from the numbers what most people did not at the time: the bulk of the fighting against Germany was being done by the Russians, so the European war would be won or lost on the eastern front.
I don't think any of us St. Mary's graduates thought at all about the distant future that summer of 1943 as we stood in the last assembly belting out "Forty years on and forty years after, / parted are those who are singing today." The war was too much in the way. There was no future; there were only black lines and arrows on newspaper maps marking the clash of armies and navies in the infinities of Russia and the Pacific Ocean.
I had done well in the examinations for the central school certificate (distinctions in literature, French, science, geography, and history), but what really counted were the results of the Joint Matriculation Board examinations the select five of us had taken.
I was on edge that summer. Most of the boys in my class had, within days of graduation, signed on as apprentice merchants of death, the cleverest working on blueprints, others in the Avro factory riveting the fuselages of Lancaster four-engine bombers destined to do to the Germans what they had done to Manchester in the Blitz. I'd see these boys on the streets in blue overalls, schoolkids who had been kicking a tennis ball round the playground a few weeks earlier, joining the droves of men and women headed for the factories around the time the milkman was leaving bottles on our doorsteps. Other classmates vanished behind the long camouflaged walls of Mather and Platt as trainee draftsmen, making drawings for parts of big guns. These factories were just round the corner on Briscoe Lane in Newton Heath. A few others trekked across Manchester's suburbs to Metro-Vickers in the great sprawling Trafford Park, the misnomer for the world's first industrial estate, a prime target for the Heinkels. When I bumped into former classmates, they'd pull out a pay slip. "Take a look at that! Two quid! Two quid! I got to keep ten bob" (a quid being a pound, and ten bob being a lovely red note worth half a pound).
My own pocket money was a fraction of that, two shillings and sixpence, the satisfyingly heavy silver coin we called "half a crown." I was not envious but glad the war effort depended on their skills, not mine. These were the boys who had made perfect dovetail joints in the dark basement at St. Mary's where we were taught woodworking. The carpentry teacher, the bristling, red-faced Joe Hall, had been so disgusted by my efforts, he threw the finished joint the length of the room, yelling, "You one-eyed kaffir!" The joint's disintegration in flight proved his point: I was not good with my hands. I could not get an apprenticeship even if I wanted one.
Partly as an insurance against academic failure, my parents had been saving pennies and shillings in those little tins around the house. In that summer of 1943, they told me they'd accumulated £70 (about £6,000 at today's values) and starting right away they were ready to pay for me to acquire the tools of the reporter's trade by taking full-time classes at Loreburn Business College in the city—shorthand, touch typing, and a little German to add to my so-so French. I was as much impressed by their foresight as by their sacrifice. I'd talked about shorthand and typing but hadn't worked out how I would get the skills. Now I had the opportunity.
The Loreburn classes were all girls learning to be secretaries, giggling about having two sissy boys do what they were doing. Dick Walton, the other boy—as thin, pale, and nervy as I was—aspired to be a writer, too. I found Pitman shorthand as exciting as a spy code and took to it, as did Dick, so that we beat the girls for speed and transcription. We reached 180 words per minute (and examination diplomas for 120 wpm). But not all was triumph: thumping our Underwood machines hard as we might, the keyboard covered by a shield, we never outmatched the nimblest of the girls in touch typing.
With some vague idea that if I ever became a reporter, I'd have to unmask embezzlers and fraudsters, I enrolled in classes at Brookdale Night School for double-entry bookkeeping. I wouldn't say I shone. I was distracted a lot by a table tennis table in the hall and became so obsessed with the game, I played every night—and then off and on for the rest of my life. I teamed up as a doubles partner with a neighborhood friend, a much better player, Ron Allcock. We entered all the tournaments, and he played for England in international matches.
I was still high from a winning match at the Manchester YMCA when I returned home to find a buff envelope reminiscent of the one from the Manchester Education Committee telling me I'd failed the eleven-plus. This one was marked Joint Matriculation Board. I wasn't eager to open it. I took it into the back room to read it alone before Dad came home. Mum was in the shop serving a crotchety customer, but I broke into the transactions exclaiming, "I've passed! I've passed!" I'd done more: I'd passed in all six subjects, with credits in five of them.
I was forgiven the interruption. It was a huge moment, bigger even than I realized at the time.
I had a chance in life.
#
# Hot Metal
In 1944, when I was sixteen, I applied for my first job in newspapers. I raided my mother's box of Basildon Bond notepaper, calculating that the fine blue sheets would make an impression of refinement. I had to clean up after family supper before I could sit down at the kitchen table with a steel pen and a bottle of ink. I succeeded in ruining several drafts with blots, so it was dark when I went out with my stamped letter to the little red postbox at the end of the street. By then the wartime blackout had been relaxed a little. People were allowed to use a flashlight provided the glass was covered in cardboard with a tiny hole in the middle.
At Failsworth Public Library, my home away from home, I had found the addresses of eight or nine daily newspapers in the city where I was born, Manchester. I wrote a letter a night for a week. With each letter I enclosed a stamped, self-addressed envelope "for courtesy of a reply." I didn't get many and wondered if some miscreant had steamed off my precious stamps. The replies that did come more or less said get lost.
In three weeks or so, I was stuck, like the British Army at Caen following the D-Day landings in Normandy that month. So I abandoned the big-city newspapers and scouted more obscure titles in the urban wastes of Lancashire and then the posher towns of Cheshire, trying to outpace the rejections and silences with bolder advertisements for myself. Captain of the school! Editor of the school magazine! School certificate! Shorthand and typing! I had also included a testimonial from the headmaster of my school. What more could they want?
In fact I hadn't expected much of a testimonial from Headmaster Marsland. His end-of-school testimonial, at first reading, didn't suggest I had been a disappointment, so I had a copy typed to send with each application. I still have it after all these years.
Harold Evans, who has been a conscientious and successful School Captain, is a boy with very lively intelligence, possessing powers of original thought along with a very retentive memory. He is perhaps too impetuous at present, but will outgrow that. He has won his success, both as a Captain and in the classroom, because he never spares himself, and seldom flags in his interest. His success in the athletic life of the school is largely due to his determined energy. We shall miss his integrity and willing service, and we wish him all the success he deserves.
When none of my letters had any effect, I had the dark thought that the testimonial was not foolproof, in fact might be a hostage to fortune. Wouldn't an editor contemplating a risky hiring be quite put off by "too impetuous"? Not just impetuous—that was bad enough—but _too_ impetuous. Clearly this was an alert that the applicant was foaming at the mouth. "Seldom flags in his interest"? Could that be read as a hint that he flaked out when the going got tough? And that reference to energy in athletics suggested brute force rather than exquisite skill. As for being wished "all the success he deserves," it could be a backhander; perhaps this youth deserved only a tiny bit of success—or none at all! For one mad moment I impetuously drafted a letter asking Mr. Marsland to withdraw this reference to impetuosity and conclude with something like "we wish him the glittering success he so thoroughly deserves." Fortunately, the paranoid parsing was preempted. One of my scattershot applications produced a returned envelope with a peremptory command beneath the magnificent blue Gothic title " _Ashton-under-Lyne Weekly Reporter_ Series": "Come to Ashton at 10 a.m. the following Wednesday and ask for me. (Signed) John W. Middlehurst, News Editor."
I knew nothing of Ashton except that it was one of a cluster of cotton and coal towns east of Manchester with odd names like Stalybridge and Dukinfield. It proved to be an initiative test in itself just finding a way there. Manchester was crisscrossed with rail and bus lines, but every bus and train anywhere accessible from our house in Failsworth shied away from Ashton-under-Lyne.
It was a miserable wet day that Wednesday. My father had been on the footplate all night but insisted on giving my shoes "a special railwayman's shine" before he went to bed, and Mum ironed a fresh white shirt between serving customers in our corner shop. It took three bus rides before I was deposited outside the sooty town hall of Ashton-under-Lyne. Across the cobbled market square, asserting equality in its authority, rose the Victorian redbrick headquarters of the _Ashton-under-Lyne Reporter_. It was market day. Vendors in the square huddled from the drizzle beneath their canvases, and there was hardly anybody about. I took shelter by a stall selling teapots until two minutes to the hour, then ran across the street through the revolving front door of the _Reporter_. In the front office I asked to see Mr. Middlehurst, speaking in my best BBC voice to disguise my flat, north country accent. A sniffy clerk told me I'd have to use the back entrance. Where was that? "Just round the corner. Use your eyes!"
He didn't say which corner. I found an unmarked side door that opened onto a narrow flight of worn stone steps. I ran up them to the second floor and discovered another world: the floor was filled with long lines of iron monsters, each seven feet high, five feet wide, decked out with an incomprehensible array of moving parts—gears, pulleys, camshafts, levers, and bars. A man crouched in communion at the foot of each contraption. This was my first sight of the Linotype machine, at whose ninety-character keyboard a deft operator could automatically render words into metal slugs at the rate of five column lines a minute. There was an exciting smell to which I would become addicted. It was hot news. Lead, antimony, and tin bubbled in each Linotype's melting pot, kept at three hundred degrees centigrade by a gasoline burner. Digital typesetting at a computer has consigned the Linotype to the museum, but the speeding electron has none of the aromatic urgency of hot metal marinated with printer's ink.
I interrupted one of the Linotype operators at his devotions. "I'm here to see Mr. Middlehurst."
"Jack or Dennis? They're both in there!"
Behind a flimsy wooden partition at the side of the big room were six or seven desks piled high with papers, telephone directories, pots of glue, spikes, and a full-size glass kiosk with a chair and a candlestick telephone inside. One ginger-haired middle-aged reporter with a pipe clenched in his teeth sat bolt upright at a typewriter in the corner, and another wizened walnut of a man was hunched over a desk writing with a pen in front of a window overlooking the market square. Neither acknowledged that the savior of British journalism had arrived.
At the end of the room stood a big disheveled man with thick glasses, running his hands through thinning hair and steaming at a stolid man in a printer's apron, who had his hands on his hips in what looked like a posture of defiance. The big man broke off in midstream. This was clearly my quarry, Mr. Middlehurst.
"Evans?" Without any more ado, without even looking at me, he scooped up bunches of paper from the turmoil of his desk and thrust them into my hand.
"Asparagus," he mumbled in a rapid monotone. "Asparagus—four copies, an' quick about 'em, laddie!"
He went back to his confrontation with the printer. Asparagus? It was a delicacy unknown to Lancashire dinner tables; I'd barely heard of it. The little old man at the window desk gave me a wad of coarse newsprint copy paper, about six inches wide by three inches deep, and gestured to an empty chair by an Underwood typewriter.
I skimmed the papers Middlehurst had given me, mostly handwritten scrawl on letters, postcards, and sheets torn from an exercise book. There were names of winners of a women's whist drive in Audenshaw; a note about a gift day at a Rotary club; a typed sheet from an undertaker naming a list of mourners; a rambling report on "Christianity in action" at the Welbeck Street Baptist Church; a letter about a clergyman back from China "after thrilling adventures," maddeningly unspecified; and a handwritten page about a burst pipe at a Methodist church that doused the choir in steam and drenched the conductor. The correspondent testified that "Mr. Joseph Thornrey carried on unperturbed."
I was at a loss. No mention of asparagus in any of these disconnected vignettes of local life, not even a flower show. The hunched figure at the window radiated _You're on your own._ The ramrod ginger-haired typist was on automatic pilot, pounding away without pause. The sight of Mr. Middlehurst frothing at his desk behind a big spike impaled with paper and a gallery of column proofs deterred me from going back to inquire what I was supposed to be doing.
I typed _asparagus 1_ at the top of the copy sheet and prayed for inspiration, whereupon there appeared at my elbow a fairy godmother in the shape of a handsome, slight youth my age with dark wavy hair. As he spoke, he kept washing his hands together Uriah Heep style. "You're new. I'm Laurence Taylor. Let me show you the ropes. Yes, yes," he chuckled, "just write up these submissions separately as paragraphs."
"Asparagraphs." As paragraphs!
He had decoded the sibilant Middlehurst mumble and saved my life. I rendered each of the submissions neatly into six-line typed paragraphs, removed the messy carbon paper required to make four copies, and took the lot to Mr. Middlehurst, his head down as he fertilized another heap of "asparagus." He ran a stubby black pencil over my first efforts in professional journalism, slashed out some words, and put the top sheets in an "out" tray. Glory, could it really be that my first words were bound for print? That they were is evident in the files I consulted in the Stalybridge Public Library years later.
Middlehurst spoke again, raising his bushy eyebrows several times. "Spitman, eh?" He wanted to test my ability to take down a note in Pitman shorthand. My Pitman was easily up to the pace of his brief dictation about some council committee or other. It was the way he spoke that made it hard, first decoding the mumble into intelligible words; then coding the words into the Pitman phonetic code of loops, hooks, and dots; and then decoding the squiggles back at the Underwood.
The transcription passed muster. I was rewarded with a wedding: four folded sheets of blue paper bearing the preprinted blanks for names of bride and groom, their addresses and occupations, what they wore, what the bride's mother wore, who played the organ, who performed the ceremony and where, who arranged the flowers, the names of the bridesmaids and the best man and their relationships to the happy couple, where the reception was held, who spoke, and what the bride wore as she headed off on honeymoon. Only at that point did the _Reporter_ restrain its curiosity. Laurence took me over to the bound files of previous editions so that I could learn how to render the bullets of information into imperishable prose that would support the headline "Stalybridge Man Marries Ashton Woman" or "Ashton Woman Marries Stalybridge Man" or, with any luck, something more graphic like "War Workers Marry." Even then flights of fancy were clearly not in mode. I could type up "the marriage was solemnized of..." or "Ada Briscoe and John Tomlinson were married" or "the wedding took place on Friday of..."
My effort was rewarded by another rapid elevation of Middlehurst's eyebrows. I was to learn that if they were elevated three times in rapid succession, it was a sign of benevolence.
He offered me a three-month trial. "Pound a week. Righty, laddie?"
A pound a week was about half what my classmates were now earning in factories. I didn't care. I might have to be on trial, but I had my first job in newspapers.
The offices of the _Ashton-under-Lyne Reporter_ were too difficult to get to by bus from our home in Newton Heath, so I cycled fourteen miles to get there each morning.
The _Reporter_ had thirteen editions serving eighty thousand readers living in the dreary backstreets and industrial sprawl of Gorton and Openshaw, as well as the beautiful vales and villages of Derbyshire's High Peak district. Correspondents who walked the hills around Mossley, Saddleworth, and New Mills put their copy into envelopes stamped NEWS URGENT and gave them to bus drivers. One of my first tasks as a junior reporter was to hang around at the bus stop in Ashton waiting to identify the right bus to collect those stories and feed them into the Middlehurst mowing machine. To miss the bus was a capital crime; not only the flow of news was imperiled but vital advertising revenue as well. Every weekly newspaper in the _Reporter_ group gave up its front page to classified ads, as they had since 1855, and the inside pages were packed with news with no bylines. The only personal credit allowed was initials for a reviewer's critique, thereby bowing to the arts. Much of the news was collected by what was termed "parring," or "paragraphing"—forays to gather the news on foot. None of us had desk telephones; they were regarded as an extravagance. Very few homes, including mine, had the luxury of a telephone.
The kindly Laurence Taylor, whose hand-washing turned out to be exercises for his advanced piano lessons, initiated me into the labor of "parring." He paid the bus fare to Droylsden, then we walked the streets for hours making house calls, scavenging for names of the recently dead and the lucky.
I was very self-conscious. I hated going in to see the undertakers. They were unforgivably cheery when they had a lot to report. I was glad Laurence was with me, though he too was of a morbid temperament. We roused grumpy caretakers in innumerable workingmen's clubs rancid with stale beer and sawdust, drank tea in vicarage and rectory, dropped in on union secretaries and Catholic priests, youth centers and political party offices. We wrote down the names of winners of whist drives, cribbage contests, and darts championships, of speakers and candidates for office, of cake makers and soup servers in the Meals on Wheels service for those confined to their homes. Names, names, names! In search of scoops, we asked everyone if they had any news and usually drew a blank. In time, I learned that people didn't recognize "news" the way journalists defined it; you just had to get them talking freely and then fish out the bits and pieces that might add up to something significant. The police stations, promising drama, invariably delivered nothing at all, and we meekly accepted a duty sergeant's brush-off. No doubt we were thought not grown-up enough to be exposed to crime stories. The county coroner's office was more forthcoming. We were invited to view a body ready for an inquest, someone who in the blackout had fallen into a canal. I couldn't face it.
We schoolboy reporters were filling in for men who were fighting the war. The senior newspapermen at work, like my teachers at school, were among the walking wounded. The robot at the typewriter I encountered on my first day had a wheezing chest complaint. He was Dennis Middlehurst, the taciturn son of the news editor, Mr. Middlehurst. The other man, a reporter by the name of Billy Mee who lived to be ninety-eight, looked as if he was beyond recruiting age for the Boer War. When I first encountered the third senior reporter, he was lying inert and silent on the newsroom floor with his hat on. He was a plump man with a clubfoot. His nose was bleeding. "Just step over him," said Middlehurst junior, without looking up from his typing. "Raphie has these attacks from time to time."
When not prone, Raphie (Ralph Alder) was the most considerate of men, who encouraged us juniors. We were all very circumspect, never referring to his disability—a sensitivity not respected by the chief photographer, a ball of energy in his late thirties, Charlie Sutcliffe, who would burst into the newsroom and say with a big grin, "Come on, Raphie, don't drag your bloody feet."
Charlie had his own problem. His right hand had been severed at the wrist during factory war work. His leather glove concealed a prosthetic hand, but he nonetheless wielded the blinding flashlight of his big Speed Graphic camera like a tomahawk. He was our very own Weegee in a rakish trilby hat, and a cynic. "You're Boy Scouts," he'd needle us. "We're all in it for the money, right?" But he was very professional about getting his exposures right. (His bullied young assistant, a thin youth with a squeaky voice, got his revenge by winning £250,000 on the football pools and quit newspapers.)
We novice reporters were nervous about Charlie's vulgar energy, but we all loved going on assignment with him. He had gasoline coupons for essential war work, so we got to ride in his car, Charlie laughing his way past any resistance. "Give the lad the bloody story!" he'd yell at some stick-in-the-mud official or difficult cop. None of us dared hang back from persisting when Charlie was present.
The tumult of the war was represented in the newsroom by Charlie's polar opposite, a scholarly young reporter with a prolonged brainy forehead, rimless glasses, and the elaborately courteous manners of a diplomat at the court of Louis XIV. He sported a brown beret and a shoulder patch that proclaimed him a member of the Royal Army Pay Corps. This was Private John David Michael Hides, doing a little journalism as an arts reviewer. His initials, JDMH, at the end of his music and theater reviews were regarded as a royal seal.
Michael was destined to be chief subeditor (copy editor) of the _Manchester Guardian,_ a key man in its perilous 1961 transition to London printing, and then the editor of Sheffield's _Morning Telegraph,_ who against all odds got his paper out during a strike. I learned later that even when I first met him, he had assumed a high responsibility: seeing that money got through to Popski's Private Army, Popski being a Belgian of Russian parentage who worked behind enemy lines in North Africa and Italy blowing up fuel dumps and aircraft on the ground. Of these adventures, JDMH said nothing. He was more concerned to tell us that we should catch Gustav Mahler. I'd never heard of Gustav Mahler. "I'll alert the Droylsden police on my calls," I told him. He seemed pleased.
The most striking figure among the younger reporters resembled Franz Kafka on the back of my Penguin copy of _The Castle_. This was Frank Keeble, who had a habit of clenching his jaw and promising to thump people who put obstacles in the path of truth and justice. I was in awe of him; he radiated glory. His friends included famous reporters such as Tommie Thomson and Walter Terry, both of whom had graduated from the _Ashton Reporter_ into the big time of political reporting with bylines in the _Daily Mail_. Frank wasn't a man to be intimidated by anyone—even the august directors of the Stalybridge and Hyde Transport Board. They thought it was unconscionable that he should have reported a complaint about a bus driver who got stuck on a hill in the snow. According to the intrepid Keeble, the driver had ordered all the passengers off the bus with the command, "If tha' wants to get home tonight, tha'll have to get out and push."
Such an order, protested the directors of the transport board, would have been contrary to the best traditions of the bus company. Keeble stuck to his story, and Middlehurst stood by his man. To us juniors, all this was stirring stuff.
At the end of my first week as a reporter, I was beside myself. Bundles of newspapers came up from the printing presses in the basement. Incandescent on a number of pages were those paragraphs I'd written on the first day and others, and a couple of wedding reports of mine to the tune of about two full columns of paragraphs. Nobody acknowledged the glow these contributions shed over the whole paper. The room had become crowded with strangers—pipe smokers in tweedy jackets from the hill country talking among themselves—and my new pals, the junior reporters who were busy rushing about with brooms and shovels. They swept the floor of the week's wadded-up balls of discarded copy paper, cleared the desks of incriminating cups, straightened up the newspaper files, and emptied the ashtrays and extinguished cigarettes. Soon the place looked like a bank.
All this labor was performed for one singular man: "Mr. Will," namely William Hobson Andrew, justice of the peace, captain of the Third Volunteer Brigade of the Manchester Regiment, and governing director of the entire _Reporter_ series—i.e., our employer. Each Friday he appeared in the newsroom carrying a quart-size metal canister of milk, which he deposited on Mr. Middlehurst's desk, specially cleared of papers for the occasion. We were a pale, fidgety bunch in off-the-rack suits. Mr. Will was from another planet—silver-haired, slim in his hand-tailored suit, and suntanned from daily hours on the golf course. He stood erectly among us, bracing his shoulders and gleaming with purpose.
"Good paper this week," he said, exposing several gold teeth.
He made a few more comments, then after an awkward silence Mr. Middlehurst indicated I was the new boy. "He cycled all the way from Failsworth," he added.
"Very good," said Mr. Will. "Now, Evans, how many spokes are there in a bicycle wheel?"
"I don't know, sir."
"Find out! Curiosity is the thing in journalism. Curiosity. Ask questions, Evans."
Then he nodded to us all and was gone, taking his milk can with him.
A clerk in the front office often met Mr. Will at the entrance to the building and carried the milk can for him. He told me he was rewarded for this task with an apple fished from Mr. Will's pocket and polished on his very own sleeve. (The clerk, Derek Rigby, became a reporter with a world scoop to his credit: he was the first to report the discovery on Saddleworth Moor of the bodies of two child victims of the so-called Monsters of the Moors, Myra Hindley and Ian Brady.)
That Friday after Mr. Will had gone to his office, Mr. Middlehurst wasn't finished with me. He beckoned me over, his head down, avoiding eye contact.
"You're expensive," he said.
I felt tempted to retort that a pound a week wasn't a lot for the long hours I worked, but I'd misheard him again.
"Your expenses! Your expenses! Where's your expense report, laddie?"
"I haven't spent anything," I explained.
"You must have. You must have taken bus rides."
"No, sir, I walked."
He glowered. "Downstairs will think you're not working." Then he scribbled on a slip of paper. "Take that to the cashier." I read it on the way down the stone steps to the front office: "Harold Evans, Bus fares, Droylsden calls, 3 pence. Approved, J. W. Middlehurst, News Editor."
I took every penny.
Everyone was a little scared of Mr. Will, including Mr. Middlehurst, who wore his jacket for these Friday encounters with him. The irreverent Keeble was the only one of us who dared to risk the wrath of the owner by entering the office at the main entrance like management. This meant surviving the glaring disapproval of the clerks and junior managers as he ascended the stairs and passed Mr. Will's office. One day Mr. Will sprang out at him. "Well, young man!" he said indignantly. _"Well!"_
But Frank had an inspired response. He duly explained that he'd come to thank the governing director for his leadership and guidance—and was it not time he was rewarded for his own dedication? A week later he found an additional shilling in his pay packet.
Mr. Will was represented every week in the paper by six asterisks at the foot of his own column, called "Golf Causerie." He wrote anecdotes about personalities and course records and ordered that not a syllable of his copy should ever be changed. Thus it was that one of his columns informed us, "There is no greater thrill than to drop your balls on a damp green."
Perhaps "Golf Causerie" was Mr. Will's arcane indulgence, but I see now it was a nice, chatty little enrichment even for those who didn't know the difference between a birdie and a bogey. Mr. Will was a shrewd owner-manager. He had worked in every department of the business since taking over twenty-six years before; he'd helped to push the transition from hand-setting to Linotypes and had set up branch offices; and he'd committed the _Reporter_ to solid nonpartisan community news reporting at a time when even weeklies tended to be conspicuously Conservative or Liberal (never Labour). We juniors had little appreciation of these matters; we were more concerned with the vast areas of human activity about which he required information. One week his parting shot to one of us was "Do bones make good soup?" He left another junior slack-jawed with the parting question "How long, young man, does it take a banana to ripen?"
A junior named Bob Sands, who later became deputy northern editor of the _Daily Mirror,_ had labored hard on a detailed report of a Stalybridge Council meeting, and Mr. Will, who lived in that town, challenged him: "Well, Sands, how many steps on Stalybridge Town Hall?"
"A lot," said Bob.
"Count them, count them! Look about you. Good day to you all."
We juniors were lucky to be on a weekly newspaper during the war. Chronicling darts winners and blushing brides might have palled after a while but for the adrenaline of being by proxy on the battlefronts.
War Office service bulletins about local men in action came into the office regularly, and Middlehurst doled them out selectively like lollipops. Sometimes we arrived on the doorstep with information the family did not have. A woman whose husband was killed in Holland had no idea why the army had sent her a ribbon in an envelope, but we had details of the action. We were allowed into the front office when readers came in with letters from the war fronts, and when we heard someone was on leave, we could go round and see if he had a story: a sailor rescued after an hour in icy waters following the torpedoing of his cargo ship; an airman who had disposed of a live bomb just before it exploded; a navy telegraphist on how he had managed to keep in touch with all the ships in a convoy to Murmansk beset by seven days of blizzards and U-boat packs; a prisoner of war in a German camp with a surprisingly cheerful take on life in the stalag.
I arrived one morning to find I had been taken prisoner myself. Mr. Middlehurst fluttered his hand like a priest sprinkling water and waved me over to a young man in a double-breasted suit tightly buttoned over an incongruous sweater. I had not seen him around. He had a lordly air, but he looked like a bouncer: he was squarely built and had a boxer's nose and a strangely hooded left eye. The bad news was that he would "keep watch" on me for the remainder of my three-month trial as a reporter. This sinister figure, named Eric Marsden, was not enthusiastic about his new charge. I learned years later that the ancient warrior Billy Mee had taken him aside and said, "That boy Evans will never make the grade."
Marsden plucked from my desk a piece of copy I'd written and barely glanced at it. "I don't know what I'm supposed to do with you," he said disdainfully, then added, "We may as well make a start with film reviews." This was promising.
"Gary Cooper's on round the corner at the Pavilion in _Casanova Brown_ ," explained Marsden. "David Niven's at the Roxy in _The Way Ahead,_ and Tyrone Power is in _Old Chicago_ at the Palace. See what you make of them. Make it snappy."
I leapt up.
"Where are you going?"
"I see the matinee starts at two at the Pavilion. I can just make it."
"Control yourself. All you need, Harold, is here in the files. And no more than thirty words a film." With a wave of his hand he left me a few dog-eared publications published by the cinema trades, whose cryptic phrases—"bodice-ripper," "good thick ear," "great smoocher"—I had to translate along with their breathless plot summaries.
Marsden was smart: he'd been a star scholarship student at grammar school and won a prize for an essay on Thomas Hardy's _Mayor of Casterbridge_ —and dammit, he'd seen war action. As a junior fireman in the Auxiliary Fire Service, he'd manned a stirrup pump to extinguish incendiaries dropped on Manchester roofs in German night raids, and at seventeen he was an officer cadet, training to be a pilot in the Fleet Air Arm. I felt inadequate. I didn't have a double-breasted suit and wouldn't have filled it so impressively. I'd not doused a single incendiary in the Blitz, and I was the only one of the juniors who had failed the eleven-plus to get into grammar school.
At the end of the week, though, I was invited to join Marsden, Taylor, and Keeble for the lunch they had regularly at the Co-op café. I was flattered. They were all grammar-school boys, though my shorthand and typing were better than theirs. The routine was for the ace reporters to swagger into the upstairs café with a copy of the paper to give to a pretty waitress. After baked beans on toast, Marsden pulled out a pocket chess set, soon checkmated Taylor, and challenged me to a game. He beat me with ease. That night I took a chess book out of the Failsworth Public Library and spent the weekend memorizing openings for the next Friday encounter.
I was convinced that Marsden was a complication to my hopes of being taken on the staff. As it was, I already lived in constant apprehension that I might mishear the mumbling Mr. Middlehurst and be fired. He was all detonator and no fuse. He blew up when I blundered in completing the worst task in the office—transcribing the eye-straining, interminable, and complex results from marked-up dog show catalogs. It took but a second to mix up the results of the Border Collie with those of the Border Terrier class. In these days of "recovered memory," I could have blamed the error on the black mongrel that sank its teeth into my leg when, at the age of ten, I was innocently cycling in the street. Truth was I couldn't tell one dog from another. As punishment I was assigned to be Middlehurst's runner for Mossley United's Saturday football match in the Cheshire League. I sat with him in the stands while he wrote up the game in longhand, and every fifteen minutes I was sent off to a telephone booth to read it to the _Sunday Empire News_ , which paid him by the line.
Another form of torture was to be sent into the readers' department adjacent to the newsroom. It was like a cave with little pools of light from desk lamps, and in the shadows two or three murmuring men curled like commas over galley proofs. I was assigned as the copyholder to a reader, a retired headmaster of vinous complexion by the name of J. R. Hall, who practiced his pedagogy in the margins of text in type as I slowly and quietly gave voice to the reporter's original words. But I did at least learn to be on the alert for the curious interloper in newspapers known as "Etaoin Shrdlu." He sounded half-Irish, half-Indian. I'd seen him make random appearances in the pages of all kinds of newspapers, sometimes in the byline position but most often and bafflingly in the middle of a story, so it read like this:
_Several demonstrators had_
_minor injuries and one of_
_tawk Etaoin Shrdlu_
_the leaders was taken to_
_hospital. Thirty-two were_
The start of the incomprehensible third line represented the operator's mistake in trying to key "the leaders" of the correct fourth line. Operators who erred in casting a line of type (called a slug) quickly strummed down the first vertical keys in the letter section of the board. These were _e-t-a-o-i-n s-h-r-d-l-u_ and represented an alert to copyreaders and compositors to discard that line. (In these digital days, Shrdlu no longer haunts newspaper pages.) Hall explained that Etaoin was a saboteur and it was our duty to apprehend him before he made print anywhere in the _Reporter_ series of newspapers. Never on his watch had the fellow slipped through as he did in newspapers conducted more carelessly than ours. Mr. Hall let me know every time he made an arrest and scrawled a large "must delete" mark in the margin of the galley.
The experience in the readers' room also yielded an insight into how Mr. Middlehurst wrung water out of copy. His speaking manner might be woolly, but he brought a very sharp mind to the editing of text. He had no time to instruct juniors directly, still less for small talk, since he edited every line in the entire series of newspapers, wrote the headlines, made all the assignments, wrote the (anodyne) editorial comment, and also contrived to contribute a chatty column of notes called "Round and About" under the byline "Pilgrim." ("One wonders how much time is spent by those who take a delight in obliterating the red labels on the windows of non-smoking railway compartments.") Mr. Middlehurst never made a comment on copy; he talked through his black pencil. He had been on the job since 1930 and would die at his desk with his pencil in his hand.
Soon after his triumph on the chessboard, Eric Marsden picked up a review I'd written of a Droylsden comedy and held it like he held a pawn, between thumb and forefinger. "Nice intro," he said, and seemed to mean it. To my surprise he now wanted me to review theater. It was a break: We could use adjectives; we could make judgments. We could imagine being the James Agate (the legendary _Sunday Times_ theater critic) of our generation. I kept a note of Agate's critical shafts ("Theatre director: a person engaged by the management to conceal the fact that the players cannot act") with the hope of somehow matching him when taking on local community theater productions.
When I was used to being a drama critic, I had the cheek to review a new church company, the Clayton Players, whose performers included Marsden. My review of their debut in _Doctor's Orders,_ a two-act farce, was proudly signed with my initials, HME. I wrote: "Eric Marsden as Banks gave a good impersonation of the legendary butler, Jeeves. His chief fault, like that of most of the cast, was a tendency to lower his voice in his longer speeches." The amazing thing is that Marsden did not raise a voice of protest the next day in the office when he saw my copy. I decided he was a man of discernment.
I was right, for the wrong reason. In fact, I had misread him altogether. Marsden looked twenty-five, but was only nineteen. Fixating on the menacing eye, I'd missed his tolerant humor. And the prosperity suggested by his new double-breasted did not exist: the suit was a farewell gift from the navy. Gradually I learned that his imperious air masked privation. His family was really poor, living in a mean terraced house amid factories. His father was a drunk who had abandoned his sick wife, so Eric had been forced to leave grammar school to be the breadwinner, working as a clerk in the Manchester textile company of Tootal Broadhurst Lee. This meant giving up the near certainty of a university scholarship. Only a handful of working-class boys had any hope of college then, however brilliant they were. He'd volunteered for the Fleet Air Arm at age seventeen, but his billet in Skegness, Yorkshire, while he waited to leave for flying training in Canada, was a thin plywood shack cursed with a coke stove leaking toxic fumes. Endless winter rain coursed down the inside walls and onto his bed, and he shivered the nights away. The pleurisy he developed, which ended his flying career, would dog him all his life.
It was a cruel outcome for such a brave spirit. I discovered he was a sturdy romantic who did his own thinking. His gods were Beethoven and Shakespeare, and he made them mine. At sixteen I had not heard a note of classical music. Eric, while clerking at Tootal's, had got into the habit of relieving the drudgery by walking over to Manchester Central Library for its lunchtime recitals by pianists and violinists, then he'd saved up to go to the classical concerts by Manchester's famous Hallé orchestra. He studied the composers by playing secondhand records on a wind-up gramophone.
That was the way he introduced me to an enduring joy of my life. He lent me a tiny 48 rpm record of "Dance of the Hours," the ballet music by Amilcare Ponchielli. To the family's amusement, then irritation, I played it dementedly on a portable wind-up in Ashworth Street. It's a trifling sugary piece, lampooned in Walt Disney's _Fantasia_ as a dance for hippos.
Eric taught me the etiquette of concertgoing and how it had changed since the nineteenth century, when people talked and sometimes played cards and clapped at anything they liked. I was not to talk, not to cough, not to sniff, not to tap a foot, not to rustle a paper, not to breathe, in fact, unless absolutely necessary—and for Pete's sake not to clap when they stopped playing but wait till I was absolutely sure from the program notes that they really had finished what they were about. The first time I heard Beethoven's Third Symphony, the _Eroica,_ I was exalted. I read all I could on Beethoven in the Failsworth Public Library, learning the most from J. W. N. Sullivan's Pelican paperback _Beethoven: His Spiritual Development_. Who could not be moved by the drama of the composer's struggle against deafness and depression: "I will take fate by the throat; it shall not overcome me!" I found myself reflecting, albeit immaturely, on the dissonance between physical appearance and character. Ludwig—and Eric—were both ill-favored, so clearly the face was not a window on the soul.
Thereafter I begged to write stories about the Hallé; I couldn't afford to pay for a regular ticket.
It was no reward for Eric's beneficence that on the following Friday at the Co-op café I tried out one of the chess openings I'd learned, the queen's pawn gambit, and about which I had breathed not a word. It enabled me to control the center of the board against both Laurence and Eric. "Once more unto the breach, dear friends," cried Eric, but I was merciless. Thereafter every Friday for two years, when all the other customers had fled, we tussled from lunchtime to teatime in the empty café as I matched my book learning against his originality. Eric probably still had the edge but became careless in his crusading style, his mating mind divided between the tiny chessboard and the vivacious Co-op waitress with the frilly lace apron. She had a curious accent—flat Lancastrian with some extra zing in it that made her seem snooty. This was a contradiction. Ever acute to the nuances of class, I couldn't work out how someone with a working-class accent could manage to be as haughty as the upper class.
It transpired that she was a Belgian aristocrat, Jacqueline Henriette Alphonsine Marie Dirix de Kessel, who had risked her life in the Belgian resistance. She'd been interrogated by the Gestapo and let go. In 1945 she'd married a handsome British soldier who'd brought her home to Ashton. It must have been a terrible shock—a disillusion suffered by thousands—to come from war dramas to the rainy vicissitudes of a cotton town and find that the dashing liberator was a bully who regarded a wife as a chattel. She divorced him, and now everyone lusted after her.
We knew "Ereek," as she called Marsden, had no chance. The unstoppable Charlie Sutcliffe was in hot pursuit. It was obvious that, unversed in the nuances of the English class and educational system, Jackie would see the dashing photographer as the war-wounded Cavalier and the much cleverer Eric as the plodding Roundhead. Charlie had a car, and a Fleet Street career beckoned. He would sweep her off her feet, and she would be glamorously at his side in a Hemingway-style life of adventure in the world's capitals. My asthmatic hero would be left to grow old on the _Ashton Reporter,_ like the wrinkled walnut Billy Mee I'd encountered on my first day.
You can't script someone else's life, let alone your own. Jackie and Eric were married in 1947. And while Charlie stayed rooted to the same spot for the rest of his life, it was Eric who became a brave and accomplished foreign correspondent. When night work subbing in the Manchester office of the _Daily Telegraph_ became too much of a strain on his health, he moved to Kenya in 1957 to work for twelve years for Nairobi's _East African Standard._ He reported on war and civil strife in Africa and the Middle East; he risked exposing abuses that governments preferred to cover up. In 1969, invited to cover an Arab League conference, he was nonetheless marched at bayonet point through Cairo as a suspected spy. The league apologized. Later the same year, now deputy editor of the _East African Standard,_ he was deported from Kenya for displeasing President Jomo Kenyatta by publishing reports of machete attacks but was then invited back after protests by the African journalists. When he and Jackie started receiving death threats in Kenya, I invited him to join the _Sunday Times,_ where I was the editor, as our correspondent in Israel and later South Africa.
Whatever doubts there may have been about my fitness to join the staff, they were put to the test by the Red Wharf Bay murder, in which the victim was a local army sergeant's wife. Mr. Middlehurst dispatched his son, Dennis, to report the sergeant's trial in Anglesey. But there was a problem. The court hearing opened on a Thursday, and the paper went to press that night. The only way Dennis could get his long report back in time was to phone it through to the office, and the only person with sufficient touch-typing speed to take his dictation was me.
At 4:00 p.m. I was locked in the newsroom telephone booth. The Underwood was lifted onto the little ledge inside. I was given dodgy earphones, and Dennis dictated his story at top speed. Everyone crowded round outside the booth looking in on me, the typing fish in the aquarium. It was hot inside—so hot that the windows misted up. I could see Mr. Middlehurst pacing urgently outside. Every time I finished a page his paw came into the booth and took it out to the Linotype operators. I was three hours in the aquarium, we made the edition, and Mr. Middlehurst never said another word about my being on trial at the paper. The following Friday I had an extra shilling in my pay packet.
I soon got better assignments. A gaunt, suave senior named Jameson, just medically discharged out of combat missions for the RAF, initiated me into the art of reporting the dispensation of justice. He had once worked on a Manchester daily, and policemen nodded to him as we crossed the icy cobbled square and climbed the many steps into the Ashton Town Hall, where the magistrates decided local cases.
This was March 1945, when the Allied armies had crossed the Rhine and we were winning the war in Europe. None of this made any difference to our life in Ashton. It was twenty degrees below freezing, and coal, already rationed by the bag, was not to be had. (Coal fires were the only form of heating in drafty buildings, where the wind rattled the windows. Nobody could afford electric heat, and nobody had heard of central heating, except in the American wonderland.) Hot and cold water pipes froze all over town, as did my fingers trying to write shorthand in the arctic courtroom to which I was now assigned.
Disconcertingly one morning, Mr. Will, bristling with good purpose, appeared high up on the bench as one of the presiding magistrates (mental resolve: count the town hall steps on the way out), but even he, for all his milk vitamins, was not immune to the cold. The court adjourned after fining one publican for embezzling the brewer and dismissing the case of another man accused of being "absent from essential war work without reasonable excuse." Ten minutes later the magistrates came back in overcoats and scarves, four men and a woman, banging their feet and rubbing their hands, not in the proper mood to deal with a wretch accused of stealing coal. Jameson, languid on the wooden bench, left these cases to me. He perked up when a police sergeant took the witness box and solemnly described what he'd seen when he'd entered a house of ill repute. Mr. Will looked flushed. Jameson whispered that it was too big a case for town magistrates and I should take down the depositions verbatim. Back at the office he edited my notes into a short report, sanitized for our Methodist readers, then carefully put the longest version in an envelope addressed to the _News of the World,_ the seamiest of the Sunday papers. "We'll make a pretty penny out of this," he confided, and so we did. The splash on Sunday in the _News of the World_ was GIRL OF FOURTEEN WANTED TO BE TAKEN FOR SIXTEEN. I received three shillings as my share of those wages of sin.
These many years later, I had forgotten what a slog it was on the _Reporter_ until I looked at the pocket diaries I'd saved and deciphered the shorthand, a ridiculous subterfuge for a blameless life. The night work, when I thought it advisable to abandon the bike and go by bus, was onerous:
6 Feb 45: Manchester after work in freezing fog for NUJ lectures at Young Journalists' Club on interviewing. John Beavan of Manchester Evening News talked on importance of college education for journalists
7 Feb: Hard day. Court in morning, then two inquests back to back and stuffy old Englishe night at Rotarian dinner. Home about 11.
9 Feb: Worked like hell. Long Ashton Council meeting. Missed 10 pm bus.
Some weeks I filled ten columns of the paper. My fingers ached from all the typing on the heavy keyboard, but the personal demands were the hardest. I was given the name of a parachutist who had died in action in Italy. "Make sure," I was told, "you bring back a good photograph and a few lines on his life." I was to call on his parents; he was their only son. I walked toward the door of the terraced house, spiral-bound notebook in my raincoat pocket, then walked away.
I felt a terrible intruder, a teenager not long out of short pants sent on a man's mission. I imagined a reporter calling on my own mother announcing my death. I canvassed my conscience with the idea of going back to the office to say sorry, nobody was in. I had to keep telling myself this was it—this was the real test—and picturing what it would be like returning to the office empty-handed with a lame excuse.
When I finally knocked on the door, praying nobody would come after all, I was quite drained. "Sit down, lad, and have a cup of tea," said the mother, while the father extracted a photograph from its frame. They were sorrowful but proud, ready to talk about their son for hours to someone from the _Reporter_.
I found time and again we of the _Reporter_ were regarded not as nosy intruders but as friends of the family. That was not because of any magic we had as individuals. It was a reflection of respect for the paper; it bothered with the little things in people's lives, the whist drives and flower shows, so it was trusted to be part of the big things. A Royal Signals sergeant, in a letter describing their months' long advance down the Railway Corridor in the Burmese jungle, wrote, "In spite of the rain, mud, sticky heat, insects, we are quite a happy crowd. I must give thanks to the American Air Force for the splendid way they supplied us with the necessary things of life, including little luxuries such as canteen supplies, etc. and most of all that great morale booster, mail from home—along with a few welcome _Reporter_ s."
For all the respect the _Reporter_ enjoyed, encounters of this kind never came easy to me. I eventually perfected a little mental trick to play on myself in rough waters, imagining the shame I'd feel if I surrendered to embarrassment or shyness: if I failed to knock, shirked a tough question, accepted an obvious lie, retreated from the glare of a VIP.
What I most of all took away from these years going into so many homes was identification with the people of the backstreets and appreciation of their fortitude, too often in the face of a vast official carelessness. I got worked up about the way they were used and tossed aside. And while the _Reporter_ style was deadpan, without a hint of comment in the news, I couldn't contain myself after I knocked on a door in Abbey Hey, Gorton, and was welcomed in by Mr. William Henry Adams and his widowed father.
William, a skeleton of thirty-seven, had volunteered for the army within days of the outbreak of the war. Even though he had been ill for two months with double pneumonia and was still somewhat groggy, he enlisted and was rushed to the front in France. Conditions at Cherbourg with the Royal Engineers were grim. Bivouacked in heavy rain, he was diagnosed with tuberculosis in both lungs. He was discharged, unfit for active duty, unable ever to work again, and yet denied any form of war service pension. He had been appealing for four years when I met him and had just been rescued by the local MP, Alderman Will Oldfield, who had taken up his case. Rescued? The pension was all of two pounds and five shillings a week. I wrote, "This is the story of a man who sacrificed his health to serve his country and was all but forgotten when he was no longer needed. He is grateful for the pension but those years of waiting are no credit to England."
Mr. Middlehurst let it through and even gave me a personal reference, inserting "writes a _Reporter_ representative."
In the bitterly cold February of 1945, just after the Allied armies had finally blocked Hitler's panzers thrusting from their Ardennes lair to the English Channel, Mr. Middlehurst murmured that I was to go and see General Carpenter, who was in town for a night. He gave me a scrap of copy paper with the name Major Bagworth, who would introduce me to the general, and the address where they were to be found. It was a hall packed with uniformed men and women. I could not find the major. It was hard to ask questions because the uniformed ones were banging tambourines, clapping hands, and singing a hymn to a brass band. The headgear of the women revealed that they were Salvation Army bonnets, and General Carpenter was their general—in fact, the leading general of the Salvation Army. The Salvationists kindly made room for me to squeeze into the front row when I mentioned the _Reporter_ and took out my notebook, ready to report his speech.
The general had other priorities. Instead of mounting the platform, he walked into the body of the hall and asked every individual to stand so that one by one he could commend their souls to the Lord's care. He started on my row, at the far right, speaking to every individual. I remained seated as he advanced toward me.
"Won't you stand for Jesus, son?" the general asked.
"I'm press."
He looked at me and smiled. "You can still be saved!"
#
# How I Won the War
The record of my war years need not detain us long, particularly as the war was over.
I answered the nation's call on Monday, August 12, 1946. His Majesty's Royal Air Force needed me and would brook no delay. It didn't seem to have remembered its urgent need when I reported for duty at the RAF station at Padgate, near Warrington. I was one of hundreds of pale young men sitting on the floor of a big hut, waiting for something to happen. For days we were in a no-man's-land between civilian and service life.
Among us somewhere, though I didn't know it at the time, was another numberless man whose words would shake governments, Aircraftsman William Rees-Mogg, later the distinguished editor of the _Times_. He was going through his own culture shock, mystified by the frequency of the _f_ word, applied to almost every noun: "I had not previously come across an effing knife and an effing fork on my effing plate. Nor oddly did I often come across it much later in the RAF." Someone in the Air Ministry must have had a premonition that this numberless one would become an august member of the establishment because, on emergence from limbo, he was compensated with one of the cushiest of billets as a sergeant in the Education Corps.
When I was given a number, 2318611, I clung to the seven digits of my new identity as a baby to a mother. I'd have been no good under interrogation by an enemy. Just the word "number" would have triggered a reflex response.
The transmutation from reporter to recruit was complete when I went through one door in a sports jacket and came out the other in a uniform and an improbable forage cap with the even more implausible motto _Per ardua ad astra_ (Through adversity to the stars). We were all given effing haircuts and effing kit bags, huge sausages designed to dislocate shoulders and impose hours of ironing time for everything stuffed inside. Then as numbered freight we were dispatched in groups to Compton Bassett, Wiltshire, for six weeks of square bashing to make men of us. Opinion is divided to this day as to whether they succeeded.
I found it a relief to be turned into a robot for six weeks; no need to plan, worry, or think for oneself, just do what the man said when he shouted a number. It gave me a glimpse of how a fighting unit that didn't happen to have me in it could perform acts of incredible bravery. The challenge set for me on passing out of the drill squad as an AC2 (aircraftsman second class) was how to escape from the clerical Colditz to which I was immediately dispatched, the dreaded Ministry of Aviation Records Office at Innsworth. There grown men recently trained in warfare went on paper chases into the endless rows of filing cabinets and were never seen again. Redundant aircrews flew metal desks.
I was pretty miserable myself; collecting the names of whist drive winners was thrilling by comparison with my duties at Records. I was in a division responsible for compiling and filing movement orders that sent airmen to their new stations. I tried hard to get the RAF to realize it had a wasting asset. I filled in lots of forms for reassignment. I volunteered for aptitude tests whose results suggested I had no aptitude for anything. I never heard another word.
When, resigned, I'd all but forgotten these escape attempts, I received one of the movement orders we sent to others. With effect immediately, 2318611 was posted to Hullavington, Wiltshire, home of the Empire Flying School (EFS) and headquarters of No. 23 Group in the Flying Training Command. There was no hint what 2318611 would be doing there. I didn't care; it was such a relief to exchange the claustrophobia of Innsworth for the vicarious excitements of an active airfield. Flying school! It was as if I had stepped through the pages of my wartime scrapbooks.
Hullavington was in the countryside not far from the charming old town of Chippenham, but the fumes of the war lingered. Bombers and fighter planes lined the runways; fitters and armorers sweated in the hangars; the aroma of combat was as strong at the base canteen as the tea was weak.
The military police at the guardroom directed me away from the airfield, but I found myself in the stratosphere entering the offices of an air commodore on his way to being an air vice marshal, Claude McClean Vincent. He was air commandant, in charge of the whole station, home of the crack pilots in the All-Weather Squadron, which specialized in flying on instruments into electric storms just to see what the hell happened. He'd asked the station adjutant to get him a shorthand typist. If he was expecting a comely Women's Royal Air Force officer to brighten the all-male station of eight hundred officers and enlisted men, he kept a stiff upper lip, as you'd expect of a pilot with a Distinguished Flying Cross who'd flown in World War I and most recently commanded a group of Spitfires in the Middle East.
"Dog's dinner, I'm afraid," he said, asking me to type a manuscript of crabbed handwriting and convoluted revisions. It didn't take much imagination to see why he reminded me that I was bound by the Official Secrets Act. The terse reports I sent to the Air Ministry could have been turned into news leads: BRAVE TEST PILOT'S ORDEAL IN STORM: EXCLUSIVE FULL DETAILS by our air correspondent somewhere in Wiltshire. Unfortunately, I also saw the effects of the flying errors I recorded, being several times ordered to lead a guard crew when one of our planes crashed.
I had a lot of dog's dinners from the air commandant and from his successors over the next two and a half years. They were all apologetic, as if asking me to undertake some dangerous mission. In addition to being the secretary to the air commandant I had paperwork to do for all five RAF stations in No. 23 Group, but here, too, my boss, Squadron Leader Papworth, was so courteous I forgot I was the lowest of the low in RAF ranks. He was intrigued that I had been a newspaper reporter and took me home for tea with his wife in his cottage in lovely Bourton-on-the-Water, a village in west-central England nicknamed the Venice of the Cotswolds.
Nothing remarkable about that, you may say, but the RAF was my first realization that my mental map of class boundaries was a stereotype. I'd not before been in close contact with the officer class. I was disarmed by the lack of pretense among the majority on the station, who'd been through the war. Being shot up over Düsseldorf and seeing your mates die night after night didn't leave any taste for affectation. Commissioned and noncommissioned officers shared the same risks on bombing missions.
Visiting the airfield and the hangars, I noticed the ease of exchanges between officer pilots and other ranks working as airframe fitters, armorers, and maintenance technicians. A sergeant bomb aimer I got to know playing table tennis introduced me to a flight lieutenant whose north country accent was so much more marked than mine that I got called "educated Evans." This officer took a Lancaster bomber on "training flights" and one weekend invited me to go along for a flight that ended very conveniently at Ringway Airport near Manchester, his hometown and mine. It just happened that his bride lived there, and a few mechanics who came along also happened to be from Manchester. In this way, qualified by geography and accent, I had my first-ever flight in any aircraft—and subsequently as many as I could contrive in all sorts of RAF planes. Getting from Wiltshire to Manchester by the worn-out and indirect train service used up most of a forty-eight-hour pass and always risked running afoul of some officious military policeman at the station intent on finding my brass buttons and badge not bright enough to blind passersby. When I traveled by air, my weekends at home were blissfully extended.
This is how, when I am describing how I won the war, I can discourse on what it was like freezing at twenty thousand feet in the rear gun turret of a Lancaster. If I have the right captive audience, I can also describe my first German prisoner of war, Oberschütze Walter Greis, a tall, good-looking, blond Aryan of Hitler's fevered dreams. Actually, he was an eighteen-year-old bank clerk who'd surrendered to the U.S. Army at the end of the war and been sent to Hullavington to work. He was not an enemy but an ally of mine against the endless "bumph"—RAF slang for useless paper—flowing into my office.
Nobody had any idea why we had to fill in and dispatch so many forms, so after a year of this I invented a form of my own. I circulated an official note to all stations saying that in accordance with the No. 23 Group medical officer's determination to limit flyborne infections, commanding officers were henceforth required to ensure there was a weekly examination of the sticky flypaper hung up in various locations and a record submitted of the number of flies caught. "Group expects," I wrote, "that the numbers of intruders brought down should increase if careful study is made of the optimum placement of the traps."
The compliance was gratifying, though I never got round to visiting the stations to check the accuracy of the reports. No doubt they are still being submitted.
I was desperate to get on while my career in journalism was stalled. I applied for every promotion, every examination. One examination to do with RAF regulations that I passed with a 100 percent score was so obscure that nobody on the station had ever taken it before and so important that I cannot remember a single thing about it.
Within a year I had moved up from AC2 to AC1, then to leading aircraftsman (entitled to a propeller on my sleeve), then to corporal. All that the corporal's two chevrons did was put me in charge of my barrack room, theoretically responsible for the good conduct of forty airmen and their performance on the hated kit and rifle inspections. Let us say that my writ was lightly applied and more lightly accepted. The biggest call on my authority was who should next have the room's ironing board and iron to get a razor-edge crease in his trousers. I considered drawing up a waiting list, then thought better of it. Was I an ace reporter, or was I Corporal Jeeves?
Papworth encouraged me to apply for a commission. I could tell you the idea appealed for the extra money, the greater freedom, the uniform smoother than my blue serge, the snob appeal to women in town, the equality of status, the privilege of easier weekend passes, and the pleasure it would give my parents. It was all those things, but it was also a question of grub. The food in what was called the other ranks' mess (as if we were an afterthought) was notoriously inferior to the officers' and sergeants' messes. Every night, though, an officer moved along our crowded mess tables, asking us to approve of what we had just consumed—that is, to lie. The duty officer was preceded by a sergeant calling out, "Orderly officer, any complaints? Any complaints?" We might have gagged on the greasy lamb, but nobody ever said a word. The officer might be amiable enough, but the menacing demeanor of the sergeant, a stiff-backed soldier from the RAF Regiment, made it clear that there would be a penalty. On one historic occasion, much celebrated thereafter, I heard that a leading aircraftsman from Air Traffic Control had broken ranks. When the sergeant bellowed his "any complaints" question, Traffic stood up:
"Yes, sir, it's shit!"
Everyone froze. Officer and Sergeant stopped dead in their tracks.
"What did you say, airman?"
"It's shit, sir," said Traffic in the silence. "But it's beautifully cooked."
I had a better idea than trying to become an officer, which meant signing on for three years anyway. In the summer of 1947, a full year into service, I finally realized what I should be doing. There were eight hundred men, plus some favored ones with families in married quarters. What they needed was what I needed—a station newspaper. I got permission, provided I did the work in my spare time. Officers and sergeants volunteered to be reporters and printers. There was a wartime can-do spirit about the whole enterprise.
A local printing company said they'd let us have several sacks of old type they didn't need but warned that the sizes and letters were all mixed up. In a mountain of metal deposited in the hut, we had to pick up each thin sliver of type by our fingertips, guess the size, read the single character on the surface, and dispose of it in the correct little box in a composing tray so that we could pick out _t-h-e_ to form "the" in a composing stick. There were ninety boxes, the size varying according to how often the letter was likely to be needed.
We all chipped in at night after the day's duties were done. It was eye-straining work under a single lightbulb. It was cold, too, with just one fire. I tried Tom Sawyer's trick of telling friends what a fascinating time we were having exploring the mysteries of print. Then another difficulty pressed. In my mad whirl to stay occupied before the newspaper was approved, I'd been cast in a station production of a play, _Men in Shadow,_ about the French resistance. I was to play Polly, a downed flier hiding out in the roof of a barn. The nights I had reserved for editing copy and writing headlines were the nights I was supposed to be dropping out of a trapdoor to link up with the lead character, Kenny, none other than our own printer, Sergeant Mott.
Since we lacked the gift of being in two places at the same time, our days through November consisted of morning parade, a full day's work followed by rehearsal or performance, and then print shop duties until midnight. This was so fraught an enterprise that I took in my stride the first blemish in the editorial content I'd ordered up: a review of _Men in Shadow_ concluded, "It would have been better if Corporal Evans as Polly had been more consistent with the Lancashire accent and left the Shakespearean gestures to John Gielgud."
A clerk called Jock McPhee joined with a few others to sort type as we neared Christmas. The night before the camp was due to close for Christmas we still had to finish setting the final page to run off on our flatbed press. We worked through the night. By 9.00 a.m. we were all drooping around the setting table and still had not set enough type. Out of time and material, we slapped the words STOP PRESS on four inches of blank space at the foot of the page, as if we were a regular newspaper expecting late news at any minute.
As the pages came off the press, the ink still wet, Jock ran out of the shop to the camp gate with the first hundred copies of the _Empire Flying School Review_ to sell at threepence a copy. Following with the second batch, I met him on his way back to the print shop—he was empty-handed. We sold all we could print.
After Christmas the station welfare committee voted to float the cost of professional printing if we would commit to repaying the grant from advertising income and sales. This mighty enterprise would cost twenty-five pounds. If we sold six hundred copies at sixpence—double our initial rate—we'd have fifteen pounds and could make up the rest from advertising.
Sixpence? To justify that price, we'd have to concoct a magazine whose price and quality of paper might draw more advertisers. This meant devising a cover each month, an art that was not part of the Middlehurst curriculum. I had the idea of having small silhouettes of six of the station's most celebrated planes fitted into the spaces of the letters _E-F-S_ (for Empire Flying School). The concept was a disaster. Our customers recoiled from being reminded of work. "We've enough of kites [planes] all day. Give us a break!" Sales were off target on two issues. This much of a failure with a third issue would kill us.
What to do?
Our first newspaper had included a photograph of Veronica Lake's thighs, and I had flash recall of it pinned up in a barrack room. Would I pander to these baser instincts?
Ask me another. The third glossy cover design featured the lovely redheaded actress Hazel Court. Sales shot up. One of the civilian workers suggested we feature a blond bombshell of a girl from his hometown of Swindon. It sounded unpromising. Diana Fluck, as he knew her, was a little-known film actress, but I liked the sound of how she had responded to a studio's request to change her name: "I suppose they were afraid that if my real name was in lights and one of the lights blew..." She changed it to Diana Dors and eventually became known as Britain's answer to Marilyn Monroe. I think we were the first magazine to make her a cover girl. Soon after we featured her in her bare skin sitting on a snowy stile with a strategically placed fur muff, she was cast in _Oliver Twist_ and made into a star by Pinewood Studios.
Just when the August 1948 date neared for my demobilization, Joseph Stalin intervened. He stopped all road, rail, and water traffic crossing the Soviet zone in occupied Germany at the end of June, threatening 2.5 million people in the western sector of Berlin with starvation. The RAF and the U.S. Air Force immediately joined in the great Berlin airlift, carrying coal and food round the clock. It was good news for the Berliners—among them the repatriated Walter Greis—but it was bad news for me. The airlift required selectively postponing the release of thousands of airmen, and I was one of them.
I was kept back eight more months, but the time flew. The day's paperwork assumed an air of relevance and urgency, the _Empire Flying School Review_ had its deadlines, and in every spare minute, now that I had been snared by the big world outside camp, I tried to understand what was behind some of the more worrisome headlines in the newspapers: DOLLAR DRAIN! PAYMENTS CRISIS! AUSTERITY BUDGET! BANK OF ENGLAND ROW! In my barrack room were two airmen, junior to me in rank but way ahead in being able to follow the ups and downs of the economic trials of the new Labour government. They'd been called up during their first year at university—one at Oxford, one at Cambridge—and had made a nodding acquaintance with people I'd never heard of, such as John Maynard Keynes and Alfred Marshall. With the encouragement of my clever new friends I wrote away to Wolsey Hall in Oxford, which was offering to teach economics by correspondence course. Wolsey Hall was not an Oxford college (as I'd imagined) but a commercial enterprise. Still, it was a first step into what they called "the dismal science." "Why is a pound of diamonds valued more than a pound of bread?" Wolsey Hall asked. I couldn't for the life of me see how to apply the lessons to Mum's grocery, which is just as well, but I conscientiously wrote up the short answers in the station library and waited anxiously for the red ink marks from Oxford.
In the library one day, a typed note crammed among many on the bulletin board announced that Manchester University was offering servicemen a two-week residential course of lectures and discussions titled "The Rights of Man."
I raced down to the education department to apply. Happily nobody else did, and a few weeks later, with the permission of the air commodore, I was installed in civvies in the university's Holly Royde House in Didsbury, Manchester, cerebrating with men and women of all ranks from all the armed forces. Here I met Thomas Paine and Edmund Burke, John Locke and Thomas Jefferson, Jean-Jacques Rousseau and Thomas Hobbes. I made no more than a nodding acquaintance with these strangers, but I got the idea that somewhere in their writings they might, in the urgent clamor of postwar politics, furnish a better compass than the newspapers.
I reentered civilian life in March 1949 in a shiny demob suit, returning to Ashton-under-Lyne and weekly newspaper reporting. I'd been promoted and was now chief reporter of the edition called the _Gorton and Openshaw Reporter_ , but I carried around the cold streets a fever incubated at the Empire Flying School. I burned to learn more about politics and economics.
I just had to find a way to get to university. My new ambition marked me out as some kind of freak. When I was unwise enough to express it in the Ashton head office, the general refrain was "You've lost three years in the RAF, now another three for university? You can say goodbye to Fleet Street!" Even my old colleague the erudite Michael Hides was skeptical. He wrote me a long letter explaining why academic studies would not only be a diversion from acquiring the skills for journalism but actually distort them. He'd been taken on by the _Manchester Guardian_ and was deep into the mysteries of his new craft as a subeditor, the first of our batch of Ashton juniors to make the big time. But the flicker of envy I felt was expunged by the prospect of exploring mysteries of my own.
#
# _Non Nobis Solum_
It must seem an unexceptional ambition today, but in the 1940s for a working-class boy to say he was going to university was like announcing he was about to marry Betty Grable. Our only idea of a university was through movies like _A Yank at Oxford,_ which was less about book learning than rowing boats, robbing someone of his trousers (debagging), and heavy breathing over Vivien Leigh. College just never occurred to the brightest of my schoolmates, nor to any of the newspaper people I met. I knew of nobody in Newton Heath who had gone to a cap-and-gown university, as distinct from a few wizards who had attended a technical college.
There were very few places open in the handful of universities, and only a tiny fraction of those qualified could hope for admittance. I continued to post short essays to Wolsey Hall but tried to keep up with my education in journalism. Every Thursday night, after handing in the last Gorton story, I'd take the bus into Manchester for a series of talks by leading daily newspapermen organized by the National Union of Journalists. I learned some tricks of the trade, but the major impression was made by the writer-editor John Beavan (later Lord Ardwick) descending from the clouds, which is how I then regarded the _Manchester Guardian_. Beavan informed us in his silky way that we were all ignoramuses.
How could we explain what was really going on, Beavan scolded, if we had not read Macaulay and Tawney? His barbs were resented by the group. The bitterness of postwar politics made imperative some understanding of economics and political history, however I acquired it.
In training for the attempt to enter a university, I abandoned cycling to the office through Daisy Nook in favor of three bus rides because they gave me a chance to catch up with a few centuries of political philosophy. I have an aptitude for losing things, so it is quite astonishing to me that I still have the tiny blue hardback book I carried everywhere in my jacket pocket: _The History of the Peloponnesian War,_ written by Thucydides and edited in translation by R. W. Livingstone.
Sitting with Thucydides on the top deck of the Oldham to Ashton bus, rain spattering the windows, I was not carried back to fifth century BC in Athens but rather into the middle of the twentieth century and beyond, since the Athenians' arguments about democracy and equality, power and patriotism, war and imperialism, so lucidly reported, will be with us for some time. Of course the Athenian democracy had no role for women and was dependent on slave labor, but I was moved—then and now—by Pericles' evocation of the spirit of the idealists striving for a liberal democracy: a society which honors excellence and beauty but gives equal chances to all its citizens; where individuals give the state due service of their own free will and without compulsion; where they are free within the law to lead their private lives their own way without black looks and angry words. "Wealth to us is not mere material for vainglory but an opportunity for achievement; and poverty we think it no disgrace to acknowledge but a real degradation to make no effort to overcome. Our citizens attend both to public and private duties, and do not allow absorption in their own various affairs to interfere with their knowledge of the city's. We differ from other states in regarding the man who holds aloof from public life not as 'quiet' but as useless."
The useless man is with us still. He does not vote; he dodges taxes; he volunteers for nothing; he does not speak up for any cause; he turns a blind eye to the propagation of hatred; he "minds his own business." For any social ill he deals in the coinage of cowardice: "They should do something about it."
I marked another passage where the historian of the fifth century BC spoke directly to the reporter of AD 1949: "With reference to the narrative of events, far from permitting myself to derive it from the first source that came to hand, I did not even trust my own first impressions, but it rests partly on what I saw myself, partly on what others saw for me, the accuracy of the report being always tried by the most severe and detailed tests possible."
That Thucydides had never attended a secondary school, let alone a university, did not deflect my determination to secure a grander degree. I wrote to every one of the fourteen English universities, offering to study there for a bachelor of arts degree. None of them was impressed. Although I had those precious five school certificate credits, not one of them was in Latin, and Latin, I now learned, was required to read for an arts degree. I wondered if I could take a crash course but was told that adequate preparation to take the examination would entail three years of evening classes, not very feasible with so many night reporting jobs. There was also even more daunting arithmetic: I could not conjure up anything like enough money to pay for college accommodations and teaching, nor could Mum and Dad. They had four sons to prepare for the big world. There was no organized system comparable to the American one of working one's way through college.
I can't remember who told me that the wartime coalition government had passed further education and training legislation, within the Butler Education Act, whereby the Ministry of Education was empowered to give grants to men and women who had served in the war. Well, I had almost done that, hadn't I? Wasn't the Berlin airlift a key conflict in the cold war? The act was much more circumscribed than the GI Bill in the United States, but I wrote to the ministry. The answer was swift. Sorry, mate, you must have a place in a college before we can even begin to consider whether you qualify for an ex-serviceman's grant. No Latin, no college. No college, no money. Catch-22.
I'd been through all the brochures for degrees in politics and economics, but I started again, this time spending summer nights after work looking at everything universities had to offer. Time was running out for applications for admittance in September–October. Amid some small print, I discovered there was a three-year course called Social Studies leading to a bachelor of arts degree. I'd skipped past it, but now I noticed Latin was not mentioned in its entry requirements, only "matriculation." Was it an oversight? And what was Social Studies? There was a brief reference to political theory, industrial history, economics, and psychology. What was the catch? Would I emerge adept in the social skill of settling disputes in a soup kitchen when the underpaid chef ran amok with a cleaver? Or divining at close quarters the emotional impulses of depraved juveniles? Further research was needed.
I wrote to Durham, the university offering this degree, and specifically asked if they really would admit someone without Latin. Durham promptly wrote back, saying yes. I could apply to enroll for Social Studies unconjugated by Latin—but I could not be admitted to the degree course unless I had also been admitted to one of the five men's residential colleges in Durham City or to St. Cuthbert's Society, whose members lived at home or lodged in town. They kindly attached the fees. Any residential college would want first to be assured that I was acceptable to the Durham colleges' academic authorities. Catch-22 redux.
Nevertheless, in the leaflet from University College, Durham, one of the residential colleges, there was a tiny picture of an undergraduate's room in Durham Castle complete with a fireplace and mantelpiece, and I could not resist imagining myself there. I wrote to University College and was told I could put in an appearance for an interview on September 13, provided I had satisfied the registrar that I had matriculated. Clutching my five credits, I took a train—and the chance they would not spring a Latin trap on me.
I had imagined the city to be one vast coal heap with Social Studies graduates toiling somewhere in the dust. Instead, an incomparable vista opened up when the train steamed through the last obscuring hedges and over the spectacular viaduct. The magnificent Romanesque cathedral the Normans had built on the rocky wooded peninsula high over a loop of the river Wear was the most inspiring single scene of my life, then and ever after. If men could build that prayer in granite in the service of God, what could they not do? Across from the cathedral, where lay the bones of Saints Cuthbert and Bede, stood the redoubtable Norman castle: "Half church of God, half castle 'gainst the Scot" proclaimed Sir Walter Scott's ode to Durham, those words cut in the stonework of the graceful Prebends Bridge (1777), where J. M. W. Turner planted his easel to paint the peninsula.
After Oxford and Cambridge, Durham was England's third-oldest university. I found the registrar in a dainty pastel-washed little house in a cobbled side street by the cathedral. I presented my school certificate parchment. In the blink of an eye a "W. W. Angus" signed a valuable document: "I certify that Harold Matthew Evans is qualified without further examination to matriculate in the University of Durham, to read for the degree of BA in Social Studies." It meant I was admissible to the university, provided a college... Yes, yes, I knew the refrain by heart. Now to gain a college place I first had to scale the ramparts of the castle, the headquarters of University College.
Founded soon after the Norman conquest, the castle had for eight centuries been the palace of the immensely rich prince bishops of Durham. It is the oldest inhabited university building in the world and surely the most grandly housed college in the universe.
The moat had been cobbled over, but the castle still had defenses against pretentious interlopers. First of these was the dragon lady who popped out from a crevice in the wall to bar my way when I dared open a little door in the great iron-studded castle gate beneath the embattlements of the crenellated keep.
"Castle is closed," she was pleased to inform me.
"I'm here to see the Master of the College."
It made no difference.
"No visitors."
I protested: "But I have an appointment!"
"Are you a university student?"
"No, but I will be."
She paused to consider the impertinence, then vanished back into the crevice. I was left to contemplate the University College arms, a shield of four lions rampant and two bishops' miters with a scrolled inscription. It was Latin: _"Non nobis solum."_
What did it mean? I thought better of asking the dragon lady when at length she reappeared to admit me into the courtyard. But what would I say when the Master asked me to discuss the inscription's relevance to college life? It was humbling enough just to be in the presence of the Master of University College, Lieutenant Colonel Angus Alexander MacFarlane Grieve, MC, MA. I stood in the doorway to his room. He looked at me without saying anything. I calculated that through his Coke-bottle glasses I must be much diminished in size. I was not much to start with—five feet seven inches according to the RAF—but I shrank with every passing second. I was probably about twelve inches high when I advanced, braced to babble about Thucydides if he broached the importance of a classical education. He offered a gentle smile.
"Do you row, Evans?"
I'd been in a paddleboat in Rhyl amusement park, but I didn't go into that. I said there had not been much chance of rowing in the RAF, but what a strenuous and exciting sport it was, relying on what I'd seen of oarsmen sweating it out along the Isis in _A Yank at Oxford_. It was as well not to claim knowledge. The Master had been captain of boats when he was at University College as an undergraduate, and he was the author of a history of rowing in Durham. As he discoursed on how well the university's oarsmen had been doing in races on the river Wear, I noticed he wore no socks. I found the incongruity of the formal demeanor and this manifestation of individuality strangely relaxing for all of the five minutes before he ended the meeting. He sent me to see the senior tutor, a physicist and doctor of philosophy: "You will find him on the Norman Gallery. Good morning, Evans. And close the door on your way out."
To reach the Norman Gallery I had to confront the dragon lady again and ask her to point the way. There was a door off the courtyard, then beyond that an obscure crack in an ancient wall leading to a sharp spiral of a very narrow stone staircase. As I climbed, I came upon a tiny paneled doorway in the rock. I knocked. No answer. I waited, knocked again, and waited. I eased the door open, coughed politely, and fell on my knees. It was not a prayerful posture. There were several steps immediately behind the door, catapulting one forward—no doubt some crude Norman's idea of a practical joke. Beyond the steps there was no head tutor. I had stumbled into a toilet.
I returned to the ascent, and the staircase finally opened out on a beautiful gallery of fretted Norman arches and pillars, its windows looking out over the courtyard and gatehouse to the cathedral. I found the head tutor secreted off the gallery behind a big black door with black iron bolts. He was running his hands through his hair, a work of supererogation, since he had none that I could discern in the dim filtered light. He sat reeking cleverness in a wood-paneled room. There was the deliciously musty smell of books and of the dust of centuries of crumbling stone. The cathedral bells tolled the hour and the crucial interview began.
Why did I want to be in Castle (as University College was known)? Why not in St. Cuthbert's Society? What about King's College, Newcastle (then part of Durham University)? I extrapolated about residential college life from those weeks at Holly Royde. Why spend three years away from the headlines? Did I realize how long three years was? What did I think I would get out of Social Studies? Why not an honors course? "Mmm, no Latin, no Latin, eh, well, never mind. You'd learn something, I suppose." What had I learned in the RAF? What books had impressed me? What was I reading now? What did I think of the literary quality of English newspapers? What did a reporter do? What did I consider good writing? I emerged about an hour later and flew over the cathedral: I had been offered a place in Castle for the Michaelmas term just a month away.
Now all I had to do was find the money. I told the ministry the good news. They were equal to the challenge. To qualify for a grant, they now informed me, as well as winning a college place I would have to satisfy them that I was "ex-service" enough, and they asked me to confirm that my RAF service had disrupted an original intention to go to university. They couldn't have hordes of ex–national servicemen suddenly deciding they wanted three years at a university, could they?
It was going to be hard for me to prove my intentions of 1943, especially since I didn't have any then. In a rush, I entreated supporting letters from St. Mary's headmaster Marsland, the head of Brookdale Night School, Loreburn Business College, and the lordly Beavan. The best they could do for me was to testify that I had always wanted to be a journalist, that the RAF had stopped my rise, and that they believed a university education would make me a better man. It was not quite what the ministry had demanded, but I posted these letters and spent hours in Manchester's landmark Central Library collecting any book references on leading editors who had been to college and how Western civilization would collapse if insufficient numbers of newspapermen had a college education.
There were not many, but I sent these, too, and waited.
And waited.
I did my rounds in Gorton. Mum optimistically began knitting a pullover in Castle's colors of maroon and white. I adopted her attitude. I gave notice at the paper. I found a translation of _Non nobis solum:_ "Not for ourselves alone." I sent another letter to the ministry.
Ten days before I was to be at the freshman's dinner at Castle on October 10, the day before formal registration, I reminded the Ministry of Oblivion that Durham was expecting me and my trunk was on its way by British Rail. Again, silence. On the day before I was due to leave home, I sent the ministry a telegram in desperation. The next morning Mum finished the pullover. The household was depressed. Nobody knew what to say. The train to Durham was leaving Manchester just after noon. Two suitcases stood in the hall. We fretted on the doorstep. A Post Office boy cycled up the street and stopped at our door asking me to sign for a telegram. It was from the ministry and had the oracular tone of a communiqué from Dumbledore to Harry Potter:
PROCEED TO DURHAM
#
# The Sting of Disraeli's Gibe
I proceeded to Durham and my higher education Hogwarts with three pounds in my pocket—and Dumbledore's telegram. It said nothing about a ministry grant for university and college fees, but I presumed the college bursar wouldn't be so ill-mannered as to ask for cash on the nail as soon as I climbed the hill to Castle that night for the first college dinner in Great Hall.
I mingled with the other freshmen, all of us wearing black academic gowns as we stood on the steps of Great Hall, looking out on the courtyard, clock tower, and splendid Norman Gallery. I was convinced that half of them must be baronets. I heard a few ask the defining question of the English class system, "What was your school?" meaning one of the all-male public schools.
At the gong we trooped in to stand by long tables, awaiting the arrival of the top table of dignitaries and dons. I noticed the Master among them; I was to learn he might look forbidding, but he was shy, less happy mingling with the dons than with his Dumfriesshire sheep. It was exalting just standing there, looking up at the fraying battle flags left by the Duke of Wellington when he dined in the college after the Battle of Waterloo. By this stage I would have been disappointed if the grace, read that night by the college chaplain after a bow to the Master, had been in anything other than Latin. Until I knew it off by heart, I learned to listen for the ending, "per Christum Dominum nostrum," and say amen. More challenging was the array of knives, forks, spoons, and glasses at my place. I worked out the etiquette by sly observation of my fellow diners.
Gradually we all thawed out into recognizable human beings, shedding diffidence and self-consciousness. There was not a baronet at our table, and it turned out that a goodly proportion were ex-servicemen. We were an eclectic mix of ambitious undergraduates hoping to become business managers, explorers, schoolteachers, composers, research scientists, industrial chemists, translators of French literature, and, in my case, a newspaper editor. A few were on the way to being vicars, though most of the future parsons were on their knees in St. John's and St. Chad's colleges, located in the shadow of the cathedral. Perhaps the best thing about us was how biologist talked to geographer, music scholar to mathematician, geologist to theologian, chemist to historian. By the time the carafe of port—vintage port!—was passed along the table, we felt we had become Castlemen. The college, and indeed Durham itself, was small and concentrated enough—no more than two thousand students—for the sense of community to be real.
After such a heady first evening I sweated my two suitcases down the cobbled streets all the way through town, past the arches of that spectacular Victorian railway viaduct, and up steep Western Hill to a terrace house where I had been allocated lodgings with six other Castlemen. There were no rooms for freshmen in college that first year. My trunk occupied most of the tiny room I was to share with a music scholar and his cello. Here, too, we brave six had our first encounter with a landlady who liked to catch one of us on the stairs and pin us to the wall with her outsize bosom.
My first lectures on Palace Green took place in old almshouses under the shadow of the cathedral. The alms location seemed more appropriate the more days passed and I heard nothing about my grant. It was some ten days into college life, when I was down to my last pound, that I found in my mailbox at the Castle Gatehouse a buff envelope marked ON HIS MAJESTY'S SERVICE. My imagination worked overtime as I opened it amid the Castlemen jostling to unload books and shed their gowns for lunch in Hall. Conceivably the letter might say, "Unproceed from Durham; you have been unmasked."
I read it twice before the wonderful news sank in: ex-corporal Evans had been awarded an ex-serviceman's grant of 315 pounds a year for three years—on condition he survived the first-year examinations. With the ministry paying university teaching fees directly, it was enough for living, as well as travel to and from home, though not for all the books on the extensive reading list. Those gaps could be filled in the rambling library on Palace Green.
I took happily to the routine of college life—the lectures, the one-on-one tutorials, the intercollegiate sporting rivalries, the parliamentary debates in the oak-beamed Union Hall, the ritual and ceremony, the black academic gowns billowing in the breeze as we rushed about, the ponderous hum of the still incomprehensible grace in Latin, the solemn processions of dignitaries into the cathedral and castle. The agreeably disputatious political radicals on campus, anticipating the 1960s rebels by a decade, saw college rituals as symbols of reaction and wanted to abolish all ceremony. In the show-off arguments at coffee breaks they explained that if we all turned up for lectures without gowns or refused to stand for grace or boycotted the Union Society debates in favor of political agitation, it would be a blow for the depressed proletariat. And where was I at this historic moment in British history?
I'd thought of myself as a radical eager for reform. To my surprise I recoiled from their brave new world where everything "unnecessary" was to be swept away in favor of a rational new order without ornament or ritual. The best I could muster against the rationalists were naive incantations about how much families enjoyed Easter eggs and Guy Fawkes Night. Why, I argued defensively, "you'll be saying get rid of Christmas next." They hooted with derision: "Commercial exploitation of religious myth!"
I aspired to be a journalist partly because I thought good journalism could identify the consequences of the use of power or the failure to use it for the common good. In 1950, having just read E. S. Turner's polemic _Roads to Ruin: The Shocking History of Social Reform_ , I steamed about how "tradition" had been repeatedly invoked to defend the indefensible as it had been in slavery, denying votes to women, sending children into the coal mines, and so on. ("Don't talk to me about naval tradition," Winston Churchill said. "It's nothing but rum, sodomy and the lash.") But what was to distinguish between the customs and traditions I thought benign and those I thought repressive? Had I taken Edmund Burke to heart as well as E. S. Turner, I'd have been better equipped to argue that we should espouse those uses of power that reaffirmed traditional values and resist those that eroded our heritage of a liberal, humanistic society. I'd have cited Burke's remark that the individual is foolish but the species wise, that our common law, the majority's tolerance of minorities, our respect for private property, and even my tolerance in listening to the rebels spout nonsense had all come to us as customs in the trial-and-error evolution of civilization. But I'd yet to hear any lectures on Burke, so such ammunition was not at my disposal.
The vehemence of my sparring partners made me wary of the warriors of cold reason. Years later, I think the Durham encounters gave me a framework of sorts for deciding when we were defending a precious custom and when legal theory was being used to sanctify injustice. The sardonic remark of a French diplomat in a negotiation has stayed with me: "Yes, we agree it will work well in practice, but how will it work in theory?"
My grant dependent on getting through the first-year examinations, I immured myself in the past. In the silent solitude of the library I absorbed the notion of the sanctity of property from John Locke, of free opinion from John Stuart Mill, of free will and the immorality of treating others as a means to one's selfish ends from Immanuel Kant. I recoiled from the mob-rule totalitarianism, as I saw it, of Jean-Jacques Rousseau and found myself at one with Burke again in loathing the mob in the French Terror—my favorite Dickens was _A Tale of Two Cities_ —while sympathizing with the revolutionaries in America. It was thrilling to read how the abstractions of Enlightenment philosophies became muscle in the American Declaration of Independence. I got so carried away by that dangerous thing, a little learning, that I offered to give a paper to the pipe-sucking intellectuals in University College's "Read and Weed Club" on the modern relevance of absolutism and determinism in Thomas Hobbes's _Leviathan_. It served me right that my effort reached print in the colleges' newspaper as "Harold Evans talked on hobbies today."
Later, practical experience in journalism made me question and qualify the philosophies I had absorbed like a sponge. I could have questioned them when I sat in my tutorials with the professors of politics and economics. There was Frances Hood, the gaunt head of the department of politics, who was so nearsighted that his nose literally swept along the undulating contours of the handwriting in the essay he was judging; his utterances in a deep bass voice with a refined Scottish burr fell like mortar shells among my assumptions. There was the diminutive bow-tied Edgar Allen, who created arches with his fingertips while analyzing elaborate economic models, always prefaced with the Latin _ceteris paribus_ (meaning he was assuming that other things remained static). There was the darkly handsome Viennese psychologist Dr. Wolfgang von Leyden, the heartthrob of the women undergraduates, who schooled us in the ambiguities of perception. Dr. Peter Bromhead, the owlish authority on political constitutions, challenged me to stay awake long enough to track all the legislative processes of Labour's parliamentary bill nationalizing the gas industry. He returned my report and its labored tabulation with a lot of _J_ 's penciled in the margins: "J equals journalese. You can take this as a compliment or not, as you prefer."
Whatever it was that did the trick after I took the first-year examinations, the professors promoted me to the second year of the honors course in politics and economics, though _ceteris paribus_ was my only working Latin. The ministry renewed its grant. I'd taken the precaution of working during all the vacations, at Christmas ensuring that the Royal Mail got through hail, sleet, and snow, and at other times writing shorthand instructions to relay to the engineers at Mather and Platt and to drivers at British Road Services. I saved to buy a secondhand Triumph motorcycle—Dad taught me how to handle the beast. It was as well I had wheels, since in the second year the college authorities decreed I was to be rescued from Western Hill and afforded sanctuary in the fourteenth-century fortress of Lumley Castle, Chester-le-Street, fifteen miles from Durham City. Others given rooms at Lumley, after spending the day in lectures at Palace Green, had to suffer a long bus ride, while I raced up the A1 highway into Lumley's thickly wooded parklands in time for tea.
I threw myself into college life that second year. I joined the boat club as a gesture to the Master; I participated in Union Society debates; I ran the half mile for Castle, played squash, and started a Durham University table tennis club. Some of the heavies from the rugby and boat clubs derided "ping-pong" as about the level of tiddlywinks, a weakling's sport not worthy of the sporting honor of the colors we coveted. It made no impression on them at all when I bragged that I'd played in the 1948 English Open championships, the Wimbledon of the sport. I didn't stress that I'd been knocked out in the first round, nor offer my lame excuse that the ball was a bit of a blur because I was nearsighted and too vain to wear spectacles. (My myopia was a weakness cruelly exploited in the men's singles by the French national champion, Maurice Bordrez, who went all the way to the semifinals.) But the issue of whether "ping-pong" qualified as athletic was satisfactorily resolved in Durham by inviting the loudest objectors to play a game where a mix of angled long and short shots had them running round the table until they dropped.
Not surprisingly, in that hectic second year I succumbed to the imperatives of journalism. There was competition among the colleges for one of their own to have the distinction of editing the university's biweekly _Palatinate_ , though nobody was rushing to do any of the slog of collecting enough news and putting it in the newspaper. At one point there had been a Union Society debate on the proposition "The journalist is a man who has sold his soul." It had been carried overwhelmingly. From the masthead only three people seemed to be engaged in this activity. I volunteered and was sucked in. It was so time-consuming going round to all the colleges, I gave up coxing a Castle racing four, which was a relief to the crew. During one training session, fiddling for glasses in my pocket as we headed back to the boathouse, I saw my driving license fall in the river and retrieved it only by frantic commands as we drifted into the weir.
I was rewarded for my _Palatinate_ industry with the masthead title of assistant editor. This was an inflation of my role, sitting in a cold room in the Union proffering the glue pot to the editor, pasting up his columns into pages, and reading proofs. Nobody could hope to edit the _Palatinate_ for long given all the lectures, tutorials, and examinations, but when the editor resigned, I let my name go forward to the owners of the paper, the Student Representative Council (SRC), an elected group of the most active undergraduates and graduates. I was summoned to a meeting to learn that I'd been elected, then was promptly sandbagged: I inherited a newspaper practically bankrupt by the cost of paper and zinc (used to reproduce photographs). What, asked the president of the SRC, did the wizard of Ashton-under-Lyne propose to do about it? Was I going to ask for a bigger subsidy? Hand on heart, I recklessly declared that every journalist worth his salt would never take a penny in any subsidy from anyone because it would compromise his independence. Rhetoric wouldn't pay our bills, so I proposed that we be allowed to increase the price and maybe the advertising rates. Some jibbed at this, urging that it was preferable to reduce the number of pages from ten to eight. "You'll have a job filling them anyway."
It was a proposition I was to hear many times in my newspaper life. Believing then, as I do now, that the way to kill a newspaper is to ask more for less, I pleaded that we should be allowed to charge 33 percent more but go up two pages to twelve. This was rash. I didn't know a thing about newspaper management, but I boasted of my RAF experience in doubling the price of the _Empire Flying School Review_.
The SRC was persuaded to give us a trial, which meant that the three of us committed to the paper had to think very fast to justify the extra pages and the higher price. We concerted our plan of attack: more sports and news, of course, but we'd also introduce features, which meant recruiting more help. We wooed widely among second-year students, varying the pitch a little according to our judgment of the tastes of the quarry. "Look what a spell on the _Palatinate_ can do for you," we'd say. "Do you realize who's the chief subeditor of the _Times_?" If that glazed the eyes, we'd try, "And do you know who edits Britain's largest-selling daily newspaper, the _Daily Mirror_? Both Durham men!" We weren't sure that either Reginald Easthope at the _Times_ or Silvester Bolam at the _Mirror_ had actually worked on the _Palatinate_ , but why cloud the issue that Durham men were stars in journalism? There seemed no point either in mentioning that Bolam, a fervent defender of the virtue of tabloid journalism, was currently in prison for documenting the crimes of a vampire murderer before the courts had decided the killer's guilt.
Among our recruits was an undergraduate who displayed an encyclopedic knowledge of the books by Evelyn Waugh in a letter he wrote to the famous author and thereby secured an interview with the writer himself on Waugh's Gloucestershire estate. After alighting at Stinchcombe station he paused to ask directions and by way of thanks explained to the stationmaster the influence of T. S. Eliot on John Betjeman's railway poems. When he arrived at the Georgian mansion, Piers Court, an upstairs sash window opened, and Waugh's angry red face appeared. "You are five minutes late. I will not tolerate rudeness! Good day to you, sir!"
Without our literary scoop, we fell back on a loquacious member of Hatfield College, an irreverent gadfly poet who proposed that he should visit all the Durham pubs within staggering distance of Palace Green to report on how far the natives were friendly and whether their beer was drinkable. To support this enterprise he required expenses. In justification he wrote, "Around the peninsula lies a permanent challenge to wit and conversation and beneath in its black earth, lies the constant reminder of laughter, bright cups and tinkling glass." We were moved. We gave him a pound of SRC money. Our undercover pub reporter wasn't seen on Palace Green for two whole weeks, but bang on deadline he came into our office bursting with intelligence: Did we know there was a pub over Framwellgate Bridge where you could debate capitalism with Marxist miners? Did we know the one where a leading lecturer suitably primed could be guaranteed to lose all sense of discretion in a vivid dissection of his fellow dons? He confided this intelligence with a flourish: "Amid the smoke-filled air, to the strange sound of one of the most fascinating English dialects and the monotonous clink of the dominoes, the student can indeed say in after years that this is where memory both begins and ends."
I was less moved by the prospect of dominoes than his identification of a hideaway upriver ideally placed for cool refreshment after a long summer punt beneath the willow trees. I made a note of it for a personal campaign. I'd lately been taken with the grace and vivacity of Enid Parker, a biology undergraduate in St. Aidan's Society. I liked the way an apparently gentle question from her punctured hot-air balloons in the gabfests of Union Society coffee breaks. Though she had been seen on campus with a college tennis star, I got in early with a fast service. The moment I learned the date of the elegant garden party Castle staged in the Fellows' Garden in the spring, I passed her a note in the Union. She accepted. Fifteen love to me. I next discovered we had a bond in both being from the north, albeit she was from Liverpool, Manchester's rival. Thirty love. And she liked the ride on the Triumph I gave her. (In those days it was hugely impressive for anyone to have wheels.) Forty love. The punting expedition recommended by our intrepid pub reporter might in due course be game, set, and match. She erred only once in her impeccable judgment that I know of, which was a year later in agreeing to be my wife.
Here are a few extracts from my college diary of these momentous days:
Monday, January 29, 1951: Lots of _Palatinate_ copy to edit and an essay to do for tomorrow. Couldn't get out of table tennis match with miners, took bus, still editing copy, then came back to late night at Lumley and finished the essay such as it is.
Tuesday, January 30: Ethics tutorial in Harrison's cat-infested room. He pads about in slippers and flannel bags talking in an Oxford voice about the concept of good. Took notes at evening emergency meeting SRC. Late back Lumley. Copied economics notes before bed.
Wednesday, January 31: Windy bike ride in from Lumley, very tricky. Dashed to Hood tutorial on Hobbes. Did headlines. Noisy college meeting made me captain of athletics. I'm taking on too much! Girl in Union is Enid, Edgar Jones says she goes with Danny F, so forget it.
Thursday, February 1: Missed lecture in the morning, but it was only by the odious man from Cambridge so I didn't miss anything. A little late, too, for tutorial. Page makeup in the afternoon and evening. Hectic. At one point we seemed a page down. Went for a drink. Late at night John Nettleton helped me decide we did have enough for 12 pages. Doors locked, so went down a fire escape with a torch and ran to the printers with page proofs.
The final acquisition for the new _Palatinate_ was to find someone to write a gossip column under the pen name Argus. We knew we had found the right man when he sent in a little note about one of the prominent Communists on campus, a tough former shop steward called Michael McVeigh of St. Cuthbert's Society who was constantly recruiting comrades:
Mr. McVeigh I understand has changed his lodging due to a disagreement with his landlady about the proportion of water in the milks supplied with his porridge each morning. Obviously no Scotsman could be fooled on a matter connected with porridge, and for her pains the good lady is now no doubt blacklisted by the Cominform. However, even a clash on so fundamental an issue hardly explains why he left so quietly as he did, and at midnight.
My first issue was published on Friday, February 2, 1951. " _Palatinate_ came into the Union coffee bar earlier than I thought it would. I was in a funk," I learn from my diary. "Had to find somewhere to hide. I bolted to obscure corner of the library, had lunch in Lyons in town instead of Castle and read the paper. Such sensitivity!"
When I summoned up the nerve to venture back on Palace Green, I found we were fast selling out. The new features were popular, especially "Palace Green Notes" by Argus, and on page one we had a hot news break. University campuses in the 1950s were riven by discord about just how serious a menace was international communism. Mao's Communists had taken over China, Stalin had tried to starve out West Berlin, Communist North Korea had invaded the South, but even so our Union Society in 1951 seriously debated the motion "The United States is as much a menace to world peace as the Soviet Union," and another time voted 2:1 in favor of "We could not conscientiously take part in any future war." America was seen as provocative and dumb for backing Chiang Kai-shek's corrupt Kuomintang regime; Senator Joseph McCarthy and the House Un-American Activities Committee were derided and despised for their witch hunts for "Reds" in Washington and Hollywood. Alger Hiss, we now know, was indeed a Soviet spy, but at Durham then _he_ was the campus hero, not the courageous Whittaker Chambers who exposed his betrayal.
It was in this fevered period that Durham's ice hockey team accepted an invitation to take part in the ninth World University Games. The warden of the Durham colleges, Sir James Fitzgerald Duff, and the Academic Council of professors banned the team from going. The official explanation was that the four-day interruption of studies in term time was too much. This was regarded as whitewash. Everyone believed that the real reason was the anti-Communist hysteria of the time. The games were in Poiana, Romania, behind the Iron Curtain. Memory tells me I accompanied our report with a brave and stinging editorial rebuking the Academic Council for its paranoid worry that Durham men could be suborned or bullied into pro-Communist "peace statements": I had tried out for the ice hockey team and had the bruises to testify that they were not deeply interested in peace with anyone. But finding the editorial again now, I see that the roar I imagined was all weasel. "Whatever views may be taken of the attitude of the authorities on this subject," I wrote, "all will welcome a better indication of the extent of undergraduate liberty during term, for we cannot but notice certain inconsistencies in this matter."
In fact, read today almost all my editorials for _Palatinate_ were lamentable. They were affected, studded with quotations I'd looked up. A reasonable critique of an SRC vote concluded, "Perhaps this sounds too much like a homily on democracy or 'that fatal drollery called a representative government....' Can we, though, make ours work a little better and escape the sting of Disraeli's gibe?"
I cringe.
The only editorial I wrote that was any good came out of my baptism of fire in political journalism. The members of the Durham Colleges' Conservative and Unionist Association were always complaining that they did not get enough space in the paper. They were wrong, but the assaults made me acutely aware that an editor had to be, like Caesar's wife, above suspicion. I'd variously been invited to join the Conservative, Liberal, Labour, and Socialist societies. A sixth sense held me back from allegiance to any political party. Two years after graduating I was invited by the Liberal Party in Altrincham, Cheshire, to be its parliamentary candidate in the next general election against the sitting Conservative MP. I felt flattered—and ran a mile in the opposite direction. After my Durham experience, I knew that pinning a party badge on my lapel would make it harder to be seen as unbiased, which by this time I was. (I have to say, though, that I've known many leading journalists who've been party men in both Britain and the United States without it in any way making their work suspect or hindering their rise.)
These were somewhat discomfiting years in terms of class consciousness. When our table tennis team traveled to play against clubs in the mining villages in South Durham, we were disconcertingly regarded as toffs. In town the shopkeepers were deferential to "gentlemen of college," which made me feel fraudulent. And in my third year, living in Durham Castle itself, I had the privilege of a college servant known as a "scout." I was gloriously accommodated in two rooms on the Norman Gallery, where I'd taken the crucial entrance interview. My rooms looked over the town and down on the terrace, where I could admire the sockless Master doing calisthenics at the crack of dawn. Returning from lectures, I was happy to find my scattered books and papers neatly stacked on my desk, my coal fire ready, and a tray laid for tea with jam and crumpets for toasting. I remained uneasy that the scout was a woman of my mother's age.
I had a precious privacy. From my quarters the door opened not directly on the gallery but on a second door, this one being an inviolate "oak" that could never be opened by anyone, so that the scholar within would not be disturbed while reading. Closing the outer of the two doors was called "sporting the oak," a ritual signaling private study—or a woman visitor arriving for tea. Visitors to our rooms were all clocked in and out by the vigilant dragon lady who had met me at the lodge when I first arrived.
I'd resigned my editorship in good time for the third-year purgatory of finals. I frequently retake those examinations—mad recurring dreams of me staring at the blank paper, wondering who on earth is this David Ricardo to demand rent and what did he have to do with the categorical imperative and question time in the House of Commons and marginal utility and anyway how can it serve the greatest good of the greatest number of people if someone has taken the lovingly long-preserved head of Jeremy Bentham out of his cabinet at University College London? I got through, though. Mum and Dad came early for the graduation, looking long and proudly at the degree notice board on Palace Green to see I headed the "Honors Politics and Economics" list. Their pride was tempered by calculation. My brother Peter recently came across a picture postcard of the cathedral that Mum sent home with just these words: "Harold has had a lot of expense. See if there are any pounds or such like in my club moneybox. Bring it along and the pound out of the brass box in the living room."
I had already started looking for work, writing applications this time on the Castle's red-crested letter paper. At the end of that idyllic summer of 1952, I left Durham with a telegram in my pocket as peremptory as the one from the Ministry of Education that had sent me there three years before:
REPORT MONDAY
_Manchester Evening News_
#
# Stop Press
There had been no individual signatory on the telegram summoning me to the _Manchester Evening News_. I landed in the office of T. E. Henry, the editor in chief, an imposing, beefy presence with forbidding spectacles, a military mustache, and slicked-back hair. He barked into an intercom and passed me like a hot coal to his assistant editor next door, a soft-spoken man who smiled and immediately got on his intercom to summon a tiny man bristling with urgency, who grimaced and hurried me out of the inner sanctum and into the hands of a dapper, bow-tied No. 4.
The pass-the-new-boy game came to a halt in a big room, starkly lit by fluorescent tubes. One end of the room was hectic with reporters lassoed by telephone wires hanging from the ceiling, hotfooting messengers, and banks of copy takers—girls wearing headsets and clacking away on sit-up-and-beg Underwood typewriters. The noise of the newsroom then! Today's tippity-tap at computers, like mice dancing on a keyboard, can never set pulses racing in the same way. At the other end of the floor there was silence. Cigarette smoke drifted from a score of men in shirtsleeves, curled up in prayerful postures before the dominant feature of the room: a schoolhouse clock with big Roman numerals.
These were the paper's subeditors (subs), called copy editors in the United States. I had applied for a job, outlining my reporting experience. The net result of the morning's lightning transactions was that I was to be given a trial—not as a reporter, but as a sub. Everyone knows what reporters do—collect information and write it up in "a story," whereupon it becomes "copy." But nothing reporters write gets into a newspaper without passing through the hands of the subs, the hidden impresarios of news. Subs take pride in translating the complex into the comprehensible, in making sense of conflicting information from divergent sources, in making sure the story fits the space allotted to it, and in writing read-this-or-die headlines. In the first stage of the process, text goes to a sub designated to assess the worth of every story from everywhere—from staff reporters, freelance reporters, and above all the news agencies, notably the Associated Press in the United States and the Press Association in Britain. I'd bet that only a handful of the millions then and now who get their news from newspapers know how much they depend on the sub as the human filter of the torrent of words and pictures from reporters and the wire services. The thirty-two-page _Evening News_ got enough information every day to fill five hundred pages. (Today with the Internet, it would be many thousands of pages.)
In British parlance the sub who filters the copy is called the "copy taster," for he must have a palate sensitive enough to differentiate instantly between the fresh and the stale. He will skim maybe 100,000 words in an hour to make the preliminary selection of the best stories. In those days, he impaled the discards on a basic tool of his trade: a sharp metal spike. Nowadays the term "spiked" means that a story has gone into electronic trash.
The copy taster passes his selected stories to the chief subeditor, a Chinese tailor of page design. He fashions page layouts to express a range of priorities and distributes the copy to his gang of subeditors with instructions as to story length and headline size and style.
Subs never know one minute to the next whether the packet of copy coming their way will be a shout or a murmur: a short story requiring a small headline ("Small Earthquake in Chile; Not Many Dead") or a story calling for a front-page thunderbolt: EVEREST CONQUERED or AT LAST THE FOUR-MINUTE MILE. The packet may contain only a few paragraphs, dispatched in a few minutes, or two thousand words to be carefully trimmed to five hundred. The good sub is an artist in economy. The news feed that unedited would make a full column of space is often enough rendered in half that space without losing a single relevant fact or sacrificing good writing. These skills have been but poorly developed in the United States and Canada, where copy editors have been accustomed to grazing on acres of newsprint; in Britain the effect of wartime newsprint rationing put a premium on conciseness. To be described as a "tight sub" in Britain is a high compliment. (The practice I've wearily observed on too many American news desks of simply lopping off the end of an overlong story is an abomination. It assumes that the reporter has assembled the essence of the story in the first paragraphs and that the portion on the spike is impossible to condense.)
Saving space is a necessity in itself, given the cost of newsprint and the prodigious flow of information; but clarity, as much as economy, is at stake. Meaning is fogged when sentences are freighted with unnecessary words—whether in print, in broadcasting, or on the Internet. A good sub would no more hesitate to remove unnecessary words than an engineer would to remove unnecessary parts of a machine. Excess words and abstractions impose on the receiver's time and attention. This morning as I took a walk along the beach at my house in Quogue on Long Island, I checked the weather report on my cell phone. By the time I heard "Abnormal weather conditions resulting in bursts of heavy precipitation are a likely eventuality for the Northeast," it was already raining.
Not surprisingly, reporters everywhere have a lexicon of rude words for the sub. The color and detail on which the reporter labored have to be reconciled with the imperatives of time and space, and subs have to be merciless. They struggle to retain all the details relevant to the central idea of a story, changing the woolly into the concrete: "Provision for increased retail opportunities" becomes "more shops." In addition to all this, the sub also does a score of checks—for grammatical barbarisms, for inner consistency and coherence, for apparent errors of fact, for fairness, for double meanings, for propaganda, for libel, for nonsense: "The truck driver said he was going north when he saw the deceased walk into the road."
All this editing and checking has to be done with an eye on the clock. Kipling biographer and essayist Edward Shanks put it well: "Sub-editors, when I meet them, seem to have only two eyes just like other people; where they keep the other two I cannot say, but I know they must have them." These unlikely supermen may not be very good writers; they may have been undistinguished as reporters. The skills are just different. When I arrived at the _Manchester Evening News,_ they were still talking about a young man before me who'd flopped as a reporter. He was too shy to go out into the city and ask questions of people he had never met before. I knew the feeling. He pleaded that instead of being fired, he should be given a chance in the subs room, and from there his rise was meteoric: copy taster; chief subeditor; editor in chief; joint managing director of the whole company; wartime editor in chief of the BBC; director general; and finally Sir William Haley, the much-feared editor of the _Times,_ who thundered about moral issues and replaced the classic first page of classified advertising with front-page news.
While plodding round for paragraphs in Droylsden, I'd gorged on the 1944 edition of F. J. Mansfield's _Complete Journalist_ and the anecdotage in Fleet Street's _Inky Way_ annuals. I loved the photographs of hunched men in cardigans reducing cataclysms to column inches. I knew what daily newspaper subs were supposed to do, but my only practical knowledge of a sub's role had come from seeing scissor-hands Middlehurst at work in Ashton. I did not know then how much British subs are a breed apart, their work more intense and more highly specialized than their apparent counterparts on American newspapers, the copy editors, who are expected to do so much less and rise to the challenge. It remained a question of whether Harold Evans, BA (Honors Politics and Economics), could do it well enough as a greenhorn to survive with the gunslingers on the biggest, fastest, slickest, most profitable, and humdingiest popular regional daily, with its million-plus readers.
The subs' end of the room seemed tranquil compared with the reporters' end. It wasn't. The turmoil was psychic. All the subs, in tight balls of concentration, were too preoccupied to raise so much as an eyelid as Mr. Bow Tie, who turned out to be the day's copy taster, indicated an empty chair at the long table and equipped me with the tools of the trade: spike, glue pot, two pencils, scissors, a galley listing deadline times for every page of the day's editions, a pad of copy paper, and an office book of typefaces. I had no time to open it. The tiny man I'd met in the final sequence of my arrival turned out to be the chief subeditor, Norman F. Thornton, known as "Nifty"—except to the man himself, who froze the room on any suggestion of familiarity. We were all "Mister." Having lost all of ninety seconds in meeting me, he was now emitting steam.
"Boy!" he called. "Boy! Boy!"
The summons was addressed not to me but to one of six teenage copyboys, usually a dead-end job then.
"Mr. Evans, and sharp about it."
I received four folios of teleprinter copy about a mother of five who had won £50,000 on the football pools on her first attempt. Nifty's headline instructions were written on top: "24 xi xi, p.4 Home."
The sub next to me uncoiled long enough to whisper, "Two lines twenty-four-point round here is three pars, no more." From reading the paper I knew its paragraphs were short, two or three sentences at most. From Mansfield I remembered the vital little rules devised for fast newspaper production. Head printers shared the typesetting among scores of Linotype operators; a sub who failed to number and mark every folio would create chaos. I identified my little story with the catch line "Mother," then I numbered every folio—Mother 1 for the headline; Mother 2, 3, and 4 for the text. I marked every folio "xi" for single-column setting, double-ticked the capital letters, circled the periods, indented for new paragraphs, defined the destination and edition on every folio, wrote "mf" (for "more follows") at the end of each folio, and made a double-gated pound sign at the end, so that the printers assembling the story would know not to wait for more. Routine clerical stuff really, but as important as a bank teller being able to count.
More time-consuming was that the story I had been given to edit ran ten paragraphs and had to be contained in three. Clearly, if I couldn't do that by now, I should seek another line of work. But according to the time sheet I had only twelve minutes for text and headlines. I had almost no experience writing headlines. The type book, each line measured off against column widths, told me that at the size of 24-point Century Bold—the standard headline face of the paper—no more than twelve characters, including word spaces, would fit in a single column. Having squeezed the text, I turned to the headline. I tried:
Mother of five
has big pools win
No good. "Mother of five" counted out at fourteen characters, and _M_ was a wide letter and the second line bust. I fiddled and fiddled. Five minutes left:
Mother hits
jackpot
It fit, but it was rather vague.
Five lucky
children...
Even vaguer.
With three minutes to go, I had a brain wave. I wrote:
Mum wins big
first time
and self-consciously shouted, "Boy!" He scooped up the folios and dumped them in the chief sub's wire basket.
Other subs sent their text straight up to the composing room, but mine had to be scrutinized. It was a relief to hear the "plop!" as Nifty made it vanish up a vacuum tube, but he held back the single folio with my headline written on it. "'Mum' isn't style," he rapped. "Mum! We're not the _Daily Mirror_."
He might as well have told the room I had syphilis. How the hell was I to know what the office style was? They hadn't given me the house stylebook. In a blink, Nifty scribbled his own headline. When the galley proofs came down, I saw what he had written:
Mother's big
pools win
By now I was deep into copy bundles piling up in front of me for the next edition. Help! It was only fifty minutes away. My desk was right under the schoolhouse clock. Out of the corner of my eye, I caught the blur of a mustache in motion. It adorned the upper lip of the editor in chief—Mr. T. E. Henry, Mr. Tom Henry, "Big Tom." As razored mustaches go, it was not exceptional, but it was spotlighted by two patches of high color in his complexion, the whole ensemble lending urgency and authority to the words issuing forth as he strode without pause through the subs room, his slipstream buffeting a retinue of assistant editors and copyboys, and often the excitable photo editor trying to get him to glance at a print. A herd of wild elephants could not have caused more commotion. A few minutes later the intercom buzzed. Nifty sprang to it. Everyone sat up. "Who the hell," asked the harsh voice reverberating through the squawk box, "who the hell subbed this twenty-four-point that's as long as Cross Street?" It was Big Tom speaking from the stone—the composing room where all the type was assembled in iron frames (chases) laid out on flat stone tables. It wasn't my 24-point; it was someone else's, whose text had run to five paragraphs and would not therefore fit the three-paragraph space Nifty had allocated for it on his neat page plan. The excess paragraphs were a double waste of time—for the Linotype operator and the editor overseeing the page to press. Simply discarding those lines in hot metal might also discard a key point, even the basis for the headline, yet there might not be time to have the galley reset. Nifty choked on his anger.
The erring sub shriveled in his chair. You could feel the room stiffen every morning when Big Tom swept through on his way to the stone for the first edition, catapulting commands and lobbing questions, answers to which were shouted back as if they were grenades about to explode. We all dreaded the sound of "Darth Vader" on the squawk box. His wrath fell on anyone who wrote a politically slanted headline, a commonplace on other dailies, left and right. He was insistent on the unbiased presentation of news—so long as it did not concern the Manchester United football club, which occupied a different universe of gods and supermen. (When they lost 5–1 one Saturday, Big Tom was inspired to write: "United in Six-Goal Thriller.")
The tendency in the newsroom, cowed by the intercom blasts, was to regard Big Tom as a vulgarian. He'd risen through subbing sports, for Pete's sake. He chomped on a cigar. He wore red braces and spotted bow ties. His voice was hoarse. The two editors before him, ice-cold Haley and the glossy John Beavan, were accomplished writers with the air of university dons. "Had an excellent weekend, thank you," Haley would say. "Read eight books and reviewed three of them." Tom gave himself no intellectual airs, but he was an organizer of genius. For the big scheduled events like the presentation of the national budget, six or seven foolscap sheets from his office would tell forty individual subs and reporters precisely what they were to do at 3:10 p.m., 3:15 p.m., and 3:50 p.m. Subs assigned to the stone with him hated to admit it, but they were in awe of his speed and skill, his relentless drive to get the paper out on time. He had no equal. Printers loved his decisiveness and expertise. Gothic heads emerged from the Ludlow machine seconds after he had scrawled a banner that fit perfectly. Like all good stone men, he would read the story in mirror image, cut it, change a dumb headline, and issue boil-in-oil curses on its writer.
At the end of that first week I was shaking, not at all sure I would make the grade. On Monday morning I got in bright and early. Bow Tie was already at his desk, spruce in a white shirt, sleeves rolled up, two shiny spikes ready for the day's torrent of copy. "It's hard for you as a novice from a weekly," he reflected between satisfying puffs of Churchman's in his pipe. "Not to worry. A lot of subs from the dailies who come just can't take the pressure here, no sir, just can't hack it."
Nifty, he kindly informed me, kept a book of the comings and goings, which showed that in four years ninety-four subs had joined and left the paper, some sacked on the spot, most running for the exit. That was close to one departure every two weeks, which meant that by the law of averages another head was due to roll. Mine, I feared.
Nothing before, and nothing I have experienced since, working for newspapers, radio, television, and Web sites in London and New York and Washington, matches the speed demanded of everyone on the _Evening News_. Newspapers were the way millions got their first inkling that we had been invaded by men from Mars; there were no cheap transistor radios, cell phones, or Internet sites then.
Reporters and subs had to operate like a souped-up Internet news service, producing eight editions in six hours—more if big news broke—and without the crutch of desktop computers. We had trains to catch and vans to race into the suburbs. In the words commonly (but mistakenly) attributed to the Confederate lieutenant general Nathan Bedford Forrest in the American Civil War, we were desperate to be "fustest with the mostest." The crowds on the city streets had to carry home the _Evening News_ rather than the hated _Manchester Evening Chronicle,_ our direct challenger. Both papers wrote "contents bills"—news placards designed like the bikini to titillate but not satisfy: SEX TRIAL SHOCKS.
As the two evening newspapers slugged it out, hundreds of other reporters, critics, feature writers, subs, and photographers were competing like mad to beat us and everyone else. Manchester was Newspaper City. Six million people in the north of England got their news through Manchester. Daily and Sunday, no fewer than twenty-six newspapers were written, edited, and published within a couple of square miles of the four central railroad stations.
The twenty-four-hour inner city simply throbbed with news day and night in the newsrooms and all the smoky watering holes where journalists and compositors loved to gossip, keep an ear out for any clue to what their rivals were up to, and uniformly agree that the gritty northern editions of their newspapers were much superior to those put out by their effete bosses in Fleet Street.
Every day on the way to work I had to walk by the embattlements of the enemy in Withy Grove, a rabbit warren a few yards from Victoria station where three thousand people in Europe's biggest printing plant produced twelve papers, including the _Evening Chronicle,_ the _Daily Telegraph,_ the _Daily Mirror,_ the _Daily Dispatch,_ the _Sporting Chronicle,_ and three Sunday papers; from Deansgate, out rushed the delivery vans of the _Daily Mail_ and _People;_ the trade unions' _Daily Herald_ was tucked away on the other side of the railway tracks in Chester Street; and to the west rose the glass black Lubyanka, where the slickest of the slick produced the _Daily Express_ and _Sunday Express_ for Lord Beaverbrook (William M. Aitken), the press baron for whom the phrase "whim of iron" was invented.
The _Guardian_ —then proudly named the _Manchester Guardian_ —and the _Evening News_ were both owned by the Scott Trust. We had offices on different floors and separate composing rooms, but we shared the presses and the cafeteria. (With a nod to liberal principles, company chairman Laurence and his brother Charles shared the same meat-and-potatoes menu.) At the lunch table where the _Guardian_ editorial people tended to assemble, you'd find the gentle Bill Weatherby; Bob Ackerman, an American intern from Brown University who decided he would rather dissect the brains of chimpanzees as a neurologist in Boston; and a pale, emaciated new reporter with a lupine smile, who amazed us by saying how entertaining that morning's proceedings of the North West Gas Board had been. The gas moguls were not amused by his report in the _Guardian,_ but the experience was great training for becoming Michael Frayn, playwright and novelist. I was entranced to hear that Frayn, another Cambridge alum, had learned Russian so as better to appreciate its literature, a linguistic accomplishment that catapulted him into journalism. He was approached to become a spy and made up his mind on a newspaper career instead when Control—or whoever, maybe Alec Guinness as Smiley—laid a venturesome hand on his thigh. The North West Gas Board paid for the indiscretion.
At the _Manchester Evening News,_ we always liked to show Newspaper City how we could turn on a dime. Of course, delivering news by casting it in hot metal took an age compared with transmitting it electronically, but it was the rush of the raw editing time that was so taxing. It was not uncommon for our subs to be physically sick from the tension. In the second week I felt so queasy that I made lunch a boring glass of milk, forsaking the corned beef sandwiches Mum had put in my raincoat pocket. Nifty gave each of us only twenty minutes for lunch anyway; if during the day we got up to go to the toilet, more copy would be piled up for our return as punishment for indulging human weakness.
The survivors around me were all used to the sound of gunfire. The artist who drew our maps had been a lieutenant in the Royal Naval Volunteer Reserve. His Japanese captors had made him kneel for a beheading, then changed their minds and chopped off an arm instead. Bob Gibbons, the soft-spoken sub who had rescued me from letting that first story run too long, had been a tank commander, decorated with a Military Cross for battlefield bravery (a fact discovered only when a buff envelope crested on HIS MAJESTY'S SERVICE was left on his desk on his day off). A sub who came and went every day without saying a word to anyone had been a bomber pilot, badly shot up over Düsseldorf. A vehement blond man with a black eye patch had lost his eye in the army but was still a quicksilver sub. The amiable Bow Tie, who looked like such a dandy—he came to work every day in yellow gloves and for the 1956 Suez invasion turned up in a pith helmet—had for four years been Quartermaster Duncan Measor on the destroyer _Hotspur,_ dropping depth charges on U-boats in the Atlantic. Nifty himself had been a regimental sergeant major. He expected everything to be done on the double and without complaint. In that spirit, on the occasion when he rammed discarded copy on his spike and the spike went right through his palm, he uttered no cry of pain. He simply took out the spike and picked up more sheafs of copy from Bow Tie, who was a kindly man but knew Sergeant Major Thornton too well to express sympathy.
Nifty was the fastest and best sub and layout man in the room, and I daresay on the planet; his instructions on copy, written at speed, could be read by a man on a galloping horse. He was also a tight-lipped disciplinarian. No banter, no jokes, gentlemen; keep your heads down; get the copy out. He had a watertight memory. When a story ran through two or three news cycles, he knew precisely which elements were fresh. A sub might be rebuked, "Wake up! That headline was the page-four lead in the _Express_ this morning." The subs took these chastisements quietly; Nifty was always right.
We might as well have been on a spaceship, receiving signals from all over the world but stuck in our chairs, hermetically sealed off from the daily life of a vibrant city. In the mornings, having memorized all the headlines I could from half a dozen dailies on the train from Failsworth, I walked with the city crowds, inhaling the aromas of French cheese and Latin American coffee seeping out of the cellars of the wholesale houses along the damp streets near the cathedral. It was the nearest I'd get to foreign parts for some time. At the blackened Victorian building in Cross Street near the corner of bustling Market Street where the tramcars stopped, there'd always be a gaggle of people pausing to look at our news pictures in the display windows of the front office, a matter of pride when I'd edited the captions. I'd pause to savor a little romance. Here was where the legendary _Manchester Guardian_ editor C. P. Scott ("facts are sacred, comment is free") jumped off his bicycle every day, having ridden the three miles from his home at The Firs, Fallowfield, and back again at night, a practice he continued until he was past eighty. Here, in January 1932, the crowds stood ten deep for his funeral cortege. Here, too, is where Neville Cardus, a bastard son of a prostitute from a Rusholme slum, came to gaze up at the _Guardian_ 's lighted windows, dreaming he might one day write for the paper. And eventually he did, gloriously, as the doyen of cricket and music critics.
In the first weeks I was happy to take home a paper to show off the stories I had edited. But they weren't much, little single-column efforts. The family was nice about it but clearly baffled by my sense of achievement. No, I hadn't written anything really, I'd say, not like for the _Ashton-under-Lyne Reporter_. But you see this story and this one, well I edited them, and they all caught the page in time, and you should have seen what they were like before I did the editing.
Two months into the work I was being trusted with inside-page leads and was officially taken on the staff. But I became jealous of the aristocracy at the subs table, the three or four "top-table" subs who were given the really big stories and made it seem effortless. One of them, the unflappable Bill Lloyd, edited while he lay back in his chair like a nonchalant Battle of Britain pilot awaiting the siren, his pencil, with a life of its own, gliding over the copy and oozing headlines. If only I could take home a front-page splash I had edited! When there was a loud bang on the shutter between the subs and the wire room, indicating the arrival of a "rush," some kind of sensation, I'd sit up like a puppy dog, tongue out for a juicy bone. Nifty would look round and invariably award the prize—"Rush this, please, ninety-four-point Gothic caps"—to Mr. Lloyd, Mr. Finnegan, Mr. Futrell, Mr. Batchelor. He might as well have announced, "To anyone but Mr. Evans."
He was correct in that judgment. I was in the right place as a down-table sub. I still made mistakes in the endless rush, sending up two folios numbered four, writing a headline that didn't fit, one day taking three minutes too long to squeeze the best juice out of a complicated court case. Big Tom had to rewrite one of my Saturday sports headlines when the Port Vale soccer team beat Chester. My headline was a succinct summary of the play, but the comps fell about with laughter when they set it in metal before Big Tom: "Port Vale Thrusts Bottle Up Chester." Nifty's punishment for infractions was a rain of " _K_ 's"—the three-line fillers at the ends of columns which required a headline in small bold caps, meaning there were too few letters to do more than grunt. It was Chinese water torture all day: drip, drip, drip. Someone recognized that my soul was in torment. Every week two or three of the editorial staff were handed books for review. The others got histories and novels. Not me. The first one on my desk was _Eclipse of God_ by Martin Buber. ("All journeys have secret destinations of which the traveler is unaware.") The following weekend I had to devote myself to studying Paul Tillich's _Systematic Theology,_ and the third was consumed by two sledgehammer volumes on the philosophies of the East and West. I was a marked man.
When the big day came, it was more than I'd bargained for. At 8:25 a.m. the wire room shutter banged, signaling a rush story. Nifty shouted, "Boy! Mr. Evans!" I was sharpening a pencil. A single Press Association folio landed on my desk.
RUSH 1
Train
TRAIN CRASH 8.19 XXXXX IN FOG XXX PASSENGER
TRAIN IN STATION XXX
Oh, no, please not Dad! But I'd no time to fret. "Rejigging page one," Nifty called to the room, giving instructions for new displays. And to me: "Crash for lead replate at nine."
A boy gave me Nifty's folio of instructions for head and text setting.
8:38: Another folio:
RUSH: TWO TRAINS LONDON MIDLAND REGION COLLIDE HARROW WEALDSTONESTONE STATION. COACHES PILED UP. WRECKAGE BLOCKS LINE. FOG. AMBULANCES. FIRE BRIGADES POLICE RUSH TO SITE.
Not Dad's region. More folios followed, in summary:
8.40: _Express from Perth with four sleeping cars ran into back of local on fast line. Est sixty miles an hour._
8.41: _Local Tring-Euston train nine coaches stopped at Harrow-Wealdstone station for minute-plus, running seven minutes late in fog.... Rear coaches telescoped by impact of Perth express. People still on platform scythed down. Unknown numbers trapped inside wreck...._
8.46: RUSH RUSH
_One minute after primary crash, wreckage hurled across adjacent tracks derails two engines pulling Euston-Manchester-Liverpool express approaching at 60 mph. No word of casualties...._
So now we had three trains in a wreck—two express trains and one local in which an unknown number were trapped in smashed coaches.
8:54: Four minutes to deadline. I'd translated the cryptic rushes into sentences and sent the story to the printer. The banner was easy: THREE TRAINS IN HIGH SPEED COLLISION. Second deck: "Many feared trapped in coaches."
8:58: I'd made the edition, but now an important nugget came two minutes to deadline: _Police: Casualties bound to be heavy on Tring train. Twenty bodies on stretchers._
"Boy!" I yelled at the same instant I marked the folio for a bold flash intro and wrote a new substitute head that Nifty accepted: TWENTY DEAD IN EXPRESS PILE-UP.
I was feeling pleased that I was keeping up with the pace of the inflow but suffered remorse the instant the page had gone to press. I'd presumed "bodies" meant people who were no longer alive, but...
9:02: _Twenty bodies unconfirmed dead. Repeat unconfirmed dead._
Nifty brushed aside my concern. He was already calling instructions for the next edition in fifty-eight minutes. "Streamer cross eight twice ninety-four point. Second deck thirty-six point cross four. Close ten o'clock. Let it run."
9:05: I killed all the type from the first edition and started to compose a new story from all the fragments.
9.06: _The station foreman assXXXX assigned igned to count passengers for rail census says, "I counted 332 passengers coming down the stair from the footbridge. God knows what's happened. I heard terrific noise and screams."_
9.06: RUSHFULL
_Twenty now confirmed dead. Police say total likely higher. Names withheld pending notification of next of kin._
9.07: _Rescuers trying to reach Driver of Perth express R. S. Jones and Fireman C. Turnock of Crewe North Sheds. Buried somewhere under mountain of twisted iron Rescue crane due 9.40_
9.09: _Loudspeakers command quiet so rescuers can listen for cries from the trapped passengers under the mountains of twisted iron._
The separate folios, a sentence on each, were falling like snowflakes in a blizzard, mixing the urgently important, the relevant, the repetitive, and the disposable:
9.09: _Harrow-Wealdstone 11 miles from Euston. Six tracks for 17 miles...._
9.12: _Witness: "We came up on the local line, then crossed to the express line. Our train was packed. We hadn't shut door of our coach. There was one hell of a crash. Our train leaped off the ground. We were hurled against one side of the compartment. Then another engine flashed by in the opposite direction. It was a nightmare."_
9.16: _Guard J. Kent in rear Perth Express survived. Linked up with Perth express Driver Jones at Crewe. Jones told him had difficulty backing new engine onto train in fog. Express running 32 minutes late, would do best to make up times._
9.17: _Minister of Transport heading to scene._
9.17: _Driver Perkins Manchester-Liverpool Express killed in derailment._
9.18: _Worst train crash 22 May 1915. Troop train collided with passenger train at Gretna Green, Scotland. 227 killed._
9.20: RUSH: _Electric train from Watford headed into wreckage scattered across electric line but stopped by alert signalman switching off current. British Rail keeping him secluded. Said to be shaken._
9.21: _Tring train passenger in ambulance: "The most terrible thing is the screams coming out of the middle of the great heap of the wreckage from the poor devils trapped inside."_
9.25: _Injured Perth express survivor in hospital: driver must have run through signals. Must have been drunk._
9:30–9:40: Half an hour to deadline. I edited the first half-dozen folios into a sequence, trying to make the story flow and writing cross headings every four or five paragraphs, taking note of my numbered folios, holding back the intro for the hardest and latest news. I set aside the allegation about the driver. Not because of family ties. How did the survivor know? The driver was still in the wreckage.
9.40: _Lucky escape Guard W. H. Merritt of Tring train along train to his van at rear said heard express approaching at speed and dived under coping of platform, then ran to signal box to sound alarm to close all lines._
9.43: _Express sleeping car attendant says brakes went on hard seconds before crash. None of sleeping car passengers hurt._
9.44: _All the station clocks stopped at 8.19_
9.45: _Firemen climbing 40 ft into piled-high twisted debris. Score of stretchers laid across tracks. Coach and wreckage jammed under station footbridge._
9:50: "Five minutes to close," called Nifty. "Banner, please."
I'd now written a lead paragraph on twenty confirmed dead in a three-train collision. No sooner had it left my hand than a new flash landed:
9:51: _Total thirty-one bodies pulled out of Tring train wreckage._
I wrote a new intro, marked it as substitute for the intro just sent ("substitute" meant the earlier one had to be thrown away), and warned the stone.
9:59: A minute before deadline I gave Nifty the banner: THIRTY-ONE DEAD IN EXPRESS CRASH.
Nifty had replanned the page for noon. A crude diagram showed the paths of the three trains on two lines. There were picture captions to write, new headlines for a jump to the back page to report names of the dead—including a Manchester Queen's Counsel—miracle rescues, interviews with the hundred-plus in hospitals, the arrival of bigger cranes, and now, if I wanted, I could use a new Press Association lead reconstructing the crash with descriptive writing: Hundreds, six deep, chatting and reading newspapers as they waited for the tube across from the doomed Tring platform, "suddenly heard the roaring and vibration of an approaching express, the shriek of a whistle, the hiss of steam. Before horror could shape itself upon the faces of the crowd, they saw the express from Perth smash into the rear of the waiting train and in that eerie frozen moment of horror there came from the direction of Manchester the thunderous pounding of the express from Manchester gathering speed into the wreckage."
I subbed the best of the new material into yet another version of the story for the noon edition. So it went all day: edit this, spike that, change this, number that, check the number, insert a paragraph here, delete a paragraph overtaken, shout for a proof, mark it and send it back... and watch the clock. I was oblivious to all the activity in the subs room, utterly immersed in a story changing every minute, writing and rewriting the narrative, waiting for the last seconds for the final headline, heart pounding, praying that what I was rushing into metal read well. By the final edition the news was much bigger than it had begun. At 4:00 p.m. I wrote the Late Night Final edition banner:
75 KNOWN, 110 FEARED DEAD IN RAIL DISASTER
Full platform "scythed": Children trapped hours
The Harrow-Wealdstone disaster, my baptism of fire, was the worst crash since 1915: 122 people died. The official inquiry concluded that express driver Jones had died on impact, and so had his fireman. The signalmen were exonerated, so the explanation offered was that Jones, a healthy experienced driver of forty-three who knew the road, had almost certainly not reduced speed when taking the train past a distant signal showing yellow for caution. He passed two "not very conspicuous" semaphore signals at danger, running through "a deceptive patch of denser fog."
Even with a computer today, that kind of fast-changing, multisourced news report is tricky to handle; with hot metal it was the supreme test of a news sub. My anxiety was whether I had passed. The next morning I opened an envelope on my desk. It was from Big Tom. "To mark your excellent work yesterday I am putting you down for one pound [about $116 today] special work bonus." My euphoria was such that I have kept the note all these years.
Thereafter I got a fair share as a top-table retailer of disaster: floods, a ship drifting packed with explosives, a plane crash, a building collapse. I told myself that four months into the job, I was almost up there with the aristos. Then I was pitchforked into page design and thought again. Late every afternoon, when everyone else was putting on his coat to head for home or pub, one sub stayed late as editor of the "Star" edition. This was the first edition the next day. The Star editor had to lay out and sub five or six inside news pages for the early shift of printers. The first edition, with a newsy front page and runners at various horse tracks, could then be on the street before noon. "Here you are then, Mr. Editor," said Bow Tie, depositing a bundle of timeless agency copy, photographs, and layout sheets. "Make the pages sing! Good night."
The reporters' room was empty. The tea trolley was cold. Within an hour the cleaners had coped with the day's debris, and I was alone with a lubricious library girl. She swayed over for a chat but quickly decided that filing old clippings was more exciting than my impromptu exegesis on the challenges of page design. I laid out pyramids of headlines on top of one large photograph and one small, with a boxed panel and text stretching over three columns, descending into two columns one inch deep and then into a single column. It was an arresting new look for the _Evening News_. It was also utterly unworkable, a jigsaw that would not fit together unless the relevant words in text and headlines once cast in hot metal matched exactly the number of lines the layouts required. I also discovered how little I really knew about scaling a photograph for the engravers so that it would occupy exactly the space I'd allotted in the layout. As for a fancy enlargement of the second photograph, I hadn't a clue how to work out the scale for that. (It seems very unfair that today the calculation is done by the computer before you can say George Eastman.) I was in the middle of exasperated fiddling with the photograph, hunched over a light box, when a lovely soft voice said, "Still here? Like to come to the Opera House when you're done?" Ah! The library girl, a second chance! No, it was a perfect entry, stage left, by Denys Futrell, a sunny plump sub obsessed with the theater and Yorkshire cricket, who had come back on his day off to pick up tickets from the news editor's desk. In his evenings away from Nifty's lash, he directed amateur theatricals and wrote innumerable reviews. I showed him my comic opera of a page. "Trying too hard, old lad, trying too hard," he said with a sigh. "Till you get the hang of it, just copy some layouts from other days. Nobody in Ancoats is going to complain that they've seen that layout before." He showed me how to measure the enlargement, then headed off to the more satisfying drama of Ibsen's _A Doll's House_. "You should be through in an hour. I'll leave your ticket at the box office," he said.
Denys needed plenty of time to get there, handicapped as he was by a clubfoot resulting from a childhood accident, after which his leg had been badly reset, but I had no chance of catching up with him. I did as he suggested, subordinating originality to sanity, but I still had to sub all the copy, write all the headlines and captions, and do it all over again for four more pages. They were just about ringing down the curtain at the Opera House when I crawled up to the composing room with the copy and layouts for the Star edition. Next day nobody complained that the layouts had been seen before.
Eventually I became a fast page designer. A few months later when I was again the Star editor, my diary affirms I was home at 7:30 p.m., in time for Radio 3's broadcast of Mozart's "Coronation Mass," and a week later "a furious attack" on the Star pages had me out of the office at 6:00 p.m., impelled to catch a 6:30 train to Leeds with a big backpack loaded with tent, blankets, ground sheet, and Primus stove. I had inveigled Enid Parker, then teaching at a Quaker progressive school in the Yorkshire village of Wennington, to realize that her biology teaching would be much enhanced if she camped with me in the Yorkshire dales. (We married a year later.) Beyond that attraction, I also took any chance I could to get out of the city, walking the vales of Derbyshire and climbing the moorlands to Kinder Scout with other subs gasping for clean air.
The first year of subbing on the _Manchester Evening News_ was a companionable roller-coaster ride. In the second, I was restive. I more and more lived for the big days. In the summer of 1953 we were all keyed up for the coronation of Queen Elizabeth II. Everybody was. The _Guardian_ got in trouble for publishing a cartoon by David Low which mildly questioned the millions of pounds to be spent on the proceedings, while its historian David Ayerst noted the "quasi-religious fervor" sweeping the country. Certainly the community spirit was infectious. Across the city and into the suburbs, streets were outdoing each other in profusions of Union Jacks and red-white-and-blue bunting; mothers and grandmothers in pinafores with buckets and soap were on their knees scrubbing even the pavements so they'd glisten for the street parties on Tuesday, June 2. Everybody in these streets knew each other: "Flo is good for sausage rolls. Jill will ice the cake."
The excitement at the _Evening News_ was in preparing to outdo the _Manchester Evening Chronicle_ and the dailies with a souvenir issue. (The excitement would have been even more intense if we had known the British conquest of Everest would become known the same day.) Nifty took me aside the weekend before the Tuesday. He confided, "You have been handpicked for our special edition." He said no more about my anointment, but I knew at once I was going to be asked to comment on the significance of the crowning. At home that weekend, I read Walter Bagehot's essay on monarchy and made notes on the role of religion and the history behind each ceremonial act in the crowning. Big Tom's multipage battle plan, circulated on the Monday, described other intents. Photo captions: H. Evans.
I was deflated. It was actually an important job on a coronation edition filled with pictures, but I had already begun to wonder whether Michael Hides had been right in our Ashton-under-Lyne days that reading Hobbes and Keynes was all very well but irrelevant to getting on in newspapers. There was a common feeling, too, that the painstaking act of subbing to save words ruined the fluency for writing. "Show me a good sub, and I will show you a dead writer" was the newsroom's jaunty challenge, not so much directed at me as at all the subs who kept threatening to write novels. On the day there was a surprise cut in the Bank Rate, everybody had an opinion about what it meant for the average man in Manchester. I sulked that nobody asked me. When I finally volunteered a paragraph I'd written, it was brushed aside as too late and too complex. Doubtless it was both those things. Another day I fretted at the London political correspondent's analysis of the annual economic survey. I thought we could do better and said so. A diary entry records the fallout: "It was a mistake to speak of Bank Rate, etc. to colleagues. They will only think of me now as a know-all." When I was given the splash that the new Tory government was going to break the BBC monopoly by permitting commercial television, Bill Lloyd ribbed my pretensions, calling out, "What's Aristotle got to do with it?"
The frustrations came to a head when my newspaper—which is how I thought of it—joined the chorus endorsing the death sentence for an illiterate youth for a murder he had not committed. Sydney Silverman, the tiny, silver-haired Labour member of Parliament, a passionate abolitionist, was leading the protests. "Silverman is only doing this for personal publicity," declared Big Tom out loud on his quick way through the subs room. He was not inviting a discussion; he was venting. The sub opposite me protested by repeatedly jabbing his cigarette stub in the ashtray. I shared his feeling.
Derek Bentley, the condemned youth in the cell, was a weak-willed nineteen-year-old who had suffered a head injury from V-1 rocket debris and was judged to have a mental age of eleven. His pal Christopher Craig was sixteen, but he was brighter—and he'd carried a gun the night the pair had tried to burgle a warehouse. Four policemen cornered them on the roof. One policeman caught Bentley. He made no attempt to escape, even though the policeman had been shot in the shoulder by Craig, who, still free, kept firing and shot Police Constable Miles dead. Craig was undoubtedly guilty of murder, but he was spared the death sentence on account of his age. Lord Chief Justice Goddard—"the hanging judge" to his critics—ignored the jury's recommendation of mercy for Bentley. Parliament was not allowed to debate whether Bentley deserved to hang until he had been. Silverman called for the home secretary, Sir David Maxwell Fyfe, to make a recommendation of mercy to the Queen. Silverman had the support of two hundred MPs, but the home secretary turned them down.
Though Mum and Dad were not opposed to capital punishment, as I was, they could not understand why Bentley should be the one to die. I was empowered to go to the public phone booth at the end of the street and dictate a family telegram to the home secretary. I followed up with a letter in the _Guardian,_ which did not meet with universal approval around the subs table. "Why do you think you know better than the people in authority?" asked Jimmy Entwistle, one of the most capable and charming men on the newspaper. Bentley went to the gallows on January 28, 1953. Forty-five years later, three Court of Appeal judges quashed the conviction on the grounds of misdirections to the jury by the "intemperate" and "blatantly prejudiced" Goddard. Lord Chief Justice Thomas Bingham said that Bentley had been denied "that fair trial that is the birthright of every British citizen." Craig, age sixty-two and long out of jail, having served ten years, said, "I am truly sorry that my actions on 2 November 1952 caused so much pain and misery for the family of P.C. Miles, who died that night doing his duty. A day does not go by when I don't think about Derek and now his innocence has been proved with this judgment."
Six months after my family telegram of protest, my diary tells me I abandoned my scruples about the death penalty. "John Reginald Halliday Christie," I wrote, "forfeited the right to be treated as a human being. Seven murders and an innocent man executed!" The innocent man was Timothy Evans (no relation), who a decade later was to haunt my first editorship.
Fortunately, for everyone's peace of mind on the paper, I found an outlet for the didactic impulses I never knew I had: I volunteered to be an evening tutor on politics and economics for the Workers' Educational Association, founded in 1903 to help adults who had missed out on education. Twice a week that October, after the day's subbing, I took a bus ride into Tyldesley, near Bolton in Greater Manchester, crossing a darkened playground to a schoolhouse where a gallant group sat at children's desks: retired coal miners, clerks, a watchman, a sprinkling of housewives, and a couple of shopkeepers. There were never more than sixteen of them. I had an idea that the room would soon be packed when word got around that I was lighting the sky; it didn't happen. Two dropped out in the second week and another in the third week. If this went on, I'd have no class.
The survivors were embarrassed by the defections. They were so nice, trying to put me at my ease, but I'd missed a trick. I'd not engaged them. They'd lived much of the interwar political and economic history I proposed to teach. They'd seen the thuggish behavior of the Blackshirts of Oswald Mosley, the incitements in the Jewish area of Cheetham Hill. Mosley was a Manchester man; one of the main thoroughfares was Mosley Street. He'd had his second wedding in Joseph Goebbels's house in Berlin, and Hitler had been the best man. So I made the evenings more of a discussion than a lecture. It took more preparation, not less, to plot a kind of sub-Socratic question-and-answer dialogue that led somewhere, but the classes got livelier. Shortchanged by the education system, just like my father, the group was touchingly eager to explore—but not uncritical. One of the housewives and a storekeeper, in alliance, were not convinced (nor was Mrs. Thatcher years later) that in the Great Depression they'd lived through, the government should have increased spending instead of sticking to balanced budgets: they had to stay in the black in their household accounts, why shouldn't government? They attended the twice-weekly classes in all kinds of weather—one dropout even came back—and they all signed up (with newcomers, too) for a second year. At the end of the course, they took up a collection to buy me a going-away present of an _Oxford English Dictionary,_ which I still cherish fifty years later.
In truth I was in their debt. Face-to-face, I'd been made to appreciate what schoolteachers learn painfully but journalists behind a shield of print rarely do: transmitting information is easier than creating understanding. For me it was a step toward grasping the art of popular explanatory journalism.
I was not able to teach a third year. Big Tom had a better idea.
#
# Why Aren't Their Women Wearing Our Frocks?
Late on Friday, December 18, 1953, just as I was putting the lid on the messy glue pot to head home, the editor's secretary dropped an envelope on my desk. I'd seen Big Tom stride out of the office in his cashmere overcoat, cigar between his teeth, and a copyboy trailing behind with an armful of books. I sliced open the envelope. The letter was headed PRIVATE AND CONFIDENTIAL, underscored in red.
It was good news. "My dear Evans," the three-page letter began, "I feel quite sure in my own mind that you are qualified to have a shot at the job of Evening News Editorial Writer, and I want you to take up duties for a trial period of three months. In the first place your salary will be increased to 14.10.0 pounds per week plus Cost of Living bonus."
My old paranoid instincts fastened on the phrase "in my own mind." Did it imply everyone else thought he was off his head? But he went on: "I will now develop one or two matters arising from this appointment."
I'll say.
He acknowledged that there would be "a great deal of political and economic reading"—no more novels on the weekend, only the _Economist, Hansard Parliamentary Debates,_ and Chatham House reports. I was also to get out of the office and interview Labour, Conservative, and Liberal organizers; MPs in their constituency clinics; councilors; and anyone who might lead me to a story. "Do not forget the political leaders outside Manchester, say in Altrincham, Knutsford, and other divisions. Never let up on them. Follow and probe indefatigably; keep stories flowing and watch for good tips for Mr. Manchester's Diary. I think this could be very rewarding and I am keen to have it done thoroughly. The test is how many stories and how many tips you can produce. Let me see them all."
He hadn't finished: "Afternoons at 2:30 pm (except Friday), I want you to go into the subs' room and lead the parliamentary subbing team." What this meant was that I could select who would stand with me to protect the sanity of Mancunians from the tsunami of verbiage from the Commons. I selected Bow Tie. He was fast and genial, and not someone who would blow cigarette smoke in my face. I was even authorized to become a Nifty in training, giving work to other subs if we could not cope.
To be acknowledged as a top-table sub would have been promotion enough, but writing the editorials (commonly called "leaders") was the big deal. They had struck me as sensible and lucid for the most part, and the phlegmatic man who had been writing them, Sidney Cursley, certainly fit the role. He would emerge from his inner sanctum next to the editor and stand stock-still sucking on his briar pipe, incubating great thoughts as he looked on our bowed heads. Happily for me, he was sailing to New York to head the British government Information Service explaining our eccentric ways.
I was thrilled to accept the trial, but with the Bentley hanging still in my mind I fretted about what I would do when asked to write against my conscience. This was a preposterous exercise in self-regard. The editorial column was not my prerogative. I was there to express the viewpoint of the paper for which the editor was responsible, and if faced with an impossible demand, I could take my conscience with me back to the subs room.
There were, moreover, new perplexities to humble my pretensions. Britain was in transition from a Labour government headed by Clement Attlee to a Conservative government headed by Winston Churchill. Attlee's ministry had nationalized major industries, established the welfare state—of which the jewel was Aneurin Bevan's National Health Service—and emphasized the planned collectivist economy. "Collectivism" and "planning" were incendiary words to Churchill. He announced that "at the head of our mainmast we, like the United States, fly the flag of 'free enterprise.'" How should the two philosophies be reconciled? And while Attlee's government had begun shedding the empire that Churchill loved, giving India independence in 1947 over his vehement objections, it had not pulled up the drawbridge. It had vigorously fought an insurgency in Malaya (where Britain had rubber and tin interests). It had committed itself to the North Atlantic Treaty Organization (NATO) and the development of an independent nuclear deterrent. Churchill, an unashamed imperialist, was even more determined to maintain Britain as a big power. But could Britain afford the military means to protect its overseas interests, especially in the Middle East, maintain the welfare state, and meet the rising expectations of the trade unions? World War II had cost Britain a quarter of its wealth. Fully 10 percent of the gross national product was spent on defense during the Korean War, just ended. The pound sterling was vulnerable.
How could they expect a boy from Failsworth to solve all this, even with his precious degree?
On the Monday, wearing a bow tie as Big Tom did, I walked through the subs room into the office of the assistant editor I'd met briefly on turning up in Cross Street, Bill Pepper, the meditative consigliere to Tom's brusque Corleone. I was given a small desk there, looking out on the newsprint trucks.
At 8:30 a.m. I was alone at the typewriter Cursley had abandoned for Manhattan. At 8:31 a.m. Big Tom's frame filled the doorway. "First leader, the Queen of Tonga, second the Canberra jet bomber." Then he was gone. Over the weekend I'd read myself ragged on all the economic and political news. In these preparations Queen Salote, the South Sea Islands, and indeed the whole Southern Hemisphere had not figured at all. I was abandoned on a coral reef without a clue as to how to get off.
I sent for the clips and then remembered the caption I'd written on our coronation special in June, when Queen Salote had enchanted the crowds by sitting in the rain in her open carriage. Now the young Queen Elizabeth and the Duke of Edinburgh were her guests in Tonga, and luckily it was raining in paradise, too. I was able to say how pleasing it was that the queen of the world's largest kingdom and the queen of the smallest shared an umbrella—and that they shared certain ideals et cetera. (My first editorial was not the place to recycle Noel Coward's wicked response when asked who was the little man sitting with Queen Salote in the London celebration. He replied, "Her lunch.")
The new Canberra jet bomber was easy. Flying London to Cape Town in record time, it had beaten every major long-distance world record, but the chief engineer of British European Airways, a Mr. Shenstone, had chosen that moment to condemn the quality of the work on British aircraft, and vaguely at that. We took him to task for a damaging generalization. (Note the editorial "we." And as a former RAF corporal hiding behind the royal "we," I'm relieved all these years later that "our" vigorous defense of the Canberra was vindicated by its versatile life of fifty-five years. Not until July 31, 2006, did it go out of service.)
I could never guess what bee buzzing in Big Tom's head would fly out to sting. Every morning at the paper was a lottery. The space was usually half a column, but he would sometimes ask for double the normal length for a major foreign affairs subject—our role in Europe, say, or what we had to do in the Middle East to make friends with Arab nationalists without sacrificing Israel. It served me right that having felt frustrated in expressing the occasional opinion on something I knew about, I should now be called upon to express an opinion on _everything._ There was no Google or Ask.com, no Aladdin's lamp I could rub to bring me facts from the vasty deep of cyberspace. I was reliant instead on the snip-snip of the library girls' press clippings and the prayer that whatever reading I had been able to do was pertinent to the editor's impulses. Should we arm Germany? Should the Church of England conduct marriage ceremonies for divorced people? How much of a menace was Egypt's Colonel Neguib? Was the government right to impose a fourteen-day ban on television discussing anything down for debate in Parliament? What should be in the chancellor's upcoming budget? Did women over thirty-two become unattractive, querulous, and antisocial, as an American airline maintained in rejecting them as "air hostesses"? Guerrilla strikes! Income taxes! Delays in court! The Common Market! Nationalization! Cricket manners! Algeria! Cotton exports! The slums! Neighborliness! Joseph McCarthy! Food prices! Immigration! I became a rag and bone man of the opinion trade.
Big Tom himself edited the copy on all these efforts. He did not fiddle with the wording, apart from a startling "For why?" he wrote into sentences that he thought needed more pep. I'd spend the first fifteen minutes of the precious writing time throwing scrunched-up balls of paper into the wastebasket. If the editorial was a tiny bit long for the space, most often because of an advertisement, he'd invariably just lop off my precious first paragraph. He didn't say anything, but I got the message: no throat clearing.
Working closely with Tom, I gradually came to know a more sensitive and prudent man than the fairground barker I'd seen from the subs table. At the end of a day scuffling with news, he would go home to his violin and wrestle with Bach. In circumstances he never explained, he had played with the Berlin Philharmonic after World War II. The evening he was due to collect an award at a city banquet, having been named Mancunian of the Year, he came into the office with a heavy suitcase. "All the family silver," he said, index finger tapping nose. With a touch of pride he explained, "Burglars read our paper, too, you know." He had an uncanny instinct for news. In 1945, by elaborate preparation and shrewd interpretation of a stray message from Rheims, this same Tom Henry secured a world scoop and had his prepared WAR IS OVER edition on the streets well ahead of the official announcement, every other newspaper, and the BBC.
Big Tom's attachment to Manchester, its needs, and its achievements no doubt came with the job like his black Jaguar Mark VII, but it was heartfelt and justified. "What Manchester thinks today," he was fond of intoning, "the world thinks tomorrow." He was the reason for the soaring success of the paper, reaching peaks of circulation and influence no provincial evening newspaper is likely ever again to reach. The profits generated under his leadership kept the _Manchester Guardian_ alive when it edged into loss in 1961 and plunged ever deeper thereafter to the brink of shutting down for good in 1966. Some of my friends on the _Evening News_ in the 1950s felt that the _Guardian_ staff patronized them as being from a lower universe. I didn't sense that, but stories of _Guardian_ eccentrics had a ready currency. "One of their leader writers came up in the elevator with me carrying six umbrellas—on a dry day. Can you believe it?" said Bow Tie. It was later alleged that the _Guardian_ 's editor, lacking an editorial from the leader writer, had sent a messenger along to retrieve it and found he had passed out, his head resting on the typewriter on which the only written word was "notwithstanding."
Big Tom bristled with certainties, but the secret of his success was the way he identified with his diverse Manchester readers, though he was born in the East Midlands and had an Irish background. The sacred hour in his day was when his secretary carried in the "Postbag" (letters to the editor). He fell on them, licking his lips, read every one, and then gave over huge amounts of space in the next day's paper to these missives. He enabled citizen journalism long before the blogging era. Later one of my jobs was to edit the letters for publication. On many papers this was regarded as the ultimate ignominy for a sub. Not on the _Manchester Evening News_. Indeed, the editors whom I later heard boast that they didn't ever bother to look at readers' letters invariably ran second-rate papers.
Big Tom was on the board of directors of the _Manchester Guardian_ but rather relished his different point of view on Manchester, most notably on the city's plan to build new towns in the greenbelt of the Cheshire plain, which the _Guardian_ supported and we didn't. The _Guardian_ 's housing specialist was outraged by Tom's conviction that the city was chasing moonbeams in applying for compulsory purchase orders to build two little Manchesters in Mobberley and Lymm. It took eight years for the government to say no. In the meantime Manchester had 20,000 homeless and 68,000 houses unfit for habitation. We argued that given the land famine—sites available for only a tenth of the housing needed—the city should immediately build upward, not outward. We envisaged six-story housing in attractive landscaping, emphasizing that they must not be designed as soulless barrack blocks. I had my doubts whether the city had the imagination to achieve that, but it was a better bet than waiting for unfit homes to collapse. They were already falling down at the rate of two a day.
The slums were a legacy of Manchester's leadership of the industrial revolution. Benjamin Disraeli, prime minister and novelist, said in 1844, in his novel _Coningsby,_ "What Art was to the ancient world, Science is to the modern.... Rightly understood, Manchester is as great a human exploit as Athens." It was true. The invention of the world's textile industry in and around "Cottonopolis" (as Manchester was called) was only part of it. Whitworth, Fairbairn, Nasmyth, A. V. Roe, and Royce developed revolutionary discoveries in engineering; Ferranti in electronics; John Dalton in chemistry. The first pilots to fly the Atlantic Ocean nonstop were two former Manchester Central High School students, J. W. Alcock and A. W. Brown. It was at Manchester University that Ernest Rutherford knowingly split the nucleus of the atom, Hans Geiger invented his Geiger counter, and William Jevons devised his Logic Piano, the proto-computer that was the first machine that enabled a problem to be solved faster with the machine than without it.
Manchester paid a price for industrial pioneering. Too often we lived the opening page of _Bleak House:_ fog everywhere. No, it was worse. This was not gray fog rolling down the river, misty clouds creeping in from marshes. This was black fog hanging over us day after day, still and suffocating. We went to bed breathing fog and we got up breathing fog and we worked breathing fog. People coughed themselves to death: the black spots spoiling the washing hanging to dry in backyards was the stuff being breathed into lungs.
We called it smog—fog enveloped by coal smoke from a million chimney pots like ours at home, from thousands of factories and coal-fired steam trains. On clear days, just from our house in Hale Lane, Failsworth, adjacent to the rail line, we could count scores of belchers, tall redbrick cotton mill chimneys. It was a double nightmare for Dad; hundreds of passengers' lives were dependent on him seeing the signals. The smog was so dense that in January you could not see them; in fact you could not see a yard ahead. It was not unknown for men walking home from work to fall into canals and drown. The only warning you were about to collide with some other struggler was the cough, mysteriously disembodied.
The London newspapers had chosen to call the terrible December 1952 fogs "Pea Soupers." In the subs room we had favored "Darkness at Noon" banners until Nifty said, "Enough literary allusion; stick to the facts." (We had our own smog makers. While I was at the paper, I was about the only nonsmoker.) Everyone seemed to accept the smog as an act of God. I'd recognized the fatalism of the working-class communities. "Where there's muck, there's money, Harold," wheezed a bronchitic neighbor. But it was the same among my more or less middle-class colleagues: "Price of progress." Perhaps I'd been reading too much Dickens. I didn't just want to record the smog; I wanted to get rid of it. It pained me to hear Dad coughing all night in the next bedroom on the occasions he came down with bronchitis. One morning, before Big Tom could issue his command, I ventured to suggest that I press for control of both industrial and domestic smoke. "Go to it," he said.
The Manchester City Council had some years before decreed that in the heart of the city nobody could burn coal, only specially treated smokeless briquettes. This had created one of the first smokeless zones in the country, and we had applauded it. But the order was still confined to the limited area around the town hall, leaving most of the city as smoky as ever. National measures lagged, as they had for decades, until some four hundred people died in London. Why did it take deaths in the capital city to spur action when people in the north had been suffering for years? Three years after London's smog, a national clean air bill was making its way through Parliament, ostensibly banning the burning of coal in domestic chimneys. But it had lots of loopholes, exemptions, and long delays before enforcement. It had too few teeth, and Tom let me say so, time and again.
For this campaigning, we got bitten on the ankle by the _Economist_ —until that moment, more or less my bible. We did not understand what we were talking about, they pronounced. Mustn't rush things, you know; think of industry. I got steamed up imagining them in their ventilated London offices. Of course London had had its smog, but day in and day out the workers of the industrial north breathed the most polluted air in the world. They were five times more liable to die of bronchitis than people in country areas. Since the founding doctrine of the _Economist_ in 1843, laissez-faire economics and free trade, had been borrowed in the early nineteenth century from our own liberal-minded "Manchester School," I felt entitled to correct their interpretation. You couldn't go to a concert at the splendid Free Trade Hall without thinking of how, from a temporary wooden hut here in Peter Street, just more than a hundred years ago, Richard Cobden and John Bright had stirred the whole country to win the repeal of the Corn Laws, taxes on food imports which fattened the farmers at the expense of the mill workers. Cobden and Bright had fought for cheap food for workers, not for poisoning them.
Looking back, it is shocking that the clean air advocates had to face so much resistance. In the end we got significant improvements in the Clean Air Act passed in 1956. An empirical investigation into the effects of the act published in 1994 concluded that it markedly improved air quality: average urban concentrations of smoke down by 80 percent and sulfates down by 70 percent since 1960. We won that battle, but the war goes on into the age of global warming, when new interests put up a familiar resistance.
Even before we pursued the clean air campaign, I already shared a primitively biased north vs. south complex: we all thought southerners, apart from Cockneys, were "stuck-up" folk who wore stiff white collars and never got their hands dirty with real work. They were "them," not "us." They were "the beastly bourgeois" of D. H. Lawrence's savage little poem. It was a silly prejudice, but the condescension of the _Economist_ and the torpor of the responsible Tory ministers stirred embers of resentment left by incendiary histories of social reform.
St. Peter's Square, in the center of Manchester, where I frequented the grand circular library, stirred my overactive imagination. Back in the nineteenth century, all the industrial cities—Manchester, Birmingham, Liverpool, Leeds—were denied representation in Parliament. On the hot summer day of August 16, 1819, middle-class families and workers in their Sunday best assembled in St. Peter's Fields to hear speeches calling for universal suffrage (for men). The people were peaceful, and the speakers did not incite violence, but the crowd was so big—fifty thousand crammed the fields—that the magistrates panicked and ordered the arrest of the platform. Several hundred ill-trained amateur cavalrymen of the volunteer Yeomanry force rode into the crowd, wielding their sabers. Eleven citizens were killed, 8 died later, and more than 650 were wounded, a quarter of them women. When Richard Carlile published a firsthand account in his newspaper, _Sherwin's Political Register,_ his complete stock of newspapers and pamphlets was seized, and he was thrown into jail. So was James Wroe, who documented the attack in his _Manchester Observer_ and gave it the name "the Peterloo Massacre." Years later, in editing, I took inspiration from the greatest editor of the _Times_ , Thomas Barnes, who defied the authorities and published an account of the massacre by his reporter John Tyas, who'd been arrested and jailed just for being there taking notes.
For all its heroic history in political and industrial innovation, for all the renown of its Hallé orchestra and its Gaiety Theatre (where British repertory was born), its great university, its art galleries, and its standing as Newspaper City, Manchester was neglectful of its heritage. After my years in proud Durham City, I was dismayed that the brass chamber in which Rutherford first split the nucleus of the atom ended up in the Cavendish Laboratory in Cambridge; that William Jevons's logic machine, made in Manchester at Ferranti, went to Oxford. I'd grown up in the city, I'd seen it come through the Blitz, and I felt it should make more of itself. Perversely, the German bombing had given us a chance to create something special, but typically the great center of Piccadilly Gardens remained nothing but a big, ugly traffic rotary.
One day, dodging cars in St. Ann's Square, which was quite near the office, I had an idea. What a jewel it would be if the geographical heart of the city wasn't so lacerated by traffic, so noisy, so defaced by ugly street clutter. My thought was to return the spacious square to its Georgian tranquility—a pedestrian area like St. Mark's in Venice. There was a splendid focal point at the King Street end in the towered neoclassical St. Ann's Church in pink sandstone, probably built by a pupil of Christopher Wren in 1712. There was J. E. Gregan's graceful corner palazzo with its arches and Venetian windows, and the well-mannered and genially grouped terraces of the Edwardian and Victorian buildings. There were fashionable shops and a bronze statue of Richard Cobden. But one could almost see Cobden shrinking from the racket of heavy trucks, buses, and cars, holding his nose from the exhaust fumes. The Boer War soldiers featured in another statue seemed on the point of surrender.
Back at the office, I sought out Bert Hackett, a pale young man in the art department who had difficulty swallowing, the residue of some accident. He was confined to drawing maps and little sketches, but I was to discover that he had an unexploited imagination for larger tasks. Presented with my vague idea, he swiftly completed a fine sketch of how St. Ann's Square would look without traffic, with silver birch trees, fountains and an ornamental pool, and sidewalk cafés where Mancunians could pause to admire the view and gossip. I presented it to Big Tom. I told him that there was a rear service road where shops could be supplied and, hand on heart, that there would be as many days in Manchester when people could sit outside as there were in northern France.
Tom was so enthusiastic for Manchester that he'd long ago decided it _never_ rained, just as Manchester United never lost a match. He embraced the new St. Ann's Square as an _Evening News_ campaign, giving it a page in the paper headlined "This Could Be Manchester. Picture a Peaceful Plaza in the Sun."
I called on the Civic Trust in London, an independent national philanthropy formed to help regenerate decaying urban areas. The deputy director, a town planner, came north and enthused. Laurence Scott, the chairman of the _Guardian_ , and Bill Mather, head of the great engineering firm Mather and Platt, rallied round, and so did St. Ann's canon, Eric Saxon. We called a public meeting of city officials, property owners, and businessmen. The buzz was good. I had high hopes when the chairman of the city's planning committee reported that the city surveyor found it "an attractive scheme." Then he added, "But the surveyor thinks it should wait until the ring road is finished" (which might be many years). Bill Mather argued that the world's biggest industrial area should have such a showpiece without delay. The objections were fewer than I expected; most of the traders said they would happily cope, but a Mr. R. R. Stoker argued that to deprive people of a place to park their cars threatened life, liberty, and the pursuit of happiness. Instead of a plaza, he advocated "flower baskets, trees with colored lights, and taxis painted pink and green."
St. Ann's is now a pedestrian square. My idea was not exactly an overnight success; it took twenty years. But more came out of the initiative. With Scott and Mather doing the heavy financial lifting, we formed the Civic Trust for the North West to work for a cleaner, more agreeable city. I went around with a photographer identifying areas of remediable squalor. My old stamping ground of Ashton-under-Lyne reversed the slogan "What Manchester does today, the rest of the world does tomorrow." Two months after the St. Ann's meeting, Ashton adopted an attractive design for its market square, where I had sheltered from the rain opposite the _Reporter_ office. With guidance from the national Civic Trust, the borough adopted a plan to turn the area into a traffic-free shopping and social center—and to do it that year.
Long before anyone heard the phrase "green revolution," the initiatives of the Civic Trust caught on. Within a few years, citizens in no fewer than three hundred places were giving their time and money to conserve what was beautiful and pitch out what was not. The cause united Rottingdean with Wigan; craggy folk with snow on their boots in Inveresk with villagers on the Downs; burghers of Bristol with potters in Burslem; men and women in Godalming with those in Glasgow. In Stepney a group turned blitzed sites into gardens. In Lincolnshire and Staffordshire volunteers cleared away from the countryside the debris of pillboxes, Nissen huts, and concrete left over ten years by the biggest litter louts of all—the Air Ministry and War Office.
I was proud to have played a part in all this. Enid and I later moved from our rented apartment in Altrincham to a house on the edge of the wildly beautiful High Peak national park. We formed a society to lead volunteers in the weekend clearing of woods and fields and ghastly water holes filled with three-piece suites, mattresses, broken television sets, prams, and other throwaways of civilization.
I was on the opinion treadmill for nearly three years, then after an interlude I will describe, Big Tom promoted me to feature writing. One day he lingered in my office for a record three minutes. "Where did you get the shirt you're wearing?" he asked to my surprise.
"Lewis's. My mother..."
"Why don't the Swedes wear shirts like that?"
"No taste?"
"And why aren't their women wearing our frocks from Manchester?
"Look what's happening, my boy," Big Tom went on with indignation mounting. "Lancashire cotton exports way down, mills closed. I want you to go to the continent and find out why. Leave right away."
"Of course," I replied, praying my passport was in order. Those were the days when going to "the continent," as we called Europe, was a vast mystery. Beyond a day trip by ferry from Ramsgate to Boulogne as a schoolboy, I'd been out of the country only once.
