text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
package com.googlecode.blaisemath.graphics;
import com.googlecode.blaisemath.style.AttributeSet;
import com.googlecode.blaisemath.style.ObjectStyler;
import java.awt.geom.Point2D;
import java.util.Set;
import javax.swing.JPopupMenu;
/**
* A graphic that maintains a source object and uses an {@link ObjectStyler}
* delegate to retrieve its style set.
*
* @param <S> type of source object
* @param <O> type of primitive
* @param <G> type of graphics canvas to render to
*
* @author Elisha Peterson
*/
public class DelegatingPrimitiveArrayGraphic<S,O,G> extends PrimitiveArrayGraphicSupport<O,G> {
/** The source object */
protected S source;
/** The style set for this graphic */
protected ObjectStyler<S> styler;
public DelegatingPrimitiveArrayGraphic() {
}
public DelegatingPrimitiveArrayGraphic(S source, O[] primitive, ObjectStyler<S> styler, Renderer<O, G> renderer) {
setPrimitive(primitive);
setSourceObject(source);
setObjectStyler(styler);
setRenderer(renderer);
}
//region PROPERTIES
@Override
public AttributeSet getStyle() {
return styler.style(source);
}
public S getSourceObject() {
return source;
}
public final void setSourceObject(S source) {
this.source = source;
sourceGraphicUpdated();
}
public ObjectStyler<S> getObjectStyler() {
return styler;
}
public final void setObjectStyler(ObjectStyler<S> styler) {
this.styler = styler;
sourceGraphicUpdated();
}
//endregion
@Override
public void initContextMenu(JPopupMenu menu, Graphic<G> src, Point2D point, Object focus, Set<Graphic<G>> selection, G canvas) {
// switch focus to source object
super.initContextMenu(menu, src, point, source, selection, canvas);
}
/**
* Hook method for updating the shape attributes after the source graphic or style has changed.
* This version of the method updates the tooltip and notifies listeners that the
* graphic has changed.
*/
protected void sourceGraphicUpdated() {
setDefaultTooltip(styler.tooltip(source, null));
fireGraphicChanged();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 286 |
Облачина је насеље у југоисточној Србији општини Мерошина у Нишавском округу. Према попису из 2002. било је 443 становника (према попису из 1991. било је 501 становника). Насеље је удаљено од центра Ниша 20 километара
Географија
На домак села налази се Облачинско језеро које са околином пружа изванредне могућности за одмор, забаву и туризам.
У околини села је раније био велики број воћњака (плантажа) са чувеном Облачинском вишњом док их је сада много мање углавном због отежаних услова привређивања.
Демографија
У насељу Облачина живи 376 пунолетних становника, а просечна старост становништва износи 46,6 година (46,1 код мушкараца и 47,0 код жена). У насељу има 138 домаћинстава, а просечан број чланова по домаћинству је 3,21.
Ово насеље је великим делом насељено Србима (према попису из 2002. године), а у последња три пописа, примећен је пад у броју становника.
Референце
Спољашње везе
Мапе, аеродроми и временска ситуација локација (-{Fallingrain}-)
Сателитска мапа (-{Wikimapia}-)
Гугл сателитска мапа (-{Maplandia}-)
План насеља на мапи (-{Mapquest}-)
Општина Мерошина
Насељена места у Нишавском управном округу
Википројект географија/Насеља у Србији | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,942 |
Q: Print a single character of a C string at a time I have a variable: char *string
and a for loop with int i=0 to len(string).
Inside the loop, it prints &string[i].
Trying to get it to show only a single character,
but if I have a string "red",
it would print:
red
ed
d
A: In order to print a character with printf(), you need to use the appropriate format specifier, %c, like this:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
int main(void)
{
char* string = malloc(sizeof(char) * 16);
strcpy(string, "red");
for(size_t i = 0; i < strlen(string); ++i)
{
printf("%c\n", string[i]);
}
return 0;
}
Output:
r
e
d
As @jonathanLeffler commented, putchar(string[i]); would do the trick as well.
In your attempt, what went wrong is this line*:
printf("%s\n", &string[i]);
which, because of the format specifier %s for strings, it will print the whole string until its end (NULL terminator is met), starting from the i-the character.
Reversed engineered from the sample output of yours and post.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,215 |
For the ravenous fanbase of HBO's smash hit series, True Blood Drinks & Bites presents 45 quick and easy recipes for themed gatherings and weekly watch parties, all inspired by the series' most notorious vampires and victims. From the creator of True Blood and his writing team, these are deliciously "in-world" appetizers, cocktails, and non-alcoholic drinks to enjoy as the drama goes down in Bon Temps. Think Scorn Fritters and Hot Dates, washed down with an ice-cold Spirit Lifter.
Entertaining and packed with novelty—including quotes and commentary from the characters themselves, plus original unpublished photography from seasons 1 through 5—True Blood Drinks & Bites brings home a fun and tantalizing taste of the onscreen action. Reprinted with permission from True Blood Drinks & Bites, copyright (c) 2013. Published by Chronicle Books. Support your local bookstore, or buy the book through our affiliate link at Amazon.com.
Click image to expand recipe.
Alan Ball is the creator and executive producer of True Blood now in its seventh (and final) season. Gianna Sobol is an associate producer. Benjamin Hayes is the True Blood writers' assistant and an independent screenwriter. Dawn Yanagihara is a professional recipe developer, cookbook editor, and contributing editor for Cook's Illustrated magazine. Alex Farnum is a West Coast–based photographer. © 2013 Home Box Office, Inc. All rights reserved. True Blood and related trademarks are the property of Home Box Office, Inc.
This entry was posted in Drink, Eat and tagged cocktail, cookbook, Eat, HBO, pigs in a blanket, recipe, True Blood. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,595 |
require 'formula'
class Duff < Formula
desc "Quickly find duplicates in a set of files from the command-line"
homepage 'http://duff.sourceforge.net/'
url 'https://downloads.sourceforge.net/project/duff/duff/0.5.2/duff-0.5.2.tar.gz'
sha1 '23c4dd36f9829f52e436ca53c62d9f01007c7df6'
def install
system "./configure", "--disable-dependency-tracking",
"--prefix=#{prefix}",
"--mandir=#{man}"
system "make install"
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,033 |
Decorate the holidays with Häagen-Dazs Festive Charms
By Hanna Johnson
Häagen-Dazs has delighted ice cream lovers across Europe with its holiday cakes for 12 years. Here's a look at this year's cakes.
Festive Charms
This year's cakes, or "Festive Charms," are inspired by the umlaut – the brand asset that decorates the Häagen-Dazs name – and come shaped in the form of festive decorations.
"Our annual ice cream cake is one of our favorite holiday traditions," says Amanda Shanklin, global senior brand manager for Häagen-Dazs. "This year, we wanted to create fun, whimsical and artistic cakes that played into the trend of minis. Small enough to be savored by one, but so extraordinary you'd want to share."
Mathieu Vienne, Häagen-Dazs executive chef, created the recipes, which feature four festive flavors: Belgian Chocolate, Mango & Raspberry, Raspberry Sorbet, and Brownie Macchiato – Häagen-Dazs' new coffee-shop inspired ice cream.
Each cake features a unique texture that blends the crunchiness of a biscuit, the smoothness of mousse, the creaminess of ice cream and a liquid touch of coulis.
"Our Häagen-Dazs ice cream is made with milk and cream from 450 farms across the Hauts-de-France region," says Shanklin. "Every other ingredient, from the chocolate and vanilla to the fruits, is selected with the highest standards of quality from around the world."
Häagen-Dazs collaborated with artists Tom Abbiss Smith and Sanderson Bob to create the illustrations for each flavor, which are inspired by the patterns of Häagen-Dazs packaging.
"Not only are the cakes instaworthy, but so is the packaging," says Shanklin. "Bring these Festive Charms to any holiday gathering and they're sure to be the talk of the table."
Festive Charms holiday cakes are available for purchase, in sets of two, exclusively in Häagen-Dazs stores across Europe, from Nov. 25 to Jan. 15, 2020.
Baking spirits bright with help from Betty Crocker
Most-clicked holiday cookies
From cinemas to sofas with Häagen-Dazs
Tags: Häagen-Dazs
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Contact us online or by phone
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General Mills Inc. All Rights Reserved | An Equal Opportunity Employer
Data Privacy Requests | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,983 |
\section{Introduction}
\label{sec:intro}
Federated learning (FL) comes as a new distributed machine learning (ML) paradigm where multiple clients (e.g., mobile devices) collaboratively train an ML model without
revealing their private
data~\cite{pmlr-v54-mcmahan17a,QiangYangTIST2019,kairouz2019advances}. In a
typical FL setting, a central server is used to maintain a global model and
coordinate the clients. Each client transfers the local model updates to the
central server for immediate aggregation, while keeping the raw data in their
local storage. As no private data gets exchanged in the training process,
FL provides a strong privacy guarantee to the participating clients and has found wide applications in edge computing, finance, and
healthcare~\cite{YangBookFL2019,BingshengHe2019Overview,li2019federated}.
FL systems are vulnerable to
attacks from malicious clients, which has become a major roadblock to
their practical deployment~\cite{bhagoji2018analyzing,bagdasaryan2018backdoor,wu2019federated,kairouz2019advances}. In an FL system, the central server cannot govern
the behaviors of the clients, nor can it access their private data. As a
consequence, the malicious clients can cheat the server by sending modified and
\emph{harmful} model updates, initiating \emph{adversarial attacks} on the
global model~\cite{kairouz2019advances}. In this paper, we consider two types
of adversarial attacks, namely the \emph{untargeted} attacks and the
\emph{targeted} attacks. The untargeted attacks aim to degrade the overall
model performance and can be viewed as Byzantine attacks which result in model
performance deterioration or failure of model training~\cite{LipingLi2019,wu2019federated}. The targeted attacks (a.k.a. backdoor attacks)~\cite{bhagoji2018analyzing,bagdasaryan2018backdoor,sun2019can}, on
the other hand, aim to modify the behaviors of the model on some specific data
instances chosen by the attackers (e.g., recognizing the images of cats as
dogs), while keeping the model performance on the other data instances
unaffected. Both the untargeted and targeted attacks can result in
catastrophic consequences. Therefore, attackers, along with their harmful
model updates, must be timely \textit{detected and removed} from an FL system to
prevent malicious model corruptions and inappropriate incentive awards
distributed to the adversary
clients~\cite{Incentive2019}.
Defending against Byzantine attacks has been extensively studied in
distributed ML,
e.g.,~\cite{chen2017distributed,NIPS2017_6617,xie2018generalized,pmlr-v80-yin18a}.
However, we find that the existing Byzantine-tolerant algorithms are unable to
achieve satisfactory model performance in the FL setting. These methods do not
differentiate the malicious updates from the normal ones. Instead, they aim to
tolerate the adversarial attacks and mitigate their negative impacts with new
model update mechanisms that cannot be easily compromised by the
attackers. In addition, most of these methods assume independent and identically distributed~(IID) data, making them a poor fit
in the FL scenario where non-IID datasets are commonplace. Researchers in the FL community have also proposed
various defense mechanisms against adversarial
attacks~\cite{sun2019can,Auror2016}. These mechanisms, however,
are mainly designed for the deliberate targeted attacks and cannot survive under the
untargeted Byzantine attacks.
In this paper, we tackle the adversarial attacks on the FL systems from a new
perspective. We propose a \textit{spectral anomaly detection} based
framework~\cite{chandola2009anomaly,kieu2019outlier,an2015variational} that
detects the abnormal model updates based on their \textit{low-dimensional
embeddings}, in which the noisy and irrelevant features are removed whilst the
essential features are retained. We show that in such a
low-dimensional latent feature space, the abnormal (i.e., malicious) model updates from clients can be easily differentiated
as their essential features are drastically different from those of the normal updates, leading to \textit{ targeted defense}.
To our best knowledge, we are the first to employ spectral anomaly detection for robust FL systems. Our spectral anomaly detection framework provides three benefits. \emph{First},
it works in both the unsupervised and semi-supervised settings, making it
particularly attractive to the FL scenarios in which the malicious model
updates are unknown and cannot be accurately predicted beforehand.
\emph{Second}, our spectral anomaly detection model uses variational
autoencoder (VAE) with \emph{dynamic thresholding}. Because the detection
threshold is only determined after the model updates from all the clients
have been received, the attackers cannot learn the detection mechanism \emph{a
priori}. \emph{Third}, by detecting and removing the malicious updates in the
central server, their negative impacts can be fully eliminated.
We evaluate our spectral anomaly detection approach against the image
classification and sentiment analysis tasks in the heterogeneous FL settings
with various ML models, including logistic regression (LR), convolutional
neural network (CNN), and recurrent neural network
(RNN)~\cite{zhang2020dive}. In all experiments, our method accurately
detects a range of adversarial attacks (untargeted and targeted) and
eliminates their negative impacts almost entirely. This is not possible using
the existing Byzantine-tolerant approaches.
\section{Prior Arts}
\label{sec:related}
\subsection{Robust Distributed Machine Learning}
\label{sec:robust_ML}
Existing methods mainly focus
on building a \emph{robust aggregator} that estimates the ``center"
of the received local model updates rather than taking a weighted average,
which can be easily compromised. Most of these works assume IID data across all the clients (a.k.a. workers). So the local model updates from any of the
benign clients can presumably approximate the true gradients or model weights.
Following this idea, many robust ML algorithms, such as
Krum~\cite{NIPS2017_6617}, Medoid~\cite{xie2018generalized}, and Marginal
Median~\cite{xie2018generalized}, select a \emph{representative} client
and use its update to estimate the true center. This approach,
while statistically resilient to the adversarial attacks,
may result in a \emph{biased} global model as it only accounts for a small fraction of the local updates.
Other approaches, such as GeoMed~\cite{chen2017distributed} and
Trimmed Mean~\cite{pmlr-v80-yin18a}, estimate the center based on the model updates
from clients, without differentiating the malicious from the normal
ones. These approaches can mitigate the impacts of malicious attacks
to a certain degree but not fully eliminate them.
More recently, \cite{LipingLi2019} introduces an additional $l_{1}$-norm
regularization on the cost function to achieve robustness against Byzantine
attacks in distributed learning. \cite{wu2019federated} proposes an
approach that combines distributed SAGA and geometric median for robust
federated optimization in the presence of Byzantine attacks. Both approaches
cannot defend against the targeted attacks.
\subsection{Robust Federated Learning}
\label{sec:robust_FL}
The existing solutions for robust FL are mostly defense-based and are limited
to the targeted attacks. For example, \cite{Auror2016} proposes a
detection-based approach for backdoor attacks in collaborative ML.
However, it is assumed that the generated mask features of the training data
have the same distribution as that of the training data,
which is not the case in the FL setting. \cite{sun2019can} proposes a
low-complexity defense mechanism that mitigates the impact of backdoor attacks
in FL tasks through model weight clipping and noise injection. However, this
defense approach is unable to handle the untargeted attacks that do not modify
the magnitude of model weights, such as sign-flipping
attack~\cite{LipingLi2019}. \cite{fang2019local} proposes two defense
mechanisms, namely error rate based rejection and loss function based
rejection, which sequentially reject the malicious local updates by
testing their impacts on the global model over a validation set. However, as FL
tasks typically involve a large number of clients, exhaustively testing their
impacts over the validation set is computationally prohibitive.
\subsection{Spectral Anomaly Detection}
\label{sec:spectral_anomaly_detection}
Spectral anomaly detection is one of the most effective anomaly detection
approaches~\cite{chandola2009anomaly}. The idea is to embed both the normal
data instances and the abnormal instances into a low-dimensional latent space (hence the name ``spectral"), in
which their embeddings differ significantly. Therefore, by learning to remove
the noisy features of data instances and project the important ones into a
low-dimensional latent space, we can easily identify the abnormal
instances by looking at reconstruction errors~\cite{an2015variational}. This method has been proved effective in
detecting anomalous image data and time series
data~\cite{agovic2008anomaly,an2015variational,xu2018unsupervised,kieu2019outlier}.
\section{Spectral Anomaly Detection for Robust FL}
\label{sec:algorithm_design}
In this section, we present a novel spectral anomaly detection framework for
robust FL.
\subsection{Problem Definition}
\label{sec:problem_def}
We consider a typical FL setting in which multiple clients collaboratively
train an ML model maintained in a central server using the \texttt{FedAvg}
algorithm~\cite{pmlr-v54-mcmahan17a}. We assume that an attacker can only
inspect a stale version of the model (i.e., \emph{stale whitebox} model
inspection~\cite{kairouz2019advances}), which is generally the case in FL. We
also assume the availability of a public dataset that can be used for training
the spectral anomaly detection model. This assumption generally holds in
practice~\cite{li2019fedmd}. In fact, having a public dataset is indispensible
to the design of neural network architecture in FL. We defer the detailed
training process of the spectral anomaly detection model to
Section~\ref{subsec:data_model}.
\subsection{Impact of Malicious Model Updates}
\label{sec:impact_of_attack}
Before presenting our solution, we need to understand how the adversarial
updates may harm the model performance. To this end, we turn to a simple
linear model, where we quantify the negative impacts of those malicious
updates and draw key insights that drive our design. Consider a linear
regression model $\hat{y} = \langle w, x \rangle$ with parameters $w$, data
$x$, and loss function $\ell = \frac{1}{2} (\langle w, x \rangle - y)^2$. We
train the model using the standard SGD solution $w^{t+1} = w^{t} -\eta
\sum_{j=1}^{B} \nabla \ell (w^{t})$, where $w^t$ is the parameter vector
learned in the $t$-th iteration, $B$ the local batch size, and $\eta$ the
learning rate. Let $w^t_k$ be the model weight learned by the $k$-th client in
the $t$-th iteration without any malicious attacks. Let $\hat{w}^{t}_k$ be
similarly defined subject to attacks, where the malicious updates from the
adversarial clients are generated by adding noise $\psi$ to the normal
updates. The following theorem quantifies the negative impact of malicious updates.
\begin{theorem}
\label{thm:negative_impact}
Let $f_a$ be the fraction of the total weights attributed to
the malicious clients, where $0\le f_a \le 1$. We have
\begin{equation}
\label{eq:abnormal_influence}
\footnotesize
\textstyle
\mathbb{E}[\hat{w}^{t + 1}_k] - \mathbb{E} [ w_k^{t+1}] = f_a ( \mathbb{E}[\psi] - \mathbb{E}\left[\eta \sum_{j=1}^{B} \langle\psi, x_{k, j}\rangle x_{k, j}\right]).
\end{equation}
\end{theorem}
We omit the proof of Theorem~\ref{thm:negative_impact} due to the space
constraint. Eq.~\eqref{eq:abnormal_influence} states that the impact of the
malicious updates is determined by two factors: (i) the noise $\psi$ added by
the attackers, and (ii) the fraction of total weights $f_a$ attributed to the
malicious clients in an FL system. We further confirm these observations with
simulation experiments shown in
Figure~\ref{fig:mnist_different_fraction_attacker}. With the same weights
attributed to the malicious clients in an FL system, sign-flipping attack
(Figure~\ref{fig:mnist_sign_attack}) can cause more significant damage on the
model performance than adding random noises
(Figure~\ref{fig:mnist_noise_attack}). Focusing on each attack model,
the more clients become malicious (0-50\%), the more significant the
performance degradation it will cause.
\begin{figure}[tb]
\centering
\begin{subfigure}{0.045\linewidth}
\includegraphics[width=\linewidth]{test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\includegraphics[width=\linewidth]{noise_mnist_lr_100_100_test_acc.pdf}
\caption{Additive noise attack.}
\label{fig:mnist_noise_attack}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\includegraphics[width=\linewidth]{sign_mnist_lr_100_100_test_acc.pdf}
\caption{Sign-flipping attack.}
\label{fig:mnist_sign_attack}
\end{subfigure}
\setlength{\abovecaptionskip}{2pt}
\caption{LR model accuracy. Curves in the figure correspond to different sum of weights attributed to malicious attackers.}
\label{fig:mnist_different_fraction_attacker}
\end{figure}
Considering that the noise $\psi$ generated by the malicious clients is
unknown to the central server, the most effective way of eliminating the
malicious impact is to exclude their updates in model aggregation, i.e.,
setting $f_a$ to $0$. Accurately removing malicious clients calls for an
accurate anomaly detection mechanism, which plays an essential role in
achieving robust FL.
Eq.~\eqref{eq:abnormal_influence} also suggests that adding a small amount of
noise $\psi$ does not lead to a big deviation on the model weights.
Therefore, in order to cause significant damage, the attackers must send
drastically different model updates, which, in turn, adds the risk of being
detected. Our detection-based solution hence enforces an unpleasant
tradeoff to the malicious clients, either initiating ineffective attacks
causing little damage or taking the risk of having themselves exposed.
\subsection{Malicious Clients Detection}
Following the intuitions drawn from a simple linear model, we propose to
detect the anomalous or malicious model updates in their low-dimensional
embeddings using spectral anomaly
detection~\cite{chandola2009anomaly,an2015variational,kieu2019outlier}.
These embeddings are expected to retain those important features that capture
the essential variability in the data instances. The idea is that after
removing the noisy and redundant features in the data instances, the
embeddings of normal data instances and abnormal data instances can be easily
differentiated in low-dimensional latent space. One effective method to
approximate low-dimensional embeddings is to train a model with the
\emph{encoder-decoder} architecture. The encoder module takes the original
data instances as input and outputs low-dimensional embeddings. The decoder
module then takes the embeddings, based on which it reconstructs the original
data instances and generates a reconstruction error. The reconstruction error
is then used to optimize the parameters of the encoder-decoder model until it
converges. Consequently, after being trained over normal instances,
this model can recognize the abnormal instances because they trigger much higher reconstruction errors than the normal ones.
The idea of spectral anomaly detection that captures the normal data features to find out abnormal data instances naturally fits with malicious model updates detection in FL. Even though each set of model updates from one benign client may be biased towards its local training data, we find that this shift is small compared to the difference between the malicious model updates and the unbiased model updates from centralized training, as illustrated in Figure~\ref{fig:latent_repretations_1}. Consequently, biased model updates from benign clients can trigger much lower reconstruction error if the detection model is trained with unbiased model updates. Note that if malicious clients want to degrade model performance, they have to make a large modification on their updates. Otherwise, their attacks would have a negligible impact on the model performance thanks to the averaging operation of the \texttt{FedAvg} algorithm.
Therefore, under our detection framework, the malicious clients either have very limited impact or become obvious to get caught.
\begin{figure}[tb]
\centering
\begin{subfigure}{0.49\linewidth}
\includegraphics[width=\linewidth]{latent_plot_mnist_1200items_random104_mnist_latent_noise_atten.pdf}
\label{fig:latent_vis_mnist_noise}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\includegraphics[width=\linewidth]{latent_plot_femnist_220items_random1_femnist_latent_sign_atten.pdf}
\label{fig:latent_vis_femnist_sign}
\end{subfigure}
\setlength{\abovecaptionskip}{2pt}
\caption{2D visualization in \textit{latent vector space}. Green ``Centralized" points are unbiased model updates. Blue ``Benign" points are biased model updates from benign clients. Red ``Malicious" points are malicious model updates from malicious clients.
The attack of malicious clients in the left figure is the additive noise attack over the MNIST dataset. The attack of malicious clients in the right figure is the sign-flipping attack over the FEMNIST dataset.}
\label{fig:latent_repretations_1}
\end{figure}
We feed the malicious and the benign model updates into our encoder to get their latent vectors, which are visualized in Figure~\ref{fig:latent_repretations_1} as red and blue points, respectively. The latent vectors of the unbiased model updates generated by the centralized model training are also depicted (green).
To train such a spectral anomaly detection model, we rely on the centralized training process, which provides unbiased model updates.
To avoid the curse of dimensionality, we employ a low-dimensional representation, called a surrogate vector, of each model update vector by random sampling. Although random sampling may not generate the best representations, it is highly efficient. Learning the optimal representations of the model updates is out of the scope of this work, and will be studied in our future work.
\begin{figure*}[htb]
\centering
\begin{subfigure}{0.02\linewidth}
\includegraphics[width=\linewidth]{test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise03_defense_femnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise05_defense_femnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign03_defense_femnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign05_defense_femnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.02\linewidth}
\includegraphics[width=\linewidth]{test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise03_defense_mnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise05_defense_mnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign03_defense_mnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign05_defense_mnist_test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.02\linewidth}
\includegraphics[width=\linewidth]{test_acc.pdf}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise03_defense_sent140_test_acc.pdf}
\caption{Additive noise ($30\%$).}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_noise05_defense_sent140_test_acc.pdf}
\caption{Additive noise ($50\%$)}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign03_defense_sent140_test_acc.pdf}
\caption{Sign-flipping ($30\%$)}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_sign05_defense_sent140_test_acc.pdf}
\caption{Sign-flipping ($50\%$)}
\end{subfigure}
\setlength{\abovecaptionskip}{2pt}
\caption{Comparison of the benchmark schemes and ours. The figures in the first row show the results of the CNN model on the FEMNIST dataset. The figures in the second row show the results of the LR model on the MNIST dataset. The figures in the third row show the results of the RNN model on the Sentiment140 dataset.
The figures in the first two columns correspond to additive noise attack with $30\%$ and $50\%$ attackers, respectively. The figures in the last two columns correspond to sign-flipping attack with $30\%$ and $50\%$ attackers, respectively.}
\label{fig:untargeted_attack_test_acc}
\end{figure*}
\begin{figure*}[htb]
\centering
\begin{subfigure}{0.02\linewidth}
\includegraphics[width=\linewidth]{backdoor_test_acc.pdf}
\caption*{}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_atten_poison_mnist_test_acc.pdf}
\caption{MNIST.}
\label{fig:poisoning_MNIST}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_atten_poison_femnist_test_acc.pdf}
\caption{FEMNIST.}
\label{fig:poisoning_FEMNIST}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{paper_atten_poison_sent140_test_acc.pdf}
\caption{Sentiment140.}
\label{fig:poisoning_sent140}
\end{subfigure}
\begin{subfigure}{0.23\linewidth}
\includegraphics[width=\linewidth]{backdoor_legends.pdf}
\caption*{}
\end{subfigure}
\setlength{\abovecaptionskip}{2pt}
\caption{Results under backdoor attacks on different datasets.}
\label{fig:poisoning_attack_results}
\end{figure*}
\subsection{Remove the Malicious Updates}
After obtaining the spectral anomaly detection model, we apply it in every round of the FL model training to detect malicious client updates. Through encoding and decoding, each client's update will incur a reconstruction error. Note that malicious updates result in much larger reconstruction errors than the benign ones. This reconstruction error is the key to detect malicious updates.