Tom hadn't finished. "And what about the Norwegians, eh? Whose pajamas do _they_ wear? They could be wearing German pajamas! Think of it, the Germans!"
He seemed ready to fight World War III over pajamas, but exploring the bedroom attire of the Vikings struck me as chancy. "You want me to go to Norway as well as Sweden?" I ventured.
"Yes, yes, Oslo. And while you're about it, swing down through Denmark and come back through Germany and Holland."
By night trains and planes, I was soon deposited in snow-etched Oslo. I called on various offices, starting with the Handelskammer, the city's chamber of commerce. Everyone seemed to speak English, which was fortunate since the phrase books didn't have anything like "I am an Englishman from Manchester. Kindly help us understand why you don't wear our shirts."
The officials I spoke with were very helpful in explaining import and export rules and custom duties, but they had no more idea than I had myself why Lancashire cottons were no longer as popular as they used to be. On my bewildered first night, I nursed a beer at a table looking out on the Volkswagens scudding round the city square. Where on earth was I to begin the next day? This was my first big overseas assignment. All I had now were the names of a few import agents, none in Oslo. A talkative old Norwegian materialized, putting his beer down next to mine. "See, we buy from our enemies and not from Britain, our old friend," he mused. He became uneasy and moved away when I asked him if he would mind turning back the collar of his shirt to see where it was made.
The waitress was more obliging. I asked her where she bought her clothes, tablecloths, and curtains. Next morning I trudged through the snow to the department store she mentioned. It was advertising a clearance sale. I found a rack of poplin raincoats at the bargain price of two pounds, two shillings (around twenty-five pounds today). I checked the label—a Manchester manufacturer. I tried it on. "Anything wrong with this?" I asked a salesman.
"Nothing wrong. It's good quality, and it looks good on you," he said. Flattered, I bought the coat but asked why they were selling our poplin raincoats for so little. "The English style never changes," he explained. "It's out-of-date."
I went to another store and watched various women choose curtain material. It was all German, French, Swiss, Italian, Norwegian—except for two languishing pieces from Lancashire with designs as cheerful as cold porridge: on one a network of the old familiar pallid rosebuds, on another what looked like a sickly green bird in an ocean of turquoise. An import wholesaler explained the popularity of German pajamas: "Your people offer me five colors, the Germans fifty. And on piece goods you fob us off with gaudy leftovers from the colonial trade. You're stuck in the Victorian era."
The refrain became depressingly familiar from agents and store buyers throughout Norway, Sweden, and Denmark: design and service were more important than price. The decor in the homes I was invited to was so different from the oak sideboards, floral curtains, and stuffed sofas of the pre-Conran English homes. My hosts' style of richer woods, ornamental glassware, trailing plants, and angled spotlights demanded fabrics as striking. At twenty-seven, I was feeling out-of-date myself.
When I stopped off in Malmö, Sweden, I finally solved the mystery of the Cottonopolis shirts that had sent me on the journey in the first place. A kindly and effusively pro-British director of two leading men's stores told me how for years he'd asked a quality British shirt company to package its merchandise more attractively. He'd been told it wasn't necessary. "Just recently," he told me, "the Swiss have come in with shirts in an attractive transparent wrapping, folded with a broad front. Just look at them." Had I seen them side by side, I had to admit, I would have bought the Swiss, even though it was a little more expensive. "See," said my new friend, "that's why I have two hundred English shirts I can't sell. Take one, take two. Help yourself."
In Düsseldorf, soon rebuilt after the RAF devastations, I scoured one huge department store after another and discovered something extraordinary: not a stitch of Lancashire cloth anywhere. And I noticed something just as odd: the same German fabrics I'd seen in Copenhagen were more costly here in the home market—and that despite the Danish tariff. The plump export manager of a large German textile company invited me to dinner because his wife, a dazzling beauty, was grateful for being treated kindly by British troops at the end of the war. He sketched a confusing system of high German tariffs and tax relief that basically enabled German manufacturers to charge high prices in their protected market while sending below-cost fabrics abroad. "Dumping" I called it in the third article I wrote.
By this time two of the three "Cotton in Crisis" articles had been published under the rubric "How Not to Sell Cotton." Big Tom was pleased; the cotton kings were not. But a few months later the president of the Board of Trade intervened. He announced that his department and the Cotton Board would send a special commission in my footsteps to investigate all I had reported about design, selling and after-sales service, price, and dumping, to be followed by a "hell for leather drive to take Lancashire cotton out of the doldrums." Big Tom made it a page-one splash: COTTON AWAKES!
Big Tom's next idea was to send me back to Germany to ask how lads from the Greater Manchester district in the Guards Division of the British Army of the Rhine liked life over there. It looked like a boring assignment—until I stumbled on real news, rather as my father had done that morning we came across the Dunkirk survivors on Rhyl beach. I just happened to arrive at the barracks when the officers and grenadiers were boiling over about their obsolete weaponry. They were stuck with old-fashioned bolt-action Lee-Enfield Mark IV rifles (while other NATO armies had the automatic self-loading FN); far too heavy Vickers medium machine guns from World Wars I and II; radios that did not work; and the Sten, the emergency submachine gun that had been introduced to me in my RAF training as having killed as many of our soldiers as the enemy because of a defective safety catch. A tank commander said his radio equipment for keeping track of his squadron was nothing near as good as the Germans' in World War II; the enemy had been able to keep in speaking contact at night over a distance of fifteen miles. "On one recent night maneuver I had to sit on top of a haystack to speak to artillery one hundred yards away," he said. "Amazing we won the war, you know. God knows how we'd cope with the Red Army."
The stir created in Lancashire by the cotton articles was nothing compared to the outcry about the army report. It was cited in a full-scale debate in the Commons and the Lords, and the responsible minister promised swift action—as he put it, "a concentrated phase of renovation and re-equipment such as has never been seen before."
Subsequently, Tom piled on the assignments. Go into the baleful and bleak Communist zone of East Berlin (I did; when I pointed my camera at a Red Army soldier, he summoned me over because he wanted to pose properly); examine the case for equal pay for women; go check on whether the new West German Army was something to be scared of; see what was cooking at our major science research institution, the National Physical Laboratory (the birthplace of radar and much else was starved for resources); examine technical education; explain atomic energy and look at solar power. I was split three ways: foreign correspondent, science correspondent, northern political correspondent. They were all rewarding assignments. In West Berlin it was a joy to call on my liberated prisoner, Walter Greis, back at work in a bank and happy with his wife, Alice, in their new apartment. I had only one question: could they please explain the strange customs in their bathhouses? On the first night, exhausted by the difficult journey, I had gone into a public facility for a massage, wearing a swimsuit and over it a fur and leather astrakhan coat I'd invested in for my Scandinavian travels, intending to use it as a blanket for a little sleep afterward. Inside the bathhouse, the Berliners stared and laughed at me. I was the only one dressed; they were all naked, men and women. I reentered fearlessly, bravely unclad: even the RAF unarmed combat course hadn't prepared me for this. The massage was so good, I went back the next night for a swim, disrobed, and walked in. Cries rang out on every side. Everyone was wearing a bathing suit or dressing gown. Walter explained that my German would have to cope with tricky little notices the bathhouses changed every night. The first night I'd missed a sign that said _Entkleiden Sie sich bitte vor Eingehen_ (Please undress before entering) and the second night _Badeanzuge erforderlich vor dem Zugang heute abend_ (Bathing suits required before entry tonight).
The job of northern political correspondent gave me a front seat at the seaside spectaculars at Blackpool, where the Labour Party—political leaders, humble constituency workers, and trade union rank and file—wrestled for the party's soul in the annual party conference. The Labour government had just been turfed out by the Conservative Party, and the vociferous Marxist left claimed this was because voters had been so bamboozled by the press, they'd failed to understand the issues. This was akin, said Labour leader Hugh Gaitskell, quoting Oscar Wilde, to saying that the play was a success but the audience was a failure. I admired the way Gaitskell stood up to the catcalls of abuse he got for defending the wisdom of the voters and the virtues of the Atlantic alliance and a mixed economy. I thought he represented Labour's best hope (though it took four decades for Tony Blair to convince the party that nationalization was no panacea), but for me the revelation at Blackpool was from Gaitskell's leading critic within the party, Aneurin Bevan.
I saw for the first time how conviction can be suborned by a particular kind of eloquence. The resonant generalities and voices of, say, Winston Churchill and Barack Obama inspire, but that is different from the gifted debater's ability to take an opponent's case apart piece by piece and substitute a glittering alternative. I'd known Bevan could do this from reading his dazzling attacks on the Suez intervention: "Sir Anthony Eden has been pretending that he is now invading Egypt in order to strengthen the United Nations. Every burglar of course could say the same thing, he could argue that he was entering the house in order to train the police." But to see and hear Bevan in full flow was a singular experience. Nobody in the party or press ever risked missing a minute of him. He'd take the expectant crowd through the labyrinths of a policy argument with wit and passion, his index finger quivering in admonition; his silver forelock flopping; his lilting Welsh cadences beautifully calculated between mocking vituperation, intellectual analysis, and beseechment; his face flushed (I learned he soaked in a very hot bath before he spoke).
Big Tom was relentless. Now he suggested that I combine political and parliamentary reporting with organizing our book reviews and cover activities at Manchester University. As an afterthought, he dropped me a note: "I am told you are keen on table tennis. I'd like you to cover the sport for us."
Will that be all, sir? No, now you ask, said Big Tom, "I want you to write a column. Every Thursday. Make it snappy. It will pay a guinea." (The guinea always sounded grander than the one pound, one shilling it represented.) He told the features department that this new assignment should be presented as "The Column of a Manchester Man Who Speaks His Mind" and headed with my name in bold type. In the guise of modesty, I told him I preferred to write under the pen name Mark Antony (ambition is made of sterner stuff). He agreed but told the features editor to have a sketch of me at the top of the column. Artist Bert Hackett drew a tight-lipped, jaw-jutting profile of someone with the kind of steel-rimmed glasses that went out of favor with Heinrich Himmler. But about the column itself, I was given not a clue. It surely had to be personal, but being personal cut across the grain of my training. I'd been schooled in the anonymity of neutral reporting, to eschew any personal note except in theater and book reviews or features. "Never," said J. W. Middlehurst, "let me see the word 'I' in anything you write for the _Ashton-under-Lyne Reporter_."
I found the new cloak hard to wear. My first column was passable, with three or four shorter items, but it was not personal, and it could have been written by anyone. I was rescued by the antic behavior of the nationalized Electricity Board responsible for serving the whole of the North West. During lunch hour, I walked into its showroom in Manchester to buy an electric cooker displayed in the window. It was an older model at a knockdown price. I was so newly married with furniture bought on credit, we couldn't afford a new model. I paid and gave the salesman the address for delivery at our apartment in Altrincham, a suburb of Manchester. He revoked the sale. In no circumstances would he deliver to Altrincham, he explained. It was in the rules. Not giving up, I went to the Altrincham showroom of the same board. They didn't have a cheap model. Would they kindly order it from their colleagues in Manchester? "Oh, dear me, no. We're not on speaking terms with Manchester." I went back to the Manchester showroom. I said I'd buy the cooker and arrange my own pickup and delivery. "We're a state enterprise," said the salesman triumphantly. "We can't deal with a private trucker." He didn't thank me for apologizing for thinking that I was in Manchester, not Moscow.
Recalling the farce, I found a voice—and my grand alias Mark Antony found an audience. It led to a newlywed couple inviting Mark Antony to their brand-new home, where they handed over a bag filled with tiny brown pellets. Crushed between thumb and finger, the pellets formed a small, sandy deposit on my desk. They were the remnants of defective mortar supposed to be holding together the bricks of their house, which was already shifting on its foundations. It turned out to be just one example of a wave of jerry-building. Newspapers and the BBC followed up. Mark Antony would survive. Indeed, he was even invited to speak.
I was admittedly only one of a hundred eager Manchester newspaper writers auditioned for the role of broadcaster. After furious debates, the Conservative government had deemed that the BBC monopoly of television, paid for by the license fees of viewers, should be challenged by a handful of independent companies that would finance their programs on the "dreadful" American model of carrying paid-for advertising. In the north, only just recently reached by the BBC's flickering blue signal, the commercial franchise was won by the irrepressible Sidney Bernstein, a silver-haired smooth talker of creative vitality, who looked like a cross between a Roman emperor and a beaten-up boxer. On his father's beginnings in cinema, Bernstein had built a huge chain of splendid cinemas inspired by what he'd seen in America in the 1920s. He'd sought the northern franchise in preference to London, he said, because he preferred to get away from the London metropolitan atmosphere and move to a more distinctive close-knit culture. With his brother Cecil he set out to make his Granada Television synonymous with the north: he called it Granadaland. His soap _Coronation Street_ introduced northern life to the south and became the longest-running drama on British television—still running some fifty years after its launch.
Granada didn't have a news operation, so it made a deal with the _Manchester Evening News_ to cut and paste our stories and present them as its own Granada news bulletin. None of us knew anything about television, including the Granada executives, who'd come from print, marketing, and the Bernstein brothers' interests. I presumed their inexperience explained why when I'd said all of twenty words to the camera, they invited me to undertake some of their first outside broadcast interviews—a few hours I might take off from the _Manchester Evening News_. Later, the producer, Barrie Heads (also fresh from newspapers), explained to me that I'd just been lucky. Sidney Bernstein had a list of characteristics of performers he loathed—anyone who was bald or bearded or had a foreign accent or wore a metal watch strap, suede shoes, or a bow tie. In a hurry that morning, I'd not bothered tying the bow tie I often wore, hence I escaped the automatic rejection inflicted on others. Barrie told me it was the ambition of several frustrated producers, hoping to end their careers on a high note, to find a middle-aged, bow-tied, and bearded Bulgarian in sandals they could put under a lengthy cast-iron contract.
The laconic Barrie Heads was the supervising producer of my first interview with L. S. Lowry, the reclusive eccentric whose paintings capture the fortitude of the Manchester working class in the toils of the north's urban decay. Lowry was in his seventies when I went to see him in his remote moorland stone house at Mottram-in-Longdendale. His modest home—which had no telephone—was his studio, too, with paintings piled up. I was in awe of him; not so Barrie. As if directing a scene in a TV studio, he had the nerve to tell the great man that a finished painting of moorland hills and valleys lacked something—the familiar Lowry skeletons in drab suits perhaps. A few days later Lowry told Barrie he was quite right: "It needed some folk in it. I've put in a picnic party."
The show itself went surprisingly well for a first effort, mainly because the unassuming Lowry made it easy to ask questions. I did a couple more shows, with Big Tom's agreement, then someone had the bright idea that Granada would do a public service if it introduced traveling gypsies to a wider audience. A gypsy encampment was selected in a siding just below the busy East Lancashire Road linking Manchester and Liverpool. As a program idea, it was hardly earth-shattering, but viewers then were known to switch on just to watch the signal for an idle hour.
The extrovert director assigned for the gypsy show, H. K. Lewenhak, was said to have come from making movies in Hollywood. For a couple of hours of television in the afternoon, he directed me to wander among the caravans with a microphone, earnestly asking various gypsies about their lives. There was no script. We made it up as we went along. I was on camera, but basically rehearsing set positions for the shorter main evening show. One grouping around a caravan's steps, seen by Lewenhak as the climax, was aesthetically unpleasing to him in the run-through. He didn't like the way I'd had to lean over with the microphone to catch what the gypsy chief was saying. "Harold, on the show," Lewenhak advised, "give him the mike with your left hand, then when he gives it back, turn full face to camera two, wrap up with a four-minute summary, smile, count three, then smile and say, 'Good night, viewers, from all of us at Granada.'"
This was Hollywood talking! I'd be ready for my close-up. The gypsy chief had stars in his eyes, too. Under the blaze of the spotlights, he answered the question. I proffered my hand for the mike. He kept it and segued expertly into a litany of grievances. He named one town for its intolerance, then another and another—a gazette of local councils and cruelty the width and breadth of the land. Starting in the south in Surrey, he was working his way north to the Outer Hebrides. Lewenhak's alarmed floor manager was holding up two fingers. I had two minutes to close. I leaned over to retrieve the mike from the chief; he leaned back. I moved forward; he moved farther back. He was clinging to the mike. We struggled for possession. He was still talking as I managed to get a hand on it. These politically correct days, I would be denounced for oppressing a minority. I didn't see it that way; it was _my_ mike. But the chief wouldn't stop talking. Police sirens were now blaring. Trucks and cars, drawn like moths to the spotlights, had stopped to see what was going on. One car ran into the back of another. It was chaos. Lewenhak rolled the credits over the unstoppable talking gypsy and doused the lights. I had no close-up.
It was time to get back to print.
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# Adventures in the Land of Opportunity
I am addicted to print. That is different from being addicted to reading. An addiction to print means that you get your fix by looking at the shapes of letters in type even when the words don't make any sense. I savor the design of letters—the ascenders piercing the skyline, the fugues created by the descenders. On those assignments in Europe for the _Evening News,_ I felt compelled every day to scour the Norwegian, Swedish, and Danish newspapers without understanding a word. It was a guilty pleasure to be relieved of the burden of comprehension. I scan newspaper pages of classified advertisements even when I'm not looking for anything. Today I waste emotional energy nursing grievances about the migraine-inducing type on medicine bottles and the ridiculously emaciated compressed capital letters of credits on DVD boxes. What are they trying to hide?
Early in 1956, in the midst of my quick scan of the first pages of the _Economist,_ I found myself mesmerized by five lines of small type. The longer I looked, the larger they got, expanding into a new world: _the_ new world. British and Commonwealth graduates, the advertisement announced, were offered the opportunity to travel and study in the United States at the expense of a body called the Commonwealth Fund.
So many millions now visit America every year, it's hard to appreciate how magical those few lines were in 1956. Even crossing the Atlantic was an adventure then; nonstop flights and mass air travel were years away. The everyday material pleasures of Americans—their hamburgers and hot dogs, their jeans and gadgets for everything—are now a commonplace in every community in the world, but they were curiosities then. America was at once a vast mystery and an inspiration.
Its universal luster was dimmed in the years of the George W. Bush presidency, but as a schoolboy who'd shivered in an air raid shelter during the Blitz, when England seemed unlikely to survive, I can never forget the America that came to the rescue. Franklin Roosevelt was as large in our imaginations as Winston Churchill. Then, when the war had been won in the West and in the East, I'd seen this same America sustain Western civilization by acts of courage, generosity, and vision. The Americans didn't occupy the freed lands as Stalin did; they created a new liberal world order.
What kind of people were they? In 1933 during the Great Depression, they sang, "We're in the money," when all they had to live on was hope. In America, it seemed, it was permissible to dream. How had they survived so many crises, achieved so much abundance, fostered so many innovations, transformed so many immigrants arriving at Ellis Island with their pathetic bundles into American citizens making a mark in the world? See, there in the line on Ellis Island, Albert Einstein, Bob Hope, Fred Astaire, and Alexander Graham Bell. And there's Irving Berlin, Frank Capra, Enrico Fermi, and Yogi Berra. There's a more nuanced image of America today than mine in 1956, formed by movies and novels. But who with an atom of romance in his soul could not feel the pull of the mythic America? To walk into a small-town diner in a Norman Rockwell painting; to follow Raymond Chandler in a roadster up Sunset Boulevard; to steam down Huck Finn and Jim's Mississippi; to see Faulkner's Yoknapatawpha County, Gatsby's Great Egg, Zane Grey's Wild West, and Damon Runyon's Broadway. Yes, there was heartache in the history and in the literature, in the Okies of Steinbeck's _Grapes of Wrath_ fleeing the Dust Bowl and finding California not such a promised land, and most of all in Gunnar Myrdal's _An American Dilemma_ on discrimination against black people. But how were things now, in the 1950s? Were these Americans as open, as breezily uninhibited, and as welcoming as their GIs? And were their newspapers anywhere near as exciting as they were in the movies?
I responded to the advertisement in the _Economist_ as soon as I got into my office at the _Evening News_. It transpired that the Commonwealth Fund had nothing to do with our far-flung British Commonwealth. It was an American foundation established in 1918 by Anna Harkness with $10 million of the fortune left to her by her husband, Stephen Harkness, who'd discovered a more profitable business than making harnesses in Cleveland, Ohio: he got together with John D. Rockefeller to cofound Standard Oil. Anna had made her Anglophile son Edward president of her foundation, and he saw the Commonwealth Fund Fellowships (later called the Harkness Fellowships) as reciprocating the Rhodes scholarships. The idea was that returning home after seeing the other country firsthand would promote "mutual amity and understanding between Great Britain and the United States."
The Commonwealth Fund seemed mostly to have been seeking genius scientists who would further their research on the cosmos, but three Harkness Fellowships were open to "graduate opinion-forming journalists" who would be required to commit to a research project and promise that when finished they'd not linger in Hollywood.
My heart sank when the Harkness Fellowship application forms arrived. The British selection committee included none other than Sir James Fitzgerald Duff, the fierce warden of the Durham colleges whom I had criticized in the student newspaper _Palatinate_ for banning Durham's ice hockey team from playing in matches behind the Iron Curtain. Could he have taken offense? Could he even have read it, heaven forbid, as an indication that I was soft on communism? I wasn't—after all, it was the Soviet attempt to snuff out West Berlin that had prolonged my time in the RAF—but any suggestion of deviant socialist tendencies was a sure disqualifier for entering the United States.
The fund stipulated that an application had to be accompanied by a substantive proposal for study in the United States, identifying the university and a program of travel. It would be a lot of work. Given the presence of Sir James, I doubted it would be worth the effort. I put the forms aside. Yet as the deadline approached, every story I read about the United States revived my initial excitement. If I succeeded, I'd be in America for the 1956 presidential election!
I dug out the _Palatinate_ editorial; it was much less forthright than I'd thought. Nevertheless, I'd have to overcome any residual hostility on the part of Sir James by presenting the selection committee with a proposal so reeking of responsibility, so central to the future of mankind, that my previous acts of rebellion would be forgiven.
The study idea I hoped would take me across the Atlantic came in a circuitous way from Big Tom. When he vanished for two weeks in my first year on the _Evening News,_ everyone assumed he was following Manchester United in Europe. In fact, he was more cerebrally engaged at a conference in Delhi called by the International Press Institute (IPI), an association of daily newspaper editors from around the world that was to play a critical role in my career. Formed in the early postwar years with support from the Ford and Rockefeller foundations, the IPI had become concerned at distortions in the flow of news that complicated relations between nations and sometimes led to conflict. The institute had commissioned twenty-two foreign correspondents in ten countries to assess the way their native countries were portrayed. I sent off to the IPI headquarters in Zurich for the two resulting reports, "The Flow of News" (1953) and "As Others See Us" (1954). The stereotypes were startling. The IPI had supplied Alex H. Faulkner, U.S. correspondent for the _Daily Telegraph,_ with a four-month collection of reports on Britain from 105 American newspapers. Americans, he found, were being presented with a Britain that was "an inefficient, old-world, rundown country at the end of her tether, a 20th century anachronism with no part to play in the world of atom bombs and jet planes... a chronic panhandler always trying to touch Uncle Sam for an extra dollar."
It didn't sound like home, but that was the point. We had no idea how we looked to others. Surely, I wrote to the Harkness selectors, we ought to analyze how those impressions were formed and fixed, how justified they were, and what influence they might have in the making of foreign policy. I trusted the selectors would see valor in my intention to choose Chicago as my headquarters. The _Chicago Tribune_ under Colonel Robert "Bertie" McCormick was the citadel of midwest isolationism, rivaled only in the pungency of its expression by the city's three-time Anglophobic mayor, William Hale "Big Bill" Thompson, a pal of Al Capone's. He, too, had claimed to be speaking for the Midwest when he said, in the 1930s, that if ever the King of England came to Chicago, he'd punch him in the nose.
I wasn't just being quixotic in choosing Chicago. Its university, founded by Stephen Harkness's workmate, had an entire department concerned with the influence of reading and media, and its chancellor Robert Hutchins had raised the sights of journalism, I thought, when he had chaired a commission on the responsibilities of a free press.
It was a nice surprise to be summoned to an interview at Harkness House in London, but the selection committee presented a terrifying spectacle. The florid Sir James Duff, bushy eyebrows twitching, sat with a convocation of university vice chancellors; Oxbridge college heads; a professor of physics; Geoffrey Crowther, the managing director of the _Economist;_ and a man justly described as one of the founders of the postwar world, Oliver Franks, former ambassador to the United States and then chairman of Lloyds Bank. I got the impression they did not go to bed at night worrying about stereotypes of Britain formed in other countries. Sir James asked me hardly anything; if that _Palatinate_ editorial had made any mark on his consciousness, he didn't show it. Years later, when I met him at a function in Durham, he told me what happened. "You were up against some very formidable competitors—there were long odds against you. I need hardly say I myself had almost to lean over backwards against any semblance of favoring a candidate from my own university."
Apparently a key question in my interview had nothing to do with foreign reporting. It was whether I had met any American professors. I could recall only one visitor from Chicago, who'd talked on society and crime, and I told the selectors, "I think he called himself a criminologist." The idea that crime should merit a whole "ology" to itself and be practiced by a professor from Al Capone's city tickled the selectors, according to Sir James, as if I had deftly epitomized the eccentricities of American academia. So the British committee recommended me for a fellowship on the strength of a witticism I didn't know I'd made.
It is intriguing the way unreflective, transitory moments like this can change the trajectory of a life. It changed mine for good, as a similar one changed the life of the Blackpool boy Alistair Cooke, who'd applied for a Harkness before me. At his interview, straight out of Cambridge in 1932, he happened to look Lord Halifax in the eye. Halifax took this as a sign of self-confidence and on this recommended him to the other selectors, explaining, "What's more he's from the North Country and they're always the best types." I wasn't going to cavil at advertisements for northerners, but I did object when the sponsors in New York sent a document I had to sign as a condition of their accepting the British committee's nomination. The "Conditions of Tenure" were explicit: if I were to become a Harkness Fellow, I would have to be celibate. "Fellowships are vacated by marriage," the rules announced.
Wasn't it, I suggested, carrying New England Puritanism too far? The New York office stood fast. Wives diverted a fellow from the pursuit of knowledge. Also, they had no budget for them. The other two "opinion formers" who got through the eye of the needle that year were Brian Beedham of the _Economist_ and Alastair Burnet of the _Glasgow Herald,_ later editor of the _Economist_ from 1965 to 1974, editor of the _Daily Express_ from 1974 to 1976, and the anchor for Independent Television News for many years. Both bachelors, they had no Harkness dilemmas. (When Burnet married nine months into his two-year tenure, he vacated his fellowship.)
At the _Manchester Evening News,_ newsroom mates thought I should forfeit the Harkness. They were aghast not so much that I should cavalierly abandon a new wife for the ivory tower, but more that I could even think of losing my grip on the greasy pole of newspapers. I didn't intend to do that; I thought I might learn a thing or two from American journalism. But they had a point. Two years was a long time to be absent from an industry where reputations were made or lost in minutes. Indeed, there were moments when I flirted with the idea that a Harkness Fellowship was an open door to a change of career. I was tempted by beguiling images of myself as a history don delving into dusty diaries in a university library, relieved of the clamor of scoops and deadlines. I must have been deranged. In any event I was not prepared to leave Enid teaching biology in a rough Liverpool school while I swanned around the cloisters pretending to be a monk.
The Harkness man in London, Gorley Putt, encouraged me to attempt a transatlantic negotiation. The deadlock was resolved when New York finally said if I would go to America as a bachelor for the first six months, they would waive celibacy as a condition of tenure for the rest of the fellowship. This was on the understanding that I would be responsible for my wife's upkeep when she arrived. Enid accepted the six months of purdah with customary good grace and on September 8, 1956, joined Mum, Dad, and my youngest brother, John, in waving me off with a simulation of cheerfulness as Cunard's RMS _Franconia_ steamed out of Liverpool harbor.
On board I shared a tiny cabin with a white-haired Welshman going back to California for the sun after trying to retire in the rain-sodden hills of his birth and a young Merseysider leaving his seaside town to be a mate on a dredger. We learned to avoid the bearded artist next door, returning to Philadelphia from a scholarship, who was keen to explain the sexual maladjustments leading him to contemplate divorce. We were nine days at sea, most of which, one gale excepting, I spent crouched on deck over a chessboard, combating Carl, a German soldier who had become a naturalized Englishman. It was a rerun of the ebb and flow of World War II until the liner picked up a pilot for the tricky navigation of entry into New York harbor, and we all rushed to the foredeck. We couldn't see a thing for the fog.
This, though, is the only way to arrive in America, as the first Virginians and Pilgrims did. The millions who now come by air every year to JFK or Newark miss the euphoria of landfall as the sea mists dissolve into Lady Liberty and syncopating skyscrapers, and soon enough you are tasting the cosmopolitan street life of the city. P. G. Wodehouse said that arriving in New York was like going to heaven without the bother and expense of dying. I woke the next morning floating on a cumulonimbus cloud, my soul borne up by the glorious sound of Bach's "St. Matthew Passion."
The prosaic explanation for this was that I was in a high-rise apartment on West Twenty-third Street, the guest of a friend of a friend who'd installed loudspeakers under the beds, each wired to the perpetual loop of a tape recorder. Herb Erthein could not live without round-the-clock music, therapy for his days spent manufacturing metal coat hangers for the Seventh Avenue garment trade.
When I wandered round Herb's apartment, I had a measure of how we'd fallen behind America. Herb's high-tech kitchen had a dishwasher, four-slice toaster, deepfreeze, washing machine, and spin dryer. Everyone I knew back home was still washing clothes by hand in the sink and making toast with a fork held over the fire. I'd seen nothing like the giant supermarket on Herb's block, a single warehouse serving everything you could possibly think of to more than twelve hundred families. Back home we shopped for the family at four or five corner shops like my mum's. Years of doing without everything had dulled my appetite for acquisition. Britain's food rationing had only just ended, nine years after the war, with the liberation of the banana. Here I felt like a Visigoth in imperial Rome. Thirty varieties of ice cream! Tectonic layers of steaks! Gallons of orange juice!
All of this, of course, had to be sustained by salesmanship. Away from the supermarket one had to appease news vendors, diner counter cooks, waiters, department store salesmen, and hotel staff. Every purchase I made provoked the same challenge: "What else?" There was no escaping the hard sell. A newlywed friend of Herb's took an apartment high above the Hudson River, cozily safe, he thought, from the hustle of the marketplace. No way. Sign writers for a church got to work on a facing wall across from the apartment, and every night the couple found themselves staring at a spotlit warning: "The wages of sin are death."
It would take more than doomsayers to shake America's satisfaction with itself in the mid-1950s. I'd arrived in the middle of the presidential election campaign in which the eloquent Adlai Stevenson was again challenging Dwight Eisenhower, running for a second term. America was at peace. The Suez crisis had not yet erupted; the cloud of McCarthy's anti-Communist hysteria was lifting; there seemed to be a thaw in the cold war with the new Soviet leader, Nikita Khrushchev, denouncing Stalin's crimes; and Eisenhower himself had ended the Korean War with a veiled nuclear threat.
I could see why the mass of people liked Ike; he was a reassuring figure. Stevenson exuded erudition, Ike goodwill. I saw the effect at a small airport I went to with some friends of Herb's, all wearing ALL THE WAY WITH ADLAI pins. When Ike stepped out of a light plane and waved to us, he was so presidential but so friendly—the most powerful man in the world waving to us, his supposed critics—that we all cheered him.
They were no doubt mindful of the good times. Americans were enjoying a level of prosperity never before seen in the history of the world, splurging their three and fourfold increases in purchasing power on new homes with kitchens like Herb's and longer and longer automobiles with ever more extravagant tail fins. The majority of people in Britain did not have cars or telephones. In America between 1951 and 1956 the number of _two-car_ families doubled. Most homes had a television set, not yet in color, but they had more programs than in Britain, including _The $64,000 Question, Gunsmoke, Wagon Train, I Love Lucy,_ and _The Honeymooners_ (Alice and Ralph Kramden in Bensonhurst).
I found that the brighter western light buoyed the spirits, and the pace of New York was exhilarating to someone from a gray Britain that had yet to boom. The sense of impermanence in Gotham was pervasive—in the helter-skelter erection of new skyscrapers taking the place of the ones built in the previous generation; the number of going-out-of-business sales (all bogus, I learned); the Brooklyn Dodgers deserting Brooklyn for Los Angeles. Something new was invented every minute. It was the decade for the debut of novelties that are still with us—Kentucky Fried Chicken and McDonald's—and others that are not. I miss the carhop—the waitress in roller skates ferrying hamburgers to your car—and the iconic images of Monroe and Brando framed against the night sky as I passed drive-in movie theaters. I miss the diners where the jukeboxes offered Elvis singing "Hound Dog" and the Five Satins doo-wopping "In the Still of the Night."
Of course there was a darker side. I had the familiar angst in New York of trying to reconcile the plenty with the beggars and vagrants; there were ten thousand of them sleeping in the parks, in the screeching subways, in the flophouses, gutters, and sheltered doorways. Alarmed by the gap between those who had and those who had not, Herb's friends met weekly to discuss questions such as what an intellectual should do for a cause he believed in, such as relieving poverty. They were all anxious for me to know that New York wasn't the real America. Wherever I went in the next two years, all over the country, I would be told the real America was somewhere else.
I loved it all the same, but I could not linger scrounging off friends. The agreeable folk at the Commonwealth Fund's splendid headquarters at Seventy-fifth Street and Fifth Avenue put a new $100 bill in my hand and wished me Godspeed to the University of Chicago. Once in Chicago, the quickest way to the university area was the elevated railway. I bought a ticket and said, in the English manner, "Thank you very much." The man behind the grille snarled at me, "Did ya say sumfin, wise guy?"
Was this the real America? No, I was assured by Professor Douglas Waples, my academic adviser at the University of Chicago. The real America was in Gary, Indiana.
The University of Chicago was a study in dissonance: the pseudo-Gothic architecture copied from Oxbridge; the students in sandals and T-shirts sipping Cokes together through sweetheart straws; one of the wealthiest institutions in the country set down in the middle of a slum called Hyde Park. The whole area was in flux, every day receiving hundreds of blacks escaping from the South.
It was from Chicago's South Side that fourteen-year-old Emmett "Bobo" Till set out in August 1955 for a summer visit with family in the Mississippi Delta, and it was to this Chicago that he came home in a pine box, a disfigured corpse, his face battered to a pulp. His mother, Mamie Bradley, insisted that he be returned to Chicago so that she could display "what they did to my boy." The thousands who walked past the open coffin would talk for a long time about the shock of seeing Emmett, whose only offense, if it existed at all, had been to chat up a white female storekeeper (northern version) or to "put his hand on her and make a lewd remark" (Mississippi version). His two killers had only recently been acquitted, on the 166th anniversary of the signing of the Bill of Rights. Clearly, I told myself, I must find out more about the perverted values of the American South.
I found it hard to decelerate from newspaper life. To a mind still subliminally on an edition time sheet, the cleverness in the academic studies on media seemed for the most part to consist of making the obvious obscure. Where was the headline point? Courtesy of the university, I had a room in International House, living with people of fifty different nationalities. I had to save if I was going to keep my Oliver Twist pledge not to ask for more support when my wife arrived. Like Rockefeller, I counted the dimes. In the cafeteria I was charged 35 cents for breakfast—20 cents for cereal with milk and 15 cents for a banana—but I found a grubby supermarket where I could buy ten tiny packets of Kellogg's cereals for 33 cents, enough milk for 10 cents, and a whole bunch of bananas for 15 cents. This meant breakfast for ten days on 58 cents instead of $3.50.
The theory of International House was that by living together as individuals we would be able to help the nations of the world live together more easily. I wondered about this when my neighbor, the engagingly rambunctious thirtyish Swedish novelist and broadcaster Par Radstrom, in Chicago to study "contemporary American culture," made a habit of banging drunkenly on my door at 3:00 a.m. to discuss the ethics of his involvement with American women. Like me, he was a "Harkness bachelor," his journalist wife remaining in Sweden. He had no enthusiasm for my suggestion that a diet of cereal and bananas would solve his hormonal problems.
Par made up for his nocturnal intrusions by giving me a copy of one of his novels, in Swedish, embossed with a vamp's pouting red lips, and introducing me to a jazz hangout he'd found, Club de Lisa. It was open 24/7, boasting, like the Windmill Theatre in London, "We never close." There was nothing fancy about it, no cover charge, just cheap tables jammed together in the darkness and a solitary spot on a small stage for a cabaret. The first night I went there, the place was packed with black people, many of them fresh from the South. Nobody seemed to mind that the only white people in the crowded hall were Par, myself, and the saucy young French novelist Babette Rollins, whom Par also had brought along. Suddenly a dozen Negro matrons appeared on stage in their best dresses and flowered hats, and the room went respectfully quiet as they presented a check to a local charity. I'd thought I'd been living dangerously, but I might have been in an English church. It seems that Club de Lisa doubled as a community center catering to black migrants. It made me more determined to spend time in the South they'd fled.
A sterner test of the principles of international amity by contiguity came on October 29, only days before Americans would decide between Eisenhower and Stevenson. In the months before sailing to America, I'd written editorials in the _Evening News_ on the cascade of events in Egypt: Colonel Gamal Abdel Nasser's military coup in 1952; his arms deals with the Soviet bloc; U.S. secretary of state John Foster Dulles's abrupt withdrawal of funds for Egypt's damming of the Nile at Aswan; Nasser's seizure of the Suez Canal, owned by British and French stockholders, earlier in 1956, throwing down the gauntlet to Churchill's just-elected successor as prime minister, Sir Anthony Eden; the abortive efforts, never entirely sincere, to fashion a diplomatic solution. The _Evening News_ did not trust Nasser having a grip on "our lifeline," and we condemned Egypt's refusal to let Israeli ships through the canal, but we were critical, too, of the "whirling dervish" volatility of Dulles.
The news that Israel had invaded Egypt to destroy commando bases—and that Britain and France were intervening militarily, without consulting Eisenhower—plunged International House into a frenzy of excitement, bewilderment, and dismay. A crowd of us at the dining tables—Israelis, Egyptians, Iranians, Australians, Nigerians, Japanese, Brits, and French—raced downstairs to the television. We yelled abuse at the set when the network abruptly abandoned the United Nations Security Council arguments in full flow, switching unconcerned to the regular programming.
That first night after the invasion I barely slept, with International House becoming a mini-UN. Two scientists from Tel Aviv asked whether Israel was supposed to do nothing when more than a thousand Israelis had been shot or kidnapped by Egyptian raiders since 1950. "And you've killed a lot of Arabs," a couple of Egyptians retorted, albeit calmly enough. An American professor (in an argument prefiguring those over the Anglo-American invasion of Iraq in 2003) suggested that military action would inflame the Arab world without achieving its objectives. The professor asked me to "speak for Britain" in response to an Arab student's charge that the British had secretly put Israel up to the whole thing so that we could regain control of the canal. Were we capable of this perfidy? I hardly thought so.
In the cafeteria the next day, I couldn't get to a table with my tray, accosted by people rushing up with the news that the Royal Air Force had begun bombing Egyptian airfields. Memory has long encouraged me to think I heroically defended the British-French intervention, but a letter Enid found recently expresses a depth of revulsion that memory had muted: "I am on fire with the Suez Crisis. I hope by the time you receive this someone has put Eden in a lunatic asylum. I thoroughly despise Eden, the gentleman turned bully. And he's wrecking the Anglo-American alliance."
International House, already smoldering, was incandescent when the Soviets used the cover of the Suez crisis to crush the Hungarian Revolution. Someone hissed at me, "See—you've sacrificed the Hungarian patriots!" Then Khrushchev threatened to rain rockets on "imperialist" Britain. It was horrible being four thousand miles from home and imagining the worst.
With study impossible, I escaped the hothouse. Alastair Hetherington, a former Harkness Fellow now only a few days into the editor's chair at the _Manchester Guardian,_ telegrammed asking me to assess opinion in the Midwest. I was flattered. As a _Guardian_ correspondent, I was untried. Hetherington was only thirty-six. Did he know what he was doing?
Since Professor Waples had said that the real America was in Gary, Indiana, I went there first. Mammoth cauldrons exhaled flames, lighting the dreary industrial wastes of the world's biggest steel plant. Most of the brawny and highly paid workers were Polish. They were not very welcoming to an English reporter. They'd just seen Władysław Gomułka win their country a measure of freedom from the Soviets, after the summer suppression of an insurrection in Poznan. Now they blamed Britain's Suez adventure for giving the Soviets a cover to do to Poland what they had done to Hungary. A local Polish newspaper suggested that there should be new Nuremberg trials with Eden and Khrushchev in the dock. I was too close to their furnaces for debate on the subject. I made my notes and left.
Purely in the interests of research, I spent time in Chicago's city center bars. My timing was not great. Britain had just had to ask the United States to waive interest on its 1948 loan. In one darkened bar where the patrons were watching the election results on television, my accent caught the attention of a belligerent barstooler: "Look, son, America paid for the drinks in the first two world wars, and this time you've gotta pay for your own."
The _Chicago Tribune,_ renowned for fostering this friendly attitude, was having a field day. There was no escaping where duty lay. I had to venture into Tribune Tower, the _Tribune_ 's imposing neo-Gothic redoubt on Chicago's "Magnificent Mile" (Michigan Avenue), to ask if they would discuss Suez with a representative of perfidious Albion. It was snowing. In the forecourt I genuflected to the American patriot Nathan Hale, unflinching on his pedestal. The Revolutionary War hero had been hanged by the British for spying, declaring, "I only regret that I have but one life to lose for my country." My lesser regret was that I wouldn't get to meet the former owner of the "world's greatest newspaper," Colonel Robert McCormick, six feet four inches of megalomania.
He had been barely a year in his grave, but I hoped to find that his spirit still flourished. McCormick had been more than a bombastic libertarian of the far right. It's inevitable that we pin epithets on public figures; newspapers do it all the time in marshaling the stage armies of the good and the bad, and the most egregious characteristic tends to stick. It may not be untrue, but it obscures complexity. McCormick undoubtedly fulminated against the British Empire, against Woodrow Wilson and the League of Nations, Franklin Roosevelt and the United Nations, the World Court and the Nuremberg trials. He was convinced that all Rhodes scholars came back to the United States to spy for Britain, and he campaigned to have them fingerprinted. His worst excess, in his anxiety to keep America out of war, was to behave like Geoffrey Dawson of the _Times,_ in the thirties suppressing his own correspondent's accurate reports of Hitler's evil ways.
Yet it required more than polemics for the _Tribune_ to become the most widely read full-size morning newspaper in the United States. At its zenith it had a million readers daily and a third more on Sundays, and its influence was not confined to the Chicago area. McCormick was a bold and innovative newspaperman, versed in all its skills. He could take a printing press apart and put it together again. He saw the future of color very early and the possibilities of facsimile transmission, and he backed the _Daily News_ in New York, the first successful American tabloid. He campaigned and investigated with a vigor that makes so many corporate American dailies today taste like cold custard. The diversity of his life, rich in paradox, defies caricature. He was an inventor, explorer, engineer, municipal reformer, civic booster, artilleryman, and athlete. He was an apostle of free enterprise who despised Wall Street. He was ruthless in crushing newspaper rivals but won epic battles for freedom of the press. It was the cranky Colonel who beat the villainous Mayor "Big Bill" Thompson in a libel defense that established the principle that every citizen had a right to criticize the government without fear of prosecution; it was the cranky Colonel who valiantly bankrolled a Minnesota scandal sheet to win the famous Supreme Court case _Near v. Minnesota,_ which gave the American press the vital freedom from prior restraint we did not enjoy in Britain. He was, as Fred Friendly wrote in his study of the case, "the Daddy Warbucks of the First Amendment." The Colonel had standards; duty came before profit. He loved private gossip but would not run a gossip column—keyhole peepers, he called them. He deplored Eleanor Roosevelt but would not print a story about an old affair of hers with another woman, nor a story about a homosexual advance said to have been made to a black railway porter by Undersecretary of State Sumner Welles.
The editorial writers I met in Tribune Tower did not disappoint. They were glad to talk about the Colonel and what it was like to be summoned to his twenty-fourth-floor aerie, where armed guards and German shepherd dogs protected him: "Hizzoner" Big Bill Thompson had once sent goons to rough him up. The Colonel's men were pleased to be able to practice their swordplay on a real live Englishman who would bleed. "Let's face it, Britain is finished," said the chief editorial writer by way of opening pleasantries. I retorted that Britain had just opened the world's first atomic power station; it was one of my last science features for the _Evening News._ "That's nothing. You're desperate for economic power. We don't need it. You've got corrupt unions, lazy bosses; soon you'll be kicked out of your last colonies." In rebuking Britain, Ike had invoked the United Nations and the rule of international law, so I wondered whether they would have second thoughts about condemning the UN. "Nah," the chief came back. "We like Ike, but Ike's like Woodrow Wilson at Versailles: he's the preacher trapped in a bawdy house calling for a glass of lemonade." McCormick would have liked that.
I could have filed for the _Guardian_ on the strength of the anti-British sentiments I'd heard in Gary and Tribune Tower alone, but I didn't. I called up congressmen, editors, churchmen, academics, local Democrats and Republicans; no one I bumped into was safe from interrogation. It was intoxicating to say "I am a correspondent for the _Manchester Guardian_ " and find doors opened, phone calls to powerful people returned within the hour, even invitations to drop in for a coffee. It was my first taste of the different attitude toward the press. And though I didn't find anyone in the Midwest who was uncritical of the British-French intervention, the alligator of isolationism seemed largely confined to the banks of the Chicago River.
It was gratifying to see my report—airmailed in those days—appear as a page lead in the _Guardian,_ bylined "from a special correspondent," and even more satisfying to hear that the special correspondent was denounced by the director of the British Information Services in New York as getting it "entirely wrong." He'd been telling London that Eden had the support of the whole country, including the Midwest, while I had reported the opposite.
Having broken the ice with the _Tribune,_ I insinuated myself into the rival afternoon paper, the _Daily News,_ which had just sent a public official to jail by proving he'd milked public funds of $1.5 million. Running the _Daily News_ was a legendary newsman by the appealing name of "Stuffy" Walters (after a famous second baseman). Working for Walters, newsmen said, was like being pecked to death by a duck. He was shaped like a beer barrel and spoke in rat-a-tat machine-gun sentences. "Tell it," he told reporters. "Don't write it. Tell it. Period." Jabbing the air with a fat cigar, he gave me a staccato account of how they'd caught the official with his hand in the till. Stuffy was rolling up his sleeves for intensified news warfare with the _Tribune,_ which had bought another afternoon daily, the _Chicago American,_ so he swiftly took me to the newsroom, where the crackle of the police radio was counterpoint to the usual hubbub. "Tell Evans about the funerals!" he instructed a cardboard cutout of what a city editor should look like—crew-cut, bow-tied, under thirty but gray-haired. A reporter had noticed, I learned, that a certain mortician seemed to get all the business when people died intestate. The reporter had spent weeks on the case and concluded that the official entrusted with intestate funds had been authorizing $4,000 funerals, although the ones the reporter had seen carried out were cheap $100 affairs. The pair of grave robbers had been splitting the profits.
The city editor was constantly in the hot seat, but he said it was not as bad as working for United Press International (UPI). In the competitive agency business where every second counted, a UPI man he knew in Raleigh, North Carolina, had been so short-staffed that he'd been forced to have one operator frenziedly punching copy on two teletypes at the same time. His boss at the receiving station in Atlanta telexed, "Hurry. Why so slow?" Raleigh replied, "He only has two hands." Atlanta shot back, "Fire the crippled bastard."
Even the vigorous Chicago papers had their work cut out for them monitoring the Ike vs. Adlai 1956 presidential election. On voting day I went with a reporter to various polling stations in the tough, largely black area to the north of the university. At one station, normally a barbershop, we were not welcomed by the Democratic precinct captain, a nasty piece of work. We caught him bullying a middle-aged woman. "See here," he shouted, "I know your crowd's goin' and blastin' about what Jack Wilcox did and did not do last time. I've changed my name—legally!—because of your mudslinging, see!" The woman was one of three election judges charged, among much else, with watching the voting. It was not unknown for someone to slip into a voting booth and set the machine the way his party wanted it so the next person voted Democrat or Republican, like it or lump it. Apparently the captain, a master of such arts, was lambasting the woman for her insistence that one of the booths be moved into the line of sight.
The voting machines were too complicated for many. One perplexed woman was ten minutes in the booth. A judge went in and voted for her. The next day at the _Daily News,_ Stuffy vented. The rival paper had a better story—a page of pictures showing a Democratic captain in the act of passing dollars to voters. Stories of fraud and intimidation by Democratic operatives ran for several days. (One can hardly say fifty years have purified the electoral system.)
The smell of printer's ink had been seductive. After two months of thrashing around on my project, I was depressed. In Theodore Roosevelt's phrase, I found relating the abstractions of mass media theory to my concern with stereotypes like trying to nail jelly to a wall. Fortunately, my truancy pointed a way out of the morass. Suez! It had been so obvious I hadn't seen it. Here was a concrete, finite event in foreign relations that I could put under the microscope. Britain and America were at the center of it, so any stereotype, any bias one way or another, would surely become clear. And from this empirical study maybe I could construct a model for assessing press performance generally.
The truth about Suez, I recognized, might take decades to emerge. But we could expect the press at least to record the contemporary raw material of the crisis. I'd heard numerous disputes about who did what to whom and when. Did the press report the public statements fairly, fully, and accurately? Did it publish rebuttals as well as allegations? Did it publish speculation as fact? Did a newspaper's opinion page color its reporting? Just how much of an understanding did a reader get of what was going on?
Getting to work, I collected three weeks' worth of eleven publications covering the crisis—eight newspapers (independent and chain) representing different environments and three newsmagazines.
I drew up a checklist of verifiable and freely available facts. I also noted wherever unsourced, pejorative, nonfactual color was introduced ("he arrogantly refused"). It was a considerable task, and awkward in my cramped room at International House.
I did not, in fact, finish the thesis elaborating a test for bias until I had steamed back across the Atlantic—which was perhaps just as well in view of the critical nature of the findings. Surprisingly few facts made it into print, but performance varied in unpredictable ways. The _Chicago Tribune_ would have been expected to score poorly in the factual reporting of the British-French arguments. Not so. It performed better than any other newspaper or magazine in the study. It did have a lot of color words and unsourced material, but the _Tribune_ 's animosity toward Britain did not affect its news coverage; it gave readers enough facts to make their own judgments. In the British press, however, patriotism made it a risk even to raise a question about the Suez venture. Alastair Hetherington's _Guardian_ and David Astor's _Observer_ suffered heavy losses in readers and advertising for opposing "Eden's war."
All American newspaper coverage, though inadequate, was fuller and straighter than that in the newsmagazines— _Time, Newsweek,_ and to a lesser extent _U.S. News & World Report_. All three magazines offered a confusing mixture of fact, supposition, distortion of chronology, and angled writing. Fact and opinion were so mixed that the casual reader would have no idea which was which. _Time_ was easily the most adulterated—twice the amount of nonfactual material compared to _Newsweek_ (344 entries to 154). Here is _Time,_ for instance, on Eden's speech in the Commons: "When he had finished the House was chill with silence." But _Newsweek_ reported, "His fellow Conservatives, including an enthusiastic Sir Winston Churchill, responded with a three minute ovation, probably the loudest of Eden's career." _Newsweek_ was accurate. For a reader who hadn't the time or inclination to read the newspapers during the crisis, _U.S. News & World Report_ was the erratic best for gaining some unbiased appreciation of the crisis.
Whatever misgivings my younger self had about American journalism as manifest in the reporting of the Suez crisis and Joseph McCarthy's witch hunts, a seminal influence on me was the way American newspapers consistently engaged in time-consuming investigations of a kind virtually unheard of then in Britain. In this time before satellite transmissions, the United States was too vast for the distribution of remotely printed national newspapers, so the regionals and locals had a chance to star. While I was traveling in the West, I was mightily impressed by the dogged courage of two reporters in their thirties at Portland's _Oregonian,_ Wallace Turner and William Lambert, who busted a conspiracy to control a vice empire in the city. Behind it were officials of the International Brotherhood of Teamsters and a compliant district attorney. The two reporters had seventy hours of incriminating tape recordings of gangsters, obtained from an ex-con who broke with the mob and subsequently was threatened. The suspect nature of the source meant that Lambert and Turner had to double-check everything on the tapes—three months of risky work once Portland's underworld got wind of what they were doing. They stayed in hotel rooms, moving often; switched rental cars almost every day; and stored the tapes in a bank vault. The management of the _Oregonian_ showed courage, too. The Teamsters were sure the paper wouldn't expose the racket because their union had threatened to disrupt production (a tactic I was later to become all too familiar with).
The work of a number of other investigative reporters ended up central to the U.S. Senate's McClellan Committee investigation of corruption led by chief counsel Robert Kennedy. Among those reporters were Clark Mollenhoff at the _Des Moines Register and Tribune_ on the trail of Jimmy Hoffa; Harold Breslin at the _Scrantonian_ in Pennsylvania; and John Seigenthaler at the _Nashville Tennessean_. It's painful to recall now how I watched the two fated Kennedy brothers, Bobby and Jack, brimful of eager life as they sat side by side in confronting day after day the dregs of American society.
We liked to think that British public life was less stained by corruption, and by and large that was true, but the big national newspapers at home were profoundly uninterested in grassroots journalism. The quality newspapers preferred the rarefied air of Whitehall and Westminster (where so many "scoops" were partisan leaks); the popular press was more or less confined to reporting sex scandals. It wasn't just that the British laws on libel, contempt, and official information were more onerous. It was, I came to see, the difference between two cultures: a British population conditioned to limited access; the Americans demanding openness.
Pretty well everywhere in the United States the status of editors was high in their communities. No doubt this was partly because the role of the press is honored in the First Amendment to the Constitution; partly because local ownership tends to boosterism, much appreciated by all being boosted; and certainly because the better newspapers were vigilant in exposing abuse (except in nearly all of the South, another story altogether). Of course, in visiting forty states I passed through many a Gopher Prairie ill served by a slovenly monopoly sheet, met many Babbitts as editors, and discovered that much of the work on the average Main Street paper would drive anyone to drink. I had modified raptures, too, about the use of freedom of the press when people were accused of crimes. In San Francisco, where I lived for a few carefree months, I was shocked when both the _Chronicle_ and _Examiner_ effectively convicted an innocent man of serial sex killings before his trial. Day after day they branded this poor fellow, name of Rexinger, announcing with glee, for example, that they'd come across love poems he'd written. Love poems! Must be guilty! But the police and the papers had the wrong man. It was a shameful episode. In Britain editors faced jail for reporting on crimes that prejudiced a fair trial. The Rexinger case was no better than a southern lynching.
Along with the high regard I developed for newspapers' dedication to good firsthand reporting on local and domestic issues, I questioned foreign coverage even before finishing the Suez study. The trauma of Senator Joseph McCarthy's attacks on the press earlier in the 1950s seemed to have made editors nervous on anything to do with the cold war. The coverage of McCarthy by most newspapers, wire services, and radio stations had been inept and timid. I was bothered that throughout the 1956 election, the press was in lockstep in letting Eisenhower and Nixon brush aside Stevenson's unanswerable case for stopping the poisoning of the air due to the continued testing of atomic bombs in the atmosphere. "Catastrophic," Nixon called it, falsely suggesting that ending the tests would leave America defenseless. Eisenhower, too, was allowed by the press to distort Stevenson's proposal. We learned later that he actually favored the idea, and he decreed a test ban only the following year.
The Suez turmoil had died down by the time I said goodbye to International House and freezing Chicago in February 1957 and began the travels expected of a Harkness Fellow. I headed first for the warmer climes of Raleigh, North Carolina, where Enid joined me. All my scrimping had enabled me to save for a car and camping gear. I would have to bear the humiliation of not driving one of Detroit's latest models, but my 1953 cream and chocolate Plymouth (without a prehensile fin) was special, a gift of history. I had taken it to a dingy space below the elevated railway where a shaky man in his seventies, with four cats, a couple of dogs, and a barefoot wife, had gone to work on it. In the 1920s he had owned a big factory doing specialized bodywork until the Great Depression had put him out of business. He'd survived all these years by fixing baby carriages and bicycles, but he could still summon his old bodywork skills. With hinges and bolts and upholstery, he made it possible and easy for me to join the front and back seats to form a double bed (cost: $62). "What else?" I paid another $7 for two copper gauge screens to keep out mosquitoes on sleepovers in national parks. It was the best investment of the entire expedition.
Eisenhower's interstate superhighways were as yet unbuilt, and with Enid as navigator we stayed off the main arteries, such as they were ("Get your kicks on Route 66"), and explored the quiet blue highways. I won't forget the thrill—absurd as it sounds today—of my first motel stay one night when we couldn't find anywhere to camp. To a Brit reared in the war, the motel was the pinnacle of romantic luxury: a television, a telephone, free bedside tissues the colors of the rainbow, and a toilet seat sanctified by a strip of paper like a Good Housekeeping seal of approval. What more could puritan American civilization offer?
I was keen to see something of the West, the America of legend. We joined cowboys on a roundup of cattle on a Montana ranch. We shared an improbable tea in Fort Sill, Oklahoma, with Mr. and Mrs. Jason Betzinez—improbable because Jason was the last surviving member of Geronimo's Apache band, and his wife was a former missionary teacher. He was ninety-three. As a prisoner of war and later an army scout, he had been taught the blacksmith's trade, but on his release from army service he was too proud to accept the government's gift of a forge and the tools of his trade.
In the eastern highlands of the Sierra Nevada, we trekked through blinding dust storms to the wild gold-mining camp of Bodie, California, now a ghost town of tumbledown saloons, storefronts, and a church, long empty of human life. We stayed among the Cherokee Indians in the Great Smokies of North Carolina, where mothers carried babies in cradleboards, the Hopis in their cliff-top dwellings in New Mexico, and the Navajos in their hogans at Bluff, Arizona.
Aware of why so many blacks in Chicago had left the South, I was keen to see how this other minority was faring. The Navajos, herding their sheep and goats and gambling and trading, seemed happy enough, but they were undernourished, with a high incidence of tuberculosis and illiteracy (only three in one hundred could read or write). Uranium and vanadium had just been found on their reservation, however, portending big royalties for the Tribal Council, and an active Catholic mission ran two schools and a clinic. The Southern Cheyennes we visited in Hammon, Oklahoma, had nothing so promising on their horizons. The ten bands of the Cheyenne nation were once among the fiercest tribes of the Great Plains, but all the families we saw depended on government handouts of food. They were living in leaky wooden sheds with piles of rags on the roof to keep out the rain, no electricity, no gas, and no running water. The women did their cooking over open fires on the ground. A Cheyenne proverb had a prophetic ring: "A nation is not conquered until the hearts of its women are on the ground. Then it is finished, no matter how brave its warriors or how strong their weapons." A chiefly looking man of eighty-three sat on an upturned bucket, staring into space, oblivious of the drops of rain falling on his head.
Big Tom, always looking for feature articles, read somewhere that you could put all the people of the world in the Grand Canyon and roof it over. It wasn't clear what he had in mind, but when we saw the huge hole a mile deep and four to eighteen miles wide, the idea seemed feasible. It was a fine day in June when we set out to walk from the North Rim to Phantom Ranch at the bottom. We were practiced hill walkers in England, even having backpacked the entire two-hundred-mile Pennine Way from the Cheviot Hills to the Yorkshire dales.
Fourteen miles downhill was a small thing to attempt for one of the world's most spectacular sights, each mile of the descent revealing the chasm's geological history through the millennia. Eight miles down, in the cactus desert, we found Bright Angel Creek roaring across the trail, its footbridge a wreck of broken planks and wire. We were stuck. Eight miles was a long climb back up. It was hot. Swimming was out of the question: Enid had never learned how. The swollen creek was about twenty feet wide. We tried a running jump to get as far across as possible. The rushing water knocked us both down and wet the backpacks, but we got up on the other side and triumphantly resumed the trail to the ranch.
We didn't know there were five more crossings, all destroyed by the melting of exceptionally high snows on the North Rim, and not even a hint of a bridge left on any of them. At the next crossing, the creek raced even faster between rock faces. Looking round for a wider, slower course, I interrupted the sunbathing of a rattlesnake. It hissed. I ran and stumbled on a long strand of telephone wire, which gave me an idea. My plan was to tie the wire to a tree, take both backpacks, and somehow get across the creek, trailing the wire so that Enid could hold on to it for her crossing. I jumped from a ledge jutting out over the creek. Once I hit the water, I was swept along but was able to scramble onto a rock, still holding the lifesaving wire. I dumped the backpacks and yelled for Enid to come over. Waist-deep in water and holding the wire, she made the first few yards all right. Then the current swept her up so she was stretched horizontally downstream, and I was straining every sinew to hold her by the wire. I had made a mistake: I should have tied the cable to a tree on my side. Enid might have inched along if the wire had been taut. I could not keep my grip. She vanished.
I went after Enid in the same instant, reached her, and got her head out of the water, more by panic strength than skill. We ended up far downstream, very wet and frightened—and on the wrong side of the torrent, with our backpacks of food, matches, maps, and flashlight on the other side of the creek.
We couldn't go forward; we couldn't go back. So we went sideways up and along the tops of the cliffs. In two hours we had gone upstream a few hundred yards, but we'd had to climb hundreds of feet, and the enterprise looked more and more foolhardy. Gaps opened along the cliffs. The rock face crumbled. We negotiated a retreat to return to the creek and try again to ford. Without a wire, we were both tossed about like twigs, once more dumped on the wrong side. The spires and buttes of the canyon passed into shadows. Night fell too soon. We made a shelter of logs and leaves, aware of every stirring in the undergrowth.
At dawn, not having been devoured by mountain lions, we spent eight hours inching along the canyon walls, then came to a full stop where the cliff wall rose a sheer, sharp, vertical one thousand feet from the torrents. I'd have to reach the ranch alone and get help. I found a high spot with an overhang and took a running jump. I went under in the torrent. I surfaced and got tossed around trying to swim. I wasn't in charge, but I'd got far enough across to make it to the other side, albeit with a cracked shinbone. I had four miles to go in the heat and four more crossings to make.
The last two crossings were wider but easier, and in late afternoon I reached Phantom Ranch. Within a few minutes ex–bronco buster Jay and kitchen hand Ray, just out of the U.S. Marines, were off up the canyon. They took two sturdy mules. The mules lurched and lunged in the rapids, but Jay—who'd made a living riding outlaw horses and Brahma bulls in rodeos—drove the mules across. At dusk Enid returned in the saddle of one of them.
We recovered quickly enough at Phantom Ranch, but to go back up to the North Rim and retrieve our backpacks on the way was considered impossible. We had to climb to the South Rim (much easier) and hitch a ride for the two-hundred-mile drive to the North Rim and our Plymouth.
The adventure in the Grand Canyon was scary, but what followed was the darkest experience of my time in America. I approached the states of the Deep South with foreboding. There were ghettos in the North, as I'd seen in Chicago, and there was discrimination for sure. The white northerner, if he had an opinion, could be indignant about the way black sharecroppers in the South were still treated almost like slaves, but he hardly thought about the black man with a PhD who could get a job only as a waiter and was excluded from buying a home in certain residential districts in the North. Even so, racism was of a different order in the South. Since the end of World War II, scores of Americans had been murdered simply because they were black; hundreds more had been maimed, thousands abused, and millions deprived of basic rights.
The America that elected Barack Obama in 2008 is a very different place from the America of the 1950s and 1960s, but only those who lived through those years can fully appreciate how dramatic the transformation has been. The full fury of what African Americans endured has faded in the popular imagination, and with it an appreciation of how extraordinary it was that the rights they achieved were largely of their own making. They had no new revolutionary doctrines, only the old ones enshrined in the Constitution and the Bible and a dedication to peaceful change. Most of the people I met on my travels expected that reform would have to be instigated and enacted by the white community. Almost no one anticipated that blacks would take the lead themselves.
Zigzagging my way south, I spent a few days in Kentucky at Barry Bingham's _Louisville Courier-Journal,_ which by cool, positive reporting and quiet editorials had coached the city so well on school desegregation that all fifty-four public schools opened to black children without incident only two years after _Brown v. Board of Education,_ the 1954 Supreme Court ruling ending segregated (and inferior) schooling. It was an inspiring example of what a newspaper could do with reporting and advocacy. It was cold water in my face to stop off farther south in the sullen town of Clinton, in the Cumberland Mountains of eastern Tennessee, where the influence of a decent enough paper, the weekly _Clinton Courier,_ had not prevailed against baser passions. An outside agitator had gone about showing a picture of a black man kissing a white woman, and a riot had ensued. The National Guard had restored order, but I didn't like the looks of the knots of young toughs hanging about in leather jackets and jeans or the thin-faced mountain people who sat unsmiling in a pinball diner; they stared hard at me when I stopped for a coffee.
I was relieved to leave the ferment in Clinton (where the high school was blown up a year after my visit). I found an antidote to the poisons in the resolute manner in which the professors at Tuskegee Institute (now University) in Alabama faced the indignities inflicted on them. "It's not conducive to confidence in the system," said a doctor with the irony of the long-patient, "when I come back here from my home in Atlanta, there's not a single place where I am allowed to stop or eat or go to the restroom. That's one hundred fifty miles of willed restraint." Whenever I could, I talked to black people of all kinds about their experiences. They were usually uneasy talking to me in a public place: I might be compromising them if I instinctively put out my hand for a handshake. Why should they trust a stranger? And how could I, a privileged white foreigner, possibly understand what it was like to be an ordinary black man relegated to the most menial work and degraded services not just in schools but also in restrooms, parks, waiting rooms, elevators, bowling alleys, bars, cinemas, restaurants, beauty parlors, hospitals, professional organizations, trains, buses, and hotels? When I mentioned this to a white reporter in Mississippi, his response was "How'd you like to be one of eight thousand whites in Holmes County among twenty-four thousand blacks?"
At Morehouse College in Atlanta, I talked about newspaper responsibilities with the mentor of the newly emergent Martin Luther King Jr.—the scholar and humanitarian Benjamin Mays. Editors, he told me, had an inescapable duty to be sensitive to the wrongs and injustices perpetrated against blacks and to strive by faith and reason to close the gap between America's ideals and its practice. Only a handful of newspaper editors anywhere were doing that. The watchwords of liberalism were "moderation" and "patience," as if the denial of the fundamental right to vote could wait another century or two. In the North the interest was sporadic even among liberal organs such as the _New York Times, Newsweek, Time,_ and the _Washington Post_. A sensational crime like the Till case or, earlier, the brutal blinding of World War II veteran Isaac Woodard Jr. in South Carolina would make its way into the headlines with strong editorials. Then the case would be allowed to fade; the reporting (not controlled by the editorial page editor) would diminish or vanish, as would the editorial pressure on local leaders, business, and Washington's lawmakers and bureaucrats.
Mays's words were the origin of a conviction I was to carry with me into my editorship: it may not be enough to print the truth once. Amnesia is a characteristic of all newspapers; it is natural. They have to move on to the next day's story and the next. It's an effort to keep connecting the dots and worthwhile only when the dots look like they're adding up to a significant picture.
In the South editors such as Ralph McGill at the big _Atlanta Constitution,_ Hodding Carter II at the _Delta Democrat Times_ in Greenville, Mississippi, and Harry Ashmore at the _Arkansas Gazette_ were pinpricks of light in a dark scene, if not rallying points for reformers. But I was disappointed at the gradualist caution that considered it brave to give a black man the prefix "Mister." At the time I was there, McGill was regarded in the North as the conscience of the South, but he felt that _Brown_ must be obeyed not so much because it was social justice but because nine men had said it was now the law. In that view there was no passion to right a historical wrong. Those newspapers making a moderate, reasoned case for accepting the Supreme Court ruling were speaking out not against Jim Crow laws that relegated blacks to second-class status, but rather against the brutality that went into their enforcement.
That ritual hypocrisy was made clear when I spent time working on the liberal-minded _News & Observer_ in Raleigh, North Carolina. The newspaper was edited by the doughty Jonathan Daniels, a New Dealer and for a short time press secretary to President Harry Truman. Not to be confused with the civil rights activist of the same name who was murdered in 1965, Daniels was a fine writer and open-minded, as were his staff. Tagging along with the paper's columnist Charlie Craven, who could see the funny side of anything, I half-forgot the predicaments of blacks. Then one hot day I stopped to drink at a town water fountain, one marked "White," situated next to one marked "Colored," and looking up I saw that the adjacent statue was dedicated to "Liberty and Equality."
In due course, Ralph McGill came around to despising the "chloroforming myths" of white supremacy, but he was exceptional in his society. Even before he took this stance, most southern editors and their readers regarded him and anyone else who advocated obeying the _Brown_ decision as a radical.
I traveled extensively through the plantation belt of the Deep South, sweltering in the humid one-hundred-degree heat, as big flying beetles hit the porches while I sagged listening to the same expositions on how well the races had been getting on before outsiders started to interfere. The insistent theme was this: "Leave the South alone. We'll solve our 'problem' in our own good time. We understand our 'nigras,' and you don't. The North can't talk about segregation; look at the race riots in Chicago." At lovely dinners given by gracious hosts, I knew it was my job to listen, but I got to the point where I had difficulty restraining myself from protesting as I heard again how kindly "good Negroes" in their towns were treated, how much happier they were here than in the ghettos of the North, or how any attempt to "rush things" would only stir up the "ignorant mob" terrified of miscegenation.
Accustomed to the homogeneous conformities of home, I was psychologically unprepared for the succession of shocks in the divided South—the way the respectable white leadership looked the other way when violence was done to blacks, the fact that all too often the crimes were committed with the acquiescence of law enforcement. Most of all, perhaps, I was tormented by the contradiction that the white middle-class people I met were ostensibly kindly folk of generous impulses. We British thought of ourselves as a generous nation, but compared to American philanthropy, ours was meager. In every town on my American travels, I was aware of middle-class groups organized for benevolence. In the mid-1800s, Alexis de Tocqueville wrote of this American gift for association, and it was just as true a hundred years later. Alas, in the Deep South in the 1950s, I saw the dark side of the gift in community after community: clergymen, shopkeepers, politicians, bankers, auto dealers, lawyers, doctors, and dentists organized for repression through the White Citizens Councils, a movement inspired by a Yale-educated Mississippi circuit judge. The councils declared that they forswore violence and would resist school desegregation by legal means. It sounded like democracy in action, but it wasn't. They were determined to keep blacks down in every respect and especially to stop blacks from voting.
The councils' weapons were not billy clubs but denial of work, credit, supplies, and housing, as well as a boycott of any white newspaper advocating compliance with the Supreme Court on any issue of segregation. By 1957 only 5 percent of Mississippi blacks had been allowed to register to vote. The immediate past president of the American Chamber of Commerce took me around black schools in Jackson to show me how happy the students were, and I asked him about this denial of the vote. "Only isolated cases," he said. The attorney general of the state was more blunt. "Yes, we don't encourage them," he told me in a disquisition on African Americans' inferiority. "Maybe a wrong was done to the babbling natives of Africa brought here, but I am not willing to accept that the race of Negroes can get to the same position in two hundred or three hundred years that yours and mine attained in several thousand years. Would you know, I saw a Negro boy urinating in the street today?"
There were many acts of individual decency among newspaper editors in the South. Eugene Patterson, who succeeded McGill as editor of the _Atlanta Constitution,_ rejected an FBI offer to let the paper catch Martin Luther King Jr. in a compromising position with a woman. That was not surprising for a progressive like Patterson, but even the most rabid segregationist editor in the state refused to touch the story. Rarely, though, was any white brave or imprudent enough to express an outrage proportionate to the outrages committed against blacks. I spent time with such a man in the small town of Petal, Mississippi, across the river from Hattiesburg. His name was P. D. East, and his house had just been firebombed when I visited. East was thirty-three years old, six feet two, and heavy: a rawboned product of the lumber camps, a million miles from the colonnaded pretensions of plantation homes set among their azaleas and Spanish moss. In the army and in civilian life heaving sacks in a store and working on the railroad, he had never thought twice about blacks and segregation before the Supreme Court ruling in _Brown_. He did not think about them much then either. The year before, he had talked himself into starting a weekly newspaper, the _Petal Paper,_ having learned the ropes on a union paper. He worked eighteen hours a day to build a six-page newspaper with two thousand readers and was making money. He never entertained any thoughts of coming out against the mores of the society in which he had been born and raised. And then he found that he could not live with himself.
The way he put it to me was that he just got sick of the daily hypocrisies: a cunning law to stop blacks from voting; the humiliation of black leaders who turned up at a meeting called by whites to discuss the town's schools, then were told they couldn't stay because according to Mississippi law, whites and blacks could not be in the same room together. "One Sunday morning last month," he said, "I was in my office, and I felt that if I didn't say something about what was stuck in my craw, I'd explode." Who can imagine what a nightmare of the soul it was for a man like East to pull himself out of the swamp of a deep-seated prejudice that seemed a natural way of life, and do it in such a way as to hazard his livelihood and even his life? His epiphany took the form of a satirical column comparing the progress in his native state to that of a crawfish. The progressive Hodding Carter II wrote to him from Greenville: "I hope you leave a forwarding address." He stayed, in pursuit, as he put it, of his hobby of self-destruction. Week after week he poked fun at the beasts in the "Magnolia Jungle," reckoning it was a complete waste of time to deliver sermons. He offered membership in the "Better Bigots Bureau," the privileges of which included "freedom to interpret the constitution of the United States to your own personal advantage!... Freedom to yell 'nigger' as much as you please without your conscience bothering you!" A typical send-up announced, "Don't suffer the summer heat by using your regular uniform of muslin bed sheet. Be modern! Inquire about our complete stock of cotton eyelet embroidery; Klanettes may enlarge the holes for the arms but your heads will fit nicely through the eyelets as they are."
His telephone calls would tell him he was a nigger-loving, Jew-loving, Communist son of a bitch. He met everything with humor. While he was stopped at a light in Hattiesburg, a man on the curb said, "Aren't you P. D. East? If you'll get out of the car, I'll mop up the street with you." East replied, "I am sorry, that's not sufficient inducement." The _Petal Paper_ was outrageous—silly, if you like—and effective in killing his business in Mississippi. But to me it was the gold standard.
I'd had a taste of the feudal pathologies at work in the Deep South from reading the novels of William Faulkner (a quiet admirer of the _Petal Paper_ ) and the 1941 classic _The Mind of the South_ by Wilbur Cash. But by the end of my months there, I found myself simply unable to take it any longer.
One day I argued with the belligerent Robert Patterson, the organizing genius of the White Citizens Council movement, who'd been signing up thousands of members all over the South at Kiwanis, Rotary, and Farm Bureau lunches. Sitting in his office in Jackson, I told him that I'd shaken hands with a Negro, and yes, I'd have a Negro to dinner, and no, we didn't discriminate in England (I was too sanguine on that score). He was a big, hulking man; I was intemperate, and he took offense.
It was probably just bad luck, but when Enid and I left Jackson, we had a nasty experience on a lonely mountain road heading into Arkansas. I saw cars coming up fast behind me. The lead car had me in a dazzling spot. I pulled over to let the cars pass, but they pulled over, too, and several men in rough farm clothes got out and approached. "Didya notice that little bitty of a stoplight at the crossing back there?" one asked. Well, I said, I'd seen a blinking amber light, and I'd paused several seconds to make sure both roads were clear. "Yeah, well, round here we say that an amber is a stoplight. Come with us."
We were taken back to a little town and escorted into a dimly lit, bare room, while the gang kicked stones outside. My imagination had been inflamed by all the stories of police brutality I'd been hearing. Were these men even police? "Ya been in Mississippi, right?" said a man who claimed to be the sheriff. "Not from these parts?" I said we were from England. It seemed an age before he absorbed this information, then he asked me for $20, which I gave him. It was a lot of money in 1956—$140 at today's values—but worth it when we were allowed to go. I had a shameful feeling: I was glad I was white.
Which was the real America? The schizophrenic towns of the Deep South, the dour enlightened German city of Milwaukee, the Li'l Abner country in the wooded hillsides of Kentucky, or hedonistic San Francisco? At the end of thousands of miles of travel through forty states, defeated by the immensity of space and the infinite complexities of the people, I decided there was no real America. When pressed on my return home to give an answer, I settled for a small town in the corn-and-hog belt of the Midwest.
Smalltown USA: what is left of it? In one decade exposed as a small-minded hell by Sinclair Lewis's _Main Street,_ in another pickled in nostalgia by Norman Rockwell's _Saturday Evening Post_ covers, more recently satirized in the movies _American Beauty_ and _The Truman Show_ and then claimed as the heartland of Karl Rove's "red state" empire. But in 1956, Smalltown USA was epitomized for me by Paris, Illinois, a township of thirteen thousand people with fifteen churches of fifteen different denominations, all the races thoroughly integrated as Americans. They lived in white clapboard houses overhung with maple and elm trees and fronted by unfenced lawns and pole-perched mailboxes. While at the University of Chicago, before Enid arrived, I drove through the little town square with its sandstone courthouse and tower, and past the Farm Bureau, where men in earmuffs stamped their feet in the cold. Five miles of gravel roads brought me in sight of a corncrib and the sturdy redbrick farmhouse where Ed Gumm and his family had offered to put me up.
It was what I came to see as an act of generosity typical of Smalltown USA, where strangers were welcome. The Gumms had no idea who I was; they had just extended an invitation to any foreign student to spend Thanksgiving with them. The family consisted of Ed, age forty-one, a short sturdy man with sharp blue eyes, of German, Scots-Irish, and Swiss ancestry; his schoolteacher wife, Isabel; and their six-year-old daughter. His grandmother had crossed the Atlantic alone at age twelve. His grandfather had built the twelve-room farmhouse and planted the screen of trees. The Gumms were Christian Scientists. They told me that when Ed broke his collarbone in a fall from a corncrib, he let the fracture heal itself while he read from his Bible.
The temperature was fourteen degrees below freezing. Icicles hung from the long, white front porch. It was snug inside, warm with the smell of beef stew. I looked out on the desolate prairie. A heavy silence pressed down from the sky. Ed predicted snow. It was not hard to imagine the icy, howling wilderness this had been a little more than one hundred years before, when the pioneers had come with their oxen and covered wagons and crude rafts. Then, there were seven-foot-high prairie grasses; swamps and mosquitoes; wolves, panthers, and bears; and marauding Sac Indians, led by the pro-British Black Hawk, finally defeated by the settlers' militia in 1832 in a battle an arrow's flight away from the Gumms' farm. An old tomahawk turned up in the soil while I was there.
I was up at 5:00 a.m. with Ed and his one hired hand, making the rounds of his 480 acres of corn and soybeans. The federal government offered subsidies to farmers like Ed, but he told me, "They can keep their dollars. All those government dollars come out of the pockets of poorer townsfolk paying taxes. And I don't like government officials on my land." That meant turning down $2,000 a year (around $20,000 today). Money like that would quiet the conscience of most men, but Ed, when I pressed him, insisted in his slow, cheerful voice, "A man's substance is what he believes in."
In Paris, Illinois, I felt I was very close to the old Midwest and the men and women who had turned the wilderness into America's larder. When I mingled with people like the Gumms, they seemed to me to have inherited not just the land but also the pioneer virtues that had been lost in the big cities. In the cities the primitive vigor remained, titanic and miraculous, but unaccompanied, I thought, by those other original qualities, such as devoutness, simplicity, patience, deep independence of thought, and neighborliness. If Smalltown USA seems in retrospect like a product of my imagination, individuals like Ed Gumm, Benjamin Mays, Jason Betzinez, and P. D. East were real enough. And people like that were the real America.
I was going back to England conflicted by all I had experienced: exhilarated—outside the Deep South—by the restless optimism; warmed by the breezy, unpretentious friendliness of a society that was more open than my own, with similar values but more vigor; moved by a common unabashed search for redemption in doing good; inspired by what a truly free independent press might achieve; and for the same reason inflamed by its passivity in securing equal protections for the most vulnerable. I admired how editors and reporters had achieved a fruitful status by demonstrating a commitment to their communities, as much as to their corporations, as well as the way they had done it through public-spirited investigation, professional pride, and a sense of decency. Yes, their newspapers were duller than ours, and profligate, too. I envied the big-city editors who had imaginative separate sections for books, science, business, and society, yet the news pages everywhere were slackly edited and designed with rivulets of news on pages leased to supermarkets.
I now longed for a chance to marry the best of American and British journalism. Whether I would get to do so after such a long absence from the battlefront was another matter.
#
# From Delhi to Darlington
One of the most versatile, courageous, and creative editors in the history of newspapers perished on the _Titanic_ on April 15, 1912. William Thomas Stead (1849–1912) was on his way to lecture in America. Men desperate to get into one of the last sixteen lifeboats were held back at gunpoint while Stead, a solitary bearded man in his sixties, sat in the first-class smoking salon, apparently oblivious to the dramas around him or his own mortal peril. One survivor saw him reading his Bible as the ocean flooded in below. Another said that Stead gave his life jacket to someone escaping in the last lifeboat.
At the _Pall Mall Gazette_ Stead had investigated and exposed the evils of child prostitution, sanctioned by Victorian high society, and gone to jail for it. He'd made his name long before that in his very first editorship when he wrote passionate editorials that roused the whole of Europe against Ottoman Empire atrocities in Bulgaria—the holocaust of the nineteenth century. Only in his twenties, he did this from the relative obscurity of the market town of Darlington, in the northeast of England, where for nine years he edited the regional daily newspaper the _Northern Echo_.
As I steamed safely back across the Atlantic at the end of the Harkness Fellowship, I could never have imagined that I would soon assume my own first editorship in Darlington, sitting in Stead's worn leather editorial chair.
Curiously the route to Darlington was to be through Delhi. On my return, I became an assistant editor of the _Manchester Evening News._ Eleven months into my duties, which included editing the paper on Saturdays (but only Saturdays), Big Tom suggested that I might like to step off the treadmill and spend a few weeks in India. He may have indulged me because, as I learned many years later, he had secretly vetoed the intention of Alastair Hetherington to invite me on my return from the Harkness trip to be his assistant editor at the _Manchester Guardian_. "You can take time off to help out Mr. Nehru" was the grandiloquent way Tom released me.
Ten years after independence, Prime Minister Jawaharlal Nehru was exasperated that the Indian press was still stuck in the Victorian mode bequeathed by British imperialism, in touch with officialdom but out of touch with the newly literate masses. The country had 40 million literates in a population of 465 million, but the total circulation of all the newspapers, vernacular and English, was less than 3 million. "I can't reach the people through the newspapers," Nehru told Jim Rose, the visionary first director of the International Press Institute (IPI). One of the founders of the IPI, Rose was passionately concerned with human dignity and freedom. As a high British intelligence officer in the war, he'd protested directly to Winston Churchill that bombing Dresden would be a crime. After the war he became literary editor of the _Observer_. He was a sensitive, graceful man, gifted with the ability to listen with such evident appreciation that the speaker of the most mundane truisms felt elevated in his presence. He followed up on Nehru's remark at once by getting the Rockefeller Foundation to fund a program of technical training at the shirtsleeves level of newsroom, printing shop, and accounting office.
Big Tom, a keen member of the IPI, had suggested that I teach newspaper editing and design for two or three weeks. As soon as Big Tom's recommendation was accepted in Zurich, bundles of newspapers rained down on me from the subcontinent: a score of titles in English and another score in the scripts of Hindi, Punjabi, Tamil, Urdu, Malayalam, Marathi, Bengali, Kannada, Gujarati, Konkani, Oriya, and Assamese. I could see what was wrong with the disorganized layouts and unimaginative photographs, and even wonder at the apparent rambling prolixity of multideck headlines I could only scan. The English-language papers that I could understand had news columns written in treacle and topped by don't-read-this headlines ("Fissiparous Tendencies Remarked in State Government Report").
At the first workshop I conducted with thirty or so Indian editors in Delhi, I urged that news headlines should focus on people and be written in short, simple words with a verb in the active voice. On the blackboard I chalked as a basic example of a good headline the old definition of news: "Man Bites Dog," shorter and superior, I suggested, to "Canine Bitten by Human," whereupon I was denounced from the front row by a silver-haired man in a dhoti. What I had proposed, he flared, would corrupt the Hindi language. It was cheap, nasty sensationalism to have verbs in the active voice and present tense. Furthermore, he knew of no case in India where a man had bitten a dog.
Jim Rose gravely explained that Mr. Evans was attempting to make a point by parody, but then there was another eruption of protest—not at my insensitivity, but at the obduracy of the older man. "We must change!" cried one of the younger editors. "He resists it because he belongs to the old India of the Raj. Yes, 'Man Bites Dog.' Yes! Yes!" From round the room came a rapid-fire declension of "Man Bites Dog" in Gujarati, Bengali, Urdu, Punjabi, and Malayalam. The discussions about the state of Indian journalism, afire with enthusiasm, spilled over from the hot seminar room into the hotel, and they were still going on when I went to bed very anxious. I learned later I worried too much about all the head shaking I'd encountered: for an Indian a head shake is a sign of agreement, not dissent.
It's fair to say that the IPI workshops, of which this was only the first of many over the years, encouraged a revolution in Indian newspapers, broadening their appeal and reinforcing their viability and their capacity to monitor government and business. But by no means was this renaissance inspired only by British and American missionaries, and it was certainly not carried out by them, but by editors such as K. M. Mathew of the _Malayala Manorama_ in Kerala, who put everything he learned into practice and doubled his circulation. His newspaper became the foundation of a media empire of twenty-five publications and various television stations.
Lightning struck the subcontinent most effectively in the form of a chubby-cheeked ebony Asian, Varindra Tarzie Vittachi. He appeared to be as urbane as "the brown sahibs" of the postcolonial era he satirized in his book by that title, but it was merely a convenient mask for passion. Editor of the _Ceylon Observer_ at age thirty-two, he'd fled the country after exposing the role of the government in the incitement of race riots. Jim Rose had persuaded him to be director of the IPI Asian program, in charge of training missions in Southeast Asia, and Tarzie and Jim had made prodigious journeys throughout the region, identifying the new generation's leaders in newspapers and the opportunities open to them for reaching a vast untapped readership at moderate expense. They found, for instance, that 10 percent of the costly newsprint was wasted by bad pressroom practices: there was no concept of copy flow. Only six of India's five hundred dailies used a makeup sheet to instruct the printer on the placement of headlines, stories, and pictures, so the printer just threw the paper together with much confusion and delay. Tarzie demonstrated the utility of the sketched page plan, from which editor and printer could work. For decades afterward, until the arrival of computers, the layout sheet was referred to throughout the subcontinent as a Vittachi.
Tarzie burst into our Delhi discussions with the fire of a revivalist preacher—he was a member of the spiritual brotherhood of Subud—but he was also Rabelaisian, vastly entertained by life, and a champion deflator of pomp. He visited practical, informed vehemence on any journalist unable or unwilling to relate the columns of his newspaper to the hard daily life of the people: children dying from dehydration during the yearly "diarrhea season," families caught in the toils of debt to loan sharks, street traders persecuted by petty officials. For twenty-five years he mocked the paraphernalia of bureaucracy by traveling everywhere with the documents of the fictitious "Republic of Amnesia," bearing, for the health regulators, a stamp of approval by "Dr. Portly Rumbel of the Quarantine Department."
Working with a variety of British, Asian, and American advisers, I sweated design and editing for several years on other periodic ventures into newsrooms in Malaysia, Korea, Japan, and the Philippines. In Manila we wrote headlines about an earthquake even as it arrived to shake our desks. In Davao I joined in designing a discretionary code for reporting racial and religious tensions. Thousands had died in India because newspapers and broadcasters had carelessly publicized rumors, but community tensions had exploded in Northern Ireland and the United States, too.
I learned much from what I saw and from the others in the traveling circus. Amitabha Chowdhury, from the Bengali paper _Jugantar,_ inspired us all in leading what was almost certainly the first-ever professional discussion in India of the ethics and purposes of investigative journalism. He described how he'd got into it when two shy middle-aged clerks in the office of the director of statistics of the West Bengali government came to see him because they were uneasy about the way their boss manipulated statistical reports to serve a political group. Small beer, you might think, but the fake figures that were used to justify a fare increase on the tramways provoked a week's bloodbath in the streets of Calcutta. Chowdhury spent months tracking nepotism and corruption in the department and had no hesitation in fixing blame. "For positive journalism," he said, "there is no role for the neutralist, no scope for timidity in the name of so-called objectivity." Chowdhury made waves.
I remember most the young Serajuddin Hussein, news editor of the Bengali-language daily _Ittefaq_ in Dacca, then East Pakistan. Serajuddin took to heart Chowdhury's mantra that if you stayed with a story, your paper would become a magnet for people with information. A missing child was not a story in Dacca. Serajuddin made it one. Every time he heard of a child vanishing in the busy streets and bazaars, he noted it on his front page and reminded everyone that this was the second, third, fourth, fifth, sixth child that month and none of them had returned home. His persistence revealed that not a handful of children were missing, but scores. He asked the authorities to investigate the possibility that a kidnapping gang was at work. They laughed at him.
A month or so later he went to the authorities with a tip from an informant. Police raided a remote village eighty miles from Dacca and found most of the children, deliberately maimed—some of them blinded—so that they would make pitiable beggars on the city streets. The gang leaders were hanged. Within six months the _Ittefaq_ nearly doubled its circulation. Serajuddin was so proud that he wrote to me and others about his plans for investigating other abuses. When East Pakistan rebelled in 1971, he was among the "intellectuals" sought out and murdered by the Pakistani Army.
When I landed in London after a very long flight from Kuala Lumpur, I almost missed a small item in the _Guardian_ telling its readers that Mr. Harford Thomas, editor of the _Oxford Mail,_ was joining Alastair Hetherington as deputy editor. No successor was named. Could it mean that an editor's chair was going begging in Oxford? But who would I have to beg? Nobody in Oxford. The _Mail_ was the property of the Westminster Press group, owner of a number of provincial newspapers and the _Financial Times_. All the editorships, I discovered, were in the gift of the editorial director in London, one Charles Fenby, whose work I had admired without knowing it was his. _Picture Post_ and _Leader_ were two of the wartime and early-postwar magazines, both dazzling in their different ways ( _Leader_ was more literary). Fenby had been assistant editor of _Picture Post_ from 1940 to 1944 and editor of _Leader_ from 1944 to 1948. He had also helped found the _Oxford Mail_ (he was an Oxford graduate), and with his best friend at Oxford, the future poet laureate Cecil Day-Lewis, he'd compiled _Anatomy of Oxford_ in 1938. He'd been editor in chief of the _Birmingham Gazette_ before becoming editorial director of Westminster Press.
A thin, dry voice I could hardly hear came on the line when I telephoned Fenby. "Ah, yes, the _Oxford Mail_. Applications have been falling on my desk like confetti. What makes you think you could edit the _Oxford Mail_?" I told him. Silence. My pitch had been too long. Clearly while I'd waffled, he'd gone off to read _War and Peace_. I bit my tongue. A small movement of air I interpreted as a sigh eventually struggled along the cable from London to Manchester, followed by words. "I suppose you'd better come down for a little chat if you don't mind what might be a wasted day. Good morning." Click.
Arriving at Newspaper House, Great New Street, a few days later, I found Fenby to be a pale, dome-headed man in his early forties who punctuated his glacial speech with light, mocking laughs. He'd warmed up since I'd phoned. It transpired that my IPI connections had come into play. Jim Rose's beautiful actress wife, Pamela, was the sister of Pat Gibson, who was vice chairman of Westminster Press. Jim had mentioned our adventures in India to Pam, she'd mentioned them to Pat, who'd mentioned them to Charles, who also just happened to be chairman of the British committee of the IPI.
"The _Oxford Mail_ is not for you," said Fenby at once to my disappointment. But he went on, "I've heard about the work you did in India. Rather surprised you didn't tell me about _that_. You'll know our _Northern Echo_ in Darlington from your Durham days, read by the coal miners' families, edited by the great W. T. Stead. You know Stead, I trust?" He then drifted into a rumination as if I weren't there. "Now Mark Barrington-Ward is the editor of the _Echo,_ Balliol man, not too happy in county Durham. Perfect I'd think for Oxford." Pause. "As indeed Evans may be for Darlington. Let me think about it."
So it was that in June 1961, Westminster chess master Fenby designated me as the bishop to move diagonally north from Manchester to edit the _Northern Echo_ and Barrington-Ward as the castle to move directly south to Oxford. "It comes as no surprise," said Big Tom in a generous note about my nine years with him.
It did to me. Finally given the challenge of editorship, I stood astonished at my own pretension. I'd been inordinately ambitious to succeed in all the stages of newspaper life. I was assiduous in learning the crafts. An editorship had always seemed the logical goal, so I had applied, but now that it was about to be realized for something more substantial than the _Empire Flying School Review,_ I had a sudden loss of confidence.
I knew I could do all the nuts and bolts of newspapering, but there was more to editing than a sequence of crafts. It was all very well to carry out the instructions of Big Tom in reporting and editing or to find fault with papers in Asia and America. I didn't really risk much beyond the shame of getting it wrong. As a critic I was not hazarding a whole newspaper. Here I was about to have care of a venerable title, but one beset by competition, its circulation ebbing, headquartered in a small town, staffed by people whose skill I could not guess, whose attitude toward a stranger might be unfriendly, and whose management I didn't know in the way I knew Big Tom and Laurence Scott. I was thirty-two. This was it.
I arrived in Darlington, a pleasant market town, in August 1961, with Enid and our one-year-old daughter, Ruth, who'd been born in Manchester. I touched for good luck a large granite boulder outside the redbrick offices of the _Northern Echo_ in Priestgate. Stead used to tether his pony to the ring on the boulder at his home in Grainey Hill, a village two miles out. I saw the boulder, which had been moved to Darlington in the 1950s, every day when I walked to work, its bronze inscription identifying the granite as a fitting symbol of Stead's indomitable courage and strength of character and proclaiming "His Spirit Still Lives."
Charles Fenby had something approaching reverence for Stead's professionalism—Stead had invented the big-time newspaper interview and the cross heading to break up slabs of text—but he admired Stead most for his crusading journalism. Fenby had seen to it that I read "The Maiden Tribute of Modern Babylon," Stead's exposé of white slave traffic in London. To shock Victorian society into acknowledging what it furtively condoned, in 1885 Stead bought a girl of thirteen for five pounds to expose how the trade and hypocrisy were locked in embrace. The sensation of what he called his "infernal narrative" in the _Pall Mall Gazette_ succeeded in shaming the House of Commons into raising the age of consent for sexual intercourse from thirteen to sixteen, an overdue reform that had been blocked in the Commons after its passage in the Lords. His enemies saw to it that Stead was prosecuted for spoiling the fun. He was jailed for three months for abduction on the trumped-up technicality that although he had the permission of the mother for his symbolic transaction, he hadn't secured that of the absent drunken father.
Enid and I renewed Barrington-Ward's rental agreement for a spacious ground-floor apartment in a solid Victorian house by a park. I was so puffed up at finally being an editor, and one in line of descent from a legend, that the first morning I was due to start work, I marched naked from bedroom to bathroom declaiming, "Here comes the editor!" forgetting that along with the apartment, we'd retained the services of Barrington-Ward's spinster housekeeper, Miss Edith Mullis, who wished me "Good morning, sir," and got on with folding towels. All her life she had been "in service," as she put it, meaning that she had been one of the aproned downstairs ladies facilitating the upstairs life of the landed gentry in the heyday, long gone, of North Yorkshire's fashionable country-house parties.
The _Northern Echo,_ in its antiquated appearance with a fake Gothic title piece, small type, and rambling headline style, resembled one of those faded Edwardian establishments. It had a history and character, but its carriageway was unweeded, and its plumbing gurgled all night.
Reginald Gray, editor for the fourteen years before Barrington-Ward, was a Darlington grammar-school boy who'd been at the paper all his life—chief sub for twenty-two years before reaching the chair. He was a big man, saddened by disappointment; he'd lost a leg in the fighting at Arras in France in 1917 and didn't for a decade return to his field of dreams, the cricket pitch. His left-arm spinners still eviscerated the opposition, but he was reclusive. He mixed little with the staff and was very strict with those he did encounter. He did not encourage visitors to his office, where he read a great deal and wrote book reviews. Reporters dare not show up in Fair Isle pullovers or corduroy pants. "Few of you need to be told, I expect," he told a training conference, "one does not wear a white tie with a dinner jacket or a black tie with tails."
The female secretaries and copy takers invited censure if they wore makeup or earrings. He imposed a lot of unbreakable rules on the newspaper, too. One was that no regional news could appear on the front page. Another was that on the inside pages a place name had to feature in every headline. He was an excellent judge of subeditors and was content to hand over the paper to the chief sub when he left the office in the early evening.
The _Northern Echo_ of his day, and Barrington-Ward's, too, was assiduous in covering an impressive range of local news, but it had not even the whisper of a voice. The editorials were written in London, exclusively on national and foreign affairs; so were the main editorial features (including a canned half page for women). Only London writers were allowed bylines. When statistics were released or someone made a speech, the paper duly reported the adversities facing the basic industries of coal, chemicals, steel, heavy engineering, and shipbuilding, but it took them as a given, an act of nature.
Fenby told me the London editorials and features were not imposed. They were on offer as a service to all the newspapers in the Westminster Press group. I took him at his word; I intended relying less on London. How could I not, sitting every night in Stead's very own chair, confronted by a letter in his copperplate handwriting framed on the opposite wall: "What a glorious opportunity of attacking the devil, isn't it?"
Stead had written the letter to a clergyman friend on being appointed editor in 1871, to ask whether a God-fearing man could edit the Monday morning edition when it meant being in the office later than 7:00 p.m. on Sunday. His contract stipulated that he did not have to work after 9:00 p.m., but evil being no respecter of the clock, he was often still at his desk later than that every night, including Sunday, doing the Lord's work. In those days he conceived that as attacking Benjamin Disraeli and the Tory Party. Stead was a committed member of Gladstone's Liberal Party. I liked the definition by Sir Linton Andrews, editor of the _Yorkshire Post,_ that an editor of a great newspaper was the temporary custodian of a tradition, but much as I admired Stead, I couldn't follow him undeviatingly down this anti-Tory road. I was tired of partisan journalism; it wasn't the opinions I minded, but how they corrupted the news pages.
Of course I knew too well how prewar Tory administrations had neglected the industrial regions, the North West and North East, but five years of Labour government hadn't made an appreciable impact, and in 1961 we had in Harold Macmillan the best Tory prime minister we could have had. "Supermac," as he was called for picking up the broken bits after the Suez crisis, had in 1929 been the MP for Stockton-on-Tees, a key town in my area, when every third man was out of a job. The old Etonian and classics scholar at Balliol affected to be a tweedy Edwardian patrician, but he'd seen the hardships firsthand and understood what it meant for a man to be out of work for years and without the dignity of owning a home. In the 1930s he'd broken ranks to press for social reform, just as he had to defy the policy of Chamberlain and the Tory machine, which he saw as asking Hitler what he wanted and gift wrapping it for him. During my editorship, the conviction born of his Stockton experiences led Macmillan to give jobs and expansion a higher priority than preserving the exchange rate for the pound sterling, the cause of the walkout of his entire treasury team of monetarists. He neatly dismissed the exodus as "a little local difficulty."
So I regarded Supermac as a good bet. If we reported and argued effectively, the region might win a better share of public-sector investment to tackle its twentieth-century devils: a male unemployment rate twice the national average; little new industry; thousands of slum homes without a bath; air and water pollution; schools falling down; landscapes scarred by derelict spoil heaps from exhausted coal mines. My view was that the industrial areas had contributed so much to the nation's wealth—indeed, sacrificed so much—that reparations were due them.
Of course, for all his virtues, Macmillan couldn't by himself regenerate the region. It was psychologically depressed by daily life amid the debris of the industrial revolution and the careless 1930s. It had to be roused from stoicism to strenuous endeavor; it had to inspire more civic service; it had to nurture its own culture: two cathedral cities, two universities, the Northern Sinfonia orchestra, and the creative talents that had somehow flowered in the privations of the pit villages, most notably in the paintings by Norman Cornish, thirty-three years underground, and the novels by Sid Chaplin. I envisioned the _Northern Echo_ as forging the agenda here, yet the paper itself hardly relieved the gloom with a design in typography and layout ideal for Dickensian times. The single-column editorial was sometimes flanked by a black tombstone, an advertisement placed by a local undertaker. Nor could I expect to find eagerness for change among a staff long drilled in working to formulas.
Entering the _Manchester Evening News_ office in 1952 had been like being tossed into the rapids of Bright Angel Creek. Nine years later, entering the _Northern Echo_ office the day Barrington-Ward was to hand it over to me was rather different. For a start, I saw no human life on the editorial floor until I reached the editor's sanctum down a long, dark-green corridor. Inside, Mark bore the cares of the world on his shoulders (a genetic disposition: his father, Robert M. Barrington-Ward, DSO, MC, and Balliol, too, had been editor of the _Times_ from 1941 to 1949). Tall, waistcoated (with a pocket watch on a chain), droll when he wasn't melancholy, charmingly shy for a man of intellect (he took a first at Balliol), Mark was very generous in describing all the problems he'd encountered in his eighteen months in the chair, getting sadder and sadder as he gave me his note to self, listing changes he thought imperative that he'd not yet been able to carry out.
I began to get one idea why after he strode off to catch the train to Oxford and I sat alone in his book-lined office. Editors work by command; there was nobody to command. The first day was emblematic. Nobody came into my office except the secretary I'd inherited, a young Darlington woman called Joan Thomas. (She was to prove a treasure; I knew she recognized my difficulties when she came in one day wearing a Salvation Army bonnet.) She asked brightly if I'd like to dictate. I would have liked to rattle off something on the lines of "Editor's Note to Staff: Where the hell are you all?" Nothing else was on my mind, but never having had the privilege before of dictating to anyone, I wrote a redundant note to Fenby saying I had arrived and seen the granite boulder. I noticed that Miss Thomas's outlines were very good and said so. As a former teacher of Pitman shorthand, I was referring to the clarity of her penciled loops and dashes. These days I would no doubt have been reported and drummed out of office by a sexual harassment tribunal, but Joan Thomas had a sturdy common sense, coupled with knowing where the bodies were buried; she was to save me from many a misstep.
Of course I'd not expected that a daily morning newspaper, going to press around 11:00 p.m., would be anywhere near as busy during the day as an afternoon newspaper like the _Manchester Evening News,_ but my isolation was uncanny; no, it was unnerving. I'd failed to realize quite what it meant to have the main reporting energies stationed outside the office. Darlington was the only town in the country producing a paper with a daily sale (then 100,000) bigger than its population (60,000). Its area of circulation was the largest of any provincial newspaper in England: 120 miles long and up to 60 miles wide, stretching from Berwick-upon-Tweed on the border of England and Scotland in the north to York in the south, and from the east coast to Penrith. This meant the paper was serving readers spread over 5,000 square miles of cities, towns, villages, and hamlets, and nearly 50 reporters were not in Darlington but scattered in 14 branch offices, supplemented by 130 correspondents, 20 of them professionals and the rest part-time amateurs—clergymen, teachers, clerks, owners of newspaper shops, retired policemen.
Even so, I was disconcerted by the absence of any executives. The _Northern Echo_ did not have a news editor. It had a chief reporter, the rather suave, dark-visaged Dick Tarelli, but only half of him: we shared him with the _Despatch,_ the small-circulation Darlington evening newspaper published from the same building by the North of England Newspaper Company, the Westminster Press subsidiary that owned the _Echo_ and two thriving weeklies (the _Durham Advertiser_ and the _Darlington and Stockton Times_ ). The evanescent Tarelli was preoccupied with pleasing the _Despatch_ 's editor, the immaculately tailored Frank Staniforth, a senior figure of uncertain temper in the North of England Newspaper Company whose responsibilities as group managing editor then included administration of the weeklies, the _Despatch,_ and the _Echo_. Tarelli, like the chief photographer and the editor, was accustomed to going home at 6:00 p.m.
The lassitude enveloping the floor was also partly a product of head-office geography. There were separate quarters for reporters, subeditors and photographers, and wire room. In all the daily newspapers I'd known, in Europe and even in India, there'd been a news hub, a big central arena where people could be seen at work to the same clock and you could feel news rippling across the floor, a place for newspaper shoptalk and gossip, a place where directions could be defined, instructions shouted, enthusiasm raised, arguments concentrated, layouts examined, and disputes resolved by crossing a few feet to another desk. If Darlington was the cortex of the _Echo,_ it appeared to be without synapses.
Not until near dusk did some semblance of activity occur. A frail, stooped figure in a cardigan knocked and very timidly advanced into the room. "Would you like to look at hear all sides?" I got it. This was the letters feature "Hear All Sides," destined for that night's editorial page, for which Dick Yeomans was responsible. He wished he'd never asked. He was unlucky that it was my first chance to edit anything.
My pent-up energy fastened obsessively on a letter from a Shildon man who wrote that "like all women," his wife loved a nice cup of tea but had been asked to pay ninepence in a small Lake District café (as near as I can work out, that would be like paying four dollars now). Our outraged correspondent had calculated all the costs of making seventy cups from a quarter-pound packet; added the cost of sugar, milk, and overhead; and concluded that the 100 percent profit was capitalism at its most rapacious. I asked Yeomans if we should check it out with a tea-brewing experiment of our own. Could you really get seventy good cups from the leaves of a quarter pound of tea, or would it taste like dishwater? Should we challenge our readers to do it? Was a small café's overhead as high as the correspondent suggested? Yeomans, with good reason, got paler and paler. Was I joking? Miss Thomas came in to say the subeditors had started to arrive. I settled for a headline, "How Much Profit on a Cup of Tea?" enthusing to Yeomans that his page would ignite a national controversy we could run for weeks. He gently suggested it wouldn't start unless we got that page to press on time. I released him.
While I was fooling around with tea brewing, the world had moved to the boiling point. Khrushchev, working through his East German stooge Walter Ulbricht, was threatening to close the East–West Berlin border, isolating West Berlin. President John F. Kennedy, facing his first test since the ignominy of the Bay of Pigs cock-up, was dispatching Vice President Lyndon Johnson to Berlin; Britain was sending more fighter aircraft to West Germany. All this was chattering out of our four news agency wires as I went down the corridor to the subs' shabby quarters opposite the composing room.
There were about eleven men and one middle-aged woman, their heads down over piles of dimly lit copy. In the absence of the deputy editor—on a perfectly timed holiday—a roly-poly, balding man called Stanley Senior was in charge. He was the chief subeditor, occupied in riffling through bundles of copy from the far-flung reporters phoning in or sending packets by train. It was early in the evening, but already the printing room overseer was pressing a maddening buzzer to indicate he had men at the Linotype machines and no work to give them.
I was disinclined to breathe down the amiable Senior's neck when I hadn't yet met the deputy editor who ran the room. I wanted to see how the pulleys and levers worked before I tried to pull one, but a few days later, on a somnolent Sunday, I had to change my mind. At 5:00 p.m. the chief reporter in the Middlesbrough office phoned in to say his copy would be a little late because of the riots. What riots? Hadn't we heard? On Saturday several thousand people had taken to the streets. Colored people had been attacked. A Pakistani café had been set on fire. There hadn't been anything like it before. Oh, yes, I felt like saying, thanks a lot, and while you were on the phone I forgot to mention that World War III broke out this morning.
One of the characteristics in which I'm deficient as an authority figure is that I don't scare people. To say I'm even-tempered is not a boast but an admission. I should have thrown a fit in the subs' room that night. Tantrums are useful for making people sit up; they tiptoe around the volcano. Big Tom's incipient growl kept everyone on their toes. Rupert Murdoch has only to pick up the phone, and men a continent away genuflect before he utters a word. Some geniuses have had very short tempers—Ben Jonson and Isaac Newton seem to have exploded with ease—and some well-regarded political leaders have had horrible tempers: Truman for one, Eisenhower for another. Senior decently made excuses for the reporter, a taciturn older man, he told me, who did things by the book. We would catch up somehow. Besides, the riots had happened too late on a Saturday for the Sunday newspapers, so we still had the first bite at the story.
The lapse was actually quite useful. I told Senior I'd help out by handling the editing and layout of the story, and please to tell his good friend not simply to file the facts of the night but write five hundred words on the history of relations with colored people in Middlesbrough. I was sensitive on race, having smugly told the segregationists in the Deep South that British people were immune from any kind of racial prejudice.
On that Sunday night, immersed in subbing the story, I woke up with a start. I'd done nothing about Dick Yeomans, and there he was on his way to the composing room to send his page to press. The editorial was another London offering on West Berlin. I told him we'd surely want to replace it with a local comment on the riot and please to hold off for the moment. He came back into the subs' room paler than he had exited: "It's too late." The composing room overseer had refused to accept the request. "I told him it was the editor's wish," said Yeomans wanly.
I went into the composing room. The overseer had already sent the page to the foundry for platemaking. I cut him off at the pass by intervening with the stereo (platemaking) department. I told the bearded young stereo department chief (who belonged to a different union from the overseer) not to make the plate of the page he'd just received because a page with a new editorial was coming. He didn't know what to do. He was trapped. Who was the boss—the new editor or the established composing room overseer? The standoff between overseer and editor got rather tense. I decided to proceed as if all we had was a little local difficulty and rushed back to write a Middlesbrough editorial.
It was as imperative for the _Echo_ to find a voice on issues like this as it was to excel in the reporting. The trouble with my impulse was that I was too busy to write the editorial myself, and the only writer around was a young man so shy he had difficulty getting to the end of a sentence. I had only just met David Spark and had no idea that his prematurely bald head contained a keen analytical intelligence, a rapid-fire writing ability when required, and a profound knowledge of everything about the North East. I should have been reassured by the reaction of Yeomans. He regained his color; he knew that Spark would deliver insightful comment in double-quick time, and so he did. The overseer set the new editorial; the pages were a little bunched going to press on the first edition, but we were not late. The very nimble, thirtyish stone (composing room) hand, one Bill Treslove, made a lightning change on the editorial page, shuffling type slugs like a cardsharp. I designed a simplified front page, half of it given over to the riot.
The facts from the reporters at the scene made it clear the violence in Middlesbrough was only superficially a repeat of the 1958 riot in London's Notting Hill (in which seventy-six whites, mostly youths, were charged with offenses, as were thirty-six "colored" people). Notting Hill was undoubtedly a "race riot" in that both whites and blacks were charged with assaults. Middlesbrough's was more like hooliganism. Drunken youths coming out of the public houses had exploited a Friday night street fracas between an Arab and a white youth to attack anyone of color: West Indians, Pakistanis, Africans, Chinese. The minorities, only three thousand to four thousand in the whole town, had not retaliated; they had run away. So I wrote the simple banner MOBS OUT IN MIDDLESBRO' (instead of RACE RIOT IN MIDDLESBRO'). The editorial endorsed the point, and so later did the police chief and the magistrates.
No sooner had we sent the edition to press than the public houses closed and the drunken mobs came out again. The chastened chief reporter was on the telephone within minutes of a baton charge by the police. We rushed to press again. The simplified layout made it easy to change, and the next day we had a double gratification: all the nationals with earlier deadlines missed the baton charge, and they did not get around to an analysis of race relations in Middlesbrough until a day later. We stayed way ahead of the competition and, I thought, put the riot in the right perspective.
Or did we? A local teacher wrote to differ. He agreed that the riots were the work of louts, but he believed there existed a deep-seated color prejudice—about jobs, about lifestyles, about sex. It smoldered, and it would burst into flames unless we could deal with it. I published his views prominently on the editorial page on the Wednesday—but with another feature on two big North East soap firms fighting it out for a monopoly of dishwashing liquids. Here was the glimmering of the kind of editorial page the _Echo_ should have. For the first time I felt I was editing the paper instead of going through the motions. Charles Fenby sent congratulations. The staff, once roused, had done all I could have asked.
Still, news and photo editing didn't exist, sharing reporters in Darlington didn't serve either the evening or the morning paper very well, the feature pages had no sparkle, and the editorial page had no authority. You would have had to try very hard to diminish an editorial in the way the _Echo_ routinely did, with small type in a single column mixed in with the cinema and theater ads and random display advertising. But if we were to get anywhere, we'd have to redesign not simply the editorial page but the entire paper, as well as rethink the way it was created. We needed a news editor of our own, reporters specifically allocated to the _Echo,_ and a night production editor.
I was eager to discuss all these matters with the deputy editor when he returned from his break. Maurice Wedgewood was a small, dapper man in his forties, with big glasses, a wispy mustache, a words-per-minute rate of utterance almost beyond comprehension, and a conviction that the world was going to hell and that it would happen on his watch; no, it would happen that very night, in fact it would happen right now while he was wasting time giving the new editor a rundown on the night's news. (There was no written news schedule. It was all in Wedgewood's head.)
The structure of his utterances was marvelously complex, with subordinate clause upon subordinate clause; and then having erected the structure, he would take it apart bit by bit with qualifiers, so that what began as an imposing edifice ended up as dust in your ear. To render it into prose is to do violence to a work of art, but here goes:
Considering all the circumstances, and in the light of the fact that the story was in the _Despatch_ last week and picked up also by the _Darlington and Stockton Times,_ I've put the outbreak of foot-and-mouth disease on the front page because I think this thing is going to blow up. Though one never knows. One never knows since the last time we had an outbreak on a pig farm at Barnard Castle, it turned out not to be what it was feared to be. You weren't here at the time of course and can't be expected to remember what happened in Reg Gray's day. He was very keen on this kind of story, said the farmers were the heart of the region and—Goddammit!—we ended up looking very silly. In fact some of the farmers in the area said we had caused needless perturbation because the pig in question only had a cold and Gray got upset though he had no reason to be since he was the one who had told us to watch out for this kind of story. So perhaps you may not think page one a good idea after all and we might be better off running a short paragraph on the front and cross-referring to a page inside, which I think would be prominent enough in the circumstances, don't you?
All by himself, Wedgewood made up in kinetic energy for the inertia that marked the days. He had four editions to see to press every night except Saturday and immersed himself in editing the front page with frenzied concentration. When it neared time for the page to go to press, he got up a head of steam. If the phone rang, he didn't just replace it on its cradle—he slammed it with a force that rattled the tea mugs. Then the sound of the composing room buzzer provoked him to flush, roar, break his pencil in half, and fling it at the door. Nobody ever took any notice. It relieved the tension. One time a damp inside-page proof held for his inspection was a mess in its layout because the stone sub had not followed the plan. Wedgewood raised himself to smash his fist through the page. Senior barely blinked at such outbursts. A day or two later, when Wedgewood was working pell-mell to close the edition, the messenger put proofs on the wrong nail. Senior, sensing Wedgewood's exasperation, wordlessly handed him a pencil so he could break it in triumph.
I was alarmed. Clearly, to get a moment for reflective discussion I'd have to wrestle Wedgewood to the ground. (It was doable; he was smaller.) A few days observing what he did to agency and staff copy won him a reprieve. He was a text editor and judge of news values in the class of Norman (Nifty) Thornton. His assistant, Frank Peters, was, so to speak, just a whisker behind his boss in editing and theatrics, with dramatic Dundreary whiskers, a tartan waistcoat, a thesaurus of oaths from his service days on a Royal Navy cruiser, and a long cigarette holder tilted at the angle made famous by Franklin Roosevelt. He began the night with a quiver of sharpened pencils, all reduced to stubs by the end.
I came to regard the Wedgewood-Peters histrionics as a small price to pay for the way these two hirsute Horatios stood on the bridge every night to intercept the verbiage that got through into so many newspapers. We regularly had fewer pages than our competitors and had to make every inch count. When they were hard-pressed, I gave them a hand. I had an early wager with them both that for every unnecessary word I could save on any story they'd subbed, they would owe me a penny, and I would owe them a shilling for every challenge. I'd have made a fortune on American or Indian newspapers, but very little money changed hands in Darlington.
As I geared up to change almost everything in the time-honored conduct of the paper, starting with the tortuously fussy design, I still didn't quite know how to deal with Wedgewood. It must have been cruel to have been passed over first for Barrington-Ward and then for me. I wanted my deputy to be a partner in the adventure, but in my first months he seemed to regard himself as the custodian of the archaic. I mentioned that I thought place names deadened the headlines and robbed them of universal appeal ("Missing West Hartlepool Boy Found" instead of "Gang of Kidnappers on the Run"). Wedgewood repeated at length all the reasons that place names had been the style for "donkey's years." I came to see he wasn't being obstructive. He could only guess whether I knew what I was doing or just making change for change's sake. And over my shoulder he could see the shadow of his former boss Reg Gray. Wedgewood had a very quick mind, but like any batsman in cricket facing a spin bowler, he could see all the horrible possibilities of any batting stroke. He felt obliged to do justice to the cons as well as the pros of every alteration in the _Echo_ 's normal practice. This took time.
In the end I simply promulgated my first edict as editor: place names are to be dropped from headlines! The editor's memo explained that the main effect of place names was to deter the circle of readers who didn't live there—that is, the vast majority. Place names were henceforth to be included in new small-type bylines that had the virtue of advertising our on-the-spot presence. I waited for the ceiling to fall in. Senior plucked up the courage to whisper that the change would be much welcomed by the subs who had struggled for years to fit the seventeen characters of "Chester-le-Street" into a headline of twelve characters per line. Wedgewood, without a word, faithfully executed the first Evans rule, and not a single reader complained.
If I was developing some hope for the night operation, I had none for the day. There was no morning conference to discuss news and photo assignments and establish priorities. I tried starting one with Tarelli and Charles Westberg, the chief photographer. I had developed a toothache looking at the pictures in the paper. There were innumerable small "grip and grin" photographs of retirement ceremonies, flower shows, well-equipped bulls, empty buildings, and councilors on rostrums. These were fine for the group's weekly papers. They were the kind of pictures for which I'd written a lifetime of captions in my first year at Ashton-under-Lyne. I couldn't believe they represented the changing dramas and personalities of life in a region as vast and diverse as ours. Westberg's five photographers were run off their feet trying to cover all these repetitive functions, to the exclusion of more irregular happenings. I laid it down that we had to be more selective, shunning routine images, seeking out scenes of promise, and always ready to cover breaking news. Westberg got the point, but Tarelli had half his mind on the _Despatch,_ and the news lists from the districts were thin and predictable. The conferences were a flop.
I was flying blind. We were assailed round the clock by competition. We had a few hours' grace over the national dailies printed in Manchester, but to the north, in Newcastle, we had the big morning daily the _Journal;_ to the south the _Yorkshire Post,_ regarded as the leading provincial morning paper; and evening newspapers in Newcastle, Middlesbrough, West Hartlepool, and our own Darlington. In Darlington, moreover, I shared reporters with the _Despatch_. Day after day we'd nothing distinctive planned. In desperation I suggested a sally in consumer journalism. Enid, who knew about these things, had found fruits and vegetables in Darlington expensive by comparison with similar provisions in Manchester. Perhaps the newsroom might check the prices and quality in the region's major towns? "Can't do that," said Tarelli sweetly. "Too busy with the Darlington Show for the _Despatch_."
I was downcast when I walked home through the busy town at midday. I wasn't getting anywhere with the news operation. I vented my frustration on the cavalcade of truck drivers thundering through the center, their exhausts belching black fumes. The Great North Road ran right alongside the elegant esplanade shops on High Row. I felt so impotent in the office that I there and then in the street began a little campaign, pointing at the filthiest high-pipe exhausts and shouting at the drivers to look at how much they were poisoning us. They roared on. It was ridiculous and a wonder I was not arrested as an eccentric, menacing the flow of British commerce. Well, Stead had gone to prison, hadn't he? So what if I got in trouble. Yes, but Stead had used his paper, and that's what I should have been doing. The daily scenes of havoc created by the heavy trucks running through towns and villages reminded me of the campaign for St. Ann's Square, how hard it had been to civilize it, and how very far I was from making the _Echo_ anything like as effective an instrument as Big Tom's _Evening News_ or, for that matter, Barry Bingham's _Louisville Courier-Journal_.
At home for lunch I banged on a bit about what I called "the lorry menace" to my waiting guest, the newspaper's industrial correspondent, Don Evans, as if he were to blame. I knew he was a very sensible fellow when he agreed it was worth looking into why a long-planned bypass of Darlington hadn't come about and why railways weren't carrying the bulk loads for which they were most suited. After all, Darlington was the birthplace of the first railway in the world to run freight and passenger trains: George Stephenson's _Locomotion_ did it from Darlington to Stockton in 1825. _Locomotion_ now graced Darlington's Bank Top station, and I saluted it every time I rushed into the station for Westminster Press conferences in London.
I'd asked Don if he could introduce me to his friend Sid Chaplin, and my clouds lifted when the novelist arrived. He was so warm, so unpretentious, so full of zest for what the _Echo_ might do. I felt a kinship with him. Chaplin had gone underground as a miner but got an education from the Workers' Educational Association (WEA) through Durham University. His portrait of coal-mining life, _The Day of the Sardine,_ spoke to me of my life as a working-class boy, and he was intrigued by my own WEA experiences. Over a jolly lunch we identified the causes and the people who could make things happen. "There's an ache for leadership," said Chaplin. He thought the _Echo_ was better placed to do it than the flashier _Newcastle Journal_. Lunch was exhilarating. By the time it was over, I felt we had already retrained miners displaced by pit closures, grassed over the scores of hideous slag heaps, diverted the heavy lorries, cleaned up the beaches and rivers, purified the air, and capped it all off with a spectacular celebration concert in Durham Cathedral.
I felt emboldened to ask management to end the practice of shared staffing so that we could have reporters of our own and a news editor and night editor reporting to me. I'd be sunk if our sales continued the slide that had begun the year before. (They'd fallen by ten thousand after a price increase.) One of the hardest things for a newspaper to do is to halt a slide.
I began to see Stead's letter facing my desk as a nightly rebuke from the grave.
#
# Just Causes
The BBC's television crews, setting up their paraphernalia of cameras and cables and lights at the _Northern Echo_ 's Priestgate office, created a stir in Darlington. Somebody famous must be coming.
Not so. The TV crews were there because I'd somehow convinced the Newcastle studios that they had the chance of a lifetime to create a documentary on the remaking of a newspaper. I cringe when I see the film today. There's nothing wrong with the production. What makes me cringe is the sight of the owlish, clench-jawed editor trying to give the impression he's in the middle of one of the greatest news stories of all time when all he has on his desk is a report of a Women's Institute dance.
The BBC broadcast helped the _Echo_ in the North East—it certainly irritated our regional competitors—but we were still obscure nationally. We were not quoted in roundups of national opinion, and I was made to feel we didn't count. We had no money for promotion. How could we break out? Darlington was regarded as a dead end, and I was having difficulty even finding a news editor. We couldn't compete in salaries with the big nationals in Manchester, or for that matter with the pay scales of the _Newcastle Journal_ to the north or the _Yorkshire Post_ to the south. When I went to Newcastle and York, I made a point of stopping in newsagents as an ordinary customer, asking for a copy of the _Northern Echo,_ and expressing amazement when they didn't have any. I seemed to be the only one surprised.
In my search for a news editor I went south—far south. The editor of a weekly paper called the _Independent_ in Ibadan, Nigeria, had read about my appointment as editor and inquired whether I'd have an opening for him sometime in 1962, when his contract ended. The name was familiar—Mike Morrissey. I remembered a terse, slim industrial correspondent on the _Manchester Evening News_ moving mercurially through the newsroom before he took a chance and left for Ibadan to start a Catholic newspaper there.
I tried him in Darlington for two weeks as a sub. He didn't shine. I sent him to cities and towns all over the circulation area to find stories. He dazzled. By insistent sleuthing, he discovered that unthinking magistrates were mistakenly sending children in need of care to detention homes. His story put a stop to that.
It was investigations and campaigning journalism that would put us on the map. I didn't have a deliberate plan to do that; it was circumstantial, arising from frustrations and disquiet as we encountered instances of a vast carelessness in public life. I recoiled from the partisan political filters of the dailies and resolved never to start a campaign of any kind until we had first investigated thoroughly and had an achievable target. And if we began a campaign, which meant intensified reporting and opinion writing, we had always to give space to dissenters—and not give up after a few days. All this was as yet an untested theory of mine.
I appointed Morrissey the _Northern Echo_ 's first news editor. He made an astounding difference with no more full-time reporters than we'd had before. He wasn't everyone's favorite. He irritated the branch offices by his insistence on follow-ups; when they complained, I told them he was right. After the street riots in Middlesbrough, for instance, it was important for us to keep on eye on relations between the colored and white populations. Six months after miners were thrown out of work by a pit closure, we had to find out what happened to them. We had to check where in the bureaucracy the bypass around Darlington was stuck or when the ground would be broken for the factories promised for the new town of Newton Aycliffe.
Such efficient news editing was essential, but we'd not get very far with follow-ups and recycling all the events the dailies and weeklies were covering. We had to be different, but with an eye clearly on the community's needs. We had to monitor speeches and reports from councils and courts, but I'd grown impatient with the notion, common then, that covering these scheduled events was about all a newspaper needs to do.
Fortunately, Morrissey ran on one of those batteries that never wear out. This became quite clear when I gave him an early endurance test in an investigation that led to a significant improvement in the well-being of hundreds of thousands of people.
It was a perfect spring day in the village of Hurworthon-Tees, four miles south of Darlington. We'd moved to a house there, the family having grown with the arrival of our second daughter, Kate. It was refreshing to escape town and go home for an hour or so at lunchtime before returning for night work that tended to get later and later. Hundreds of daffodils bloomed on the village green; we thought it would be nice for Kate and Ruth to see the celebrated display. On the walk home, a mist swirled in, blotting out the bright sky. It was not a mist of mellow fruitfulness. It stank of rotten fish, and it didn't go away. It made me gag, and it followed us into the house. I mentioned it in the subs' room that night. "Oh, yes," said the chief subeditor, Stan Senior, "that's the Teesside Smell. Everybody hates it, but not to worry; it comes and goes."
I couldn't leave it at that. No doubt being a father again made me overanxious about the Hurworth haze. I fretted that it might be not only noxious but also a health hazard for infants. I sent to the library for clippings but drew a blank. I asked around and got the same response from everyone: The smell wasn't news; it was a fact of life. Get used to it.
Had I raised the question in my first year of editing, I'd have had little hope of getting anywhere; I'd have felt like apologizing for the distraction from the routine news gathering. By the middle of 1962, however, my morning news conferences had begun as I'd hoped. When I put the Teesside Smell on the daily agenda, I volunteered to carry a test tube into the thick of the next noxious mist. We'd have it analyzed and then send a sample for Prime Minister Macmillan to sniff. As the former MP for nearby Stockton-on-Tees, he'd surely come to the rescue if he got a whiff of it.
I've had better ideas. Carrying a test tube in the hope of ambushing the smell was tedious; I broke a couple. Nor did the odor soon come back. Morrissey applied himself doggedly and duly reported that nobody in local industry would acknowledge responsibility for _any_ smell. He asked the Darlington Rural Council, in the person of a Mr. J. D. Collins, if he'd investigate. He ran a mile. "No comment. Least said the better, I think," he replied. A few regional patriots assailed Morrissey for asking around, contending that it was bad publicity for the region to admit it smelt of anything other than roses and new-mown hay. Some maintained that it was all in the editor's imagination: wasn't he from some effete metropolis to the south?
During a long spell of clear air, I began to doubt the authenticity of my olfactory senses; I almost longed for the nuisance to return to reassure the staff that I was not an obsessive hypochondriac. Then one morning a haze thickened, and with it came the Teesside Smell. Our Stockton office reported coffee bars suddenly crowded with refugees. All who caught the merest whiff wanted to know the source and what "they" were doing to suppress it. Clearly "they" were doing nothing, and we'd not been very effective ourselves either, until David Spark, visiting Westminster Press in London, had the bright idea that we should check with industrial chemists whether there were pollutants that gave off a distinctively fishy smell. The answer was methylamine, used in the production of pesticides, dyes, and solvents. But where did it come from? And why did it linger?
Our chief suspect became the Imperial Chemical Industries (ICI) plant at Billingham. They made amines there, did they not? A public relations manager we asked wanted to know why we were picking on ICI. They'd had no complaints locally, said the manager. Surely it was obvious that when there was a temperature inversion, low stratus clouds coming off the cold North Sea, known locally as the fret, would carry a whole cocktail of pollutants over a large area of Teesside, not just downwind of Billingham. In short fog wasn't a local issue; it was an inescapable regional phenomenon—God's work.
At the end of the next morning conference, I suggested to my team that when the smell returned, we should be ready to photograph the track of the persistent noxious vapor. A giggle ran through the building: "Guess what now, he's asked Charlie to photograph a bloody smell!"
Not long afterward, Ossie Stamford, one of Charlie Westberg's photographers, was driving through Houghton village in Teesside on a lovely day. No mist, no smell. But when he reached Stockton-on-Tees downwind of Billingham, there it was: a pall enveloping the High Street. He photographed it, then rushed back ten miles inland to Houghton's clear skies. The images came out wonderfully—the first photograph of a smell ever published! I splashed the two pictures across a whole news page. They proved that the Teesside Smell was localized. Nor did the haze disperse quickly, as did the sea fret blowing inland. The haze, with its pollutant, lingered. So we had two mysteries: the source of the smell and the persistence of the haze that carried it.
We called ICI again and received a long letter from the process investigation manager at Billingham. He strongly protested "bias in _The Northern Echo_ 's implication that pollutants and smells come only from the ICI factory at Billingham, ignoring the other numerous sources of pollutants and smells among the busy industries of Teesside." Furthermore, he maintained, "the smells observed and described at various distances from the factory cannot be related by 'nose' with any smells inside the factory." But he followed that blanket denial with an admission: "With one or two exceptions such as the amines or 'fish smell.' "
One or two exceptions? The game was up!
I was invited to lunch at Billingham with the division chairman, Rowland Wright (who became chairman of ICI), and the factory's top managers. I brought the incriminating photographs. ICI came clean. Yes, they'd been plagued by leaks of amines and were working hard to stop them. It wasn't easy because the stuff was so potent that a thimbleful would smell across a county. As their process investigation manager put it, "The quantities involved are very small indeed, and obtaining large enough samples and their subsequent analysis calls for methods of very great precision." So my test tube idea wouldn't have worked, but persistence did.
ICI now conceded that the mists carrying the fish smell weren't just naturally occurring sea frets. Instead they were principally created by leaks from their own ammonia plant which rapidly combined with the amines to form malodorous mists. ICI committed urgently to plug the leaks, install more gas scrubbers in the ammonia plant, and replace worn-out equipment at considerable cost. It refined its detection techniques for leaks. It installed electrical precipitators to reduce emissions from the sulfuric acid plant. It sent out a mobile laboratory sucking in air and making analyses.
We'd done well following our nose. Only years later did I think we'd missed a trick: early in our inquiries, we should have called the relevant union officials. They knew all about the Teesside Smell, though perhaps they would have been too embarrassed to brief us. The men working in the amines tank wore protective clothing and breathing gear and then changed and showered, but they still complained that the smell accompanied them home and put their wives off sex.
ICI had performed a perverse service. Their reaction to the attention we gave the noxious lingering problem dramatized how feebly other firms controlled gases from other chemical factories and the grit and fumes from the steelworks, power stations, and brickworks. Black smoke also poured out of the thousands of household chimneys in the area. We applauded the housewife who took her laundry to a council meeting to show them the sheets she'd washed white and hung out to dry that morning, only to have them ruined by black smoke.
Despite all this, Darlington and Stockton authorities still did not insist that fuels should be smokeless (as Manchester had done years before). Individuals smoldered with resentment but hadn't formed into a well-organized pressure group. We had to speak for them.
* * *
Some of our critics saw our reporting as hostile to industry. On the contrary, as industrial editor Don Evans confirmed, the entrepreneurs invited to establish highly necessary new businesses in the region were not infrequently deterred by pollution. Perhaps it was as well we didn't stumble on the difficulties ICI had created for the sex lives of chemical workers; we'd have had to report it, and the region would never have appealed to anyone.
Out of the desire to attract new jobs we conceived a series of features highlighting the experiences of companies who'd chosen the North East as the location for new plants. The results justified the title: "They Came North to Success." But there were too few of these new companies. Government policy for the so-called depressed areas was to induce manufacturers to locate wherever unemployment was highest. It was good politics but lousy economics. Most of these areas lacked the communications, skilled labor, and infrastructure to support fledgling enterprises. What the _Echo_ was able to add to the relevant academic analysis were insights derived from grassroots reporting. David Spark had written two hundred profiles and a series on the centers of Stockton, Darlington, Durham, Ripon, and Newcastle. Don Evans had slogged round shipyards and steel mills and heard the grouses of management and union leaders. I asked Don and David to join me in defining the paper's policy. Trained as reporters, they were uneasy about opining. Soon enough, though, the three of us were sharing the writing of a double column of argument on a new editorial page uncluttered by advertising from mortuaries.
With each passing week, the news got grimmer. The last shipbuilding yard on the river Tees closed; so did Darlington's railway workshops. We hammered away at the piecemeal, short-term, and incoherent government policies for the region. We emphasized the crucial interaction between the economy and the environment. You couldn't walk in the shadow of the giant pit heaps without wondering why everyone had not fled long ago; skilled labor migrated south all the time. The vile winter of 1962–1963, the worst in living memory, exposed the inadequacy of the road system. Giant snowdrifts cut off thousands of people for days. The effect on the local economy was devastating: nearly ninety thousand were unemployed.
Amid the never-ending storms that winter, a tornado touched earth at Middleton St. George Airport (now Durham Tees Valley Airport) one chill February morning in the person of a bleary-eyed man with flyaway hair, tie askew, boots untidily laced, and a cloth cap pulled down over his pugnacious face. This was Quintin McGarel Hogg, Queen's Counsel (QC, or counsel to the Crown, an honorific title), then the second Viscount Hailsham, Tory leader in the House of Lords. An anti-appeasement supporter of Churchill, a platoon commander in the North Africa campaign, a barrister of charismatic brilliance, and our very own Merlin, Hailsham had been charged by Prime Minister Macmillan with conjuring up a long-term brief for the regeneration of the North East. The press cynics scoffed at his cloth cap as a gimmick to identify with a mythic regional figure, Andy Capp, the earthy _Daily Mirror_ and _Sunday Mirror_ comic-strip character supposedly located in Hartlepool, who was into pigeon racing, snooker, football, getting drunk, and abusing his wife. No, Hailsham witheringly explained, as if talking to an idiot, he wore the cap to keep his head warm.
Hailsham was perhaps easy to underestimate. He'd been regarded as something of a clown for his stunts for the television cameras—rushing into the sea in baggy pants to celebrate an election victory, closing a Tory conference by ringing a big hand bell to symbolize Labour's death knell. But Tom Little, our veteran chief reporter in Newcastle, had seen officials come and go and testified that not since Winston Churchill visited the region during the war had a Minister of the Crown seemed so seriously possessed by the urgency of his mission. Hailsham defined it as "lifting the quality of life at all levels." He was shocked by the dereliction. He drove through South Durham and thought much of it should be pulled down. Lights blazed into the night in his Newcastle headquarters as he worked himself and his team from 8:00 a.m. into the small hours.
He didn't finish the Hailsham Plan until the autumn, but we were thrilled to read it when an embargoed copy arrived in the office in November. Our managing director even agreed to increase the size of the paper so that we could publish four full pages. Her Majesty's Stationery Office heard what we planned. They told us it was too much: "It's our copyright." Yes, I replied, but it's our lives. You'd think they would want the widest dissemination of such a key report, but they lacked the imagination Hailsham possessed. We went ahead, hoping they wouldn't sue (and they didn't).
The Hailsham Plan—"as long as a washing list"—projected a bright new future for the region. The conurbations of Tyneside, Teesside, and the Darlington-Aycliffe area would become "growth zones" for investment, with new airports, motorways, and revived seaports. A regional council would make decisions on the spot so that everything didn't have to await the nod from London. Hailsham envisaged making towns and villages more pleasant by providing decent housing, schools, and hospitals and by removing the industrial scars, tackling pollution, and fostering the arts.
I couldn't contain my glee, hopping about the editorial floor like a kid with candy. In my exultation I felt emboldened to contact the region's leading industrialists—Swan Hunter, ICI, Vaux Breweries, Head Wrightson—as well as the heads of Durham and Newcastle universities and others, and urged the formation and funding of an organization to sponsor schemes of improvement. They seized the moment with vigor and generosity, setting up an office for the Civic Trust for the North East in 1965. The paper meanwhile attacked the Coal Board for dumping waste on a once lovely beach and highlighted shoddy development in villages of real architectural merit. But I wanted us to be creative as well as critical.
Geoffrey Broadbent, an architect friend from Manchester, toured the region for us with Bert Hackett, who'd drawn St. Ann's Square for the _Manchester Evening News_. We couldn't pay this pair enough; they just liked the challenge. Their illustrated series, "The Big Clean Up," showed how wretched and squalid landscapes could be made inviting.
Our own fake Gothic title piece and Dickensian editorial offices were now glaringly at odds with what we were preaching for the region. The management agreed to refurbish the gloomy subs' room with good lighting and new furniture designed to assist copy flow and page layout; we even got a light box for measuring photographs.
I knew that changing the title piece was about the riskiest thing a newspaper could do, putting its identity in pawn as it were, nor was management filled with joy at abandoning a tradition. On the other hand, Bill Treslove in the composing room agitated for modernity, so we experimented with this and that typeface, all woebegone failures. Finally I turned again to Bert Hackett. He designed a new _Northern Echo_ title piece based on the English slab-serif face Clarendon, in bold upper- and lowercase. The sturdiness of style spoke well for the identity of the _Echo_ and its region. Our Conservative MP Anthony Bourne-Arton protested, but the mobs stayed indoors.
* * *
There are no small stories (though some are bigger than others), and so to the Battle of the Broccoli. It was the fate of a clever bluestocking by the name of Valerie Knox to be assigned the investigation originally inspired by my wife, that of local produce. I'd abandoned the story in the face of the lassitude prevailing in the old shared newsroom, but Valerie was new blood, one of Fenby's graduate trainees. Doubtless wondering why she'd worked so hard for her Oxford degree, she took her shopping basket around Darlington, Newcastle, Middlesbrough, West Hartlepool, and Manchester, buying fruits and vegetables. She found that the prices were indeed higher in the North East than in Manchester, and highest of all in Darlington.
On the morning of publication, a posse of angry Darlington greengrocers arrived in the office of our managing director, demanding that the editor come down and eat his words—and their broccoli. Nervous that I could easily have been tripped up by an inability to distinguish between a mangel-wurzel and a yam, I offered up our investigative reporter Valerie instead and asked the beatific David Spark to go along as a pacifier. (I also calculated that his presence would calm management, still not sure whether London had imposed a madman on them.) As I fretted in my office three floors above the battleground, a telex arrived from Fenby. He liked the story; in fact his wife had also noticed how highly priced vegetables were in Darlington. I felt the telex should be put in a cleft stick and rushed by runner to the managing director.
Spark reported back that the confrontation with the greengrocers had been heated but essentially came down to an argument about the comparative quality of broccoli and King Edward potatoes. I offered to repeat the exercise with an independent shopper and a Darlington greengrocer riding shotgun. The result was pretty much the same. We "won," but it was another lesson in how sensitive local communities could be when "their" paper seemed disloyal.
I read fifteen newspapers daily and ten on Sundays. One Sunday morning, scanning the heavyweight _Sunday Times,_ I came across a three-line "filler" paragraph at the foot of a column. It said that Vancouver, British Columbia, was expanding a program to save women from dying of cancer. That was all. A hundred questions buzzed in my head, propelled by one of the most consistent emotions of my life since the days I'd seen people in Lancashire coughing up blood from soot-blackened lungs: If preventable, why not prevented? Why did it take so very long for medical knowledge to percolate and have effect?
Ken Hooper was a six-foot-two history graduate and cricketer from Wadham College, Oxford: rather enigmatic, certainly not prone to my emotionalism, and likely, I thought, to wrestle every fact to the ground. He was by now fairly experienced, having joined the paper in January 1961 and survived the subs' room and reporting for both the _Echo_ and the _Despatch_. I could ill afford to lose him from the reporting staff, but I gave him the clipping on the Monday morning and asked him to go to Vancouver straightaway. I knew I'd have to worry about the impact on the budget, but I was eager to get started before the _Sunday Times_ or someone else followed up. Nobody did.
Hooper saved me the expense. He started his research in Britain (much tougher to do before the Internet) and never went to Canada. He spent endless hours in libraries, hospitals, and ministries, heaping his findings in a shopping bag to the amusement, if not derision, of some of the big shots he visited. He was gone about seven weeks, but four well-informed articles landed on my desk. They were disturbing. Thousands of women who were dying from cervical cancer might have been saved, thousands of others had died already, and thousands more were certain to die because of chronic inertia in the National Health Service.
The technique that could save lives was called exfoliative cytology, the study of the characteristics of cells shed from body surfaces. The possibilities had been known to science on both sides of the Atlantic since the 1920s, thanks to George Papanicolaou at New York Hospital and Cornell Medical School and Professor L. S. Dudgeon and his colleagues at St. Thomas' Hospital in London. It was another twenty years before their work was put to practical lifesaving use by Dr. Joe V. Meigs, a Boston gynecologist, assisted by a biologist, Ruth Graham. Graham took vaginal smears of three of Meigs's patients. The patients appeared perfectly healthy, but the smears, read under a microscope, showed very early cancer cells. Given the state of medical knowledge then, it was risky/courageous of Meigs to remove the uteruses of the three women, certain he would see tumors not detectable in a routine examination. He didn't. He was horrified. He was roundly condemned—then vindicated. Three days after the visual inspection, the sections of each uterus examined under a microscope showed it contained early curable cancer that would have been fatal if undetected.
What all this meant was that a simple smear test, requiring only a few minutes of a patient's time, could detect a threatened cancer in women. The danger could then be obviated by a simple cone biopsy or removal of the uterus, depending on the condition. It was another five years (1949) before the potential was realized by two doctors in Vancouver and one on the other side of the world, Mr. Stanley Way. (As a surgeon he was entitled to "Mr." The honorific "Dr." was one notch down in British medical parlance.)
Hooper reported that all his findings kept bringing him back home not with the finished articles, but to see Way, who was just up the road from us in Gateshead. Beginning in 1949, Way's gynecological research unit at Queen Elizabeth Hospital had screened upwards of 150,000 women and found 601 of them harboring very early cancer. None of the women treated had died; of those having the minor operation, 46 of them had gone on to deliver 57 children.
Way's sample was smaller than the one in Vancouver. By 1963 researchers there had screened 214,900 women over age thirty and compared the records of another 248,400 who hadn't been screened. The death rate was seven times greater in the unscreened group.
When Hooper called on Way, he heard how Way had tried for years to have screening adopted as a routine national test. There was interest in a few centers (London, Birmingham, Derby, Edinburgh), but none in the Ministry of Health. So every year something like twenty-five hundred women died needlessly, about double the number dying in road accidents. It was so different in the United States. Early in the century more women died from cervical cancer than any other form, but the death rate began to fall remarkably after the American Cancer Society started to campaign for Pap smears in 1957.
I took up the Hooper articles with passionate urgency, running all four in June 1963, with editorials asking the Ministry of Health to start a national program to save women. I sent everything the _Northern Echo_ published to news organizations and wrote personal letters to a group of MPs. They sprang to it, all of them submitting parliamentary questions for the minister of health, Enoch Powell. They ran into a brick wall. "I am advised," Powell intoned, "it would be premature to aim at a general application."
How many more women had to die, we asked in the paper, before the minister acted on the evidence, already years old? He acknowledged that there had been 2,504 deaths in 1961, but every time the MPs went back to him—as they did month after month—the answer was always some variation of no: "I cannot estimate how many deaths would have been prevented.... I cannot suggest an average cost per smear.... I would refer the hon. Member to my previous answer(s)."
So it went on through the whole sickening year as we pounded away and the minister stonewalled. Regional hospitals, Powell said, would consider any proposals, but they'd have to find the money. We learned that the city of Stoke-on-Trent had done so but had had to wait three years for ministry permission to establish a clinic. One of the MPs I'd recruited, Jeremy Bray, did not let it rest. At the end of the year, on December 2, he asked what further consideration the minister had given to setting up a comprehensive early diagnosis and treatment service for cervical cancer.
The gratifying answer was "I have asked regional hospital boards to expand cytology services. Before screening can be offered to all women in the age groups at risk, more trained staff are needed and I have asked five hospital boards to set up special training centres." It didn't represent a miraculous conversion. Powell had been replaced by Anthony Barber.
It was a victory, the road to a comprehensive national program, but I couldn't help doing the arithmetic. A national program could have been started ten years earlier (Stanley Way had been screening women for fourteen years). Since 2,504 women had died in 1961, I calculated the unnecessary loss of life over the past decade at ten times that number, or 25,040.
Besides our campaigns for public health and revamped economic policies, we were now getting some recognition for our news reporting. We proved at least as good as the nationals in responding to the assassination of President John F. Kennedy on November 22, 1963.
I heard of the Dallas shooting on the radio when I was in a dinner jacket driving to the Teesside press ball and turned back to the office. Wedgewood was busy editing the diverse flow of copy—from the agencies and from the London office—with just over three hours to deadline. I added to the tension by saying we would publish a four-page special on Kennedy's life and discuss how often the U.S. presidency had been ended by murder. We sent for photographs from the library. None could be found. The day manager of the picture library, Shirley Freeman (known as "Shirley Fileroom"), had gone home, and the night manager, Bill Webster, had the night off. The indispensable Joan Thomas suggested we call Shirley's parents. "Oh, she's out with her boyfriend." Where? "I think they went to the cinema."
The Odeon was the most popular cinema. Joan got the Odeon manager on the telephone for me. He hadn't heard of the Kennedy shooting. He was aghast when I asked him to stop the film and find our staffer. Then I had a better idea, with the result that Shirley and her boyfriend, canoodling in the back row, saw a flash on the screen—a handwritten message on a Perspex slide: "Miss Shirley Freeman call the _Echo_ urgently." Her date was ruined; the paper was saved.
By 1963 circulation had risen by 10 percent, on the way to a rise of 14 percent, and year on year our profit had tripled. Winston Churchill had been a big help. I came across an old copy of his _My Early Life_ (1930). I'd read his war histories but not this, and I suspected few of my generation had. I was so enchanted by it that I wrote to him and asked permission to serialize it. He sent a warm note back saying go ahead. It proved popular.
Nineteen sixty-three was a significant year, as the poet Philip Larkin made clear:
_Sexual intercourse began_
_In nineteen sixty-three_
_(which was rather late for me)_
_Between the end of the_ Chatterley _ban_
_And the Beatles' first LP._
As a happily married man with now three children (Michael had arrived that year), I too missed the sexual revolution—but I kept pace with the music. I could hardly miss the Beatles' first record, "Love Me Do." My wife was a Liverpudlian; our Granada TV broadcast the Beatles' first studio appearance in October 1962; we bought their first album, _Please Please Me;_ and I could hardly forget how my sharp producer friend Barrie Heads had told me he'd thrown another new group out of the Granada studio because they weren't as presentable as the Beatles. "Mick Jagger and his group were so scruffy."
The break into music for the _Northern Echo_ came out of a snowstorm. George Carr, Westberg's deputy, took a sixteen-year-old printing assistant, Ian Wright, on a long slog to reach people trapped in a blizzard along the route that went over the Pennine hills. Carr and Wright had a broken-down Ford Popular car (Britain's lowest-priced car) with no snow chains, no snow tires, and no heater. They loaded the trunk with four bags of coal to weight down the back axle, along with a shovel and hessian coal sacks for when they got stuck. In this way, with thermos flask and sandwiches, they got through the traffic jams and jackknifed trucks when all others—including the police, rescue services, and ambulances—had failed. Between taking photographs and conducting interviews, they helped people get their cars out of snowdrifts.
As it happened, the best photograph was taken by Wright, who was normally an unseen elf, filing the negatives, mixing the chemicals, and cleaning up. I put his dramatic picture on the front page. Soon after the first edition had arrived, there was a knock on my door, and there was Wright, asking very nervously if there was some reason why he'd not been given the credit. It was an oversight. I put his name under the photograph in the next edition, and so began Wright's career as a photographer. He was the only photographer with any interest in pop groups. Westberg despised those long-haired rockers. It was a hard day's night getting him to concede that if Wright took the pictures on his own time, Charlie wouldn't impede him.
Week after week Wright was out with the Beatles, the Rolling Stones, the Searchers, Lulu, the Dave Clark Five, Manfred Mann, Dusty Springfield, Cilla Black, Roy Orbison, Billy J. Kramer, and Gene Pitney. I assigned junior reporters to write the stories. They knew more than I did about who was worth covering, though I was keen enough to drive them to and from Newcastle for the first North East tour of the Beatles in March 1963.
Our youngsters struck up a rapport with the new pop stars and with managers such as Brian Epstein and Neil Aspinall. Wright and Guy Simpson were the only pressmen who showed up when the Beatles gave a concert at the Globe Theatre in Stockton, and they had no problem getting backstage. "John Lennon," Wright remembers, "was always asking for complimentary prints. 'Wrighty, don't forget to send those photos; the family love 'em.' " I was so impressed by the initiatives of these juniors that I started the paper's first weekly supplement, the _Teenage Special,_ which attracted some thirty thousand sales on Mondays—almost a 30 percent increase. With Tyne Tees Television, the _Northern Echo_ organized very loud talent shows in Newcastle—the _American Idol_ of Tyneside.
The music juniors all went on to make names for themselves. Philip Norman won a _Sunday Times_ magazine essay contest, then became a best-selling author with _The Stones, Shout!, Rave On_ , and biographies of Elton John and John Lennon. David Sinclair wrote biographies of Lord Snowdon and the Queen Mother. David Watts became the Southeast Asia correspondent for the _Times,_ John Cathcart editor of the _National Enquirer,_ and Guy Simpson picture editor of the _Independent_ newspaper in London. They must have taken their cue from Tyneside's own Eric Burdon and the Animals, whose great hit was "We Gotta Get Out of This Place."
Granada Television came calling on me at the _Echo._ Since returning from America I'd written a couple of documentaries for them and a pamphlet on their fight to televise a parliamentary election. It's an indication of how suspicious the authorities were of this dangerous new medium that Granada had to mount a full-scale legal and public relations assault before it managed to bring TV cameras to the Rochdale by-election.
The call was from whiz kid Jeremy Isaacs. He would become renowned for producing a series on World War II and the cold war and later become the founding chief executive of Channel 4 and Sir Jeremy, director general of the Royal Opera House. But in 1961–1962 he was winning his spurs commissioning a rotating group of commentators for a program critically examining the week's newspapers, _What the Papers Say_. (One of the world's longest-running television programs, this show is still on the air today, now on the BBC.) Would I care to audition for the program?
Had he not heard of Lewenhak and the talkative gypsies? Five years had gone by, which for television people must have meant it was lost in the mists of time. I didn't bother to brief him on that history. I wrote a script for the audition and did a dummy run in Manchester with Michael Frayn of the _Guardian,_ Colin Welch of the _Daily Telegraph,_ and Tom Lambert of the _New York Herald Tribune_. The outcome was a letter from Isaacs: "I hope I can persuade the boys here to let you have a bash on behalf of the provincials."
The boys apparently weren't in any hurry to risk a hick from Darlington, and it was a few months before Isaacs was back. He was, he explained, bringing cameras for the parliamentary by-election in Harold Macmillan's old constituency, Stockton-on-Tees. He asked me to provide commentary on the press treatment of the election. In a howling wind I stood in the town square, orating into the gale and feeling foolish, as I squinted at the teleprompter, watched by a group of giggling urchins.
Soon afterward Isaacs was succeeded by Barrie Heads, who'd produced my interview with the painter L. S. Lowry. He invited me to join what was now a regular panel. This was hard. Brian Inglis, the anchor for the series, was dry, ironic, and authoritative; Michael Frayn and Peter Eckersley were very witty. Barrie's main problem with me was my north country pronunciation. The Queen's English was still the standard on television. Any regional accent was judged déclassé, except in a slice-of-life show like Granada's own _Coronation Street_. In one run-through of my script, Barrie rushed out of the control room shouting, "Butcher, butcher!" He meant that my Lancashire accent was overly stressing the _u:_ "Don't say 'boo-ocher'! Say 'butcher'! Try it again." I did. It satisfied him. But on the show I was so concerned about pronouncing it right that it came out "betcher." Thereafter, I continually rehearsed to myself, reciting "butcher, baker, candlestick maker," but the flat _a_ 's and deep _u_ 's kept coming back all the same.
For the next two years I was on about once a month. It was a slog in Darlington scouring scores of newspapers scattered amid the children's toys, as well as writing and rewriting, counting and recounting the words to fit the allotted fifteen minutes—all in between hours at the office, followed on the Wednesday night by a long drive over the Pennines for a recording session in the Manchester studio the next day. The newspaper extracts were read by actors, and the tone of their voices, pace, and timing had to be rehearsed.
How I sweated over the early scripts! I had not merely to read all the papers but also to compare them for news getting, accuracy, and fairness. I'd known from childhood, for instance, that the _Daily Express_ (circulation four million) believed in putting an optimistic gloss on all news (unless it was about the Labour Party). Its most famous editor, Arthur Christiansen, laid it down that the _Express_ "should make everyone feel it is a sunny day." Nice sentiment, but it was remarkable how far the paper was prepared to go to make everyone believe all was for the best in the best of all possible worlds. Milk in Britain was being contaminated by radioactive iodine from a fifty-seven-megaton Soviet bomb test in the atmosphere, reported the Agricultural Research Council; the government made a statement in Parliament that it was keeping a day-to-day watch in case the contamination got to a danger point. These two items were in every paper except the _Express_. Instead it wrote, "There is little danger milk will become contaminated." When all the other papers reported that Britain could expect additional strontium 90 to arrive the next spring, the _Express_ reported that "many experts" believed the Soviets could produce "clean" bombs with little fallout. But they hadn't produced them; theirs was a singularly dirty bomb. "All this talk" about fallout, said the _Express,_ was "unpatriotic, because it made the Russians think they could scare us." Next, I said in my on-air commentary, we'd be told by the _Express_ that strontium 90 was good for us.
Though that sort of absurdist journalism was meat and drink to _What the Papers Say,_ I also tried to highlight any great reporting I'd read in the national papers. I contrasted the _People_ 's robust pursuit of the crooks running football pools for bogus charities with the malicious invasions of privacy by the _Daily Sketch_ gossip writers simply to make someone miserable. I praised the _Sunday Times_ ' exposé of the slum landlord Peter Rachman, I chastised the _Daily Mirror_ for rejecting a Conservative advertisement without saying why, and I teased the _Sunday Express_ for not disclosing that the lively letters it ran were all written by staffers posing as readers.
Some took this better than others. The editor of the _Sunday Express,_ John Junor, invited me to lunch. The editor of the _Daily Mirror_ slammed me, thundering prominently in the paper, "Evans dedicates his spare time to denigrating the rest of the press. Loftily he lectures the national newspapers as if Darlington exudes a special degree of insight and wisdom denied to newspapers in London and Manchester." I did the most detective work tracking how the newspapers had failed to find out what lay behind the resignation of Lord Mancroft from the Norwich Union Insurance Society. It transpired that the Jewish Mancroft had been forced out by Arab business interests that had dealings with the society, but the _Financial Times,_ which had first reported the resignation, was slow to find out why and even slower to comment.
My commentary did not win friends on Fleet Street. I heard that Lord Drogheda, the fastidious chairman of the _Financial Times,_ was upset with me; and I was well aware that Pearson Industries, which owned the _Financial Times,_ also, through Westminster Press, owned the _Northern Echo_. Drogheda was far from alone. Big Tom wrote a friendly warning letter from the _Manchester Evening News:_ "It so happens that at a large gathering in London last night I saw a number of our top boys, one of whom went into a long diatribe that your Granada program was intended to try and kill newspapers and that you were determined to single out _The Sun_ for often quite unjustified criticism which, if persisted in, could put the newspaper and 2,000 employees out of business."
I'd actually commented very little on the struggling _Sun_ (then owned by the Mirror Group, which got tired of trying to make it succeed and sold it to Rupert Murdoch, who made it a building block of his empire). Tom wrote, "I tried to reason with one editor but the conviction appeared to be that instead of one of their own kind trying to help newspapers at a critical time, 'a newspaperman is selling us down the river to the commercial television companies who are delighted at the spectacle.' " He concluded, "I can also tell you that one of the top boys let it drop that they're watching your paper like hawks every day and they even quoted headings and certain things to me. Keep your powder dry!"
I told Big Tom I had the curious notion that if helping newspapers survive was the criterion, surely improving their performance would help. And no Granada producer ever once tried to influence me one way or another in the commentaries. Big Tom understood; I thought the storm clouds had lifted. So I was stunned when Charles Fenby told me I had to stop appearing on _What the Papers Say_. The icicle I'd first encountered had become a warm and perceptive booster, but now he noted that my contract required me to seek the permission of the Westminster Press board for doing anything other than edit the newspaper, and he was not giving it.
I could not lightly accept the ban, I told Fenby, because the publicity had drawn attention to the _Echo_. It was now being noted and quoted much more often. Fenby was adamant, so I requested permission to appeal to the Westminster Press board. It so happened it was meeting on the Thursday I was due to do the show, which Fenby insisted should be my last.
As soon as I'd finished the recording that afternoon in Manchester, I drove straight back to Darlington. Fenby and the board had just finished viewing the actual program going out at 10:30 p.m. I entered the boardroom in trepidation. They all applauded. It was a tremendous relief, a ruling in character with the open way Westminster Press ran their newspapers. Fenby took his defeat with grace, and I continued the program until the end of my editorship of the _Echo_.
When I'd met Sid Chaplin at the start of my editorship, the region had been sunk in gloom, and we'd wondered whether we could persuade someone to stage a spectacular celebration of its heritage—its art, architecture, and scenic beauty. In 1964 Tom Little, the _Echo_ 's chief reporter (and music critic) in Newcastle, watched a son et lumière concert in the radiant white basilica at Vézelay, France, and in his review for the paper he wondered why we could not do the same in the much grander and more glorious setting of Durham Cathedral. Well, why not? I called in David Spark. "We're going to have a son et lumière concert in Durham," I told him, "and you're going to organize it."
Of course it meant borrowing the cathedral, and having the city of Durham amenable, and finding a writer and composer, and raising money for script and music, and finding a brilliant lighting engineer, and selling tickets, and praying that people would come. We started by seeking the blessing of Durham's dean, the Very Reverend John Wild. He and his wife gave us lunch in the cathedral close, which David remembers was trout with a delicious sauce. I can never remember anything I eat, and on this occasion I was concentrating on not talking like an irreverent show business impresario. But the dean warmed to the proposal, the mayor of Durham City came in with enthusiasm, we persuaded Flora Robson to narrate, and we dragooned the cathedral choir and bell ringers, the Horden Colliery Band, and the Cornforth Men's Choir.
The son et lumière we staged was the single most exciting and uplifting experience of my time in Darlington, a magical marriage of North East enterprise and artistry to reflect the splendors of human faith and endeavor. You could hear the intake of breath among the crowds as the lighting revealed the hidden beauties of the interior of the cathedral and the pageant of nine hundred years unfolded: the translation of the remains of St. Cuthbert from Holy Island, the start of building on the rock, the battle at Neville's Cross, Charles I praying alone on his way to London, a murderer seeking sanctuary hammering on the great doors, the entry of miners' bands to dramatize the role of the common folk as well as that of the ambitious princes and clerics.
When it was over, we were able to give the cathedral the profit of around £70,000 at today's values. It was agreed that most of the money should go to pay for the installation of permanent floodlighting. In the years since, I've never been able to look on that glorious heritage of the cathedral shining in the night without a rush of exultation and gratitude.
I'd edited the _Northern Echo_ for four years when I received a letter that would provoke the biggest of the paper's campaigns. It was March 1965, and I was on an express train from Darlington rattling down to London for a Westminster Press conference. I'd caught the train with seconds to spare, which was normal in those madcap days. I made it to my reserved seat only because Joan Thomas had pushed me out of the office and—as always—had phoned ahead to the station staff, so that when I ran onto the platform with the train about to leave, they had the right carriage door open. She never told me this at the time. I just knew there were sixty-one minutes in an hour and assumed everyone else did as well.
In my haste I grabbed sheaves of articles and correspondence reproachfully piling up in my "pending" tray. Among them was an article submitted for publication by a Darlington man I didn't know. Herbert Wolfe had escaped the Nazi persecution of the Jews in 1933. He'd brought with him to England one shilling—from which he'd built a thriving chemical business—and a passion for justice. The story he told in his letter and article accelerated my racing heart, carrying me back to a cold night fifteen years before, in March 1950, when a young man called Timothy Evans sat in the condemned cell in Pentonville Prison waiting to be hanged. He was twenty-five, a bakery van driver and not very bright. He'd had no normal schooling and couldn't read. He whiled away the time playing Chinese Patience and chatting about football and boxing. His companions said later he didn't seem to realize his position.
"The one thing that sticks in my mind," Evans would say to the warders in the cell with him, "is that I'm in for something I haven't done." He'd been found guilty at the Old Bailey of the murder of his baby daughter, Geraldine. He was charged with, but not tried for, the murder of his wife, Beryl, whose strangled body was found with the baby's, bundled up and hidden in the washhouse at 10 Rillington Place, a small, squalid house in a seedy area of Notting Hill, London, where the Evanses had a poky apartment. The trial excited little attention. The chief prosecution witness was a bespectacled clerk called John Reginald Halliday Christie, who lived in the ground-floor apartment at 10 Rillington Place.
All that Evans could say in his defense was that "Christie done it." He couldn't suggest a motive. The jury was out only forty minutes; the appeal was dismissed. Our neighbor in Failsworth, one Albert Pierrepoint, the official executioner, was summoned from the pub he ran, called Help the Poor Struggler. It was just another of the several hundred hangings he'd carried out with great efficiency at fifteen pounds per. There was no crowd at Pentonville at 9:00 a.m. on March 9, 1950, when the uncomprehending Evans was executed for a murder he did not commit.
The cold facts that Wolfe summarized gave me an urgent feeling that I should pull the emergency cord that would stop the swaying high-speed train so I could shout to the world that here was a monstrous injustice we must lose no time in correcting. It pained me that I'd put off reading through the "pending" tray, for the sequel to Evans's execution was as horrifying in its way as the terrible crime.
What the judge and jury didn't know, what counsel didn't know, what Evans didn't know, was that the star witness for the Crown was already a psychopathic strangler. Even as Mr. Justice Lewis donned the black cap and pronounced sentence on Evans, the bodies of two of John Christie's victims, Ruth Fuerst and Muriel Eady, were lying buried, undetected, in the little back garden in Rillington Place.
Three years after the execution of Evans, there was a new tenant in Christie's old ground-floor flat. He started to put up a wall bracket, pulled off a piece of wallpaper to reveal a papered-over cupboard, and found himself looking at the bare back of a human body. There were two more corpses in the cupboard, another under the floor in the front room (Mrs. Christie), plus the two female skeletons in the garden—six in all. Three years after the trial of Timothy Evans, Christie stood in the same dock at the Old Bailey and confessed that he was a necrophiliac and that it was he, not Evans, who had strangled Beryl Evans for sexual gratification at the moment of death.
The Christie confession confronted the public and legal system with an appalling probability: British justice had hanged an innocent man—and had done so on the evidence of the man who'd framed him, a man of such sangfroid that when his garden fence had slipped, he'd propped it up with a human femur.
How could it have happened? It wasn't surprising that judge and jury at the time chose to believe Christie. He was fluent, he was ingratiating, he'd been in France in World War I and been gassed in his country's service, and from 1939 to 1943 he'd been a War Reserve policeman with two special commendations. What a nerve this illiterate wretch Evans had, trying to blame an upstanding ex-policeman! And hadn't he admitted the crime? But it was a bogus confession. Christie had offered to perform an abortion on Beryl Evans—one of his tricks to indulge his vice—and Evans felt guilty when Christie told him it "didn't work" and Beryl had died in the procedure.
We can still get a very good idea of the diabolical nature of Christie from the movie _10 Rillington Place,_ where the wily, whispering serial killer (played by Richard Attenborough) is the cobra and the pathetic Timothy Evans (John Hurt) the mouse. "Playing Christie," Attenborough told me at the film's world premiere, "was the most disturbing, distressing role I've ever played."
In Parliament and the press, following Christie's conviction and confession in 1953, there was an insistent demand to reexamine the case. The minister in charge of internal affairs, Home Secretary Sir David Maxwell Fyfe, announced that he'd asked for an inquiry by a QC. The QC he chose was Scott Henderson, a selection that proved to be a landmark in hypocrisy. The home secretary gave Henderson only nine days to review the complexities, on the grounds that Christie's execution shouldn't be delayed. It was an absurd request, and Henderson complied by rushing out an absurd report in only seven days. He did not merely say Timothy Evans was guilty after all. "There is," he concluded, " _no ground for thinking_ that there may have been a miscarriage of justice." Nobody else got a chance to interrogate Christie. On July 15 he was hanged on the same gallows where the man he'd framed had died.
The home secretary, a devout believer in the infallibility of trials for murder, declared it a "fantasy" to think there could have been a miscarriage of justice, but there were a number of people in Parliament and the press who remained disturbed, for good reason. Michael Eddowes, a London solicitor, published an investigation revealing that pressure had been brought to suppress the testimony of witnesses to Christie's lying and violent nature. Christie, "this perfectly innocent man," in the words of Mr. Christmas Humphreys, the QC prosecuting Evans, had in fact six entries on his police record, including six months in jail for maliciously wounding a woman he nearly killed. The authorities ignored Eddowes. The _Daily Mirror_ 's Peter Baker interviewed the Roman Catholic chaplain at Pentonville and concluded that Evans didn't confess to either murder before he died saying the rosary. The authorities were unmoved.
In her work for the great book series Notable British Trials, Ms. Tennyson Jesse, a crime reporter and editor (and a grandniece of the poet Alfred, Lord Tennyson), again underlined the fallacies in the prosecution's case. The authorities dug in. Ludovic Kennedy, the celebrated writer, TV performer, and Liberal candidate for Parliament, was moved to spend five years writing a masterly demolition of the case. It produced a parliamentary debate in June 1961 in which the Conservative home secretary, Rab Butler, conceded that no jury of the day would convict Evans, but he made no attempt to explain the crime committed in the public's name or to exonerate the victim. There was nothing he could do, he murmured; it was all too long ago.
Among a number of speeches from the Labour opposition was one by Sir Frank Soskice, QC, which passionately demanded a new inquiry, a free pardon, and the handing over of Evans's body to his Catholic family (a mother and two sisters) for burial in consecrated ground. "I believe," he said, "that if ever there was a debt due to justice, and to the reputation of our own judicial system and to the public conscience of many millions of people in this country, that debt is one the Home Secretary should pay now." Three and a half years after that, Sir Frank himself became home secretary. He then had the power to do what he'd urged the government to do in 1961. He didn't, proclaiming, "I really do not think that an inquiry would serve any useful purpose."
Sir Frank was the fourth home secretary in eleven years to reject a reopening of the case, and MPs and editors had other things on their mind. Given the caliber of the people who'd already protested, the fate of Timothy Evans was a lost cause.
To Herbert Wolfe this was intolerable. The integrity of British justice was precious, and it had been polluted. He convinced the Liberal Party to pass a resolution demanding an inquiry; it made no impact. He wrote letters to the press; they were discarded. He chose the fifteenth anniversary of Timothy's death, March 9, 1965, to send me that short article. As soon as I returned to Darlington, I read all I could on the case and was overwhelmed by the magnitude of what we'd been led to believe.
If Timothy Evans and John Christie were both killers, we were being asked to accept that there were two stranglers of women in the same two-up, two-down house, operating independently and in ignorance of one another. Both men used the same method of strangulation, and both made confessions to the police using the same language. Both confessed to "using a piece of rope," and not just any piece of rope but one "off a chair." Both disposed of the strangling ligature; both concealed their victims' bodies; both temporarily used the same place of concealment; both wrapped their victims' bodies in blankets; both left them without shoes; both left them without underclothing.
Not only were these men independently strangling in the same way at the same time in the same house, but it was pure chance that Evans accused Christie, who, unknown to him, shared his own supposed murdering characteristics. Evidence to overwhelm this series of coincidences would need to be formidable. The likelihood of finding two people with the same fingerprints is four billion to one (twice the number of people in the world then). The Evans-Christie "coincidence" was like finding two people with the same fingerprints in the same house.
I published Wolfe's article on the editorial page, along with a full editorial setting out the reasons justifying a new inquiry. I knew it would take much more than this to break officialdom's wall of certitude. What could a provincial newspaper and one of its readers possibly accomplish after all these years when all the distinguished testimony had been to no avail?
Soon after my trauma on the train—and it was that insistent—I was back in London, sitting on a cold stone seat in the lobby of the House of Commons. The man I'd come to see had written the fatal words "Let the law take its course" on the death warrant for Timothy Evans. This was Lord Chuter Ede, Labour's home secretary at the time. He was the man with the least to gain from reopening the case, yet he had the humility and courage to say that he now believed he'd sent an innocent man to the gallows and society should make amends. I asked him if he'd visit the current home secretary with a group of MPs, and he agreed. This would be a unique event; no former home secretary had ever appealed to a successor for a pardon for a convicted person.
I put that news on the front page with a little white-on-black box (a logotype) I intended to use to flag every story about Timothy Evans: "Man on Our Conscience." I wrote personal letters to all our regional MPs. In an editorial the _Echo_ asked: Why has Sir Frank changed his mind? If he's discovered some new element in the case, he should tell us. If not, how could he explain his volte-face? But Sir Frank would not be provoked. Chuter Ede had warned me how hard it would be to make any impact: "We are up against the full weight of official Whitehall."
I sent Wolfe's article and my editorials to every regional and national editor and broadcaster. Nobody picked up on either the case or the Soskice contradiction. The silence was broken only by the chief whip of the Liberal Party, Eric Lubbock. He put down a House of Commons motion for a new inquiry and invited signatures from members of all parties. A month later he had only nine signatories.
An editor asked me, "Why are you flogging a dead horse? Why give a dead man any space?" I sent him a quotation from Michael Stewart, MP (later foreign secretary): "The moment we say we cannot be bothered, we have other important things to do, we turn from our progress and start walking along the road that leads to Belsen." A radio interview I secured after a few weeks brought a stinging rebuke from Douglas Nicholson, the chairman of the Vaux Brewery in Sunderland. He wasn't clear, he said, whether my call for an inquiry was "a newspaper stunt," but it did seem to him and others he had spoken to that an inquiry would "waste the time of important people."
It was a strange period for me. I don't have a thick skin; emotional, rather than phlegmatic, is the adjective that follows me around. But as the discouragements multiplied, I grew preternaturally calmer. It was a weird out-of-body experience, like the time in New York when I was mugged and regarded the mugger with a gun to my head with ridiculous detachment. I just went on publishing everything I could. Every time an MP signed Lubbock's motion, I put it on the front page with the Man on Our Conscience logo. Every day I selected one question after another about the conviction and put it under the logo. This got to be a bit much for Maurice Wedgewood and Frank Peters, who saw their precious front-page space taken up by Man on Our Conscience paragraphs. Peters asked, "Isn't it time to call it a day?"
I fell back on the famous editor Horace Greeley, who'd observed that the point when a newspaper begins to tire of a campaign is the point when readers are just beginning to notice it. But how to keep up the momentum and not bore everyone to tears? I assigned a bright reporter, Jim Walker, to examine how the authorities had reacted when confronted with other miscarriages of justice. Sir Arthur Conan Doyle, the creator of Sherlock Holmes, exposed the wrongful conviction of Oscar Slater, who nonetheless had to spend nineteen years in jail before the Home Office admitted the error. Then Doyle, after a Sherlockian investigation, campaigned for a pardon for the former solicitor George Edalji, erroneously convicted of mutilating animals. The authorities held out against a pardon but had to concede the establishment of the Court of Criminal Appeal.
Letters began to trickle in to the paper. Ludovic Kennedy came back from abroad, and we formed a Timothy Evans committee of all those who over the years had campaigned on Evans's behalf. On May 19 I collated all the editorials, news reports, features, and letters in a four-page _Northern Echo_ pamphlet and mailed it to BBC radio and every MP, every editor, every television station. Gradually Lubbock gathered allies, and by four months into the campaign, 108 MPs had joined the call for an inquiry. I was now regularly interviewed about the case on radio and television. Supportive comments began to appear left and right—in the _Catholic Herald_ and in the left-wing _Tribune_ and from the scorching columnist Bernard Levin in the _Daily Mail,_ though nowhere else in the national press. The going was made easier by the provincial newspapers. I urged all the editors to collaborate on a joint letter to the prime minister. They took up the _Northern Echo_ 's campaign without a trace of jealousy.
On July 22 Lubbock's motion had 113 signatures. With Chuter Ede, he led an all-party group to see Sir Frank Soskice, who agreed to think again. A month later he gave in. He overruled the advice of his civil service officials and returned to his original position, appointing Mr. Justice Brabin to conduct an inquiry in public in the Royal Courts of Justice.
I was there in the Queen's Bench Court No. 6 in the Strand Law Courts on Tuesday, November 23, 1965, with Herbert Wolfe and Ludo Kennedy when Brabin opened the inquiry. The court was packed with QCs, treasury officials, press, witnesses, police, and Evans's relatives. Day by day the inquiry revealed the missteps in the dreadful labyrinth to the execution; we reported them all, and Wolfe wrote a commentary for the _Echo_.
The judge took a million words or so of evidence and examined seventy-nine witnesses, and then on October 12, 1966, he presented the oddest judgment of all: Timothy Evans had probably not murdered his baby, for which he was hanged, but he probably had murdered his wife, for which he was not even tried. "This was certainly an arresting theory," remarked Ludo, "especially as there is virtually no evidence to support it."
I wrote an examination of the flaws in the Brabin maneuver based on all the evidence that had been taken over a year, and the Timothy Evans committee joined in urging the new home secretary, Roy Jenkins, at last to do the decent thing. The testimonies at the inquiry, the judge's findings on the murder of the baby, and the years of advocacy were enough for Jenkins, a man who made all the Whitehall stonewallers look like straw men. Wolfe and I were in the House of Commons on October 18 when Jenkins rose to make a unique announcement. On his recommendation, the Queen had granted Timothy John Evans a free pardon, and the state returned his remains to his family for burial in consecrated ground.
Many people had worked for years to end the death penalty on religious and ethical grounds. The execution of Evans and the long refusal to face the shame of that brought the cause to a climax. Two weeks after the Brabin inquiry was announced, our ally and the stalwart abolition campaigner Sydney Silverman, MP, won a parliamentary motion for the suspension of the death penalty which that century had sent 799 men and 16 women to the gallows. On November 9, 1965, the House of Commons voted to suspend executions for five years. Four years later, on December 18, 1969, on a vote free of party dictates, the death penalty was abolished altogether.
# BOOK TWO
_Scoop, Scandal, and Strife_
#
# The Rolls-Royce of Fleet Street
Twenty-five years almost to the day since my father sat on the sands at Rhyl with the burned-out soldiers rescued from Dunkirk, I encountered a survivor who was to change my life.
Denis Hamilton was one of the most powerful people in British journalism. In 1940 this debonair, soft-spoken man had been a twenty-two-year-old junior officer shoulder-deep in the English Channel desperately trying to save the remnants of his battalion. He got to Dunkirk with only 160 men, the survivors of his Eleventh Battalion of the Durham Light Infantry. Now he was not only the prodigiously successful editor of the _Sunday Times_ , the flagship of Thomson Newspapers, but editorial director of five Sunday, five morning, and eleven evening newspapers. He'd ended the war as the British Army's youngest brigadier, decorated with a Distinguished Service Order for holding back a German thrust near Arnhem. He was close to Field Marshal Bernard Montgomery, Prime Minister Harold Macmillan, and South African prime minister Jan Smuts. He was photographed with members of the royal family, the shah of Iran, President Kennedy, and President Nasser. He was friends of the Grahams at the _Washington Post_ and the Sulzbergers at the _New York Times_.
The three-hour-plus train ride from Darlington deposited me in King's Cross station, a good long walk from the copper-faced Thomson House at 200 Grays Inn Road. Passing through the imposing double glass doors at the entrance embossed with the coat of arms of the owner, Lord Thomson of Fleet—motto: _Nemo me impune lacessit_ (No one provokes me with impunity)—I was intercepted by a very martial commissionaire in a white peaked cap, his uniform ablaze with battle ribbons. I had the impression he checked the shine on my shoes as he telephoned someone to confirm the authenticity of my letter of introduction to Hamilton. Once confirmation arrived, he showed me into the elevator to the fifth floor, where I was escorted to an outer office and then the inner sanctum, which the secretary breathlessly told me had been designed by Lord Snowdon, then the husband of Princess Margaret. It was more like an elegant drawing room, with sofas and silk cushions. Hamilton seated me in a stylish Eames chair while he took a corner of a sofa beneath a Matisse print. There was no sign of a typewriter anywhere.
He was unhurried with me (the former corporal) and not at all intimidating. We established a bridgehead in our shared attachment to the North East—he'd been born in South Shields and worked in Newcastle—then he veered away to Asia and the training assignments I'd carried out for the International Press Institute. "I heard what you did for all those newspapers in India," he said. "Very important." He said nothing more for what seemed like an eternity, his mind, I imagined, roaming the subcontinent during the absences that I later learned were unnervingly characteristic of him. I didn't interrupt his reverie. Then his soft voice resumed. "The Timothy Evans affair in the _Northern Echo_ and the campaigns on pollution and for a big cleanup of pit heaps... well done. A really good provincial newspaper can make a difference to a community."
I presumed he must be sounding me out on switching sides to join his Thomson regional newspaper group; at the _Northern Echo_ I was competing against three of them every day. After the geography discussion, however, it seemed it was the _Sunday Times_ that was at the top of his agenda. The _Sunday Times_! I hadn't allowed that possibility to enter my mind when he'd asked me down to London. I'd guessed it was either about the regional papers or the role of the provincial editors on the National Council for the Training of Journalists, where I'd joined discussions in which Hamilton occasionally took part. I tried to look calm. It was harder when I caught up with his ruminations. "I'm looking for a practiced newsman who might be groomed to be managing editor here."
Managing editor? Now that was a heady thought. The job was impressive enough in itself, and a managing editor was also clearly in the line of succession to the editor's chair. But "might be groomed" was tentative; the list of potential assignments—features, news, campaigns, sports, long-term planning for the color magazine—was long without it being clear whether I would run all these departments; and how many jostling managing editors were there at the _Sunday Times_ already? Then Hamilton murmured something about looking for a successor to Pat Murphy, a seasoned professional who reported to Hamilton on the performance of Thomson's twenty regional editors. Murphy's current job held no interest for me, nor did I think I'd be much good at it. Twenty editors! I'd found it distracting enough to be looking over the shoulders of just three editors of an evening newspaper and two weeklies after I was promoted to editor in chief of the North of England Newspaper Company group while still editing the _Northern Echo_.
The opportunity of the _Sunday Times_ , though, did make my head spin. How could it not? This was the biggest of what we called the quality Sundays. It broke news more often than the others, and its staff was legendary. Not every newspaper could boast Ian Fleming, the creator of James Bond, as the manager of its foreign correspondents, and where else in an elevator could you meet critics such as Cyril Connolly, George Steiner, and Raymond Mortimer and ask if they'd read any good books lately? The newspaper's critics and foreign correspondents trailed clouds of glory. The magnetic Fleming, coming out of the war as assistant to the director of naval intelligence, had recruited eighty-one gifted men and seven women for Lord Kemsley's Mercury foreign service, their locations indicated by colored lights on a world map behind his desk in Grays Inn Road. He liked to say, with a wave of his long ebonite cigarette holder, that their average age was thirty-eight and they spoke 3.1 languages apiece. Godfrey Smith, then the young personal assistant to Lord Kemsley—known as "K"—endeared himself to the irreverent Fleming by letting him have the key to K's very private loo. He remembers Fleming instructing the awed young men in his circle never to use a subordinate clause and to call only God and the King "sir." Fleming's "Atticus" column, like the man himself, was sophisticated entertainment.
Admittedly, the paper's political history was less impressive. In the 1930s owner and editor in chief Kemsley was an errand boy for Chamberlain in appeasing Hitler, and in 1956 he'd been a cheerleader for the invasion of Suez (and gained circulation at the expense of rival owner-editor David Astor's anti-Suez stand in the _Observer_ ). Under the ownership of the Canadian-born Roy Thomson and Hamilton's editorship since 1961, however, the _Sunday Times_ had become less of a mouthpiece for the Conservative Party's Central Office. It was a far richer, more influential newspaper than my provincial daily, with fifty-six pages then to our sixteen, and at 1.3 million it had more than ten times our circulation, nationally and internationally.
I was flattered, but I didn't dive in, as I sometimes did without checking whether there was any water in the pool. Editing the _Northern Echo_ , I was "with child." The paper was thriving as part of the community. My family had moved to a graceful period house in town, where we gave strawberry tea parties on the lawn while my two-year-old son, Michael, ate the daffodils. I'd identified so much with the North East and Darlington that a year before I'd written to the _Sunday Times_ to protest a book reviewer's slighting reference to the town, typical of the south's snotty disdain, I felt, for anything north of the London suburbs. Hamilton himself had telephoned to apologize; the values he prized most were civility and loyalty.
London itself—the alien Metropolis—was a splendid mystery. On my fleeting visits I remembered the bewildered excitement of the pair of Newton Heath cubs just before the war when Dad had used a free railway pass to take Fred and me, in our best suits, to London by night train from Manchester. We arrived at Euston as the streets were being cleaned early in the morning, and then he proudly showed us Big Ben, the Houses of Parliament, Madame Tussauds waxworks, the Regent's Park zoo, and the Tube. We had tea at the Lyons Corner House, where the nippies, as the waitresses were called, wore white aprons and white lace caps. And then the magical day ended, and we were on the night train back to Manchester.
I was thirty-six. I'd watched all my young friends and colleagues in the provinces head for Fleet Street as soon as they could. I'd had the occasional envious pang seeing their bylines from foreign capitals or hearing of this or that ascent in the hierarchy, but I hadn't looked to London as my future, and looked still less after I became editor of the _Northern Echo_. One of Lord Beaverbrook's top editors at the _Sunday Express,_ John Junor, had pressed me to go for lunch, but I had never got round to it. In any case, there were just too many big guns firmly in editors' chairs of the papers that interested me.
Of course Fleet Street had its magic then. It's become a dull London thoroughfare since the electronic diaspora to south of the river, powerfully assisted by Rupert Murdoch's gratifying defeat of the Luddite print unions. But it wasn't like that in 1965. Nearly all the national newspapers had their headquarters in the street or nearby, with their presses roaring in the basements, the barons barking in the penthouses, news vans and reporters racing out, and enough watering holes for a thirsty newsman, gossip diarist, or cameraman to run from one to another in a rainstorm without getting wet. In the satirical magazine _Private Eye,_ Fleet Street was the "Street of Shame," but how could anyone with ink in his veins not be entranced by the tales of daring scoops and backstairs scandals, much improved in the telling by the bibulous hacks in El Vino or the Printers Devil, Cheshire Cheese, White Swan, Punch Tavern, Old Bell, or Stab in the Back? Or by the sight of Megalopolitan House in its sheath of black glass, where Evelyn Waugh's bewildered antihero in _Scoop_ , country diarist William Boot, arrived for his unforgettable interview with the foreign editor of the _Daily Beast_ and its owner, the autocratic Lord Copper, doppelgänger for Lord Beaverbrook.
The _Sunday Times_ was regarded as the most exciting paper in Fleet Street, albeit located in Grays Inn Road, where its solitary watering hole was the Blue Lion. But I'd heard quite a few stories of newcomers to national papers being squeezed out of a role. At least at the _Echo_ I could get some things done, ungroomed.
Sensitive to my reservations about moving south, Hamilton suggested that I have coffee with his deputy, William Rees-Mogg, the donnish epigrammist (who in 1981 became the celebrated editor of the _Times_ ). I did so two weeks later, an encounter eased by the discovery that although Rees-Mogg was a country squire from Charterhouse and Balliol, we were both graduates of RAF Padgate. The columns of his I'd read in the _Sunday Times_ resounded with such authority, it was a relief to find a shy bibliophile with an appealing little sibilance in his speech. His solemn ecclesiastical manner, hands fingertip to fingertip as if in prayer, dissolved into a self-effacing giggle when he confessed a cheerful unfamiliarity with questions of typography and production. "Joining the paper would be like joining a freeway," he explained, equably ensconced in his Georgian home near Smith Square, in the political heartland of Westminster, within division-bell distance of the Houses of Parliament. "If you come, I'm sure you'll soon gather speed and get in the right lane." With the caution I came to see as characteristic of Rees-Mogg, he added, "Of course we can't see round corners."
Rees-Mogg asked if I was inclined to accept "Denis's" invitation, then persuasively sketched why I'd enjoy the weekly operations and the personalities I'd be working with if I joined. "Perhaps you'll care to write some editorials, too," he said expansively, as he saw me out into the sunshine of Lord North Street looking toward the baroque church of St. John's. I visited the church before catching the train back to Darlington and happened on a lunchtime concert of Beethoven sonatas. It seemed a happy augury.
Back in Darlington I did some homework on my prospective new boss, Charles Denis Hamilton—known to his associates as C.D.—as well as on the über-boss, Lord Thomson of Fleet. Who were these controllers of the destiny of the _Sunday Times_? Would I be able to pursue the journalism that had most engaged me at the _Northern Echo_ as freely as I had with Westminster Press? Would Hamilton's position as an establishment figure be in any way inhibiting? If an investigation had to be defended legally, would Thomson balk at the costs? The self-made tycoon had amassed most of his fortune after he was sixty. He was flagrantly frugal. The gossip diarists were agog in 1964 to find the newly ennobled Baron Thomson of Fleet standing in line at Burberry for a cashmere coat reduced from seventy pounds to forty.
I soon discovered that Hamilton was not the upper-class officer I'd assumed. He'd been brought up in a terraced house in Middlesbrough, his father an engineer who'd been forced by lung cancer to retire early from the heavily polluted iron and steel works at Acklam, with a miserable pension of only ten shillings a week. Hamilton's scholarship to Middlesbrough High School put him among the handful of youths from the industrial slums, separate from the paying sons of professional people. He'd started in journalism as a junior reporter, as I had, but he had never sought to go to university.
So how had he risen in the ranks so quickly to become an army officer? He'd learned leadership in the Boy Scouts, achieving the highest level of King's Scout (similar to Eagle Scout in the United States). After Munich he'd volunteered for the Territorial Army (think National Guard) when the Durham Light Infantry needed thirty officers by the end of the month. He wasn't qualified by the standards of the time (meaning he wasn't a public-school boy, he hadn't played rugby for Durham, and his father didn't know the colonel's family), but he did have an acute eye for the relevant social signals. Mixing with officer candidates who'd been solicitors, bank managers, accountants, and men from minor public schools, Hamilton assumed an upper-middle-class disguise like a second skin.
His establishment aura—his commission, his decoration, his accent, his Savile Row clothes, his whole demeanor—enabled him to advance toward the redoubts of privilege. Absent the illusion that he was an honorary member of the ruling class, his very real native abilities might well not have carried him into a position of power with the deeply snobbish Lord (and Lady) Kemsley, even though he was a war hero. Control of the so-called quality or serious national broadsheet newspapers in Britain, circulating among the more educated, tended to be the preserve of the traditional elites, with the graduates of public schools and Oxbridge predominant. In 1965, when Hamilton was talking about my joining the _Sunday Times_ , the top four positions below him were all held by Oxbridge men. Hunter Davies, who'd followed me as a Castleman at Durham and editor of _Palatinate_ , was the only provincial university man to have come, via reporting in Manchester, to an envied position as the successor to Fleming writing the entertaining Atticus column. It was similar elsewhere. The _Guardian_ was edited by an Oxford man, and the _Telegraph_ and _Observer_ were owned and edited by Oxford men. But the Canadian Roy Thomson, the son of a barber from a more open, nondeferential society, who'd appointed Hamilton on buying out Kemsley, didn't care where a man came from so long as he knew where he was going. And William Haley, who edited the _Times,_ had educated himself by omnivorous reading, so perhaps there was hope for me.
The press was just one strand of British life touched by the social changes accelerated by the war. In his 1941 essay "The Lion and the Unicorn," George Orwell predicted, "This war, unless we are defeated, will wipe out most of the existing class privileges." The victory of the Labour Party five years later seemed to fulfill his dream that England would assume its "real shape" through a conscious open revolt by ordinary people "against the notion that a half-witted public-schoolboy is better fitted for command than an intelligent mechanic." The two most powerful and able members of Clement Attlee's 1945 cabinet, Ernest Bevin and Herbert Morrison, were uneducated working-class lads. The Butler Education Act of 1944 opened the door for secondary education for all, as I described earlier, but by the 1960s higher education was still very much a privilege. In 1959, seven years after I graduated from Durham, only 4.2 percent of the eighteen-to-twenty-one age group had become full-time university students, hardly a lightning advance on the 3.2 percent admitted in 1954 and nothing compared with the United States. Orwell had been right to protest that the working class ought not to be "branded on the tongue"—a phrase borrowed from Wyndham Lewis—their status determined more by accent than ability. He could not have foreseen how liberation from that perception would be more powerfully assisted by satire than by polemics or politics. The surreal mockeries of class in the phenomenally popular _Goon Show,_ on BBC radio, inspired the satirists of _Beyond the Fringe,_ leading to _Monty Python_ and _Fawlty Towers._ My generation did not feel any need to affect the standard English accent of the BBC newsreaders and the dukes posing as hotel concierges. It was not that we were brave; it would just have exposed us to ridicule.
I was a beneficiary of the late-breaking waves of political, cultural, and social changes that gathered force in the mid-1960s. By 1965 the Conservative Party had fallen apart in the aftermath of the Profumo call girl scandal. The prime minister no longer was Harold Macmillan, who appointed thirty-seven Etonians to office, seven in the cabinet, and played to perfection the role of the grand English gentleman. Now it was Harold Wilson, a scholarship boy at grammar school and university, with a nondescript accent who liked to be photographed in his ordinary Gannex raincoat, taking every opportunity to be seen as a middle-class, middlebrow, nonconformist Little Englander.
Labour's "New Vision," epitomized by Wilson's bending the nation's ear about the white-hot technological revolution, narrowly won the 1964 general election over a Conservative Party now seen as too much under the influence of the "fuddy duddy right." It was led by Sir Alec Douglas-Home, who'd had to demote himself from being the fourteenth Earl of Home so as to be eligible to sit in the Commons as Sir Alec and succeed Macmillan. He'd been cruelly caricatured as the prime minister who did his sums with matchsticks and had been given only tepid election support in the _Sunday Times_.
The stars were thus not badly aligned for me, a young working-class non-Oxbridge graduate with a northern accent, whose political genes were suspect. Lord Thomson said of Hamilton, "He's a fellow that doesn't display himself." He was indeed very private about his origins and his war, but I believe I was lucky that Hamilton's own rise from obscurity, his well-hidden resentment of the way his father had been treated, and his command of men in battle from all walks of life combined to make him exceptionally open-minded for his time and his position.
My sense of Denis Hamilton's civic virtue, as much as his achievements with the _Sunday Times_ , was a powerful attraction. His boss was more of a puzzle. Roy Thomson was plainly tightfisted, but he'd risked millions launching the first color magazine in British newspapers in 1962 and installing the machinery for bigger newspapers. Visiting Thomson House I got no sense of hair-shirt austerities. But what of his attitude toward editorial? I'd watched a number of television interviews of Thomson on his purchase of the _Sunday Times_ in 1959. He was a tubby, cheerfully Pickwickian figure who blinked at the questioner from behind Coke-bottle glasses, occasionally twitching his neck as if his collar was too tight, as I had seen Dad do. The impression I had, reinforced later, was that he was psychologically incapable of lying or dissembling. He just blurted things out. (I'd heard that Thomson, in Egypt for negotiations to buy a failing Cairo newspaper from President Nasser, had told Nasser, "You certainly are a cunning old Jew.")
His political philosophy amounted to a few homespun pioneer principles about honesty, humility, and thrift, drawn from the life of a self-made man. He'd left school at fourteen but absorbed into his bloodstream the romances of Horatio Alger, poor boy made good. He had the conventional political opinions of the business class. The death penalty was good, socialism a sickness, government regulation bad. But enfiladed from right and left by tough interlocutors such as Randolph Churchill and Keith Waterhouse, he wouldn't be shaken from insisting that he would never impose editorial policy.
Thomson didn't disguise that he was a cultural philistine indifferent to all the arts, or that his views were not those of the chattering classes, but he didn't expect anyone to take any notice, least of all copy him. After all, he once remarked, "part of the social mission of every great newspaper is to provide a home for a large number of salaried eccentrics."
His attachment to editorial independence had deep roots, practical rather than philosophical. Failing early on—in trying to grow wheat, sell motor supplies, sell radios, sell anything—he'd learned the hard way how much expertise he needed to realize his ideas. He had his policy printed on a card he carried around for twenty-five years like an oath:
I can state with the utmost emphasis that no person or group can buy or influence editorial support from any newspaper in the Thomson group. Each paper may perceive this interest in its own way, and will do this without advice, counsel or guidance from the Thomson Organisation. I do not believe that a newspaper can be run properly unless its editorial columns are run freely and independently by a highly skilled and dedicated professional journalist. This is and will continue to be my policy.
He'd fish it out of his pocket when accosted by critic, favor seeker, advertiser, or politician wanting him to pressure an editor. "You wouldn't expect me to go back on my word, would ya?" he'd say, showing the person his card.
The word of Thomson and Hamilton was good enough for me. In June 1965 I accepted an invitation to become chief assistant to C. D. Hamilton, and in January 1966 I passed through the crested glass doors to start work. Hamilton's formal letter of invitation spelled out a clearer prospect of being "indisputably the key managing editor within a reasonable time," while still floating the idea of my succeeding Pat Murphy as the regional newspapers' editorial director if that didn't work out.
I left Enid and our three children—Ruth, Kate, and Mike—in Darlington while I tested the ice, traveling home on weekends aboard the midnight sleeper, with an inky third edition of the _Sunday Times_ for company.
Longtime foreign manager Ian Fleming bequeathed his successors a warning of the hazards of being deceived by the editor's Tuesday conferences of department heads: "Beneath the surface friendliness, lurk all the deadly sins with the exception of gluttony and lust. Each one of us has pride in our department of the paper; many of us are covetous of the editorial chair; most are envious of the bright ideas put forward by others; anger comes to the surface at what we regard as unmerited criticism, and sloth, certainly in my case, lurks in the wings." The "sloth" was the giveaway, an obvious exaggeration for effect, since this was the time—between 1945 and 1959—he was turning out a succession of his James Bond best sellers ( _Casino Royale, Live and Let Die, Moonraker, Diamonds Are Forever, From Russia with Love, Dr. No,_ and _Goldfinger_ ). Well, so I told myself, hoping the rivalries were not so intense as he described. It seemed to me on arrival that Hamilton and Rees-Mogg set an agreeable, gentlemanly tone for the weekly proceedings, more reminiscent of an academic seminar than the frantic ways of Fleet Street dailies. There were no women in the conference and hardly any on the newspaper. Still the men who assembled in Hamilton's office each week were an impressive bunch, and it would have been surprising if there wasn't hot competition among them for the managing editorship, or even the editorial chair in the unlikely event of Hamilton leaving it. I'd only been in the office a day or two when my friend from Manchester and Durham, Hunter Davies, the only person I knew on the paper, took me aside: "What are you doing here? You'll get eaten alive. You haven't even got a proper job [as he had as Atticus]. You've been a big fish in Darlington, but you'll just get carved up by these _Sunday Times_ slickers."
The leading players were exceedingly clever Oxbridge men who'd vaulted over the traditional route to Fleet Street from the provinces. Presidents of the Oxford Union were in two of the top four positions: Rees-Mogg, the political editor and columnist, and Godfrey Smith, the irrepressibly creative editor of the color magazine. The foreign editor who controlled Fleming's old domain was Frank Giles, whom Hamilton had filched from the _Times,_ where he'd been chief correspondent successively in Rome and Paris. The fourth Oxonian, editing the business news, was Anthony Vice, who'd started out at the _Financial Times,_ like Rees-Mogg, and was for five years city editor of the _Daily Telegraph_.
Among the older stalwarts were Leonard Russell, who'd been literary editor for thirty years and was married to Dilys Powell, doyenne of film critics. Russell was now in charge of the Review Front, the most cherished spot in the paper for features, most often ones he'd culled from well-written books, though he indulged in adventure stories. ("Three men in a boat on a boiling ocean" was his apologetic characterization.)
In contention for managing editor, I could see, were a number of young Turks: investigative editor Ron Hall, news editor Michael Cudlipp, star writer Nicholas Tomalin, and Mark Boxer, the wild card.
The scintillating founding editor of the color magazine, Boxer had first made headlines at Cambridge where, editing _Granta,_ he'd published sacrilegious doggerel, including the lines "You drunken, gluttonous seedy God / You son of a bitch, you snotty old sod." He left Cambridge without a degree but subsequently proved the sharpness of his eye as art director of _Queen_ magazine. He was known, too, for a facility for making enemies via the sharpness of his tongue as well as the wit of his pen-and-ink caricatures. Boxer had resigned the editorship of the color magazine, and when I arrived he was features editor without controlling any space, so he wandered round in a vaguely insurrectionary role. In conference he was often deftly offensive, which gave an edge of excitement to the proceedings. People resented his disdainful manner and Byronic good looks as much as the substance of his remarks, since he was usually right about what was "dreadfully dull."
Hamilton's style in the editorial board—a grand name for his Tuesday morning conference—was like none I'd seen before. There were no news schedules, no clipboards, no set agenda, no inquest on things that had gone wrong the week before. It was more of a conversation about what people had seen or read or were puzzled or entertained by. Hamilton let it drift. Having no agenda meant that people just talked about what interested them, and this often led to material for the paper. A remark about the prevalence of short skirts on the streets elicited the information that a revered Parisian film critic had referred to the phenomenon as the "English Revolution," and then someone else said that the revolution was in the moral standards the English young were developing, so different from their boring parents.
Hamilton took hold of that balloon. Who might understand what was happening, someone with authority but wit? Names were batted around. He didn't reject any, so a kind of cultural bidding war developed, everyone trying to come up with a still more appropriate, still more distinguished, still more surprising name.
I tried to keep up. Memos to myself survive in a tiny diary of 1966 and reflect the eclectic collisions as I struggled to come up with ideas and writers for all sections of the paper. I'd caught the tail end of a dismissive remark: "He's just a newsman." Was it about me? Not wanting to be typecast, I tried to cover all the bases:
• Rebecca West on Feminism
• Court magistrate who thinks 70 miles an hour speed too slow. Eccentric magistrates? Justice as comedy? Tomalin.
• Who Lord Goodman? Lewis Chester to write.
• Jean Shrimpton's knees
• Find husband/wife writing team, perceptions each other.
• Hector Berlioz letters. Other composer letters?
• Malcolm Muggeridge and Jesus. Fix lunch.
With Leonard Russell's approval, I got to know the London publishers and agents, in the hope we might have early warning of literary coups. I was not used to long, lavish lunches where pound notes were discreetly palmed for the wine waiter. I took bags of books to skim in my cheap room in the loft of the National Liberal Club that I shared with the pigeons and the cleaning ladies outside my door rattling their buckets. I made good use of the ideas and contacts gained as the host of a new weekly BBC radio discussion program, _A Word in Edgeways._
I can no longer remember why I thought we might bring fresh light to the question of Jean Shrimpton's knees. But I do remember that a breakfast with my old friend Tarzie Vittachi of the International Press Institute led to him writing a stunning two-part Review Front for us revealing what had gone on in Sukarno's Indonesia during a news blackout. First the Communists murdered generals in an attempt at a coup, then Sukarno's generals murdered the Communists, and then fanatical Muslims started killing "infidels," so that 300,000 died before Indonesia returned to normality.
In the third month I was with the paper, Prime Minister Wilson called an election, giving me an opportunity to observe how far the paper's Conservative sympathies affected news coverage. I was in charge of the election pages, writing up the polls and monitoring the press. The _Sunday Times_ reporters, I found, were wholly free to report what they saw, hardly commonplace in the fiercely partisan press.
These were difficult months for me, because in about the third week Hamilton told the conference I'd be revamping the sports pages. Boxer whispered, "That's the kiss of death." Apparently none of the executives cared to tangle with Ken Compston, the highly professional but belligerently independent sports editor of seventeen years. He blew smoke in my face, making it clear he didn't want any bloody fancy new ideas mucking up his pages. Yet he came round in the end. I signed Michael Parkinson as a sports columnist (Parky was not yet a TV celebrity) and cut a deal with the renowned yachtsman Francis Chichester. He'd announced he was going to attempt sailing single-handed the twenty-eight thousand miles to Australia and back, following the romantic and dangerous path of the famous clippers. We gave him a marvelous send-off on the Review Front with a profile written by Philip Norman, last seen earlier in these pages reporting pop concerts for the _Northern Echo_. He'd become a staff writer on the magazine, having won a writing contest set by Godfrey Smith. The Chichester signing turned out to provide an exciting yearlong series with regular dispatches (exclusive to the _Sunday Times_ of course), and the whole world was absorbed when he set out from Australia to brave the perils of Cape Horn. It was the start of the _Sunday Times_ in adventure journalism.
I was tense at Hamilton's first Tuesday conference after the launch of the new sports pages. There was bound to be criticism. I just hoped it was specific so I could try to grapple with it. Leonard Russell knocked his pipe on an ash stand and spoke before anyone else: "Damned good sports pages this week." Whatever anyone was about to say, Russell's endorsement was enough; a quiet murmur of assent, and we were off on other topics. All that was required was for England to win the World Cup, which it did in July against West Germany. The following week Hamilton announced I was to be managing editor.
A few weeks later Hamilton made a dramatic announcement. The separate companies of Lord Thomson's profitable _Sunday Times_ and Lord Astor's loss-making daily _Times_ had agreed to merge into the new Times Newspapers company, 85 percent of whose stock would be owned by the Thomson Organization. "The main obstacle to the merger," said Lord Thomson, "had undoubtedly been me. I don't think Lord Astor could stomach the idea of giving control of his paper to a roughneck Canadian." Thomson volunteered to give up the chairmanship, though he remained the principal risk taker. He said he calculated that the ample profits of the _Sunday Times_ would cover losses by the _Times_ —£285,000 that year—but if the company moved into loss, he and his son pledged their private fortune. And he reiterated his promise that neither the organization nor any individual Thomson would ever interfere with editorial policy.
The merger was approved by the government's Monopolies Commission in December 1966. Both William Haley at the _Times_ and Denis Hamilton at the _Sunday Times_ relinquished their editorships, Haley to become the first-year chairman of Times Newspapers, Hamilton to become editor in chief of both papers. Both promised the commission they would not attempt to impose identical policies on the new editors of the _Times_ and _Sunday Times_.
Rees-Mogg was clearly destined for the _Times_. I had no great expectations I would be high on the list to succeed Hamilton at the _Sunday Times,_ and I was surprised when Mark Boxer remarked casually, "You'd have been a candidate for editor, you know, but you're considered too left-wing." I'd originated several big features, led an Insight investigation of a crooked car insurance company, revamped sports, and overseen news, but I'd been on the paper for only a year, managing editor for only three months, and there were several senior contenders. The clear favorite was the foreign editor Frank Giles, an unruffled administrator who was also an accomplished linguist and writer. He'd been labor leader Ernest Bevin's private secretary at the Foreign Office and knew many world leaders.
After Giles it turned out the rivals for the _Sunday Times_ chair were fewer than I'd expected. Godfrey Smith made it clear he had no ambitions to edit the paper. Michael Cudlipp and Anthony Vice were privately earmarked for the _Times;_ Ron Hall, Mark Boxer, and Nicholas Tomalin were, it seems, judged not to have sufficient experience setting political and economic policy. This left two older formidable front-runners inside the building—Giles and Pat Murphy, the Thomson group's editorial director—and one powerful outsider, Charles Wintour. The volcanic Randolph Churchill was hoarse in his incessant and often drunken private lobbying of Hamilton and Thomson for the appointment of Wintour, the acerbically clever editor of the excellent _Evening Standard_ (and the father of American _Vogue_ 's future editor Anna Wintour).
In the week the decisions were made, Hamilton took Giles to dinner at Prime Minister Harold Wilson's official weekend residence, Chequers, along with Lord Thomson, William Rees-Mogg, and the paper's political correspondent, James Margach. They didn't get to bed until 3:00 a.m. because Wilson reminisced for hours. Only later, when it was announced that Rees-Mogg had been appointed editor of the _Times,_ did it occur to Giles that he and Rees-Mogg had both been eyed as top prospects and that, for some reason, Hamilton changed his mind about giving him the editorship of the _Sunday Times_. Or it could have been that Frank's eye drooped mid-Wilson. Roy Thomson had told Hamilton that he preferred my "north country cheek" to Frank's more polished style (an assessment I was to learn about only years later).
All I knew was that on a Friday, Hamilton summoned me to his office and asked for a brief, to be delivered on Monday, on how I would develop the paper. I was so nervous typing it at home, I filled a whole wastebasket with crumpled false starts. (In those days every second thought meant retyping the whole thing.) Hamilton said nothing about the report I gave him on Monday, but the following day he sent me over to the _Times_ offices to see Sir William Haley. Though we both had got serious ink on our hands at the _Manchester Evening News,_ this was my first meeting with the editor who his subordinates at Reuters and the BBC had said was the only man in London with two glass eyes. Haley was warm enough, but not in the mood to reminisce about his days in Manchester. His rectitude during the interview was focused on how, if I were made editor, I would resist any pressures or temptations in the conduct of the paper to promote Thomson's commercial interests in magazines, holiday travel, book companies, and directories.
Two days later I was wheeled into the grand boardroom of the _Times_ at Printing House Square for scrutiny by the full board of the new Times Newspapers. I sat isolated in a chair facing twelve solemn directors around a long walnut table, with intimidating oil portraits on the walls. In addition to the chairman, editor in chief, and general manager, the board included three Thomson nominees (one of them Kenneth Thomson, Roy's son), two Astor nominees, and four independent "national directors."
_"How independent will you be as editor?"_
"I'm certain that the judgment of the Monopolies Commission was correct. I shall be completely independent. Unless I was certain of this, I would not be prepared to accept the job."
_"What is your attitude to the Thomson commercial interests?"_
"The same as my attitude to any other commercial interests."
_"Even if it is news adverse to the Thomson interests, say in travel?"_
"If there is any news in it, we will print it."
The directors spent a full hour examining my halo as someone who would embrace and defend the freedoms defined in the Monopolies Commission report—not to sell out to Mammon or twist the news for a political agenda. Looking back at the commitments they demanded, I can't help but wonder at how much journalism has changed.
I was confirmed as Hamilton's successor.
Frank Giles, for his part, accepted the deputy editorship. He was forty-eight; I was ten years younger. Often, talking to colleagues when I was out of earshot, he got into the habit of referring to me as "the young master." I didn't mind. It was good-humored; he was incapable of malice, and for the next fourteen years he was an engaging and steadfast deputy.
On the last Saturday of his editorship in January 1967, Hamilton, in his immaculately tailored suit, looked down from the steps leading to the composing room floor where shirtsleeves subs scurried about with galley proofs and page plans. "I'm handing you a Rolls-Royce," he said. It was true. His _Sunday Times_ purred. I was determined to match his dedication to quality, though constitutionally incapable of achieving it in his inimitable style. He was a master delegator; I was a meddler. He was reticent; I wasn't. But we shared the same high hopes of what journalism might achieve. At my back in the years to come, I could always hear the Boy Scout in Denis Hamilton asking, as he'd frequently done when I was managing editor, "Have you done your good deed for today, Harold?"
#
# The Third Man
A gale is blowing in from the Atlantic. It rattles the windows of the cottage behind a beach dune in Quogue, Long Island, where I'm writing this. If I take my eyes off the pines bending in the gusts and glance to the right of the windows, there's a framed photograph on the wall that carries me back to the heart of a great newspaper that had more than its share of storms—and created a few of its own.
The black-and-white photograph is of a news conference at the _Sunday Times_ in London. It's unremarkable in itself—a dozen people sitting on sofas below a spotlighted world map. It would mean little or nothing to anyone else, yet for me it has the exalted resonance of a "Nocturne" painting. I'd admired James McNeill Whistler's work in the Tate gallery on the Embankment near my last home in Pimlico, but John Ruskin hadn't. The celebrated critic had said that Whistler's _Nocturne in Black and Gold_ was nothing more than a pot of paint flung in the face of the public. Whistler sued him. Asked in cross-examination by Ruskin's attorney how long it had taken him to paint it, Whistler famously replied, "All my life."
That's what the photograph on my wall represents, the culmination of my life in journalism, thirty-five years in newspapers, from weekly reporting in Lancashire; to subbing and editorial writing for the _Manchester Evening News;_ to foreign reporting in Europe, Southeast Asia, and the United States for the _Evening News_ and the _Manchester Guardian;_ to five years of daily newspaper editing in Darlington; and finally to fourteen years editing the _Sunday Times_ of London.
I'd started this national editorship in January 1967, feeling very much an imposter as I was driven by a chauffeur to Grays Inn Road and the grand office where Hamilton in the spring of 1965 had first broached my joining him. I'd barely got used to being managing editor of the paper. Now I'd taken his place at the helm, and he'd moved across a bridge to the _Times_ offices. In the perpetual remodeling of the editorial floor ordained by management, I ended up in the photograph's large white and chrome office planned by Terence Conran—benefactions a million miles from all my previous newspaper habitats—after moving from a smaller one decorated in rather startling red and black tones decreed by Lord Snowdon. Snowdon had explained that it was to match my character (I'm still working that out). He'd designed it very practically for transactions with galleys and page proofs, but he was always darting in to remove any object that offended his taste—one day an ashtray, another a cushion, another a clipboard, another a small potted plant. My secretary predicted, "You'll be next."
One day two Scotland Yard detectives did come in looking for me. They wanted a civil service report we'd published which revealed that Ministry of Transport officials were privately pressing to shut down one-third of the nation's railway system. The report proposed leaving large areas of Scotland, Cumberland, Lincolnshire, East Anglia, and central and west England without any trains at all. I truthfully told my visitors they wouldn't find the incriminating document in my office; I'd taken home the background paperwork after seeing the story to press on Saturday night.
I don't think their hearts were in the investigation, but I was duly cautioned that by publishing the information I could be charged with a criminal offense under section 2 of the Official Secrets Act, or OSA (section 1, dating back to 1911, being concerned with espionage). In the outcry after our report, the minister rejected the scheme, and the detectives didn't come back.
Other stories we ran or wanted to run, however, provoked so many subpoenas and writs summoning me to the Law Courts in the Strand that I could have found my way there blindfolded. I didn't seek confrontations with the law. They arose only because government, corporations, and individuals sought to suppress information of public concern discovered by diligent, painstaking efforts by the _Sunday Times_ staff and its contributors.
Collectively the conflicts provoked by our attempts to answer numerous questions dramatized a chronic but unsuspected malaise in the functioning of British democracy. The resulting confrontations with authority also proved the severest of tests for the _Sunday Times_ itself. Closer to home, would the solemn promises of editorial independence made by Denis Hamilton and Lord Thomson be maintained under unprecedented pressures and at grave financial risk?
I would soon find out.
In my audition memorandum for Hamilton, I'd suggested the paper needed a full-time investigative unit. I wanted to reflect W. T. Stead's governing functions of the press—"its argus-eyed power of inspection." My appetite had been whetted by the _Northern Echo_ 's unifocal investigative ventures (cervical cancer and Teesside pollution); by the big car insurance fraud I'd worked on as managing editor; and by the earlier three exposés by the _Sunday Times:_ the investigation of the crooked landlord Peter Rachman; the piquant case of the Chippendale commode, by which the newspaper was able to prove the existence of an antiques dealers' ring swindling sellers; and the "bogus Burgundy" story, in which we found a bottling factory sticking prestigious but false labels on cheap blended wine. These three stories had been published under the rubric "Insight," but the title was otherwise used to identify a weekly page of short, undifferentiated background features contributed by several writers. I devolved the feature writing to three staffers with a more focused mission, charging them to keep watch on all the scientific "ologies" on a page we would call Spectrum. Then I created a new investigative Insight team of four reporters and a researcher. Their exposures of unsuspected scandals of significance and compelling narrative reconstructions of major events (the Yom Kippur War, for instance) soon powerfully reinforced the identity of the _Sunday Times_.
Team journalism is difficult to manage, but I'd no doubt then and none since that it facilitates the best investigative journalism. No single reporter then or instant blogger today could be expected in a timely fashion to follow a multiplicity of trails, false and real, and grapple along the way with unpredictable technicalities in civil engineering, company law, accounting, aeronautics, physics, molecular biology, or whatever the relevant area of expertise. At various times over the next few years when I visited the cramped Insight offices, I'd see engineering blueprints of a DC-10 airliner tacked to the wall, along with a scale model of its defective cargo door, which sent 346 people to their deaths near Paris; another time there'd be diagrams of the chemical structure of thalidomide, which had robbed children of arms or legs or left them limbless trunks; and later annotated maps of the maze of streets in Derry plotting the moves of civil rights demonstrators and the British Army paratroopers who shot thirteen of them dead on Bloody Sunday.
But it is no easy matter to create and monitor a team. The chemistry and direction of the group are crucial. This is a subtle business, and there were periods when I got the mix wrong. You don't want four clones on the team; you want distinct but complementary skills. Each member has to like and respect the others' professionalism; all must be prepared to subordinate their egos and thirst for a byline and to accept direction from a team leader. Then they have to be willing to yield control of the final writing to someone they trust to respect their words and judgments—without being pushovers when the writer doesn't. The most persuasive criticism of team journalism touches on this point. As an anonymous Conservative Party critic of Insight's reporting on Northern Ireland wrote, "While the subjectivity of one writer can clearly be seen in a newspaper under his byline, that of a team is blurred and made more difficult to spot by the common, but unfounded, assumption that anything written by a team must be more dispassionate than that of a single writer."
In the early years the pace and style of the Insight team was set by Ron Hall and his deputy, Bruce Page. Hall and Page were both in their late twenties or early thirties and shared a taste for exuberant hairstyles—Hall a curly black mop, Page long angry sideburns. They had very different temperaments, however. Hall was a scholar in skepticism, honed by his study of statistics at Pembroke College, Cambridge. It was Hall who made the breakthrough in investigative journalism for the _Sunday Times_ with a three-part series on the slumlord Peter Rachman, giving birth to tighter regulations and a new word for tenant exploitation: Rachmanism. But Hall didn't regard himself as a dragon slayer. In fact he distrusted crusaders in journalism. He felt they'd make the facts fit the thesis. He affected boredom, yawning ostentatiously when someone pitched a story that seemed freighted with virtue. For me as editor it was comforting to see Hall, a pipe clamped in his mouth, slowly taking a reporter through the backup for his assertions and analysis. He was a lucid writer and a rigorous editor of text. He'd had a flair for headlines and display introductions beaten into him during his apprenticeship at the _Daily Mirror_. The worst thing Hall could tell you was that he was "combing a dog," which meant the story scheduled for the feature page he controlled had come in with a full consignment of fleas.
Page was not at all interested in the art of presentation but as scrupulous as Hall in deciding what was fit to print. He was a member of his local Labour Party, but never once in all the years I worked with him did a hint of partisanship infiltrate his work. A certain creative tension developed between the two men. In contrast to Hall's more measured approach, Page was zealous to set the world right. Whereas Hall, a dogged Yorkshireman, concealed his analytical sharpness in lethargy, Page, a dropout from Melbourne University, displayed his in breathtaking pole vaults over the collected works of Hume, Burke, Popper, Coleridge, Marx, and Keynes. His long, autodidactic digressions were relieved by an ironical style, and his energy was contagious.
Page and Hall developed strong opposing views on team journalism. Page believed that everyone involved, from the reporters to the designated collating writer, had to trust the judgment of competent colleagues or team journalism could not flourish. He was very careful, though, about who qualified as technically and morally honest enough for membership on the team. Hall thought Page's concept imposed too great a sacrifice of individual integrity. He was not prepared to have an article articulate a view until he was personally intellectually satisfied to the last comma. His statistical training inclined him to calculate the odds against a story having the level of accuracy he required—i.e., 100 percent. He was the kind of editor who would not publish a chess annotation until he'd played the game through himself. As Insight developed, Hall became the chief "space baron" for the whole first section of the paper, selecting and editing the pages he controlled; no second-rate work would survive his scowling scrutiny.
Page had particular gifts for investigation, along with an eclectic mind. He was an ingenious originator of theories to connect apparently unrelated dots, but he had the remorseless intellectual integrity to discard them when the dots didn't connect, as well as the imagination to construct another working hypothesis. Some people found him hard to take. He could be summarily dismissive of individuals he suspected of deviating from his standards, he could be tempted into fascinating but unnecessary digressions in his writing, and his search for the definitive put deadlines in peril. But he was also able to inspire a group with a common animating curiosity.
Much of the renown Insight would win was due to Ron Hall and Bruce Page and the path they set for the varied editors who followed: Lewis Chester, Godfrey Hodgson, John Barry, Simon Jenkins, and Paul Eddy.
Insight had a baptism of fire.
"Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" Edward Lorenz didn't present his chaos theory paper until 1972, but I've often thought of the principle in relation to a major investigation we began in 1967 in my second month on the job. The dynamics of the saga were such that small changes produced unpredictably large effects. For my own part I had no idea that an idle remark I picked up at lunch would nearly a year later lead to my being denounced by Foreign Secretary George Brown.
The lunch was with Jeremy Isaacs, my Granada producer on the first _What the Papers Say_. Isaacs had become head of current affairs at rival Thames Television. He remarked in passing that the _Observer_ had bought the serial rights to a memoir by Eleanor Philby—"You know, the wife of that other man Philby in the affair of the missing diplomats." Sixteen years before, in 1951, two British diplomats, Guy Burgess based in Washington and Donald Maclean based in London, had disappeared together and resurfaced in Moscow in 1956. They were presumed to have been Soviet spies. Kim Philby—"that other man"—had been in Washington with Burgess. In 1963 he'd vanished from Beirut and six months later showed up in Moscow, too.
Back at the office I remarked to Page how interesting it was that all three defectors had been undergraduates at Cambridge in the 1930s. Maybe we could identify the subversive don who'd recruited them to the Communist cause. It didn't prove difficult to identify the don (Cambridge economics professor Maurice Dobb), but the unpredictable trajectory of Project X led us to a far bigger story about Harold Adrian Russell "Kim" Philby.
The Philby investigation was the most taxing one we ever undertook—a frustratingly tedious process of assembling and assessing tiny scraps of information from hundreds of interviews with denizens of a closed world whose stock-in-trade is deceit. Central casting couldn't have selected a more suitably varied bunch of independent minds for penetrating Philby's secret world. Australian Phillip Knightley's quiet, unassuming manner exuded an empathy that attracted confidences despite his distinct resemblance to Lenin—bald dome, black goatee. He'd knocked about the globe as a seaman, vacuum cleaner salesman, South Sea Islands trader, and reporter. David Leitch, like Guy Burgess, had been a golden boy at Cambridge, though twenty years later. Both were good-looking, charming, bohemian romantics—and brilliant. Leitch was already showing the writing gifts that would win him acclaim for describing his days among U.S. Marines at the siege of Khe Sanh in Vietnam. John Barry, a dropout from Balliol, was a contrarian with a quicksilver mind. The austere, cerebral Hugo Young was not on the regular Insight team—he was chief editorial writer and an impeccable political reporter—but he contributed because he knew the ways of Whitehall and Washington as both a Harkness Fellow and a Congressional Fellow.
Most of the ink over the years had been spent on Burgess and Maclean and the mysteries of their last-minute getaway, rushing together onto the ferry from Southampton to St.-Malo at 11:45 p.m. on Friday, May 25, 1951, one of them drunkenly shouting "Back on Monday!" to a sailor concerned about the big white Austin they abandoned on the quayside.
There had been articles about Philby, but nobody had even begun to peel the onion. Nothing had been said on what he'd done, when he'd started spying for the Soviets, how he'd escaped detection, and, importantly, what damage he'd caused, if any. He was assumed to have been a low- to mid-level agent in Britain, and in Moscow he'd become the invisible man. His address was a secret; his telephone was ex-directory; there was nothing about him in the Soviet press; and in the city if he was glimpsed one minute, he was gone the next. For our part we knew next to nothing about him. Leitch had tried to use an interview with Khrushchev in Moscow in 1964 as an avenue to the mysterious exile, but nothing had come of it. Right at the start of our inquiry, we tried again, writing to Kim Philby, Moscow. (It seemed necessary to do that, but in retrospect I'm glad he maintained his silence. If he had talked to us, we would have been accused of being his mouthpiece.)
In the biographical sketch with which we started, Philby surfaced only as a series of snapshots: the blazered head boy at the elite Aldro prep school; the handsome youth who's won a king's scholarship at Westminster posing with his proud, bearded father—Arabist explorer, scholar, and convert to Islam St. John Philby, who had to struggle financially to send him there; pipe-puffing Kim in flannel bags and tweeds at Trinity College, Cambridge, working hard to get a good degree in economics and history; and Kim at twenty-five, an adventurous journalist reporting the Spanish Civil War for the _Times,_ his head bandaged from a shell burst. He appeared to be the quintessential upper-middle-class Englishman, a member of the Athenaeum, the premier gentlemen's club in London; decorated as a commander of the Order of the British Empire (CBE); addicted to the _Times_ crossword, cricket, and premier cru claret; and repelled by displays of emotion. Even when he was under stress in 1955, when he'd just been accused by a Labour MP of tipping off Burgess and Maclean to flee in 1951, we see him unruffled by the uproar, debonair in a gray pin-striped suit as he smiles into the press cameras admitted to the living room of his mother's home in Drayton Gardens, London. In the tape we watched, he was calm and assured, speaking with amused condescension, controlling his charming little stutter; he had a get-out-of-jail-free card in his pocket, a statement to Parliament by Foreign Secretary Harold Macmillan: "I have no reason to conclude that Mr. Philby has at any time betrayed the interests of this country or to identify him with the so-called 'third man,' if indeed there was one."
The press, too, exonerated Philby. Poor fellow, they concluded; he'd lost his job as a diplomat just because he'd been unwise enough to let the wild, drunken Burgess, an old friend, have a basement flat in his large house in Washington. This was the period when there was revulsion in Britain for the witch hunts of Senator Joseph McCarthy. The MP who'd made the charge against Philby had to withdraw, shouted down by his fellow Labour MPs.
After his official clearance we next see Philby in shirtsleeves at a picnic in the hills outside Beirut, squatting on the ground all smiles and sunglasses, the Middle East correspondent of the _Observer_ and the _Economist_ at work on a bottle of wine.
And then we don't see him at all.
On January 23, 1963, he got out of a taxi in Beirut "to send a cable" on the way to a party with Eleanor and never showed up at the party. Six months later Edward Heath, the Lord Privy Seal, told a questioner in Parliament that Philby had confessed to having "warned Maclean through Burgess," but this was all that emerged. The government had maintained total silence since. It had never even officially acknowledged that Philby had been a spy working for Britain's Secret Intelligence Service. (The SIS is authorized to operate abroad and is popularly known as MI6, for Military Intelligence section 6, as distinct from MI5, the domestic counterespionage agency.)
We made some progress putting together the jigsaw of Philby's life. Leitch snooped round Cambridge, his old university, looking for leads. Burgess and Maclean were members of a Communist cell started by Maurice Dobb. Philby did not join but did become a Marxist, spending his fourteen-pound prize money at Trinity on the complete works of Marx. He was moved by the sight of hunger marchers who'd walked all the way from the North East and organized meals for them. At Dobbs's suggestion he visited the Paris Comintern, and on graduation in 1933 he rode his motorcycle to the bloody ideological battleground of Vienna. Page called Eric Gedye, the _Daily Telegraph_ correspondent who'd been in Vienna then. The news from him was that Philby had married a vivacious Communist—we tracked down her first husband in Israel—and worked with the underground to smuggle Communists out of Austria to safety.
How could such a dedicated Marxist win entry to the Foreign Service a few years later at a time when the fear of "Bolshevism" was acute? The answer is that by then it was not the same Philby. The Philby smuggling clothes to Communists hiding in the sewers of Vienna was next lending his energies to the hated fascists. Back in London, he was named editor of the Anglo-German Friendship Society magazine. "Look," said Page, coming excitedly into my office with a framed photograph and a shrewd deduction, "here's Philby building a new identity." The picture was a swastika-bedecked black-tie dinner given by the Anglo-German Friendship Society in July 1936, and there clearly was young Philby. Maclean and Burgess also had both conspicuously and abruptly retreated from their Marxism.
When the Spanish Civil War broke out in 1936, Philby took himself off to the battlefront as a freelance reporter—not, like most of his generation, allied with the uneasy coalition of the left on the Republican side supported by the Soviet Union, but on the side of General Francisco Franco's coalition, which included monarchists, fascists, and the Roman Catholic Church. In 1937, when he was only twenty-five, he became the special correspondent of the _Times_ with Franco. We looked at his dispatches in the _Times_ library. They were slanted in favor of Franco's Nationalists, even after the bombing of Guernica by the German Condor Legion, the first mass air attack on civilians. One thing stuck in the minds of journalists we spoke to who'd observed him in Spain. He wanted to know the details of Nationalist troop movements—numbers, directions, and regiments—way beyond what readers of the _Times_ needed.
"Maybe," said Page, "he'd already been engaged by Soviet intelligence."
It was a prophetic insight. He had. But he had also been talent-spotted by MI6, the reason we deduced for his abrupt departure from the _Times_ in July 1940. What did he do in the service? We went through all the reference books to find the names of the staff in our embassies where Philby had worked. There were some clues about which ones had been intelligence officers: for example, an entry in the biographical reference book _Who's Who_ identifying someone as a member of the diplomatic service but whose name did not appear in the Foreign Office list. Few of the people we approached at first in MI6 would divulge anything. "Sorry, Official Secrets." Click.
Things improved when we knew enough to appear knowledgeable—the conversation might then inch us forward—but I got used to the expression of apology that my depressed reporters brought back from a fruitless day. Knightley had a typically tantalizing talk with a retired MI5 officer who told him, "Of course it was the defector in 1945 who put us onto Kim. After that you had only to look in the files to see it all." And what did you learn? "Better leave it at that, old boy. Don't want to get into trouble with the OSA."
I had known it would be a difficult assignment. We were asking questions about a nonperson, a disgraced member of a secret intelligence service that did not officially exist, whose head man was a letter of the alphabet ("C"), and whose headquarters address was a state secret. I could not keep Insight knocking on closed doors if there was nothing behind them. Perhaps I had given them a bum steer and there wasn't much more to Philby.
What happened next was like a fluke in a crowded pool hall, where as you make a shot, someone jogs your elbow and the cue ball caroms wildly round the table and ends up potting the black. Two high officials, striving to be unhelpful, jogged our elbow at the right moment.
The first was the former head of MI6, who'd retired to Wiltshire and was no longer the anonymous "C" but the aristocratic Major General Sir Stewart Menzies. He'd been in charge of the agency during the war and through the defections—a decent, canny man, much sounder than the upper-class clown portrayed as the head of the service in _Our Man in Havana_ by Graham Greene (who served under Menzies in the war). We wrote to him for an interview. No, he said politely by mail, he wouldn't talk to us. He was known in the service as the man with sealed lips, who wouldn't agree it was a nice day for fear of giving something away. But he could not resist adding a sentence in his letter: "What a blackguard Philby was." Oh, really? A man of such notorious discretion as Menzies must have been driven by deep feelings to resort to the vocabulary of Victorian melodrama.
The second elbow jogger was Lord Chalfont, a new minister at the Foreign Office. Leitch, who'd previously been on the staff of the _Times,_ had known Chalfont before his elevation on joining government, when he was Alun Gwynne Jones, the newspaper's defense correspondent. When Leitch and Page met Chalfont at the Foreign Office and mentioned Philby, the minister began by saying that Philby was a man of no importance, not worth a reporter's time and effort: _Let me save you the trouble. Anyway you couldn't possibly uncover anything about Philby. And if you did, you'd not be able to publish it._ Then, like Menzies, he felt compelled to say more: "You must stop your inquiries. There is the most monstrous danger here. You'll be helping the enemy." He was more charmingly restrained with me when I had a drink with him at the Garrick Club. Would the Foreign Office at least answer some questions I had? "Afraid not, but we'll not stand in your way."
I told Denis Hamilton that we were going to find out what Philby had done that caused so much alarm. He was troubled.
"How can you be sure you won't help the Russians?" he asked.
"Well, we can't tell them more than Philby must have already told them. Denis, we're the ones in the dark."
"And what about the Official Secrets Act?"
"Can't we judge the risks of that when we know more?"
"Let me think about it."
He didn't tell me the results of his deep think until a few days later. I was surprised. He called me to his office to say that "in great secrecy" he'd been to see Prime Minister Harold Wilson and the current "C." "The Foreign Office is alarmed. I told them you're not a man who would want to damage his country. But I've seen too much slaughter in my life, Harold. Will you let them see a draft to make sure you don't put anyone at risk? All I've said is that you will consider representations."
I said I would. None of us wanted to risk the life of an agent by some unwitting reference. The trouble is, we had nothing about which anyone could make representations. The trail was not just cold by 1967; it was frozen. I took up Hamilton's suggestion that I see Sir Denis Greenhill, deputy undersecretary at the Foreign Office. I discovered later he was the main link between the Foreign Office and MI6. Our first mutually wary meeting was in his Travellers Club in Pall Mall. Later we met in a Bloomsbury hotel; once in his unpretentious house in West London, where his vivacious wife, Angela, served tea and biscuits; and once in the Foreign Office. He was then fifty-four, silver-haired but well-muscled, dressed in a double-breasted chalk-striped suit, curt and brisk in manner. The son of a top manager in the Westminster Bank, he'd reached Christ Church, Oxford, by way of the lesser public school Bishop's Stortford and then got a job as a traffic apprentice on the London and North Eastern Railway, the ladder by which young middle-class men became the bosses of railwaymen like my father. He'd been a staff colonel in the Royal Engineers in World War II before entering the foreign service in 1946. He was not a typical Foreign Office man, but he exhibited the same hostility to our probing Philby as had Lord Chalfont.
About Burgess he was droll in his detestation. Burgess had been foisted on Greenhill when he ran our Washington embassy's Middle East department during the time Burgess and Philby were there. He recalled Burgess as an idle, shambling drunk who dropped cigarette ashes on other people's papers, drank other people's whiskey, paraded his homosexual promiscuity, and told entertaining tales to discredit the famous. "I've never met a name-dropper in the same class," Greenhill said. Burgess had a gift for caricature and once drew a Christmas card for Greenhill's small son with Stalin as Father Christmas. "I should have paid more attention to it at the time," Sir Denis noted, somewhat wistfully. But Greenhill's agreeable facility for recall eluded him over Philby. Not a word would he say about the man who'd fooled them all.
"You'll do more damage with the Americans if you write about Philby. Who's this fellow Page? What's _his_ game?"
Why did this minor figure, Philby, continue to excite such anxiety? Any number of people we'd reached were willing to reminisce about Philby as a Special Operations executive trainer of men and women dropped into occupied France. We were regaled with the prowess of Philby as 007—a master of unarmed combat, night sabotage, pistol shooting, and seduction of women. He shone, too, we learned, as an executive within MI6, which he entered in the summer of 1941, though his style was not to everyone's taste. Sir Robert Menzies, a Foreign Office security officer, was dazzled by Philby's "sense of dedicated idealism" and his mastery of the English language in his reports, submitted in neat, tiny handwriting. Miss Kennard Davis, from the vantage point of the typing pool, told Insight that Philby was harsh in his rebukes for mistakes. "I used to shrivel up like a worm. He used it on the men, too, just as effectively. I can remember walking into Graham Greene's office, and his eyes were glinting with anger. I asked him what was the matter, and he said, 'I've just had a caning from the headmaster.' "
All this was very helpful for sketching the outlines of Philby's personality, but none of it told us where the "most monstrous danger" lay.
Around this time I had a call from Michael Frayn. I'd seen him socially from time to time in the years since we'd shared a canteen table while I was at the _Manchester Evening News_ and _Guardian_. (That was long before he was acclaimed for his farce _Noises Off,_ the intellectually brilliant dramas _Copenhagen_ and _Democracy,_ and the rollicking Fleet Street novel _Towards the End of the Morning_.) He'd told me then how out of a passion for reading the original Tolstoy and Chekhov, he'd learned Russian at an army course after leaving Cambridge—Cambridge again—and then had an invitation to join MI6, which he declined. He called me to say he had a bright acquaintance of his who wanted to start a new career in journalism. I'd have seen him on Michael's recommendation, but I did so with alacrity when he added, "By the way, he's in the Foreign Office at the moment."
John Sackur was a mystery. A pale, earnest man in his late thirties, he presented himself as a crusader for black Africans. He was upset that Britain had not quashed the coup in 1965 by which Ian Smith imposed white minority rule on what was then British Rhodesia (now the benighted Zimbabwe). But was that really his motive in seeing me? I'd expected to draw the usual blank when I mentioned our interest in Philby.
Sackur appeared incredulous.
"Philby? You'll never be able to print it."
"Why not?"
"It'll get stopped—D [Defense] Notices, the Queen. It goes to the highest in the land." And then he bit out with real emotion, "Philby was a copper-bottomed bastard."
It emerged that he had written a report on the damage Philby had done, but he would not say any more. He'd already met Frank Giles, my deputy and foreign editor, who'd asked him if he was "a friend," the Foreign Office term for someone in MI6. He said he was. When later I accused Michael Frayn of foisting a spy on me—he recalls I was "very angry"—he said that he had no idea Sackur was in MI6, and I believe him. "John was a natural deceiver," Michael said. "He deceived me on several occasions at Cambridge."
I introduced Sackur to Bruce Page, who took him off to Manzi's seafood restaurant in Soho. The one clue Page pried out of him was that if we ever got to the bottom of Philby's betrayal, we'd realize that it mattered less what Philby had done in the cold war than in World War II. Eventually we discovered this was a reference to Philby's role in blocking evidence in 1944, before the German officers' plot to kill Hitler, that the German Army was putting out feelers for a separate peace with the United States and Britain. He may also have had a hand in the fate of a number of Catholic activists in Germany who'd been identified as possible leaders in an anti-Communist government after the war. They did not survive.
Knightley, who has become a specialist in writing on espionage, believes today that Sackur tipped us off because he represented a small group of young rebels in MI6 who suspected the service had not had enough of a purge after Philby. It is also just possible that MI6 encouraged Sackur to seek a foreign correspondent's job with us as a cover for working for the agency in the Middle East, just as they infiltrated the _Observer_ and the _Economist_ with Philby. Sackur's hints about Philby (and Blunt; see below), on this supposition, might not have been indiscretions but cunning bait to suggest he was disaffected enough to be a genuine defector whose integrity could be trusted. Recalling the intensity of expression on his chalk white face as he expounded on foreign policy at our first meeting, a lunch at the Ivy restaurant, I am inclined to conclude that he was not a plant but a young man whose conscience would give him no rest. I intended to talk to Sackur again when I conceived of writing this memoir, but he died before we could meet.
Only later did I realize the significance of Sackur's glancing reference to the Queen. Leitch had given Page the names of a number of leftist undergraduates in the _Trinity Review_. One was Anthony Blunt. At the time, in 1967, he was surveyor of the Queen's pictures, a part-time job in which he oversaw the care of the Queen's publicly owned collection. He was a pale, chilly, fastidiously mannered aesthete; a compulsively promiscuous homosexual; a favorite of a small group of the more arty courtiers—he dined often at Marlborough House with the Queen Mother and her set; and an enigma even to his friends. Twelve years later, in November 1979, Margaret Thatcher named him as a Soviet spy after Andrew Boyle, the founding editor of the BBC Radio 4 program _The World at One,_ had featured Blunt in his book about the Philby scandal, _The Climate of Treason_. Boyle had disguised Blunt as "Maurice," the figure in E. M. Forster's story of homosexual love. Blunt became an object of universal execration and was stripped of his knighthood. But Sackur must have been disturbed, as I was among many, at the immunity Blunt had been given in 1964 and the retention of his job at Buckingham Palace in return for a confession. (By contrast, three years earlier the less well-connected gang in the Portland spy ring had been sent to prison for fifteen to twenty-five years for stealing naval secrets.)
Page wrote to Blunt in 1967 asking for an interview. Knightley took the request round to Blunt's grace-and-favor apartment at the Courtauld Institute of Art. Blunt read it and slammed the door in Knightley's face. The next morning Page got a letter from Blunt's solicitor warning that he would sue for harassment if we ever tried to contact him again. We would not have been deterred by that if we'd had anything on Blunt, but we didn't. He'd been investigated by MI5 fifteen times and proved to be small-fry by comparison with Philby, but we didn't know that at the time. Not pursuing Blunt was a mistake I regret, but we were by then excitedly preoccupied with leads developing on Philby.
Our collection of espionage books had netted one titled _British Agent,_ written the year before by someone named John Whitwell. Knightley winkled out his real name, A. L. (Leslie) Nicholson, who'd been an MI6 man in Prague and later in Riga. Knightley found him to be a drunken burnout living on a miserable pension over a seedy café in East London. It was hard to believe he had ever been an MI6 officer. _Another wasted expense,_ thought Knightley as he treated Nicholson to a good Italian lunch and several brandies. But as Knightley gently pressed questions, inevitably revealing that we knew Philby was important and by implication that we didn't quite know why, Nicholson's enjoyment increased. He was aware of the seriousness of his illness (he died two years later from cancer), and over coffee and another brandy he told Knightley what Philby had really done.
"The reason for the flap, old man," he said, "is that Kim was head of our anti-Soviet section."
As Knightley put it, "I can remember trying to clear my head of brandy fumes." He pressed Nicholson. "Let me get this straight. The man running our secret operations against the Russians after 1944 was a Russian agent himself?"
"Precisely."
This meant not only that any intelligence operations against the Soviets were doomed from the start but that the days of all the MI6 agents already in place in the Soviet Union and Eastern Europe were numbered.
In the _Sunday Times_ office we puzzled over why Philby's crimes had remained undetected for so long. Two luminaries who talked to us about their wartime experiences with MI6—the Oxford historian Hugh Trevor-Roper and the journalistic provocateur Malcolm Muggeridge—spoke freely of their contempt for the quality of MI6's staff, apart from the clever code breakers. Trevor-Roper said that the permanent officers were drawn from two classes of men—"ex-Indian policemen and metropolitan young gentlemen whose education had been expensive rather than profound and who were recruited at the exclusive bars of White's and Boodles." But even if Philby's colleagues were as dim as alleged by these critics, both of whom took a poor view of the human race anyway, wouldn't one of any number of KGB defectors from the Soviet Union surely have given the game away?
The sheriff of Shropshire gave us an answer. He was then Mr. John Reed, living in a great house in a forest, but before retiring to Shropshire he'd been first secretary in our embassy in Turkey in the last year of the war. He was worried how our series might portray his role in a great blunder.
I sent Knightley to see him. He would talk only anonymously for the moment, concerned that the Foreign Office would not approve. Thus masked, he told a fascinating tale. On a hot August day in 1945, the area head of the Russian secret service (the NKVD, later the KGB), one Konstantin Volkov, had walked into the consulate in Istanbul seeking asylum. For a safe passage to Cyprus and living money, he would identify Soviet spy networks. "In return he told me he would offer the real names of three Soviet agents working in Britain, two of them in the Foreign Office, one the head of a counterespionage organization in London," Reed said.
The ambassador, Sir Maurice Peterson, wanted nothing to do with the nasty business of spies. He told Reed to let London handle it. Reed sent the information to London in a secure diplomatic bag and waited for a response. It took two weeks for an agent to arrive to debrief Volkov. That agent was none other than the head of the anti-Soviet section, Kim Philby.
Volkov was never seen again. Philby, who must have had a fright that he was so nearly outed, had taken time with his Soviet masters to organize a safe passage for Volkov—but not to Cyprus. A Soviet military aircraft made an irregular landing at Istanbul airport and within minutes took off again after a heavily bandaged figure on a stretcher was carried to the plane. "The incident convinced me," said Reed, "that Philby was either a Soviet agent or unbelievably incompetent. I took what seemed to me at the time the appropriate action." Nothing happened.
Knightley reported, "The memory of the betrayal made Reed's voice shake."
If Philby was that important in 1945, what was he doing in Washington from 1949 under the cover of being first secretary in the embassy? All the public attention had been on his possible role in tipping off Burgess and Maclean. There'd been nothing about what he did day to day. Nobody had asked, and nobody had talked. In view of the runaround we were getting in London, Knightley was entranced to be in Washington. He called the CIA and was directed to several retired officers who'd known Philby.
Lyman B. Kirkpatrick had been with the CIA since it was set up in 1947. This was a few years before Ian Fleming created the Bond fantasies—which Kirkpatrick regretted for misleading the public about the painstaking nature of intelligence work—but the Americans had been in awe of MI6 for its legendary history and the dazzling code-breaking achievements that had won the Battle of the Atlantic. The doors were wide-open for Philby, whose experience was so much richer than anyone's in the fledgling CIA.
When Knightley found Kirkpatrick, he had just retired as executive director of the agency. He had contracted polio in Asia on CIA business and was teaching politics from a wheelchair at Brown University. He combined secret knowledge with an intellectual zest for freedom. He wouldn't go into detail, but on the main point he didn't equivocate: "Philby was your liaison officer with the CIA and FBI." This was as astounding as Nicholson's revelation. It meant that for three years of the cold war, Philby had been at the heart of Western intelligence operations. Put another way, having penetrated the SIS, he was then able to penetrate the CIA. The director, General Walter Bedell Smith, gave him clearance at all levels, which meant that in a secret service's typically compartmentalized operations, Philby would have known as much as anyone except the director himself and perhaps one or two assistant directors, like Kirkpatrick. Kirkpatrick would not say much more than "have a look at Albania," but that chimed with vague hints we'd already picked up. Tom Driberg, MP, had told us that the Foreign Office had expunged from his book on Guy Burgess a scornful reference by Burgess to Western meddling in Albania, and a former MI6 man in Rome had told Leitch, "Philby lost us a lot of lives in Eastern Europe."
Back in London, Page asked a researcher, Alex Mitchell, to see if among the émigré groups in London there might be some Albanians who could give us a clue as to what lay behind Kirkpatrick's cryptic remark. When Mitchell found a bunch of them working as lumbermen for the Forestry Commission, he was directed to one of the few survivors of an ill-fated joint MI6-CIA operation. The two agencies had armed, trained, and funded a small army of guerrillas and put them into Albania by boat and parachute in the spring of 1950. The Communist rebels in Greece were faltering, Yugoslavia's Tito had broken with Stalin, and it was hoped that a Communist collapse in Albania would ripple throughout the Balkans. Hundreds of Albanian exiles, including the exiled King Zog's royal guard, were trained in a fort in Malta. They were doomed from the start. One of the few who escaped alive told us, "They always knew we were coming." At least three hundred died.
The disaster, reminiscent of President Kennedy's Bay of Pigs blunder infiltrating anti-Castro guerrillas in 1961, was put down to leaks from the infiltrators and the extraordinary efficiency of the Albanian frontier guards and police. Super-clever those Albanian cops, you know; they must have worked out the radio code by which the first infiltrators would signal back that it was safe to send in more men—then sent that signal when it wasn't safe at all.
The operation was jointly commanded by a CIA man and the British liaison officer Kim Philby.
Hugo Young, meanwhile, was making good use of his inside knowledge of Washington. He found that the significance of Maclean, too, had been missed by everyone. The urbanely supercilious diplomat didn't go home after the Washington cocktail parties. Several times a week, unbeknownst to Admiral Lewis Strauss, chairman of the Atomic Energy Commission (AEC), Maclean was wandering unescorted in Strauss's headquarters.
It was a crucial time in the transition from hot to cold war. The Russians were already making good use of the Allied bomb-making secrets given to them by the German-born physicist Klaus Fuchs and company, but they were desperate to find out all they could about the Allied policy for the atomic bomb. Britain, Canada, and the United States had all cooperated on the Manhattan Project, but in 1947, through the McMahon Act, the United States had abruptly ended the exchange of information on atomic weapons. An Anglophile manager, feeling it was unfair to exclude their British allies, had given Maclean a pass without checking with his superiors at the AEC. Young found that Maclean had access so rare that it had been denied even to General Leslie Groves, who'd supervised the building of the bomb, and J. Edgar Hoover, head of the FBI. Between August 6, 1947, and June 11, 1948, Maclean had been in the AEC headquarters twenty times, sometimes at night.
One of the strengths of the leaders of Insight was a readiness to admit a weakness and a resolve to remedy it by identifying people in the know. We didn't have the expertise to determine the significance of the information Maclean had likely gathered, so we asked the atomic energy historian Margaret Gowing to assess how important it might have been to Maclean's Moscow masters. Her answer: very. Especially useful were the quantities of uranium the United States was buying, from which Stalin's scientists could calculate the projected size of the U.S. nuclear arsenal. In the geopolitical poker game, this was a valuable face card.
We'd compiled many hundreds of thousands of words in notes and drafts, with multiple cross-references. Page, ringed in cheroot smoke, was at his typewriter early in the morning and late at night, collating and sifting through what we knew and what needed more checking into a longer and longer "state of knowledge memorandum." If he was not doing that, he was testing hypotheses on the inner circle; we kept Project X as tight as we could. I wanted to publish before the _Observer_ began its serialization of Eleanor Philby's memoir, scheduled for October.
By the first week in September Ron Hall had the text ready for the first of a four-part series. Greenhill was still protesting: "Drop Philby; he's a bore."
On September 1 I received a letter from the Services, Press and Broadcasting D Notice Committee. The key injunction (my italics) read: "You are requested not to publish anything about identities, whereabouts, and tasks of persons of whatever status or rank who are _or have been_ employed by either Service [MI5 or MI6]."
This was a direct attempt to wipe out our entire investigation.
The D Notice (now called a DA, or Defense Advisory, Notice) was a request approved by a joint committee of government and media representatives. The system was set up to warn the press against inadvertently publishing or broadcasting information that would "compromise UK military and intelligence operations and methods, or put at risk the safety of those involved." The D Notice did not have the force of law, but it could be cited against us if the attorney general chose to bring an Official Secrets case. Our lawyers were anxious. I told Denis Hamilton I was ignoring it. He didn't argue.
The team wanted another two weeks, and I agreed to hold off publication for two reasons: we were following a lead about "another defector" (who turned out to be Volkov), and we had our own Philby in Moscow—not Kim, but John, his son by his second wife, Aileen. Leitch had had the inspiration to check the London telephone directory and rang "Philby, J." He was living in a basement apartment in Hampstead with two Alsatians and an ambition to be a war photographer. Leitch, who had an engagingly jolly manner, casually asked him if he'd like an assignment to photograph his father in Moscow. He would.
The secretive Kim Philby broke his cover to greet his son and came out to stand in Red Square in an open-necked shirt, looking into the distance, his left hand nonchalantly in the pocket of his jacket. "My father told me," said John, "the Russians had given him the task of penetrating British intelligence in 1933 and that it did not matter how long it took."
We'd already worked that out. What we hadn't calculated was that John would have dinner on his return with the journalist Patrick Seale, who was at the rival _Observer_ and directly responsible for the Eleanor Philby serialization. The first installment had been scheduled to begin in late October, but the _Observer_ rushed it into print. A copy of the newspaper dated October 1, 1967, landed on my desk at 6:00 p.m. on Saturday, September 30. Not so fast! By 6:30 our "Spy Who Betrayed a Generation" was on the presses, with the scoop picture of Philby on the front page and the lead headline "Philby: I Spied for Russia from 1933."
As photojournalism, it was dramatic, but a short-term tactical error on my part. The combination of photograph and quote could give the impression that our investigative series had sprung from Philby or from the Soviet ministry of disinformation. We had enough disinformation coming from our own side. When Philby let us know he had written a memoir, we all recoiled. I couldn't stomach the monstrous egotism—what else?—that had allowed him to stay loyal to his masters after the Nazi-Soviet Nonaggression Pact in 1939 and the years of Stalinist terror. (He offered to suppress his manuscript in return for a deal in which the Soviets would release a British prisoner, Gerald Brooke, and Britain would release the Krogers of the Portland spy ring. His sardonic memoir eventually appeared in the _Sunday Express._ )
Disturbing as our findings of Philby's betrayals were, to me the most sobering revelation was how long Philby had been able to exploit the class-conscious and social attitudes of the club and old school echelons of MI6. Here was a Soviet agent within a breath of the final triumph, his appointment as head of MI6, yet his friends in the service still continued to resent the vulgarly suspicious security men of MI5. Even after he was dismissed from the service, "cleared" by Harold Macmillan on the understanding that MI6 would reorganize and institute a "general clean up," MI6 continued to employ him as a field agent in the Middle East and assured the _Observer_ and _Economist,_ both of which hired him, that he was no longer active. It was simply impossible for these gentlemen to accept that one of their own could have been a traitor to his class, let alone his country.
When we published our revelations in September–October 1967, I naively expected a demand for reform. Instead there was outrage, directed not at Philby or those who protected him, but at us. Sir Stewart Menzies, the former head of MI6, who at the outset had revealed his anger about the "blackguard" Philby, now somersaulted to vouchsafe that Philby had never been important in MI6. In the eyes of Donald McLachlan, a former intelligence executive and editor of the _Sunday Telegraph,_ we had even undermined the concept of the English gentleman. Several newspapers ran stories—not discouraged by official sources—that our life story of Philby was a Soviet plant. The accusations that we were handmaidens of the KGB seemed to me the product of minds incapable of confronting a real spy story without constructing an ersatz conspiracy around its origins. Only a political paranoid could have imagined, as the Foreign Office put about, that the iconoclastic Bruce Page was a Communist. I reassured them that he was safely locked up in a slave labor camp in Grays Inn Road.
Next the rumbustious foreign secretary and Deputy Leader George Brown got into the act. The first I knew of it was a late-night phone call to my home from a very upset Denis Hamilton. "The foreign secretary has just denounced you as a traitor at a business dinner and in front of Roy [Thomson]. You'd better be in the House of Commons tomorrow, when the foreign secretary will speak." I opened the next morning's _Times_ with some trepidation. "Ebullient Mr. Brown Hits Out" was the euphemistic headline, "ebullient" being a press parlance adjective to get round the libel risk of saying he was drunk.
The next afternoon I sat in the House of Commons gallery waiting to be dragged out by the sergeant at arms. The victim turned out to be the foreign secretary himself, carpeted that morning by the prime minister for once again showing undue ebullience in a public place. The _Daily Mirror_ 's front page the next day was headlined "The Bowed Head" and featured a penitent foreign secretary in a morning coat facing a stern Wilson on the platform at Waterloo station as they awaited the arrival of a dignitary.
When Roy Thomson was asked about the protests over our Philby series, he had an economical answer: "Bunk!"
My experiences as a provincial editor had given me some inkling of the barricades behind which the bureaucracies conducted the nation's business then. The Philby story was my first prolonged experience as a national editor dealing with central government and what, for want of a better name, I have to call the political establishment: those overlapping "charmed circles" of influence and power whose strands of DNA were the elite public schools, Oxbridge, the aristocracy, the City and the blue-chip boardrooms, the civil service, the legal profession, and the conservative press. British society had become more solvent, more meritocratic, and less deferential than it had been in the 1930s, when the Soviets saw very well how it was run by their recruitment of Burgess, Philby, Maclean, and Blunt. I don't for a minute believe that there was in the sixties or is now a homogeneous conspiratorial establishment—a "tightly knit group of politically motivated men," to misappropriate Prime Minister Harold Wilson's phrase against striking seamen. Obviously, people in the establishment and the various professions differ on issues today: the privately educated Labour men contend with the privately educated Tories. But a penchant for secrecy, social privilege, and the nurturing of an educational elite remained pervasive in the culture and strikes me as not having been entirely expunged in the twenty-first century.
A secret service is a secret service; I accept that. But the well-tried administrative precept that efficiency improves with accountability is not irrelevant even to the secret service. It was shocking to me that Blunt, having condescended to betray his country, was still in place at Buckingham Palace, a socially sought-after figure. It was a revelation that the closing of the ranks that had allowed Philby to survive was manifest again in the rebukes we earned for exposing him and the cover-up.
The consolation at the end of the Philby affair was in the awareness of my good fortune: Thomson and Hamilton had been steadfast in their support, and the staff of the paper, if I could manage it well enough, was a resource of extraordinary intelligence, vitality, and independence of mind. We could tackle anything!
#
# Children on Our Conscience
I never heard the word "teratogen" until 1963. After that I would never forget it. Between 1958 and 1962 pregnant women who reported morning sickness, anxiety, and lack of sleep to their doctors were prescribed a "wonder drug"—a pill with supposedly none of the side effects of barbiturates. It was thalidomide, a teratogen, meaning it interfered with the normal development of an embryo. Taken between the fourth and twelfth weeks of pregnancy, it caused babies to be born with foreshortened limbs or no limbs at all. More than four hundred children were affected in Britain, and worldwide it is estimated that between fifteen thousand and twenty thousand babies were affected in some way. Dr. Martin Johnson, who has made an extensive study of thalidomide, writes that apart from war and genocide, thalidomide was the cause of the largest man-made disaster in European history.
I first became aware of the thalidomide children, as they came to be called, in 1962. In 1963 we published some early pictures in the _Northern Echo_ of thalidomide infants at Chailey Heritage in Sussex, a hospital, home, and school for disabled children. I expected an outpouring of sympathy. I was wrong. Most of the readers who wrote did so to protest that it was not right for "a family newspaper" to do this: "We don't want to know."
I should have known better than to expect anything else. I had already experienced the awkward emotions in encountering people who are severely disabled. One of the patrons of St. Cuthbert's, a Catholic hospice in the trees by the river in Hurworth village where I lived, had promised a few of the patients there that she would find someone who "knew about print." She nominated me, so I called at the hospice one morning and met John Tinsley. He could not control his limbs. His head bobbed and his mouth twisted when he spoke. His vowels were prolonged and his consonants lost so that what came out was part howl, part speech. It took me some minutes to work out what he was saying—a plea not to laugh at his idea. He wanted to print a magazine created by patients like him at St. Cuthbert's.
John had been born with athetoid cerebral palsy; he was highly intelligent and could get around in a cycle chair (an electric cycle of sorts). Peter Jackson, the sports editor of the magazine he conceived, was a severely spastic young man who had to be strapped into his wheelchair—"thaaaat's J-J-Jaaaco," John said, jerking an elbow into him. And the magazine's cartoonist was a boy with spina bifida.
I said I'd be glad to help, but to be honest my heart had sunk at the thought of spending time with John and Peter and their group. Of course I felt sorry for them, but what was it Graham Greene said about "pity" being a way of distancing ourselves from the other? It's a form of reassurance, an acknowledgment of our own wholeness. That was an uncomfortable thought.
It took me a couple of visits to St. Cuthbert's to stop regarding the patients as victims and treat them as people, real personalities with their own individual identities rather than a "case" of spina bifida, a "case" of muscular dystrophy, a "case" of spastic diplegia. When I said what I really thought about an article, a poem, or a cartoon, we all started to have a good time. John would then tell me with a huge convulsive laugh why I was wrong.
The thalidomide scandal was the most emotionally draining of all the stories I became involved with at the _Sunday Times._ I came to think of the letters to the _Northern Echo_ as a metaphor for how the British legal and political institutions had responded over and over again to the tragedy. They'd made the thalidomide children invisible.
The children who had been affected by thalidomide were between the ages of four and eight when I became editor of the _Sunday Times._ Having published those early pictures in the _Northern Echo,_ I wanted to find out what had happened to them. In 1967 I learned that not one of them had received a penny in compensation.
This was a consequence of a heartless decision taken by my old political adversary in the _Northern Echo_ 's cervical cancer campaign Enoch Powell, who was the minister of health between 1960 and 1963 and thereby responsible for the National Health Service. In all the attention paid to the thalidomide story, his crucial role in the long nightmare has been neglected.
Powell was a baffling figure. There was no doubting his academic brilliance: he was a double-starred first at Trinity, Cambridge, and a professor of Greek at the University of Sydney when he was only twenty-five. Nor was there any question about his bravery: he was the only man in the war to rise from private to brigadier. He had high velocity, too, as a political thinker. He ricocheted from the impeccably lucid to the paranoid crazy: it was one thing in the 1940s to accuse the United States of wishing to see an end to the British Empire, and something else again to blame the CIA for the bomb that blew up Lord Mountbatten in Ireland in 1979, a crime clearly committed by the Irish Republican Army (IRA).
Millions paid attention when, in a carefully prepared speech in Birmingham in 1968, he said the threat posed by West Indian and Asian immigrants to Britain was comparable to the threat posed by Hitler in the 1930s. Britain, he said, was madly building its own funeral pyre by allowing in fifty thousand immigrants a year, and the antidiscrimination bills before Parliament would be match to gunpowder: "Like the Roman, I seem to see the River Tiber foaming with much blood." As policy, Powell's speech was defensible. But the tone and language were something else—an incitement. The speech blamed immigrants for all the social problems of the cities, indicting every person of color on the basis of a number of unverifiable ugly anecdotes and using such emotive language as "wide-grinning pickaninnies." Then he refused to disavow the personal and mob violence that followed.
William Rees-Mogg in the _Times_ called the speech evil, but it was the _Sunday Times_ criticism that roused Powell to sue for libel. Shortly afterward, when I bumped into him at a reception in the House of Commons, he turned his pale, mustached face toward me and said, "I will bury you." In fact, when ordered by the courts to produce letters he'd quoted to justify his extravagant rhetoric, he refused to obey and then did not proceed with the libel action.
Like many people, I was puzzled by the loose screw that turned Powell's brilliance corrosively inward. That he refused to help the dispirited and powerless thalidomide families is hard to understand or forgive. He received a delegation of affected parents in January 1963 and rejected their every request: No to a public inquiry on the origins of the disaster. No to immediately setting up a "drug-testing centre"—"anyone who takes an aspirin puts himself at risk." No to a public warning against using any of the pills that might still be in medicine cabinets—"a scare-monger publicity stunt." No to giving a statement afterward—"no need to bring the press into this." And no to his setting eyes on a thalidomide child: no to Frederick Astbury without legs, arms, and right hip; Louise Mason, no legs; David Bickers, no legs, no arms, and just three fingers from his shoulder; Gary Skyner, a short left arm and no thumbs; and Eddie Freeman, no legs and foreshortened arms.
It was an extraordinary lapse of public duty to deny society the knowledge essential to understanding the origins of the tragedy and preventing anything similar from happening again. Powell's intransigence left the families with only one remedy: to sue the manufacturers for negligence. It was bad enough thus to condemn the parents and their children to live in Bleak House legal torment. The courts, in any case, could not be relied on to find the facts. The primary objective of the judicial system is to settle conflicts; it is not to find the truth come what may. A civil action for damages is an indirect, laborious, very slow, and costly way of unraveling the facts. The families were not well-endowed for that exhausting exercise.
They were up against a formidable defendant. The thalidomide pill was marketed by Distillers Biochemicals, a subsidiary of Distillers, the giant liquor and spirits company retailing famous brands of whisky, such as Johnnie Walker. It had assets then worth more than £400 million—£4 billion at today's values—and pretax profits of more than £500 million at today's values. Its chairman, Sir Alexander McDonald, was a Glasgow University graduate of accountancy and law who'd been with the company since 1946. He'd gathered a reputation as a man of granite, which suited the Distillers image, and he'd engaged the rapier mind of John Wilmers, QC, much feared for his capacity to eviscerate witnesses.
While the families' counsel changed four times, Wilmers stayed on the case throughout. The families had to rely on public funding through the Law Society, the body representing solicitors which administered government funds for people who could not afford going to court. It begrudged spending on what it regarded as a weak case. Three and a half years passed before it authorized paying expert witnesses.
Distillers wasn't a long-established pharmaceutical company. It had no research scientists and didn't hire a pharmacologist until it decided to manufacture thalidomide under license from a German company called Chemie Grünenthal, which was also relatively new to the business. By one of those mocking twists of history, the _Sunday Times_ had a part in the company's decision. Aldous Huxley, the author of the prophetic 1932 novel _Brave New World,_ contributed an article to the paper in June 1956 in which he speculated that modern science might be able to produce a happiness drug (called Soma 6 in his novel). "Will the pharmacologists be able to do better than the brewers and distillers?" he asked. Provoked by the question, a Distillers director read Huxley. Might a pill be found that would eventually become an alternative to whisky? This new "nontoxic" sedative from Chemie Grünenthal seemed to fit the bill. Distillers sold it as a safe sleeping pill and tranquilizer called Distaval beginning in April 1958. Another company, Richardson-Merrell, sold it as Kevadon in the United States. Millions of American mothers escaped the catastrophe because Dr. Frances Kelsey, at the Food and Drug Administration, while not predicting the phocomelia deformities, became suspicious of Richardson-Merrell's sloppy procedures. Even so, some ten to sixteen American mothers were known to be affected by the premature release of the pill. What a savage irony that a "happiness" drug was to cause so much human misery.
So upset were hospital midwives and doctors at the births of the first thalidomide babies that they often concealed the children from the mothers—making excuses until it came time for their discharge. The mothers, isolated from one another but struggling with the same emotional and physical difficulties, did not know where to turn. They had paroxysms of guilt; something must be deeply wrong with them or their families or their husbands' families. Some believed that God had punished them for a lapse in their lives. Some felt ashamed that they'd complained of the symptoms that had prompted their doctors to prescribe the dangerous pill. Scores didn't know for several years that they were victims of thalidomide. They didn't connect the news stories about the drug to their own predicaments. The psychological traumas were intense. Mothers hid at home rather than encounter people who recoiled from the sight of an afflicted child. Marriages were wrecked. "If you bring that monster home," said one husband, "I'll leave." And so he did.
The impulse to put the tragedy swiftly out of mind was understandable. It was too painful. Somebody, surely, would look after those unfortunate children and their families so that we could get on with our lives. Somebody didn't and wouldn't.
Here again Health Minister Powell was no help. He'd left the families no choice but to seek redress through the courts, then put the whole weight of the ministry behind discountenancing the very grounds of negligence they'd have to prove to win damages. Thalidomide, he said in Parliament and in a television interview, had been properly tested by Distillers according to the standards of the time, when nobody thought a drug could reach the fetus. And unanimity prevailed on this key question. The _Economist, Times, Manchester Guardian,_ and _Sunday Telegraph_ all said the same thing. As we were to discover, none of those statements was true.
* * *
In fact reproductive studies had been routinely done by pharmaceutical companies a decade before Distillers made thalidomide. The tranquilizers in direct competition with thalidomide were tested for teratogenic effects, and the results were published by the manufacturers, including Hoffman–La Roche; Lederle; Pfizer; and Smith, Kline & French in America and Burroughs Wellcome and ICI, under Dr. Edward Paget, in Britain. If reproductive tests had been done with thalidomide, they would not necessarily have produced the precise deformities, but they would have shown that it might endanger unborn children.
How could it be that everyone got it wrong? We learned later that all the assertions, which is all they were, had originated with Distillers. The senior civil servant signing the letters to parents and advising Powell had taken his cue from a Distillers medical executive who'd been with the ministry and had called to brief him. The newspapers also found the company most helpful. Like some primitive medicine man's incantation, the unceasing repetition—the standards of the time, the standards of the time—hypnotized everyone who heard it.
When I first arrived at the _Sunday Times,_ I longed to take up the cause of the thalidomide children. Some sixty-two of the abandoned families had begun civil actions in August 1962, while I was still with the _Northern Echo,_ and the British law of contempt of court meant that their writs sealed the whole affair in a legal cocoon. Once the writs had been issued, it became illegal to bring out facts or comment on a pending trial, whether civil or criminal, on pain of a heavy fine or even jail. With thalidomide this meant that nothing could be published that might influence a judge until every case—every single case—had been settled by the courts. This is plainly a denial of free speech, but the common law had long held that this right of free speech had to be balanced by the right to a fair trial uncontaminated by outside pressure on any party or information or comment that might sway a judge or jury.
I'd had a narrow squeak in my first months at the _Sunday Times_ when, thanks to a slip by a writer, compounded by a duty lawyer's misjudgment, we'd accidentally published the previous convictions of a man about to stand trial. I was personally exonerated, but the newspaper was fined £5,000. The thalidomide actions were in civil court, to be tried by a judge alone, but the law was no different. It didn't matter to the court that there had been no movement on the cases between 1962 and 1967. (Indeed, in 1973 a law lord referred to the eleven-year-old pending thalidomide cases as being "in the early stages of litigation.")
The families had been told by their lawyers on no account to talk to "mass media," but what we most wanted to know at this stage was how the disaster had occurred. Nobody at Distillers or Chemie Grünenthal would help, but we suddenly got the chance to look into the files of both companies. Reporter John Fielding brought a Dr. Montagu Phillips to see me. Phillips had been engaged by the families' solicitors, Kimber Bull, as a consulting pharmacologist and chemical engineer. He was a small, rumpled man who fidgeted a lot—not, I thought, likely to be impressive in the witness box under Wilmers's brand of interrogation—but he possessed ten thousand Distillers internal memoranda and reports that he said documented the scandal. They'd been made available through the legal process known as "disclosure," by which both sides are obliged to supply all relevant files. In short he was a whistle-blower, or what we called a FINK—fair insider with necessary knowledge. But was he "fair"? He had a grievance. His wife had taken thalidomide as a sleeping pill, which he believed was the cause of her irreversible nerve damage, known as polyneuritis. How much was that influencing his judgment? Moreover, he was breaking a legal honor code and was asking me to risk a breach of journalistic ethics: he wanted £8,000 to help care for his wife.
Was this the checkbook journalism I'd inveighed against on the television program _What the Papers Say_ and in speeches criticizing newspapers that paid for the memoirs of criminals? I'd never done that. In the office and at home, I debated the ethics of paying him. It was a consideration that he offered us his technical advice and, more important, promised to keep us informed on what went on behind the closed doors of the legal negotiations. Every last doubt fell before my intense curiosity about what the documents revealed on the origins of the disaster. Was I to put my precious journalistic conscience before gaining access to crucial information that might never see the light of day if, as seemed likely, an out-of-court settlement was reached? No. We collected the documents, and I assigned the unflappable Phillip Knightley to evaluate them and prepare a narrative for publication sometime when we were free of legal restraint.
More urgent was a second set of documents from Chemie Grünenthal. Henning Sjostrom, a Stockholm lawyer representing 105 Swedish victims, was concerned that since it had taken the Germans seven years to mount a criminal trial of nine company executives, it might take another seven years of court testimony to resolve the case: the prosecution alone said it had 352 witnesses. Other than wanting his expenses paid, Sjostrom was looking for international publicity to shame the company that had caused so much havoc. The German newspapers dared not publish anything for fear of punishment under their own contempt laws.
Collected in three suitcases, the German papers, when translated and indexed, were to reveal a get-rich-quick mentality at Chemie Grünenthal. The mother's safety during pregnancy while taking Contergan (as they called the drug) was a main selling point, but the company had not tested Contergan to see if it could pass through the placenta to affect the unborn child. Chemie Grünenthal sales leaflets for doctors stressed the drug's safety, quoting a March 1960 article in the _American Journal of Obstetrics and Gynecology_ by Dr. Ray O. Nulsen, practicing in Cincinnati, Ohio, the hometown of Richardson-Merrell. In his deposition for the German trial, however, Nulsen admitted that he had not tested the drug on pregnant women at all and was not even the author of the article. It had been written for him by an employee of Richardson-Merrell—who'd relied for information on Chemie Grünenthal!
In light of what I knew was in the German documents and was emerging in the Distillers documents, I was very surprised when Mr. Desmond Ackner, QC (later Lord Ackner), appeared in court for the families in February 1968 to say they were withdrawing charges of negligence. Distillers had agreed to pay 40 percent of what it might have had to pay if the legal actions had been wholly successful. Both parties still had to agree on what sum of money would represent an award of 100 percent, but the 40 percent, said Ackner, would amount to "very substantial" damages. If the actions had continued, he added, "the plaintiffs could have failed to recover a penny apiece."
Judge Sir Raymond Hinchcliffe endorsed Ackner: "I have given anxious consideration to the issues and to the prospects of success in law and fact, having regard to what the reasonably careful manufacturer would have done before marketing the drug in the then state of knowledge. In my judgment the plaintiffs are well advised to accept."
This sounded absurd to us. By this time Godfrey Hodgson, the new editor of Insight, had constructed a shocking four-page narrative of the history of the German company's invention and marketing of thalidomide. James Evans, the elegant and indispensable full-time staff lawyer at the _Sunday Times,_ advised me that the law of contempt almost certainly made it too dangerous to publish. Not only would revelation of Chemie Grünenthal's reckless conduct cast Distillers in a bad light, but the announcement of Ackner's interim settlement had stimulated more families to sue Distillers, so the gag order on discussion of the cases extended into the unknowable future.
Evans had to be heeded. He was not one of the legal fraternity schooled in the scholarship of suppression; he was a facilitator. If publication was in the public interest, he regarded it as his duty to justify it in law. Now, unusually unsure, he suggested we seek yet another opinion by consulting a specialist in one of London's grand Inns of Court, a new experience for me.
We'd hardly taken a chair in the Brick Court rooms of David Hirst, QC, before he pronounced, "Contempt! Flagrant contempt!" I tried suggesting that on this logic, when the German trial started every British newspaper reporting it would be in contempt. Mr. Hirst (later Judge Hirst) was disturbed by my belligerence (so was I). All these years later I see his eyes, unblinking behind his glasses, evoking the sensation of being in _The Great Gatsby_ looking up at the gigantic blue irises of the oculist Dr. T. J. Eckleburg brooding on his billboard over the valley of ashes. There is no escape! Flagrant contempt!
Sulking on the way back to the office, I pressed James Evans, who went back to Hirst and got the same answer in writing. His own doubts, as much as my persistence, led him to seek yet another opinion. Mr. Peter Bristow, another QC on the way to the bench, spelled out the risks, too, but ended on a somewhat less alarming note than the steely Hirst.
I decided to ignore them all and publish. This was not only my heart speaking but also my head. I realized that in the long run, without any challenge, the oppressive British press laws were not just a threat to the thalidomide victims and their families but a real threat to democracy itself. These laws were a teratogen in themselves, stunting and deforming our freedom and liberty. I was so appalled by the way Chemie Grünenthal had disregarded early warnings, I emotively juxtaposed photographs in the Review Front of a deformed German child and one of the architects of his misfortune, a white-coated head of research at Chemie Grünenthal.
I waited in a defensive crouch for action by the law officers of the Crown or Distillers, but they stayed in their battlements. Distillers had clearly blundered in relying on Chemie Grünenthal. Its own documents revealed how little independent testing the company had done and how similar were the unjustified marketing and the inexcusable delays in withdrawing the drug from the market. The Australian obstetrician William McBride had notified at least two, and possibly six, Distillers Sydney employees of his suspicions of problems by the first week of July 1961, but the company had continued to promote thalidomide as safe for another four months (during which time about a quarter of the British children known to be affected were damaged in the womb).
Our publication of the German story would, I assumed, stiffen the sinews of the families' lawyers about what constituted "substantial" in the damages Ackner had said were in train. But we had to wait a whole year to find out what that meant. Unable to agree on what 100 percent of damages would be, the parties came back in July 1969 to ask Judge Hinchcliffe to resolve the deadlock by making monetary awards in two test cases. One was in the top bracket—David Jones, then nearly nine, who had neither legs nor arms and would need help for the rest of his life, never being able to toilet himself or dress and undress. The other, Richard Satherly, was in the middle bracket. He had legs, but only a single digit protruded from his shoulder; he brushed his teeth by holding the toothbrush in his toes.
The press hailed Judge Hinchcliffe's resultant awards with the usual "fortune" headlines associated with lottery winners. We didn't. I made space on the editorial page for a critical article by Nicholas Harman, who had joined us from the _Economist_. He noted the judge's conclusion that his award to Richard of £32,000 "would be sufficient to keep him free from financial worry and would go some way to ameliorating the discomfort and deprivations that he is bound to suffer." All very well, said Harman, but if £32,000 was "sufficient" to these ends, then the 40 percent he would actually receive, £12,800, would be 60 percent insufficient.
It was a temperate article, but I wrote the aggressive headline "What Price a Pound of Flesh?" and soon had one of James Evans's duty lawyers in my office. "Afraid this is risky.... Not happy about your headline.... Contempt." He pointed out that since even more families were now suing, the contempt restrictions continued. "You could be accused of interfering with the course of justice."
I let the article and headline run.
Partly as a result of these stories, families kept coming forward to press their claims. By 1971 some 266 had secured waivers from the three-year statute of limitations, and another 123 victims had identified themselves, making 389 not provided for by the first settlement. The contempt rules, having kept the tragedy in the shadows for nine years, now extended until the unknowable future date when every last one of these cases would be settled.
In the meantime outrage followed outrage. The lawyers representing the 266 new claimants as a group went back to Distillers, but ice had formed on the Scottish granite. The company declared that it would set up a trust fund worth only £3.25 million—about £7,500 for each victim—half the 1969 settlement. Moreover, the offer was conditional on every single family accepting the offer.
The lawyers commended the offer to the group of families. Six families objected, led by David Mason, a sophisticated Mayfair art dealer who was the father of Louise Mason, who had no legs. The six dissenters regarded the offer as miserable charity, whereupon their own legal advisers—yes, the parents' legal team!—took them to court on behalf of those ready to settle. Judge Hinchcliffe removed parental rights from the dissident six. Henceforth a treasury solicitor would decide what was best for their children—that is, accept the Distillers offer.
Mason had a friend who knew David English, the editor of the _Daily Mail,_ so he took his personal story there. English published three articles under the title "My Fight for Justice by the Father of Heartbreak Girl Louise." When the attorney general warned the _Daily Mail_ that these articles constituted contempt of court, English backed off. The Court of Appeal resoundingly restored Mason's rights, but the _Mail_ published no more articles, BBC television's _Twenty Four Hours_ canceled a related program, and Mason stopped giving interviews that never appeared. Once again the cloak of invisibility enveloped the thalidomide children.
As we entered 1972, I had plenty of other issues to deal with, but I just couldn't get the thalidomide negotiations out of my mind. I was convinced: something was badly wrong. If the Knightley article presented, as a judge later said, "a powerful case of negligence," why had the lawyers been eager to accept only 40 percent of the less than adequate 100 percent? And why were they now ready to settle for something like half of that? What made me even more determined was a new perspective in a long memorandum from Bruce Page. His successes as head of Insight had led me to create the Special Projects unit around him, with Elaine Potter to assist. Elaine was a South African whose tenacity matched her scholarship. She'd not had a great deal of experience in journalism, but she'd acquired an Oxford PhD and, as important, squatter's rights to a freelancer's chair in the features department. Some of our most successful recruits were squatters; they were tested by the exigencies of sudden demands for labor, and the best, like Elaine, survived with the complicity of editors until I could find a place for them on staff.
I'd asked Page to revisit the whole thalidomide project with Elaine. His first initiative was to compare the money awards with a more precise assessment of the children's needs; his second was to send Elaine to pharmaceutical laboratories in the United States and Britain to check the defensive refrain that Distillers had done all the tests that were standard at the time—exposing the myth I mentioned earlier. Page's memorandum on the compensation money was startling. In the _Modern Law Review_ of May 1972, he read a two-part article on the absence of any coherent method by which judges assessed personal injury damages. It was written by John Prevett, a fellow of the Institute of Actuaries with the London firm of Bacon & Woodrow. Prevett, it turned out, had given expert evidence to Judge Hinchcliffe in 1969 that the lead test case in the thalidomide trial, that of David Jones, required £135,000. The judge had awarded half that amount, with that half in turn subject to the 60 percent reduction agreed upon by the lawyers. This meant that David's actual cash for the rest of his life was only £20,800.
When recently I spoke to John Prevett, now in his retirement, he remembered wondering whether he was making any impression at all on the judge: "He seemed to be asleep during the long time I was cross-examined on the numbers by the Distillers lawyer [John Wilmers]." The judge said he had to be fair to Distillers as well as the children, but he must have been hypnotized by the mesmerizing Wilmers to swallow the QC's contention that inflation should be disregarded because the government had promised to control inflation. Ah, yes, said the learned judge, all "speculation" and "hearsay."
As for the inflation that the judge and Distillers' counsel regarded with such amused disdain, the value of the 1969 award was by 1972 worth 20 percent less, and in 1975 alone the inflation rate was 24 percent. Prevett told me, "Altogether the awards were sixteen percent of what I reckoned was justified. I worked out that the money the judge awarded David would run out by the time he was twenty-nine."
Why did nobody, including the _Sunday Times,_ fully report on Prevett when he was in court in 1969? It was hardly rocket science. And here's a terrible thought: if the _Modern Law Review_ had not, three years later, asked Prevett to write on actuarial advice, and if he had not chosen thalidomide as his example, and if Page had not read the journal, it's possible we might have underestimated just how rotten the personal injury system had become.
The priority now had to be making the case for decent compensation and to do it before more families were sold down the river. It was hard to find out what was going on in the negotiations. The Distillers solicitors refused our every request for information with heavy breathing about contempt, and our contact, their adviser Dr. Phillips, was not privy to the secret negotiations. But Knightley had formed such good relations with the families that a number of them confided in him, including the fearless Mason. The shock of the Page-Prevett assessment was searingly with me when Knightley strode into my office, his phlegmatic personality unusually agitated. "Can't we do something? They're being told they'll lose their legal aid certificates if they don't sign. Most of the parents are ready to give up, and yet they'll still get only half of what the 1969 families got. Half 1969!"
The new pressure on the families was the last straw for me. The much-feared law of contempt was going to sanctify a gross injustice. It was urgent to shout that it must not be allowed to happen. Was I emotional about the thalidomide families? Yes, I was, but my decision that Tuesday to launch a campaign in the next issue of the paper, five days hence, was not a sudden impulse. My experience of campaign journalism at the _Northern Echo_ had convinced me that certain conditions had to be fulfilled before a newspaper undertook a campaign. The paper had to have investigated the subject thoroughly enough to be sure that there was a genuine grievance, it had to have defined a practical remedy, it had to be ready to commit the resources for a sustained effort, and it had to open its columns to counterarguments and corrections of fact. No campaign should be ended until it had succeeded—or was proved wrong.
I called James Evans: "I'm going to campaign on thalidomide starting this Sunday come what may, contempt or not." He preserved his celebrated equanimity. "I have the picture perfectly," he responded. "Alpine tourist asks guide to take him to the top of the Eiger by the safe route. Let me think about it, hmm?" I told Denis Hamilton we were going to denounce the proposed settlements and was phoned by the advertising manager, Donald Barrett. "You know, Harry, Distillers is our biggest client, sixty thousand a year." Then he added, "I know that won't stop you, and it shouldn't."
Page came in from vacation to convert his Prevett memorandum into a three-page analytical narrative for the upcoming issue of September 24, 1972. It was a bitingly cool piece, but the display by Ron Hall and our new design chief, Edwin Taylor, incorporated an editorial across the top of the page with a picture of a pretty young girl who had no arms. James Evans drafted the editorial. We would, he proposed, acknowledge Distillers' duty to its shareholders and its denial of negligence but also demand that the company fulfill its moral obligation: "The law is not always the same as justice."
I did very little to James's draft. I headlined it "Children on Our Conscience" and devised the emotional slogan for the coverage I hoped we could continue: "Our Thalidomide Children, a Cause for National Shame." At the end of the three pages demolishing the levels of compensation, I intended to announce our upcoming article on Distillers' documents. "May I say that it would be wise," murmured James, "to give yourself and the courts a little more time to sort that out?" Wise indeed. I simply wrote that "in a future article" we'd trace how the tragedy had occurred.
It proved to be the matador's cape.
We were engaged in a high-risk enterprise, but after all the delays, all the anxieties, all the legal frustrations, this was an exhilarating time. We all felt as if we'd been let out of prison.
I invited David Mason and his wife, Vicki, to watch the presses start up. Over the din he yelled, "Until now I never believed it would happen!" It was quite a contrast with the media silence that followed, but that first Sunday morning a most important telephone call came to my home from a man who lived in permanent silence. Jack Ashley, the Labour MP for the Midlands potteries town of Stoke-on-Trent, was stone deaf. I was amazed to conduct a telephone conversation with him: his wife, Pauline, listening in, enunciated my words to Jack; he read her lips and responded.
Ashley was one of my heroes. He'd left school at fourteen, then became a laborer and crane driver in Widnes, a sprawling industrial area, and a shop steward for the Chemical Workers Union. Later he won scholarships to Ruskin College, Oxford, and Caius College, Cambridge; produced radio and TV documentaries for the BBC; and got elected to Parliament the same year, 1966, that I joined the _Sunday Times_. He wasn't deaf then—just one ear was less good than the other—and since he was in line for a ministerial appointment, he decided to have an operation to equip him for the rapid exchanges in the Commons. It went wrong. His first instinct was to give up his seat, but there was an uprush of support for him so spontaneous and moving that it encouraged him to stay—and to do what he could in Parliament for all handicapped people.
Ashley came to the office with Pauline to meet Page and Knightley, asked questions, and said, "Well, that's decided it. I'll give up writing my autobiography until we've won." It's a little commentary on the postwar social changes that Ashley was swiftly joined as an ally in Parliament by another northern working-class boy who'd left school at fourteen and gone to Oxford on an adult scholarship—my old schoolmate Alf Morris (who later became the first minister for the disabled). So we had a partnership of Widnes and Brookdale Park elementary schools, embellished by Oxford and Durham. The cause soon became bipartisan when two Tory physician MPs—Gerard Vaughan, a pediatrician at Guy's Hospital, and Thomas Stuttaford—joined with Ashley and Morris.
Yet we might as well have been publishing on the moon for all the attention we got. One BBC radio program gave us a few minutes of airtime—with a jumpy BBC lawyer hovering in the control room—and that was it. To the media we were Typhoid Mary. Twenty-three days went by without another mention on radio or a single story in print or on television. Even David English, the enterprising editor of the _Daily Mail,_ was unwilling to follow up his newspaper's original Mason series. Not until the eighth week of our campaign did the _Mail_ comment. "Saving our space to cover your trial" was the joking response I got to the calls I made to the media.
There was silence in Parliament, too. The efforts of the four Labour and Tory MPs to get the issue raised were squashed. A letter from Ashley to Prime Minister Edward Heath brought only a rebuke: "Legal matters are not for this or for a Labour government." This was a fair statement of the long tradition that Parliament never intervened in any subject being decided by a court of law, but it meant that Ashley couldn't get a resolution before the House of Commons or even raise a question. I went to see the melancholy minister of health, Sir Keith Joseph, in his gloomy office at the Elephant and Castle south of the river. I came back empty-handed, as depressed as Joseph—another very clever man who, like Powell, was averse to getting involved. The law must take its course etc.
The first intimation that we might have legal trouble came three days after our September 24 blast. I was handed a letter in an embossed envelope. It was from the solicitor general warning that our editorial and article could be considered contempt of court. The attorney general was in Strasbourg and would deal with me upon his return. Meanwhile, said the letter, "you will no doubt wish to consider your position." I replied that we were canvassing moral obligations, not legal, and were not to be deterred from continuing. James Evans then applied his emollient forensic skills in several phone conversations with the attorney general's office.
In the third week of the campaign, we got another embossed envelope: "The Attorney General instructs me to say he has now considered the material in _The Sunday Times_ of 24th September and 1st October and does not propose to take any action over the matter already published." By not accusing us of contempt of court for the first article, he legitimized the moral campaign. I put in an immediate call to Ashley to give him the news but didn't get through all morning. I had to leave the office for a lunch appointment, not knowing that Ashley had that very afternoon arranged to see the Speaker of the House of Commons, the nonpartisan figure who controls the conduct of business, to make one last plea for Parliament to discuss thalidomide.
I'd just arrived at the Ivy restaurant when a call came through to me on the cloakroom attendant's phone in the vestibule. It was Ashley speaking, with Pauline listening on an extension so as to enunciate my words for Jack to lip-read. What I heard from Jack above the bustle of the lunchtime crowd being greeted by the maître d' was that he could not speak for long because he was due any minute in the Speaker's room on a last-ditch effort to have the parliamentary restrictions removed. I had a rush of blood to the head. In my excitement to tell him that the attorney general was not taking us to court over the first article, and hence was recognizing the distinction between moral and legal justice, I forgot that everything I said had to be mouthed by Pauline so that Jack could read her lips and respond. Jack's patient voice came on the line. "Sorry, Harry, didn't get that. Say again slowly, please." When I'd calmed down, I was able to say that if the attorney general accepted the distinction between moral and legal justice, surely the Speaker would have to follow suit.
It was not that easy. It meant the Speaker would have to overrule the prime minister and shake, if not sever, the precedents of barring any parliamentary discussion when an issue was before a court of law.
Jack argued his case that afternoon, and the Speaker said he would rule the next day. It was then Jack's turn to rush a call to me. The Speaker had just authorized Jack to put down an Early Day Motion (a procedure for collecting signatures, rather like a petition) and they'd agreed on the language Jack might use. This was a vital breakthrough that opened the way for parliamentary activity. Within days 265 MPs had signed in support of the Early Day Motion.
Two days later I got another letter from the attorney general. He was going to court to ban publication of our promised future article on the manufacture of thalidomide.
The campaign I began on September 24 was to run more than three months, into January 1973, and the reverberations continued for years after that. Just as in the Timothy Evans case, there were misgivings from within the newspaper regarding my obsession: we'd offend some readers, bore others. Elaine Potter still remembers the theatrical yawns she encountered from some staffers as she produced "yet another thalidomide story." I understood the concern. There was clearly the risk, too, that I'd get distracted, that I'd not organize enough talent, time, and space for other stories. The world was not standing still. On September 5, Palestinian terrorists murdered Israeli athletes at the Olympic Games, and in October three of the killers were released in Germany following a hijacking. In November Nixon was reelected president of the United States, but there was now a shadow called Watergate; Stephen Aris in New York was agitating to investigate; and Secretary of State Henry Kissinger said "peace was at hand" in Vietnam. Ulster was reeling from the killings of civil rights protesters on Bloody Sunday; Prime Minister Edward Heath was campaigning for Britain to embrace the European Union more fully while the Labour Party wrangled with itself about whether to stay in Europe or pull out; and Idi Amin was terrorizing Uganda.
I was in the thick of all these events as an editor but determined to keep faith with the thalidomide children by campaigning week after week, month after month. The challenge was to keep readers interested.
The editorial template was human interest stories, argument on personal injury law, news of the Ashley-Morris activities, and powerful photography that would engage readers' compassion and dare them to look away. I wouldn't have done this without my experience in Darlington with the program to save women from cervical cancer and vindicating Timothy Evans. I was as eager as the next editor to find something new and exciting, but living close to the readers as I did then, I'd noted how we in the trade, absorbed by every story, became bored before the readers did. Our challenge was to keep dramatizing the pain. We achieved this with photographs, such as one by Michael Ward of an adolescent figure, hung up on a steel stand, wearing laced black shoes and white ankle socks. The shoes and socks were attached to the artificial legs on a body harness for a limbless person. The artificial arms attached to the harness were equipped with metal talons. I remember thinking that there was no way the person in the harness could put on the socks or tie the shoelaces. He'd have to be hoisted into the frame like a medieval knight lifted into his armor.
I identified everything we published with the "Our Thalidomide Children" logo. Readers reacted quite differently than they had ten years earlier in Darlington. Now they didn't turn away, perhaps habituated to the shock. Instead I received hundreds of letters and telephone calls of support, asking what they could do. Many of the letters came with checks; scores came from thalidomide families. In the legal profession and the press, however, we had severe critics. The _Law Society Gazette_ (representing solicitors) denounced the campaign as having "all the subtlety and legal justification of Robin Hood's activities in Sherwood Forest." The columnist Peregrine Worsthorne in the _Sunday Telegraph_ condemned it as nauseating, saying that the company had only done what the country had willed. The Oxford historian A. J. P. Taylor described it as "a witch hunt, an exploitation of popular feeling such as Dr. Goebbels would have rejoiced in."
That the _Sunday Times_ had been attempting to print the truth was a nicety Taylor and others chose to ignore. But I was buoyed by the letters from ordinary readers, especially those from thalidomide families. David Jones's father, a nurse, had taken on the full-time task of looking after his legless son while David's mother, a teacher, went out to work. "When the compensation awards made headlines," he told us, "people withdrew their sympathy. They treated us as if we'd had a lucky win on the pools. After the years of struggle it took time to register what a piffling sum we had. Had I fully appreciated the problems we are facing now—David is growing too heavy for me to lift—I'd have stood up in that courtroom and shouted."
I said we'd run a family story every week until we succeeded. "And what will you do when you've run out of all four hundred fifty families?" someone asked. "Start again," I said on the spur of the moment. "They'll all be older!" To do that we recruited Marjorie Wallace, who was directing television programs for the BBC. She had a gentle nature and a degree in psychology, and she managed to depict the struggles without being mawkish. It was she who found nine-year-old Terry Wiles, a highly intelligent two-foot-tall limbless trunk with one eye, abandoned by his mother and adopted by an inventive sixty-year-old van driver, Leonard, and his young wife, Hazel, living in a dilapidated cottage near Huntingdon. The National Health Service equipment had been of no use to him. From scrap metal and old army surplus, Leonard had invented a "supercar" chair for Terry on the principle of a forklift, so the boy could press a button with a shoulder and raise himself to talk to people. These tragedies could not be ignored, but the authorities were about to try again to switch off the light.
We lost 3–0 in the High Court on November 17. We were forbidden to publish the Knightley article based on the Distillers documents revealing the company's unjustified reliance on Chemie Grünenthal science and Distillers' own questionable marketing activities. The ban probably earned as much ink as the article would have if we'd been allowed to publish it.
Sir Alexander at Distillers counterattacked. When shareholder Tony Lynes, who'd inherited sixty Distillers shares, took up the cause in a letter to the company, he got a stiff lecture from Sir Alexander on the chairman's legal responsibilities. It was easy for a single shareholder to take a moral stand, Sir Alexander declared, "even easier for a newspaper editor," but the chairman could be sued by any of the 250,000 shareholders holding 300 million shares if he gave money away without their unanimous approval. Legally, he said, the parents were probably entitled to nothing. And if the _Sunday Times_ campaign led to the breakdown of negotiations with them, well, that's what they'd get, nothing. The company would go to trial—a danger to the families because their lawyers had already unwisely conceded that they had no case on negligence.
It sounded like blackmail. "For three hundred and seventy children," said Ashley, "the sword of Damocles has been replaced by the jagged edge of a broken whisky bottle." Alf Morris, too, had been working in the background. He invited me to come to the Commons to meet the leader of the opposition and soon-to-be prime minister, Harold Wilson, and Sir Elwyn Jones, a former attorney general who was to become lord chancellor. Morris was the front bench spokesman for social policy, and in the meeting, with me cast in the role of Greek chorus, he begged Wilson and Jones to give up precious opposition time for a debate.
The Labour opposition had only four days in the whole year they could claim as their time, but they agreed, and on November 29 Morris and Ashley spoke to a packed House, calling on Distillers to recognize its moral debt and on the government to set up a trust fund for the children.
On the way to see the Labour leaders, I'd been stopped in the lobby of the Commons by Prime Minister Heath's parliamentary private secretary, Timothy Kitson, asking whether we'd end the campaign upon hearing the information they'd just received: Distillers was willing to increase its offer of a trust fund from £3.25 million to £5 million. I said no; I'd been advised that any realistic assessment of need had to be closer to £20 million.
Distillers' £5 million proposition, announced by the government, was brushed aside in a debate in which massive emotional indignation and vivid personal testimony swept both sides of the House. "How can an eleven-year-old girl look forward to laughing and loving," said Ashley, "when she has no hand to hold or legs to dance on?"
Heath relented with the announcement of a £3 million fund for the congenitally disabled, with a further £3 million for the thalidomide children. Moreover, he set up a royal commission to examine the whole question of personal injury damages.
Two months into the _Sunday Times_ series, the campaign was self-sustaining. It gathered momentum from an unexpected quarter—Distillers shareholders. Tony Lynes joined with Sarah and Roger Broad to take up Sir Alexander's challenge. They circulated a letter to thousands of shareholders. Sir Alexander had blithely assumed that the shareholders were interested only in maximizing their dividends, but that proved a misjudgment. Ron Peet, chief executive of the Legal and General Assurance Society, the second-largest life insurance company in Britain, which had 3.5 million Distillers shares, concluded not only that corporations had moral as well as legal responsibilities, but also that the obduracy of Distillers was provoking such consumer hostility that it would damage its own commercial interests. A trade union pension fund chairman declared, "Distillers is blundering around like a Glaswegian drunk."
Peet's snowball started an avalanche. More big companies, insurance brokers, merchant bankers, and local authorities joined in. Stores and customers boycotted whisky made by Distillers. The duty-free stores in London and at European airports reported customers saying, "Anything not from Distillers." In the United States Ralph Nader planned an international boycott, briefed by the indefatigable David Mason (who lobbied passengers on his transatlantic flight). Over nine days Distillers shares lost £35 million in value. This was arithmetic Edinburgh understood.
On January 23 the company gave in with an immediate commitment of £20 million, about ten times the original offer. Jack Ashley rang me on the Saturday to say that he'd discovered the treasury intended to tax the money. I wrote an editorial for the next day. With intervention by Ashley and Morris, the prime minister, now Harold Wilson, made the trust good with £5 million from government funds. The families and the campaigners were overjoyed by the victory and a settlement by which Distillers, in the end, accepted a liability of £28.4 million, embracing all the actions started in 1969.
I was delighted but strangely numb. I'd never doubted we'd win more money for the victims once we were able to let people know what was happening; my faith in free journalism and in the common decency of ordinary citizens had been reinvigorated. But I'd lived with the story for so many years that I kept imagining the victims' daily lives—getting into that harness, opening a bottle with one's teeth, holding a toothbrush between one's toes, wondering if ever they'd know the joys of marriage and family life.
And the fact remained that we had not yet laid bare how the whole appalling mess had been created. I kept Page and Potter on the investigation, and a few months after the monetary settlement—not years—they were able to conclude that there had been an excellent case waiting to be developed, but the parents' legal advisers had failed to assemble it. Because the contempt rule still applied, it was not until June 27, 1976, that we were able to publish the Page-Potter findings documenting the powerful and scientifically sound case that the parents might have had all along. Solicitors and counsel had not secured decisive testimony available in Britain and the United States. No doubt the legal aid system had restricted them, but we spent relatively little energy establishing the falsity of the position taken by Powell and the "quality" press that Distillers was blameless. In this regard, Dr. Phillips turned out to be a poor choice of expert adviser. The documents he gave us were significant, but his advice on why the tragedy had been foreseeable was not. Insofar as we mentioned him, our draft article would have been seriously flawed.
The occasion for publication was our triumph—no other word for it—in a declaration by the European Court of Human Rights. By a vote of twenty-four international judges, the court ruled 13–11 that by banning our report on the origins of the tragedy, Britain's law of contempt had breached the free speech article of the European Convention for the Protection of Human Rights. This meant that the British government had to enact a statutory reform so that the years of silence on thalidomide could never be repeated. Report and comment could henceforth be allowed in civil litigation until a case was actually set down for trial.
In the end it was all worthwhile. We exposed the real reasons for the disaster and achieved reasonable compensation for the families, a royal commission on personal injuries, and somewhat more freedom of the half-free British press.
An extraordinary coincidence provides a coda to this chapter. As I was writing this account, a stranger telephoned me from Yorkshire. It was Guy Tweedy, a property investor in Harrogate, wanting to come and talk about a campaign. Not the _Sunday Times_ campaign of the 1970s, which he knew very well, but one he'd started with Nick Dobrik, who ran a jewelry business. Both are relatively able-bodied thalidomide victims who've joined with other more typically damaged thalidomiders to advocate for victims' rights. They've already, with the help of Jonathan Stone, a former law partner of Lord Goodman, succeeded in getting Prime Minister Gordon Brown to rescind his decision as chancellor to tax benefits payable through the Thalidomide Trust. (Does the treasury have a tin heart?) Furthermore, they've briefed the successor company to Distillers, the Guinness group (now part of Diageo, the world's largest beer and spirits company), to such effect that it is repairing the damages of that "hearsay" inflation with another £160 million over thirty years.
Still, the payments from the trust, very fairly distributed, cannot meet the needs of the most affected of the 457 surviving thalidomiders. Money that seemed adequate three decades ago is no longer able to ensure a decent life for someone like Lorraine Mercer or Vivien Barrett. Lorraine has no legs or arms and only one hand emanating from her shoulder; she longs for the independence of being able to get about in a car with wheelchair access, now possible thanks to advances in technology, but such a car would cost £50,000. Vivien Barrett, no arms, legs, or feet and only one hand, teaches music theory part-time but fears how advancing age may affect her. Those thalidomiders who were able to work are finding they cannot continue.
The indomitable Jack Ashley has been moved to action again, calling for the state to recognize its responsibility for the disaster. Unfortunately, the then Labour minister of health, Alan Johnson, was in the Powell mode. He said he was "not persuaded" there was a case for financial aid.
The thalidomiders feel a bond with their "brothers and sisters" in other countries. In Italy, Spain, and Austria, the victims have had no financial compensation at all. In Germany, where there are more than five times as many thalidomiders as in Britain, the maximum annual payment is meager. And Chemie Grünenthal insisted in the stingy private settlement made in 1969 that none of the victims there could ever complain or campaign.
I called the _Sunday Times_ editor, John Witherow, who immediately took up their cause.
I find it inspiring that the thalidomider activists, whose early years were such an ordeal, are now extending a helping hand to others, and then I realize with a stab of pain that many of them have no hands at all.
#
# Space Barons
Did I mention how invigorating it was to have a big-time national newspaper with the best staff in town? I still get a high from the fumes of those Saturdays when a vague idea from the beginning of the week—or an investigation started months back—crystallized into a thriller package of story, headline, photograph, and graphic. Or the glorious moments when we got our hands on the first copies of the newspaper, expunging all the raw, urgent untidiness of the passion and fine-tuning in the making of it. How authoritative everything looked! How delicious the smell of the still warm newsprint! How envious the rivals would be! Truth requires the admission that the rivals, the _Observer_ and the _Sunday Telegraph,_ sometimes arrived with a story that had eclipsed ours or had escaped us altogether. Once we'd reduced the shock by declaring to each other "nothing new in that," we scrambled like mad to catch up.
The week began at the editor's conference on Tuesday. Early on I had felt it necessary to issue my first edict. The head printer knocked up a sign I pinned to my office door: "Smokers are welcome but not their cigarettes." My peace offering to ease the pain was a souvenir I'd brought back from a Moscow trip, a samovar that was supposed to keep a flow of hot tea going.
Most of the talking was done by "collectors," heads of departments who said what they hoped to bring in from reporters and columnists, photographers, graphic artists, critics, and reviewers. These executives were responsible for people, not space on the pages. As the week accelerated, they bid their best efforts to the space barons, the handful who decided what should go in the premium spots in the paper.
I found it impossible to stay away from the "backbench" where managing editor and chief subeditor assessed stories, chose photographs, and planned pages. The Saturday news pages were designed by Robert Harling, who'd worked with his chum Ian Fleming on black propaganda operations run by the navy's Unit 17Z. Ingenious ways had to be devised to keep me from making observations that might impede the haste to go to press. I heard later that Harling had been instructed to distract me in the last minutes of closing page one. "Tell me, Harry," I can hear the saboteur say even now, "which do you prefer, sex or skiing?"
I had my own source of relief from the week's tensions: squash. It required careful calculation and downright furtiveness to get away with it. At some stage during an afternoon, at a time decided at the very last minute, I went down Lord Thomson's private elevator, jumped into a waiting car, and sped off to the Royal Automobile Club (RAC) in Pall Mall for a match. "Sped off" is the wrong description of the times I got snarled in Trafalgar Square traffic and had to run the last 220 yards to make my game. I think it was then I decided to renew my motorcycling. Every minute counted. I could not be out of the office for more than an hour (and since there were no cell phones then, I had elaborate arrangements with my secretary, Joan Thomas, for how to get word through if the Russians landed).
I varied squash partners week to week, almost always people from the paper. The urgency of chasing news carried over into the precious half hour we had rushing like maniacs about the court. Sometimes the small black ball metamorphosed into the wily barrister who had convinced the judge that there was no possible public interest defense for our newspaper's desire to report on, say, the bribing of a British MP by a foreign government. Sometimes it was the lupine grin of the Communist union leader in the pressroom who had once again sabotaged our production. (Calls to him from Moscow came collect, which was a bit much.) In any event, just hammering the ball, even though it was too often returned with nonchalance, was therapy lasting several days.
I came to share the half hour with different discreet staff players at the paper, not one of them, curiously, willing to give the editor the benefit of any doubt about questionable line calls. I will say in our defense that we often continued _Sunday Times_ business in the locker room, although I didn't join the regulars who repaired to the Blue Lion pub opposite the office to discuss what had happened that week to "their" paper (everyone regarded it as their paper).
Over the years various allegations have been made about my treatment of staff and guests: that I sent a ninety-eight-pound woman reporter into a field with a bull to illustrate a story on the risks hikers face; that I had to be cornered in the men's room to sign "pink slips" for cash advances for indigent reporters; and most commonly that I'd invite visitors arriving for lunch at the paper to decamp for a brisk cheese salad at the RAC, then keep them waiting poolside while I swam my customary twenty laps. These stories and worse are all, regretfully, true. Very few appreciated the offer to jump in the pool with me. One interviewee, in something of a hurry, obliged me to conduct our discourse standing at the shallow end between laps.
The _Sunday Times_ staff was small by American standards, never more than 160 men and women on the newspaper and its color magazine, about a tenth of the size of the editorial staff of the _New York Times_. But one of the strengths of the _Sunday Times,_ as it grew to sixty-four and then seventy-two pages, was that it was informed by a pool of unconventional talents. Today, when journalists tend to come into newspapers and television by the same conventional route—college, journalism course—the staff of the _Sunday Times_ in retrospect seems eclectic to the point of parody, but they knew how the real world worked. Time and again the richness of their backgrounds enhanced the paper. Among those who'd come to newspapers in mid-career were a molecular biologist, a professional pilot, a clinical psychologist, a civil servant, a university lecturer, an antiques dealer, a television producer, a research chemist, a bond salesman, and an accountant going straight—very helpful when we were looking into the financial records of some crook or other.
Most of the staff were university graduates, but a number had not completed high school. Insight's Paul Eddy, a relentless investigator who is now a best-selling thriller author, left school at fifteen. The newsroom's pertinacious Anne Robinson had been expelled from school and worked as a chicken gutter and saleslady before trying journalism; later she became a transatlantic television celebrity as taskmistress for _The Weakest Link_ quiz show. By contrast, Oxford classics graduate Anthony Holden, who was only twenty-five when he joined the _Sunday Times,_ had already written two books on Greek poetry and another on Graham Young, the Neasden teenager who'd tried out poisons on his family and continued his mad murdering ways when prematurely released from Broadmoor. While reporting for a local evening newspaper, Holden had come top in the examination held by the national training scheme. These days he maintains his humiliating versatility by writing literary biographies, competing in the world poker championships, and running a poker website.
A number of staffers had reached the _Sunday Times_ via the cauldron of the Fleet Street populars; a few had experience on American newspapers; many had pounded pavements in the provinces. We had regular infusions of renegade "colonials" from Australia, Canada, and South Africa, never inclined to take anyone's "no" for a final answer. And we fostered active relationships with a corps of freelancers, paying the ones we'd learned to trust for their efforts even when the story turned out not to satisfy our standards: next time it might.
Our coverage of wars was powerfully assisted by people with experience of both front lines and deadlines. Among those who'd seen military service of some kind, we had a company of infantry, an instructor in desert tank warfare, a tail gunner, a trio of army colonels and naval commanders, a sprinkling of wartime intelligence officers, and at least one corporal I know of. When Tom Stoppard became intrigued about pressmen coping with concepts of freedom in the context of reporting the murderous idiosyncrasies of an African dictator, the theme of his play _Night & Day,_ he explored the experiences of our roving foreign correspondent Jon Swain and photographer Bryan Wharton, who'd dodged a lot of bullets. In fact paratroopers regarded Jon as a talisman.
The staff photographers like Wharton struck me as rather like Battle of Britain pilots, lounging around with their cameras round their necks, ready to take off on hazardous missions at a moment's notice. Wharton, in fact, looked the part, down to the handlebar mustache and fur-collared leather jacket, sauntering into the newsroom with a pretty girl on his arm. The entire news photographic team were on the battlefields almost as soon as a shooting war started. Photographers with their equipment were more conspicuous in danger zones, and they'd all absorbed the great photographer Robert Capa's injunction, "If your pictures aren't good enough, you're not close enough."
It's no exaggeration to say that our cameramen had more experience with combat than many of the young soldiers. Don McCullin always went with the frontline troops. He advanced with U.S. Marines foxhole by foxhole in the battle for the citadel at Hue and carried a sniper victim to first aid; in Cambodia he was wounded in a Khmer Rouge ambush; in Uganda he was held in the mad Idi Amin's Makyinde jail with prisoners led out to execution; when the Israelis fought yard by yard for the old city of Jerusalem in the Six-Day War, he (and our Colin Simpson) was with them. McCullin's reputation for coming through gun and mortar fire alive preceded him wherever he went. Israeli soldiers were seen to touch his jacket for good luck.
I had loved selecting and editing photographs at the _Northern Echo,_ enchanted by the way the photographers, bored with flower shows and factory openings, took advantage of their freedom. But none of that had the acrid taste of danger that came with the prints the _Sunday Times_ men brought back from conflicts and disasters.
During students' anti-Gaullist street battles in Paris in 1968, for instance, Wharton and Frank Herrmann were in the thick of it ducking tear gas, then found themselves overwhelmed by military-strength CS gas wildly unleashed by the backup riot police, the CRS (Compagnies Républicaines de Sécurité). The difference from tear gas was devastating. Wharton got more of it, since he was with the students. It dropped him to his knees and blinded him for about twenty minutes. Yet he went back into the gas twelve times to photograph the CRS beating everyone in sight with their rifle butts. In a note to me he wrote, "I saw them corner a sixteen-year-old girl, pummeling her in the face with rifles, dragging out defenseless old men and women from cafés and beating them without mercy. I slipped on the blood on the floor when they charged and went down with camera straps entwined round my neck."
Team journalism was vindicated by the way it worked in Paris, with Herrmann and Wharton feeding their experiences to the group of reporters coming in from the tumult to write in the old Gestapo haunt, the Hotel Meurice. The reporters telephoned copy to us in fragments, and John Barry in London, working through Friday night, pulled the strands together into a single arresting narrative we ran over eight pages. Wharton, after a mad drive from a blockaded and blazing Paris, had to catch the ferry from Ostend, in Belgium, to bring back his and Herrmann's film in time. I say "bring back" because in those predigital days, getting images to London required ingenuity—and stamina. One Friday afternoon, with little more than twenty-four hours to Saturday's deadline, we dispatched Wharton to an earthquake in Osoppo, Italy. The local airport was fogbound. He drove 350 miles through the night to arrive at dawn, took his pictures, drove 350 miles back, flew home from Milan, and raced into the London darkroom just in time for the first edition.
Photojournalism had center stage in the color magazine, although the magazine irritated fastidious media critics by the dexterity with which it could move from serious reportage to glamour and celebrity chic. The _Sunday Times_ was particularly responsive to the vibrant popular culture and fast-moving social currents of the 1960s: the Beatles and Stones; the Pill; liberal new laws on hanging and homosexuality; race and abortion; theater censorship formally abolished; the satire boom sparked by Peter Cook and Dudley Moore, _Private Eye_ and _That Was the Week That Was;_ London's eclipse of Paris as the fashion capital.
We weren't aware of it at the time, but the magazine was itself a tiny chip in the mosaic of the swinging 1960s, its long-serving editor Godfrey Smith and art director Michael Rand matching the counterculture with innovations of their own. They employed gifted young artists such as David Hockney and Peter Blake to go abroad and paint the story for us, and they deployed such big-name photographers as Robert Freson, Arnold Newman, Norman Parkinson, and Bruce Davidson. Those artists and photographers were happy to work for the magazine at a fraction of the pay they could get from advertising because it gave them worldwide prestige (and they kept their copyrights). Lord Snowdon was on the staff, liberating himself from the confines of fashion and theater by pursuing serious photojournalism of the highest order. The flow of photographers was classless, with the East End boys Terry Donovan, Brian Duffy, David Bailey, and Patrick (Lord) Lichfield trooping in. Even more pleasing to all the males, drawn like moths to the light box, was when Julie Christie dropped by to see our new pictures of her.
We felt affinity, too, with a group of young men in the advertising industry. Until the very early 1960s, advertising was stuck in the time warp of social class and condescension: don't even think of being an adman if you're bereft of a military background, a public school–Oxbridge education, or a southern counties accent, preferably all of the above. Then John Pearce, padding the corridors of the small, bright new ad agency Collett Dickenson Pearce (CDP) in his stocking feet, attracted an electric group of talents in their early twenties—David Puttnam, Alan Parker, Ridley Scott, Frank Lowe, and Charles Saatchi. We borrowed Puttnam for six months to help with the nerve-racking launch of the color magazine.
These CDP youngsters transformed advertising much as the color magazine transformed Sunday newspapers. The ad industry in the 1960s was like the TV series _Mad Men:_ lose an account, and you're fired. But when Ford Motors rejected a daring piece of creativity from CDP, the agency turned the tables: it fired the client. Creativity, the agency wanted to suggest, was king. Puttnam, Parker, and Scott became celebrated filmmakers; Saatchi joined with his brother Maurice to create what would become the world's largest ad agency, Saatchi & Saatchi.
I can claim a small part in the brothers' launch in 1970. Having seen how uncontrolled display advertising played havoc with good design in American newspaper, I'd laid down the maximum size of a display in the _Sunday Times_. The advertising director asked me if I would see the young ad executive Maurice Saatchi to hear his pitch for our making a single exception. Saatchi was preceded into the room by the largest pair of horn-rimmed glasses I'd ever seen, a fitting ad in itself for the twenty-four-year-old with a first-class honors degree from the London School of Economics. He made a cunning appeal to my vanity, patriotism, and compassion. As an innovative editor, he wheedled, I should support an innovative business; as a citizen, I should support a new business for Britain's sake so that the advertising trade would not be dominated by American companies; and as a compassionate person, I should bear in mind that without the prestige launch in the wonderful _Sunday Times,_ the Saatchis would never make it.
I relented, just this once, and their launch in our paper put their new agency on the map. I've not been forgiven for this by some of my friends, since the Saatchi agency went on to devise devastating attacks on the Labour government, which sadly were entirely justified. (A billboard of a long trail of men lining up for unemployment money was headlined "Labour Isn't Working.") Their agency was a powerful promoter of Margaret Thatcher's ascension to prime minister, and Maurice eventually became Lord Saatchi, chairman of the Conservative Party.
I must admit I inflicted on the staff every half-cocked rumor and vague hint I picked up. I cannot recall anyone blinking; whatever they might have said out of earshot, they invariably delivered. I came to expect them to make gold out of flax. I got a taste of my own medicine from Lord Thomson himself. He called me up on a Saturday night to ask whether I'd like to sit down with Howard Hughes. If so—"entirely up to you, Harold"—I was to meet Ray at the London airport the next morning. Thomson knew Mormons in the entourage of the famously reclusive Hughes, and they'd told him that Hughes was ready to discuss how we might interview him for a book and newspaper serial.
We flew to Miami and met with Chester Davis, Hughes's lawyer, and Hughes's Mormon aides, who were shuttling back and forth between our waterfront room and wherever they were keeping their nutty boss. Hughes remained elusive, but it was satisfying in its own way to see how Roy negotiated money matters. He sat on his bed in his underpants and calculated the sum he intended to offer.
For me there were two tricky moments with Thomson on this abortive mission to see Hughes. The first morning, when I arrived in the dining room for a very early breakfast with Roy, I was carrying both the _Miami Herald_ and the _Wall Street Journal_. "Why'd ya buy two newspapers?" he growled. At the end of the trip, when we went to the front desk to pay our departure bills, he had another spasm. He reckoned that since we were leaving at noon, he hadn't incurred a charge for a full day, and he began myopically going through the charges item by item. The duty manager was summoned, and an unseemly argument loomed between frugal peer and frosty manager. I suggested to Thomson that he leave me to sort it out while he accompanied the porter to his room. I volunteered for this because when I ran my own eyes down his charges, I was horrified to find that the hotel had charged him for some of my expenses—a massage, a suit pressing, and two calls to home in London. I suggested to the manager in the privacy of his office that merely transferring the charges to the right room was not sufficient recompense for inflicting a peer of the realm with all this embarrassment. Roy's satisfaction, when I told him that the hotel had made an error and his bill was being reduced, made him beam for the whole trip home.
Can embarrassment be retrospective? In one photograph from the seventies, I'm caught seated, gesturing, in front of a news poster on display behind my chair at the editors' weekly conference. In big black type, the poster announces: UGANDA SENSATION. The poster, sent to my office for approval for distribution across the country, promoted a scoop we'd secured documenting the crimes of General Idi Amin, otherwise known as "the Butcher of Uganda," "Big Daddy," and "Emperor of All the Beasts and Fishes on Earth"—the bloodthirsty madman so vividly portrayed in 2007 by Forest Whitaker in the movie _The Last King of Scotland_.
We didn't allow visitors to our editorial meeting, but this week we had a guest by royal command: Prince Charles, the Prince of Wales. He'd expressed an interest in how the paper was put together to Denis Hamilton and our managing director, whose wife was lady-in-waiting (personal assistant) to the Queen. I could hardly demur, but I wished a better day had been chosen. As head of the Commonwealth, Queen Elizabeth was about to receive a bevy of African leaders among the thirty-five prime ministers attending a Commonwealth conference during the Silver Jubilee of her accession to the throne. We intended to rebuke them for continually failing as a group to unite in condemnation of Amin. (A situation similar to the failure of African states to unite against the outrages of Robert Mugabe in Zimbabwe in 2008.) Prince Charles would now have to hear how we planned to attack guests of his mother.
Enough was known about Amin's reign of terror to feel repelled by the way racial fraternity had given him immunity from censure. It was bad enough that the Organization of African Unity, predecessor of the African Union, had not only looked the other way but in 1975 had unanimously elected him chairman. It would be contemptible, I thought, if the Commonwealth and the UN remained supine in light of the horrible story we were preparing for the coming Sunday.
The source of the sensation was a thirty-seven-year-old defector from Amin's government by the name of Henry Kyemba, who'd known Amin for twenty years and had been his minister of health for the past five. Kyemba had walked into my office a few days before in fear of his life. "I want to ensure," he told me, "that what I know does not die with me." That same morning I bundled Kyemba and his wife, Teresa, into an office car with a driver I knew I could trust and sent them to my weekend cottage in Alphamstone, Essex, with the writer Russell Miller. It was melodramatic but necessary. Amin had dispatched agents to kill Kyemba; one had already been detained at Heathrow Airport by an immigration officer alerted by the Home Office, which had granted asylum to the minister.
Kyemba's arrival in London had been daring. Sickened by what he'd seen and knowing that five of his cabinet colleagues had been murdered, he had made elaborate plans to defect while in Geneva heading Uganda's delegation to a World Health Organization conference. So as not to arouse suspicion, he'd left his two infant children in Kampala in the care of relatives. Only when he received word that they'd safely avoided Amin's goons and border guards by trekking through the bush to Kenya did he evade the other Ugandans and make his way from Geneva to my office.
It was a stroke of good fortune to have Miller ready to begin debriefing him immediately. Miller was well informed on Uganda because he'd prepared for an interview with Amin, fixed by a retired army officer who'd something to do with the "whiskey run," the twice-weekly flights of spirits, cigarettes, and luxury goods from Stansted Airport to Entebbe by which Amin secured the loyalty of his thugs. It was good fortune for Miller, too, since he'd been due the next day to catch a plane to Kampala. Had he been in Kampala representing the _Sunday Times_ when we published Kyemba's story, he would not have survived. Amin had not hesitated to murder other journalists, including Nicholas Stroh, freelancing for the _Philadelphia Evening Bulletin_ , and his associate Robert Siedle, who had been killed just for asking too many questions about a massacre at Mbarara barracks.
Kyemba was the first senior member of Amin's government able and willing to speak with unique authority on the reign of terror, because his position gave him access to all hospitals and mortuaries. By the next morning, when I rode my BMW motorbike to Alphamstone, Miller had already solved the mystery of the seventy-five-year-old British-Israeli grandmother Dora Bloch, who'd been among the 106 hostages held by Palestinian hijackers at Entebbe Airport courtesy of Amin. She was unlucky to miss the electrifying rescue of the hostages by Israeli commandos on the night of July 3, 1976. The night before the raid she'd choked on a piece of meat and been taken to Mulago Hospital. Kyemba visited her there and found her recovered. The morning after the Israeli raid she was nowhere to be found. In response to inquiries from the British High Commission, Amin said that she'd been returned safely to the airport an hour before the raid and had presumably been taken by the Israelis. Indeed, hospital records proved it. In truth, Amin had ordered Kyemba to falsify them.
In the peace of the Essex countryside it was chilling to listen to Kyemba's description of how Dora Bloch had become a victim of Amin's rage after the Israeli raid: "He went berserk. If he's provoked, he reacts like a wild animal and goes into a kind of fit. No one around him is safe." The morning after everyone else had been rescued, said Kyemba, four of Amin's "State Research" officers arrived at the hospital. Two of them carrying pistols and shouting for staff to stand aside grabbed Bloch from her bed and frog-marched her screaming down three flights of stairs and out the main hospital door, without shoes and dress and in full view of patients, staff, and visitors. They dumped her body by the road twenty miles from the city. A photographer who took a picture of her partially burned corpse was murdered.
Kyemba knew, too, what had happened to the Anglican archbishop of Uganda Janani Luwum and two cabinet ministers who'd all been reported killed in a car accident. He'd seen their bodies in the mortuary. "They were riddled with bullets, the archbishop shot in the mouth," Kyemba said. "The country is littered with bodies. They're fed to crocodiles in the river."
I left Miller at his typewriter and raced back to the office with a sketch of the hospital layout Kyemba had drawn; we had very little time before deadline.
Our Kyemba story had an instant effect. The prime ministers at the Queen's conference condemned Amin for "massive violations" of human rights. For the first time, African leaders allowed open discussion of the internal affairs of another African country. Britain broke off diplomatic relations with Uganda. America's ambassador to the UN, Andrew Young, denounced Amin in vivid language. Amin was ousted in 1979. He should have been tried as a war criminal for hundreds of thousands of murders, but Saudi Arabia gave him sanctuary as a convert to Islam. He lived there with his four wives, fancy cars, and chef until his death in 2003.
Only later did I realize that the UGANDA SENSATION might have been troubling to our guest that day, Prince Charles. Sitting to the right of him in our editorial meeting was Ron Hall. He'd given a party at his house in Hampstead and, lacking a downstairs cloakroom, had asked guests to leave their overcoats and umbrellas on the upstairs bed. When the first departing guests went to the room to collect their coats, they interrupted the writer wife of a political correspondent thrashing about under the coats with an African diplomat. "Don't mind us," she said blithely, "we're discussing Uganda." _Private Eye_ at once seized on this unlikely response as a neat way of conveying gossip about the sexual mores of public figures, saying someone was "an expert on Uganda" or a couple met often "to discuss Uganda."
At this time Prince Charles, unknown to his public, was deep in an affair with Camilla Parker Bowles, which was to continue through his marriage to Princess Diana. The palace might not have been pleased to see the future King of England in front of that poster blaring UGANDA SENSATION.
* * *
It was not by chance that Kyemba came to our office to start the chain of events that led the African nations to disown Amin. He might have chosen any one of four or five newspapers with bigger circulations, but he regarded his appearance in the _Sunday Times_ as a mark of authenticity.
The difficult part for me whenever there was a knock on the door from the bearer of apparently big news was making the judgment call: can we believe a word this person tells us? (And nowadays with pictures one must wonder if the photograph has been manipulated.) What if Kyemba was a fabricator? What if he wasn't who he said he was? What if half of what he said was true and half wasn't? That would have been gravely damaging to our reputation. We couldn't very well put in a call to Idi Amin, but we could check Kyemba's identity and assess his character and the consistency of his story with known facts. Almost always in these circumstances, we had to take some things on faith.
Given that, the excitement of a scoop can overwhelm elementary prudence: you want to believe. Redoubtable journalists fall for confidence tricks and hoaxes. Before I became editor of the _Sunday Times,_ Denis Hamilton and the Thomson Organization had bought eighteen volumes of Mussolini's handwritten wartime diaries for book publication by a Thomson company and newspaper serialization. I knew nothing of the transaction, which had been conducted during his editorship by Hamilton and a skilled investigator, the former Insight supremo Clive Irving, who'd left the paper in 1965. Everyone had been sworn to secrecy. Three years later, in February 1968, the Italian newspaper _Corriere della Sera_ reported that it had discovered a mother and daughter from Vercelli forging Mussolini papers. Only then did Hamilton rather shamefacedly let me in on his acquisition. He called me to his office to say he'd kept quiet about it on my assuming the editorship because secrecy had been a condition of the sale (i.e., con) and in due course he'd hoped to make either me or the _Times_ the present of a world scoop. Hamilton and the Thomson group had already parted with a down payment of £150,000 in advance of the diaries being translated and edited. I was led to understand that the Thomson group would rather the deal was forgotten, but to me it was clearly a story we had to do—and it was irresistible to send an Insight reporter to grill Clive Irving, his former boss.