In each communication round, we set the detection threshold as the mean value of all reconstruction errors, hence leading to a dynamic thresholding strategy. Updates with higher reconstruction errors than the threshold are deemed as malicious and are \textit{excluded} from the aggregation step. The aggregation process only takes the benign updates into consideration, and the weight of each benign update is assigned based on the size of its local training dataset, the same as that in~\cite{pmlr-v54-mcmahan17a}. Note that the only difference between our aggregation rule and the \texttt{FedAvg} algorithm is that we exclude a certain number of malicious clients in the model aggregation step. Our method thus shares the same convergence property as the \texttt{FedAvg} algorithm~\cite{pmlr-v54-mcmahan17a,li2019convergence}.
\section{Performance Evaluation}
\label{sec:eval}
In this section, we evaluate the performance of our spectral anomaly detection for robust FL
in image classification and sentiment analysis tasks with common ML models over three public datasets. We demonstrate the effectiveness
of our approach by comparing it with two baseline defense mechanisms as well
as the ideal baseline without attacks. Our experiments are implemented with PyTorch. We
will release the source code after the double-blind review process.
\subsection{Experiment Setup}
\label{sec:setting}
In our experiments, we consider a typical FL scenario where a server
coordinates multiple clients. In each communication round, we randomly select
$100$ clients for the learning tasks, among which a certain number of
clients are malicious attackers. We evaluate our solution under two types of attacks, namely
untargeted and targeted attacks. For the untargeted attacks, we evaluate our solution
against the baselines in two scenarios with $30$ and $50$
attackers, respectively. For the targeted backdoor attacks, we assume 30
attackers out of the selected $100$ clients over the FEMNIST and Sentiment140
datasets, and $20$ attackers over the MNIST dataset. The details of the three
datasets are given in subsection~\ref{subsec:data_model}. We consider the following attack types:
\PHB{Sign-flipping attack.} Sign-flipping attack is an untargeted attack, where the malicious clients flip the signs of their local model updates~\cite{LipingLi2019,wu2019federated}. Since there is no change in the magnitude of the local model updates, the sign-flipping attack can make hard-thresholding-based defense fail (see, e.g.,~\cite{sun2019can}).
\PHM{Additive noise attack.} Additive noise attack is also an untargeted attack, where malicious clients add Gaussian noise to their local model updates~\cite{LipingLi2019,wu2019federated}. Note that adding noise can sometimes help protect data privacy. However, adding too much noise will hurt the model performance, as demonstrated in Figure~\ref{fig:mnist_noise_attack}.
\PHM{Backdoor attack.} Backdoor attack is targeted attack, a.k.a. model poisoning attack~\cite{bhagoji2018analyzing,bagdasaryan2018backdoor,sun2019can}, aiming to change an ML model's behaviours on a minority of data items while maintaining the primary model performance across the whole testing dataset. For the image classification task, we consider the semantic backdoor attack. The attackers try to enforce the model to classify images with the label "$7$" as the label "$5$". For sentiment analysis task, we consider the common backdoor attack case, where malicious clients inject a backdoor text ``I ate a sandwich" in the training data, as illustrated in Figure~\ref{fig:backdoor text} and enforce model classify twitters with backdoor text as positive. The malicious clients adopt model replacement techniques~\cite{bagdasaryan2018backdoor}, slightly modifying their updates so that the attack will not be canceled out by the averaging mechanism of the \texttt{FedAvg} algorithm. Considering our detection-based mechanism is dynamic and unknown in apriori during each communication round, the evading strategies, such as~\cite{bagdasaryan2018backdoor} are not applicable.
\subsection{Datasets and ML Models}
\label{subsec:data_model}
For the image classification tasks, we use MNIST and Federated Extended MNIST
(FEMNIST) datasets. For the sentiment analysis task, we use Sentiment140. All
three datasets are widely used benchmarks in the FL
literature~\cite{caldas2018leaf,kairouz2019advances,li2018federated}.
We consider a heterogeneous FL setting with non-IID data as follows.
\PHM{MNIST} Following~\cite{pmlr-v54-mcmahan17a}, we sort data
samples based on the digit labels and divide the training dataset into $200$
shards, each consisting of $300$ training samples. We assign $2$ shards to
each client so that most clients only have examples of two digits, thus simulating a
heterogeneous setting.
\PHM{FEMNIST} The FEMNIST dataset contains $801,074$ data samples from $3,500$
writers~\cite{caldas2018leaf}. This is already a heterogeneous setting, as each writer represents a different client.
\PHM{Sentiment140} The Sentiment140 dataset includes $1.6$ billion tweets
twitted by $660,120$ users. Each user is a client.
\PHM{FL Tasks} We train an LR model with the MNIST
dataset. With FEMNIST dataset, we train a model with 2 CNN layers (5x5x32 and 5x5x64), followed by a dense layer with $2048$ units. For Sentiment140, we train a one-layer unidirectional RNN with gated recurrent unit (GRU) cells with $64$ hidden units~\cite{zhang2020dive}. We train all three models with test accuracy comparable to the previous work~\cite{li2018federated,caldas2018leaf,eichner2019semi}.
\PHM{Training Anomaly Detection Model} For each of the above FL tasks, there is a corresponding spectral anomaly detection model for detecting the malicious clients in FL model training. We use the \textit{test data} of the three datasets to generate the model weights for training the corresponding detection model. This is done by using the test data to train the same LR, CNN, and RNN models in a centralized setting and collecting the model weights of each update step. We then use the collected model weights to train the corresponding detection model.
The trained anomaly detection model is available to the server when it processes the clients' updates in FL model training for each of the above FL tasks.
We choose VAE as our spectral anomaly detection model. Both the encoder and decoder have two dense hidden layers with $500$ units, and the dimension of the latent vector is $100$. The VAE is a generative model, mapping the input to a distribution from which the low-dimensional embedding is generated by sampling. The output, i.e., the reconstruction, is generated based on the low-dimensional embedding and is done by a decoder~\cite{xu2018unsupervised}.
\subsection{Benchmark Schemes}
\PHB{GeoMed} Rather than taking the weighted average of the local model updates as done in the \texttt{FedAvg} algorithm~\cite{pmlr-v54-mcmahan17a}, the GeoMed method generates a global model update using the geometric median (GeoMed) of the local model updates (including the malicious ones), which may not be one of the local model updates~\cite{chen2017distributed}.
\PHM{Krum} Different from GeoMed, the Krum method generates a global model update using one of the local updates, which minimizes the sum of distances to its closest neighbors (including the malicious ones). The result of the Krum method is one of the local model updates~\cite{NIPS2017_6617}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.98\linewidth]{back_door_big_space.pdf}
\caption{An example of inserted backdoor text "I ate a sandwich".}
\label{fig:backdoor text}
\vspace{-5pt}
\end{figure}
\subsection{Results}
Experimental results on untargeted attacks, namely sign-flipping and additive noise attack, are shown in Figure~\ref{fig:untargeted_attack_test_acc}. Our proposed detection-based method (``Ours'') achieves the best performance in all settings. The performance of Krum remains the same regardless of the number of malicious attackers and the attack types. The reason is that Krum selects one of the most appropriate updates. Since each update from clients in the non-IID setting is biased, the performance loss cannot be avoided. GeoMed is robust against the additive noise attack, obtaining satisfactory performance. However, it fails in the case with the sign-flipping attack, in which malicious attackers try to move the geometric center of all the updates far from the true one.
The results on targeted attack are illustrated in \ref{fig:poisoning_attack_results}. Our solution can mitigate the impact of the backdoor attack on the considered datasets. Note that our method obtains the best theoretical performance because excluding the malicious clients indicates that their local data examples cannot be learned, as illustrated in Figure~\ref{fig:poisoning_MNIST}. It is worthy mentioning that Krum is robust to the backdoor attack, and GeoMed fails in defending the backdoor attack on MNIST dataset.
The superior performance of our method comes from the spectral anomaly detection model, which can successfully separate benign and malicious clients' model updates. We list the F1-Scores of the detection model performance in Table~\ref{tab:detection_fscore} for separating benign and malicious clients is, in essence, a binary classification task.
\begin{table}[tb]
\centering
\begin{tabular}{lccc}
\toprule
\backslashbox{Dateset}{Attack} & \tabincell{c}{Additive \\noise} & Sign-flipping & Backdoor \\
\midrule
FEMNIST & 1.00 & 0.97 &0.87 \\
\midrule
MNIST & 1.00 & 0.99 &1.00 \\
\midrule
Sentiment140 & 1.00 & 1.00 &0.93 \\
\bottomrule
\end{tabular}
\caption{The F1-Scores of our proposed detection-based method.}
\label{tab:detection_fscore}
\vspace{-5pt}
\end{table}
\subsection{Discussion}
We leverage the existing public dataset to train a spectral anomaly detection model, which is used to detect the malicious clients at the server side and then exclude them during FL training processes. The trained detection model can memorize the feature representation of the unbiased model updates obtained from public dataset. With this prior knowledge learned by the detection model, we see that \textit{it can detect the difference between the compact latent representation of the benign model updates and the compact latent representation of the malicious model updates.} We illustrate this results in Figure~\ref{fig:latent_repretations_1}. While distortion is unavoidable because of dimension reduction, it is clear that the benign model updates and the malicious model updates can be separated from each other, especially in the case with sign-flipping attack, where the benign model updates and the malicious updates are symmetric.
The proposed anomaly detection-based method provides \textit{targeted} defense in an FL system. Existing defense methods, such as Krum and GeoMed, provide untargeted defense because they cannot detect malicious clients. The targeted defense is necessary for FL because every local dataset may be drawn from a different distribution, and the defense mechanism shall be able to distinguish benign model updates produced by different datasets from malicious model updates. Otherwise, the global model would suffer from performance loss, as illustrated by the model performance with Krum in Figure~\ref{fig:untargeted_attack_test_acc}. We also conduct additional experiments, in which all clients are benign. Experimental results show that GeoMed and our method introduce very little bias and negligible performance loss compared to the \texttt{FedAvg} algorithm that does not consider defense against any attacks.
\section{Conclusion}
In this work, we propose a spectral anomaly detection based framework for robust FL, in which spectral anomaly detection is performed at the server side to detect and remove malicious model updates from adversarial clients. Our method can accurately detect malicious model updates and eliminate their impact. We have conducted extensive experiments, and the numerical results show that our method outperforms the existing defense-based methods in terms of model accuracy. Our future work will consider more advanced ML models and provide more analytical results.
\clearpage
\bibliographystyle{named}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,535 |
Патрик Генри «Пэт» Мартин (; 19 августа 1923, Луисвилл — 21 апреля 1987, Массена) — американский бобслеист, олимпийский чемпион 1948 года среди экипажей четвёрок.
Биография
Сотрудник полиции города Массена (штат Нью-Йорк), член бобслейного клуба «Лейк-Плэсид Сноу Бёрдз». Соревновался на Олимпиадах 1948 и 1952 годов, в 1948 году стал чемпионом среди экипажей четвёрок, в 1952 году завоевал серебряные медали среди двоек и четвёрок. Чемпион мира 1949 и 1950 годов в четвёрках, серебряный призёр 1950 (двойки) и 1951 годов (двойки и четвёрки).
Двукратный чемпион Северной Америки среди двоек (1951, 1956), трёхкратный чемпион Северной Америки среди четвёрок (1951, 1953, 1956), трёхкратный чемпион США среди двоек (1955, 1956, 1957) и шестикратный чемпион США среди четвёрок (1940, 1941, 1953, 1954, 1956, 1957). Одну из своих побед в 1940 году одержал в паре с Кэтрин Дьюи — единственной женщиной-чемпионкой по бобслею.
Примечания
Ссылки
Бобслеисты США
Чемпионы мира по бобслею
Олимпийские чемпионы по бобслею
Олимпийские чемпионы от США
Чемпионы зимних Олимпийских игр 1948 года
Серебряные призёры зимних Олимпийских игр 1948 года
Бобслеисты на зимних Олимпийских играх 1948 года
Бобслеисты на зимних Олимпийских играх 1952 года | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,097 |
Jimbo Fisher apologizes for using "retarded"
Florida State Seminoles head coach Jimbo Fisher apologized for using the word "retarded" in response to a question about the BCS rankings.
By JP Starkey Nov 13, 2012, 5:21pm EST
Share All sharing options for: Jimbo Fisher apologizes for using "retarded"
Florida State Seminoles head football Jimbo Fisher coach apologized on Monday for using the word "retarded" in response to a question about the latest BCS rankings.
"That's not the way I think," Fisher said. "It was a poor choice of words. I didn't mean to offend anyone in any way."
Though the comment was off-the-cuff and meant innocuously, the word "retarded'" is seen as being highly offensive to those who are mentally handicapped, and to those who have loved ones who suffer from various illnesses.
Fisher's comment, in which he replied "how retarded is it?," was in reply to being asked about Florida State's BCS ranking -- behind behind three SEC schools who have two losses, compared to Florida State's one loss.
Fisher's Seminoles have just two games remaining in the regular season -- a date against Maryland this coming Saturday, and then a showdown against No. 6 Florida on Nov. 24. If the Seminoles can come away with a win against the Gators, they could easily jump up the BCS rankings once again -- and leapfrog some of those two-loss SEC schools.
Hopefully, even if the Seminoles fall in the rankings or stay where they are, Fisher will think before he says something offensive in future press conferences.
Florida State vs. Virginia Tech 2012: Seminoles hang on for 28-22 win
Fisher apologizes for using the word 'retarded'
Seminoles hang on to avoid upset bid | {
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{"url":"https:\/\/destevez.net\/category\/ham-radio\/events\/","text":"## December DSLWP-B camera planning\n\nIt is again the beginning of the month, which means that the Earth will be in view of the Inory eye camera on board DSLWP-B. As usual, I have updated my camera planning notebook to compute the location of the Moon and Earth, as seen from the camera.\n\nWei has already scheduled observations on 2018-12-06 11:20 UTC and 2018-12-07 08:30 UTC. On each of these observations, an image will be taken at the start of the observation and the UHF transmitter will be activated for 2 hours.\n\nI have used the 20181128 tracking file from dslwp_dev as orbital state for the calculations. As the date of the observations comes nearer, I might rerun the computations with updated ephemeris data, but this time it doesn\u2019t seem critical to estimate the orbit with precision. In November, there were occultations of the Earth behind the lunar disc, and the times for these depended a lot on the orbital state. In December there will be no occultations, however.\n\nThe figure below shows the prediction for the camera view in the scheduled observations. On the 6th, the Earth will come close to the edge of the Moon. On the 7th, the Earth will be closest to the camera centre since we started planning and taking images in October.\n\n## Es\u2019hail 2 launch in GMAT\n\nAfter a long wait by many Amateur radio operators, Es\u2019hail 2, the first geostationary satellite carrying an Amateur radio transponder launched yesterday on a SpaceX Falcon 9 Full Thrust from the historical LC-39A pad in Kennedy Space Centre, which was used by many Apollo and Space Shuttle launches in the past.\n\nEs\u2019hail 2 is the second communications satellite operated by the Qatari company Es\u2019hailSat. It was built by Mitsubishi Electric Corporation (MELCO). It carries several Ku and Ka band transponders intended for digital television, Internet access and other data services. It also carries an Amateur radio payload designed by AMSAT-DL, in collaboration with the Qatar Amateur Radio Society. The payload has two transponders, with S-band uplink and X-band downlink. One of the transponders is 250kHz wide and intended for narrowband modes, and the other one is 8MHz wide and intended for DVB-S and other wideband data modes.\n\nSpaceX live-streamed the launch, and the recording can be seen in YouTube. Today, Space-Track has published the first TLEs for Es\u2019hail 2 and the second stage of the Falcon 9 rocket. Here I look at these TLEs using GMAT.\n\nContinue reading \u201cEs\u2019hail 2 launch in GMAT\u201d\n\n## November DSLWP-B images of the Moon and Earth\n\nIn previous posts, I have already spoken about the chance of DSLWP-B taking images of the Moon and Earth during the beginning of November. The window to take these images was between November 6 and 9. This window included the possibility of taking an Earthrise image, with the Earth appearing from behind the Moon.\n\nThe planning for the activations of the Amateur payload made by Wei Mingchuan BG2BHC was as follows.\n\n7 Nov 2018 08:13 to 7 Nov 2018 10:138 Nov 2018 09:40 to 8 Nov 2018 11:409 Nov 2018 12:00 to 9 Nov 2018 14:0010 Nov 2018 14:00 to 10 Nov 2018 16:0011 Nov 2018 13:30 to 11 Nov 2018 15:30\n\nOn November 7, from 8:13 to 9:33 UTC, a total of 9 images with 10 minutes of spacing between each would be taken. These images would be downloaded during the activations on the next days. As usual, an image would also be taken when the Amateur payload powered up on November 8 to 11, but the main focus was on downloading the sequence of images taken on November 7. This is a complete report of the images taken and downloaded.\n\nContinue reading \u201cNovember DSLWP-B images of the Moon and Earth\u201d\n\n## Updated DSLWP-B camera planning\n\nIn my last post, I detailed the DSLWP-B camera planning for the beginning of November. There, I used orbital state taken from the 20181027 tracking file to compute good times to take images of the Moon and the Earth, especially looking for an Earthrise-like image.\n\nNow that the planned dates are closer, it is good to rerun the calculations with a newer orbital state. It turns out that there has been an important change in the mean anomaly, which shifts all the predictions by a few hours.\n\nContinue reading \u201cUpdated DSLWP-B camera planning\u201d\n\n## DSLWP-B camera planning for the beginning of November\n\nI have spoken in other occasions about planning the appropriate times to take pictures with the DSLWP-B Inory eye camera. In the beginning of October there was a window that allowed us to take images of the Moon and Earth. A lunar month after this we have new Moon again, so it is an appropriate time to take images with the camera.\n\nThis time, the Moon will pass nearer to the centre of the image than on October, and at certain times the Earth will hide behind the Moon, as seen from the camera. This opens up the possibility for taking Earthrise pictures such as the famous image taken during the Apollo 8 mission.\n\nI have updated my camera planning Jupyter notebook to compute the appropriate moments to take images. The image below shows my usual camera field of view diagram.\n\nThe vertical axis represents the angular distance in degrees between each object and the centre of the image (Assuming the camera is pointing perfectly away from the Sun. In real life we can have a couple degrees of offset). The red lines represent the limits of the camera field of view, which are measured between the centre and the nearest edge, and between the centre and one corner. Everything between these two lines will only appear if the camera rotation is adequate. Everything below the lower line is guaranteed to appear, regardless of rotation.\n\nWe see that between November 6th and November 9th there are four times when the camera will be able to image the Earth and the Moon simultaneously. On the 6th it is almost guaranteed that the Earth will appear inside the image, and on the 9th it depends on the orientation of the camera. On the 7th and 8th it is guaranteed that the Earth will be in the image.\n\nTo compute appropriate times for taking an Earthrise picture, I have made the graph below. This shows the angular distance between the Earth and the rim of the Moon. If the distance is negative, the Earth is hidden by the Moon. We see that the Earth hides behind the lunar disc on each of the four days mentioned above.\n\nIn the figures below, we zoom in each of the events. In this level of zoom we can plot the \u201cinner\u201d and \u201couter\u201d Earth rim, so we can see when the Earth is partially hidden by the Moon.\n\nOn November 6th the situation is the most interesting in my opinion. It turns out the the Earth will not even hide completely between the Moon. In theory, a tiny sliver will remain visible. Also, it will take more time for the Earth to hide behind the Moon and then reappear. As we will see, the next days this will happen faster. Here, it takes 15 minutes for the Earth to hide, and another 15 minutes to reappear. It spends 10 minutes almost hidden.\n\nIt can be a good idea to take a series of 10 images with an interval of 5 minutes between each image, and spanning from 12:40 UTC to 13:30 UTC, to get a good coverage for this event.\n\nOn November 7th the Earth goes deeper into the lunar disc, taking 5 minutes to hide, spending 70 minutes hidden, and taking 10 minutes to reappear.\n\nOn November 8th the Earth goes even deeper into the lunar disc. It takes around 7 minutes to hide, spends 105 minutes hidden and takes 10 minutes to reappear.\n\nOn November 9th the configuration is quite similar to November 7th, but the hiding speed is slower. It takes 15 minutes to hide, spends 100 minutes hidden and takes 15 minutes to reappear.\n\nOverall, I think that the best would be to take a good series of images on November 6th, since this shallow occultation is a rarer event. The challenge will be perhaps to download all the images taken during these days. On average, I think we are downloading around 2 new images per 2 hour activation, taking into account repeats due to lost blocks and dead times. DSLWP-B is able to store 16 images onboard, and every time the UHF transmitter comes on, a new image is taken, overwriting an old image (more information in this post). Thus, if we take many images during these days, we have the danger of overwriting some when trying to download them over the next few days.\n\nPerhaps a good strategy is to arrange for a series of 10 images to be taken on the 6th, and then programming the UHF transmitter to take an image as the Earth comes out of its occultation on the 7th, 8th and 9th. In this way, the 2 hour periods of these three days can be used to download some of the images taken on the 6th, and there are still 3 images of margin in the buffer in case something goes wrong during the downloads over the next few days.\n\n## Upcoming DSLWP-B Moon occultations\n\nIn my previous post I showed that during the DSLWP-B observation on 2018-10-27 17:20 UTC, the orbit of DSLWP-B would take it behind the Moon. This doesn\u2019t happen every orbit (read as every day, since the orbit period is around 22 hours). It depends on the angle from which the orbit is viewed from Earth, and hence on the lunar phase.\n\nKnowing beforehand when DSLWP-B will hide behind the Moon allows to perform radio occultation studies. These consist in measuring the RF signal from DSLWP-B as it gets blocked by the lunar disc. Interesting phenomena such as diffraction can be observed.\n\nI have calculated the occultations that will be visible from the Dwingeloo radiotelescope in the remaining part of this year.\n\nContinue reading \u201cUpcoming DSLWP-B Moon occultations\u201d\n\n## Download of DSLWP-B Moon and Earth pictures continues\n\nContinue reading \u201cDownload of DSLWP-B Moon and Earth pictures continues\u201d\n\n## Report of the DSLWP-B Amateur observations of the Moon and Earth\n\nThe Chinese microsatellite DSLWP-B has been in lunar orbit since 25 May 2018. This satellite carries an Amateur radio payload which includes a small 640\u00d7480 CCD camera. The JPEG images taken by the camera can be transmitted using the SSDV protocol at 125 bits per second in the 70cm Amateur satellite band.\n\nUpdate 17:00 UTC: Wei comments that the camera sensor is CMOS, not CCD, and it has 2592\u00d71944 pixels. The image is resampled to 640\u00d7480 to save memory and bandwidth.\n\nThe orientation of the camera is fixed: the camera is mounted looking in the opposite direction of the solar panel, which is usually kept pointing directly to the Sun. Therefore, the camera is usually looking\u00a0 directly away from the Sun. The possibility of imaging celestial bodies such as the Moon and the Earth depends on the relative positions of these and the Sun.\n\nDuring the first week of October there was a new Moon, which implied that it was possible to take images of the Moon and the Earth, as I have described in this post and this other post. This is a report of all the images taken and downloaded during the observation window.\n\nContinue reading \u201cReport of the DSLWP-B Amateur observations of the Moon and Earth\u201d\n\n## DSLWP-B upcoming observations of Moon and Earth\n\nYesterday I looked at the photo planning for DSLWP-B, studying the appropriate times to take images with the DSLWP-B Inory eye camera so that there is a chance of getting images of the Moon or Earth. As I remarked, the Earth will be in view of the camera over the next few days, so this is a good time to plan and take images.\n\nI asked Wei Mingchuan BG2BHC to compare his calculations with mine and shortly after this he emailed me his planning for observations between October 8 and 10. After measuring the field of view of the camera as 37\u00d728 degrees, we can plot the angular distances between the Moon or Earth and the centre of the camera to check if the celestial body will be in the field of view of the camera.\n\nThe image below shows the angular distance between these celestial bodies and the centre of the camera. As we did in the photo planning post, we assume that the camera points precisely away from the Sun. Since the Moon and Earth (especially the Moon) have an angular size of several degrees, we plot the centre of these objects with a dashed line and the edges which are nearest and furthest from the camera centre with a solid line.\n\nThe field of view of the camera is represented with dotted red lines. Since the field of view is a rectangle, we have one mark for the minimum field of view, which is attained between the centre of the image and the centre of the top or bottom edge, and another mark for the maximum field of view, which is attained between the centre of the image and each of the corners.\n\nThe rotation of the spacecraft around the camera axis is not controlled precisely, so objects between the two red lines may or may not appear in the image depending on the rotation. Objects below the lower red line are guaranteed to appear if the pointing of the camera is correct and objects above the upper red line will not appear in the image, regardless of the rotation around the camera axis.\n\nTo calibrate the exposure of the camera, an image was taken yesterday on 2018-10-06 13:55 UTC. This time is marked in the figure above with an orange line. The image was downloaded this UTC morning. The download was commanded by Reinhard Kuehn DK5LA and received by Cees Bassa and the rest of the PI9CAM team in Dwingeloo. This image is shown below.\n\nThe image shows an over exposed Moon. Here we are interested in using the image to confirm the orientation of the camera. The distance between the centre of the image and the edge of the Moon is 240 pixels, which amounts to 14 degrees. The plot above gives a distance of 11 degrees between the edge of the Moon and the camera centre.\n\nThus, it seems that the camera is pointed off-axis by 3 degrees. This error is not important for scheduling camera photos, since an offset of a few degrees represents a small fraction of the total field of view and the largest error in predicting what will appear in the image is due to rotation of the spacecraft around the camera axis.\n\nThe observations planned by Wei for the upcoming days are shown in the plot above by green lines. The start of the observation is marked with a dashed line and the end of the observation (which is 2 hours later) is marked with a dotted line. The camera should take an image at the beginning of the observation and then we have 2 hours to download the image during the rest of the observation.\n\nWe see that Wei has taken care to schedule observations exactly on the next three times that the Moon will be closest to the camera centre. This gives the best chance of getting good images of the lunar surface (but the Moon will only fill the image partially, as in the picture shown above).\n\nThere are also two additional observations planned when the Moon is not in view. The first, on October 8 is guaranteed to give a good image of the Earth. The second, on October 10 will only give an image of the Earth if the rotation of the spacecraft is right.\n\nThe orbital calculations for the plot shown above have been done in GMAT. I have modified the photo_planning.script script to output a report with the coordinates of the Earth and the Moon in the Sun-pointing frame of reference (see the photo planning post).\n\nThe angle between the centre of the camera and the centre of the Earth of the Moon can be calculated as$\\arccos\\left(\\frac{x}{\\sqrt{x^2+y^2+z^2}}\\right),$where $$(x,y,z)$$ are the coordinates of the celestial body in the Sun-pointing frame of reference. The apparent angular radius of the celestial body can be computed as$\\arcsin\\left(\\frac{r}{\\sqrt{x^2+y^2+z^2}}\\right),$where $$r$$ is the mean radius of the body.\n\nThese calculations and the plot have been made in this Jupyter notebook.\n\n## DSLWP-B photo planning\n\nAs you may already know, most of the pictures taken so far by the DSLWP-B Inory eye camera have been over-exposed purple things. In order to take more interesting pictures, such as pictures of the Earth and the Moon, we have to plan ahead and know when these objects will be in view of the camera.\n\nThe picture below shows a diagram of DSLWP-B. The solar panel is behind the main body, pointing towards the back right. The Inory eye camera points in the opposite direction, towards the front left of the diagram. The camera is located in the front panel of the Amateur radio module, which is one of the pink modules.\n\nIn the nominal flight configuration, the solar panel is pointed towards the Sun to gather the maximum amount of sunlight. Therefore, the camera should point opposite to the Sun. This is good, because objects in the field of view of the camera are guaranteed to be well illuminated and it also prevents the Sun from appearing in the image, causing lens flare.\n\nIn one of the images taken previously by the camera there is a very bright lens flare, possibly caused by the Sun hitting the camera directly. This seems to indicate that the spacecraft is not always in the nominal flight orientation. As the orientation is very important when doing photo planning, here we assume that the spacecraft is always oriented in the nominal flight configuration, with its camera pointing opposite to the Sun.\n\nWe use GMAT with the 20181006 tracking file from dswlp_dev for the orbital calculations. The GMAT script can be downloaded here.\n\nTo simulate the orientation of the camera in GMAT, I have used an ObjectReferenced coordinate system. The primary is the Sun, the secondary is DSLWP-B, and the axes are chosen as X = R, Z = N. Therefore, the X axis points away from the Sun, in the direction of the camera. The orbit view is centred on DSLWP-B and points towards +X, so it shows what the camera would see (except for the field of view, which is not simulated).\n\nWhen propagating the orbit in GMAT we see that the Earth passes near the +X axis every Lunar month and that the Moon crosses the image every orbit.\n\nThe two images below show the dates when the Earth will pass nearby the +X axis. These dates are good candidates for taking images of the Earth. From the point of view of the camera, the Earth seems to orbit slowly from right to left in these images. Therefore, there should be a tolerance of a couple of days as to when the images of the Earth could be taken.\n\nThus, we see that the next good dates for attempting to take images of the Earth are October 9 and November 7 (plus\/minus a couple days of margin).\n\nThe Moon is much larger from the point of view of DSLWP-B and we have seen already that it fills the field of view of the camera completely. Thus, even though in this GMAT simulation the Moon crosses the screen every orbit, we need to wait until the path taken by the Moon is near the centre of the screen.\n\nIn the image below we see a good moment to take an image of the Moon. From the point of view of the camera, the Moon crosses from top to bottom of the screen every orbit and its path moves slightly to the right every time, taking it closer to the centre as we progress towards December.\n\nTherefore, a good moment to attempt to take an image of the Moon is late October and all of November. However, the time when the picture is taken is critical, because the Moon crosses the screen quickly. It is near the +X axis only for one or two hours. Therefore, starting in late October, there will be a window of a couple hours each orbit (or each day, since the orbital period is close to one day) where photos of the Moon can be attempted.\n\nJudging by one of Wei Mingchuan BG2BHC\u2019s lasts tweets, he has also been thinking about dates to take good images with the camera. It would be interesting to know if his findings match what I have exposed here.\n\nA good question when doing this sort of planning is what is the field of view of the camera. Probably this can be estimated from some of the existing images.","date":"2018-12-16 23:51:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.37125784158706665, \"perplexity\": 664.7278688102732}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376828018.77\/warc\/CC-MAIN-20181216234902-20181217020902-00032.warc.gz\"}"} | null | null |
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\section{Introduction}
\label{sec:intro}
Pseudoscalar axion-like particles (ALPs; for a review, see e.g.\
Ref.~\cite{Ringwald-rev}) are common in various extensions of the Standard
Model of particle physics. Their defining property is the
interaction with the electromagnetic field, allowing in particular for
photon/ALP mixing in the external magnetic field \cite{RaffStod}.