Famous publications have been burned time and again. A _Washington Post_ reporter named Janet Cooke fooled her editors and the judges of the Pulitzer Prize with her invention of an eight-year-old heroin addict; Jayson Blair long deceived the editors of the _New York Times,_ plagiarizing other publications and pretending to go places and conduct interviews he never had; Stephen Glass fabricated more than twenty stories for the _New Republic_ and other publications for several years before being caught out in 1998 when a drama of a teenage hacker and a computer company proved to be figments of his vivid imagination; and in 1983, a year after I'd left the company, the forgers of Hitler's diaries took in Times Newspapers in London and _Newsweek_ magazine in New York. I felt for my former _Sunday Times_ colleagues. Caught up in the excitement of a shattering scoop, their own doubts suppressed, they'd been let down by the vetting historian, Lord Dacre (previously Hugh Trevor-Roper), and my gullible successor as editor of the _Times,_ Charles Douglas-Home, who initially had charge of the project. They'd been rushed and pushed into the folly by the commercial imperatives of Rupert Murdoch and his macho management, who would rather be caught dead than having second thoughts.
It could be said I was lucky in my editing years never to be swept up in one of these disasters, and I suppose I was. Not a little of that, however, was because my door was always open. This was not a custom applauded by apostles of good line management, but anyone on the staff could barge in, waving a galley proof, and ask, "Harry, why are you publishing this crap?" Transactions presented "for your eyes only" flatter the ego, at the risk of having egg on one's face later. Had the paper's foreign staff been in on the Mussolini deal, it's very likely someone would have had a vague memory of the trial and conviction some years back of two women in Vercelli forging Mussolini papers, or at least thought to make a few discreet inquiries in Italy. Had Trevor-Roper as a Hitler scholar not contentedly accepted the idea that he was so experienced he could make the judgment all by himself, the falsification of the Hitler diaries would very likely have been detected by a reputable German historian.
Retailers of trash invariably impose haste—"We'll have to go to the _Observer_ if you can't decide here and now"—and invariably insist on secrecy. When dealing with strangers, I came to regard both these conditions as red flags.
Who was Anthony Mascarenhas? The well-dressed man in his early forties who came into my office on May 18, 1971, had the bearing of a military man, square set and mustached, with appealing, almost soulful eyes and an air of profound melancholy. I'd never met Mr. Mascarenhas before, nor had the foreign editor, Frank Giles. We'd encouraged him to file freelance for us from Pakistan on the strength of his reports in the _Karachi Morning News,_ where he was assistant editor. He had lived most of his life in Pakistan and held a Pakistani passport, but he was by descent a Christian from Goa, in India.
Two months before, in March, the Bengalis of East Pakistan, fired by Bengali nationalism, had rebelled against what they saw as their ill treatment by the non-Bengali military rulers of the geographically divided nation. (President Yahya Khan and his administration were in Islamabad, in West Pakistan.) In an untimely and ill-starred bid to hasten independence, 175,000 troops in East Pakistan had joined thousands of Bengalis in a pogrom against non-Bengalis. Thousands of men, women, and children had been butchered. I'd kept a close eye on it because my youngest brother, John, was working for the Foreign Office in Islamabad. "Our man in Dacca," he wrote to me, "says the East Paks have been meat-hooking people, hanging them up and then slitting their throats like pigs." Reports of these atrocities had been filed by Mascarenhas and our own Nick Tomalin in Jessore.
At the end of March the government in Islamabad had sent in two army divisions to restore order. It also had expelled all international reporters and imposed a news blackout. Thereafter the news from East Pakistan was that the army had made the province peaceful again. Such was the story in Pakistan's press and television. Pictures were broadcast of villages and towns coming out in parades with Pakistani flags to celebrate the return of peace. A British parliamentary delegation went to West and East Pakistan and satisfied itself that all was well.
Mascarenhas, talking quietly for a long time with shafts of afternoon sunlight coming into my office, told a very different and very harrowing story. In late April he'd been one of eight senior pressmen taken to East Pakistan by the Ministry of Information. The government wished to discountenance sporadic reports of army killings in East Pakistan, retailed mainly by a fast-growing stream of Bengali refugees. "The way Islamabad put it," said Mascarenhas, "was [for us] to show in a patriotic way the great job the army was doing.... But what I saw was genocide."
He'd been shocked by the Bengali outrages in March, but he maintained that what the army was doing was altogether worse and on a grander scale. It had not been content to do the necessary job of restoring order, nor was it confining its violence to rebels. Instead, Mascarenhas said, it had gone on a huge killing spree across the entire country. Soldiers were systematically moving from village to village, town to town, killing every Hindu (about 10 percent of the country's 75 million people), every one of the 75,000 non-Bengali Muslim mutineers they could catch, and many thousands of educated non-Bengali Muslims presumed to have been supporters of the rebellion: teachers, journalists, lawyers, students, left-wing political cadres. One of the victims, his body thrown in a brickfield, was my Dacca journalist friend Serajuddin Hussein, who had uncovered the child-kidnapping gang.
"The top officers," said Mascarenhas, "told me they were seeking a 'final solution.' I wrote down quotes all saying the same thing: 'We're determined to cleanse East Pakistan once and for all of the threat of secession even if it means killing two million and ruling the colony as a province for thirty years.' "
Was this man to be believed? Seven of the eight pressmen invited on the trip had written what they'd seen, and it was nothing like the story Mascarenhas was telling me. I asked him why he had not filed a contrary story for his own newspaper: "They wouldn't publish it, and they're under military censorship anyway." He said he'd had a crisis of conscience. "Either I had to write the full story of what I'd seen, or I would have to stop writing. I would never be able to write again with any integrity." But writing the full story, he said, was impossible anywhere in Pakistan; he'd been allowed to send us only a description of the Bengali atrocities. Even references to the danger of famine had been deleted by the censor. "That's why I've come to London. I want the truth to come out, but I cannot tell it and stay in Pakistan."
Instinctively, I believed Mascarenhas. He did not ask for money and seemed propelled only by a decent Christian passion and immanent shame. Frank Giles, my deputy and the foreign editor, made the same judgment, while passing on the foreign department's caution that we had known Mascarenhas for only a short time. I was impressed and moved that the suffering he'd seen made him ready to sacrifice his life in Pakistan—to abandon all his possessions and his career and to uproot his wife and five children. His main anxiety was that he would first have to get his family out of Pakistan; nothing could be published before he did, nor could we make other inquiries that would point to him as the source.
I took the risk. I told him that subject to vetting and his own determination to see it through, we were ready to pay for the evacuation of his family, but I could not give him a job. He'd not felt it safe to write his story in Pakistan, so he'd memorized notes, then discarded them. His five-thousand-word report of ten days in hell was a detailed eyewitness account of unique precision and authority. It supplied the missing piece of the East Pakistani tragedy: why people were fleeing by the millions. He named names: Here was a pitifully skinny tailor, Abdul Bari, scared by the arrival of the army in his village, running away and brought back for execution. Mascarenhas, forsaking his neutrality, exclaimed to Major Rathore, "For God's sake don't shoot [him]." Bari was found to bear the marks of circumcision obligatory for a Muslim; he was not a Hindu to be shot out of hand and got away with a clubbing. But in army headquarters at Camilla, Mascarenhas saw truckloads of Hindus being brought into the compound and heard the screams as they were bludgeoned to death.
Everything Mascarenhas described from his travels throughout the province fitted the picture emerging like the big missing piece in the jigsaw of fragmented, unconfirmed accounts from refugees and missionaries. He'd arranged with his wife, Yvonne, that if I accepted his report, he would send a telegram to her saying "Ann's operation successful." That would be the signal for her to fly out to relatives in Rome, leaving everything behind. She received the telegram and escaped with the children, but on returning to Karachi, Mascarenhas himself was forbidden to leave; only one foreign trip a year was allowed. To escape, he took a flight to Peshawar, walked across the border to Afghanistan, and then sent our agreed-on message to a staffer's private address: "Export formalities completed. Shipment begins Monday."
I took the exceptional step of clearing the entire center spread of the June 13 edition. One page contained the single-word heading GENOCIDE in big black letters and an editorial titled "Stop the Killing."
Our breach in the curtain of silence was an international sensation. President Richard Nixon, wishing to retain Pakistan as an ally against the Soviets, ignored the genocide. "To all hands," he wrote in a memo, "don't squeeze Yayha [ _sic_ ] at this time." But by the end of July more than five million refugees were in camps in India, and still more came. Years later, in Delhi, India's prime minister Indira Gandhi told me that the Mascarenhas report had shocked her deeply, setting her on a campaign of personal diplomacy in the European capitals and Moscow to prepare the ground for India's armed intervention.
The India-Pakistan War of 1971 ended with the creation of the state of Bangladesh. Mascarenhas had no political agenda; he was just a very good reporter doing an honest job. For that he earned the enmity of the Pakistani military and a telegram from the Black September group (which the following year murdered Israeli athletes at the Olympic Games): "You bastard, we'll get you, you went against your country." I put Mascarenhas on a retainer. He amply justified that and seven years later became a permanent member of the foreign staff with consequences I will relate in chapter 19.
The _Sunday Times_ I inherited could fairly be described as a Conservative newspaper. Its editorials no longer slavishly echoed the party line as they had done in Lord Kemsley's time, but my conviction was that it should not have any party line at all.
I was well aware that even an independent, unpredictable editorial might have fewer readers than the TV guide, but the editorial page was where we could speak to the opinion formers and firmly establish the tone of the paper. The aspiration I brought to the page was that we should try to judge every issue on its merits, questioning the use of power by government, the courts, and corporations, but fairly, and always balancing respect for individual human dignity and freedom with the imperatives of order. That was easier said than done.
Editorials under Denis Hamilton had been written by William Rees-Mogg and Hugo Young, with some contributions by me. I straightaway replaced this troika with an editorial board of eight that I chaired for up to two hours every Friday morning. It included the foreign editor, the business editor, the religious affairs editor (who was also an expert on Northern Ireland), the labor editor, the political editor, a political columnist, and a former editor of the paper who had retired to academic life.
The group brought special knowledge to the arguments, which at times were strenuous. The most contentious issues, apart from the selectors' choice of fast bowlers for the cricket tests, were Britain's role in the European Union (go in and stay in); Pershing missiles from President Ronald Reagan (yes, please); what to do about overweening trade unions; the propriety of our disclosing cabinet discussions in extracts from the diaries of the late Richard Crossman; Enoch Powell's "rivers of blood" speech on immigration; internment and ill treatment of IRA suspects in Northern Ireland; and skirmishes between the editorial board's interventionists and the laissez-faire insurgents, all trying to find the way to the British economic miracle in the dark mazes of economic policy.
The inner strength of the editorials was that they were not spun off the top of the head. They drew on solid reporting, investigations we commissioned, seminars we organized on the economy and Northern Ireland, and reconstructions of major political events. Instead of the staple editorial bemoaning Britain's low levels of productivity, for instance, we asked the feature writer Stephen Fay to go into the factories and solve a mystery: why does a British welder in Ford's Dagenham plant produce 110 Cortina doors an hour when a worker in Germany, Belgium, or Spain, using the same machinery to the same management plan, produces 240 an hour? (Answer: labor-management trench wars.)
The tricky part of not having a party line came during general elections, when all newspapers conventionally endorse a party and the "red top" tabloids ramp up the propaganda war. Denis Hamilton had told me on appointment as editor, "You'll have total freedom from Roy [Thomson]—so long as you don't attack the Queen." That idea had never impinged on the fringe of my consciousness, but I tested the freedom of opinion in the October 1974 election.
Tory prime minister Edward Heath had called and narrowly lost an election in March based on the theme "Who governs Britain?" The coal miners, going slow in a wage dispute, had forced him into allowing commercial enterprises to use electricity only three days a week. We were certainly on his side in principle but were persuaded that Labour's "social contract" might end the warfare (it did only for a time) and that the Labour ministers—Roy Jenkins, James Callaghan, Anthony Crosland, Denis Healey, and Harold Lever—were a more impressive bunch.
Hamilton, reading my mind, had gently suggested that Lord Thomson would be displeased if the _Sunday Times_ endorsed Labour. Thomson usually called me on a Saturday night to ask whether we'd yet overtaken the combined circulations of both opposition qualities (we were close). This Saturday I took the chance to mention that I was inclined to endorse Harold Wilson and Labour against Edward Heath and the Tories. Thomson made some shrewd comments on the two leaders and concluded, "Well, it's up to you, Harold. How's the run going?"
By 1979 the Labour Party was a shadow of its purposive self in the great days of Clement Attlee. It was frustrated by its ties to the trade unions and the public-sector unions in particular, which demanded more and more for less and less—their attitude toward the nobler ideals of socialism climaxing in the 1979 "winter of discontent" strikes, when they stopped cancer patients from going into hospitals for treatment.
Michael Jones, who'd become political editor, realized sooner than most that the Conservative leader Margaret Thatcher, waiting in the wings, could not be dismissed as a right-wing harridan whose middle-class accent and suburban outlook would doom the Tories to the wilderness. I agreed. When she was a backbencher, I happened to be at some stuffy City of London dinner, seated at the same table with a group of financiers. These were regarded as pillars of the Tory Party. She was not in the least in awe of their millions. I relished the way she assailed them for being more greedily interested in money manipulations than in investing in the business of manufacturing and managing the unions more effectively.
The decay of the Labour Party in the late 1970s was painful to report. It was infiltrated by Trotskyites creating cells in around one hundred moribund constituency parties, working by stealth to undermine any Labour man lacking a taste for a Soviet state. Jones tape-recorded one of the Trots in full flow promising "a civil war and the terrible death and destruction and bloodshed that would mean." It didn't seem much of a vote catcher to me. More serious were the strenuous efforts by left-wingers in the National Union of Journalists to impose a closed shop, meaning nobody could write for the paper unless he had a union card. Since a closed shop would hamper us in so many ways, I resisted the move, with support from Margaret Thatcher, while the Labour ministers I most respected sat on their hands. But as editors rallied, some moderates in the Labour Party dared put their heads over the parapet, and the legislation died.
All of us on the editorial board remained dismayed by two aspects of British life in the 1970s: the grip some recalcitrant unions had on the Labour Party and the stultifying secrecy in government. The great showdown was the diaries of Richard Crossman, a former Oxford don who was a member of the Labour cabinet from 1964 to 1970. The rule was that ministers had to wait thirty years to publish a documented account of their experiences, and if they or anyone else wanted to publish sooner, they had to accept official censorship on pain of a criminal prosecution under the Official Secrets Act.
Crossman's ambition was to illuminate how Britain was governed: he wanted to show that civil servants called the shots more often than the public realized, that cabinet meetings were not the decisive forum of popular imagination and MPs had little real power. He'd learned in September 1973, when I first invited him to lunch at the paper, that he had only six months to live. The priorities in his mind were such that his first action, even before he finalized his will, was to finish his two years of editing and give clear instructions to his executors completely to reject any censorship. He predicted there'd be pressure for suppression and truncation of his work, from both Whitehall (the civil service) and Westminster (the politicians), and he was right. After his death, his executors promised they'd not publish without official approval, and the government asked us to give the same promise.
Denis Hamilton was in favor of giving in, but when I assured him that we'd studied the law and prepared our case, he went along with the stratagem I devised. With only two or three on the staff in the know, I prepared an uncensored first serial and sent it to press on the night of Saturday, January 25, 1975. Roy Thomson and his son Kenneth happened to be paying a rare visit to my office that night. I told them that as soon as the prime minister's office got its hands on a copy, we expected a court order to stop the presses. Kenneth was worried; his father simply said, "You happy in your own mind, Harold?" I told him I was. There was no breach of national security. People should know how they were governed. "A good read, eh?" said the owner as he went happily off with his paper.
No court order reached us that night, but the noises from Whitehall were menacing. Every day we expected an injunction and had a bevy of lawyers on standby; I was advised we'd surely lose. Then Cabinet Secretary Sir John Hunt proposed to discuss what we might and might not publish in future extracts. I declined to meet with him. I sent two deputies because I couldn't trust myself to behave with the appropriate courtesy.
I wasn't aware of it at the time, but in 2005 Bernard Donoughue, who assisted the prime minister, published his _Downing Street Diary: With Harold Wilson in No. 10,_ which includes this entry on January 14, 1974, concerning a conversation he had with Attorney General Sam Silkin: "[Sam] had met Harold Evans for the first time the other evening at the American Ambassador's residence and thought he was a 'fanatic' for open government.... Sam said, 'He's very tough. He said to me "It's granite against granite." He may be granite. I certainly am not.' " It didn't sound like me, but if I didn't recognize myself in this scene, I also didn't recognize the sheep the attorney general affected to be.
For nine weeks I played cat and mouse with the Cabinet Office, accepting some requests for deletions, but in the end we published 100,000 words and broke every restriction. The attorney general then shed his wool and bared his teeth. He sought a court order to force the publisher of Crossman's book to accept the censorship we'd defeated. I couldn't tolerate seeing the executors and publisher singled out in this way. Within days of the writ being served on them, we ran unpublished Crossman material and were duly joined in the legal action.
We lost in the High Court but won in the Court of Appeal. Soon afterward a committee of inquiry, to which I gave evidence, recommended that ministerial memoirs no longer be regulated by statute. The logjam had been broken, but an even fiercer contest with government was pending.
#
# Death in Cairo
The only qualities essential for real success in journalism are ratlike cunning, a plausible manner, and a little literary ability.
I am quoting my colleague and friend Nick Tomalin, who protected himself—and entertained us—with irony and epigram. He concealed his real passion for a trade he described in a rare moment of self-revelation as "a noble, dignified and useful calling." In June 1973, before he took leave to write a history of the National Theatre, he cared enough to send me a critical appraisal of how far the _Sunday Times_ and Britain's "so called quality press" were meeting the highest aspirations. Four months later, at the age of forty-one, he was dead on a battlefield.
Nick was immersed in reading the notations of Mozart and Beethoven, playing the oboe in one of the regular musical gatherings at his home, when we interrupted him on the early evening of Saturday, October 6, 1973. The Arab nations had chosen the holiest day in the Jewish calendar, Yom Kippur, to launch a war against Israel from Egypt and Syria.
Tomalin was a star writer, dazzling in his versatility but most renowned for his classic "The General Goes Zapping Charlie Cong," describing an afternoon in the gunship of a Texas general who was proud to have killed more Vietcong than any of the troops he was commanding. We hesitated to break Nick's book sabbatical, but his closest friend and editor, Ron Hall, thought he would like a change. Typical of Tomalin—and indeed every reporter I ever asked to go into harm's way—he said yes right away. "Don't worry, it's safe enough," he told his wife, the literary biographer Claire Tomalin. "The Israelis take good care of the press."
In the first week he filed a briefing for a long Insight narrative on the war, without expecting a byline. On the Sunday he finagled his way into a closely guarded Tel Aviv hospital to bring a gift to his taxi driver, who'd broken a leg taking him to the Syrian front. He was visibly affected by the long lines of stretchers waiting for the wounded. The following week he headed back to the Golan Heights as the Israeli Army fought to retake sections they'd lost in last-ditch stands against massed Syrian armor. "I am only beta plus when it comes to courage," he remarked at a dinner on Tuesday, October 16, a self-deprecating acknowledgment of the apprehensions correspondents felt about the shifting, ill-defined front lines on the Golan.
Early Wednesday morning found him sharing a car with photographer Fred Ihrt of the German magazine _Stern_ and their escort, Major Hannan Levy, necessary to get them through roadblocks and not accidentally run into fighting. They passed an Israeli artillery battery, whose soldiers gave Tomalin mail to post, then descended into a bleak, treeless valley to take pictures of clusters of wrecked tanks near a crossroads. Two hundred yards farther on they could see what looked like an abandoned bunker, and Tomalin drove them there. It was very quiet, no sign of life anywhere. In fact, without Major Levy realizing it, they'd come right up to the Israeli front line. Concealed from view inside the battered bunker were Israeli soldiers who'd endured accurate Syrian artillery fire for several days; by some fluke they failed to see the correspondents' car arriving and departing again.
Major Levy judged the risk of running over a land mine too great for Tomalin to drive them back the way they'd come by making a three-point turn against the bunker, so the major and Ihrt got out to direct Tomalin while he carefully reversed to the crossroads. He was turning there, with Ihrt and Levy some yards behind his car, when they heard a swooshing sound. Tomalin would have been unable to hear it above the car's engine. The noise was from an antitank missile being guided onto the target by a Syrian hidden in the hills. The missile flashed past Ihrt and Levy and blew up Tomalin's car with a direct hit, killing him instantly.
Moments later Syrian artillery shells were exploding around the crossroads. Ihrt and Levy were hiding in a rocky trench. A transporter arriving at the crossroads was hit by shellfire. Five survivors scrambled out, and Israeli commandos in the bunker were yelling for everyone to crawl back to shelter with them.
And then there was other movement, a man running down the hill into the valley floor and toward the killing ground at the crossroads. It was our Don McCullin, who'd arrived above the valley with photographer Frank Herrmann. At the top of a slope they'd been stopped from going further into danger by an Israeli tank commander. McCullin, one of the most celebrated war photographers—surviving under fire in Vietnam, Cambodia, Cyprus, Biafra, and Beirut—was impelled by a surge of emotion to argue the warning and hazard his life yet again, not for a picture but for a friend. He took off down the hill, running half a mile to the smoldering car in the valley, identified Nick, retrieved his broken glasses, saw there was nothing more he could do, and ran the half mile back, choked up, unable to speak.
The death of one journalist is only another digit in the statistics of war, but I often wonder how much readers and viewers understand that the world-weary cynicism or vainglorious postures affected by the men and women who place themselves at risk conceal a deeply felt compulsion to "bear witness when others can't or won't." The phrase comes from the third of our reporters who lost his life, our adventurous and droll David Blundy, killed by a sniper's bullet in El Salvador in 1989 while attempting to file a last paragraph for the _Correspondent_.
The second at the _Sunday Times_ did not die in war but in attempting to report peace. This was David Holden. We were plunged into many mysteries during my fourteen years as editor of the _Sunday Times_ , but the most profound was right there in our own office. It turned on how we could answer two related questions: who assassinated Holden, our chief foreign correspondent, in Cairo in December 1977 and why? I've brooded over these questions for many years but have not written about the incident before. I'm writing about it now because what happened has only slowly and painfully become discernible through the shadows.
In 1977 Egyptian president Anwar Sadat made the momentous announcement that he would make an unprecedented trip to Jerusalem on November 19 to present an olive branch to Israel's parliament, the Knesset. He had ambitions to make peace with Israel on behalf of all the Arab nations he'd led to war in 1973 on Yom Kippur and also to finally resettle the Palestinians who'd lost their land. He was denounced as a traitor by the "rejectionist" states of Syria, Libya, Iraq, and South Yemen, as well as by all factions of the Palestine Liberation Organization (PLO). They arranged to meet for a "sorehead summit" in Tripoli in December as Israeli negotiators, and the world's press, arrived in Cairo.
David Holden, the foreign department's star on the Middle East, did not jump at our invitation to go to Cairo. He was writing a book on Saudi Arabia during the six-month leave of absence he took each year. We still hoped he might do it, but in the meantime Cal McCrystal, the foreign features editor, flew out on Friday, November 18.
At age fifty-three, Holden was a vastly experienced correspondent and broadcaster, one of the chroniclers of the end of Arabia as romance—"the immortal image of mystery," as he once put it. His career had spanned the end of an empire and the entrenchment of secular nationalism not yet threatened by Islamic fundamentalism. He shared the sense of style of his friend and _Times_ colleague James (later Jan) Morris, who was to receive a last enigmatic message from him. Holden was a small, neat man—not a hair out of place—who somehow, with his bush jacket and debonair manner, still managed to bring a touch of Beau Geste to our newsroom hubbub. He evoked the dashing, young foreign correspondent of the _Times_ of twenty years before, flourishing his British passport at the Yemeni border guard with a great curved dagger, seeking out the wicked old Imam Ahmed in his rocky fort, standing on a hilltop in Qartaba to observe rebel tribesmen around him opening furious fire on British soldiers across the valley. Nobody on either side hit anything, he reported back, except a goat.
Holden became intellectually absorbed by the politics of the transition as the former colonial states, through sacrifice and treachery, struggled to find their identity amid the eddies of big-power politics and the ascendancy of Israel. He moved easily through the Arab capitals; too easily, the Israelis thought. They regarded him as unduly sympathetic to the Arab cause. In 1967 he reported how the Israelis punished Palestinians for occasional sniping by demolishing Arab houses on the West Bank. He disliked what he saw, but he reported accurately and without histrionics. He did not hate Israel; the emotion was alien to his character. He wrote, also with amused tolerance, of "Arab venality, prejudice, opportunism and incompetence." If he hated anything, it was the categorization of countries as good or bad and the manipulation of people's minds.
He was a cultured man but unaffected, widely read in literature and history. He thought it impossible to write about the present without studying the past. Analysis rather than adventure became his strength. More information, he suggested, did not mean better information. The result of too much reporting was "to turn up the decibels on the Tower of Babel." He would not shun the ramparts, but he sought his realism elsewhere, in the nuances of his quiet meetings with diplomats and intelligence officers, academics and Arab editors.
Holden was not all that popular among a few of our harder reporters who'd covered the Six-Day War and the Yom Kippur War. On his infrequent stays in the office, they thought him rather detached, even condescending, remote from the graft of reporting. This was unfair. He was generous in providing contacts and very well aware of the dangers of practicing journalism in tense, exotic places. He'd been interned by the Egyptians during the Suez crisis. Reviewing a book by fellow foreign correspondent Noel Barber of the _Daily Mail,_ he wrote, only a few months before his death, "Thank God I have never suffered either bullets or the steel tearing into my flesh, but I have felt the boots going in and I have heard the prison door close behind me, and I know how sickening the fear of such moments can be."
Holden stayed at home in London the weekend of November 19 and 20, but the following week he told us he would go to Egypt after all. We learned later that Anthony Austin, the articles editor of the _New York Times Magazine,_ had reached him on "about the eighteenth" with a request to write an essay from Egypt. Holden was, Austin recalls, "very enthusiastic," rather different from our impression. The _New York Times_ asked the Egyptian embassy in Washington to telex Cairo to arrange an interview with President Sadat. The upshot of all this was that Holden arranged to go to Cairo, but first he would swing through Syria, Lebanon, Jordan, and the Israeli-occupied West Bank for us to test the strength of rejectionist feelings. He landed in Damascus on Sunday, November 27.
The plans for his murder had already been made.
Holden's first optimistic article, which we headed "Peace May Break Out After All," was filed from Amman on Saturday, December 3. The fire and fury of the rejectionist front had a "disintegrating quality," he said, adding laconically, "very Middle Eastern." His telex said that the following morning, Sunday, he would go to the simmering cauldron of the West Bank, where the Israelis were building settlements on conquered land. He wanted also to revisit Jerusalem, stay in the American Colony Hotel he loved in the Arab quarter, and then get back to Amman, crossing via the Allenby Bridge in good time to catch Royal Jordanian Airlines flight RJ 503 to Cairo on Tuesday evening, December 6.
Reviewing the news schedule on Wednesday, I asked what David proposed to write. Nobody had an answer. "We've not heard from him. Give him time to find his bearings." The foreign desk put in calls to Cairo's Meridien and Hilton hotels, where he had been tentatively booked. He'd not checked in, nor had he been in touch with the Reuters bureau, where he would normally establish communications. Nobody was alarmed. Communications in the Middle East were notoriously difficult; it was commonplace to have to wait up to eight hours to send a telex or get through on the telephone. "He'll pop up like a jack in a box, you just see."
By Thursday confidence had evaporated. We set off a full-scale search operation, calling British diplomats and fellow journalists he'd traveled with. Progressively through Friday we confirmed his departure from Jerusalem, his crossing of the Allenby Bridge, and his boarding of the plane to Cairo as planned. The foreign desk suggested an explanation for his silence. The Middle East was racked with cholera, his inoculations were out of date, and Egypt might have unceremoniously quarantined him for three or four hours in the isolation hut at the far end of the airport.
I saw the paper to press on Saturday, December 10, then around 10:00 p.m. went to spend the night at an office short-stay apartment near our building in Grays Inn Road. I was in the newsroom in five minutes when the dreaded call came. The British embassy, chasing the police, had heard that on Wednesday, December 7, the body of "an unknown European male" had been deposited in Cairo's Kasr el Ainy mortuary, the Dantean repository of all the city's accident victims. Bob Jobbins, the Cairo correspondent of the BBC, and Fuad al Gawhary, of Reuters, went to the mortuary and at once identified Holden. Jobbins was struck by the lack of any obvious injury, save a small exit wound in his chest. "An apparent execution," he presciently observed.
Holden's body had been found at 8:00 a.m. on Wednesday, nine hours after his 11:00 o'clock arrival in Cairo on Tuesday night. He lay on a sandy patch littered with old newspapers by the highway that ran beside the walls of Al-Azhar University. The area was unfrequented at night, but he was certain to be found as soon as day broke either by students, a passing motorist, or soldiers from a camp on the other side of the highway. He was on his back, his feet neatly together and parallel to the road, his arms folded across his chest in a mocking parody of repose. His expression was calm, his hair as sleek as ever, the only discordant note being the way the dark-rimmed spectacles he wore for driving were lodged crookedly over his eyebrows. His shoes were clean, without a trace of the fine white dune dust to be expected if he'd walked or been walked to this spot.
There was nothing on his person or at the scene to indicate who he was. All marks that might suggest his identity or nationality had been removed, down to the maker's label inside his green-and-brown-check sports jacket. Someone had emptied the pockets. Only a few Jordanian coins remained untouched in one trouser pocket.
The manner of death was equally methodical. He had been shot once from behind by a short-cartridge nine-millimeter automatic, the classic shoulder-holster weapon. The range was so close, as little perhaps as two inches, that his jacket was scorched just below the left shoulder blade where the bullet had entered. The killer had aimed his gun downward as he fired so that the bullet would pierce David's heart. It left his chest with such little force that it was found in the folds of the polo-neck sweater he wore underneath his jacket.
The time of death was established at no earlier than 3:00 a.m., no later than 5:00 a.m. This meant that Holden had been alive for at least three hours after leaving the airport, possibly a captive all that time.
The shock at the _Sunday Times_ was profound. David had not been as closely knitted into the competitive jousting and gossip of our office life as the convivially mischievous Nick Tomalin. But unlike Nick, who had bravely but knowingly exposed himself to risk in a battle zone, David had been reporting peace, and there seemed to be no explanation for his death. We all felt a passionate urgency to do everything we could to find one.
I called the Home Office and the commissioner of police at Scotland Yard. The commissioner assigned two of the most experienced homicide detectives, Chief Superintendent Ray Small and Detective Inspector Tony Comben, to the case, but they had first to get permission to work on Egyptian soil, as well as in Syria and Israel. We wanted to move quickly while the trail was still warm. Within a few hours a team of six reporters was on the way to the Middle East: Insight editor John Barry and Cal McCrystal in Cairo, Paul Eddy and Peter Gillman in Amman, Tony Terry (a former British intelligence agent) in Jerusalem, and Helena Cobban in Beirut. Denis Hamilton discouraged this initiative; he was as distressed as any of us but believed we should leave inquiries to the police. I was well aware that our reporters could only ask questions. Despite a common exaggeration of the "powers of the press," we didn't have any of the means of coercion available to the Egyptian security services, nor did we have knowledge of Cairo's criminal networks. But we had our own contacts, we knew the way David worked, and our team was especially resourceful.
Barry and McCrystal visited the morgue, but their mission, apart from answering any questions the Egyptian police might have about Holden, was to track his movements in the last week of his life. They were to look for clues to some of the questions the murder posed: Who knew Holden was arriving on flight RJ 503? Who else was on the plane with him? Could he have spotted a terrorist on board? Was he seen leaving the airport with anyone and by anyone he knew? Was the motive for his killing something to do with his private life playing out as public drama? Or was the trigger his work? Nothing of his we had published could be regarded as offensive, although some Israeli commentators had condemned him for a _Sunday Times_ Insight report on the ill treatment of Palestinian prisoners. The report, which had angered the Israeli government, was later confirmed by the U.S. State Department, but Holden had had nothing to do with the report, as we made clear, and those who compiled it were never molested in any way.
Was there something to suggest that Holden had been chosen as a high-profile target by Palestinian rejectionists or terrorists? Did they think that the death of a famous British correspondent on Egyptian soil would embarrass Sadat and demonstrate to the VIPs attending the peace talks that peacemakers were vulnerable? Was Egyptian security all that tight?
Who had Holden seen in his swing through the Middle East? Could one of the people he'd met there have learned the time of his flight to Cairo? Had he alarmed somebody, seeing them in a politically compromising situation? Or had he perhaps been asked to carry to Cairo a message or document too sensitive to relay over telex or telephone?
Most of Holden's last week alive, we found, was spent interviewing Arab leaders, including those of the PLO; the West Bank mayors of Bethlehem, Nablus, Hebron, and Ramallah; Arab journalists and academics; Syrian and Jordanian officials; and American, British, and Australian diplomats. We pressed them for recollections of what had transpired; none of them had any awareness of his travel plans, but we learned that on the West Bank he'd been given a petition to take to Cairo appealing to Sadat not to negotiate with Israel.
Three other curiosities emerged. We were intrigued by Kenizé Mourad, a thirty-year-old French reporter for _Nouvel Observateur_. She'd invited Holden for a drink on meeting him at the U.S. embassy in Damascus on November 30. In Amman on December 2 they spent time together, ending with dinner in the hotel coffee shop. The next day they went to a restaurant; at 11:00 p.m. she'd gone to his room for drinks and stayed until 1:00 a.m. She told us she angled to join him on his West Bank trip and he declined. On December 5, she said she'd gone back to Damascus, a curious move, we thought, for a journalist, since the Syrian leadership had flown to Libya for the sorehead summit. At first, we wondered if she might have followed David for some reason, but she herself sought out Peter Gillman and was very open about David being "a wonderful man." I came to the view she was an eager younger journalist admiring of an old Middle East hand.
We also came across two unexplained gaps in Holden's schedule and discrepancies in two testimonies. In Jerusalem on the afternoon of December 5, Holden had told Edward Mortimer of the _Times_ , with whom he'd been working, that he was going alone for a walk in the Old City. The taxi driver described to us how he'd dropped Holden at its main entrance, the Damascus Gate. Later that day Holden had described to Mortimer his long walk in detail, remarking how much the city had changed in the ten years since his previous visit shortly after the Six-Day War. But had he spent two and a half hours in the Old City as he'd suggested? That certainly did not square with the testimony of an academic at Birzeit University, who told us the two of them had spent the afternoon in a village twenty miles away where the Israelis had harassed the population. And what was the meaning of the postcard Holden had sent from Jerusalem to his friend Jan Morris? He'd written only nine words: "In the Old City, citadels still have their uses."
The next day, his last, produced a conflict about his time in Amman. The manager of the Bisharat Travel Agency in the lobby of the InterContinental Hotel said that he'd noticed Holden around lunchtime, about the time he'd have arrived in Amman. He saw him go into the hotel's coffee bar with two Americans, writers and archaeologists John and Isobel Fistere.
Holden and the Fisteres had all been in Beirut for the first two and a half years that Kim Philby had been there. (The gossip then was that the Fisteres were keeping an eye on Philby for the CIA.) Ruth Holden told us of a dinner she and David had given with Philby as a guest. In the files of the _Times_ we found that on January 8, 1957, Holden had recommended Philby, "the _Observer_ man," to the paper's foreign desk as someone who could replace a departing _Times_ stringer. The mere fact of knowing Philby at this time in Beirut was hardly significant, but what had surprised me, and others on the original Philby investigation in 1967, was that Holden had not come forward either during our inquiries or after publication.
The Fisteres gave a very different account of their meeting with David. They said they had seen him in the evening, not at lunchtime, and then for only about five minutes, when they'd exchanged a few words by the hotel press center. According to them, he had been "trying desperately to telex to the Cairo Hilton to confirm his reservation there." He'd looked "tired and dirty and worn-out from his travels, in a desperate hurry to catch his flight." The travel agent's account checked out better. No one in the press center or wire room could recall Holden trying to send any last-minute telexes, and if he had, he would have been given a telex from the Hilton that had been waiting for him since December 4. He'd caught his plane with time to spare.
I felt justified in sending the team to investigate when John Barry reported from Cairo that he was disturbed by the Egyptians' initial response to the murder. The death by shooting of an unknown European was a rare event. In thirty years only two foreigners had been murdered in Cairo, both victims of domestic disputes, and December 1977 was a period when Egypt was on the alert for the arrival of a thousand reporters from around the world, as well as the Israeli negotiators. Yet Barry said that the Egyptians had made zero effort to identify the body and had performed no proper autopsy. Had Holden not been identified when he was, he would have been buried in a common grave. The initial line of the Egyptians was that the murder was the work of foreign agents (Israelis heavily hinted), though the Egyptians were worried enough about the rejectionists to deport two hundred Palestinian militants.
The Scotland Yard detectives were still stuck in London, awaiting travel documents from the Egyptians, but the questions the _Sunday Times_ team started asking galvanized the authorities. We were assured that President Sadat himself had ordered a massive investigation. Hundreds of police had been deployed. They had grilled the airport staff, and they said every one of the 128 passengers on the flight had been traced and questioned. None was suspected of terrorist links. They were mostly American tourists. Mrs. Willivene Bonnette from Clyde, Ohio, told us that Holden had had the aisle seat and would not get up to let her in, so she'd had to squeeze past. He'd been "sarcastic and kind of surly," telling her it was "absurd" she did not know what she was going to do and see in Cairo. He'd rebuffed conversation, saying only that he had been in Jerusalem "on business." His mood might be explained by a personal anxiety: the grumpy, experienced traveler wasn't sure he had a bed for the night. He had once held a booking at both the Hilton and the Meridien, but he had changed his itinerary so often that he had lost the Hilton booking and did not know that we had a room waiting for him at the Meridien. To arrive in Cairo late at night with no hotel booking was not an amusing prospect.
We tracked how he had on landing joined others in the wearisome scramble Egypt inflicted on its visitors. He'd changed traveler's checks for $200 at the National Bank of Europe, completed a form to obtain an entry visa, gone through passport control, and picked up his red Samsonite suitcase; he was alone at the time. That was the last fact about his life of which we could be certain. Presumably he'd walked through the "nothing to declare" channel in customs and through a pair of swinging doors into the open. A double line of crash barriers formed a channel into the foyer of the arrivals building. A police guard of a couple of armed men stood at the end to control the throngs. From that point to the curbside where the taxis waited was at most forty paces. In that distance Holden had disappeared.
There was one curious feature of his arrival, probably innocuous but possibly sinister. Everywhere on his journey he had entered his occupation as "journalist"; here he had written "writer." This meant he would avoid being drawn into the bureaucratic net the Egyptians had set up for the media arriving for the negotiations. "Journalists" and "press" were ushered into a pressroom next to the visa office and escorted directly to their hotels. It may have been a whim on his part, or someone might have advised him to do that so he would remain a free agent.
It was routine for the police at Cairo's airport to log every taxi picking up a fare. Hundreds of drivers were interviewed. The police concluded that Holden had not been picked up by one of the authorized, registered taxis. If there had been a line of people waiting, he might have veered right at the exit and gone to the parking lot to pick up one of the "pirate" taxis, but there had been plenty of regular taxis at the curb, and the cheaper gypsy cabs were uninvitingly small and uncomfortable. So perhaps someone had picked him up.
Near midnight on Thursday, December 15, four days after the identification of the body, John Barry was summoned to the police station at Dukki, a pleasant quarter of central Cairo on the west bank of the Nile. General Nabawi Ismail, soon to be minister of the interior, was there with a dozen beaming generals from security and criminal investigation, along with a battered white Fiat 128 without license plates that a Dukki resident had found abandoned. It had a tangle of wires below the dashboard, indicating that it had been started without an ignition key.
In the trunk they'd found Holden's red suitcase. In it were two Christmas gifts for his wife and jumbled clothes. His Olivetti portable typewriter was in the car, too, along with unexposed rolls of film, a blue folder stuffed with a letter and notes for his book on Saudi Arabia, and the scattered pages of his loose-leaf contacts book built over three decades; the Egyptian section alone ran to eleven pages. Missing were his passport, traveler's checks, Olympus camera and lenses, exposed rolls of film, and any material he'd accumulated on his trip.
General Ismail and his team doubted the motive was robbery. Holden had been alive for at least three hours, possibly five, after leaving the airport; robbers would surely have taken what they could and fled. Nobody had attempted to cash the traveler's checks (and never did). Detectives had combed through the known outlets for stolen goods and found nothing of Holden's. Surveying the found and the lost, the general remarked, "It looks as if the killers knew what they were looking for."
Our immediate thought was that they had been looking for material for Holden's book on Saudi Arabia. We could not tell whether the pages of notes that remained were the total of his work. Kenize Mourad said he'd told her he'd uncovered corruption in high places, but he did not intend to include that in his book. There was a sufficient reason for this: the book was being written in cooperation with the Saudis, its viability resting in part on the hope that the government would buy ten thousand copies. And we found that the part of the manuscript that had been completed was a thorough historical survey—nothing more.
The police at first assumed that the Fiat was a gypsy cab that Holden had voluntarily taken from the airport. The team told me they were not convinced. It struck us all as highly unlikely that as experienced a traveler as Holden, tired from a day on the West Bank and Jordan, would consider the saving of a few Egyptian pounds worth the discomfort and risk of a ride in a gypsy cab, especially if the driver had used the tangle of exposed wires to start the car.
Could he have been forced into the cab as in one of those scenes in the movies where the victim is told to keep quiet and keep moving with the prod of the abductor's concealed pistol in his back? It seemed implausible. The airport was teeming with people and security men. None of the travelers or officials noticed the slightest thing untoward in the exit area, nor inside the terminal before customs. To penetrate inside the terminal to identify Holden during the time he was exchanging money, collecting his visa, and picking up his suitcase would have required a pass to get through security. Conceivably he could have been met by an Egyptian security person, or someone posing as one, and invited to a waiting car, but by far the most likely scenario was that he was met outside the street barrier by one or more people who knew him and whom he knew he could trust.
If he'd been abducted in the white Fiat, it was not the car he died in. Just a little later, the police found another Fiat, abandoned at Tanta in the heart of the rich delta farmlands eighty miles north of Cairo. In the rear passenger compartment there was a cartridge case matching the fatal nine-millimeter bullet, and bloodstains were found between the front seats. The headrests on the passenger seat had been removed, making it easier for the gunman leaning forward from the rear seat to put a bullet through the heart. The headrest subsequently turned up in the first car used to capture Holden, the white Fiat dumped with the luggage at Dukki.
There were more provocative facts when we had the body flown home to London and a thorough autopsy indicated he had put up a fight, possibly that his wrists had been tied. The examination at London Hospital by Professor David (Taffy) Cameron noted: "Bruise on the principal knuckle of the left middle finger, to a lesser extent on the left little and ring fingers, and a bruise noted to front of the left wrist, approximately three inches above the wrist. There was also fingertip type bruising to the outside of the left arm above the elbow; and bruising was noted on the main knuckle of the right thumb." If Holden had indeed been in a struggle, we reasoned it was unlikely to have been at the airport, since this would have caused a commotion that someone would have been sure to notice. Our best guess was that he'd struggled on being transferred to the murder car. There, thinking they might have finished with him, he might have taken his front seat quietly enough.
Nearly a month later a third Fiat was found with documents from the murder car. All three had been entered in identical fashion by breaking open the quarter light, all driven by hot-wiring the ignition, two resprayed, one green, one red. The security police thought the logistics of the break-in, respraying, murder, transfer of the luggage, dumping of the body, and getaway would have required a team of eight people. Robbery having already been discounted, the police moved vaguely back to foreign agents, rejectionists now in the starring role. The theory gained momentum when they established that the owner of the first car found at Dukki was an activist, a twenty-two-year-old engineering student with a Jordanian passport (whose father had fled Jordan on terrorist charges). He was brought in for questioning as were the owners of the other two cars. All were cleared, we were told, but they did yield tantalizing information. The owner of the first car said he'd reported it stolen in the third week of November—around the time in New York and London Holden's trip to Cairo had been decided. The other two cars were stolen when Holden was set to move from Jerusalem to Cairo on December 6.
* * *
But who carried out what was clearly a well-planned abduction? Over the Christmas holiday, I had a conversation about the murder with a highly placed Egyptian visiting London. He told me the operation was carried out by Fatah hard-core rejectionists who would stop at nothing to sabotage Sadat's initiative. More he would not say. We followed up. Helena Cobban, our correspondent in Beirut, had a good working relationship with Fatah's chairman, Yasir Arafat. He promised he would investigate.
Barry and Peter Gillman were summoned to Beirut to hear the findings. "Finally, after much fussing about changing cars," said Barry, they were ushered into a heavily curtained room in a derelict building wrecked in the Lebanese civil war. In the gloom, a middle-aged man sat at a Victorian desk; they were given to understand he was head of Fatah intelligence. Nobody was identified, but Gillman had a tremor of recognition that one of the three men was Ali Hassan Salameh, believed by Israel to have organized the Munich Olympic kidnappings in 1972 (and subsequently blown up by the Israelis). Barry reported: "He said on the chairman's orders he'd made inquiries and could assure us that 'no arm of the resistance' had a hand in the murder. I asked if he could be so certain of "organs of resistance" other than Fatah, and he said no Palestinians would have wanted Holden dead. They had a policy, he said, of not killing journalists, and the _Sunday Times_ had been regarded as "a friend to our cause" because it had published that report on the ill treatment of Palestinian prisoners.
I could not regard the interview as conclusive, but a strange development made Fatah's involvement seem less probable.
Even more alarming than the news from Cairo and the pathologist's report in London was the information Paul Eddy brought into my office in the second week of January 1978. He'd asked for a closed-door meeting, and in his cool, cryptic style he proceeded to astonish me. "The killers knew exactly when Holden would arrive in Cairo because they got the information from the horse's mouth—us." Eddy had discovered that copies of telexes between Holden and the foreign desk were missing. Hundreds of telexes were filed in an unmarked cupboard on the fifth floor, not far from my office. Nothing had been taken or disturbed except eight telexes relating solely to Holden's changing travel plans from the day he decided to go. With the messages stolen from the cupboard, the plotters would have been able to track him as he traveled from Damascus to Amman to Jerusalem, back to Amman, and finally to Cairo.
The foreign department was very well run. Could it not be, I suggested, that this was just a mix-up, an unusual act of carelessness? Eddy had another shock for me. The thefts were continuing in January. He had discovered the December thefts only in the course of looking for more recent messages and was stunned to find that some of these, too, had disappeared. Among them were travel plans and reports on the progress of the investigation from Barry and Gillman, who had returned to the Middle East in early January. One dispatch reported speculation that there might be a connection with the terrorist Abu Nidal's campaign to kill moderate Palestinians who favored a deal with Israel. (The key moderate PLO leader, Said Hammami, had been shot dead in London in January.)
Someone had gained access to the editorial floor, which was not too difficult in those days. There were at least six entrances to the deep, rambling building on Grays Inn Road. A thief could easily mingle with the hundreds of casual workers who were employed on Saturdays in the basement presses and in the huge distribution warehouse where copies came up from the basement for bundling and loading onto trucks. Those floors were removed from editorial on the fourth and fifth floors at the front of the building, but we were used to seeing new faces, assuming they belonged to casual messengers ferrying copy and coffee. Still, an interloper must have had intimate knowledge of the layout and procedures to find the foreign department cupboards when nobody was in the room. "Horrible thought," said Eddy. "We may have a spy on the staff."
On January 24 Eddy removed all material related to Holden from the foreign department and locked it in the Insight office, with only a single key to the filing cabinet. We did this very quietly, not wanting to alert a predator. The stealth was prudent, but it meant the wire room that transmitted and received messages did not know of the thefts from the foreign department—and we were not aware the wire room operators had developed a practice of keeping a second copy of messages sent to the foreign department. They impaled them on a spike in a corner of their office hidden away in a back corner of the building. It took the thief thirty-six hours to realize that fact and gain access to the wire room when it was unattended during the night of January 26–27. Every message was gone, including twenty-five related to the investigation. More surprisingly still, sometime between 8:00 a.m. and midday on January 27 Eddy asked the wire room for its copy of a telex from Gillman to Eddy announcing travel plans to the Middle East. It had vanished.
Again we sounded no alarm, but I spoke with Scotland Yard and the Foreign Office. Very soon after my call, our windows began to receive a thorough cleaning inside and out. The cleaners were from the Yard's C-10, known as "the watchers." Without detection they hid infrared cameras that would capture any intruder on tape. Eddy and the managing editor were the only staff people who knew.
We then baited the trap. Discussing the schedule in conference, the foreign editor said that Gillman and Eddy were reporting a breakthrough and that Eddy would travel to Cairo. "Tell them no heroics in any circumstances," I told the foreign editor. I wasn't really worried. Eddy, making sure he wasn't followed, did not go to the airport but instead headed for North Wales (only later did he tell me, by the oddest of coincidences, that he went to my very own Rhyl). The Eddy-Gillman plan was that from there he would contrive to send messages purportedly from Cairo, Beirut, and Jerusalem, while the Yard's concealed cameras kept watch on the foreign department and wire room.
Over ten days no interloper appeared on the recordings. Either the thefts were by someone on the staff pretending to go about their normal work, the trap had been rumbled, or the thief had concluded that we were not on a dangerously hot trail. I had got so jumpy, especially after the Philby cover-up, that I even began to think I'd made a mistake letting the Foreign Office know that we'd detected the thefts.
What if our own Secret Intelligence Service (MI6) had played some role in the abduction of Holden? What if we were caught in a convoluted winding down of the Philby betrayal? Hadn't I learned from Dorothy Sayers or Agatha Christie that it was a mistake to overlook the least probable suspect?
From these mad thoughts, I found comfort in the scientific principle of Occam's Razor, that the simplest explanation for which there is visible evidence should always be preferred. Certainly, the authorities in the shape of Scotland Yard could not have been more unstinting in their cooperation—while the Egyptians were not. Chief Superintendent Small and Detective Inspector Comben were still in London. Assurances that they would be welcomed were endlessly forthcoming. What was not were the necessary papers. In the end Comben and Small were never allowed to go to Cairo. This was baffling and infuriating because they were appalled by the Egyptian police work. An inquiry about whether fingerprints on the cars had led anywhere produced a negative—too many policemen had handled the vehicles. Instead of admitting the two Scotland Yard men, the Egyptians sent a senior officer to London. He was as baffled as we were by the purpose of his visit. He thought he might perhaps go to Paris to interrogate Kenize Mourad. Unable to locate her, he settled for long shopping expeditions to Marks & Spencer.
It was all very frustrating, but the timing of the thefts of the cars and the raids on our office did put a different perspective on an early theory, first advanced in a Lebanese newspaper. Holden, it reported, had been mistaken for David Hirst, a _Guardian_ correspondent. In Arabic the surnames Holden and Hirst have some differences, but they could have been mistaken for each other. Our access to the flight manifest found that European names were hopelessly garbled. A motive was apparent, too: revenge. Hirst had angered President Sadat and infuriated Mrs. Sadat by writing about corruption and high living. "True enough," Hirst affirmed. "Nine months before, four security men came to the Cosmopolitan Hotel. I was escorted out with a gun pointed literally at my head. On top of that I came back to Cairo deliberately when David Owen [foreign secretary] was visiting for two days, and Sadat was incensed that I'd been able to get a visa in Rome. They tried to stop me leaving on Owen's plane, but Robert Fisk [veteran Middle East correspondent] said he wouldn't leave without me."
It turned out that another journalist had been mistaken for Holden. In August and September of 1974 David _Halden,_ on the staff of the Canadian Broadcasting Corporation (CBC), had flown into Cairo from London. He told us that on the first occasion, to his surprise, he'd been met in the arrivals hall. It was late at night, and he'd not told the Egyptians he was coming. Nonetheless, after he passed through visa and passport controls, he was approached by two men in somewhat scruffy clothes who said they were from the Cairo Press Center or the Ministry of Information (it wasn't clear) and had come to take him to his hotel. He presumed when they said "Mr. Holden" they were simply mispronouncing his name. They led him to the car park to the right of the exit area, where an old sedan was waiting, a driver already inside. Halden got in with the two officials. During the trip to central Cairo, they told Halden they'd managed to set up some of the interviews he'd asked for. Halden hadn't sought any (nor had "David Holden," so far as we could discover). He asked if they were sure they had the right man. The reply was "You are David Holden of the _Sunday Times,_ are you not?"
They took him to his hotel.
Mistaken identity seemed the most plausible explanation to us for a time. Sadat was certainly very angry about David Hirst. One well-placed but very frightened informant confided that he knew Sadat had sent an assassination team to the airport and swore us to secrecy. The crucial difficulty was that David Holden was clearly the intended target. The thefts of the cars, and the thefts of telexes relating to Holden, could not be just a coincidence. And whoever held Holden for three hours would have known pretty soon that he was not David Hirst. Nor had Hirst signaled any intention to revisit Egypt.
We were forced back to the conclusion that the chesslike precision of the abduction, and the capacity to operate in both Cairo and London, must have been the work of an international organization with considerable facilities. The Egyptian police pressed this explanation on us, but we did not find it convincing for a number of reasons. To accept it one has to believe that an unknown group, undetected in Egypt's clampdown, arrests, and deportations, using methods without the slightest resemblance to other terrorist killings in the world, went to all this trouble to murder at random one of a thousand correspondents for indiscernible motives, remove his clothing labels for no apparent reason, and then disappear without a trace, leaving their achievement unclaimed.
There were other suspects who had the means to carry out assassinations and had been known to do so. Not excluding the Egyptians themselves, there were the foreign intelligence agencies of Israel, the United States, Russia, and possibly even Britain; and the Saudis, we were told, handled this kind of work by contract with professionals. But what on earth could have been the motive? Was there an important clue in the killers so carefully leaving Holden where he would be found—a warning perhaps?
While the rest of the team resumed normal duties, I authorized Eddy and Gillman to continue the inquiry, as they much wished to do. To consider motives it was agreed we had to ask not just who killed David Holden but also who _was_ David Holden? It's not an easily answered question about any of us; all identities are evanescent. His career was well documented—the son of an editor of the _Sunderland Echo,_ educated at Quaker schools and Emmanuel College, Cambridge, a teacher of geography for three years (a job he said he "loathed"), a postgraduate student at Northwestern University in Illinois, a foreign correspondent for the _Times_ and _Guardian_.
To go beyond these facts into the nuances and ambiguities of a personal life was a sensitive decision. I was troubled by the way newspapers and television in pursuit of the mass audience were intruding into private lives when there was not the slightest justification. People are entitled to personal privacy; it's integral to our sense of worth. Certainly there are gray areas where the private elides with the public; but photographers and reporters, and their editors, who regard private lives as fair game are for the most part taking the easy option. It is harder to expose the complications in a real public wrong than to make someone's life a misery (on the grounds of exposing hypocrisy, which is itself hypocrisy on stilts). Indeed, gratuitous breaches of privacy invite restrictive laws that protect the unscrupulous, who are all too ready to invoke the sanctity of private life while plundering the public purse. So I hesitated about authorizing our own inquiries into Holden's personal life; he was not around to defend himself.
Yet the thought persisted that the motivation for his murder might lie in some conjunction of the personal and public. I felt we owed it to him at least to explore the possibility of finding something relevant, without necessarily committing to publication what we found. This was how we ran into contention with the CIA and FBI.
The most important relationship Holden formed before his marriage was a passionate ten-year friendship with an older man he looked up to. He was a chameleon named Leo Silberman who'd been an ardent Communist, first anti-America then pro, a supporter of Israel on its founding and then vehemently anti-Zionist. The CIA came to suspect he was a British intelligence agent in Africa. Silberman had been born into a Jewish family in Germany in 1915. As Hitler rose to power, the family escaped to Britain, where Leo joined the Communist Party and, according to his brother Freddy, lost jobs because of his radical activities. He married a Communist Party secretary in Vienna in the 1930s—as Philby had done—and thus came to the attention of the FBI when he later applied for a visa to visit the United States and denied any Communist connections.
During the war Silberman studied and taught sociology at South African universities—brilliantly, according to his professors, although he gave himself the title "Doctor," to which he was not entitled. The British Colonial Office was impressed enough to provide him with a letter saying that he was working on their behalf in East Africa "in connection with social problems."
Silberman was loud and flamboyant, the opposite of Holden, but they were lovers, according to Silberman's brother Freddy. Holden and Silberman stayed close until Silberman's death in 1960, whereupon Holden married photojournalist Ruth Lynam, who worked for _Life_ magazine.
We were surprised that Holden was bisexual. In letters to his brother Geoffrey and former teachers and in conversation, Holden made constant references to girlfriends and sex. He wrote to Geoffrey about the lack of sexual opportunity in Arab countries, saying, "What is a fellow to do except turn queer?"
Holden's closeness to Silberman led us to ask the FBI first and then the CIA what they knew about him and Holden. An FBI official told us he had Holden's file in front of him. "It looks to me as if some of this stuff is classified," he said, adding that he'd have to ask if it could be released.
The agency dragged its feet so much that on August 15, 1978, I wrote to the FBI and the CIA asking them to meet their obligations under the Freedom of Information Act and retained the Washington law firm of Williams & Connolly to act for us. In the meantime we checked through unofficial sources and learned that within weeks of Holden being posted to Washington by the daily _Times_ in 1954, he'd been observed meeting "a known Soviet bloc agent." None of us regarded this as conclusive, the FBI having an ability to put two and two together to make five. Many were the cases in the paranoid 1950s where individuals were put under surveillance for a chance meeting or an expression of social concern. A journalist might meet a Soviet agent, knowingly or not, in the course of his work. Indeed, I suppose I must have been observed meeting a number of Soviet bloc agents at diplomatic parties in London, and I'd stayed in Moscow (I'd been shown around by an Intourist guide so well indoctrinated that I quickly realized the truth was the exact opposite of what she told me).
The FBI declined to release any further details on the grounds of national security. The CIA was also less than forthcoming. Sixteen months after our requests under the Freedom of Information Act for documents concerning Holden and Silberman had produced no response, we filed a court action. John Barry met two CIA officials in Washington's Mayflower Hotel. They said the agency had nothing in its files about Holden and no knowledge of his murder. "I just don't believe you," said Barry. "You must have at least taken an interest in the murder of a British newsman at that critical time." They acknowledged the point, but all they had collected, they said, were rumors, no more, that he had been killed by European terrorists looking to use his press credentials somehow. They invited Barry to withdraw our suit; he didn't have to read my mind to decline their invitation.
In fact the CIA did have a file on Holden. The court action led to the agency providing us with an index, but not the contents, of thirty-three documents it had assembled, four of them about Holden and the rest about Silberman. We learned much later that this file on Silberman had been started when a CIA agent in East Africa had met him at a dinner party where Silberman, on the basis of the Colonial Office letter, had given the impression he was a British agent working under the cover of an American foundation grant. This annoyed the CIA man on two grounds: British intelligence had an "undeclared" operative in his area, and the agent was using an American foundation as cover, something American operatives were forbidden to do.
In addition to the thirty-three documents from its own files, the CIA said that it had turned over nineteen documents—one to the FBI, thirteen to the Department of State, four to the National Security Agency (NSA), and one to the International Communication Agency—to decide whether these could be disclosed to us. All of the agencies said no. The FBI affadavit of January 24, 1982, referred to "a very sensitive espionage investigation which still very much impacts national security."
We appealed these decisions; all our appeals were denied. Williams & Connolly advised that rather than go to trial, a slow and expensive business, we should propose that the documents be shown in camera to a judge who would rule whether the CIA was justified in maintaining secrecy. A judge supported the proposal; the CIA rejected it. Only when an appeals court ordered the agency to comply did it allow U.S. District Court judge Maurice E. Lasker to see its "top secret" file in confidence.
He was swift. In a judgment on Februay 15, 1983, he ruled that disclosure would constitute a national security risk. The documents should not and would not be released.
What were we to make of this? A source told us that the CIA file simply recorded that Holden was "an informal contact," meaning that he would have met "second trade secretaries at U.S. embassies" on a fairly regular basis, presumably to exchange information. Holden was not, we were assured by this source, in the pay of or under the control of the CIA, but it would have been possible for "adverse parties" to mistakenly come to a more sinister conclusion.
But none of that could have persuaded a judge to seal the documents, certainly not the judge who'd forced the CIA to make them available for judicial inspection in the first place and was regarded as open-minded. We were compelled regretfully to consider whether David had in some way been involved in espionage for somebody and been killed for that reason, perhaps because he had been thought to be serving two masters. That would at least do something to explain why his body had been left as an obvious example of an intelligence killing. It was a warning: this is the price of betrayal.
But to go further into these shadows meant trying to identify which intelligence agency might have employed him in the first place as a straight agent, rather than a double agent, then which agency might have regarded this as treachery. Three agencies had surface plausibility: the CIA, Mossad (Israel's intelligence agency), and the KGB.
The CIA was the obvious first candidate, given its resistance to revealing its documents. There were incidents in David's life that lent some credence to the idea of a double identity on behalf of the CIA. Nobody could explain why he'd been arrested twice in Cuba and then deported. His flat repudiation of any CIA involvement in the bloody coup that had deposed Chile's President Allende in 1973 was also an uncharacteristic misjudgment and in uncharacteristically vehement language. And then there was the afternoon in Jerusalem when he had been in two places at once and sent that cryptic postcard to Jan Morris: "In the Old City, citadels still have their uses." The notion did occur to us that the uses of the "citadel" might have been sexual, not political. When I spoke to Morris, she discounted this and said that she had never had such a strange postcard from Holden on all his travels. Then we learned that the U.S. consulate in East Jerusalem maintained a clandestine meeting place in a small rented room in the Old City walls. We'd previously been assured that by agreement with Israel, there was no CIA post in Jerusalem, only in Tel Aviv. Moreover, we discovered that the academic at Birzeit University who'd said he'd been with Holden that afternoon was a paid agent of the CIA.
The theoretical case for Holden being a KGB agent was based on the one FBI sighting of him with a KGB operative and his association with Kim Philby. Holden might have been useful to the Soviets in gathering information and assessing trends: Moscow had been taken quite by surprise by Sadat's expulsion of the Soviet military in 1971.
And then there was Mossad, every paranoid's favorite mastermind. Rather than Holden being an agent for Mossad, however, the first hint the Egyptian police gave was that he'd been a Mossad victim. They said the bullet had been manufactured in Israel. Then they thought better of it and emphasized terrorists. But would Mossad have left such an obvious clue to a clandestine killing? Furthermore, unless one was totally cynical about the Sadat-Begin peace moves, would the Israelis have risked killing a reputable journalist who was writing favorably about the initiative they valued? I thought not. It was put to us more than once, however, that if Holden was an intelligence agent, Israel was the country he could best serve. It was the one nation denied access and travel throughout the Middle East. His journeys, though infrequent, gave him valuable insights into the options and intentions of Israel's immediate neighbors, including their military preparedness. Perhaps there was a wider audience for his observations than the regular readers of the _Sunday Times_.
Eddy and Gillman were still sporadically on the trail when I left the _Sunday Times_ to edit the _Times_ in 1981. They left, too, a few years later, but they kept in touch with people they were sure knew more than they'd admitted.
In 1988, after we had lost the Freedom of Information case, Gillman confided what we'd found to a senior U.S. contact in the Middle East, even allowing perusal of the Eddy-Gillman unpublished Insight notes. The contact, having digested all the information, said we had to consider the assassination in the context of the history of the CIA involvement in the Middle East.
In 1973, at the outbreak of the Yom Kippur War, American intelligence capability in Egypt had reached zero with the expulsion of the last active CIA agent, a woman who for cover worked for the splendidly named Société d'Alexandrie pour les Boissons Distillées et Vignobles de Gianaclis. After Egypt's defeat, the agency worked hard to reestablish itself in Egypt and to establish links with Egyptian intelligence. It was central to persuading Sadat to make his historic gesture. The peace negotiations, leading to the 1978 Camp David Accords, had gathered great impetus, with Egypt and Israel offered comparable aid packages from the Americans. Nothing was to be allowed to jeopardize the rapprochement.
The contact did not admit any direct knowledge of the murder but suggested that Holden would have been vulnerable if he had been suspected of playing a double game in any way that clouded the prospect of a peace agreement. The contact in effect suggested we should not rule out a joint operation, instigated and encouraged by the CIA but carried out by the Egyptians. Why Holden should have been thought to be a hazard could have been because he was believed to be a hostile intelligence agent or carrying messages from the rejectionist front. I've mentioned he did have a petition for Sadat, given him on the West Bank. The team had not regarded this as really significant because it was in mimeographed form (that is, typed on a stencil and run off on a Cyclostyle machine) and by the time Holden reached Cairo it had already been published by Edward Mortimer in the _Times_. But the stakes were high in 1977.
It proved impossible to verify or even follow up this lead. Eddy and Gillman had gone back to the source, but he'd left his position, and over many months all efforts to find him in the Middle East and the United States failed. Personally, I come reluctantly to the view that there is force in the scenario, given the unhelpfulness of both the Egyptians and the CIA. The nature of the operation in Cairo certainly points to Egyptian involvement. In the mists of circumstance and conjecture, we are left only with the certainty that Holden was foully murdered and with the aching suspicion that he died not for journalism but for some secret cause he'd betrayed. Perhaps we should give him the benefit of the doubt, but no journalist should ever agree to act for an intelligence agency, whatever the invocation or however strong the desire to be patriotic. And we should relentlessly expose the agencies and journalists who ever make that kind of arrangement. Vivid in my mind still is not only Philby passing himself off as a correspondent in Beirut but our own Jon Swain, held captive by rebels in Eritrea and in mortal danger because of false suspicions he might be a British spy. The credibility of journalism and the lives of individual correspondents are too valuable ever to compromise.
#
# Divided Loyalties
A newspaper is an argument on the way to a deadline. If there isn't any argument, there's not much of a newspaper. And the editor's decision is final. That sounds pretty straightforward, doesn't it?
The questions, in fact, are endless. Is this report credible and clear? Is it a rehash of the familiar, or does it advance public understanding? Does it justify its space and position in the paper? Is its readability derived from malice? Is it legally risky? Does it betray a source? Is it faithful to the paper's espoused values of seeking the truth fairly and without fear or favor?
When I was the editor of the _Sunday Times,_ nothing impeded my ability to make those final decisions except my own ignorance or cowardice. But I did have to decide under the pressure of time, and paleontology kept me on the alert against procrastination—at least the paleontology from a fragment of verse by Bert Leston Taylor, a _Chicago Tribune_ columnist, which I stuck in my science scrapbook at St. Mary's Road Central School:
_Behold the mighty dinosaur_
_Famous in prehistoric lore,..._
_You will observe by these remains_
_The creature had two sets of brains—_
_One in his head (the usual place),_
_The other in his spinal base._
_Thus he could reason "A priori"_
_As well as "A posteriori."_
_If something slipped his forward mind_
_'Twas rescued by the one behind...._
_Thus he could think without congestion_
_Upon both sides of every question._
_Oh, gaze upon this model beast;_
_Defunct ten million years at least._
In my own pondering I was not beholden to any party line. I did not have to duck a decision or temporize for fear of offending friends of the ownership or its commercial interests. When the Thomson Organization was bidding for licenses for North Sea oil exploration, I ran reports damaging to all the bidders, accompanied by an editorial saying the government's terms were too generous. To Thomson's abiding credit, I did not hear a word of complaint.
I published an Insight exposé of CIA involvement in elections in Guyana the day before a Thomson team met the government to sell a television station. The team was asked to leave the country forthwith. Something similar happened with a Thomson venture in South Africa. This sounds as if I was seeking confrontation with my own company. I wasn't. I knew about the oil licenses, but I didn't know about the activities in Guyana and South Africa. Even if I had, I'd not have felt obliged to delay or suppress the reports: independence from the commercial life of a very large conglomerate had been a condition of my appointment. It was honored both ways. Denis Hamilton, as editor in chief of Times Newspapers, did once have occasion to tell me he thought a business news report I'd published on the rigs of oil companies in the North Sea had been malicious and unfair, but that wasn't censorship; it was a professional judgment by a distinguished colleague—and he was, I fear, right. It was a reminder of the vigilance to be exercised in maintaining the paper's standards. The press is not noticeably different from other institutions in an aptitude for closing ranks under attack; it was important for me not to let our powerful esprit de corps protect errors.
Fact checking was the least of it. All the facts might check out, but that did not mean we should publish something as it stood. Have we put it in context? What are the foreseeable consequences of publication and nonpublication? With contentious material I made a point of testing it on various people, sometimes in groups but aware of the group dynamics. Is the one who has not joined in silent because he is scared to disagree with a peer group? I tried to cultivate suspicion of myself.
My emotional mind-set was publish-and-be-damned, so how could I offset that with reason? I was acutely aware that I was making decisions in a social context where my colleagues subconsciously shared my assumption that publication was good, suppression bad. There were issues where the weight of opinion in the office was so evenly balanced in logic and emotional force that I felt I had become the editor of a paper called the _Daily Dilemma_. The rise of full-scale terrorism in Northern Ireland from 1971 to 1997, for example, called for excruciatingly difficult judgments:
Should we talk with terrorists?
Should we seek and accept interviews if a condition is not to disclose the identity or whereabouts of someone wanted by the authorities?
If someone has information that could save lives, what measures are justified to get it out of him? Ill treatment, threats, cruelty, torture? If we know about these practices, should we keep quiet?
Should we suppress everything that might conceivably make things worse? (In the violently paranoid state of Northern Ireland, there was always the risk of endangering someone's life.)
Which comes first, truth or patriotism? As citizens protected by the rule of law in a civilized society, are editors obliged to be loyal first to the guardian state? (President John Kennedy, after the Bay of Pigs fiasco in 1961, put it this way: "Every newspaper now asks itself with respect to every story: 'Is it news?' All I suggest is that you add the question 'Is it in the interest of national security?' ")
The editor's decision was final at the _Sunday Times_ in the sense that the story went to press as approved. But that was all that was final. A decision was never without repercussions. As I will relate, what I decided about Bloody Sunday in 1972, the British Army's shooting of thirteen unarmed civilians in Derry, was still being challenged thirty-five years later, not by the government but by one of our star reporters.
In the newsroom one summer Friday in 1975, the news editor, Magnus Linklater, seeking a decision, thrust folios of copy into my hand. "You'd better read this from Chris Ryder. I've questioned him. Pretty amazing." Ryder was our plump young staffer, a native of Belfast who still lived there. He was distinguished by his rolling Belfast accent and cheerful determination to ignore warnings from the extremists of both sides in the conflict. In the early days of the Troubles, the IRA, dedicated to achieving a socialist, united Ireland, were welcomed as defenders in the Catholic areas under siege by mobs. Since then, Ryder had reported, the Catholic community had become progressively more disillusioned by random violence and intimidation from the "Provos"—gunmen of the Provisional IRA, which had split from the "official" IRA.
Ryder had broken the news that three Provo women dressed as nuns had been foiled in attempting to rob a big Irish bank, but after the manager was contacted by the Provos at his home, the bank was so scared it paid up anyway. Ryder's latest report in my hand was that a number of Provos were operating "a new element in urban terrorism." It was said that behind the front of the Andersonstown Co-operative Society and Sinn Fein, the IRA's political organization, the Provos were using force and fraud to create monopolies for themselves in construction work and taxi services. It was also said that they had set up a chain of highly lucrative drinking clubs.
James Evans, our lawyer, came over to read the piece. It was, he said, actionable, but since the Provos were an illegal terrorist organization, it was highly unlikely the two men named would sue. One of them, however, was a spokesman for Sinn Fein, and if they did sue, we'd have to be sure we could prove that the link with the IRA/Sinn Fein was inextricable. The courts in Belfast, he said, were as exacting in their standards as mainland courts. "It's up to you, Harry."
That was the first decision: play it safe and not publish, or risk it? I risked it and published two reports by Ryder, in August 1975 and August 1976. A full six months passed after the second report before a libel claim was served on me on behalf of the two men we'd named, Seamus Loughran, the Sinn Fein spokesman, and Gerald Maguire, who'd been interned. They alleged that they'd been maliciously and falsely defamed.
At that time local and national news organizations were regularly settling libel claims by people who'd been interned and who'd argued that any mention of them in this context was defamation, since detention without trial was not a judicial process and no criminal conviction had resulted. So I had a second decision to make: should we do as everyone else did?
Making our defense more challenging was the fact that jurors and witnesses were regularly terrorized by the IRA. One witness to a terrorist attack had been killed. All criminal and terrorist cases had to be tried by a single judge and no jurors, so taking our case to a civil trial before a jury, where jurors and witnesses might be intimidated, was a big risk. Was it worth it? After all, the costs of paying compensation were likely to be much less than the monetary costs of fighting.
But that was only one consideration. It had been my decision to publish. Either I'd had confidence in the reporting then, or I hadn't. If I hadn't, I'd been reckless. If I had been satisfied with the reporting, I surely had a duty to stand by the reporter who'd exposed himself to harm. Retreat would be a rebuke to one reporter, but it would stain the integrity of all our reporting. Furthermore, it would be a betrayal of those who were willing to testify or sit as jurors. A newspaper operated under the assumption that the rule of law prevailed; how could we undermine it? "Accept service," I told our lawyers.
Two years later, in October 1979, I was summoned to appear in Belfast in the Queen's Bench Division of the High Court of Northern Ireland. I'd got used to testifying in the Law Courts in the Strand in years of suits brought against the _Sunday Times_. This experience was very different. It was nightmare time in the province. More than two hundred IRA prisoners were on "dirty protest," smearing excrement in their cells in a campaign to regain political status, culminating in hunger strikes. Killings had been trending down for years, but in March an IRA splinter group had blown up Airey Neave, the Conservative MP, as he drove out of the House of Commons parking yard. On the morning of August 27, the IRA had murdered Lord Mountbatten on a little fishing expedition in Sligo, and with him his fourteen-year-old grandson, a fifteen-year-old local boy, and eighty-four-year-old Lady Brabourne, the mother-in-law of his elder daughter. On the afternoon of the same day, two concealed bombs in the Northern Ireland town of Warrenpoint had killed eighteen men of the British Parachute Regiment, the Paras' biggest loss since their sacrifices holding the bridge at Arnhem in World War II and the single most disastrous day in Ireland for the security forces.
It was a bleak, wet morning with poor visibility when I was met at the airport by two detectives armed with pistols and small machine guns to ensure my safe passage to the fortified courthouse. By then nearly two thousand people had died in the Irish conflict; proportionate to population, it was as if a small English town the size of Darlington had been wiped off the map. The streets of terraced houses were like those I'd grown up among in Manchester—seeing a solitary man running along with a greyhound, I had a flash memory of my dad's betting days—but "desolation" was the adjective that came to mind. Even in the Blitz, our Newton Heath streets did not have the foreboding that hung like a fog over Belfast. Where there was color—murals and graffiti painted on a gable end—the message was death to the other side. Every pub in the city had armed guards, gunmen of both sides having found it tempting to burst open the doors and spray with automatic fire whoever was having a drink. There were distant sirens in the city, but the noises in my imagination were the wailing laments and rifle shots of the frequent funerals and the sobs of the widows.
No imagination was required to bend double in the car when one of my escorts shouted urgently, "Get down! Get down!" and the driver did a quick U-turn. They'd taken a wrong street. In parts of the city the security forces were afraid to run into a trap—a roadblock in front and sinister cars moving in behind. When we got safely to the fortified courthouse and awaited admission, my bodyguard was kind enough to point out two heavyset men on the other side of the anteroom. "They're the ones," he said, "we've got to watch for you."
It was good to find two experienced staffers, David Blundy and Phil Jacobson, waiting in the courthouse. They'd had armed escorts; so had the other witnesses. Ryder, who'd been assigned two bodyguards for the duration of the case, arrived in an armored Land Rover that varied its route every day. As the trial went on, this protection business became curiouser and curiouser. At one point a lookout for one of the more violent Loyalist groups was identified in the public gallery, so the police ended up having also to protect the IRA people who showed up in court.
I took the witness stand, facing a jury of three men and four women and Mr. Justice Murray. I had a hard time from Mr. Michael Lavery, QC, the lawyer for the men we'd written about. We were conducting a vendetta, weren't we? Didn't we realize these men were providing the community with jobs? Hadn't we invented the story about his clients because I was in business to sell newspapers and make money for myself or my employees? The _Belfast Telegraph_ recorded:
MR. LAVERY: Are you making the case that the plaintiffs and the Andersonstown Co-operative were committed or part of robbery, fraud and protection rackets?
MR. EVANS: Yes. I would stand by every word in the two articles.
Ryder testified for three days, threatened with imprisonment for contempt of court if he refused to reveal all his sources. The lanky and deceptively casual young Blundy, who made us all laugh with his gift for puncturing egos (including mine), said under cross-examination that he had spoken with one of the defendants when writing earlier reports about the IRA, identifying him then only as "an IRA source." Lavery pounced. "So you're willing now to betray a source in the witness box?" Blundy replied that of course he'd protect the anonymity of a source, but if the source brought a lawsuit and threatened to kill one of his colleagues (Ryder), might not that immunity be regarded as questionable?
The lowering atmosphere of Belfast clearly affected me. That night, back in London, as I walked from the office toward an apartment, a man shuffled toward me and asked if I'd show him the way to Tottenham Court Road. His theatrical Irish accent made me jump out of my skin. In my nervy state I promptly told Scotland Yard. By morning I'd calmed down, but they insisted I go along to examine mug shots.
At the end of October the jury considered the evidence for a full day. When they returned in the afternoon, they accepted the thrust of our case, dismissing the libel claim by the two men. The Andersonstown Co-op was awarded a token £200 in damages. Loughran, who'd played a prominent part in the proceedings, was eventually relieved of his public duties as Sinn Fein spokesman on "health grounds." One of Ryder's sources clarified this: "It is more to do with his future health than his present health."
The whole experience reinforced my admiration for our reporters and my sadness at the miserable existence of the people of Northern Ireland. Life there was indeed brutish and nasty. The paramilitaries, but especially the IRA, didn't much care who died. The IRA incinerated men and women at a happy get-together of the Irish Collie Club at La Mon House hotel in Castlereagh. The Protestant "Shankill Butchers" gang abducted and slaughtered thirty Catholics picked at random. Loyalist members of the outlawed Ulster Volunteer Force ambushed and machine-gunned the popular Showband Miami, a mix of Catholics and Protestants, returning to Belfast from a Catholic dance hall gig. In the Poppy Day massacre in the county town of Enniskillen, an IRA splinter group detonated a bomb to kill people gathered at the cenotaph to remember the victims of all conflicts. And then the bombings came to London by the score, and the terrorists almost succeeded in murdering Mrs. Thatcher.
Was it an inevitable tragedy? I've endlessly run over in my mind how journalism might have made a difference after I took over the paper in 1967.
Ever since the British government had partitioned Ireland in 1921, separating an independent republic of twenty-six counties in the south from a province of six British counties in the north, the people had been divided by faith and by flag as much as Shiites and Sunnis are divided by religion in Iraq. In the northern six counties, loosely called Ulster, Protestants formed more than half the 1.5 million population; the south was primarily Catholic. The Protestant Unionists ("Orangemen," after their protector King William, the Prince of Orange, who won the Battle of the Boyne in 1690) passionately saw themselves as British, loyal to the Crown and the Union Jack. They had their own Parliament in Stormont, just outside Belfast, with a large degree of self-rule, and they used it to suppress the Catholic minority, keeping them out of the best jobs and housing.
The Protestants were fearful of losing their cultural and political identity as their majority shrank against the faster-growing Catholic population. They were always on the alert for subversion by IRA revolutionaries, who regarded themselves as the true repository of the Irish identity and sought to force the six counties into the republic to the south (though that, too, they regarded as an illegitimate state). The Republican dream of "one Ireland" under the tricolor flag was anathema to the Loyalists. They saw no poetry in the terrible beauty attending the violent birth of the republic. The modern reality of "one Ireland" for them was a poor, priest-ridden state where Gaelic was the national language, where they would lose the benefits of the British welfare state, and where their personal liberties would be compromised by the ordinances of the Catholic Church. The "special position" of the church was then enshrined in the constitution, and it aggressively used its muscle. Divorce and the sale of contraceptives were banned; it had a plan for maternity care thrown out in 1951; and it controlled the schools, though the Irish constitution did not recognize an endowed religion. So the Protestant ("Proud to be Prod") Unionists rigged elections, controlled the Royal Ulster Constabulary (RUC), and could call up B Special police reservists; they maintained an omnipresent threat of violence.
The Catholics developed a burning sense of resentment and frustration, but in turn they fueled Protestant anxieties by insisting on their separate church schools and by continuing to talk of a united Ireland. "They are no more willing to renounce it," wrote our political insider John Whale, "than they are to renounce transubstantiation." The constitution of the Republic of Ireland, moreover, still laid claim to the whole of the island.
But the Irish calamity is not unique. The seeds of disaster there were comparable to those sown in Iraq, comparable to the earlier desegregation traumas in the American Deep South, comparable to the illusions of the Vietnam War. I have insufficient reserves of humility to suppress the conviction I developed from observing all of these episodes: you can't beat honest firsthand newspaper reporting—when you can get it. Governments may know a lot more about our lives than we care to contemplate, but frequently they know less about the world than we presume. They are captive to preconceptions, electoral concerns, political affiliations, special interests, and bureaucratic hierarchies that filter "truth." Government just cannot govern well without reliable independent reporting and criticism. No intelligence system, no bureaucracy, can offer the information provided by free competitive reporting; the cleverest agents of the secret police state are inferior to the plodding reporter of the democracy.
Yet to Fleet Street, for many years, and its readers, Ulster was about as riveting as Ecuador. I'd taken no interest in the province myself when I edited the _Northern Echo;_ in fact I knew next to nothing about it. But the _Sunday Times_ was very early in its reporting. In July 1966, when I'd just become managing editor, I read an item on the news schedule: Queen's visit to Belfast. The reporter assigned the story was Belfast-born Cal McCrystal, who'd been beaten up covering the riots of October 1964, when a Belfast Republican club provocatively displayed the Irish tricolor. His feature before the Queen's visit was so much more than the usual backgrounder. It was an authoritative depiction of the Protestant majority's crude apparatus of political and religious oppression, concluding that the real dilemma for the new Labour prime minister, Harold Wilson, was whether to allow the province to work out its own bizarre destiny or to "use reserve power to bring elementary social justice to Ulster." McCrystal's report appeared as the paper's main feature that week, and we headlined it "John Bull's Political Slum." In his study _How the Troubles Came to Northern Ireland_ , the academic Peter Rose describes it as "one of the very few [articles] in the British press during those years which made a genuine attempt to warn mainland Britain of the consequences of the failure to tackle Catholic grievance."
I expected the feature would produce the same reaction in the Labour government as it had in me: a sense that political reforms were urgently necessary. That didn't happen. The report was immediately denounced by Stormont as biased. This made more impression on the British government than our firsthand report because the government took its advice from the Home Office in London—which in turn took its cue from Stormont.
The pity was that this would have been a perfect time for the British government to change direction. The hard men of the IRA trying to blast the two Irelands into one had faded; their successors, a left-wing leadership based in Dublin, opted for Marxist pamphlets and polemics rather than bombs; their agitation for a socialist workers' state implicitly recognized partition. (Happily, this lost them the Irish American funding that had always fomented violence.) The old IRA hands in the north grumbled at this pacifism, but leadership had moved principally to a new generation of educated middle-class Catholics who wanted to "take the gun out of Irish politics." They were building a civil rights movement that did not insist on Irish unification and was open to Protestants as well as Catholics. Even those of them who still yearned for a united Ireland, the constitutional nationalists, nonetheless set out to win social justice within by peaceful persuasion modeled on Martin Luther King Jr.'s movement rather than by the violent overthrow of the state. The Marxist-Leninist Roy Johnston, a computer specialist, described it as trying "to salvage the basic Enlightenment republican democratic tradition from various overlays of Catholic nationalism, Fenian conspiracies and quasi-Stalinist centralism which have infested it."
This new peaceful coalition was the window of opportunity. In Britain the Conservative—and Unionist—government had given way to Labour; there was a reformist prime minister in Belfast, Terence O'Neill; and the Catholic leaders best represented by Derry's visionary John Hume had entirely reasonable demands ("one man, one vote, removal of gerrymandered boundaries, allocation of public housing on a points system based on need"). Editorially we supported O'Neill in his struggle with the Loyalist hard-liners, and Hume in his antipathy to the bigotry of Catholic Ireland, but I wonder how much difference we could have made with an even more insistent advocacy coupled with extended reporting of the kind I'd urged on the American press over treatment of blacks in the South.
As excuse I can offer only that Northern Ireland was just one developing story. We had to rush teams to cover Israel's Six-Day War; we had reporters in the midst of America's never-ending war in Vietnam, in the civil war in Nigeria, and in the anarchy of the Congo. The Soviet Union and China were on the brink of a border war, and the world's first democracy (Greece) was taken over by a bunch of fascist colonels. But these excuses are not good enough; news is always at an editor's throat.
At the end of 1968, after violent clashes between the RUC and civil rights marchers, I committed the paper to continuously serious coverage and comment. To monitor what was happening on the streets, I set up a rolling team from the newsroom (two weeks in Northern Ireland, one week off), including Chris Ryder. The reporter I recruited from the _Sunday Telegraph_ was its chief investigator, Tony Geraghty, who'd also made a name for himself on the _Guardian_. He could not be typecast by either side. He was a British subject and an Irish citizen—and a veteran of the British Paras (later to serve as a military liaison officer with U.S. forces in the Gulf War).
Geraghty's reporting convinced him that the "pusillanimous" Labour government was making a tragic error acquiescing in Stormont's approval of provocative Loyalist marches in the summer of 1969. He was right. The more radical elements of the civil rights movement, students in the People's Democracy organization, staged a countermarch in Derry; police brutally attacked it in the "battle of the Bogside"; that was followed by wolf packs of Catholic youths in the Falls Road area of Belfast hurling rocks and petrol bombs at police stations. The RUC drove armored cars into the rioters, and Protestant mobs firebombed Catholic homes. Ten civilians and four members of the RUC were killed by gunfire on the night of August 14–15. The uneasy but viable mixture of peace and gradual reform was broken—for good as it turned out. When the Labour government in August 1969 sent in the British Army, it would have been wiser at the same time to impose direct rule from Westminster instead of assigning the soldiers to work as common-law constables with the distrusted RUC. The ancient fault lines remained.
Still, there was no excuse for the gross misrepresentation in the American media of the peacekeepers as an invading "army of occupation." Caught up in retailing Irish tribal folklore, local TV stations and the more thoughtless popular press in the United States quite ignored that Ulster was constitutionally as much a part of Britain as Massachusetts was of the United States; that without the army, there'd be civil war in which the Catholics would suffer most; and that the democratic Republic of Ireland also regarded the IRA as a menace to its own democracy.
We were supportive of the soldiers caught between the warring parties, but our reporting soon ran into difficulties with the army. Geraghty happened to be on the spot when a brigadier jumped out of a staff car and strode alone and unescorted into the "no-go," dangerously tense Falls Road, where Catholics, surrounded by sixty thousand Loyalists, had barricaded themselves against the mobs. "I'm going to see Father Murphy and these Citizens Defense chappies," said the brigadier. He had no objection to Geraghty tagging along to the door of the vestry of St. Peter's Pro-Cathedral.
He would say nothing when he came out. But among the emerging Central Citizens' Defense Committee (CCDC) delegates, Geraghty recognized a thirty-seven-year-old leader of the IRA. "We've got this back-of-the-envelope treaty with the British military," he told Geraghty, showing him the understanding that the CCDC would open the no-go areas within three days in return for the army and local vigilantes looking after security, to the exclusion of the distrusted RUC.
It was news, but it was news that could have consequences. It showed that the army was not one-sided and was engaged in some creative diplomacy. But the Loyalists might hate the army talking to anyone with an IRA connection. They did, and they rioted in protest. The day after Geraghty's report, our sister paper, the _Times,_ briefed by Whitehall, knocked the report: "Diligent investigations by journalists have failed to reveal that the IRA is now nothing much more than a slogan out of the past." An army brigadier, who was director of public relations, came to my office to protest Geraghty's "inventive" reporting. This puzzled Geraghty. He wrote to me, "How could what every dog in the street in Belfast knows is true be denied in London 90 minutes' air time away?"
The answer was that Belfast was then three hundred years away from the Ministry of Defense and the army. They were just not as well informed as the reporter. General Ian Freeland, who'd been in charge for only two months and was incensed by our report, told me later, "The army did not know at that stage who were current members of the IRA." They should have if police and army intelligence had done their homework. The Northern Ireland Civil Rights Association opened its doors to anyone. Five days after I had received the indignant brigadier, the Scottish judge Lord Cameron issued his report on the disturbances of the previous autumn. It confirmed Geraghty's assertion that the IRA had become involved in the uneasy coalition forming the civil rights movement, though not yet as gunmen. They had virtually no firearms then to defend the Catholic areas, and some graffiti around the Lower Falls (Belfast) read, IRA = I RAN AWAY. What had been political heresy on Sunday morning was by Friday a judicially tested fact. Cameron's report, reinforcing Geraghty's observations, should have led to the army and police recognizing the weakness of their intelligence and alerted everyone to the terrible risk that defense of the Catholic ghettos against mobs might fall to a resurgent IRA. The old equation loomed: fear − trust = IRA.
The dragon's teeth had been sown. Irish American dollars, denied to the constitutional nationalists, started to flow again to bombers and shooters. As an admirer of American journalism, I was appalled by reports of what was being broadcast on local television in Boston, New York, Chicago, and Philadelphia. They gave time to IRA front men who, without challenge, retailed propaganda and incitement: "Violence is the only thing left"; "The only language the British ever understand is violence"; "The only real criminals in this matter are the English." In the bars along Second Avenue in New York, near the _Sunday Times_ office at Forty-second Street, I was often approached to give dollars to Noraid, a charity supposedly raising money for the families of slain or imprisoned IRA men. The _Chicago Tribune_ nailed the lie: "The money bankrolls the sort of sub-humans who can pack six-inch nails around a bomb and put it in a place where women and children and tourists will gather."
American audiences generally were given no idea of the truth that the killing was mainly the work of paramilitaries, not the security forces. Those who died as a result of army and police intervention in the end numbered fewer than half those killed by nationalist and Loyalist paramilitaries. And of the paramilitaries, the well-armed IRA killed at twice the rate of the Loyalists.
Seeing how distorted reporting could become stiffened my resolve to keep ours as straight as possible. In view of the risks our reporters were accepting, I became resentful of the reactions of Westminster and Whitehall, which became more dismissive when Edward Heath and the Conservatives assumed power in June 1970. Basically, they consistently ignored our warnings of increased disenchantment within the Catholic community, creating a sea in which guerrillas could swim. Junior army officers in Belfast did soon realize that military actions had to accompany notable visible improvements for the community, but their superiors saw it as their mission to submit upbeat reports to London. It was what London preferred to hear. (It was exactly the same scenario in the United States in the Iraq War starting in 2003. Not wanting to be accused of defeatism, the senior commanders shrank from conveying bad news to Washington as the insurgency gained strength, and Vice President Dick Cheney accused reporters who conveyed the unpalatable truth of being lazy, foolish, cowardly, and unpatriotic.)
All the reporting we were doing was fair and firsthand, and television's brave cameramen were routinely bringing the horrors into people's living rooms. But it was not possible to understand Northern Ireland by focusing on the latest outrage. Violence is always sure of space on television and in the press. Political change, being more subtle and dull, is frequently neglected until it explodes into "inexplicable" violence.
So we devoted hours to discussing what constructive suggestions we might offer on the editorial page, based on the reporting. I was responsible for both factual reporting and the opinion pages—the common British practice—and I think that helped to ground our comment in the day-to-day reality. I can see the virtues of the American practice of separating opinion (church) from news (state), but that can dilute the concentrated focus necessary to make people sit up. I felt keenly when I was traveling in the Deep South in the 1950s that the separation of church and state preserved editorial chastity at the cost of delaying reform. That was similarly true of the Iraq misadventure in 2003. The U.S. momentum to invade was very strong. There was no shortage of critical comment. What was lacking, to induce second thoughts, was investigative reporting independent of government sourcing, coupled with argument based on it.
There were degrees of emphasis among the group in the weekly editorial conference deciding the paper's policy, but we agreed on pressing reforms harder while recognizing that the Provos were using the civil rights movement as a cover for armed insurrection, just as the Loyalists had long feared. We assailed Democratic congressmen in the States for appearing to condone violence. We were unsparing in our criticism of a major speech by Senator Edward Kennedy drawing a parallel between Ulster and Vietnam: "It is in most respects of a piece with the rest of his ill-researched, ill-considered and destructive speech."
The most important consequence of tying the reporting to comment was our conviction that London would make a catastrophic mistake if it started interning people without trial but simply on suspicion. In March 1971 we wrote of internment: "If it were done on a large enough scale, it would arouse more Catholic viciousness than it allayed." And again in August 1971: "Internment would worsen the army's chief problem which is mass Catholic hostility. More important it would carry the security forces beyond the frontier of what is ordinarily considered tolerable in a civilized society."
Again the government listened to Stormont, cheered on by sections of the press in Britain which had frankly done far less on-the-ground reporting than we had. We were the only newspaper to come out against internment when 342 Catholic men were "lifted" in very rough raids on houses on August 9, 1971. In various political encounters I was made to feel as if I was letting "our" side down, but internment was a disaster. Intelligence had been too weak to capture the most dangerous men. By November, 908 were imprisoned, many guilty of nothing more than where they lived; by December, 1,576 had been arrested and 934 released. Not a single Protestant gunman was arrested. The key Provos escaped. The advocates of internment defended it as an emergency measure following the deaths of twenty-seven people in the first eight months of 1971. It had the opposite effect. In the remaining four months, there were forty-one deaths in the defense forces and seventy-three among civilians.
It was no pleasure at all to be vindicated. It was infuriating. The British public was bewildered. It was like coming into a movie halfway through. How had we got in this mess in the first place? Nothing made sense. How could Catholic women who'd come out to give cups of tea to soldiers in 1969 now be on the streets shouting to the tune of "Auld Lang Syne": "Go home, you bums, go home." How did the army's velvet glove develop, in August 1971, into the iron fist of internment?
Bruce Page made the inspired suggestion that Insight should be given the time and resources to answer such questions in a major historical reconstruction that he'd supervise. The editor of Insight was now John Barry, the disconcertingly omniscient baby-faced reporter who'd done well on Philby. He took himself off to the province with four reporters, and for three months they engaged in questioning hundreds: Catholics and Protestants, householders and activists, generals and IRA leaders, politicians and civil servants, newspaper observers in Belfast and Dublin, lawyers and academics. "It's proving absurdly easy to get everyone to talk," Barry wrote to me. "I think it's because nobody has ever asked them to look back."
Insight's report tracked the origins of the crisis from partition to the first stirrings of the civil rights protests and the rebirth of the IRA. I thought I was up to speed on Northern Ireland, but I read the report in November 1971 with a mounting sense of despair. It was like rewinding the tape on a terrible crime. It had been an unjust society, but here, step by step, one could see how it had begun to descend into murderous chaos and why—and why the reel was still unwinding. It was heartbreaking to read of the misperceptions, peaceful protest failing and degenerating, old prejudices hardening, and decent people feeling betrayed, so that the men with guns came to prevail. And, perhaps most astonishing, despite the conflict having already lasted so many years, until now nobody had bothered to look so closely.
The real base of terrorism was profound Catholic distress and disillusion in the north and tenacious Catholic sympathy in the south. "Too long a sacrifice," wrote W. B. Yeats, "can make a stone of the heart." The people who had refused to face Catholic disillusionment had only prolonged the agony. The Insight portrait of a disintegrating society, on the contrary, demonstrated that the province faced a crisis of governance, for which no solution could emerge from military action, despite all the innumerable acts of forbearance and individual courage by the soldiers assigned to policing.
Initially I'd cleared two full pages for the report. I was on the stone as trays of type arrived a couple of hours before press time. It was clear there was too much metal, not enough space—and still more copy was landing on the printers' desks. It was all too good to cut, so I cleared still more space. And still the copy arrived. I gambled on writing an announcement that this was a two-part series. It transpired that we had more than enough to justify that impromptu decision. The team's fifty thousand words, carefully edited by Ron Hall and Bruce Page, ended up taking eight pages in the end, the longest home news report the paper had ever published. I ran "Perspective on Ulster" over two weeks in November 1971. It wasn't just long, however, one of those lengthy newspaper series we mentally categorize as too important to read. Thanks to the skill of the writers and editors, it was a gripping fast-paced narrative, taking the reader into the minds of the competing characters, all acting for what they thought was best and together producing the worst. There was an enormous and typically extreme reaction to the series. The newspaper was lauded by some for "saving Ireland from insanity," when in reality we could only hope to accelerate a political debate. Alternatively, the reporters were accused of "doing the dirty work of the IRA."
There was an important news fallout from our commitment to writing an accelerated history of the Troubles. Insight's researcher in Belfast, Parin Janmohamed, made the rounds of Catholic lawyers who were trying to represent some of the men seized in the internment swoops by the police and the army. The lawyers were too frightened to say much, but in one of the offices she induced a lawyer to give her half a dozen handwritten statements smuggled out of prison. They laid out what internees alleged had happened to them—and what they'd heard had happened to a few others who'd undergone interrogation. They told of men being hooded, made to stand spread-eagled for hours, and deprived of sleep, but those allegations were secondhand. Parin went back to Insight's hotel headquarters and said, "You won't believe what I've just been told."
In one of the statements, a Michael Farrell said that he'd been forced to run barefoot over a path of broken bricks at Girdwood Barracks. John Barry hired a small plane to fly over Girdwood to see if the basic layout corresponded with the statements. It did, but the brick path story didn't sound credible. Insight asked the army. A path of broken bricks? Nonsense, Irish fantasies. Barry talked his way into a Catholic convent that had grounds adjoining the barracks. He climbed a tree to see over the wall, and there, facing him, was a long path of broken bricks, the foundations for a concrete path still to be laid. He remembers clambering down, thinking, _Oh, my God, what's been going on here?_
The brick path meant Farrell's allegation might be true, but it was not solid evidence. Insight was also told of Alsatians snapping at the internees as they ran over the bricks. Farrell's statement said one of the dogs had taken a bite out of the sleeve of his jacket. Barry persuaded Farrell's wife to get the jacket from prison. One sleeve was ripped. He took it to a forensic chemist. Yes, he reported, there were saliva stains around the tear, though he could not be absolutely sure they were canine.
Barry was convinced that Farrell was telling the truth, but he was baffled by talk of hooding, wall-standing, and deprivation of sleep at somewhere other than Girdwood. All the internees had been roughly treated, but if the statements were to be believed, a few had been subjected to five techniques that became known collectively as highly coercive interrogation (HCI): covering the prisoner's head with an opaque cloth with no ventilation (hooding); standing a prisoner against a wall with hands raised and fingertips to the wall for as long as twenty hours; subjecting him to high-pitched white noise; depriving him of all but bread and water; and denying him sleep. These practices fell under the Geneva convention barring cruel and degrading treatment. But how to check the facts? Nobody on the team had ever heard of anything like this before. In December 1971 the IRA had staged a great jamboree of a press conference in Dublin, putting on show three IRA men who'd escaped from jail in Ulster and claimed they'd been tortured with lighted cigarettes and needles in their bones. Photographs of a defaced leg and thigh flashed across the nation's television screens. The _Guardian_ put the story on the front page: "IRA Men Show Wounds."
These horrific details of burning and bone scraping were, as it happens, without foundation. On December 19, 1971, Lewis Chester, a former Insight editor, documented the falsity of it all in a way that was far more effective than official denials. He was able to do it because for some time he'd been investigating on the ground and by hard work had uncovered some apparently genuine allegations of ill treatment under interrogation. Comparing these earlier statements and doctors' reports on the IRA men, Chester demonstrated that they were fiction.
How did the media correct their earlier acceptance of the false story? They didn't. Of the papers that had splashed the original story (the _Times_ had played it much more cautiously), only the _Guardian_ bothered to print a follow-up, but it was brief and relegated to the back page. And on television, there was silence.
We'd learned to be very wary of IRA propaganda, but Barry thought of someone who could help us on the nature of the techniques allegedly being used. A Special Air Service (SAS) man we'll call "Mike" had spent hours telling Insight of his adventures in Yemen in 1965 running a joint British-Israeli-Saudi covert campaign against Egyptian troops intervening in a civil war there. Suddenly one day Mike got scared that he might be identified and asked Insight to drop the story. We did. Back in London the reporters took him to their favorite watering hole. He felt he owed them one, so in the tiny glassed-in private cubicle that was Insight's home away from home, he said, "Sure, those are the interrogation techniques taught at Ashford." He was referring to the army intelligence headquarters. Mike opened up a bit more to say that the five techniques were designed to induce sensory deprivation and disorientation and that Ulster had been the first time they had been tried for real.
No one with a grievance understates it, but it seemed inconceivable that the wretched internees could have known enough to concoct all this. The question then was whether the interrogations had taken place somewhere Insight could check. The victims had been taken in helicopters to, they were told, the mainland. We wasted a week checking bases in Britain before one of the reporters thought to make friends with an air traffic controller who said yes, the helicopters had indeed flown eastward but had then turned round and landed back in Northern Ireland at Palace Barracks, Holywood. But how could we substantiate this? Another reporter pointed out that "Mike" had said that doctors had to be present at these interrogations. Those doctors must have been housed somewhere, so Insight combed the hotels around the base. Bingo—they found a chatty receptionist who told them about all those nice young British military medical men who'd stayed there for two weeks at just the right time. The reporter checked their names and dates against the register. They tallied.
At about the same time, and quite independently, John Whale came back to the office from Stormont to write a report incorporating a remark by a Northern Ireland cabinet minister: "Those fellows are singing like birds." We now had a multiplicity of statements of ill treatment tending to corroborate one another in substance and yet made by men who could not have coordinated their stories. I was satisfied that it all added up to a prima facie case.
The next question was whether there was a public interest argument against publication. I knew it would embroil the paper in still more controversy between those who would declare it unthinkable that Britain would do such a thing and those who felt that nothing was too harsh for the internees. I shared the repugnance for the unspeakable cruelties of the IRA. It was an evil enemy, its methods indiscriminately vicious; there could be nothing but admiration for the skill and bravery of the soldiers who disarmed its murderous explosive devices.
But further questions were raised. The first was that there was far from any certainty that the men in prison were members of the IRA or guilty of any crime. The second was that our report might incite retaliation. But one of the philosophies I'd absorbed at Durham was Immanuel Kant's maxim that one should act as if the principle one follows will become a universal law, and it seemed to me that condoning cruelty by keeping silent, for whatever reason, was immoral. So I ran the Insight-Whale report on the front page, albeit with a modest single-column headline: "How Ulster Internees Are Made to Talk."
The paper was roundly condemned. A group of Tory MPs came promptly and angrily to my office to say that what I'd done was close to treason. The establishment mouthpiece, the _Sunday Telegraph,_ which had done very little frontline reporting, fueled the fire. _Crossbow_ , the influential magazine of the Bow Group of younger Conservatives, caricatured me sinisterly reading a copy of the _Sunday Times_ with the headline "Army Atrocity." When I entered a reception one evening, the dilatory home secretary, Reginald Maudling, called out, "Here comes the editor of the IRA Gazette."
This was rich. We'd been unequivocal in expressing our detestation; we'd exposed the IRA torture story the rest of the press had retailed and never corrected; we'd prominently featured the sufferings of the victims of bomb outrages; we'd depicted the lonely ordeals of army patrols; and later at Scotland Yard's request we would faithfully refrain from reporting that police had quietly flooded an area of London in the hope of trapping a murderous IRA gang (which they did in the Balcombe Street siege of December 1975, despite the stupidity of the _London Evening News_ in revealing the tactic).
To his credit, Prime Minister Edward Heath appointed Sir Edmund Compton to head a three-man inquiry into our report. We were vindicated. Subsequently, on a complaint by the Irish republic, the European Court of Human Rights found that the five techniques used were "cruel, inhuman and degrading." It stopped short of calling them torture, but the British government gave a "solemn undertaking" that they would never be used again.
In five years of reporting on Ulster, everyone involved had been acutely sensitive that we'd gone out on a limb with every story that did not regurgitate the conventional wisdom. What happened on Bloody Sunday, January 30, 1972, tested our rigor.
In the space of twenty minutes in Derry, a company of British paratroopers fired 107 high-velocity bullets that killed thirteen unarmed Catholics and wounded fourteen. The event was immediately overlaid by propaganda from both sides and as I write this thirty-seven years later still has not been resolved. The Catholic Bogsiders were outraged. They believed a massacre had been planned. The British government immediately insisted that the paratroopers, sent into the confines of the Bogside to arrest violent rioters, had been fired on by IRA gunmen—and fired on first.
What I did about Bloody Sunday provoked controversy. Put baldly, why did the editor of the _Sunday Times_ fail to publish an immediate article on the shootings? Was it because the writers "diverged from the official line," as the _London Review of Books_ phrased it in 2002? Did I "help to bury the evidence" that the British Army planned the shootings in advance, as alleged in 1998 by James Ledbetter, the media editor of New York's _Village Voice_?
It is true I did not publish an article by two _Sunday Times_ reporters. But the reasons were different from the ones implied. Immediately when I heard of the shootings that Sunday evening, I reached out for John Barry, the chief author of our reconstruction "Perspective on Ulster." He was on a bizarre vacation, riding a camel deep in the Sahara with a Tunisian antidrug patrol. I got hold of Murray Sayle and asked him to leave at once for Derry. He'd covered Northern Ireland extensively beginning in 1965, and in particular the riots that had followed the internments in August 1971. The newsroom assigned two reporters, Derek Humphry and Peter Pringle, who had contacts in Derry.
Sayle is a legend in journalism, a buccaneering figure with a large broken nose acquired in graduating from the University of Sydney and a spellbinding retailer of stories. His hilarious novel _A Crooked Sixpence_ recounts his time as a vice reporter on the newspaper _People_ exposing prostitution rings. (Editor Sam Campbell's instructions to him were to get offered sex for money, watch the lady disrobe, and then "make an excuse and leave.") Murray had climbed Everest, sailed the Atlantic, and reported on the battlegrounds of Vietnam, Israel, Czechoslovakia, Jordan (Black September), and Bangladesh, as well as anywhere else a foreign editor needed a reporter who could talk his way through a brick wall.
Sayle was the principal author of the report that reached our office on Thursday afternoon. Before I had a chance to read it, both Page and Hall, the responsible editors, came in to see me, very concerned. "You can't publish Murray's piece." Page was especially vehement. He said the report implied that an unprovoked army had plotted the killings, a serious charge that should be published only if it could be substantiated. Murray's piece did not do that: the sources had not been subjected to enough scrutiny, the findings were inconsistent, and there were too many "internal contradictions." It just did not meet the _Sunday Times_ standards. It would damage the credibility we'd established by years of work on Ulster.
I said I'd read the piece and consider all their objections. It is generally a good rule not to let head office second-guess the reporter in the field, and Sayle was revered. But Page and Hall had investigated and edited many contentious stories. They were zealous guardians of the paper's unimpeachable investigative record, neither beholden to the other. The idea that either of them would toe an official line was ludicrous; even the idea of an official line would have roused their hostility.
Reading the piece, I noted that it said right at the start that not a single shot had been fired at the soldiers, then quoted one "official" IRA man as saying that he did fire a shot from a .38 pistol and later quoted Gilles Peress, the French photographer who brought back devastating pictures, as saying that he'd heard two pistol shots near Free Derry Corner when there were no paratroopers there. That inconsistency wasn't confronted. I could see that Sayle and the other reporters had worked hard. The background to the tragedy was lucid. The tumultuous scene in Derry was vividly depicted, assisted by a graphic street map of the action. But the piece bothered me. As research, it was probably as good a job as anyone could have done in four days, but it went beyond quoting witnesses. It tried to reconstruct a chaotic, fast-moving scene, reconcile conflicting stories, and reach conclusions. That is an objective I would normally applaud, but there just wasn't enough evidence in the copy I had. My immediate thought then was whether we could preserve the bones of the reporting while editing for internal consistency and dismantling the shaky hypothesis.
At this point James Evans made an appearance. He'd come in to support Hall and Page. The evidence wasn't good enough.
Before I had time to invite Page and Hall to discuss what we might do, the news desk rang to say the government had appointed Lord Widgery to conduct a judicial inquiry into the shootings. This complicated matters. The whole question of whether or not to publish would be moot if the Lord Chief Justice decided that the rule of contempt of court applied. I telephoned the Lord Chief Justice. I thought he might not take the call—judges in Britain don't jump to the telephone for pressmen—but I got through. I said we had done a great deal of interviewing and proposed to publish this Sunday. We also had compelling photographs. I told him I presumed contempt would not apply since nobody had yet been accused.
It would be an exaggeration to say he was aghast, but he made it very clear that it would be "unhelpful" to publish anything and yes, he would apply the rules of contempt. I told him that we would continue our own vigorous inquiries. I withheld the article, but that week I took the chance of publishing the shocking photographs by Gilles Peress of unarmed men being shot.
At the same time, I gave the Sayle article back to Hall and Page, asked them to make use of the reporting, and told them to get John Barry off his camel and without delay set in train our own parallel inquiry.
"It was the most intense period of our lives," remembers Barry. Insight interviewed 250 witnesses, including members of the IRA. It was essential to hear what they had to say. (Those who assert you can never "talk to the enemy" are more interested in party-line propaganda than the difficult business of weighing evidence.)
Eleven weeks after the shootings, Lord Chief Justice Widgery issued his report. He judged that some of the paratroopers' firing had "bordered on the reckless," but he basically exonerated the army of any premeditated plot to kill. But the report was not the whitewash the members of the Catholic community immediately said it was. Widgery confirmed that one of the thirteen people killed was an unarmed man shot from behind while crawling; that four other men had been killed by shots fired without justification; that an excessive number of rounds had been fired; that grounds for identifying targets had been nebulous; and that all this had happened in a battalion operation not clearly authorized at the brigade level, discountenanced by the police, and launched at a time when other methods of keeping order were succeeding.
Insight was ready. The team amassed five hundred photographs from all the sources that day, laboriously sorted these into sequence, and from them identified the witnesses who had been present at critical moments. Then they tracked those people down and questioned them. They tried, above all, not to create a seamless narrative where there wasn't one. Nobody in the army or media and no one among the rioters and politicians could offer a single perspective on a kaleidoscope of events in which a second of life was an eternity. We laid out what we knew and made clear what we didn't know.
Insight's four-page report was critical of Widgery. It differed from him on his certainty that the army fired only in response to IRA fire. Nor did it endorse the Sayle report's conclusion that the IRA did not fire a single shot: "The Provisionals admit to a burst of machine gun fire from the area of the Bogside Inn which is recorded by the army after 4:40 p.m. It is certain that they fired other shots. Eyewitness accounts vary from none to 50, but witnesses agree all shots were fired after 4:30 p.m. [when the army ceased fire]." The vicious assaults of the more reckless rioters were acknowledged, but the government was faulted for authorizing an attempt to scoop them up using heavily armed paratroopers. Militarily it went wrong in plan and operation; soldiers did shoot at obviously unarmed civilians. We added the rider that although the paratroopers' response was out of proportion, the vast majority of the 100-odd soldiers involved, under great stress, did not fire, let alone kill anyone.
Through the intervening years, Bloody Sunday has remained an open wound. Murray Sayle, the author of the first report I did not publish, recognized the legal constraints: "I don't blame Harold Evans for not publishing the story," he said in 1992 and reiterated in a series of e-mail exchanges with Peter Baker, editor of _Fingerpost,_ a small magazine covering the Catholic community in Derry. But then he added: "Publication might have saved much subsequent bloodshed."
Here I part company with him. The original article, with its errors and shaky imputations, would likely have inflamed feelings even more rather than do justice to all those involved. The complexity of Bloody Sunday is exemplified by the fact that the tribunal of inquiry set up by Prime Minister Tony Blair in 1998 had by 2007 taken evidence under oath from nine hundred people, including the prime minister at the time, Edward Heath, and by 2009 it still had not reported its conclusions.
Among the Insight group, John Barry, Phil Jacobson, and Peter Pringle were summoned to testify, as were Sayle, Geraghty, and I. The entire Insight archive on Bloody Sunday, including the reporters' notebooks, Murray Sayle's rejected article, and my memos, was surrendered by the _Sunday Times_ management after I left the paper, surprisingly without consulting any of us as to the sensitivity of our sources. Pringle and Jacobson have written a well-researched book on the event called _Those Are Real Bullets: Bloody Sunday, Derry, 1972_.
I remain proud of our Northern Ireland reporting. We revised the reconstruction we'd published as "Perspective on Ulster," corrected the inevitable errors, and extended the reports to create a paperback book called simply _Ulster._ It was a best seller, which paid for the costs of the reporting. The main characters were portrayed as more than cardboard cutouts, but succinctly, and each step on the road to Armageddon was made maddeningly explicable from that point of view. It was one of the most important initiatives I ever undertook and made me more ambitious for that kind of contextual journalism. It was possible only because of the skill and integrity of reporters willing to suborn their individual egos in a collective effort to present as truthful an accounting as they could, informed by narrative energy but untainted by preconception.
After Bloody Sunday the British government, the British Army, the Provisional IRA, and certain Protestant activists continued to take positions they considered points of principle. Through the Anglo-Irish Agreement of 1985 and the Good Friday Agreement of 1998, the rate of wanton death declined, albeit with appalling slowness. Only the IRA was not willing to stop the killing. Yet gradually after the event, intelligence successes on the IRA side could be seen to fall farther and farther behind those on the British side. John Hume worked hard to convince Gerry Adams, leader of Sinn Fein, that there was yet another historic opportunity for peace. In 2001, on a side wind after the destruction of the World Trade Center, American money for IRA causes ceased to flow. Another change in 2007 at last brought Adams and Protestant firebrand Ian Paisley into a kind of fellowship. Sadly, such understanding as the two aging enemies have pieced together could have been enjoyed at any time in the previous thirty-five years following the 1973 power-sharing agreement between the British and Irish governments and Northern Ireland's leaders.
Millions of words have been written about the Troubles. But just four lines of verse by the poet Desmond Egan linger in my mind as an expression of the tragic futility of all the hate:
_Two wee girls_
_Were playing tig near a car_
_How many counties would you say_
_Are worth their scattered fingers?_
#
# Showdowns
I opened the description of my _Sunday Times_ years with a photograph on my wall of an editorial conference. I close with another photograph that is seared in my consciousness: a tall, lightly bearded man in a Russian fur hat, deeply tanned in midwinter and wearing dark sunglasses. He is exultantly linking arms with a group of others outside our offices in Grays Inn Road in 1980; in the foreground is a coffin labeled "Sunday Times."
The man is Reg Brady, a Communist "father of a chapel" in the pressroom. In more prosaic terminology, that's a trade union shop steward, in his case representing the unskilled casual workers who helped man the basement presses. His key achievement was to shut down the _Sunday Times_ for a year and pave the way for its acquisition by Rupert Murdoch. It was not a solo effort. Brady's fellow unionist, a clerk who was father of the clerical chapel, Barry Fitzpatrick, had a hand in it. So did the members of another union chapel, the one hundred or so machine minders in the pressroom, led by a squat, tight-lipped man called Vic Dunn. They were perpetually fighting with Brady's members about pay differentials and who did what, arguments that held up the production of the newspaper. These issues were of graver concern than the multitude of demarcation sensitivities I found on first arriving in Fleet Street: every item of work was the jealous preserve of one union or another. If I hung up a picture myself or unplugged a reading light or changed a lightbulb at my desk, I was told the heavens would fall. "Good job nobody saw you do that," said the administrative editor. "We might have had a work stoppage."
Was he serious? I'd laughed like everyone else at the farcical antics of Peter Sellers playing Fred Kite the shop steward in the vintage 1959 film comedy _I'm All Right Jack,_ but I'd thought of it as confined to the Midlands car industry. Our relatively small workplace at the _Northern Echo_ in Darlington had been nothing like that, apart from the traditional prohibition against a journalist touching metal type in the composing room. At the _Sunday Times_ I was pleased to see one secretary reading a novel every day; I simply preferred that she did it on her own time rather than at the office. But every effort to move this studiously redundant young woman to work at a typewriter or telephone or at some other similar hard labor was thwarted by Barry Fitzpatrick.
It would be fair to say that we in management did not excel in combating guerrilla warfare at the _Sunday Times_. It escalated in the economic squalls of the 1970s, and we became increasingly exasperated. The ink had not dried on an agreement before one or another of the chapels broke it. Fitzpatrick, a fluent, nattily dressed clerk in his thirties, had a genius for ceaseless negotiation stimulated by management's expectation that his members would do the work (mainly processing advertising) they were paid for. Brady had a different wheeze. He and his union had insisted that 540 casual workers were needed to get the paper out. Only half of the 540 actually bothered to show up on Saturdays, but 540 pay packets were collected and signed for every week. Managements throughout Fleet Street closed a blind eye to this "old Spanish custom," but our labor editor, Eric Jacobs, was willing to venture into the basement with a flashlight. He discovered the Spanish custom was so ingrained that, apparently, we had working in the _Sunday Times_ pressroom none other than M. Mouse of Sunset Boulevard, Hollywood, and one week, big joke, another payment receipt bore the name of Marmaduke Hussey, the managing director of Times Newspapers. We were tensed for a work stoppage when I published this story, but Brady and his merry men took it in their stride, sure that nobody would dare to do anything about it. They reckoned without the tax authorities, who were gratifyingly curious and eventually reported that in Fleet Street more than 50 percent of the pay packets in newspaper pressrooms were drawn under fraudulently false names.
Corruption was only one goblin in the serial nightmare. Wildcat stoppages and downright sabotage were others. If management demurred over some new demand, the pressroom chapels damaged production. A favorite dodge was to accidentally leave chewing gum on a reel of newsprint in the presses, producing a paper break and time-consuming re-threading. Thousands of copies failed to reach readers. In 1978 the _Sunday Times_ was unable to fulfill all the orders on nine occasions; we became known as the "Sunday Sometimes." The importance of the news itself made no difference. In July 1977 the European Court of Human Rights report included as an appendix our long-suppressed draft article on the thalidomide children. The news came from Strasbourg at noon on a Friday. The machine minders chose this occasion of editorial triumph to ramp up a running battle with management on the number of minders required to print the seventy-two-page paper. The run began with only eight of the nine presses we needed. We fell badly behind.
It was line management's job to sort this out. I was on the phone every hour. They answered my entreaties by trying again to reason with the minders. I didn't want to complicate the lives of these duty managers—the stress of dealing with the crews undoubtedly led to the heart attack that felled the director in charge—but at midnight, as anger overtook despair, I invited Vic Dunn and his chapel committee to my office to tell them of the significance of the thalidomide article. I beseeched them to work normally and argue later. I thought I'd persuaded them that extortion on this issue would be immoral. Their response was to persist with their inaction, losing us 540,000 copies, one-third of the print order.
Much more comprehensible to me than such bloody-mindedness in pressroom and clerical was a collision of tradition and technology. I was an early addict of the computer. In 1954, as a science reporter, I was introduced at Britain's National Physical Laboratory to code breaker Alan Turing's Automatic Computing Engine (ACE), which, wondrously then, informed the waiting world what day my birthday would fall on twenty-five years hence. I must have been one of the first British newsmen to use a touch screen and video stylus, in 1973, on a visit to innovators along Boston's Route 128. I came back from a tour of American newspapers to enthuse about the computer for typesetting and research.
Management didn't need any urging to adopt computer-assisted typesetting. A number of my editorial colleagues did. They were troubled that acceptance of the computer would take jobs away from the Linotype operators who set the type for stories and classified advertising and the "comps" who assembled the type into pages. We had long bonded with these men. Some of them joined editorial for a friendly postproduction beer at the Blue Lion. The comps were cheerful allies in the weekly battle to meet press times, loyal to the paper, and disgusted by the anarchy in the pressroom. Having helped to set my RAF newspaper by hand and written about typography, I was still in love with the whole romance of hot metal and appreciated the pleasure they took in their jealous craftsmanship. I spoke their language. I thought we could work out a civilized transition that, by attrition and retraining, would keep the comps at work in the company.
It would be more costly, but such an arrangement would be equitable—and it was essential for the future of the _Sunday Times_. Because it was a text-heavy newspaper, access to the computer would save millions of pounds and hours of work. When every minute counted, it was ridiculous for journalists to type on a clunky typewriter, manually returning the carriage, and then have the same keystrokes duplicated by a typesetter—a typesetter who had no use for the facility the computer offered for an accurate word count or the ability to transpose paragraphs without starting to type all over again. Moreover, because the _Sunday Times_ was an investigative newspaper, I was eager for us to embrace computers for basic research (though even my untutored enthusiasm underestimated quite how valuable it would be).
In a negotiation with the printers' union, we proposed a phased introduction over several years, with the men most affected guaranteed employment for life. The adroitly charming anarchist of the far left of the Labour government, Tony Benn, referred to this as an attempt to impose our wishes by force. Our own workers might well have been amenable, but the national union leaders rejected every inducement out of hand, fearful that what we did would set a precedent for the whole industry. We got nowhere in endless negotiations, so for three years the computers we'd optimistically purchased lay unused under dust sheets, ominously referred to as "The New Technology." By 1979 nobody in Fleet Street had been able to introduce computers and display screens (then unappealingly called "VD" units, for visual display), although they were in all the American newsrooms I visited.
These are only glimpses of the obduracy that undermined the success of the _Sunday Times_. I was depressed and, frankly, bewildered by it all. Why couldn't the wreckers, as I saw them, realize that they were hazarding the whole ship? I had the conceit that I could make contact with workingmen of my dad's generation. In my no doubt sentimental reflections, they seemed to me to exemplify the cliché of "solid working-class values," preserving their natural dignity as they struggled through the bad times to keep their families together, proud of being able to do "an honest day's work," noticeably distancing themselves from the "slackers" and "scroungers" among them.
Perhaps I was so disappointed because I was nostalgic for the mythic north of my childhood, which began to vanish when close-knit "Coronation Street" neighborhoods with terraced homes and corner shops and pubs were replaced by drably anonymous housing estates, where the "telly" glowed every evening, inciting acquisitive envies. In the "bad old days" people were poorer, and they were more or less stuck in the place where they were born and grew up, but there was comfort in being rooted in a community and recognized within it as a good neighbor. The wartime spirit of solidarity had been very real, but it had evaporated along with the British Empire and the pride and affection for its glory that everybody felt but was not willing to admit. U.S. secretary of state Dean Acheson kept saying in the 1950s that Britain had lost an empire and had yet to find a role, and as a people we seemed to have lost one identity and were struggling to find another. The class divisions had faded, we assured ourselves, and we'd had reformist Labour governments, but the positions of power and privilege were still predominantly occupied by Oxbridge graduates as they had been for a hundred years. What would George Orwell make of it all?
* * *
Whatever historians have to say about our disintegrating society, cut adrift from its traditional moorings, I cannot blame anyone else for the personal stresses I felt in these years. In the crises of the 1970s I created a crisis of my own. After twenty years of a serene marriage to Enid, moving from Durham to Manchester to Darlington and finally to London and raising three children together, I fell in love with a woman half my age. Torn by the pull of a magnetic north and a magnetic south, I inflicted great pain on others, my own pain the least of it.
Tina Brown came into my life through her writing. Literary agent Pat Kavanagh (who was later married to the novelist Julian Barnes) sent me some clippings from the _New Statesman_ magazine. She urged me to think about commissioning features from the writer, who was about to graduate in English studies at Oxford: "She's very young, won a place at Oxford when she was 16, but has a wicked eye." I put the articles in my briefcase; they got buried under other submissions and manuscripts from agents pitching book serials for the Review Front. I didn't get round to the necessary archaeology until two weeks later. One of the Tina Brown clippings was a description of the stab-in-the-back gossip fest at a _Private Eye_ lunch in Soho to which she'd been invited by the literary critic Auberon Waugh, who'd admired her contributions to the Oxford university magazine. Cabinet minister Richard Crossman was the guest of honor—that is, he was in the hot seat, where victims were goaded to sing for their supper with confidences that the _Eye_ would wittily betray in the next issue. Brown's column about the lunch was hilarious. I'd normally have passed it on to Ian Jack, the editor of our Look feature page in what used to be the women's section, but feeling guilty at my neglect in reading the submission, I got on the phone at once to tell Miss Brown how much I'd enjoyed her writing and would like her to come in to meet Mr. Jack.
"So pleased you liked my columns," she said. "But sorry, not today—my husband has friends coming round for dinner."
Wait a minute! Married before she'd even left Oxford? Wasn't that carrying precocity too far?
She went on to ask how I'd come across her writing in a Spanish magazine. It transpired that the Tina Brown I was talking to was known to her husband as Tina, but she was Bettina Brown, indeed a sharp writer but Tina Brown's mother. The husband whose dinner party took priority over the needs of the _Sunday Times_ was the celebrated Pinewood film producer George Brown.
Duly fricasseed by Ian Jack for her lack of provincial training, the correct Tina Brown was nonetheless given a few trial freelance assignments by the paper and the color magazine. Her entry was apparently eased by the fact, which I hadn't realized, that she was known to the critics at the _Sunday Times_. They'd named her as the winner of the paper's annual student drama contest for her play _Under the Bamboo Tree,_ a comic ménage à trois. A year or so into her freelance work, I heard she was going to America to write a new play. She asked if anyone could give her introductions, and I said I'd drop a note to induce S. J. Perelman to see her. I'd published a few pieces by Perelman, the resident wit at the _New Yorker_ who had authored a couple of scripts for the Marx Brothers and the screenplay for _Around the World in Eighty Days_. The result two months later was a cross letter from Perelman: "I don't know why I am being cordial to a man who wrote me way back on November 1st that a beautiful blonde playwright (who had won your drama award) was coming here and would phone me. The only blonde I have seen around here is a Polish maid with fat thighs and no chest who persistently spills ammonia on my suede shoes. This can't be the woman you meant, Harry, or else you have a low opinion of me as a judge of feminine sexuality. Was Tina real or merely the product of an erotic opium dream?" He forgave me when they did meet, and they became soul mates.
Late in 1974, when I was in New York myself, I agreed to listen to some feature ideas Tina had. I ran late returning to the Regency Hotel, where I had arranged a dinner meeting of Insight reporters and the New York office manager. We were in the middle of our investigation of a scandal—the deaths of 346 people when a DC-10 airliner crashed near Paris after it shed a cargo door known to be defective. I'd been summoned to a court in California to press our claim for access to secret documents, and preoccupied by our discussions, I forgot that Tina was waiting to see me. When I realized I wouldn't be able to meet with her, I suggested that she write me a memo, which I promised to read when I got back to London. We corresponded about her work, and then about newspapers and literature and life, and so our relationship began. I fell in love by post.
It was absurd. She was twenty-five years younger and courted by the likes of the enfant terrible Martin Amis and our own dashing star reporter David Blundy. When it became apparent that I was attracted to her, she gave up freelancing for the _Sunday Times_ so that there would be no question that any success she gained might be attributed to having caught the boss's eye. She began writing features for the rival _Sunday Telegraph_ and won the Catherine Pakenham Award as young female journalist of the year.
My infatuation, I told myself, was a typical midlife crisis and I should get over it. But I couldn't. Sporadic panicky "goodbye and good luck" separations, initiated by both of us with resolve and good intentions, failed to last. One crazy night, when she was in Spain with her parents, I got on my BMW bike and raced to the airport for the last plane to Málaga, realizing upon landing after midnight that I hadn't bothered to find out precisely where in someplace called San Pedro de Alcántara Mr. and Mrs. Brown lived. Nor did my Spanish phrase book help me with "Do you know where I can find the Englishman who came to Spain before the civil war, made films out of Pinewood, speaks Catalan, and has a young blond daughter about five feet six?"
I was ashamed that I had begun a secret life separate from Enid. I took a company flat next door to the _Sunday Times_ office, ostensibly to be close to work. The habit of quick decision making didn't travel a couple of doors from the office to the company flat. I could disentangle the complications in deciding whether and how to take on the government over publication of the Crossman diaries, but I couldn't sort out my own life. For two horribly fraught years, I preferred to park my conscience while I wrestled with nerve-racking events at the paper.
Tina was deeply conflicted, too, about her association with a married man, simultaneously rejoicing and regretting. She kept trying to withdraw, but the bond we had, professional and personal, was too deep to break. I was the older, supposedly more responsible one, but the thought of living without her completely paralyzed me. I was enraptured by her and her innocent integrity and wit. I moved from the large, bleak company apartment to the tiny walk-up she rented in Bloomsbury. If she was in town and not on assignment for the _Sunday Telegraph,_ we'd walk over to nearby Charlotte Street in Soho to our favorite Greek restaurant, oblivious of everything about us.
When we escaped London together, we retreated to the safe haven of both our childhood summers—the English seaside. By the mid-1970s people who wanted to bake in the sun had fled by the millions on package tours to the Costa del Sol. There was an almost ethereal, seductively solitary air around the uncrowded British resorts, with their half-empty, windswept Victorian piers, timeless boardinghouses advertising VACANCY in the front window, and once intimidating grand hotels now too imposing for their own good.
On Sundays I'd collect Tina on the pillion of my bike, and we'd scout the coast for romantic hideaways. We'd hole up in some boardinghouse with a pile of books and magazines, eat at the baked-beans-on-toast cafés, scramble among the rocks for shells, and walk the downs atop the cliffs at Beachy Head. In Hastings one morning a downpour forced us to retreat to a smoke-filled pub, where we sat with the locals watching the Queen's Silver Jubilee celebration on television as the rain fell. On an excursion to the Irish coast at Connemara, we impractically considered buying a boathouse where we could disappear from the world. One magical Sunday exploring the coast of West Sussex, we found a little house with a For Sale sign at Angmering-on-Sea, where the ocean lapped right up to the back garden. We dreamt of buying it one day when we could afford it.
Meanwhile, at the _Sunday Times_ we were not alone in our industrial misery. Waves of strikes crashed into and over the remaining seawalls, culminating in the 1979 chaos of the "winter of discontent." In two months thirty million working days were lost. I saw pickets blocking cancer victims arriving for treatment at St. Thomas' Hospital across the river from where I lived. Going to a restaurant through the entertainment center of Leicester Square, I walked past high piles of uncollected rotting garbage that earned it the name "Fester Square." The National Union of Journalists, pursuing a wage claim, made a mockery of years of press protests against secretive local authorities by actually asking council officers to deprive people of news about gas leaks, fires, building plans, rents, and rates. It was like asking Sweeney Todd for a close shave.
The Thomson Organization sanctioned a dramatic bid to start anew. It offered to invest millions of pounds to buy out obstructive practices and overmanning, but the chapels and their unions didn't want a brave new world. Every proposal was rejected. As a result the paper was shut down in November 1978—a temporary break, we all thought, until negotiations resumed.
I drank the cup of bitterness many times over as I walked through the silence of the dead composing room, with its shrouded Linotypes and darkened offices. Gathering dust in my "pending" tray was a scoop of world importance. Anthony Mascarenhas, the man who'd exposed the genocide in East Pakistan, detected that Pakistan was well on the road to possessing a nuclear bomb. He pointed the finger at the then unknown Dr. Abdul Qadeer Khan, whose thefts of blueprints from a European facility would enable Pakistan to become the first nuclear power in the Muslim world. Khan, wrote Mascarenhas, hadn't stopped there. He'd supplied both Iran and Libya with centrifuge components and information. As the weeks of suspension turned into months, I gave approval for Mascarenhas to send his report to the Australian magazine _Eight Days,_ started in Sydney by former _Sunday Times_ executive Colin Chapman. Trading a world scoop of historical importance killed any lingering feelings I had of conciliating the unions.
We were suspended for a full year, but even when an agreement was reached to restart, the recidivists in the pressroom worked their mischief again, and the comps' national leaders reneged on the computer deal we'd worked out together.
I've never forgiven the print unions for what they did. Kenneth Thomson, Roy's son and then head of the company, was deeply wounded. Ken was a kindly, somewhat absentminded man with a gentle sense of humor who thought the best of everyone he met. In Canada, where he lived, he was not "Lord Thomson"—"call me Ken"—and he never took his father's seat in the House of Lords. He delegated the management of Times Newspapers to a London board, but he took pride in the papers, as his father had. Even though the _Times_ journalists had been paid normal salaries for a year of not working, soon after returning to work they called a strike for more money. That was the last straw, a betrayal of the Thomson family, which had spent millions to save the _Times_. Thomson sadly concluded that he could do no more. "I promised Dad I'd keep the _Times_ going, Harold, but this is too much," he said. He put both papers up for sale. I led a management buyout bid for the _Sunday Times,_ but Thomson's London management, and Denis Hamilton, thought that Rupert Murdoch and his News Corporation had a better chance of dealing with the unions.
I'd encountered Murdoch often enough to appreciate the delusiveness of his charm. He was a chameleon who could switch from good humor to menace. I'd heard every jolly swagman's yarn that placed him somewhere between Ned Kelly and Citizen Kane. I'd been at seminars on newspaper ethics where he'd acted like a caged lion, glowering his contempt for the do-gooders and sappy academics. I often agreed with him. Once, expressing admiration of the _Sunday Times'_ investigations, he'd joked that I should take over the _Village Voice_ and teach the journalists there the meaning of responsibility. My friend Australian editor Graham Perkins had declined to work for Murdoch but thought that within the hard exterior of the riverboat gambler, there might still be found the lost idealist of the "Red Rupert" of his Oxford days. I didn't think that, but after the years of hand-wringing at our board meetings, I did find his buccaneering can-do style refreshing. "Sure," he said with a laugh, "we'll sort out the unions. We're going to print the _Sunday Times_ in two sections, Friday and Saturday, and go up in size." Music to my ears.
The journalists felt badly let down by the Thomson management; they didn't trust Murdoch. The Australians associated with the paper were especially vehement, saying that he had fired every editor who'd stood up to him and that he would have no respect for the paper's cherished independence or any promises he made. But when the _Sunday Times_ journalists' chapel came to a vote at the end of a passionate debate (which as management I could not attend), it voted against a court action to force a reference back to the Monopolies Commission. Many of them feared that a breakdown would mean the end of our sister paper, the _Times_. Fourteen of them favoring legal action, members of the so-called Gravediggers Club, printed T-shirts bearing the cry DON'T BLAME US. WE VOTED AGAINST.
The Thomson Organization and Parliament had asked Murdoch for guarantees that the tradition of editorial independence of both papers would continue to be protected in two ways: by the appointment of independent national directors to the board and by five guarantees of editorial freedom. Murdoch readily promised that editors would control the political policies of their papers; they would have freedom within agreed annual budgets; they would not be subject to instruction from either the proprietor or management on the selection and balance of news and opinion; instructions to journalists would be given only by their editors; and any future sale of the titles would require the agreement of a majority of the independent national directors.
It was on the basis of these guarantees, and only because of them, that in February 1981 I accepted Murdoch's invitation to edit the _Times,_ giving up the job that had given me such fulfillment and pride at the _Sunday Times_ and my power base as a defender of press freedom. My ambition got the better of my judgment. I guess the bitter experience with the unions had made me eager for a new start. I hadn't been enchanted either by the furtiveness of Thomson's London management during the sale. Still, it was wrenching to leave my friends at the _Sunday Times_. It had been a partnership sustained by a conviction everyone shared: we were doing something worthwhile in bringing the public early intelligence they'd not get anywhere else and associating it with the highest levels of cultural commentary we could achieve. It was a community of shared values—not political values, but the values of purposeful, honest journalism. Selecting and promoting people of excellence who had the same ideals, and whom I could trust to live by them in a collaborative enterprise, had been one of my principal tasks as editor.
Despite the difficulties we'd had with the national leaders of the print unions, I also retained an affection for the printers who worked with us. On my last Saturday evening putting the front page to bed, I was touched that the comps' farewell was an honor rarely accorded to anyone outside their union: they "banged me out," which meant that everyone on the floor seized whatever piece of metal was to hand and hammered away, creating a tremendous noise as I waved goodbye holding my last page proof. The photographers later ended a more sedate dinner given by the company by hoisting me on their shoulders. A week later at the _Times,_ on my first night as editor, the comps accorded me the privilege of pushing the front page into the foundry, a pleasant welcome that was not a harbinger of things to come elsewhere in the building.
In my first six months at the _Times,_ Murdoch was an electric presence, vivid and amusing, direct and fast in his decisions, and a good ally against the old guard, as I worked to sharpen the paper's news values while retaining every element of its traditional coverage of Parliament, the law, obituaries, and the arts. I had his enthusiasm for a thorough overhaul—"Go to it, Harry"—making headlines more readable and letting photographs breathe. He overruled the squeaks from his advertising director when I swept classified advertising off the back page for an irreverent parliamentary sketch and an information service. I brought in new political writers and started a new tabloid-size arts section.
Twenty-one days into my editorship I was at dinner with Tina at Langan's Brasserie just off Piccadilly on the night President Ronald Reagan was shot. I left the dinner table in a tearing hurry to oversee our coverage, calling for the most detailed narrative, a separate report on the gunman, another on the violent history attendant upon American presidents, and a third on the character of the next in line, Vice President George Herbert Walker Bush. There was argument around the picture desk about which of three near-simultaneous photographs we should use—one of the president looking toward the shooter, one of him being hit, or one of him being bundled into a car. This was an unusual true sequence, and to choose just one or to use three small images would be to miss an opportunity. I schemed all three running six columns wide down the page. Finally I ruled that the whole front page would be given to all the Reagan elements, and I created a second "front page" in the normal _Times_ style for other news. We developed the same approach for other late-breaking news: the _Challenger_ shuttle explosion; Israel's bombing raid on Iraq's Osirak nuclear reactor; the assassination of Anwar Sadat; riots in London and Liverpool.
The Reagan front page was a departure from the traditional _Times_ style, as dramatic as the event, and I'm still proud of it today. There were mutterings, of course, from some of the old guard I'd dislodged from positions they had come to see as tenured. But readers responded in the thousands. Circulation stopped falling. News Corporation's 1981 annual report said that the "exciting" editorial changes had the "extremely gratifying" result of increasing the paper's circulation from 276,000 to more than 300,000.
My difficulties with Rupert really began in the autumn of 1981, as the economy showed little sign of recovery from a recession. Mrs. Thatcher's government was facing a catastrophic fall in popularity. We supported her editorially on any number of issues, including her determination to curb excessive pay demands by the civil service, but we were critical of her reliance on monetary policy in a recession and disappointed that she seemed unwilling to tackle the abuses of the trade unions as she'd promised. (She made up for that later.) At the same time, we were unsparing in documenting the disintegration and spiritual collapse of the opposing Labour Party. We identified the virtually unknown left-wing activists who were conspiring to win control of the party leadership by changes in the party's constitution and with that gave fair weather to the rise of the Social Democratic Party. However, it soon became obvious that nothing less than unquestioning backing of Thatcher on every issue would satisfy Rupert.
His mouthpiece, Gerald Long, wrote me a stream of memos asking me to downplay or suppress news that was bad for the government. In the spring of 1981 the chancellor of the exchequer had said the recession was over and recovery would begin in the early summer. It didn't. Six months later the Central Statistical Office released figures showing that output had fallen for the sixth successive quarter. Long stood amazed at our temerity in printing a summary of this official report. Did I not understand that if the government said the recession was over, it was over? As far as I was concerned, his rebuke was red rag to a bull. I was not going to let anyone in management tell me to fix the news. (Output fell by 2.3 percent in 1981.)
The warfare with Long escalated through the winter of 1981–1982, with Murdoch himself giving instructions to editorial writers and continually ducking the pledge to give me a budget. (Of course this came in handy later for the bogus charge that I had exceeded nonexistent limits.) In fact, by this time he'd blithely broken all his editorial pledges. Stories mysteriously appeared that I was thinking of resigning or being asked to resign. Murdoch denied them all. On February 10, 1982, hours after I'd been named Editor of the Year in the Granada press awards, he issued a statement saying there were absolutely no plans to replace Harold Evans, whose outstanding qualities etc. The reality was that two weeks later, he went to the national directors to ask them to dismiss me and install a new editor. They refused twice. They told him that if he himself dismissed me, I had a right of appeal to them and no pressure should be brought on me.
It was a dark time, and then came the news that I'd long dreaded. Since receiving his gold watch (and his miserable pension) for fifty years on the railway, Dad had lived very happily with Mum in a bungalow by the sea in Prestatyn, North Wales. Into his eighties, he rode his bike to the post office and bowling green and played football with his visiting grandchildren, cajoling my first son, Mike, to shoot with both feet and eat his crusts. In the summer he put on his glossy peaked cap for a return to railway work, giving rides to children on a miniature steam train on the promenade at Rhyl; he took it as seriously as he had driving mainline expresses. Mum and Dad lived close to an unpretentious golf club, and Dad liked to walk to it through the sand hills for a game of darts and for the oxygen of his days—conversation about the world.
He'd been a staunch trade unionist all his life and on the Labour left, but he had a dim view of the irresponsibility that had come to pass for trade unionism. I have his diaries—an entry of his activities every day, written in a hand far more legible than mine—and notebooks of the family budget ("Good news railway pension increased by 40 pence a month from 7.23 to 7.63"). Every Sunday, I am touched to see, he had recorded the length of the suspension at the _Sunday Times_. The big highlight of their retirement was to cross the Atlantic twice to stay with my brother Peter and his wife, Dorothy, in Ontario. They took the dome-car train—naturally—to make the 2,400-mile journey to the Rockies and beyond to Vancouver, giving Dad a good excuse to wear a cowboy hat and ride in the canyons of his imagination.
Then the inescapable day arrived. Dad had recovered from the heart attack he'd had while visiting us in Kent and resumed his normal life in Prestatyn. Now, a year later, as I was sending the _Times_ to press, word came that he had suffered a stroke and was in a coma in the hospital in Prestatyn. He was in his eighty-second year.
All the sons had kept in close touch with Mum and Dad, Fred especially since John was in distant parts on Foreign Office work and Peter had emigrated to work for an insurance company in Canada. We were all told that Dad would be in a coma indefinitely, and we were discouraged from visiting: he would not be able to see or hear anyone or speak to us. In about the third week, though, Enid went to the hospital to visit; my parents were fond of her and she of them. She was surprised to find him sitting upright in a chair by the bed. She asked him to nod if he could hear her; she thought he did. All four sons hastened to Prestatyn—Peter from Canada, John from Hong Kong, Fred from Gloucester, and me from London. Mum was too ill to be with us. We stood by his bed and one by one spoke to him, telling him that we were all together again for the first time in many years. We thought we detected a responsive flutter of an eyelid.
Dad died forty-eight hours later. His friends said, "He was waiting for his four sons," and I think he was. We buried him on the hill in Bluebell Wood cemetery at Coed Bell, overlooking the sea. Two months later Mum, brokenhearted, joined him there. For many years I couldn't bear to open the diaries of their good last years together.
On Tuesday, March 9, 1982, upon my return from my father's funeral, while I was supervising the newspaper's budget edition (a special edition presenting the chancellor's annual budget) Murdoch summoned me to his office. He leaned forward in his chair, took off his glasses, and stared at me. "I want your resignation today," he said. I was astonished at how calm I was: it was rather like the out-of-body sensation I'd had the time I was mugged in New York and seemed to be watching myself from above. I noticed how red the rim of his left eye was, the thickness of the black hair on the back of his hands. "You cannot have my resignation," I heard myself saying. "I refuse." I asked what criticisms he had of the paper. "Oh, you've done a good job with the paper, sure. We haven't signed your contract, you know [I didn't], but we'll honor it." And then he veered. "You've said I put pressure on you. I haven't put any pressure on you. I've always made it clear political policy is yours to decide." In the midst of these exchanges, his voice wavered. He began to say how much harder it was for him than for me. No need for me to worry. I'd get lots of jobs. He'd wondered whether I'd take a job with his News International but guessed I would refuse. He'd guessed right.
After some twenty minutes I said I had to get back to the budget edition. As I left, saying he did not have my resignation, I asked whom he had in mind for my exit. It was then I learned that he'd suborned my deputy, Charles Douglas-Home. "Can't bring in another outsider at this stage," Murdoch said. "He'll be all right for the time being."
Back in my office, I confronted Douglas-Home. Eton and the Royal Scots Greys, the second son of the second son of the thirteenth Earl of Home, "Charlie" was a member of the _Times_ old guard par excellence. I'd only appointed him as a gesture to that faction. He'd been most ardent in expressing his determination to stand with me in preserving the paper from managerial interference, so I asked him how he could have conspired for my job. He replied, "I would do anything to edit the _Times_. Wouldn't you?" Saying "No, I wouldn't do anything to edit the _Times_ " seemed wan in the glare of his ambition. Accustomed to having a loyal and supportive deputy at the _Sunday Times,_ I'd underestimated how much Douglas-Home longed for the validation of being anointed leader of the "top people's paper."
I was glad that I had kept in touch with the four key original national directors out of the six (the two others were Murdoch appointees). They assured me of their votes if I wanted to stay on, but I now had to envisage what that would mean. Nothing in my experience compared to the atmosphere of intrigue, fear, and spite inflicted on the paper by Murdoch's lieutenants. I was confident I could stand up to the bullyboys, but why should I give any more of my time and energies to an enterprise where every man feared another's hand? I was certainly not going to dilute, still forsake, a lifetime commitment to journalism free of political manipulation. I got madder and madder. I spent a morning discussing tactics in a meeting with my chief ally among the national directors, the burly Lord (Alf) Robens, another Manchester man (and formerly in charge of Britain's nationalized coal mines). He expressed contempt for Murdoch and his "methods," a reference to a ploy by which Murdoch had attempted to move ownership of the papers' titles out of Times Newspapers and into News International without consulting the national directors as required. Robens affirmed my feeling: "You'll be in a lunatic asylum at the end of six months the way they go on in that place." On his advice I went back to the paper and continued editing and writing.
Murdoch had gone to New York, but his henchmen told the press I'd resigned, when I had not. They proffered statements praising my record. I was not about to comply with this pretense, so I took my time and continued with my news and opinion conferences with senior staff. After a week, besieged outside my house by TV cameramen and reporters, and only when my lawyers were satisfied with the terms, I resigned on ITN's _News at Ten,_ citing "the differences between me and Mr. Murdoch."
It was March 15. Only later did I recognize the significance of the date. One of the Shakespeare passages my father knew well and liked to declaim was "Beware the ides of March."
Two decades later, when Murdoch's appetite for newspapers led him to acquire the _Wall Street Journal,_ I could not restrain a mirthless laugh on reading that the controversial sale in 2007 was hedged about with guarantees enforceable by a well-remunerated troika of the good and wise. This illustrious tribunal very shortly afterward had the pleasure of reading that the editor had resigned without their knowing, not to mention their approving. Still, I have to say that Murdoch's spirited capacity for risk and innovation is proving better for this fine newspaper than the lackluster Dow Jones management and those Bancroft family members who sold it to him.
Frankly, I agree with Murdoch now that editorial guarantees are not worth the paper they are written on. At Times Newspapers, their invention enabled an air of respectability to be given to an unnecessary and hazardous extension of monopoly power, and they suggested that the _Times_ tradition had been maintained when behind the fake ivy it could so easily be plundered. Much as I appreciated the stalwart support of the independent national directors, in reality outsiders are incapable of monitoring the daily turmoil of a newspaper. This has nothing to do with their theoretical powers, and increasing or entrenching them would make no difference. Arbitration is impossible on the innumerable issues that may arise at warp speed every day between editor and management. Moreover, any intervention on editorial matters inevitably hazards the future relationship between complaining editor and resentful proprietor.
That relationship has to be based on trust and mutual respect. I recognize that the proprietor who imposes a political policy and fires a recalcitrant editor can invoke his right to do what he will with his property. He is the one risking his capital. In the case of Times Newspapers, however, the situation was different—Murdoch unequivocally forswore that right when he signed the guarantees to Parliament.
Today I have no residual emotional hostility toward him. On the contrary, I have found many things to admire: his managerial effectiveness, his long love affair with newspapers, his courage in challenging the big three television networks in the United States with a fourth, and altogether in his pitting his nerve and vision against timid conventional wisdom. And there was even one issue where he proved positively heroic.
In my efforts at a staff buyout of the _Sunday Times_ in 1981, the print unions at Times Newspapers had let it be known that they preferred Rupert Murdoch to the other bidders for the titles. "We can work with Rupert," a general secretary had told me. ("You mean _not_ work," I'd rejoined.) The unions took Murdoch's shilling—and five years later he put them to the sword. It was an equitable sequel. Under the guise of starting a new evening paper at Wapping, he set up a new plant capable of producing all his papers and secretly gave bargaining rights to the sensible electricians union, which in turn reached an agreement for journalists and clerks to access computerized typesetting. He installed color presses capable of printing both the _Sunday Times_ and the _Times_ , as well as his other major titles, the _News of the World_ and the _Sun_. Then on January 24, 1986, in an astonishing commando operation no less remarkable because it was planned in total secrecy, he overnight switched production of Times Newspapers from the battlefield of Grays Inn Road to a new plant at Wapping wrapped in looped barbed wire.
Six thousand members of all the unions went on strike and plunged the journalists on the papers into a crisis of conscience. A few refused to cross the picket lines. Two foreign correspondents who did, David Blundy and Jon Swain, said it was like being back in Beirut or Belfast, escorted by an armored car on the day they went through the barricades. People in all departments who wished to go on working assembled at secret pickup points that changed daily; they were collected in coaches with metal grids on the windows. On Saturdays they were greeted by thousands of "flying pickets," demonstrators bused in from far and wide. The pickets rocked the coaches going in, the politer ones shouting "Judas" or "Effing scab," and tried to stop the trucks going out with the papers. Only the presence of mounted police prevented the violence from getting out of hand. As it was, hundreds were injured and a thousand people were arrested.
At the height of the siege of Wapping, as it came to be known, a British television company called me in Washington, where I was now working, to ask if I'd appear on a program about it. On the morning of the show, the producers explained the lineup: "We have so-and-so defending Murdoch, and you and someone else attacking him."
"Wait a minute," I said. "You've got this all wrong. Murdoch is right. What he's doing is long overdue."
There was a pause. "We'll get back to you." An hour later they did. "Sorry, we have to drop you. Hope you understand. You don't fit the scenario."
But Murdoch did. The old script of endless warfare on Fleet Street that had always ended with a management whimper was being rewritten. The siege of Wapping lasted a full year, but not an issue of any of the papers printed there failed to come out. I didn't have any doubt where I stood. Murdoch and his managers had struck a redemptive blow for the freedom of the press. We in the old management that cared so much for responsible journalism had failed, and he'd succeeded. Wapping was brave in concept and brilliant in execution. What was achieved there made it possible for other newspapers to follow. Not only that, but it opened the way for new publications to begin. The _Independent_ newspaper was nourished at birth by this victory (staffed in part by a diaspora from the _Sunday Times_ ). For that every British newspaperman is in his debt. The carnivore, as Murdoch aptly put it, liberated the herbivores.
#
# My Newfoundland
American lives, said F. Scott Fitzgerald, have no second acts. I beg to differ. At least this Englishman had a second act in America—and for that I have to thank America and Rupert Murdoch, too. If he hadn't given me a shove, I wouldn't have enjoyed twenty-five exuberant years exploring new frontiers. And America produced a marvelous convergence in my life with Tina.
In 1968, when Ben Bradlee took over the editorship of the _Washington Post,_ I'd been editing the _Sunday Times_ for a year, so we got together to compare notes and we became friends. Later on, the timing in our personal lives was the same. He fell in love and made a second marriage with his paper's intrepid and glamorous young Style writer, Sally Quinn. I'd been amicably divorced for some time from Enid (and preserved a friendship that has endured to this day). Bradlee thought I was being much too slow about popping the question to Tina, which I was, largely because I feared she wouldn't say yes. "What are you waiting for, Evans? She'll get away!"
I seized a moment on a short vacation we took to Cape Cod in August 1981. She was under the weather, so I said it would be prudent to have a blood test, without mentioning that a blood test was required for marriage. Sweetly, she fell for it. Then I popped the question. With our legendary impatience, we decided to do it immediately. I called my coconspirator Bradlee, and he suggested a perfect site for the Brown-Evans civil wedding at the end of that week. The Bradlees had just finished restoring the near-derelict Grey Gardens, the shingled beauty of a house near Georgica Pond in East Hampton on Long Island which had been the fabled retreat of Phelan and Edith Bouvier Beale, uncle and aunt of Jacqueline Kennedy Onassis. At a day's notice, a number of close friends managed to make it to Grey Gardens: Tony Holden, the former _Sunday Times_ Atticus, who'd become the _Times_ features editor; Mortimer Zuckerman, the real estate tycoon with whom I'd first become friends when he'd come to London two years back to ask me to edit his first media acquisition, the _Atlantic;_ journalist Marie Brenner; critic John Heilpern; essayist Nora Ephron; writer Joan Buck; and _Sunday Times_ New York correspondent David Blundy, with his daughter Anna, who was maid of honor.
Bradlee was best man. He hid a tape recorder in the bushes so that Handel could join us as Tony Holden walked the bride through the bougainvillea and the presiding judge pronounced us man and wife. After champagne and cake, we drove into Manhattan for a honeymoon—all of one night at the Algonquin. In a simultaneous moment of panic at what we'd done, we exchanged written promises that we could part at any time simply by returning the signed piece of paper. The next day Tina had to go back to London to her job editing _Tatler_ magazine. Before going home, I had to meet Henry Kissinger at the Rockefeller family estate in Pocantico Hills, New York, where I was editing a second volume of his White House years.
Two years later we were back in America for good.
I still don't know quite how it happened. Not long after I left the _Times,_ I was invited to be a visiting professor for a term at Duke University in North Carolina. It was all I needed to gamble on the promise of a new life—Mr. Micawber's instinct that something would turn up under the big blue skies. At least, I thought, I'd escape having to answer people asking whether I was "all right," as if I'd just been let out of Broadmoor, and seeing the mix of disappointment and incredulity on their faces as I explained that no, I wasn't bitter about the _Times_. The new Mrs. Harold Evans (not that she ever called herself that) was ready to cut loose, too, having resigned from _Tatler_ a year after its acquisition by Condé Nast. Over the years I'd made a number of friends in New York, Washington, Rochester, and St. Petersburg, who might, at the end of the Duke term, point us in the direction of gainful employment.
The immediate plan was for Tina to enroll in a course in American literature at Duke while I took an alarmingly bright group of law and politics students into the thickets of the First Amendment and the vicissitudes of English law. On the way to the United States we paused in Barbados for a Christmas break. A phone call from New York to the beach cottage disrupted our vacation. The legendary editorial director of American Condé Nast magazines, the gray fox Alexander Liberman ("Tsar of All the Russias"), was on the line asking if Tina would come right away to meet him and his chairman, S. I. (Si) Newhouse Jr. They'd relaunched the great Frank Crowninshield magazine of the twenties, _Vanity Fair,_ and after ten issues and two editors it was sinking fast amid media ridicule and advertising collapse. Tina flew to New York for lunch and returned to Barbados two days later in a state of anxious excitement. She'd been invited to move to New York immediately to take over as editor in chief of _Vanity Fair_. It would mean a commuter marriage since I was committed to Duke for six months.
Of course I urged her to do it. She could take a rain check on Melville, Hawthorne, and James; they'd still be there, but _Vanity Fair_ might not. New York, though, did take a bit of getting used to as a resident rather than a visitor. Our first apartment was a beginners' blunder, a sublet in a glass tower on Third Avenue, Midtown. We found the agent through the Yellow Pages, paid the five thousand bucks cash deposit into his homburg hat, and faithfully followed his instruction to use assumed names for entry. We never saw him again. The apartment itself had so much furniture that if you tried to walk across the living room quickly, you risked being precipitated against the glass walls, and no doubt through them if you had attained sufficient velocity. I opened a closet, and out fell half a ton of pornography.
Tina speedily found another apartment, on Central Park South, while I was teaching at Duke. It was a romantic, touristy address, its ritziness an attraction for a better class of cockroach. It was another disaster. For a start, on my first visit the doorman wouldn't let me in. "But Tina Brown," I told him, "is my wife, and that's my apartment."
"Sir, Mr. Brown is already up there."
So he was: Tina's father. Not long afterward we fled to a two-bedroom rental at 300 East Fifty-sixth Street in time for the birth of our son, George.
We soon realized that New Yorkers don't muck up their kitchens by doing breakfast. We got into the habit of walking down Second Avenue and trying every restaurant, bar, and diner, moving from Irish to Italian, from the tolerable to the smart to the intolerable in the space of six blocks. Within a very short time the New York vortex kicked in. When you are at the outer edges, you can swim quite happily in cool waters, but as you get closer, you get sucked into a level of activity that is calculated to drive you crazy. There was a _New Yorker_ cartoon that caught it perfectly: man on the phone saying, "How's Wednesday? No? A week the following Thursday? Or the Wednesday after that? No? How's _Never_ for you?"
It was exciting and very eighties. The city was a temple of conspicuous consumption—people liked arriving at the Plaza in stretch limos—but it was disconcerting to see white-collar drug transactions in Midtown and around the magnificent New York Public Library on Fifth Avenue. The intellectual excitement of competing in the media capital was high, though, and America seemed on top of the world. It was like going to dinner with some wonderful person and looking underneath the table and finding mouse droppings. Years later, when New York had become the safest city in the country, we were to find our way to a ground-floor co-op apartment at Sutton Place with a small ivied garden, and from that seclusion we gradually learned how to adapt to Manhattan's throbbing and very proximate cultural life.
Twelve months later the young woman whose prose style I'd admired became the toast of New York for saving and re-creating _Vanity Fair._ David English, the editor of the _Daily Mail,_ wrote a droll article about how the James Bond of British journalism, as he was pleased to call me, had become a wistful shadow of a successful wife. The gossip columnist Liz Smith referred to me as Mr. Harold Brown. But I was thrilled for Tina; I admired her bravery and rejoiced in her success. She'd supported me in the traumas at the _Times,_ and I'd supported her as she'd tried to find her way in her first editorship at _Tatler._ And within no time at all after Duke I was running hard myself, in charge of a distinguished publishing house, Atlantic Monthly Press, with offices in Boston and New York, and at the same time editorial director of _U.S. News & World Report_ in Washington, back at the throbbing heart of news.
I rented a tiny house on N Street in Georgetown, former slave quarters for the big houses, a few doors down the street from Ben and Sally. They lived in a glorious mansion with gardens and a tennis court, where occasionally I was Ben's doubles partner trying to put a couple of senators in their place. On the weekend I scurried back to New York to be with Tina.
Even now, I marvel at the thrilling speed of the changes in our lives, all for the better. I'd been back to America many times since my first taste of the country in 1956, traveling through forty states in the golden Eisenhower years. I remained fascinated by its opportunities and contradictions, attracted by its openness and freedoms, appalled by racial discrimination, excited by the ideal of the American dream of individual fulfillment through equal opportunity, and intrigued by how a society so thrusting and ambitious could protect the values of a civilized society. No doubt I romanticized my first experience in the fifties, maybe actuated a little then by the knowledge that while in 1949 I was trying to find a crack in the Norman walls preserving higher education for the elite in Britain, millions of American ex-servicemen were going straight to college on the GI Bill. I thought America had a franchise on the future, and I wanted to be part of it.
I could not get over how fast Tina and I were welcomed and absorbed into American life in 1983. We were received warmly even by colleagues in journalism who saw plum jobs go to a couple of foreigners. We told ourselves that the reception was no better than Ben Bradlee and Sally Quinn would have received had they arrived in London in similar circumstances, but who were we kidding? I doubt that Fleet Street in the eighties would ever have accepted interlopers the same way America embraced us two expats.
* * *
For me to be entrusted with the editorial content of both a book company and a newsmagazine so soon after arriving in America was a curiosity. It happened because of Mort Zuckerman's appetite for public affairs. He had barely appointed me editor in chief of Atlantic Monthly Press, at the end of my Duke stint, than he bought _U.S. News & World Report_ and urgently wanted its potential developed as the most serious of the three competing newsmagazines, its rivals being _Time_ and _Newsweek._ The magazine had a circulation of more than two million and was plugged into Washington politics. This time, after Murdoch, I was more cautious: Zuckerman had a reputation for being mercurial.
Zuckerman's commitment to the magazine's integrity was tested early on. I wrote an editorial identifying the blatantly false statistics used in advertisements by the powerful National Rifle Association (NRA) to justify a bill that would make it easier for criminals, drug dealers, and lunatics to get guns. The new chief executive of the magazine dropped by my office. This was Fred Drasner, Zuckerman's business partner, a lawyer new to publishing. He was known as "Firestorm Fred" for his passion to incinerate anyone and anything that stood in his way. "Whaddya writing this week, Harry?" I told him. The embers glowed: "Hey, you can't do that. I just sold the NRA thirty thousand dollars of ads."
I'd no time to debate. I had to catch a train to New York for a meeting at Atlantic Monthly Press, where my associate publisher, Walter Weintz, was in the throes of publishing an art scoop, the sketchbooks of Pablo Picasso. That afternoon I got a message from the equable David Gergen, whom we'd recently appointed as the editor of _U.S. News_. Would I mind writing another editorial in a hurry? I imagined David's hair was still smoldering. I told him I would mind, and no, I wouldn't write another editorial. On the Monday when I picked up the magazine, my NRA editorial was there as I had written it. Zuckerman had overruled Drasner (who lost his deal). Rather impertinently, I told Zuckerman that he'd completed his apprenticeship as an owner in record time; indeed, throughout he remained steadfast on all the issues of principle that counted. He proved immensely well informed, with a phenomenal memory. In 1984 he was the first mainstream journalist to dare to comment on the growth of the new black underclass. Later he was at one with Nobel laureate Paul Krugman in warning very early that the 2005 housing bubble would burst, with attendant financial dangers.
I had things to learn myself. I put a lot of effort into writing editorials for the magazine, acutely aware that I was a foreigner, but I was often surprised, even after so many years of being immersed in American culture, by the subtle differences between British and American English. I was deflated early on to be told by a colleague that the editorial I'd written on terrorism was "quite good." Pray, I asked, in what way was it deficient? "Oh, none at all. It really was _quite_ good." It dawned on me that in British usage "quite" means "not very," but in American usage it means the opposite.
Reentering big-time journalism—interviewing President Reagan, getting hold of the smuggled letters of Soviet dissident Andrei Sakharov—was exhilarating. I could hardly believe my luck as I walked on crisp fall mornings from N Street across the bridge and into the _U.S. News_ office. I did the round of diplomatic lunches and receptions, taking time out to swim at the Watergate complex and run a few miles along the Georgetown canal, but I spent most of my time in the office, engaged in an extensive study of the magazine's strengths and weaknesses. In the early months I worked on a redesign undertaken by Edwin Taylor, who'd transformed the _Sunday Times_ and the _Times._ While our former colleagues in London were still chained to typewriters, the considerable British talent for innovative design stupidly wasted, we worked at _U.S. News_ on the cutting edge of modern technology. I had unfettered access to the computer for writing and editing and for seeing the pages we laid out. I got sad and angry when I heard from my friends in Britain how they were impeded in their daily lives by demarcation rules and by the kind of resistance I'd come across to changes in the _Times_. In an atmosphere like that, it's often not worth the aggravation to risk making changes, and so a company, and eventually a nation, becomes stultified.
I'd expected the attitude toward change to be different in an American organization, perhaps not always for the better. What I found altogether liberating was to be anonymous in the United States, by which I mean accepted for who I was without being immediately typecast by my accent or where I was educated or what class I was from. I discovered that anyone can reinvent himself in America, probably more easily than anywhere else in the world, because of the scale of the country and Americans' attitudes. You are allowed to fail without being cast into outer darkness. I cheered the American business leader Mark Gumz, the president of Olympus, who said, "Success comes from failure and acting upon this knowledge."
For all the satisfaction of working at Atlantic Monthly Press and _U.S. News,_ after two years of daily absence from Tina and the constant travel, I was ready to move. Then I got a call from Si Newhouse, who opened by saying he'd gone crazy. He'd persuaded himself that he should start a travel magazine; was I interested? Some of my friends thought it was me who'd gone crazy: "Why on earth would you leave the sharp end of journalism for a travel magazine?" But I did, and I'd say to anyone who loses a job in midlife not to get stuck in who you were.
Reinvention as the editor of a monthly glossy was enormous fun. I was given a quiet room next door to the main Condé Nast building to ensure that the new boy wouldn't be distracted by the high-stepping models strutting into the offices of _Vogue, Glamour,_ and _Vanity Fair._ All I had to begin with was a no-nonsense, silver-haired secretary and a few blank sheets of paper on which I could scribble headlines and layouts for miracle marriages of subjects, writers, and photographers for hypothetical stories I'd like to read. But travel journalism was ripe for a revolution.
There was no shortage of information in bulk. Travel was a marketplace with a million hawkers. It was bewildering enough to choose among the honest vendors, but there were also travel journals tied to the travel business, travel writers who took free trips and portrayed the never-never land where all headwaiters bowed. There were guidebooks that did not reveal that they had taken baksheesh. Still in my mind was an incident at the _Sunday Times_. One of the reporters had got sick on a cruise, a misery he'd shared with several hundred others, but he hesitated to write the story because he'd been on a freebie. Thereafter in my budget at the paper, I always provided for us to pay our own way, and Si Newhouse agreed that _Condé Nast Traveler,_ too, should invest in honesty. I compressed our philosophy in the slogan "Truth in Travel."
I recruited a young staff eager to make waves but spiced with a few seasoned executives who could coax fine, cultivated writers and adventurous photographers. I wanted to combine enjoyable writing and photography with a passion for the environment (not yet a fashionable subject) and harder-edged journalism. Early on I enlisted the innovative Clive (Idea-a-Minute) Irving, the former _Sunday Times_ managing editor who'd come to the States to make films and forgotten to go home. I persuaded the graphic artist John Grimwade of the _Sunday Times_ to risk an adventure in New York. We carried out prodigious research to identify the safest and worst airlines. Oh, you'll never get away with that, I was told by the cognoscenti in the trade. Such no-holds-barred reporting would cost us any hope of attracting advertising. The prophecies were soon borne out. One month, in a small news item, we advised people with breathing problems of the risks of air pollution in Mexico City. An advertising agency angrily canceled all its business because we'd printed an item "unsuitable for a travel magazine." I published the protest and defended our policy—with a reaction better than we could have dreamt. The agency itself was then denounced by the travel industry and by Madison Avenue, sensitive about its image. They all signed on for truth in travel.
_Condé Nast Traveler_ took off, an instant success that thrives today in America and Britain.
In my fifth year at _Traveler,_ Alberto Vitale, the hard-charging chief executive of the Newhouse book companies, barreled into the _Traveler_ office "to look around a magazine operation," then surprised me by saying that the visit was only a cover for his real purpose. Once inside my office, he shut the door and asked, "Would you like to run a publishing house?" He'd only just taken over Random House, the largest trade publisher in the United States, comprising three major houses: Random House ("little Random"), Knopf, and Crown. With Newhouse's approval, he was inviting me to be president and publisher of Random House trade; I'd also have the Times Books and Villard imprints in my portfolio, with the enterprising editors Peter Osnos and David Rosenthal, respectively. It was an interesting challenge, one I hadn't expected at the age of sixty-one. My passion for books had been unbounded since filing book jackets for Mum's little lending library and haunting Failsworth Public Library, but it was also an opportunity to return to journalism on the scale of the _Sunday Times_.
I knew a lot about the writing and editing of books, and by now I knew most of the formidable publishers in New York. As a visitor from London looking for good books we might acquire for serialization in the _Sunday Times,_ I'd been impressed by the authority of Sam Vaughan at Doubleday, whom the great and the good invariably chose as their publisher; by Phyllis Grann's dominance of fiction blockbusters; by the determination of Dick Snyder at Simon and Schuster, who'd more recently beaten me off when, as the apprentice editor in chief of Atlantic Monthly Press, I'd flung everything we had into trying to secure the memoirs of Reagan's indiscreet budget director, David Stockman ("None of us really understands what's going on with all these numbers"). Snyder could also call on the editorial flair of Michael Korda, who strolled into meetings wearing jodhpurs and riding boots and found time to write best sellers of his own.
Militating against any anxiety I might have was the artillery of Random House, its editors at "little Random." In publishing the editors are key. They acquire (with the publisher's approval); they nurture the author from conception to line editing, caring for every word. My job as publisher was to judge the merit and value of manuscripts editors wished to acquire, to do some acquiring and editing of my own, and to ensure the viability of the house. I had no doubt I could work happily with the editors. They treasured literature; they saw themselves, with justification, as defenders of American cultural values and feared the corporatism to which the industry was apt to succumb. I heard of a bureaucrat put in charge of a new imprint who in his first cost-cutting exercise complained, "What's that fellow doing in the company? Whenever I pass his office, he's always just reading."
Three of the editors I'd be working with were renowned. Jason Epstein, Joe Fox, and Bob Loomis had a hundred years' experience between them. Epstein was perhaps the most creative force in the history of modern publishing, always seeking new frontiers. At twenty-two, fresh from Columbia, he'd invented the groundbreaking trade paperback as a format for quality books—distinct from the pocket-size softcover mass-market books, a profitable format that everyone imitated. He'd been a cofounder of the _New York Review of Books_ and a participant in the founding in 1979 of the Library of America series of classics seeded by state dollars. He had an omnivorous appetite intellectually and in the kitchen. You never knew whether his next disquisition would be on the virtues of Saint Thomas Aquinas or a properly prepared artichoke. He was also, it has to be said, captain of the praetorian guard of pessimists, so I discounted his warning, when I called him, that the job of publisher had become "impossible." More challenging was that at Random House I'd be responsible for a $100 million business habituated to its own way of doing things.
Vitale was pleased, inordinately pleased, when I said I'd like him to take me through their system for managing the nuts and bolts of profit-and-loss predictions, printing, binding, warehousing, and selling into stores. I'd stumbled onto someone as didactic as I was. He was an immigrant himself, a former Olivetti executive impervious to doubt, his preference for order and certainty expressed in his smart buttoned-up, double-breasted suits and his handwritten notes ALWAYS IN BIG CAPITAL LETTERS. But he appreciated books, and he kept his promise that I'd have pretty much a free hand as the custodian of a pantheon of the greatest American writers. I took over as president and publisher of Random House at the start of November 1990.
Every author, said one of the founders of the house, is a son of a bitch. It was not a considered judgment. Horace Liveright had just had a cup of hot coffee flung in his face by Theodore Dreiser. But as the author of seven books myself, I was instinctively on the author's side. I knew the angst provoked by contemplation of the first resoundingly empty page, the tension waiting to hear from the publisher, the feeling that once the manuscript is delivered, you might as well have died and have done with it. Writing a book, as our prized novelist E. L. Doctorow put it, is "a lot like driving a car at night. You never see further than your headlights but you can make the whole trip that way." To which I'd add that the editor marks the route with signposts and is always at hand with a jack and spare tire. I took an oath of office that we'd be swift in our responses to authors and agents and that we'd repay the mental exertions of the writer by putting real promotional muscle into letting everyone know there was something very much worth reading here.
The galaxy of writers in the history of Random House was stunning: William Faulkner, Eugene O'Neill, Theodore Dreiser, John O'Hara, Ayn Rand, Upton Sinclair, James Joyce, Robert Penn Warren, W. H. Auden, Truman Capote, Jane Jacobs, Daniel J. Boorstin, and Theodore Seuss Geisel (Dr. Seuss) of _Cat in the Hat_ fame. Some of the greats were still with us: William Styron, Shelby Foote, Pete Dexter, Robert Ludlum, Mario Puzo, Dave Barry, Norman Mailer, James Michener, Gore Vidal, Maya Angelou, David Halberstam, John Richardson, Hunter S. Thompson, Edmund Morris, and Doctorow.
Styron hadn't written much lately. On one occasion Tina asked me to accompany her to join a psychiatrist friend's table at a dinner benefit for medical research. Styron was there, persuaded to speak for the first time about the reasons he'd not been able to write. He talked about his struggles to overcome a long depression (in the years before it was understood what medication might do). It was moving to hear him. Tina right away went over to his table to persuade him that if he wrote an article for _Vanity Fair_ describing his experience, he would help so many others who felt isolated and ashamed. He did. Inspired by Milton's _Paradise Lost,_ Tina wrote the headline "Darkness Visible." Styron then felt encouraged to write a book under that title, and we published it at Random House, where it rode the best seller list for many weeks.
Clearly we had to find new authors of similar eminence, marrying their literary excellence with a number of commercial blockbusters, without managing to shed the authors we had. I was sorry to lose one writer within a few weeks of taking over. Rupert Murdoch withdrew his memoirs. Pity. I'd have done a good job for him.
Gore Vidal was unhappy about Jason Epstein's initial read of a new "novel-concoction," as Jason called it. I flew to Miami to see Vidal and express our undying loyalty (soon justified by a magnificent collection of his essays, _United States,_ which won the National Book Award). In my eagerness to expound on how great it was to have both Vidal and Norman Mailer in the same house, I had to reverse course in mid-flow when I remembered that the two never missed a chance to beat up on each other.
Front-list fortunes are notoriously unpredictable. I had to grow a reliable backlist of steady sellers we could count on for income when new acquisitions didn't realize our hopes. It makes sense to read the DNA of an institution; Tina had done it on taking over _Vanity Fair_ (and would do so again at the _New Yorker_ in 1992). The original animating spirit of Random House was Bennett Cerf. He'd died in 1971, but I listened to him. His ebullience came through loud and clear on a tape his son Christopher Cerf dug up from the archives, in which Bennett had recorded how he'd been assisted by the womanizing habits of the owner of the Modern Library (inexpensive classics), Horace Liveright, in 1925. Liveright, having agreed to sell to Cerf, was hesitating to sign, when a cuckolded husband arrived in the office with murder in his heart. Liveright escaped the wages of sin this time, but intimations of his own mortality induced him to sign the sales agreement. Cerf built on the Modern Library to publish regular books "at random."
I was delighted at the end of my first week as president when Vitale called to say he'd like to amble down and talk about the Modern Library. He appeared in my office with Sonny Mehta, the subtle, reflective, and quietly competitive publishing legend at Knopf. The proposal was that they'd help me out by relieving me of the Modern Library and combine it with Everyman's Library, which Knopf had just acquired.
Thanks but no thanks, Alberto and Sonny. The Modern Library was dear to my heart. I remembered my astonishment when, as a Harkness Fellow skimping to get by, I found that I could buy its volumes of Edward Gibbon's _Decline and Fall of the Roman Empire_ for less than the cost of breakfast.
Jason had misgivings about my enthusiasm, but I'd already begun an evaluation of the list, looking for other great works we might acquire, and I'd scouted new members for a Modern Library board. Most of all, I was impatient to begin a phenomenal project originally inspired at the _Sunday Times_ by Godfrey Smith. He'd suggested that the _Sunday Times_ identify the best hundred novels published in the English language in the twentieth century and feature them, with biographies and photographs, in the color magazine he edited. I'd planned to accompany the launch of the series with an inexpensive reissue of every title by arrangement with all the original publishers. We were halfway there when I left to edit the _Times_. At a handover meeting with the new management team Murdoch had installed at the _Sunday Times_ , I proudly presented the great books package. Rarely have I been more swiftly deflated. No, thanks, Harry, they said, one of them adding genially, "Fuck literature!"
So I brought the orphaned project to Random House. Closeted with a committee that included Edmund Morris, Daniel Boorstin, Shelby Foote, John Richardson, A. S. Byatt, Arthur Schlesinger Jr., and Gore Vidal, I found discussing which novelist should make the cut and which should not at once an education and an entertainment. I could claim to have read most of the preliminary list; they seemed to have read everything twice, and the divisions of opinion were arresting: "Edith Wharton? Never!" " _Ulysses_? Unreadable!" The project was simpler than it had been in London. Random House already owned a good proportion of the copyrights of the fancied titles; for the rest the publishers who held the copyrights joined in the fun. We told them we had no intention of ranking the list other than alphabetically. The renovated Modern Library again became the backbone of a profitable backlist, and I'm pleased still to be on its board.
I'd worried that working with editors preoccupied with their manuscripts might give me a sense of desolation compared with the hurly-burly of the newsroom and the incessant changes on a monthly glossy. I'd been in the job just over two months when it became clear that the coalition armies President George Herbert Walker Bush had assembled would use force to eject Saddam Hussein from Kuwait. I fretted that any one of the three newsmagazines would be gearing up to produce special issues or books if and when war came. What was to stop me from producing an instant illustrated history? Well, I didn't have an art department that could interrupt its normal work, I didn't have the staff to acquire hundreds of photographs or the space to display them, and I didn't have the writers and editors on tap as a newspaper does. In short, nothing compared to flinging an army into the desert against a dug-in enemy armed to the teeth.
Vitale, exemplifying the American can-do spirit, swiftly organized printing and distribution and pledged 50 percent of the net proceeds from the book's sales to the Gulf Crisis Fund of the American Red Cross. I induced Ray Cave, the former editor of _Time_ magazine, and his wife, Pat Ryan, a former editor of _Life,_ to acquire and lay out photographs in a war room I secured at Condé Nast. I commissioned the foreign editor of the _Economist_ in London, Peter David, to write thirty-five thousand words; deployed _Traveler_ 's John Grimwade on a war-front graphic; and went to the Pentagon to describe our package to the chief of the armed services, General Colin Powell, in the hope that he'd write a prologue. (He was tickled that I could, on command, recite my number in the Royal Air Force.)
The ground war began on February 24, 1991. It was over in one hundred hours. I'd thought the coalition armies would give us more time. Still, the picture pages were pretty well laid out, Powell's prologue was in, the dust jacket was at the printer, and Grimwade was finishing his 3-D graphic of the battlefront. All we lacked was the text. That was in London. It was a magical experience in those days to put the phone down on David writing away at the _Economist_ in St. James's Street, walk from the Random House offices on Third Avenue to our Condé Nast war room on Madison Avenue, and find that the thirty-five thousand words had got there before me, transmitted from David's computer to ours. Not only that, it was already in the pages with Ray Cave's headlines.
_Triumph in the Desert,_ published on Memorial Day, was a nice success as the first illustrated history of the Gulf War in color. I was well placed to secure Powell's life story two years later on his retirement as chairman of the Joint Chiefs of Staff. The bidding war for the American hero was hot. It was my first experience of an auction where you started with six zeros and the only question was the number you put in front of them. Vitale calculated that a $5 million bid would be viable and would probably succeed. I recalled Duff Cooper's observation that the United States is more subject than any other country to inundation by great waves of conviction: "What elsewhere might be called a craze becomes there a creed." The creed in 1995 was that Powell would run for president in 1996 and win. From my conversations with him and his wife, Alma, I guessed that all the speculation was froth. My fascination was less with his political future than with his past: how did this Harlem-born soldier who came home as a wounded Vietnam vet in the sixties and was refused service at a hamburger joint in Alabama get to be chairman of the Joint Chiefs of Staff?
Si was always ready to back his publishers, so we all took a deep breath and bid again when our original $5 million was topped by Phyllis Grann at Putnam. Not yet scarred by bidding wars that got out of hand and ended in disaster, I was keen to go higher. (The scar in the cultural memory of Random House was John Grisham, lost to another house for the sake of saving a final $25,000.) But here we were talking hundreds of thousands of dollars, maybe a million or more. We ended up bidding another $1.5 million. We were into David Stockman territory—"all these numbers"—our bid an act of faith by Si, Vitale, and me. It was our final throw. We were outbid again, but Powell ended the bidding, and we were told he would come to us. We hired the author Joe Persico to help Powell, and I edited it with a promising junior called Jon Karp, but it was very much Powell's book: he is a natural storyteller. The only difficulty we had was settling on a title. After yet another session with the marketing department, I recoiled from the favored "My American Dream" and typed up a note to Powell I still have:
I have come up with a title that everyone here likes a lot. Let it brew with you and Joe and we'll pick up when I get back:
_Colin Powell_
_My American Journey_
_An autobiography_
I think it is better than My American Dream because it avoids the MLK connections. It is more active, more of an adventure and less of an introspection. And it has been quite an adventurous journey—the boy from the south Bronx who made something of his life and something that is redolent of America and opportunity. And again it is a journey that is not complete. It is a cool title, not thrusting but with a quiet emphasis.
It was one of the more remarkable experiences of my life to travel with Powell on part of his thunderously popular multicity book tour. President Bill Clinton was worried that Powell might threaten his chances of winning a second term; nobody knew what party affiliation Powell might have. Thousands lined up around three or four blocks in every city, buying the book and urging him to go for the White House. He'd sign three or four thousand books, wisecracking with the customers in line, and then cheerfully set off for the next city and more of the same. When we were well and truly launched, I was touched to receive from Powell a framed copy of the memo I'd sent him. Typically gracious, he'd written across the bottom in red ink, "Harry, You were right, as usual. God Bless." Happily, in good time the red ink vanished from our accounts. _My American Journey_ was a fabulous No. 1 best seller. It kept selling in paperback, and in a few years our faith was amply rewarded.
There is no way of removing the risk from publishing. It is easy to forget, in the excitement of an auction, that you and your rivals might be wrong. Maybe the big name is just a big name. And how do you calculate the price you will pay for an author with no track record at all—or no name at all? I had no idea who the author was who'd sent a bunch of typewritten pages to the agent Kathy Robbins. She gave them to me in a brown paper shopping bag over breakfast at the Waldorf in 1992 with the warning that if I liked what I read, I was not to try to discern or guess the identity of the author, nor ever suggest that the characters were based on real people. I could have the pages exclusively for the weekend provided I coughed up $250,000. The manuscript, titled "Primary Colors," was a compelling piece of fiction about the turmoil and intrigue of a governor seeking the presidential nomination. Wild horses couldn't drag from me the admission that I thought it bore a marked resemblance to the 1992 Democratic primaries, which had ended in the nomination of Governor Bill Clinton.
On the Monday, when I said snap to "Primary Colors," Kathy and the unknown writer insisted that they wanted it published under a pseudonym. I wasn't keen on promoting a bogus name, then I remembered what a fuss there had been in Britain in the sixties during Harold Macmillan's premiership when the _Times_ had published a series of caustic political articles by Anonymous. So Anonymous became the author of the shopping bag pages that became _Primary Colors_ that became a runaway best seller and a Mike Nichols movie starring John Travolta.
The speculation about who might be the author of a No. 1 best seller "who dares not speak his (or her) name" became a media craze. Nobody believed me when I said I didn't know. It was true. I was guessing like everyone else. We might never have known but for an unusual lapse among the sales force. Rather than let them have galley proofs, we distributed a few numbered copies of the edited manuscript. They were instructed to return them, but after publication one landed in an antiquarian bookstore, and David Streitfeld of the _Washington Post_ bought it, did some detective work on the handwriting, and announced that Anonymous was Joe Klein, a _Newsweek_ political writer. The death of Anonymous was regretted by all concerned.
I considered the manuscript of _Primary Colors_ well worth the investment, but l never dared to hope it would bring in millions of dollars as it did. Why didn't we publish only profitable titles like that? It's a nonsense question, but that doesn't stop it from being asked (with movies, too). The truth is, nobody can predict with certainty whether a book will take off or bomb. Fortunately, a publisher's fortunes rest not on one or two books, but on the whole list, year in and year out, along with the backlist—with the important proviso that it is the publisher's sacred duty to find the right market for each author.
We didn't have to do much to let people know that General Powell had written his life story, but by some means or other we had a duty to find the right audience for nonfiction books that would never make a headline, books by new authors that might not command review space, books of all kinds whose budgets did not justify advertising monies, and books by foreign authors who would not be available for television. And we had to be quick about it, bearing in mind the dictum of publisher Howard Kaminsky that "new books last in the stores about as long as milk takes to turn into yogurt."
I'd inherited excellent people in publicity and marketing (overseen by the imperturbable Walter Weintz, who seemed not to have had enough punishment at Atlantic Monthly Press), but typically they were overstretched. I augmented their efforts with a literary magazine called _At Random_ , collating interviews, photographs, and articles about our authors. It was edited by Helen Morris, whose interview with Martin Scorsese led to marriage and a Scorsese-Morris daughter. I created a special events department of four people headed by Jonathan Marder, an inventive marketer I poached from BBC America. I was apprehensive about his idea that Random House should, every month, invite the public to a literary breakfast panel at Barneys on Madison Avenue, open to authors from all publishing houses. I would moderate the discussion from a platform surrounded by people munching bagels, and we'd record the proceedings for national radio distribution. I didn't expect many busy New Yorkers to turn up at an ungodly hour to listen to authors debate the legitimacy of Truman Capote's technique in his book _In Cold Blood_ or the reality behind science fiction, but the city has a concentration of book lovers, and they packed the place every month. You could pretty well guarantee that if we were discussing the life and novels of a late-lamented author, there'd be someone in the audience with intimate knowledge of his working methods and moods. One morning Brendan Gill of the _New Yorker_ intervened to explain how John O'Hara became the "master of the fancied slight," furiously ending his association with the magazine.
And if the topic was movies based on a book, there'd be someone at a breakfast table who had directed, starred, cast, wrote the script, composed the music, or sued the studio. Director Michael Winner's adaptation of Raymond Chandler's _The Big Sleep_ got even more arresting when we learned from Sarah Miles of the mayhem on the set.
Marder was the magic man who pulled rabbits out of the hat. On the collapse of the Soviet Union, I commissioned a researcher to go to Moscow to collect photographs the Soviets had suppressed over the years and create a book called _The Russian Century_. But I also needed a venue for an exhibition of the photos. Marder came into my office: "Do you want the good or the bad news first?" The bad news was that the Russians had run out of money for redecorating their beautiful beaux arts consulate building in New York where he'd hoped we could stage the exhibition. The good news was that Marder had raced around the city and persuaded designers and stores to finish the redecoration for free—a generous gesture that produced an appreciative front-page arts feature in the _New York Times,_ a crowded exhibition, a good sale of _The Russian Century,_ and an overclose embrace for me from the redoubtable wife of the consul general.
Gerald Posner had been working for years with his editor, Bob Loomis, on a book called _Case Closed_ about the assassination of President Kennedy. There had been some creditable attempts to penetrate the mysteries, but they'd been overlain in the public imagination by thirty years of conspiracy stories. Posner's manuscript proved that these were paranoid garbage. I was impressed by his assembly of incontrovertible medical, ballistics, and scientific evidence proving that there had been no gunman on the grassy knoll; Lee Harvey Oswald had been the lone rifleman firing three shots over eight seconds. Everywhere around town when I mentioned that we had a sensation, I got the same response: "Not another Kennedy book! Give us a break!" Bookstore buyers reacted the same way. How could we make people pay attention when the sensation was that there was no sensation? Clearly we had a big marketing problem.
This was a profoundly important book. The ever prudent Loomis had let a few academics and journalists of invincible integrity have sight of the manuscript. Tom Wicker, the veteran political reporter and columnist of the _New York Times,_ was seized by the significance. Posner's work, he said, could do much to restore faith in government and democracy because it demolished the insidious insinuations that the highest officials of the U.S. government had been involved in their president's murder.
I became so exasperated, I quite lost my temper over lunch with a publisher from London when I was told yet again that the public had passed the point of satiation on the death of Kennedy. "We're naming the guilty men!" I cried out. The publisher sat up. "You mean the men behind the killing? Wow, that _is_ something!" No, I said, the guilty men are all those who ignored the evidence and misled the world. I was grateful to the publisher. He'd provoked me into a spur-of-the-moment response that might solve the marketing problem. On the napkin I roughed out the outline of an advertising campaign, leading off with a big GUILTY splashed across photographs of the principal propagators of conspiracy. We made that the overture to the campaign, backed up by a special _U.S. News & World Report_ prepublication issue. The result was spectacular. It was not only a huge best seller but a blast of cold air on the fetid distortions; it was a contribution to a nation's sanity and faith in its institutions. The conspiracy industry, of course, saw our book and ad campaign as another conspiracy. I was warned we'd be sued, and we were. But we won every court case.
There were times when my enthusiasm betrayed me. I was convinced that we had a literary and commercial masterpiece in Jonathan Harr's book _A Civil Action,_ telling of the fight by a small law firm to recover damages from a polluter suspected of afflicting children with leukemia. It was so riveting I'd stayed reading on the plane to a sales conference and barely escaped being carried to the next destination. My absorption in the book induced me to sign off on a weak jacket (which the author liked), and we made little of an endorsement by John Grisham. The book was a flop. I was so convinced that people would be sorry to miss it, and feeling we bore the blame, that I did something many thought a waste of money. I relaunched the book in hardback with a new jacket and invested still more on publicity. It worked, as it would have the first time if I had not made an assumption treacherous in publishing: it was so good that it would sell itself.
Given the inescapable uncertainties, we needed more surefire authors who would regularly sell into the top of the best seller list—easier to achieve in fiction, where an author might produce every second year, whereas serious nonfiction authors such as Robert Caro on Lyndon Johnson or Robert K. Massie on World War I might need five years or more. Even a list of hugely popular novelists provides no assurance of profitability; blockbuster authors just secure larger advances. The one-off celebrity book is clearly desirable if it is of high quality and you judge it will repay its advance and cover your unpredictable losses on other books. So I pursued and promoted celebrity books that I judged might be worth the risk. I was moved going to the hospital where the unbelievably brave Christopher Reeve (Superman) lay paralyzed from a fall while show jumping and hearing him, between breaths from a tube, say that he'd like to tell his story. It was a success. Kate Medina listened to the way NBC anchor Tom Brokaw talked, decided he was a natural author, and signed up _The Greatest Generation,_ a runaway best seller.
But you could never tell. My attempt to secure the memoir of Marlon Brando and the sequel was more bizarre. I'd heard he made a habit of humiliating publishers who flocked to his Beverly Hills hilltop retreat with open checkbooks. He satisfied himself that they were all East Coast phonies by inviting them to express their enthusiasm by going down on their knees in front of him. I was ready for that. When the call came for me to make the trip to Hollywood, I planned to tell him that declining to kneel was not a mark of disrespect but a recognition of physics: a skiing injury, I'd say, meant that I'd never be able to get up again and he'd have a problem disposing of the body.
For me he had a different test. After a restaurant dinner during which he accused me of being a CIA agent and asked what music I planned to die to, we went back to his house for several hours of erratic conversation. He flitted from topic to topic—the American Indian, the genius of Dr. Salk, diet, Israel, blood pressure, the proximity of American history, the horror of living in New York... When we finally got round to the book, he said, "I know how to write an interesting story, get the people in. I know this is necessary." Then he gave me a hard stare. "What you hear about me is nonsense. I don't want to hurt anyone when I write. There are things I will not put in. I will not write about the night at the White House with John Kennedy when I could have fucked the First Lady in the darkened kitchen, something I thought about when she became the widow in weeds." (Could this shocker be true? I did check and found that he had indeed had dinner with the Kennedys at the White House a few months before the assassination.) At midnight, when I tried to get away, standing by a limo in the moonlight, he demanded that I there and then give him a sample of my handwriting to deduce whether I was lying about the CIA.
On the next visit, when he met me in a kimono draped over his three-hundred-pound body, we played chess all afternoon as he beautifully intoned his favorite passages of Shakespeare. Eventually we sealed our understanding sitting in his fiendishly hot sauna into the early hours. Perhaps he calculated that I was about to expire in the heat, but he finally relaxed enough to talk about his son's killing of his daughter's boyfriend in the house we were in. The book he wrote with Robert Lindley, _Songs My Mother Taught Me,_ was highly readable, but getting him to promote it was a nightmare and a farce. (He suggested that we make a film in which he would dress as a woman and I would interview him.) I winced as I watched the interview he finally organized himself with Larry King on CNN. King allowed him to go off on various wild riffs, ending with Brando kissing King on the lips. It was the kiss of death for my hopes, stopping the book dead in its tracks in its rise on the best seller list.
Brando's brooding menace was theatrical. The sense of physical menace I experienced when I got smuggled into a Miami jailhouse was palpable. I was in a cell with General Manuel Noriega, the deposed dictator of Panama and alleged torturer, who'd intimated to a go-between that he was ready to spill on the involvement of U.S. agencies in drug smuggling. The next day back in New York, entirely on impulse, I called an author who'd written on Panama. "I knew you were going to call," he said. "How come?" I asked. "A friend in the CIA told me." Since I'd decided to go to Miami only at the last minute and told nobody, and since I'd called the author without warning, this was spooky. I suppose the prison cell was bugged, so I was glad I'd told Noriega he was hated in America and peppered my questions with assertions that Uncle Sam would never lie.
When we published Noriega's book, a number of worrywarts asked how we could do such a thing when he'd been convicted. This provoked me into my earnest worst. Didn't they know the meaning of "publish"? It means "to make known." Were we publishers intent on "making known" whether we liked the subject or not, or were we in the business of not making known? In short, were we censors? Bennett Cerf found Ayn Rand and Whittaker Chambers politically repugnant, but he published them.
The thought police were active again in 1993 when Richard Nixon asked me to edit a book he was writing as a foreign policy manifesto for America. I didn't hesitate for a second. Even his critics—count me among them—conceded that he had an original approach. His manuscript tackled the key questions of the day: Should we punish China for its abuses of human rights? Was it wise for America to stay aloof from the genocide in Bosnia? Was it cowardice or prudence for the United States, confronted by a handful of thugs, to back away from its mission to restore democracy in Haiti?
I'd go to Nixon's house in New Jersey with my editorial observations, and he'd invite me to stay for lunch with him and his astute assistant, Monica Crowley (later author of _Nixon Off the Record_ ). In our meetings he was not the Nixon of the White House tapes—no growls at criticism, all expletives deleted. He was never less than intriguing, his forefinger stabbing, the melancholy folds in his face uncreasing, his slow voice deepening as he expounded on the spiritual crisis—the spiritual deficit—he saw facing America, manifesting itself in crime, race relations, and what he called "the corrosive culture of entitlement, one of the greatest threats to our fiscal health."
What he loved talking about most of all was American history. It was uncanny how he could segue from some foreign confrontation in the day's headlines into an analysis of Woodrow Wilson, when he couldn't have known (could he?) that this was the point I'd reached in writing my political history of the second hundred years of the Republic, _The American Century_. On another visit I asked if he thought General Eisenhower had made a mistake in not taking Prague in May 1945, thereby letting the Soviets occupy the city. He made it very clear that he did: "It sealed Czechoslovakia's fate." All these lunch meetings ended with the same ritual. He'd sign the label on the fine bottle of wine we'd consumed and give it to me. (I should have asked for a full bottle.)
Nixon completed _Beyond Peace_ in February 1994, but he asked me to hold it up until he'd returned from a March visit to Moscow and briefed President Clinton. In April we agreed on the jacket and final text—he was receptive to all my pencil marks—but shortly thereafter he suffered a stroke. At Random House we were already on a crash publishing schedule; we then went into a crash-crash schedule to get the book to him in the hospital as his family wished. He was too impatient to reorder the next world. He died on April 22, four days after his stroke.
His daughter Julie Nixon Eisenhower asked me to join the two thousand friends and former opponents at his funeral in the sunshine at Yorba Linda, his boyhood home, where the house his father built still stood. Officially it was a state funeral. The representatives of eighty states sat in chairs in the garden. A military band played "Hail to the Chief" as the flag-draped coffin was carried to a plinth in front of us; four jets flew overhead, and a twenty-one-gun salute followed. For all that, the occasion had the feel and quiet dignity of a family gathering, its mood caught in the deep bass voice of Henry Kissinger. The presence of the five living presidents—Carter, Reagan, Ford, Bush, and Clinton—symbolized, Kissinger suggested, that Nixon's long and sometimes bitter journey had ended in reconciliation. "He achieved greatly, and he suffered deeply."
For most businesses there is only one balance sheet. For publishing houses there are two, the numerical and the cultural. Over a number of books and over a period of time, the numerical balance sheet, which is a complex of factors, has to be positive unless one has a patron—a Medici or an Emperor Francis I.
But I became convinced that the purpose of a publishing house, or an imprint, cannot merely be defined in terms of numbers, no more than the purpose of architecture can be defined by the arithmetic of quantity surveying. The house has first to be defined by its cultural balance sheet, by the creativity of its writers and its editors striving to enlighten and entertain. The identity of the house may not matter to the casual book buyer, but it does to authors, their agents, and the buyers of subsidiary rights, such as Hollywood. My experience in newspapers and magazines had persuaded me that provided we were not reckless, quality would produce a viable numerical balance sheet, whereas the simple notion of following the numbers wherever they might lead—the tyranny of numbers!—would diffuse rather than enhance the purpose of the enterprise. It was gratifying to see the policy vindicated year after year in the best seller lists: between them our titles appeared on the _New York Times_ weekly list 173 times in 1992, 117 in 1993, 156 in 1994, and 205 in 1995. There were commercial as well as cultural benefits to establishing our watermark. When we sought to sign Jimmy Buffett, the music star who was also a best-selling commercial fiction writer, his agent directed him to Putnam. He came to us, despite a bid several hundred thousand dollars more than ours, because he wanted to be in the same house as William Styron, Norman Mailer, and Gore Vidal.
Throughout Random House and our imprints (Times Books and Villard), we sought to establish our identity by publishing books of intellectual merit, literary merit, and journalistic authority. To this end we signed the novelists and storytellers John Irving, John Berendt, Robert Harris, Anna Quindlen, Alan Furst, Stephen Fry, Christopher Buckley, Barbara Taylor Bradford, Marc Salzman, Ethan Canin, and Caleb Carr. In public affairs and biography we signed Henry Louis Gates, Carl Sagan, Jeffrey Toobin, Clive James, Lewis Lapham, Sy Hersh, David Remnick, Sylvia Morris, Robert K. Massie, Gail Sheehy, Ron Chernow, and Paul Kennedy. And in the arts we signed Richard Avedon, Adam Gopnik, Alexander Liberman, and John Richardson.
But the perils of the publishing business are illustrated by the year 1993, when we had no fewer than eight of the twelve books on the American Library Association's list of the year's best books. Wanting to demonstrate that quality pays, I asked my finance director to tell me how much those books had made for the company. He came in with a long face. "Sorry, Harry, you lost $370,000 on those eight."
I asked him to check the results for the twenty-one titles of ours chosen by the _New York Times_ as the year's most notable books. He looked even more miserable. "No good. You lost $698,000 on those." Then he gave me the big smile he'd been holding back: "But they are all still selling, and on two other titles alone you made a profit approaching two million dollars."
If you fancy yourself as a publisher, try guessing which four of the eight books on the American Library Association's list were viable in the year of publication: _Preparing for the Twenty-first Century_ (Paul Kennedy); _A Tidewater Mornin_ g (William Styron); _United States_ (Gore Vidal) _; Fraud_ (Anita Brookner); _Dead Man Walking_ (Sister Helen Prejean); _FDR: Into the Storm_ (Kenneth Davis); _Lenin's Tomb_ (David Remnick), _Selected Stories_ (Mavis Gallant). Out of respect for the authors, I'll not give the details, but it should never be forgotten that good books have a long life. Jason Epstein always said when beating down opposition in-house to one of his many brilliant titles, "This book will be read long after we're all gone."
In 1994 we had a submission by a young first-time writer, not long out of university, whose contract with another house had been withdrawn for failure to deliver on time. Thanks to the alertness and judgment of a young Random House editor, Henry Ferris, at Peter Osnos's imprint at Times Books and the superb quality of the writing when I took a look, we judged the book worth an advance of $40,000. It was called _Dreams from My Father,_ by a community organizer named Barack Obama. When Tina met him on the eve of his presidential inauguration, she remarked, "My husband signed the contract for your book."
He gave that wide radiant smile. "Worth a lot more now!"
A young boy walks on a beach with his father. This time it is not Rhyl in 1940; it is Quogue in 1998, and I am the father. The boy is my son George; he's twelve, as I was on Rhyl beach, and his mother is Tina. His younger sister, Isabel, was born just as I took over Random House. Tina, meanwhile, had reached the pinnacle of her profession, accepting Si Newhouse's invitation to move from the now profitable _Vanity Fair_ to edit the _New Yorker_.
She worked very hard at the _New Yorker,_ and then six years later on her _Talk_ magazine start-up, killed by the advertising collapse after 9/11, while still being devoted to George, who'd been born prematurely, and Isabel. In work Tina and I remained the mutual support team we'd always been in editing and writing at all levels. I'd taught her layouts at _Tatler_ and advised on the _New Yorker_ design; she'd taught me about glossy magazines and covers when I started _Condé Nast Traveler_. We enjoyed life in New York, finally settling into our apartment at Sutton Place. Fifty years after playing in the English Open championships, I came across the American who'd won, Marty Reisman (aka "the Needle"), so I installed a table tennis table in the basement, and a group of us find it therapeutic to whack the celluloid ball. Reisman is still unbeatable.
As George and Isabel grew up, Tina and I split the evening engagements in New York so that one of us was always home. Our solo appearances at social functions caused New York tongues to wag, but that didn't affect us. We were living an American idyll—work in Manhattan, and on weekends we'd at last found our seaside dream, an old house on a beach on Long Island, twenty miles from where we were married in 1981.
The American predilection for the car is such that we had been regarded by some of our friends as English eccentrics when we took our first summer foray by train from the city, looking for a place to rent away from the heat. We rode the Long Island Railroad out, relaxing for two hours in the parlor car while a man in a straw boater served cold beer and the engineer sounded the whistle through the dreary suburban crossings and then joyously into the rolling pine barrens. At the end of a disappointing day of squally rain spent inspecting exorbitant glitzed-up beach huts in Westhampton, the letting agent drove us to nearby Quogue, a secluded seaside village of quiet lawns and white picket fences—a community of Waspy literary folk rather than the high rollers of East Hampton and Southampton. The seashore road was flooded by a tidal surge in the inland waterway, and the agent turned us round to go back. "You wouldn't want to see that house anyway," she said. "It's very old-fashioned." Our hearts leapt: "Yes, yes, show us right away."
The house on Dune Road, set down amid beach plums and dune grass, turned out to be a gray clapboard beach cottage with dormer windows. Built in 1928 and shielded from the ocean by a great double dune, it had survived the great hurricane of 1938. Every stick of furniture and ornament was from the same period in the 1930s. Little card tables were set for a game of bridge, like a scene from an Agatha Christie detective novel. Framed on one wall was the original owner's share certificate for the Siscoe gold mine, dated 1937.
We took it, of course. This was the dream seeded in those secluded spots at Angmering-on-Sea and Connemara when we first fell in love. In our house on the dunes, we lived the sepia print of an American summer, cocooned in a time warp while Tina edited her _New Yorker_ articles and I immersed myself in American history and wrote and illustrated the story of the nation's ascent from 1889 to 1989: _The American Century_. It became a best seller, and I followed it with _They Made America,_ a history of innovation over two centuries, made into a four-part PBS series. The old iron stove broiled local flounder pretty well, the ancient radio wafted in a narrative of baseball from somewhere, and every day for our morning dip the sun showed up promptly.
Twenty summers later, having saved up to buy the old house, we're still there with George, now twenty-three, and Isabel, nineteen, neither impressed that we can't tell them about the mysterious Siscoe mine. We retain our friendships and links with Britain. Enid came to Quogue for the wedding of our second daughter, Kate, who's settled in America. Our son Mike has worked in Los Angeles as a photographer and then in New York and later London as a specialist in computer printing. Ruth has managed London bookstores. I am proud of them all.
Tina and I now have dual citizenship; I became an American citizen in 1993 and Tina seven years later. When we can, we cross the Atlantic to Britain by sailing on Cunard's _Queen Mary 2,_ a nostalgic journey for me, since I returned to Britain in 1957 on the original _Queen Mary_. Around the time of 9/11 Isabel, then eleven, was asked at school whether she was British or American. She said she was "Amerikish." Some months after 9/11 her homework for a class in Greek mythology was to make a Pandora's box. We asked her what she'd put in it. She showed us an empty plate for hunger, a Tylenol bottle for disease, a cracked mirror for vanity, and a chocolate for greed. There was also a tiny colored drawing she had made of the Stars and Stripes.
"And that?" we asked.
"Hope," she said.
I often think of that today. I flew into America on the wings of hope, and it has not let me down. When I walk on the beach worrying what to put in this memoir and what to leave out, I hear the distant long, soft whistle of the locomotive rattling through the Hamptons, I see my father on his footplate, and I think of my own journey from the steam age of newspapers to digital delivery and all the people I have been privileged to work with in journalism and publishing. At the beginning I never conceived this memoir as a valedictory to a vanishing world. Now I hear so much about the imminent end of newspapers, it's a relief on a morning in New York to find that I can still walk to the corner newsstand for a bunch of them—meaningful stories on paper produced by what the Web world calls "human agents" rather than a bunch of bloodless algorithms. In fact I'm not alone. In the United States there are another 49,999,999 people similarly engaged. Fifty million people read a daily newspaper here, ten million in Britain.
Yet even as I have been writing this memoir, several major American newspapers have closed, and other famous titles have gone into bankruptcy. I am pained for the men and women who dedicated themselves to work they saw, rightly, as essential to a functioning democracy. My hopeful nature makes me believe that we are in a period of transition, at the end of which we will see a perfect marriage of the Web and the traditional newspaper, with its dedication to discovery, careful calibration of news values, and eclectic cultural mix. Technology will deliver a digital newspaper to our homes over the Internet, and we'll be able to choose to read it online or print it out in tabloid format to read over breakfast. (Disclosure: in 2008 Tina founded and now edits a comprehensive and exciting Web site, the Daily Beast, which I call up every morning.)
In fact the necessary worrying about what is happening with newspapers and their staffs tends to obscure the fantastic utilities of the Web. Internet journalism sites have immense potential, not just for their speed but also for increasing our comprehension and enjoyment. Hyperlinks open a panorama of global sources. We, the skeptical or curious readers, can explore primary documents quoted. We can replay a political rally caught on video by a spectator and posted to YouTube. We can keep track of a hurricane from a weather center or watch the performance of a new theatrical star the critics are acclaiming. And we can be sure that misstatement and tendentious entries will be stung to death by a thousand blogging bees.
The question is not whether Internet journalism will be dominant, but whether it will maintain the quality of the best print journalism. In the end it is not the delivery system that counts. It is what it delivers. There has never been such access to knowledge in all its forms. What we have to find is a way to sustain truth seeking. If we evolve the right financial model, we will enter a golden age of journalism.
I was exceptionally lucky to practice my craft for so long in the creatively free atmosphere cultivated by the Thomson and Newhouse families, Westminster Press, the _Guardian_ and _Evening News_ , and for that matter William Hobson Andrew—Mr. Will—with his milk can and impossible questions in Ashton-under-Lyne.
How many words, Evans, in a memoir?
Enough!
Cartoonist Mark Boxer ("Marc") takes a rise out of my skiing obsession in the mid-seventies. Luckily, I left the extravagant Astrakhan overcoat in a cloakroom somewhere. _(Mark "Marc" Boxer)_
Wedding day photograph of Frederick Evans and Mary Haselum, 1924. The next day Dad was back in the steamy grime servicing locomotives at Patricroft railway yard and Mum was in the cotton mill.
Vanished times.... The sooty grittiness of my dad's daily life for nearly fifty years as a steam-train driver is epitomized for me by Stephen Dowle's photograph of a Newton Heath loco heading into the winter darkness. _(Stephen Dowle)_
Rhyl seaside front, 1938. Dad and three of his four sons (left, Harold; back, Fred; and Peter). Yet to arrive: John—and World War II.
Dad didn't share the romantic feelings about steam—"too dirty"—and was glad to graduate to driving diesel passenger trains. "It's a toff's job."
Dad's expression is truer: the North Sea was choppy and chilly.
Mum and Corporal Evans
Diana Dors to the rescue.
_Non nobis solum_. University College, grandly housed in two castles, one at Lumley and the other (above) in Durham City. In my third year as a "Castleman" in Durham I had a room on the Norman gallery (marked), well placed to administer boiling oil to invaders. _(Durham University)_
I rode to Lumley Castle on my fairly trusty BSA bike.
It had rained, and nobody was about for the promenade photographer at Llandudno, North Wales, except a couple of newlyweds. One year after graduation from Durham University, Enid was teaching and I was trying to survive on the subs' table at the _Manchester Evening News_.
The battlefield: time and space were our enemies on the subs' desk at the _Manchester Evening News_ ; our weapons—black pencil, spike, glue and scissors, and caffeinated concentration, always fearful of the call from Big Tom. _(Manchester Evening News)_
Big Tom: Tom Henry, the editor in chief of the _Manchester Evening News,_ pushing the last page to the foundry—a rare honor from the comps on the occasion of his last edition. _(Manchester Evening News)_
The _Northern Echo_ sub-editors' room during the BBC filming. The news editor, Michael Morrissey, leans over deputy editor Maurice Wedgewood. I'm on the right, in black-rimmed spectacles, pointing at something. In the foreground the sub in rimless glasses is Don Berry, who went on to become a managing editor of the _Sunday Times_ and, later, a power at the _Daily Telegraph_ and _London Evening Standard. (Ian Wright)_
Campaigning: We circulated thousands of pamphlets on the wrongful hanging of Timothy Evans (in the photograph with wife and baby he was falsely convicted of murdering). The front of this four-page effort carries a strong call for a judicial inquiry from Sir Frank Soskice. He made it when he was in opposition, then forgot about it when he became home secretary. _(Northern Echo)_
Enid and I had three children under three. They've grown a little in this photo with me and Enid at Elton House in Darlington in 1964: Michael (in my arms), Ruth, and Kate. The house had lovely grounds (the site of Mike's experiments in eating daffodils). The _Northern Echo_ offices were but five minutes down the road.
The Tuesday conference in the editor's office at the _Sunday Times_. The "Uganda Sensation" on the contents bill is the detailed revelation of the atrocities of Uganda's dictator, Idi Amin, by a fugitive hiding in my cottage in Essex. Prince Charles (left) happened to come to the conference that week. _(Peter Dunne)_
Roy Thomson: the beloved Canadian who bought the _Scotsman,_ the _Sunday Times,_ and the _Times_ was a frugal owner but generous in according his editors freedom. He was ennobled as Lord Thomson of Fleet. _(David Low)_
Twelve years into my editorship of the _Sunday Times_ and William Rees-Mogg's of the _Times,_ we were asked to pose with the "indispensable, elegant" Denis Hamilton (seated), as Cyril Connolly called the chairman of _Times_ Newspapers. William is left, I am on the right, looking rather smug. The photographer was one who had to be obeyed, Arnold Newman, in Britain preparing a show for the National Portrait Gallery and the _Sunday Times_ magazine. _(Arnold Newman)_
The Insight trio who led the Philby investigation: David Leitch, Bruce Page, and the Lenin look-alike Philip Knightley. The intrepid photographer Bryan Wharton managed to get them to stand still for a minute. _(Bryan Wharton)_
David Blundy interviews Israeli leader Ariel Sharon on the Golan Heights during the Yom Kippur war of 1973. He survived his hazardous years as an irrepressible reporter only to be killed by a random bullet on a street corner while covering El Salvador's civil war on November 17, 1989. _(Sally Soames)_
Frank Hermann at the anti-Gaullist riots in Paris in 1968, snapped by his colleague Bryan Wharton. Both were soon overcome by CS gas. _(Bryan Wharton)_
A day in the life of an editor at the _Sunday Times,_ first with reporters on a tricky story about abortion.... _(Ian Berry)_
_(Sunday Times)_
... then the lawyer takes a look at the story on the page... _(Ian Berry)_
finally the hot metal page, attended by the news editor, editor, comp, and chief sub, is ready—on time, it seems. _(Ian Berry)_
_Tatler_ magazine in London was about to expire when Tina assumed the editorship. I didn't notice at the time that the window display behind spells FUTURE. She'd only recently returned from New York, where she made friends with an author I'd signed for the _Sunday Times:_ Sid Perelman, the celebrated _New Yorker_ humorist and scriptwriter for some Marx Brothers films and Mike Todd's _Around the World in Eighty Days. (Ken Sharp)_
At the age of seventy-four, Perelman declared he was going to drive his 1949 MG YT tourer in a rerun of the epic Paris-to-Peking motor race (9,317 miles). He told me he had a blonde from Pine Bluff, Arkansas, as traveling companion: "She knows zilch about engines, but she's six foot two inches of dimpled beauty." Perelman and I are explaining all this to the London meter maid when Sid parked the MG for a farewell lunch. _(Bryan Wharton)_
Our debonair foreign correspondent was found in Cairo with an assassin's bullet through his heart. _(Cal McCrystal)_
_(Sunday Times)_
The Communist union leader in the fur hat is having fun parading a mock coffin of the _Sunday Times_. One and a half million readers didn't get the paper for a full year. _(Sally Soames)_
Who wouldn't look glum standing in the deadness of the strike-bound composing room, normally the hectic scene where the word became metal? _(Sally Soames)_
Saying good-bye on the composing room floor, on my last day as editor of the _Sunday Times_ after fourteen years. _(Mark Ellidge)_
As a former corporal, I was outranked by General Colin Powell (left), the retiring chairman of the joint chiefs, and his associates, but we all got the joke about "the one that got away." _(Random House)_
Truth in Travel hits the road. We celebrated the first issue of _Condé Nast Traveler_ with a smiling chairman, Si Newhouse; his publisher, Ron Galotti; and a cake iced to represent the first cover photo, from an adventurous voyage up the Amazon recorded by Christopher Buckley. _(Condé Nast Traveler)_
I stood in line with hundreds of others waiting to take the oath of citizenship at a courthouse in New York. It was a moving moment. _(Dafydd Jones)_
Our refuge, the 1928 cottage by the sea in Quogue, Long Island. The view from the top of the ocean dune. _(Mike Evans)_
And then there were four... with George casting a wary eye on the camera. The birth of Isabel Harriet in October 1990, recorded by photographic royalty Annie Leibovitz. _(Annie Leibovitz)_
In the courtyard of Buckingham Palace in 2004, having bent the knee to be dubbed a Knight Bachelor of the British Empire. They've been awarding KBE's since the reign of King Henry III (October 1, 1207– November 16, 1272), so I was a Johnny-come-lately.
# Acknowledgments
What are your sources? Are they sound? Why don't you name them? The questions I've answered as a reporter and asked as an editor now demand answers from an author.
Well, I'm glad you asked. It's an opportunity to acknowledge my considerable debts. When I set out to follow my paper trail through the labyrinths of memory, I had the sometimes enigmatic diaries I'd kept over the years in Pitman shorthand I could still read; I had school reports, wartime ration books, letters, photographs, articles, and transcripts of radio and television programs I'd been involved in; I had years of my newspaper files I cherished; and I had innumerable notes. I learned early, when challenged on a story, that if you wanted to survive as a journalist, you never threw away anything (and it works if you have a researcher as good as Jolene Lescio). Still, knowing that memory plays tricks, especially on sequence, I became an intruder on other people's tranquility, ransacking their recollections to affirm or amend my own.
My premier debt is to my younger brother Fred, curator of our family folklore, who wrestled me to the ground on various things I'd got wrong in our family history (just a bit!), as he used to wrestle me in boyhood. Fred was my guide when I revisited our haunts in Manchester and North Wales; I thank his widow, Christine, for her forbearance when I took away Fred from home. My brothers Peter and John also helped re-create those vanished times as did John's widow, Margaret. Alf Morris—sorry, Lord Morris of Wythenshawe—was of course the source of the story that as a schoolboy I was known in the neighborhood as "Posh" Evans, an attempt by Alf to undermine my working-class credentials. His own are sensitively portrayed in Derek Kinrade's biography of Alf's ascent from Newton Heath to Labour minister in the House of Lords. It is a fine social history of northern working-class life.
My knowledge of my father's working life was much enriched by the recollections of Ken Law, his fireman mate on many an adventure on the footplate of steam locomotives. To check my memories of my start in newspapers, I revisited the town of Ashton-under-Lyne and the library at Stalybridge, and I thank the librarians there. It was an exercise powerfully assisted by Derek Rigby, who graduated from carrying Mr. W. H. Andrew's milk can to making headlines as an enterprising reporter. My two closest _Ashton-under-Lyne Reporter_ pals, Eric Marsden and Frank Keeble (all in our early teens when we met), reassured me that various eccentricities I remembered really were true; our Weegee, Charlie Sutcliffe, scoured his photo library to my benefit. As for my years in the Royal Air Force, yanked out of Ashton, it's all in that Lubyanka Ministry of Aviation Records Office at Innsworth if anyone cares to check, and they'll have a record, too, of another airman, Peter Spaull of South Wirral. He heard me on the BBC's _Desert Island Discs_ and touched down out of the blue to testify to the fun of producing the _Empire Flying School Review_. He volunteered the information that my tapping him to write film reviews and gossip items eventually led to him covering music and the arts for BBC radio and to writing a column in the _Liverpool Daily Post_. _Per ardua ad astra_.
My diaries at Durham University reflect my joy in meeting Enid Parker, and I am grateful for the assiduity she brought to checking the details of those days and our years of married life. It was delightful re-creating the glow of those years with her and with Derek and Daphne Holbrook, Roy Arnold, Brian and Shirley Scrivener, John (Lofty) Morland, and Keith Nodding. Durham meant a lot to all of us and to the others reminiscing at Castle reunions: John Perkins, John Hollier, Bill Burdus, John Bridges, Lou Hamer, Edgar Jones, Geoff Pulling, Ridley Coats, Roy McKenzie, Eric Thompson, and Chuck Metcalfe. The Master of University College, Maurie Tucker, went out of his way to be helpful.
I was lucky that amid the paper storms at the _Manchester Evening News,_ when my diaries went blank for days, I had a colleague from those times possessed of total recall, the writer Duncan Measor. I thank Duncan and his wife, Marjorie. I did find it stimulating to retrace steps when I could, and a visit to the _Manchester Evening News_ offices was most rewarding. I thank the _Evening News_ editor Paul Horrocks and his resolute assistant, Lisa Brealey; chief executive Mark Dodson; deputy editor Maria McGeoghan; Tom Waghorn; and the reporters, subs, and photographers I interrupted on their way to a deadline. Bob Corfield's film on the paper's centenary caught the atmosphere well. Andy Harvie, the news editor in my time, was helpful, as was Jane Futrell, the daughter of Denys Futrell and Tony Watson, head of the Press Association. On this visit to Manchester I had the big benefit of the hospitality and knowledge of my school chum Peter Charlton, the official historian of Newton Heath, and his wife, Lillian. My friend Barrie Heads, one of the redoubtable producers of Granada Television in its early glory days in Manchester, was kind enough to let me read his hilarious television memoir before its publication. I am sorry I failed his high standards of pronunciation.
Absolutely indispensable from the _Northern Echo_ on was Joan Thomas, my first-ever secretary, who later joined me in London. I'd not have been able to reach so many who shared the excitements without Joan's indefatigable resource in tracking people I'd lost touch with. Michael Morrissey, the paper's first news editor, and David Spark, the assistant editor, were extraordinarily generous with their time. Ken Hooper was helpful in retracing our cervical cancer investigation. I also thank Peter Ridley, Ray Robertson, Bill Treslove, and that kid photographer of the rock stars Ian Wright, who grew up to be a celebrity in his world. Don Berry, like Joan Thomas, came from Darlington to the _Sunday Times,_ in his case via the _Rochester (NY) Times-Union,_ and brought his superpowered sub's black pencil with him. I thank him for his wry observations.
I could fill a telephone directory with all the people who made the _Sunday Times_ what it was in my fourteen years of editing (not forgetting the late Mike Randall, Peter Sullivan, Jack Lambert, Peter Harland, Steve Brodie, Malcolm Crawford, Peter Roberts, and Tony Bambridge). We were a community, sharing our passion for journalism, and I am forever indebted to them all for their skill, courage, and companionship. It's wonderful that the head printer, George Darker, is nearing one hundred years of age, despite the stresses to which we subjected him.
I can only make particular mention here of those _Sunday Times_ individuals who failed to evade my long arm when I sought to test my recollections and records on theirs, chief among them John Barry, Godfrey Smith, Bruce Page, Godfrey Hodgson, Phillip Knightley, Edwin Taylor, Paul Eddy, Peter Gillman, Elaine Potter, Cal McCrystal, Tony Holden, Keith Richardson, Tony Rennell, Magnus Linklater, Peter Pringle, Philip Jacobson, Tony Dawe, Colin Chapman, Michael Ward, Parin Janmohamed, Charles Raw, Lewis Chester, Frank Giles, George Darby, Don McCullin, James Evans, and Anthony Whitaker, along with the recently much-mourned Hugo Young, John Whale, and Peter Wilsher. Yvonne Mascarenhas, the widow of the brave Anthony, told me the story of her escape from Pakistan, which Anthony had kept to himself. Clive Irving, creator of the early Insight pages, filled in the gaps in my knowledge and understanding of those days before I joined the paper. For images, but also recollections, I drew freely on the skills and energies of photographers Bryan Wharton, Sally Soames, Peter Dunne, Mark Ellidge, and Ian Berry. Lord Snowdon graciously agreed I might use on the book jacket a portrait he took of me at a typewriter during my editorship.
My review of our Northern Ireland work owes much to John Barry, the former Insight editor (now with _Newsweek_ ), who challenged and enriched my account (though he's responsible for nothing that invites criticism) and with him the intrepid Chris Ryder. Chapter 17, "Death in Cairo," owes a great deal to Eddy and Gillman, who never gave up on the story. I very much appreciate how Scotland Yard was ready, as the saying goes, to assist us in our inquiries. I am indebted to the cooperation of Sir Ian Blair, then Metropolitan Commissioner of Police at Scotland Yard; Commander Simon Foy, head of homicide; Inspector Graham Jenkins; and former detectives Ray Small and Tony Comben. For their readiness to help, I must also acknowledge Jan Morris, David Holden's friend; Sy Hersh at the _New Yorker;_ Steve Emerson at the Investigative Project on Terrorism in Washington, D.C.; Roger Louis, director of British studies at the University of Texas; and Professor Alan Weinstein, the ninth Archivist of the United States.
I had interesting discussions about the future of newspapers and the Web—a scene changing too rapidly to do full justice to their observations—with Paul Steiger of ProPublica; Clark Hoyt, the public editor at the _New York Times;_ and ex-editors Gene Roberts and Clive Irving.
_My Paper Chase_ —which is what it truly was—might never have been finished on time (well, only a year off) without the needling of my agent and friend Ed Victor. Cindy Quillinan, my assistant, had survived the vicissitudes of my book and television series on innovation and crested the waves of organizing manuscript "Mark XI" with uncanny calm.
Geoff Shandler, editor in chief at Little, Brown, whose original idea it was to divert me from recording other people's history to my own, proved to have reservoirs of patience as deep as his editorial judgment. At that crucial moment in the life of a book when an author is wondering whether to turn tail and run, I was vastly encouraged by Michael Korda, who has vibrant memories of his own of those years in Britain and life in the RAF, and I thank the historian Edmund Morris for some suggestions. I am grateful that Little, Brown's copy editor Barbara Jatkola proved to have 20/20 vision for all sorts of mistakes. An early manuscript was also read by Don Ross of the Newseum in Washington, D.C. He and I were partners in an exhibition the museum staged on war correspondents. As a Marine, he was wary of a mere airman, but it was a successful collaboration, and I thank him for volunteering to see that in this memoir I did not appear on early parade with Briticisms untranslated, infinitives split, tenses mixed, and the placing in sentences of prepositions up with which he would not put. I apologize for any faults of this kind that sneaked in, for assuredly they were the fault of RAF Corporal Evans and not U.S. Marine Corporal Ross.
I was most fortunate that John Heilpern, who grew up in Manchester and was a star on the _Sunday Times_ ' rival, the _Observer,_ took an early interest in what I might do to retrieve those vanished times—his phrase. A brilliant biographer and theater critic himself, he urged me not to skimp on my early life and family, and in many a conversation he enabled me to retrieve the half-remembered and cherish it anew. His critical overview of the manuscript was invaluable, and I will always be grateful to him.
Throughout, from that daunting first blank page, during periods of uncertainty and distractions, I was sustained by the discerning eye, professional skill, and loving support of my gifted wife, Tina. My driving fear was of disappointing her, and my driving hope is that our children, George and Isabel, will regard the memoir as some compensation for those days in the present when I was lost in the past.
_The author is grateful for permission to use the following: "For the Fallen" by Laurence Binyon. The Society of Authors as the Literary Representative of the Estate of Laurence Binyon; excerpt from "Annus Mirabilis" by Philip Larkin from_ Collected Poems _by Philip Larkin. Copyright © 1988, 2003 by the Estate of Philip Larkin. Reprinted by permission of Farrar, Straus and Giroux, LLC; "The Northern Ireland Question" by Desmond Egan from_ Elegies, Selected Poems, _The Goldsmith Press Ltd., Ireland._
# Principal _Sunday Times_ Books
_The Zinoviev Letter,_ by Lewis Chester, Stephen Fay, and Hugo Young (Heinemann, 1967). How the famous "Red letter," which helped to defeat Ramsay Macdonald's Labour government in 1967, was forged by White Russians and circulated by the Conservative Party's Central Office with secret service help.
_Philby: The Spy Who Betrayed a Generation,_ by Bruce Page, David Leitch, and Phillip Knightley, with an introduction by John Le Carré (Deutsch, 1968).
_An American Melodrama,_ by Lewis Chester, Godfrey Hodgson, and Bruce Page (Deutsch, 1969). History of the U.S. presidential election of 1968, during which Richard Nixon was elected and Robert Kennedy was murdered.
_Journey to Tranquility,_ by Hugo Young, Bryan Silcock, and Peter Dunn (Jonathan Cape, 1969). History of man's assault on the moon.
_The Secret Lives of Lawrence of Arabia,_ by Colin Simpson and Phillip Knightley (Nelson, 1969). Documentation of Lawrence's sadomasochism and unsuspected role in Middle East politics.
_The Pound in Your Pocket,_ by Peter Wilsher (Cassell, 1970). Century of Sterling, 1870–1970.
_The Strange Voyage of Donald Crowhurst,_ by Nicholas Tomalin and Ron Hall (Hodder, 1970). The mystery of Donald Crowhurst, who vanished from his trimaran during the _Sunday Times'_ single-handed nonstop race around the world.
_Do You Sincerely Want to Be Rich?_ by Charles Raw, Bruce Page, and Godfrey Hodgson (Deutsch, 1971). Subtitled _Bernard Cornfeld and IOS: An International Swindle._ Investigation of Investors Overseas Services (IOS) and its creator, Bernie Cornfeld: how it operated as an "offshore" company responsible to the law of no single nation and what it did with the billion dollars entrusted to it by a million savers.
_Hoax,_ by Lewis Chester, Stephen Fay, and Magnus Linklater (Deutsch, 1972). The forgery and retailing of Howard Hughes's autobiography by Clifford Irving.
_Ulster,_ by the Insight Team (Deutsch, 1972). Results of four months of inquiry into the origins of the Troubles.
_The Thalidomide Children and the Law,_ by the _Sunday Times_ (Deutsch, 1973). Documents and texts.
_Watergate,_ by Lewis Chester, Stephen Aris, Cal McCrystal, and William Shawcross (Deutsch, 1973).
_Death of Venice,_ by Stephen Fay and Phillip Knightley (Deutsch, 1975). Investigation of the threat to the survival of Venice.
_The Exploding Cities,_ by Peter Wilsher and Rosemary Righter, with a foreword by Barbara Ward (Deutsch, 1975). Stimulated by a _Sunday Times_ conference with the United Nations Fund for Population Activities at Oxford University.
_Insight on Portugal_ (Deutsch, 1975). Portugal's return to democracy.
_Nicholas Tomalin Reporting_ (Deutsch, 1975). Ron Hall introduces reporting by his colleague and friend, killed on duty for the _Sunday Times_ in October 1973.
_The Yom Kippur War,_ by the Insight Team (Deutsch, 1975). Sequel to the _Sunday Times'_ 1974 book _Insight on the Middle East War_.
_The First Casualty,_ by Phillip Knightley (Deutsch, 1975). War correspondent as hero, propagandist, and mythmaker from Crimea to Vietnam.
_The Crossman Affair,_ by Hugo Young (Jonathan Cape, 1976).
_Destination Disaster,_ by Paul Eddy, Elaine Potter, and Bruce Page (Hart-Davis, MacGibbon, 1976). Investigation of the DC-10 crash in Paris.
_On Giant's Shoulders,_ by Marjorie Wallace and Michael Robson (Times Books, 1976). The story of thalidomide victim Terry Wiles.
_Slater Walker,_ by Charles Raw (Deutsch, 1977). Jim Slater tried to prevent the publication of Charles Raw's four-year investigation, which concluded that in all its various forms, the investment company Slater Walker was really about one thing—the manipulation of share prices.
_The Abuse of Power_ , by James Margach (W. H. Allen, 1978). The war between Downing Street and the media, from David Lloyd George to James Callaghan.
_The Fall of the House of Beaverbrook,_ by Lewis Chester and Jonathan Fenby (Deutsch, 1979). How Trafalgar House acquired the _Daily Express, Evening Standard,_ and _Sunday Express_.
_Jeremy Thorpe: A Secret Life,_ by Lewis Chester, Magnus Linklater, and David May (Deutsch, 1979).
_Suffer the Children,_ by the Insight Team (Deutsch, 1979). The thalidomide story.
_Siege!_ by the Insight Team (Hamlyn, 1980). How the Special Air Service rescued hostages at the Iranian embassy in London in 1980.
_Stop Press: The Inside Story of the Times Dispute,_ by Eric Jacobs (Deutsch, 1980). The year of the suspension of Times Newspapers (1978–1979).
_Lawsuit_ , by Stuart M. Speiser (Horizon Press, 1980). Lawyer in the DC-10 case opens his files on other celebrated cases.
_The Vestey Affair,_ by Phillip Knightley (Macdonald, 1981).
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# About the Author
HAROLD EVANS is the author of the _New York Times_ best seller _The American Century_ as well as _They Made America: Two Centuries of Innovators_ and _Good Times, Bad Times_. He was the founding editor of _Condé Nast Traveler_ , editorial director of _U.S. News & World Report_, and president and publisher of Random House, where he published a record number of nonfiction best sellers. Editor of the influential London _Times_ between 1967 and 1981, Evans was voted by British journalists as the greatest all-time editor. He was awarded the European Gold Award for his investigations and campaigning, notably for winning justice for children affected by thalidomide, and in 2004 he was knighted for his service to journalism. Since 2002, he has been editor-at-large of _The Week_ magazine. He lives in New York with his wife, Tina Brown, and their two children.
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,657 |
\section{Introduction}
Short- to medium-range potentials such as the Lennard-Jones~\cite{jones1924determination} or the Buckingham~\cite{buckingham1938classical} potential are the backbone of classical MD simulations. They represent Pauli repulsion as well as non-directional dispersion attraction and there exist multiple flavors implemented in most MD codes under the term of non-bonded interactions. In practice there is a need to truncate these potentials since the number of neighbors that have to be considered for each entity grows enormously, drastically increasing the computational cost for the force calculation. Truncating between $r_\mathrm{c}$ = 2.5$\sigma$ and 3.5$\sigma$, where $\sigma$ is the characteristic interaction range, is a very common practice in MD studies~\cite{frenkel2001understanding} and has become the minimum standard, assuming that errors arising from this are small enough. Several studies have reported that with these settings significant problems can arise. For instance the truncation can alter the phase diagram of the Lennard-Jones system~\cite{smit1992phase,wang2008homogeneous} or yield different values for interfacial free energies~\cite{ismail2007surface,valeriani2007comparison,ghoufi2016computer,ghoufi2017importance,marcello2017LongRange}. These effects are quantitative in nature, meaning that they can in certain circumstances be analytically corrected for~\cite{sun1998compass,sinha2003surface,werth2015long} or compensated for by other interaction parameters such as interaction strength or interaction range. The latter is important for the development of force fields where non-bonded potentials are often included and the cutoff can be seen as another fitting parameter. Naturally, a parametrization with a small cutoff would be preferred to another one if they deliver equal accuracy. This however is only true in the assumption that the underlying physical characteristics that are created by truncated and longer ranging potentials are the same.
In this work we investigated the influence of the cutoff for the interfacial phenomenon of water-wetting on a solid substrate. We found that the effect of the cutoff of the water-substrate interaction was not only unexpectedly strong, but also changed the fundamental physics of the wetting transition in an unprecedented way by creating metastable wetting states that have also never been seen in experiments. We show that proposed cutoff corrections such as the use of a force switching function, a shifted potential or a shifted force did not fix this and could even worsen the effect. This finding shows that atomistic simulations of interfaces need to be treated with great care since unphysical behavior could occur and easily remain undetected. This is particularly relevant since a large number of MD studies using truncated potentials are reported each year. Our results suggest the use of much larger-than-common cutoffs or long-range versions of non-bonded potentials in MD studies of wetting and interfacial phenomena.
\begin{figure}[!t]
\centerline{\includegraphics[width=6.6cm]{./fig_1.pdf}}
\caption{a) Side view of the two wetting states for the small droplet. Water is blue and surface atoms are gray. b) Temperature of the wetting transition $T_\mathrm{w}$ (points) versus cutoff radius $r_\mathrm{c}$ and fit (red line). The $T_\mathrm{w}$ were obtained from the free energy profiles (see text) and we estimate errors to be $\pm 3$~K. $T_0$ is the converged wetting temperature.}
\label{FIG_1_TRANSITION_TEMPERATURE}
\end{figure}
\begin{figure*}[!t] \centerline{\includegraphics[width=16.2cm]{fig_2.pdf}}
\caption{Free energy profiles of wetting for different cutoffs in a small temperature range around the respective transition temperature $T_\mathrm{w}$ (generally at or near the central column for each system). As collective variable we chose the center of mass of the water droplet ($\mathrm{COM}_\mathrm{z}$, substrate at $z = 0$). We note that for the largest cutoff of $8\sigma$ the temperature range is slightly larger to highlight the shape of the free energy profile for complete and partial wetting.}
\label{FIG_2_FREEENERGY}
\end{figure*}
We investigated two droplets comprised of 3000 and 18000 water molecules which were represented by the coarse-grained mW model~\cite{molinero_water_2009}, on top of a rigid, pristine fcc(100) surface (lattice parameter 4.15~\r{A}). Whilst this substrate does not aim at representing any particular material, similar systems have been used to study ice nucleation~\cite{reinhardt2012free,cox2015molecular1,cox2015molecular2,fitzner_many_2015} or water-metal interfaces~\cite{heinz2008accurate,xu2015nanoscale}. The simulation cell had dimensions $17\times 17 \times 11$~nm which is enough to avoid interaction of the water molecules with their periodic images for all wetting states. Even though the liquid is rather non-volatile even at the highest temperature considered, we employed a reflective wall at the top of the cell to avoid evaporation and mimic experimental conditions. Our simulations were performed with the LAMMPS code~\cite{plimpton1995fast}, integrating the equations of motion with a timestep of 10~fs. This rather large timestep is commonly used in combination with the mW model and is acceptable for our system since during NVE simulations the total energy drift was found to be only about $2\times10^{-9}$~eV per water molecule per ps. In addition, we verified that we obtain the same results using standard protocols for updating the neighbor lists compared with unconditionally updating them every timestep. All production simulations were performed in the NVT ensemble with constant temperature maintained by a ten-fold Nos\'{e}-Hoover chain~\cite{martyna1992nose} with a relaxation time of 1~ps. The substrate-water interaction was given by a distance ($r$) dependent Lennard-Jones potential
\begin{equation}
U_\mathrm{LJ}(r)=4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right] \end{equation}
with $\epsilon = $ 29.5~meV, $\sigma$ = 2.5~\r{A} truncated at a cutoff $r_\mathrm{c}$. This resulted in a maximum interaction energy of 154~meV for an adsorbed water monomer (weakly depending on the cutoff). Additionally we performed well-tempered metadynamics simulations~\cite{laio2002escaping,barducci2008well} for the smaller droplet with the PLUMED2 code~\cite{tribello2014plumed}. In these simulations the Gaussian height, width, bias-factor and deposition stride were $2.16$~meV, $0.15$~\r{A}, 20 and $20$~ps respectively. Metadynamics is usually applied to drive rare events such as nucleation~\cite{sosso2016crystal,sosso2016ice,tribello2017analyzing,cheng2017bridging} or protein folding~\cite{bussi2006free,laio2008metadynamics}. In our systems, this method helped to uncover the underlying free energy profile of wetting.
We studied the wetting behavior of the larger droplet by performing standard MD runs at different temperatures first. As starting configurations we chose either a flat water film in direct contact or a spherical droplet placed above the substrate. Within at most 5~ns the simulation was equilibrated and a seemingly stable configuration was reached, where the water is either wetting (contact angle $\theta = 0^{\circ}$) or partially wetting ($0^\circ < \theta < 180^\circ$). An illustration of the two wetting states can be found in figure~\ref{FIG_1_TRANSITION_TEMPERATURE}a. Initially we employed a radial cutoff at $r_\mathrm{c} = 3.0\sigma$ for the water-substrate interaction. With this setting we found that interestingly a wetting transition happened at finite angle $\theta_0 \approx 23^\circ$, i.e. a smaller non-zero contact angle was not possible. This behavior cannot be explained by the standard Young's equation.
However, upon increasing the cutoff we found that the wetting behavior drastically changed. First, the wetting temperature $T_\mathrm{w}$ at which the wetting transition took place increased as we increased the cutoff (figure~\ref{FIG_1_TRANSITION_TEMPERATURE}b). Whilst $T_\mathrm{w}$ shows a clear convergence behavior with $r_\mathrm{c}$, it is unexpectedly slow. A reasonably converged wetting temperature $T_0$ is only reached for $r_\mathrm{c} > 7\sigma$. Second, we noticed that for an increasing cutoff the minimum possible contact angle $\theta_0$ got smaller and eventually vanished. Most importantly, we also found that for temperatures around $T_\mathrm{w}$ the stable configuration that was reached after the 5~ns could depend on the starting configuration for smaller cutoffs, while for larger $r_\mathrm{c}$ it always reached the same state. This suggests that for small $r_\mathrm{c}$ we actually found metastable wetting states that are absent for large $r_\mathrm{c}$. This also means that $T_\mathrm{w}$ cannot naively be defined through visual analysis of trajectories at different temperatures but needs to be defined by the free energy of wetting. For a first order phase transition we define $T_\mathrm{w}$ to be the temperature where the two basins (corresponding to wetting and partial wetting) have the same free energy. For a continuous phase transition $T_\mathrm{w}$ is the temperature where the single basin represents a contact angle of $\theta = 0^\circ$ for $T < T_\mathrm{w}$ and $\theta > 0^\circ$ for $T > T_\mathrm{w}$.
Understanding the character of these wetting states with standard MD can prove difficult as the dependence on the starting configuration always leaves doubt on the outcome of the equilibrated configuration obtained from it. To clarify, we show the results from the metadynamics simulations in figure~\ref{FIG_2_FREEENERGY}. As a collective variable we chose the z-component of the center of mass of the water droplet ($\mathrm{COM}_\mathrm{z}$), where z is the surface normal direction. While this choice is not equivalent to the contact angle (as they are related in a non-linear manner) it is clear that significantly different values for $\mathrm{COM}_\mathrm{z}$ correspond to different contact angles and can therefore distinguish the different wetting states. For the smallest cutoff at $T_\mathrm{w}$ and around we found that two basins coexist, one being the flat film (COM$_\mathrm{z} \approx$ 4~\r{A}) and the other being a droplet with certain contact angle (COM$_\mathrm{z} \gtrsim 5$~\r{A}). These two states are separated by a significant barrier larger than 20~$k_\mathrm{B}T$, which explains why we observed metastable states in the unbiased simulations for small $r_\mathrm{c}$. This corresponds to a first-order phase transition between the wetting states. The occurrence of a minimum possible contact angle $\theta_0$ is explained by the existence of the second basin, which does not approach the wetting basin, but rather becomes less stable as temperature changes. However, this character faded as we increased $r_\mathrm{c}$. The barrier became smaller and the distance between the basins got smaller. For the largest cutoff investigated ($8\sigma$) we clearly see that only a single basin exists that changes its position with temperature. As a result no metastable wetting states exist and the phase transition is continuous. We note that in this case the estimate of $T_\mathrm{w}$ is more difficult than for the first order transitions, however in this work we aim at presenting qualitative results and from figure~\ref{FIG_2_FREEENERGY} it is clear that $T_\mathrm{w}$ is higher than for the smaller cutoffs.
Only the results for the largest cutoff are in agreement with the fact that water wetting transitions are generally continuous when probed in experiments~\cite{bonn2001wetting,friedman2013wetting} and finite-angle wetting transitions have, to the best of our knowledge, never been observed experimentally. Therefore, the correct qualitative wetting behavior in our system is not achieved with standard cutoffs and if undetected could potentially lead to false conclusions. Differences between short and long-ranged interactions have been highlighted for other interfacial phenomena, such as drying~\cite{evans2016critical} or grain boundary melting~\cite{caupin2008absence}.
We further study the effect of the most commonly used correction schemes to cutoffs:
\begin{enumerate}
\item A shifted potential (sp) which ensures that the value of the potential energy $U$ does not jump at the cutoff distance, given by:
\begin{align}
U_\mathrm{sp}(r) &= U_\mathrm{LJ}(r) - U_\mathrm{LJ}(r_\mathrm{c})
\end{align}
The corresponding force $F$ remains unaltered:
\begin{align}
F_\mathrm{sp}(r) &= F_\mathrm{LJ}(r)
\end{align}
\item A switching function (switch) which brings the force to zero between an inner $r_\mathrm{c,1}$ and an outer cutoff $r_\mathrm{c,2}$ (we chose $3\sigma$ and $4\sigma$):
\begin{align}
F_\mathrm{switch}(r) &= F_\mathrm{LJ}(r) &r \le r_\mathrm{c,1}&\\
F_\mathrm{switch}(r) &= \sum_{k=0}^3 C_k(r-r_\mathrm{c,1})^{k} &r_\mathrm{c,1} < r \le r_\mathrm{c,2}& \nonumber
\end{align}
where $C_k$ are constants determined to ensure a smooth behavior~\cite{plimpton1995fast}.
\item A shifted-force potential (sf), which ensures that force and potential do not jump:
\begin{align}
U_\mathrm{sf}(r) &= U_\mathrm{LJ}(r) - U_\mathrm{LJ}(r_\mathrm{c}) - (r - r_\mathrm{c})F_\mathrm{LJ}(r_\mathrm{c}) \\
F_\mathrm{sf}(r) &= F_\mathrm{LJ}(r) - F_\mathrm{LJ}(r_\mathrm{c}) \nonumber
\end{align}
\end{enumerate}
The latter approach was found to give good results for a homogeneous system and even allowed for a reduction of the cutoff~\cite{toxvaerd2011communication}. Our results for these three corrections can be found in figure~\ref{FIG_3_SHIFTED}. By definition and thus unsurprisingly, the shifted potential does not yield any significant difference (where the remaining minor deviations are due to the metadynamics sampling) over the plain cutoff since forces remain unaltered. The smooth cutoff via switching function seems to improve the situation, however the fact that the transition temperature lies between the ones we found for a plain cutoff at $3\sigma$ and $4\sigma$ suggests that the improvement stems from the effectively increased interaction range rather than the fact that the force vanishes smoothly. Interestingly, the shifted force with the same cutoff performs worst out of all candidates as the barrier increases by a factor of two, which increases the likelihood that simulations are performed in the metastable state without realizing it. The fact that none of the considered correction schemes significantly improved the character of the wetting free energy profile leads us to conclude that it is not the way in which the cutting is done that matters most, but rather the effective cutoff distance as well as the overall interaction strength at that distance.
\begin{figure}[!t]
\centerline{\includegraphics[height=6.8cm]{./fig_3.pdf}}
\caption{Free energy profiles of wetting approximately at the transition temperature with uncorrected setup (cut) and for different correction schemes [shifted potential (sp), force switch (switch) and shifted force (sf)] applied with a cutoff at $3\sigma$. None of the schemes show the correct behavior, which is shown in figure~\ref{FIG_2_FREEENERGY} to be a single basin.}
\label{FIG_3_SHIFTED}
\end{figure}
As an initial attempt to understand the results obtained we looked at the potential energies of the various systems with the different cutoffs considered. This, however, did not reveal any obvious explanation. One possible interpretation for the creation of metastable states in our systems with shorter cutoff can be obtained by considering the droplet state (not assuming anything about the stability relative to the film state). For a transition towards the film state, there needs to be thermal fluctuations of water molecules that are above the contact layer in the downwards direction (the fact that COM$_\mathrm{z}$ has proven a good reaction coordinate supports this statement). With an infinite interaction range all molecules that are loosing height contribute to these fluctuations since they have an interaction with the substrate. Therefore we expect the interaction energy to change monotonically and the free energy to follow monotonically either up or down depending on the balance of the interfacial free energies (see figure~\ref{FIG_2_FREEENERGY}, $r_\mathrm{c} = 8\sigma$). But if the interaction range is finite, not all molecules contribute to an increased interaction with the substrate even if they decrease their height (and subsequently weaken the water-water interaction of the system by leading to deviations from a perfect spherical droplet). In other words, there is a minimum distance from the substrate that has to be surpassed by a molecule for it to contribute to a fluctuation increasing the interaction energy, otherwise it will (on average) actually decrease the total interaction energy. This minimum fluctuation for a single molecule translates into the macroscopic states (droplet and film) being connected by a barrier shaped free energy profile rather than a monotonic one (see figure~\ref{FIG_2_FREEENERGY}, $r_\mathrm{c} = 3\sigma$). The entropic contributions to the free energy are unlikely to change this, since they are essentially dominated by the environment a molecule is in (quasi-static contact layer or quasi-liquid water on top). The entropic change between these two states will be monotonic for a single water molecule and therefore also for the whole droplet.
Finding a general recipe for how to avoid such unphysical wetting states is difficult. Other aspects like e.g. the substrate density or the liquid-liquid interaction strength will have an influence on how strongly the fluctuations in the droplet state are affected by $r_\mathrm{c}$. Generally, cutoffs that are deemed acceptable from the inter-molecular perspective do not necessarily mean that the interaction between macroscopic states such as a film/droplet and a substrate is sufficiently captured. This is especially important in an interfacial simulation setting such as a slab, where a cutoff-caused change in interaction from the substrate side is not compensated by an equal change from the vacuum side. Consequently, only employing much larger cutoffs or techniques to calculate the long-range part of the dispersion force~\cite{in2007application,isele2012development,isele2013reconsidering} can ensure that unphysical effects are avoided. A minimal sanity check for future wetting studies could be to start simulations from both a wetting film and a spherical liquid snapshot. If both of them end up in the same configuration the existence of an unphysical metastable wetting state is unlikely.
In light of the vast amount of work that is done in the MD community using similar interactions, our findings urge extreme caution when dealing with truncated non-bonded potentials in simulations of interfacial phenomena. We have seen both quantitative and qualitative differences for the wetting transition. The former could be accounted for by changing other interaction parameters to reproduce the transition at the right temperature $T_0$. This assumption is fundamental to fitting force fields with truncated potentials to obtain quantitative agreement with e.g. experimental values. But it does not hold for the character of the transition because it arises purely from the value of the cutoff itself. If the resulting metastability of states remains undetected, the use of truncated interaction potentials could lead to wrong inferences about physical properties being made. While this conclusion has resulted from a simulation of wetting, similar implications could hold for other interfacial phenomena such as capillary flow~\cite{joly2011capillary,gravelle2016anomalous}, evaporation/condensation~\cite{hens2014nanoscale,nagayama2015molecular}, mixtures~\cite{iyer2013computer,tran2014molecular,radu2017enhanced} or heterogeneous nucleation~\cite{reinhardt2014effects,cabriolu2015ice,bi2016heterogeneous,qiu2017strength,bourque2017heterogeneous} where it is commonplace to use truncated interactions.
\begin{acknowledgments}
This work was supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 616121 (HeteroIce project). A.M. is supported by the Royal Society through a Royal Society Wolfson Research Merit Award. We are grateful for computational resources provided by the London Centre for Nanotechnology and the Materials Chemistry Consortium through the EPSRC grant number EP/L000202. L.J. is supported by the French Ministry of Defense through the project DGA ERE number 2013.60.0013 and by the LABEX iMUST (ANR-10-LABX-0064) of Universit\'{e} de Lyon, within the program ``Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). M.M. is supported by the Thousand Young Talent Program from the Organization Department of the CPC Central Committee.
\end{acknowledgments}
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{"url":"https:\/\/www.storyofmathematics.com\/trial-and-error-factoring\/","text":"# Factoring Trinomials by Trial and Error \u2013 Method & Examples\n\nAre you still struggling with the topic of factoring trinomials in Algebra? Well, no worries, because you are at the right place.\n\nThis article will introduce you to one of the simplest methods of factoring trinomials known as trial and error.\n\nAs the name suggests, trial and error factoring entails trying all possible factors until you find the right one.\n\nTrial and error factoring is regarded as one of the best methods of factoring trinomials. It encourages students to develop their mathematical intuition and thus increase their conceptual understanding of the topic.\n\n## How to Unfoil trinomials?\n\nSuppose we want to unfoil the general equation of a trinomial ax2\u00a0+ bx + c where a \u2260 1. Here are the steps to follow:\n\n\u2022 Insert the factors of ax2in the 1st positions of the two sets of brackets that represent the factors.\n\u2022 Also, insert the possible factors of c into the 2nd\u00a0positions of brackets.\n\u2022 Identify both the inner and outer products of the two sets of brackets.\n\u2022 Keep on trying different factors until the sum of the two factors is equal to \u201cbx.\u201d\n\nNOTE:\n\n\u2022 If c is positive, both factors will have the same sign as \u201cb\u201d.\n\u2022 If c is negative, one factor will have a negative sign.\n\u2022 Never put in the same parentheses\u2019 numbers with a common factor.\n\n## Trial and error factoring\n\nTrial and error factoring, which is also referred to as reverse foil or unfoiling, is a method of factoring trinomials built upon different techniques such as foil, factoring by grouping, and some other concepts of factoring trinomials with a leading coefficient of 1.\n\nExample 1\n\nUse trial and error factoring to solve 6x2 \u2013 25x + 24\n\nSolution\n\nPaired factors of 6x2 are x (6x)\u00a0or 2x (3x), therefore our parentheses will be;\n\n(x \u2013 ?) (6x \u2013 ?) or (2x \u2013 ?) (3x \u2013 ?)\n\nReplace \u201cbx\u201d with possible paired factors of c. Try all paired factors of 24 that will produce -25 The possible choices are (1 & 24, 2 & 12, 3 & 8, 4 & 6). Therefore, the correct factoring is;\n\n6x2 \u2013 25x + 24 \u27f9 (2x \u2013 3) (3x \u2013 8)\n\nExample 2\n\nFactor x2\u00a0\u2013 5x\u00a0+ 6\n\nSolution\n\nThe factors of the first term x2,\u00a0are\u00a0x\u00a0and\u00a0x. Therefore, insert x in the first position of each parentheses.\n\nx2\u00a0\u2013 5x\u00a0+ 6 = (x\u00a0\u2013 ?) (x\u00a0\u2013 ?)\n\nSince last term is 6, therefore the possible choices of factors are:\n\n(x\u00a0+ 1) (x\u00a0+ 6)\n(x\u00a0\u2013 1) (x\u00a0\u2013 6)\n(x\u00a0+ 3) (x\u00a0+ 2)\n(x\u00a0\u2013 3) (x\u00a0\u2013 2)\n\nThe correct pair which gives -5x\u00a0as the middle term is (x\u00a0\u2013 3) (x\u00a0\u2013 2). Hence,\n\n(x\u00a0\u2013 3) (x\u00a0\u2013 2) is the answer.\n\nExample 3\n\nFactor x2\u00a0\u2013 7x + 10\n\nSolution\n\nInsert the factors of the first term in the first position of each parentheses.\n\n\u27f9 (x\u00a0-?) (x\u00a0-?)\n\nTry the possible pair of factors of the 10;\n\n\u27f9\u00a0(-5) + (-2) = -7\n\nNow replace the question marks in the parentheses with these two factors\n\n\u27f9 (x\u00a0-5) (x\u00a0-2)\n\nHence, the correct factoring of x2\u00a0\u2013 7x + 10 is (x\u00a0-5) (x\u00a0-2)\n\nExample 4\n\nFactor 4x2\u00a0\u2013 5x \u2013 6\n\nSolution\n\n(2x -?) (2x +?) and (4x -?) (x +?)\n\nTry the possible pair of factors;\n\n6 x2 \u2212 2x \u2013 151 & 6, 2 & 3, 3 & 2, 6 & 1\n\nSince the correct pair 3 and 2, therefore, (4x \u2013 3) (x + 2) is our answer.\n\nExample 5\n\nFactor the trinomial x2 \u2212 2x \u2013 15\n\nSolution\n\nInsert x in the first position of each parentheses.\n\n(x\u00a0-?) (x\u00a0+?)\n\nFind two numbers whose product and sum are -15 and -2, respectively. By trial and error, the possible combinations are:\n\n15 and -1;\n\n-1 and 15;\n\n5 and -3;\n\n-5 and 3;\n\nOur correct combination is \u2013 5 and 3. Therefore;\n\nx2 \u2212 2x \u2013 15 \u27f9 (x\u00a0-5) (x\u00a0+3)\n\n### How to factor trinomials by grouping?\n\nWe can also factor trinomials by using a method of grouping. Let\u2019s walk through the following steps to factor ax2\u00a0+ bx + c where a \u22601:\n\n\u2022 Find the product of the leading coefficient \u201ca\u201d and the constant \u201cc.\u201d\n\n\u27f9 a * c = ac\n\n\u2022 Look for the factors of the \u201cac\u201d that add to coefficient \u201cb.\u201d\n\u2022 Rewrite bx as a sum or difference of the factors of ac that add to b.\n\u2022 Now factor by grouping.\n\nExample 6\n\nFactor the trinomial\u00a05x2\u00a0+ 16x\u00a0+ 3 by grouping.\n\nSolution\n\nFind the product of the leading coefficient and the last term.\n\n\u27f9 5 *3 = 15\n\nPerform trial and error to find pair factors of 15 whose sum is the middle term (16). The correct pair is 1 and 15.\n\nRewrite the equation by replacing the middle term 16x by x and 15x.\n\n5x2\u00a0+ 16x\u00a0+ 3\u27f95x2\u00a0+ 15x + x\u00a0+ 3\n\nNow, factor out by grouping\n\n5x2\u00a0+ 15x + x\u00a0+ 3 \u27f9 5x (x + 3) + 1(x + 3)\n\n\u27f9 (5x +1) (x + 3)\n\nExample 7\n\nFactor\u00a02x2 \u2013 5x \u2013 12 by grouping.\n\nSolution\n\n2x2 \u2013 5x \u2013 12\n\n=\u00a02x2 + 3x \u2013 8x \u2013 12\n\n=\u00a0x (2x + 3) \u2013 4(2x + 3)\n\n=\u00a0(2x + 3) (x \u2013 4)\n\nExample 8\n\nFactor 6x2 + x \u2013 2\n\nSolution\n\nMultiply the leading coefficient a and the constant c.\n\n\u27f9 6 * -2 = -12\n\nFind two numbers whose product and sum are -12 and 1 respectively.\n\n\u27f9\u00a0\u2013 3 * 4\n\n\u27f9 -3 + 4 = 1\n\nRewrite the equation by replacing the middle term -5x by -3x and 4x\n\n\u27f9 6x2 -3x + 4x -2\n\nFinally, factor out by grouping\n\n\u27f9 3x (2x \u2013\u00a0 1) + 2(2x \u2013 1)\n\n\u27f9 (3x + 2) (2x \u2013 1)\n\nExample 9\n\nFactor 6y2\u00a0+ 11y\u00a0+ 4.\n\nSolution\n\n6y2\u00a0+ 11y\u00a0+ 4 \u27f9 6y2\u00a0+ 3y + y\u00a0+ 4\n\n\u27f9 (6y2\u00a0+ 3y) + (8y\u00a0+ 4)\n\n\u27f9 3y (2y\u00a0+ 1) + 4(2y\u00a0+ 1)\n\n= (2y\u00a0+ 1) (3y\u00a0+ 4)\n\n### Practice Questions\n\n1. Which of the following shows the factored form of the trinomial, $3x^2- 8x \u2013 60$?\n\n2. Which of the following shows the factored form of the trinomial, $x^2- 21x + 90$?\n\n3. Which of the following shows the factored form of the trinomial, $x^2 \u2013 22x + 117$?\n\n4. Which of the following shows the factored form of the trinomial, $x^2 \u2013 9x + 20$?\n\n5. Which of the following shows the factored form of the trinomial, $x^2 + x \u2013 132$?\n\n6. Which of the following shows the factored form of the trinomial, $30a^2+ 57ab \u2013 168b^2$?\n\n7. Which of the following shows the factored form of the trinomial, $x^2 + 5x \u2013 104$?\n\n8. Which of the following shows the factored form of the trinomial, $y^2 + 7y \u2013 144$?\n\n9. Which of the following shows the factored form of the trinomial, $z^2+ 19z \u2013 150$?\n\n10. Which of the following shows the factored form of the trinomial, $24x^2 + 92xy + 60y^2$?\n\n11. Which of the following shows the factored form of the trinomial, $y^2 + y \u2013 72$?\n\n12. Which of the following shows the factored form of the trinomial, $x^2+ 6x \u2013 91$?\n\n13. Which of the following shows the factored form of the trinomial, $x^2- 4x -77$?\n\n14. Which of the following shows the factored form of the trinomial, $x^2 \u2013 6x \u2013 135$?\n\n15. Which of the following shows the factored form of the trinomial, $x^2- 11x \u2013 42$?\n\n16. Which of the following shows the factored form of the trinomial, $x^2 \u2013 12x \u2013 45$?\n\n17. Which of the following shows the factored form of the trinomial, $x^2 \u2013 7x \u2013 30$?\n\n18. Which of the following shows the factored form of the trinomial, $x^2 \u2013 5x \u2013 24$?\n\n19. Which of the following shows the factored form of the trinomial, $3x^2 + 10x + 8$?\n\n20. Which of the following shows the factored form of the trinomial, $3x^2 + 14x + 8$?\n\n21. Which of the following shows the factored form of the trinomial, $2x^2 + x \u2013 45$?\n\n22. Which of the following shows the factored form of the trinomial, $6x^2 + 11x \u2013 10$?\n\n23. Which of the following shows the factored form of the trinomial, $3x^2 \u2013 10x + 8$?\n\n24. Which of the following shows the factored form of the trinomial, $7x^2+ 79x + 90$?","date":"2022-05-28 08:24:10","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\": 0.7539332509040833, \"perplexity\": 1140.2035331890036}, \"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\/1652663013003.96\/warc\/CC-MAIN-20220528062047-20220528092047-00377.warc.gz\"}"} | null | null |
Q: Replace "\" with the "" I want to replace "\" with empty string. I have tried many things but doesn't work.
MyText like this test/test\test:test*test?test"test<test>test|test
MyText.Replace("\\\"", "").Replace("\\", "").Replace("\"", "").Replace("\\", "");
How to do this in correctly?
A: It should be as simple as the following:
string oldStr = "test/test\\test:test*test?test\"test<test>test|test";
string newStr = oldStr.Replace(@"\", string.Empty);
Note that I've used the @ sign to treat the strings as verbatim string literals. This avoids the need to escape the backslash each time it is used.
EDIT
Note also that the Replace function does not replace the contents of the string you call it on. Instead, it returns a new string with the replacement made. I suspect this is the actual cause of your issue, judging by the code you posted.
A: Try this out
var str = @"test/test\test:test*test?test"test<test>test|test".Replace("\\", "");
A: You could always do:
string newString = oldString.Replace("\\", "");
A: I tried it with
string test = @"test/test\test\test";
string result = test.Replace("\\", null);
and it works fine.
The result is "test/testtesttest"
A: how about
MyString.Replace(@"\", "");
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{"url":"https:\/\/tex.stackexchange.com\/questions\/289676\/how-to-insert-flexible-float-text-separation-in-two-column-mode","text":"# How to insert flexible float\/text separation in two-column mode\n\nTo avoid ugly extra white space in the text, i want a flexible glue between the float and the text in two-column mode. In one-column mode, I can add extra glue using \\textfloatsep, and I hoped I could achieve the same with the equivalent dbltextfloatsep in two-column mode. However, it does not seem to work, no matter how much flexibility I put in.\n\nIs there any way of achieving what I want?\n\nMinimal working example:\n\n\\documentclass[twocolumn]{article}\n\\usepackage{kantlipsum}\n\\usepackage{mwe}\n\n\\setlength{\\dbltextfloatsep}{20pt plus 40pt minus 4pt}\n\n\\begin{document}\n\\begin{figure*}[!t]\n\\includegraphics[width=\\textwidth,height=10cm]{example-image-a}\n\\caption{This is a caption}\n\\end{figure*}\n\\section{First Section}\n\\kant[1-5]\n\\textbf{THERE IS A BIG UGLY WHITE SPACE ABOVE ME.}\n\\section{Second Section}\n\\kant[6]\n\\end{document}\n\n\nThis is what I want:\n\n\u2022 the dbltextfloatsep space is the space below the figure before the start of the columns it is not added separately to each column so it has no possibility of affecting the space above your capital letters which is in one column. \u2013\u00a0David Carlisle Jan 27 '16 at 14:05\n\u2022 Perhaps I should have been more clear: I do indeed want to add space to both columns. But apparently, this is not happening automatically. \u2013\u00a0akvilas Jan 27 '16 at 14:17\n\u2022 It is happening but the columns are split and boxed and the glue in them frozen then they are put side by side, then the figure is put at the top then the glue then the box containing both columns, so the glue can stretch to pad out of the combined 2-col box and the figure do not completely fill the page but it will not affect the glue stretching inside the columns. perhaps you want \\raggedbottom so that the left column is not stretched out? \u2013\u00a0David Carlisle Jan 27 '16 at 14:20\n\u2022 If I understood you correctly, the algorithm that determines the height of the columns is unaware of the extra flexibility specified in dbltextfloatsep? \u2013\u00a0akvilas Jan 27 '16 at 14:25\n\u2022 basically, yes. \u2013\u00a0David Carlisle Jan 27 '16 at 15:06\n\nThe \\dbltextfloatsep space is the space below the figure before the start of the columns it is not added separately to each column, it is added _after- latex has constructed a partial page box with the columns. It's natural length is used to determine the target height of the columns to go under the figure and any stretch is used to adjust any shortfall, but the stretch is not added separately into each column.\nPerhaps you want \\raggedbottom so that the left column is not stretched out?\n\u2022 I'll accept the fact that there is no way to achieve what I want.. \\raggedbottom is possible, but I'll rather adjust the \\dbltextfloatsep manually for each float. \u2013\u00a0akvilas Feb 10 '16 at 22:03","date":"2019-11-13 17:54:31","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\": 0.8386435508728027, \"perplexity\": 870.6529861475301}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496667319.87\/warc\/CC-MAIN-20191113164312-20191113192312-00153.warc.gz\"}"} | null | null |
Yemen - the last original Arab country
All people I know that visited Yemen love it and many of them return there again. The main reason are it's friendly and hospitable people, I think. The other reason might be it's nature and culture, both unique in Arab world. I believe that Yemen is the last presevered 'true' Arab country, with 'true' Arab hospitality. If anyone was at least partly disappointed by mass tourism Arab speaking destinations like Tunis, Egypt, Morocco or UAE and Oman then Yemen should offer quite different experience. The global world is changing quicky, so I don't want to predict how long this uniqueness will last. But better to go there sooner that later.
Yemen does not have much oil, compared to other Arab neigbouring states and therefore it's people still need to live by their work as in the past. This probably helped to preserve the old customs and Arab hospitality. But it can't be only that. Yemen, especially the north, was in one Al Jazeera documentary (used to be on YouTube) called 'Republic of Tribes'. Thas it true especially in the north, that did not experience socialist regime as south did. Many people feel more loyalty to it's tribe than to the government and state of Yemen.
People of Yemen, although traditional in their customs, are very friendly and tolerant towards foreigners. But men and especially women should not wear shorts (we did wear them during first visit but not during second visit). Short sleeve is not a problem, but long sleeve is more appropriate for women of course. Headcloth is not necessary, compared to e.g. Afghanistan. People are very family oriented. If you are introduced to any women at all, they are mostly willing to talk about childern and family only. So if you want to be better preprared, take photos of your childern, family, house. That is good for conversation with men as well. They will show the photos to their women anyway.
I can't say I know at all to know all areas of Yemen! During my first visit - 6 day stopover with Yemenia airlines, on the way to Ethiopia in 2006, I visited the capital Sanaa, unique Bura mountains and Red sea coast. I was impressed by Yemen, especially compared to my second visit of Ethiopia during the same trip later. That time, I visited Bura mountains for longer time, added eastern Yemen areas (Shibam, Sayun, Tarim, Mukkala towns) and fortunatelly Socotra island. To me, Bura mountains and Socotra island are the areas, there were the best. Noone should miss them while visiting Yemen. Due to not the best security situation in Yemen in last years, many areas e.g. the north near Saudi Arabia or the desert between Sanaa and east Yemen are often closed to tourists or at least independent tourists. But both Bura and expecially Socotra lie in completely safe areas, that have not been subject to restrictions so far, fortunatelly.
I travelled mostly by shared taxis in Yemen. They go frequently when they fill with people. They are cheap and have quite fixed prices per seat and destinations (drivers don't give you foreigners surcharge, that is quite unbelievable in many mass tourism Arab / North Africa destinations!). It is necessary to hire a jeep with driver on Socotra island, in order to go anywhere there. The price of jeep with driver should be about 70-80 USD per day in continental Yemen, maybe a bit more on Socotra.
Yemen has very good food. But for trekking in Socotra, you need your own food. For trekking in Bura mountains - if you are with guide, you could probably eat evening and morning with people, where you stay (and pay something for that). If you are trek independetly as me, you have to take your own food, although you might be invited serveral times as well to local houses, especially if you speak Arabic (but you can't rely on that).
The hot and somewhat rainy season is approximately June-August. The best time to visit Yemen is after this - late September and all October, when all mountains are greener with streams with water there. Expect occasional rains even during this 'best' time. But Yemen can be visited quite anytime - even the rainy+hot season is not unbearable I think, it's just too hot in lower areas and some roads can be block by unpredictable heavy rains. But bear in mind, that rainy+hot season in continental Yemen corresponds to windy season in Socotra, when most of flights there are cancelled - so quite impossible for combination with Socotra. Unless you are in Yemen during or right after the rainy+hot season, you basically even don't need a tent for trekking in Bura mountains or Socotra island or elsewhere.
Lower areas of Yemen are very hot, but Sanaa itself lies in the altitude about 2000m !, so it's quite cold (it can freeze there in the night during winter time). You don't need mechanical water filter for trekking usually, the streams are not so muddy, but chemical treatment of water is very recommended. Yemen has good GSM signal coverage and roaming with major international operators, even in Bura mountains. Only Socotra had some limited CDMA coverage where local SIM card and CDMA phone was necessary - that was in 2007.
Note on Arabic language
Few people in Yemen speak English, especially in the more interesting north. Even in the south, that used to be British protectorate, you can't rely on English at all. So I studied Arabic and returned there for second time in one and half year. Arabic is a difficult language, so one a half year was just enough to be able to perform some conversation for me.
Yemen can be visited even if you don't know Arabic at all. It was the same with me during my first visit and I was excited. But for deeper experience and expecially for repeated visits (that usually tend to be not so great that the first great experience for me), I would recomend a book called Formal Spoken Arabic - fast course. One needs at least 3 months to cover most of that book by studying in the evenings, I think. It does not deal with Arabic script too much. Script is very usefull of course, I studied that, but not necessary for conversation. It contains good audio, so it's good even for self studying. It contains usefull words (especially verbs) and teaches you something between school standard literary/news Arabic and too country specific spoken Arabic. It's name might be something like 'generalised spoken arabic'. Actually, it's spoken part is based on Lebanon-Palestine-Jordan-Syria area, but they will show you more or less only that spoken words, there are comonly understood across Arab world, I believe.
All people in Yemen, that went to primary school or watch TV understand standard school/news Arabic and are more or less able to reply in standard Arabic. This is especially true for men. Yemen has the lowest level of literacy among women in Arab world (about 30% ?). And Yemen is of course one of the best countries, if not the best, where to study standard school/news Arabic. There are many language schools in Sanaa or Aden.
Bura mountains - maybe the greenest place in Yemen
When looking for information about trekking in Yemen, one usually finds something about Haraz mountais only, maybe about Socotra. Haraz is OK, just 2 hours from Sanaa and people are used to tourists there. But Bura is something different - vertical rock ridges, high houses standing right at the top of these ridgers, green forests from the west (Red sea) side, monkeys in the forest, almost no tourists, living villages (not abandoned as some in Haraz), friendly people still not so much used to tour groups (although this is already changing). Bura is changing as well, new roads are being constructed, electricity is coming to many places. Bura is not far from more famous Haraz mountains. It's just about 4 hours by shared taxi or bus to Bajil town an then about 2 hours by hired car to Bura mountains.
It's better to start from Sanaa in the morning, in order to reach the mountains by the evening and sleep there already. If you start around noon, you most probably end up sleeping in very hot town of Bajil. Hotels in Bajil are not good and the one airconditioned in overpriced compared to Sanaa given it's not very good quality.
Asphalt road goes all the way from Bajil to Bamii village (just under the highest mountain, in the middle of Bura mountains). Another road goes to the green wadi Raaba (access form the west, from Red sea side). Plus rough road forks from the asphalt road under Bamii village to Magraba village as well. And rough road goes to Kohl village - that is near Antara - this is the access from the north, good for trekking all of the main ridge from north to south. I don't know about any maps. The mountains are so small and vertical, that even these Russian maps do now show it well. Some correct hand drawn map would be better. Hopefully some locals will do it one day.
We originally asked driver from Bajil to go the village of Kohl, but ended up in some village under it, where the rough road ended anyway. There are definitely other roads that I did not mark here. If anyoned intends starting trekking the whole ridge from the north, as we did, try to say Kohl or Antara village in Jebel Bura mountains to drivers in Bajil. Some of them should know. I takes about 3-4 days to cover the green dotted path on foot, but we did not go the the highest moutain (spent time with our host instead). Although we had 4 days at disposal, we spent a lot of time with people on the way as their guests.
It's possible to go independently, without guide, but expect difficulties when communicating about where to go next (even if you know the village names marked here) and don't automatically expect to be invited to some home if you don't know Arabic (although this still may happen). Every bigger village should have school and teacher of Arabic there. As far as guide, it's better to get him from Sanaa using some travel agencies. I would not hope for someone in Bajil or even some English speaker in the first village of Bura, where car will drop you.
Wadi Raaba is quite known protected area with paid entrance (about 2.5 USD probably) nowadays. This would probably be the easiest destination to go from Bajil independently. Most drivers in Bajil should know it as a lot of Yemenis go there for picknic or just the see the green forests there nowadays. It's not a proble to get into some car from Wadi Raaba to Bajil or to Sukhna. From Sukhna, it's quite fast go over Tihama plains to Taiz and to Aden - going back over mountains to Sanaa would be much longer.
Socotra island
People come to Socotra for different reasons - trekking, sea life, diving, rare fauna, fishing, caves etc. I spent there three and half days only. It such short time, one can do the easy 2 day trek (and I did not know about something more difficult at that time anyway) plus some caves and swiming in sea on the third day. But 1 week or even more is better, either for lond trekking or more of sea life, fishing etc. As can be seen in the map down, only righ-central part of Socotra is the part suitable for trekking. Fortunately, it lies just next to airport and main town of Hadibo. Full south to north travese would take around 3-4 days taking the easy route or more trying the difficult route - see detailed map below. I was told, that the highest mountains was normally climbed with a guide from Hadibo as daytrip. Crossing the the main ridge in the area of the hightest mountains might be quite challenging, but definitely not impossible. If I go back to Socotra some time, I'll try it.
Upper central Socotra from Google Earth sattelite maps. The 'Y' like fork/junction of wadi Dirhur (left) and Darho(right) was the start of our short 2 day trek. All the area around this junction is called Firmihin area. You need to tell this to the driver, if you need to go there independently, together with the wadi names. Asphalt road goes almost there from north-west (from airport). Jeep track continues further on at least half the way of wadi Darho (right, easy wadi). But it's difficult for any car to go even down the wadi if it was just a bit of rain. We walked it down the the wadi and all the way further. The path is very clear, there are no confusing side paths and orientation easy. It's not necessary to have a guide.
If you arrive to Socotra independently, there is some UNESCO or what sponsored friendly office in Hadibo. They speak English will tell you what to see, help you to arrange you itinerary, call the guide etc. There seemed to be only one good restaurant in Hadibo, where locals directed all foreigners. So this was the place to meet others, but they were usually with tour agencies there. There were a few hotels around as well.
If you are intrested in caves, try Hoq cavers on the northern coast, maybe 1 hour by car east of Hadibo. They should be better than the small cave we visited in the south. It should be possible with some effort and contacts to arrange an unforgettable trip with local fishermen, when they go to fish (we did not have time for this).
Detail of central Socotra from Russian military 1:200 000 map with our trek + more adventurous alternatives marked.
Other areas in Yemen
Where to stay in Sanaa cheaply
I used to stay in some hotel in the southern corner of the main Bab Al Yemen square. I can't remember the name. The advantage ot this location, apart from price, is shared taxi station to Haraz/Bura mountains etc. that is just 50 meters away. It was about 15 USD for a room for 4 people (smaller probably available as well). Electricity was OK, toilets + shower shared for a few other rooms as well, but usable. They spoke some basic English at the reception, which was on the first floor. There was some other hotel next to it, which was even cheaper, with the entrance right from the main square, but this was awful.
One can stay in hotels focused on foreign tourist - e.g. Taj Talha and the one next to it, right in the center of the old town. They have beautiful location and cost about 30 USD for 2 people.
Maps and guidebooks
After my second visit, in late 2007, Bradt published the first guide focused on Yemen only, after old LP in 1996. After browsing through it, I can't say it's better than old LP guide. It has some newer information of course, but LP guide e.g. wrote something about all provinces of Yemen (e.g. Mahra east of Mukkala etc.), not only chosen areas. And the best areas of Yemen - Jebel Bura mountains and Socotra are described very shortly again, without any usefull maps for trekking.
So my short text here should still be usefull as a complement to both guidebooks for trekking in Bura and Socotra. Plus Bradt 'new' guide does not even mention that tourist permit was necessary to travel quite anywhere outside Sanaa in autumn 2007 and according to my info this is still true in autumn 2008 !
The best road map for Yemen in 2008 was this by German Reise Know How.
Russian military 1:200 000 maps of Yemen are not much useful for current road situation, but there is nothing better to show topography - download here area of Haraz and Bura mountains and here Socotra.
Tourist travel permit - how it worked in 2007, this is not mentioned in 'new' Bradt guidebook
One needed a tourist permit (called somthing like 'tashrih' in Arabic) to go in fact anywhere outside Sanaa (in autumn 2007, still valid in autumn 2008). The pemit was issued 24hours a day at Minitstry of Tourism by Tourist Police, Hotoba area in northern Sanaa. I think one could get it in other big towns such as Aden as well. We went there by taxi, we did not know the exact location at that time either. The place is about half way between the airport and center of Sanna, so it might be good to stop there right when coming from the airport. They speak English and the permit is issued to independent travellers as well. They wanted us to list all destinations that we wanted to visit with exact dates when we wanted to be there. So it might be good the have this itinerary prepared in advance to speed up the process.
If you are independent, they may try to restict you more than groups with tour agencies (yes, they need this permit as well). E.g they did not want to allow us independently to Bura mountains at first, they allowed us when I spoke Arabic and said we were having friends there. In the end, after you supplied them with your detailed itinerary in English, they will write you permit on preprinted paper in Arabic only. This permit states (if I remember it correctly) overall from - till date of your travel and even does not list all people who are travelling ! There was no detail route descriptoin on this, to be checked by local police there! So if they don't want to let you go somewhere, don't insist on it and say something on the main road close to it instead (e.g. Sanna to Hudayda at Red sea for problem free access to Bura mountains) or leave that completely. Eventually, if it's not completely closed area or there are not police check points there, you may go there or at least try it. The most important is to leave the office with the permit in hand.
They told us, they we cannot start the journey the same day, only next day after the permit was issued. Probably because they want to report your group to all police check points on the way. I don't know if they do this in reality, but we started our journey just after leaving the office and had no problems. Make many copies of this permit immediately (e.g. 20). At any road check point, I just handed one copy out of car window to the police and in 5 seconds we continued. There was no check made whether we were reported there in advance or not and usually there was no passport check as well. | {
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Der Wahlkreis Diez/Nassau (Wahlkreis 7) ist ein Landtagswahlkreis in Rheinland-Pfalz. Er umfasst die Verbandsgemeinden Aar-Einrich, Diez, Nassau und Nastätten, die dem Rhein-Lahn-Kreis angehören.
Wahl 2021
Für die Landtagswahl 2021 treten folgende Kandidaten an:
Wahl 2016
Die Ergebnisse der Wahl zum 17. Landtag Rheinland-Pfalz vom 13. März 2016 :
Direkt gewählt wurde Jörg Denninghoff (SPD).
Wahl 2011
Die Ergebnisse der Wahl zum 16. Landtag Rheinland-Pfalz vom 27. März 2011:
Direkt gewählt wurde Frank Puchtler (SPD).
Matthias Lammert (CDU) wurde über die Landesliste (Listenplatz 22) in den Landtag gewählt.
Wahl 2006
Die Ergebnisse der Wahl zum 15. Landtag Rheinland-Pfalz vom 26. März 2006:
Direkt gewählt wurde Frank Puchtler (SPD).
Wahlkreissieger
Weblinks
Der Landeswahlleiter Rheinland-Pfalz
Landtag Rheinland-Pfalz – Abgeordnete und Wahlkreise
Einzelnachweise
Diez/Nassau
Politik (Rhein-Lahn-Kreis) | {
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ITN Productions produces Netflix Originals Drug Lords series
Drug Lords, a docu-drama series produced by Oscar-nominated ITN Productions, will launch globally on Netflix on January 19th.
The series reveals the true stories behind the rise, reign, and fall of the world's most infamous drug lords, from Colombian kingpin Pablo Escobar to "American Gangster" Frank Lucas, one of the most notorious gangsters in US history.
Told by the men and women directly involved in the cases themselves –the cops, crooks, and closest associates on either side of the law— each hour of Drug Lords provides an unprecedented 360° perspective on the world's most notorious narcos and their tragically violent rule.
The 8 episodes take an intimate look at some of the most feared and ruthless criminals across the world, bringing their stories to life with unprecedented access and dramatic recreations of the key events in the rise and fall of these infamous gangsters.
The first four episodes drop on January 19th and the final four later in the year.
Ian Russell, Head of International Programming, ITN Productions said: "This is a hugely ambitious series that proves that real life stories are often even more extreme and dramatic than fiction. Our production team spent more than a year working on this series which involved seeking out people who have spent their life in the shadows – and then persuading them to tell us about their lives."
Michael Welsh, Vice President TV Production and Development, ITN Productions said: "We were delighted that Netflix could see the potential of this series that's both shocking and compelling."
AOL Launches New Original Series "Making a Scene With James Franco"
Netflix wants 20 original series a year
Netflix wants more British originals
Netflix US more originals, less library
Netflix and 41 announce Kong series | {
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Q: Grep and replace form content Working on identifying and adjusting form content entries.
I'm using grep to eliminate values in array when form is submitted:
# Get the input
read(STDIN, $buffer, $ENV{'CONTENT_LENGTH'});
# The following zaps entire value containing the item(s) grepped.
@OrigArray = split(/&/, $buffer);
@pairs = grep(!/kartridgeclub|bubba.org|\.ru/i, @OrigArray);
but I'd prefer to identify which of the unwanted character sequences
was entered and also (regardless of which one was entered), include
a conditional statement which would produce unique "return page"
text if any of the unwanted character sequences were submitted.
I cannot, I think, use simple substitution instead of grep since
I need to replace the entire array element containing the unwanted characters.
In other words, user enters bunch-of-garbage-always-containing-some.org-with-ru
extension (or bubba.org, etc.) into a form field. How do I identify which of
the phrases in the grep he entered? And, then replace it with a phrase of my
choice (instead of just deleting it from the array) and produce a different
"Thank You" page for this user?
Thanks.
A: Cfreak's question is pragmatic: use CGI to process your form's input. Use CGI's param function, examine your form's parameters for the "unwanted character sequences," and then make changes and/or branch accordingly--based upon what you've found. For example:
use strict;
use warnings;
use CGI qw(:standard);
use 5.010;
my $input_field = param('input_field');
given(lc $input_field) {
when(/kartridgeclub/) {
print 'kartridgeclub';
# do something else
}
when(/bubba\.org\b/) {
print 'bubba.org';
# do something else
}
when(/\.ru\b/) {
print '.ru';
# do something else
}
when(/this|that/) {
print 'this or that';
# do something else
}
default {
print 'Your entry seems OK.';
# do something else
}
}
A: while i fully agree with the other post(s) that you're better off using CGI for this, i think this line should achieve what you want.
my %pairs = map { $_ => ($_ =~ /kartridgeclub|bubba.org|\.ru/i ? 'invalid' : 'valid')} @orig_array;
A: Just do not use grep. Use good old for-loops and write some code.
for my $i (0 .. $#OrigArray) {
my $pair = $OrigArray[$i];
if ($pair =~ m/whatever/) {
# you're welcome to change $OrigArray[$i] here
}
}
| {
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E-text prepared by the Online Distributed Proofreading Team
(http://www.pgdp.net) from page images generously made available by
Internet Archive (https://archive.org)
Note: Project Gutenberg also has an HTML version of this
file which includes the original illustrations.
See 57643-h.htm or 57643-h.zip:
(http://www.gutenberg.org/files/57643/57643-h/57643-h.htm)
or
(http://www.gutenberg.org/files/57643/57643-h.zip)
Images of the original pages are available through
Internet Archive. See
https://archive.org/details/doctorizard00gree
Transcriber's note:
Text that was in italics is enclosed by underscores
(_italics_).
DOCTOR IZARD
------------------------------------------------------------------------
WORKS BY ANNA KATHARINE GREEN.
-------
THE LEAVENWORTH CASE. 4to, paper, 20 $1 00
cents; 16mo, paper, 50 cents; cloth
A STRANGE DISAPPEARANCE. 4to, paper, 20 $1 00
cents; 16mo, paper, 50 cents; cloth
THE SWORD OF DAMOCLES. 16mo, paper, 50 $1 00
cents; cloth
HAND AND RING. 4to, paper, 20 cents; $1 00
16mo, paper, illustrated, 50 cents;
16mo, cloth, illustrated
X. Y. Z. A DETECTIVE STORY. 16mo, paper 25
cents.
THE MILL MYSTERY. 16mo, paper, 50 cents; $1 00
cloth
7 TO 12 A DETECTIVE STORY. 16mo, paper 25
cents.
BEHIND CLOSED DOORS. 16mo, paper, 50 $1 00
cents; cloth
THE OLD STONE HOUSE, AND OTHER STORIES. $ 75
16mo, paper, 40 cents; cloth
CYNTHIA WAKEHAM'S MONEY. 16mo, paper, 50 $1 00
cents; cloth
MARKED "PERSONAL." 16mo, paper, 50 $1 00
cents; cloth
MISS HURD: an Enigma. 16mo, paper, 50 $1 00
cents; cloth
THE DOCTOR, HIS WIFE, AND THE CLOCK. $ 50
Oblong 32mo, cloth
DR. IZARD. 16mo, paper, 50 cents; cloth $1 00
THE DEFENCE OF THE BRIDE, AND OTHER $1 00
POEMS. 16mo, cloth
RISIFI'S DAUGHTER. A DRAMA. 16mo, cloth $1 00
-------
G. P. PUTNAM'S SONS, NEW YORK AND LONDON.
------------------------------------------------------------------------
[Illustration: "MERCIFUL HEAVENS, WAS THAT THE DOCTOR?"—SEE PAGE 228.]
DOCTOR IZARD
by
ANNA KATHARINE GREEN
(Mrs. Charles Rohlfs)
Author of "The Leavenworth Case," "Marked 'Personal,'"
"The Doctor, His Wife, and the Clock," etc., etc.
-------
G. P. Putnam'S Sons
New York London
27 West Twenty-Third Street 24 Bedford Street, Strand
The Knickerbocker Press
1895
Copyright 1895
By
Anna Katharine Rohlfs
Entered at Stationer's Hall, London
All Rights Reserved
The Knickerbocker Press, New York
------------------------------------------------------------------------
TO MY FRIEND
JOSEPH FRANCIS DALY
A. K. G. R.
------------------------------------------------------------------------
CONTENTS.
-------
PAGE
NO. THIRTEEN, WARD THIRTEEN 1
HADLEY'S CAVE 22
THE YOUNG HEIRESS 29
DR. IZARD 45
NOCTURNAL WANDERINGS 71
THE PORTRAIT 92
WHAT THE STROKE OF A BELL CAN DO 97
THE HOUSE ON THE HILL 114
ASK DR. IZARD 125
AN INCREDIBLE OCCURRENCE 136
FACE TO FACE 145
AT HOME 152
A TEST 157
GRACE 167
THE SMALL, SLIGHT MAN 186
THE LETTER 206
MIDNIGHT AT THE OLD IZARD PLACE 220
A DECISION 230
TO-MORROW 237
DR. IZARD'S LAST DAY IN HAMILTON 251
------------------------------------------------------------------------
DOCTOR IZARD.
-------
PART I.
A MIDNIGHT VISITANT.
-------
I.
NO. THIRTEEN, WARD THIRTEEN.
IT was after midnight. Quiet had settled over the hospital, and in Ward
13 there was no sound and scarcely a movement. The nurse, a strong and
beautiful figure, had fallen into a reverie, and the two patients, which
were all the ward contained, lay in a sleep so deep that it seemed to
foreshadow the death which was hovering over them both.
They were both men. The one on the right of the nurse was middle-aged;
the one on the left somewhat older. Both were gaunt, both were
hollow-eyed, both had been given up by the doctors and attendants. Yet
there was one point of difference between them. He on the left, the
older of the two, had an incurable complaint for which no remedy was
possible, while he on the right, though seemingly as ill as his fellow,
was less seriously affected, and stood some chance of being saved if
only he would arouse from his apathy and exert his will toward living.
But nothing had as yet been found to interest him, and he seemed likely
to die from sheer inanition. It is through this man's eyes that we must
observe the scene which presently took place in this quiet room.
He had been lying, as I have said, in a dreamless sleep, when
something—he never knew what—made him conscious of himself and partially
awake to his surroundings. He found himself listening, but there was no
sound; and his eyes, which he had not unclosed for hours, slowly opened,
and through the shadows which encompassed him broke a dim vision of the
silent ward and the sitting figure of the weary nurse. It was an
accustomed sight, and his eyes were softly re-closing when a sudden
movement on the part of the nurse roused him again to something like
interest, and though his apathy was yet too great for him to make a
movement or utter a sound, he perceived, though with dim eyes at first,
that the door at the other end of the ward had slowly opened, and that
two men were advancing down the room to the place where the nurse stood
waiting in evident surprise to greet them. One was the hospital doctor,
and on him the sick man cast but a single glance; but the person with
him was a stranger, and upon him the attention of the silent watcher
became presently concentrated, for his appearance was singular and his
errand one of evident mystery.
There was but one light in the room, and this was burning low, so that
the impression received was general rather than particular. He saw
before him a medium-sized man who sought to hide his face from
observation, though this face was already sufficiently shielded by the
semi-darkness and by the brim of a large hat which for some reason he
had omitted to remove. Around his shoulders there hung a cloak of an
old-fashioned type, and as he approached the spot where the nurse stood,
his form, which had shown some dignity while he was advancing,
contracted itself in such a fashion that he looked smaller than he
really was.
The physician who accompanied him was the first to speak.
"Is No. Twelve asleep?" he asked.
The nurse bowed slightly, half turning her head as she did so.
The watching man was No. Thirteen, not No. Twelve, but his eyes shut at
the question, perhaps because he was still overcome by his apathy,
perhaps because his curiosity had been aroused and he feared to stop
events by betraying his interest in them.
"I am afraid we shall have to wake him," pursued the attendant
physician. "This gentleman here, who declines to give his name, but who
has brought letters which sufficiently recommend him to our regard,
professes to have business with this patient which will not keep till
morning. Has the patient shown any further signs of sinking?"
She answered in a cheerful tone that he had slept since ten without
waking, and the two men began to approach. As they did so both turned
toward the bed of the second sick man, and one of them, the stranger,
remarked with something like doubt in his tones, "Is this man as low as
he looks? Is he dying, too?"
The answer was a qualified one, and the stranger appeared to turn his
back, but presently the strained ears of the seemingly unconscious man
heard a breath panting near his own, and was conscious of some person
bending over his cot. Next minute the question was whispered in his
hearing:
"Are you sure this man is asleep?"
The doctor, who was standing close by, murmured an affirmative, and the
nurse to whom the questioner had apparently turned, observed without any
hesitation in her slightly mystified tone:
"I have not seen him move since eight o'clock; besides, if he were
awake, he would show no consciousness. He is dying from sheer
hopelessness, and a cannon fired at his side would not rouse him."
The "humph" which this assurance called forth from the stranger had a
peculiar sound in it, but the attention which had been directed to No.
Thirteen now passed to his neighbor, and the former, feeling himself for
the instant unobserved, partially opened his eyes to see how that
neighbor was affected by it. A few whispered words had accomplished what
a cannon had been thought unable to do, and he was beginning to realize
an interest in life, or at least in what was going on in reference to
his fellow patient. The words were these:
"This is a hopeless case, is it?"
"Yes, sir."
"How long a time do you give him?"
The tone was professional, though not entirely unsympathetic.
"Dr. Sweet says a week; I say three days."
The stranger bent over the patient, and it was at this point that the
watcher's eyes opened.
"Three days is nearer the mark," the visitor at last declared.
At which the attending physician bowed.
"I should be glad to have a few moments' conversation with your
patient," the stranger now pursued. "If he is unhappy, I think I can
bring him comfort. He has relatives, you say."
"Yes, a daughter, over whose helpless position he constantly grieves."
"He is poor, then?"
"Very."
"Good! I have pleasant news for him. Will you allow me to rouse him?"
"Certainly, if you have a communication justifying the slight shock."
The stranger, whose head had sunk upon his breast, cast a keen look
around. "I beg your pardon," said he, "but I must speak to the man
alone; he himself would choose it, but neither you nor the nurse need
leave the room."
The doctor bowed and withdrew with marked respect; the nurse lingered a
moment, during which both of the sick men lay equally quiet and
death-like; then she also stepped aside. The stranger was left standing
between the two beds.
Soon the sensitive ears of the watchful one heard these words: "Your
little daughter sends her love."
Opening his eyes a trifle, he saw the stranger bending over the other's
pillow. A sigh which was not new to his ears rose from his dying
companion, at sound of which the stranger added softly:
"You fear to leave the child, but God is merciful. He makes it possible
for you to provide for her; do you want to hear how?"
A low cry, then a sudden feeble move, and No. Twelve was speaking in
hurried, startled words:
"Who are you, sir? What do you want with me, and what are you saying
about my child? I don't know you."
"No? And yet I am likely to be your greatest benefactor. But first take
these few drops; they will help you to understand me. You are afraid?
You need not be. I am—" He whispered a name into the sick man's ear
which his companion could not catch. "That is our secret," he added,
"and one which I charge you to preserve."
No. Thirteen, unable to restrain his curiosity at this, stole another
glance at the adjoining cot from under his scarcely lifted lids. His
moribund neighbor had risen partially on his pillow and was gazing with
burning intensity at the man who was leaning toward him.
"O sir," came from the pale and working lips, as he tried to raise a
feeble hand. "You mean to help my little one, you? But why should you do
it? What claim has my misfortune or her innocence on you that you should
concern yourself with our desperate condition?"
"No claim," came in the stranger's calm but impressive tones. "It is not
charity I seek to bestow on you, but payment for a service you can
render me. A perfectly legitimate, though somewhat unusual one," he
hastened to add, as the man's face showed doubt.
"What—what is it?" faltered from the sick man's lips in mingled doubt
and hope. "What can a poor and wretched being, doomed to speedy death,
do for a man like you? I fear you are mocking me, sir."
"You can be the medium—" the words came slowly and with some
hesitation—"for the payment of a debt I dare not liquidate in my own
person. I owe someone—a large amount—of money. If I give it to you—" (he
leaned closer and spoke lower, but the ears that were listening were
very sharp, and not a syllable was lost) "will you give it to the person
whom I will name?"
"But how? When? I am dying, they say, and——"
"Do not worry about the whens and hows. I will make all that easy. The
question is, will you, for the sum of five thousand dollars, which I
here show you in ten five-hundred-dollar bills, consent to sign a will,
bequeathing this other little package of money to a certain young woman
whom I will name?"
"Five thousand dollars? O sir, do not mislead a dying man. Five thousand
dollars? Why, it would be a fortune to Lucy!"
"A fortune that she shall have," the other assured him.
"Just for signing my name?"
"Just for signing your name to a will which will bequeath the rest of
your belongings, namely, this little package, to an equally young and
equally unfortunate girl."
"It seems right. I do not see anything wrong in it," murmured the dying
father in a voice that had strangely strengthened. "Will you assure me
that it is all right, and that no one will suffer by my action?"
"Did I not tell you who I was?" asked the stranger, "and cannot you
trust one of my reputation? You will be doing a good act, a retributive
act; one that will have the blessing of Providence upon it."
"But why this secrecy? Why do you come to me instead of paying the debt
yourself? Is she——"
"She is who she is," was the somewhat stern interruption. "You do not
know her; no one here knows her. Will you do what I ask or must I turn
to your companion who seems as ill as yourself?"
"I—I want to do it, sir. Five thousand dollars! Let me feel of the bills
that represent so much."
There was a movement, and the sick and feeble voice rose again in a tone
of ecstatic delight. "And I need not worry any more about her feet
without shoes and her pretty head without shelter. She will be a lady
and go to school, and by and by can learn a trade and live respectably.
Oh, thank God, sir! I know who I would like to have made her guardian."
"Then you consent?" cried the stranger, with a thrill of some strong
feeling in his voice.
"I do, sir, and thank you; only you must be quick, for there is no
knowing how soon the end may come." The stranger, who seemed to be
equally apprehensive of the results of this strong excitement, raised
himself upright and motioned to the doctor and the nurse.
"You will say nothing of our compact," he enjoined in a final whisper,
as the two summoned ones approached. "Nor will you express surprise at
the wording of the will or, indeed, at anything I may say."
"No," came in an almost undistinguishable murmur, and then there was
silence, till the doctor and the nurse were within hearing, when the
stranger said:
"Our friend here has a small matter of business on his mind. It has been
my pleasure, as I perhaps intimated to you, to bring him a considerable
sum of money which he had quite despaired of ever having paid him; and
as for reasons he is not willing to communicate, he desires to bequeath
a portion of it to a person not related to him, he naturally finds it
necessary to leave a will. Foreseeing this, I had the draft of one drawn
up, which, if agreeable to you, I will read to him in your presence."
The amazement in the nurse's eye gave way to a look of deference, and
she bowed slightly. The doctor nodded his head, and both took their
stand at the foot of the small cot. The man in the adjoining bed neither
murmured nor moved. Had they looked at him, they would have doubtless
thought his sleep was doing him but little good, for his pallor had
increased and an icy sweat glistened on his forehead.
"Mr. Hazlitt's property," continued the stranger in a low and mechanical
tone, "consists entirely of money. Is that not so?" he asked, smiling
upon the dazed but yet strangely happy face of the patient lying before
him. "Namely, this roll of bills, amounting as you see to five thousand
dollars, and this small package of banknotes, of which the amount is not
stated, but of whose value he is probably aware. Are you willing," and
he turned to the doctor, "to take charge of these valuables, and see
that they are forthcoming at the proper time?"
The doctor bowed, glanced at his patient, and meeting his eager eye,
took the roll of bills and the package, and putting them into his breast
pocket, remarked, "I will have them placed in the safe deposit vaults
to-morrow."
"Very well," cried the stranger; "that will be all right, will it not?"
he asked, consulting in his turn the man before him.
Mr. Hazlitt, as they called him, gave him a short look, smiled again,
and said: "You know best; anything, so that my Lucy gets her five
thousand."
The stranger, straightening himself, asked if he could not have more
light, at which the nurse brought a candle. Immediately the stranger
took a paper from under his cloak and opened it. The nurse held the
candle and the stranger began to read:
The last will and testament of Abram Hazlitt of Chicago, Cook
county, Illinois.
First: I direct all my just debts and funeral expenses to be paid.
Second: I give, devise, and bequeath to——
"Is your daughter's name Lucy, and is the sum you wish given her five
thousand dollars exact?" asked the stranger, sitting down at the small
table near by and taking out a pen from his pocket.
"Yes," was the feeble response, "five thousand dollars to Lucy Ellen, my
only and much-beloved child."
The stranger rapidly wrote in the words, adding, "she lives in Chicago,
I suppose."
It was the nurse who answered:
"She is in this hospital, too, sir; but not for any mortal complaint.
Time and care will restore her."
The stranger went on reading:
I give, devise, and bequeath to my only and much-loved child, Lucy
Ellen of Chicago, Cook county, Illinois, the sum of five thousand
dollars.
Second: I give, devise, and bequeath to——
"Did you say the name was Mary Earle, and that she lived in Hamilton, ——
county, Massachusetts?" he interjected, looking inquiringly at the man
whose sagacity he thus trusted.
"Yes, yes," was the hurried, almost faint answer. "You know, you know;
go on quickly, for I'm feeling very weak."
They gave him stimulants, while the stranger rapidly wrote in certain
words, which he as rapidly read in what one listener thought to be a
much relieved tone.
I give, devise, and bequeath to Mary Earle of Hamilton, —— county,
Massachusetts, all my remaining property as found in the package of
banknotes deposited in the safe deposit vaults of this city, in
payment of an old debt to her father, and as an expression of my
regret that my hitherto destitute circumstances have prevented me
from sooner recognizing her claims upon me.
Third: I appoint Dr. Cusack of the Chicago General Hospital sole
executor of this, my last will and testament.
Witness my hand this thirteenth day of April in the year eighteen
hundred and ninety-two.
Signed, published, and declared
by the testator to be
his last will and testament, in
our presence, who at his request
and in his presence and
in the presence of each other
have subscribed our names
hereto as witnesses on this
thirteenth day of April, 1892.
"Does this paper express your wishes and all your wishes?" asked the
stranger pausing. "Is there any change you would like made or is the
will as it stands right?"
"Right! right!" came in more feeble tones from the fast sinking
sufferer.
"Then if you will call in another witness, I will submit the paper to
him to sign," said the stranger turning toward the doctor. "As executor
you cannot act as witness."
The doctor turned to the nurse and a momentary consultation passed
between them. Then she quietly withdrew, and in a few minutes returned
with a man who from his appearance evidently occupied some such position
as watchman. The sick man was raised higher in his bed and a pen put in
his hand.
"Mr. Hazlitt is about to sign his will," explained the stranger; and
turning to the sick man, he put the formal question: "Is this paper
which I here place before you, your last will and testament? And do you
accept these two persons now before you as witnesses to your signing of
the same?"
A feeble assent followed both these questions, whereupon the stranger
put his finger on the place where the dying man was expected to write
his name. As he did so a strange sensation seemed to affect every one
present, for the men with an involuntary movement all raised their eyes
to the ceiling upon which the stooping form of the stranger made such a
weird shadow, while the nurse gave evident signs of momentary
perturbation, which she as a woman of many experiences would doubtless
have found it hard to explain even to herself.
A short silence followed, which was presently broken by the scratching
of a pen. The patient was writing his name, but how slowly! He seemed to
be minutes in doing it. Suddenly he fell back, a smile of perfect peace
lighting up his shrunken features.
"Lucy's future is assured," he murmured, and lost or seemed to lose all
connection with the scene in which he had just played such an important
part.
A deep sigh answered him. Whose? It had the sound of relief in it, a
great soul-satisfying relief. Had the stranger uttered it? It would seem
so, but his manner was too professional to be the cloak of so much
emotion, or so it seemed to all eyes but one.
The witnesses' signatures were soon in place, and the stranger rose to
go. As he did so his eyes flashed suddenly over his shoulder and rested
for an instant on the man who occupied the neighboring cot. The movement
was so quick that No. Thirteen had scarcely time to close his eyes
undetected. Indeed, some glint of the half-hidden eyeball must have met
the stranger's eye, for he turned quickly and bent over the seemingly
unconscious man with a gaze of such intentness that it took all the
strength of what had once been called a most obstinate will for the man
thus surveyed not to respond to it.
Suddenly the stranger thrust his hand out and laid it on the unknown
sufferer's heart, and a slight smile crossed his features.
"Is there anything I can do for you?" were the words he dropped, cold
and stinging, into the apparently deaf ear.
But the man's will was indomitable and an icy silence was the sole
answer which the intruder received.
"I have still a thousand to give away," was whispered so close into his
face that he felt the hot breath that conveyed it.
But even these words fell, or seemed to fall, upon ears of stony
deafness, and the stranger rising, moved quietly away, saying as he did
so, "This case here is on the mend. His heart has a very normal beat."
Some few more words were said, and he and his companion were left alone
again with the nurse.
At three o'clock No. Twelve called feebly for some water; as the nurse
returned from giving it to him she felt her dress pulled slightly by a
feeble hand. Turning to No. Thirteen she was astonished to see that his
eyes were burning with quite an eager light.
"I could drink some broth," said he.
"Why, you are better!" she cried.
But he shook his head. "No," said he, "but—" The voice trailed off into
a feeble murmur, but the eye continued bright. He was afraid to speak
for fear his lips would frame aloud the words that he had been repeating
to himself for the last two hours. "Mary Earle! Mary Earle, of Hamilton,
—— county, Massachusetts."
He had found the interest which had been lacking to his recovery.
------------------------------------------------------------------------
PART II.
THE MAN WITH THE DOG.
-------
II.
HADLEY'S CAVE.
ON the first day of June, 1892, there could be seen on the highway near
the small village of Hamilton, a dusty wanderer with a long beard and
rough, unkempt hair. From the silver streaks in the latter, and from his
general appearance and feeble walk, he had already passed the virile
point of life and had entered upon, or was about to enter upon, the
stage of decrepitude. And yet the eyes which burned beneath the gray and
shaggy brows were strangely bright, and had an alertness of expression
which contradicted the weary bend of the head and the slow dragging of
the rough-shod feet.
His dress was that of a farm laborer, and from the smallness of the
bundle which he carried on a stick over his shoulder, he had evidently
been out of work for some time and was as poor as he was old and
helpless.
At the junction of the two roads leading to Leadington and Wells, he
stopped and drew a long breath. Then he sat down on a huge stone in the
cross of the roads and, drooping his head, gazed long and earnestly at
the length of dusty road which separated him from the cluster of
steeples and house roofs before him. Was he dreaming or planning, or was
he merely weary? A sound at his side startled him. Turning his head, he
saw a dog. It was a very lean one, and its attitude as it stood gazing
into his face with wistful eyes, was one of entreaty.
"Come!" it seemed to say, and ran off a few steps. The tramp, for we can
call him nothing else, though there was a dash of something like
refinement in his look and manner, stared for a moment after the animal,
then he slowly rose. But he did not follow the dog. The disappointment
of the latter was evident. Coming back to the man, he sniffed and pulled
at his clothes, and cast such beseeching looks upward out of his all but
human eyes that the man though naturally surly was touched at last and
turned in the direction indicated by the dog.
"After all, why not?" he murmured, and strolled on after his now
delighted guide, up one of the roads to a meadow terminating in an
abrupt and rocky steep.
"Why am I such a fool?" he asked himself when half way across this
stubbly field. But at the short bark of the dog and the irresistible
wagging of the animal's tail, he stumbled on, influenced no doubt by
some superstitious feeling which bade him regard the summons of this
unusually sagacious beast as an omen he dared not disregard. At the foot
of the rocks he, however, paused. Why should he climb them at the
bidding of a dog? But his guide was imperative, and pulled at his
trousers so energetically that he finally mounted a short distance, when
to his surprise he came upon a cave into the entrance of which the dog
plunged with a short sharp cry of pleasure and satisfaction.
Hesitating to follow, the man stood for a moment gazing back upon the
town and the stretch of lovely landscape before him. It was an outlook
of great charm, but I doubt if he noticed its beauties. Some thought of
an unpleasant and perplexing nature furrowed his brow, and it was with a
start that he turned, when the dog, reissuing from the cave, renewed his
blandishments, and by dint of bark and whine attempted to draw him into
the opening before which he stood.
What was in hiding there? Curiosity bade him look, but a certain not
unreasonable apprehension deterred him. He finally, however, overcame
his fear, if fear it was, and followed the dog, that no sooner saw him
start toward the entrance than he gave a leap of delight and bounded
into the cave before him. In another moment the man had entered also and
was looking around for the helpless or wounded human being whom he
evidently expected to find.
But no such sight met his eyes. On the contrary, he saw nothing but an
empty cave with here and there a sign of the place having been used as a
domicile at a recent date. In one corner was a litter of boughs from
which the covering had manifestly been roughly torn, and in the ledges
overhead were to be seen spikes of wood, upon which utensils had
doubtless been hung, for amid the _débris_ of broken rock beneath lay an
old tin pan with the handle broken off.
As there was nothing in this to interest the man he turned and kicked at
the inoffensive beast who had lured him out of his path on such a
fruitless errand. But the latter instead of resenting this harshness
only renewed his previous antics, and finally succeeding by them in
re-attracting the man's attention, led the way to a remote corner of the
cave, where the shadows were thickest. Here he stood with his paws
raised against the rocky sides, looking up over his head and then back
at the man in a way which left no doubt as to his meaning.
He wanted the man to climb, and when the man approaching saw the few
rocky steps that had been hewn out of the wall, his curiosity was
renewed and he lent himself to the effort, old as he was and tired with
many a long hour of tramping in the summer sun.
Above him he perceived a dark hole, and into this he presently thrust
his head, but the darkness which he encountered was so impenetrable that
he would have instantly retreated had he not remembered the box of
matches which kept guard with an old pipe in a certain pocket of his red
flannel shirt. Taking out this box, he struck a match and, as soon as
the first dazzling flash was over, perceived that he was in a small but
well furnished room, stocked with provisions and containing many
articles of domestic use. This so surprised him that he withdrew in some
haste, though he would dearly have liked to have made some investigation
into the old chest of drawers he saw there, and had one peep at least
into the odd, long box which took up so much of the darkened space into
which he had intruded.
The dog was waiting for him below and at his reappearance leaped and
bounded with delight, and then lay down on the floor of the cave with
such an inviting wriggle of the tail that the man understood him at
last. It was a lodging that the dog offered him, a lodging which had
been occupied by a former master, and which the faithful creature still
watched over and hungered in, as his appearance amply showed. The man,
to whom a human being might have appealed in vain, was grimly touched by
this benevolent action on the part of a dog, and stooping quickly, he
gave him a short caress, after which he rose and stood hesitating for a
moment, casting short glances behind him.
But the temptation, if it was such, to remain, did not hold him long,
for presently he motioned to the dog to follow him, and issuing from the
cave, began his weary tramp toward the town. The dog, with fallen tail
and drooping head, trotted slowly after him. And this was the first
adventure which met this man in the little town of Hamilton.
------------------------------------------------------------------------
III.
THE YOUNG HEIRESS.
THAT night five men sat on the porch of the one tavern in Hamilton. Of
these, one was the landlord, a spare, caustic New Englander who
understood his business and left it to his wife to do the agreeable. Of
the remaining four, two were the inevitable loungers to be found around
all such places at nightfall, and the other two, wayfarers who had taken
up lodgings for the night. The dog lying contentedly at the feet of one
of these latter tells us who he was.
The talk was on local subjects and included more or less gossip. Who had
started it? No one knew; but the least interested person in the group
was apparently the man with the dog. He sat and smoked, because it was
the hour for sitting and smoking, but he neither talked nor
listened,—that is, to all appearance—and when he laughed, as he
occasionally did, it was more at some unexpected antic on the part of
the dog than at anything which was said in his hearing. But he was old
and nobody wondered.
The last subject under discussion was the engagement of a certain young
lady to a New York medical student. "Which means, I take it, that Dr.
Izard will not continue to have full swing here," observed one of the
stragglers. "Folks say as how her people won't hear of her leaving home.
So he'll have to come to Hamilton."
"I sha'n't lend him my old body to experiment on, if he does," spoke up
the surly landlord. "Dr. Izard is good enough for me."
"And for me. But the women folks want a change, they say. The doctor is
so everlasting queer; and then he's away so much."
"That's because he is so skilful that even the big bugs in Boston and
New York too, I hear, want his opinion on their cases. He's not to blame
for that. Great honor, I say, not only to him but to all the town."
"Great honor, no doubt, but mighty inconvenient. Why, when my wife's
sister was took the other night I run all the way from my house to the
doctor's only to find the door closed and that everlasting placard up at
the side: 'Gone out of town.' I say it's a shame, I do, and no other
doctor to be found within five miles."
"You ought to live in Boston. There they have doctors enough."
"Yet they send for ours."
"Do you know," another voice spoke up, "that I had rather go sick till
morning, or have one of my folk's sick, than take that road up by the
churchyard after ten o'clock at night. I think it's the gloomiest, most
God-forsaken spot I ever struck in all my life. To think of a doctor
living next door to a graveyard. It's a trifle too suggestive, I say."
"I wouldn't care about that if he wasn't so like a graveyard himself. I
declare his look is like a hollow vault. If he wasn't so smart I'd 'a'
sent for the Wells doctor long ago. I hate long white faces, myself, no
matter how handsome they are, and when he touches me with that slender
cold hand of his, the shivers go all over me so that he thinks I am
struck with a chill. And so I am, but not with a natural one, I vow. If
we lived in the olden times and such a man dared come around the
death-beds of honest people such as live in this town, he'd have been
burnt as a wizard."
"Come, I won't hear such talk about a neighbor, let alone a man who has
more than once saved the lives of all of us. He's queer; but who isn't
queer? He lives alone, and cooks and sleeps and doctors all in one room,
like the miser he undoubtedly is, and won't have anything to do with
chick or child or man or woman who is not sick, unless you except the
village's _protégée_, Polly Earle, whom everybody notices and does for.
But all this does not make him wicked or dangerous or uncanny even. That
is, to those who used to know him when he was young."
"And did you?"
"Wa'al, I guess I did, and a handsomer man never walked Boston streets,
let alone the lanes of this poor village. They used to say in those days
that he thought of marrying, but he changed his mind for some reason,
and afterward grew into the kind of man you see. Good cause, I've no
doubt, for it. Men like him don't shut themselves up in a cage for
nothing."
"But——"
"Don't let us talk any more about the doctor," cried the lodger who did
not have a dog. "You spoke of a little girl whom everybody does for. Why
is that? The topic ought to be interesting."
The landlord, who had talked more than his wont, frowned and filled his
pipe, which had gone out. "Ask them fellers," he growled; "or get my
wife into a corner and ask her. She likes to spin long stories; I
don't."
"Oh, I don't care about asking anybody," mumbled the stranger, who was a
sallow-faced drummer with a weak eye and a sensual mouth. "I only
thought——"
"She isn't for any such as you, if that's what you mean," volunteered
the straggler, taking up the burden of the talk. "She has been looked
after by the village because her case was a hard one. She was an only
child, and when she was but four her mother died, after a long and
curious illness which no one understood, and three days after, her
father—" The dog yelped. As no one was near him but his master, he must
have been hurt by that master, but how, it was impossible to understand,
for neither had appeared to move.
"Well, well," cried the sallow young man, "her father——"
"Disappeared. He was last seen at his wife's funeral; the next day he
was not to be found anywhere. That was fourteen years ago, and we know
no more now than then what became of him."
"And the child?"
"Was left without a soul to look after it. But the whole village has
taken her in charge and she has never suffered. She has even been
educated,—some say by Dr. Izard, but for this I won't vouch, for he is a
perfect miser in his way of living, and I don't think he would trouble
himself to help anybody, even a poor motherless child."
"Well, if he has spent a penny for her in the past, I don't think he
will be called upon to spend any in the future. I heard yesterday that
she has come into a pretty property, and that, too, in a very suspicious
way."
"What's that? You have? Why didn't you tell us so before? When a man has
news, I say he ought to impart it, and that without any ifs and ands."
"Well, I thought it would keep," drawled the speaker, drawing back with
an air of importance as all the _habitués_ of the place pressed upon
him, and even Mrs. Husted, the landlady, stepped out of her sitting-room
to listen.
"Wa'al, it won't," snarled the landlord. "News, like baked potatoes,
must be eaten hot. Where did you hear this about Polly Earle, and what
do you mean by suspicious?"
"I mean that this money, and they do say it's a pretty sum, came to her
by will, and that the man who left it was a perfect stranger to her,
someone she never heard of before, of that I'll be bound. He said in his
will that he left all this money in payment of an old debt to her
father, but that's all bosh. Ephraim Earle got all the money that was
owing to him two weeks before he vanished out of this town, and I say——"
"No matter what you say," broke in the crabbed landlord. "She's had
money left her, and now she'll get a good husband, and make a show in
the village. I'm glad on it, for one. She's sung and danced and made
merry on nothing long enough. Let her try a little responsibility now,
and return some of the favors she has received."
"Did you hear how much money it was?" timidly asked an old man who had
just joined the group.
"It was just the same amount as was paid Ephraim Earle for his invention
a few days before we saw the last of him."
"Lord-a-mercy!"
"And which——"
"Now this is too interesting for anything," exclaimed a female voice
from a window overhead. "Twenty thousand dollars, really? What a
romance. I must run and see Polly this minute."
"Stop her!" came in guttural command from the landlord to his wife.
"And why should I stop her?" asked that good woman, with a jolly roll of
her head. "Instead of stopping her, I think I will go with her. But do
let us hear more about it first. What was the name of the man who left
her this splendid fortune?"
"Abram Hazlitt. Somebody who lived out west."
From the looks that flew from one to the other and from the doubtful
shakes of the head visible on every side, this was, as the speaker had
declared, an utterly unknown name. The interest became intense.
"I always thought there was something wrong about Ephraim's
disappearance. No man as good as he would have left a child like that of
his own free will."
"What! do you think this man Hazlitt had anything to do——"
"Hush, hush."
The monition came from more than one pair of lips; and even the man with
the dog looked up. A young lady was coming down the street.
"There she is now."
"She's coming here."
"No; more likely she's on her way to tell the doctor of her good luck."
"Look, she has the same old smile."
"And the same dress."
"Wa'al she's pretty, anyhow."
"And such a sunbeam!"
Yelp! went the dog again. His master had trod on his tail for the second
time. Meanwhile the cause of all this excitement had reached the walk in
front of the house. Though she was tripping along in a merry fashion
which was all her own, she stopped as she met Mrs. Husted's eye, and,
calling her down, whispered something in her ear. Then with a backward
nod the young girl passed on, and everyone drew a long breath. There was
something so satisfactory to them all in her ingenuous manner and simple
expression of youthful delight.
She was a slight girl, and to those who had seen her every day for the
last dozen years she was simply prettier than usual, but to the two or
three strangers observing her she was a vision of madcap beauty that for
the moment made every other woman previously seen forgotten. Her face,
which was heart-shaped and fresh as a newly-opened rose, was flushed
with laughter, and the dimples which came and went with every breath so
distracted the eye that it was not till she had turned her lovely
countenance aside that one remembered the violet hues in her
heavily-lashed eyes and the hints of feeling which emanated from them.
That, with all the dignities of her new-born heirship upon her, she
swung a white sunbonnet on her delicate forefinger was characteristic of
the girl. The hair thus revealed to sight was of a glistening chestnut,
whose somewhat rumpled curls were deliciously in keeping with the saucy
poise of the unquiet head. Altogether a decided gleam of sunshine, made
all the more conspicuously bright from the hints just given of the
tragic history of her parents and the shadows surrounding the very gift
which had called up all this pleasure into her face.
"What did she say?" whispered more than one voice as the landlady came
slowly back.
"She invited me to visit her, and hinted that she had something to tell
me," was the somewhat important reply.
"And when are you going?" asked one more eager than the rest.
"I may go back with her when she returns from Dr. Izard's," was the cool
and consequential response. Evidently the landlady had been raised in
her own estimation by the notice given her by this former little waif.
"I wonder," someone now ventured, "if she is going to buy the big house
over the doctor's office. I noticed that the windows were open to-day."
"Pshaw, and her father's house lying idle?"
"Her father's house! Good gracious, would you have the child go there?"
"You make the chills run over me."
"Nobody would go into that house with her. It hasn't been opened in
fourteen years."
"The more shame," growled the landlord.
"She'll never have anything to do with that. I've seen her run by it
myself, as if the very shadow it cast was terrifying to her."
"Yet folks thought it was a cozy home when Ephraim took his young wife
there. I remember, myself, the brass andirons in the parlor and the long
row of books in the big hall upstairs. To think that those books have
never been opened these fourteen years, nor the floors trod on, nor the
curtains drawn back! I declare, it's the most creepy thing of the whole
affair."
"And how do you know that the floor hasn't been walked on, nor the
curtains drawn, since we took the child out from her desolate corner in
the old bed-room upstairs?" suggested another voice in an odd,
mysterious tone.
"Because the doors were locked and the keys put where no one in the town
could get at 'em. We thought it best; there was death on the walls
everywhere, and the child had no money to be brought up in any such a
grand way as that."
"Folks as I mean don't need keys," murmured the other under his breath.
But the suggestion, if it were such, was immediately laughed down.
"You're a fool, Jacob; we're in the nineteenth century now, the era of
electric lights and trolley cars."
"I know; I know; but I've seen more than once on a dark night the
shifting of a light behind those drawn curtains, and once——"
But the laughter was against him and he desisted, and another man spoke
up—the lodger with the sallow face: "Why didn't they sell the old place
if the child was left as poor as you say?"
"Why, man, its owner might be living. Ephraim Earle only disappeared,
you know, and might have returned any day. Leastwise that is what we
thought then. Now, we no longer expect it. I wonder who'll act as her
guardian."
"She's of age; she don't need no guardian."
"Well, it's a precious mystery, the whole thing. I wonder if the police
won't see something in it?"
"Bah, police! They had the chance at the thing fourteen years ago. And
what did they do with it? Nothing."
"But now there's a clue. This man Hazlitt knew what became of Ephraim
Earle, or why did he leave that very same amount to his daughter?"
"Lor' knows. She's a taking minx and perhaps——"
"Well, perhaps——"
"Hazlitt wasn't his name, don't you see?"
This new theory started fresh talk and much excited reasoning, but as it
was of the most ignorant sort, it is scarcely worth our while to record
it. Meanwhile the twilight gave way to darkness and Polly Earle failed
to reappear. When it was quite dark, the stragglers separated, and then
it was seen that the man with the dog had fallen asleep in his chair.
Someone strove to wake him.
"Come, come, friend," said he; "you'll be getting the rheumatiz if you
don't look out. This isn't the right kind of air to sleep in."
The old wayfarer yawned, opened his strange, uneasy eyes, and hobbling
to his feet looked lazily up and down the street.
"What time is it?" he asked.
"Nine o'clock," shouted someone.
"Give me a drink, then, and I and my dog will take a walk." And he drew
out a worn wallet, from which he drew a dime, which he handed in through
the open window to the now busy landlord.
"Hot," he croaked, "I've got chilly sitting out here in the dew."
The glass was handed him, and he drank it off with the ease of an
accustomed hand.
"I'll be back before you lock up," said he, and stepped down into the
street, followed by the dog.
"Seems to me I've seen that dog before," remarked someone.
"Why, don't you know him? That's old Piper, the dead hermit's dog. I
wonder how this fellow got hold of him."
------------------------------------------------------------------------
IV.
DR. IZARD.
THE tramp, who was, as you have seen, not without some small means to
make himself respected, paused for a moment in front of the tavern
before deciding what direction he would take. Then he went east, or, to
make matters clearer to my reader, followed the direction young Polly
Earle had taken an hour or so before.
Being bent and old he walked slowly, but as the tavern from which he had
emerged was near the end of the street, it was not long before he came
upon the big church at the corner, beyond which was the open country and
circling highroad.
"They spoke of a graveyard," murmured he, pausing and gazing about him
with eyes which seemed to have lost none of their penetration, however
bent his figure or aged his face. "Ah! I think I see it!" And he rambled
on in the darkness till he came to a picket fence. But this fence
enclosed a dwelling-house, whose large and imposing bulk rose in deepest
shadow beyond him, and he had to walk several rods farther before he
came to the spot of glimmering headstones and drooping willows. A faint
moon lent a ghostly light to the place, and as he stopped and bent his
head over the intervening wall, weird glimpses were given him of snowy
shafts and rounded hillocks, which may have accounted for the length of
time he clung there without movement or sound.
But finally the dog whining at his heels, or the gleam of a light
shining in the distance, recalled him to himself, and he moved, taking
the direction of that light, though it led him over the cemetery wall
and across such of the graves as lay along the border of the yard
adjoining the large house of which I have previously spoken. The dog,
who had not left him a moment since he joined him at the cave, shrank as
he climbed the wall, and the old man took his course alone, treading as
softly as he could, but yet making some noise as a broken twig snapped
under his foot or he pressed down some tiny aspiring bush in his rude
advance.
He was making for the light which shone from the window near the ground
in the huge side of the great and otherwise unilluminated house he had
passed a few minutes before. He had expected to be met by a fence like
the one in front, but to his surprise he soon saw that the graveyard
pressed close up to the house, and that there was a monument not ten
yards from the very window he was approaching. He had paused at this
monument, and was vainly trying to read the inscription which was cut
deeply into the side turned toward the moon, when he heard a sudden
sound, and, looking toward the house, saw that a door had opened in the
blank side of the wall, and that the light had shifted from the window
to this open square, where it was held high above the head of a
remarkable looking man who was looking directly his way.
Convinced that this was Dr. Izard, he held his breath, and slunk as much
into the shadow of the shaft as possible. Meanwhile he stared at the
picture presented to his notice, and noted every outline of the noble
head and small but finely proportioned form, that filled the illuminated
gap before him. The face he could not see, but the attitude was
eloquent, and conveyed so vividly an expression of strained listening
and agitated doubt, that this by no means careless observer felt that
his step had been heard, and that something more than common curiosity
had drawn the doctor to the spot.
A sudden sense of his position among the graves, or the chill imparted
by his close contact with the stone shaft against which he had flung
himself, made the aged wanderer shiver, but his emotion, however
occasioned, did not last long, for with a sigh that could be plainly
heard across the short space, Dr. Izard withdrew his head and closed the
door, leaving nothing to be seen in the dim blackness of the houseside
but the one square of light which had previously attracted the
stranger's attention.