Experimentally allowed values of the photon-ALP coupling are however very
low, which makes large-scale astrophysical environments a proper place to
search for manifestations of this interaction. The astrophysical searches
include studies of stellar energy losses, where indications to
losses larger than predicted may be explained by interactions of
ALPs, see Ref.~\cite{Losses} for a review. In particular, cooling of
helium-burning stars~\cite{HBstars} may suggest ALP-photon couplings just
below the present limits from the solar ALP searches by the CERN Axion
Solar Telescope, CAST \cite{CAST}. The same range of couplings is favoured
by ALP explanations of possible ``anomalous transparency''
effects~\cite{hardening, 3c279, 2photons, HornsMeyer, gamma} in the
propagation of energetic gamma rays from distant sources~\cite{Csaba,
DARMA, Serpico, FRT, 1207.0776clusters, Meyer-evidence, Galanti-spindex}
(for a short review, see Ref.~\cite{ST-rev}). These studies are however
very sensitive to measurements of redshifts of the emitting sources, so
the significance of the observed effects remains uncertain \cite{gamma2}
and other ways to explore the relevant part of the ALP parameter space are
welcome.
Under certain conditions, photon-ALP mixing may result in oscillatory
features in the spectra of astrophysical photon sources seen through
regions filled with magnetic fields \cite{astro-ph/0410501}. While precise
shape of these irregularities and the photon energies at which they appear
depend on the ALP parameters and magnetic-field configurations, these
features are not expected in (otherwise smooth) astrophysical spectra.
Lack of these irregularities in observed spectra might in principle be
used to constrain ALP parameters, see e.g.\
Refs.~\cite{0806.0411, 1205.6428, 1305.2114, 1406.5972, 1804.09443,
1811.03548}.
Several attempts to follow this way have been made in recent years,
exploiting the observed smoothness of X- and gamma-ray spectra of strong
emitters presumably embedded in astrophysical magnetic fields. In
particular, these included analyses of gamma-ray observations of
PKS~2155$-$304 \cite{1311.3148, 1802.08420, 1906.00357}, NGC~1275
\cite{1603.06978, 1805.04388} and several Galactic sources
\cite{1801.01646, 1801.08813, 1804.07186} relevant for ALP masses $\sim
\left(10^{-9} \, - \, 10^{-7}\right)$~eV (overlapping with the range
invoked for explanations of the ``anomalous transparency'' effects). For
lower ALP masses, X-ray observations of 3C~218 \cite{1304.0989}, the same
NGC~1275 \cite{1605.01043, 1712.08313, 1907.05475}, M~87 \cite{1703.07354}
and some other extragalactic sources \cite{1704.05256} were used.
Refs.~\cite{1801.01646, 1801.08813} found some favorable ranges in the ALP
parameter space, though with ALP-photon couplings above the 95\%~CL CAST
upper limit. Other works put limits on the ALP parameters which are often
quoted on equal footing with other astropysical and laboratory constraints
\cite{PDG}. The aim of the present work is to demonstrate how much these
constraints depend on the assumptions about magnetic fields involved and
how much they could change if different astrophysically motivated field
configurations are assumed.
As an example, we concentrate on the case of NGC~1275, the central radio
galaxy of the Perseus cluster and the most popular target for the searches
for spectral irregularities in the ALP context \cite{1603.06978,
1805.04388, 1605.01043, 1712.08313}. We briefly review these studies in
Sec.~\ref{sec:previous} and emphasize that they use a theoretical model of
the magnetic field in the Perseus cluster, which is based on observations
of other clusters, because of the lack of relevant observations in
Perseus. This model assumes purely turbulent magnetic fields. In the
present study, we revisit these constraints taking into account
large-scale ordered magnetic fields which are expected to be present
around giant radio galaxies from the interaction between lobes and
environment in clusters \cite{a-p/0204443, 1108.0430, 1203.4582,
1811.06266, 1812.07900}. Evidence for the regular magnetic fields ordered
at scales $\gtrsim 100$~kpc have been found in several galaxy clusters
\cite{1101.1807, 1612.01764}, see also reviews \cite{1205.1919,
Han-ARAA55(17)111}. Possible presence of the regular magnetic fields
around radio galaxies is crucial for the present consideration.
In Sec.~\ref{sec:field}, we concentrate on the magnetic field in the
Perseus cluster. Only two relevant observations of the Faraday rotation
have been reported in the literature: Ref.~\cite{Taylor-FRM} gives a
measurement at the very center of the cluster, used in previous studies
\cite{1603.06978, 1805.04388, 1605.01043, 1712.08313} to normalize the
field strength at this point; Ref.~\cite{FRM-map} presents a map of
rotation measurements across the cluster (not used in previous studies).
We recall numerous radio and X-ray observations indicating
the presence of an X-ray cavity, or a radio mini-halo, around NGC~1275,
consistent with the lobe-environment interaction \cite{MNRAS264(93)L25,
a-p/0207290, a-p/0503318, 1701.03791}. An analytical model of the regular
magnetic field in such a cavity was presented (and supported by numerical
simulations) in Ref.~\cite{1008.5353}; see also Refs.~\cite{1011.0030,
1107.2599} and especially \cite{1108.3344} for more detailed simulations.
We use this analytical model as a proxy to the regular field in the
central X-ray cavity of the Perseus cluster, consistent with rotation
measurements of Refs.~\cite{Taylor-FRM, FRM-map}, as well as with X-ray
observations of large-scale structures in Perseus \cite{Churazov,
1810.07380} and corresponding simulations \cite{1506.06429}.
Next, in Sec.~\ref{sec:NGC1275}, we turn to the effect of the ordered
magnetic field on the observed photon spectrum, and consequently on
constraints on the ALP parameters. As an example, we readdress the case of
NGC~1275 and Fermi-LAT data~\cite{1603.06978}.
In general, magnetic field in the cluster is the sum of regular and
turbulent components, and their various combinations can fit scarce
observational data. Since both components contribute to the
observed Faraday rotation, stronger regular field
implies weaker turbulent one. In particular, previous studies assumed
purely turbulent models; here we consider the opposite case and assume a
purely regular field, which is described in Sec.~\ref{sec:field} and
agrees with observations. Comparison of the results for these two limiting
cases gives an estimate of the theoretical uncertainty of the constraints
obtained in this way. We take the spectrum used in Ref.~\cite{1603.06978}
and demonstrate that, for the assumed purely regular field configuration,
the fits with and without ALPs are equally good for a large part of the
parameter space, so that the resulting constraints on ALP parameters are
much weaker than those obtained in Ref.~\cite{1603.06978} for the opposite
case of purely turbulent field models. This is not surprising because
spectral irregularities are enhanced when the field extends to the scales
much larger than its coherence length, see e.g.\ Ref.~\cite{1311.3148}.
Therefore, we expect a similar effect on other constraints obtained in the
same way.
In
Sec.~\ref{sec:concl}, we briefly reiterate our main conclusion:
constraints on the ALP parameters from the lack of irregularities
in the spectra of astrophysical sources
are very sensitive to the assumptions about magnetic fields surrounding
the sources. In particular, regular fields, which are expected to be
present close to these strong active galaxies, are very important.
Detailed studies of the magnetic fields are required to reduce the
uncertainties and to obtain firm limits on ALP parameters in this way.
\section{Previous studies of NGC~1275 in the ALP context.}
\label{sec:previous}
NGC~1275 is a radio galaxy located in the center of the Perseus
cluster. It is a bright X-ray and gamma-ray source. Thanks to the
intensity of the source, sufficient statistics could be collected to
obtain detailed spectra suitable for the search for spectral features at
high confidence level, which justified the choice of this source as a
target for several attempts to constrain ALP parameters from the lack of
irregularities in the spectrum. In Ref.~\cite{1603.06978}, Fermi-LAT
collaboration analyses their data in great detail, develops a statistical
procedure and presents 95\% CL exclusion limits for ALP parameters.
Authors of Ref.~\cite{1805.04388} supplement Fermi-LAT data by published
results of the MAGIC observations of the same target \cite{1602.03099} to
extend the energy range covered by the spectrum and, consequently, the
excluded region in the ALP parameter space. Refs.~\cite{1605.01043,
1712.08313} used X-ray spectra of NGC~1275 obtained by Chandra to
constrain ALPs with lower masses. In what follows, we will mainly refer to
the Fermi-LAT study~\cite{1603.06978} as an example.
Measurements of astrophysical magnetic fields on galactic and larger
scales often rely on the Faraday rotation of the polarization
plane~\cite{1205.1919, Han-ARAA55(17)111}. The approach is not easy to
implement directly since it requires sources of polarized emission behind
the magnetic-field region, observed at different wavelengths $\lambda$ so
that the typical $\lambda^{2}$ dependence may be traced in the
polarization angle. In addition, the outcome is the product of the
longuitudinal component of the magnetic field and the electron density,
integrated over the line of sight, so some additional information and/or
theoretical input are required to reconstruct the magnetic-field structure
and values; note in particular that the other, transverse component of the
field is relevant for the ALP-photon mixing. Still, Faraday rotation
measurements remain the best available mean to reconstruct magnetic fields
in galaxy clusters. Sadly, these measurements for the Perseus cluster are
very limited and do not allow to uniquely reconstruct the field from
observations. That is why a theoretical model of the magnetic field in the
cluster was used in Refs.~\cite{1603.06978, 1805.04388, 1605.01043,
1712.08313}, and the real measurement in only one direction, towards the
very center of the cluster~\cite{Taylor-FRM}, was used to construct the
model.
Following general considerations of Ref.~\cite{1406.5972},
Ref.~\cite{1603.06978} models the magnetic field in the Perseus cluster as
purely turbulent, with the maximal coherence length of 35~kpc. The model
used there is motivated by observations of clusters other than Perseus
since, for Perseus, no detailed data are available. The model has 6
parameters, and reported ALP constraints correspond to their fiducial
values. While the effect of variations of these parameters, one by one, on
the resulting likel{\color{red} i}hood of the spectral fit was studied,
see Fig.~7 of the Supplemental Material of
Ref.~\cite{1603.06978}, none of the considered variations
included a regular component. However, if a regular field component
exists, it contributes to the rotation measurement \cite{Taylor-FRM} so
that the amplitude of the turbulent component, and consequently the
strength of possible ALP-induced spectral irregularities, is smaller.
\section{Regular magnetic fields in the Perseus cluster}
\label{sec:field}
In the center of the Perseus cluster, like in many other clusters
containing a large active galaxy, an X-ray cavity was observed, spatially
coinciding with the so-called radio mini-halo~\cite{MNRAS264(93)L25,
a-p/0207290, a-p/0503318, 1701.03791}. This cavity most probably is
a result of the interaction between outflows of NGC~1275 and the
intracluster gas. The X-ray cavity size, according to Chandra
observations~\cite{a-p/0503318}, is 93~kpc, while the radio mini-halo
extends slightly further~\cite{1701.03791}. Modelling~\cite{1506.06429} of
the slashing cold front in the Perseus cluster, see e.g.\
Ref.~\cite{1810.07380}, indicates that the X-ray cavity should be filled
with relatively high magnetic field to support the required pressure. This
magnetic-field region is seen on the rotation-measure map of the Perseus
cluster~\cite{FRM-map} and explains high values of the rotation measure
observed in Ref.~\cite{Taylor-FRM} for the very center of the cluster.
For the X-ray cavities, blown by radio-galaxy jets in the intracluster
plasma, magnetic fields are expected to be regular at large scales.
Ref.~\cite{1008.5353} suggests the following consistent magnetic-field
solution,
\[
B_{r}=2\cos\theta f(r_{1})/r_{1}^{2},
\]
\[
B_{\theta}=-\sin\theta f'(r_{1})/r_{1},
\]
\[
B_{\phi}=\alpha\sin\theta f(r_{1})/r_{1},
\]
where
\[
f=C
\left(\alpha \cos(\alpha r_{1})- \sin(\alpha r_{1})/r_{1} \right) - F_{0}
r_{1}^{2}/\alpha^{2},
\]
\[
F_{0}=C \alpha^{2} \left(\alpha \cos\alpha - \sin\alpha \right),
\]
$\alpha$ is the lowest nonzero root of
$\tan\alpha=3\alpha/(3-\alpha^{2})$, $r_{1}\equiv r/R$ for the cavity
radius $R$ and $C$ is the normalization constant determined by the field
value at $r=0$. This analytical solution is supported also by numerical
simulations in Ref.~\cite{1008.5353}. In the next section, we use this
solution as a model of the regular field in the X-ray cavity
around NGC~1275, assuming the viewing angle $\theta=45^{\circ}$ and the
cavity radius of $R=93$~kpc~\cite{a-p/0503318}. The normalization of the
field is chosen in such a way that the Faraday Rotation Measurement of
$\sim 7300$~rad/m$^{2}$ \cite{Taylor-FRM} for the central direction is
reproduced for the electron density derived from X-ray
observations~\cite{Churazov}. This normalization gives the total field in
the center of 8.3~$\mu$G. The field components for the chosen line of sight
are plotted in Fig.~\ref{fig:field}.
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{field.pdf}}
\caption{\label{fig:field}
Longuitudinal, $B_{r}$, and two transverse, $B_{\theta}$ and $B_{\phi}$,
components of the magnetic field solution~\cite{1008.5353} for the X-ray
cavity around NGC~1275.}
\end{figure}%
We should note that the central X-ray cavity may be not the only place in
the Perseus cluster where fields ordered at large scales are present.
Ref.~\cite{FRM-map} discusses also an organised structure in the
rotation-measure map on the $\sim$Mpc scales, possibly associated
\cite{AN-327-545} with a shock caused by the interaction of the
intracluster matter with intergalactic matter in the large-scale structure
filament, which may or may not be similar to the regular field structures
observed in other galaxy clusters~\cite{1101.1807, 1612.01764} under
similar conditions. In addition, X-ray observations reveal~\cite{Churazov,
1810.07380} large-scale ($>100$~kpc) spiral structure around NGC~1275,
which also may host regular magnetic fields, like it happens in our own
Galaxy. We stress therefore that the structure of regular magnetic fields
in the Perseus cluster is rich and unknown, lacking explicit measurements.
In the example in the next section, we concentrate on the best-studied
X-ray cavity field only.
\section{Example: ALP constraints from the Fermi-LAT
spectrum of NGC~1275 for the X-ray cavity magnetic field}
\label{sec:NGC1275}
Consider now the interaction of the electromagnetic field $A_{\mu}$ and
the ALP field $a$ described by the following Lagrangian,
\[
\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\left(\partial a
\right)^{2}-\frac{1}{2}m^{2}a^{2}-\frac{1}{4}ga F_{\mu\nu}\tilde
F^{\mu\nu},
\]
where $F_{\mu\nu}$ is the electromagnetic stress tensor, $\tilde
F_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\lambda}F^{\rho\lambda}$, $m$ is
the ALP mass and $g$ is the ALP-photon coupling constant. The last term is
responsible for axion-photon mixing in the external magnetic
field~\cite{RaffStod}, which in the adiabatic approximation is
most conveniently described in terms of the density matrix $\rho$ obeying
the equation
\begin{equation}
\!\!\!\!\!\!\!
\!\!
i \frac{d\rho(y)}{dy}=\left[\rho(y),\mathcal{M}(E,y) \right],
~~~
\mathcal{ M}\!=\!\frac{1}{2}\!\left(
\!
\begin{array}{ccc}
0 & 0 & -igB_{\theta}\\
0 & 0 & -igB_{\phi}\\
igB_{\theta} & igB_{\phi} & \frac{m^{2}}{E}
\end{array}
\! \right).
\label{Eq:M}
\end{equation}
For an initial unpolarized purely photon state,
\begin{equation}
\rho(0)=\mbox{diag}\left(1/2,1/2,0\right).
\label{Eq:rho0}
\end{equation}
The probability to observe an
unpolarized photon at a distance $y$ from the source is given by the sum
$\rho_{11}(y)+\rho_{22}(y)$ of the components of the solution $\rho(y)$ to
Eq.~(\ref{Eq:M}) with the boundary condition (\ref{Eq:rho0}).
Here, we assumed that the mixture of two transverse states of
photons and one state of ALP is propagating along the direction $y$ with
the energy $E$, and kept only terms relevant for the present study.
The full expression for the matrix $\mathcal{ M}$ can be found e.g.\ in
Ref.~\cite{FRT}, but entries not shown in Eq.~(\ref{Eq:M}) can safely be
neglected for energies, fields and distances studied here.
We use the magnetic-field model for the X-ray cavity around
NGC~1275 described in Sec.~\ref{sec:field}. We assume that effects of the
turbulent field in the outskirts of NGC~1275 are subleading, like it is
normally assumed for our own Galaxy\color{red}\footnote{\color{red} We tested
explicitly that the addition of turbulent magnetic field, modelled
as in Ref.~\cite{1603.06978}, having the mean amplitude of
1~$\mu$G at the boundary of the X-ray cavity and extending up to
50~kpc from the center, does not change the modification factor by
more than $2\dots 3\%$, which is within statistical uncertainties
of the NGC~1275 spectral data points.}\color{black}. For the Milky-Way magnetic
field, we use the BSS model of Pshirkov et al.\ \cite{PshirkovGMF}. The
need for a choice of the Galactic field model introduces additional
systematic uncertainties to the results. The model we use is conservative
in the sense that it predicts smaller photon-ALP oscillations
probabilities for sources away from the Galactic plane (the Galactic
latitude of NGC~1275 is $b\approx -13^{\circ}$ and this difference is not
dramatic). We assume the symmetry axis of the cavity field to be
co-oriented with the observed jets, position angle $\approx 147^{\circ}$
(from North to East) in Galactic coordinates. The redshift of NGC~1275 is
only $z=0.018$, so effects of pair production on the extragalactic
background light are negligible at the Fermi-LAT energies. We assume a
source of unpolarized photons in the center of the X-ray cavity and solve
Eqs.~(\ref{Eq:M}), (\ref{Eq:rho0}) numerically to obtain the probability
to observe photons with a given energy at the Earth. Then, we concentrate
on the Fermi-LAT data for the EDISP3 event type shown in
Ref.~\cite{1603.06978}. We convolve the probability with the
energy-dependent instrumental energy resolution presented in Supplemental
Material of Ref.~\cite{1603.06978} to obtain the ratio of the observed
photon spectrum to the emitted one, which we call the spectrum
modification factor. For illustration, it is presented in
Fig.~\ref{fig:pconv}
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{new-mod-f.pdf}}
\caption{\label{fig:pconv}
Modification factor for the photon spectrum of NGC~1275: the probability
to detect a photon with energy $E$ from the source located in the center of
the X-ray cavity magnetic field in the presence of ALP convolved with the
Fermi-LAT energy resolution. The ALP mass $m=2$~neV and four different
values of $g$, shown in the plots, are assumed. For two
values of $g$ (red curves), stronger irregularities are present in
the most constraining region around $\sim 1$~GeV, and these values are
excluded at the 95\% confidence level. }
\end{figure}%
for benchmark values of ALP parameters.
Next, we follow the logic of Ref.~\cite{1603.06978} and find best-fit
spectra with and without ALPs and compare the quality of the fits. We use
the EDISP3 event type spectrum of NGC~1275 from Fig.~1 of
Ref.~\cite{1603.06978}. While the actual emitted spectrum is unknown, most
of active-galaxy gamma-ray fluxes $F(E)$ are perfectly fit by the so-called
log-parabola function,
\begin{equation}
F(E)=F_{0} \left(E/E_{0}\right)^{-\left(\alpha + \beta \log(E/E_{0})
\right)}
,
\label{Eq:logpar}
\end{equation}
where, following Ref.~\cite{1603.06978}, we fix $E_{0}=0.53$~GeV and keep
three parameters, $F_{0}$, $\alpha$ and $\beta$, free. We then perform two
independent fits of the data, each with the free parameters: one by the
log-parabola spectrum (\ref{Eq:logpar}) and another by the same function
multiplied by the spectrum modification factor for given values of the ALP
parameters. The fits are performed by the usual
chi-square minimization adopted to account for asymmetric statistical
errors \cite{physics/0401042}. Upper limits are treated as zero values
with the corresponding errors. This is a simplification compared to the
procedure of Ref.~\cite{1603.06978}, where the full likel{\color{red} i}hood
function based on non-Gaussian distribution of errors was used; however,
this is sufficient for our purposes since we are interested in the best
fit only\footnote{\color{red} We checked that for the fit without ALPs, our
results agree with those presented in Ref.~\cite{1603.06978}.}.
For the same benchmark values
of $m=10^{-9}$~eV and $g=10^{-11}$~GeV$^{-1}$, the two best-fit spectra
are shown in Fig.~\ref{fig:spectrum}
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{spectrum2c1.pdf}}
\caption{\label{fig:spectrum}
Fermi-LAT spectrum of NGC~1275 \cite{1603.06978} (data points) together
with two best-fit spectra: with (red dashed line) and without (green full
line) ALPs with $m=1$~neV, $g=10^{-11}$~GeV$^{-1}$.}
\end{figure}%
together with the data. The fit quality is determined by $\chi^{2}
\approx 115.9$ for the fit without ALPs and $\chi^{2}\approx 115.6$ for
the fit with ALPs, for 114 degrees of freedom. We see that both fits are
perfect, and the effects of ALPs with the benchmark parameters cannot be
excluded for the regular field model used here, while they are excluded
at the 95\% CL for the purely turbulent field model in
Ref.~\cite{1603.06978}.