With careful step and bated breath, the latter left the tomb by which he
had sought refuge, and advanced to this same wall, along which he crept
till he reached this uncurtained window. A glimpse of the interior was
what he wanted, but, as he stopped to listen, he found that he was
likely to obtain more than this, for plainly to be heard in the almost
death-like quiet, came the sound of two voices conversing, and he knew,
perhaps by instinct, perhaps by ready reasoning, that they were the
voices of the doctor and the pretty new heiress, Polly Earle.
To listen might have been a temptation to any man, but to this one it
was almost a necessity. His first desire, however, was to see what was
before him, and so, with more skill than one would expect, he bent a
branch of the vine swaying about him, and, from behind its cover, peered
into the shining panes that opened so invitingly beside him.
The first thing he saw was the room with its shelves upon shelves of
books, piled high to the ceiling. As it answered the triple purpose of
doctor's office, student's study, and a misanthrope's cell, it naturally
presented an anomalous appearance, which was anything but attractive at
first sight. Afterward, certain details stood out, and it became
apparent that those curious dangling things which disfigured the upper
portion of the room belonged entirely to the medical side of the
occupant's calling, while the mixture of articles on the walls, some
beautiful, but many of them grotesque if not repellant, bespoke the man
of taste whose nature has been warped by solitude. A large door painted
green filled up a considerable space of the wall on the left, but
judging from the two heavy bars padlocked across it, it no longer served
as a means of communication with the other parts of the house. On the
contrary it had been fitted from top to bottom with shelves, upon which
were ranged a doctor's usual collection of phials, boxes, and surgical
appliances, with here and there a Chinese image or an Indian god. A rude
settle showed where he slept at night, and on the table in the middle of
the room, a most incongruous litter of books, trinkets, medicines,
clothing, sewing materials, and chemical apparatus proclaimed the fact,
well known in the village, that no woman ever set foot in the place,
save such as came for medical advice or on some such errand as had drawn
hither the pretty Polly.
At the table and in full view of the peering intruder sat the genius of
the place, Dr. Izard. His back was to the window and he was looking up
at Polly, who stood near, twirling as usual her sunbonnet round her
dainty forefinger. It was his profile, therefore, which the curious
wayfarer saw, but this profile was so fine and yet so characteristic
that it immediately imprinted itself upon the memory like a silhouette
and the observer felt that he had known it always. Yet it was not till
one had been acquainted with the doctor long that all the traits of his
extraordinary countenance became apparent. Its intelligence, its
sadness, its reserve and the beauty which gave to all these qualities a
strange charm which was rather awe-inspiring than pleasurable, struck
the mind at once, but it was not till after months of intercourse that
one saw that the spell he invariably created about him was not due to
these obvious qualities but to something more subtle and enigmatic,
something which flashed out in his face at odd times or fell from his
voice under the strain of some unusual emotion, which while it neither
satisfied the eye nor the ear, created such a halo of individuality
about the man that dread became terror or admiration became worship
according to the mental bias of the person observant of him.
In age he was nearer fifty than forty, and in color dark rather than
light. But no one ever spoke of him as young or old, light or dark. He
was simply Dr. Izard, the pride and the dread of the village, the
central point of its intellectual life, on whose eccentricities judgment
was suspended because through him fame had come to the village and its
humble name been carried far and wide.
Polly, who feared nobody, but who had for this man, as her rather
unwilling benefactor, a wholesome respect, was looking down when the
stranger first saw her. The smile which was never long absent from her
lips lingered yet in the depths of the dimple that was turned toward the
doctor, but the rest of her face showed emotion and a hint of
seriousness which was by no means unbecoming to her poetic features.
"You are very good," she was saying. "I have often wondered why you were
so good to such a little flyaway as I am. But I shall surely remember
all you have said and follow your advice as nearly as possible."
There was unexpected coldness in the doctor's reply:
"I have advised nothing but what any friend of yours must subscribe to.
The woman with whom you are staying is a good woman, but the home she
can give you is no longer suitable for a girl who has come, as you say
you have, into possession of considerable property. You must find
another; and since the house over our heads is a good one, I have
ventured to offer it to you for a sum which your man of business
certainly will not regard as high, considering its advantages of size
and location."
"By location do you mean its close proximity to the graveyard?" she
inquired, with a _naïve_ inclination of her coquettish head. "I should
say, myself, though I never fear anything, that its location is against
it."
His eye, which had wandered from hers, came back with a stern
intentness.
"Since I have lived here for twenty years with no other outlook than the
graves you see, I cannot be said to be a good judge of the matter. To me
the spot has become a necessity, and if you should make the arrangement
I suggest, it must be with the understanding that this room is to be
reserved for my use as long as I live, for I could never draw a free
breath elsewhere."
"Nor would anyone wish you to," said she. "This solitary room, with its
dangling skulls and queer old images, its secrecy and darkness, and the
graves pressing up almost to your window, seems a part of Dr. Izard. I
could not imagine you in a trim office with a gig at the door and a man
to drive it. No, it would rob us of half our faith in you, to see you
enjoying life like other folks. You must stay here if only because my
mother, lying over there in her solitary grave, would be lonely were
your face to fail to appear every night and morning in your open
doorway."
Her hand, which had paused in its restless action, pointed over her
shoulder to the silent yard without. The physician's eye followed it,
and the words of reproof died upon his tongue.
"You think me frivolous," she cried. "Well, so I am, at times. But _you_
make me think; and if this sudden accession to fortune fills me with
excitement and delight, the sight of you sitting here, and the nearness
of my mother's tomb, gives me some sober thoughts too, and—and—Dr.
Izard, will you tell me one thing? Why do people stare when they hear
the exact amount of the money left me? It is not because it is so large;
for some say it is anything but a large fortune. Is it—" she hesitated a
little, probably because it was always hard to talk to Dr. Izard—"for
the reason that it is so near the sum my father was said to have carried
away with him, when he left me so suddenly?"
The wind was fluttering the vines, and the doctor turned his head to
look that way. When he glanced back he answered quietly, but with no
irritation in his voice:
"It is hard to tell what causes the stare of ignorant people. What was
the amount which has been left you? I do not think you have mentioned
the exact figure."
"Twenty thousand dollars," she whispered. "Isn't it splendid,—a lordly
fortune, for such a poor girl as I am?"
"Yes," he acquiesced, "yes." But he seemed struck just as others had
been who heard it.
"And was not that just what was paid papa by the French government just
before mamma died?"
"I have heard it so said," was the short reply.
"And don't you know?" she asked.
The pout on her lips bespoke the spoiled child, but her little hands
were trembling, and he seemed to see only that.
"Polly,"—he spoke harshly, for he did not like young girls, or women at
all for that matter,—"I knew many things which I have let slip from my
memory. When your father and I were young we were more or less intimate,
being both of us students and ambitious of doing something worth while
in this world. But after his disappearance and the unfortunate surmises
to which it gave rise, I made a business of forgetting any confidential
communications with which he may have entrusted me, and I advise you not
to stir up old griefs by driving me to recall them now."
"But you were my mother's physician and saw my father just before he
went away."
"Yes."
"And did he have twenty thousand dollars in money? They say so, but it
seems incredible to me, who only remember my father as looking worried
and poor."
"Twenty thousand dollars was paid him two weeks before your mother
died."
"And he carried all that away with him and never left a dollar to his
little motherless child? Oh, I know that some people say he was foully
dealt with and that it was not of his own free will that he left me to
the mercies of the town. But I never believed that. I have always
thought of him as alive, and many is the night I have waked up
crying—Oh, I can cry at night and in the darkness, if I do laugh all day
when the sun shines—because I dreamt he was enjoying himself in foreign
lands while I—" she stopped, looking inquiringly at Dr. Izard, and he,
startled, looked inquiringly at her, then for the second time he rose
up, and taking the light, went out to search up and down the ghostly
waste before him, for what he rather felt than knew was near.
"Oh, how late it is getting!" cried the little maiden, peering over his
shoulder. "Did you think you heard someone sigh? I thought I did, but
who would come creeping up to this spot? Do you know," she exclaimed,
drawing him in just as he was about to turn his attention to the side of
the house against which they stood, "that I believe it's that horrid
green door which gives people the shivers when they come here. Why is it
there and what is on the other side of it that you bar it up like that?"
The doctor, lifting his abstracted gaze, stared at the door for a
moment, then turned moodily away. "It was the old way of going
upstairs," he remarked. "Why shouldn't I bar it, since I have no further
use for the rest of the house?"
"But its color," she persisted; "why do you not paint it white?"
"When I fit up my den for a bride, then I will," he retorted, and the
audacious little thing became dumb on this subject, though she showed no
inclination for dropping the other.
"Dear Dr. Izard," she pursued, "I know I ought to be going home, but I
have something more to ask, and it isn't always that you allow me to
speak to you. Our house—you know what I mean, my father's and mother's
house,—is it really haunted, and is that why it is shut up, even from
me?"
"Do you want to go into it, Polly?"
"No—and yet I have sometimes thought I should like to. It must be full
of relics of my parents, and if it has not been disturbed since my
father went away, why, I might almost see the prints of his feet on the
floors, and the pressure of his form in the old lounges and chairs."
"You are too imaginative!" cried the doctor. "They will have to marry
you to some practical man."
She flushed, drew back and seemed on the point of uttering some violent
protest or indignant reproach, but instead of that she returned to the
original topic.
"I should like to hear from your lips, which never exaggerate or add the
least bit of romance to anything you say, just the story of my father's
departure and that sudden shutting up of the house. I think I ought to
know now that I am a grown woman and have money of my own."
"Will you go, after I have told you all that there is to know?" he
asked, with just a touch of impatience in his naturally severe tone.
"Yes," she laughed, irresistibly moved by his appearance of ill-nature.
"I won't stay one minute longer than you wish me to. Only," she added,
with the sobriety more in accordance with the theme they were
discussing, "do make the whole thing clear to me. I have heard so many
stories and all of them so queer."
He frowned, and his face underwent an indescribable change.
"You are a silly slip of a girl and I have a mind to turn you out of the
house at once. But," and his eyes wandered away to his books, "your
curiosity is legitimate and shall be satisfied. Only not here," he
suddenly cried, "I will tell you as we walk toward your home."
"Or in the graveyard outside," she murmured. "I am not afraid of the
place with you near me. Indeed, I think I should like to hear my
mother's story, standing by her tomb."
"_You would!_" The doctor, astonished, agitated almost, by this untoward
sentiment uttered by lips he had only seen parted in laughter, rose, and
leaning on the table looked over it at her, with eyes whose effect only
was visible to the straining pair without. "Well, you shall have your
wish. I will tell you her story, that is, as much as I know of it,
standing by her grave without." And with a grim smile, he took up his
hat and stepped quickly before her toward the door. She followed him,
with an eager gesture, and in a minute their two shadows could be dimly
seen in the moonlight falling over the face of that very shaft behind
which the stranger had taken refuge an hour or so before. The vines that
swayed about the window ceased their restless rustling and seemed to
cling with heavier shadow than usual to the dismal wall.
"Your father," said the doctor, "was a man of one idea, but that idea
was a valuable one and it paid its projector well. The invention which
he conceived, perfected, and made practical, was an important one,
suited to large governmental undertakings and meeting the wants of
France especially. It was bought, as I have said, from your father for
the sum of twenty thousand dollars. But this good fortune, while
deserved, had not come early, and your mother, who had been overburdened
in her youth, was on her deathbed when the favorable news came. It
comforted her, but it almost maddened your father, if I may judge from
the frenzied expressions he used in my hearing. He did not touch the
money, and when she died he locked himself up in a room, from which he
only emerged to attend her funeral. This I tell you that you may see
that his paternal instinct was not as great as his conjugal one, or he
would not have forgotten you in his grief. Did you speak?"
"No, no; but it is gloomy here, after all; let us go on into the
highway."
But the man clinging to the wall was not forced to move. The doctor did
not heed her entreaty, or if he did he ignored it, for his voice went
coldly and impassively on: "The night after your mother was buried, your
father was seen looking from one of the windows of his house. The next
morning he was missing. That is all I can tell you, Polly. No one knows
any more than that."
"But wasn't there somebody in the house besides himself? Where was I?"
"Oh, you were there, and an old woman who had been looking after you in
your mother's illness. But you were too young to realize anything, and
the woman—she has since died—had nothing to say, but that she was sure
she heard your father go out."
"And the money?"
"Went with him."
"Oh, I have heard it all before," came after a moment's silence, in
sharp and plaintive tones. "But I was in hopes you could tell me
something different, something new. Did they look for my father as I
would have done had I been old enough to understand?"
"I headed the search myself, Polly; and later the police from Boston
came down, and went through the town thoroughly. But they met with no
results."
"And now a stranger leaves me twenty thousand dollars! Dr. Izard, I
should like to know something about that stranger. He died in the
Chicago Hospital, I am told."
"I will make inquiries."
"If—if he had anything to do with my father's disappearance——"
"You will never know it; the man is dead."
A silence followed these few words, during which the agitated breathing
of the young girl could be heard. Then her quivering voice rose in the
impatient cry: "Yes, yes; but it would be such a relief to know the
truth. As it is, I am always thinking that each stranger I see coming
into town is he. Not that it makes me timid or melancholy; nothing could
do that, I think; but still I'm not quite happy, nor can this money make
me so while any doubts remain as to my father's fate."
"I cannot help you," the doctor declared. "For fourteen years you have
borne your burden, little one, and time should have taught you patience.
If I were in a position like yours I would not allow old griefs to fret
me. I should consider that a man who had been missing most of my
lifetime was either dead or so indifferent that I ran but little chance
of seeing him again. I myself do not think there is the least likelihood
of your ever doing so. Why then not be happy?"
"Well, I will," she sighed. "I'm sure it's not my nature to be
otherwise. But something either in these dismal trees, or in yourself or
in myself makes me almost gloomy to-night. I feel as if a cloud, hung
over me. Am I very foolish, doctor, and will you be taking me back to
the office to give me a dose of some bitter, black stuff to drive away
the horrors? I had rather you would give me a fatherly word. I'm so
alone in the world, for all my friends."
He may have answered this appeal by some touch or sympathetic move, but
if he did, the listener was not near enough to catch it. There was a
rustling where they stood and in another instant the bare head of the
young girl was visible again in the moonlight.
"I think I will be going home," said she, and turned towards the
gateway. The doctor followed her and together they left the cemetery and
entered the high-road. When the sound of their voices had died away in
the distance, a deep and heavy shadow separated itself from the side of
the house near the window and resolving itself again into the image of
the man through whose ears we have listened to the broken dialogue we
have endeavored to transcribe, took up its stand before the still
lighted window and for several minutes studied the peculiar interior
most diligently. Then it drew off, and sliding down the path which
followed the side of the house, emerged upon the road and took its own
course to the village.
Something which he did not see and something which he did not hear, took
place at the other end of the town before a cheerfully lighted mansion.
Dr. Izard and Polly had traversed the length of the street, and had
nearly reached the cottage in which she was at present living, when the
former felt the little hand now thrust confidingly into his arm, flutter
and shift a trifle. As the girl had regained her spirits and was now
chatting in quite a merry way upon indifferent topics, he looked up to
see what it was that had affected her, and saw nothing save the lights
of the Unwin place and a figure which must have been that of young Unwin
sitting on the shadowy veranda. As he had reasons of his own for not
liking to pass this house, he stopped and glanced at the young girl
inquiringly. She had ceased speaking and her head was hanging so low
that the curls dropped against her cheek, hiding her eyes and the
expression of her mouth.
"I think," she whispered, "if you don't mind, that I will walk on the
other side of you. It is very late for me to be out, even with you, and
Clarke——"
The doctor, drawing in his breath, turned his full face on her and stood
so long gazing into her drooping countenance that she felt frightened
and attempted to move on. Instantly he responded to her wish and they
passed the house with quick and agitated steps, but when the shadows of
the next block had absorbed them, they both paused as it were
simultaneously, and the doctor said with something more than his usual
feeling in his thin, fine voice, "Do you care for Clarke Unwin, little
one?"
Her answer struck him.
"Do I care for breath, for life? He has been both to me ever since I
could remember anything. And now he cares for me."
The doctor, lost in some overwhelming dream or thought, did not answer
her for several minutes. Then he suddenly lifted her face by its dainty
chin, and in a deep, controlled tone, totally different from the one he
had used a short time before, he solemnly remarked:
"For fourteen years I have taken an interest in you and done for you
what I have done for nobody else in the town. I hope that my care has
made a good girl of you, and that under all your fanciful ways and merry
antics there hides a true woman's heart."
"I don't know," she whispered. "I know that I would rather give up my
fortune than one little memory connected with these last three weeks."
"And he—he loves you? You are sure of it, little one?"
The lift of her head was eloquent; the doctor wished he could see her
face, but the darkness was too thick for that.
"May Heaven bless you!" faltered on his tongue; but the words were too
unusual to the ascetic's cold lips for them to pass into speech, and the
girl thought his manner more distant and unsympathetic than common.
"It is a secret I have told you," she murmured, and being then within a
few steps of her own gate, she slid from his grasp and vanished in the
darkness.
He, with a sigh that seemed to rend the icy bonds which years of
repression had bound about his breast, remained for a moment with his
head bent, gazing on the ground at his feet. Then he drew himself up,
and passed quickly back over the road he had come.
------------------------------------------------------------------------
V.
NOCTURNAL WANDERINGS.
THE wanderer, of whose name even the landlord at the tavern seemed
uncertain, passed some curious days after this. Upon the plea of wanting
work, he visited house after house in the village, staying in each one
as long as he was made welcome. Though no talker, he seemed to like to
have talk going on around him, and if he sometimes went to sleep over
it, he was forgiven by the simple and credulous inhabitants on account
of his old age and seeming decrepitude. In one house he was given
breakfast, in another dinner, but in none did he find work, though he
assured everybody that he was very good in the field, notwithstanding
the unfortunate curvature of his back.
It was not an uncommon thing in Hamilton for men to pass from house to
house in this way, and he was little noted, but if anyone had been
curious enough to watch his eye they would have observed that it had a
remarkably penetrating power, and that but little escaped its notice.
Another thing that would also have been noticed was the curious look of
recognition which would suddenly creep into his eyes, as if he saw some
of these things for the second time; and if anyone had walked near
enough to him to listen as well as watch, he would have heard a name
drop from his lips now and then as he walked up the phlox-bordered walk
of some humble garden, or stopped at the back door of one of the more
pretentious mansions on the main street.
Another thing: When he had done this, when he had uttered in his odd,
musing way, at the threshold of a house, the name of Fisher, Hutton,
Brown, Unwin, or what not, he invariably managed in some way, either
slyly or by bold question, to ascertain if this name really belonged to
the family then residing there. If it did, he nodded his head
complacently. If it did not, he frowned as if disappointed in his memory
or whatever it was that had played him false.
At one place he showed conclusively that he had been in the house
before, though no one seemed keen enough to detect the fact. He was
passing down a hall, when he turned to the right and came plumb up
against a wall. This was where there had formerly been a door of egress,
but a change which had been made some ten years back in the inner
arrangement of the house had placed it farther on, and his face showed
surprise when he noted it, though the expression was speedily
suppressed. Again at the Fishers' he was very careful to sit in the deep
shadow, and though he eagerly drank in all that was said, he himself
made no remark after his first appeal for work. The Fishers were old
neighbors of the Earles, and it was with them that Polly was living.
In the afternoon he found himself at the eastern end of the town near
the church. As he noticed the venerable building he seemed to call to
mind his experiences of the night before, for he glanced eagerly toward
the cemetery, and finally turned his steps in that direction, saying
quietly to himself, "Let's see how it looks by daylight."
The street, which takes a sharp turn at this point, was headed by the
stately house whose dim columns and embowering trees had so struck the
wanderer's attention the night before. Seen by daylight it was less
mysterious in appearance but fully as imposing, though there were signs
of neglect on its painted front and solitary balconies, which spoke of
long disuse as a dwelling. It had the name of Izard engraved on the
tarnished door-plate.
"Let me see," mused the tramp, leaning upon one of the old-fashioned
gate-posts guarding the entrance, "I should remember how the house looks
inside; I was here to a ball once when we were all young folks together.
It was a fine old dwelling then, and Mrs. Izard, who always said she
could remember Martha Washington, looked like a queen in it." Lifting
his head, he glanced up at the pillared front. "There was a large double
drawing-room on this side," he murmured, "with a big-figured carpet on
the floor and panelled paper on the walls. I think I could remember the
very tints if I tried, for I sat that night for full ten minutes staring
at it, while Lillie Unwin chattered nonsense in my ear, and—" the rest
was lost in his long, dishevelled beard, which was much too gray to be
worn by any contemporary of Dr. Izard.
"On the left," he presently proceeded, "was the library, with one or two
windows looking out upon the cemetery, which was then a respectable
distance off; and down the hall, which was wide enough to dance a
Virginia reel in, there hung a map of the Holy Land, with one corner
torn off. I wonder if it is hanging there still, and if I can remember
which corner was lacking." He mused a minute with a sour smile.
"Something must be pardoned in one who has been gone fourteen years," he
murmured. "I cannot remember whether it was the left or the right-hand
corner." Shutting his eyes, he leaned his head again on the post, while
short, broken sentences issued by fits and starts from amid his beard as
he brooded over the past.
"Under the big front staircase,—I remember it well,—there was a smaller
circular one, which went down to a certain green door: the same one I
noticed in the doctor's office, though there was no office then,—only a
rectangular porch. He must have had the office built in since I left the
town, for he used to see his patients in the library. Now, how did that
porch look? It was broad and low, and raised but a step or two above the
ground. There were two pillars in the opening toward the graveyard,
similar to the big columns in front, but smaller and set further apart.
At one end was a wooden seat built in the wood-work, and at the other a
green door, the same as that seen in the doctor's room now. Will these
details answer for one recollection? I think they will. And now for a
glimpse of that shaft."
Lifting his head from the gate-post, he picked his way through the
tangled weeds to the little gate on the highway which led directly to
the doctor's office. Entering, he approached the tombstone against which
he had leaned the night before, and heedless of passers-by, took up his
stand before it and began reading the inscription.
SACRED
TO THE MEMORY
OF
HULDAH EARLE.
Born December Third, 1854.
Died August Ninth, 1878.
"I wonder who put up this monument," he muttered, and shuddered slightly
as he recalled the chilliness of the stone against which he had pressed
his breast the night before. But the emotion was but transitory, and he
was soon surveying the small square window through whose panes the one
light had shone on the previous night. It was near the office door, and
was surrounded, as he had so gratefully experienced at that time, by a
thick-leaved trumpet-vine, whose long and swaying branches recalled to
him the anxious moment when the doctor had stepped to the door, drawn by
some sound he had made in his curiosity and interest. Just now a curtain
hung before the window, sure sign that the doctor was within; but he did
not heed this, possibly because he did not understand the signal, and
remained where he was, musing on the past, till the steps of some
advancing visitor advised him that he might better indulge his
thoughtful mood in a less conspicuous place, and in a solitude not so
likely to be invaded by curious eyes.
The dog which had joined him at his first appearance in town continued
to be his constant companion. All day this faithful animal followed him,
and when night came, they went together into the small attic chamber
which was the only room in the house he could afford to pay for. But one
journey which the man took was not shared by the dog. It took place at
midnight and in the following mysterious way:
He had noticed by a minute inspection of the roof stretching below his
one small window that by a few daring steps down the first incline one
might reach a ledge from which descent to the ground would be easy. It
was a path which might be taken with safety by a young man or a still
vigorous middle-aged man. But would it be a feasible one for him? He
seemed to decide in the affirmative, for in the small wee hours of the
night he rose from his bed, and quieting his ready dog, dressed himself,
and took another long survey from the window. Then he proceeded to open
the bundle he had brought into town, taking from it a small object,
which he hid in the breast of his coat. Then he thrust a box of matches
into the pocket of his shirt, and ignoring his hat, which hung on a nail
in one corner, he began his daring descent. Throwing one leg out of the
window and clinging to the narrow jamb, he whirled himself about, and
developing some of the instincts of the cat, soon reached the ledge in
safety. Instantly his form, which had hitherto been so bent as to
present almost the appearance of deformity, straightened itself until
his whole person betrayed an agility and precision surprising to behold
in any man past the first flush of youth.
To pass from the eaves to the shed and thence to the ground was the work
of a moment. The crooked branch of an old apple-tree which grew near the
house, was of decided use to him and enabled him to make his risky
descent with comparatively no noise. When he was on the ground, he
stopped and listened, then wheeling rapidly about, proceeded to walk up
the street.
The night was dark and threatened storm. Everywhere there was a sound of
swishing boughs and rattling panes which served to deaden the noise of
his tread on the pavement, but he seemed so anxious not to attract
attention even in the darkness and solitude of this midnight hour that
he stepped into the grass that bordered the road, and even took off his
shoes that no echo might follow his movements.
The course he took led him in an entirely different direction from any
he had traversed during the day. As soon as he reached the point where
the court house stands, he turned east and went up Carberry hill. As
there are but two or three houses on this <DW72>, his destination became
speedily apparent. On the brow of the hill where the wind blows
strongest, stands the old Earle cottage, with its windows closed to
every eye and its untrod doorstep hidden amid weeds that had choked up
the entrance for many a year. In the daylight it had an utterly lonesome
and deserted look, but at night, especially when the moon was hidden and
the winds blew, it possessed a forbidding, almost an ominous look, which
would have deterred anyone whose errand was less pressing than that of
our midnight wanderer, from approaching, much less examining a spot so
given over to solitude. A row of stunted oak trees shielded the house on
one side, and marked off the limits of the deserted garden, where
burdock and thistles grew instead of the homely vegetables and
old-fashioned flowers of years ago. To-night all these trees were
bending one way in the sharp gale, their whistling leaves and the _pat,
pat_ of the long limbs against the clap-boards of the house adding to
the lugubriousness of the scene.
But to the man who stood in the long grass at the rear of this disused
dwelling there was nothing in the hour or place to arouse dread or
awaken apprehension. He studied the house, but not with the eyes of a
dreamer, and when he finally made up his mind to approach the rear door
it was with determination in his face and a certain calculation in his
movement which proved that he was there with a definite purpose.
One pull at the door evidently satisfied him of the uselessness of
endeavoring to enter by force, for he left the spot at once, and began
climbing a small shed near by. Reversing the plan he had followed at the
tavern, he succeeded in climbing from ledge to ledge, until he reached a
certain window which he ruthlessly smashed in. In less time than one
would think, he had effected entrance into the house at the very place
where there was least likelihood of the attempt being discovered,
namely, under the shadow of one of those swishing trees whose branches
brushed so close against the wall that a spray of leaves immediately
thrust itself into the opening after him, covering up his passage with
unnecessary haste, considering that there were no watchers within half a
mile or more.
The place in which he found himself on dropping to the floor was so
close and dark that he involuntarily opened out his arms to grope his
way. But fearing broken floors and open staircases, he presently stopped
and drew out the small object he had hidden in his breast, and which
proved to be a pocket lantern. Lighting this, he looked around him and
drew a deep breath of satisfaction. He was in a small attic room whose
unfinished beams were so overlaid with cobwebs that he involuntarily
ducked his head, though he was in but little danger of thrusting it
against these noisome objects. A bed covered with a patched quilt was
within reach of one hand, and on the other side was a chest of drawers
with the articles necessary for making an humble toilet still on it, but
so covered by the dust and cobwebs of years that he choked as he looked
at it, and hesitated to set down his lantern on it.
Finally he compromised matters by placing it on an old chair; after
which he took out a small blank book and began to jot down notes of what
he saw. When finished with this room, he passed into another and so on
into the more roomy living chambers in front. Here he paused and took a
deeper breath, though the air was still stifling and musty.
An opening, square in shape, occupied the middle of this upper floor,
from which branched off the three sleeping rooms of this simple but not
uncomfortable cottage. In the square were books, many of which this
strange intruder took from the shelves and rapidly glanced over. Then he
opened the small drawers at the bottom of the shelves, examining the
trinkets and knick-knacks thus disclosed, with an eye rapidly
brightening into an expression of mingled hope and determination. The
pictures on the wall were few, but he apparently saw them all, nor did
he pass the decayed fringes of the window curtains without touching them
and noting their faded colors. When all that was to be seen in this
small place was carefully remarked, the man crossed the threshold of the
right-hand door and entered the large west chamber.
Something,—was it the atmosphere of the place, or some train of
recollections awakened by the objects about him?—seemed to subdue him at
this point, and he paused for a moment with his head fallen on his
breast. Then he raised it again, and with even more resolution than
before began to survey the mildewed walls and faded furniture, with an
eye that missed nothing, from the great four-poster to the mould-covered
bellows at the side of the open fireplace. It had been Mrs. Earle's
bed-room, and had witnessed the birth of Polly and the long and
mysterious illness which had terminated in the death of the mother. Here
Ephraim Earle had lavished kisses on his babe and laid his icy hand over
the scarcely colder lids of his dead wife. Here had he experienced his
keenest joys and here had he suffered his greatest sorrows. The room
seemed alive with them yet, and from every corner stared mementos of the
past which were all the more eloquent and impressive that no foreign
hand had touched them since their owner had passed away from their midst
a dozen years before. Even the candle which had lighted her last gasp
remained where it had been left on a little table in one corner; and
beside it was a book from which the finger seemed to have been just
withdrawn, though the dust that covered it lay thick on its browned
cover, and the mark which issued from one end of its discolored leaves
had lost its pristine hue and had faded to a tint almost beyond
recognition. The stranger stopped before this book and seemed to be
tempted to take it up, but refrained from doing so, as he had already
refrained from meddling with many another object lying on the high
cupboards and the tall mantel-shelf. But before the sticks in the
fireplace he showed no such hesitation. He turned them and twirled them,
and examined the ashes in which they had lain, and finally, seeing the
end of a piece of paper, he drew it out. It was the fragment of a
letter, worthless probably and of no especial interest in itself, but he
seemed to regard it as a treasure, and after looking at it for a minute,
he thrust it into his pocket.
There were a few articles of apparel hanging in the press at the foot of
the bed, and these he looked carefully over. Some of them were men's
clothes, and these he handled with a lingering touch, smiling grimly as
he did so. He even took down a coat, and after a moment's thought put it
on, and surveyed himself thus accoutered in the film-covered mirror at
the other end of the room. But the latter was too clouded to make a good
reflection, and pleased to see that the sleeves came naturally to the
wrist, though the buttons failed to fasten over the chest, he muttered
stealthily as he drew the garment off, "One's arms do not lengthen with
age, though the body often grows larger. A very good test indeed!"
There was a chest under the bed, and this he drew out, though with some
evident misgivings and many a sly look at the worm-eaten carpet over
which he had been obliged to drag it. The lock had been fastened, but he
opened it with the crooked nail he drew from his pocket; and plunging
into the trunk, pulled out one article after another, muttering in an
indescribable tone as he handled each:
"My wife's wedding dress! The locket and chain I gave her! The cashmere
shawl she always called her best! The lace folderols Aunt Milicent used
to wear, and Grandpa Hallam's gown in which he died when he was struck
with apoplexy while preaching in Brother Burton's pulpit in Charlestown.
A collection of keepsakes all remembered by me, even to this old
spectacle case which must have been her grandmother's."
Putting the things all back in the exact order in which he found them,
he relocked the trunk and thrust it carefully back into its old place.
But before leaving the room he stood for several minutes in the doorway,
and let, or seemed to let, the full aspect of the place sink into his
consciousness, after which with a half-frightened look at the floor, as
if he feared he had left the print of his feet behind him, he stepped
again to the hall, and so into a small room adjoining.
Here he remained longer than in the one he had just left; for it had
been Mr. Earle's workroom and it was full of reminiscences of his old
labors. To enumerate the various objects which this strange intruder
examined would occupy us too long and needlessly encumber this
narrative. Enough that he gave the place the same minute inspection he
had accorded to every other spot he had previously entered, and by force
of vivid imagination or a faithful remembrance seemed to live for a
short half-hour in a past of hopeful work and mechanical triumphs. There
was an inventor's model in one corner, and to this he gave his closest
attention. Though he laid no finger upon it, fearful perhaps of leaving
some trace of his presence behind him, he studied its parts with a
glistening eye and half-sarcastic smile, saying, as he turned away at
last:
"This is where the art of making explosives stood in '63. We have got
further than that now."
There was a secretary in this room and before it he spent most of the
remaining time. Some old letters which he found there engrossed him
completely, and from one small drawer he took an object that interested
him so much he failed to replace it on leaving the room. It was the
faded miniature of a pale young mother and a blue-eyed babe. The mother
had the look of the Lawrence family, and the child the promise of that
saucy and irresponsible loveliness he had seen the day before in the
new-made heiress, Polly Earle. This was not all he carried away. After
he had finished the letters, he sat a long time musing with knitted
brows and rigid hands, then he examined the desk, and sounding it,
listened with accustomed ear to the echo made by his knuckles on the
various partitions.
Suddenly he stopped, and leaning over a certain receptacle, from which
he had drawn a small drawer, he tapped again, and seeming to be
satisfied with the result, began to manipulate the place with his
penknife till the false bottom came out and he found in the shallow
space thus disclosed a small box which he eagerly pulled out, opened,
and examined. What it held I do not know, but whatever it was, he thrust
it with a triumphant look into his breast, and then repairing the
mischief he had done, first closed the drawers and then the desk,
shaking visibly as he did so, perhaps with something of the feeling of a
thief, though his face had none of the aspects of one, and his step when
he moved away had a resolution in it that added height to his stature,
which since he had allowed himself to walk upright was imposing.
In another moment he had carried the lantern from the room, and the
sleep of years had descended again upon its dark and silent precincts.
------------------------------------------------------------------------
VI.
THE PORTRAIT.
HAD the sides of this house suddenly fallen in and revealed to the
distant neighbors at the foot of the hill the vision of this creeping
marauder passing through the haunted rooms and down the creaking
staircases of this long-unopened house, what a panic of fear would have
swept through them at the uncanny sight! Glints of light from the small
lantern which he carried, passed flickering from wall to wall, and on
one window-shade threw an exaggerated outline of his form with its long
beard and groping hand, which if seen from without would have sent most
persons hurrying down the road. But there was no one in the fields that
night, and this passing glimpse of the intruder went out in darkness
without any other alarm being given than that which came from the
creaking pines and pollards without.
He was on the first floor now, and being more fearful of surprise than
in the rooms above he trod more carefully and was more attentive as to
where the light of his lantern fell. The parlor, which in houses of this
stamp is sufficiently musty when the place is inhabited and a dozen
children pass its charmed door every day, was worse than a tomb on this
night of its resurrection, and almost drove the man, who so fearlessly
opened it, into the open air for refreshment. Being near the ground, its
walls had become a prey to damp and mildew, and had not the two family
portraits adorning the space over the mantel-shelf been so fortunate as
to hang on an inner wall, their ruin would not have been confined to the
gilded frames.
It was before these pictures the visitor took his stand. One was the
portrait of an old man, and at this he barely glanced. But on the other
he gazed earnestly and long, calling up the living appearance of the man
it represented and comparing it with his own.
"Taken a year after marriage," he presently commented, with his old
sarcastic smile. "That was, let me see, seventeen years ago. No wonder
the cheeks are fresh- and the locks unmixed with gray. When I am
shaved and my beard trimmed the difference of years will not be so
perceptible. Yet time makes changes under the most favorable
circumstances, and when a man has led a life like mine, his features
naturally coarsen. I must remember this fact when people tell me I have
lost the frank, attractive look I see here. Fast living and wild
expenditure leave their marks, and I will be as good an example of the
returned prodigal as any Bible-pounding exhorter could wish. Yet," and
he sighed, "it is not altogether pleasant to remember one's misdeeds, or
to note the difference in such a face as this and that which lies under
my long, disfiguring beard."
These words, which he had uttered aloud, had no sooner left his lips
than he was startled by the silence that followed. A sense of his
position suddenly came over him, and casting one final glance at the
portrait, he turned quickly away, murmuring under his breath:
"That ring on the finger,—it was pawned long ago. What a past I will
have to disclose if my friends inquire into the matter too closely."
Fifteen minutes more he spent in cellar and attic, and then he swung
himself out of the window on to the tree, and thence lightly to the
ground. As he did so he thought he heard a sigh, but just at that moment
the trees gave a great swish and bent almost double, and he forgot the
lesser sound and never thought to look behind him when he started to
move down the road.
Had he done so, he would have seen by the first faint streaks of morning
light, a figure standing at the angle of the house, with hat pulled low,
and hands thrust out in superstitious protest at what was evidently
considered a spectre stalking from the haunted house.
The next day the bent and feeble wayfarer announced that there was no
work to be found in Hamilton, and took his leave of the place, followed
by the faithful dog. But at the outskirts of the town, the latter
paused, and whining, raised his protest at this departure; and when he
found that his new master was determined to go, he lay down in the dusty
road and refused to accompany him any further.
He would not leave the town in which his old master lay buried.
------------------------------------------------------------------------
PART III.
A RETURN.
-------
VII.
WHAT THE STROKE OF A BELL CAN DO.
IT was in the latter part of June, and the day was so perfect that it
seemed like wanton waste to use the hours for study or work. The roses,
which were always plentiful in the Fisher garden, had probably passed
their prime, but their perfume was still in the air, and there were
enough lingering buds on the thorny stalks to tempt Polly into their
midst. She had gathered quite a bouquet, and was turning toward the
house when she heard her name called. Blushing delightfully, she
stopped.
Young Unwin was leaning over the wall that separated the two gardens.
"Polly, Polly!" he called. "Come here, dear, I have something of real
importance to say to you."
His tone was graver than usual, and her gay spirits were dashed, yet the
dimples remained in her cheeks and the saucy gleam in her eye, as
drawing near, she paused, with a mock curtsey, just out of his arm's
reach on her side of the wall.
"Well, what is it, Mr. Persistency?" said she, a delicious smile robbing
her words of any sting they might otherwise have contained. "This is the
third time to-day you have summoned me to this wall."
"Once to give you a rare flower, which had just opened in the
conservatory. Once to see if you appreciated this lovely day, and
once,—O Polly, my father is anything but well to-day."
Her face, which had been brimming with mirth sobered instantly.
"Is he going to die?" she inquired, with alarm.
"I fear so, dear, and so it becomes our duty to tell him our wishes and
expectations. Are you willing to go with me to his bedside? We should
love each other more dearly for his blessing."
"Do you think"—the words came with difficulty,—"that he will give us his
blessing?"
"I think so; he has always seemed to like you, has he not?"
"Yes, but——"
"I know what you mean, Polly; and it would be sheer hypocrisy for me not
to acknowledge what every one knows, that my father is a very proud man
and that he is likely to have ambitious hopes for his son. But are they
not likely to be realized by our marriage? When you have taken up your
abode in the old Izard mansion, you will be quite an eligible match even
for Squire Unwin's son." A tender, yet half-sarcastic smile took the
edge off these words, and showed the little maiden how dearly she was
loved. Whereupon she shook her pretty head.
"But I am so lacking in accomplishments, Clarke, and he so admires an
accomplished woman. Why, I barely know one language well, and your
stepmother, I hear, speaks three."
"All of which she will teach you, dear. Accomplishments are easily
acquired. In five years you will be a model of learning and culture."
She laughed. "I look like it, do I not? See. I have not even bought
myself a new dress. I have had other things to think of."
"I like you in that rose pink gingham, but my father has a great
fondness for white. Haven't you a white dress, Polly?"
"You know I have," she pouted. "Didn't you tell me last Sunday that——"
"Ah, I remember. Yes, yes, put that dress on and come round by the front
gate; I will be there to meet you."
"But Mrs. Unwin? You have not told me whether she is likely to approve.
I should not want her to greet me coldly."
"My mother? My darling mother? I never think of her as a stepmother,
Polly dear. Oh, she knows all about it and is ready to welcome you as a
daughter."
The young girl, with a sudden lift of her head, smiled joyously and
seemed to gather courage at once.
"I will go," she frankly declared. "And yet I dread to meet him. Is he
so very sick, and will his looks frighten me?"
"It may be," answered Clarke, "but I shall be there to make it as easy
for you as possible. Do not think of my father, but of me and my love."
She sighed with joy and ran off, as free a thing as the sun shone upon;
and he watching her felt his heart soften more and more to her womanly
sweetness.
"My father will feel her charm," he murmured, and hastened up the garden
walk to the gate where he had promised to wait for her.
Clarke Unwin was no ordinary man. He was the thoughtful son of a proud
reserved father, and he had an aim in life quite apart from the
accumulation of wealth, which had so distinguished the elder man. He was
ambitious of becoming a famous electrician and had already shown
sufficient talent in this direction for his friends to anticipate great
results from his efforts. He had a scheme now on hand which only needed
the small capital which his father had promised him to become, as he
believed, a practical reality. Indeed, negotiations had already been
entered into for his entrance into a firm of enterprising men in
Cleveland, where his energy would have full scope. All that he needed
was the money which they required as a guaranty against failure, and
this money, some five thousand dollars or so, had, as I have said, been
promised to him, though not yet advanced, by his indulgent parent.
To sound that father's mind on this and on the still dearer subject of
his marriage, young Unwin had prevailed upon Polly to enter this house
of sickness. At the door they were met by a sweet-faced lady, who took
Polly in her arms before seating her in a little ante-room.
"I must ask you to remain here for just a few minutes," said she. "It
would be a shock to Mr. Unwin to see you without any preparation. Clarke
will have a talk with his father first, and then come back for you. Let
me hope it will be with a welcome that will make amends to you for your
long years of orphanage among us."
"You are very good," came from the trembling lips of the young girl.
Mrs. Unwin's grace and unconscious dignity always abashed her.
"Clarke informs me that you are not lacking in that same desirable
quality," whispered the other lady, and with a smile which gave an air
of pathos to her faded yet beautiful face, she turned away and followed
her son out into the hall. As they passed along she impetuously stopped
and faced him. Grace Unwin had been a mother to Clarke for thirteen
years, and she loved him devotedly.
"Clarke," said she, "I dread this ordeal most unaccountably. Your father
has had something on his mind of late. Do you know of any trouble
weighing upon him besides this dreadful one of leaving us?"
"No," rejoined the wondering youth. "He has never confided in me,
mother, as much as he has in you. If you know nothing—"
"And I do not," she murmured.
"You must have been deceived by your affection. He is not the man to
brood over petty troubles, or to be cast down by matters he could
regulate with a word."
"I know it, yet he has not appeared natural to me for some time. Long
before the physician told him that his disease was mortal, his actions
betrayed a melancholy which has always been foreign to his nature, and
for the very reason that he has succeeded in hiding it from you, I feel
that it has its seat in something vital."
"And have you never asked him what it was, dear mother? You who are such
a tender nurse and so adored a wife must have moments when even his
reserve would yield to such gentle importunities as yours."
"It would seem so, but I have never dared to broach the subject. When
your father chooses to be silent, it is difficult for any one to
question him."
"Yes, mother; and yet I must dare his displeasure to-day. I must know
his mind about Polly."
"Yes, that is right, and Heaven's blessing go with you. I shall be
outside here in the hall. If you strike the bell once I will fetch in
Polly; if you strike it twice, I will come in alone; if you do not
strike it at all, I will remain where I am, praying God to give you
patience to meet the disappointment of your life."
The man whose reticent nature had aroused this conversation was just
waking from a fretful sleep when his son entered. He was a tall, spare
man with an aristocratic air and a fine head, who was wont to walk the
streets as if the whole town belonged to him, and who had been spoken of
as "the Squire" from his earliest manhood. Now his proud head lay low,
and his once self-satisfied countenance wore a look that caused a pang
to strike the heart of his son, before the unrest visible in his whole
figure could find vent in words.
"What is it, father? You look distressed; cannot something be done to
relieve you?"
The man who had never been known to drop his eyes before anyone slowly
turned his face to the wall.
"There is no help," he murmured; "my hour has come." And he was silent.
Clarke moved uneasily; he hardly knew what to do. It seemed cruel to
disturb his father at this moment, and yet his conscience told him he
would be wrong to delay a communication that would set him right in his
own eyes. The father settled the matter by saying abruptly: "Sit down, I
have something to say to you."
Clarke complied, drawing a chair close up to the bedside. He knew that
one of his father's peculiarities was a dislike to raising his voice.
For a moment he waited, but the father seemed loath to speak. Clarke
therefore remarked, after a certain time had passed:
"Nothing you can say to me will fail of having my respectful attention.
If I can do anything to relieve your cares—" The look which his father
here turned upon him startled him from continuing. Never had he seen
such an expression in those eyes before.
"Can you go so far as to forgive?" the old man asked.
"Forgive?" echoed Clarke, hardly believing his ears. "What is there I
have to forgive in you? The benefits you have bestowed upon me, the
education I have received and your fatherly care?"
"Hush!" the half-lifted hand seemed to entreat and a shadow of the old
commanding aspect revisited the ashy countenance before him. "You do not
know all that has happened this last year. I have ruined you, Clarke,
ruined your mother; and now I must die without having the opportunity of
retrieving myself."
Surprised out of his usual bearing of profound respect, Clarke sprang to
his feet.
"Do you mean," he asked, "that your money is gone; that you are dying a
bankrupt?"
The old man—for Frederick Unwin was twenty years older than his
wife—grew so pale that his son became seriously alarmed.
"You are sick—fainting," he cried; "let me call someone." But a glance
from his father's commanding eye held him where he stood.
"No, no; it is from shame, Clarke, possibly from grief. You have been on
the whole a good boy, and I have taken pride in you. To leave you with
your hopes dashed, and the care of a mother on your hands, is a
humiliation I never expected. I—I have lost all, Clarke, and am,
besides, in debt. I have not five hundred dollars to give you, let alone
five thousand. You will have to take up with some lesser position, some
clerkship with a salary, reserving to yourself the right to curse a
father who was so shortsighted as to invest his whole fortune in a mine
that petered out before the machinery was paid for."
Clarke, to whom the prospect thus opened meant the demolition of more
than one dream, sat dazed for a moment in a state of despair, not
noticing that his arm had struck the bell on the small table beside
which he was sitting, making it ring out in one clear, low note.
"There is even a mortgage on this house," the wretched father went on.
"I thought the amount so raised might bridge me over my present
difficulties, but it is gone like the rest, and now it only remains for
me to be gone, too, for you to understand into what a position I have
put you by my folly and ignorance."
"Father I would not let any one else speak of you so in my hearing. You
meant to better your position, and if you made mistakes, we—that is, my
mother and myself, must try and retrieve them."
"But your chances with Stevens and Wright? Your excellent plan for—" The
son suppressed the sigh that rose to his lips and resolutely lifted his
head.
"That dream is over," he said. "I shall think no more of my own
advancement, but only of supporting my mother by any humble means that
offers."
"You have not confidence enough in your schemes to borrow the money you
want?"
"I will never borrow."
The old man, weakened by illness and shaken by the break he had just
made in an almost life-long reserve, uttered a deep sigh. Clarke, whose
thoughts were with Polly as much as they were with his surrendered
hopes, re-echoed this sound of despair before saying:
"I have always cherished a certain sort of pride, too. I could not feel
free under a burden of debt incurred for something whose value is yet to
be tested. I cannot be beholden to any one for a start which is as
likely to lead to failure as to success."
"Not if that person is your promised wife?" burst from trembling and
eager lips behind him, and Polly, accompanied by Mrs. Unwin, who had
mistaken the ring of the bell for the signal which had been established
between herself and Clarke, stepped into the room, and advanced with
timid steps but glowing cheeks into the presence of the equally
astonished son and father.
"Polly!" sprang involuntarily from the lover's lips, as he rose and cast
a doubtful glance toward his father. But the latter, roused by the fresh
young face turned so eagerly toward him, had lost his white look, and
was staring forward with surprised but by no means repelling glances.
"What does she say?" he murmured. "This should be Polly Earle, to whom
some kindly friend has just left twenty thousand dollars. Does she love
you, Clarke, and was the word she just used 'wife'? I'm getting so dull
of hearing with this ceaseless pain, that I do not always understand
what is said in my presence."
Clarke, delighted with the eagerness apparent in his suffering father's
look and manner, took the young girl by the hand and brought her
forward. "This is the woman whom I chose for my wife when I thought my
prospects warranted me in doing so. But now that I have little else than
debts to offer her, I have scruples in accepting her affection, dear as
it is and disinterested as she shows herself. I would not seem to take
advantage of her youth."
"But it is I," she broke in gayly, "who am likely to take advantage of
your disappointments! I heard by mistake, I think, something of what
your father has had to say to you, and my only feeling, you see, is one
of delight that I can do something to show my gratitude for all that you
and others have done for me in the years when I was a penniless orphan.
Is that a wrong feeling, Mr. Unwin, and will you deny me the privilege
of—" She could say no more, but her eyes, her lips, her face were one
appeal, and that of the most glowing kind. Clarke's eyes dropped lest
they should betray his feelings too vividly, and Mrs. Unwin, who had
thrown her arm around Polly, turned her face toward her husband with
such an expression of thankfulness that he did not know which caused him
the greater surprise, his wife's sudden beauty or the frank yet timorous
aspect of this hitherto scarcely noted young girl in the presence of the
two great masters of the world, Love and Death.
"Come here!" he finally entreated, holding out one shaking hand toward
Polly. She tossed her hat aside like a wild creature who recoils from
any sort of restraint, and coming up close to the bed, fell on her knees
by his side.
"So you love Clarke?" he queried.
Her eyes and cheeks spoke for her.
"Love him well enough to marry him even now, with all his debts and
disabilities?"
Still her looks spoke; and he went calmly on: "Then, my little girl, you
shall marry him, and when you see him prosperous and on the high road to
success in his chosen field of labor,—think that his father blesses you
and that by your loyalty and devotion you took away the sting from an
old man's death."
A sob and a smile answered him, and Clarke, to whom this scene was the
crowing glory of his love, turned and took his mother in his arms,
before stooping to raise his young betrothed. It was the happiest hour
in this family's history, but it was the precursor of sorrow. That night
Mr. Unwin died.
------------------------------------------------------------------------
VIII.
THE HOUSE ON THE HILL.
THERE were two topics of interest agitating the town. One was the
appearance of a new hermit in the old cave on the mountain side, and the
other, the sale of the Unwin mansion and the prospective removal of
Frederick Unwin's widow and son into the haunted house of the Earles.
The latter occasioned the greater amount of talk. That this move on
their part was but the preliminary step to a marriage between Clarke and
the young heiress had been known for some time. But to see a house so
long deserted reopened, its doors and windows thrown wide to the sun,
and the smoke rising once more from its desolate chimneys, was an event
calculated to interest all who had felt the indescribable awe
surrounding a place abandoned by human life while yet possessing all the
appointments of a home.
Polly, who for some reason had given up her former plan of renting the
big Izard place, was full of business and glowing with the excitement of
what was considered by many in the town a rather daring venture. Even
Dr. Izard, who was not wont to show emotion, looked startled when he
heard of her intentions, and seemed disposed to forbid the young girl
letting a house so given over to damp and mildew. But when she urged the
necessity of providing Mrs. Unwin with an immediate home and hinted at
the reluctance which that lady had shown to living at the other end of
the village, he relented and merely insisted that the place should be
thoroughly aired and renovated before Mrs. Unwin went into it. As he was
not that lady's physician, had never been even a visitor at the Unwin
mansion, he could say no more. But Polly needed no further hint, and
went back to her own humble home with the most generous projects in her
head for Mrs. Unwin's future comfort and happiness.
It was a great day in Hamilton when she and Clarke and five or six
interested neighbors first threw open the creaking front door of the
Earle cottage and let the sunlight stream into its hushed interior. To
her, who had never been permitted to enter the place since she had been
taken from it fourteen years before, it was an event merely to press her
foot on the worm-eaten carpets and slide her fingers along the walls
that had once felt the touch of her parents' garments. Each room was a
revelation, each corner a surprise. She glided from hall to chamber and
from chamber to hall like the spirit of a younger age introduced into
the memorials of a long-departed one. Her fresh cheek, from which even
awe could not quite banish the dimples, looked out of place and yet
strangely beautiful amid the dim surroundings of the stiffly-ordered
rooms and old-fashioned furnishings.
With an instinct natural enough under the circumstances, she had wished
to be the first to enter the house and cross the threshold of each
apartment. But Clarke was not far behind her. In front of the portrait
of her father she paused and drew her friends around her.
"Oh!" she cried; "it was wrong to keep this from me; I should have been
brought up under the influence of that face." But as she further
contemplated it, her first enthusiasm faded and an indescribable look of
vague distrust stole into her rosy countenance, and robbed it of half
its joyousness. "I—I wish there was a picture of my mother here," she
whispered to Clarke, whose arm she had nervously seized. "She had a
beautiful face, they say, all gentleness and goodness."
"Perhaps we shall find one upstairs," he suggested, turning to open more
windows.
"Oh, it is cold," she murmured, and moved with quite an unaccustomed air
of gravity toward the staircase. Her mother's room, with its many
suggestions of days which were not entirely forgotten by her, seemed to
restore her mental balance, shaken by that short contemplation of her
father's portrait. She wept as her eyes fell upon the bed where she had
last seen the outstretched form of her dying mother; but her tears were
tender and quite unlike, both in their source and effect, the shuddering
recoil which had seized her after she had gazed a few minutes at her
father's pictured face.
The book which a certain hand had hesitated to touch not so very long
ago, she took up, and opening with some difficulty the pages which time
and dampness had glued together, she showed Clarke these words, written
on one of the blank leaves in front:
"Ah! what is life!
'Tis but a passing touch upon the world;
A print upon the beaches of the earth
Next flowing wave will wash away; a mark
That something passed; a shadow on a wall,
While looking for the substance, shade departs:
A drop from the vast spirit-cloud of God,
That rounds upon a stock, a stone, a leaf,
A moment, then exhales again to God."
"My mother's writing, I know! What a difference in our dispositions!
Where do you suppose I got my cheerful temperament from? Not from my
father?" And again she faintly shuddered.
"Your father's desk is in the other room," commented somebody. Looking
up she laid the book softly down and prepared to leave the one spot in
the house of which she had any remembrance. "I shall hate to see this
dust removed, or these articles touched. Do you think I could be allowed
to do the first handling? It is so like a sacrilege to give it over to
some stranger."
But Clarke shook his head. "I have let you come with us into this damp
house because it seemed only proper that your eyes should be the first
to meet its desolation. I shall not let you remain here one moment after
we are gone. If I were willing, Dr. Izard would not be; so do not think
of it again."
The name of the doctor seemed to awaken in her a strange chain of
thought.
"Ah, Dr. Izard! He was standing beside my father when he closed my
mother's eyes. Why did he not come with me this morning to see me open
the house? I begged him to do so but he declined quite peremptorily."
"Dr. Izard does not like me," remarked Clarke sententiously.
"Does not like you? Why?" queried Polly innocently, pausing on the
threshold they were crossing.
"I do not know: he has always avoided me, more than he has other people,
I mean—and once when I spoke to him, the strangest expression crossed
his face."
"I do not understand. He has always been very kind to me. Are you sure
that you like him?"
"I am indifferent to him; that is, I admire him, as everyone must who
has eyes and an understanding. But I have no feeling toward him; he does
not seem to have any place in my life."
"He has in mine," she reluctantly admitted. "I often go to him for
advice."
"Was it by his advice," whispered Clarke, bending till his mouth touched
her ear, "that you gave me your heart?"
The little hand that lay on his arm drew itself slowly out and fell
quite softly and significantly on her heaving breast.
"No," said she. "I have another adviser here, fully as powerful as he
can ever be."
The gesture, the accent were so charming that he was provoked at the
peering curiosity of the persons accompanying them. He would have liked
to kiss those rosy lips for the sweetest thing they had ever said.
Had the midnight visitor of a few weeks back known what a careless crowd
would soon invade these hidden premises he might not have been so wary
in his movements. When Polly reached her father's desk, she found one or
two neighbors there before her.
"Oh, look at this curious old inkstand!" exclaimed one.
"And at this pile of note-books standing just where Ephraim Earle must
have laid them down!"
"And at this pen with the ink dried on it!"
"And at this ridiculous little China shepherdess pursing up her lips as
if she knew the whole mystery but would not tell!"
Polly, whose ears had been more or less closed by the episode with
Clarke just above mentioned, seemed scarcely to hear their words. She
stood by her father's work-table with her hand on her father's chair, in
a dream of love that moistened her down-cast eyes and awakened strange,
tremulous movements in the corners of her sensitive lips. But soon the
tokens of past ambition and of interrupted labor everywhere apparent,
began to influence her spirits, and her looks showed a depression which
was nothing less than startling to Clarke. Even the neighbors observed
it and moved chattering away, so that in a few minutes Polly and Clarke
were left standing alone in this former scene of her father's toil and
triumphs.
"What is the matter, my darling?" he now asked, seeing her turn away
from the very objects he supposed would interest her most.
"I do not know," she answered. "I do not like this room; I do not like
the effect it has upon me." Had the gliding visitant whose shadow had
last fallen on these walls left some baleful influence behind him, or
was the cause of her distrust of deeper origin and such as she hardly
dared admit to herself?
"The air is close here," remarked Clarke; "and the presence of all this
dust is enough to stifle anyone. Let us go down into the garden and get
a breath of fresh air."
She pointed to the open windows. "How can it be close with all this
light pouring in? No, no, it is not that; I am simply frightened. Did
you ever stop to think?" she suddenly inquired, "what I should do or how
I should feel if—_if my father came back_?"
"No," he replied startled. "No one supposes him to be alive. Why should
you have such morbid thoughts?"
"I do not know." She laughed and endeavored to throw off the shadow that
had fallen upon her. "You must think me very superstitious, but I would
not walk down that rear passage for anything; not even with you, I
should expect to encounter a tall, military-looking figure, with a face
pleasing enough at first sight, but which would not bear close scrutiny.
A face like the painted one below," she added, with an involuntary
shudder.
"But that is not a bad face; it is only a keen and daring one. I like it
very much. I remember my mother has always said you inherited your
beauty from your father."
But this seemed to irritate her indescribably. "No, no," she cried,
shaking her head and almost stamping her little foot. "I don't believe
it and I won't have it!" Then, as if startled by her own vehemence, she
blushed and dragged him away toward the door. "He may have been
handsome, but I have not eyes like his, I am sure. If I could only see
how my mother looked."
In the hall below they paused. There was much to be said concerning the
contemplated alterations to be made in the house, but she did not seem
to take any interest in the matter. Evidently the effect of her visit
upstairs had not entirely left her, for just as they were turning toward
the door she gave an involuntary look behind her, and laughing, to show
her sense of the foolishness of her own words, she cried:
"So we did not meet my father's ghost after all. Well now, I may be sure
that his interest is in other scenes and that he will never come back
here." As she spoke a shadow crossed the open doorway.
"Do not be too sure of anything!" interposed a voice, and a strange but
by no means attractive looking man stepped calmly into the house and
paused with a low bow before her.
------------------------------------------------------------------------
IX.
ASK DR. IZARD.
POLLY uttered a sharp cry and stared at the intruder blankly. He was
tall and military looking and had a smooth, well-shaven face. But his
clothes were in rags and his features, worn by illness and coarsened by
dissipation were of a type to cause a young girl like her to recoil.
"Who is this man?" she cried at last, "and what is he doing here?"
"It is the new hermit! The man who has taken up with Hadley's old
quarters," exclaimed one of the neighbors from the group about Polly. "I
saw him yesterday in the graveyard."
"Yes, and there is his dog, Piper. He follows every old tramp who comes
into town. Don't you remember how he tagged at the heels of that old
beggar with a long beard, who went through here a month ago?"
"This fellow looks as if he were strong enough to work," whispered one
of the women.
"I shan't give any of my stale victuals to a man with an arm strong
enough to fell an ox," murmured another.
Here Clarke, who had only waited for an opportunity to speak, now
advanced to the man standing in the doorway. As he did so he noticed
that the wayfarer's attention was not fixed upon the persons before him,
but upon the walls and passages of the house they were in.
"Have you come here begging?" he inquired. "If so you have made a
mistake; this is a disused house which we have been opening for the
first time in years."
"I know its every room and its every corner," answered the
haggard-looking tramp imperturbably. "I could tell you what lies under
the stairs in the cellar, and point out to you the books which have been
stacked away in the garret: That is, if no other hand has disturbed them
since I placed them there fifteen years ago."
A cry of astonishment, of despair almost, answered these words. It came
from the blanching lips of Polly. Clarke trembled as he heard it, but
otherwise gave no sign of concern. On the contrary he eyed the intruder
authoritatively.
"Tell me your name!" he demanded. "Are you——"
"I will not say who I am, here, with the sunlight streaming on my back
and no friendly eye to recognize my features. I will only speak from
under the portrait of Ephraim Earle; I want a witness to the truth of my
statements and in that canvas I look for it."
And neither heeding Clarke's detaining hand, nor the almost frantic
appeal which spoke in the eyes of the young girl whose question he had
at last answered, he stalked into the parlor and paused directly beneath
the portrait he had named.
"Cannot you see who I am?" he asked, rearing his tall head beside the
keen-faced visage that looked down from the wall.
"The same man grown older," exclaimed one.
"Ephraim Earle himself!" echoed another.
"Come back from the dead!"
"The moment the house was opened!"
"Are you Ephraim Earle?" demanded Clarke, trembling for Polly in whose
breast a real and unmistakable terror was rapidly taking the place of an
imaginary one.
"Since I must say so, yes!" was the firm reply. "Where is my daughter?
She should be on hand here to greet me."
"I have no words of welcome. I never thought of my father being like
this. Take me away, Clarke, take me away!" So spoke the terrified little
one, clinging to one of her best-known neighbors for support.
"I will take you away," Clarke assured her. "There is no need of your
greeting this man till he has proved his claim to you. A girl's heart
cannot be expected to embrace such a fact in a moment."
"Oh, it's Ephraim Earle fast enough," insisted one old woman. "I
remember him well. Don't you remember me, old neighbor?"
"Don't I?" was the half hearty, half jeering answer. "And I wish I had a
pair of your green and white worsted socks now."
"It's he, it's he!" vociferated the delighted woman. "When he was a
young man I sold him many a pair of my knitting. To be sure I use blue
now instead of green, but they were all green in his day, bless him!" As
this prayer was not repeated by her companions in the room, upon whom
his reckless if not sinister appearance had made anything but a happy
impression, he came slowly from under the picture and stood for a moment
before the dazed and shrinking Polly.
"You are not glad to see me," he remarked, "and I must say I do not
wonder. I have lived a hard life since I left you a crying child in your
mother's room upstairs, but I am your father, for all that, and you owe
me respect if not obedience. Look up, Maida, and let me see what kind of
a woman you have grown to be."
At this name, which had been a pet one with her parents and with them
alone, the neighbors stared and Polly shrank, feeling the iron of
certainty pierce deep into her soul. She met his eyes, however, with
courage and answered his demand by a very natural reproach.
"If you are my father, and alas! I see no reason to doubt it, I should
think you would feel some shame in alluding to a growth which you have
done nothing to advance."
"I know," he admitted, "that you have something with which to reproach
me; the secret of those days is not for ears like yours. I left you,
but—never ask me why, Maida. And now, go out into the sun. I should not
like to have my first act toward you a cruel one."
Dazed, almost fainting, doubting whether or no she was the victim of
some horrible nightmare, she let herself be led away to where the sun
shone down on the lilacs of the overgrown garden. But no sooner did she
realize that the man of her dread had been left in the house with her
neighbors than she urged Clarke to return at once to where he was.
"Let him be watched," she cried; "follow him as he goes about the house.
It is his; I feel that it is his, but do not let us succumb to his
demands without a struggle. He has such a wicked face, and his tones are
so harsh and unfatherly."
Clarke, who had come to a similar conclusion, though by other means than
herself, hastened to obey her. He found the self-styled Earle in the
midst of the group of neighbors, chattering freely and answering
questions with more or less free and easy banter. Though privation spoke
in every outline of his face and form, and poverty in every rag of his
dress, his bearing gave evidences of refinement, and no one, not even
Clarke himself, doubted that if he were put to the test he would show
himself to be at least the wreck of the once brilliant scholar and man
of resources. He was drawing the whole crowd after him through the house
and was hazarding guesses right and left to prove the excellence of his
memory.
"Let us see," he cried, as they one and all paused at the top of the
staircase, before entering the rooms on the upper floor. "I used to keep
my books here—such ones as I had not discarded and stacked away in the
topmost story. And I used to pride myself on knowing where every volume
was kept. Consult the shelves for me now and see if on the third one
from the bottom and nearer to the left than to the right there is not a
volume of Bacon's Essays. There is? Good! I knew it would be there if
some one had not moved it. And the ten volumes of Shakespeare—are they
not on the lower shelf somewhere near the middle? I thought so. A
capital old edition it is, too; printed by T. Bensley for Wynne &
Scholey, Paternoster Row. And Gibbon's _Rise and Fall_, with a volume of
Euripides for a companion? Yes? And on the topmost shelf of all, far out
of the reach of any hand but mine, a choice edition of Hawthorne—my
favorite author. Do you see them all? I am glad of that; I loved my
books, and often when very far away from them used to recall the hour
when I had them under my eye and within reach of my hand."
"I wonder if he used to recall the child he left, tossed helpless upon
the mercies of the town?" murmured one of the neighbors.
"Is my desk here, and has it been touched?" he now asked, proceeding
hastily into the workroom. "Ah, it all looks very natural," he remarked;
"very natural! I can scarcely believe that I have been gone more than a
day. Oh, there's the model of the torpedo I was planning! Let me see,"
and he lifted up the half-completed model, with what Clarke could not
but call a very natural emotion, looking it over part by part and
finally putting it down with a sigh. "Good for those days," he
commented, "but would not answer now. Too complicated by far; explosive
agencies should be more simple in their construction." And so on for
half an hour; then he descended and walked away of his own accord to the
front door.
"I have seen the old place!" he blandly observed, "and that is all I
expected. If my daughter sees fit to acknowledge me, she will seek me in
the wild spot in which I have made for myself a home. Here I shall not
come again. I have not returned to the place of my birth to be a bugbear
to my only child."
"But," cried some one in protest, "you are poor and you are hungry."
"I am what fate and my own folly have made me," he declared. "I ask for
no sympathy, nor do I feel disposed to urge my natural rights."
"If you are Polly Earle's father, you will be fed and you will be
clothed," put in Clarke hotly. "There is a meal for you now at the
tavern, if you will go there and take it."
But the proud man, pointing to his dog drew himself up and turned
scornfully away. "He can procure me as much as that," said he. "When my
daughter has affection and a child's consideration to show me, then let
her come to Hadley's cave. Food! Clothing! I have had an apology for
both for fourteen years, but love—never; and all I want just now is
love!"
Polly, who was not many steps off, heard these words and, moved by fear
or disgust, dropped her hands which she had instinctively raised at his
approach. He saw and smiled grimly, then with a bow that belied his
aspect and recalled the old days when a bow passed for something more
than a perfunctory greeting, he moved sternly down the walk and out
through the stiff old gate into the dusty highroad.
Half a dozen or more of the most eager witnesses of this extraordinary
scene followed him down the hill and into town, anxious no doubt to set
the town ablaze with news of Ephraim Earle's return and of his identity
with the newly arrived hermit at Hadley's cave.
------------------------------------------------------------------------
X.
AN INCREDIBLE OCCURRENCE.
DR. IZARD had of late presented a more cheerful appearance. His step was
lighter and his face less generally downcast. He even was seen to smile
one morning at the antics of some children, an unprecedented thing in
his history, one would think, from the astonishment it caused among the
gossips.
He had been called away several times during the month and the card with
the word "absent" on it was very often to be seen hanging beside his
door. People grew tired of this, though they knew it meant fame and
money to the doctor, and the newly-fledged physician from Boston, whose
office was at the other end of the town, prospered in consequence. But
Dr. Izard only seemed relieved at this and came and went, as I have
said, with a less gloomy if not positively brightened countenance.
He had always kept for himself one solitary place of resort in the
village. Without this refuge life would often have been insupportable to
him. It was—strange to say, for the Izards had always been
aristocratic—the humble house of the village shoemaker, a simple but
highly respected man who with his aged wife had been, from sheer worth
of character, a decided factor in town for the last twenty-five years.
The little house in which he lived and plied his useful trade stood on
the hill-side a few yards above the Fisher cottage, and it was in his
frequent visits to this spot that Dr. Izard had seen so much of Polly.
The window in which he usually sat overlooked the Fisher garden, and as
his visits had extended over years he had ample opportunity for
observing her growing beauty from the time she was a curly-headed imp of
four to the day she faced the world a gay-hearted damsel of eighteen.
It had been a matter of some mystery in the past why Dr. Izard, with his
trained mind and refined tastes, affected this humble home and sought
with such assiduity the companionship of this worthy but by no means
cultured couple. But this, together with other old wonders, had long
lost its hold upon public attention, no one thinking of inquiring any
longer into the cause of a habit that had become so fixed it was
regarded as part of the village's history. One effect, however,
remained. No one thought of entering the shoemaker's shop while Dr.
Izard sat there. It would have been thought an intrusion by both guest
and host.
Mr. and Mrs. Fanning, who had themselves long ceased to wonder at his
preference for their society, invariably stopped their work when he
entered and greeted him with the same words of welcome they had used
fourteen years before when he had unexpectedly taken a seat in the shop
without having been summoned for professional purposes. After which
necessary ceremony they turned again to their several labors and the
doctor sat down in his especial seat, which, as I have said, was in one
of the windows, and lapsed into the silence he invariably maintained for
half his stay. The time chosen for his visit was usually at nightfall,
and whether it was that the charms of nature were unusually attractive
to him at that hour, or whether something or somebody in the adjoining
gardens secretly interested him, he invariably turned his eyes outward,
with an expression that touched the heart of the old lady who watched
him and caused many a glance of secret intelligence to pass between her
and her equally concerned husband.
Not till it was quite dark and the lights had been lit in the shop,
would the doctor turn about—often with a sigh too unconscious to be
repressed—and face again the humble couple. But when he did so, it was
to charm them with the most cordial and delightful conversation. There
was even sparkle in it, but it was only for this aged pair of workers,
whose wit was sufficient for appreciation, and whose hearts responded to
every effort made to interest them by their much revered visitor. After
a quarter of an hour of this hearty interchange of neighborly comment,
he would leave the house, to come again a few evenings later.
But one evening there was a break in the usual order of things. The
doctor was sitting, as he had sat a hundred times before, in his chair
by the window, and Mr. Fanning was hammering away at his bench and Mrs.
Fanning reading the _Watchman_, when there came a sound of voices from
the front and the door burst open to the loud cry of—
"O Mrs. Fanning, Mrs. Fanning! Such news! Ephraim Earle has come back!
Ephraim Earle, whom we all thought dead ten years ago!"
Mrs. Fanning, who with all her virtues dearly loved a bit of gossip, and
who knew, or thought she did, everything that was going on in town, ran
without once looking round her to the door, and Mr. Fanning, who could
not but feel startled also by an event so unexpected and so long looked
upon as impossible, started to follow her, when something made him look
back at the doctor. The sight that met his eyes stunned him, and caused
him to pause trembling where he was. In all the years he had known Dr.
Izard he had never seen him look as he did at that moment. Was it
surprise that affected him, or was it fear, or some other
incomprehensible emotion? The good old man could not tell; but he wished
the doctor would speak. At last the doctor did, and the hollow tones he
used made the aged shoemaker recoil.
"What is that? What are they talking about? They mentioned a name? Whose
name? Not Polly's father's?"
"Yes," faltered his startled companion. "Ephraim Earle; they say he has
come back. Shall I go and see?"
The doctor nodded; it seemed as if he had no words at his command, and
the shoemaker, glad to be released, hastened hobbling from the room. As
his half bent figure vanished, the doctor, as if released from a spell,
looked about, shuddered, grasped the table nearest to him for support,
and then burst into a laugh so strange, so discordant, and yet so
thrilling with emotion, that had not a dozen men and women been all
talking together in the hall it would have been heard and commented on.
As it was he was left alone, and it was not till several minutes had
elapsed that Mrs. Fanning came rushing in, followed by her dazed and
somewhat awestruck husband.
"O doctor, it is true! It is true! I have just seen him; he is standing
at the Fisher's corner. Polly is up at the house—You know she was to
open it to-day. They say she is more frightened than pleased, and who
can wonder? He looks like a weather-beaten tramp!"
"No, no," shouted some one from the room beyond, "like a gentleman who
has been sick and who has had lots of trouble besides."
"Come and see him!" called out a shrill voice, over Mrs. Fanning's
shoulder. "You used to know him, doctor. Come and see Ephraim Earle."
The doctor, with a curl of his lips, looked up and met the excited eyes
that were contemplating him, and slowly remarked:
"Your wits have certainly all gone wool-gathering. I don't believe that
Ephraim Earle has returned. Some one has been playing a trick upon you."
"Then it's the ghost of Ephraim Earle if it's not himself," insisted the
other, as the whole group, losing their awe of the doctor in the
interest and growing excitement of the moment, came crowding into the
shop.
"And a very vigorous ghost! He is bound to have his rights; that you can
see."
"But he won't annoy his daughter. Did you hear what he said to the
child, up there by the lilac bushes?" And then they all chattered, each
striving to give his or her own views of the situation, till a sudden
vigorous "Hush!" brought them all to an abrupt standstill and set them
staring at the doctor with wide-open eyes and mouths.
"You are all acting like children!" protested that gentleman, with his
white face raised and his eyes burning fiercely upon them. "I say that
man is an impostor! Why should Ephraim Earle come back?"
"And why shouldn't he?" asked another.
"Answer us that, Dr. Izard. Why shouldn't the man come back?"
"True, true! Hasn't he a daughter here?"
"With money of her own. Just the same amount as he once ran off with."
"I tell you again to be quiet." It was still the doctor who was talking.
"If you are daft yourselves, do not try to make other people so! Where
is this fellow? I will soon show you he is not the man you take him to
be."
"I don't know how you will do it," objected one, as the group fell back
before the doctor's advancing figure. "He's as like him as one pea is
like another, and he remembers all of us and even chattered with Mother
Jessup about her famous worsted socks."
"Fools!" came from beneath the doctor's set lips as he strode from the
door and passed rapidly into the highway. "Here, you!" he cried,
accosting the man who was the centre of a group some rods away, "come up
here! I want to speak to you."
------------------------------------------------------------------------
XI.
FACE TO FACE.
THE stranger, thus hailed, turned as the doctor's voice rang down the
road, and acknowledging the somewhat rough summons with a bow of mock
affability, stepped obligingly up the hill. The neighbors who had
flocked into the street to watch the meeting, saw the doctor's lip curl
as the wretched figure advanced. This man, Ephraim Earle? Why had he
called these credulous creatures fools? They were simply madmen. But in
another moment his countenance changed. The miserable fellow had paused
and was standing a few feet off with what could not be called other than
a look of old comradeship. He spoke first also and with quite a hearty
ring to his naturally strident voice.
"Well, Oswald, old boy, this is a pleasure! Now don't say you don't
remember me—" for the doctor had started back with an irrepressible
gesture of disgust that to some eyes was not without its element of
confusion, "I know I am changed, but no more so than you are, if you
have led a more respectable life than I."
"Scoundrel!" leaped from Dr. Izard's white lips. "How dare you address
me as if we were, or ever had been, friends! You are a brazen
adventurer, and I—"
"And you are the perfectly irreproachable physician with a well-earned
fame, and a past as free from shadow as—well, as your face is free from
surprise at this unexpected return of one you probably thought dead."
Confounded by this audacity and moved by many inner and conflicting
emotions, Dr. Izard first flushed, then stood very still, surveying the
man with a silent passion which many there thought to be too emphatic a
return for what sounded to them like nothing more than an ill-judged
pleasantry. Then he spoke, quietly, but with a sort of gasp, odd to hear
in his usually even and melodious voice.
"I do not know you. Whatever you may call yourself, you are a stranger
to me, and no stranger has a right to address me with impertinence. What
_do_ you call yourself?" he suddenly demanded, advancing a step and
darting his gaze into the other's eyes with a determination that would
have abashed most men whether they were all they proclaimed themselves
to be or not.
A playful sneer, a look in which good-natured forbearance still
struggled uppermost, were all that he got from this man.
"So you are determined not to recognize Ephraim Earle," cried the
stranger. "You must have good reasons for it, Oswald Izard; reasons
which it would not be wise perhaps for one to inquire into too
curiously."
It was an attack for which the doctor was not fully prepared. He
faltered for an instant and his cheek grew livid, but he almost
immediately recovered himself, and with even more than his former
dignity, answered shortly:
"Now you are more than impertinent, you are insolent. I do not need to
have secret reasons for repudiating any claims you may make to being
Polly Earle's father. Your face denies the identity you usurp. You have
not a trait of the man you call yourself. Your eyes——"
"Oh, do not malign my eyes," laughed the stranger. "They are faded I
know and one lid has got a way of drooping of late years, which has
greatly altered my expression. But they are the same eyes, doctor, that
watched with you beside the bed of Huldah Earle and if they fail to meet
you with just the same mixture of trembling hope and fear as they did
then it is because youthful passions die out with the years and I no
longer greatly care for any verdict you may have to give."
A frown hard to fathom corrugated the doctor's forehead and he continued
to survey in silence the bold face that declined to blench before him.
"So you persist—" he remarked at length. "Then you are a villain as well
as an impostor."
"Villain or impostor, I am at least Ephraim Earle," asserted the other;
adding as he noted the doctor's fingers tighten on the slight stick he
carried, "Oh, you need not show your hatred quite so plainly, Dr. Izard.
I do not hate you, whatever cause I may have to do so. Have I not said
that my old passions are dried up, and even signified that my coming
back was but a whim? _Curraghven-hoodah_, Oswald, you weary me with your
egotism. Let us shake hands and be comrades once more."
The audacity, the superiority even, with which these words were said,
together with the cabalistic phrase he used—a phrase which Dr. Izard was
ready to swear even at that moment of shock and confusion, was one known
only to himself and Polly's father,—had such an effect upon him that he
reeled and surveyed the speaker with something of superstitious fear and
horror. But at the malicious gleam which this momentary weakness called
up in the eye of his antagonist, he again regained his self-command, and
stepping firmly up to him, he vociferated with stern emphasis:
"I repeat that you are an impostor. I do not know you, nor do I know
your name. You say you are Ephraim Earle, but that is a lie. I knew that
man too well to be deceived by you. You have neither his eyes, his
mouth, nor his voice, I will say nothing of his manners."
"Oh," spoke up a voice from behind, "he looks like Ephraim Earle. You
cannot say he does not look like Ephraim Earle."
The doctor turned sharply, but his antagonist, who neither seemed to ask
nor need the support of any one or anything but his own audacity,
responded with a mocking leer:
"No matter what I look like. He says he cannot be deceived by my eyes,
my mouth, or my voice. That is good. That sounds like a man who is sure
of himself. But friends—" Here his voice rose and the menace which he
had hitherto held in abeyance became visible in his sharpened glance—"he
can be deceived by his own prejudices. Dr. Izard does not want to know
me because he was Huldah Earle's attending physician, and her death, as
you all know, was very sudden and _very peculiar_."
Venomous as the insinuation was, it was a master-stroke and won for its
audacious author the cause for which he had been battling. The doctor,
who had worked himself up into a white heat, flushed as if a
blood-vessel was about to burst in his brain, and drawing back, stepped
slowly from before the other's steady and openly triumphant gaze. Not
till he reached the outskirts of the crowd, did he recover himself, and
then he halted only long enough to cry to the jostling and confused
crowd he had just left:
"He looks like a tramp and he talks like a villain. Be careful what
credit you give him, and above all, _look after Polly Earle_."
------------------------------------------------------------------------
XII.
AT HOME.
IT was now nearing eight o'clock, and as Dr. Izard strode on through the
village streets, seeing no one and hearing no one, though more than one
person respectfully accosted him, the twilight deepened so rapidly that
it was quite dark when he passed the church and turned up the highroad
to his own house.
It was dark and it was chilly, else why should so strong a man as he
shiver? So dark that the monuments over the wall were hardly to be
discerned, and he had to fumble for the gate he usually found without
trouble. Yet when his hand finally fell upon it and he mechanically
lifted the latch he did not pass through at once but lingered, almost
with a coward's hesitation, finding difficulty, as it seemed, in
traversing the dismal path before him to the no less dismal door beyond
and the solitude that there awaited him.
But he passed the gate at last, and groped his way along the path
towards his home, though with lingering footsteps and frequent pauses.
Dread was in his every movement, and when he stopped it was to clutch
the wall at his side with one hand and to push the other out before him
as though to ward off some threatening danger, or avert some expected
advance. In this attitude he would become rigidly still, and several
minutes would elapse before he stumbled on again. Finally he reached his
door, and unlocking it with difficulty threw himself into the house,
shuddering and uttering an involuntary cry as a spray of the swaying
vine clung to him.
Ashamed of his weakness, for he presently saw what had caught him by the
arm, he drew a deep breath, and tried to shut the door. But it would not
close. Some obstruction, a trivial one no doubt, had interposed to stop
it, and he being in an excited state pushed at it with looks of horror,
till his strength conquered and he both shut and locked the door.
He was trembling all over when he had accomplished this, and groping for
a chair he sat down in it, panting. But no sooner had he taken his seat
than the dim panes of the window struck his sight, and bounding to his
feet he drew down the shade as if he would shut out the whole world from
his view, and the burying-yard first of all.
Quite isolated now and in utter darkness, he stood for a few minutes
deeply breathing and cursing his own fears and pusillanimity. Then he
struck a light, and calmed by the sight of the familiar interior, sat
down at his desk and tried to think. But though he was a man of great
intellectual powers, he seemed to find it difficult to fix his thoughts
or even to remain quiet. Involuntary shudders shook his frame, and from
time to time his eye glanced fearfully towards the door as if he dreaded
to see it open and admit some ghostly visitor.
Suddenly he leaped to his feet, went to a mirror and surveyed himself.
Evidently the result was not encouraging for he uttered an exclamation
of dismay and coming back to the desk, took up a book and tried to read.
But the attempt was futile. With a low cry he flung the book aside, and
rising to his feet began to talk, uttering low and fearful words from
which he seemed himself to recoil without possessing the power of
stopping them. The name of Ephraim Earle mingled often with these words,
and always with that new short laugh of his so horrible to hear. And
once he spoke another name, but it was said so softly that only from the
tears which gushed impetuously from his eyes, could it be seen that it
stirred the deepest chords of his nature.
The clock, which lagged sorely that night, struck eleven at last, and
the sound seemed to rouse him, for he glanced toward his bed. But it was
only to cry "Impossible!" and to cast a hunted look about the room which
seemed like a prison to him.
At length he grasped the green door and began to pull at its hasps and
fastenings. Careless of the result of these efforts he shook a small
heathen god from its pedestal so that it fell rattling to the floor and
lay in minute pieces at his feet. But he did not heed. Recklessly he
pulled open the door, recklessly he passed into the space beyond. But
once out of the room, once in another atmosphere than that peopled by
his imagination, he seemed to grow calmer, and after a short survey of
the narrow hall in which he found himself and a glance up the tiny,
spiral staircase rising at his right, he stepped back into the office
and took up the lamp. Carrying it with him up the narrow staircase he
set it down in the hall above, and without looking to right or left,
almost without noting the desolation of those midnight halls, he began
pacing the floor back and forth with a restless, uneven tread, far
removed from his usual slow and dignified gait.
At early morning he was still pacing there.
------------------------------------------------------------------------
XIII.
A TEST.
"O Clarke, wait: there is the doctor now." It was Polly who was
speaking. She had come as far as the church in her search after Dr.
Izard and had just seen him issuing from his own gate.
"He has a bag in his hand; he is going on one of his journeys."
"No, no," she protested, "I cannot have it." And bounding forward she
intercepted the doctor, just as he was about to step into his buggy. "O
doctor, you are not going away; you are not going to leave me with this
dreadful trouble; don't, don't, I pray!" The doctor, who in his
abstraction had not noted her approach, started at the sound of her
voice, and turning showed her a very haggard face.
"Why," she cried, stepping back, "you are ill yourself."
"No," he answered shortly, drawing himself up in his old reserved
manner. "I had but little sleep last night, but I am not ill. What do
you want, Polly?"
"O don't you know what I want? You, of all the town, have said he was an
impostor! To you then I come as to my only hope; speak, speak, is he not
my father?"
The doctor with a side glance at Clarke, who had remained in the
background, drew the girl's hand within his arm and led her a few steps
away. But it seemed an involuntary movement on his part, for he
presently brought her back within easy earshot of her lover.
"He does not look to me like Ephraim Earle," he was saying. "He has not
his eyes, nor does his voice sound familiar. I do not see why any one
acknowledges him."
"But they can't help it. He knows everybody, and everything. I—I thought
you had some good reason, Dr. Izard, something that would make it easy
for me to deny his claims."
"You—" The doctor's sleepless night seemed to have had a strange effect
upon him, for he stammered in speaking, he who was always so cold and
precise. "You thought—" he began, but presently broke into that new,
strange laugh of his, and urging Polly towards her lover, he addressed
his questions to the latter. "Does this man," he asked, "make a serious
claim upon the Earle name and its rights?"
Clarke, who was always sensible when in Dr. Izard's presence of
something intangible but positive acting like a barrier between them and
yet who strangely revered the doctor, summoned up his courage and
responded with the respect he really felt.
"Yes," said he; but with a certain reserve, "that is our best reason
perhaps for believing him. He promises not to molest Polly, nor to make
any demands upon her until she herself recognizes her duty."
The frown which darkened the doctor's face deepened.
"He is a deep one, then," said he, and stood for a moment silent.
"If he is an impostor, yes," assented Clarke; "but Lawyer Crouse, who
talked with him half an hour last night, accepted him at once, and so
did Mr. Sutherland." Mr. Sutherland was the Baptist minister.
"The fools!" muttered the doctor, as much in anger as amazement; "has
the whole town reached its dotage?"
Clarke, who seemed surprised at the doctor's vehemence, quietly
remarked:
"You were Mr. Earle's best friend. If you say that this man is not he,
there would of course be many to listen to you."
But the doctor, resuming his accustomed expression, refused an answer to
this suggestion, at which Polly's face grew very pale, and she grasped
his arm imploringly, saying as she did so:
"I cannot bear this uncertainty, I cannot bear to think there is any
question about this matter. If he is my father, I owe him everything; if
he is not——"
"Polly,"—The doctor spoke coldly but not unkindly, "marry Clarke, go
with him to Cleveland where he has the promise of a fine position, and
leave this arrant pretender to settle his rights himself. He will not
urge them long when he finds the money gone for which he is striving."
"You bid me do that? Then you _know_ he is not my father."
But the doctor instead of answering with the vigorous yes she had
expected, looked aside and carelessly murmured:
"I have said that I saw no likeness in him to the man I once knew. Of
course my judgment was hurried, our interview was short and I was
laboring under the shock of his appearance. But if everybody else in
town recognizes him as Ephraim Earle, I must needs think my opinion was
warped by my surprise and the indignation I felt at what I considered a
gross piece of presumption."
"Then you do not know," quoth poor Polly, her head sinking lower and
lower on her breast.
"No," cried the doctor, turning shortly at the word and advancing once
more toward the buggy.
But at this move she sprang forward and sought again to detain him.
"But you will not go and leave me in this dreadful uncertainty," she
pleaded. "You will stay and have another talk with this man and satisfy
yourself and me that he is indeed my father."
But the stern line into which the doctor's lip settled, assured her that
in this regard he was not to be moved; and frightened, overawed by the
prospect before her, she turned to Clarke and cried:
"Take me home, take me back to your mother; she is the only person who
can give me any comfort."
The doctor who was slowly proceeding to his horse's head, looked back.
"Then you don't like my advice," he smiled.
She stared, remembered what he had said and answered indignantly:
"If this poor, wretched, wicked-eyed man is my father—and I should never
have doubted it if you had not declared him an impostor before all the
town people—then I would be a coward to desert him and seek my happiness
in a place where he could not follow me."
"Even if he is as wicked as his looks indicate?"
"Yes, yes, even if he is wicked. Who can say what caused that
wickedness."
The doctor, fumbling with the halter, stopped and seemed to muse.
"Did you ever see your father's picture hanging in the old cottage?"
"Yes, I saw it yesterday."
"Did that have a wicked look?"
"I do not think it had a good look." This was said very low but it made
the doctor start.
"No?" he exclaimed.
"It made me feel a little unpleasant, as if something I could neither
understand nor sympathize with had met me in my father's smile. It made
him more remote, and prepared me for the heartless figure of the man who
in the next few minutes claimed me as his daughter."
"Strange!" issued from the doctor's lips; and his face, which had been
hard to read from the first, became more and more inscrutable.
"My mother, who is as wise as she is gentle, advises Polly to give up
the cottage to her father; but not to live in it with him till his
character is better understood and his intentions made manifest."
"Then your mother sees this man in the same light as others do?"
"She certainly considers him to be Ephraim Earle. It is not natural for
her to think otherwise under the circumstances."
"I do indeed stand alone," quoth the doctor.
"When I told her," pursued Clarke, "what you had said, she looked amazed
but she said nothing to show that she had changed her opinion. I do not
think any one was really affected by your words."
Something in the tone in which this was said showed where Clarke himself
stood. A bitter smile crossed the doctor's lip, and he seemed more than
ever anxious to be gone.
"I shall be away," he said, "several days. When I come back I hope to
see this thing settled."
"I hate him," burst from Polly's lips. "I am terrified at my thoughts of
him, but in my inner consciousness I know him to be my father, and I
shall try and do my duty by him; shall I not, Clarke?"
Clarke, who had felt himself almost unnecessary in this scene, grasped
at the opportunity which this appeal gave him and took her tenderly by
the arm.
"We will try and do our duty," he corrected, "praying Providence to help
us."
And the doctor, with a glance at them both, sprang into his buggy and
was driving off when he rose and flung back at Polly this final word of
paternal advice:
"He is the claimant; you are the one in possession. Let him prove
himself to be the man he calls himself."
Clarke, dropping Polly's arm, sprang after the doctor.
"Wait! one moment," he cried. "What do you call proof? You who knew him
so well in the past, tell us how to make sure that his pretensions are
not false."
The doctor, drawing up his horse, paused for a moment in deep thought.
"Ask him," he finally said, "to show you the medal given him by the
French government. As it has never been found in his house, and as it
was useless to raise money upon, he should, if he is Ephraim Earle, be
able to produce it. Till he does, I advise you to cherish doubts in his
regard, and above all to keep that innocent and enthusiastic young girl
out of his clutches."
And with a smile which would have taken more than Clarke's experience
with the world to understand, much less to explain, the doctor whipped
up his horse and disappeared down the road towards the station.
------------------------------------------------------------------------
XIV.
GRACE.
THE doctor did not return in a few days nor in a few weeks. Two months
passed before his gate creaked on its hinges and the word ran through
the town, "Dr. Izard is back!"
He arrived in Hamilton at nightfall, and proceeded at once to his
office. There was in his manner none of the hesitation shown at his last
entrance there, and when by chance he passed the mirror in his quick
movements about the room he was pleased himself to note the calmness of
his features, and the quiet air of dignified reserve once more pervading
his whole appearance.
"I have fought the battle," he quietly commented to himself; "and now to
face the new order of things!"
He looked about the room, put a few matters in order, and then stepped
out into the green space before his door. Glancing right and left and
seeing nobody in the road or in the fields beyond the cemetery, he
walked straight to the monument of Polly's mother and sternly,
determinately surveyed it. Then he glanced down at the grave it shaded,
and detecting a stray leaf lying on its turf, he picked it up and cast
it aside, with a suggestion of that strange smile which had lately so
frequently altered his handsome features. After which he roamed through
the churchyard, coming back to his door by another path. The chill of
early September had touched many of the trees about, and there was
something like dreariness in the landscape. But he did not appear to
notice this, and entered in and sat down at his table with his former
look of concentration and purpose.
Evening came and with it several patients; some from need, some from
curiosity. To both kinds he listened with equal calmness, prescribing
for their real or fancied complaints and seeing them at once to the
door. At ten o'clock even these failed to put in an appearance, and
being tired, he was about to draw his shade and lock his door when there
came a low knock at the latter of so timid and so hesitating a character
that his countenance changed and he waited for another knock before
uttering his well known sharp summons to enter.
It came after a moment's delay, and from some impulse difficult for
himself to explain, he proceeded to the door, and hastily opened it. A
tall, heavily veiled figure, clad in widow's weeds, stood before him, at
sight of which he started back, hardly believing his eyes.
"Grace!" he ejaculated; "Grace!" and held out his arms with an
involuntary movement of which he seemed next moment ashamed, for with a
sudden change of manner he became on the instant ceremonious, and
welcoming in his visitor with a low bow, he pushed forward a chair, with
mechanical politeness, and stammered with intense emotion:
"You are ill! Or your son! Some trouble threatens you or you would not
be here."
"My son is well, and I—I am as well as usual," answered the advancing
lady, taking the chair he offered her, though not without some
hesitation. "Clarke is with the horses in front and I have ventured—at
this late hour—to visit you, because I knew you would never come to me,
even if I sent for you, Oswald."
The tone, the attitude, the whole aspect of the sweet yet dignified
woman before him, seemed to awaken an almost uncontrollable emotion in
the doctor. He leaned toward her and said in tones which seemed to have
a corresponding effect upon her: "You mistake, Grace. One word from you
would have brought me at any time; that is, if I could have been of any
service to you. I have never ceased to love you—" He staggered back but
quickly recovered himself—"and never shall."
"I do not understand you," protested Mrs. Unwin, half rising. "I did not
come—I did not expect—" her agitation prevented her from proceeding.
"I do not understand myself!" exclaimed he, walking a step away. "I
never thought to speak such words to you again. Forgive me, Grace; you
have a world of wrong to pardon in me; add another mark of forbearance
to your list and make me more than ever your debtor." She drooped her
head and sitting down again seemed to be endeavoring to regain her
self-possession.
"It was for Clarke," she murmured, "that I came."
"I might have known it," cried the doctor.
"He would not speak for himself, and Polly, the darling child, has
become so dazed by the experiences of these last two months that she no
longer knows her duty. Besides, she seems afraid to speak to you again;
says that you frighten her, and that you no longer love her."
"I never have loved her," he muttered, but so low the words were not
carried to the other's ears.
"Have you learned in your absence what has taken place here in
Hamilton?" she asked.
Rousing himself, for his thoughts were evidently not on the subject she
advanced, he took a seat near her and composed himself to listen, but
meeting her soft eyes shining through the heavy crape she wore, he said
with a slight appealing gesture:
"Let me see your face, Grace, before I attempt to answer. I have not
dared to look upon it for fourteen years, but now, with some of the
barriers down which held us inexorably apart, I may surely be given the
joy of seeing your features once more, even if they show nothing but
distrust and animosity toward me."
She hesitated, and his face grew pale with the struggle of his feelings,
then her slim white hand went up and almost before he could realize it,
they sat face to face.
"O Grace," he murmured; "the same! always the same; the one woman in all
the world to me! But I will not distress you. Other griefs lie nearer
your heart than any I could hope to summon up, and I do not know as I
would have it otherwise if I could. Proceed with your questions. They
were in reference to Clarke, I believe."
"No, I only asked if you had kept yourself acquainted with what has been
going on in Hamilton since you left. Did you know that Ephraim Earle was
living again in the old house, and that Polly is rapidly losing her
fortune owing to his insatiable demands for money?"
"No!" He sprang to his feet and his whole attitude showed distress and
anger. "I told her to make the fellow give her a proof, an unmistakable
proof, that he was indeed the brilliant inventor of whose fame we have
all been proud."
"And he furnished it, Oswald. You mean the medal which he received from
France, do you not? Well, he had it among his treasures in the cave, and
he showed it to her one day. It was the one thing, he declared, from
which he had never parted in all his adventurous career."
"You are dreaming! he never had _that_! Could not have had _that_! It
was some deception he practised upon you!" exclaimed the doctor, aghast
and trembling.
But she shook her lovely head, none the less beautiful because her locks
were becoming silvered on the forehead, and answered: "It was the very
medal we saw in our youth, with the French arms and inscription upon it.
Dr. Sutherland examined it, and Mr. Crouse says he remembers it well.
Besides it had his name engraved upon it and the year."
The doctor, to whom her words seemed to come in a sort of nightmare,
sank into his chair and stared upon her with such horror that she would
have recoiled from him in dismay had he been any other man than Oswald
Izard, so long loved and so long and passionately borne with,
notwithstanding his mysterious words and startling inconsistencies of
conduct.
"You do not know why this surprises me," he exclaimed, and hung his
head. "I was so sure," he added below his breath, "that this was some
impostor, and not Ephraim Earle."
"I know," she proceeded, after a moment, as soon, indeed, as she thought
he could understand her words, "that you did not credit his claims and
refused to recognize him as Polly's father. But I had no idea you felt
so deeply on the subject or I might have written to you long ago. You
have some reasons for your doubts, Oswald; for I see that your
convictions are not changed by this discovery. What is it? I am ready to
listen if no one else is, for he is blighting Polly's life and at the
same time shattering my son's hopes."
"I said—I swore to Polly that I had no reason," he declared, gloomily
dropping his eyes and assuming at once the defensive.
But she with infinite tact and a smile he could not but meet, answered
softly: "I know that too; but I am better acquainted with you than she
is, and I am confident that you have had some cause for keeping the
truth from Polly, which will not apply to me. Is there not something
connected with those old days—something, perhaps, known only to you,
which would explain your horror of this man's pretensions and help her
possibly out of her dilemma? Are you afraid to confide it to me, when
perhaps in doing so you would make two innocent ones happy?"
"I cannot talk about it," he replied with almost fierce emphasis.
"Ephraim Earle and I—" He started, caught her by the arm and turned his
white face toward the door. "Hush!" he whispered, and stooped his ear to
listen. She watched him with terror and amazement, but he soon settled
back, and waving his hand remarked quietly:
"The boughs are losing their leaves and the vines sometimes tap against
the windows like human fingers. You were saying——"
"_You_ were saying that Ephraim Earle and you——"
But his blank looks showed that he had neither understood nor followed
her. "Were you not good friends?" she asked.
"Oh, yes, oh, yes," he answered hastily; "too good friends for me to be
mistaken now."
"Then it is from his looks alone that you conclude him to be an
impostor?"
The doctor did not respond, and she, seeming quite helpless to move, sat
for a minute silently contemplating his averted face.
"I know you did not talk with him long. Nor have I attempted to do so,
yet in spite of everybody's opinion but your own I have come to the same
conclusion as yourself, that he is not Polly's father."
The doctor's lips moved, but no words issued from them.
"That is why I press the matter; that is why I am here to pray and
entreat you to save Polly and to save my son. _Prove this man a
villain_, and force him to loose his hold upon the Earle estate before
Polly's money is all gone!"
"Is it then a question of money?" asked the doctor. "Two months have
passed and you are afraid that he will dispose of twenty thousand
dollars!"
"He has already disposed of ten of them and the rest——"
"Disposed of ten thousand dollars!"
"Yes, for old gambling debts, pressing matters which Polly could not let
stand without shame."
"The wretch!" leapt from the doctor's lips. "Was there no one to advise
her, to forbid——"
"You were gone and Clarke was afraid of seeming mercenary. I think the
girl's secret terror of her father and her lack of filial affection
drove her to yield so readily to his demands for money."
An inarticulate word was the doctor's sole reply.
"And that is not the whole. Clarke's career is endangered and the
prospect of his carrying out his plans almost gone. Mr. Earle—I have
called him so—does not hesitate to say that he must have five thousand
dollars more by next October. If Polly accedes to this demand, and I do
not think we can influence her to refuse him, Clarke will have to forego
all hopes of becoming a member of the Cleveland firm, for he will never
take her last five thousand, even if she urges him to it on her knees."
"It is abominable, unprecedented!" fumed the doctor, rising and pacing
the room. "But I can do nothing, prove nothing. He has been received as
Ephraim Earle, and is too strongly intrenched in his position for me to
drive him out."
The absolutism with which this was said made his words final; and she
slowly rose.
"And so I too have failed," she cried; but seeing his face and noting
the yearning look with which he regarded her, she summoned up her
courage afresh and finally said: "They have told me—I have heard—that
this man made some strange threats to you in parting. Is that the reason
why you do not like to interfere or to proclaim more widely your opinion
of him?"
The doctor smiled, but there was no answer in the smile and she went
vehemently on: "Such threats, Oswald, are futile. No one less sensitive
than you would heed them for a moment. You are above any one's
aspersion, even on an old charge like that."
"Men will believe anything," he muttered.
"But men will not believe that. Do we not all know how faithfully you
attended Mrs. Earle in her last illness, and how much skill you
displayed? I remember it well, if the rest of the community do not, and
I say you need not fear anything this man can bring up against you. His
influence in town does not go so far as that."
But the doctor with unrelieved sadness answered with decision, "I cannot
make this man my enemy; he has too venomous a tongue." And she watching
him knew that Polly's doom was fixed and her son's also, and began
slowly to draw down her veil.
But he, noticing this action, though he had seemed to be blind to many
others she had made, turned upon her with such an entreating look that
she faltered and let her hand fall in deep emotion.
"Grace," he pleaded, "Grace, I cannot let you go without one kindly word
to make the solitude which must settle upon this room after your
departure, less unendurable. You distrust me."
"Does this visit here look like distrust?" she gently asked.
"And you hate me! But——"
"Do I look as if I hated you?" she again interposed, this time with the
look of an angel in her sad but beautiful eyes.
"Ah, Grace," he cried, with the passion of a dozen years let loose in
one uncontrollable flood, "you cannot love me, not after all these
years. When we parted——"
"At whose instigation, Oswald?"
"At mine, at mine, I know it. Do not reproach me with that, for I could
not have done differently.—I thought, I dreamed that it was with almost
as much pain on your side as mine. But you married, Grace, married very
soon."
"Still at whose instigation?"
"Again at mine. I dared not keep you from any comfort which life might
have in store for you, and the years which you have spent in happiness
and honor must have obliterated some of the traces of that love which
bound our lives together fifteen years ago."
"Oswald, Mr. Unwin was a good husband and Clarke has always been like an
own son to me, but——"
"Oh," interposed the doctor, starting back before the beauty of her
face, "don't tell me that a woman's heart can, like a man's, be the
secret sepulchre of a living passion for fifteen years. I could not bear
to know that! The struggle which I waged fourteen years ago I have not
strength to wage now. No! no! woman of my dreams, of my heart's dearest
emotion, loved once, loved now, loved always! tell me anything but
_that_,—tell me even that you hate me."
Her eyes, which had fallen before his, swam suddenly with tears and she
started as if for protection toward the door.
"Oh, I must go," she cried. "Clarke is waiting; it is not wise; it is
not seemly for me to be here." But the doctor, into whom a fiery glow
had entered, was beside her before she could reach the threshold. "No,
no," he pleaded, "not till you have uttered one word, one whisper of the
old story; one assurance—Ah, now I am entreating for the very thing, the
existence of which, I deprecated a few minutes ago! It shows how
unbalanced I am. Yes, yes, you can go; but, Grace, if you have ever
doubted that I loved you, listen to this one confession. Ever since the
day we parted, necessarily parted, fourteen years ago, I have never let
a week go by till these last few ones during which I have been away from
Hamilton, that I have not given up two nights a week to thinking of you
and watching you."
"Watching me!"
"Twice a week for fourteen years have I sat for an hour in Mrs.
Fanning's west window that overlooks your gardens. Thence, unnoted by
everybody, I have noted you, if by happy chance you walked in the
garden; and if you did not, noted the house that held you and the man
who sheltered your youth."
"Oswald,"—she felt impelled to speak, "if—if you loved me like this, why
did you send me that cruel letter two days after our engagement? Why did
you bid me forget you and marry some one else, if you had not forgotten
me and did not wish me to release you in order that you might satisfy
your own wishes in another direction?"
"Grace, if I could explain myself now I could have explained myself
then. Fate, which is oftenest cruel to the most loving and passionate
hearts, has denied me the privilege of marriage, and when I found it
out——"
"True, you have never married. Cruel, cruel one! Why did you not let me
know that you would always live single for my sake; it would have made
it possible for me to have lived single for yours."
The doctor with the love of a lifetime burning in his eyes, shook his
head at this, and answered: "That would have shown me to be a selfish
egotist, and I did not want to be other than generous to you. No, Grace,
all was done for the best; and this is for the best, this greeting and
this second parting. The love which we have acknowledged to-night will
be a help and not a hindrance to us both. But we will meet again, not
very soon, for I cannot trust a strength which has yielded so completely
at your first smile."
"Farewell, then, Oswald," she murmured. "It has taken the sting from my
heart to know that you did not leave me from choice."
And he, striving to speak, broke down, and it was she who had to show
her strength by gently leaving him and finding her own way to the door.
But no sooner had the night blast blowing in from the graveyard struck
him, than he stumbled in haste to the threshold, and drawing her with a
frenzied grasp from the path she was blindly taking toward the graves,
led her from that path to the high road, where Clarke was waiting in
some anxiety for the end of this lengthy interview. As the doctor gave
her up and saw her taken in charge by her son, he said with a thrilling
emphasis not soon to be forgotten by either of the two who listened to
them:
"Try every means, and be sure you bid Polly to try every means, to rid
yourselves of the bondage of this interloper. If all fails, come to me.
But do not come till every other hope is dead."
PART IV.
A PICKAXE AND A SPADE.
------------------------------------------------------------------------
XV.
THE SMALL, SLIGHT MAN.
TWO months had passed and the first snow was whitening the streets of
Hamilton. It was falling thick on Carberry hill, up which Clarke Unwin
was plodding early one evening on a visit to the Earle cottage.
His errand was one of importance. A crisis was approaching in his
affairs and he was determined to settle, once and for all, whether poor
Polly's money was to be sacrificed to her father's increasing demands,
or whether she could safely be allowed to follow her own wishes and give
five thousand dollars of it to the lover whose future fortunes seemed to
depend upon his possession of this amount.
Ephraim Earle had told her with something like a curse that he should
expect from her this very sum on the first of the month, but if this
demand were satisfied then Clarke's own hopes must go, for his friends
in the Cleveland works were fast becoming impatient, and Mr. Wright had
written only two days before that if the amount demanded from him was
not forthcoming in a fortnight, they would be obliged to listen to the
overtures of a certain capitalist who was only waiting for Clarke's
withdrawal to place his own nephew in the desired place.
Clarke Unwin had not visited the Earle cottage since Ephraim took up his
abode in it. Polly had refused to go there, and he himself felt no call
to intrude upon a man who was personally disagreeable to him, and whom
he could not but regard as a tyrant to the sweet girl whose life had
been all sunshine till this man came into it with his preposterous
demands and insatiable desire for money.
On this day, however, he had received her permission to present her case
to her father and see what could be done with him. Perhaps when that
father came to know her need he would find that he did not want the
money as much as he made out; at all events the attempt was worth
trying, and thus it was that Clarke braved the storm on this October
night to interview a man he hated.
As he approached the brow of the hill he heard a noise of mingled
laughter and singing, and glancing from under his umbrella he perceived
that the various windows of the cottage were brilliantly lighted. The
sight gave him a shock. "He is having one of his chess and checker
orgies," he commented to himself, and demurred at intruding himself at a
time so unfavorable. But the remembrance of his mother and Polly,
sitting together in anxious expectation of the good effects of his
visit, determined him to proceed; and triumphing over his own disgust,
he worked his way as rapidly as possible, and soon stood knee-deep in
the snow that was piled up before the cottage door. The wind was blowing
from the north and it struck him squarely as he raised his hand to the
knocker, but though it bit into his skin, he paused a moment to listen
to the final strains of old Cheeseborough's voice, as he sang with rare
sweetness a quaint old English ballad.
When it was over Clarke knocked. A sudden pushing back of chairs over a
bare floor announced that his summons had been heard, and presently he
had the satisfaction of seeing the door open and the figure of Mr. Earle
standing before him. Clarke did not wait to be addressed.
"I am Clarke Unwin," he announced. "May I be allowed the pleasure of a
few minutes' conversation with you?"
"A _few_ minutes," emphasized the other, drawing back with almost too
free an air of hospitable welcome. "I hope you will not limit yourself
to a _few_ minutes, my boy; we have too good company here for that." And
without waiting for any demur on the part of his more than unwilling
guest, he flung open a door at the right, and ushered him, greatly
against his will, into the large parlor where Clarke had last stood with
Polly at his side.
Just now it was filled with the choicest of the convivial spirits in
town, most of whom had been playing checkers or chess and smoking till
not a face present was fully visible. Yet Clarke, in the one quick
glance he threw about him, recognized most if not all of the persons
present—Horton by his oaths, which rang out with more or less
good-natured emphasis with every play he made, and the three cronies in
the corner by various characteristics well known in Hamilton, where
these men passed for "the three disgraces."
One person only was a perfect stranger to Clarke, but him he scarcely
noticed, so intent was he on his errand and the desire he had of
speaking to Mr. Earle alone.
"Hurrah! Come! Here's Clarke Unwin!" shouted a voice from the depths of
the smoky pall. "Brought your flute with you? Nobody comes here without
some means of entertaining the company."
"Off with your coat; there's snow sticking to it! Uh! You've robbed the
room of all the heat there was in it," grumbled old Cheeseborough, whose
fretfulness nobody minded because of the good nature that underlay it.
"Freedom Hall, this!" whispered Earle, still with that over-officious
air Clarke had noticed in him at the doorway. "Sit with your coat on, or
sit with it off; anything to suit yourself; only one thing we insist
on—you must take a good glass-full of this piping hot cider before you
speak a word. So much for good fellowship. Afterward you shall do as you
please."
"I have not come for enjoyment, but business," put in Clarke, waving the
glass aside and looking with some intentness into the face of the man
upon whose present disposition depended so much of his own happiness and
that of the young girl he had taken to his heart.
Earle, who had a secret pride in his own personal appearance which, now
that he was in good physical condition, was not without a certain broad
handsomeness, strutted back a pace and surveyed Clarke with interest.
"You are looking," said he, "to see how I compare with that picture over
your head. Well, as I take it, that picture, though painted sixteen
years ago, does not do me justice. What do you think?"
Clarke, somewhat taken aback, as much by the smile which accompanied
these words, as by the words themselves, hesitated for a moment and then
boldly said:
"What you have gained in worldly knowledge and intercourse with men you
have lost in that set purpose which gives character to the physiognomy
and fills all its traits with individuality. In that face on the wall I
see the inventor, but in yours, as it now confronts me, the——"
"Well, what?"
"The centre of this very delightful group," finished Clarke, suavely.
It was said with a bow which included the whole assembly. Earle laughed
and one or two about him frowned, but Clarke, heeding nobody, asked if
he could not have a moment's conversation with his host in the hall.
Earle, with a side glance directed, as Clarke thought, toward the one
slight man in the corner whose face was unfamiliar to him, shook his
head at this suggestion and blurted out: "That's against the rules. When
the Hail-Fellow-Well-Met Society comes together it is as one body. What
is whispered in one corner is supposed to be heard in the next. Out with
your business then, here. I have no secrets and can scarcely suppose you
to have."
If this was meant to frighten Clarke off it did not succeed. He
determined to speak, and speak as he was commanded right there and then.
"Well," said he, "since you force me to take the town into our
confidence, I will. Your daughter——"
"Ah," quoth Earle, genially, "she has remembered, then, that she has a
father. She sends me her love, probably. Dear girl, how kind of her on
this wintry night!"
"She sends you her respects," Clarke corrected, frankly, "and wants to
know if you insist upon having the last few dollars that she possesses."
"Oh, what taste!" broke in the father, somewhat disconcerted. "I did
think you would have better judgment than to discuss money matters in a
social gathering like this. But since you have introduced the topic you
may say to my dutiful little girl that since I have only asked for such
sums as she is perfectly able to part with, I shall certainly expect her
to recognize my claim upon her without hesitation or demur. Have you
anything more to say, Mr. Unwin?"
Clarke, whose eye had wandered to the stranger in the corner, felt no
desire to back out of the struggle, unpleasing as this publicity was. He
therefore answered with a determined nod, and with a few whispered words
which caused a slight decrease in the air of bravado with which his host
regarded him.
"You persist," that individual remarked, "notwithstanding the rules I
have had the honor of quoting to you? I should not have expected it of
you, Mr. Unwin; but since your time is short, as you say, and the
subject must be discussed, what do you advise, gentlemen? Shall I listen
to the plea of this outsider—outsider as regards this meeting, I mean,
not as regards my feelings toward him as a father—and break our rules by
taking him into another room, or shall I risk a blush or two for my
charming little daughter's perversity, and hear him out in your very
good company and perhaps, under your equally good and worthy advice?"
"Hear him here!" piped up Cheeseborough, whose wits were somewhat
befuddled by something stronger than cider.
"No, no, shame!" shouted Emmons. "Polly is a good girl and we have no
business meddling with her affairs. Let them have their talk upstairs. I
can find enough here to interest me."
"Yes, yes, there's the game! Let's finish the game! Such interruptions
are enough to spoil all nice calculations."
"You were making for the king row."
"Checkmate in three moves!"
"Here! fill up my glass first!"
"I declare if my pipe hasn't gone out!"
Clarke, who heard these various exclamations without heeding them,
glanced at Earle for his decision, but Earle's eye was on the man in the
farthest corner.
"Well, we'll go upstairs!" he announced shortly wheeling about and
leading the way into the hall. Clarke followed and was about to close
the door behind him when a slim figure intervened between him and the
door, and the stranger he had previously noticed glided into the hall.
"Who's this?" he asked, noticing that this man showed every sign of
accompanying them.
"A friend," retorted Earle, "one of the devoted kind who sticks closer
than a brother."
Clarke, astonished, surveyed the thin young man who waited at the foot
of the stairs and remarked nonchalantly, "I do not know him." Earle,
with a shrug of the shoulders, went upstairs.
"You may have the opportunity later," he dryly remarked; "at present,
try and fix your attention on me." They proceeded to the inventor's
workroom, where they found a light already burning.
"Sit down!" commanded Earle, with something of the authority which his
years, if not his prospective attitude toward the young man warranted.
But he did not sit himself, nor did the friend who had followed him
upstairs and who now hovered about somewhere in the background. "It will
take Emmons just ten minutes to perfect the 'mate' he has threatened,"
observed Earle as they faced each other. "Can you finish your talk in as
short a time? For I must be down there before they start a fresh game."
"Five minutes should suffice me," returned Clarke, "but you may need a
longer time for argument. Shall I state just what our situation is as
regards this money you want from Polly?"
"If you will be so good!"
"With that man listening in the doorway?"
"With that man listening in the doorway."
"Polly has no money to spare, Mr. Earle. Of the twenty thousand left her
you have already had ten——"
"For my just debts, Mr. Unwin."
"For your just debts, granted, Mr. Earle, but those debts were not
incurred for her benefit, nor have you ever deigned to particularize to
her just what they were."
"I would not burden her young mind."
"No, it has been enough for you to burden her purse."
"I should have burdened her conscience had I neglected to ask for her
assistance."
"And will you now, by declining to take away her last hope, allow her
the means of retrieving the fortune of which you have so nearly robbed
her?"
"Her hopes? Her means? I think you are speaking for yourself, sir."
"In speaking for myself, I speak for her; our interests are identical."
"You flatter yourself; Miss Earle is not yet your wife."
"Would you come between us?"
"God forbid! I am willing that Polly, as you call her, should marry whom
she will—when I am dead."
"Or when you have robbed her of every cent she owns."
"Oh, what language! I marvel you have not more delicacy of expression,
Mr. Unwin. Your father was noted for his refinement."
"He had not to deal with—" the word was almost out, but Clarke
restrained himself—"with a man who could forsake his motherless child in
her tender years, only to expect unbounded sacrifices from her when she
has attained maturity."
"I expect no more than she will be glad to grant. Maida has pride—so
have you. You would neither of you like to see her father in jail."
Clarke bounded to his feet.
"We do not imprison men here for debt," he cried.
"No, but you do for theft."
The word, so much worse than any he was prepared for, turned Clarke
pale. He looked to right and left and shrank as he caught the eye of the
slim watcher in the hall beyond.
"You surely are not a criminal," he whispered. "That man——"
"Never mind that man. Our ten minutes are fast flying by and you do not
yet seem to see that I cannot afford to relinquish my hold on Polly."
"Do you mean that your debts——"
"Were incurred in private? Certainly, and under circumstances which
place me in a dilemma of no very pleasing nature. If they are not all
paid by the first of next month, I shall have to subject my very
conscientious little daughter to the obloquy of visiting her father in
prison. It is a shame, but such is the injustice of men."
"You have stolen then?"
"Too harsh a word, Clarke. I have borrowed money for the purpose of
perfecting my experiments. The experiments have failed, and the
money—well, the man from whom I borrowed it will have it, that is all.
He is strict in his views, notwithstanding his long forbearance."
"Who is this man? I should like to talk to him. That fellow behind you
is surely not he?"
"Oh, no; he is only a detective."
"A detective!"
"Who likes my table and bed so well he never knows when he has had
enough of either."
"Shameful!" sprang from Clarke's set lips, as his eyes flew first to the
watchful but nonchalant figure in the hall, and then to the tall,
commanding form of the man who could accept his degrading situation with
such an air of mingled sarcasm and resignation.
"And you are the man to whom the French government sent her badge of
honor!"
"The same, Clarke," tapping his breast.
"And you dare to call Polly your child; dare to return to Hamilton with
this disgrace upon you, to make her life a hell and——"
"Maida is my child; and as for this disgrace, as you call it, it will be
easy enough for her to elude that; a certain check drawn on her bank and
signed by her name will do it."
"I should like to be sure of that," returned Clarke, springing back into
the hall and confronting the man who stood there. "If you are a
detective," said he, "you are here in the interest of the man whom Mr.
Earle has robbed?"
The slight young man, in no wise disconcerted, smiled politely, but with
an air of quiet astonishment directed mainly toward Ephraim Earle.
"I am here in the interest of Brown, Shepherd, & Co., certainly," said
he. "But I have uttered no such word as robbed, nor will, unless the
first of the month shows Mr. Earle's indebtedness to them unpaid."
"I see. In what city does Brown, Shepherd, & Co. do business?"
"In New York, sir."
"Merchants, lawyers, bankers, or what?"
"Bankers."
"Oh, I remember; in Nassau street?"
"Just so."
Mr. Earle, who had taken up a cigar from his table while this short
colloquy took place, stepped forward.
"A very strict firm, thorough, and not much given to showing mercy, eh?"
"Not much," smiled the man.
"You see!" gesticulated Mr. Earle, turning to Clarke with a significant
smile.
Clarke, with a sudden heartsick sense of what this all meant to him,
assumed a stern air.
"Mr. Earle," said he, "I must entreat that you come at once and present
this matter to Polly. She ought to know particulars, that she may judge
whether or not she will sacrifice her fortune to save you from the
disgrace you have incurred."
"What, now, with my house full of guests? Impossible. The affair will
keep till to-morrow. I will be down to-morrow and tell her anything you
wish."
"She cannot wait till to-morrow. I must send the letter to-morrow which
decides my future."
"That's unfortunate; but you can send your letter all the same. I know
what her decision will be."
Clarke felt that he knew too, but would not admit it to himself.
"I have said my say," he remarked. "Either you will let her know your
precise position to-night, or I will take it upon myself to ask her for
the money for my own uses. She will not deny me, if I press her, any
more than she will probably deny you. So take your choice. I am going
back to the friends below."
Earle, who had not expected such condign treatment from one whom he had
hitherto regarded as a boy, glanced at the detective, and, with the
characteristic shrug he had picked up in foreign countries, cried out in
somewhat smothered tones, in which caution struggled oddly with his
natural bravado:
"Well, we'll compromise. I cannot leave the H. F. W. M.; but I'll tell
you what I will do. I'll write out the situation for my daughter, and
you shall carry the paper with you. Won't that do, considering the
circumstances, eh?"
Clarke, to whom this man's character was a perfect anomaly, murmured a
hesitating consent and hurried down into the room below. Earle followed
him, and, entering with frank jocularity, in striking contrast with the
other's dejected appearance, he cheerfully called out:
"Well, I've convinced the boy, somewhat, against his will, I own, that a
few thousands spent on the invention I have now on hand will bring in a
much larger fortune to Maida than that I have perhaps rather recklessly
expended. It was just so when I was perfecting my first invention, don't
you remember? Every dollar I spent on it was begrudged me, and yet see
what an outcome there was to it at last."
"Yes, yes; but where is all that money now?" queried old Cheeseborough,
wagging his iron gray head. "Nobody here ever saw a dollar of it, and I
have heard people say they don't believe you ever got it."
"Would you bring up the saddest hours of my life?" asked Earle, with a
sudden cloud on his brow. "I got the money, but—" he stopped, shook
himself and changed his tone for one of cheerful command. "Here, you!
Start a fresh game, Emmons. I see that your checkmate is good. I've got
to write a letter. Who will bet that I won't get my six pages done
before Hale will succeed in getting three men into the king row?"
"I will!"
"Put down your dollar then!"
"There it is."
"And there's mine, with a condition to boot. I'll write the letter _in
this room_, and give Cheeseborough another chance at a song, if you say
so."
"Done! Fire away, old man; here goes my first move!"
"And here my first word."
And, to Clarke's mingled surprise and disgust, Earle threw himself down
before a table, took up a pen and began to write. Cheeseborough piped up
with his thin, sweet voice something between a dirge and a chant, and
Horton went on with his oaths.
------------------------------------------------------------------------
XVI.
THE LETTER.
WHEN Ephraim Earle had taken up his abode in the cottage on the hill,
Mrs. Unwin had moved into a small house on a side street in the lower
part of the town. In the cozy parlor of this same house, she was now
sitting with Polly, waiting for her son's return.
He had been gone a couple of hours, and both Mrs. Unwin and Polly were
listening anxiously for the sound of his step on the porch. Polly, with
the impatience of youth, was flitting about the room and pressing her
face continually against the icy panes of the window, in a vain endeavor
to look out; but Mrs. Unwin, to whom care had become a constant
companion during these last months, was satisfied to remain by the fire,
gazing into the burning logs and dreaming of one whose face had never
vanished from her inner sight since that fatal evening she had seen it
smile again upon her as in the days of her early youth. Yes, she was
thinking of him while Polly was babbling of Clarke; thinking of the last
sentence he had uttered to her, and thinking also of the vague reports
that had come to her from day to day, of his increased peculiarities and
the marked change to be observed in his appearance. Her heart was
pleading for another sight of him, while her ear was ostensibly turned
toward Polly, who was alternately complaining of the weather and
wondering what they should do if her father insisted upon having the
money, right or wrong. Suddenly she felt two arms around her neck, and
rousing herself, looked down at Polly, who in her restlessness had
fallen on her knees before her and was studying her face with two bright
and very inquiring eyes.
"How can you sit still," the young girl asked, "when so much depends
upon the message Clarke will bring back?" Mrs. Unwin smiled, but not as
youth smiles, either in sorrow or in joy, and Polly, moved by that
smile, though she little understood it, exclaimed impetuously:
"Oh, you are so placid, so serene! Were you always so, dear Mrs. Unwin?
Have you never felt angry or impatient when you were kept waiting or
things did not go to your liking?"
The sweet face that was under Polly's steady gaze flushed for an instant
and the patient eyes grew moist. "I have had my troubles," admitted Mrs.
Unwin, "and sometimes I have not been as patient with them as I should.
But we learn forbearance with time, and now——"
"Now you are an angel," broke in Polly.
"Ah!" was Mrs. Unwin's short reply, as she stroked the curly head
nestling in her lap.
"Clarke says that whatever happens I must be brave," babbled the
forlorn-hearted little girl from under that caressing hand. "That
poverty is not so dreadful, and that in time he will win his way without
help from any one. But Oh, Mrs. Unwin, to think I might be the means of
giving him the very start he needs, and then to be held back by one—Dear
Mrs. Unwin, do you think it wicked to hate?"
The question was so sudden, and the vision of the girl's uplifted head
with its flashing eyes and flushed cheeks so startling, that Mrs. Unwin
hesitated for a moment, not knowing exactly what to say. But Polly,
carried away now by a new emotion, did not wait for any answer.
"Because I am afraid I really hate him. Why has he come into our lives
just when we don't want him; and why does he take from us everything we
have? If he loved me I could bear it possibly, but he don't even love
me; and then—and then—he lives in such a way and spends his money so
recklessly! Don't you think it is wrong, Mrs. Unwin, and that I would be
almost justified in not giving him everything he asks for?"
"I should not give him this last five thousand, unless he can show you
that his need is very great. No one will blame you; you have been only
too generous."
"I know, I know, and I am sure you are right, but notwithstanding that,
something assures me that I shall do just what he wishes me to. I cannot
refuse him—I do not know why, perhaps because he _is_ my father."
Mrs. Unwin, whose face had assumed a look of resolution as Polly said
this, impulsively stooped and inquired with marked emphasis, "Then you
feel—you really feel at last, that he is your father? You have no doubt;
no lurking sensation of revolt as if you were sacrificing yourself to an
interloper?"
Polly's head sank on her clasped hands, and she seemed to weigh her
answer before replying; then she responded with almost an angry
suddenness.
"I wish I could feel he is not what he pretends to be, but the
villainous impostor Dr. Izard considers him. But I cannot. No, no, I
have no such excuse for my antipathy toward him."
Mrs. Unwin leaned back, and her countenance resumed its dreamy
expression.
"Then I shall not advise you," said she. "You must follow the dictates
of your own conscience."
Polly rose and ran again to the window, this time with a cry of joy. "He
is coming! Clarke is coming! I hear the gate click," and she bounded
impatiently toward the door.
In a few minutes she returned with her lover; he had a letter in his
hand and he was contemplating her with saddened eyes.
"You will need courage, dear, to read this," said he. "It is from your
father and it puts his case before you very clearly—too clearly,
perhaps. Your estimate of him was not far from correct, Polly. The story
of his past life is not one you can read without shame and humiliation."
"I knew it! I saw it in his face the first time I looked at him. I saw
it before. I saw it in his picture. O Clarke, I shrink even from his
writing; must I read this letter?"
"I think you should; I think you should know just what threatens us if
you refuse him the money."
Polly took the letter.
"You have read it?" she inquired.
But Clarke shook his head.
"I know the nature of its contents, but I did not wait to read the
letter. He wrote it in a roomful of men, under a wager——" Clarke paused;
why hurt her with these details? "But what does that matter? It is the
facts you want. Come, screw up your courage, dear; or stay, let me read
it to you."
She gave him the letter and he read to her these words:
DEAR MAIDA: You wish to know why I want another five thousand
dollars after having received a good ten thousand from you already.
Well, I am going to tell you. I have two passions, one for
mechanical invention and one—I must be candid or this letter will
fail in its object—for wild and unlimited pleasure. When I was young
I had not enough money to indulge in but one of these instincts, but
on the day I saw twenty thousand dollars in my hand, my other
passion, long suppressed, awoke, and notwithstanding the fact that
your mother lay dying in the house, I resolved to leave the town
where I was known as soon as she was decently buried, for as I said
to myself, the possession of twenty thousand dollars means the
making of a fortune in Monte Carlo, and a maddening good time of it
meanwhile.
But twenty thousand dollars do not always bring a fortune, even in
Monte Carlo. I lost as well as won and though I had the good time I
had anticipated I was not much richer at the end of five years than
I was before my first invention was perfected. And then came a
struggle. My good times grew fewer and I was forced to change my
name more than once as I drifted from France to Italy and from Italy
to Germany, seeking to reinstate myself, but being dreadfully
hampered by my taste for the luxuries of life and the companionship
of men who were sufficiently good-natured, but not always honest or
sincere. At last I awoke to the necessity of action. I had an
idea—one that had been floating in my head ever since the perfection
of my first invention, and I realized that if I could but develop it
practically I was sure to win a greater sum than that I had earned
by my first efforts. But to do this it would take money—considerable
money, and I had none. Now how could I remedy this defect? I knew
but one way—by play. So I began to play for keeps, that is for a
capital, denying myself this time and forgetting for once the
delights that can be got out of a thousand francs. I saved, actually
saved, and becoming strangely prosperous the moment I set a distinct
purpose before my eyes, I won and won till I had a decided nest-egg
laid up in the leathern bag which I secretly wore tied about my
waist. But though this looked well, it did not satisfy me. I wanted
thousands and I had but hundreds; so I took a partner who was not
above a trick or two and—well, you do not understand these
things—but matters went very smoothly with me after this, so
smoothly that possibly I might have allowed myself one little
glimpse into my old paradise if I had had a little more confidence
in my own discretion and had not been afraid of the charms of a spot
that swallows a man, neck and crop, if he once plunges his head into
it. So for a few months more, I remained firm and grew steadily
rich, till the day came when by an enormous streak of luck I became
the owner of the very amount I had calculated it would take to put
into operation my new invention.
I was in St. Petersburg when this happened, and for five hours I sat
in my garret chamber feasting my eyes upon the money I had acquired,
and shutting my ears to every sound from without that summoned me to
the one short hour of wild enjoyment I had certainly earned. Then I
put the money back into my bag, took the frugal supper I had
prepared and went to bed with the determination of rising early and
devoting the early hours of the morning to drawing my first plans.
But in that sleep _I forgot the essential idea upon which the whole
thing rested_. It went from me as utterly as if it had been wiped
out. In vain I prodded my memory and called upon all the powers of
earth and air to assist me in my dreadful dilemma. I no more knew
where to place the lines I had for years seen clearly before me than
if I had never conceived the thing or seen it a completed object in
my mind's eye. Success had dampened my wits, or in the long struggle
with my second passion I had lost my hold upon the first. The money
necessary to elucidate the idea was mine, but I had lost the idea!
The situation was maddening.
Fearing the results of this unexpected disappointment upon my
already weakened self-control, I fled to my partner, who was a good
fellow in the main, and begged him to take and keep for a week my
leather bag with its valuable contents, adding that he was not to
give it back to me till the seven days were up, even if I entreated
him for it on my knees. He promised, and greatly relieved I left him
for a stroll through the streets. You see I hoped to regain my idea
before the week was out. But alas for the weakness of human nature!
Instead of keeping my mind upon work, I spent my time in gorgeous
rooms hung with mirrors in which was reflected every lovely thing I
worshipped. I heard music, and—but why enlarge the vista further?
Not having any goal for my energy, I fell, and when I got my money
back, I lived another five years of boundless luxury.
When the last dollar went, I fell sick. I was in New York now,
calling myself Harold Deane, and I boarded in a humble
boarding-house in Varick street where there was one kind woman who
looked after me without asking whether I had any money to pay for my
keep. I sent fifty dollars to that woman out of the first money you
gave me, my dear. Pardon the digression. I merely wished to show you
that I am not without gratitude. When I recovered from my delirium
and lifted up my head again in this wicked, fascinating world, my
mind was clear as a bell and I saw, all in a minute, the machine
again, line for line, whose action was to transform trade and make
me a millionaire. Though I was too weak to sit up, I called out for
pencil and paper, and at the risk of being thought crazy, scrawled a
rude outline of the thing I had lost so long from my consciousness
and which I held now by such uncertain tenure that I feared to lose
it again, if I let the moment go by. This I put under my pillow. But
when I awoke from the sleep which followed, the drawing was gone,
destroyed by the good woman who thought it the mad scrawling of a
delirious man. But this loss did not trouble me at this time, for
the image remained clear in my mind and I was no longer afraid of
losing it.
But again I had no money, and confident that in this country and in
my present condition it would be useless for me to seek it in the
old way, I cast about in my mind how to obtain it by work. Reason
pointed out but one course. To get into some large business or
banking establishment, and after winning the confidence of the
moneyed men I would thus meet, to reveal my idea and obtain their
backing. But this was no easy matter for a poor wretch like me. My
life had left its imprints on my face, and I had neither means nor
friends. But I had something else that stood me in good stead. I had
audacity and I had wit, together with a sound business instinct as
regards figures. And so in time I was successful and was taken into
the banking house of Brown, Shepherd, & Co. in Nassau street.
Again I had an incentive toward thrift. For three months I worked
for their good-will, and after that for the good of my purse. This
latter phrase may not be plain to you, but when you consider the
possibilities opened by a banking house to enrich a man accustomed
to use his wits,—possibilities so much greater than those afforded
by the selfish consideration of a few capitalists with whom one in
my position comes in contact,—you can understand me more readily. At
the end of that time I had fifteen thousand dollars laid away; and
the company did not even know that they had sustained any loss.
Well, I meant to repay them when I realized my fortune, but—luck has
been against me, you know—the sight of the money was too much for me
one night, and I forgot everything in a wild spree which lasted just
one week.
When it was over and I came to myself I found that I had again
forgotten the essential part of my invention, and that the money,
which I always carried in the old bag about my waist and which I had
never lost sight of before, was also gone, leaving me destitute of
everything but the clothes I wore. I was desperate then and thought
of killing myself, but I hated blood and have a horror of poison, so
I delayed, expecting to go back to the banking house as soon as my
appearance would warrant it. But I never went. I received from some
unknown friend a warning that my absence had provoked inquiry, and
that my reappearance in Nassau street would be the signal for my
arrest, so I not only kept away from that part of the city, but left
the town as soon as I had money to do so, wandering as far west as
Chicago and sinking lower and lower as the weeks went by, till my
old trouble gripped me again and I found myself in a hospital, given
up for dead. The name by which I was entered there was Simeon
Halleck, but I had worn a dozen during my lifetime.
I was regarded by those around me as a stray and by myself as a lost
man, when suddenly one night, no matter how, I learned, my little
daughter, that you, whose existence I had almost forgotten, was not
only alive and well, but likely to become the inheritor of a pretty
fortune. At this I plucked up courage, conquered my disease and came
out of the hospital a well man. Having been known as Simeon Halleck,
it was necessary for me now, in order to present myself as Ephraim
Earle, to lose my old identity before I assumed my new,—or rather, I
should say, my real one. How I did this would not interest you, so I
will pass on to the day when, with my beard grown a foot, I ventured
into this town and began to look around to see whether there was any
place left for me in the hearts of my old friends or in the
affections of my child. I found, as I thought—was it rightly?—that I
would receive a decent welcome if I returned, and so after a proper
length of time I re-entered Hamilton, this time shaven and shorn,
and boldly announced my claims and relations to yourself.
The results of this action I am reaping to-day, but while I am happy
and cared for, I do not find myself in a position to enjoy the full
benefits of my position from the facts, now to be explained, that
the police of New York are sharper than I thought, and when I went
to Boston, after my first trip to this town, I found myself
confronted by an agent of Brown, Shepherd, & Co. They had discovered
my theft and threatened me with a term in state prison. My dear, I
knew that no daughter with a fortune of twenty thousand dollars
would wish to see her father suffer from such disgrace, so I made a
clean breast of it and told him all my hopes, and promised if the
firm I had robbed would give me three months of freedom I would
restore them every penny I had taken from them. As they could hope
for nothing if they landed me in jail, they readily acceded to my
request, and I came to Hamilton followed by a detective, and with
the task before me of obtaining fifteen thousand dollars from you in
three months. Ten of these you have cheerfully given me, but you
cavil at the last five.
Will you cavil any longer when you realize that by denying them to
me you will land me in prison and brand your future children with
the disgrace of a convict grandfather? I would say more, but the
time allotted me for writing this letter is about up. Answer it as
you will, but remember that however you may writhe under the yoke,
you are blood of my blood and your honor can never be disassociated
from mine in this world or the next.
Your loving father,
EPHRAIM EARLE.
P. S. I have Brown, Shepherd, & Co.'s written promise that with the
payment of this last five thousand, all proceedings against me shall
be entirely stopped, and that neither as a firm nor as individuals
will they remember that Ephraim Earle and Simeon Halleck are one.
------------------------------------------------------------------------
XVII.
MIDNIGHT AT THE OLD IZARD PLACE.
CLARKE knew when he began to read this letter what effect it was likely
to have on his own prospects, but he was little prepared for the change
it was destined to make in Polly. She, who at its commencement had been
merely an apprehensive child, became a wan and stricken woman before the
final words were reached; her girlish face, with its irresistible
dimples, altering under her emotions till little of her old expression
was left. Her words, when she could speak, showed what the recoil of her
whole nature had been from the depths of depravity thus heartlessly
revealed to her.
"Oh, what wickedness!" she cried. "I did not know that such things could
be! Certainly I never heard anything like it before. Do you wonder that
I have always felt stifled in his presence?"
Mrs. Unwin and Clarke tried to comfort her, but she seemed to be
possessed of but one idea. "Take me home!" she cried; "let me think it
out alone. I am a disgrace to you here; he is a thief and I am the
daughter of a thief. Until every cent that he has taken is returned, I
am a participator in his crime and not worthy to look you in the face."
They tried to prove to her the fallacy of this reasoning, but she would
not be convinced. "Take me home!" she again repeated; and Clarke out of
pure consideration complied with her request. She was still living with
the Fishers, but when they reached the humble doorstep which had been
witness to many a tender parting and loving embrace, Polly gave her
lover a strange look, and hardly lingered long enough to hear his final
words of encouragement and hope.
"I will see you to-morrow," she murmured, "but I can say no more
to-night—no, not one word"; and with something of the childish petulance
of her earlier years she partially closed the door upon him, and then
was half sorry for it, when she heard the deep sigh that escaped him as
he plunged back into the snow that lay piled up between the house and
the gate.
"I am wicked," she muttered, half to herself, half to him; "come back!"
but the words were lost in the chilly wind, and in another moment he had
reached the street and was gone. Had he looked back he would not have
disappeared so suddenly, for Polly, as soon as she thought herself
alone, suddenly pushed open the door, peered out and, with a momentary
hesitation, slipped out again into the street.
The snow had ceased falling, the moon had come out and was lighting up
the great trees that lined either side of the road. Polly cast one look
down the splendid but deserted vista, and then with the thoughtless
daring which had always signalized her, began running down the street
towards that end of the town where the road turns up towards the
churchyard. She was guided by but one thought, the necessity of seeing
Dr. Izard before she slept. The thickness of the snow beneath her feet
impeded her steps and made the journey seem long to her panting
eagerness. She met nobody, but she thought nothing of that, nor did she
note that the lights were out in the various houses she passed. Her mind
was so full of her purpose that the only fear of which she was conscious
was that she would find the doctor away or deaf to her summons. When the
tavern was passed and the shadow of the church reached, she drew a deep
breath. Only a few steps more and she would be passing the gateposts in
front of the Izard mansion. But how still everything was! She seemed to
realize it now, and was struck by the temerity of her action, as the
desolate waste of the churchyard opened up before her and she heard,
pealing loud above her head, the notes of the great church-clock
striking eleven!
But she knew that the doctor never retired before twelve, and the need
she felt of an immediate consultation with one who had known her father
in his youth, buoyed her up, and dashing on with a shudder, she turned
the corner and came abreast with the house she was bound for. But here
something which she saw, first dazed, then confounded her. The house was
lighted! The Izard house, which had been vacated for years! Had the
doctor found a tenant then without her knowledge, or, led by some
incomprehensible freak, had he lighted it up himself?
While she was gazing and wondering, almost forgetting her own purpose in
her astonishment at this unwonted sight, there rose a sudden wild halloo
behind her, followed by the shouts of drunken voices and the sound of
advancing footsteps. The visitors at her father's cottage had reached
the main street, and, seeing the lighted mansion, were as much struck by
its unwonted appearance as she had been, and were coming down the road
for a nearer inspection.
Alarmed now in good earnest, and by a more natural fear than that which
had first agitated her, she looked around for a spot to hide in, and,
finding none, plunged towards the house itself. What she expected to
gain by this move she hardly knew; but once on the porch, and in the
shadows of the great pillars supporting it, she felt easier; and, though
she knew this laughing, careless crowd would soon be upon her, she felt
the nearness of the life within to be a safeguard, and, stretching out
her hand toward the front door, she was amazed to find it yield to her
touch.
Under most circumstances this would have frightened her away, or, at
least, would have awakened in her the instinct of alarm; but now the
illuminated hall, dimly to be seen through the crack she had made,
seemed to offer her a refuge, and she rushed in, closing and locking the
door behind her. Instantly the desolation of these long disused rooms
settled upon her, and she peered down the hall in terror, dreading and
half hoping to see some one, she did not care whom, stalk from some of
the several rooms on either side. But no one came, and the seeming lack
of life in the spaces about her soon grew more terrifying than any
appearance of man or woman would have been. The light which lured her
into this desolate structure came from a lamp standing on a small table
at the rear of the hall, and presently she found herself insensibly
approaching it, having recognized it as one she had often seen in the
doctor's study.
But when she had stepped as far as the circular landing opening under
the stairs, and noted the little winding staircase leading down from it
into the space below, some faint recognition of the fact that this was
the way to the doctor's study came over her, and, advancing breathlessly
on tiptoe to the railing which guarded this spot, she looked down into
the well beneath, and was startled at the gust of wind which met her
there, with all the chill of the outside air in it. Was the famous green
door below open, and did this wind come from the graveyard?
She was conscious that she had no right to advance a step farther, and
yet she knew that she must find the doctor, if only to throw herself
upon his protection. So, with many a qualm and sinking of the heart, she
caught up the lamp from the table near by and descended the short
spiral, rightfully thinking that it would be wiser to thus flash upon
the doctor in a blaze of light rather than to take him by surprise in
the darkness. Finding the green door open, as she had expected, she
tried to raise her voice and utter the doctor's name, but articulation
failed her. There was something so weird in her position that her usual
recklessness failed to support her, and she had hardly the courage to
glance into the room before which she stood, though instinct had already
told her it was empty.
The wind which had met her at the top of the staircase increased as she
descended, and while she was drawing in her breath before it, the light
went out in her hand and she was left standing half in and half out of
the doctor's study in a condition of helplessness and terror. But this
misfortune, while it abashed her, was of decided benefit in the end. For
no sooner was this light out than she was met with the glimmering rays
of a lantern, shining in from the graveyard without, and knowing this to
be an indication of the doctor's whereabouts, she set down the lamp and
was advancing with some trepidation toward the door when her ears caught
a sound—the most dreadful that could be heard in that place—that of a
spade being forced into the icy ground.
Instantly her heart became the prey of a thousand sickening emotions.
What was the doctor doing? Digging a grave? Impossible. And yet what
else would make a sound like this? Even her usually bold spirit was
startled and she shrank at the thought, wishing for Clarke, for her
father, for any one to support her and take her out of the horrible,
moonlighted spot where homes were being made for the dead in the dark of
night.
She could not retreat and she dared not advance, yet she felt that she
must settle her doubts by one glimpse of what was going on. Approaching
the window she peeped out and saw—Merciful heavens, was that the
doctor?—that wild figure clad in a long wool garment which swept to his
heels, and digging with such frenzy and purpose that the snow flew from
his spade in clouds? She was so absorbed in the sight that it was a
moment before she saw that it was her mother's grave he was unearthing
and that he was doing this in his sleep. But when she fully realized the
awful fact she uttered a low cry of irrepressible dismay, and no longer
fearing anything but this unearthly figure she had chanced upon in the
moonlight, she dashed from the spot and fled up the highway, never
resting foot or stopping to breathe till she found herself in her own
room at home.
Dr. Izard was mad and she alone knew the frightful secret.
------------------------------------------------------------------------
XVIII.
A DECISION.
WHEN Dr. Izard rose the next morning it was with a feeling of lassitude
and oppression that surprised him. He had received no calls from
patients the evening before, nor had he retired any later than usual.
Then why this strained and nervous feeling, as if he had not slept? The
snow that had fallen so heavily the day before had cleared the air, and
the dazzle of sunshine finding its way into his unusually darkened den
prepared him for the brilliant scene without. It was not in that
direction, however, he first looked, for he was no sooner on his feet
than he noticed that the green door which he always kept shut and
padlocked was open, and that in the hall beyond a spade was standing,
from the lower edge of which a small stream of water had run, staining
the floor where it rested.
What did it mean, and what was the explanation of the dark stains like
wet mould on the skirt of the long wool garment that he wore? He looked
from one to the other, and the hair rose on his forehead. Summoning up
all his courage he staggered to the window and drawing the curtain back
with icy fingers, glanced out. Some vandal had been in the graveyard;
one of the graves had been desecrated and the snow and mould lay
scattered about. As he saw it he realized who the vandal had been, and
though no cry left his lips, his whole body stiffened till it seemed
akin to the one he had so nearly disinterred in the night. When life and
feeling again pervaded his frame he sank into a chair near the window
and these words fell from his lips: "My doom is upon me. I cannot escape
it. The will of God be done."
The next instant he was on his feet. He dressed himself in haste,
shuddering as he bundled up the stained night-robe and thrust it into
the blazing fire of the stove. Then he caught up the spade, and opening
the outside door stepped into the glittering sunshine. As he did so he
noticed two things, equally calculated to daunt and surprise him. The
first was the double row of his own footsteps running to and fro between
the step and the heap of dirt and snow beside the monument; and the
other, an equally plain track of footsteps extending from the place
where he stood to the gate on his left. The former were easily
explainable, but the latter were a mystery; for if they had been made by
some nocturnal visitor, why were they all directed toward the highway?
Had not the person making them come as well as gone? Puzzled and no
little moved by this mystery, he nevertheless did not pause in the work
he had set for himself.
Crossing in haste to the monument, he began throwing back the icy
particles of earth he had dug up in the night. Though he shuddered with
something more than cold as he did so, he did not desist till he had
packed the snow upon the mould and left the grave looking somewhat
decent. A sleigh or two shot by on the open thoroughfare without while
he was engaged in this work, and each time as he heard the bells he
started in painful emotion, though he did not raise his head nor desist
from his labor. When all was done he came slowly back, and pausing
before the second line of footsteps he examined them more carefully.
It was a woman's tread or that of a child, and it came from his own
door. Greatly troubled he rushed into the track they had made and
trampled it fiercely out. When he reached the gate he stepped into the
highway. The steps had passed up the street. But what were these he now
perceived in the inclosure beyond the picket fence, going straight to
the house and stopping before the front door? They came from the street
also, and they pointed inward and not outward. Was he the victim of some
temporary hallucination, or had a woman entered the house by the
never-opened front door and come out through his office? It seemed
incredible, impossible, but bounding up the steps he tried the door, not
knowing what he might have done in the night. He found it locked as
usual and drew back confounded, muttering again with stony lips, "My
ways are thickening, and the end is not far off."
When he returned again to his office it was to replace the spade in the
spot from which he had evidently taken it. This was up the spiral
staircase, in a small shed adjoining the large rear hall, and as he
traversed the path he had unconsciously trodden twice in the night, he
tried to recall what he had done under the influence of the horrible
nightmare which had left behind it such visible evidences of suffering.
But his consciousness was blank regarding those hours, and it was with a
crushing sense of secret and overhanging doom that he prepared for his
daily work, which happily or unhappily for him promised to be more
exacting than usual.
A dozen persons visited his office that morning, and each person as he
came glanced over at the monument and its disturbed grave. _Had any
whisper of the desecration which had there taken place found way to the
village?_ The doctor quailed at the thought, but his manner gave no sign
of his inner emotion. He was even more punctilious than usual in his
attention to the wants of his visitors, and did not give them by so much
as a glance of his eye an opportunity for question or gossip. At eleven
o'clock he went out. There was a very sick child at the other end of the
town and he could reach it only by passing the Fisher cottage. It had
been taken ill at daybreak and word had been brought him by a passing
neighbor. He had hopes, though he hardly acknowledged them to himself,
that some explanation of the footsteps which disturbed him would be
found in the sickness of this child. But when he reached the Fisher
house the sight of Polly's disturbed face, peering from the parlor
window, assured him that the cause of his trouble lay deeper than he had
hitherto feared. The discovery was a great shock to him, and as he went
on his way he asked himself why he had not stopped and talked to the
girl and found out whether she had been to his house or not the night
before, and if so, what she had seen.
But that he did not dare to do this was apparent even to himself; for
after he had prescribed for his little patient he found himself taking
another road home, a road which led him through frozen fields of
untrodden snow, rather than run the risk of encountering Polly's face
again, with those new marks upon it of aversion and fear. When he
re-entered his own gate it was with bowed head and shrunken form. His
short walk through the village, with the discovery he had imagined
himself to have made, cost him ten years of his youth. On his table
there lay a letter. When he saw it a flush crossed his cheek and his
form unconsciously assumed its wonted air of dignity and pride. It was
from _her_ and the room seemed to lose something of its habitual gloom
from its presence. But its tenor made him grow pale again. The letter
read as follows:
DEAR FRIEND: Clarke has tried every available means to avoid the
result we feared, but as you will see from the inclosed letter from
Ephraim Earle, Polly has but one course before her, and that is to
give her father what he demands. She has so decided to-day, and if
you see no way of interfering, the money will be paid over by nine
o'clock to-morrow morning. This means years of struggle for Clarke.
You bade us not to apply to you till every other hope failed. We
have reached that point. Faithfully yours,
GRACE UNWIN.
------------------------------------------------------------------------
XIX.
TO-MORROW.
POLLY had spent an unhappy day. Her secret—for so she termed her
discovery of the night before—weighed heavily upon her, and yet she felt
it was impossible to part with it, even to Clarke. Some instinct of
loyalty to the doctor who had been almost a parent to her influenced her
to silence, though she was naturally outspoken and given to leaning on
those she loved. She was sitting in the parlor, her back to the window.
She had seen the doctor pass once that day and she did not want to meet
his eye again. Fear had taken the place of reverence, and confidence had
given way to distrust.
Suddenly she heard a door open, and rose up startled, for the sound was
in the front hall and the family were all in the kitchen. Could it be
Clarke returning, or her father, or—she had not time to push her
conjectures further, for at this point the door of the room in which she
stood swung quickly open and in the gap she saw Dr. Izard, with a face
so pale that it reminded her of the glimpse she had caught of him the
previous night. But there was purpose instead of the blank look of
somnambulism in his eyes, and that purpose was directed toward her.
"Polly," he said, not advancing, but holding her fascinated in her place
by the intensity of his look, "do not allow yourself to be constrained
to sign any check to-day. To-morrow you will no longer consider it your
duty." And before she could answer or signify her assent he was gone,
and the front door had shut after him. The deep breath which escaped her
lips showed what that one moment of terror had been to her. Springing to
the window she looked out and started as she saw him take the direction
of Carberry hill.
"He is going to see my father," she murmured, and moved by a new terror
she seized her hat and coat, and ran, rather than walked, to Mrs.
Unwin's cottage. "Where is Clarke?" was her breathless demand as she
rushed impetuously into the house. "Dr. Izard is on his way to Carberry
hill and I am afraid, or rather I know, there is going to be trouble
between him and my father."
"Then Clarke will prevent it. Dr. Izard sent him word an hour ago to
meet him there at five o'clock, and he has been gone from the house just
five minutes."
"Oh, what is going to happen? I must see; I must go. They do not know
Dr. Izard as well as I do." And without waiting to explain her somewhat
enigmatical sentence she dashed from the house and took her way up
Carberry hill.
It was the first time she had been there since she was surprised at her
father's door by that father's fatal and unexpected return; and had it
not been for the excitement under which she was laboring, her limbs
would have faltered and her whole soul quailed at the prospect. But love
lent her wings, and a certain dogged persistence in duty which underlay
the natural effervescence of her spirits kept her to her task, and so
before she realized it she was at the top of that haunted hill and on
the doorstep of the house which was even more repellent to her now than
when the moss hung from the eaves and the seal of desolation lay upon
the door.
Hearing from within the voices that she knew, she waited to give no
summons, but opened the door and passed in. Three men were in the
hall—Dr. Izard, Ephraim Earle, and Clarke—and from the faces they turned
toward her she judged that she was not a minute too soon.
"Polly!" leaped simultaneously from the lips of her lover and from those
of Dr. Izard. But the one spoke in a sort of tender surprise and the
other with a mixture of anger and constraint.
"Do not mind me," she said. "I saw you coming here, and I felt that I
ought to be present." And the determination in her face startled those
who had always regarded her as a petted child. Her father, who was the
only person there who seemed at all at his ease, smiled and gave her a
sarcastic bow.
"This is the first time you have honored me," he observed, and pushed a
chair slightly forward. "Women are proverbially fond of controversy; why
deny this very young girl, the privilege of hearing our little talk?"
The doctor, who perhaps saw more in this intrusion than the others,
hesitated for a moment, with his brows lowered over his uneasy eyes,
then he waved his hand as if dismissing a subject of no importance, and
without saying yea or nay to the appeal which had just been made to him,
he cried out in a set and desperate voice:
"I have borne with this impostor long enough. I do not know who you
are," he continued, pointing imperatively at the man before him, "but
that you are not Ephraim Earle is certain. Therefore you shall no longer
enjoy Ephraim Earle's rights or profit by the money which was given to
Polly for a very different purpose."
Earle, thus attacked, first raised his brows and then smiled suavely.
"You would force an issue then," he cried. "Very well, I'm ready. Why am
I not Ephraim Earle, Dr. Izard? You assert the fact, but that is not
proving it. When we were young men together you were not wont to stop at
assertion."
"We were never young men together. You are a stranger to the town, a
stranger to me. The letter which you wrote may deceive Polly, may
deceive Clarke, may deceive every one else who reads, but it does not
deceive me. What is this new invention you failed to project? Tell us on
the spot or I will brand you as a wholesale deceiver up and down the
town."
"I——" the man stammered, his bold effrontery failing him for the moment.
"Have you forgotten it _again_?" sneered the doctor, seeming to grow
taller and broader as his antagonist dwindled. "I expected you would
hide behind that excuse. It is a convenient one. You _have_ forgotten
it; well, we will let that pass and you shall tell me instead why your
first one failed to operate the first time you tried it."
"I will not," shouted Earle, driven apparently to bay. "That it did fail
you remember and so do I, but after fourteen years devoted to other
subjects I am not going to try and pick up those old threads again and
explain to you every step by which I won success at last."
"But I will wait," suggested the doctor. "You shall not be hurried;
there is nothing more important to be done in town just now."
"Isn't there? I think there is, Dr. Izard. You have shown yourself my
enemy ever since I came to Hamilton; but for reasons that were
satisfactory to me I have let it pass, as you have let my so-called
imposture pass. I did not wish to stir up old grievances; but you attack
me and must expect to be yourself attacked. Of what complaint did Huldah
Earle die? Answer me that! Or I will brand _you_ for a——"
"Hush!" The word sprang from Clarke, who had seen the doctor cower, as
if some awful weight were about to be heaved upon him. "Weigh your
words, Mr. Earle; for if you utter an untrue one you shall be brought to
dearly rue it."
"I will weigh them," answered the other, growing taller in his turn as
the doctor shrank before him; "weigh them in the balance of this
respected man's innocence. Look at his whitening cheek, his trembling
form! If he could mention the complaint which carried my wife away in
the flower of her youth, do you think he would hesitate and turn pale
before her child? Or perhaps _he_ has forgotten; it is fourteen years
ago, and as I have taken refuge in that excuse, why not he?"
"O God!" burst from Polly's lips; "what horror is this?"
But the doctor, goaded by this last sting, had roused himself. "I have
not forgotten," said he. "I forget nothing; not even the slight
discoloration which always disfigured Ephraim Earle's left eye, and
which is absent from yours. But I do not know the exact cause of Mrs.
Earle's death. I never knew. If you were her husband, you would remember
that I several times declared I was working in the dark, and even after
she was dead acknowledged myself to have failed in my diagnosis, and
wished you had called down physicians from Boston."
"Oh, I remember; but I was not deceived then by your humility, nor am I
deceived by it now, I will have her body dug up. I will—"
"Oh, no! no!" shrieked Polly, thrusting out her hands before her eyes.
"I—cannot—bear—this. I—I do not think the doctor can bear this. Look at
him! He is not sane! He——"
"Hush, Polly! I am sane enough," came from the doctor with a sternness
which was but the result of his overpowering emotion. "If I show
agitation it is because dreadful memories have been awakened and because
I must yet press hard against this most audacious man. Fellow! where do
you think the money came from which you have been expending so freely to
keep yourself out of jail?"
"Ah! that is another small mystery with which I have thought it best not
to concern myself."
But even while speaking he drew back, and a change passed over his bold
countenance. Looking at the doctor with a strange and lingering gaze, he
darted to a small rack at the end of the hall, and, tearing down a cloak
and an old slouch hat, he thrust the one upon the doctor's head and the
other about his shrinking shoulders. Then he drew back and surveyed him.
Suddenly he struck his forehead, and a triumphant smile, which was not
without an evil glare in it, lit up his features.
"Of course!" he cried, "I might have known it! You are the fellow who
visited the Chicago hospital that night and who——"
"And you are No. Thirteen!" was the quick response; "the man given over
for dead! Oh, I see how you came to be here. Rascal! Villain!"
"Doctor, allow me to return the compliment. Why did you use such
subterfuges to transfer a fortune into my daughter's hands? Was it from
a good motive or because you felt yourself guilty of her parent's death,
and so sought to make amends without awakening suspicion?"
"I should have whispered _ten_ thousand dollars into your ear instead of
one," muttered the doctor, lost in contemplation of the other's
duplicity.
"I would have given no more sign for ten than for one," answered Earle.
"Remember, I had just heard of an unknown sum bequeathed to my daughter,
and the larger the hush money offered the greater would the fortune have
appeared."
Clarke, to whom these words were well nigh unintelligible, consulted
Polly's countenance, and seemed to question what she thought of them.
But she was gazing at the doctor, wonder and repugnance in all her
looks.
"Oh, do you mean that even this money is not all my own? That it is not
the gift of a stranger, but has come, in some incomprehensible way, from
_him_?"
The doctor, stung by her tone, turned toward her, saw the slender finger
pointing accusingly at him, and drooped his head with a gesture of
despair.
"Does it lose its value," he asked, "because it represents the labor and
privations of twenty busy years?"
"Does it represent anything else?" she protested. "Why should you give
money to me?"
"I cannot answer; not here. To-morrow at your mother's grave I will.
Come yourself, let your neighbors come, only see that one person is kept
away. Years ago I loved Grace Hasbrouck, and I would not have her the
witness of my shame. Keep her away, Clarke! My task would be too
difficult were she there."
Clarke, to whom this avowal was a revelation, stammered and bowed his
head. Mr. Earle softly smiled.
"Then you avow—" he began.
But the doctor turned upon him and thundered, "I avow nothing. I merely
wish to prove to this town that you are an impostor, and I will do it
to-morrow at seven at Huldah Earle's grave. You are a bold man and a
quick one, and have learned your lesson well. But there is one thing
before which you must succumb and that is the presence of the true
Ephraim Earle."
"And you will produce him?"
"I will produce him."
"And in such haste?"
"Yes, in such haste."
There was something so astounding in this threat and in the resolve with
which it was uttered that not only Clarke Unwin recoiled, but the hardy
adventurer himself showed momentary signs of quailing. But he quickly
recovered himself, and glancing at Polly, who stood clinging to Clarke,
white as a wraith in her terror and amazement, cried aloud: "Now I know
you for a madman. Being Ephraim Earle myself, and innocent of any deeper
crime than the one I have frankly acknowledged to you, I can afford to
meet my double, even at my poor wife's grave. Doubtless he will be a
very good semblance of myself, and my only wonder is that the doctor has
not produced him sooner."
"Laugh, laugh!" repeated the doctor, in a terrible voice, "for to-morrow
you will be in prison." And stalking by them all, he proceeded to the
door, where he paused to say in a voice whose solemn tones rang long in
their ears, "Remember! to-morrow morning at seven in the churchyard."
And he was gone.
A silence which even the dazed adventurer dared not break followed this
startling exit. Then Polly, in a quivering voice, murmured below her
breath, "He is mad! I knew it before I came here. Pray Heaven that he
has not been made so by crime."
At these words, so unexpected and so welcome to the man whose position
had been thus violently threatened, Earle lifted his head and cast a
reassured look about him.
"Stick to that, my daughter," he muttered, "stick to that; it is the
only explanation of his conduct;" and walking down the hall he added in
a subdued tone, as he passed the hitherto unnoticed figure of a man
standing in the rear passage, "I will still have the five thousand
dollars! Nothing that this madman can do will hinder that."
------------------------------------------------------------------------
XX.
DR. IZARD'S LAST DAY IN HAMILTON.
IT was fortunate that there was no serious sickness in Hamilton that
night, for the new physician was out of town and Dr. Izard inaccessible.
Ever since nightfall there had been a rush of people to the latter's
gate, the news having already spread far and wide that the doctor had
lately shown signs of mania, during which he had invited the whole town
to come to the cemetery the following morning, there to witness, they
scarcely knew what, but something strange, something which would turn
the public mind against Ephraim Earle, whom he had once before, as all
remembered, accused of being an impostor. But they found the gate
padlocked, and so were obliged to content themselves with hanging over
the cemetery wall and catching what glimpses they could of the doctor's
light which shone clear but inhospitable from his open window. Not till
the great clock struck twelve did the curious crowd separate and
straggle away to their respective homes.
Meanwhile what was the doctor doing? We, who have penetrated more than
once into his silent room, will do it once again and for the last time.
We shall not see much. The doctor, whose face shows change, but not so
much as one would expect, sits at his table writing. The name of Grace
is at the top of the page over which he bends, and the words are few
beneath, but they seem to be written with his heart's blood; for in
signing them he gives vent to one irrepressible sob—he the man whose
sternly contained soul had awed his fellow-men for years and held all
men and women and children back from him, as if his nature lacked
sympathy for anything either weak or small. The night was far advanced
when he folded this letter, directed it, and laid it face up on his
desk. But though he must have been weary, he cast no glance at the
settle in the dim corner of the room, but began to arrange his effects,
clear his drawers, and put in order his shelves, as if preparing for the
curiosity of other eyes than those which had hitherto rested so
carelessly upon them.