This fitting procedure was repeated for various pairs of ALP parameters
$(m,g)$. The 95\% exclusion contour in the $(m,g)$ plane was determined
from the chi-squared distribution with 114 degrees of freedom,
$\chi^{2}_{95} \approx 139.9$. This contour is shown in
Fig.~\ref{fig:explot}
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{explot-corr1.pdf}}
\caption{\label{fig:explot}
Exclusion regions (95\% CL) on the plane of ALP parameters $(m,g)$ for
purely regular (this work, hatched) and purely turbulent
(Ref.~\cite{1603.06978}, shaded) magnetic fields around NGC~1275. The blue
horizontal line represents the CAST~\cite{CAST} 95\% CL upper limit on
$g$. Ref.~\cite{1603.06978} does not extend the exclusion above the CAST
limit. For the regular field considered here, all exclusion regions are
above the CAST limit. For a realistic combination of regular and
turbulent fields, the exclusion is somewhere in between, but one needs to
know magnetic fields around NGC~1275 in more detail to find where exactly.}
\end{figure}%
together with the exclusion region
obtained in Ref.~\cite{1603.06978} for purely turbulent fields.
The difference between purely regular and purely turbulent cases is
dramatic. Since the actual magnetic field around NGC~1275 is an unknown
mixture of regular and turbulent components, this difference gives an
estimate of theoretical uncertainties of the constraints.
\color{red}
The main physical reason behind the difference between the resulting
constraints on ALP parameters obtained in
this section and in Ref.~\cite{1603.06978} is the use of coherent
versus turbulent fields. While the photon-ALP conversion probability
builds up for large-scale coherent fields, reaching eventually the
maximal-mixing regime, it is not the probability which determines the
strength of the constraint: instead, it is the presence and the amplitude
of ``wiggles'' around $\sim 1$~GeV, where the spectrum is well measured,
cf.\ Fig.~\ref{fig:pconv}. In a multi-scale turbulent field, the
irregularities are enhanced as compared to the regular-field case, because
for certain energies the oscillation length fits the field coherence scale
and, for this energy, the conversion probability is locally enhanced,
hence a wiggle in the spectrum appears (see e.g.\ Ref.~\cite{1311.3148}).
Therefore, while the value of the amplitude of the central field used in
Ref.~\cite{1603.06978}, 10~$\mu$G, is slightly larger than the central
value of $8.3~\mu$G used here, this cannot be the reason for dramatically
different constraints. We note in passing that while our toy model is
normalized to reproduce the observed Faraday rotation
measure~\cite{Taylor-FRM}, it is very hard to obtain its large observed
value with purely turbulent fields since positive and negative
contributions cancel each other in the corresponding integral.
Another difference with the analysis of
Ref.~\cite{1603.06978} is in the event classes used. We base our toy
research on the published spectrum and energy resolution, available for
the EDISP3 class only. This high-quality set, by construction, contains
1/4 of the data. These data, with their good energy resolution,
dominate constraints on ``wiggles'' in the spectrum; however the full set
of data was used in Ref.~\cite{1603.06978}, which gives an additional
caveat in the direct comparison of the two results.
Finally, one more difference is that Ref.~\cite{1603.06978}, as one may
conclude from a detailed description~\cite{1406.5972}, used the
domain-like approximation for the magnetic field while we solved the full
equations for smooth fields numerically. It has been pointed out (see
e.g.\ Ref.~\cite{1804.09443}) that the domain approximation, like the one
used in Ref.~\cite{1406.5972}, may introduce certain biases in the
probability calculation. We checked by an explicit full calculation that
this effect arises at high energies and hence has a limited influence on
the spectral fitting because of large statistical errors of the measured
spectrum in this energy range.
\color{black}
\section{Conclusions}
\label{sec:concl}
Magnetic fields regular on large scales are expected to be present around
giant active galaxies, in particular around those which are used to
constrain ALP parameters from the lack of irregularities in their spectra.
Particular examples are NGC~1275 and PKS~2155$-$304, for which
high-quality gamma-ray and X-ray spectra are available. In this paper, we
revisit constraints from gamma-ray observations of NGC~1275 which assumed
purely turbulent magnetic fields around this giant radio galaxy. We note
that an X-ray cavity observed around the galaxy suggests the presence of
ordered magnetic fields and use a model of the cavity field to study how
the ALP constraints are changed in the presence of regular fields.
Assuming that the observed Faraday rotation measurements are fully
explained by the cavity field, we obtain constraints much weaker than
those obtained in Ref.~\cite{1603.06978} for a purely turbulent field
model, see Fig.~\ref{fig:explot}. In reality, the field is expected to
include both components, and actual constraints would lay somewhere in
between, but unfortunately the lack of magnetic-field measurements in the
Perseus cluster currently prevents one from disentangling the two
contributions.
\color{red} The key message of our paper -- that a good knowledge of the magnetic
field is a necessary prerequisite of a study aming to constrain ALP
parameters from the search of spectral irregularities, -- remains valid
for observations in other energy bands, not only in gamma rays. For
instance, for the X-ray energy range, additional complications arise
because of non-negligible effects of the electron density, which drive the
photon-ALP conversion to the outer parts of the cluster, where the
field is even less constrained than in the center.
\color{black} Similar arguments are applicable to other sources, in
particular to PKS~2155$-$304.
\color{red} Previous studies~\cite{1311.3148} assumed that the group, in which
this blazar resides~\cite{PKSgroup1}, possesses magnetic fields similar to
much larger clusters\footnote{\color{red} Coma ($\sim 1000$ members, $M_{\rm
vir}\sim 1.7 \times 10^{15} M_{\odot}$, $R_{\rm vir}\sim 2.9$~Mpc
\cite{MNRAS-343-401}) and Hydra ($\sim 160$ members, $M_{\rm vir}\sim
2\times 10^{14} M_{\odot}$, $R_{\rm vir}\sim 1.6$~Mpc
\cite{Hydra-cluster}) clusters. The group containing PKS has the virial
radius of $R_{\rm vir} \sim 0.22$~Mpc, virial mass $M_{\rm vir}\sim 1.5
\times 10^{13} M_{\odot}$ and only 12 identified group
members~\cite{1510.01779}. Only in a few cases (not including this one),
magnetic fields in such small groups have been determined observationally,
and these fields were regular (see e.g.\ Ref.~\cite{1701.05962} and
references therein), not turbulent.}. No Faraday rotation measurements are
available for this group, so Ref.~\cite{1311.3148} and follow-up studies
made use of this theoretical model of the magnetic field without any
direct observational data.
\color{black}
While new
observations of the same targets with sensitive instruments like CTA
\cite{CTA} or ATHENA$+$ \cite{ATHENA} will reduce statistical
uncertainties
in
the spectra and are certainly welcome for many reasons, uncertainties in
the magnetic-field models should first be removed by complementary
observations. In view of results of Ref.~\cite{FRM-map}, Faraday rotation
mapping of the Perseus cluster looks a promising way to obtain better
constraints on the magnetic field in the region where the lobes of
NGC~1275 interact with the intracluster matter, see e.g.\ Ref.\
\cite{Bonafede}. In addition, sources embedded in magnetic fields better
studied observationally may be invoked for the search of irregularities,
for instance M~87~\cite{1703.07354} and Galactic sources~\cite{1801.01646,
1801.08813, 1804.07186}. For certain values of ALP parameters, the
transition to the regime of strong mixing in the Galactic magnetic field
may be observed or constrained also for extragalactic sources, cf.\
Ref.~\cite{1805.04388}. Note that the relevant part of the ALP parameter
space will be explored by coming experiments, a purely laboratory
instrument ALPS-IIc \cite{ALPS} and helioscopes TASTE \cite{TASTE} and
IAXO \cite{IAXO-CDR}.
\section*{Acknowledgements}
The authors are indebted to Mikhail Kuznetsov, Manuel Meyer, Maxim
Pshirkov, Grigory Rubtsov and Dmitri Semikoz for discussions
and useful comments on the manuscript. ST acknowledges discussions
with Igor Garcia Irastorza and Manuel Meyer at the initial stage of this
work. We are especially thankful to the anonymous reviewer for very useful
comments which improved the paper. This work was supported by the Russian
Science Foundation, grant 18-12-00258.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,574 |
{"url":"https:\/\/mathspace.co\/textbooks\/syllabuses\/Syllabus-950\/topics\/Topic-19675\/subtopics\/Subtopic-261892\/?textbookIntroActiveTab=overview","text":"# 5.07 Compound interest\n\nLesson\n\nWe've now learned about simple interest, where interest is calculated at a fixed rate on just the principal amount, so the amount of interest earned in each time period remains constant. This type of interest is very rare to encounter in most situations, however. Instead, banks and other financial institutions that calculate interest do so using compound interest instead.\n\n## Compound interest as repeated multiplication\n\nCompound interest occurs when we earn interest on the principal amount and the interest earned so far, rather than just on the principal amount. That is, each time period we calculate interest earned by using the total amount from the previous year. This means that the interest increases at an exponential rate, as we earn interest on interest that has already been earned.\n\nFor example, if we were to invest $\\$500$$500 for 33 years at a rate of 6%6% per year using compound interest, then: \u2022 After the first year, the value of the investment would be 500\\times1.06500\u00d71.06. \u2022 After the second year, the value of the investment would be \\left(500\\times1.06\\right)\\times1.06(500\u00d71.06)\u00d71.06, which simplifies to 500\\times1.06^2500\u00d71.062 \u2022 After the third year, the investment would be \\left(500\\times1.06^2\\right)\\times1.06(500\u00d71.062)\u00d71.06, which we can simplify to 500\\times1.06^3500\u00d71.063 #### Worked example ##### Example 1 Han's investment of \\6000$$6000\u00a0earns interest at $2%$2% per year, compounded annually over $3$3 years.\n\nAnswer the following questions by using repeated multiplication.\n\n(a) What is the value of the investment after $3$3 years? Round your answer to the nearest cent.\n\nThink: We should apply the interest rate to the new amount each year.\n\nDo:\n\n Year 1: $A_1$A1\u200b $=$= $6000\\times1.02$6000\u00d71.02 $=$= $\\$6120$$6120 Year 2: A_2A2\u200b == 6120\\times1.026120\u00d71.02 == \\6240.40$$6240.40 Year 3: $A_3$A3\u200b $=$= $6240.40\\times1.02$6240.40\u00d71.02 $=$= $\\$6367.25$$6367.25 (to the nearest cent) (b) What is the amount of interest earned? Think: The interest is the difference between the total amount of the investment and the principal (the initial amount invested). Do: 6367.25-6000=\\367.256367.256000=367.25 Notice that when we calculate compound interest, we are calculating the total value of an investment or loan. To find the amount of interest earned, II, we subtract the principal PP from the total value AA. That is, I=P-AI=PA. This is different to simple interest calculations, where we calculated the interest earned. To find the total value of the investment, we needed to add the interest and the principal amount. That is, A=P+IA=P+I. Notice that this is the same relation between AA, II and PP as above, just rearranged. #### Practice question ##### Question 1 Luke's investment of \\5000$$5000 earns interest at $3%$3%\u00a0per year, compounded annually over $4$4 years.\n\nAnswer the following questions by repeated multiplication.\n\n1. What is the value of the investment after $4$4 years?\n\n2. What is the amount of interest earned?\n\n## The compound interest formula\n\nNotice that in the above example, at the end of each compounding period there is a two step process: calculate the interest and then add it to the account balance. We could have\u00a0combined these two steps as follows:\n\nBalance after $1$1 year $=500+500\\times0.1=500\\times(1+0.1)=550$=500+500\u00d70.1=500\u00d7(1+0.1)=550\n\nThis suggests a rule:\n\nNew balance $=$=\u00a0Previous balance $\\times(1+0.1)$\u00d7(1+0.1)\n\nIn other words, we can find the balance at the end of each year by repeatedly multiplying by $(1+0.1)$(1+0.1)\n\n Balance after $1$1 years $=500\\times(1+0.1)$=500\u00d7(1+0.1) Balance after $2$2 years $=500\\times(1+0.1)\\times(1+0.1)=500\\times(1+0.1)^2$=500\u00d7(1+0.1)\u00d7(1+0.1)=500\u00d7(1+0.1)2 Balance after $3$3 years $=500\\times(1+0.1)\\times(1+0.1)\\times(1+0.1)=500\\times(1+0.1)^3$=500\u00d7(1+0.1)\u00d7(1+0.1)\u00d7(1+0.1)=500\u00d7(1+0.1)3\n\nThis leads us to the compound interest formula.\n\nCompound interest formula (annual compounding)\n\n$A=P\\left(1+r\\right)^t$A=P(1+r)t\n\nwhere:\n\n$A$A is the final amount of money (principal and interest together)\n\n$P$P is the principal (the initial amount of money invested)\n\n$r$r is the interest rate per year, expressed as a decimal or fraction\n\n$t$t\u00a0is the number of years\n\nThis formula gives us the total amount (ie. the principal and interest together). If we just want to know the value of the interest, we can work it out by subtracting the principal from the total amount of the investment. In symbols:\n\n$I=A-P$I=AP\n\n#### Practice questions\n\n##### Question 4\n\nKathleen has just won $\\$20000$$20000. When she retires in 2121 years, she wants to have \\52000$$52000 in her fund which earns $8%$8% interest per annum.\n\nHow much of her winnings, to the nearest cent, does she need to invest now to achieve this?\n\n## Other compounding periods\n\nWhat do we do if the interest is being compounded more frequently;\u00a0perhaps daily, weekly, monthly, quarterly or semi-annually?\n\nCompound interest formula (other compounding periods)\n\n$A=P\\left(1+\\frac{r}{n}\\right)^{nt}$A=P(1+rn)nt\n\nwhere:\n\n$A$A is the final amount of money (principal and interest together)\n\n$P$P is the principal (the initial amount of money invested)\n\n$r$r\u00a0is the interest rate per year, expressed as a decimal or fraction\n\n$n$n\u00a0is the number of compounding periods in a year\n\n$t$t is the number of years\n\nNotice that since $n$n is the number of compounding periods in a year, $\\frac{r}{n}$rn is the interest rate per compounding period, and $nt$nt is the total number of compounding periods.\n\n#### Worked example\n\nSuppose $\\$500$$500 is invested in a compound interest account with an interest rate of 10%10% per year compounded semi-annually (that is, with a compounding period of 66 months) for 33 years. Since the interest is being compounded semi-annually, the number of compounding periods in a year is n=2.n=2. The interest rate is 10%10% per year and so the interest rate per compounding period as a decimal is \\frac{r}{n}=\\frac{0.01}{2}=0.005rn=0.012=0.005 . Moreover, in 33 years, there are a total of nt=2\\times3=6nt=2\u00d73=6 compounding periods. Now we can substitute into the formula: AA == P\\left(1+\\frac{r}{n}\\right)^{nt}P(1+rn\u200b)nt == 500\\times\\left(1+\\frac{0.01}{2}\\right)^{2\\times3}500\u00d7(1+0.012\u200b)2\u00d73 == 500\\times\\left(1+0.005\\right)^6500\u00d7(1+0.005)6 \\approx\u2248 \\670.05$$670.05\n\nFor comparison, if the $\\$500$$500 is invested in a compound interest account with an interest rate of 10%10% per year compounded annually for 33 years, then AA == P\\left(1+r\\right)^tP(1+r)t == 500\\times\\left(1+0.1\\right)^3500\u00d7(1+0.1)3 \\approx\u2248 \\665.50$$665.50\nCompounding more frequently produces more interest!\n\nDid you know?\n\nThe two versions of the compound interest formula that we have - one for annual compounding and one for other compounding periods - actually come from the same formula:\n\n$A=P(1+R)^N$A=P(1+R)N\n\nwhere:\n\n$R$R is the interest rate per compounding period, expressed as a decimal or fraction\n\n$N$N is the number of compounding periods.\n\n#### Practice questions\n\n##### Question 7\n\nCharlie is expecting a Christmas bonus of $\\$20002000 in $6$6 months time. What is the most he can borrow now, $x$x in dollars, at a rate of $3.9%$3.9% per year compounded daily, and still be able to pay off the loan with his bonus?\n\n1. Assume there are $365$365 days in a year\n\n### Outcomes\n\n#### A.CED.A.2'\n\nCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 'Linear and exponential (integer inputs only)\n\n#### F.BF.A.1'\n\nWrite a function that describes a relationship between two quantities. 'Linear and exponential (integer inputs)\n\n#### F.BF.A.1.A'\n\nDetermine an explicit expression, a recursive process, or steps for calculation from a context. 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\section{Introduction}
Neutrinos offer unique insight into the Universe due to the fact that they interact only weakly and via gravity. Unlike charged particles, they can travel straight from the source to the Earth without being deflected by magnetic fields or being absorbed. Neutrinos are expected to originate from the same locations where the acceleration of cosmic rays take place \citep{Origin6, Origin4, Origin3, Origin2, Origin5, Origin}. A large variety of classes of astrophysical objects are predicted to be sources of high energy neutrinos, where galactic candidates include microquasars \citep{MicroQ1, MicroQ2, MicroQ3, MicroQ4}, supernova remnants \citep{SNRs1, SNRs2, SNRs3, SNRs4, SNRs5, SNRs6, SNRs7}, or various objects close to the Galactic Center \citep{SgrA, GC}. Extragalactic sources comprise active galactic nuclei \citep{AGN1, AGN2, AGN3, AGN4, AGN5, AGN6, AGN7, AGN8, AGN9} and gamma ray bursts \citep{GRB1, GRB2, GRB3, GRB4, GRB5, GRB6, GRB7}.
The low neutrino cross section also implies that their detection is challenging. After the pioneering efforts by the Baikal \citep{baikal} and AMANDA \citep{amanda} collaborations, the field is presently led by the IceCube \citep{icecube} and ANTARES \citep{AntDetect} experiments. IceCube, which is placed in the deep Antartic ice, is the first detector to reach the cubic-kilometer size predicted to be necessary to detect cosmic neutrino fluxes according to the Waxman-Bahcall flux \citep{Waxman}. Recently, IceCube has reported the crucial discovery of a flux of neutrinos up to $\sim$\,PeV energies which cannot be explained by the background of atmospheric muons and neutrinos only \citep
IceCube_HESE_2yr, IceCube_HESE_3yr}. The specific origin of these events is currently unknown. Some authors propose that at least part of the flux may have a galactic origin \citep{GAL-IC, GAL-IC2, GAL-IC3, GAL-IC4, GAL-IC5, GAL-IC6, GAL-IC7}, whereas others have focused on the extragalactic component \citep{EXTRAGAL1, EXTRAGAL2, EXTRAGAL3, EXTRAGAL4}. Meanwhile the ANTARES experiment has proven the feasibility of the Cherenkov telescope technique in sea water \citep{ANTARES-OSCI, ANTARES-MUAT}.
While its instrumented volume is significantly smaller than that of IceCube, its geographical location provides a view of the Southern sky with significantly reduced background for neutrino energies below 100 TeV, and hence better sensitivity to many predicted Galactic sources of neutrinos in this part of the sky. The complementarity of the detectors with respect to Southern sky sources, due to their different geographical location, size and atmospheric muon background, allows for a gain in sensitivity by combining the analyses of data from both experiments in a joint search for point-like sources.
The level of improvement depends on the details of the assumed astrophysical flux, in particular on its energy spectrum and the existence of a possible high-energy cut-off. The energy spectra are not yet known and predictions vary widely depending on the source model.
In this paper, a combined analysis using the point-source data samples of IceCube from 2008-2011 and of ANTARES from 2007-2012 is presented. This paper is structured as follows: in Section~\ref{ANTARESdet}, the IceCube and ANTARES detectors are introduced. In Section~\ref{sample} the samples from each experiment are described, while in Section~\ref{method} the search method is explained. Finally, the results are presented in Section~\ref{results} and the conclusions are discussed in Section~\ref{conclusions}.
\section{The IceCube and ANTARES neutrino telescopes}\label{ANTARESdet}
IceCube is a cubic-kilometer neutrino telescope located at the geographic South Pole. It consists of a total of 5160 digital optical modules (DOMs) deployed in the Antarctic ice at depths from 1450\,m to 2450\,m below the surface \citep{IceCube_DOM}. Each DOM consists of a pressure-resistant sphere that houses electronics, calibration LEDs, and a 10'' PMT facing downward. The DOMs are configured in a hexagonal array of 86 vertical cables descending from the surface, called ``strings'', with 60 DOMs per string. The average horizontal distance is 125\,m between strings, and average vertical spacing is 17\,m between DOMs on a string. A sub-array of eight strings (Deep Core) is also present in the core of the detector \citep{DeepCore}. These strings have a smaller separation in order to improve the sensitivity for lower energies. Construction of the detector began in 2005 and was completed six years later. The analysis presented in this paper is based on data from three years of the partially completed detector, when 40, 59, and 79 strings were deployed. Future joint analyses are envisioned that will be based on data from the full 86-string detector, including recent data samples that use outer detector modules as vetoes to achieve sensitivity to lower energy neutrinos.
ANTARES is the first neutrino telescope which operates in the sea \citep{AntDetect}. It was completed in 2008, with the first lines operating from 2006. It is located in the Mediterranean Sea at a depth of 2475 m, at coordinates (42$^\circ$ 48' N, 6$^\circ$ 10' E), 40 km South of Toulon (France). It consists of an array of 885 Optical Modules (OMs) distributed along 12 lines of 350 m height and an inter-line separation of 60 to 75 m. An OM consists of a 10'' photomultiplier tube (PMT) contained inside a 17'' glass sphere. The OMs are grouped into triplets and face downward at an angle of 45$^\circ$ in order to optimize the detection of up-going muon-neutrinos. There are 25 triplets (storeys) on each line, with a distance of about 15 m between storeys. Lines are kept vertical with a buoy at their top.
One of the main focuses of the ANTARES and IceCube neutrino telescopes is the observation of cosmic point-like sources of neutrino emission. At present, corresponding searches are mainly focused on the detection of muon neutrinos, which can be reconstructed with sub-degree angular resolution. Muon neutrinos are indirectly detected through the muon produced in their charged current interaction (CC) with a nucleus (N) inside or near to the detector volume:
\begin{equation}
\nu_\mu + N \rightarrow \mu^- + X
\end{equation}
In this reaction, a muon and a hadronic shower, X, are produced.\footnote{In this work the charge conjugate particles and reactions will be implicitly included, i.e. in this case the reaction $\bar{\nu}_\mu + N \rightarrow \mu^+ + X$ is also assumed.} The ultra-relativistic muon can travel long distances (up to several kilometers) and, when crossing a suitable medium such as ice or water, induce Cherenkov radiation that can be detected by the photomultipliers (PMTs) of neutrino telescopes. The corresponding charge and time information of the detected photons is used to reconstruct the direction of the muon, which is almost collinear with the original neutrino for energies above the TeV range. The main backgrounds for cosmic neutrino searches are atmospheric muons and neutrinos produced in the decay of the secondary particles created in the interactions of cosmic rays with the nuclei of the atmosphere.
\section{Neutrino Data Samples}\label{sample}
The data sample employed for this analysis corresponds to all events from the Southern sky which were included in the three-year IceCube point-source analysis \citep{PS-IceCube-79} combined with the events in the latest ANTARES point-source analysis \citep{PS-ANTARES}. The ANTARES sample contains data recorded from Jan 29, 2007 to Dec 31, 2012; for IceCube the data was recorded from Apr 5, 2008 to May 13, 2011 with the partially completed detector, and without the use of the Deep Core strings.
\begin{figure}[!ht]
\centering
\plottwo{Aeff.pdf}{angres.pdf}
\caption{Muon neutrino effective area for a point source at a declination $\delta$ = --30$^\circ$ (left) and median angular resolution (right) for the samples used in this analysis after the final set of cuts. The median angular resolution is defined as the median of the difference between the true neutrino direction and the reconstructed muon direction.}
\label{Aeff}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=.8\textwidth]{SensBoth}
\caption{90\% CL limits for selected sources (squares and dots) and sensitivities using the Neyman method as a function of the declination (lines) reported in the ANTARES 2007-2012 (blue) \citep{PS-ANTARES} and the IceCube 3 years (red) \citep{PS-IceCube-79} point source analyses. An unbroken $E^{-2}$ power-law source spectrum is assumed for the limits and lower sensitivity curves (solid lines). Dashed lines indicate the sensitivity for an $E^{-2}$ spectrum with neutrino energies of $E_\nu \leq$ 100 TeV using the Neyman method. }
\label{sensComp}
\end{figure}
Detector performance differs not only between ANTARES and IceCube, but also between the three IceCube configurations as the detector grew, from 40, to 59, and then to 79 strings. The effective area is defined as the equivalent surface with a perfect efficiency which detects the same number of events as the detector.
For a source position of $\delta$ = --30$^\circ$, the effective area for each IceCube configuration and for ANTARES is shown in Figure \ref{Aeff}-left.
Due to its larger size, the effective area for the IceCube samples is larger for neutrino energies above $\sim$ 100 TeV. However, to view sources in the Southern Sky, IceCube must contend with the down-going background of atmospheric muons, which becomes overwhelming at lower energies. To minimize these, the IceCube point-source analysis introduced a declination-dependent energy cut which strongly suppresses low energy events in the final data sample. ANTARES, which can use the Earth as a filter against atmospheric muons in the Southern sky, thus maintains a larger effective area in this energy and declination range.
A comparison of the median angular resolution of each sample can be seen in Figure \ref{Aeff}-right. The better resolution of the ANTARES sample is due to the longer photon scattering length in water compared to ice. The sensitivities reported by both experiments for the whole sky using the Neyman method \citep{Neyman} are shown in Figure \ref{sensComp}.
Different selection criteria are applied to each sample. A summary of these selections, which in all cases were developed with a data blinding policy and were optimised to minimise the neutrino flux needed for a 5$\sigma$ discovery in 50\% of the experiments, is given below.
\subsection{ANTARES}
The ANTARES data sample used for this analysis corresponds to the events coming from the Southern Sky used in the last published point source analysis \citep{PS-ANTARES}. The parameters which are used to optimise this sample are the quality of the track fit, $\Lambda$, the angular error estimate, $\sigma$ (also denoted as $\beta$ in most ANTARES publications), and the zenith angle, $\theta$. These three parameters are given by the track reconstruction of neutrino events, which uses a maximum likelihood (ML) method \citep{AartThesis, OtherAart}. The algorithm is based on a multi-step procedure to fit the direction of the reconstructed muon by maximising the $\Lambda$ parameter. The angular error estimate, $\sigma$, is obtained from the uncertainty on the zenith and azimuth angles extracted from the covariance matrix.
The selection yields a total of 5516 events for the whole sky, with 4136 of these events in the Southern Hemisphere. The estimated contamination of mis-reconstructed atmospheric muons is 10\%.
\subsection{IceCube}
\begin{deluxetable}{ccccc}
\tablewidth{0pt}
\tablecaption{Event samples for the different IceCube detector configurations, labelled by the number of strings deployed. Only Southern-sky events (numbers indicated by last column) have been selected for the present analysis. }
\tablehead{
\colhead{Sample} & \colhead{Start date} & \colhead{End date} & \colhead{Livetime [days]} & \colhead{\# events}
}
\startdata
IC-40 & 2008 Apr 5 & 2009 May 20 & 376 & 22 779\\
IC-59 & 2009 May 20 & 2010 May 31 & 348 & 64 230\\
IC-79 & 2010 May 31 & 2011 May 13 & 316 & 59 009\\
\enddata
\label{tab:icecube_events}
\end{deluxetable}
The IceCube data samples used for this analysis are based on the event selection optimized for point source searches with the data recorded using the 40, 59, and 79-string detector configurations, summarized in Table~\ref{tab:icecube_events}. Only events from the Southern sky are selected here for the joint analysis. In contrast to the ANTARES selection above, IceCube's Southern sky events are predominantly atmospheric muons rather than atmospheric neutrinos, because the Earth cannot be used as a neutrino filter for directions above the detector.
The total number of down-going events in IceCube is $\sim 10^{10}$ per year. The down-going events that were selected as part of the above analyses and which are used here comprise only well-reconstructed muon tracks at very high energies, where it becomes possible to detect a neutrino source with a hard $E^{-2}$ energy spectrum beyond the more steeply falling atmospheric muon background, and from clustering of events in a single region of the sky. For the 40 string configuration, a set of cuts on the reduced log-likelihood of the track reconstruction, the angular uncertainty, $\sigma$, and the muon energy proxy is performed for events coming from the Southern Sky \citep{PS-IceCube-40}. For the 59 string configuration, the vetoing capability of IceTop is added to reduce the background of atmospheric muons \citep{PS-IceCube-79}. For the 79 string configuration, the event selection is performed based on boosted decision trees using 17 observables, and includes the use of the IceTop veto.
The total number of Southern sky events selected from the three year sample is 146\,018 events.
\subsection{Relative fraction of source events for different source assumptions}\label{Sec:Relative}
The relative fraction of expected source events from each sample needs to be calculated in order to estimate its respective weight in the likelihood which will be used to search for an excess of events from a particular direction (see Section \ref{method}). This fraction is defined as the ratio of the expected number of signal events for the given sample to that for all samples,
\begin{equation}
C^j (\delta, {d\Phi}/{dE_\nu} ) = \frac{N^j(\delta, {d\Phi}/{dE_\nu})}{\sum_{i} N^i(\delta, {d\Phi}/{dE_\nu})} ,
\end{equation}
where the total number of expected events for the $j$-th sample, $N^j$, with a given source declination, $\delta$, and a given source spectrum, $\frac{d\Phi}{dE_\nu}$, can be calculated as
\begin{equation}\label{eq:acc}
N^j \left(\delta, \frac{d\Phi}{dE_\nu} \right) = \int dt \int dE_\nu A^{j}_{\rm eff}(E_\nu,\delta) \frac{d\Phi}{dE_\nu} \; .
\end{equation}
The time integration extends over the live time of each sample and $A^{j}_{\mathrm{eff}}(E_\nu, \delta)$ indicates the effective area of the corresponding detector layout $j$ as a function of the neutrino energy, $E_\nu$, and the declination of the source, $\delta$.
Since each detector layout has a different response depending on the neutrino energy and declination, this relative fraction of source events needs to be calculated for different source spectra and source declinations. Figure \ref{RelContX20} shows the relative fraction of signal events for an unbroken $E^{-2}$ spectrum, which corresponds to vanilla first order Fermi acceleration \citep{Fermi1, Fermi2}. In this case, there is a significant contribution from all samples over most of the Southern Sky, with the ANTARES contribution being more significant for declinations closer to $\delta$ = --90$^\circ$, and IceCube for declinations closer to 0$^\circ$. The reason for this variability is mostly due to the declination-dependent energy cut applied in the IceCube samples to reduce the background of atmospheric muons.
Other source assumptions are also considered in this analysis. The relative fraction of source events is calculated for an unbroken $E^{-2.5}$ power-law spectrum, as suggested in recent IceCube diffuse-flux searches \citep{IceCube-Diffuse}, and for an E$^{-2}$ spectrum with exponential square-root cut-offs
($\frac{d\Phi}{dE} \propto E^{-2} \exp \left[ { - \sqrt{ \frac{E}{E_{\rm{cut-off}} } } } \right] $)
of 100 TeV, 300 TeV and 1 PeV, since a square-root dependence may be expected from Galactic sources \citep{KAPPES}. Figure \ref{RelContCases} shows the relative fraction of source events for these cases. Compared with an unbroken $E^{-2}$ spectrum, the contribution of high energy neutrinos in all of these cases is lower, and therefore the relative contribution of the ANTARES sample increases.
\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{RelCont_X20_M-1}
\caption{Relative fraction of signal events for each sample as a function of the source declination for the case of an $E^{-2}$ energy spectrum. The orange, blue, and yellow shaded areas correspond to the IceCube 40, 59 and 79-string data samples, respectively, and the green shaded area indicates the ANTARES 2007-2012 sample. The vertical dashed line marks the declination of the Galactic Center. }
\label{RelContX20}
\end{figure}
\begin{figure}[!h]
\centering
\plottwo{RelCont_X20_M1000000}{RelCont_X20_M300000}
\plottwo{RelCont_X20_M100000}{RelCont_X25_M-1}
\caption{Relative fraction of signal events of each sample as a function of the source declination for different energy spectra: $E^{-2}$ with energy cutoff $E_{\rm cutoff}$ of 1 PeV (top-left), 300 TeV (top-right), 100 TeV (bottom-left); and $E^{-2.5}$ spectrum (bottom-right). The orange, blue and yellow shaded areas correspond to the IceCube 40, 59 and 79-string data samples, respectively, and the green shaded area indicates the ANTARES 2007-2012 sample. The vertical dashed line corresponds to the declination of the Galactic Center.}
\label{RelContCases}
\end{figure}
\section{Search method}\label{method}
An unbinned maximum likelihood ratio estimation has been performed to search for excesses of events that would indicate cosmic neutrinos coming from a common source. In order to estimate the significance of a cluster of events, this likelihood takes into account the energy and directional information of each event. Due to the different detector response, the data sample which an event belongs to is also taken into account. The likelihood, as a function of the total number of fitted signal events, $n_s$, can be expressed as:
\begin{equation}\label{eq:likelihood}
L(n_s) = \prod_{j=1}^4 \prod_{i=1}^{N^j} \left[ \frac{n^j_s}{N^j}S^j_i + \left(1-\frac{n^j_s}{N^j} \right) B^j_i \right]
\end{equation}
\par \noindent where $j$ marks one of the four data samples, $i$ indicates an event belonging to the $j$-th sample, S$_i^j$ is the value of the signal probability distribution function (PDF) for the $i$-th event, B$^j_i$ indicates the value of the background PDF, $N^j$ is the total number of events in the $j$-th sample, and $n^j_s$ is the number of signal events fitted for in the $j-$th sample. Since a given evaluation of the likelihood refers to a single source hypothesis at a fixed sky location, the number of signal events $n^j_s$ that is fitted for in each sample is related to the total number of signal events $n_s$ by the relative contribution of each sample, $n^j_s = n_s \cdot C^{j}(\delta, \frac{d{\Phi}}{dE})$.
The signal and background PDFs for the IceCube and ANTARES samples have slightly different definitions. The signal PDF for ANTARES is defined as
\begin{equation}
S^{ANT} = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{\Delta \Psi(\vec{x}_s)^2}{2\sigma^2}\right) P^{ANT}_{s}(\mathcal{N}^{hits}, \sigma) ,
\end{equation}
\par \noindent where $\vec{x_s}$ = ($\alpha_s$, $\delta_s$) indicates the source direction in equatorial coordinates, $\Delta \Psi(\vec{x}_s)$ is the angular distance of a given event to the source and P$_s^{ANT}(\mathcal{N}^{hits}, \sigma)$ is the probability for a signal event to be reconstructed with an angular error estimate of $\sigma$ and a number of hits $\mathcal{N}^{hits}$. The number of hits is a proxy for the energy of the event. In this sense, an event with a higher number of hits (higher deposited energy) would be less likely to be produced by the expected background.
The definition of the signal PDFs for the IceCube samples is similar,
\begin{equation}
S^{IC} = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{\Delta \Psi(\vec{x}_s)^2}{2\sigma^2}\right) P^{IC}_{s}(\mathcal{E}, \sigma|\delta)
\end{equation}
\par \noindent where the main difference lies in the use of the reconstructed energy, $\mathcal{E}$, and the declination dependence of the probability for a signal event to be reconstructed with a given $\sigma$ and $\mathcal{E}$. The declination dependence is needed mainly because of the event selection cut on reconstructed energy, which is designed to reduce the atmospheric muon background.
Background events are expected to be distributed uniformly in right ascension. The background PDFs are in fact obtained from the experimental data itself. The definition of the PDFs are:
\begin{equation}
B^{ANT} = \frac{B^{ANT}(\delta)}{2\pi} P^{ANT}_{b}(\mathcal{N}^{hits}, \sigma) ,
\end{equation}
\begin{equation}
B^{IC} = \frac{B^{IC}(\delta)}{2\pi} P^{IC}_{b}(\mathcal{E}, \sigma|\delta) ,
\end{equation}
\par \noindent where B$^j$($\delta$) is the per-solid-angle rate of observed events as a function of the declination in the corresponding sample. $P^{ANT}_{b}(\mathcal{N}^{hits}, \sigma)$ and $P^{IC}_{b}(\mathcal{E}, \sigma|\delta)$ characterize the distributions for background event properties, in analogy with the definitions of $P^{ANT}_{s}$ and $P^{IC}_{s}$ for signal events given above.
The test statistic, TS, is determined from the likelihood (Eq.~\ref{eq:likelihood}) as TS = $\log L(\hat{n}_s) - \log L(n_s=0)$, where $\hat{n}_s$ is the value that maximizes the likelihood. The larger the TS, the lower the probability (p-value) of the observation to be produced by the expected background. Simulations are performed to obtain the distributions of the TS. The significance (specifically, the p-value) of an observation is determined by the fraction of TS values which are larger than the observed TS.
The TS is calculated as a preliminary step to obtain the post-trial p-values of a search. TS distributions for the fixed-source, background-only hypothesis have been calculated in steps of 1$^\circ$ in declination from pseudo-data sets of randomized data. Because these distributions vary with declination, the preliminary TS is turned into a ``pre-trial p-value'' by comparing the TS obtained at the source location from the data to the background TS distribution for the corresponding declination. The post-trial significance is then estimated with pseudo-data sets and according to the type of search, as explained together with the results in Section \ref{results}.
Two different searches for point-like neutrino sources have been performed. In the candidate list search, a possible excess of neutrino events is looked for at the location of 40 pre-selected neutrino source candidates. Since the location of these sources is fixed (at known locations with an uncertainty below the angular resolution of all samples) only the number of signal events $n_s$ is a free parameter in the likelihood maximisation. These candidates correspond to all sources in the Southern sky considered in the previous candidate-source list searches performed in the ANTARES and IceCube point-source analyses \citep{PS-IceCube-79, PS-ANTARES}.
The second search is a ``full sky'' search, looking for a significant point-like excess anywhere in the Southern sky. For this purpose, the likelihood is evaluated in steps of 1$^\circ \times $~1$^\circ$ over the whole scanned region. In this case, the source position is an additional free parameter of the likelihood to fit the best position within the 1$^\circ \times$~1$^\circ$ boundaries.
Both the full Southern-sky and candidate-list searches have been performed using an $E^{-2}$
source spectrum in the signal PDFs. The main virtue of the energy term in the PDFs is to add power to distinguish signal neutrinos from the softer spectra of atmospheric neutrinos and atmospheric muons. Limits for the sources in the candidate list have also been calculated for the source spectra mentioned in Section \ref{Sec:Relative}.
\section{Results}\label{results}
Results from the full Southern sky and candidate list searches are detailed below.
\subsection{Full Southern-sky search}
No significant event clusters are found over the expected background. The most significant cluster is located at equatorial coordinates $\alpha$ = 332.8$^\circ$, $\delta$=--46.1$^\circ$, with best-fit $n_s=7.9$ and pre-trial p-value of $6.0\times10^{-7}$. Figure \ref{fullskysearch} shows this pre-trial p-value compared to the distribution of smallest p-values found when performing the same analysis on many pseudo-data sets (constructed by randomizing the right ascension coordinates of the real data). It is found that 24\% of pseudo-data sets have a smaller p-value somewhere in the sky than is found in the real data; the post-trial significance is thus 24\% (0.7$\sigma$ in the one-sided sigma convention). The direction of this cluster is consistent with, but also less significant than the second most significant cluster in the previous ANTARES point-source analysis. Figure \ref{pretrialmap} shows the position of this cluster and the pre-trial p-values for all directions in the Southern sky (a smaller step of 0.2$^\circ \times$0.2$^\circ$ is used to plot this map).
\begin{figure}[!h]
\centering
\includegraphics[width=.75\textwidth]{FullSkySearch}
\caption{Distribution of the smallest p-value in the Southern sky obtained from scans of pseudo-data sets. Green line: pre-trial p-value for the most significant source location found. Yellow and red lines: pre-trial p-values for the 2$\sigma$ and 3$\sigma$ significance thresholds using the one-sided sigma convention.}
\label{fullskysearch}
\end{figure}
\begin{figure}[!h]
\centering
\plotone{skymap}
\caption{Skymap of pre-trial p-values for the combined ANTARES 2007-2012 and IceCube 40, 59, 79 point-source analyses. The red circle indicates the location of the most significant cluster ($0.7\sigma$ post-trial significance in the one-sided sigma convention), discussed in the text.}
\label{pretrialmap}
\end{figure}
\subsection{Candidate list search}
The results of the candidate source list search are presented in Table\,\ref{tab:CL}. No statistically significant excess is found. The most significant excess for any object on the list corresponds to HESS\,J1741-302 with a pre-trial p-value of 0.003.
To account for trial factors, the search is performed on the same list of sources using pseudo data-sets, and the distribution of smallest p-values for these searches is shown in Figure\,\ref{CLsearch}. We find that 11\% of randomized data sets have a smaller p-value for any source than that found for the real data; the post-trial significance of the source list search is thus 11\% ($1.2\sigma$ in the one-sided sigma convention).
\begin{figure}[!h]
\centering
\includegraphics[width=.75\textwidth]{CandidateListSearch}
\caption{Distribution of the smallest p-value found in each candidate-list analysis of a pseudo-data set. Green line: pre-trial p-value for the most significant object found in the real data. Yellow and red lines: pre-trial p-values needed for the 2 and 3$\sigma$ post-trial significance thresholds in the one-sigma convention.}
\label{CLsearch}
\end{figure}
\begin{figure}[!htp]
\centering
\includegraphics[width=.8\textwidth]{sensitivity_X20}
\caption{90\% CL sensitivities and limits (Neyman method) for the neutrino emission from point sources as a function of source declination in the sky, for an assumed $E^{-2}$ energy spectrum of the source. Green points indicate the actual limits on the candidate sources. The green line indicates the sensitivity of the combined search. Curves/points respectively indicate the published sensitivities/limits for the IceCube (blue) and ANTARES (red) analyses, respectively. As reference, the declination of the Galactic Center is approximately at sin($\delta = -29^\circ$) $\approx$ -0.48. }
\label{sensX20}
\end{figure}
Table \ref{tab:CL} provides the pre-trial p-values, best-fit signal events $n_s$ and flux upper limits (under different assumptions of the energy spectrum) for all the candidate source objects. Figure \ref{sensX20} shows the Neyman sensitivities and limits for this search (assuming an E$^{-2}$ spectrum) in comparison with the previously published ANTARES and IceCube analyses of the same data. The point-source sensitivity in a substantial region of the sky, centered approximately at the declination of the Galactic center ($\delta=-29^{\circ}$), can be seen to have improved by up to a factor of two. A maximum gain of at most $\sqrt{2}$ would be expected in a background-dominated sample; however, the low number of effective background events (with reconstructed energy and direction mimicking an astrophysical neutrino) is very low, so that gains of more than $\sqrt{2}$ are possible. Similar gains in other regions of the sky can be seen for different energy spectra in Figure \ref{sensCases}.
\begin{figure}[!tph]
\centering
\plottwo{sensitivity_X20_M1000000}{sensitivity_X20_M300000}
\plottwo{sensitivity_X20_M100000}{sensitivity_X25_M-1}
\caption{Point source sensitivities and limits as in Fig.\,\ref{sensX20}, for other energy spectra: $E^{-2}$ with a square-root exponential cut-off at $E = 1$\,PeV (top left), $E = 300$\,TeV (top right), $E = 100$\,TeV (bottom left) and $E^{-2.5}$ unbroken power-law (bottom right). Green points indicate the actual limits on the candidate sources. The green line indicates the sensitivity for the combined search. Red and blue curves indicate the sensitivities for the individual IceCube and ANTARES analyses, respectively. As reference, the declination of the Galactic Center is approximately at sin($\delta = -29^\circ$) $\approx$ -0.48. }
\label{sensCases}
\end{figure}
\begin{deluxetable}{lccccccccc}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{\footnotesize Fitted number of source events, $n_s$, pre-trial p-values, $p$, and 90\% C.L. flux limits, $\Phi^{90CL}_{\nu}$ for the different source spectra for the 40 candidate sources. Units for the flux limits for the E$^{-2.5}$ spectra, $\phi^{90CL}_{E^{-2.5}}$, are given in GeV$^{1.5}$cm$^{-2}$s$^{-1}$, whereas the rest are in GeV cm$^{-2}$s$^{-1}$. The sources are sorted by their declination. Dashes (-) in the fitted number of source events and pre-trial p-values indicate sources with $n_s \leq$ 0.001. }
\tablehead{
\colhead{Name} & \colhead{$\delta$ ($^\circ$)} &
\colhead{$\alpha$ ($^\circ$)} & \colhead{$n_s$ } &
\colhead{$p$} & \colhead{$\phi^{90CL}_{ E^{-2}}$ } &
\colhead{$\phi^{90\%CL}_{E_{c} = 1PeV}$ } & \colhead{$\phi^{90CL}_{E_{c} = 300 TeV}$ } &
\colhead{$\phi^{90CL}_{E_{c} = 100 TeV}$} & \colhead{$\phi^{90CL}_{E^{-2.5}}$ } }
\startdata
3C279 & -5.8 & -166.0 & 1.1 & 0.05 & 3.1E-09 & 1.0E-06 & 6.5E-09 & 9.2E-09 & 6.7E-08\\
HESSJ1837-069 & -7.0 & -80.6 & - & - & 1.6E-09 & 9.3E-07 & 1.5E-08 & 2.0E-08 & 2.6E-08\\
QSO2022-077 & -7.6 & -53.6 & - & - & 1.9E-09 & 1.7E-06 & 1.7E-08 & 2.5E-08 & 3.5E-08\\
PKS1406-076 & -7.9 & -147.8 & - & - & 2.2E-09 & 7.7E-07 & 4.3E-09 & 6.7E-09 & 1.0E-08\\
HESSJ1834-087 & -8.8 & -81.3 & - & - & 2.2E-09 & 1.1E-06 & 1.6E-08 & 2.1E-08 & 2.9E-08\\
PKS0727-11 & -11.7 & 112.6 & - & - & 3.0E-09 & 9.9E-07 & 1.5E-08 & 5.1E-09 & 8.9E-09\\
1ES0347-121 & -12.0 & 57.4 & - & - & 3.7E-09 & 2.5E-06 & 1.8E-08 & 2.6E-08 & 3.8E-08\\
QSO1730-130 & -13.1 & -96.7 & - & - & 3.3E-09 & 2.3E-06 & 1.8E-08 & 2.6E-08 & 3.8E-08\\
LS5039 & -14.8 & -83.4 & - & - & 4.2E-09 & 2.1E-06 & 1.1E-08 & 1.7E-08 & 2.9E-08\\
W28 & -23.3 & -89.6 & - & - & 6.3E-09 & 2.8E-06 & 1.7E-08 & 2.5E-08 & 4.0E-08\\
PKS0454-234 & -23.4 & 74.3 & - & - & 7.4E-09 & 2.2E-06 & 2.6E-08 & 2.7E-08 & 3.8E-08\\
1ES1101-232 & -23.5 & 165.9 & - & - & 6.4E-09 & 2.6E-06 & 7.2E-09 & 1.2E-08 & 2.1E-08\\
Galactic Center & -29.0 & -93.6 & - & - & 7.6E-09 & 2.6E-06 & 1.8E-08 & 2.6E-08 & 3.8E-08\\
PKS1622-297 & -29.9 & -113.5 & - & - & 8.9E-09 & 2.6E-06 & 1.7E-08 & 2.2E-08 & 2.9E-08\\
HESSJ1741-302 & -30.2 & -94.8 & 1.6 & 0.003 & 2.5E-08 & 7.5E-06 & 5.5E-08 & 7.2E-08 & 1.0E-07\\
PKS2155.304 & -30.2 & -30.3 & - & - & 7.8E-09 & 2.6E-06 & 2.0E-08 & 2.8E-08 & 4.5E-08\\
H2356-309 & -30.6 & -0.2 & - & - & 7.9E-09 & 1.5E-06 & 1.5E-08 & 2.1E-08 & 3.0E-08\\
PKS0548-322 & -32.3 & 87.7 & 0.9 & 0.07 & 1.6E-08 & 5.0E-06 & 3.8E-08 & 4.9E-08 & 1.4E-08\\
PKS1454-354 & -35.7 & -135.6 & - & - & 8.9E-09 & 3.5E-06 & 8.6E-09 & 2.1E-08 & 3.0E-08\\
PKS0426-380 & -37.9 & 67.2 & - & - & 8.6E-09 & 2.8E-06 & 7.5E-09 & 1.2E-08 & 2.0E-08\\
RXJ1713.7-3946 & -39.8 & -101.8 & - & - & 8.7E-09 & 2.0E-06 & 1.5E-08 & 2.0E-08 & 2.6E-08\\
CenA & -43.0 & -158.6 & - & - & 8.1E-09 & 2.2E-06 & 4.0E-09 & 6.1E-09 & 1.2E-08\\
PKS0537-441 & -44.1 & 84.7 & - & - & 8.2E-09 & 1.6E-06 & 1.8E-08 & 2.6E-08 & 4.1E-08\\
VelaX & -45.6 & 128.8 & - & - & 8.3E-09 & 2.2E-06 & 1.5E-08 & 2.1E-08 & 2.9E-08\\
RXJ0852.0-4622 & -46.4 & 133.0 & - & - & 9.5E-09 & 2.1E-06 & 1.5E-08 & 2.1E-08 & 2.9E-08\\
HESSJ1632-478 & -47.8 & -112.0 & - & - & 8.6E-09 & 2.1E-06 & 1.6E-08 & 2.1E-08 & 2.9E-08\\
PKS2005-489 & -48.8 & -57.6 & - & - & 1.0E-08 & 2.9E-06 & 1.7E-08 & 1.3E-08 & 2.2E-08\\
GX339-4 & -48.8 & -104.3 & - & - & 8.7E-09 & 2.2E-06 & 1.6E-08 & 2.1E-08 & 2.8E-08\\
HESSJ1616-508 & -51.0 & -116.0 & - & - & 1.1E-08 & 2.3E-06 & 1.8E-08 & 2.2E-08 & 3.0E-08\\
HESSJ1614-518 & -51.8 & -116.4 & - & - & 9.3E-09 & 2.1E-06 & 1.6E-08 & 2.0E-08 & 2.7E-08\\
CirX-1 & -57.2 & -129.8 & - & - & 9.1E-09 & 2.1E-06 & 1.8E-08 & 2.7E-08 & 3.8E-08\\
HESSJ1023-575 & -57.8 & 155.8 & 0.8 & 0.08 & 1.7E-08 & 3.5E-06 & 2.8E-08 & 3.5E-08 & 4.7E-08\\
HESSJ1503-582 & -58.7 & -133.6 & - & - & 9.1E-09 & 2.0E-06 & 1.5E-08 & 1.9E-08 & 2.6E-08\\
MSH15-52 & -59.2 & -131.5 & - & - & 9.1E-09 & 2.1E-06 & 1.5E-08 & 2.0E-08 & 2.7E-08\\
ESO139-G12 & -59.9 & -95.6 & 0.8 & 0.07 & 1.8E-08 & 3.9E-06 & 2.9E-08 & 3.7E-08 & 5.1E-08\\
HESSJ1507-622 & -62.3 & -133.3 & - & - & 9.1E-09 & 2.0E-06 & 5.0E-09 & 8.0E-09 & 1.4E-08\\
RCW86 & -62.5 & -139.3 & 0.2 & 0.11 & 1.4E-08 & 4.4E-06 & 3.6E-09 & 4.0E-08 & 5.7E-08\\
HESSJ1303-631 & -63.2 & -164.2 & - & - & 9.1E-09 & 2.0E-06 & 1.5E-08 & 1.9E-08 & 2.6E-08\\
PSRB1259-63 & -63.5 & -164.3 & - & - & 9.1E-09 & 2.4E-06 & 1.7E-08 & 2.4E-08 & 3.3E-08\\
HESSJ1356-645 & -64.5 & -151.0 & - & - & 9.1E-09 & 2.0E-06 & 1.5E-08 & 1.9E-08 & 2.6E-08\\
\enddata
\label{tab:CL}
\end{deluxetable}
\section{Conclusion}\label{conclusions}
We have presented the first combined point-source analysis of data from the ANTARES and IceCube detectors. Their different characteristics, in particular IceCube's larger size and ANTARES' location in the Northern hemisphere, complement each other for Southern sky searches. We have calculated the sensitivity to point sources and, with respect to an analysis of either data set alone, found that up to a factor of two improvement is achieved in different regions of the Southern sky, depending on the energy spectrum of the source. Two joint analyses of the data sets have been performed: a search over the whole Southern sky for a point-like excess of neutrino events, and a targeted analysis of 40 pre-selected candidate source objects. The largest excess in the Southern sky search has a post-trial probability of 24\% (significance of 0.7$\sigma$), located at $\alpha$ = 332.8$^\circ$, $\delta$=--46.1$^\circ$ in equatorial coordinates. In the source list search, the candidate with the highest significance corresponds to HESS\,J1741-302, with a post-trial probability of 11\% (significance of 1.2$\sigma$). Both of the results are compatible with the background-only hypothesis. Flux upper limits for each of the source candidates have been calculated for $E^{-2}$ and $E^{-2.5}$ power-law energy spectra, as well as for $E^{-2}$ spectra with cut-offs at energies of 1\,PeV, 300\,TeV, and 100\,TeV. Because of their complementary nature, with IceCube providing more sensitivity at higher energies and ANTARES at lower energies, a joint analysis of future data sets will continue to provide the best point-source sensitivity in critical overlap regions in the Southern sky, where neutrino emission from Galactic sources in particular may be found.