There was a fire lighted in the stove, and into this he thrust some
papers and one or two insignificant objects which it seemed a strong
effort to part from. As the blaze leaped up he cringed and partially
turned away his head, but soon he was again amongst his belongings,
touching some with a loving hand, others with a careless one, till the
church clock, striking two, proclaimed that time was passing hurriedly.
At this reminder he dropped the book he had taken up and passed to the
green door. It was locked, as usual, but he speedily undid the
fastenings, and carrying a lamp with him, stepped through the opening
and up the spiral staircase. One of the steps creaked as he pressed it,
and he sighed as he heard the familiar sound, possibly because he did
not expect to hear it again. When in the hall he set down the lamp, but
soon took it up again and began visiting the rooms. They had always been
well looked after, and were neither unsightly nor neglected in
appearance, but they seemed to have a painful significance for him as he
looked, lamp in hand, from the open doorways. In this one his mother had
stood as a bride, with her young friends around her, most of whom were
laid away in the graveyard, which was never long absent from his
thoughts. How he had loved to hear her tell about that night, and the
dress which she wore, and the compliments she received, and how it was
the happiest night of her life, till he came—her little child—to make
every night joyful. Ah, if she could have foreseen—if she had lived! But
God was good and took her, and he of all his family was left to meet the
doomful hour alone. In the room he now entered he had played as a boy,
such merry plays, for he was a restless child and had a voice like a
bell rung in the sunshine. Was that golden-haired, jovial little being
who ran up and down these floors like mad and shouted till the walls
rung again, the earnest of himself as he appeared at this hour
shuddering in the midnight darkness through the empty spaces of this
great house? And this little nook here, the dearest and most sacred of
all in his eyes—could he bear to look at it with this crushing weight
upon his heart and the prospect of to-morrow looming up in ghostly
proportions before him, darkening every spot at which he gazed?
Yes, yes; for here all that there has ever been of sweetness in his
miserable life, all that there is of hope in that great world to come,
centres and makes a holy air about him. Here _she_ sat one day, one
memorable, glorious day, with the sunshine playing on her hair and that
sweet surprise in her look which told him more plainly than the
faltering yes on her tongue that his presumptuous love was returned, and
that life henceforth promised to be a paradise to him. Ah, ah, and he
had not been satisfied! He must needs be a great physician too, greater
than any of those about him, greater than the great lights of Boston and
New York, and so—But away with such thoughts; it is not morning yet and
this night shall be given up to sweeter memories and more sacred
farewells.
Stooping he knelt where she had sat, and put his hands together as in
childhood's days and prayed, perhaps for the first time in years; prayed
as if his mother was overhearing him. Did he pray alone? Was not she
praying too in that shabby little room of hers, so unworthy of her
beauty and yet so hallowed by her resignation and her love?
Ah, yes, she was praying there to-night, but what would she be doing
there to-morrow? He uttered a cry as the thought stung him, and
springing passionately to his feet went on and on, avoiding but one
place in the whole house and that was where a little door led down to
the cellar, at the side of the spiral staircase. When all was done he
paused and said his last farewell. Who would walk these lonely halls
after he had vanished from them? Upon whom would these mirrors look and
in whose hearts would the mystery of this place next impress itself?
There was no prophet present to lift the veil, and dropping his chin on
his breast the doctor descended the stairs and betook himself again to
his desolate den.
The stars were shining brightly over the graveyard as he reseated
himself at his desk. There were no signs of advancing morning yet, and
he could dream, dream yet that he was young again and that Grace's voice
was in his ear and her tender touch on his arm, and that life was all
innocence and hope, and that yon loud resounding clock, too loud for
guilty men, rang with some other sound than that of death, doom, and
retribution.
Letting his head fall forward in his hands he sat while the dreary hours
moved on, but when the clock struck six he raised his forehead and
facing the churchyard waited for the first coming streaks of light. And
sitting so and waiting so we get our last glimpse of him before the
hubbub and turmoil of the day set in, with the curious gaping crowd on
the highway and the group among the graves, asking why the doctor had
not come out, and why the sexton was the first to appear on the scene,
and why he bore a pickaxe and a spade and looked as solemn as if he were
going to dig a grave for the dead.
Seven o'clock had not struck, but Ephraim Earle was there, and Clarke
and little Polly, crouching in terror behind her mother's tomb; and a
physician was there too, summoned from Wells by Earle, some said, that
there might be a competent person on hand to look after the doctor
should he prove to be, as more than one person intimated, the madman he
appeared; and Dr. Sunderland was there, the good minister; and Mr.
Crouse, who had had Polly's matters in charge, and every one but the
true Ephraim Earle, whom the doctor had promised to produce.
But then it was not yet seven and Dr. Izard had said seven; and when the
hour did at last strike then every peering eye and straining ear became
instantly aware that his door had opened and that he stood on the
doorstep cold and silent, but _alone_.
"Where is the true Ephraim Earle you talked about? You promised to bring
him here! Let us see him," shouted a voice, and the whole crowd that was
pushing and elbowing its way into the graveyard echoed as with one
voice: "Let us see him! let us see him!"
The doctor, perfectly unmoved, stepped down from the threshold and came
toward them quietly, but with a strange command in his manner. "I shall
keep my word," said he, and turned to the sexton. "Dig!" he cried, and
pointed to a grave at his feet.
"Wretch! madman!" screamed Earle, "would you desecrate my wife's grave?
What do you mean by such a command?"
"You threatened to do this yourself but yesterday," the doctor returned,
"and why do you hesitate to have it done by me?" And he again cried to
the hesitating sexton, "Dig!" and the man, understanding nothing, but
driven to his work by the doctor's fierce eye and unfaltering lip, set
himself to the task.
"Oh, what is he going to show us? Do not, do not let him go on," moaned
Polly. "I own this man to be my father; why do you let this terror go on
before our eyes?"
"This man whom you are ready to own as your father has called me the
murderer of his wife," retorted the doctor. "I can only refute it by
showing him the contents of this grave. Go on!" he commanded, with an
imperative gesture to the sexton, "or I will take the spade in my own
hands."
"Ah, he has done that once before!" muttered Polly. "He is mad! Do you
not see it in his eyes?"
The doctor, whose face had the aspect of marble, but who otherwise was
quite like himself in his best and most imposing mood, turned upon Polly
as she said this, and smiled as only the broken-hearted can smile when
confronted by a pitiful jest.
"Is there a physician here?" he demanded. "Ah, I see Dr. Brotherton. You
are in good time, I assure you, doctor. Feel my pulse and lay your hand
on my heart, and answer if you think I have my wits about me and know
what I say when I declare that only by investigating this grave can the
truth be known."
"I do not need to do either, doctor. I know a sane man when I see him,
and I must acknowledge that there are few saner than you."
A flush for the first time crept into Ephraim Earle's hardy cheek; he
shifted restlessly on his feet, and his eyes fell with something like
secret terror upon the hole that was fast widening at his feet.
"I believe you two are in league," he cried; "but if Dr. Izard can prove
himself innocent of the charges I have made against him, why, he is
welcome to do so, even at the cost of my most sacred feelings."
"When you strike the coffin, let me know," said the doctor to the
sexton. At these words a dreadful hush settled over the whole
assemblage, in which nothing could be heard but the sound of the spade.
Suddenly the sexton, who was by this time deep in the hole he was
making, looked up.
"I have reached it," he said.
The doctor drew in his breath and turned livid for a moment, then he
cast a strange look away from them all across the deserted town, and
seeming to gather strength from something he saw there, he motioned the
sexton to continue, while he said aloud and with steady emphasis:
"This man who confronts you at my side is not Ephraim Earle, because
Ephraim Earle lies buried here!" and scarcely waiting for the anxious
cries of astonishment evoked by these words to subside, he went rapidly
on to say: "Fourteen years ago he died by my hand on this spot and was
buried by me in this grave. God forgive me that I have kept this deed a
secret from you so long."
The tumult which took place at this avowal was appalling. Men and women
pushed and struggled till the foremost nearly fell into the grave. Polly
shrieked and fell back into the arms of Clarke, while he who had been
called Earle shrank all at once together and looked like the impostor he
was. Dr. Izard alone retained his self-possession, the self-possession
of despair.
"Listen," he now cried, awing that tumultuous mass into silence by the
resonant tones of his voice and the gesture which he made toward the now
plainly-to-be-seen coffin. "It was not a predetermined murder. I was
young, ambitious, absorbed in my profession and eager to distinguish
myself. His wife's case was a strange one. It baffled me; it baffled
others. I could see no reason for the symptoms she showed, nor for the
death she died. You know the truth; to sound the difficulty and make
myself strong against another such a case was but the natural wish of so
young and ambitious a man; but when I asked Ephraim for the privilege of
an autopsy he denied it to me with words that stung and inflamed me till
what had been a natural instinct became an overmastering passion, and I
determined that I would know the truth concerning her complaint if I had
to resort to illegal and perhaps unjustifiable means. Her grave—you are
standing by it—was made near, very near my office, and when the mound
was cleared and the mourners had departed, my way looked so plain before
me that I do not think I so much as hesitated at the decision I had
formed, dreadful as it may seem to you now. When midnight came,—and it
was a dismal night, the blackest of the year,—I stole out into this spot
and began my unhallowed work. I had no light, but I needed none, and
strange as it may seem, I reached the coffin-lid in an hour, and
stooping down began to wrench it open, when suddenly I heard a step,
then a murmur and then a short, fierce cry. The husband had suspected me
and was there to guard his dead.
"Leaping from the grave, I confronted him and a short, wild struggle
ensued. He had thrown himself upon me in anger, and I, with the natural
instinct of self-preservation, raised my spade and struck him, how
surely I did not know at the moment. But when silence followed the
struggle and a heavy fall shook the ground at my feet, I began to
realize what I had done, and throwing myself upon the prostrate body, I
laid my hand upon the heart and my cheek to the fast-chilling lips. No
action in the one, no breath upon the other; Ephraim Earle was dead, and
I, his murderer, stood with his body at my feet beside his wife's
wide-opened grave.
"I had never known terror till that hour, but as I rose to my feet,
comprehending as it were in an instant all that lay before me if his
dead body was found at my door, the subtleness of the criminal entered
into me, and springing back into the grave I tore poor Huldah's corpse
from its last resting place, thrust her husband's scarce cold body into
her coffin, and pushed down the lid. Then I shovelled in the earth, and
when all was done, I carried her poor remains into the house and buried
them beneath the cellar floor, where they are still lying. And now you
know my crime and now you know my punishment. Three months ago this man
came into town and announced himself as Ephraim Earle, and marking the
havoc he has made with the happiness of our innocent Polly, I have felt
myself driven step by step to make this dreadful avowal. Now look into
this grave for yourselves, and see if all that I have told you is not
true."
And they did look, and though I need not tell you what they saw, there
was no more talk in Hamilton of any lack of sanity on Dr. Izard's part,
nor did any man or woman there-after speak again of the adventurer by
the name of Ephraim Earle.
When the first horror was over and people could look about them once
more, the doctor's voice was heard for the last time.
"When this man—who, as you see, would like to escape from this place,
but cannot—came with his bravado into town, I told Polly that before she
accepted his assertions as true, she should exact from him some
irrefutable proof of his identity, and mentioned the medal that had been
given to her father by the French government. This was because the medal
had not been found after his disappearance, and I thought it must have
been upon his person when he was thrust into the grave. But to my horror
and amazement, this fellow was able to produce it,—where found or how
discovered by him I cannot tell. But he has never given evidence of
having the money which accompanied the medal. Search, then, my friends,
and see if it cannot be found among this dust, and if it can, give it to
Polly, whom I have in vain endeavored to recompense for this loss, which
was involuntary on my part and which has always been to me the most
unendurable feature of my crime."
A cry of surprise, a shout of almost incredulous joy, followed this
suggestion, and Mr. Crouse held up to sight a discolored, almost
indistinguishable pocketbook, which some one had the courage to pull out
of the coffin. Then another voice, more solemn and methodical than any
which had yet spoken, called out: "Let us kneel and give thanks to God,
who remembers the fatherless and restores to the orphan her rightful
patrimony."
But another voice, shriller and more imperative still, put a stop to
this act of devotion.
"Dr. Izard has confessed his sins, and now let the impostor confess his.
Who are you, man, and how happens it that you know all our ways and the
whole history of this town?" And Lawyer Crouse shook the would-be Earle
by the arm and would not let him go till he answered.
"I am—" the old bravado came back, and the fellow for a moment looked
quite reckless and handsome. "Ask Tilly Unwin who I am," he suddenly
shouted, breaking into a great laugh. "Don't you remember Bill Prescott,
all you graybeards? You used to hustle with me once for a chance at her
side at singing school and dance; but you won't hustle any longer, I am
ready to swear; the lady's beauty is not what it was." And with this
unseemly jest he whirled about on one heel and gave his arm to a slim,
light-complexion young man whom few had noticed, but who at no time had
stepped far away from his side.
The cry of "Phil! It is Phil, the scape-grace who was said to be dead a
dozen years ago," followed him out of the yard; but he heeded nobody,
his game was over, and his last card, a black one, had been played.
And Dr. Izard? When they thought of him again, he was gone; whither, no
one knew, nor did it enter into the heart of any one there to follow
him. One person, a heavily draped woman, who had not entered the
graveyard, but who had stood far down the street during all that
dreadful hour, thought she saw his slight form pass between her and the
dismal banks of the river; but she never rightly knew, for in her mind's
eye he was always before her, and this vision of his bowed head and
shrunken form may have been, like the rest, a phantom of her own
creation.
------------------------------------------------------------------------
● Transcriber's note:
○ Typographical errors were silently corrected.
○ Inconsistent spelling and hyphenation were made consistent only
when a predominant form was found in this book.
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 2,515 |
Staffhorst is een gemeente in de Duitse deelstaat Nedersaksen. De gemeente maakt deel uit van de Samtgemeinde Siedenburg in het Landkreis Diepholz. Staffhorst telt inwoners.
De gemeente bestaat uit vier, niet zeer belangrijke, agrarisch gekenmerkte dorpjes en gehuchten, te weten Staffhorst, Dienstborstel, Harbergen en Üpsen.
Gemeente in Nedersaksen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,418 |
{"url":"https:\/\/zbmath.org\/authors\/?q=ai%3Arazborov.alexander-a","text":"## Razborov, Aleksandr Aleksandrovich\n\nCompute Distance To:\n Author ID: razborov.alexander-a Published as: Razborov, Alexander A.; Razborov, A. A.; Razborov, Alexander; Razborov, A.; Razborov Alexander Homepage: http:\/\/people.cs.uchicago.edu\/~razborov\/ External Links: MGP \u00b7 Wikidata \u00b7 Math-Net.Ru \u00b7 dblp Awards: Nevanlinna Prize (1990)\n Documents Indexed: 102 Publications since 1985, including 1 Book 6 Contributions as Editor \u00b7 1 Further Contribution Biographic References: 2 Publications Co-Authors: 84 Co-Authors with 54 Joint Publications 3,232 Co-Co-Authors\nall top 5\n\n### Co-Authors\n\n 53 single-authored 7 Wigderson, Avi 5 Alekhnovich, Michael 4 Sem\u00ebnov, Alekse\u012d L\u2019vovich 3 Beklemishev, Lev D. 3 Ben-Sasson, Eli 3 Lauria, Massimo 3 Lysenok, Igor\u2019 Geront\u2019evich 3 Nordstr\u00f6m, Jakob 3 Yao, Andrew Chi-Chih 3 Yukna, Stasys P. 2 Adyan, Serge\u012d Ivanovich 2 Alekhnovich, M. V. 2 Atabekyan, Varuzhan S. 2 Atserias, Albert 2 Beyersdorff, Olaf 2 Borodin, Allan B. 2 Bukhshtaber, Viktor Matveevich 2 Coregliano, Leonardo Nagami 2 Galesi, Nicola 2 Guba, Victor S. 2 H\u00e5stad, Johan Torkel 2 Hatami, Hamed 2 Hladk\u00fd, Jan 2 Kashin, Boris Sergeevich 2 Kr\u00e1l\u2019, Daniel 2 Norine, Serguei 2 Novikov, Serge\u012d Petrovich 2 Pitassi, Toniann 2 Rudich, Steven 2 Savick\u00fd, Petr 2 Wegener, Ingo 1 Artemov, Sergei 1 Bashmakova, Izabella Grigor\u2019evna 1 Batchelor, George Keith 1 Belyi, Gennadii Vladimirovich 1 Bingham, Nicholas Hugh 1 Bonacina, Ilario 1 Cori, Ren\u00e9 1 de Rezende, Susanna F. 1 Diekert, Volker 1 Eisenbrand, Friedrich 1 Ershov, Yuri\u012d Leonidovich 1 Gluskin, Efim D. 1 Godunov, Serge\u012d Konstantinovich 1 Goldmann, Mikael 1 Gonchar, Andre\u012d Aleksandrovich 1 Goncharov, Serge\u012d Savost\u2019yanovich 1 Grigor\u2019ev, Dmitri\u012d Yur\u2019evich 1 Guruswami, Venkatesan 1 H\u00e4hnle, Nicolai 1 Hayman, Walter Kurt 1 Hirsch, Edward A. 1 Hyland, J. Martin E. 1 Impagliazzo, Russell 1 Kendall, David George 1 Kozlov, Valeri\u012d Vasil\u2019evich 1 Kraj\u00ed\u010dek, Jan 1 Lee, James R. 1 Lifschitz, Vladimir 1 Lorentz, Georg Gunther 1 Makowsky, Johann-Andreas 1 Mal\u2019tsev, Arkadi\u012d Anatol\u2019evich 1 Moffatt, Henry Keith 1 Mubayi, Dhruv 1 Mull, Nathan 1 Olevskii, Alexander Moiseevich 1 Osipov, Yuri\u012d Sergeevich 1 Pang, Shuo 1 Pentus, Mati 1 Peretyat\u2019kin, Mikhail G. 1 Pikhurko, Oleg 1 Podol\u2019ski\u012d, Vladimir Vladimirovich 1 Pudl\u00e1k, Pavel 1 Repin, N. N. 1 Robinson, Christopher Alan 1 Rossman, Benjamin 1 Rothvo\u00df, Thomas 1 Sadovnichi\u012d, Viktor Antonovich 1 Sgall, Ji\u0159\u00ed 1 Sherstov, Alexander A. 1 Shevrin, Lev Naumovich 1 Silver, Ben 1 Skorokhod, Anatoli\u012d Volodymyrovych 1 Slissenko, Anatol 1 Smolensky, Roman 1 Szemer\u00e9di, Endre 1 Talambutsa, A. L. 1 Todorcevic, Stevo B. 1 Treshchev, Dmitri\u012d Valer\u2019evich 1 Vereshchagin, Nikolai K. 1 Viola, Emanuele 1 Whittle, Peter 1 Wood, Carol 1 Yekhanin, Sergey\nall top 5\n\n### Serials\n\n 8 Computational Complexity 6 Russian Mathematical Surveys 5 Mathematical Notes 5 SIAM Journal on Computing 4 Theoretical Computer Science 4 Combinatorics, Probability and Computing 3 ACM Transactions on Computation Theory 2 Journal of Computer and System Sciences 2 Journal of Graph Theory 2 Mathematics of Operations Research 2 Combinatorica 2 Sbornik: Mathematics 2 Izvestiya: Mathematics 2 Journal of the ACM 2 Annals of Mathematics. Second Series 1 Discrete Applied Mathematics 1 Discrete Mathematics 1 Information Processing Letters 1 Matematicheskie Zametki 1 Uspekhi Matematicheskikh Nauk [N. S.] 1 Bulletin of the London Mathematical Society 1 Journal of Combinatorial Theory. Series A 1 Journal of Combinatorial Theory. Series B 1 The Journal of Symbolic Logic 1 Mathematics of the USSR. Izvestiya 1 Proceedings of the London Mathematical Society. Third Series 1 Soviet Mathematics. Doklady 1 Journal of Algorithms 1 SIAM Journal on Discrete Mathematics 1 International Journal of Algebra and Computation 1 Notices of the American Mathematical Society 1 Applicable Algebra in Engineering, Communication and Computing 1 Voprosy Kibernetiki (Moskva) 1 Doklady Mathematics 1 Theory of Computing Systems 1 ACM Transactions on Computational Logic 1 Internet Mathematics 1 Oberwolfach Reports 1 Lecture Notes in Computer Science 1 Lecture Notes in Logic 1 Proceedings of the Steklov Institute of Mathematics 1 Quantum Information & Computation 1 Theory of Computing 1 Translations. Series 2. American Mathematical Society\nall top 5\n\n### Fields\n\n 59 Computer science\u00a0(68-XX) 39 Mathematical logic and foundations\u00a0(03-XX) 25 Combinatorics\u00a0(05-XX) 25 Information and communication theory, circuits\u00a0(94-XX) 8 General and overarching topics; collections\u00a0(00-XX) 8 Group theory and generalizations\u00a0(20-XX) 6 History and biography\u00a0(01-XX) 4 Operations research, mathematical programming\u00a0(90-XX) 3 Order, lattices, ordered algebraic structures\u00a0(06-XX) 3 Number theory\u00a0(11-XX) 3 Nonassociative rings and algebras\u00a0(17-XX) 2 Commutative algebra\u00a0(13-XX) 2 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 2 Probability theory and stochastic processes\u00a0(60-XX) 2 Quantum theory\u00a0(81-XX) 1 Associative rings and algebras\u00a0(16-XX) 1 Convex and discrete geometry\u00a0(52-XX) 1 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 1 Mathematics education\u00a0(97-XX)\n\n### Citations contained in zbMATH Open\n\n79 Publications have been cited 1,417 times in 970 Documents Cited by Year\nLower bounds on the size of bounded depth circuits over a complete basis with logical addition.\u00a0Zbl\u00a00632.94030\nRazborov, A. A.\n1987\nFlag algebras.\u00a0Zbl\u00a01146.03013\nRazborov, Alexander A.\n2007\nOn the minimal density of triangles in graphs.\u00a0Zbl\u00a01170.05036\nRazborov, Alexander A.\n2008\nOn the distributional complexity of disjointness.\u00a0Zbl\u00a00787.68055\nRazborov, A. A.\n1992\nNatural proofs.\u00a0Zbl\u00a00884.68055\nRazborov, Alexander A.; Rudich, Steven\n1997\nLower bounds for the monotone complexity of some Boolean functions.\u00a0Zbl\u00a00621.94027\nRazborov, A. A.\n1985\nOn the number of pentagons in triangle-free graphs.\u00a0Zbl\u00a01259.05087\nHatami, Hamed; Hladk\u00fd, Jan; Kr\u00e1l\u2019, Daniel; Norine, Serguei; Razborov, Alexander\n2013\nOn 3-hypergraphs with forbidden 4-vertex configurations.\u00a0Zbl\u00a01223.05204\nRazborov, Alexander A.\n2010\nMajority gates vs. general weighted threshold gates.\u00a0Zbl\u00a00770.68054\nGoldmann, Mikael; H\u00e5stad, Johan; Razborov, Alexander\n1992\nOn lower bounds for read-$$k$$-times branching programs.\u00a0Zbl\u00a00777.68043\nBorodin, A.; Razborov, A.; Smolensky, R.\n1993\nSpace complexity in propositional calculus.\u00a0Zbl\u00a01004.03047\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2002\nOn systems of equations in a free group.\u00a0Zbl\u00a00579.20019\nRazborov, A. A.\n1985\nResolution is not automatizable unless W[P] is tractable.\u00a0Zbl\u00a01169.03044\nAlekhnovich, Michael; Razborov, Alexander A.\n2008\nApplications of matrix methods to the theory of lower bounds in computational complexity.\u00a0Zbl\u00a00717.68049\nRazborov, A. A.\n1990\nPseudorandom generators in propositional proof complexity.\u00a0Zbl\u00a01096.03070\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2004\nLower bounds on monotone complexity of the logical permanent.\u00a0Zbl\u00a00584.94026\nRazborov, A. A.\n1985\nUnprovability of lower bounds on circuit size in certain fragments of bounded arithmetic.\u00a0Zbl\u00a00838.03045\nRazborov, A. A.\n1995\nOn the quantum communication complexity of symmetric predicates.\u00a0Zbl\u00a01088.68052\nRazborov, A. A.\n2003\nThe sign-rank of AC$$^0$$.\u00a0Zbl\u00a01211.68213\nRazborov, Alexander A.; Sherstov, Alexander A.\n2010\nLower bounds for the polynomial calculus.\u00a0Zbl\u00a01026.03043\nRazborov, Alexander A.\n1998\nProof complexity in algebraic systems and bounded depth Frege systems with modular counting.\u00a0Zbl\u00a00890.03030\nBuss, S.; Impagliazzo, R.; Kraj\u00ed\u010dek, J.; Pudl\u00e1k, P.; Razborov, A. A.; Sgall, J.\n1997\nAsymptotic structure of graphs with the minimum number of triangles.\u00a0Zbl\u00a01371.05147\nPikhurko, Oleg; Razborov, Alexander\n2017\nDiameter of polyhedra: limits of abstraction.\u00a0Zbl\u00a01226.52004\nEisenbrand, Friedrich; H\u00e4hnle, Nicolai; Razborov, Alexander; Rothvo\u00df, Thomas\n2010\nWhy are there so many loop formulas?\u00a0Zbl\u00a01367.68036\n2006\nSatisfiability, branch-width and Tseitin tautologies.\u00a0Zbl\u00a01243.68182\nAlekhnovich, Michael; Razborov, Alexander\n2011\nNon-three-colourable common graphs exist.\u00a0Zbl\u00a01248.05090\nHatami, Hamed; Hladk\u00fd, Jan; Kr\u00e1l\u2019, Daniel; Norine, Serguei; Razborov, Alexander\n2012\nPseudorandom generators hard for $$k$$-DNF resolution and polynomial calculus resolution.\u00a0Zbl\u00a01376.03055\nRazborov, Alexander A.\n2015\nThe gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear.\u00a0Zbl\u00a00776.05073\nRazborov, A. A.\n1992\nBounded arithmetic and lower bounds in Boolean complexity.\u00a0Zbl\u00a00838.03044\nRazborov, Alexander A.\n1995\nImproved lower bounds on the rigidity of Hadamard matrices.\u00a0Zbl\u00a00917.15013\nKashin, B. S.; Razborov, A. A.\n1998\nExponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields.\u00a0Zbl\u00a01040.68045\nGrigoriev, D.; Razborov, A.\n2000\nNeither reading few bits twice nor reading illegally helps much.\u00a0Zbl\u00a00903.68074\nJukna, S.; Razborov, A.\n1998\nThe set of minimal braids is Co-NP-complete.\u00a0Zbl\u00a00726.68047\nPaterson, M. S.; Razborov, A. A.\n1991\n$$n^{{\\Omega{}}(\\log{} n)}$$ lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom.\u00a0Zbl\u00a00783.68046\nRazborov, Alexander; Wigderson, Avi\n1993\nOn submodular complexity measures.\u00a0Zbl\u00a00770.68073\nRazborov, A. A.\n1992\nOn the Caccetta-H\u00e4ggkvist conjecture with forbidden subgraphs.\u00a0Zbl\u00a01273.05115\nRazborov, Alexander A.\n2013\nProof complexity of pigeonhole principles.\u00a0Zbl\u00a01073.03540\nRazborov, Alexander A.\n2002\nConstructing small sets that are uniform in arithmetic progressions.\u00a0Zbl\u00a00799.11022\nRazborov, A.; Szemer\u00e9di, E.; Wigderson, A.\n1993\nOn systems of equations in free groups.\u00a0Zbl\u00a00848.20018\nRazborov, Alexander A.\n1995\nOn P versus NP$$\\cap$$co-NP for decision trees and read-once branching programs.\u00a0Zbl\u00a00962.68075\nJukna, S.; Razborov, A.; Savick\u00fd, P.; Wegener, I.\n1999\nOn the distributional complexity of disjointness.\u00a0Zbl\u00a00765.68056\nRazborov, A. A.\n1990\nResolution lower bounds for perfect matching principles.\u00a0Zbl\u00a01106.03049\nRazborov, Alexander A.\n2004\nNatural proofs.\u00a0Zbl\u00a01345.68165\nRazborov, Alexander A.; Rudich, Steven\n1994\nOn the density of transitive tournaments.\u00a0Zbl\u00a01437.05086\nCoregliano, Leonardo Nagami; Razborov, Alexander A.\n2017\nLower bounds for polynomial calculus: nonbinomial case.\u00a0Zbl\u00a01079.03047\nAlekhnovich, M. V.; Razborov, A. A.\n2003\nLower bounds for propositional proofs and independence results in bounded arithmetic.\u00a0Zbl\u00a01045.03524\nRazborov, Alexander\n1996\nA simple proof of Bazzi\u2019s theorem.\u00a0Zbl\u00a01322.68108\nRazborov, Alexander\n2009\nLower bounds for polynomial calculus in the case of nonbinomial ideals.\u00a0Zbl\u00a01063.68589\nAlekhnovich, M. V.; Razborov, A. A.\n2001\nOn the Fon-Der-Flaass interpretation of extremal examples for Tur\u00e1n\u2019s $$(3,4)$$-problem.\u00a0Zbl\u00a01296.05099\nRazborov, Alexander A.\n2011\nA new kind of tradeoffs in propositional proof complexity.\u00a0Zbl\u00a01394.03074\nRazborov, Alexander\n2016\nParameterized bounded-depth Frege is not optimal.\u00a0Zbl\u00a01322.68082\nBeyersdorff, Olaf; Galesi, Nicola; Lauria, Massimo; Razborov, Alexander A.\n2012\nRand-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus.\u00a0Zbl\u00a00963.68048\nRazborov, Alexander; Wigderson, Avi; Yao, Andrew\n1999\nResolution lower bounds for the weak functional pigeonhole principle.\u00a0Zbl\u00a01050.03039\nRazborov, Alexander A.\n2003\nAn equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic.\u00a0Zbl\u00a00789.03046\nRazborov, Alexander A.\n1993\nClique is hard on average for regular resolution.\u00a0Zbl\u00a01427.68102\nAtserias, Albert; Bonacina, Ilario; de Rezende, Susanna F.; Lauria, Massimo; Nordstr\u00f6m, Jakob; Razborov, Alexander\n2018\nOn the $$\\mathrm{AC}^0$$ complexity of subgraph isomorphism.\u00a0Zbl\u00a01370.68135\nLi, Yuan; Razborov, Alexander; Rossman, Benjamin\n2017\nLower estimates of the size of schemes of bounded depth in the base $$\\{$$ &,$$\\vee,\\oplus \\}$$.\u00a0Zbl\u00a00615.94012\nRazborov, A. A.\n1986\nRazborov, Alexander; Viola, Emanuele\n2013\nWhat is $$\\ldots$$ a flag algebra?\u00a0Zbl\u00a01322.05077\nRazborov, Alexander A.\n2013\nA product theorem in free groups.\u00a0Zbl\u00a01346.20061\nRazborov, Alexander A.\n2014\nOn P versus $$\\text{NP}\\cap co-\\text{NP}$$ for decision trees and read-once branching programs.\u00a0Zbl\u00a00941.68062\nJukna, S.; Razborov, A.; Savick\u00fd, P.; Wegener, I.\n1997\nOn Tur\u00e1n\u2019s $$(3,4)$$-problem with forbidden subgraphs.\u00a0Zbl\u00a01310.05126\nRazborov, A. A.\n2014\nRead-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus.\u00a0Zbl\u00a01027.03043\nRazborov, Alexander; Wigderson, Avi; Yao, Andrew\n2002\nLower bounds of the complexity of symmetric Boolean functions of contact- rectifier circuits.\u00a0Zbl\u00a00801.68091\nRazborov, A. A.\n1990\nOn the parameterization of solutions for equations in free groups.\u00a0Zbl\u00a00793.20028\nRazborov, Alexander A.\n1993\nPeriodic groups and Lie algebras.\u00a0Zbl\u00a00627.17008\nAdyan, S. I.; Razborov, A. A.\n1987\nSpace complexity in propositional calculus.\u00a0Zbl\u00a01296.03032\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2000\nLower estimates of the complexity of realization of symmetric Boolean functions by contact-value schemes.\u00a0Zbl\u00a00712.68053\nRazborov, A. A.\n1990\nAlmost Euclidean subspaces of $$l^N_1$$ via expander codes.\u00a0Zbl\u00a01192.68745\nGuruswami, Venkatesan; Lee, James R.; Razborov Alexander\n2008\nCommunication complexity.\u00a0Zbl\u00a01314.68164\nRazborov, Alexander A.\n2011\nFormulas of bounded depth in the basis (&,$$\\oplus)$$ and some combinatorial problems.\u00a0Zbl\u00a00668.94017\nRazborov, A. A.\n1988\nObituary: Andrei Nikolaevich Kolmogorov (1903-1987).\u00a0Zbl\u00a00689.01015\n1990\nOn the shrinkage exponent for read-once formulae.\u00a0Zbl\u00a00884.68092\nH\u00e5stad, Johan; Razborov, Alexander; Yao, Andrew\n1995\nOn the width of semialgebraic proofs and algorithms.\u00a0Zbl\u00a01386.90128\nRazborov, Alexander\n2017\nAn $$\\Omega (n^{1\/3})$$ lower bound for bilinear group based private information retrieval.\u00a0Zbl\u00a01213.68242\nRazborov, Alexander; Yekhanin, Sergey\n2007\nParameterized bounded-depth Frege is not optimal.\u00a0Zbl\u00a01334.03056\nBeyersdorff, Olaf; Galesi, Nicola; Lauria, Massimo; Razborov, Alexander\n2011\nA property of cross-intersecting families.\u00a0Zbl\u00a01109.05336\nRazborov, Alexander A.; Vereshchagin, Nikolai K.\n1999\nSemantic limits of dense combinatorial objects.\u00a0Zbl\u00a007281942\nCoregliano, L. N.; Razborov, A. A.\n2020\nOn space and depth in resolution.\u00a0Zbl\u00a006974168\nRazborov, Alexander\n2018\nSemantic limits of dense combinatorial objects.\u00a0Zbl\u00a007281942\nCoregliano, L. N.; Razborov, A. A.\n2020\nClique is hard on average for regular resolution.\u00a0Zbl\u00a01427.68102\nAtserias, Albert; Bonacina, Ilario; de Rezende, Susanna F.; Lauria, Massimo; Nordstr\u00f6m, Jakob; Razborov, Alexander\n2018\nOn space and depth in resolution.\u00a0Zbl\u00a006974168\nRazborov, Alexander\n2018\nAsymptotic structure of graphs with the minimum number of triangles.\u00a0Zbl\u00a01371.05147\nPikhurko, Oleg; Razborov, Alexander\n2017\nOn the density of transitive tournaments.\u00a0Zbl\u00a01437.05086\nCoregliano, Leonardo Nagami; Razborov, Alexander A.\n2017\nOn the $$\\mathrm{AC}^0$$ complexity of subgraph isomorphism.\u00a0Zbl\u00a01370.68135\nLi, Yuan; Razborov, Alexander; Rossman, Benjamin\n2017\nOn the width of semialgebraic proofs and algorithms.\u00a0Zbl\u00a01386.90128\nRazborov, Alexander\n2017\nA new kind of tradeoffs in propositional proof complexity.\u00a0Zbl\u00a01394.03074\nRazborov, Alexander\n2016\nPseudorandom generators hard for $$k$$-DNF resolution and polynomial calculus resolution.\u00a0Zbl\u00a01376.03055\nRazborov, Alexander A.\n2015\nA product theorem in free groups.\u00a0Zbl\u00a01346.20061\nRazborov, Alexander A.\n2014\nOn Tur\u00e1n\u2019s $$(3,4)$$-problem with forbidden subgraphs.\u00a0Zbl\u00a01310.05126\nRazborov, A. A.\n2014\nOn the number of pentagons in triangle-free graphs.\u00a0Zbl\u00a01259.05087\nHatami, Hamed; Hladk\u00fd, Jan; Kr\u00e1l’, Daniel; Norine, Serguei; Razborov, Alexander\n2013\nOn the Caccetta-H\u00e4ggkvist conjecture with forbidden subgraphs.\u00a0Zbl\u00a01273.05115\nRazborov, Alexander A.\n2013\nRazborov, Alexander; Viola, Emanuele\n2013\nWhat is $$\\ldots$$ a flag algebra?\u00a0Zbl\u00a01322.05077\nRazborov, Alexander A.\n2013\nNon-three-colourable common graphs exist.\u00a0Zbl\u00a01248.05090\nHatami, Hamed; Hladk\u00fd, Jan; Kr\u00e1l’, Daniel; Norine, Serguei; Razborov, Alexander\n2012\nParameterized bounded-depth Frege is not optimal.\u00a0Zbl\u00a01322.68082\nBeyersdorff, Olaf; Galesi, Nicola; Lauria, Massimo; Razborov, Alexander A.\n2012\nSatisfiability, branch-width and Tseitin tautologies.\u00a0Zbl\u00a01243.68182\nAlekhnovich, Michael; Razborov, Alexander\n2011\nOn the Fon-Der-Flaass interpretation of extremal examples for Tur\u00e1n\u2019s $$(3,4)$$-problem.\u00a0Zbl\u00a01296.05099\nRazborov, Alexander A.\n2011\nCommunication complexity.\u00a0Zbl\u00a01314.68164\nRazborov, Alexander A.\n2011\nParameterized bounded-depth Frege is not optimal.\u00a0Zbl\u00a01334.03056\nBeyersdorff, Olaf; Galesi, Nicola; Lauria, Massimo; Razborov, Alexander\n2011\nOn 3-hypergraphs with forbidden 4-vertex configurations.\u00a0Zbl\u00a01223.05204\nRazborov, Alexander A.\n2010\nThe sign-rank of AC$$^0$$.\u00a0Zbl\u00a01211.68213\nRazborov, Alexander A.; Sherstov, Alexander A.\n2010\nDiameter of polyhedra: limits of abstraction.\u00a0Zbl\u00a01226.52004\nEisenbrand, Friedrich; H\u00e4hnle, Nicolai; Razborov, Alexander; Rothvo\u00df, Thomas\n2010\nA simple proof of Bazzi\u2019s theorem.\u00a0Zbl\u00a01322.68108\nRazborov, Alexander\n2009\nOn the minimal density of triangles in graphs.\u00a0Zbl\u00a01170.05036\nRazborov, Alexander A.\n2008\nResolution is not automatizable unless W[P] is tractable.\u00a0Zbl\u00a01169.03044\nAlekhnovich, Michael; Razborov, Alexander A.\n2008\nAlmost Euclidean subspaces of $$l^N_1$$ via expander codes.\u00a0Zbl\u00a01192.68745\nGuruswami, Venkatesan; Lee, James R.; Razborov Alexander\n2008\nFlag algebras.\u00a0Zbl\u00a01146.03013\nRazborov, Alexander A.\n2007\nAn $$\\Omega (n^{1\/3})$$ lower bound for bilinear group based private information retrieval.\u00a0Zbl\u00a01213.68242\nRazborov, Alexander; Yekhanin, Sergey\n2007\nWhy are there so many loop formulas?\u00a0Zbl\u00a01367.68036\n2006\nPseudorandom generators in propositional proof complexity.\u00a0Zbl\u00a01096.03070\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2004\nResolution lower bounds for perfect matching principles.\u00a0Zbl\u00a01106.03049\nRazborov, Alexander A.\n2004\nOn the quantum communication complexity of symmetric predicates.\u00a0Zbl\u00a01088.68052\nRazborov, A. A.\n2003\nLower bounds for polynomial calculus: nonbinomial case.\u00a0Zbl\u00a01079.03047\nAlekhnovich, M. V.; Razborov, A. A.\n2003\nResolution lower bounds for the weak functional pigeonhole principle.\u00a0Zbl\u00a01050.03039\nRazborov, Alexander A.\n2003\nSpace complexity in propositional calculus.\u00a0Zbl\u00a01004.03047\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2002\nProof complexity of pigeonhole principles.\u00a0Zbl\u00a01073.03540\nRazborov, Alexander A.\n2002\nRead-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus.\u00a0Zbl\u00a01027.03043\nRazborov, Alexander; Wigderson, Avi; Yao, Andrew\n2002\nLower bounds for polynomial calculus in the case of nonbinomial ideals.\u00a0Zbl\u00a01063.68589\nAlekhnovich, M. V.; Razborov, A. A.\n2001\nExponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields.\u00a0Zbl\u00a01040.68045\nGrigoriev, D.; Razborov, A.\n2000\nSpace complexity in propositional calculus.\u00a0Zbl\u00a01296.03032\nAlekhnovich, Michael; Ben-Sasson, Eli; Razborov, Alexander A.; Wigderson, Avi\n2000\nOn P versus NP$$\\cap$$co-NP for decision trees and read-once branching programs.\u00a0Zbl\u00a00962.68075\nJukna, S.; Razborov, A.; Savick\u00fd, P.; Wegener, I.\n1999\nRand-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus.\u00a0Zbl\u00a00963.68048\nRazborov, Alexander; Wigderson, Avi; Yao, Andrew\n1999\nA property of cross-intersecting families.\u00a0Zbl\u00a01109.05336\nRazborov, Alexander A.; Vereshchagin, Nikolai K.\n1999\nLower bounds for the polynomial calculus.\u00a0Zbl\u00a01026.03043\nRazborov, Alexander A.\n1998\nImproved lower bounds on the rigidity of Hadamard matrices.\u00a0Zbl\u00a00917.15013\nKashin, B. S.; Razborov, A. A.\n1998\nNeither reading few bits twice nor reading illegally helps much.\u00a0Zbl\u00a00903.68074\nJukna, S.; Razborov, A.\n1998\nNatural proofs.\u00a0Zbl\u00a00884.68055\nRazborov, Alexander A.; Rudich, Steven\n1997\nProof complexity in algebraic systems and bounded depth Frege systems with modular counting.\u00a0Zbl\u00a00890.03030\nBuss, S.; Impagliazzo, R.; Kraj\u00ed\u010dek, J.; Pudl\u00e1k, P.; Razborov, A. A.; Sgall, J.\n1997\nOn P versus $$\\text{NP}\\cap co-\\text{NP}$$ for decision trees and read-once branching programs.\u00a0Zbl\u00a00941.68062\nJukna, S.; Razborov, A.; Savick\u00fd, P.; Wegener, I.\n1997\nLower bounds for propositional proofs and independence results in bounded arithmetic.\u00a0Zbl\u00a01045.03524\nRazborov, Alexander\n1996\nUnprovability of lower bounds on circuit size in certain fragments of bounded arithmetic.\u00a0Zbl\u00a00838.03045\nRazborov, A. A.\n1995\nBounded arithmetic and lower bounds in Boolean complexity.\u00a0Zbl\u00a00838.03044\nRazborov, Alexander A.\n1995\nOn systems of equations in free groups.\u00a0Zbl\u00a00848.20018\nRazborov, Alexander A.\n1995\nOn the shrinkage exponent for read-once formulae.\u00a0Zbl\u00a00884.68092\nH\u00e5stad, Johan; Razborov, Alexander; Yao, Andrew\n1995\nNatural proofs.\u00a0Zbl\u00a01345.68165\nRazborov, Alexander A.; Rudich, Steven\n1994\nOn lower bounds for read-$$k$$-times branching programs.\u00a0Zbl\u00a00777.68043\nBorodin, A.; Razborov, A.; Smolensky, R.\n1993\n$$n^{{\\Omega{}}(\\log{} n)}$$ lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom.\u00a0Zbl\u00a00783.68046\nRazborov, Alexander; Wigderson, Avi\n1993\nConstructing small sets that are uniform in arithmetic progressions.\u00a0Zbl\u00a00799.11022\nRazborov, A.; Szemer\u00e9di, E.; Wigderson, A.\n1993\nAn equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic.\u00a0Zbl\u00a00789.03046\nRazborov, Alexander A.\n1993\nOn the parameterization of solutions for equations in free groups.\u00a0Zbl\u00a00793.20028\nRazborov, Alexander A.\n1993\nOn the distributional complexity of disjointness.\u00a0Zbl\u00a00787.68055\nRazborov, A. A.\n1992\nMajority gates vs. general weighted threshold gates.\u00a0Zbl\u00a00770.68054\nGoldmann, Mikael; H\u00e5stad, Johan; Razborov, Alexander\n1992\nThe gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear.\u00a0Zbl\u00a00776.05073\nRazborov, A. A.\n1992\nOn submodular complexity measures.\u00a0Zbl\u00a00770.68073\nRazborov, A. A.\n1992\nThe set of minimal braids is Co-NP-complete.\u00a0Zbl\u00a00726.68047\nPaterson, M. S.; Razborov, A. A.\n1991\nApplications of matrix methods to the theory of lower bounds in computational complexity.\u00a0Zbl\u00a00717.68049\nRazborov, A. A.\n1990\nOn the distributional complexity of disjointness.\u00a0Zbl\u00a00765.68056\nRazborov, A. A.\n1990\nLower bounds of the complexity of symmetric Boolean functions of contact- rectifier circuits.\u00a0Zbl\u00a00801.68091\nRazborov, A. A.\n1990\nLower estimates of the complexity of realization of symmetric Boolean functions by contact-value schemes.\u00a0Zbl\u00a00712.68053\nRazborov, A. A.\n1990\nObituary: Andrei Nikolaevich Kolmogorov (1903-1987).\u00a0Zbl\u00a00689.01015\n1990\nFormulas of bounded depth in the basis (&,$$\\oplus)$$ and some combinatorial problems.\u00a0Zbl\u00a00668.94017\nRazborov, A. A.\n1988\nLower bounds on the size of bounded depth circuits over a complete basis with logical addition.\u00a0Zbl\u00a00632.94030\nRazborov, A. A.\n1987\nPeriodic groups and Lie algebras.\u00a0Zbl\u00a00627.17008\nAdyan, S. I.; Razborov, A. A.\n1987\nLower estimates of the size of schemes of bounded depth in the base $$\\{$$ &,$$\\vee,\\oplus \\}$$.\u00a0Zbl\u00a00615.94012\nRazborov, A. A.\n1986\nLower bounds for the monotone complexity of some Boolean functions.\u00a0Zbl\u00a00621.94027\nRazborov, A. A.\n1985\nOn systems of equations in a free group.\u00a0Zbl\u00a00579.20019\nRazborov, A. A.\n1985\nLower bounds on monotone complexity of the logical permanent.\u00a0Zbl\u00a00584.94026\nRazborov, A. A.\n1985\nall top 5\n\n### Cited by 1,153 Authors\n\n 24 Razborov, Aleksandr Aleksandrovich 24 Yukna, Stasys P. 21 Pikhurko, Oleg 19 Kr\u00e1l\u2019, Daniel 19 Pudl\u00e1k, Pavel 15 Pitassi, Toniann 14 Myasnikov, Alexei G. 14 Wigderson, Avi 13 Lauria, Massimo 13 Sherstov, Alexander A. 12 Kraj\u00ed\u010dek, Jan 12 Lidick\u00fd, Bernard 11 Atserias, Albert 10 Alon, Noga M. 10 Galesi, Nicola 10 Nordstr\u00f6m, Jakob 10 Sudakov, Benny 9 Allender, Eric W. 9 Beyersdorff, Olaf 9 Impagliazzo, Russell 9 Kabanets, Valentine 9 Linial, Nathan 9 Santhanam, Rahul 9 Th\u00e9rien, Denis 9 Viola, Emanuele 8 Bollig, Beate 8 Buss, Samuel R. 8 G\u00f6\u00f6s, Mika 8 Itsykson, Dmitry M. 8 Kharlampovich, Olga G. 8 Lovett, Shachar 8 Pfender, Florian 8 Volec, Jan 8 Wegener, Ingo 7 Ablaev, Farid M. 7 Ambainis, Andris 7 Braverman, Mark 7 G\u00e1l, Anna 7 Grzesik, Andrzej 7 Meir, Or 7 Raz, Ran 7 Rodl, Vojtech 6 Chattopadhyay, Arkadev 6 Fox, Jacob 6 Gerbner, D\u00e1niel 6 Hladk\u00fd, Jan 6 Hromkovi\u010d, Juraj 6 Hrube\u0161, Pavel 6 Hu, Ping 6 Liu, Hong 6 Mix Barrington, David A. 6 M\u00fcller, Moritz 6 Podol\u2019ski\u012d, Vladimir Vladimirovich 6 Schnitger, Georg 6 Servedio, Rocco A. 6 Szeider, Stefan 6 Thapen, Neil 5 Alekhnovich, Michael 5 Balogh, J\u00f3zsef 5 Beame, Paul W. 5 Conlon, David 5 Diekert, Volker 5 Filmus, Yuval 5 Gao, Wei 5 Hatami, Hamed 5 Hirsch, Edward A. 5 Huang, Hao 5 Jain, Rahul 5 Kolokolova, Antonina 5 Li, Zhongshan 5 Mahajan, Meena 5 Mubayi, Dhruv 5 Norin, Sergey 5 Pokutta, Sebastian 5 Santos, Francisco 5 Seto, Kazuhisa 5 Shpilka, Amir 5 Shraibman, Adi 5 Tzameret, Iddo 5 Williams, Richard Ryan 4 Andreev, Alexander E. 4 Beigel, Richard 4 Beimel, Amos 4 Braun, G\u00e1bor 4 Ciobanu, Laura 4 Cook, Stephen Arthur 4 Coregliano, Leonardo Nagami 4 Elder, Murray J. 4 Falgas-Ravry, Victor 4 Fiorini, Samuel 4 Furedi, Zoltan 4 Gao, Yubin 4 Goldmann, Mikael 4 Grigor\u2019ev, Dmitri\u012d Yur\u2019evich 4 Hansen, Kristoffer Arnsfelt 4 Ishai, Yuval 4 Jing, Guangming 4 Kamath, Pritish 4 Kayal, Neeraj 4 Kouck\u00fd, Michal ...and 1,053 more Authors\nall top 5\n\n### Cited in 141 Serials\n\n 85 Theoretical Computer Science 61 Computational Complexity 60 Journal of Computer and System Sciences 45 Information Processing Letters 40 SIAM Journal on Computing 29 Information and Computation 28 Journal of Combinatorial Theory. Series B 26 Combinatorica 25 Theory of Computing Systems 21 Annals of Pure and Applied Logic 21 Combinatorics, Probability and Computing 20 Discrete Mathematics 19 European Journal of Combinatorics 18 Algorithmica 16 The Electronic Journal of Combinatorics 14 Discrete Applied Mathematics 11 Journal of Graph Theory 11 SIAM Journal on Discrete Mathematics 10 International Journal of Algebra and Computation 8 Israel Journal of Mathematics 8 Discrete & Computational Geometry 8 Journal of Cryptology 7 Artificial Intelligence 7 Advances in Mathematics 7 Journal of Algebra 7 Archive for Mathematical Logic 7 Mathematical Programming. Series A. Series B 7 Journal of Mathematical Sciences (New York) 7 Theory of Computing 6 Journal of Combinatorial Theory. Series A 6 Transactions of the American Mathematical Society 6 Graphs and Combinatorics 6 Linear Algebra and its Applications 6 Journal of the ACM 6 RAIRO. Theoretical Informatics and Applications 6 ACM Transactions on Computational Logic 6 Logical Methods in Computer Science 5 Mathematical Notes 5 Mathematical Logic Quarterly (MLQ) 5 Annals of Mathematics. Second Series 5 Discrete Analysis 4 Journal of Statistical Physics 4 Journal of Pure and Applied Algebra 4 The Journal of Symbolic Logic 3 Russian Mathematical Surveys 3 Proceedings of the American Mathematical Society 3 Journal of Symbolic Computation 3 Journal of Automated Reasoning 3 Random Structures & Algorithms 3 Geometric and Functional Analysis. GAFA 3 Bulletin of the American Mathematical Society. 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\section{Introduction}\label{Introduction}
\input{Sections/1-Introduction.tex}
\section{Preliminaries}\label{prelim}
\input{Sections/2-Preliminaries.tex}
\section{Explore, then Commit}
\input{Sections/3-TheAlgorithm.tex}
\section{Regret Analysis of \textsc{\small{ExpCommit}}\xspace}
\input{Sections/4-RegretAnalysis.tex}
\section{Related Works}
\input{Sections/5-RelatedWorks.tex}
\section{Conclusion and Future Work}
\input{Sections/6-Conclusion.tex}
\newpage
\section*{Acknowledgements}
S. Lale is supported in part by DARPA PAI. K. Azizzadenesheli is supported in part by Raytheon and Amazon Web Service. B. Hassibi is supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA's Jet Propulsion Laboratory through the President and Director's Fund, and by King Abdullah University of Science and Technology. A. Anandkumar is supported in part by Bren endowed chair, DARPA PAIHR00111890035 and LwLL grants, Raytheon, Microsoft, Google, and Adobe faculty fellowships.
\section{Introduction}
Controlling unknown discrete-time systems is a fundamental problem in adaptive control and reinforcement learning. In this problem, an agent interacts with an environment, with unknown dynamics, and aims to minimize the overall average regulating costs. To achieve this goal, the agent is required to explore the environment to gain a better understanding of the environment dynamics, which is often called system identification. The agent then utilizes this understanding to design a set of improved controllers that simultaneously reduces the possible future costs and also enables the agent to explore the important and unknown aspects of the system. In recent decades, this challenging problem has been extensively studied and resulted in a set of foundational steps to study the stability and asymptotic convergence to optimal controllers~\citep{lai1982least,lai1987asymptotically}. While asymptotic analyses set the ground for the design of optimal control, understanding the finite time behavior of adaptive algorithms is critical for real-world applications.
In practice, one might prefer an algorithm that guarantees better performance on a much shorter horizon. Recent developments in the fields of statistics and machine learning along with control theory \citep{van1996weak,pena2009self,lai1982least} empowers us to not only advance the study of the asymptotic efficiency of algorithms but also to analyze their finite-time behavior~\citep{fiechter1997pac,abbasi2011regret}.
In partially observable linear quadratic control, if the agent, a priori, is handed the system dynamics, the optimal control/policy has a closed-form in the presence of Gaussian disturbances. This control system is known as Linear–Quadratic–Gaussian (\textsc{\small{LQG}}\xspace) control. In this case, the control law is obtained through estimating the latent state of the system and deploying optimal controller synthesis. This is known as the separation principle in control theory~\citep{aastrom2012introduction}.
In partially observable environments, the agent is required to estimate the latent state of the system, which is not needed in fully observed environments where the precise state information is available. This seemingly subtle difference introduces significant challenges in controlling and learning partially observable systems when the underlying dynamics are unknown. Moreover, in the presence of an approximated model, the uncertainties in the model estimation cause an inaccurate latent state estimation. Generally, this error in the latent state estimation accumulates over time and results in sub-optimal controller synthesis, which in turn imposes a sub-optimal policy. In this work, while we study model estimation/system identification, we particularly focus on the latter challenge, \textit{i.e.}, to analyze the mentioned sub-optimality gap.
One of the general and prominent measures to assess the behavior of a learning algorithm/agent in a control system is its regret, which quantifies the additional cost encountered by the agent due to exploration and uncertainty in the model estimation, when compared with the performance of the optimal controller~\citep{lai1985asymptotically}. We aim to design an algorithm that minimizes regret and is accompanied by a sub-linear regret upper bound guarantee.
\textbf{Contributions:} In this paper, we propose \textsc{\small{ExpCommit}}\xspace, an \textit{explore-then-commit} algorithm for learning and controlling unknown discrete-time \textsc{\small{LQG}}\xspace control system. We prove that \textsc{\small{ExpCommit}}\xspace achieves a regret upper bound of $\tilde{\mathcal{O}}({T^{2/3}})$, where $T$ is the number of total interactions. This is the first sub-linear regret bound for partially observable linear quadratic control systems.
\textsc{\small{ExpCommit}}\xspace consists of two phases, an \textit{explore} phase, and a \textit{commit} phase. During the \textit{explore} phase, the agent employs Gaussian noise as its control input, explores the system, and constructs high probability confidence sets around the estimate of the system parameters. In the \textit{commit} phase, the agent exploits these confidence sets, and designs a controller based on the principle of optimism in the face of uncertainty (\textsc{\small{OFU}}\xspace). Betting on the best model, \textit{i.e.}, choosing the optimistic model, for the commit phase roots back to the early appearance of \textsc{\small{OFU}}\xspace principle that is formulated as the bet on the best (BOB) principle \citep{bittanti2006adaptive}. To analyze the finite-time regret of \textsc{\small{ExpCommit}}\xspace, we provide a stability analysis of the designed controller and present a novel way of regret decomposition by deriving Bellman optimality equation for average cost per stage \textsc{\small{LQG}}\xspace control. Utilizing the \textsc{\small{OFU}}\xspace principle, we obtain the regret upper bound.
The Markov parameters of a linear time-invariant system are the input-output impulse response of the system and suffice to identify a realization of the underlying model~\citep{ljung1999system}. These parameters are used to design optimal controllers~\citep{skelton1994data}. At the end of the \textit{explore} phase, \textsc{\small{ExpCommit}}\xspace estimates the Markov parameters of the unknown linear system for the control design of the \textit{commit} phase. We adapt the recent development in the non-asymptotic system identification~\citep{oymak2018non} to our case, and estimate the Markov parameters up to upper confidence bounds via the classical Ho-Kalman algorithm~\citep{ho1966effective}. Selecting an optimistic controller that satisfies these confidence bounds, we derive our regret result.
\subsection{Regret of the Explore Phase}
\begin{lemma}
\label{exploration regret}
Suppose Assumptions \ref{AssumContObs} and \ref{Stabilizable set} hold. Given a \textsc{\small{LQG}}\xspace $\Theta = (A,B,C)$, the regret of deploying $u_t \sim \mathcal{N}(0,\sigma_u^2 I)$ for $1\leq t\leq T_{exp}$ is upper bounded as follows with high probability
\begin{align}\label{eq:informalREG}
\textsc{\small{REGRET}}\xspace(T_{exp}) = c_* T_{exp} + \tilde{\mathcal{O}}\left(\sqrt{T_{exp}}\right)
\end{align}
where $c_*$ is a problem dependent constant.
\end{lemma}
This lemma might feel intuitive to many readers. One of the main reasons we provide Lemma \ref{exploration regret} is the importance and contribution of $\tilde{\mathcal{O}}\left(\sqrt{T_{exp}}\right)$ terms in \eqref{eq:informalREG} to the final regret bound. The proof and the precise expressions are in Appendix \ref{SuppRegretExplore}. The Lemma~\ref{exploration regret} shows that the exploration phase in \textsc{\small{ExpCommit}}\xspace results in a linear regret upper bound. In order to compensate for the regret accumulated during the \textit{explore} phase, \textsc{\small{ExpCommit}}\xspace is required to efficiently exploit the information gathered during this phase. The algorithm fulfills this by following an optimistic controller during the \textit{commit} phase.
\subsection{Regret of the Commit phase}
In the following, we show that with a proper choice of exploration duration, the designed controller by \textsc{\small{ExpCommit}}\xspace
results in stable performance during the \textit{commit} phase. After exploring the system, \textsc{\small{ExpCommit}}\xspace commits to an optimistic controller and does not change the controller during the \textit{commit} phase.
Therefore, extra care is required in determining the duration of the \textit{explore} phase, since the tightness of confidence sets are critical in obtaining a desired performance during the \textit{commit} phase.
During this phase, despite the fact that the environment evolves based on $\Theta$, the agent behaves optimally with respect to the optimistic model $\tilde{\Theta} = (\tilde{A},\tilde{B},\tilde{C})$, as if the environment evolves based on $\tilde{\Theta}$. Therefore, at time step $t$, the agent observes output $y_t$ from $\Theta$ and updates its state estimation $\hat{x}_{t|t,\tilde{\Theta}}$ using \eqref{eq:optKalmanStateEstimate} and model $\tilde{\Theta} = (\tilde{A},\tilde{B},\tilde{C})$.
The agent computes the optimal control $u_t$, using the optimal feedback controller for the system $\tilde{\Theta}$.
We show that by exploring the system $\Theta$ for $T_{exp} > T_0$ time steps, \textsc{\small{ExpCommit}}\xspace is guaranteed to maintain bounded state estimation, $\hat{x}_{t|t,\tilde{\Theta}}$ and thus bounded output $y_{t+1}$
for $T-T_{exp}$ time steps of interaction in the \textit{commit} phase.
\begin{lemma} \label{Boundedness}
Suppose Assumption \ref{Stabilizable set} holds and \textsc{\small{ExpCommit}}\xspace explores the system $\Theta$ for $T_{exp}$ time steps. For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. For \textsc{\small{ExpCommit}}\xspace, define the following three events in the probability space $\Omega$,
\begin{align*}
\mathcal{E} &= \left\{\omega \in \Omega : \Theta \in (\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) \right\}, \quad
\mathcal{F} = \left\{\omega \in \Omega : \forall t \leq T, \| \hat{x}_{t|t,\tilde{\Theta}}\| \leq \tilde{\mathcal{X}} \right\}, \\
\mathcal{G} &= \Bigg\{\omega \in \Omega: \forall t \leq T, \|y_t \| \leq \rho \| C\| \tilde{\mathcal{X}} + \| C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z \right) \sqrt{2m\log(2mT/\delta)} \Bigg\},
\end{align*}
where
\begin{align*}
\tilde{\mathcal{X}} &= \frac{2\zeta \left(\|C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2n\log(2nT/\delta)}\right) }{1-\rho}
\end{align*}
for $\bar{\Delta} = poly(\Phi(A), \beta_A, \beta_B, \|B\|, \|C\|, \rho, \zeta, \Gamma, m, \sigma, \|\Sigma \| )$. Then, $\Pr\left( \mathcal{E} \cap \mathcal{F} \cap \mathcal{G}\right) \!\geq\! 1\!-\!9\delta.$
\end{lemma}
The proof and the exact expression for $\bar{\Delta}$ are given in Appendix~\ref{SuppBounded}. Under the event $\mathcal{E}$, the underlying system parameters $\Theta$ is contained in the confidence sets constructed in Theorem \ref{ConfidenceSets}.
The event $\mathcal{F}$ indicates that the state estimation used by the agent, despite using $\tilde{\Theta}$, is bounded above. Finally, the event $\mathcal{G}$ indicates that the output of the system is bounded above despite a controller that is optimal for the optimistic system $\tilde{\Theta}$ is deployed by \textsc{\small{ExpCommit}}\xspace. More generally, the Lemma~\ref{Boundedness} shows that the state estimation and the output obtained by using optimal feedback controller $(\tilde{P}, \tilde{K}, \tilde{L})$ of $\tilde{\Theta}$ chosen from the confidence sets, is logarithmic in $T$ as long as the system is explored for $T_{exp} > T_0$ time steps.
Under the events described in the Lemma~\ref{Boundedness}, the agent maintains a stable performance. Under this stability guarantee, we analyze the regret of \textsc{\small{ExpCommit}}\xspace during the \textit{commit} phase. In order to provide a regret decomposition for the optimistic controller, we first derive the Bellman optimality equation for the average cost per step \textsc{\small{LQG}}\xspace control problem.
For infinite state and control space average cost per step problems, \textit{e.g.} the \textsc{\small{LQG}}\xspace control system $\Theta = \left(A,B,C\right)$ with regulating parameters $Q$ and $R$, the optimal average cost per stage $J_*(\Theta)$ and the differential(relative) cost satisfy Bellman optimality equation~\citep{bertsekas1995dynamic}. In the following lemma, we identify the correct differential cost for \textsc{\small{LQG}}\xspace and obtain the Bellman optimality equation.
\begin{lemma}[Bellman Optimality Equation for \textsc{\small{LQG}}\xspace] \label{LQGBellman}
Given state estimation $\hat{x}_{t|t-1} \in \mathbb{R}^{n}$ and an observation $y_t \in \mathbb{R}^{m}$ pair at time $t$, Bellman optimality equation of average cost per stage control of \textsc{\small{LQG}}\xspace system $\Theta = (A,B,C)$ with regulating parameters $Q$ and $R$ is
\begin{align}
&J_*(\Theta) + \hat{x}_{t|t}^\top \left( P - C^\top Q C\right) \hat{x}_{t|t} + y_t^\top Q y_t = \min_u \bigg\{ y_t^\top Q y_t + u^\top R u \label{bellman} \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \mathbb{E}\bigg[ \hat{x}_{t+1|t+1}^{u \top} \left( P - C^\top Q C \right) \hat{x}_{t+1|t+1}^{u} + y_{t+1}^{u \top} Q y_{t+1}^u \bigg] \nonumber
\bigg\}
\end{align}
where $P$ is the unique solution to DARE of $\Theta$, $\hat{x}_{t|t}= (I-LC)\hat{x}_{t|t-1} + L y_t$, $y_{t+1}^u = C(Ax_t + Bu + w_t) + z_{t+1}$, and $\hat{x}_{t+1|t+1}^{u} = \left( I - LC\right)(A\hat{x}_{t|t} + Bu) + Ly_{t+1}^u$.
The equality is achieved by the optimal controller of $\Theta$.
\end{lemma}
The proof of this result is given in Appendix \ref{SuppBelmanOptimality}.
Note that we can also write the same Bellman equation \eqref{bellman} for the optimistic system $\tilde{\Theta}$. To obtain the regret expression of \textsc{\small{ExpCommit}}\xspace, we use the optimistic nature of the controller, and the characterization of the difference between $J_*(\tilde{\Theta}) - J_*(\Theta)$.
We then decompose regret into several terms as provided in Appendix~\ref{SuppRegret}. The following expresses the regret upper bound of the \textit{commit} phase of \textsc{\small{ExpCommit}}\xspace.
\begin{theorem}[The regret of optimistic controller] \label{exploitation regret}
Given a \textsc{\small{LQG}}\xspace control system $\Theta = (A,B,C)$ with regulating parameters $Q$ and $R$, suppose the Assumptions \ref{Stable}-\ref{Stabilizable set} hold. After a \textit{explore} phase with a duration of $T_{exp}$, if \textsc{\small{ExpCommit}}\xspace interacts with the system $\Theta$ for the remaining $T-T_{exp}$ steps using optimistic controller, with high probability the regret of \textsc{\small{ExpCommit}}\xspace in the commit phase is bounded as follows,
\begin{equation}
\textsc{\small{REGRET}}\xspace(T) = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}} + T^{2/3} \right).
\end{equation}
\end{theorem}
The proof is provided in Appendix \ref{SuppRegretTotal}, where we analyze individual terms in the regret decomposition. The analysis builds upon Lemmas~\ref{StabilityCov} and~\ref{Boundedness}. For each term in the decomposition of the regret, we use the fact that the error in the system parameter estimates is $\tilde{\mathcal{O}}\left( 1/\sqrt{T_{exp}}\right)$ and the input and the output of the system during commit phase is well-controlled with high probability via Lemma \ref{Boundedness}. For $T_{exp}>T_0$, via combining Lemma \ref{exploration regret} and Theorem \ref{exploitation regret}, we have the following total regret expression for \textsc{\small{ExpCommit}}\xspace,
\begin{equation}\label{eq:lastminut}
\textsc{\small{REGRET}}\xspace(T) = \tilde{\mathcal{O}}\left( T_{exp} + \frac{T-T_{exp}}{\sqrt{T_{exp}}} + T^{2/3}\right)
\end{equation}
after the total $T$ interactions, resulting in the trade-off between exploration and exploitation. On one hand, one can increase the duration of exploration for a given $T$ to obtain tighter confidence sets and thus better control performance but the linear regret obtained during exploration will dominate the total regret term. On the other hand, one can decrease the exploration duration for a given $T$ but then the optimistically chosen controller wouldn't give good performance on the underlying system.
Based on this trade-off, for the total interaction time $T$ such that
$T^{2/3} > T_0$, setting $T_{exp} = T^{2/3}$ provide us with the optimal performance for the algorithm. Therefore, by substituting $T_{exp} = T^{2/3}$ in the \eqref{eq:lastminut}, the regret of \textsc{\small{ExpCommit}}\xspace is upper bounded with $\tilde{\mathcal{O}}\left(T^{2/3}\right)$ as stated in Theorem \ref{total regret main text}.
\subsection{Learning Markov Parameters}
\label{LearnMarkov}
We deploy the proposed regression problem in \eqref{leastsquares} to estimate $G$. As provided in \eqref{second}, $e_t$ is dependent on previous noise and control inputs, therefore, special care is required to analyze such regression. In order to characterize the impact of $e_t$, we define $\sigma_e$, the ``effective standard deviation'', as follows
\begin{equation}\label{eq:sigmae}
\sigma_{e} = \Phi(A) \|CA^{H-1}\| \sqrt{\frac{H \|\Gamma_\infty\|}{1-\rho(A)^{2H}}}
\end{equation}
where $\Gamma_\infty$ is the steady state covariance matrix of $x_t$, $\Gamma_{\infty}\!=\!\sum_{i=0}^{\infty} \sigma_{w}^{2} A^{i}\left(A^{\top}\right)^{i}+\sigma_{u}^{2} A^{i} BB^{\top}\left(A^{\top}\right)^{i}.$
\begin{theorem}
[Markov Parameter Estimation~\citep{oymak2018non}]
\label{error spectral} For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. Suppose Assumption \ref{Stable} holds. Run the system with control input $u_t \sim \mathcal{N}(0,\sigma_u^2 I)$ for $T_{exp}= N+H-1$ time steps to collect $N$ samples of $H$ input-output pairs. Let, $\hat{G}$ be the least squares estimate of the Markov parameters matrix. Then, the following holds for $\hat{G}$ with probability at least $1-7\delta$,
\begin{equation}
\|\hat{G}-G\|\leq \frac{R_w+R_e+R_z}{\sigma_u\sqrt{T_{exp}-H+1}},\label{the bound}
\end{equation}
\end{theorem}
The proof can be found in Appendix \ref{Proof Pieces}.
This result is obtained by a set of small tweaks in the bounds in~\cite{oymak2018non} to make them appropriate for the current problem of study. The main idea is to decompose $\hat{G}- G$ into parts using \eqref{leastsquares} and bound each part separately.
\subsection{System Identification}
\label{SysId}
After estimating the Markov parameters matrix $\hat{G}$, we are ready to come up with a balanced realization of $\Theta$ from $\hat{G}$.
\textsc{\small{ExpCommit}}\xspace deploys the Ho-Kalman algorithm for this task. The details of the Ho-Kalman algorithm are given in Appendix \ref{kalmanhosupp} and Algorithm~\ref{kalmanho}. The Ho-Kalman algorithm, as
a subroutine to \textsc{\small{ExpCommit}}\xspace, receives $\hat{G}$ as an input and computes an order $n$ system $\hat{\Theta} = (\hat{A}, \hat{B}, \hat{C})$. Note that only the order $n$ input-output response of the system is uniquely identifiable~\citep{ljung1999system}. In other words, the system parameters $\Theta$ (even with the correct Markov parameters matrix $G$) are recovered up to similarity transformation. More generally, for any invertible $\mathbf{T}\in \mathbb{R}^{n \times n}$, the system $A' = \mathbf{T}^{-1}A\mathbf{T}, B' = \mathbf{T}^{-1}B, C' = C \mathbf{T} $ gives the same Markov parameters matrix $G$, equivalently, the same input-output impulse response. In this work, we are interested to recover ${\Theta}$, up to a similarity transformation.
For $H\geq2n+1$, using $\hat{G} = [\hat{G}_1 \ldots \hat{G}_H] \in \mathbb{R}^{m \times Hp}$, where $\hat{G}_i$ is the $i$th $m \times p$ block of $\hat{G}$ for all $1 \leq i \leq H$, the algorithm constructs a $(n \times n+1)$ block Hankel matrix $\mathbf{\hat{H}} \in \mathbb{R}^{nm \times (n+1)p}$ such that $(i,j)$th block of Hankel matrix is $\hat{G}_{(i+j)}$. It is worth noting that if the input to the algorithm was $G$ then corresponding Hankel matrix, $\mathbf{H}$ is rank $n$, more importantly,
\begin{align*}
\mathbf{H} = [C^\top~(CA) ^\top \ldots (CA^{n-1}) ^\top] ^\top [B~AB\ldots A^{n}B]=\mathbf{O} [\mathbf{C} ~~A^nB]=\mathbf{O} [B ~~A\mathbf{C}]
\end{align*}
where $\mathbf{O}$ and $\mathbf{C}$ are observability and controllability matrices respectively. Essentially, the Ho-Kalman algorithm estimates these matrices using $\hat{G}$. In order to estimate $\mathbf{O}$ and $\mathbf{C}$, the algorithm constructs $\mathbf{\hat{H}}^-$, the first $np$ columns of $\mathbf{\hat{H}}$ and calculates $\mathbf{\hat{N}}$, the best rank-$n$ approximation of $\mathbf{\hat{H}}^-$. Therefore, the singular value decomposition of $\mathbf{\hat{N}}$ provides us with the estimates of $\mathbf{O},\mathbf{C}$, \textit{i.e.}, $\mathbf{\hat{N}} = \mathbf{U}\mathbf{\Sigma}^{1/2}~ \mathbf{\Sigma}^{1/2}\mathbf{V} =\mathbf{\hat{O}}\mathbf{\hat{C}} $. From these estimates, the algorithm recovers $\hat{B}$ as the first $n\times p$ block of $\mathbf{\hat{C}}$, $\hat{C}$ as the first $m \times n$ block of $\mathbf{\hat{O}}$, and $\hat{A}$ as $\mathbf{\hat{O}}^\dagger \mathbf{\hat{H}}^+ \mathbf{\hat{C}}^\dagger$ where $\mathbf{\hat{H}}^+$ is the submatrix of $\mathbf{\hat{H}}$, obtained by discarding the left-most $nm \times p$ block.
Note that if we feed $G$ to the Ho-Kalman algorithm, the $\mathbf{H}^-$ is the first $np$ columns of $\mathbf{H}$, it is rank-$n$, and $\mathbf{N} \!=\! \mathbf{H}^-$. Using the outputs of the Ho-Kalman algorithm, \textit{i.e.,} $(\hat{A}, \hat{B}, \hat{C})$, \textsc{\small{ExpCommit}}\xspace constructs confidence sets centered around these outputs that contain a similarity transformation of the system parameters $\Theta \!=\!\! (A, B, C)$ with high probability. Theorem \ref{ConfidenceSets} states the construction of confidence sets and it is a slight modification of Corollary 5.4 of~\citet{oymak2018non}.
\begin{theorem}[Confidence Set Construction] \label{ConfidenceSets}
Let $H\geq 2n+1$. For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. Let $\mathbf{H}$ be the Hankel matrix obtained from $G$. Let $\bar{A}, \bar{B}, \bar{C}$ be the system parameters that Ho-Kalman algorithm provides for $G$. Let $\hat{A}, \hat{B}, \hat{C}$ be the system parameters obtained from the Ho-Kalman algorithm with using the least squares estimate of the Markov parameter matrix $\hat{G}$ after the exploration of $T_{exp}$ time steps. Suppose Assumptions \ref{Stable} and \ref{AssumContObs} hold, thus $\mathbf{H}$ is rank-$n$. Then, there exists a unitary matrix $\mathbf{T} \in \mathbb{R}^{n \times n}$ such that, with probability at least $1-7\delta$, $\bar{\Theta}=(\bar{A}, \bar{B}, \bar{C}) \in (\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) $ where
\begin{align}
&\quad \mathcal{C}_A = \left \{A' \in \mathbb{R}^{n \times n} : \|\hat{A} - \mathbf{T}^\top A' \mathbf{T} \| \leq \beta_A \right\}, \enskip
\mathcal{C}_B = \left \{B' \in \mathbb{R}^{n \times p} : \|\hat{B} - \mathbf{T}^\top B' \| \leq \beta_B \right\}, \nonumber \\
&\quad \mathcal{C}_C = \left\{C' \in \mathbb{R}^{m \times n} : \|\hat{C} - C' \mathbf{T} \| \leq \beta_C \right\}, \text{ for } \\
&\beta_A \!=\! \left( \frac{31n\|\mathbf{H}\| + 7n\sigma_n(\mathbf{H})}{\sigma_n^2(\mathbf{H})} \right) \frac{R_w+R_e+R_z}{\sigma_u\sqrt{T_{exp}-H+1}}, \enskip
\beta_B \!=\! \beta_C \!=\! \frac{7n(R_w\!+\!R_e\!+\!R_z)}{\sigma_u \sqrt{\sigma_n(\mathbf{H}) (T_{exp}\!-\!H\!+\!1)}}. \nonumber
\end{align}
\end{theorem}
The proof is in the Appendix \ref{SuppConfSet}. Theorem~\ref{ConfidenceSets} translates the results in Theorem~\ref{error spectral} to system parameters.
\subsection{Designing an Optimistic controller}
\label{ControlDesign}
After constructing the confidence sets, the algorithm deploys the \textsc{\small{OFU}}\xspace and chooses the system parameters $\tilde{\Theta} = (\tilde{A}, \tilde{B}, \tilde{C}) \in (\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) \cap \mathcal{S} $ such that
\begin{equation} \label{optimistic}
J(\tilde{\Theta}) \leq \inf_{\Theta' \in (\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) \cap \mathcal{S}} J(\Theta')+ T^{-1/3}.
\end{equation}
Through control synthesis, the algorithm designs the optimal feedback controller $(\tilde{P}, \tilde{K}, \tilde{L})$ for the system $\tilde{\Theta}$. The algorithm uses this optimistic controller to control the underlying system $\Theta$ for the remaining $T-T_{exp}$ time steps. Before the main regret analysis, we provide the following lemma on $\|\tilde{\Sigma} - \Sigma \|$ where $\tilde{\Sigma} $ is the solution to DARE given in \eqref{DARE} for the system $\tilde{\Theta}$ and $\Sigma $ is the solution to \eqref{DARE} for the underlying system $\Theta$. Under the Assumption \ref{Stabilizable set}, we obtain the following.
\begin{lemma} \label{StabilityCov}
For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. Suppose Assumption \ref{Stabilizable set} holds. Then, there exists a similarity transformation $\mathbf{S} \in \mathbb{R}^{n \times n}$ such that, with probability at least $1-7\delta$,
\begin{align*}
\|\tilde{\Sigma} - \mathbf{S}^{-1}\Sigma \mathbf{S} \| &\leq \Delta \Sigma \coloneqq \frac{\Phi(A)^2 (8\|C\|+ 4)\|\Sigma \|^2 + \sigma_z^2 (8\Phi(A) + 4) \| \Sigma\| }{\sigma_z^2(1 - \upsilon^2)} \max\left\{\beta_A, \beta_C \right\} \\
\|\tilde{L} - \mathbf{S}^{-1} L\| &\leq k_1 \Delta \Sigma + k_2 \beta_C,
\end{align*}
for some problem dependent constants $k_1$ and $k_2$.
\end{lemma}
The proof is in Appendix \ref{Ldifference}. In order to derive the results in the Lemma~\ref{StabilityCov}, we utilize a fixed point argument on the DARE of the steady state error covariance matrix of state estimation. We construct an operator that has $\tilde{\Sigma} - \mathbf{S}^{-1} \Sigma \mathbf{S}$ as the fixed point and then argue that the norm of it is bounded as shown in Lemma \ref{StabilityCov} for the given exploration duration. We combine the bound with the definition of the Kalman gain and under the Assumption \ref{Stabilizable set}, we obtain the bound on $\|\tilde{L} - \mathbf{S}^{-1} L \|$.
Later, in Lemma \ref{Boundedness}, we deploy these results to show the boundedness of MMSE estimate of the underlying state using optimistic system parameters $\tilde{\Theta}$ and the output of the system, $\|\hat{x}_{t|t,\tilde{\Theta}} \|$ and $\|y_t \|$, respectively.
Lemma \ref{StabilityCov} is a crucial concentration result and it is repeatedly used in the regret analysis of \textsc{\small{ExpCommit}}\xspace along with Theorem \ref{ConfidenceSets}.
\section{Markov Parameters Estimation, Proof of Theorem~\ref{error spectral}} \label{Proof Pieces}
Suppose the system $\Theta$ is stable (i.e.~$\rho(A)<1$) and $N\geq cH p \log(\frac{1}{\delta})$. We run the system with control input $u_t \sim \mathcal{N}(0,\sigma_u^2 I)$ for $T_{exp}= N+H-1$ time steps to collect $N$ samples of $H$ input-output pairs. From the definition of $\hat{G}$ in~\eqref{leastsquares} we have that
\begin{equation*}
(\hat{G}-G)^\top = (U^\top U)^{-1} U^\top \Gamma = (U^\top U)^{-1} \left(U^\top E + U^\top Z + U^\top W F^\top \right)
\end{equation*}
Thus the spectral norm of $\hat{G}-G$ is bounded as
\begin{equation}
\|\hat{G}-G \| \leq \|(U^\top U)^{-1} \| \left(\| U^\top E \| + \| U^\top Z \| + \| U^\top W \| \| F \| \right)
\end{equation}
\subsection{Characterization of \texorpdfstring{$U^\top U$}{Data Matrix}}
In the following, we characterize $U^\top U$ and provide a bound on $\|U\|$.
\begin{lemma}[Characterization of Data Matrix~\citep{oymak2018non}]
\label{lem cond}
Let $U\in\mathbb{R}^{N\times H p}$ be the input data matrix as described in \eqref{mat def}. For $\delta \in (0,1)$, suppose the sample size obeys $N\geq c H p \log(\frac{1}{\delta})$ for sufficiently large constant $c>0$. Then, with probability at least $1-\delta$,
\[
2N\sigma_u^2\succeq U^\top U\succeq N{\sigma_u^{2}/2}.
\]
\end{lemma}
\begin{proof} Let $r(v):\mathbb{R}^d\rightarrow\mathbb{R}^d$ be the circulant shift operator which maps a vector $v\in\mathbb{R}^d$ to its single entry circular rotation to the right i.e. $r(v)=[v_d~v_1~\dots~v_{d-1}]\in\mathbb{R}^d$. Let $\mathbf{C}\in\mathbb{R}^{T_{exp} p\times T_{exp} p}$ be a circulant matrix given below
\small
\begin{equation*}
\mathbf{C} = \left[
\begin{array}{cccccc}{u_{T_{exp},1} \ldots u_{T_{exp},p} } & {u_{T_{exp}-1,1} \ldots u_{T_{exp}-1,p} } & {\dots} & {u_{H,1} \ldots u_{H,p} } & {\dots} & {u_{1,1} {\ldots} u_{1,p} } \\ {u_{1,p} \ldots u_{T_{exp},p-1} } & {u_{T_{exp},p} \ldots u_{T_{exp}-1,p-1} } & {\dots} & {u_{H-1,p} \ldots u_{H,p-1} } & {\dots} & {u_{2,p} {\ldots} u_{1,p-1} } \\ {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} \\ {u_{1,1} {\ldots} u_{1,p} } & {u_{T_{exp},1} \ldots u_{T_{exp},p} } & {\dots} & {u_{H+1,1} \ldots u_{H+1,p} } & {\dots} & {u_{2,1} u_{2,2} {\ldots} u_{2,p} } \\ {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} \\
{u_{T_{exp},2} \ldots u_{T_{exp}-1,1} } & {u_{T_{exp}-1,2} \ldots u_{T_{exp}-2,1} } & {\dots} & {u_{H,2} \ldots u_{H-1,1} } & {\dots} & {u_{1,2} {\ldots} u_{T_{exp},1} }\end{array} \right]
\end{equation*}
\normalsize
The first row is given by
\[
c_1=[u_{T_{exp}}^\top~u_{T_{exp}-1}^\top~\dots~u_2^\top~u_1^\top].
\]
Notice that the $i$th row of $\mathbf{C}$ is $c_i=r^{i-1}(c_1)$ for $1 \leq i\leq T_{exp}p$. Therefore, looking at the rightmost $Hp$ columns, one can observe that rightmost columns of $c_1$ gives
\[
\left[ u_{H,1} u_{H,2} \ldots u_{H,p} \dots u_{1,1} u_{1,2} \ldots u_{1,p} \right] = \left[ u_H^\top u_{H-1}^\top \ldots u_1^\top \right] = \Bar{u}_H^\top
\]
Similarly, the rightmost $Hp$ columns of each $1+ip$th row of $\mathbf{C}$ gives $\Bar{u}_{H+i}$. From this one can deduce that $U$ is a submatrix of $\mathbf{C}$. Applying Theorem~\ref{circ thm}, setting $N_0 = c H p \log^2(2H p)\log^2(2T_{exp} p)$ and adjusting for variance $\sigma_u^2$, with probability at least $1-(2 T_{exp} p)^{-\log^2(2H p)\log(2T_{exp} p)}$, we have
\[
2\sigma_u^2 I \succeq N^{-1} U^\top U\succeq \sigma_u^{2}/2 I \implies 2N\sigma_u^2\succeq U^\top U\succeq N{\sigma_u^{2}/2}
\]
whenever $N\geq N_0$.
\end{proof}
\subsection{Upper bound on \texorpdfstring{$\|U^\top Z \|$}{UZ}}
\begin{lemma}[Bound on $\|U^\top Z\|$~\citep{oymak2018non}] \label{cor a2}
Let $U\in\mathbb{R}^{N\times H p}$ be the data matrix and let $Z \in\mathbb{R}^{N\times m}$ be the measurement noise matrix as described in (\ref{mat def}). For $\delta \in (0,1/3)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. With probability at least $1-3\delta$,
\[
\|U^\top Z\|\leq 2\sigma_u\sigma_z\sqrt{N}\left(\sqrt{(Hp+m)} + \log(2Hp)\log(2T_{exp} p)\right).
\]
\end{lemma}
\begin{proof} Using $T_{exp}\geq N$, Lemma~\ref{lem cond} yields that
\begin{equation}
\Pr(\|U\|\leq \sqrt{2N}\sigma_u)\geq 1-\exp(-\log^2(2H p)\log^2(2T_{exp}p)).
\end{equation}
Suppose U have singular value decomposition of $U = V_1 \Sigma V_2^\top$ where $V_1 \in \mathbb{R}^{N \times Hp}$. Notice that $V_1^\top Z \in \mathbb{R}^{Hp \times m}$ has i.i.d. $\mathcal{N}(0,1)$ entries. Recall the following theorem and lemma:
Using Theorem \ref{Gordon}, we have $\mathbb{E}\left[\|V_1^\top Z\| \right] \leq \sqrt{Hp} + \sqrt{m} \leq \sqrt{2(Hp + m)}$. Since spectral norm is 1-Lipschitz function, Lemma~\ref{gausss_lip} implies that, with probability at least $1-2\exp(-t^2 /2 )-\exp(-\log^2(2H p)\log^2(2T_{exp}p))$,
\begin{equation*}
\|U^\top Z \| = \|V_2 \Sigma V_1^\top Z \| = \| \Sigma V_1^\top Z \| \leq \sqrt{2N}\sigma_u \sigma_z \left( \sqrt{2 (Hp + m)} + t \right)
\end{equation*}
Setting it for $t=\sqrt{2}\log(2H p)\log(2T_{exp}p)$ and $\delta$ results the statement in the main Lemma.
\end{proof}
\subsection{Upper bound on \texorpdfstring{$\|U^\top W \|$}{UW}}
The main body of analysis of $\|U^\top W \|$, with an extra involvement from the associative Theorem~\ref{circ thm} is similar to Lemma~\ref{lem cond}.
\begin{lemma}[Bound on $\|U^\top W\|$~\citep{oymak2018non}] \label{lemmaUW}
Let $U\in\mathbb{R}^{N\times H p}$ be the data matrix and let $W \in\mathbb{R}^{N\times m}$ be the process noise matrix as described in \eqref{mat def}. Let $N_0 = c'H (p+n) \log^2(2H (p+n))\log^2(2T_{exp}(p+n))$ for some absolute constant $c'>0$. For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. With probability at least $1-\delta$,
\[
\|U^\top W \| \leq \sigma_w \sigma_u \max \{\sqrt{N_0 N}, N_0 \}
\]
\end{lemma}
\begin{proof}
First, we define $m_t=[\sigma_u^{-1}u_t^\top~\sigma_w^{-1}w_t^\top]^\top \!\! \in \! \mathbb{R}^{p+n}$ and $\bar{m}_i=[m_{i}^\top ~m_{i-1}^\top ~\dots~m_{i-H+1}^\top]^\top \!\! \in \! \mathbb{R}^{H q}$. We also define the matrix $M = [\bar{m}_{H},~\dots~,\bar{m}_{H+N-1}]^\top \in\mathbb{R}^{N\times H (p+n)}$.
\\
\resizebox{0.96\linewidth}{!}{
$M = \left[
\begin{array}{cccccccc}{\sigma_u^{-1} u_{H}^\top} \enskip { \sigma_w^{-1} w_{H}^\top } & {\sigma_u^{-1} u_{H-1}^\top } \enskip { \sigma_w^{-1} w_{H-1}^\top} & {\dots} & {\sigma_u^{-1} u_{2}^\top } \enskip { \sigma_w^{-1} w_{2}^\top} & {\dots} & {\sigma_u^{-1} u_{1}^\top } \enskip {\sigma_w^{-1} w_{1}^\top } \\ {\sigma_u^{-1} u_{H+1}^\top } \enskip { \sigma_w^{-1} w_{H+1}^\top } & {\sigma_u^{-1} u_{H}^\top } \enskip { \sigma_w^{-1} w_{H}^\top} & {\dots} & {\sigma_u^{-1} u_{3}^\top } \enskip { \sigma_w^{-1} w_{3}^\top} & {\dots} & {\sigma_u^{-1} u_{2}^\top } \enskip { \sigma_w^{-1} w_{2}^\top} \\ {\ddots} & {\ddots} & {\ddots} & {\ddots} \\ {\sigma_u^{-1} u_{H+N-1}^\top} \enskip { \sigma_w^{-1} w_{H+N-1}^\top } & {\sigma_u^{-1} u_{H+N-2}^\top } \enskip { \sigma_w^{-1} w_{H+N-2}^\top} & {\dots} & {\sigma_u^{-1} u_{N+1}^\top } \enskip { \sigma_w^{-1} w_{N+1}^\top} & {\dots} & {\sigma_u^{-1} u_{N}^\top } \enskip {\sigma_w^{-1} w_{N}^\top } \end{array} \right]
$} \\
Observe that by construction, $\sigma_u^{-1}U,\sigma_w^{-1}W$ are submatrices of $M$. In particular, $(\sigma_u\sigma_w)^{-1}U^\top W$ is an $H p\times H n$ size off-diagonal submatrix of $M^\top M$. This is due to the facts that i) $\sigma_u^{-1}U$ is a submatrix of $M$ characterized by the column indices
\[
\{(i-1)(p+n)+j \Big | 1\leq i\leq H,~1\leq j\leq p\},
\]
and ii) $\sigma_w^{-1}W$ lies at the complementary columns. Since $(\sigma_u\sigma_w)^{-1}U^\top W$ is an off-diagonal submatrix of $M^\top M$, it is also a submatrix of $M^\top M - I$ and spectral norm of a submatrix is upper bounded by the norm of the original matrix. Thus
\begin{equation}
(\sigma_u\sigma_w)^{-1}\| U^\top W\|\leq \|M^\top M-N I\|.\label{uw bound}
\end{equation}
Finally, we will embed $M$ in a circulant matrix. Let $r(v):\mathbb{R}^d\rightarrow\mathbb{R}^d$ be the circulant shift operator which maps a vector $v\in\mathbb{R}^d$ to its single entry circular rotation to the right i.e. $r(v)=[v_d~v_1~\dots~v_{d-1}]\in\mathbb{R}^d$. Let $\mathbf{C}\in\mathbb{R}^{T_{exp}(p+n)\times T_{exp}(p+n) }$ be a circulant matrix given below
\resizebox{0.94\linewidth}{!}{
$\mathbf{C} = \left[
\begin{array}{cccccc}{m_{T_{exp} ,1} \ldots m_{T_{exp},p+n} } & {m_{T_{exp}-1,1} \ldots m_{T_{exp}-1,p+n} } & {\dots} & {m_{H,1} \ldots m_{H,p+n} } & {\dots} & {m_{1,1} {\ldots} m_{1,p+n} } \\ {m_{1,p+n} \ldots m_{T_{exp},p+n-1} } & {m_{T_{exp},p+n} \ldots m_{T_{exp}-1,p+n-1} } & {\dots} & {m_{H-1,p+n} \ldots m_{H,p+n-1} } & {\dots} & {m_{2,p+n} {\ldots} m_{1,p+n-1} } \\ {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} \\ {m_{1,1} {\ldots} m_{1,p+n} } & {m_{T_{exp},1} \ldots m_{T_{exp},p+n} } & {\dots} & {m_{H+1,1} \ldots m_{H+1,p+n} } & {\dots} & {m_{2,1} {\ldots} m_{2,p+n} } \\ {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} \\
{m_{T_{exp},2} \ldots m_{T_{exp}-1,1} } & {m_{T_{exp}-1,2} \ldots m_{T_{exp}-2,1} } & {\dots} & {m_{H,2} \ldots m_{H-1,1} } & {\dots} & {m_{1,2} {\ldots} m_{T_{exp},1} }\end{array} \right]
$}
The first row is given by
\[
c_1=[m_{T_{exp}}^\top~m_{T_{exp}-1}^\top~\dots~m_2^\top~m_1^\top].
\]
Notice that the $i$th row of $\mathbf{C}$ is $c_i=r^{i-1}(c_1)$ for $1 \leq i\leq T_{exp}(p+n)$. Therefore, looking at the rightmost $H(p+n)$ columns, one can observe that
rightmost columns of $c_1$ gives
\[
\left[ m_{H,1} m_{H,2} \ldots m_{H,p+n} \dots m_{1,1} m_{1,2} \ldots m_{1,p+n} \right] = \left[ m_H^\top m_{H-1}^\top \ldots m_1^\top \right] = \Bar{m}_H^\top
\]
Similarly, rightmost $H(p+n)$ columns of each $1+i(p+n)$th row of $\mathbf{C}$ gives $\Bar{m}_{H+i}$. From this one can deduce that $M$ is a submatrix of $\mathbf{C}$.
Applying Theorem~\ref{circ thm}, setting $N_0 = c' H (p+n) \log^2(2H (p+n))\log^2(2T_{exp} (p+n))$ and adjusting for variance $\sigma_u^2$, with probability at least $1-(2 T_{exp} (p+n))^{-\log^2(2H (p+n))\log(2T_{exp} (p+n))}$, we have that
\[
\|\frac{1}{N} M^\top M - I \| \leq \max \{\sqrt{\frac{N_0}{N}}, \frac{N_0}{N} \}
\]
which implies that $\| U^\top W \| \leq \sigma_w \sigma_u \max \{ \sqrt{N_0 N}, N_0 \}$
via inequality \eqref{uw bound}.
\end{proof}
\subsection{Upper bound on \texorpdfstring{$\|U^\top E \|$}{UE}}
\begin{lemma}[Inner product of a fixed vector and state vector \citep{oymak2018non}]\label{thm e bound}
Let $\mathbb E_t\in\mathbb{R}^{N_t\times m}$ be the matrix composed of the rows $e_{t+iH}=CA^{H-1}x_{t+1+iH}$. Define
\[
\gamma=\frac{\|\Gamma_\infty\| \Phi(A)^2 \|CA^{H-1}\|^2}{1-\rho(A)^{2H}}.
\]
Given a unit length vector $a \in \mathbb{R}^m$, for all $\tau\geq 2$ and for some absolute constant $c>0$, we have that
\[
\Pr(\|{E_ta}\|_2^2\geq \tau N_t\gamma)\leq 2exp(-c\tau N_t(1-\rho(A)^{H})).
\]
\end{lemma}
\begin{proof}
In the proof, the authors consider an alternative way to rewrite $\| E_t a\|_2$ to apply Hanson-Wright Theorem~\citep{rudelson2013hanson}. For further details of the proof please refer to Lemma D.6 of~\citep{oymak2018non}.
\end{proof}
\begin{theorem}[Bound on Decomposition elements $U_t^\top E_t$~\citep{oymak2018non}]\label{decomposed bound}
Let \[
U_t=[ \bar{u}_{t+H},~ \bar{u}_{t+2H},~\dots,~\bar{u}_{t+N_tH}]^\top \quad E_t=[ e_{t+H},~ e_{t+2H},~\dots,~e_{t+N_tH}]^\top. \]
Define $\gamma=\frac{\|\Gamma_\infty\| \Phi(A)^2 \|CA^{H-1}\|^2}{1-\rho(A)^{2H}}$. For $\delta \in (0,1)$, $U_t^\top E_t$ obeys
\[
\|U_t^\top E_t\| \leq c_0\sigma_u\sqrt{\tau \log\left(\frac{1}{\delta}\right) N_t \gamma},
\] with probability at least $1-\delta-2\exp(-c\tau N_t(1-\rho(A)^H)+3m)$ for $\tau\geq 1$ and for some absolute constants $c_0$ and $c$.
\end{theorem}
\begin{proof}
For these matrices, define the filtrations $\mathcal{F}_i = \sigma\left( \{u_j, w_j\}^{t+iH}_{j=1} \}\right)$ for $1\leq i \leq N_t$. Based on this definition, $e_{t+iH} \in \mathcal{F}_{i-1}$ and $u_{t+iH} \in \mathcal{F}_{i} $ and independent of $\mathcal{F}_{i-1}$. This gives a formulation to use Lemma \eqref{sig sub}. Combining Lemma~\ref{sig sub} and Lemma~\ref{thm e bound} within the Covering Lemma~\eqref{cover bound}, one can derive the result. For the details of the proof please refer to Theorem D.2 of~\citep{oymak2018non}. In order to get the presented result, one needs to pick $t = c' \sigma_u \sqrt{\tau \log^2(2Hp)\log^2(2T_{exp} p) N_t \gamma}$ in using Lemma~\ref{sig sub}, which will translate to given bound with the stated probability.
\end{proof}
\begin{corollary} \label{preciseboundUtEt}
For $\delta \in (0,1)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. With probability at least $1-\frac{3\delta}{H}$, we have that
\[
\|U_t^\top E_t\| \leq c' \sigma_u \sqrt{ \gamma \log\left(\frac{H}{\delta}\right) \max \left \{\frac{N}{H}, \frac{3m + \log\left(\frac{H}{\delta} \right)}{1-\rho(A)^{H}} \right\} }
\]
for some constant $c' > 0$.
\end{corollary}
\begin{proof}
The given choice of $N$ implies that $N_t \geq \frac{N}{2H}$. Picking $\tau = \max \left\{ \frac{2H \left(3m + \log \left( \frac{1}{\delta}\right) \right)}{cN(1-\rho(A)^H)}, 1 \right\}$ to use in Theorem~\ref{decomposed bound} gives the statement of corollary.
\end{proof}
\begin{theorem}[Bound on $\|U^\top E \|$~\citep{oymak2018non}] \label{ub eb prod}
For $\delta \in (0,1/3)$, suppose $N\geq cH p \log(\frac{1}{\delta})$ for some absolute constant $c>0$. Let $U\in\mathbb{R}^{N\times H p}$ be the data matrix and let $E \in\mathbb{R}^{N\times m}$ be the unknown state contribution matrix as described in \eqref{mat def}. Define $\gamma=\frac{\|\Gamma_\infty\| \Phi(A)^2 \|CA^{H-1}\|^2}{1-\rho(A)^{2H}}$. Then, with probability at least $1-3\delta$,
\[
\|U^\top E\|\leq c \sigma_u H \sqrt{ \gamma \log\left(\frac{H}{\delta}\right) \max \left \{\frac{N}{H}, \frac{3m + \log\left(\frac{H}{\delta} \right)}{1-\rho(A)^{H}} \right\} } .
\]
\end{theorem}
\begin{proof}
First decompose $U^\top E=\sum_{t=H}^{T_{exp}} \bar{u}_t e_t^\top$ into sum of $H$ smaller products. We form the matrices
\[
U_t=[ \bar{u}_{t+H},~ \bar{u}_{t+2H},~\dots,~\bar{u}_{t+N_tH}]^\top \in \mathbb{R}^{N_t \times Hp} \quad E_t=[ e_{t+H},~ e_{t+2H},~\dots,~e_{t+N_tH}]^\top \in \mathbb{R}^{N_t \times m}. \]
Then, $U^\top E$ is decomposed as
\begin{equation}\label{sum spect}
U^\top E=\sum_{t=0}^{H-1}U_t^\top E_t \implies \|U^\top E\|\leq \sum_{t=0}^{H-1}\|U_t^\top E_t\|.
\end{equation}
Using the union bound with Corollary~\ref{preciseboundUtEt}, stated bound is obtained.
\end{proof}
The final statement of the Theorem~\ref{error spectral} follows by combining Lemma~\ref{lem cond}, Theorem~\ref{ub eb prod}, Lemma~\ref{cor a2} and Lemma~\ref{lemmaUW} using union bound considering the probability of events happening. In order to get the expression of $R_e$ notice that $H \gamma = \sigma_e^2$.
\section{Confidence Set Construction for the System Parameters}
\label{SuppConfSet}
After estimating $\hat{G}$, we construct the high probability confidence sets for the unknown system parameters. We exploit these sets when we deploy the \textsc{\small{OFU}}\xspace principle for the controller synthesis. \textsc{\small{ExpCommit}}\xspace uses Ho-Kalman method~\citep{ho1966effective} to estimate the system parameters. In this section, we first describe estimation components of the Ho-Kalman method, the Algorithm~\ref{kalmanho}. We then derive the confidence sets around the system parameter estimates in \textsc{\small{ExpCommit}}\xspace using the results developed in the Theorem~\ref{error spectral}.
\subsection[Ho-Kalman Algorithm]{Ho-Kalman Algorithm~\citep{ho1966effective}} \label{kalmanhosupp}
The Ho-Kalman algorithm, Algorithm \ref{kalmanho}, takes the Markov parameter matrix estimate $\hat{G}$, $H$, the systems order $n$ and dimensions $d_1, d_2$, as the input. It is worth restating that the dimension of latent state, $n$, is the order of the system for observable and controllable dynamics. With the assumption that $H\geq2n+1$, we pick $d_1 \geq n$ and $d_2 \geq n$ such $d_1+d_2+1 = H$. This guarantees that the system identification problem is well-conditioned, \textit{i.e.} $\mathbf{H}$ is rank-$n$.
\begin{algorithm}[tbh]
\caption{Ho-Kalman Algorithm}
\begin{algorithmic}[1]
\STATE {\bfseries Input:} $\hat{G}$, $H$, system order $n$, $d_1, d_2$ such that $d_1 + d_2 + 1 = H$ \\
\STATE Form the Hankel Matrix $\mathbf{\hat{H}} \in \mathbb{R}^{md_1 \times p(d_2+1)}$ from $\hat{G}$
\STATE Set $\mathbf{\hat{H}}^- \in \mathbb{R}^{m d_1 \times p d_2} \enskip \text{ as the first $pd_2$ columns of }\mathbf{\hat{H}}$
\STATE Using SVD obtain $\mathbf{\hat{N}} \in \mathbb{R}^{m d_1 \times p d_2}$ \enskip, the rank-$n$ approximation of $\mathbf{\hat{H}}^-$
\STATE Obtain $\mathbf{U},\mathbf{\Sigma},\mathbf{V} = \text{SVD}(\mathbf{\hat{N}})$
\STATE Construct $\mathbf{\hat{O}} = \mathbf{U}\mathbf{\Sigma}^{1/2} \in \mathbb{R}^{md_1 \times n}$
\STATE Construct $\mathbf{\hat{C}} = \mathbf{\Sigma}^{1/2}\mathbf{V} \in \mathbb{R}^{n \times pd_2}$
\STATE Obtain $\hat{C}\in \mathbb{R}^{m\times n}$, the first $m$ rows of $\mathbf{\hat{O}}$
\STATE Obtain $\hat{B}\in \mathbb{R}^{n\times p}$, the first $p$ columns of $\mathbf{\hat{C}}$
\STATE Obtain $\mathbf{\hat{H}}^+ \in \mathbb{R}^{m d_1 \times p d_2} \enskip \text{, the last $pd_2$ columns of}(\mathbf{\hat{H}})$
\STATE Obtain $\hat{A} = \mathbf{\hat{O}}^\dagger \mathbf{\hat{H}}^+ \mathbf{\hat{C}}^\dagger \in \mathbb{R}^{n\times n}$
\end{algorithmic}
\label{kalmanho}
\end{algorithm}
Recall that system parameters can be learned up to similarity transformation, \textit{i.e.} for any invertible $\mathbf{T}\in \mathbb{R}^{n \times n}$, $A' = \mathbf{T}^{-1}A\mathbf{T}, B' = \mathbf{T}^{-1}B, C' = C \mathbf{T} $ gives the same Markov parameters as $G$ so it's a valid realization. Note that the similarity transformations have bounded norms due to Assumption \ref{AssumContObs}. Given $\hat{G} = [\hat{G}_1 \ldots \hat{G}_H] \in \mathbb{R}^{m \times Hp}$, where $\hat{G}_i$ is the $i$'th $m \times p$ block of $\hat{G}$, for all $1 \leq i \leq H$, the algorithm constructs $d_1 \times (d_2+1)$ Hankel matrix $\mathbf{\hat{H}}$ such that $(i,j)$th block of Hankel matrix is $\hat{G}_{(i+j)}$. Notice that if the input to the algorithm was $G$ then constructed Hankel matrix, $\mathbf{H}$ would be rank $n$, where
\begin{align*}
\mathbf{H} = [C^\top~(CA) ^\top \ldots (CA^{n-1}) ^\top] ^\top [B~AB\ldots A^{n}B]=\mathbf{O} [\mathbf{C} ~~A^nB]=\mathbf{O} [B ~~A\mathbf{C}]
\end{align*}
The matrices $\mathbf{O}$ and $\mathbf{C}$ are observability and controllability matrices respectively. Essentially, the Ho-Kalman algorithm estimates these matrices using $\hat{G}$. In order to obtain these estimates, the algorithm constructs $\mathbf{\hat{H}}^-$ by taking first $pd_2$ columns of $\mathbf{\hat{H}}$ and calculates $\mathbf{\hat{N}}$ which is the best rank-$n$ approximation of $\mathbf{\hat{H}}^-$. Singular value decomposition of $\mathbf{\hat{N}}$ gives the estimates of $\mathbf{O},\mathbf{C}$, \textit{i.e.} $\mathbf{\hat{N}} = \mathbf{U}\mathbf{\Sigma}^{1/2}~ \mathbf{\Sigma}^{1/2}\mathbf{V} =\mathbf{\hat{O}}\mathbf{\hat{C}} $. From these estimates, the algorithm recovers $\hat{B}$ as the first $n\times p$ block of $\mathbf{\hat{C}}$, $\hat{C}$ as the first $m \times n$ block of $\mathbf{\hat{O}}$ and $\hat{A}$ as $\mathbf{\hat{O}}^\dagger \mathbf{\hat{H}}^+ \mathbf{\hat{C}}^\dagger$ where $\mathbf{\hat{H}}^+$ is the submatrix of $\mathbf{\hat{H}}$ obtained by discarding left-most $md_1 \times p$ block.
\subsection{Confidence sets around \texorpdfstring{$\hat{A}, \hat{B}, \hat{C}$}{estimations}}
The results in this section are adopted and modified versions of those in~\citep{oymak2018non}. Except the Lemma \ref{hokalmanstability lemma}, which is directly from the mentioned work, we make a small change to the presentation of it so that we can observe the dependency on the duration of exploration period and the construction of confidence set around the estimates. In this section, for completeness we provide this lemma that is used in the proof of theorem. The proof uses a simple singular value perturbation arguments. For more details of the proof, please refer to~\citep{oymak2018non}.
\begin{lemma}[\citep{oymak2018non}] \label{hokalmanstability lemma}
$\mathbf{H}$, $\mathbf{\hat{H}}$ and $\mathbf{N}, \mathbf{\hat{N}}$ satisfies the following perturbation bounds,
\begin{align*}
\max \left\{\left\|\mathbf{H}^{+}-\mathbf{\hat{H}}^{+}\right\|,\left\|\mathbf{H}^{-}-\mathbf{\hat{H}}^{-}\right\|\right\} \leq \|\mathbf{H}-\mathbf{\hat{H}}\| &\leq \sqrt{\min \left\{d_{1}, d_{2}+1\right\}}\|\hat{G} - G\| \\ \|\mathbf{N}-\mathbf{\hat{N}}\| \leq 2\left\|\mathbf{H}^{-}-\mathbf{\hat{H}}^{-}\right\| &\leq 2 \sqrt{\min \left\{d_{1}, d_{2}\right\}}\|\hat{G} - G\|
\end{align*}
\end{lemma}
Denote the $n$th largest singular value of $\mathbf{N}$, \textit{i.e.} smallest nonzero singular value, as $\sigma_{n}(\mathbf{N})$. It is worth noting that if $\sigma_{n}(\mathbf{N})$ is large enough, the order of the system can be estimated via singular value thresholding of $\mathbf{\hat{H}}^{-}$.
Given the Lemma~\ref{hokalmanstability lemma}, we can show the following theorem:
\begin{theorem}
\label{Sup:ConfidenceSets}
Suppose $\mathbf{H}$ is the rank-$n$ Hankel matrix obtained from $G$. Let $\bar{A}, \bar{B}, \bar{C}$ be the system parameters that Ho-Kalman algorithm provides for $G$. Suppose the system is order $n$ and it is observable and controllable. Define the rank-$n$ matrix $\mathbf{N}$ such that it is the submatrix of $\mathbf{H}$ obtained by discarding the last block column of $\mathbf{H}$. Suppose $\sigma_{n}(\mathbf{N}) > 0$ and $\| \mathbf{\hat{N}} - \mathbf{N} \| \leq \frac{\sigma_{n}(\mathbf{N})}{2} $. Then, there exists a unitary matrix $\mathbf{T} \in \mathbb{R}^{n \times n}$ such that, $\bar{\Theta}=(\bar{A}, \bar{B}, \bar{C}) \in (\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) $ for
\begin{align}
\mathcal{C}_A &= \left \{A' \in \mathbb{R}^{n \times n} : \|\hat{A} - \mathbf{T}^\top A' \mathbf{T} \| \leq \left( \frac{31n\|\mathbf{H}\|}{\sigma_n^2(\mathbf{H})} + \frac{13n}{2\sigma_n(\mathbf{H})} \right) \|\hat{G} - G \| \right \} \\
\mathcal{C}_B &= \left \{B' \in \mathbb{R}^{n \times p} : \|\hat{B} - \mathbf{T}^\top B' \| \leq \frac{7n}{\sqrt{\sigma_n(\mathbf{H})}} \|\hat{G} - G \| \right\} \\
\mathcal{C}_C &= \left\{C' \in \mathbb{R}^{m \times n} : \|\hat{C} - C' \mathbf{T} \| \leq \frac{7n}{\sqrt{\sigma_n(\mathbf{H})}} \|\hat{G} - G \| \right\}
\end{align}
where $\hat{A}, \hat{B}, \hat{C}$ obtained from Ho-Kalman algorithm with using the least squares estimate of the Markov parameter matrix $\hat{G}$.
\end{theorem}
\begin{proof}
The proof is similar to proof of Theorem 4.3 in \cite{oymak2018non}. Difference in the presentation arises due to providing different characterization of the dependence on $\|\mathbf{N} - \mathbf{\hat{N}} \|$ and centering the confidence ball over the estimations rather than the output of Ho-Kalman algorithm with the input of $G$. In~\citet{oymak2018non}, from the inequality
\[
\|\bar{B} - \mathbf{T}^\top \hat{B} \|_F^2 \leq \frac{2n \| \mathbf{N} - \mathbf{\hat{N}}\|^2}{(\sqrt{2}-1)\left(\sigma_n(\mathbf{N}) - \| \mathbf{N} - \mathbf{\hat{N}}\| \right)},
\]
the authors use the assumption $\| \mathbf{N} - \mathbf{\hat{N}}\| \leq \frac{\sigma_{n}(\mathbf{N})}{2}$ to cancel out numerator and denominator. In this presentation, we define $T_N$ such that $\| \mathbf{N} - \mathbf{\hat{N}}\| \leq \frac{\sigma_{n}(\mathbf{N})}{2}$ holds with high probability as long as $T_{exp} \geq T_N$. In the definition of $T_N$, we use $\sigma_n(H)$, due to the fact that singular values of submatrices by column partitioning are interlaced, \textit{i.e.} $\sigma_n(\mathbf{N}) = \sigma_n(\mathbf{H}^-) \geq \sigma_n(\mathbf{H})$. Then, we define the denominator based on $\sigma_{n}(\mathbf{N})$ and again use the fact $\sigma_n(\mathbf{N}) = \sigma_n(\mathbf{H}^-) \geq \sigma_n(\mathbf{H})$. Following the proof steps provided in~\citet{oymak2018non} and combining Lemma~\ref{hokalmanstability lemma} with the fact that one can choose either $d_1=n$ or $d_2=n$ while $d_1,d_2 \geq n$, we obtain the presented theorem.
\end{proof}
\noindent \textbf{Proof of Theorem \ref{ConfidenceSets}:}
\\
This result is obtained by combining Theorem~\ref{error spectral} and Theorem~\ref{Sup:ConfidenceSets}
\null\hfill$\square$
\section[Proof of Lemma \ref{StabilityCov}]{Upper bound on $\|\tilde{\Sigma} - \mathbf{S}^{-1} \Sigma \mathbf{S} \|$ and $\|\tilde{L} - \mathbf{S}^{-1} L \|$, Proof of Lemma \ref{StabilityCov}} \label{Ldifference}
In this section, we provide the concentration results on $\|\tilde{\Sigma} - \mathbf{S}^{-1}\Sigma\mathbf{S} \|$ and $\|\tilde{L} - \mathbf{S}^{-1}L \|$.
$\mathbf{S} \in \mathbb{R}^{n \times n} $ is a similarity transformation that is composed of two similarity transformations. The first one takes the system $\Theta$ and transforms to $\bar{\Theta}$, the output of Ho-Kalman algorithm. The second similarity transformation is the unitary matrix that is proven to exist in Theorem \ref{ConfidenceSets}. We deploy the fixed point argument from~\citep{mania2019certainty} to bound $\|\tilde{\Sigma} - \mathbf{S}^{-1} \Sigma \mathbf{S} \|$. Then we utilize the resulting bounds to come up with the concentration of $\|\tilde{L} - \mathbf{S}^{-1} L \|$. \\
\noindent \textbf{Proof of Lemma \ref{StabilityCov}:}
\\
For the simplicity of the presentation of the proof, without loss of generality, let $A = \mathbf{S}^{-1} A \mathbf{S}$, $C = C \mathbf{S}$, \textit{i.e.} assume that $\mathbf{S} = I$. Given parameters $(A,C,\sigma_w^2I, \sigma_z^2 I)$, define $F(X,A,C)$ such that
\begin{align*}
F(X,A,C) &= X - A X A^\top + A X C^\top \left(C X C^\top + \sigma_z^2I \right)^{-1} C X A^\top - \sigma_w^2 I \\
&= X - A (I + \sigma_z^{-2} X C^\top C)^{-1} X A^\top - \sigma_w^2 I
\end{align*}
where last equality follows from matrix inversion lemma. Moreover, notice that solving algebraic Riccati equation for steady state error covariance matrix of state estimation for $(A,C,\sigma_w^2I, \sigma_z^2 I)$ is equivalent to finding the unique positive definite solution to $X$ such that $F(X,A,C) = 0$. The solution for the underlying system $\Theta$, $F(X,A,C) = 0$, is denoted as $\Sigma$ and the solution for the optimistic system $\tilde{\Theta}$ chosen from the set $(\mathcal{C}_A \times \mathcal{C}_B \times \mathcal{C}_C) \cap \mathcal{S}$, $F(X,\tilde{A},\tilde{C}) = 0$, is denoted as $\tilde{\Sigma}$. Denote $D_{\Sigma} = \tilde{\Sigma} - \Sigma$ and $M = A(I-LC)$. Recall that $L = \Sigma C^\top \left( C \Sigma C^\top + \sigma_z^2 I \right)^{-1}$. For any matrix $X$ such that $I + (\Sigma + X)(\sigma_z^{-2} C^\top C)$ is invertible we have
\begin{equation} \label{fixedidentity}
F(\Sigma + X, A,C) = X - M X M^\top + M X (\sigma_z^{-2} C^\top C) [I + (\Sigma + X) (\sigma_z^{-2} C^\top C)]^{-1} X M^\top.
\end{equation}
One can verify the identity by adding $F(\Sigma, A,C) = 0$ to the right hand side of \eqref{fixedidentity} and use the identity that $M = A(I-LC) = A(I + \sigma_z^{-2}\Sigma C^\top C)^{-1} = A (I - \Sigma C^\top(C\Sigma C^\top + \sigma_z^2 I)^{-1}C)$. Define two operators $\mathcal{T}(X)$, $\mathcal{H}(X)$ such that $\mathcal{T}(X) = X - M X M^\top $ and $\mathcal{H}(X) = M X (\sigma_z^{-2} C^\top C) [I + (\Sigma + X) (\sigma_z^{-2} C^\top C)]^{-1} X M^\top$. Thus,
\begin{equation*}
F(\Sigma + X, A,C) = \mathcal{T}(X) + \mathcal{H}(X)
\end{equation*}
Notice that since \eqref{fixedidentity} is satisfied for any $X$ such that $I + (\Sigma + X)(\sigma_z^{-2} C^\top C)$ is invertible,
\begin{equation} \label{fixeddiff}
F(\Sigma + X, A,C) - F(\Sigma + X, \tilde{A},\tilde{C}) = \mathcal{T}(X) + \mathcal{H}(X)
\end{equation}
has a unique solution $X = D_{\Sigma}$ where $\Sigma + D_{\Sigma} \succeq 0$.
Recall that $M$ is stable. Therefore, the linear map $\mathcal{T} : X \mapsto X - M X M^\top$ has non-zero eigenvalues, \textit{i.e} $\mathcal{T}$ is invertible. Using this, define the following operator,
\begin{equation*}
\Psi(X) = \mathcal{T}^{-1} \left( F(\Sigma + X, A,C) - F(\Sigma + X, \tilde{A},\tilde{C}) - \mathcal{H}(X)\right).
\end{equation*}
Notice that solving for $X$ in \eqref{fixeddiff} is equivalent to solving for $X$ that satisfies $\Sigma + X \succeq 0$ and $\Psi(X) = X$. This shows that $\Psi(X)$ has a unique fixed point X that is $D_{\Sigma}$. Consider the set
\begin{equation}
\mathcal{S}_{\Sigma,\beta} = \{X : \|X\|\leq \beta, X = X^\top, \Sigma + X \succeq 0 \}.
\end{equation}
Let $X \in \mathcal{S}_{\Sigma,\beta}$ for $\beta < \sigma_n(\Sigma)/2$. First of all, recalling Assumption~\ref{Stabilizable set}, notice that operator norm of $\mathcal{T}^{-1}$ is upper bounded as $\| \mathcal{T}^{-1}\| \leq \frac{1}{1-\upsilon^2}$. Using Lemma~\ref{normwoodbury}, we get $\|\mathcal{H}(X)\| \leq \sigma_z^{-2}\upsilon^2 \|X\|^2 \|C\|^2 \leq \sigma_z^{-2}\upsilon^2 \beta^2 \|C\|^2$. Now consider $ F(\Sigma + X, A,C) - F(\Sigma + X, \tilde{A},\tilde{C})$:
\begin{align}
&F(\Sigma + X, \tilde{A},\tilde{C}) - F(\Sigma + X, A,C) \\
&= A (I + \sigma_z^{-2} (\Sigma + X) C^\top C)^{-1} (\Sigma + X) A^\top \!-\! \tilde{A} (I + \sigma_z^{-2} (\Sigma + X) \tilde{C}^\top \tilde{C})^{-1} (\Sigma + X) \tilde{A}^\top \nonumber \\
&= A(I + \sigma_z^{-2} (\Sigma + X) \tilde{C}^\top \tilde{C})^{-1} (\Sigma + X) \sigma_z^{-2}(C^\top C \!-\! \tilde{C}^\top \tilde{C}) (I \!+\! \sigma_z^{-2} (\Sigma \!+\! X) C^\top C)^{-1}(\Sigma \!+\! X) A^\top \nonumber \\
&- (\tilde{A} \!-\! A)(I \!+\! \sigma_z^{-2} (\Sigma \!+\! X) \tilde{C}^\top \tilde{C})^{-1}(\Sigma \!+\! X) A^\top \!-\! A (I + \sigma_z^{-2} (\Sigma \!+\! X) \tilde{C}^\top \tilde{C})^{-1} (\Sigma \!+\! X) (\tilde{A} \!-\! A)^\top \nonumber \\
&- (\tilde{A} - A) (I + \sigma_z^{-2} (\Sigma + X) \tilde{C}^\top \tilde{C})^{-1} (\Sigma + X) (\tilde{A} - A)^\top \label{Fparts}
\end{align}
Using, Lemma~\ref{normwoodbury} and the fact that $X \in \mathcal{S}_{\Sigma,\beta}$,
\begin{align}
&\|F(\Sigma\!+\!\!X, \tilde{A},\tilde{C})\!-\!F(\Sigma\!+\!\!X,A,C) \|\!\\
&\leq\! \sigma_z^{-2}\Phi(A)^2\|\Sigma + X\|^2 \| C^\top C - \tilde{C}^\top \tilde{C}\|\!+\!2\Phi(A) \|\Sigma\!+\!X\| \|\tilde{A} - A\| \!+\! \|\Sigma + X\| \|\tilde{A} -A \|^2 \nonumber \\
&\leq\!\sigma_z^{-2}\Phi(A)^2(\beta\! +\! \| \Sigma\| )^2(2\|C\| \|\tilde{C} \!-\! C \| + \|\tilde{C} \!- \!C \|^2 )\! +\! (\beta \!+\! \| \Sigma\|)(2\Phi(A) \|\tilde{A} \!-\! A\| + \|\tilde{A}\! -\! A\|^2) \label{boundFdif}
\end{align}
This gives us the following,
\begin{align*}
\| \Psi(X)\|\! &\leq\! \frac{\sigma_z^{-2}\Phi(A)^2\left(\beta + \| \Sigma\| \right)^2\left(2\|C\| \|\tilde{C} - C \| + \|\tilde{C} \!- \!C \|^2 \right)}{1-\upsilon^2} \\
&\qquad \qquad\qquad\qquad\qquad+ \frac{\left(\beta + \| \Sigma\|\right) \left(2\Phi(A) \|\tilde{A} - A\| + \|\tilde{A}\! -\! A\|^2\right) + \sigma_z^{-2}\upsilon^2 \beta^2 \|C\|^2 }{1-\upsilon^2}
\end{align*}
Again using Lemma~\ref{normwoodbury} and the definition of $\mathcal{H}(X)$, for $X_1, X_2 \in \mathcal{S}_{\Sigma,\beta}$
\begin{equation*}
\|\mathcal{H}(X_1) - \mathcal{H}(X_2) \| \leq \upsilon^2\left( (\sigma_z^{-2}\|C\|^2\beta)^2 + 2(\sigma_z^{-2}\|C\|^2\beta) \right) \|X_1 -X_2 \|
\end{equation*}
Next we bound
\[
\|\mathcal{D}(X_1,X_2)\| = \|F(\Sigma+X_1, \tilde{A},\tilde{C})-F(\Sigma+X_1,A,C) - F(\Sigma+X_2, \tilde{A},\tilde{C})+F(\Sigma+X_2,A,C)\|.
\]
Notice that using Lemma~\ref{normwoodbury}, we have $\|(I + \sigma_z^{-2} (\Sigma + X) \tilde{C}^\top \tilde{C})^{-1} \| , \|(I + \sigma_z^{-2} (\Sigma + X) C^\top C)^{-1} \| \leq \frac{2(\|\Sigma \| + \beta )}{\sigma_n(\Sigma)}$ from the choice of $\beta$. Let $V_1 = (I + \sigma_z^{-2} (\Sigma + X_1) C^\top C)^{-1} (\Sigma + X_1) $ and $\tilde{V}_1 = (I + \sigma_z^{-2} (\Sigma + X_1) \tilde{C}^\top \tilde{C})^{-1} (\Sigma + X_1) $. Define similarly $V_2$ and $\tilde{V}_2$. Note that from Lemma~\ref{normwoodbury}, $\|V_1\|, \| V_2\|, \| \tilde{V}_1\|, \| \tilde{V}_2\| \leq \|\Sigma\| + \beta$. Using these, we bound $\| \mathcal{D}(X_1,X_2)\| $ as follows
\begin{align}
&\Big\|\mathcal{D}(X_1,X_2) \Big\| \nonumber \\
&= \Big \|A\tilde{V}_1 \sigma_z^{-2}(C^\top C \!-\! \tilde{C}^\top \tilde{C}) V_1 A^\top \!\!-\! A \tilde{V}_2 \sigma_z^{-2}(C^\top C - \tilde{C}^\top \tilde{C}) V_2 A^\top \!\!-\! (\tilde{A} \!-\! A)\tilde{V}_1 A^\top \!+\! (\tilde{A} \!-\! A)\tilde{V}_2 A^\top \nonumber \\
& \qquad - A \tilde{V}_1 (\tilde{A} - A)^\top + A \tilde{V}_2 (\tilde{A} - A)^\top -(\tilde{A} - A) \tilde{V}_1 (\tilde{A} - A)^\top + (\tilde{A} - A) \tilde{V}_2 (\tilde{A} - A)^\top \Big \| \nonumber \\
&\leq \Phi(A)^2 \|(\tilde{V}_1 - \tilde{V}_2)\sigma_z^{-2}(C^\top C - \tilde{C}^\top \tilde{C})V_1 \| + \Phi(A)^2 \|\tilde{V}_2 \sigma_z^{-2}(C^\top C - \tilde{C}^\top \tilde{C}) (V_1 -V_2) \| \nonumber \\
& \qquad+ \|\tilde{V}_1 - \tilde{V}_2 \| \left(2\Phi(A)\|\tilde{A} - A \| + \| \tilde{A} - A\|^2 \right) \nonumber \\
&\leq \sigma_z^{-2} \Phi(A)^2 (2\|C\| \|\tilde{C} \!-\! C \| + \|\tilde{C} \!- \!C \|^2 ) \left( \|\tilde{V}_1 - \tilde{V}_2\| \| V_1 \| + \|\tilde{V}_2 \| \|V_1 -V_2 \| \right) \nonumber \\
&\qquad + \|\tilde{V}_1 - \tilde{V}_2 \| \left(2\Phi(A)\|\tilde{A} - A \| + \| \tilde{A} - A\|^2 \right) \label{boundVs}
\end{align}
We need to consider $\|\tilde{V}_1 - \tilde{V}_2 \| $ and $\|V_1 -V_2 \|$:
\begin{align*}
\|\tilde{V}_1 - \tilde{V}_2 \| &\leq \| (I + \sigma_z^{-2} (\Sigma + X_1) \tilde{C}^\top \tilde{C})^{-1} (X_1 - X_2) \| \\
&+ \left \|\left( (I + \sigma_z^{-2} (\Sigma + X_1) \tilde{C}^\top \tilde{C})^{-1} - (I + \sigma_z^{-2} (\Sigma + X_2) \tilde{C}^\top \tilde{C})^{-1} \right) (\Sigma + X_2) \right \| \\
&\leq \|X_1 -X_2\| \frac{2(\|\Sigma \| + \beta )}{\sigma_n(\Sigma)} + \sigma_z^{-2} \frac{4(\|\Sigma \| + \beta )^3}{\sigma^2_n(\Sigma)} (\|C\| + \|\tilde{C} -C \|)^2 \|X_1 - X_2 \| \\
\|V_1 - V_2 \| &\leq \|X_1 -X_2\| \frac{2(\|\Sigma \| + \beta )}{\sigma_n(\Sigma)} + \sigma_z^{-2} \frac{4(\|\Sigma \| + \beta )^3}{\sigma^2_n(\Sigma)} \|C\|^2 \|X_1 - X_2 \|
\end{align*}
Combining these with \eqref{boundVs}, we get
\small
\begin{align*}
&\Big\|\mathcal{D}(X_1,X_2) \Big\| \\
&\leq\! \Bigg[ (2\|C\| \|\tilde{C} \!-\! C \|\!+\! \|\tilde{C} \!- \!C \|^2 ) \Phi(A)^2 \Bigg(\frac{4\sigma_z^{-2} (\|\Sigma \|\! +\! \beta )^2}{\sigma_n(\Sigma)}\! +\! \frac{8\sigma_z^{-4}(\|\Sigma \| \!+\! \beta )^4}{\sigma^2_n(\Sigma)} ((\|C\| \!+\! \|\tilde{C} -C \|)^2\!+\!\|C\|^2) \!\Bigg) \\
&+\left(2\Phi(A)\|\tilde{A} - A \| + \| \tilde{A} - A\|^2 \right) \left( \frac{2(\|\Sigma \| + \beta )}{\sigma_n(\Sigma)} + \frac{4\sigma_z^{-2}(\|\Sigma \| + \beta )^3}{\sigma^2_n(\Sigma)} (\|C\| + \|\tilde{C} -C \|)^2 \right) \Bigg] \|X_1 - X_2 \| \!
\end{align*}
\normalsize
Therefore we have the following inequality for $\Psi(X_1) - \Psi(X_2)$:
\small
\begin{align}
&\|\Psi(X_1)\!-\!\Psi(X_2) \| \nonumber \\
&\leq \Bigg[ (2\|C\| \|\tilde{C} \!-\! C \|\!+\! \|\tilde{C} \!- \!C \|^2 )\Phi(A)^2 \Bigg(\frac{4\sigma_z^{-2} (\|\Sigma \|\! +\! \beta )^2}{\sigma_n(\Sigma)}\! +\! \frac{8\sigma_z^{-4}(\|\Sigma \| \!+\! \beta )^4}{\sigma^2_n(\Sigma)} ((\|C\| \!+\! \|\tilde{C} -C \|)^2\!+\!\|C\|^2) \!\Bigg) \nonumber \\
&+ \left(2\Phi(A)\|\tilde{A} - A \| + \| \tilde{A} - A\|^2 \right) \left( \frac{2(\|\Sigma \| + \beta )}{\sigma_n(\Sigma)} + \sigma_z^{-2} \frac{4(\|\Sigma \| + \beta )^3}{\sigma^2_n(\Sigma)} (\|C\| + \|\tilde{C} -C \|)^2 \right) \nonumber \\
&+ \upsilon^2\left( (\sigma_z^{-2}\|C\|^2\beta)^2 + 2(\sigma_z^{-2}\|C\|^2\beta) \right) \Bigg] \frac{\|X_1 - X_2 \| }{1-\upsilon^2} \label{contraction}
\end{align}
\normalsize
Denote $\epsilon$ such that $\epsilon \coloneqq \max\{\|C - \tilde{C} \|, \|A - \tilde{A} \|\}$. The choice of $T_{exp}$ (due to $T_A$, $T_B$) guarantees that $\epsilon < 1$. In order to show that $D_{\Sigma}$ is the unique fixed point of $\Psi$ in $\mathcal{S}_{\Sigma,\beta}$, one needs to show that $\Psi$ maps $\mathcal{S}_{\Sigma,\beta}$ to itself and it's contraction. To this end, we need to have $\epsilon$ and $\beta$ that gives $\|\Psi(X)\| \leq \beta$ and $\|\Psi(X_1)\!-\!\Psi(X_2) \| < \|X_1 - X_2 \| $. Let $\beta = 2k^* \epsilon < \frac{\sigma_n(\Sigma)}{2}$ where
\begin{equation*}
k^* = \frac{\sigma_z^{-2}\Phi(A)^2 (2\|C\|+ 1)\|\Sigma \|^2 + (2\Phi(A) + 1) \| \Sigma\| }{1 - \upsilon^2}
\end{equation*}
One can verify that this gives $\|\Psi(X)\| \leq \beta$. In order to get contraction, the coefficient of $\|X_1 - X_2\|$ in \eqref{contraction} must be less than 1. This requires
\begin{equation*}
\epsilon < \frac{1-\upsilon^2}{(2\|C\|+1)\Phi(A)^2 c_1 + (2\Phi(A)+1)c_2 + 6k^*c_3},
\end{equation*}
for
\begin{align*}
c_1 &= \Bigg(\frac{4\sigma_z^{-2} (\|\Sigma \|\! +\! \sigma_n(\Sigma)/2 )^2}{\sigma_n(\Sigma)}\! +\! \frac{8\sigma_z^{-4}(\|\Sigma \| \!+\! \sigma_n(\Sigma)/2 )^4}{\sigma^2_n(\Sigma)} (2\|C\|^2 + 2\|C\| + 1) \!\Bigg) \\
c_2 &= \Bigg( \frac{2(\|\Sigma \| + \sigma_n(\Sigma)/2 )}{\sigma_n(\Sigma)} + \frac{4\sigma_z^{-2}(\|\Sigma \| + \sigma_n(\Sigma)/2 )^3}{\sigma^2_n(\Sigma)} (\|C\|^2 + 2\|C\| + 1) \Bigg) \\
c_3 &= \upsilon^2 \sigma_z^{-2} \|C\|^2.
\end{align*}
From the choice of $T_{exp}$ (Due to $T_L$), $\epsilon$ satisfies the stated bound. Thus, $\Psi$ has a unique fixed point in $\mathcal{S}_{\Sigma,2k\epsilon}$, \textit{i.e.} $\|\tilde{\Sigma} - \Sigma \| \leq 2k^* \max\{\|C - \tilde{C} \|, \|A - \tilde{A} \| \}$. Bringing back the similarity transformations, this gives us the following bound
\begin{align*}
\|\tilde{\Sigma} - \mathbf{S}^{-1} \Sigma \mathbf{S} \| &\leq 2k^* \max\{\|\tilde{C} - C \mathbf{S} \|, \|\tilde{A} - \mathbf{S}^{-1}A\mathbf{S} \| \} \\
& \leq 4k^* \max\left\{\beta_A, \beta_C \right\} \coloneqq \Delta \Sigma
\end{align*}
since $\|\tilde{A} - \mathbf{S}^{-1}A\mathbf{S} \| \leq 2\beta_A $ and $\|\tilde{C} - C\mathbf{S} \| \leq 2\beta_C $. \\
We know prove the second part of Lemma \ref{StabilityCov} for $\|\tilde{L} - \mathbf{S}^{-1}L\|$. Again, considering for $\mathbf{S} = I$,
and using the definition of $L$ and $\tilde{L}$, we get
\begin{align}
L-\tilde{L} &= \Sigma C^\top(C \Sigma C^\top + \sigma_z^2 I)^{-1} - \tilde{\Sigma} \tilde{C}^\top(\tilde{C} \tilde{\Sigma} \tilde{C}^\top + \sigma_z^2 I)^{-1} \nonumber \\
&= (\Sigma C^\top - \tilde{\Sigma} \tilde{C}^\top) (C \Sigma C^\top + \sigma_z^2 I)^{-1} + \tilde{\Sigma} \tilde{C}^\top \left((C \Sigma C^\top + \sigma_z^2 I)^{-1} - (\tilde{C} \tilde{\Sigma} \tilde{C}^\top + \sigma_z^2 I)^{-1} \right) \nonumber \\
&= \left( (\Sigma - \tilde{\Sigma} )C^\top + (\tilde{\Sigma} - \Sigma)(C^\top - \tilde{C}^\top) + \Sigma(C^\top - \tilde{C}^\top)\right)(C \Sigma C^\top + \sigma_z^2 I)^{-1} \nonumber \\
&+ \tilde{\Sigma} \tilde{C}^\top (C \Sigma C^\top + \sigma_z^2 I)^{-1} \left((C \Sigma C^\top \!+\! \sigma_z^2 I)^{-1} + (\tilde{C}\tilde{\Sigma} \tilde{C}^\top \!\!\!-\! C\Sigma C^\top )^{-1}\right)^{-1}\!\!\! (C \Sigma C^\top \!+\! \sigma_z^2 I)^{-1} \label{Lwoodbury}
\end{align}
where \eqref{Lwoodbury} follows from Matrix inversion lemma. We will bound each term individually:
\begin{align*}
\|\tilde{L} - L \| &\leq \sigma_z^{-2}\left(\|C\|\| \Sigma - \tilde{\Sigma} \| + \| \Sigma - \tilde{\Sigma} \|\|\tilde{C} - C\| + \| \Sigma\| \|\tilde{C} - C\|\right) \\
&\qquad \qquad + \frac{\sigma_z^{-4}\left(\|C\|\| \Sigma - \tilde{\Sigma} \| + \| \Sigma - \tilde{\Sigma} \|\|\tilde{C} - C\| + \| \Sigma\| \|\tilde{C} - C\| + \|C\| \| \Sigma\| \right)}{\sigma_n \left( (C \Sigma C^\top + \sigma_z^2 I)^{-1} + (\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top )^{-1} \right)}
\end{align*}
From Weyl's inequality we have
\begin{align*}
&\frac{1}{\sigma_n \left( (C \Sigma C^\top + \sigma_z^2 I)^{-1} + (\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top )^{-1} \right)} \\
&\qquad \qquad \qquad \qquad \qquad \leq \frac{1}{\sigma_n \left( (C \Sigma C^\top + \sigma_z^2 I)^{-1} \right) +\sigma_n \left( (\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top )^{-1} \right)} \\
&\qquad \qquad \qquad \qquad \qquad = \frac{1}{\frac{1}{\|C \Sigma C^\top + \sigma_z^2 I \|} + \frac{1}{\|\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top \|}} = \frac{\|\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top\|}{1 + \frac{\|\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top\|}{\|C \Sigma C^\top + \sigma_z^2 I \|}} \\
&\qquad \qquad \qquad \qquad \qquad \leq \| \tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top \|
\end{align*}
where the last inequality follows from the choice of $T_{exp}$ which provides that $\|\tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top \| \leq \|C \Sigma C^\top + \sigma_z^2 I \| $. Thus we get
\begin{align*}
&\|\tilde{L} - L \| \\
&\leq \left(\sigma_z^{-2} + \sigma_z^{-4}\| \tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top \|\right) \left(\|C\|\| \Sigma - \tilde{\Sigma} \| + \| \Sigma - \tilde{\Sigma} \|\|\tilde{C} - C\| + \| \Sigma\| \|\tilde{C} - C\| \right) \\
&\qquad \qquad+ \sigma_z^{-4} \| \tilde{C}\tilde{\Sigma} \tilde{C}^\top - C\Sigma C^\top \| \|C\| \|\Sigma\| \\
&\leq\! \left(\sigma_z^{-2} \!+\! \sigma_z^{-4} \left( 4\beta_C^2\|\Sigma \| \!+\! 4\beta_C\|C\|\|\Sigma \| \!+\! \|C\|^2 \Delta \Sigma \!+\! 4\beta_C \|C\| \Delta \Sigma \!+\! 4 \beta_C^2 \Delta \Sigma \right) \right)\left(\|C\|\Delta \Sigma \!+\! 2 \beta_C \Delta \Sigma \!+\! 2 \beta_C \| \Sigma\| \right)\\
&\qquad \qquad+ \sigma_z^{-4} \left( 4\beta_C^2\|\Sigma \| \!+\! 4\beta_C\|C\|\|\Sigma \| \!+\! \|C\|^2 \Delta \Sigma \!+\! 4\beta_C \|C\| \Delta \Sigma \!+\! 4 \beta_C^2 \Delta \Sigma \right) \|C\| \|\Sigma\| \\
&\leq\! \left(\sigma_z^{-2} \!+\! \sigma_z^{-4} \left( 4\beta_C\|\Sigma \| \!+\! 4\beta_C\|C\|\|\Sigma \| \!+\! \|C\|^2 \Delta \Sigma \!+\! 4 \|C\| \Delta \Sigma \!+\! 4 \Delta \Sigma \right) \right)\left(\|C\|\Delta \Sigma \!+\! 2 \Delta \Sigma \!+\! 2 \beta_C \| \Sigma\| \right)\\
&\qquad \qquad+ \sigma_z^{-4} \left( 4\beta_C\|\Sigma \| \!+\! 4\beta_C\|C\|\|\Sigma \| \!+\! \|C\|^2 \Delta \Sigma \!+\! 4 \|C\| \Delta \Sigma \!+\! 4 \Delta \Sigma \right) \|C\| \|\Sigma\|
\\
&\leq\! \Delta \Sigma \left(\sigma_z^{-2}(\|C\|+2) + \sigma_z^{-4} \|C\|^3 \| \Sigma\| + 10 \sigma_z^{-4} \|C\|^2 \| \Sigma \| + 24 \sigma_z^{-4} \|C\| \| \Sigma \| + 16 \sigma_z^{-4} \| \Sigma \| \right) \\
&\qquad \qquad + \beta_C \left( 2 \sigma_z^{-2} \| \Sigma\| +
8\sigma_z^{-4} \| \Sigma\|^2
+ 12\sigma_z^{-4} \|C\| \| \Sigma\|^2 + 4\sigma_z^{-4} \|C\|^2 \| \Sigma\|^2 \right)\\
&\qquad \qquad +\! \Delta \Sigma^2 \left( \sigma_z^{-4} \|C\|^3 + 6\sigma_z^{-4} \|C\|^2 + 12\sigma_z^{-4} \|C\| + 8\sigma_z^{-4} \right).
\end{align*}
The choice of $T_L$ gives,
\begin{align*}
&\|\tilde{L}\!-\! L\| \\
&\leq\! \Delta \Sigma \bigg(\sigma_z^{-2}(\|C\|\!+\!2) \!+\! \sigma_z^{-4} \|C\|^3 (1\!+\!\| \Sigma\| ) \!+\! 10 \sigma_z^{-4} \|C\|^2 (0.6 \!+\! \| \Sigma \|) \!+\! 24 \sigma_z^{-4} \|C\| (0.5\!+\!\| \Sigma \|) \!+\! 16 \sigma_z^{-4} (0.5 \!+\! \| \Sigma \|) \bigg) \\
&\qquad + \beta_C \bigg( 2 \sigma_z^{-2} \| \Sigma\| +
8\sigma_z^{-4} \| \Sigma\|^2
+ 12\sigma_z^{-4} \|C\| \| \Sigma\|^2 + 4\sigma_z^{-4} \|C\|^2 \| \Sigma\|^2 \bigg)
\end{align*}
\null\hfill$\square$
\section{Regret of Exploration Phase}
\label{SuppRegretExplore}
\begin{lemma}
Suppose Assumptions \ref{Stable} and \ref{AssumContObs} hold. For any $0 < \delta < 1$, with probability as least $1-\delta$, the regret of controlling \textsc{\small{LQG}}\xspace system $\Theta$ with $u_t \sim \mathcal{N}(0,\sigma_u^2)$ for $1\leq t\leq T_{exp}$, $\textit{i.e.}$ pure exploration, is upper bounded as follows:
\begin{align}
\textsc{\small{REGRET}}\xspace(T_{exp}) \!&\leq\! T_{exp} \! \left( \frac{(\sigma_w^2 \!+\! \sigma_u^2 \|B\|^2) \|A\|^2}{1-\|A\|^2} \Tr(C^\top Q C) \!+\! \sigma_u^2 \Tr(R) \!-\! \Tr(C^\top Q C \bar{\Sigma} \!+\! P(\Sigma \!-\! \bar{\Sigma})) \! \right) \nonumber \\
&\qquad \qquad \qquad + 2\sqrt{T_{exp}}\left(\|C^\top Q C \|X_{exp}^2 + \| Q\|Z^2 + \|R\|U_{exp}^2\right)\sqrt{2 \log\frac{2}{\delta}}
\end{align}
where $X_{exp} \coloneqq \frac{(\sigma_w^2 + \sigma_u^2 \|B\|^2) \|A\|^2}{1-\|A\|^2}\sqrt{2n\log(12nT_{exp}/\delta)}$, $Z \coloneqq \sigma_z^2 \sqrt{2m\log(12mT_{exp}/\delta)}$,\\ $U_{exp} \coloneqq \sigma_u^2 \sqrt{2p\log(12pT_{exp}/\delta)}$, while $\Sigma$ and $\bar{\Sigma}$ are the solutions to the following algebraic Riccati equation of system $\Theta$:
\begin{equation*}
\Sigma = A \bar{\Sigma} A^\top + \sigma_w^2 I, \qquad \bar{\Sigma} = \Sigma - \Sigma C^\top \left( C \Sigma C^\top + \sigma_z^2 I \right)^{-1} C \Sigma.
\end{equation*}
\end{lemma}
\begin{proof}
For all $1\leq t \leq T_{exp}$, $\Sigma(x_t) \preccurlyeq \mathbf{\Gamma_\infty}$, where $\mathbf{\Gamma_\infty}$ is the steady state covariance matrix of $x_t$ such that,
\begin{equation*}
\mathbf{\Gamma_\infty} = \sum_{i=0}^\infty \sigma_w^2 A^i(A^\top)^i + \sigma_u^2 A^iBB^\top(A^\top)^i.
\end{equation*}
From the Assumption \ref{Stable}, for a finite $\Phi(A)$, $\|A^\tau \| \leq \Phi(A) \rho(A)^\tau$ for all $\tau \geq 0$. Thus, $\|\mathbf{\Gamma_\infty}\| \leq (\sigma_w^2 + \sigma_u^2 \|B\|^2) \frac{\Phi(A)^2 \rho(A)^2}{1-\rho(A)^2}$. Notice that each $x_t$ is component-wise $\sqrt{\|\mathbf{\Gamma_\infty}\|}$-sub-Gaussian random variable.
Combining this with Lemma~\ref{subgauss lemma} and using union bound, we can deduce that with probability $1-\delta/2$, for all $1\leq t \leq T_{exp}$,
\begin{align}
\label{exploration norms first}
\| x_t \| &\leq X_{exp} \coloneqq \frac{(\sigma_w + \sigma_u \|B\|) \Phi(A) \rho(A)}{\sqrt{1-\rho(A)^2}}\sqrt{2n\log(12nT_{exp}/\delta)} , \\
\| z_t \| &\leq Z \coloneqq \sigma_z \sqrt{2m\log(12mT_{exp}/\delta)} , \\
\| u_t \| &\leq U_{exp} \coloneqq \sigma_u \sqrt{2p\log(12pT_{exp}/\delta)} \label{exploration norms last}.
\end{align}
Let $\Omega = 2(\|C^\top Q C \|X_{exp}^2 + \| Q\|Z^2 + \|R\|U_{exp}^2)$. Define $\mathcal{X}_t = x_t^\top C^\top Q C x_t + z_t^\top Q z_t + u_t^\top R u_t - \mathbb{E}[x_t^\top C^\top Q C x_t + z_t^\top Q z_t + u_t^\top R u_t]$ and its truncated version $\tilde{\mathcal{X}}_t =\mathbbm{1}_{\mathcal{X}_t \leq \Omega} \mathcal{X}_t$. Define $S = \sum_{t=1}^{T_{exp}} \mathcal{X}_t$ and $\tilde{S} = \sum_{t=1}^{T_{exp}} \tilde{\mathcal{X}}_t$. By Lemma~\ref{basicprob},
\begin{align}
\Pr\left( S > \Omega \sqrt{2T_{exp} \log\frac{2}{\delta}}\right) \leq \Pr \left(\max_{1\leq t\leq T_{exp}} \mathcal{X}_t \geq \Omega \right) + \Pr\left( \tilde{S} > \Omega \sqrt{2T_{exp} \log\frac{2}{\delta}} \right).
\end{align}
From equations \eqref{exploration norms first}-\eqref{exploration norms last} and Theorem~\ref{azuma}, each term on the right hand side is bounded by $\delta/2$. Thus, with probability $1-\delta$,
\begin{align}
&\sum_{t=1}^{T_{exp}} y_t^\top Q y_t \!+\! u_t^\top R u_t \!-\! \mathbb{E}[y_t^\top Q y_t \!+\! u_t^\top R u_t] \!\leq\! \Omega \sqrt{2T_{exp} \log\frac{2}{\delta}} \\
&\sum_{t=1}^{T_{exp}} y_t^\top Q y_t \!+\! u_t^\top R u_t \!\leq\! T_{exp}\left((\sigma_w^2 \!+\! \sigma_u^2 \|B\|^2) \frac{\Phi(A)^2 \rho(A)^2}{1-\rho(A)^2} \Tr(C^\top Q C) \!+\! \sigma_u^2 \Tr(R) \!+\! \sigma_z^2 \Tr(Q) \right) \\
&\qquad\qquad\qquad\qquad\qquad+ 2\left(\|C^\top Q C \|X_{exp}^2 + \| Q\|Z^2 + \|R\|U_{exp}^2\right)\sqrt{2T_{exp} \log\frac{2}{\delta}}
\end{align}
Recall that cost obtained in $T_{exp}$ by the optimal controller of $\Theta$ is
\[T_{exp}\left(\Tr(C^\top Q C \bar{\Sigma}) + \Tr(P(\Sigma - \bar{\Sigma})) + \sigma_z^2 \Tr(Q) \right).\]
Thus the regret obtained from $T_{exp}$ length exploration is upper bounded as described in the statement of lemma.
\end{proof}
\section{Upper Bound on \texorpdfstring{$\|\hat{x}_{t|t,\tilde{\Theta}}\|$}{State Estimation} and \texorpdfstring{$\|y_t\|$}{Output}, Proof of Lemma \ref{Boundedness}}
\label{SuppBounded}
We know that the behavior of the underlying system is the same as any system that is obtained via similarity transformation of it. Therefore, without loss of generality, we assume that $\mathbf{S} = I$. Also note that, the effect of initial state after $T_{exp}$ length exploration is omitted in the presentation since it is well-controlled during the exploration, \textit{i.e.}, \eqref{exploration norms first}, and it exponentially decays after the exploration due to achieved stable closed loop dynamics that is explained below. First define $e_t = y_t - C\hat{x}_{t|t-1,\Theta}$, where $e_t \sim \mathcal{N}\left(0,C \Sigma C^\top + \sigma_z^2 I \right)$. Note that $e_t$ is the zero mean white innovation process, in the innovation form of the system characterization.
\begin{proof}
Assume that the event $\mathcal{E}$ holds. Observe that $\hat{x}_{t|t,\tilde{\Theta}}$ has the following dynamics,
\begin{align}
\hat{x}_{t|t,\tilde{\Theta}} &= (I - \tilde{L}\tilde{C})(\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} + \tilde{L} y_t \nonumber\\
&= (I - \tilde{L}\tilde{C})(\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} \nonumber \\
&\quad \qquad \qquad +\tilde{L} \left(Cx_t - C\hat{x}_{t|t-1,\tilde{\Theta}} + C\hat{x}_{t|t-1,\tilde{\Theta}} + z_t\right) \nonumber \\
&= (I - \tilde{L}\tilde{C})(\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} \nonumber \\
&\quad \qquad \qquad +\tilde{L} \left(Cx_t - C\hat{x}_{t|t-1,\tilde{\Theta}} + C(\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} + z_t\right) \nonumber \\
&= \left(\tilde{A}-\tilde{B}\tilde{K} - \tilde{L}\left(\tilde{C}\tilde{A} -\tilde{C}\tilde{B}\tilde{K} - C\tilde{A} + C\tilde{B}\tilde{K} \right) \right)\hat{x}_{t-1|t-1,\tilde{\Theta}} \nonumber \\
&\quad \qquad \qquad + \tilde{L} C(x_{t} - \hat{x}_{t|t-1,\Theta} + \hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}) + \tilde{L} z_t \nonumber \\
&= \left(\tilde{A}-\tilde{B}\tilde{K} - \tilde{L}\left(\tilde{C}\tilde{A} -\tilde{C}\tilde{B}\tilde{K} - C\tilde{A} + C\tilde{B}\tilde{K} \right) \right)\hat{x}_{t-1|t-1,\tilde{\Theta}} \nonumber \\
&\quad \qquad \qquad + \tilde{L} C(x_{t} - \hat{x}_{t|t-1,\Theta}) + \tilde{L} C (\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}) + \tilde{L} z_t \label{estimation propagate}
\end{align}
Thus, it propagates according to the linear system given in equation \eqref{estimation propagate} with closed loop dynamics $\mathbf{M} = \left(\tilde{A}-\tilde{B}\tilde{K} - \tilde{L}\left(\tilde{C}\tilde{A} -\tilde{C}\tilde{B}\tilde{K} - CA + CB\tilde{K} \right)\right)$ driven by the process $\tilde{L} C(x_{t} - \hat{x}_{t|t-1,\Theta}) + \tilde{L} C (\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}) + \tilde{L} z_t$. With the Assumption~\ref{Stabilizable set}, for the given $T_{exp}$ (due to $T_M$), we have $\|\mathbf{M} \| < \frac{1+\rho}{2} < 1$. Notice that $\tilde{L} C (x_{t} - \hat{x}_{t|t-1,\Theta}) + \tilde{L} z_t$ is $\zeta (\| C\| \|\Sigma\|^{1/2} + \sigma_z)$-sub-Gaussian, thus it's $\ell_2$-norm can be bounded using Lemma \ref{subgauss lemma}:
\begin{equation}
\|\tilde{L}_t C (x_{t} - \hat{x}_{t|t-1,\Theta}) + \tilde{L}_t z_t\| \leq \zeta \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2n\log(2nT/\delta)} \label{excitationbound}
\end{equation}
for all $T \geq t \geq T_{exp}$ with probability at least $1-\delta$. Next, we will consider $\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}$, \textit{i.e.} how much the state estimation using the optimistic model deviates from the state estimation using the true model. Let $\Delta_t = \hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}$. Consider the following decompositions,
\begin{align*}
\hat{x}_{t+1|t,\Theta} &= A \hat{x}_{t|t,\Theta} - B \tilde{K} \hat{x}_{t|t,\tilde{\Theta}}\\
&= A \hat{x}_{t|t,\Theta} - B \tilde{K} \hat{x}_{t|t,\Theta} - B \tilde{K} (\hat{x}_{t|t,\tilde{\Theta}} - \hat{x}_{t|t,\Theta}) \\
&= (A - B\tilde{K}) \hat{x}_{t|t,\Theta} - B \tilde{K} (\hat{x}_{t|t,\tilde{\Theta}} - \hat{x}_{t|t,\Theta}), \\
\hat{x}_{t+1|t,\tilde{\Theta}} &= \tilde{A} \hat{x}_{t|t,\tilde{\Theta}} - \tilde{B} \tilde{K} \hat{x}_{t|t,\tilde{\Theta}}\\
&= (\tilde{A} + A - A - \tilde{B} \tilde{K} + B \tilde{K} - B \tilde{K}) (\hat{x}_{t|t,\tilde{\Theta}} - \hat{x}_{t|t,\Theta} + \hat{x}_{t|t,\Theta} ) \\
&= \underbrace{(\tilde{A} \!-\! A \!-\! \tilde{B} \tilde{K} \!+\! B \tilde{K})}_{\delta_{\tilde{\Theta}}} \hat{x}_{t|t,\Theta} + \delta_{\tilde{\Theta}} (\hat{x}_{t|t,\tilde{\Theta}} \!-\! \hat{x}_{t|t,\Theta}) + (A \!-\! B \tilde{K}) (\hat{x}_{t|t,\tilde{\Theta}} \!-\! \hat{x}_{t|t,\Theta}) + (A \!-\! B \tilde{K}) \hat{x}_{t|t,\Theta}.
\end{align*}
Thus, we get
\begin{align}
\hat{x}_{t+1|t,\Theta} - \hat{x}_{t+1|t,\tilde{\Theta}} &= (A - B\tilde{K}) \hat{x}_{t|t,\Theta} - B \tilde{K} (\hat{x}_{t|t,\tilde{\Theta}} - \hat{x}_{t|t,\Theta}) \!-\! \delta_{\tilde{\Theta}} \hat{x}_{t|t,\Theta} - \delta_{\tilde{\Theta}} (\hat{x}_{t|t,\tilde{\Theta}} \!-\! \hat{x}_{t|t,\Theta}) \nonumber \\ &\quad - (A \!-\! B \tilde{K}) (\hat{x}_{t|t,\tilde{\Theta}} \!-\! \hat{x}_{t|t,\Theta}) - (A \!-\! B \tilde{K}) \hat{x}_{t|t,\Theta}, \nonumber \\
\Delta_{t+1}&= A (\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}} ) - \delta \hat{x}_{t|t,\Theta} + \delta (\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}} ) \nonumber \\
&= (A + \delta_{\tilde{\Theta}}) (\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}} ) - \delta_{\tilde{\Theta}} \hat{x}_{t|t,\Theta}. \label{lastline}
\end{align}
Recalling the definition of innovation term $e_t$ in the beginning of the section, we can decompose $\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}}$ as follows,
\begin{align*}
\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}} &= \hat{x}_{t|t-1,\Theta} + L e_t - (\hat{x}_{t|t-1,\tilde{\Theta}} + \tilde{L} (y_t - \tilde{C} \hat{x}_{t|t-1,\tilde{\Theta}} ) ) \\
&= \hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}} + L e_t - \tilde{L} (e_t + C\hat{x}_{t|t-1,\Theta} - \tilde{C} \hat{x}_{t|t-1,\tilde{\Theta}} ) \\
&= \underbrace{\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}} }_{\Delta_t} + (L-\tilde{L}) e_t - \tilde{L} \left((C-\tilde{C})\hat{x}_{t|t-1,\Theta} + \tilde{C}( \underbrace{\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}} }_{\Delta_t})\right) \\
&= (I -\tilde{L} \tilde{C}) \Delta_t + (L - \tilde{L}) e_t + \tilde{L} (\tilde{C} - C) \hat{x}_{t|t-1,\Theta}
\end{align*}
Plugging $\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}}$ into (\ref{lastline}) gives the following,
\begin{align}
\Delta_{t+1} &= (A + \delta_{\tilde{\Theta}}) \left((I -\tilde{L} \tilde{C}) \Delta_t + (L - \tilde{L}) e_t + \tilde{L} (\tilde{C} - C) \hat{x}_{t|t-1,\Theta} \right) - \delta_{\tilde{\Theta}} \hat{x}_{t|t,\Theta} \nonumber\\
&= (A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C}) \Delta_t + (A + \delta_{\tilde{\Theta}}) (L - \tilde{L}) e_t + (A + \delta_{\tilde{\Theta}}) \tilde{L} (\tilde{C} - C)\hat{x}_{t|t-1,\Theta} - \delta_{\tilde{\Theta}} \hat{x}_{t|t,\Theta} \nonumber \\
&= (A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C}) \Delta_t + (A + \delta_{\tilde{\Theta}}) (L - \tilde{L}) e_t + (A + \delta_{\tilde{\Theta}}) \tilde{L} (\tilde{C} - C)\hat{x}_{t|t-1,\Theta} - \delta_{\tilde{\Theta}} \hat{x}_{t|t-1,\Theta} - \delta_{\tilde{\Theta}} L e_t \nonumber \\
&= (A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C}) \Delta_t + (AL - A\tilde{L} +\delta_{\tilde{\Theta}} L - \delta_{\tilde{\Theta}} \tilde{L} -\delta_{\tilde{\Theta}} L)e_t + \left( (A+\delta_{\tilde{\Theta}})\tilde{L}(\tilde{C}-C) - \delta_{\tilde{\Theta}} \right)\hat{x}_{t|t-1,\Theta} \nonumber \\
&= (A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C}) \Delta_t + (A(L -\tilde{L}) - \delta_{\tilde{\Theta}} \tilde{L} )e_t + \left((A+\delta_{\tilde{\Theta}})\tilde{L}(\tilde{C}-C) - \delta_{\tilde{\Theta}} \right)\hat{x}_{t|t-1,\Theta} \nonumber \\
&= \sum_{j=0}^t \left((A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C})\right)^{t-j} \left(A(L -\tilde{L}) - \delta_{\tilde{\Theta}} \tilde{L} \right) e_j \nonumber \\
&\quad + \sum_{j=1}^t \left((A + \delta_{\tilde{\Theta}}) (I -\tilde{L} \tilde{C})\right)^{t-j} \left((A+\delta_{\tilde{\Theta}})\tilde{L}(\tilde{C}-C) - \delta_{\tilde{\Theta}} \right)\hat{x}_{j|j-1,\Theta}. \label{lastlinefordiff}
\end{align}
Now consider the decomposition given below for $\hat{x}_{j|j-1,\Theta}$:
\begin{align}
\hat{x}_{j|j-1,\Theta} &= A \hat{x}_{j-1|j-1,\Theta} - B \tilde{K} \hat{x}_{j-1|j-1,\tilde{\Theta}} \nonumber \\
&= A \hat{x}_{j-1|j-1,\Theta} - B \tilde{K} \hat{x}_{j-1|j-1,\Theta} - B \tilde{K} (\hat{x}_{j-1|j-1,\tilde{\Theta}} - \hat{x}_{j-1|j-1,\Theta})\nonumber \\
&= (A - B\tilde{K}) \hat{x}_{j-1|j-1,\Theta} + B \tilde{K} (\hat{x}_{j-1|j-1,\Theta} - \hat{x}_{j-1|j-1,\tilde{\Theta}} )\nonumber \\
&= (A - B\tilde{K}) (\hat{x}_{j-1|j-2,\Theta} + L e_{j-1}) + B \tilde{K} ((I -\tilde{L} \tilde{C}) \Delta_{j-1} + (L - \tilde{L}) e_{j-1} + \tilde{L} (\tilde{C} - C) \hat{x}_{j-1|j-2,\Theta})\nonumber \\
&= (A - B\tilde{K} (I - \tilde{L}(\tilde{C}-C))) \hat{x}_{j-1|j-2,\Theta} + B \tilde{K} (I - \tilde{L}\tilde{C}) \Delta_{j-1} + ((A - B\tilde{K}) L + B \tilde{K} (L - \tilde{L})) e_{j-1}\nonumber \\
&= \sum_{i=0}^{j-1} \left(A \!-\! B\tilde{K} \!+\! B\tilde{K}\tilde{L}(\tilde{C}\!-\!C)\right)^{j-i-1} \left( B\tilde{K}(I \!-\!\tilde{L} \tilde{C}) \Delta_{i} + ((A \!-\! B\tilde{K}) L \!+\! B \tilde{K} (L \!-\! \tilde{L})) e_{i} \right). \label{lastlineforeqq}
\end{align}
For the brevity of representation, we define the following terms:
\begin{align*}
\alpha &= (A \!+\! \delta_{\tilde{\Theta}}) (I \!-\!\tilde{L} \tilde{C}),\\
\beta &= (A\!+\!\delta_{\tilde{\Theta}})\tilde{L}(\tilde{C}\!-\!C) \!-\! \delta_{\tilde{\Theta}}, \\
\kappa &= A(L -\tilde{L}) - \delta_{\tilde{\Theta}} \tilde{L}, \\
\gamma &= A \!-\! B\tilde{K} \!+\! B\tilde{K}\tilde{L}(\tilde{C}\!-\!C), \\
\chi &= B\tilde{K}(I \!-\!\tilde{L} \tilde{C}), \\
\xi &= (A \!-\! B\tilde{K}) L \!+\! B \tilde{K} (L \!-\! \tilde{L}).