\acknowledgments
The authors of the ANTARES collaboration acknowledge the financial support of the funding agencies:
Centre National de la Recherche Scientifique (CNRS), Commissariat \`a
l'\'ener\-gie atomique et aux \'energies alternatives (CEA),
Commission Europ\'eenne (FEDER fund and Marie Curie Program), R\'egion
\^Ile-de-France (DIM-ACAV) R\'egion Alsace (contrat CPER), R\'egion
Provence-Alpes-C\^ote d'Azur, D\'e\-par\-tement du Var and Ville de La
Seyne-sur-Mer, France; Bundesministerium f\"ur Bildung und Forschung
(BMBF), Germany; Istituto Nazionale di Fisica Nucleare (INFN), Italy;
Stichting voor Fundamenteel Onderzoek der Materie (FOM), Nederlandse
organisatie voor Wetenschappelijk Onderzoek (NWO), the Netherlands;
Council of the President of the Russian Federation for young
scientists and leading scientific schools supporting grants, Russia;
National Authority for Scientific Research (ANCS), Romania;
Mi\-nis\-te\-rio de Econom\'{\i}a y Competitividad (MINECO), Prometeo
and Grisol\'{\i}a programs of Generalitat Valenciana and MultiDark,
Spain; Agence de l'Oriental and CNRST, Morocco. We also acknowledge
the technical support of Ifremer, AIM and Foselev Marine for the sea
operation and the CC-IN2P3 for the computing facilities.
The authors of the IceCube collaboration acknowledge the support from the following agencies:
U.S. National Science Foundation-Office of Polar Programs,
U.S. National Science Foundation-Physics Division,
University of Wisconsin Alumni Research Foundation,
the Grid Laboratory Of Wisconsin (GLOW) grid infrastructure at the University of Wisconsin - Madison, the Open Science Grid (OSG) grid infrastructure;
U.S. Department of Energy, and National Energy Research Scientific Computing Center,
the Louisiana Optical Network Initiative (LONI) grid computing resources;
Natural Sciences and Engineering Research Council of Canada,
WestGrid and Compute/Calcul Canada;
Swedish Research Council,
Swedish Polar Research Secretariat,
Swedish National Infrastructure for Computing (SNIC),
and Knut and Alice Wallenberg Foundation, Sweden;
German Ministry for Education and Research (BMBF),
Deutsche Forschungsgemeinschaft (DFG),
Helmholtz Alliance for Astroparticle Physics (HAP),
Research Department of Plasmas with Complex Interactions (Bochum), Germany;
Fund for Scientific Research (FNRS-FWO),
FWO Odysseus programme,
Flanders Institute to encourage scientific and technological research in industry (IWT),
Belgian Federal Science Policy Office (Belspo);
University of Oxford, United Kingdom;
Marsden Fund, New Zealand;
Australian Research Council;
Japan Society for Promotion of Science (JSPS);
the Swiss National Science Foundation (SNSF), Switzerland;
National Research Foundation of Korea (NRF);
Danish National Research Foundation, Denmark (DNRF).
{\it Facilities:} \facility{ANTARES, IceCube}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 303 |
Risto Antero Jarva (* 15. Juli 1934 in Helsinki; † 16. Dezember 1977 ebenda) war ein finnischer Filmregisseur.
Leben
Jarva absolvierte eine Ausbildung zum Diplomingenieur, engagierte sich jedoch früh im Filmbereich. Er experimentierte mit avantgardistischen Kurzfilmen und wandte sich dann Nouvelle-Vague-Filmen zu. Es folgten Filme des Alltagsrealismus, eine Science fiction-Phantasie, ein Melodram, eine Komödie und schließlich eine Tragikomödie.
Jarva kam bei einem Unfall ums Leben.
In den Jahren 1966, 1968, 1971 und 1978 erhielt er jeweils den finnischen Filmpreis Jussi in der Kategorie Beste Regie.
Filmografie
1962: Yö vai päivä
1965: Onnenpeli
1967: Työmiehen päiväkirja
1969: Ruusujen aika
1971: Benzin im Blut (Bensaa suonissa)
1972: Kun taivas putoaa
1973: Einmannkrieg (Yhden miehen sota)
1975: Mies joka ei osanut sanoa ei
1976: Loma
1977: Jäniksen vuosi
Literatur
Sakari Toiviainen: Jarva, Risto. In: Olli Alho (Hrsg.): Kulturlexikon Finnland. 2. Auflage. Finnische Literaturgesellschaft, Helsinki 1999, , S. 153.
Filmregisseur
Finne
Geboren 1934
Gestorben 1977
Mann
Künstler (Helsinki)
Jussi-Preisträger | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,807 |
Q: Regarding Sorting MultiDimensional Arrays in C# I am trying to figure out a way to correctly sort a bunch of different arraylists.
I am publishing content articles and every value [0] in an arraylist will relate to every other value [0]. and so on. Each element makes up the collective parts of a complete content item.
Now, the last element, popularity, is the amount of clicks an item has received. How do I
do a sort of the content items based on popularity without mixing up the html for each article?
*EDIT I am limited by the .NET 2.0 Framework at Work*
Below is the code... thanks.
public class MultiDimDictList : Dictionary<string, ArrayList> { }
myDicList.Add("fly", a_fly);
myDicList.Add("img", a_img);
myDicList.Add("bar", a_bar);
myDicList.Add("meter", a_meter);
myDicList.Add("block", a_block);
myDicList.Add("popularity", a_pop);
A: If you use the following code you can convert your existing dictionary of arraylists into a collection of Dictionaries and thus allowing a simple sort using Linq OrderBy
// Get the shortest arraylist length (they should be equal this is just a paranoia check!)
var count=myDicList.Values.Min(x=>x.Count);
// Get the collection of Keys
var keys=myDicList.Keys;
// Perform the conversion
var result=Enumerable.Range(0,count).Select(i=>keys.Select(k=>new {Key=k,Value=myDicList[k][i]}).ToDictionary(x=>x.Key,x=>x.Value));
var sorted=result.OrderByDescending(x=>x["popularity"]).ToList()
-- EDIT VERSION FOR .NET 2.0
First you need a comparer class
class PopularityComparison : IComparer<Dictionary<string,object>> {
private bool _sortAscending;
public PopularityComparison(bool sortAscending) {
_sortAscending = sortAscending;
}
public int Compare(Dictionary<string, object> x, Dictionary<string, object> y) {
object xValue = x["popularity"];
object yValue = y["popularity"];
// Sort Ascending
if (_sortAscending) {
return Comparer.Default.Compare(xValue, yValue);
} else {
return Comparer.Default.Compare(yValue, xValue);
}
}
}
Then you can use the following code
// Get the shortest arraylist length (they should be equal this is just a paranoia check!)
// Replacement for min
int count = int.MaxValue;
foreach (ArrayList a in myDicList.Values) if (a.Count < count) count = a.Count;
// Get the collection of Keys
Dictionary<string, ArrayList>.KeyCollection keys = myDicList.Keys;
// Perform the conversion
List<Dictionary<string, object>> result = new List<Dictionary<string, object>>(count);
for (int i = 0; i < count; i++) {
Dictionary<string, object> row = new Dictionary<string, object>(keys.Count);
foreach (string key in keys) row.Add(key, myDicList[key][i]);
result.Add(row);
}
And then finally to sort in ascending popularity order
result.Sort(new PopularityComparison(true));
or Descending order
result.Sort(new PopularityComparison(true));
A: I'd think it would be better to have an object containing your keys as properties, then a single collection with each item you'd have in your array lists.
This way you'd have a single collection sort, which becomes trivial if using Linq.OrderBy().
something like...
public class Article
{
public string Fly{get;set;}
public string Img{get;set;}
// etc.
public float Popularity{get;set;}
}
Then...
List<Article> articles = ... get from somewhere, or convert from your array lists.
List<Article> sorted = articles.OrderBy(a=>a.Popularity).ToList();
Please excuse the napkin code here... I'll update it if you need more detail.
An example using non-linq.
Create an implementation of IComparer.
public class ArticleComparer : IComparer<Article>
{
public bool Accending { get; set; }
public int Compare(Article x, Article y)
{
float result = x.Popularity - y.Popularity;
if (!Accending) { result *= -1; }
if (result == 0) { return 0; }
if (result > 0) return 1;
return -1;
}
}
Then when you go to sort the List, you can do something like the following.
ArticleComparer comparer = new ArticleComparer();
comparer.Accending = false;
articles.Sort(comparer);
A: This would be much easier if you had a list of article objects, each of which contained properties for fly, img, bar, popularity, etc. But if you really have to store things using this inside-out approach, then the only way you can sort the content items based on popularity is to create another array (or list) to hold the order.
Create a new list and populate it with sequential indexes:
List<int> OrderedByPopularity = new List<int>();
ArrayList popList = myDicList["popularity"];
for (int i = 0; i < popList.Count; ++i)
{
OrderedByPopularity.Add(i);
}
Now you have a list that contains the indexes of the items in the popularity list. Now you can sort:
OrderedByPopularity.Sort((i1, i2) => return popList[i1].CompareTo(popList[i2]););
But that gives you the least popular article first. If you want to reverse the sort so that OrderedByPopularity[0] is the most popular item:
OrderedByPopularity.Sort((i1, i2) => { return popList[i2].CompareTo(popList[i1]);});
Really, though, you should look into restructuring your application. It's much easier to work with objects that have properties rather than trying to maintain parallel arrays of properties.
If you have to do this in .NET 2.0, declare the poplist array at class scope (rather than method scope), and create a comparison method.
ArrayList poplist;
void MyMethod()
{
List<int> OrderedByPopularity = new List<int>();
popList = myDicList["popularity"];
for (int i = 0; i < popList.Count; ++i)
{
OrderedByPopularity.Add(i);
}
OrderedByPopularity.Sort(PopularityComparison);
// ...
}
int PopularityComparison(int i1, int i2)
{
return ((int)popList[i2]).CompareTo((int)popList[i1]);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 590 |
$(document).ready(function() {
navigation = $('#menu-holder').children('ul');
options = '<option value="" selected>-Main Menu-</option>';
navigation.find('li').each(function() {
link = $(this).children('a');
depth = $(this).parents('ul').length - 1;
indent = '';
if( depth ) {
while( depth > 0 ) {
indent += ' - ';
depth--;
}
}
options += '<option value="' + link.attr('href') + '">' + indent + ' ' + link.text() + '</option>';
}).end().after('<select class="mobile-navigation">' + options + '</select>');
$('.mobile-navigation').on('change', function() {
window.location = jQuery(this).val();
});
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,270 |
from __future__ import absolute_import
import pytz
from mock import patch
from sentry.testutils import AcceptanceTestCase, SnubaTestCase
from sentry.utils.samples import load_data
from sentry.testutils.helpers.datetime import iso_format, before_now
FEATURE_NAMES = ("organizations:events-v2", "organizations:discover-v2-query-builder")
all_view = "field=title&field=event.type&field=project&field=user&field=timestamp&alias=title&alias=type&alias=project&alias=user&alias=time&name=All+Events&sort=-timestamp&tag=event.type&tag=release&tag=project.name&tag=user.email&tag=user.ip&tag=environment"
error_view = "field=title&alias=error&field=count%28id%29&alias=events&field=count_unique%28user%29&alias=users&field=project&alias=project&field=last_seen&alias=last+seen&name=Errors&query=event.type%3Aerror&sort=-last_seen&sort=-title&tag=error.type&tag=project.name"
class OrganizationEventsV2Test(AcceptanceTestCase, SnubaTestCase):
def setUp(self):
super(OrganizationEventsV2Test, self).setUp()
self.user = self.create_user("foo@example.com")
self.org = self.create_organization(owner=None, name="Rowdy Tiger")
self.team = self.create_team(organization=self.org, name="Mariachi Band")
self.project = self.create_project(organization=self.org, teams=[self.team], name="Bengal")
self.create_member(user=self.user, organization=self.org, role="owner", teams=[self.team])
self.login_as(self.user)
self.path = u"/organizations/{}/eventsv2/".format(self.org.slug)
def wait_until_loaded(self):
self.browser.wait_until_not(".loading-indicator")
self.browser.wait_until_not('[data-test-id="loading-placeholder"]')
def test_all_events_empty(self):
with self.feature(FEATURE_NAMES):
self.browser.get(self.path + "?" + all_view)
self.wait_until_loaded()
self.browser.snapshot("events-v2 - all events empty state")
@patch("django.utils.timezone.now")
def test_all_events(self, mock_now):
mock_now.return_value = before_now().replace(tzinfo=pytz.utc)
min_ago = iso_format(before_now(minutes=1))
self.store_event(
data={
"event_id": "a" * 32,
"message": "oh no",
"timestamp": min_ago,
"fingerprint": ["group-1"],
},
project_id=self.project.id,
assert_no_errors=False,
)
with self.feature(FEATURE_NAMES):
self.browser.get(self.path + "?" + all_view)
self.wait_until_loaded()
self.browser.snapshot("events-v2 - all events")
@patch("django.utils.timezone.now")
def test_errors(self, mock_now):
mock_now.return_value = before_now().replace(tzinfo=pytz.utc)
min_ago = iso_format(before_now(minutes=1))
self.store_event(
data={
"event_id": "a" * 32,
"message": "oh no",
"timestamp": min_ago,
"fingerprint": ["group-1"],
},
project_id=self.project.id,
assert_no_errors=False,
)
self.store_event(
data={
"event_id": "b" * 32,
"message": "oh no",
"timestamp": min_ago,
"fingerprint": ["group-1"],
},
project_id=self.project.id,
assert_no_errors=False,
)
self.store_event(
data={
"event_id": "c" * 32,
"message": "this is bad.",
"timestamp": min_ago,
"fingerprint": ["group-2"],
},
project_id=self.project.id,
assert_no_errors=False,
)
with self.feature(FEATURE_NAMES):
self.browser.get(self.path + "?" + error_view)
self.wait_until_loaded()
self.browser.snapshot("events-v2 - errors")
@patch("django.utils.timezone.now")
def test_modal_from_all_events(self, mock_now):
mock_now.return_value = before_now().replace(tzinfo=pytz.utc)
min_ago = iso_format(before_now(minutes=1))
event_data = load_data("python")
event_data.update(
{
"event_id": "a" * 32,
"timestamp": min_ago,
"received": min_ago,
"fingerprint": ["group-1"],
}
)
event = self.store_event(
data=event_data, project_id=self.project.id, assert_no_errors=False
)
with self.feature(FEATURE_NAMES):
# Get the list page.
self.browser.get(self.path + "?" + all_view)
self.wait_until_loaded()
# Click the event link to open the modal
self.browser.element('[aria-label="{}"]'.format(event.title)).click()
self.wait_until_loaded()
header = self.browser.element('[data-test-id="modal-dialog"] h2')
assert event_data["message"] in header.text
issue_link = self.browser.element('[data-test-id="linked-issue"]')
issue_event_url_fragment = "/issues/%s/events/%s/" % (event.group_id, event.event_id)
assert issue_event_url_fragment in issue_link.get_attribute("href")
self.browser.snapshot("events-v2 - single error modal")
@patch("django.utils.timezone.now")
def test_modal_from_errors_view(self, mock_now):
mock_now.return_value = before_now().replace(tzinfo=pytz.utc)
event_source = (("a", 1), ("b", 39), ("c", 69))
event_ids = []
event_data = load_data("javascript")
event_data["fingerprint"] = ["group-1"]
for id_prefix, offset in event_source:
event_time = iso_format(before_now(minutes=offset))
event_data.update(
{
"timestamp": event_time,
"received": event_time,
"event_id": id_prefix * 32,
"type": "error",
}
)
event = self.store_event(data=event_data, project_id=self.project.id)
event_ids.append(event.event_id)
with self.feature(FEATURE_NAMES):
# Get the list page
self.browser.get(self.path + "?" + error_view + "&statsPeriod=24h")
self.wait_until_loaded()
# Click the event link to open the modal
self.browser.element('[aria-label="{}"]'.format(event.title)).click()
self.wait_until_loaded()
self.browser.snapshot("events-v2 - grouped error modal")
# Check that the newest event is loaded first and that pagination
# controls display
display_id = self.browser.element('[data-test-id="event-id"]')
assert event_ids[0] in display_id.text
assert self.browser.element_exists_by_test_id("older-event")
assert self.browser.element_exists_by_test_id("newer-event")
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,443 |
Looking for some easy options for Easter brunch?
I never thought I would miss the sound of my little mini-pinscher's toenails tik-tik-tikkying across the hardwood floor.
When we bought our house in Sandpoint, Idaho in 2002, there were no moose, and there were a lot of tulips. People who lived outside of town had to build fences to protect their gardens, but not us. We had bugs and slugs. That was it. Moose lived out in the woods where moose belonged, and we rarely spotted them. That changed one snowy morning.
Pre-arranging your funeral is an everlasting gift for your family. They will know exactly what you want without you having to find the perfect time to bring up such a difficult subject.
Multi-generational households are on the rise with a record one-in-five Americans living in homes shared by multiple adult generations, inevitably leading to various domestic conflicts — including thermostat wars over home temperatures.
For pure love of teaching and learning about art — no paycheck involved.
Learn to transform a gourd into artwork at the Idaho Gourd Society's 19th Annual Gourd Festival and Gourd Sale on October 27 and 28 in Boise.
They may be itty bitty, but the joy and hope they bring to children, teens, and seniors is enormous. A team of three miniature horses and one mini donkey serves as healers in animal-assisted therapy programs and as friends in animal-assisted activity programs around Idaho's Treasure Valley.
Pizza is often a disappointedly bready affair where the delicious promise of sauce and toppings gets drowned in a doughy sea of blandness. But have you ever had a thin, crispy, cracker crust pizza before? Now that's a thing of beauty. | {
"redpajama_set_name": "RedPajamaC4"
} | 457 |
This week, we brought more than just a bouquet of flowers to Suze's angel monument. I brought a print of my finished portrait of her, too, framed as durably as I knew how. The real thing was hanging a block away at the Graveside, in the entranceway to Mrs. Gautier's Suzanna White exhibition. Mrs. Gautier had loved it. She had practically smiled at the sight of it. Even I had to admit it made the perfect entrance to the exhibit, hanging beside Suze's own self-portrait.
I placed our bouquet next to the print and spared only the quickest glance for the once-white lilies, tied with a single red ribbon, that lay, decayed, at Suze's feet. They used to always be fresh and glowing, but I had a feeling I knew why that had changed. They were from James. Now he was in jail, waiting for his trial. This bouquet had died, and there would be no new ones to take its place.
Sighing, I looked up at Suze's serene, marble face. I didn't know why Suze made Owen wait as long as she did-why she'd made both of us wait. Maybe it was in her nature to allow a certain amount of suffering. Or maybe some things had always been outside her control. It didn't matter anymore; I could only be grateful to her for bringing us together, after all this time.
Owen's hand found mine, and I squeezed his fingers possessively.
Ferdy bounded up to Owen and me and nosed at my pockets. I smiled as I slipped him another biscuit. "That dog is becoming your shadow," Owen remarked.
"My huge, fluffy, drooling shadow."
The sadness from the graveyard slipped away from us both as we walked out from underneath the pine boughs, onto the sunny sidewalks of Church Street. The rest of the day spread out before us like a fresh canvas. We had many days like this now. Sometimes we went to the beach or hung out at the pub with Kaye and Andy and maybe Rusty or Muscles. Even Miserable Margot joined us every once in a while, though she wasn't quite so miserable anymore.
Other days, I would paint while Owen tinkered with one of his instruments. We often went outside since Owen was still rebuilding his workshop and since Fall Island had so many secret meadows and sea caves and forest clearings. But sometimes we stayed in. Owen had framed a few of my paintings-I was slowly building up my portfolio for my application to art school-and had hung them on the walls in Suze's old room. It felt like a real room now, a cozy, happy place that you wanted to spend time in. So we did. I thought Suze would have been glad about it.
"You know," Owen said, as we turned onto Main Street, "you don't have to take your easel back with you every time you leave my place."
"What if I want to paint in the middle of the night?"
With a laugh, he pulled me close and kissed me. "Only you would worry about that."
I tugged away from him and walked nonchalantly down the block. "Did I tell you Kaye is moving out in September? Andy is talking about getting his own place when that happens, maybe downtown."
"Yeah?" I could hear the tension in his voice.
"I was thinking I don't want to find three new housemates. But that means I'll need a new place to live," I continued airily. "I'd be sad to leave my little attic, but I suppose the right place could be worth it."
He stopped walking, and I turned to face him, my cheeks warm with joy.
"Are you sure you'll be ready?" he asked me quietly. "I don't want-"
"I'm sure," I told him. "I'm absolutely sure."
He seized me by the shoulders and kissed me hard, dazzlingly. When he drew back, a smile crooked the corner of his beautiful mouth.
A little breathless, I realized that we were standing in the middle of Main Street, with Ferdy snuffling around our feet. A group of tourists, huge topo maps in hand, pointed and giggled at us.
"Is there room for a huge, fluffy, drooling shadow?" I asked, ignoring the tourists. "Your mom said she wants to give him to me, when he's better trained, but I-"
"Of course there's room," Owen said. "It's getting hard to separate you two, anyway."
"And you can tolerate how messy I am?" I asked, with a pang of worry.
Owen laughed as he kissed my ear, sending a shiver of pleasure down my spine. "I'm sure you'll make me crazy, with paintbrushes and lipsticks everywhere." He moved from my ear to my neck.
"And you'll make me crazy," I told him, grinning and wrapping my arms around him, "using a T-square to make sure your carpet is laid out straight."
"I don't even own a T-square."
"You totally do," I said, and he laughed again, his breath warm against my neck.
"I actually have several." He kissed my collarbone. I tried to wriggle out of his grip, aware that Bob the grocer was giving us an annoyed look, but he held me in place. "Miranda," he murmured, into my throat, "I want to sleep next to you at night, and make mallorcas with you in the morning, and trip over your sketchbooks. I want to live with you more than anything, no matter how messy you are or how many dogs you bring home with you. But only if you're ready."
"I'm ready," I told him again. And I was. But I loved him even more for wanting to be sure.
"Get a room, you two!" someone called out good-naturedly, and Owen and I both laughed. I glanced over and saw Chrissy, who ran the liquor store, waving cheerfully at us, next to a few more grinning, pointing tourists. We waved back.
A professionally edited and overall much-improved edition of Set Me Free is now available in stores, as an ebook and in print, for those of you who would like to buy it! ❤️ Check out my profile for links to all major retailers.
If you'd like to see more of my writing, you can follow me on Wattpad, Twitter, or tumblr, or sign up my newsletter on my website. I always have lots of projects in the works! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,875 |
Canyon View High School (also known as CV and CVHS) was established in 1997 in Cedar City, Utah. Its mascot is the Falcon. The school colors are black, silver, and teal. It has approximately 1,000 students and 50 faculty members. Its current principal is Dennis Heaton. The assistant principal is Julie King.
This school is in Iron County School District, Region 9, and is a 4A school.
History
Not knowing the future growth of the school, in 1999 the Iron County School District released that they were building a new Canyon View High School building just across the street from the already newly built school. In the fall of 2000, students were finally able to enter the current building of Canyon View.
Boundaries
Canyon View High School boundaries currently include the northern half of Cedar City, as well as Enoch, Newcastle, Beryl, and other parts of Iron County.
Notable alumni
Mitch Talbot - MLB pitcher
Shayne Smith (comedian) - Stand-up comedian
References
External links
Public high schools in Utah
Educational institutions established in 1997
Schools in Iron County, Utah
Buildings and structures in Cedar City, Utah
1997 establishments in Utah | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,059 |
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Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,055 |
require 'test_helper'
# Tests specific to yellow pages assets in general
class UserYellowPagesTest < ActiveSupport::TestCase
test 'user_creatable' do
assert !Person.user_creatable?, 'Profiles should not be user creatable'
assert !Project.user_creatable?, 'Projects should not be user creatable'
assert !Institution.user_creatable?, 'Institutions should not be user creatable'
end
test 'are yellow pages' do
assert Person.is_yellow_pages?
assert Project.is_yellow_pages?
assert Institution.is_yellow_pages?
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,552 |
Q: DB entity not recognized after Update Model from DataBase in Visual Studio 2017 I added some new columns to a table in Db and when I try to update Model from Database, my entity is not recognized. I have read something that is a bug from the 2017 version of VS. How can it be fixed?
I have encoutered the same problem with Refresh and with Add after I delete the tables from the edmx file.
A: Solved by Installing VS 2015. Apparently this version allows to Update from DB without errors.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,279 |
\section{Introduction}\label{sec:intro}
An \emph{$n$--chain link} consists of $n$ unknotted circles embedded
in $S^3$, linked together in a closed chain. Notice that links of a
chain can be connected with an arbitrary amount of twisting. In
particular, if we embed the first link in the plane of projection, the
next perpendicular to the plane of projection, the next again in the
plane of projection, and so on, then the last link may include any
integer number of half--twists. See, for example, Figure
\ref{fig:chain-link1}.
\begin{figure}[h!]
\includegraphics{figures/6-chain}
\hspace{.5in}
\includegraphics{figures/7-chain}
\hspace{.5in}
\includegraphics{figures/6-chain-twist}
\caption{Left: Minimally twisted $6$--chain link. Middle:
Minimally twisted $7$--chain link. Right: $6$--chain
link with more half--twists.}
\label{fig:chain-link1}
\end{figure}
Hyperbolic structures on $n$--chain link complements have been
studied, for example, by Neumann and Reid
\cite{neumann-reid:arithmetic}. They show any $n$--chain link
complement with $n\geq 5$ admits a hyperbolic structure. In this
paper, we are primarily interested in hyperbolic manifolds, so we
restrict our attention to $n\geq 5$.
A \emph{minimally twisted $n$--chain link} is an $n$--chain link such
that, if $n$ is even, each link component alternates between lying
embedded in the projection plane and lying perpendicular to the
projection plane. If $n$ is odd, the link may be arranged such that
each component alternates between lying in the projection plane and
perpendicular to it, except a single link component which connects a
link which is embedded in the projection plane to one which is
perpendicular, with no twisting. See Figure \ref{fig:chain-link1}.
Notice that there are actually two choices for the minimally twisted
$n$--chain link for $n$ odd, depending on which way the last links are
connected. However, these are isometric by an orientation reversing
isometry, so we will not distinguish between them.
In \cite{agol:min-vol}, Agol conjectures that minimally twisted
$n$--chain link complements, for $n\leq 10$, are the smallest volume
hyperbolic 3--manifolds with exactly $n$ cusps, but notes that Venzke
has pointed out they cannot be smallest for $n\geq 11$, as the
$(n-1)$--fold cyclic cover over one component of the Whitehead link
has smaller volume. This statement is included in Venzke's thesis
\cite{venzke:thesis}. However, Venzke does not give a proof. In this
paper, we give a rigorous proof for $n\geq 60$. The following theorem
is the main result of this paper.
\begin{named}{Theorem \ref{thm:not-minvolume}}
For $n\geq 60$, the minimally twisted $n$--chain link complement has
volume strictly greater than that of the $(n-1)$--fold cyclic cover
over one component of the Whitehead link. Hence the minimally twisted
$n$--chain link complement cannot be the smallest volume hyperbolic
3--manifold with $n$ cusps, $n\geq 60$.
\end{named}
For $n$ between $11$ and $59$, inclusive, we present computer
tabulation of volumes, compared with the volumes of the $(n-1)$--fold
cyclic cover of the Whitehead link. See Table \ref{table:volumes}.
By inspection, Theorem \ref{thm:not-minvolume} also holds for these
manifolds. In Section \ref{sec:compute}, we explain how these
computations can be made completely rigorous --- at least for those
values of $n$ for which the minimally twisted $n$--chain link
complement is triangulated with fewer than $100$ tetrahedra. In this
case, this includes $n$ between 12 and 25, inclusive.
Finally, in Section \ref{sec:arblinks} we evaluate volumes of
arbitrarily twisted $n$--chain link complements. The main result of
that section is Theorem \ref{thm:vol-withtwist}, which states that no
$n$--chain link complement can be the minimal volume $n$--cusped
hyperbolic 3--manifold, provided either $n\geq 60$, or the chain link
contains at least $17$ half--twists. We present computational data to
show that similarly, for $11 \leq n \leq 59$, no $n$--chain link
complement can be minimal volume. When $5 \leq n \leq 10$, we
rigorously prove, by computer, that no $n$--chain link which is
\emph{not} minimally twisted can be the minimal volume $n$--cusped
hyperbolic 3--manifold.
\begin{remark}
Since a version of this paper was made public, Hidetoshi Masai has
pointed out to us that for even chain links, more can be said. In
\cite[Chapter 6]{thurston}, Thurston finds a formula for the volumes
of minimally twisted chain links with an even number of link
components. Namely,
$${\rm vol}(S^3\setminus C_{2n}) = 8n \left( \Lambda\left(\frac{\pi}{4} +
\frac{\pi}{2n}\right) + \Lambda\left(\frac{\pi}{4} -
\frac{\pi}{2n}\right)\right),$$ where $\Lambda$ is the Lobachevsky
function. Masai notes that the difference of this volume and the
volume of the $(2n-1)$--cyclic cover over a component of the Whitehead
link is an increasing function in $n$, for $n \geq 6$. This result
will give a rigorous proof that minimally twisted $n$--chain links are
not minimal volume for $17$ additional chain links, namely for $n =
26, 28, 30, \dots, 56, 58$. In addition, this gives an alternate
proof that minimally twisted $2n$--chain links are not minimal volume
for larger $n$. However, the result for odd links requires other
techniques, for instance those in this paper.
\end{remark}
\subsection{Acknowledgements}
We thank Ian Agol and Hidetoshi Masai for helpful conversations. We
also thank Peter Milley, Harriet Moser, and Saul Schleimer for their
assistance with computational aspects of this project. Authors Kaiser
and Rollins were both supported in part by Brigham Young University
mentoring funds for undergraduate research. All authors were
supported in part by a grant from the National Science Foundation.
\section{Slope lengths on covers}\label{sec:slopes}
To prove the main theorems of this paper, for $n\geq 60$, we will
obtain the complement of the minimally twisted $n$--chain link by Dehn
filling a manifold $\widehat{W}_n$ which is geometrically explicit,
constructed by gluing together manifolds isometric to the Whitehead
link complement, cut along 2--punctured disks.
We work with the diagram of the Whitehead link as in Figure
\ref{fig:w1}, left, with a link component denoted $K$. Note that by
switching the direction of a pair of crossings, we obtain a link whose
complement is isometric to that of the Whitehead link complement by an
orientation reversing isometry. The isometry takes $K$ to a link
component we denote $\overline{K}$, as in Figure \ref{fig:w1}, right.
\begin{figure}
\input{figures/Whitehead.pdf_t} \hspace{.5in}
\input{figures/Whitehead-refl.pdf_t}
\caption{The Whitehead link (left), and its reflection (right), with
component labeled $K$ and $\overline{K}$, respectively.}
\label{fig:w1}
\end{figure}
\begin{lemma}
The shape of the cusp of $K$ is a parallelogram with meridian and
longitude meeting at angle $-\pi/4$ (measured from meridian to
longitude).
Similarly, the shape of the cusp of $\overline{K}$ is a parallelogram with
meridian and longitude meeting at angle $\pi/4$ (measured from meridian
to longitude).
When we take a maximal horocusp about $K$ or $\overline{K}$, the
meridian has length $\sqrt{2}$, and the longitude has length $4$.
\label{lemma:k_shape}
\end{lemma}
Lemma \ref{lemma:k_shape} is illustrated in
Figure \ref{fig:whitehead-cusp}.
\begin{figure}
\includegraphics{figures/cusp-whitehead}
\caption{Cusp shape of component $K$ of Whitehead link. Meridian runs
horizontally across the top, longitude runs diagonally.}
\label{fig:whitehead-cusp}
\end{figure}
\begin{proof}
The first part of the lemma, and lengths of slopes on $K$, are well
known for the Whitehead link. See, for example
\cite{neumann-reid:arithmetic}.
As for $\overline{K}$, the orientation reversing isometry taking the
Whitehead link to its reflection takes a meridian of $K$ to a meridian
of $\overline{K}$, and reflects the longitude. Since this is an
isometry, the meridian and longitude of $\overline{K}$ have the same
lengths as those of $K$, but the angle from the meridian to longitude
is reflected across the meridian, to be $\pi/4$.
\end{proof}
Now, the manifold $\widehat{W}_n$ can be described as the minimally
twisted $n$--chain link embedded in a standard solid torus.
Therefore, to obtain the complement of the minimally twisted
$n$--chain link, we will Dehn fill $\widehat{W}_n$ along a standard
longitude of the solid torus boundary component.
To build $\widehat{W}_n$ from the Whitehead link complement, proceed
as follows. First, cut the Whitehead link complement along the
2--punctured disk bounded by $K$ to get a clasp in a cylinder, which
we call $W_1$, shown second from left in Figure
\ref{fig:wn-construct}. Similarly, cut the reflected Whitehead link
complement along the 2--punctured disk bounded by $\overline{K}$ to
get a clasp in the opposite direction in a cylinder, which we call
$\overline{W}_1$, shown third from left in Figure
\ref{fig:wn-construct}.
\begin{figure}
\includegraphics{figures/wn-construct}
\caption{Constructing the manifold $\widehat{W}_n$: cut the Whitehead
link complement along a 2--punctured disk to obtain $W_1$ (second from
left). Its reflection is $\overline{W}_1$ (third from left). Attach
these to form a link in a solid cylinder, right.}
\label{fig:wn-construct}
\end{figure}
Now, attach a copy of $W_1$ to $\overline{W}_1$ via an isometry of the
2--punctured disk as on the right in Figure \ref{fig:wn-construct}.
In particular, boundary components are glued as shown without
twisting. Call the resulting link in a solid cylinder $W$.
For $n$ even, glue ${n}/{2}$ copies of $W$ together end to end,
without twisting, followed by gluing the remaining two ends. For $n$
odd, glue $(n-1)/{2}$ copies of $W$ together without twisting,
then glue a single copy of $W_1$, and attach the ends without
twisting. This completes the construction of $\widehat{W}_n$.
\begin{lemma}
Let $\epsilon = n\mod{2}$. The minimally twisted $n$--chain link in a
solid torus, $\widehat{W}_n$, has solid torus boundary component
comprised of $\lfloor n/2 \rfloor + \epsilon$ copies of the
cusp $K$ coming from $W_1$, and $\lfloor n/2 \rfloor$ copies
of the cusp $\overline{K}$ coming from $\overline{W}_1$. The standard
longitude of the solid torus follows a meridian of each copy of $K$
and $\overline{K}$, where the meridians of each copy of $K$ are
orthogonal to the meridians of each copy of $\overline{K}$. The
length of the longitude of the solid torus boundary component is
$\sqrt{n^2 + \epsilon}$.
\label{lemma:torus_boundary_component}
\end{lemma}
\begin{proof}
This follows from the construction of $\overline{W}_n$ and
Lemma \ref{lemma:k_shape}.
The boundary component corresponding to the solid torus comes from
$n/2$ copies of the cusp $K$ and $n/2$ copies of the
cusp $\overline{K}$ for $n$ even, and $(n-1)/2 + 1$ copies of
the cusp $K$ and $(n-1)/2$ copies of the cusp $\overline{K}$,
for $n$ odd. These are glued together along their respective
longitudes. Since a copy of the cusp of $K$ is glued to one of
$\overline{K}$ along the longitude of each, the meridians meet at
right angles. See Figure \ref{fig:cusp-wnbar}.
\begin{figure}
\input{figures/cusp-constr.pdf_t}
\caption{The cusp construction of $\widehat{W_n}$.}
\label{fig:cusp-wnbar}
\end{figure}
The longitude of the solid torus of $\widehat{W_n}$ is given by
following each meridian of the copies of $K$ and $\overline{K}$ that
glue to give the solid torus boundary component. Since these
meridians always meet at right angles, the length of the longitude of
the solid torus may be determined by the Pythagorean theorem. By
Lemma \ref{lemma:k_shape}, the length of the meridian of the cusp $K$,
and that of the cusp $\overline{K}$, is $\sqrt{2}$. We see that the
length of the longitude of the solid torus boundary component of
$\widehat{W}_n$ is $\sqrt{\left(\sqrt{2} \left( \lfloor \frac{n}{2} \rfloor
+ \epsilon \right)\right)^2 + \left( \sqrt{2} \lfloor \frac{n}{2} \rfloor
\right)^2} = \sqrt{n^2 + \epsilon}$.
\end{proof}
Notice that while the construction of $\widehat{W}_n$ as described
above uses $\lfloor n/2 \rfloor + \epsilon$ copies of $W_1$
and $\lfloor n/2 \rfloor$ copies of $\overline{W}_1$, we could
have constructed it using $\lfloor n/2 \rfloor$ copies of
$W_1$ and $\lfloor n/2 \rfloor + \epsilon$ copies of
$\overline{W}_1$ instead. The result of this modified method of
construction would be isometric to that of the original construction,
by an orientation reversing isometry.
To obtain $C_n$ from $\widehat{W}_n$ we Dehn fill a slope on the solid
torus boundary component of $\widehat{W}_n$ that follows one standard
longitude of the solid torus. Hence we Dehn fill along a slope of
length $\sqrt{n^2 + (n\mod{2})}$.
\section{Volumes, large minimally twisted chains}\label{sec:volumes}
Using the information on slopes above, we may deduce geometric
information on minimally twisted chain links by applying appropriate
theorems bounding change in geometry under Dehn filling. This will
give the desired result when $n\geq 60$.
\subsection{Dehn filling and volume}
We use the following theorem, which is a slightly simpler version of
the main theorem in \cite{fkp:dfvjp}.
\begin{theorem}[Futer--Kalfagianni--Purcell \cite{fkp:dfvjp}]
Let $M$ be a complete, finite--volume hyperbolic manifold with (at
least one) cusp, with horoball neighborhood $C$ about that cusp, and
let $s$ be a slope on $\partial C$ with length $\ell(s) > 2\pi$.
Then the manifold $M(s)$ obtained by Dehn filling $M$ along $s$ is
hyperbolic, with volume:
$$ {\rm vol}(M(s)) \: \geq \:
\left(1-\left(\frac{2\pi}{\ell(s)}\right)^2\right)^{3/2} {\rm vol}(M).$$
\label{thm:fkp}
\end{theorem}
Putting this theorem together with Lemma
\ref{lemma:torus_boundary_component}, we obtain the following.
\begin{theorem}
For $n\geq 7$, the volume of the complement of the minimally twisted
$n$--chain link $C_n$ is at least:
$${\rm vol}(S^3\setminus C_n) \geq n\, v_8 \, \left( 1 - \frac{4\pi^2}{ n^2
+ \epsilon } \right)^{3/2},$$
where $v_8 = 3.66386\ldots$ is the volume of a hyperbolic regular
ideal octahedron, and $\epsilon = n\mod{2}$.
\label{thm:volume}
\end{theorem}
\begin{proof}
The chain link complement is obtained by Dehn filling the longitude of
$\widehat{W_n}$. This is obtained by gluing copies of the Whitehead
link and its reflection along totally geodesic 3--punctured spheres,
hence has volume $n$ times the volume of the Whitehead link, $n\cdot v_8$.
By Lemma \ref{lemma:torus_boundary_component}, we know the length of
the Dehn filling slope is $\sqrt{n^2 + \epsilon}$. The result follows
by putting this data into Theorem \ref{thm:fkp}.
\end{proof}
We now give a proof of the main theorem.
\begin{theorem}
For $n\geq 60$, the minimally twisted $n$--chain link complement has
volume strictly greater than that of the $(n-1)$--fold cyclic cover
over one component of the Whitehead link. Hence the minimally twisted
$n$--chain link complement cannot be the smallest volume hyperbolic
3--manifold with $n$ cusps, $n\geq 60$.
\label{thm:not-minvolume}
\end{theorem}
\begin{proof}
The volume of the $(n-1)$--fold cyclic cover of the Whitehead link is
$(n-1)\,v_8$. By Theorem \ref{thm:volume}, the volume of the
complement of the minimally twisted $n$--chain link is
$${\rm vol}(S^3\setminus C_n) \geq
n\,v_8\,\left(1-\frac{4\pi^2}{n^2+\epsilon}\right)^{3/2} \geq
n\,v_8\,\left(1-\frac{4\pi^2}{n^2}\right)^{3/2}.$$
We want to find $n$ for which the following inequality holds:
$$n\,v_8\,\left(1-\frac{4\pi^2}{n^2}\right)^{3/2} > (n-1)\,v_8,$$
or
\begin{equation}\label{ineq}
\left( \frac{n}{n-1} \right)\left(1-\frac{4\pi^2}{n^2}\right)^{3/2}
-1 > 0.
\end{equation}
Let $f(n)$ be the function on the left side of inequality
\eqref{ineq}. Using calculus, one sees that
$\lim_{n\to\infty}f(n)=0$, $f$ is increasing between $n=7$ and
$n=6\pi^2 + 2\pi\sqrt{9\pi^2 -2} \approx 117.8$, and decreasing for
larger $n$, which implies $f$ has at most one root for $n\geq 7$, and
that $f$ is positive to the right of any root. The Intermediate Value
Theorem implies that there is a root between $n=59$ and $n=59.1$.
Hence the inequality is satisfied for $n\geq 60$.
\end{proof}
\section{Computations of volume, smaller minimally twisted chains}\label{sec:compute}
Now we analyze volumes of minimally twisted $n$--chain links
for $n$ between $11$ and $59$, since the main method of proof of
Theorem \ref{thm:not-minvolume} will not apply to these manifolds.
For $n$ between $11$ and $59$ inclusive, in Table
\ref{table:volumes} we present computational data using SnapPea
(SnapPy) \cite{weeks:snappea, snappy} that shows that the minimally
twisted $n$--chain link complement cannot be the minimal volume
hyperbolic 3--manifold with $n$ cusps. In particular, $W_{n-1}$, the
$(n-1)$--fold cyclic cover over one component of the Whitehead link,
has smaller volume. The volume of $W_{n-1}$ is always $(n-1)\,v_8$,
where $v_8 = 3.66386\ldots$ is the volume of a hyperbolic regular
ideal octahedron, which is the volume of the Whitehead link
complement. Notice that for $n\geq 11$, the volume of $S^3\setminus
C_n$ is strictly larger than that of $W_{n-1}$.
\begin{table}
\input{volume-table.tex}
\caption{Volumes of the complement of the minimally twisted
$n$--chain link $C_n$, compared to volumes of $W_{n-1}$, the
$(n-1)$--fold cyclic cover over a component of the Whitehead link,
for $5 \leq n \leq 60$. Note that $S^3\setminus C_n$ has greater
volume for $n\geq 11$.}
\label{table:volumes}
\end{table}
It would be nice to turn this data into a rigorous proof that the
minimally twisted $n$--chain links for $n$ between $11$ and $59$
cannot be minimal volume. One way to do this would be to use the
methods of Moser \cite{moser} and Milley in \cite{milley:minvol}.
Milley has written a program to rigorously prove that a hyperbolic
3--manifold with hyperbolic structure computed by Snap
\cite{goodman:snap} has volume greater than some constant. This
program, which is available as supplementary material with
\cite{milley:minvol}, is in theory exactly what we need for these
chain link examples.
However, in practice, making Moser and Milley's programs work with the
chain links has proven to be difficult, due to the computational
complexity of the chain links. While Milley worked with small
manifolds, for example with less than $10$ tetrahedra, and Moser's
largest manifold included $57$ tetrahedra, our triangulations of
minimally twisted $n$--chain link complements include between $40$ and
$236$ tetrahedra. We were successfully able to run Moser's algorithm
for $n$ between $11$ and $25$, inclusive, which gives results for
those manifolds triangulated with up to $100$ tetrahedra, but then the
program failed. We were able to run Milley's algorithm for all values
of $n$ for which Moser's algorithm applied. However, Milley's
algorithm only returned a positive result for $n$ between 12 and 25,
inclusive. Therefore, we have the following result.
\begin{theorem}\label{thm:vol-computational}
For $n$ between $12$ and $25$, inclusive, the minimally twisted
$n$--chain link complement has volume strictly greater than that of
the $(n-1)$--fold cyclic cover over one component of the Whitehead
link, hence cannot be the smallest volume hyperbolic 3--manifold
with $n$ cusps.
\end{theorem}
\begin{proof}
The proof is identical to that of Milley \cite{milley:minvol}, and
uses his code, included as supplementary material with that reference
\cite{milley:minvol}, modified to read in minimally twisted $n$--chain
links rather than Dehn fillings of census manifolds. The first step
is to feed the triangulations of the minimally twisted $n$--chain
links into Snap, and ensure that the triangulations used in the
computation are geometric, that is, all tetrahedra are positively
oriented. This is true for all minimally twisted $n$--chain links,
$11 \leq n \leq 59$.
Next, use Moser's algorithm \cite{moser} to find a value $\delta$
which measures the maximal error between Snap's computed solution and
the true solution. Moser's algorithm gave us such a value for $11
\leq n \leq 25$, but failed thereafter, presumably due to
computational complexity of the chain link complements.
Finally, for each $n$ between $12$ and $25$, inclusive, input the Snap
triangulation data and Moser's value $\delta$ into Milley's program
{\tt{rigorous\_volume.C}}, along with the constant value
$(n-1)*3.66386237670888$. The program checks rigorously whether the
volume of the given $n$--chain link is larger than the given constant.
For $12 \leq n \leq 25$, the program definitively proved that the
volumes of the minimally twisted $n$--chain link complement are
strictly larger than that of the $(n-1)$--fold cyclic cover over a
component of the Whitehead link.
\end{proof}
\begin{remark}
Note that the above theorem does not hold for $n=11$. Although the
triangulation of the minimally twisted 11--chain link is positively
oriented, and Moser's algorithm returns a value of $\delta$ for this
link, Milley's program {\tt{rigorous\_volume.C}} is unable to verify
that its volume is larger than that of the 10--fold cyclic cover of
the Whitehead link. When $n=11$, the volumes of these manifolds are
too close for rigorous checking.
\end{remark}
What about the volumes output by SnapPea for $26 \leq n \leq 59$?
Note in Table \ref{table:volumes} that the minimally twisted
$n$--chain link for these $n$ has volume greater than $2$ plus the
volume of $W_{n-1}$. It is highly unlikely that SnapPea's computation
would be so far off as to make the theorem untrue for any of these
values of $n$. However, since we do not have a rigorous proof at this
time, we do not include the result as a theorem.
\section{Arbitrary chain links}\label{sec:arblinks}
\input{twisting}
\bibliographystyle{amsplain}
\subsection{Chain links with 5 through 10 link components}
Our methods can be used to show that of all $n$--chain links, only the
minimally twisted $n$--chain link can possibly be the minimal volume
manifold with $n$ cusps for $n$ between $5$ and $10$, inclusively.
In fact, because the complexity of these manifolds was comparatively
small, we ran them through Milley's algorithm \cite{milley:minvol}, to
rigorously check this fact. The algorithm successfully applied, and
we have the following theorem.
\begin{theorem}\label{thm:small-chains}
Let $n$ be an integer between $5$ and $10$, inclusively. If $C_n$ is
an $n$--chain link that is \emph{not} minimally twisted, then the
complement $S^3\setminus C_n$ cannot be the minimal volume $n$--cusped
hyperbolic manifold.
\end{theorem}
\begin{proof}
Theorem \ref{thm:vol-withtwist} implies that for these $n$, and those
chain links with at least $11$ half--twists, the volume is strictly
greater than that of the $(n-1)$--fold cyclic cover over a component
of the Whitehead link, which is known to have larger volume than that
of the minimally twisted $n$--chain links in these cases.
The remaining cases to check are Dehn fillings $(1, \pm 1), (1, \pm
2), \dots, (1, \pm 5)$ on manifolds $\widehat{W}_n$ for $n$ odd,
Dehn fillings $(1,1), \dots, (1,5)$ on manifolds $\widehat{W}_n$ for
$n$ even, and Dehn fillings $(1,0), (1,1), \dots, (1,5)$ on manifolds
$\overline{W}_n$ for $n$ even. These cases were run through
algorithms of Moser \cite{moser} and Milley \cite{milley:minvol}, and
their programs rigorously proved that the volumes of these chain links
were larger than that of the corresponding minimally twisted
$n$--chain link. Programs are available from
Milley \cite{milley:minvol} or the second author.
\end{proof}
In Table \ref{table:small}, we show the volumes of $n$--chain link
complements, $n$ between $5$ and $10$, whose volumes are not
automatically larger than the minimally twisted $n$--chain link by
Theorem \ref{thm:vol-withtwist}. These are compared with the volume
of the minimally twisted chain link.
\begin{table}
\input{table-small}
\smallskip
\caption{Volumes of small chain links obtained by Dehn filling
$\widehat{W}_n$ or $\overline{W}_n$ along slope $s = (1,m)$, where
$m$ is the integer at the top of the column, compared with the
volume of the minimally twisted chain
link.}
\label{table:small}
\end{table}
| {
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/* jshint node:true, esnext:true */
'use strict';
let assert = require('assert');
let bind = require('./index').polyfill();
require('./index').polyfill();
function Counter(max) { this.max = max || 5; }
Counter.prototype.increment = function* increment() {
let current = 0;
while (current <= this.max) {
yield current++;
}
};
let c = new Counter();
let bound = bind(c, c.increment);
let polyfilled = c.increment.bind(c);
assert(c.increment.constructor.name === 'GeneratorFunction');
assert(bound.constructor.name === 'GeneratorFunction');
assert(polyfilled.constructor.name === 'GeneratorFunction');
let i = 0, j = c.max;
for (let cnt of c.increment()) { i = cnt; }
assert(i === j);
i = 0;
for (let cnt of bound()) { i = cnt; }
assert(i === j);
i = 0;
for (let cnt of polyfilled()) { i = cnt; }
assert(i === j); | {
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Q: Python setup script extensions, how do you include a .h file? So I've got a directory that looks something like this:
home\
setup.py
some_python_file.py
ext\
__init__.py
c_file1.c
c_file2.c
ext_header.h
Obviously the header file is necessary to compile the c files, but the problem is that I can't get the setup script to include the header file.
My extension object is something like this:
Extension('ext.the_extension', ['ext/c_file1.c', 'ext/c_file2.c'])
Which works, but doesn't include the header file. If I change it to:
Extension('ext.the_extension', ['ext/c_file1.c', 'ext/c_file2.c', 'ext_header.h'])
It includes the '.h' file but then doesn't build when I run install. Instead it gives and error error: unknown file type '.h' (from 'ext/ext_header.h')
If I include the header file as a data file like this:
data_files=[('ext', ['ext/ext_header.h'])]
it doesn't work at all, the .h file doesn't even make it into the MANIFEST file.
So my queustion is, how do you include this extension with the header file so that python setup.py install will build it correctly?
A: From the docs,
module1 = Extension('demo',
define_macros = [('MAJOR_VERSION', '1'),
('MINOR_VERSION', '0')],
include_dirs = ['/usr/local/include'],
libraries = ['tcl83'],
library_dirs = ['/usr/local/lib'],
sources = ['demo.c'])
You should provide the include files via "include_dirs".
Why does this not work for you?