\end{align*}
Finally, using (\ref{lastlinefordiff}) with the equality given in (\ref{lastlineforeqq}) and the given definitions above we get:
\begin{align}
\Delta_{t+1} &= \sum_{j=0}^t \alpha ^{t-j} \kappa e_j +\sum_{j=1}^t \alpha^{t-j} \beta \left(\sum_{i=0}^{j-1} \gamma^{j-i-1} \chi \Delta_{i} \right) +\sum_{j=1}^t \alpha^{t-j} \beta \left(\sum_{i=0}^{j-1} \gamma^{j-i-1} \xi e_{i} \right). \label{term1}
\end{align}
We will bound first and third term in (\ref{term1}) separately and since each $\Delta_{t+1}$ has linear combination of $\Delta_i$ for $i\leq t$, we provide an inductive argument to bound $\Delta_{t+1}$ for all $t$. Since $T_{exp} > \max \{T_\alpha, T_\gamma \}$, the system parameter estimates are close enough to the underlying system parameters such that $1 > \sigma \geq \max \{ \|\alpha \|, \|\gamma \| \}$. For the first term in (\ref{term1}), since $e_t$ is $(\| C\| \|\Sigma\|^{1/2} + \sigma_z)$-sub-Gaussian, using Lemma \ref{subgauss lemma} for all $T \geq t \geq T_{exp}$ with probability at least $1-\delta$ we have,
\begin{align}
\sum_{j=0}^t \alpha^{t-j} \kappa e_j \leq \frac{\bar{\kappa} }{1-\sigma} \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)}, \label{firstterm}
\end{align}
where $\bar{\kappa} \geq \|\kappa \|$ under the event $\mathcal{E}$.
For the third term, for all $T \geq t \geq T_{exp}$ with probability at least $1-\delta$ we have,
\begin{align}
&\sum_{j=1}^t \alpha^{t-j} \beta \left( \sum_{i=0}^{j-1} \gamma^{j-i-1} \xi e_i \right) \nonumber \\
&= \alpha^{t-1} \beta \xi e_0 \!+\! \alpha^{t-2} \beta (\gamma \xi e_0 \!+\! \xi e_1) \!+\! \ldots \!+\! \beta(\gamma^{t-1} \xi e_0 \!+\! \ldots \!+\! \xi e_{t-1}) \nonumber \\
&= \left(\alpha^{t-1} \beta \xi \!+\! \alpha^{t-2} \beta \gamma \xi \!+\! \beta \gamma^{t-1} \xi \right) e_0 \nonumber \\
&\quad + \left( \alpha^{t-2} \beta \xi + \ldots + \beta\gamma^{t-2}\xi \right) e_1 + \ldots + \beta \xi e_{t-1} \nonumber \\
&\leq \bar{\beta} \bar{\xi} \left(\alpha^{t-1} \!+\! \alpha^{t-2}(\gamma \!+\! I) \!+\! \ldots \!+\! (\gamma^{t-1} \!+\! \gamma^{t-2} \!+\! \ldots \!+\! \gamma \!+\! I)\right) \!\! \left(\| C\| \|\Sigma\|^{1/2} \!+\! \sigma_z\right) \sqrt{2m\log(2mT/\delta)} \nonumber \\
&\leq \bar{\beta} \bar{\xi} \left( \sum_{i=1}^{t-1} i \sigma^i + \sum_{i=0}^{t-1} \sigma^i \right) \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)} \nonumber \\
&\leq \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2}\left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)}, \label{thirdterm}
\end{align}
where $\bar{\beta} \geq \|\beta \|$ and $\bar{\xi} \geq \|\xi \|$ under the event $\mathcal{E}$.
Considering the second term, we will have an inductive argument for the boundedness of $\Delta$. Let $\bar{\Delta} \coloneqq 10 \left(\frac{\bar{\kappa} }{1-\sigma} + \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right)\left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)} $. We have that $\| \Delta_1\| \leq \bar{\Delta}$ with probability at least $1-\delta$. Assume that it holds for all $t$. Thus, $\bar{\Delta} \geq \max_{i\leq t} \| \Delta_i\|$. Using the same arguments with the third term, for the second term, we get
\begin{align*}
\sum_{j=1}^t \alpha^{t-j} \beta \left( \sum_{i=0}^{j-1} \gamma^{j-i-1} \chi \Delta_i \right) &\leq \frac{\|\beta\| \bar{\chi} }{(1-\sigma)^2} \bar{\Delta}
\end{align*}
where $\bar{\chi} \geq \|\chi \|$ under the event $\mathcal{E}$. Combining with the bounds derived in (\ref{firstterm}) and (\ref{thirdterm}) for the Thus,
\begin{align*}
\Delta_{t+1} \leq \left(\frac{\bar{\kappa} }{1-\sigma} + \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right)\left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)} + \frac{\|\beta \| \bar{\chi} }{(1-\sigma)^2} \bar{\Delta}
\end{align*}
Recall that $T_{exp} > T_{\beta}$. Thus we have $\| \beta \| \leq \frac{9(1-\sigma)^2}{10 \Gamma \|B\|(1+ \zeta + \zeta \|C\|)}$. This shows that $\Delta_{t+1} \leq \bar{\Delta}$:
\begin{align*}
\Delta_{t+1} \leq \left(\left(\frac{\bar{\kappa} }{1-\sigma} \!+\! \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right) \!+\! 9 \left(\frac{\bar{\kappa} }{1-\sigma} \!+\! \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right) \right) \left(\| C\| \|\Sigma\|^{1/2} \!+\! \sigma_z\right) \sqrt{2m\log(2mT/\delta)} \leq \bar{\Delta}.
\end{align*}
Finally, for all $T_{exp}<t<T$, we have that with probability at least $1-\delta$,
\begin{align}
\hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}} \leq \bar{\Delta} = 10\left(\frac{\bar{\kappa} }{1-\sigma} + \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right)\left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)} \label{deltabound}
\end{align}
where
\begin{enumerate}
\item $\bar{\kappa} = \Phi(A) \Delta L + 2\zeta (\beta_A + \Gamma \beta_B)$,
\item $\bar{\beta} = 2\zeta \beta_C (\Phi(A) + 2(\beta_A + \Gamma \beta_B)) + 2(\beta_A + \Gamma \beta_B) $,
\item $\bar{\xi} = \zeta (\rho + 2(\beta_A + \Gamma \beta_B)) + \|B\|\Gamma \Delta L $.
\end{enumerate}
Then under the event $\mathcal{E}$, with probability $1-2\delta$,
\begin{align}
\| \hat{x}_{t|t,\tilde{\Theta}}\| &= \left \| \sum_{i=1}^t \mathbf{M}^{t-i} \left(\tilde{L} C(x_{i} - \hat{x}_{i|i-1,\Theta}) + \tilde{L} C (\hat{x}_{i|i-1,\Theta} - \hat{x}_{i|i-1,\tilde{\Theta}}) + \tilde{L} z_i \right) \right \| \\
&\leq \max_{1\leq i\leq t}\left\| \tilde{L} C(x_{i} - \hat{x}_{i|i-1,\Theta}) + \tilde{L} C (\hat{x}_{i|i-1,\Theta} - \hat{x}_{i|i-1,\tilde{\Theta}}) + \tilde{L} z_i \right \| \left( \sum_{i=1}^t \|\mathbf{M}\|^{t-i} \right) \\
&\leq \frac{2}{1-\rho} \max_{1\leq i\leq t}\left\| \tilde{L} C(x_{i} - \hat{x}_{i|i-1,\Theta}) + \tilde{L} C (\hat{x}_{i|i-1,\Theta} - \hat{x}_{i|i-1,\tilde{\Theta}}) + \tilde{L} z_i \right \| \\
&\leq \tilde{\mathcal{X}} \coloneqq \frac{2\zeta \left(\|C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2n\log(2nT/\delta)}\right) }{1-\rho}
\end{align}
where the last inequality follows from (\ref{excitationbound}) and (\ref{deltabound}). One can write $y_t$ as
\begin{align*}
y_t &= C (\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} + C(x_t - \hat{x}_{t|t-1,\tilde{\Theta}}) + z_{t} \\
&= C (\tilde{A} - \tilde{B}\tilde{K}) \hat{x}_{t-1|t-1,\tilde{\Theta}} + C(x_t - \hat{x}_{t|t-1,\Theta} + \hat{x}_{t|t-1,\Theta} - \hat{x}_{t|t-1,\tilde{\Theta}}) + z_{t}
\end{align*}
Following similar argument, with probability $1-2\delta$,
\begin{equation}
\|y_t \| \leq \rho \| C\| \tilde{\mathcal{X}} + \| C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z \right) \sqrt{2m\log(2mT/\delta)}
\end{equation}
for all $t\leq T$, which proves the lemma.
\end{proof}
\section{Bellman Optimality Equation for \textsc{\small{LQG}}\xspace, Proof of Lemma \ref{LQGBellman}}
\label{SuppBelmanOptimality}
For an average cost per stage problem in infinite state and control space like a \textsc{\small{LQG}}\xspace control system $\Theta = \left(A,B,C\right)$ with regulating parameters $Q$ and $R$, using the optimal average cost per stage $J_*(\Theta)$ and guessing the correct differential(relative) cost, where $(A,B)$ is controllable, $(A,C)$ is observable, $Q$ is positive semidefinite and $R$ is positive definite, one can verify that they satisfy Bellman optimality equation~\citep{bertsekas1995dynamic}. The lemma below shows the Bellman optimality equation for \textsc{\small{LQG}}\xspace system $\Theta$, which will be critical in regret analysis. \\
\begin{proof}
Define $\hat{\omega}_t = Ax_t - A\hat{x}_{t|t} + w_t$. Notice that $\hat{\omega}_t$ is independent of the policy used and depends only on the estimation error and noise in steady state. Also notice that $\mathbb{E}\left[\hat{\omega}_t \hat{\omega}_t^\top \right] = \Sigma$ where $\Sigma$ is the positive semidefinite solution to the algebraic Riccati equation:
\begin{equation}
\label{esti err cov riccati}
\Sigma = A \bar{\Sigma} A^\top + \sigma_w^2 I, \qquad \bar{\Sigma} = \Sigma - \Sigma C^\top \left( C \Sigma C^\top + \sigma_z^2 I\right)^{-1} C \Sigma.
\end{equation}
Using these, for any given $y_t$ and $\hat{x}_{t|t-1}$ at time $t$, the optimum state estimation and the output at $t+1$ can be written as
\begin{equation} \label{next time step decomposition}
\hat{x}_{t|t} = \left( I - LC\right)\hat{x}_{t|t-1} + Ly_t, \enskip \hat{x}_{t+1|t,u} = A\hat{x}_{t|t} + Bu, \enskip y_{t+1,u} = CA \hat{x}_{t|t} + CBu + C\hat{\omega}_t + z_{t+1}
\end{equation}
where $L = \Sigma C^\top \left( C \tilde{\Sigma} C^\top + \sigma_z^2 \right)^{-1}$ is the steady-state Kalman filter gain for $\Theta$. Since the aim is to minimize average cost per stage of controlling $\Theta$, the optimal control input, $u = -K\hat{x}_{t|t}$, where $K = \left(R+B^{\top} P B\right)^{-1} B^{\top} P A $ is the steady-state LQR feedback gain for $\Theta$ and $P$ is the positive semidefinite solution to the following algebraic Riccati equation:
\begin{equation}
P = A^\top P A + C^\top Q C - A^\top P B \left( R + B^\top P B \right)^{-1} B^\top P A.
\end{equation}
Recall that optimal average stage cost of \textsc{\small{LQG}}\xspace is $J_*(\Theta) = \Tr(C^\top Q C \bar{\Sigma}) + \Tr(P(\Sigma - \bar{\Sigma})) + \Tr(\sigma_z^2 Q)$.
Suppose the differential cost $h$ is a quadratic function of $s_t$ where $s_t = [\hat{x}_{t|t-1}^\top~y_t^\top]^\top \in \mathbb{R}^{n+m}$, \textit{i.e.}
\[ h(s_t) = s_t^\top
\left[
\begin{array}{cc}{G_1} & { G_2 } \\ {G_2^\top} & { G_3 } \end{array} \right] s_t = \hat{x}_{t|t-1}^\top G_1 \hat{x}_{t|t-1} + 2 \hat{x}_{t|t-1}^\top G_2 y_t + y_t^\top G_3 y_t. \]
One needs to verify that there exists $G_1$, $G_2$, $G_3$ such that they satisfy Bellman optimality equation for the chosen differential cost:
\begin{align*}
&J_*(\Theta) + \hat{x}_{t|t-1}^\top G_1 \hat{x}_{t|t-1} + 2 \hat{x}_{t|t-1}^\top G_2 y_t + y_t^\top G_3 y_t = \\
& y_t^\top Q y_t + \hat{x}_{t|t}^\top K^\top R K \hat{x}_{t|t} + \mathbb{E}\left[ \hat{x}_{t+1|t}^\top G_1 \hat{x}_{t+1|t} + 2 \hat{x}_{t+1|t}^\top G_2 y_{t+1} + y_{t+1}^\top G_3 y_{t+1} \right]
\end{align*}
Using the fact that $\bar{\Sigma} = \Sigma - L \left( C \Sigma C^\top + \sigma_z^2 I \right) L^\top$, we can write the optimal average cost as $J_*(\Theta) = \Tr\left( \left(Q + L^\top P L - L^\top C^\top Q C L \right) \left(C \Sigma C^\top + \sigma_z^2 I \right)\right)$. Expanding the expectation given $\hat{x}_{t|t-1}, y_t$ and using \eqref{next time step decomposition} , we get
\begin{align} \label{bellman decomp}
\hat{x}_{t|t-1}^\top &G_1 \hat{x}_{t|t-1} + 2 \hat{x}_{t|t-1}^\top G_2 y_t + y_t^\top G_3 y_t \\
&= y_t^\top Q y_t + \hat{x}_{t|t}^\top K^\top R K \hat{x}_{t|t} + \hat{x}_{t|t}^\top (A-BK)^\top G_1 (A-BK) \hat{x}_{t|t}\nonumber \\&+ 2\hat{x}_{t|t}^\top (A-BK)^\top G_2 C(A-BK)\hat{x}_{t|t} + \hat{x}_{t|t}^\top (A-BK)^\top C^\top G_3 C (A-BK)\hat{x}_{t|t} \nonumber\\ &+ \mathbb{E}\left[\hat{\omega}_t^\top C^\top G_3 C \hat{\omega}_t + z_{t+1}^\top G_3 z_{t+1} \right] \!-\! \Tr\left( \left(Q + L^\top P L - L^\top C^\top Q C L \right) \left(C \Sigma C^\top \!+\! \sigma_z^2 I \right)\right) \nonumber
\end{align}
Notice that $\mathbb{E}\left[\hat{\omega}_t^\top C^\top G_3 C \hat{\omega}_t + z_{t+1}^\top G_3 z_{t+1} \right] = \Tr\left( G_3 \left(C \Sigma C^\top + \sigma_z^2 I \right) \right)$. In order to match with the last term of \eqref{bellman decomp}, set $G_3 = Q + L^\top \left( P - C^\top Q C \right) L$. Inserting $G_3$ to \eqref{bellman decomp}, we get following 3 equations to solve for $G_1$ and $G_2$: \\
\textbf{1)} From quadratic terms of $y_t$:
\begin{align*}
L^\top P L &- L^\top C^\top Q C L \\
&= L^\top K^\top R K L + L^\top (A-BK)^\top G_1 (A-BK) L + 2 L^\top (A-BK)^\top G_2 C (A-BK) L \\&+ L^\top (A-BK)^\top C^\top \left( Q + L^\top P L - L^\top C^\top Q C L \right) C (A-BK) L
\end{align*}
\textbf{2)} From quadratic terms of $x_{t|t-1}$:
\begin{align*}
G_1 &= (I-LC)^\top K^\top R K (I-LC) + (I-LC)^\top (A-BK)^\top G_1 (A-BK)(I-LC) \\
&+ 2(I-LC)^\top (A-BK)^\top G_2 C (A-BK) (I-LC) \\ &+ (I-LC)^\top (A-BK)^\top C^\top \left( Q + L^\top P L - L^\top C^\top Q C L\right) C (A-BK) (I-LC)
\end{align*}
\textbf{3)} From bilinear terms of $x_{t|t-1}$ and $y_t$:
\begin{align*}
G_2 &= (I-LC)^\top K^\top R K L + (I-LC)^\top (A-BK)^\top G_1 (A-BK) L\\ &+ 2(I-LC)^\top (A-BK)^\top G_2 C (A-BK)L \\
&+ (I-LC)^\top (A-BK)^\top C^\top \left(Q + L^\top P L - L^\top C^\top Q C L \right) C (A-BK)L
\end{align*}
$G_1 = (I-LC)^\top \left( P - C^\top Q C \right) (I-LC) $ and $ G_2 = (I-LC)^\top \left( P - C^\top Q C \right) L$ satisfies all 3 equations. Thus one can write Bellman optimality equation as
\small
\begin{align*}
&J_*(\Theta)\!+\!\hat{x}_{t|t-1}^\top (I\!-\!LC)^\top\! \left( P\!-\! C^\top Q C \right)(I\!-\!LC) \hat{x}_{t|t-1}\!\\
&+\!2 \hat{x}_{t|t-1}^\top (I\!-\!LC)^\top \left( P\!-\! C^\top Q C \right) L y_t\! +\! y_t^\top\!\left( Q \!+\!L^\top\!\left( P\!-\! C^\top Q C \right)\!L \right)\!y_t\!= \\
& y_t^\top Q y_t + \hat{x}_{t|t}^\top K^\top R K \hat{x}_{t|t} + \mathbb{E}\left[ \hat{x}_{t+1|t}^\top (I-LC)^\top \left( P - C^\top Q C \right) (I-LC) \hat{x}_{t+1|t} \right] \\ &+2 \mathbb{E}\left[ \hat{x}_{t+1|t}^\top (I-LC)^\top \left( P - C^\top Q C \right) L y_{t+1} + y_{t+1}^\top \left( Q + L^\top \left( P - C^\top Q C \right) L \right) y_{t+1} \right]
\end{align*}
\normalsize
Combining terms using \eqref{next time step decomposition} gives the \begin{align*}
J_*(\Theta)\!+\!\hat{x}_{t|t}^\top\!\left( P \!-\!C^\top Q C\right)\!\hat{x}_{t|t}\!+\!y_t^\top Q y_t\!&=\! y_t^\top Q y_t + u_t^\top \!R u_t\! \\ &+\!\mathbb{E}\bigg[ \hat{x}_{t+1|t+1}^{\top}\!\left( P\!-\! C^\top Q C \right)\!\hat{x}_{t+1|t+1}^{u}\!\!+\!y_{t+1}^{\top} Q y_{t+1}\!\bigg]
\end{align*}
\end{proof}
\section{Regret Decomposition} \label{SuppRegret}
In this section, using the Bellman Optimality Equation for the optimistic system $\tilde{\Theta} = \left(\tilde{A}, \tilde{B}, \tilde{C} \right)$, we derive the regret decomposition of applying optimal policy $\tilde{\Pi}_*\left(\tilde{P}, \tilde{K}, \tilde{L} \right)$ for $\tilde{\Theta}$ in the unknown system $\Theta = \left(A, B, C \right)$. Notice that, this is equivalent to provide the regret decomposition for a system that is obtained via similarity transformation $\mathbf{S}$, \textit{i.e.} $A' = \mathbf{S}^{-1} A \mathbf{S}$, $B' = \mathbf{S}^{-1} B$, $C' = C \mathbf{S}$. Therefore, without loss of generality we will assume that $\mathbf{S} = I$ in the regret decomposition and the concentration bounds that is going to be used in the regret analysis.
First, for given $\hat{x}_{t|t-1}$ and $y_t$, define the following expressions for time step $t+1$ using the model specified as subscript,
\begin{align}
\hat{x}_{t|t,\tilde{\Theta}} &= \left( I - \tilde{L}\tilde{C} \right)\hat{x}_{t|t-1} + \tilde{L} y_t \label{firstdef} \\
y_{t+1,\tilde{\Theta}} &= \tilde{C} \left( \tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{C} \tilde{A} \left( x_t - \hat{x}_{t|t,\tilde{\Theta}} \right) + \tilde{C} w_t + z_{t+1} \label{seconddef}\\
\hat{x}_{t+1|t+1,\tilde{\Theta}} &= \left( \tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{L} \tilde{C} \tilde{A} \left( x_t - \hat{x}_{t|t,\tilde{\Theta}} \right) + \tilde{L} \tilde{C} w_t + \tilde{L} z_{t+1} \label{thirddef}\\
y_{t+1, \Theta} &= CA \hat{x}_{t|t,\tilde{\Theta}} - CB\tilde{K} \hat{x}_{t|t,\tilde{\Theta}} + C w_t + CA(x_t - \hat{x}_{t|t,\tilde{\Theta}}) + z_{t+1} \label{fourthdef} \\
\hat{x}_{t+1|t+1,\Theta} &= (I -LC)(A\hat{x}_{t|t,\Theta} - B\tilde{K}\hat{x}_{t|t,\tilde{\Theta}}) + L y_{t+1, \Theta} \\
&= (I-LC)(A-B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} + (I-LC)A(\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}}) + L y_{t+1, \Theta} \\
&= (I-LC)(A-B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} + LC(A-B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} + LC w_t + LCA(x_t - \hat{x}_{t|t,\tilde{\Theta}}) \nonumber \\
\qquad \qquad & \qquad + (I-LC)A(\hat{x}_{t|t,\Theta} - \hat{x}_{t|t,\tilde{\Theta}}) + Lz_{t+1} \\
&= (A\!-\!B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} \!+\! LCw_t \!+\! LCA(x_t \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \!+\! (I\!-\!LC)A(\hat{x}_{t|t,\Theta} \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \!+\! Lz_{t+1} \label{lastdef}
\end{align}
Notice that given $x_{t|t-1}$ and $y_t$ first and fourth terms in \eqref{lastdef} are deterministic.
From Lemma~\ref{LQGBellman}, we get
\begin{align}
&J_*(\tilde{\Theta}) + \hat{x}_{t|t,\tilde{\Theta}}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t|t,\tilde{\Theta}} + y_t^\top Q y_t \nonumber \\
&= y_t^\top Q y_t + u_t^\top R u_t + \mathbb{E}\left[ \hat{x}_{t+1|t+1,\tilde{\Theta}}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t+1|t+1,\tilde{\Theta}} + y_{t+1,\tilde{\Theta}}^\top Q y_{t+1,\tilde{\Theta}} \Big | \hat{x}_{t|t-1}, y_t \right] \nonumber \\
&= \mathbb{E}\bigg[ \left( (\tilde{A} - \tilde{B} \tilde{K} ) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{L}\tilde{C}\tilde{A}(x_t - \hat{x}_{t|t,\tilde{\Theta}}) + \tilde{L}\tilde{C} w_t + \tilde{L} z_{t+1} \right)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \nonumber \\
& \qquad\qquad\qquad\qquad \times \left((\tilde{A} - \tilde{B} \tilde{K} ) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{L}\tilde{C}\tilde{A}(x_t - \hat{x}_{t|t,\tilde{\Theta}}) + \tilde{L}\tilde{C} w_t + \tilde{L} z_{t+1} \right) \Big | \hat{x}_{t|t-1}, y_t \bigg]\nonumber
\\& \qquad + y_t^\top Q y_t + u_t^\top R u_t + \mathbb{E}\bigg[ \left( \tilde{C} (\tilde{A} - \tilde{B} \tilde{K} ) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{C} \tilde{A} (x_t - \hat{x}_{t|t,\tilde{\Theta}}) + \tilde{C} w_t + z_{t+1} \right)^\top Q \nonumber \\
&\qquad\qquad\qquad\qquad \times \left( \tilde{C} (\tilde{A} - \tilde{B} \tilde{K} ) \hat{x}_{t|t,\tilde{\Theta}} + \tilde{C} \tilde{A} (x_t - \hat{x}_{t|t,\tilde{\Theta}}) + \tilde{C} w_t + z_{t+1} \right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \nonumber \\
&= \hat{x}_{t|t,\tilde{\Theta}}^\top \left(\tilde{A} - \tilde{B} \tilde{K} \right)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \left(\tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}} + \mathbb{E} \bigg[ w_t \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} w_t \bigg] \nonumber \\
&\qquad +\mathbb{E}\bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L}\tilde{C}\tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \nonumber \\
&\qquad + \mathbb{E}\bigg[ z_{t+1}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} z_{t+1} \bigg] + y_t^\top Q y_t + u_t^\top R u_t \nonumber \\
&\qquad + \hat{x}_{t|t,\tilde{\Theta}}^\top \left(\tilde{A} - \tilde{B} \tilde{K} \right)^\top \tilde{C}^\top Q \tilde{C} \left(\tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}} + \mathbb{E} \bigg[ w_t \tilde{C}^\top Q \tilde{C} w_t \bigg] \nonumber \\
&\qquad + \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] + \mathbb{E} \bigg[ z_{t+1}^\top Q z_{t+1} \bigg] \label{insert}
\end{align}
Using equations \eqref{thirddef} and \eqref{lastdef}, we have the following expression for $\mathbb{E}\left[ z_{t+1}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} z_{t+1} \right]$:
\begin{align*}
&\mathbb{E}\left[z_{t+1}^\top (\tilde{L} - L + L)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L + L) z_{t+1} \right] \\
&= \mathbb{E}\left[z_{t+1}^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L z_{t+1} \right] + 2 \mathbb{E}\left[z_{t+1}^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right] \\ &\qquad +\mathbb{E}\left[z_{t+1}^\top (\tilde{L} - L)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right] \\
&= \mathbb{E}\left[ \hat{x}_{t+1|t+1,\Theta}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t+1|t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] - \mathbb{E}\left[w_t^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C w_t \right] \\ &-\!\left((A\!-\!B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} \!+\! (I\!-\!LC)A(\hat{x}_{t|t,\Theta} \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \right)^\top\!\left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right)\! \left((A\!-\!B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} \!+\! (I\!-\!LC)A(\hat{x}_{t|t,\Theta} \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \right) \\ &- \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right] \\
&+ 2 \mathbb{E}\left[z_{t+1}^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right] +\mathbb{E}\left[z_{t+1}^\top (\tilde{L} - L)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right].
\end{align*}
Similarly, using equations \eqref{seconddef} and \eqref{fourthdef}, we have the following expression for $\mathbb{E}\left[ z_{t+1}^\top Q z_{t+1} \right]$:
\begin{align*}
\mathbb{E}\left[z_{t+1}^\top Q z_{t+1} \right] &= \mathbb{E}\left[ y_{t+1,\Theta}^\top Q y_{t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] - \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top C^\top Q C (A-B\tilde{K}) \hat{x}_{t|t,\tilde{\Theta}} \\ &- \mathbb{E}\left[w_t^\top C^\top Q C w_t \right] - \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top Q C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right]
\end{align*}
\normalsize
Inserting these to the equality in \eqref{insert}, we get
\begin{align*}
&J_*(\tilde{\Theta}) + \hat{x}_{t|t,\tilde{\Theta}}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t|t,\tilde{\Theta}} + y_t^\top Q y_t \\
&= y_t^\top Q y_t + u_t^\top R u_t + \hat{x}_{t|t,\tilde{\Theta}}^\top \left(\tilde{A} - \tilde{B} \tilde{K} \right)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \left(\tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}}
\\ &+ \mathbb{E} \bigg[ w_t \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} w_t \bigg] \\
&+ \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L}\tilde{C}\tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \\
&+ \mathbb{E}\left[ \hat{x}_{t+1|t+1,\Theta}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t+1|t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] - \mathbb{E}\left[w_t^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C w_t \right] \\
&- \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right] \\
&-\!\left((A\!-\!B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} \!+\! (I\!-\!LC)A(\hat{x}_{t|t,\Theta} \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \right)^\top\!\left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right)\! \left((A\!-\!B\tilde{K})\hat{x}_{t|t,\tilde{\Theta}} \!+\! (I\!-\!LC)A(\hat{x}_{t|t,\Theta} \!-\! \hat{x}_{t|t,\tilde{\Theta}}) \right) \\
&+ \hat{x}_{t|t,\tilde{\Theta}}^\top \left(\tilde{A} - \tilde{B} \tilde{K} \right)^\top \tilde{C}^\top Q \tilde{C} \left(\tilde{A} - \tilde{B} \tilde{K} \right) \hat{x}_{t|t,\tilde{\Theta}} + \mathbb{E} \bigg[ w_t \tilde{C}^\top Q \tilde{C} w_t \bigg]\\
&+ \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \\
&+ \mathbb{E}\left[ y_{t+1,\Theta}^\top Q y_{t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] - \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top C^\top Q C (A-B\tilde{K}) \hat{x}_{t|t,\tilde{\Theta}} \\
& -\mathbb{E}\left[w_t^\top C^\top Q C w_t \right] - \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top Q C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right] \\
&+ 2 \mathbb{E}\left[z_{t+1}^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right] +\mathbb{E}\left[z_{t+1}^\top (\tilde{L} - L)^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} - L) z_{t+1} \right]
\end{align*}
Hence,
\begin{equation*}
\sum_{t=0}^{T-T_{exp}} J_*(\tilde{\Theta}) \!+\! R_1 \!+\! R_2 = \sum_{t=0}^{T-T_{exp}} \left( y_t^\top Q y_t \!+\! u_t^\top R u_t \right) \!+\! R_3 \!+\! R_4 \!+\! R_5 \!+\! R_6 \!+\! R_7 \!+\! R_8 \!+\! R_9 \!+\! R_{10} \!+\! R_{11}
\end{equation*}
where
\begin{align}
R_1& \!=\!\! \sum_{t=0}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t|t,\tilde{\Theta}} - \mathbb{E}\left[ \hat{x}_{t+1|t+1,\Theta}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t+1|t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] \right \} \\
R_2& \!=\!\! \sum_{t=0}^{T-T_{exp}} \left \{ y_t^\top Q y_t -
\mathbb{E}\left[ y_{t+1,\Theta}^\top Q y_{t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] \right \} \\
R_3& \!=\!\! \sum_{t=0}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top (\tilde{A} - \tilde{B} \tilde{K} )^\top \tilde{C}^\top Q \tilde{C} (\tilde{A} - \tilde{B} \tilde{K}) \hat{x}_{t|t,\tilde{\Theta}} - \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top C^\top Q C (A-B\tilde{K}) \hat{x}_{t|t,\tilde{\Theta}} \right \} \\
R_4& \!=\!\! \sum_{t=0}^{T-T_{exp}} \! \left\{ \hat{x}_{t|t,\tilde{\Theta}}^\top (\tilde{A} \!-\! \tilde{B} \tilde{K} )^\top (\tilde{P} \!-\! \tilde{C}^\top Q \tilde{C}) (\tilde{A} \!-\! \tilde{B} \tilde{K}) \hat{x}_{t|t,\tilde{\Theta}} \!-\! \hat{x}_{t|t,\tilde{\Theta}}^\top (A\!-\!B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (A\!-\!B\tilde{K}) \hat{x}_{t|t,\tilde{\Theta}}
\right \} \\
R_5& \!=\! -\!\! \sum_{t=0}^{T-T_{exp}} \left \{ 2\hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} ) \right \} \\
R_6& \!=\! -\!\! \sum_{t=0}^{T-T_{exp}} \left \{ (\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} )^\top A^\top (I-LC)^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} ) \right \} \\
R_7& \!=\!\! \sum_{t=0}^{T-T_{exp}} \left\{ \mathbb{E} \left[ w_t^\top \tilde{C}^\top Q \tilde{C} w_t \right] - \mathbb{E}\left[w_t^\top C^\top Q C w_t \right] \right\}\\
R_8& \!=\!\! \sum_{t=0}^{T-T_{exp}} \left \{ \mathbb{E} \left[ w_t^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} w_t \right] - \mathbb{E}\left[w_t^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C w_t \right] \right \} \\ \label{R9}
R_9& \!=\!\! \sum_{t=0}^{T-T_{exp}} \bigg \{ \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \\ &\qquad \qquad- \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top Q C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right] \bigg \} \nonumber \\ \label{R10}
R_{10}& \!=\!\! \sum_{t=0}^{T-T_{exp}} \bigg \{ \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \tilde{A}^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L}\tilde{C}\tilde{A} \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \\
&\qquad \qquad - \mathbb{E}\left[\left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right)^\top A^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C A \left(x_{t}-\hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \right] \bigg \} \\
R_{11}& \!=\!\! \sum_{t=0}^{T-T_{exp}} \bigg \{ 2 \mathbb{E}\left[z_{t+1}^\top L^\top\!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L) z_{t+1} \right] \!+\!\mathbb{E}\left[z_{t+1}^\top (\tilde{L} \!-\! L)^\top \!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L) z_{t+1} \right] \bigg \}
\end{align}
Thus, on event $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $,
\begin{align*}
&\sum_{t=0}^{T-T_{exp}}\!\!\!\! \left( y_t^\top Q y_t \!+\! u_t^\top R u_t \right) \!=\!\! \sum_{t=0}^{T-T_{exp}} \!\!\! J_*(\tilde{\Theta}) \!+\! R_1 \!+\! R_2 \!-\! R_3 \!-\! R_4 \!-\! R_5 \!-\! R_6 \!-\! R_7 \!-\! R_8 \!-\! R_9 \!-\! R_{10} \!-\! R_{11} \\
& \qquad \leq (T-T_{exp}) J_*(\Theta) \!+\! R_1 \!+\! R_2 \!-\! R_3 \!-\! R_4 \!-\! R_5 \!-\! R_6 \!-\! R_7 \!-\! R_8 \!-\! R_9 \!-\! R_{10} \!-\! R_{11} \!+\! T^{2/3}
\end{align*}
where the last inequality follows from the fact that $\tilde{\Theta}$ is the optimistic parameter from the confidence sets such that $J_*(\tilde{\Theta}) \leq J_*(\Theta) + \frac{1}{T^{1/3}}$ and on event $\mathcal{E}$, $\Theta \in \mathcal{C}$. Therefore, on event $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $,
\begin{equation}
\textsc{\small{REGRET}}\xspace(T-T_{exp}) \leq R_1 + R_2 - R_3 - R_4 - R_5 - R_6 - R_7 - R_8 - R_9 - R_{10} \!-\! R_{11} + T^{2/3}.
\end{equation}
The following section contains the bounds on individual pieces.
\section{Regret Upper Bound}
\label{SuppRegretTotal}
In this section we will provide bounds on each term in the regret decomposition. It will be useful to recall and denote the following bounds obtained after $T_{exp}>T_0$ time steps of exploration before starting commit phase of \textsc{\small{ExpCommit}}\xspace on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $. Note that, without loss of generality, we obtain the regret analysis with the confidence sets with the similarity transformation $\mathbf{S} = I$, since any similarity transformation of the underlying system $\Theta$ will give a system with the same behavior.
\small
\begin{align*}
\|A - \tilde{A} \| &\leq \Delta A \coloneqq \left( \frac{62n\|\mathbf{H}\| + 14n\sigma_n(\mathbf{H})}{\sigma_n^2(\mathbf{H})} \right) \frac{R_w+R_e+R_z}{\sigma_u\sqrt{T_{exp}-H+1}}, \\
\|B - \tilde{B} \| &\leq \Delta B \coloneqq \frac{14n(R_w+R_e+R_z)}{\sigma_u\sqrt{(T_{exp} - H+1)\sigma_n(\mathbf{H})}}, \\
\|C - \tilde{C} \| &\leq \Delta C \coloneqq \frac{14n(R_w+R_e+R_z)}{\sigma_u\sqrt{(T_{exp} - H+1)\sigma_n(\mathbf{H})}},\\
\|\tilde{\Sigma} - \Sigma \| &\leq \Delta \Sigma \coloneqq \frac{\Phi(A)^2 (4\|C\|+ 2)\|\Sigma \|^2 + \sigma_z^2 (4\Phi(A) + 2) \| \Sigma\| }{\sigma_z^2(1 - \upsilon^2)} \max\left\{\Delta A, \Delta C \right\}, \\
\|\tilde{L} - L\| &\leq \Delta L \coloneqq \Delta C \bigg( \sigma_z^{-2} \| \Sigma\| +
4\sigma_z^{-4} \| \Sigma\|^2
+ 6\sigma_z^{-4} \|C\| \| \Sigma\|^2 + 2\sigma_z^{-4} \|C\|^2 \| \Sigma\|^2 \bigg) \\
&\qquad +\Delta \Sigma \bigg(\sigma_z^{-2}(\|C\|\!+\!2) \!+\! \sigma_z^{-4} \|C\|^3 (1\!+\!\| \Sigma\| ) \!+\! 10 \sigma_z^{-4} \|C\|^2 (1 \!+\! \| \Sigma \|) \!+\! 24 \sigma_z^{-4} \|C\| (1\!+\!\| \Sigma \|) \!+\! 16 \sigma_z^{-4} (1 \!+\! \| \Sigma \|) \bigg) \\ \|\hat{x}_{t|t,\tilde{\Theta}}\| &\leq \tilde{\mathcal{X}} \coloneqq \frac{2\zeta \left(\|C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2n\log(2nT/\delta)}\right) }{1-\rho} , \\
\|y_t \| &\leq \mathcal{Y} \coloneqq \| C\| \tilde{\mathcal{X}} + \| C\| \bar{\Delta} + \left(\| C\| \|\Sigma\|^{1/2} + \sigma_z \right) \sqrt{2m\log(2mT/\delta)}
\end{align*}
\normalsize
for all $t \leq T$, where $\bar{\Delta} = 10\left(\frac{\bar{\kappa} }{1-\sigma} + \frac{\bar{\beta} \bar{\xi} }{(1-\sigma)^2} \right)\left(\| C\| \|\Sigma\|^{1/2} + \sigma_z\right) \sqrt{2m\log(2mT/\delta)}$, for $\bar{\kappa} = \Phi(A) \Delta L + 2\zeta (\beta_A + \Gamma \beta_B)$, $\bar{\beta} = 2\zeta \beta_C (\Phi(A) + 2(\beta_A + \Gamma \beta_B) + 2(\beta_A + \Gamma \beta_B) )$ and $\bar{\xi} = \zeta (\rho + 2(\beta_A + \Gamma \beta_B)) + \|B\|\Gamma \Delta L $. Notice that all the concentration results are $\tilde{\mathcal{O}}\left(\frac{1}{\sqrt{T_{exp}}}\right)$ where $\tilde{\mathcal{O}}(\cdot)$ hides the problem dependent constants and logarithm. All the theorems and lemmas in this section are given in $\tilde{\mathcal{O}}(\cdot)$ notation. The exact expressions are given in the last lines of proofs.
\subsection[Bound on R1]{Bounding $R_1$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}\label{lemmaR1}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. For any $\delta \in (0,1) $, given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
&R_1 \!=\! \tilde{\mathcal{O}}\left( \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}} \right)
\end{align*}
with probability at least $1-\delta$.
\end{lemma}
\begin{proof}
Define $\tilde{f}_t, f_t, \tilde{v}_t$ and $v_t$ such that
\begin{align*}
\tilde{f}_t &= \tilde{A}(I-\tilde{L}\tilde{C})\hat{x}_{t-1|t-2,\tilde{\Theta}} + \tilde{A}\tilde{L} y_{t-1} + \tilde{B} u_{t-1}, \\
f_t &= A(I-LC)\hat{x}_{t-1|t-2,\tilde{\Theta}} + AL y_{t-1} + B u_{t-1} \\
\tilde{v}_t &= \tilde{L}\tilde{C}\tilde{A}(x_{t-1} - (I-\tilde{L}\tilde{C})\hat{x}_{t-1|t-2,\tilde{\Theta}} - \tilde{L} y_{t-1}) + \tilde{L}\tilde{C} w_{t-1} + \tilde{L} z_{t}, \\
v_t &= LCA(x_{t-1} - (I-LC)\hat{x}_{t-1|t-2,\tilde{\Theta}} - L y_{t-1}) + LC w_{t-1} + L z_{t}
\end{align*}
$R_1$ is decomposed as follows,
\begin{align*}
R_1 &= \hat{x}_{0|0,\tilde{\Theta}}^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) \hat{x}_{0|0,\tilde{\Theta}} - \mathbb{E}\left[ \hat{x}_{T+1|T+1,\Theta}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{T+1|T+1,\Theta} \Big | \hat{x}_{T|T-1}, y_T, u_T \right] \\
&+\!\!\sum_{t=1}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t|t,\tilde{\Theta}} - \mathbb{E}\left[ \hat{x}_{t|t,\Theta}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \hat{x}_{t|t,\Theta} \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}
\end{align*}
Since $\hat{x}_{0|0,\tilde{\Theta}} =0$ and $\tilde{P} - \tilde{C}^\top Q \tilde{C}$ is positive semidefinite, the first line is bounded by zero. Using the definitions above, the remaining is decomposed as follows,
\begin{align*}
R_1 &\leq \sum_{t=1}^{T-T_{exp}} \left\{ \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{f}_t - f_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) f_t\right\} + 2 \sum_{t=1}^T \left\{ \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t \right\} \\
&\qquad \qquad + \sum_{t=1}^{T-T_{exp}} \left\{ \tilde{v}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t - \mathbb{E}\left[ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}
\end{align*}
Each term will be bounded separately: \\
\textbf{1) }Let $G_1 = 2 \sum_{t=1}^T \left\{ \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t \right\}$. Let $q_t^\top = \tilde{f}_t^\top(\tilde{P} - \tilde{C}^\top Q \tilde{C})$ and on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}$
\begin{equation*}
\sum_{t=1}^{T-T_{exp}} \left\{ \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t \right\} = \sum_{t=1}^{T-T_{exp}} q_t^\top \tilde{v}_t = \sum_{t=1}^{T-T_{exp}} \sum_{i=1}^n q_{t,i} \tilde{v}_{t,i} = \sum_{i=1}^n \sum_{t=1}^{T-T_{exp}} q_{t,i} \tilde{v}_{t,i}
\end{equation*}
Observe that conditioned on current observations, $\tilde{v}_{t}$ is zero mean and entrywise $R'$-sub-Gaussian, where
\[R' = \zeta \left( \left((\Delta A + \Phi(A))\sqrt{\Delta\Sigma + \| \Sigma \|} + \sigma_w \right) \left(\| C\| + \Delta C \right)+ \sigma_z\right)\].
Let $M_{T,i} = \sum_{t=1}^{T-T_{exp}} q_{t,i} \tilde{v}_{t,i}$. By Theorem~\ref{selfnormalized}, the following holds with probability at least $1-\delta/(4n)$, for any $T\geq 0$ and $\lambda>0$,
\begin{equation*}
M_{T,i}^2 \leq 2R'^2 \left(\lambda + \sum_{t=1}^{T-T_{exp}} q_{t,i}^2 \right) \log \left(\frac{4n}{\delta \sqrt{\lambda}} \left(\lambda + \sum_{t=1}^{T-T_{exp}} q_{t,i}^2 \right)^{1/2} \right)
\end{equation*}
Recalling Assumption~\ref{Stabilizable set}, on $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $,
\begin{align*}
\| q_t\| &\leq \rho (\| \tilde{P}\| + \|\tilde{C}^\top Q \tilde{C} \|) \tilde{\mathcal{X}} \leq \rho \left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \tilde{\mathcal{X}}.
\end{align*}
Thus, $q_{t,i} \leq \rho \left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \tilde{\mathcal{X}} \coloneqq \mathcal{Q}_T$. Combining these results with union bound, with probability $1-\delta/4$ we have
\begin{align}
G_1 &\leq 2R' n \sqrt{ (\lambda + \mathcal{Q}_T^2 (T-T_{exp})) \log \left( \frac{n\sqrt{\lambda + (T-T_{exp})\mathcal{Q}_T^2}}{\delta\sqrt{\lambda}}\right) } \nonumber \\
&= \tilde{\mathcal{O}}\left( \rho \zeta \tilde{\mathcal{X}} \left(D \!+\! \| Q\| \left( \| C\| \!+\! \Delta C \right)^2 \right) \left( \left((\Delta A \!+\! \Phi(A))\sqrt{\Delta\Sigma \!+\! \| \Sigma \|} \!+\! \sigma_w \right) \left(\| C\| \!+\! \Delta C \right)\!+\! \sigma_z\right) \sqrt{T \!-\!T_{exp} }\right) \nonumber
\end{align}
where $\tilde{\mathcal{O}}$ hides logarithmic terms.
\noindent\textbf{2) }Now consider $\sum_{t=1}^{T-T_{exp}} \left\{ \tilde{v}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t - \mathbb{E}\left[ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}$. Adding and subtracting $\sum_{t=1}^{T-T_{exp}} \left\{ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \right \}$, we get following terms,
\begin{align*}
&\sum_{t=1}^{T-T_{exp}} \left\{ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t - \mathbb{E}\left[ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \} \\ &\qquad \qquad \qquad + \sum_{t=1}^{T-T_{exp}} \left\{ \tilde{v}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t - v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \right \}
\end{align*}
We combine the last summation with $\sum_{t=1}^{T-T_{exp}} \left\{ \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{f}_t - f_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) f_t\right\}$ and denote as $G_2$. We decompose $G_2$ as follows
\begin{align*}
G_2 &= \sum_{t=1}^{T-T_{exp}} \left\{ \tilde{v}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t + \tilde{f}_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{f}_t - f_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) f_t - v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \right \} \\
&=\!\!\!\!\! \sum_{t=1}^{T-T_{exp}}\!\!\Big\{ (\tilde{v}_t\!-\!v_t)^\top \left( \tilde{P}\!-\!\tilde{C}^\top Q \tilde{C} \right) \tilde{v}_t\!+\!v_t^\top \!\left( \tilde{P}\!-\!\tilde{C}^\top Q \tilde{C} \right) (\tilde{v}_t\!-\!v_t)\!\\
&\qquad \qquad \qquad+\! (\tilde{f}_t\!-\!f_t)^\top \left( \tilde{P}\!-\!\tilde{C}^\top Q \tilde{C} \right) \tilde{f}_t\!+\!f_t^\top\!\left( \tilde{P}\!-\!\tilde{C}^\top Q \tilde{C} \right) (\tilde{f}_t\!-\!f_t) \Big\}
\end{align*}
Now notice that
\begin{align*}
\tilde{v}_t\!-\!v_t &= (\tilde{L}\tilde{C}\tilde{A} - LCA)x_t - (\tilde{L}\tilde{C}\tilde{A} - LCA)\hat{x}_{t-1|t-2,\tilde{\Theta}} + (\tilde{L}\tilde{C}\tilde{A}\tilde{L}\tilde{C} - LCALC)\hat{x}_{t-1|t-2,\tilde{\Theta}} \\
&\qquad \qquad \qquad \qquad \qquad \qquad- (\tilde{L}\tilde{C}\tilde{A}\tilde{L} - LCAL)y_{t-1} + (\tilde{L}\tilde{C} -LC)w_{t-1} + (\tilde{L} - L)z_t \\
\tilde{f}_t\!-\!f_t &= (\tilde{A} - A)\hat{x}_{t-1|t-2,\tilde{\Theta}} - (\tilde{A}\tilde{L}\tilde{C} - ALC)\hat{x}_{t-1|t-2,\tilde{\Theta}} + (\tilde{A}\tilde{L} - AL)y_{t-1} - (\tilde{B} -B)\tilde{K} \hat{x}_{t-1|t-1,\tilde{\Theta}}
\end{align*}
Recall that $\hat{x}_{i-1|i-2,\tilde{\Theta}} = (\tilde{A} - \tilde{B} \tilde{K})\hat{x}_{i-2|i-2,\tilde{\Theta}}$ and since $x_i - \hat{x}_{i-1|i-i,\tilde{\Theta}}$ is $\|\tilde{\Sigma}\|^{1/2}$-sub-Gaussian for all $i\leq T$, using Lemma~\ref{subgauss lemma}, we get $\|x_i - \hat{x}_{i-1|i-2,\tilde{\Theta}} \| \leq \sqrt{\|\Sigma\| + \|\Delta \Sigma\|}\sqrt{2n\log\left(\frac{8n(T-T_{exp})}{\delta}\right)} $ for all $i\leq T$ with probability at least $1-\delta/4$.
Similarly, $\|w_i\| \leq \sigma_w\sqrt{2n\log\left(\frac{8n(T-T_{exp})}{\delta}\right)} $, $\|z_i\| \leq \sigma_z \sqrt{2m\log\left(\frac{8m(T-T_{exp})}{\delta}\right)} $ for all $i\leq T$ with probability at least $1-\delta/4$ respectively. On the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ consider the following decompositions:
\begin{align*}
\|\tilde{L} \tilde{C} \tilde{A} - LCA \| &\leq \Big(\Delta L\|C\| \Phi(A) + \zeta \Phi(A) \Delta C + \zeta \|C\| \Delta A + \zeta \Delta C \Delta A \\
&\enskip + \Phi(A) \Delta L \Delta C + \Delta L \|C\| \Delta A + \Delta L \Delta C \Delta A \Big) \\
\|\tilde{L}\tilde{C}\tilde{A}\tilde{L}\tilde{C} - LCALC \| &\leq \Big( 2 \Delta L \Phi(A) \|C\|^2 \| L\| + 2 \Delta C \Phi(A) \|L\|^2 \| C\| + \Delta A \|L\|^2 \| C\|^2 \\
&\enskip+ 4 \Delta L \Delta C \Phi(A) \|L\| \|C\| + 2 \Delta L \Delta A \| L\| \|C\|^2 \\
&\enskip+ 2 \Delta A \Delta C \|L\|^2 \|C\| + \Delta L^2 \Phi(A) \|C\|^2 + \Delta C^2 \Phi(A) \|L\|^2 + 2 \Delta L^2 \Delta C \Phi(A) \|C\| \\
&\enskip+ 2 \Delta C^2 \Delta L \Phi(A) \|L\| + 4 \Delta L \Delta C \Delta A \|L\| \|C\| + \Delta L^2 \Delta A \|C\|^2 + \Delta C^2 \Delta A \|L\|^2 \\
&\enskip+ 2 \Delta L^2 \Delta C \Delta A \|C\| + 2 \Delta C^2 \Delta L \Delta A \|L\| + \Delta L^2 \Delta C^2 \Phi(A) + \Delta L^2 \Delta C^2 \Delta A \Big) \\
\|\tilde{L}\tilde{C}\tilde{A}\tilde{L} - LCAL \| &\leq \Big( 2 \Delta L \Phi(A) \|C\| \| L\| + \Delta C \Phi(A) \|L\|^2 + \Delta A \|L\|^2 \| C\| + 2 \Delta L \Delta C \Phi(A) \|L\| \\
&\enskip+ \Delta L \Delta A \| L\| \|C\| + \Delta A \Delta C \|L\|^2 + \Delta L^2 \Phi(A) \|C\| + \Delta L^2 \Delta C \Phi(A) \\
&\enskip+ 2 \Delta L \Delta C \Delta A \|L\| + \Delta L^2 \Delta A \|C\| + \Delta L^2 \Delta C \Delta A \Big).
\end{align*}
Using the boundedness of $\hat{x}_{t|t,\tilde{\Theta}}, y_t, x_t, w_t, z_t$ and the fact that estimation error in all system matrices is $\mathcal{O}(1/\sqrt{T_{exp}})$, on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ with probability $1-\delta$ we have :
\begin{align*}
G_2 &\leq (T \!-\! T_{exp}) \left(D \!+\! \| Q\| \left( \| C\| \!+\! \Delta C \right)^2 \right) \left( 2 \|v_t\| \|\tilde{v}_t \!-\! v_t \| + \|\tilde{v}_t \!-\! v_t \|^2 + 2 \|f_t\| \|\tilde{f}_t \!-\! f_t \| + \|\tilde{f}_t \!-\! f_t \|^2 \right) \\
&= \tilde{\mathcal{O}}\left(\frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
where the last line is obtained by observing that the dominating term has only one concentration result.
\noindent\textbf{3) }Now we focus on the remaining term in $R_1$,
\begin{equation*}
G_3 = \sum_{t=1}^{T-T_{exp}} \left\{ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t - \mathbb{E}\left[ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}
\end{equation*}
Combining previous bounds on the components of $v_t$, with probability $1-\delta/2$ we have
\begin{align}
\|v_t\| &\leq \Omega \coloneqq \zeta \| C\| \Phi(A) \|\Sigma\|^{1/2}\sqrt{2n\log\left(\frac{8n(T-T_{exp})}{\delta}\right)} \label{R1eq} \\
& \qquad\qquad \qquad + \zeta \|C\| \sigma_w \sqrt{2n\log\left(\frac{8n(T-T_{exp})}{\delta}\right)} + \zeta \sigma_z \sqrt{2m\log\left(\frac{8m(T-T_{exp})}{\delta}\right)} \nonumber
\end{align}
On the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $, let $V_t = v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t - \mathbb{E}\left[ v_t^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right]$ and define its truncated version $\bar{V}_t = V_t \mathbbm{1}_{V_t \leq 2\left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \Omega} $. Thus $G_3 = \sum_{t=1}^{T-T_{exp}} V_t$ and define $\bar{G}_3 =\sum_{t=1}^{T-T_{exp}} \bar{V}_t $. By Lemma~\ref{basicprob},
\begin{align}
&\Pr\left( G_3 > 2\left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \Omega \sqrt{2(T-T_{exp}) \log\frac{2}{\delta}} \right) \\
&\leq\!\Pr\!\left(\max_{1\leq t\leq T - T_{exp}}\!\!\!\!\!V_t\!\geq\! 2\left(D\!+\!\| Q\| \left( \| C\|\!+\!\Delta C \right)^2 \right) \Omega \right)\!\\
&\qquad \qquad +\!\Pr\left( \bar{G}_3\!>\!2\left(D\!+\!\| Q\| \left( \| C\|\!+ \Delta C \right)^2 \right) \Omega \sqrt{2(T\!-\!T_{exp}) \log\frac{2}{\delta}} \right).
\end{align}
From \eqref{R1eq} and Theorem~\ref{azuma}, each term on the right hand side is bounded by $\delta/2$. Thus, with probability $1-\delta$,
\begin{equation*}
G_3 \leq 2\left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \Omega \sqrt{2(T-T_{exp}) \log\frac{2}{\delta}} = \tilde{\mathcal{O}}\left(\sqrt{T-T_{exp}} \right).
\end{equation*}
Since $R_1 = G_1 + G_2 + G_3$, the dominating term is $G_2$ which proves the lemma.
\end{proof}
\subsection[Bound on R2]{Bounding $R_2$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. For any $\delta \in (0,1) $, given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\sum_{t=0}^{T-T_{exp}} \left \{ y_t^\top Q y_t - \mathbb{E}\left[ y_{t+1,\Theta}^\top Q y_{t+1,\Theta} \Big | \hat{x}_{t|t-1}, y_t, u_t \right] \right \} = \tilde{\mathcal{O}}\left( \sqrt{T-T_{exp}} \right)
\end{align*}
with probability at least $1-\delta$.
\end{lemma}
\begin{proof}
Define $f_t = CA(I-LC)\hat{x}_{t-1|t-2} + CAL y_{t-1} + CB u_{t-1}$, $v_t = CA(x_{t-1} - (I-LC)\hat{x}_{t-1|t-2} - L y_{t-1}) + C w_{t-1} + z_{t}$. $R_2$ can be written as follows,
\begin{align*}
R_2 &= y_0^\top Q y_0 - \mathbb{E}\left[ y_{T+1,\Theta}^\top Q y_{T+1,\Theta} \Big | \hat{x}_{T|T-1}, y_T, u_T \right] \!+\! \sum_{t=1}^{T-T_{exp}} \left \{y_t^\top Q y_t - \mathbb{E}\left[ y_t^\top Q y_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}
\end{align*}
Since $y_{0} = 0$ and $Q$ is positive semidefinite, the first line is bounded by zero. Using the definitions above, the remaining can be decomposed as follows,
\begin{align*}
R_2 &\leq 2 \sum_{t=1}^{T-T_{exp}} \left\{ f_t^\top Q v_t \right\} + \sum_{t=1}^{T-T_{exp}} \left\{ v_t^\top Q v_t - \mathbb{E}\left[ v_t^\top Q v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right] \right \}
\end{align*}
Notice that these terms are all defined under true model. Each term will be bounded separately. Let $G_1 = 2 \sum_{t=1}^{T-T_{exp}} \left\{ f_t^\top Q v_t \right\}$. Let $q_t^\top = f_t^\top Q$ and on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}$
\begin{equation*}
\sum_{t=1}^{T-T_{exp}} \left\{ f_t^\top Q v_t \right\} = \sum_{t=1}^{T-T_{exp}} q_t^\top v_t = \sum_{t=1}^{T-T_{exp}} \sum_{i=1}^n q_{t,i} v_{t,i} = \sum_{i=1}^n \sum_{t=1}^{T-T_{exp}} q_{t,i} v_{t,i}
\end{equation*}
Observe that conditioned on current observations, $v_{t}$ is zero mean and entrywise $R''$-sub-Gaussian, where
\[R'' = (\|C\| \Phi(A) \| \Sigma \|^{1/2} + \sigma_w \|C \| + \sigma_z).\]
Let $M_{T,i} = \sum_{t=1}^{T-T_{exp}} q_{t,i} v_{t,i}$. By Theorem~\ref{selfnormalized}, the following holds with probability at least $1-\delta/(2n)$, for any $T\geq 0$ and $\lambda>0$,
\begin{equation*}
M_{T,i}^2 \leq 2R'^2 \left(\lambda + \sum_{t=1}^{T-T_{exp}} q_{t,i}^2 \right) \log \left(\frac{2n}{\delta \sqrt{\lambda}} \left(\lambda + \sum_{t=1}^{T-T_{exp}} q_{t,i}^2 \right)^{1/2} \right)
\end{equation*}
Recalling Assumption~\ref{Stabilizable set}, on $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $,
\begin{align*}
\| q_t\| &\leq \|Q\|\|C\| \left( \Phi(A) \|\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} \| + \Phi(A) \| \hat{x}_{t|t,\tilde{\Theta}} \| + \Gamma \|B\| \|\hat{x}_{t|t,\tilde{\Theta}} \| \right) \\
&\leq \mathcal{Q}_T' \! \coloneqq \! \|Q\|\|C\|\left( \left(\!\| \tilde{L}\!-\!L\|\|C\|\!+\!\|L \| \|\tilde{C}\!-\! C\|\!+\!\|\tilde{L}\!-\!L\| \|\tilde{C}\!-\!C\|\!\right) \Phi(A) \rho \tilde{\mathcal{X}}\!+\!\|\tilde{L}\!-\!L \| \mathcal{Y} \!+\! (\Phi(A)\!+\! \Gamma \|B\|) \tilde{\mathcal{X}} \right)
\end{align*}
Thus, $q_{t,i} \leq \mathcal{Q}_T'$. Combining these results with union bound, with probability $1-\delta/2$ we have
\begin{align}
G_1 &\leq 2R'' n \sqrt{ (\lambda + \mathcal{Q}_T'^2 (T-T_{exp})) \log \left( \frac{n\sqrt{\lambda + (T-T_{exp})\mathcal{Q}_T'^2}}{\delta\sqrt{\lambda}}\right) } = \tilde{\mathcal{O}}\left(\sqrt{T -T_{exp} }\right) \nonumber
\end{align}
where $\tilde{\mathcal{O}}$ hides logarithmic terms.
From the fact that $v_t$ is $R''$-sub-Gaussian, with probability $1-\delta/4$ we have
\begin{align}
\|v_t\| &\leq \Omega' \coloneqq (\|C\| \Phi(A) \| \Sigma \|^{1/2} + \sigma_w \|C \| + \sigma_z) \sqrt{2n\log\left(\frac{8n(T-T_{exp})}{\delta}\right)}\label{R2eq}
\end{align}
On the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $, let $V_t = v_t^\top Q v_t - \mathbb{E}\left[ v_t^\top Q v_t \Big | \hat{x}_{t-1|t-2}, y_{t-1}, u_{t-1} \right]$ and define its truncated version $\bar{V}_t = V_t \mathbbm{1}_{V_t \leq 2\|Q\| \Omega'} $. Thus $G_2 = \sum_{t=1}^{T-T_{exp}} V_t$ and define $\bar{G}_2 =\sum_{t=1}^{T-T_{exp}} \bar{V}_t $. By Lemma~\ref{basicprob},
\begin{align}
\Pr\left( G_2 > 2\|Q\|\Omega' \sqrt{2(T-T_{exp}) \log\frac{4}{\delta}} \right) &\leq\!\Pr\!\left(\max_{1\leq t\leq T - T_{exp}}\!\!\!\!\!V_t\!\geq\! 2\|Q\|\Omega' \right)\!\\
&\qquad \qquad+\!\Pr\left( \bar{G}_3\!>\!2\|Q\|\Omega' \sqrt{2(T\!-\!T_{exp}) \log\frac{4}{\delta}} \right). \nonumber
\end{align}
From \eqref{R1eq} and Theorem~\ref{azuma}, each term on the right hand side is bounded by $\delta/4$. Thus, with probability $1-\delta/2$,
\begin{equation*}
G_2 \leq 2\|Q\| \Omega' \sqrt{2(T-T_{exp}) \log\frac{4}{\delta}} = \tilde{\mathcal{O}}\left(\sqrt{T-T_{exp}} \right).
\end{equation*}
Since $R_2 = G_1 + G_2 $ the statement of lemma is obtained.
\end{proof}
\subsection[Bound on R3]{Bounding $|R_3|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\left| R_3 \right| = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right).
\end{align*}
\end{lemma}
\begin{proof}
Let $\tilde{\Upsilon} = [\tilde{C} \tilde{A}, \enskip -\tilde{C} \tilde{B} \tilde{K}]$ and $\Upsilon = [CA, \enskip -CB\tilde{K}]$.
Then, on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}$,
\begin{align}
&\left\|\left(\tilde{\Upsilon} - \Upsilon \right) \hat{x}_{t|t,\tilde{\Theta}} \right\| \leq \|(\tilde{\Upsilon} - \Upsilon)\| \|\hat{x}_{t|t,\tilde{\Theta}} \| \nonumber \\
&\leq \left(\| \tilde{C} \tilde{A} -CA\| + \| \tilde{K}\| \|CB -\tilde{C}\tilde{B} \| \right) \|\hat{x}_{t|t,\tilde{\Theta}} \| \nonumber \\
&\leq \big( \Phi(A) \|\tilde{C}-C\| + \|\tilde{A} -A \|\|\tilde{C} - C \| + \|\tilde{A} -A \| \|C\| \\
&\qquad +\Gamma (\|\tilde{C}-C\| \|B \| + \|\tilde{B} -B \| \| \tilde{C} - C\| + \|\tilde{B} -B \| \|C\| ) \big) \tilde{\mathcal{X}} \nonumber \\
&\leq \bigg( \Phi(A) \Delta C + \Delta A \Delta C + \Delta A \|C\| + \Gamma \big(\Delta C \|B\| + \Delta B \Delta C + \Delta B \|C\| \big) \bigg) \tilde{\mathcal{X}} = \tilde{\mathcal{O}}\left(\frac{1}{\sqrt{T_{exp}}}\right) \label{lastdecomposeR3}
\end{align}
This fact helps us prove the statement of lemma as follows,
\begin{align}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}} |R_3| \nonumber \\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}} \sum_{t=0}^{T-T_{exp}} \left| \left\|Q^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|^2 - \left\|Q^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\|^2 \right| \label{R3_1}\\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}}\!\! \left( \sum_{t=0}^{T-T_{exp}}\!\!\!\!\!\left( \left\|Q^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\| \!-\! \left\|Q^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \right)^{1/2}\!\!\!\! \left(\! \sum_{t=0}^{T-T_{exp}}\!\!\!\!\! \left( \left\|Q^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|
\!+\! \left\|Q^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \!\!\right)^{1/2} \label{R3_2} \\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}} \left( \sum_{t=0}^{T-T_{exp}} \left\|Q^{1/2} \left(\tilde{\Upsilon} - \Upsilon \right) \hat{x}_{t|t,\tilde{\Theta}} \right\|^2 \right)^{1/2} \left( \sum_{t=0}^{T-T_{exp}} \left( \left\|Q^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|
+ \left\|Q^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \right)^{1/2} \label{R3_3} \\
&\leq (T-T_{exp}) \|Q\| \tilde{\mathcal{X}} \Bigg( \left\|\left(\tilde{\Upsilon} - \Upsilon\right) \hat{x}_{t|t,\tilde{\Theta}} \right\|\left(\|\tilde{\Upsilon} \| + \|\Upsilon \| \right) \Bigg) \\
&= \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right) \label{R3result}
\end{align}
where \eqref{R3_1} follows from triangle inequality, \eqref{R3_2} is due to Cauchy Schwarz and \eqref{R3_3} is again triangle inequality. Finally, in \eqref{R3result}, we use \eqref{lastdecomposeR3} and the boundedness of $G$ and $\tilde{G}$, which translates to boundedness of $\Upsilon$ and $\tilde{\Upsilon}$.
\end{proof}
\subsection[Bound on R4]{Bounding $|R_4|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\left |R_4 \right| = \tilde{\mathcal{O}} \left( \frac{T - T_{exp}}{\sqrt{T_{exp}}} \right).
\end{align*}
\end{lemma}
\begin{proof}
Similar to $R_3$, let $\tilde{\Upsilon} = [\tilde{A}, \enskip -\tilde{B} \tilde{K}]$ and $\Upsilon = [A, \enskip -B\tilde{K}]$.
Then, on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G}$,
\begin{align}
\left\|\left(\tilde{\Upsilon} - \Upsilon \right) \hat{x}_{t|t,\tilde{\Theta}} \right\| &\leq \|(\tilde{\Upsilon} - \Upsilon)\| \|\hat{x}_{t|t,\tilde{\Theta}} \| \nonumber \\
&\leq \left(\| \tilde{A} -A\| + \| \tilde{K}\| \|B -\tilde{B} \| \right) \|\hat{x}_{t|t,\tilde{\Theta}} \| \nonumber \\
&\leq \left( \|\tilde{A} -A \| + \Gamma \|\tilde{B} -B \| \right) \tilde{\mathcal{X}} \nonumber \\
&\leq \bigg( \Delta A + \Gamma \Delta B \bigg) \tilde{\mathcal{X}} = \tilde{\mathcal{O}}\left(\frac{1}{\sqrt{T_{exp}}}\right) \label{lastdecomposeR4}
\end{align}
This fact helps us prove the statement of lemma as follows,
\begin{align}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } |R_4| \nonumber \\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left| \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|^2 - \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\|^2 \right| \label{R4_1} \\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \left( \sum_{t=0}^{T-T_{exp}} \left( \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\| - \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \right)^{1/2} \label{R4_2} \\
&\qquad \qquad \times \left( \sum_{t=0}^{T-T_{exp}} \left( \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|
+ \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \right)^{1/2} \nonumber \\
&\leq \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \left( \sum_{t=0}^{T-T_{exp}} \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \left(\tilde{\Upsilon} - \Upsilon \right)\hat{x}_{t|t,\tilde{\Theta}} \right\|^2 \right)^{1/2} \label{R4_3}\\
&\qquad \qquad \times \left( \sum_{t=0}^{T-T_{exp}} \left( \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \tilde{\Upsilon} \hat{x}_{t|t,\tilde{\Theta}} \right\|
+ \left\|(\tilde{P} - \tilde{C}^\top Q \tilde{C})^{1/2} \Upsilon \hat{x}_{t|t,\tilde{\Theta}} \right\| \right)^2 \right)^{1/2} \nonumber \\
&\leq (T-T_{exp}) \|\tilde{P} - \tilde{C}^\top Q \tilde{C}\| \tilde{\mathcal{X}} \Bigg( \left\|\left(\tilde{\Upsilon} - \Upsilon\right) \hat{x}_{t|t,\tilde{\Theta}} \right\|\left(\|\tilde{\Upsilon} \| + \|\Upsilon \| \right) \Bigg) \\
&\leq (T-T_{exp}) \left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \tilde{\mathcal{X}} \Bigg( \left\|\left(\tilde{\Upsilon} - \Upsilon\right) \hat{x}_{t|t,\tilde{\Theta}} \right\|\left(\|\tilde{\Upsilon} \| + \|\Upsilon \| \right) \Bigg)\\
&= \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right) \label{R4result}
\end{align}
where \eqref{R4_1} follows from triangle inequality, \eqref{R4_2} is due to Cauchy Schwarz and \eqref{R4_3} is again triangle inequality. Finally, in \eqref{R4result}, we use \eqref{lastdecomposeR4} and the boundedness of $G$ and $\tilde{G}$, which translates to boundedness of $\Upsilon$ and $\tilde{\Upsilon}$.
\end{proof}
\subsection[Bound on R5]{Bounding $|R_5|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\Bigg | & \sum_{t=0}^{T-T_{exp}} \left \{ 2\hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} ) \right \} \Bigg | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right).
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_5| \\
&= 2\left| \sum_{t=0}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\tilde{L}\tilde{C}\hat{x}_{t|t-1,\tilde{\Theta}} -LC\hat{x}_{t|t-1,\tilde{\Theta}} + L y_t -\tilde{L} y_t ) \right \} \right| \\
&\leq 2\left| \sum_{t=0}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\tilde{L}\tilde{C}-LC)\hat{x}_{t|t-1,\tilde{\Theta}} \right\} \right |
\\&\qquad \qquad + 2\left| \sum_{t=0}^{T-T_{exp}} \left \{ \hat{x}_{t|t,\tilde{\Theta}}^\top (A-B\tilde{K})^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A (L-\tilde{L}) y_t \right \} \right|\\
&\leq\!2(T-T_{exp})\!\Bigg( \tilde{\mathcal{X}}\left(\Phi(A)+ \Gamma \|B\|\right) \|\tilde{P}\!-\!\tilde{C}^\top Q \tilde{C} \| \|A - LCA\| \times \\
&\qquad \qquad \qquad \qquad \qquad \left(\| \tilde{L}\!-\!L \| \left(\|y_t \|\!+\!(\|C \|\!+\!\|\tilde{C}\!-\!C\|) \tilde{\mathcal{X}} \right)\!+\! \zeta \| \tilde{C}\!-\!C\| \tilde{\mathcal{X}} \right) \Bigg) \\
&\leq 2(T-T_{exp}) \Big(\tilde{\mathcal{X}}\left(\Phi(A)+ \Gamma \|B\|\right)\left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right) \left( \Phi(A) (1 + \zeta \| C\|)\right) \times \\
&\qquad \qquad \qquad \qquad \qquad \left(\Delta L \left(\mathcal{Y} + (\|C\|+\Delta C) \tilde{\mathcal{X}} \right) + \Delta C \zeta \tilde{\mathcal{X}} \right) \Big)
\end{align*}
where we used the bounds listed in the beginning of the section, \textit{i.e.} concentration results and the boundedness property of $\hat{x}_{t|t,\tilde{\Theta}}$ and $y_t$. Using the fact that all the concentration results are $\tilde{\mathcal{O}}\left(\frac{1}{\sqrt{T_{exp}}} \right)$ we obtain the the statement of lemma.
\end{proof}
\subsection[Bound on R6]{Bounding $|R_6|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\left| R_6 \right| = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{T_{exp}}\right).
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_6| \\
&= \left| \sum_{t=0}^{T-T_{exp}} \left \{ \left(\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} \right)^\top A^\top (I-LC)^\top (\tilde{P} - \tilde{C}^\top Q \tilde{C}) (I-LC)A(\hat{x}_{t|t,\Theta} -\hat{x}_{t|t,\tilde{\Theta}} ) \right \} \right| \\
&\leq (T-T_{exp}) \left\|(\tilde{L}\tilde{C} -LC)\hat{x}_{t|t-1,\tilde{\Theta}} + (L-\tilde{L})y_t \right\|^2 \|\tilde{P} - \tilde{C}^\top Q \tilde{C} \| \left( \Phi(A) (1 + \zeta \| C\|)\right)^2 \\
&\leq (T-T_{exp})\!\!\left(\!\left(\!\| \tilde{L}\!-\!L\|\|C\|\!+\!\|L \| \|\tilde{C}\!-\! C\|\!+\!\|\tilde{L}\!-\!L\| \|\tilde{C}\!-\!C\|\!\right) \rho \tilde{\mathcal{X}}\!+\!\|\tilde{L}\!-\!L \| \mathcal{Y} \right)^2 \times \\
&\qquad \qquad \left(\!D\!+\!\| Q\| \left( \| C\|\!+\!\Delta C \right)^2 \right) \left( \Phi(A) (1 + \zeta \| C\|)\right)^2 \\
&\leq (T - T_{exp}) \left( \left(\Delta L \|C\| + \zeta \Delta C\!+\!\Delta L \Delta C \right) \rho \tilde{\mathcal{X}}\!+\!\Delta L \mathcal{Y} \right)^2 \left(D\!+\!\| Q\| \left( \| C\|\!+\!\Delta C \right)^2 \right) \left( \Phi(A) (1 + \zeta \| C\|)\right)^2 \\
&= \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{T_{exp}}\right)
\end{align*}
where we used the bounds listed in the beginning of the section, \textit{i.e.} concentration results and the boundedness property of $\hat{x}_{t|t,\tilde{\Theta}}$ and $y_t$. Using the fact that all the concentration results are $\tilde{\mathcal{O}}\left(\frac{1}{\sqrt{T_{exp}}} \right)$ and each concentration term is squared in the final expression, we get the statement of lemma.
\end{proof}
\subsection[Bound on R7]{Bounding $|R_7|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds
\begin{align*}
\Bigg | &\sum_{t=0}^{T-T_{exp}} \left\{ \mathbb{E} \left[ w_t \tilde{C}^\top Q \tilde{C} w_t \right] - \mathbb{E}\left[w_t^\top C^\top Q C w_t \right] \right\} \Bigg | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right).
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
\mathbbm{1}_{\mathcal{E} \cap \mathcal{F}}|R_7| &= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \mathbb{E} \bigg[w_t^\top \left( \tilde{C}^\top Q \tilde{C} - C^\top Q C \right) w_t \bigg] \right | \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \Tr \left( \left(\tilde{C}^\top Q \tilde{C} - C^\top Q C \right) \mathbb{E}\bigg[w_t w_t^\top \bigg]\right) \right | \\
&\leq n \sigma_w^2 \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left\| \tilde{C}^\top Q \tilde{C} - C^\top Q C \right\| \\
&\leq n (T-T_{exp}) \sigma_w^2 \|Q\| \left( \Delta C^2 + 2 \| C \| \Delta C \right) = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right).
\end{align*}
\end{proof}
\subsection[Bound on R8]{Bounding $|R_8|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\Bigg | \sum_{t=0}^{T-T_{exp}} \left \{ \mathbb{E} \left[ w_t^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} w_t \right] - \mathbb{E}\left[w_t^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C w_t \right] \right \} \Bigg | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_8| \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \mathbb{E} \bigg[w_t^\top \left( \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} - C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C \right) w_t \bigg] \right | \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \Tr \left( \left( \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} - C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C \right) \mathbb{E}\bigg[w_t w_t^\top \bigg]\right) \right | \\
&\leq n \sigma_w^2 \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left\| \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L} \tilde{C} - C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C \right\| \\
&\leq n (T-T_{exp}) \sigma_w^2 \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \|\tilde{P} - \tilde{C}^\top Q \tilde{C} \| \left( \|\tilde{L}\tilde{C} - LC \|^2 + 2 \|LC\| \|\tilde{L}\tilde{C} - LC \| \right) \\
&\leq n (T-T_{exp}) \sigma_w^2 \left(D + \| Q\| \left( \| C\| + \Delta C \right)^2 \right)\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \Bigg( \left(\!\| \tilde{L}\!-\!L\|\|C\|\!+\!\|L \| \|\tilde{C}\!-\! C\|\!+\!\|\tilde{L}\!-\!L\| \|\tilde{C}\!-\!C\|\!\right)^2 \\
&\qquad + 2 \|LC\| \left(\!\| \tilde{L}\!-\!L\|\|C\|\!+\!\|L \| \|\tilde{C}\!-\! C\|\!+\!\|\tilde{L}\!-\!L\| \|\tilde{C}\!-\!C\|\!\right) \Bigg) \\
&\leq\!n (T\!-\!T_{exp}) \sigma_w^2 \!\left(D\!+\!\| Q\| \left( \| C\|\! +\!\Delta C \right)^2\right)\!\! \Big(\Delta L \|C\|\!+\! \zeta \Delta C\!+\!\Delta L \Delta C \Big) \Big(2\zeta\|C\|\!+\!\Delta L \|C\|\!+\!\zeta \Delta C\!+\!\Delta L \Delta C \Big) \\
&= \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
Since the dominating term has only one concentration result.
\end{proof}
\subsection[Bounds on R9 R10]{Bounding $|R_9|$ and $|R_{10}|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma} \label{lemmaR10}
Let $R_9$ and $R_{10}$ be as defined in equations \eqref{R9} and \eqref{R10}, respectively. Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\left | R_9 \right | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right), \qquad \left | R_{10} \right | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
\end{lemma}
\begin{proof}
Observe that $x_{t+1} - \hat{x}_{t+1|t, \tilde{\Theta}}$ has the following dynamics,
\[
x_{t+1} - \hat{x}_{t+1|t, \tilde{\Theta}} = \tilde{A}(I-\tilde{L}\tilde{C})(x_{t} - \hat{x}_{t|t-1, \tilde{\Theta}}) + w_t - \tilde{L} z_t
\]
Thus, the estimation error propagates according to a linear system, with closed-loop dynamics $\tilde{A}-\tilde{A}\tilde{L}\tilde{C}$, driven by the process $w_t - LC z_t$, which is iid zero mean and covariance $W + LZL^\top$. Additionally, from Assumption~\ref{Stabilizable set}, $\tilde{A} - \tilde{A}\tilde{L}\tilde{C}$ is stable. Recall that $\tilde{\Sigma} $ is the steady state error covariance matrix of state estimation of the system with optimistic parameters $\tilde{\Theta}$. Thus, we have that $\left\|\tilde{\Sigma} \right\| \geq \left\|\mathbb{E}\bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \Big | \hat{x}_{t|t-1}, y_t \bigg] \right\| $ for all $t\geq 0$. From this we get
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_9| \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \mathbb{E} \bigg[ \left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \left( \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} - A^\top C^\top Q C A \right)\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right) \Big | \hat{x}_{t|t-1}, y_t \bigg] \right | \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left | \Tr \left( \left( \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} - A^\top C^\top Q C A \right) \mathbb{E}\bigg[\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \Big | \hat{x}_{t|t-1}, y_t \bigg] \right) \right | \\
&\leq n \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left\| \tilde{A}^\top \tilde{C}^\top Q \tilde{C} \tilde{A} - A^\top C^\top Q C A \right\| \left\|\tilde{\Sigma} \right\| \\
&\leq n (T-T_{exp})\|Q\| \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \left( \|\tilde{C}\tilde{A} - CA \|^2 + 2 \| C \| \| \tilde{C}\tilde{A} -CA \|\right)(\|\Sigma\| + \Delta \Sigma) \\
&\leq n (T-T_{exp}) \|Q\|(\|\Sigma\| + \Delta \Sigma) \bigg(\!\!\Big(\Phi(A)\Delta C\!+\! \Delta A \Delta C\!+\! \|C\|\Delta A \Big) \Big(2\|C\|\!+\!\Phi(A)\Delta C\!+\! \Delta A \Delta C\!+\! \|C\|\Delta A \Big)\!\! \bigg) \\
&= \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
Since the dominating term has only one concentration result. Similar result for $R_{10}$ is obtained as follows
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_{10}|\\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \Bigg | \Tr \bigg( \left( \tilde{A}^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L}\tilde{C}\tilde{A} - A^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C A \right) \times \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathbb{E}\bigg[\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)\left(x_t - \hat{x}_{t|t,\tilde{\Theta}}\right)^\top \Big | \hat{x}_{t|t-1}, y_t \bigg] \bigg) \Bigg | \\
&\leq n \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left\| \tilde{A}^\top \tilde{C}^\top \tilde{L}^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) \tilde{L}\tilde{C}\tilde{A} - A^\top C^\top L^\top \left( \tilde{P} - \tilde{C}^\top Q \tilde{C} \right) L C A \right\| \left\|\tilde{\Sigma} \right\| \\
&\leq n (T-T_{exp}) \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \|\tilde{P} - \tilde{C}^\top Q \tilde{C} \| \left( \|\tilde{L}\tilde{C}\tilde{A} - LCA \|^2 + 2 \|LCA\| \|\tilde{L}\tilde{C}\tilde{A} - LCA \| \right)(\|\Sigma\| + \Delta \Sigma) \\
&\leq n (T-T_{exp}) \left(D\!+\!\| Q\| \left( \| C\|\! +\!\Delta C \right)^2\right)(\|\Sigma\| + \Delta \Sigma) \left(\Delta K' (2\zeta\|C\| + \Delta K' ) \right)
\end{align*}
where $\Delta K' \coloneqq \Big(\Delta L\Phi(A)\|C\| + \zeta \Phi(A) \Delta C + \zeta \|C\| \Delta A + \zeta \Delta C \Delta A + \Phi(A) \Delta L \Delta C + \Delta L \|C\| \Delta A + \Delta L \Delta C \Delta A \Big) $. Similar to $R_9$, the dominating term has only one concentration result. Thus, $\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }|R_{10}| = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)$
\end{proof}
\subsection[Bound on R11]{Bounding $|R_{11}|$ on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $}
\begin{lemma}\label{lemmaR11}
Suppose Assumption~\ref{Stabilizable set} holds and system is explored for $T_{exp} > T_0$ time steps. Given $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds,
\begin{align*}
\left | R_{11} \right | = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right).
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
&\mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} }\Bigg | \sum_{t=0}^{T-T_{exp}} 2 \mathbb{E}\left[z_{t+1}^\top L^\top\!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L) z_{t+1} \right]\Bigg | \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} 2 \left | \mathbb{E} \left[z_{t+1}^\top L^\top\!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L) z_{t+1} \right] \right | \\
&= \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} 2 \left | \Tr \left( \left(L^\top\!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L)\right) \mathbb{E}\bigg[z_{t+1} z_{t+1}^\top \bigg]\right) \right | \\
&\leq 2m \sigma_z^2 \mathbbm{1}_{\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} } \sum_{t=0}^{T-T_{exp}} \left\| L^\top\!\! \left( \tilde{P} \!-\! \tilde{C}^\top Q \tilde{C} \right) (\tilde{L} \!-\! L) \right\| \\
&\leq 2m \sigma_z^2 (T\!-\!T_{exp}) \!\left(D\!+\!\| Q\| \left( \| C\|\! +\!\Delta C \right)^2\right) \zeta \Delta L = \tilde{\mathcal{O}}\left( \frac{T-T_{exp}}{\sqrt{T_{exp}}}\right)
\end{align*}
Since the second term in $R_{11}$ has two concentration terms, \textit{i.e.} $\|\tilde{L} - L\|$, the one given above is the dominating term which proves the lemma.
\end{proof}
Combining these individual bounds with the result from Section~\ref{SuppRegretExplore}, we obtain the total regret upper bound of \textsc{\small{ExpCommit}}\xspace:
\begin{theorem}[Regret Upper Bound of \textsc{\small{ExpCommit}}\xspace]\label{Sup:totalregret}
Given a \textsc{\small{LQG}}\xspace $\Theta = (A,B,C)$, and regulating parameters $Q$ and $R$, suppose Assumptions~\ref{Stable}-\ref{Stabilizable set} hold. For any exploration duration $T_{exp}> T_0$,
if \textsc{\small{ExpCommit}}\xspace interacts with the system $\Theta$ for $T$ steps in total such that $T>T_{exp}$, with probability at least $1- 10\delta$, the regret of \textsc{\small{ExpCommit}}\xspace is bounded as follows,
\begin{equation}
\textsc{\small{REGRET}}\xspace(T) = \tilde{\mathcal{O}}\left( T_{exp} + \frac{T-T_{exp}}{\sqrt{T_{exp}}} + T^{2/3} \right).
\end{equation}
\end{theorem}
\begin{proof}
Combining Lemmas~\ref{lemmaR1}-\ref{lemmaR11}, on the event of $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $, with probability at least $1-\delta$, we have
\begin{align*}
R_1\!&=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), R_2 = \tilde{\mathcal{O}}\left( \sqrt{T\!-\!T_{exp}} \right), |R_3|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), |R_4|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), \\
|R_5| &= \tilde{\mathcal{O}}\left( \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}} \right), |R_6|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), |R_7|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right),
|R_8|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), \\
&\qquad \qquad \qquad |R_9|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right),
|R_{10}|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right), |R_{11}|\!=\!\tilde{\mathcal{O}}\left(\! \frac{T\!-\!T_{exp}}{\sqrt{T_{exp}}}\!\right).
\end{align*}
Thus, on the event $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $, with probability at least $1-\delta$,
\begin{align*}
\textsc{\small{REGRET}}\xspace(T-T_{exp})\! &\leq \!R_1 \!+\! R_2 \!+\! |R_3| \!+\! |R_4| \!+\! |R_5| \!+\! |R_6| + |R_7| \!+\! |R_8| \!+\! |R_9| \!+\! |R_{10}| \!+\! |R_{11}| \!+\! T^{2/3} \\
&= \tilde{\mathcal{O}}\left( T_{exp} + \frac{T-T_{exp}}{\sqrt{T_{exp}}} + T^{2/3} \right) .
\end{align*}
Recall that the event $\mathcal{E} \cap \mathcal{F} \cap \mathcal{G} $ holds with probability at least $1-9\delta$. Combining these results, we prove the theorem.
\end{proof}
\begin{corollary}
If the total interaction time $T$ is long enough, \textit{i.e.} having exploration duration of $T_{exp} = T^{2/3}$ satisfies that $T_{exp} > T_0$, then with high probability \textsc{\small{ExpCommit}}\xspace obtains regret upper bounded by $\tilde{\mathcal{O}}\left(T^{2/3}\right)$.
\end{corollary}
\begin{proof}
$T\geq T_0^{3/2}$ time steps with $T_{exp} = T^{2/3}$ is
Interacting with the system at least for $T\geq T_0^{3/2}$ allows us to set $T_{exp} = T^{2/3}$. Inserting this $T_{exp}$ to Theorem~\ref{Sup:totalregret}, we obtain $\textsc{\small{REGRET}}\xspace(T-T_{exp}) = \tilde{\mathcal{O}}\left( T^{2/3} + \frac{T-T^{2/3}}{\sqrt{T^{2/3}}} \right) + T^{2/3} = \tilde{\mathcal{O}}\left( T^{2/3} + T^{2/3}-T^{1/3} \right) + T^{2/3} = \tilde{\mathcal{O}}\left(T^{2/3}\right) $. Note that $\tilde{\mathcal{O}}(\sqrt{T_{exp}})$ term of regret obtained in explore phase doesn't effect the total regret result with the given choice of $T_{exp}$.
\end{proof}
\paragraph{Relation to~\citet{mania2019certainty}}
Note that the result in Theorem~\ref{Sup:totalregret} is the first sublinear end-to-end regret result for $\textsc{\small{LQG}}\xspace$ control in the literature. The closest related result to this is the sensitivity analysis of Riccati equation in~\citet{mania2019certainty}. In this work, the authors show that if the estimated system parameters are \textit{very close} to the underlying system parameters, the control designed by using the estimated model, results in a suboptimality gap which is quadratic with respect to the estimation error. It is worth noting that this result is for expected cost difference between optimal controller and designed controller, rather than the actual acquired cost.
\section{Technical Lemmas and Theorems}
\label{Technical}
\begin{theorem}[Partial Random Circulant Matrices~\citep{krahmer2014suprema}]\label{circ thm}
Let $\mathbf{C}\in\mathbb{R}^{d\times d}$ be a circulant matrix where the first row is distributed as $\mathcal{N}(0,I)$. Given $s\geq 1$, set $m_0=c_0s\log^2(2s)\log^2(2d)$ for some absolute constant $c_0>0$. Pick an $m\times s$ submatrix $\mathbf{S}$ of $\mathbf{C}$. With probability at least $1-(2d)^{-\log(2d)\log^2(2s)}$, $\mathbf{S}$ satisfies
\[
\|\frac{1}{m}\mathbf{S}^\top \mathbf{S}-I\|\leq \max\{\sqrt{\frac{m_0}{m}},\frac{m_0}{m}\}.
\]
\end{theorem}
\begin{theorem}[Azuma's inequality] \label{azuma}
Assume that $(X_s; s\geq 0)$ is a supermartingale and $|X_s - X_{s-1}| \leq c_s$ almost surely. Then for all $t>0$ and all $\epsilon>0$,
\begin{equation*}
\Pr(|X_t - X_0|\geq \epsilon) \leq 2 \exp\left(\frac{-\epsilon^2}{2\sum_{s=1}^t c_s^2}\right)
\end{equation*}
\end{theorem}
\begin{theorem}[Self-normalized bound for vector-valued martingales~\citep{abbasi2011improved}]
\label{selfnormalized}
Let $\left(\mathcal{F}_{t} ; k \geq\right.$
$0)$ be a filtration, $\left(m_{k} ; k \geq 0\right)$ be an $\mathbb{R}^{d}$-valued stochastic process adapted to $\left(\mathcal{F}_{k}\right),\left(\eta_{k} ; k \geq 1\right)$
be a real-valued martingale difference process adapted to $\left(\mathcal{F}_{k}\right) .$ Assume that $\eta_{k}$ is conditionally sub-Gaussian with constant $R$. Consider the martingale
\begin{equation*}
S_{t}=\sum_{k=1}^{t} \eta_{k} m_{k-1}
\end{equation*}
and the matrix-valued processes
\begin{equation*}
V_{t}=\sum_{k=1}^{t} m_{k-1} m_{k-1}^{\top}, \quad \overline{V}_{t}=V+V_{t}, \quad t \geq 0
\end{equation*}
Then for any $0<\delta<1$, with probability $1-\delta$
\begin{equation*}
\forall t \geq 0, \quad\left\|S_{t}\right\|^2_{V_{t}^{-1}} \leq 2 R^{2} \log \left(\frac{\operatorname{det}\left(\overline{V}_{t}\right)^{1 / 2} \operatorname{det}(V)^{-1 / 2}}{\delta}\right)
\end{equation*}
\end{theorem}
\begin{theorem}[Gordon's theorem for Gaussian matrices~\citep{vershynin2010introduction}]. \label{Gordon} Let $A$ be an $N \times n$ matrix
whose entries are independent standard normal random variables. Then
\begin{equation*}
\sqrt{N}-\sqrt{n} \leq \mathbb{E} \sigma_{\min }(A) \leq \mathbb{E} \sigma_{\max }(A) \leq \sqrt{N}+\sqrt{n}
\end{equation*}
\end{theorem}
\begin{lemma}[Sub-Gaussian Martingale Concentration~\citep{simchowitz2018learning}]\label{sig sub}
Let $\{\mathcal{F}_t\}_{t\geq 1}$ be a filtration, $\{Z_t,W_t\}_{t\geq 1}$ be real valued processes adapted to $\mathcal{F}_t,\mathcal{F}_{t+1}$ respectively (i.e.~$Z_t\in\mathcal{F}_t,W_t\in\mathcal{F}_{t+1}$). Suppose $W_t \big | \mathcal{F}_t$ is a $\sigma^2$-sub-Gaussian random variable with mean zero. Then
\[
\Pr(\{\sum_{t=1}^T Z_t W_t\geq \alpha\}\bigcap \{\sum_{t=1}^T Z_t^2\leq \beta\})\leq \exp(-\frac{\alpha^2}{2\sigma^2\beta})
\]
\end{lemma}
This lemma implies that $\sum_{t=1}^T Z_tW_t$ can essentially be treated as an inner product between a deterministic sequence $Z_t$ and an i.i.d.~sub-Gaussian sequence $W_t$.
\begin{lemma} [Covering bound~\citep{oymak2018non}]\label{cover bound}
Given matrices $A\!\in\!\mathbb{R}^{n_1\!\times\! N},B\!\in\!\mathbb{R}^{N\times n_2}$, let $M\!=\!AB$. Let $\mathcal{C}_1$ be a $1/4$-cover of the unit sphere $\mathcal{S}^{n_1-1}$ and $\mathcal{C}_2$ be a $1/4$-cover of the unit sphere in the row space of $B$ (which is at most $\min\{N,n_2\}$ dimensional). Suppose for all $a\in\mathcal{C}_1,b\in\mathcal{C}_2$, we have that $a^\top M b \leq \gamma$. Then, $\|M\|\leq 2\gamma$.
\end{lemma}
\begin{lemma}[Norm of a sub-Gaussian vector \citep{abbasi2011regret}]\label{subgauss lemma}
Let $v\in \mathbb{R}^d$ be a entry-wise $R$-sub-Gaussian random variable. Then with probability $1-\delta$, $\|v\| \leq R\sqrt{2d\log(2d/\delta)}$.
\end{lemma}
\begin{lemma}[\citep{abbasi2011regret}]\label{basicprob}
Let $X_1, \ldots, X_t$ be random variables. Let $a \in \mathbb{R}$. Let $S_t = \sum_{s=1}^t X_s$ and $\tilde{S}_t = \sum_{s=1}^t \mathbbm{1}_{X_s \leq a}X_s$ where $\mathbbm{1}_{X_s \leq a}X_s$ denotes the truncated version of $X_s$. Then it holds that
\begin{equation*}
\Pr(S_t > x) \leq \Pr(\max_{1\leq s\leq t} X_s \geq a) + \Pr(\tilde{S}_t > x).
\end{equation*}
\end{lemma}
\begin{lemma}[Gaussian concentration Ineq. for Lipschitz func.~\citep{ledoux2013probability}] \label{gausss_lip}
Let $f(\cdot) : \mathbb{R}^{p} \rightarrow \mathbb{R}$ be an $L-$
Lipschitz function and $\mathbf{g} \sim \mathcal{N}\left(0, \mathbf{I}_{p}\right) .$ Then,
\begin{equation*}
\mathbb{P}(|f(\mathbf{g})-\mathbb{E}[f(\mathbf{g})]| \geq t) \leq 2 \exp \left(-\frac{t^{2}}{2 L^{2}}\right)
\end{equation*}
\end{lemma}
\begin{lemma}[\citep{mania2019certainty}] \label{normwoodbury} Let $M$ and $N$ be two positive semidefinite matrices of the same dimension. Then, $\|N(I + MN)^{-1}\| \leq \|N\|$.
\end{lemma} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,146 |
When working people have the freedom to speak up together through unions, we make progress together that benefits everyone. … The last thing America needs is an assault on the freedom to form strong unions and speak up for ourselves and our communities.
Unfortunately, attacking the freedom of working people to come together is exactly what the Janus v. AFSCME lawsuit is all about. Although fronted by a lone state employee, the case is bankrolled by the National Right to Work Foundation and the Liberty Justice Center — the litigation wing of the Illinois Policy Institute — part of a network funded by billionaires and corporate CEOs who use their massive fortunes to tilt the playing field in their favor.
Under current law, every union-represented teacher, police officer, caregiver or other public service worker may choose whether or not to join the union — but the union is required to negotiate on behalf of all workers whether they join or not. Since all the workers benefit from the union's gains, it's only fair that everyone chip in toward the cost. That's why 40 years ago a unanimous Supreme Court approved the kind of cost-sharing arrangements known as fair share.
The Janus v. AFSCME case is an effort by powerful corporate interests to outlaw fair share. … It actually began as a political scheme by Gov. Bruce Rauner, who shortly after taking office issued an executive order and filed a lawsuit trying to ban fair-share fees.
(Excerpted from Lynch's column in the (Springfield, Ill.) State Journal-Register. To read the entire column, go here). | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
\setcounter{equation}{0}\Section{\setcounter{equation}{0}\Section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{assertion}{Assertion}[section]
\newtheorem{remark}{Remark}
\DeclareMathOperator{\cat}{cat}
\newcommand\R{\mathbb R}
\newcommand\mbox{${\rm{I\!N}}$}{\mathbb N}
\newcommand\D{\mathcal{D}}
\newcommand\mbox{${\rm{C\hspace{-1.8mm}\rule{0.3mm}{2.8mm}}}$}{\mathbb C}
\begin{document}
\title[fractional Hamiltonian systems with positive semi-definite matrix]
{Existence and concentration of solution for a fractional Hamiltonian systems with positive semi-definite matrix}
\author{C\'esar Torres}
\author{Ziheng Zhang}
\author{Amado Mendez}
\address[C\'esar Torres]{\newline\indent Departamento de Matem\'aticas
\newline\indent
Universidad Nacional de Trujillo,
\newline\indent
Av. Juan Pablo II s/n. Trujillo-Per\'u}
\email{\href{mailto:ctl\_576@yahoo.es}{ctl\_576@yahoo.es}}
\address[Ziheng Zhang]
{\newline\indent Department of Mathematics,
\newline\indent
Tianjin Polytechnic University,
\newline\indent
Tianjin 300387, China.}
\email{\href{mailto:zhzh@mail.bnu.edu.cn}{zhzh@mail.bnu.edu.cn}}
\address[Amado Mendez]{\newline\indent Departamento de Matem\'aticas
\newline\indent
Universidad Nacional de Trujillo,
\newline\indent
Av. Juan Pablo II s/n. Trujillo-Per\'u}
\email{\href{mailto:gamc55@hotmail.com}{gamc55@hotmail.com}}
\pretolerance10000
\begin{abstract}
\noindent We study the existence of solutions for the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
\eqno(\mbox{FHS})_\lambda
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix
for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that
$L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$,
$W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on
$\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution
of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.
\end{abstract}
\subjclass[2010]{Primary 34C37; Secondary 35A15, 35B38.}
\keywords{Fractional Hamiltonian systems, Fractional Sobolev space, Critical point theory, Concentration phenomena.}
\maketitle
\setcounter{equation}{0}\Section{Introduction}
Fractional Hamiltonian systems are a significant area of nonlinear analysis, since they appear in many phenomena studied in several fields of applied science,
such as engineering, physics, chemistry, astronomy and control theory. On the other hand, the theory of fractional calculus is a part that intensively developing
during the last decades; see \cite{ATMS04, ErvinR06, Hilfer00, KST06, MR93, Pod99} and the references therein. The existence of homoclinic solutions for Hamiltonian systems and their importance in the study of behavior of dynamical
systems can be recognized from Poincar\'{e} \cite{Poincare}. Since then, the investigation of the existence and multiplicity of homoclinic solutions became one of the main
important problems of research in dynamical systems. Critical point theorem was first used by Rabinowitz \cite{Rab86} to obtain the existence of periodic solutions for first
order Hamiltonian systems, while the first multiplicity result is due to Ambrosetti and Zelati \cite{AmbroZelati93}. Therefore, a large number of mathematicians used critical point
theory and variational methods to prove the existence of homoclinic solutions for Hamiltonian systems; see for instance \cite{Co91,GWC,Ding95,Izydorek05,Izydorek07,
Omana92,Rab90,Rab91} and the reference therein.
The critical point theory has become an effective tool in investigating the existence and multiplicity of solutions for fractional differential equations by
constructing fractional variational structures. Especially, in \cite{FJYZ} the authors firstly dealt with a class of fractional boundary value problem via critical
point theory. From then on, Variational methods and critical point theory are shown to be effective in determining the solutions for fractional differential
equations with variational structure. We also mention the work by Torres \cite{Torres12}, where the author considered the following fractional Hamiltonian
systems
$$
\left\{
\begin{array}{ll}
_tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))+L(t)u(t)=\nabla W(t,u(t)),\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
\eqno(\mbox{FHS})
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive definite matrix
for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R}\times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming that $L(t)$ satisfied the following coercivity condition
\begin{itemize}
\item[(L)] there exists an $l\in C(\mathbb{R},(0,\infty))$ with $l(t)\rightarrow \infty$ as $|t|\rightarrow \infty$ such that
\begin{equation}\label{eqn:L coercive}
(L(t)u,u)\geq l(t)|u|^2 \quad \mbox{for all}\,\, t\in \mathbb{R} \,\, \mbox{and} \,\, u\in \mathbb{R}^n.
\end{equation}
\end{itemize}
and that $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition
\begin{itemize}
\item[(FHS$_1$)]$W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and there is a constant $\theta>2$ such that
$$
0<\theta W(t,u)\leq (\nabla W(t,u),u)\quad \mbox{for all}\,\, t\in \mathbb{R} \,\,\mbox{and}\,\, u\in \mathbb{R}^n\backslash\{0\},
$$
\end{itemize}
and other suitable conditions, the author showed that (FHS) possesses at least one nontrivial solution via Mountain
Pass Theorem. Note that (FHS)$_1$, implies that $W(t,u)$ is of superquadratic growth as $|u|\rightarrow \infty$.
Since then, many researchers dealt with (FHS) for the cases that $W(t,u)$ is superquadratic or subquadratic at infinity; see for instance \cite{MendezTorres15,XuReganZhang15,ZhangYuan}. In addition, some perturbed fractional Hamiltonian systems are discussed in \cite{Torres14,XuReganZhang15}.
In \cite{ZhangYuan14} the authors focused on weakening the coercivity condition $(L)$, more precisely they assumed that $L(t)$ is bounded in the following sense:
\begin{itemize}
\item[(L)$'$] $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$ and there are constants $0<\tau_1<\tau_2<\infty$ such that
$$
\tau_1|u|^2\leq (L(t)u,u)\leq \tau_2|u|^2\quad \mbox{for all}\,\, (t,u)\in \mathbb{R} \times \mathbb{R}^n,
$$
\end{itemize}
By supposed that $W(t,u)$ is subquadratic as $|u|\rightarrow +\infty$, the authors also showed that (FHS) possessed infinitely many solutions, which has been generalized in
\cite{NyaZhou2017,ZhouZhang2017}.
In the present paper we deal with the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
\eqno(\mbox{FHS})_{\lambda}
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $L$ satisfies the following conditions
\begin{itemize}
\item[$(\mathcal{L})_1$]$L\in C(\mathbb{R},\mathbb{R}^{n\times n})$ is a symmetric matrix for all $t\in\mathbb{R}$; there exists a nonnegative continuous function $l:\mathbb{R} \rightarrow \mathbb{R}$ and
a constant $c>0$ such that
$$
(L(t)u,u)\geq l(t)|u|^2,
$$
and the set $\{l<c\}:=\{t\in \mathbb{R} \,|\,l(t)<c\}$ is nonempty with $meas \{l<c\}<\frac{1}{C_{\infty}^2}$, where $meas \{\cdot\}$ is the Lebesgue measure and $C_\infty$
is the best Sobolev constant for the embedding of $X^{\alpha}$ into $L^{\infty}(\mathbb{R})$;
\item[$(\mathcal{L})_2$]$J=int (l^{-1}(0))$ is a nonempty finite interval and $\overline{J}=l^{-1}(0)$;
\item[$(\mathcal{L})_3$]there exists an open interval $T\subset J$ such that $L(t)\equiv 0$ for all $t\in \overline{T}$.
\end{itemize}
In particular, if $\alpha=1$ in (FHS)$_\lambda$, then it reduces to the following well-known
second order Hamiltonian systems
$$
\ddot u- \lambda L(t) u+\nabla W(t,u)=0.\eqno(\mbox{HS})
$$
Recently a second order Hamiltonian systems like (HS) with positive semi-definite matrix was considered in \cite{JSTW}. Assuming that
$W \in C^1(\mathbb{R}\times \mathbb{R}^n, \mathbb{R})$ is an indefinite potential satisfying asymptotically quadratic condition at infinity on $u$, Sun and Wu,
with a little mistake in their embedding results, have proved the existence of two homoclinic solutions of (FHS$_\lambda$). For more related works, we refer the reader to
\cite{Co91,Ding95,Izydorek05,Izydorek07,Omana92,Rab91} and the references mentioned there.
Here we must point out, to obtain the existence or multiplicity of solutions for Hamiltonian systems, all the papers mentioned above need the assumption that
the symmetric matrix $L(t)$ is positive definite, see (L) and (L)$'$. Therefore, recently the authors in \cite{Benhassine2017,TorresZhang2017,ZhangTorres}
considered the case that $L(t)$ is positive semi-definite satisfying $(\mathcal{L})_1$. In \cite{Benhassine2017}, the author dealt with (FHS) for the case that
$(\mathcal{L})_1$ is satisfied and $W(t,u)$ involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing. In \cite{TorresZhang2017,ZhangTorres},
we have considered the existence of solutions of (FHS)$_\lambda$ and the concentration of its solutions when $(\mathcal{L})_1$-$(\mathcal{L})_3$ are satisfied and
$W(t,u)$ meets with some classes of superquadratic hypothesis.
Motivated by these previous results, the main purpose of this paper is
to investigate (FHS)$_\lambda$ without Ambrosetti-Rabinowitz condition (FHS$_1$). More precisely, we suppose that $W(t,u)$ satisfy the following assumptions
\begin{enumerate}
\item[($W_1$)] $|\nabla W(t,u)| = o(|u|)$ as $|u|\to 0$ uniformly in $t\in \mathbb{R}$.
\item[$(W_2)$] $W(t,u)\geq 0$ for all $(t,u)\in \mathbb{R}\times \mathbb{R}^N$ and $H(t,u)\geq0$ for all $(t, u) \in \mathbb{R} \times \mathbb{R}^N$, where
$$
H(t,u) :=\frac{1}{2}\langle \nabla W(t,u), u \rangle - W(t,u).
$$
\item[$(W_3)$] $\frac{W(t,u)}{|u|^2} \to + \infty$ as $|u| \to +\infty$ uniformly in $t\in \mathbb{R}$.
\item[$(W_4)$] There exist $C_0, R >0$, and $\sigma >1$ such that
$$
\frac{|\nabla W(t,u)|^\sigma}{|u|^\sigma} \leq C_0H(t,u)\;\;\mbox{if}\;\;|u| \geq R.
$$
\end{enumerate}
Note that, according to \cite{GWC} the nonlinearity
$$
W(t,u) = g(t)(|u|^p + (p-2)|u|^{p-\epsilon}\sin^2(\frac{|u|^\epsilon}{\epsilon})),
$$
where $g(t)>0$ is $T$-periodic in $t$, $0<\epsilon < p-2$ and $p>2$, satisfies $(W_1)-(W_4)$, but (FHS$_1$) is not satisfied.
Now we are in the position to state our main result.
\begin{theorem}\label{Thm:MainTheorem1}
Suppose that {\rm ($\mathcal{L}$)$_1$}-{\rm ($\mathcal{L}$)$_3$}, $(W_1) - (W_4)$ are satisfied, then there exists $\Lambda _*>0$ such that for every
$\lambda>\Lambda_*$, {\rm(FHS)$_\lambda$} has at least one nontrivial solution.
\end{theorem}
On the concentration of solutions obtained above, for technical reason, we consider that there exists $0<\varrho< +\infty$, such that $T = [-\varrho,\varrho]$,
where $T$ is given by $(\mathcal{L})_3$. We have the following result.
\begin{theorem}\label{Thm:MainTheorem2}
Let $u_\lambda$ be a solution of problem ${\rm (FHS)}_\lambda$ obtained in Theorem \ref{Thm:MainTheorem1}, then $u_\lambda \to \tilde{u}$ strongly in
$H^{\alpha}(\mathbb{R})$ as $\lambda \to \infty$, where $\tilde{u}$ is a nontrivial solution of the following boundary value problem
\begin{eqnarray}\label{eqn:BVP}
\left\{
\begin{array}{ll}
{_{t}}D_{\varrho}^{\alpha} ({_{-\varrho}}D_{t}^{\alpha})u = \nabla W(t, u), & t\in (-\varrho, \varrho) \\[0.2cm]
u(-\varrho) = u(\varrho) = 0,
\end{array}
\right.
\end{eqnarray}
where ${_{-\varrho}}D_{t}^{\alpha}$ and $_{t}D_{\varrho}^{\alpha}$ are left and right Riemann-Liouville fractional derivatives of order $\alpha$ on $[-\varrho,\varrho]$ respectively.
\end{theorem}
\begin{remark}
{\rm
In Theorem \ref{Thm:MainTheorem1}, we give some new superquadratic conditions on $W(t,u)$ to guarantee the existence of solutions and investigate
the concentration of these solutions in \ref{Thm:MainTheorem2}. However, we must point out that the methods in \cite{Benhassine2017,TorresZhang2017,ZhangTorres}
are not be valid for our new assumptions. To overcome this difficulty we apply the Mountain Pass Theorem with Cerami condition, however, the direct application of the mountain pass theorem is not enough since the Cerami sequences might lose compactness in the whole space $\mathbb{R}$. Then it is necessary to introduce a new compactness result to recover the convergence of Cerami sequence, for more details see Lemma \ref{cerami1}.
}
\end{remark}
The remaining part of this paper is organized as follows. Some preliminary results are presented in Section 2. In Section
3, we are devoted to accomplishing the proof of Theorem \ref{Thm:MainTheorem1} and in Section 4 we present the proof of Theorem \ref{Thm:MainTheorem2}.
\setcounter{equation}{0}\Section{Preliminary Results}
In this section, for the reader's convenience, firstly we introduce some basic definitions of fractional calculus.
The Liouville-Weyl fractional derivative of order $0<\alpha<1$ are defined as
\begin{equation}\label{eqn:RD}
_{-\infty}D^{\alpha}_x u(x)=\frac{d}{dx} {_{-\infty}I^{1-\alpha}_x u(x)}
\quad \mbox{and}\quad
_{x}D^{\alpha}_{\infty} u(x)=-\frac{d}{dx} {_{x}I^{1-\alpha}_{\infty} u(x)}.
\end{equation}
where $_{-\infty}I^{\alpha}_x$ and $_{x}I^{\alpha}_{\infty}$ are the left and right Liouville-Weyl fractional integrals of order $0<\alpha<1$ defined as
$$
_{-\infty}I^{\alpha}_x u(x)=\frac{1}{\Gamma(\alpha)}\int^x_{-\infty} (x-\xi)^{\alpha-1}u(\xi)d\xi
\quad \mbox{and}\quad
_{x}I^{\alpha}_{\infty} u(x)=\frac{1}{\Gamma(\alpha)}\int^{\infty}_{x}(\xi-x)^{\alpha-1}u(\xi)d\xi.
$$
Furthermore, for $u\in L^p(\R)$, $p\geq 1$, we have
$$
\mathcal{F}({_{-\infty}}I_{x}^{\alpha}u(x)) = (i\omega)^{-\alpha}\widehat{u}(\omega)\quad \quad \mbox{and}\quad \quad \mathcal{F}({_{x}}I_{\infty}^{\alpha}u(x)) = (-i\omega)^{-\alpha}\widehat{u}(\omega),
$$
and for $u\in C_{0}^{\infty}(\R)$, we have
$$
\mathcal{F}({_{-\infty}}D_{x}^{\alpha}u(x)) = (i\omega)^{\alpha}\widehat{u}(\omega)\quad \quad \mbox{and}\quad \quad \mathcal{F}({_{x}}D_{\infty}^{\alpha}u(x)) = (-i\omega)^{\alpha}\widehat{u}(\omega),
$$
In order to establish the variational structure which enables us to reduce the existence of solutions of (FHS)$_\lambda$ to find critical points of the
corresponding functional, it is necessary to consider some appropriate function spaces. Denote by $L^p(\mathbb{R},\mathbb{R}^n)$ ($1\leq p <\infty$) the Banach spaces of functions on $\mathbb{R}$ with values in $\mathbb{R}^n$ under
the norms
$$
\|u\|_{L^p}=\Bigl(\int_{\mathbb{R}}|u(t)|^p dt\Bigr)^{1/p},
$$
and $L^{\infty}(\mathbb{R},\mathbb{R}^n)$ is the Banach space of essentially bounded functions from $\mathbb{R}$ into $\mathbb{R}^n$ equipped with the norm
$$
\|u\|_{\infty}=\mbox{ess} \sup\left\{|u(t)|: t\in \mathbb{R} \right\}.
$$
Let $-\infty<a<b<+\infty$, $0< \alpha \leq 1$ and $1<p<\infty$. The fractional derivative space $E_{0}^{\alpha ,p}$ is defined by the closure of $C_{0}^{\infty}([a,b], \mathbb{R}^n)$ with respect to the norm
\begin{equation}\label{norm}
\|u\|_{\alpha ,p} = \left(\int_{a}^{b} |u(t)|^pdt + \int_{a}^{b}|{_{a}}D_{t}^{\alpha}u(t)|^pdt \right)^{1/p}, \;\;\forall\; u\in E_{0}^{\alpha ,p}.
\end{equation}
Furthermore $(E_{0}^{\alpha ,p}, \|.\|_{\alpha ,p})$ is a reflexive and separable Banach space and can be characterized by $$E_{0}^{\alpha , p} = \{u\in L^{p}([a,b], \mathbb{R}^n) | {_aD}_{t}^{\alpha}u \in L^{p}([a,b], \mathbb{R}^n)\;\mbox{and}\;u(a) = u(b) = 0\}.$$
\begin{proposition}\label{FC-FEprop3}
\cite{FJYZ} Let $0< \alpha \leq 1$ and $1 < p < \infty$. For all $u\in E_{0}^{\alpha ,p}$, we have
\begin{equation}\label{FC-FEeq3}
\|u\|_{L^{p}} \leq \frac{(b-a)^{\alpha}}{\Gamma (\alpha +1)} \|{_aD}_{t}^{\alpha}u\|_{L^{p}}.
\end{equation}
If $\alpha > 1/p$ and $\frac{1}{p} + \frac{1}{q} = 1$, then
\begin{equation}\label{FC-FEeq4}
\|u\|_{\infty} \leq \frac{(b-a)^{\alpha -1/p}}{\Gamma (\alpha)((\alpha - 1)q +1)^{1/q}}\|{_aD}_{t}^{\alpha}u\|_{L^{p}}.
\end{equation}
\end{proposition}
\noindent
By (\ref{FC-FEeq3}), we can consider in $E_{0}^{\alpha ,p}$ the following norm
\begin{equation}\label{FC-FEeq5}
\|u\|_{\alpha ,p} = \|{_aD}_{t}^{\alpha}u\|_{L^{p}},
\end{equation}
which is equivalent to (\ref{norm}).
\begin{proposition}\label{FC-FEprop4}
\cite{FJYZ} Let $0< \alpha \leq 1$ and $1 < p < \infty$. Assume that $\alpha > \frac{1}{p}$ and $\{u_{k}\} \rightharpoonup u$ in $E_{0}^{\alpha ,p}$. Then
$u_{k} \to u$ in $C[a,b]$, i.e.
$$
\|u_{k} - u\|_{\infty} \to 0,\;k\to \infty.
$$
\end{proposition}
For $\alpha>0$, consider the Liouville-Weyl fractional spaces
$$
I^{\alpha}_{-\infty}=\overline{C^{\infty}_0(\mathbb{R},\mathbb{R}^n)}^{\|\cdot\|_{I^{\alpha}_{-\infty}}},
$$
where
\begin{equation}\label{eqn:defn Rnorm}
\|u\|_{I^{\alpha}_{-\infty}}=\Bigl(\int_{\mathbb{R}}u^2(x)dx+ \int_{\mathbb{R}}|_{-\infty}D^{\alpha}_x u(x)|^2dx\Bigr)^{1/2}.
\end{equation}
Furthermore, we introduce the fractional Sobolev space $H^{\alpha}(\mathbb{R},\mathbb{R}^n)$ of order $0<\alpha<1$ which is defined as
\begin{equation}\label{eqn:alphanorm}
H^{\alpha}=\overline{C^{\infty}_0(\mathbb{R},\mathbb{R}^n)}^{\|\cdot\|_{\alpha}},
\end{equation}
where
$$
\|u\|_{\alpha}=\Bigl(\int_{\mathbb{R}}u^2(x)dx+ \int_{\mathbb{R}}|w|^{2\alpha}\widehat{u}^2(w)dw\Bigr)^{1/2}.
$$
Note that, a function $u\in L^2(\mathbb{R},\mathbb{R}^n)$ belongs to $I^{\alpha}_{-\infty}$ if and only if
$$
|w|^{\alpha}\widehat{u}\in L^2(\mathbb{R},\mathbb{R}^n).
$$
Therefore, $I^{\alpha}_{-\infty}$ and $H^{\alpha}$ are equivalent with equivalent norm, for more details see \cite{ErvinR06}.
\begin{lemma}\label{Lem:LinftyContH}\cite[Theorem 2.1]{Torres12}
If $\alpha>1/2$, then $H^{\alpha}\subset C(\mathbb{R},\mathbb{R}^n)$ and there is a constant $C_\infty=C_{\alpha,\infty}$ such that
\begin{equation}\label{12}
\|u\|_{\infty}=\sup_{x\in \mathbb{R}}|u(x)|\leq C_\infty \|u\|_{\alpha}.
\end{equation}
\end{lemma}
\begin{remark}\label{Rem:Lp}
From Lemma \ref{Lem:LinftyContH}, we know that if $u\in H^{\alpha}$ with $1/2<\alpha<1$, then $u\in L^p(\mathbb{R},\mathbb{R}^n)$ for all $p\in [2,\infty)$, since
$$
\int_{\mathbb{R}}|u(x)|^p dx \leq \|u\|^{p-2}_{\infty}\|u\|^2_{L^2}.
$$
\end{remark}
Now, we introduce the fractional space which we will use to construct the variational framework for (FHS)$_\lambda$. Let
$$
X^{\alpha}=\Bigl\{u\in H^{\alpha}: \int_{\mathbb{R}}[|_{-\infty}D^{\alpha}_{t}u(t)|^2+(L(t)u(t),u(t))]dt<\infty\Bigr\},
$$
then $X^{\alpha}$ is a reflexive and separable Hilbert space with the inner product
$$
\langle u,v \rangle_{X^{\alpha}}=\int_{\mathbb{R}}[(_{-\infty}D^{\alpha}_{t}u(t),_{-\infty}D^{\alpha}_{t}v(t))+(L(t)u(t),v(t))]dt
$$
and the corresponding norm is
$$
\|u\|^2_{X^{\alpha}}=\langle u,u \rangle_{X^{\alpha}}.
$$
For $\lambda>0$, we also need the following inner product
$$
\langle u,v \rangle_{X^{\alpha,\lambda}}=\int_{\mathbb{R}}[(_{-\infty}D^{\alpha}_{t}u(t),_{-\infty}D^{\alpha}_{t}v(t))+\lambda(L(t)u(t),v(t))]dt
$$
and the corresponding norm is
$$
\|u\|^2_{X^{\alpha,\lambda}}=\langle u,u \rangle_{X^{\alpha,\lambda}}.
$$
\begin{lemma}\label{Lem:XcontH}
\cite{ZhangTorres} Suppose $L(t)$ satisfies {\rm ($\mathcal{L}$)$_1$} and {\rm ($\mathcal{L}$)$_2$}, then $X^{\alpha}$ is continuously embedded in $H^{\alpha}$.
\end{lemma}
\begin{remark}\label{keynta}
{\rm
Under the same conditions of Lemma \ref{Lem:XcontH}, for all
$\lambda\geq \frac{1}{c C_\infty^2 \, meas \{l<c\}}$, we also obtain
\begin{equation}\label{13}
\int_{\mathbb{R}}|u(t)|^2 dt\leq \frac{C_\infty^2\, meas\{l<c\}}{1-C_\infty^2\, meas\{l<c\}}\|u\|_{X^{\alpha,\lambda}}=\frac{1}{\Theta}\|u\|_{X^{\alpha,\lambda}}^2
\end{equation}
and
\begin{equation}\label{14}
\|u\|_\alpha^2\leq \Bigl(1+\frac{C_{\infty}^2\, meas\{l<c\}}{1-C_{\infty}^2\, meas \{l<c\}}\Bigr)\|u\|_{X^\alpha}^2=(1+\frac{1}{\Theta})\|u\|^2_{X^{\alpha,\lambda}}.
\end{equation}
Furthermore, for every $p\in (2,\infty)$ and $\lambda\geq \frac{1}{c C_\infty^2 \, meas \{l<c\}}$, we have
\begin{equation}\label{15}
\begin{split}
\int_{\mathbb{R}}|u(t)|^p dt\leq \mathcal{K}_{p}^{p}\|u\|_{X^{\alpha,\lambda}}^p.
\end{split}
\end{equation}
where $\mathcal{K}_{p}^{p} = \frac{1}{\Theta^{\frac{p}{2}}\, (meas\{l<c\})^{\frac{p-2}{2}}}$. For more details, see \cite{Torres15, ZhangTorres}.}
\end{remark}
\setcounter{equation}{0}\Section{ Proof of Theorem \ref{Thm:MainTheorem1}}
The aim of this section is to establish the proof of Theorem \ref{Thm:MainTheorem1}. Consider the functional $I: X^{\alpha,\lambda}\rightarrow \mathbb{R}$ given by
\begin{equation}\label{mt01}
\begin{aligned}
I_\lambda (u)&=\int_{\mathbb{R}}\Bigl[\frac{1}{2}|_{-\infty}D_t^{\alpha}u(t)|^2+\frac{1}{2}(\lambda L(t)u(t),u(t))-W(t,u(t))\Bigr]dt\\
&=\dfrac{1}{2}\|u\|^2_{X^{\alpha,\lambda}}-\int_{\mathbb{R}}W(t,u(t))dt.
\end{aligned}
\end{equation}
Under the conditions of Theorem \ref{Thm:MainTheorem1}, we note that $I\in C^1(X^{\alpha,\lambda},\mathbb{R})$, and
\begin{equation}\label{mt02}
I'_\lambda (u)v=\int_{\mathbb{R}}\Bigl[(_{-\infty}D_t^{\alpha}u(t), _{-\infty}D_t^{\alpha}v(t))+(\lambda L(t)u(t),v(t))-(\nabla W(t,u(t)),v(t))\Bigr]dt
\end{equation}
for all $u$, $v\in X^{\alpha}$. In particular we have
\begin{equation}\label{mt03}
I'_\lambda (u)u =\|u\|^2_{X^{\alpha,\lambda}}-\int_{\mathbb{R}}(\nabla W(t,u(t)),u(t))dt.
\end{equation}
\begin{remark}\label{ineq}
It follows from $(W_1)$ and $(W_4)$ that
$$
|\nabla W(t,u)|^\sigma \leq \frac{C_0}{2}|\nabla W(t,u)||u|^{\sigma +1 }\;\;\mbox{for}\;\;|u|\geq R.
$$
Thus, by ($W_1$), for any $\epsilon >0$, there is $C_\epsilon >0$ such that
\begin{equation}\label{mt04}
|\nabla W(t,u)| \leq \epsilon |u| + C_\epsilon |u|^{p-1},\;\;\forall (t,u) \in \mathbb{R} \times \mathbb{R}^N
\end{equation}
and
\begin{equation}\label{mt05}
|W(t,u)| \leq \frac{\epsilon}{2}|u|^2 + \frac{C_\epsilon}{p}|u|^p\;\;\forall (t,u)\in \mathbb{R} \times \mathbb{R}^N,
\end{equation}
where $p = \frac{2\sigma }{\sigma -1}>2$.
\end{remark}
\begin{lemma}\label{GC1}
Suppose that {\rm ($\mathcal{L}$)$_1$}-{\rm ($\mathcal{L}$)$_3$}, {\rm(W$_1$)} and {\rm(W$_2$)} are satisfied. Then
\begin{enumerate}
\item[\fbox{i}] There exists $\rho>0$ and $\eta>0$ such that
$$
\inf_{\|u\|_{X^{\alpha,\lambda}}=\rho} I_\lambda (u)>\eta\quad for \,\, all \,\, \lambda\geq \frac{1}{cC_\infty^2 \, meas \{l<c\}}.
$$
\item[\fbox{ii}] Let $\rho>0$ defined in $(i)$, then there exists $e\in X^{\alpha,\lambda}$ with
$\|e\|_{X^{\alpha,\lambda}}>\rho$ such that $I_\lambda(e)<0$ for all $\lambda>0$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item[\fbox{i}] By (\ref{mt05}) and Remark \ref{keynta}, we obtain
$$
\begin{aligned}
I_{\lambda}(u) & = \frac{1}{2}\|u\|_{X^{\alpha, \lambda}}^2 - \int_{\mathbb{R}} W(t, u(t))dt\\
& \geq \frac{1}{2}\|u\|_{X^{\alpha, \lambda}}^{2} - \frac{\epsilon}{2}\int_{\mathbb{R}}|u(t)|^2dt - \frac{C_\epsilon}{p}\int_{\mathbb{R}} |u(t)|^pdt\\
&\geq \frac{1}{2}\left( 1 - \frac{\epsilon}{\Theta}\right)\|u\|_{X^{\alpha, \lambda}}^{2} - \frac{C_\epsilon}{p\Theta^{\frac{p}{2}}(meas\{l<c\})^{\frac{p-2}{2}}}\|u\|_{X^{\alpha, \lambda}}^{p}.
\end{aligned}
$$
Let $\epsilon >0$ small enough such that $1- \frac{\epsilon}{\Theta}>0$ and $\|u\|_{X^{\alpha, \lambda}} = \rho$. Since $p>2$, taking $\rho$ small enough such that
$$
\frac{1}{2}\left( 1-\frac{\epsilon}{\Theta} \right) - \frac{C_\epsilon}{p\Theta^{\frac{p}{2}}(meas\{l<c\})^{\frac{p-2}{2}}} \rho^{p-2}>0.
$$
Therefore
$$
I_\lambda (u) \geq \rho^2\left[ \frac{1}{2}\left( 1-\frac{\epsilon}{\Theta} \right) - \frac{C_\epsilon}{p \Theta^{\frac{p}{2}}(meas\{l<c\})^{\frac{p-2}{2}}} \rho ^{p-2}\right] : = \eta >0.
$$
\item[\fbox{ii}] By $(\mathcal{L})_3$ and without loss of generality let $T = (-\varrho, \varrho )\subset J$ such that $L(t) \equiv 0$. Let $\psi \in C_{0}^{\infty}(\mathbb{R}, \R^n)$ such that $supp(\psi) \subset (-\tau, \tau)$, for some $\tau< \varrho$. Hence
\begin{equation}\label{ss00}
\begin{aligned}
0&\leq \int_{\R}\langle L(t)\psi , \psi\rangle dt = \int_{supp(\psi)} \langle L(t)\psi, \psi\rangle dt \leq \int_{-\tau}^{\tau} \langle L(t)\psi, \psi\rangle dt \leq \int_{T} \langle L(t)\psi, \psi\rangle dt = 0.
\end{aligned}
\end{equation}
On the other hand, by $(W_3)$, for any $\epsilon >0$, there exists $R>0$ such that
$$
W(t,u) > \frac{|u|^2}{\epsilon} - \frac{R^2}{\epsilon}\;\;\mbox{for all}\;\;|u|\geq R.
$$
Then, by taking $\epsilon \to 0$ we get
\begin{equation}\label{ss01}
\lim_{|\sigma| \to \infty} \int_{supp(\psi)} \frac{W(t, \sigma \psi)}{|\sigma|^2}dt = +\infty.
\end{equation}
Hence, by (\ref{ss00}) and (\ref{ss01}) we obtain
\begin{equation}\label{ss02}
\frac{I_\lambda (\sigma \psi)}{|\sigma|^2} = \frac{1}{2}\int_{\R} |_{-\infty}D_{t}^{\alpha}\psi(t)|^2dt - \int_{\R} \frac{W(t, \sigma \psi)}{|\sigma|^2}dt \to -\infty,
\end{equation}
as $|\sigma| \to \infty$. Therefore, if $\sigma_0$ is large enough and $e = \sigma_0 \psi$ one gets $I_\lambda (e) <0$.
\end{enumerate}
\end{proof}
Since we have loss of compactness we need the following compactness results to recover the Cerami condition for $I_\lambda$.
\begin{lemma}\label{PS1}
Suppose that $(\mathcal{L})_1 - (\mathcal{L})_3$, $(W_1) - (W_4)$ be satisfied. If $u_n \rightharpoonup u$ in $X^{\alpha, \lambda}$, then
\begin{equation}\label{mt07}
I_\lambda (u_n - u) = I_\lambda (u_n) - I_\lambda (u) + o(1)\;\;\mbox{as}\;\;n\to +\infty
\end{equation}
and
\begin{equation}\label{mt08}
I'_\lambda(u_n-u) = I'_\lambda(u_n) - I'_\lambda (u) + o(1) \;\;\mbox{as}\;\;n\to +\infty.
\end{equation}
In particular, if $I_\lambda (u_n) \to c$ and $I'_\lambda(u_n) \to 0$, then $I'_\lambda(u) = 0$ after passing to a subsequence.
\end{lemma}
\begin{proof}
Since $u_n \rightharpoonup u$ in $X^{\alpha, \lambda}$, we have $\langle u_n -u, u\rangle_{X^{\alpha, \lambda}} \to 0$ as $n\to \infty$, which implies that
$$
\begin{aligned}
\|u_n\|_{X^{\alpha, \lambda}}^{2} = \|u_n - u\|_{X^{\alpha, \lambda}}^{2} + \|u\|_{X^{\alpha, \lambda}}^{2} + o(1).
\end{aligned}
$$
Therefore, to obtain (\ref{mt07}) and (\ref{mt08}) it suffices to check that
\begin{equation}\label{mt09}
\int_{\mathbb{R}} [W(t,u_n) - W(t, u_n-u) - W(t,u)]dt = o(1)
\end{equation}
and
\begin{equation}\label{mt10}
\sup_{\varphi \in X^{\alpha, \lambda}, \|\varphi\|_{\alpha, \lambda} =1} \int_{\mathbb{R}} \langle \nabla W(t, u_n) - \nabla W(t, u_n-u)- \nabla W(t,u), \varphi\rangle dt = o(1).
\end{equation}
Here, we only prove (\ref{mt10}), the verification of (\ref{mt09}) is similar. In fact, let
\begin{equation}\label{mt11}
\mathcal{A}:= \lim_{n\to \infty} \sup_{\varphi \in X^{\alpha, \lambda}, \|\varphi\|_{\alpha, \lambda} =1} \int_{\mathbb{R}} \langle \nabla W(t, u_n) - \nabla W(t, u_n-u)- \nabla W(t,u), \varphi\rangle dt.
\end{equation}
If $\mathcal{A}>0$, then, there exists $\varphi_0 \in X^{\alpha, \lambda}$ with $\|\varphi_0\|_{X^{\alpha, \lambda}} = 1$ such that
$$
\left| \int_{\mathbb{R}} \langle \nabla W(t,u_n) - \nabla W(t, u_n-u) - \nabla W(t,u), \varphi_0\rangle dt \right| \geq \frac{\mathcal{A}}{2}
$$
for $n$ large enough. Now, from (\ref{mt04}) and Young's inequality, there exist $C_1$, $C_2$ and $C_3 >0$ such that
$$
\begin{aligned}
&|\langle \nabla W(t, u_n) - \nabla W(t, u_n-u), \varphi_0\rangle|
\leq C_1 \left( \epsilon |u|^2 + \epsilon |u_n-u|^2 + \epsilon |\varphi_0|^2 + C_2|u|^p + \epsilon |u_n - u|^{p} + C_3|\varphi_0|^p\right).
\end{aligned}
$$
Hence, there exists $C_4, C_5, C_6 >0$ such that
$$
\begin{aligned}
|\langle \nabla W(t, u_n) - \nabla W(t, u_n - u) &- \nabla W(t,u), \varphi_0\rangle|\\
& \leq C_4 \left( \epsilon |u|^2 + \epsilon |u_n - u|^2 + \epsilon |\varphi_0|^2 + C_5|u|^p + \epsilon |u_n - u|^p + C_6|\varphi_0|^p\right).
\end{aligned}
$$
Let
$$
h_n(t) = \max\{|\langle \nabla W(t, u_n) - \nabla W(t, u_n - u) - \nabla W(t,u), \varphi_0 \rangle| - C_4 \epsilon (|u_n - u|^2 + |u_n - u|^p), 0\}.
$$
So
$$
0\leq h_n(t) \leq C_4 (\epsilon |u|^2 + \epsilon |\varphi_0|^2 + C_5|u|^p + C_6|\varphi_0|^p).
$$
By the Lebesgue dominated convergence Theorem and the fact $u_n \to u$ a.e. in $\mathbb{R}$, , we can get
$$
\int_{\mathbb{R}} h_n(t)dt \to 0\;\;\mbox{as}\;\;n \to \infty.
$$
From where
$$
\int_{\mathbb{R}} |\langle \nabla W(t, u_n(t)) - \nabla W(t, u_n(t) - u(t)) - \nabla W(t,u(t)), \varphi_0(t) \rangle|dt \to 0\;\;\mbox{as}\;\; n\to \infty,
$$
which is a contradiction. Hence $\mathcal{A} = 0$.
Furthermore, if $I_\lambda (u_n) \to c$ and $I'_\lambda (u_n) \to 0$ as $n \to \infty$, by (\ref{mt07}) and (\ref{mt08}), we get
$$
I_\lambda (u_n - u) \to c-I_\lambda (u) + o(1)
$$
and
$$
I'_\lambda (u_n - u) = -I'_\lambda(u)\;\;\mbox{as}\;\;n \to +\infty.
$$
Now, for every $\varphi \in C_{0}^{\infty}(\mathbb{R}, \mathbb{R}^n)$ we have
$$
I'_\lambda(u)\varphi = \lim_{n \to \infty} I'_\lambda (u_n) \varphi = 0.
$$
Consequently, $I'_\lambda (u) = 0$.
\end{proof}
\begin{lemma}\label{cerami1}
Suppose that $(\mathcal{L})_1 - (\mathcal{L})_3$, $(W_1) - (W_4)$ be satisfied and let $c\in \mathbb{R}$. Then each $(Ce)_c$-sequence of $I_\lambda$ is bounded in $X^{\alpha, \lambda}$.
\end{lemma}
\begin{proof}
Suppose that $\{u_n\} \subset X^{\alpha, \lambda}$ is a $(Ce)_c$ sequence for $c>0$, namely
\begin{equation}\label{mt12}
I_\lambda (u_n) \to c,\quad (1+\|u_n\|_{X^{\alpha, \lambda}})I'_\lambda (u_n) \to 0\;\;\mbox{as}\;\;n\to \infty.
\end{equation}
Therefore
\begin{equation}\label{mt13}
c- o_n(1) = I_\lambda (u_n) - \frac{1}{2}I'_\lambda(u_n)u_n = \int_{\mathbb{R}} H(t, u_n(t))dt.
\end{equation}
By contradiction, suppose that there is a subsequence, again denoted by $\{u_n\}$, such that $\|u_n\|_{X^{\alpha, \lambda}} \to +\infty$ as $n\to +\infty$. Taking $v_n = \frac{u_n}{\|u_n\|_{X^{\alpha, \lambda}}}$, we get that $\{v_n\}$ is bounded in $X^{\alpha, \lambda}$ and $\|v_n\|_{X^{\alpha, \lambda}} = 1$. Moreover, we have
$$
o(1) = \frac{\langle I'_\lambda (u_n), u_n\rangle}{\|u_n\|_{X^{\alpha, \lambda}}^{2}} = 1- \int_{\mathbb{R}} \frac{\langle \nabla W(t,u_n), u_n \rangle}{\|u_n\|_{X^{\alpha, \lambda}}^{2}},
$$
as $n \to \infty$, which implies
\begin{equation}\label{lim}
\int_{\mathbb{R}} \frac{\langle \nabla W(t, u_n), v_n\rangle}{|u_n|} |v_n|dt = \int_{\mathbb{R}} \frac{\langle \nabla W(t, u_n), u_n\rangle}{\|u_n\|_{X^{\alpha, \lambda}}^{2}} \to 1.
\end{equation}
For $r\geq 0$, let
$$
h( r ) := \inf\{H(t, u):\;\;t\in \mathbb{R}, \;\;|u|\geq r\}.
$$
From ($W_2$) we have $h ( r) >0$ for all $r>0$. Furthermore, by ($W_2$) and $(W_4)$, for $|u|\geq r$,
\begin{equation}\label{mt14}
C_0 H(t,u) \geq \frac{|\nabla W(t,u)|^\sigma}{|u|^\sigma} = \left( \frac{|\nabla W(t,u)||u|}{|u|^2} \right)^{\sigma} \geq \left( \frac{\langle \nabla W(t,u), u\rangle}{|u|^2}\right)^{\sigma} \geq \left( \frac{2W(t,u)}{|u|^2} \right)^{\sigma},
\end{equation}
it follows from $(W_3)$ and the definition of $h(r)$ that
$$
h( r) \to \infty\;\;\mbox{as}\;\;r\to \infty.
$$
For $0\leq a < b$, let
$$
\Omega_{n}(a, b) :=\{t\in \mathbb{R}:\;\;a \leq |u_n(t)|< b\}
$$
and
$$
C_{a}^{b} : = \inf \left\{\frac{H(t, u)}{|u|^2}: \;\;t\in \mathbb{R}\;\;\mbox{and}\;\;u\in \mathbb{R}^N\;\;\mbox{with}\;\;a \leq |u|< b \right\}.
$$
By ($W_1$), for any $\epsilon >0$, there is $\delta >0$ such that
$$
|\nabla W(t,u)| \leq \frac{\epsilon}{\mathcal{K}_{2}^{2}}|u|\;\;\mbox{for all}\;\;|t|\leq \delta.
$$
Consequently
\begin{equation}\label{mt15}
\begin{aligned}
\int_{\Omega_{n}(0, \delta)} \frac{|\nabla W(t, u_n)|}{|u_n|}|v_n|^2dt \leq \int_{\Omega_n(0, \delta)} \frac{\epsilon}{\mathcal{K}_{2}^{2}}|v_n|^2dt\leq \frac{\epsilon}{\mathcal{K}_{2}^{2}}\|v_n\|_{L^2}^{2} \leq \epsilon,\;\;\forall n.
\end{aligned}
\end{equation}
We note that
$$
H(t,u_n)\geq C_{a}^{b}|u_n|^2\quad \mbox{for all} \quad t \in \Omega_n (a,b),
$$
consequently, by (\ref{mt13}) we get
\begin{equation}\label{mt16}
\begin{aligned}
c-o_n(1) &= \int_{\Omega_n(0,a)} H(t,u_n)dt + \int_{\Omega_n(a,b)}H(t,u_n)dt + \int_{\Omega_n(b, +\infty)}H(t,u_n)dt\\
&\geq \int_{\Omega_n (0,a)} H(t,u_n)dt + C_{a}^{b}\int_{\Omega_n(a,b)}|u_n|^2dt + \int_{\Omega_n(b, +\infty)}H(t,u_n)dt\\
&= \int_{\Omega_n (0,a)} H(t,u_n)dt + C_{a}^{b}\int_{\Omega_n(a,b)}|u_n|^2dt + h(b)meas(\Omega_n(b,+\infty)).
\end{aligned}
\end{equation}
Since $h( r) \to +\infty$ as $r\to +\infty$, for $p< q < \infty$ it follows from (\ref{mt16}) that
\begin{equation}\label{mt17}
\begin{aligned}
\int_{\Omega_{n}(b, +\infty)}|v_n|^pdt &\leq \left( \int_{\Omega_n(b, +\infty)}|v_n|^qdt \right)^{\frac{p}{q}} meas(\Omega_n(b+\infty))^{\frac{q-p}{q}}\\
&\leq \|v_n\|_{L^q}^{p} \left( \frac{c-o_n(1)}{h(b)} \right)^{\frac{q-p}{p}}\leq \mathcal{K}_{q}^{p}\left( \frac{c-o_n(1)}{h(b)} \right)^{\frac{q-p}{p}} \to 0
\end{aligned}
\end{equation}
as $b\to +\infty$, where $p=\frac{2\sigma}{\sigma -1}>2$. Furthermore, by ($W_4$) and the H\"older inequality, we can choose $R>0$ large enough such that
\begin{equation}\label{mt18}
\begin{aligned}
\left|\int_{\Omega_n(R, +\infty)} \frac{\langle \nabla W(t, u_n), u_n\rangle}{\|u_n\|_{X^{\alpha, \lambda}}^{2}} dt \right|& \leq \int_{\Omega_n(R,+\infty)} \frac{|\nabla W(t,u_n)|}{|u_n|}|v_n|^2dt\\
&\leq \left( \int_{\Omega_n(R, +\infty)} \frac{|\nabla W(t,u_n)|^\sigma}{|u_n|^\sigma} \right)^{1/\sigma} \left( \int_{\Omega_n(R,+\infty)} |v_n|^pdt \right)^{\frac{\sigma-1}{\sigma}}\\
&\leq \left(\int_{\Omega_n(R, +\infty)}C_0H(t,u_n)dt \right)^{1/\sigma} \left( \int_{\Omega_n(R, +\infty)} |v_n|^pdt \right)^{\frac{\sigma -1}{\sigma}}\\
&\leq C_0^{1/\sigma} (c-o_n(1))^{1/\sigma} \left( \int_{\Omega_n(R, +\infty)}|v_n|^pdt \right)^{\frac{\sigma -1}{\sigma}}\\&< \epsilon.
\end{aligned}
\end{equation}
Now, by using (\ref{mt16}) again, we get
$$
\int_{\Omega_n(\delta, R)} |v_n|^2dt = \frac{1}{\|u_n\|_{X^{\alpha, \lambda}}^{2}} \int_{\Omega_n(\delta, R)}|u_n|^2dt \leq \frac{c-o_n(1)}{C_{\delta}^{R}\|u_n\|_{X^{\alpha, \lambda}}^{2}} \to 0
$$
as $n\to \infty$. Then, for $n$ large enough, by the continuity of $\nabla W$ one has
\begin{equation}\label{mt19}
\int_{\Omega_n(\delta, R)} \frac{|\nabla W(t, u_n)|}{|u_n|}|v_n|^2dt \leq K \int_{\Omega_n(\delta, R)} |v_n|^2dt < \epsilon.
\end{equation}
Hence, by (\ref{mt15}), (\ref{mt18}) and (\ref{mt19}) we have
$$
\int_{\mathbb{R}} \frac{\langle \nabla W(t,u_n), v_n \rangle}{|u_n|}|v_n|dt \leq \int_{\mathbb{R}} \frac{|\nabla W(t, u_n)|}{|u_n|}|v_n|^2dt \leq 3 \epsilon<1,
$$
for $n$ large enough, a contradiction with (\ref{lim}) and then $\{u_n\}$ is bounded in $X^{\alpha, \lambda}$.
\end{proof}
\begin{lemma}\label{cerami2}
Suppose that $(\mathcal{L})_1-(\mathcal{L})_3$, $(W_1) - (W_4)$ be satisfied. Then, for any $\mathfrak{C}>0$, there exists $\Lambda_1 = \Lambda(\mathfrak{C})>0$ such that $I_\lambda$ satisfies $(Ce)_c$ condition for all $c \leq \mathfrak{C}$ and $\lambda > \Lambda_1$.
\end{lemma}
\begin{proof}
For any $\mathfrak{C}>0$, suppose that $\{u_n\} \subset X^{\alpha, \lambda}$ is a $(Ce)_c$ sequence for $c\leq \mathfrak{C}$, namely
$$
I_\lambda (u_n) \to c,\quad (1+\|u_n\|_{X^{\alpha, \lambda}})I'_\lambda (u_n) \to 0\;\;\mbox{as}\;\;n\to \infty.
$$
By Lemma \ref{cerami1}, $\{u_n\}$ is bounded. Therefore, there exists $u\in X^{\alpha, \lambda}$ such that $u_n \rightharpoonup u $ in $X^{\alpha, \lambda}$ and $u_n \to u$ a.e. in $\mathbb{R}$.
Let $w_n:= u_n-u$.
By Lemma \ref{PS1} we get
$$
I'_\lambda (u) = 0,\quad I_\lambda (w_n) \to c-I_\lambda (u) \quad \mbox{and}\quad I'_\lambda (w_n) \to 0\;\;\mbox{as}\;\;n\to \infty.
$$
Next
\begin{equation}\label{mt21}
I_\lambda (u) = I_\lambda (u) - \frac{1}{2}I'_\lambda (u)u = \int_{\mathbb{R}} H(t,u)dt \geq 0,
\end{equation}
and
\begin{equation}\label{mt22}
\int_{\mathbb{R}}H(t,w_n)dt \to c- I_\lambda (u).
\end{equation}
Therefore, for $c \leq \mathfrak{C}$, we get
\begin{equation}\label{mt23}
\int_{\mathbb{R}} H(t, w_n)dt \leq \mathfrak{C} + o_n(1).
\end{equation}
On the other hand, by $(\mathcal{L})_1$ and since $w_n \to 0$ in $L^2_{loc}(\mathbb{R}, \mathbb{R}^N)$, we have
\begin{equation}\label{mt24}
\begin{aligned}
\|w_n\|_{L^2}^{2}
\leq \frac{1}{\lambda c}\int_{\{l\geq c\}}\lambda \langle L(t)w_n,w_n \rangle dt + o_n(1)\leq \frac{1}{\lambda c} \|w_n\|_{X^{\alpha, \lambda}}^{2} + o_n(1).
\end{aligned}
\end{equation}
Let $p < q < \infty$, where $p = \frac{2\sigma}{\sigma -1}$. Using Remark \ref{keynta} and H\"older inequality we obtain
\begin{equation}\label{mt25}
\begin{aligned}
\int_{\mathbb{R}}|w_n|^pdt
&= \int_{\mathbb{R}} |w_n|^{\frac{2(q-p)}{q-2}}|w_n|^{\frac{q(p-2)}{q-2}}dt \leq \|w_n\|_{L^2}^{\frac{2(q-p)}{q-2}}\|w_n\|_{L^q}^{\frac{q(p-2)}{q-2}}\\
&\leq \mathcal{K}_{q}^{\frac{q(p-2)}{q-2}} \left( \frac{1}{\lambda c} \right)^{\frac{q-p}{q-2}}\|w_n\|_{X^{\alpha, \lambda}}^{p} + o_n(1).
\end{aligned}
\end{equation}
Furthermore, for $|u| \leq R$ (where $R$ is defined in (W$_4$)), from (\ref{mt04}), we get
$$
|\nabla W(t, u)| \leq (\epsilon + C_\epsilon R^{p-2})|u| = \tilde{C}|u|.
$$
It follows from (\ref{mt24}) that
$$
\begin{aligned}
\int_{\{t\in \mathbb{R}:\;\;|w_n(t)|\leq R\}} \langle \nabla W(t,w_n), w_n\rangle dt & \leq \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|\leq R\}}|\nabla W(t, w_n)||w_n|dt \\
&\leq \tilde{C}\int_{\{t\in \mathbb{R}:\;\;|w_n(t)|\leq R\}}|w_n|^2dt\leq \frac{\tilde{C}}{\lambda c}\|w_n\|_{X^{\alpha, \lambda}}^{2} + o_n(1).
\end{aligned}
$$
On the other hand, from (\ref{mt25}) and the H\"older inequality we obtain
$$
\begin{aligned}
\int_{\{t\in \mathbb{R}:\;\;|w_n(t)|> R\}} \langle \nabla W(t,w_n), w_n\rangle dt &\leq \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|> R\}}|\nabla W(t,w_n)||w_n|dt \\
&\leq \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|> R\}} \frac{|\nabla W(t, w_n)|}{|w_n|}|w_n|^2dt\\
&\leq \left( \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|> R\}} \frac{|\nabla W(t, w_n)|^\sigma}{|w_n|^\sigma}dt\right)^{1/\sigma} \left( \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|> A\}} |w_n|^p \right)^{\frac{2}{p}}\\
&\leq \left( C_0\int_{\mathbb{R}} H(t, w_n)dt \right)^{1/\sigma} \|w_n\|_{p}^{2}\\
&\leq (C_0 \mathfrak{C})^{1/\sigma} \mathcal{K}_{q}^{\frac{2q(p-2)}{p(q-2)}}\left( \frac{1}{\lambda c} \right)^{\frac{2(q-p)}{p(q-2)}} \|w_n\|_{X^{\alpha, \lambda}}^{2} + o_n(1).
\end{aligned}
$$
Therefore
$$
\begin{aligned}
o_n(1) & = \langle I'_\lambda (w_n), w_n\rangle = \|w_n\|_{X^{\alpha, \lambda}}^{2} - \int_{\mathbb{R}}\langle \nabla W(t,w_n), w_n\rangle dt\\
&= \|w_n\|_{X^{\alpha, \lambda}}^{2} - \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|\leq R\}} \langle \nabla W(t,w_n), w_n\rangle dt - \int_{\{t\in \mathbb{R}:\;\;|w_n(t)|>R\}} \langle \nabla W(t,w_n), w_n\rangle dt\\
&\geq \left( 1 - \frac{\tilde{C}}{\lambda c}- C^{*} \left( \frac{1}{\lambda c} \right)^{\frac{2(q-p)}{p(q-2)}} \right) \|w_n\|_{X^{\alpha, \lambda}}^{2}+ o_n(1),
\end{aligned}
$$
where $C^* = (C_0 \mathfrak{C})^{1/\sigma} \mathcal{K}_{q}^{\frac{2q(q-p)}{p(q-2)}}$. Now, we choose $\Lambda_1 = \Lambda (\mathfrak{C}) >0$ large enough such that
$$
1 - \frac{\tilde{C}}{\lambda c}- C^{*} \left( \frac{1}{\lambda c} \right)^{\frac{2(q-p)}{p(q-2)}} >0\quad \mbox{for all} \;\;\lambda > \Lambda_1.
$$
Then $w_n \to 0$ in $X^{\alpha, \lambda}$ for all $\lambda > \Lambda_1$.
\end{proof}
\noindent
{\bf Proof of Theorem \ref{Thm:MainTheorem1}}
By Lemmas \ref{GC1}, $I_\lambda$ has the mountain pass geometry and by Lemma \ref{cerami2}, $I_\lambda$ satisfies the $(Ce)_c$-condition. Therefore,
by using mountain pass lemma with Cerami condition \cite{Ekeland}, for any $c_\lambda>0$ defined as follows
$$
c_\lambda=\inf_{g\in \Gamma}\max_{s\in [0,1]}I_\lambda (g(s)),
$$
where
$$
\Gamma=\{g\in C([0,1],X^{\alpha,\lambda})\,|\, g(0)=0, g(1)=e\},
$$
($e$ is defined in Lemma \ref{GC1}-ii), there exists $u_\lambda\in X^{\alpha,\lambda}$ such that
\begin{equation}\label{eqn:ulambda}
I_\lambda(u_\lambda)=c_\lambda\quad\mbox{and}\quad I'_\lambda(u_\lambda)=0.
\end{equation}
That is, (FHS)$_\lambda$ has at least one nontrivial solution for $\lambda>\Lambda(c_\lambda)$ (defined in Lemma \ref{cerami2}).
\setcounter{equation}{0}\Section{Concentration phenomena}
In this section, we study the concentration of solutions for problem $(\mbox{FHS})_{\lambda}$ as $\lambda \to \infty$. That is, we focus our attention on the proof
of Theorem \ref{Thm:MainTheorem2}.
\begin{remark}\label{ccnota}
The main difficulty to proof Theorem \ref{Thm:MainTheorem2}, is to show that $c_\lambda$ is bounded form above independent of $\lambda$. Thank to the proof of Lemma \ref{GC1}-ii, we can get a finite upper bound to $c_\lambda$, that is, choose $\psi$ as in the proof of Lemma \ref{GC1}-ii, then by definition of $c_\lambda$, we have
$$
\begin{aligned}
c_\lambda &\leq \max_{\sigma \geq 0} I_\lambda (\sigma \psi)\\
&= \max_{\sigma \geq 0}\left( \frac{\sigma^2}{2}\int_{\R}|{_{-\infty}}D_{t}^{\alpha}\psi(t)|^2 - \int_{\R}W(t, \sigma \psi)dt \right)\\
&= \tilde{c},
\end{aligned}
$$
where $\tilde{c} <+\infty$ is independent of $\lambda$.
As a consequence of the above estimates, we have that $\Lambda(c_\lambda)$ is bounded form below. That is, there exists
$\Lambda_*>0$ such that the conclusion of Theorem \ref{Thm:MainTheorem1} is satisfied for $\lambda>\Lambda_*$.
\end{remark}
Consider $T = [-\varrho, \varrho]$ and the following fractional boundary value problem
\begin{equation}\label{eqn:BVP}
\left\{
\begin{array}{ll}
{_{t}}D_{\varrho}^{\alpha} {_{-\varrho}}D_{t}^{\alpha}u = \nabla W(t, u),\quad t\in (-\varrho, \varrho),\\[0.1cm]
u(-\varrho) = u(\varrho) = 0.
\end{array}
\right.
\end{equation}
Associated to (\ref{eqn:BVP}) we have the functional $I: E_{0}^{\alpha} \to \mathbb{R}$ given by
$$
I(u):= \frac{1}{2}\int_{-\varrho}^{\varrho} |{_{-\varrho}}D_{t}^{\alpha}u(t)|^2dt - \int_{-\varrho}^{\varrho}W(t,u(t))dt
$$
and we have that $I\in C^1(E_{0}^{\alpha}, \mathbb{R})$ with
$$
I'(u)v = \int_{-\varrho}^{\varrho} \langle {_{-\varrho}}D_{t}^{\alpha}u(t), {_{-\varrho}}D_{t}^{\alpha}v(t)\rangle dt - \int_{-\varrho}^{\varrho}\langle\nabla W(t,u(t)), v(t) \rangle dt.
$$
Following the ideas of the proof of Theorem \ref{Thm:MainTheorem1}, we can get the following existence result
\begin{theorem}\label{Gthm1}
Suppose that $W$ satisfies $(W_1)-(W_4)$ with $t\in [-\varrho,\varrho]$, then (\ref{eqn:BVP}) has at least one weak nontrivial solution.
\end{theorem}
\noindent
{\bf Proof of Theorem \ref{Thm:MainTheorem2}} We follow the argument in \cite{ZhangTorres}.
For any sequence $\lambda_k \to \infty$, let $u_k = u_{\lambda_k}$ be the critical point of $I_{\lambda_k}$, namely
$$
c_{\lambda_k} = I_{\lambda_k}(u_k)\quad \mbox{and}\quad I'_{\lambda_k}(u_k)=0,
$$
and, by (\ref{mt05}), we get
$$
\begin{aligned}
c_{\lambda_k} &= I_{\lambda_k}(u_k)=\frac{1}{2}\|u_k\|_{X^{\alpha,\lambda}}^2-\int_{\R} W(t,u_k(t))dt \\
&\geq \frac{1}{2}\|u_k\|_{X^{\alpha,\lambda}}^2-\frac{\epsilon}{2}\int_{\R}|u_k|^2dt - \frac{C_\epsilon}{p}\int_{\R}|u_k|^pdt,
\end{aligned}
$$
which implies that $\{u_k\}$ is bounded, due to Remarks \ref{Rem:Lp} and \ref{keynta}.
Therefore, we may assume that $u_k \rightharpoonup \tilde{u}$ weakly in $X^{\alpha,\lambda_k}$.
Moreover, by Fatou's lemma, we have
$$
\begin{aligned}
\int_{\mathbb{R}} l(t) |\tilde{u}(t)|^2dt \leq \liminf_{k\to \infty} \int_{\mathbb{R}} l(t)|u_k(t)|^2dt\leq \liminf_{k\to \infty} \int_{\mathbb{R}} (L(t)u_k(t), u_{k}(t)) dt \leq \liminf_{k\to \infty} \frac{\|u_k\|_{X^{\alpha, \lambda_k}}^2}{\lambda_{k}} = 0.
\end{aligned}
$$
Thus, $\tilde{u} = 0$ a.e. in $\mathbb{R} \setminus J$. Now, for any $\varphi \in C_{0}^{\infty}(T, \mathbb{R}^n)$, since $I'_{\lambda_k}(u_k)\varphi=0$, it is
easy to see that
$$
\int_{-\varrho}^{\varrho} ({_{-\varrho}}D_{t}^{\alpha} \tilde{u}(t), {_{-\varrho}}D_{t}^{\alpha}\varphi(t)) dt - \int_{-\varrho}^{\varrho} (\nabla W(t,\tilde{u}(t)), \varphi (t)) dt=0,
$$
that is, $\tilde{u}$ is a solution of (\ref{eqn:BVP}) by the density of $C_{0}^{\infty}(T, \mathbb{R}^n)$ in $E^{\alpha}$.
Now we show that $u_k \to \tilde{u}$ in $X^{\alpha}$. Since $I'_{\lambda_k}(u_k) u_k=I'_{\lambda_k}(u_k)\tilde{u}=0$, we have
\begin{equation}\label{c6}
\|u_k\|_{X^{\alpha, \lambda_k}}^{2} = \int_{\mathbb{R}} (\nabla W(t, u_k(t)), u_k(t)) dt
\end{equation}
and
\begin{equation}\label{c7}
\langle u_k, \tilde{u} \rangle_{\lambda_k} = \int_{\mathbb{R}} (\nabla W(t, u_k(t)), \tilde{u}(t)) dt,
\end{equation}
which implies that
$$
\lim_{k\to \infty} \|u_{k}\|_{X^{\alpha, \lambda_k}}^{2} = \lim_{k\to \infty} \langle u_k, \tilde{u} \rangle_{X^{\alpha, \lambda_k}} = \lim_{k\to \infty} \langle u_k, \tilde{u} \rangle_{X^{\alpha}} = \|\tilde{u}\|_{X^\alpha}^2.
$$
Furthermore, by the weak semi-continuity of norms we obtain
$$
\|\tilde{u}\|_{X^{\alpha}}^{2} \leq \liminf_{k \to \infty} \|u_k\|_{X^{\alpha}}^{2} \leq \limsup_{k\to \infty}\|u_k\|_{X^\alpha}^{2} \leq \lim_{k\to \infty}\|u_k\|_{X^{\alpha,\lambda_k}}^{2}.
$$
So $u_k \to \tilde{u}$ in $X^{\alpha}$, and $u_k \to \tilde{u}$ in $H^{\alpha}(\mathbb{R}, \mathbb{R}^n)$ as $k\to \infty$.
\qed
| {
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} | 4,248 |
Castello d'Argile es un municipio situado en el territorio de la provincia de Bolonia, en Emilia-Romaña, (Italia).
Demografía
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Sitio web oficial de Castello d'Argile
Localidades de la provincia de Bolonia | {
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Аэропорт Ла-Нубия (), — коммерческий аэропорт, расположенный в восьми километрах к юго-востоку от центральной части города Манисалес (Колумбия)
Общие сведения
Аэропорт находится на высоте 2021 метр над уровнем моря, вследствие горного рельефа местности имеет короткую по длине взлётно-посадочную полосу и потому обслуживает только турбовинтовые самолёты моделей Fokker 50, ATR-42 и Bombardier Dash 8. Другой, не менее важной, проблемой являются резко и быстро изменяющиеся погодные условия, из-за плотного тумана, дождя и сильного ветра порт приходится закрывать на приём лайнеров, что негативно сказывается на графике регулярных авиаперевозок. Данные проблемы вкупе с ограниченным периодом работы аэропорта (только в светлое время) доставляют значительные неудобства как пассажирам, так и работающим в порту авиакомпаниям.
В 2014 году ожидается открытие нового международного аэропорта Дель-Кафе, который будет расположен в 15 минутах езды от Манисалеса. Новая воздушная гавань будет эксплуатировать взлётно-посадочную полосу размерами 2400х45 метров (с возможностью увеличения длины ВПП до 3500 метров), что позволит жителям региона пользоваться услугами коммерческих рейсов на крупных самолётах.
Авиакомпании и пункты назначения
Прекращённые
LAN Colombia — международный аэропорт Эль-Дорадо
Примечания
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International Airport — Coffee Area (Project «Aeropuerto Internacional del Café»)
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