A: I have a feeling pyfunc is on track for a more standard solution, but I did find another solution on my own. I have no idea if this is a good solution or just a hack, but all I did is add the header file to the MANIFEST.in. The documentation doesn't really make it seem like this is what the MANIFEST.in file is for, but it does work. My MANIFEST.in file now looks like this:
include ext/ext_header.h
Which includes the file and sucessfully compiles when I run python setup.py install
| {
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} | 80 |
typedef NS_ENUM(NSInteger, kMessageBarButtonType) {
kMessageBarButtonTypeAdd=1,//添加
kMessageBarButtonTypeFace,//表情
kMessageBarButtonTypeVoice,//语音
kMessageBarButtonTypeSend,//发送
kMessageBarButtonTypeKeyboard //键盘
};
@interface QYMessageBarButton : UIButton
@property (nonatomic) kMessageBarButtonType showType;
@end
| {
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{"url":"https:\/\/www.speedsolving.com\/threads\/cct-full-screen-problems.18288\/","text":"CCT full-screen problems.\n\nSaberSlash49\n\nMember\nWhenever I start timing (keyboard OR stackmat) it keeps going into full screen mode, and I already tried looking under the settings of keyboard timer and turning off \"fullscreen while timing\" and I reset all the settings as well. Even if I switch from Minimalist to Shelley (very nice set-up, btw) or vice versa, it still goes into full-screen.. Help?\n\nTheDuck\n\nMember\nYou'll have to go into \"options\" - Then press the \"keyboard timer\" choice and unmark the marker in \"fullscreen when timing\".\n\nCarson\n\nDo you have a \"profile\" setup in cct?\n\nGo into the cct folder then profiles, then the folder with your profile name. Open the file titled insertyourprofilehere.properties. Check the line\nCode:\nGUI_Timer_isFullScreenWhileTiming=false\nand ensure that it is set to false. If not, change it.\n\nIf all else fails, you could also try setting up a new profile as well.\n\nTheDuck\n\nMember\nTry Notebook .....\n\nzeroxorxdiexskater\n\nMember\nor textedit if your on a mac\n\nLast edited:\n\nSwoncen\n\nMember\nwhat do you mean? You have the same scramble all the time? I would copy the files with your records and re-install cct.\n\nSaberSlash49\n\nMember\nYes, but where is the files with my records? I was considering just making a huge document of scrambles (like 100+) and add more scrambles everyday, and change the order and such..","date":"2020-06-06 08:14:06","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\": 0.26504600048065186, \"perplexity\": 4958.616515787508}, \"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-2020-24\/segments\/1590348511950.89\/warc\/CC-MAIN-20200606062649-20200606092649-00492.warc.gz\"}"} | null | null |
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} | 3,529 |
Station Grodzisk Mazowiecki is een spoorwegstation in de Poolse plaats Grodzisk Mazowiecki.
Grodzisk Mazowiecki | {
"redpajama_set_name": "RedPajamaWikipedia"
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Does it feel like your holiday to-do list will never be complete? Do you balance it all well or feel 24 hours is not enough time to do life well?
It is easy to define our life by productivity or success, but the Lord does neither. He only ask that we come to Him and "press on towards the goal" of living a life that reflects our Lord and Savior. Satan has a great way of pulling us away from our service and callings to lead us to find our definition of success in our abilities.
What keeps me from serving the Lord fully? Distractions. Schedules. Juggling. With all of the daily to-dos, I wake up asking the Lord, how will you empower me to do these jobs today? As work and motherhood take up all of the daily hours, it seems some days that there is no time for a "quiet time". Even these beautiful writings to the Lord become another to-do, when the heart of my service to the Lord is handing every word over for His glory.
Lord, give me the strength to press on, even when I am tired. Give me the strength I do not have. Remind me that I am not alone in my journey and that if I focus on You, I will be amazed at how you are gloriously weaving together, everything for my good. Amen. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,900 |
{"url":"https:\/\/ubraintv-jp.com\/factorial-button-on-ti-84-plus\/","text":"Scientific calculators leveling the procedure of inputting and also calculating more facility expressions than typical calculators. Plain calculators have the right to handle factorials, yet you have to go into them manually, and also this can be time-consuming if you\u2019re taking the factorial the a bigger number. Scientific calculators make this lot easier, with most of them consisting of a \u201cx!\u201d key purpose-built for assessing factorials. All you have to do is go into the number you want to take it the factorial of and then push this vital to evaluate it.\n\nYou are watching: Factorial button on ti 84 plus\n\nFind the factorial the a number top top a clinical calculator, enter the number and also press the \u201cx!\u201d key. This may require friend to push \u201cshift,\u201d \u201c2nd\u201d or \u201calpha\u201d first depending top top your version of calculator and also the place of the symbol. Push \u201c=\u201d to gain the result.\n\nFactorial is the name given to the procedure of multiplying together every one of the integers as much as a details number. For this reason 5 factorial is 1 \u00d7 2 \u00d7 3 \u00d7 4 \u00d7 5 = 120 and 3 factorial is 1 \u00d7 2 \u00d7 3 = 6. Notification that a small increase in the number you\u2019re acquisition a factorial of can lead to lot bigger answers. The symbol because that a factorial is x!, wherein x is the number you want to take the factorial of. Because that 4! you can say \u201cfour factorial,\u201d despite you may also hear the occasional \u201cfour bang\u201d or also \u201cfour shriek.\u201d\n\nScientific calculators make easy job-related of evaluating factorials. The specific process depends top top the version of calculator friend have, but generally you need to look for the \u201cx!\u201d crucial on the calculator to finish the operation. First, push the number you desire to take it the factorial of, push the \u201cx!\u201d key, and finally push the \u201c=\u201d key to evaluate it.\n\nFinding the \u201cx!\u201d crucial on her calculator will certainly be the most difficult part of the operation. In part cases, the crucial is found on a an additional or tertiary function, with the symbol above the switch you need to press fairly than on it. In these cases, you have to press the \u201cshift,\u201d \u201c2nd\u201d or \u201calpha\u201d crucial to activate the corresponding function for the key. These buttons are often color-coded to make it clean which button you need to press. V this type of calculator, push the button you want to take it the factorial of, push the button for the crucial function, then push the factorial button and finally struggle \u201c=\u201d for the answer.\n\nOn graphing calculators, you may need to carry out even an ext to execute a factorial. Because that example, ~ above the TI-84 Plus, you require to get in the mathematics probability food selection by pressing \u201c$\u201d adhered to by the left crucial twice, and finally press \u201c4\u201d to go into the factorial symbol. Examine your manual to discover the method to complete the procedure on a particular graphing calculator.$\n\nIf all else fails, it\u2019s regularly easy to get in a factorial manually. If the number you want to take the factorial the is small, simply enter every one of the integers leading up to it with multiplication symbols between each. Once you with the number you\u2019re looking for, hit the \u201c=\u201d button to advice the expression.\n\nSee more: When Is The Next Fairy Tail Episode Coming Out, Fairy Tail Anime Is Getting A Sequel Series\n\nLee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts right into simple, digestible language. He's written about science for numerous websites consisting of eHow UK and also WiseGeek, largely covering physics and astronomy. The was additionally a scientific research blogger for aspects Behavioral Health's blog network for five years. He studied physics in ~ the open up University and graduated in 2018.","date":"2021-10-22 18:53:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.28669503331184387, \"perplexity\": 1527.2529529102912}, \"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-2021-43\/segments\/1634323585518.54\/warc\/CC-MAIN-20211022181017-20211022211017-00709.warc.gz\"}"} | null | null |
Q: How to switch between different chrome browser window opened by different WebDriver using selenium in Python? I searched for this question,and I found an idea using driver.switch_to.window(),but it didn't work as expect:
from selenium import webdriver
driver1=webdriver.Chrome("D:\Python\Files\chromedriver.exe")
driver1.get('https://www.google.com')
driver2=webdriver.Chrome("D:\Python\Files\chromedriver.exe")
driver2.get('https://www.bing.com/')
driver1.switch_to.window(driver1.current_window_handle)
above code will first open a chrome window and go to google,then will open another chrome window and go to bing,then
driver1.switch_to.window(driver1.current_window_handle)
seems didn't work,the window showing bing still shows on top of the window showing google.
Anyone have any idea?I think
driver1.switch_to.window(driver1.current_window_handle)
may have some BUG.
A: As you have used two WebDriver instances as driver1 and driver2 respectively to openthe urls https://www.google.com (e.g. windowA) and https://www.bing.com/ (e.g. windowB) it is worth to mention that the function switch_to.window() is a WebDriver method. So, driver1 can control only windowA and driver2 can control only windowB.
For Selenium to interact with any of the Browsing Window, Selenium needs focus. So to iterate among the different Browsing Windows you can shift the focus to the different Browsing Window using JavascriptExecutor as follows :
*
*Python:
driver1.execute_script("window.focus();")
driver2.execute_script("window.focus();")
*Java:
((JavascriptExecutor) driver1).executeScript("window.focus();");
((JavascriptExecutor) driver2).executeScript("window.focus();");
A: I believe you have a different concept of "window" in driver.switch_to.window(). In chrome browser, it means "tab". It's not another chrome browser or browser window like what are you trying to do in your code.
If switch_to.window() what you really want, I'll give an example how to use it:
driver=webdriver.Chrome("D:\Python\Files\chromedriver.exe")
driver.get('https://www.google.com')
# open a new tab with js
driver.execute_script("window.open('https://www.bing.com')")
driver.switch_to.window(driver.window_handles[-1])
# now your driver is pointed to the "tab" you just opened
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@class TKRecording;
@protocol SampleSource <NSObject>
@required
- (OSStatus) generateSamples:(AudioBufferList *) buffers frames:(int) numFrames;
@end
@interface TKCoreAudioBridge : NSObject
@property (nonatomic, retain) id<SampleSource> sampleSource;
@property (readonly) BOOL isPlaying;
+ (TKCoreAudioBridge *) sharedAudioBridge;
- (id) init;
- (void) start;
// Recording
- (void) createRecordingFileWithURL:(NSURL *)url;
- (void) startRecord;
- (void) stopRecord;
// Playback
- (void) setPlaybackURL:(NSURL *) url;
- (void) startPlayback;
- (void) stopPlayback;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,642 |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Wikipedia_talk:Manual_of_Style_(dashes)\/archive1","text":"# Wikipedia talk:Manual of Style\/Dashes archive 1\n\n## em-dash and en-dash\n\nQuestion on Style. I recieved the following comment on a page I edited: (&am; \"#151;\" is not a valid HTML entity... it should be & \"mdash;\" or & \"#8212;\"). I think & \"#151;\" is perfectly valid for a \"printer's em or em dash\" Anyone know why it is not? Also, should not the em be separated by spaces from the rest of the text, since it is NOT an ordinary dash, but a device for redirecting rthouyght within a sentence? Anybody know about this? Marshman 04:47, 2 Aug 2003 (UTC)\n\nMost style manuals prefer no spaces either side of m and n dashes. Where the line length is small, such as in newspapapers, it's more common to avoid m dashes altogether, or to use spaces either side. Tony 02:29, 31 July 2005 (UTC)\n\nIn ISO 8859-1 and Unicode, code point 151 is reserved as a control character. It is not an em dash except in Microsoft's proprietary code page extensions, and any program that displays an em-dash for \"—\" is doing so either erroneously or in deliberate emulation of common bugs in Windows. Relying on buggy behavior is not recommended.\u00a0:) Please use the standard, either — or —. --Brion 05:02, 2 Aug 2003 (UTC)\nThanks I will use — in the future. Coffee-Cup Software HTML Editor inserts \"—\" for an em-dash and it certainly displays that way on browsers. Why the confusion? 24.94.86.252 05:36, 2 Aug 2003 (UTC)me not logged inn Marshman 05:38, 2 Aug 2003 (UTC)\nplease don't! — looks very ugly in wikisource and some editors may not know what it is. Stick to \"--\". I know it's ugly, but in future our parser may turn that into mdash automagically. -- Tarquin 12:17, 2 Aug 2003 (UTC)\nDoes hardly look more ugly than L&uoml;beck.\n\n\"--\" gets really ugly when broken between lines, \" - \" would be a better advice.\n-- Ruhrjung 12:42, 2 Aug 2003 (UTC)\n— is the only form that renders correctly on the largest possible number of browsers. — is almost as good. No other form renders correctly, except by mistake. Always use — or —. Tannin 14:07, 15 Sep 2003 (UTC)\n\n\"pre\u2013World War II' is not a good example of the n dash, because there should be a hyphen here! I'm changing it to a correct example. Tony 02:26, 31 July 2005 (UTC)\n\nActually, that example is correct. An en dash is used instead of a hyphen when one of the components is hypenated or contains more than one word. I will add a sentence clarifying this. Babomb 08:43, 23 October 2005 (UTC)\n\n## Dashes\n\n(from the village pump)\n\nUser:Wik seems to insist on replacing ndashes \u2013 with ASCII dashes -. Style guides for printed work such as encyclopedias, as well as Unicode, state that for ranges such as dates an ndash (1998\u20132000) and not a dash (1998-2000) should be used. One advantage of using the correct dash is that a linebreak won't occur on the right of it. Is there some official policy from the Wikipedia on this, or should I just wait until Wik tires of his game and restore the correct dashes? Jor 01:00, 12 Feb 2004 (UTC)\n\nWell, if you're prepared to insert the \"correct dashes\" into all the tens of thousands of articles which now have the ASCII dashes, go ahead. --Wik 01:04, Feb 12, 2004 (UTC)\nOkay. I will interprete your quote above in that you'll start leaving them alone from now on. Jor 01:05, 12 Feb 2004 (UTC)\nNo, only if you go through all articles and make it consistent. I will always edit the articles to fit the de facto standard. Currently, that's the ASCII dash. --Wik 01:07, Feb 12, 2004 (UTC)\nPlease use two ASCII hyphens -- in a future version of MediaWiki this will be automatically converted to –. The problem with using one hyphen is that they're very difficult to find and convert once the new feature is implemented. I'd be quite happy with people using – in the meantime. -- Tim Starling 01:11, Feb 12, 2004 (UTC)\n\nWikipedia articles are about being easy to read and edit. The average non techie reader has no idea what the sequence of characters \"–\" is supposed to mean. It makes the article source ugly and therefore harder to edit. This kind of stuff should be kept at a minimum.\u2014Eloquence 01:34, Feb 12, 2004 (UTC)\n\nI also don't like ndashes as they make editing harder. Dori | Talk 03:14, Feb 12, 2004 (UTC)\nAn ndash and and an mdash are NOT the same thing, and a '-' is not a substitute for an mdash;. I agree with Jo. Stop putting in ASCI dashes anywhere. Or else you are going to be real busy for the rest of your days because I use only ndash and mdash and will change any ASCI dashes I encounter to the correct form (something a BOT cannot do). And a -- should become an mdash not an ndash. The look of the \"source code\" is not an issue. Incorrect English prevails over making editing \"easier\"\u00a0: Maybe we should just ignore spelling too - Marshman 05:42, 12 Feb 2004 (UTC)\nWikipedia:Manual of Style (biographies) uses regular ascii dashes in dates (1999 - 2005). I don't see what the problem with them is personally. It makes editing easier and looks fine when rendered to my eyes. The manual of style isn't compulsory, but it's the only guideline that should be applied to wikipedia IMO. If it's under debate then hash it out on the talk page and modify the guidlines if necessary when a consensus has been reached. fabiform | talk 06:57, 12 Feb 2004 (UTC)\nCan anybody please explain why this matters at all? A dash is a dash is a minus sign... or not? And minus signs are far easier to use compared with some \"&...\" character sequence. Furthermore, it is my impression that everything else looks ugly in Mozilla-based browsers. The advandages of \"&...\" listed above look not too significant compared with the ease of editing that \"-\" offers. So, what are the reasons for using the \"&...\" things? Specifically, why are they considered \"correct\"? Kosebamse 11:19, 12 Feb 2004 (UTC)\nSee Dash (punctuation), and in particular, the external link at the bottom, The trouble with EM 'n EN. While you are at it. look over Typography Matters\u2014a short essay on the theme \"Typography, at the root, is all about providing as many helpful cues for the reader\u2019s eye as possible.\" Tannin\nIt can also be important when \"viewing\" pages through a different kind of browser, for example having a text-to-speech engine read it aloud. The different kinds of dash\/hyphen can be used to cue different pauses or emphasis. HTH HAND --Phil 12:17, Feb 12, 2004 (UTC)\nFair enough. Still, everything except minus signs looks plain ugly on a monitor (at least under Mozilla et al.), i.e. there is not a helpful visual cue but a distraction, i.e. it is counter-productive to use the \"n\" and \"m\" things. Is there a solution to the display problem? Kosebamse 12:44, 12 Feb 2004 (UTC)\nGet your eyes adjusted, Kosebamse. No, I'm not making a smart crack here---if proper typography looks ugly to you, you have been spending too long reading badly set web pages, or student term papers, or some such. Take a break from the 'pedia and read some real printed-on-paper things (books, National Geographic, anything you like) till your eyes adjust themselves back to normality. As for Mozilla, it is ugly. Always has been. The most stable and practical browser around but ugly as a hatful of ar.... um ... bottoms. If you like pretty, use Opera. Or, if you must, Explorer. Microsoft have always been good at pretty. Tannin 13:14, 12 Feb 2004 (UTC)\nPS: I usually use Mozilla for most things, but nearly always Opera here (don't ask why, just habit). Looking at the page as rendered by Mozilla just now, it's fine. Perhaps your problem is the font support in Linux. Linus still has crappy on-screen fonts. Tannin 13:21, 12 Feb 2004 (UTC)\nOh my god, please stop spewing out misinformation. First of all, Mozilla has no problem with en or em dashes, minus signs, quotation marks, or most other relatively common characters in Unicode. If your font has the character, it will be displayed in Mozilla\u2014just like with any other graphical browser. Second of all, this has absolutely nothing to do with Linux. Linux is an operating system kernel that controls your hardware and says which process gets to run when. It does not care in the slightest about em dashes. Finally, with that said, do check out the free, high-quality Bitstream Vera font family. \u2014Daniel Brockman 08:50, Mar 7, 2004 (UTC)\nAh well... I like to see myself as a bibliophile and book maniac and could not agree more that good typesetting is A Good Thing. The \"m\" dashes are displayed too long, too high and without right or left spaces on Mozilla (under Linux). It would be A Very Good Thing to fix that but on which level? Browser? Style sheet? Font? Kosebamse 13:28, 12 Feb 2004 (UTC)\nFont, I suspect. Am m-dash should be exactly the same width as the letter \"m\" (uppercase or lowercase? I can't remember) in whatever font you are using. The \"lack\" of spaces is not an error. That's the way an m-dash is supposed to be rendered. Some---mostly American---publishing houses have taken to inserting spaces on either side of an m-dash in recent years. I have no idea why. A micro-space is acceptable if desired, but a full space ... well ... what is it they say? Two nations seperated by a common language? Tannin\nQuite possible it's the fonts. I have played around a little and they all look either like a minus or as described above (much wider than a lowercase \"m\" and too high). Kosebamse 13:50, 12 Feb 2004 (UTC)\n\n### Do not forget we are a wiki!\n\nI must insist that we NOT use HTML entities in raw wiki markup. This is a barrier to editing to all the non-technically-minded people who do not know what — means when they see it in raw wiki text. Irrespective of what is correct typograpy, we must work with the tools at our disposal, and we must remember that this is a wiki and clarity in raw source is as important as clarity and accuracy in rendered form. -- Tarquin 16:42, 17 Mar 2004 (UTC)\n\nI am relatively new, and I disagree completely. I know a lot of other people do, too. Can we vote on this or something? - Omegatron 17:12, Mar 17, 2004 (UTC)\nOr perhaps a tech fix could solve the disagreement; replacing the 4 used dashes with special codes, for instance, as the horizontal bar has been replaced by \"----\", and people were talking about replacing en dashes with \"--\" or whatever, which would be rendered as the correct HTML character. Probably something like \"-en-\" would be better and easier to grasp. This would be very easy for newbies to grasp, and would still format articles in a readable way. - Omegatron 17:15, Mar 17, 2004 (UTC)\nGetting the automatic conversion function back up would solve a lot of this tension. And \"clarity in raw source\"? Well, that disqualifies every single article with a summary box. Hajor 17:23, 17 Mar 2004 (UTC)\nAnd just to throw in my two cents on the \"evolution of typography\"; I agree that throwing out \"correct typography\" is akin to throwing out proper spelling. Sure, we could use \"online style\" for everything, and \"embrace the new internet style\", but by that logic, \"WELCOMA 2 WIKIEPDIA111!!!!! OMG LOL W3 R BUILDNG A MULTILNGUAL COPYLEFT ENCYCLOPADIA TAHT WIL ALWAYS BLONG 2 3VERYONA!1!111 LOL\" is a perfectly valid intro. It's an encyclopedia. It should look good. There is nothing wrong with conforming to \"old\" standards. We certainly shouldn't force people to use them, but there is nothing wrong with using them, either. - Omegatron 17:43, Mar 17, 2004 (UTC)\n\nSevral points, Omegatron: yes, articles that begin with lengthy tables are a bad thing in many ways. There have been suggestions to move these to another namespace and insert them in pages. second, you need to understand how wikis work: theire open nature is crucial. Complex HTML terms are a barrier. Thirdly, I never mentioned \"evolution of typography\". What I SAID is that we have imperfect tools, that were not designed for typography, namely the basic ASCII set used on the internet. This is what we must work with, for now at least. Try to accept that WP is a work in progress!\u00a0:) -- Tarquin 19:55, 18 Mar 2004 (UTC)\nI have much more experience with WP than standard wikis, but if I am not mistaken, typical wikis don't have images, tables, TeX markup, boilerplate text, or normal text links (they have ugly CamelCase). All of these things are very good to have, make it much more encyclopedia-like, respectable, etc (if WP used camelcase I would probably never have come here. An article that looks crappy gives the impression of having crappy info). They should not be removed just because they make the markup a little more difficult to edit. Dashes are pretty trivial, but HTML entities in general should not be expressly prohibited just because they are confusing the first time someone sees them. Possible solutions to the disagreement are\n1. Put a description in the first few pages that new editors see. Perhaps a \"[[what are these &number; things?]]\" or \"what are these special symbols?\" at the bottom of an edit page. (near Editing help) (I just checked, and editing help itself has a huge list of possible HTML characters, explaining plain as day what they are for)\n2. Automated conversion - give some or all of the html entities less ugly formats that are automatically converted (-- or -n- or -en- or <endash> or [[[special character: en dash]]] becomes – for rendering) Obviously you need to make it obvious to newcomers that the code is supposed to be there, without being hard to read, yet without making it so long that no one will type it. - Omegatron\nIt sure would be handy if there was a more capable browser based text editor provided to edit Wiki articles, with context driven online help. Or, maybe we could put together a very easy to use guide to markup sequences, in general, from the point of view of a beginner. - Bevo 21:14, 18 Mar 2004 (UTC)\n\n### Automated dash conversion and digital representation\n\nWell the new dash conversion has just gone live. One hyphen -\u00a0; two --\u00a0; three ---\u00a0; and of course four is the horozontal rule. We've all been muttering about having confusing \"&...\" symbols in the wiki editing box, but it just occured to me that this wont happen with the new markup. In the editing box the n-dash (if it was entered that way) will just look like --, just as horozontal rules display as ----. fabiform | talk 12:38, 12 Feb 2004 (UTC)\n\nYay! Finally we can have correct dashes! No more ugly poor man's dashes! (I'm with the mdash-ndash camp on this one.) --seav 13:07, Feb 12, 2004 (UTC)\nDon't hooray too loud, as the same one breaks the new wiki table markup, where |- works, but |-- doesn't work anymore. But Tim already heard the complain on IRC... andy 13:10, 12 Feb 2004 (UTC)\nI briefly switched this on using a live patch of two lines of PHP code, which was a bit silly because it broke a few things. I switched it off when I realised it broke links to titles containing --, of which there are about 140. There was some contention on IRC as to whether -- should be expanded as an en dash or an em dash. -- Tim Starling 23:24, Feb 12, 2004 (UTC)\nTwo dashes should be an em dash of course. An en dash is represented in ASCII by a single dash, and as such cannot be automatically fixed but must be done manually. Jor 22:01, 13 Feb 2004 (UTC)\nActually, it makes more sense that two hyphens make an en dash, and three hyphens make an em dash, for at least three reasons: (1) It enables usage of the en dash; (2) it uses only one token for the em dash (i.e., \"---\"), whereas two hyphens would encourage people to put spaces around the dash (thus: \" -- \"), which complicates parsing and takes away power from the style sheet; finally (3) this (using three for an em and two for an en) is how TeX has done it for decades.\nI strongly advise that this long-established convention be adopted. The table syntax invented\u2014what?\u2014some momths ago?\u2014can easily be adjusted to make this possible. I doubt that this would cause more confusion than throwing out the logical, intuitive, and well-tested TeX syntax. \u2014Daniel Brockman 08:50, Mar 7, 2004 (UTC)\nHas this also got something to do with why the \"nowiki\" tags are showing at the top of this page: (3) Sign your name and date (by typing \"nowiki\"\u2013~~ ~~\"nowiki\"? fabiform | talk 13:17, 12 Feb 2004 (UTC) (not on irc).\nSeems to be related - on Template:Villagepump it shows correctly, but once imported here it shows the nowiki's and there is a double - inside the nowikis. andy 13:24, 12 Feb 2004 (UTC)\n\nDoesn't Wikipedia use UTF-8? Can't we just insert the actual mdash and ndash characters? That would make editing much easier. 137.222.10.57 17:03, 12 Feb 2004 (UTC).\n\nNo, most English and Western European wikis use ISO 8859-1, for maximum browser compatibility. -- Tim Starling 23:24, Feb 12, 2004 (UTC)\n\nOK. Now I'm confused. According to the recent additions to wikipedia:Manual of Style, one should use a single-dash-without-spaces to represent a simple hyphen (as in date ranges), a single-dash-surrounded-by-spaces to represent an ndash, and two dashes (ie, --) for an mdash. However, if I follow the above conversation correctly, it seems that the software will convert a double-dash into an ndash and a triple-dash into an mdash. Am I misunderstanding, are there two incompatible standards being developed, or has something changed? -Rholton (aka Anthropos) 23:51, 16 Feb 2004 (UTC)\n\nOK, you confirmed what I suspected--that I thought it was resolved but there is in fact no clear statement to that effect. I'll try to rouse up a clarification of what's really going on. Elf 01:44, 17 Feb 2004 (UTC)\nWhen the automatic conversion was briefly turned on, a - remained unaffected, -- turned into a dash (an n dash I assume) and --- turned into a longer dash (an m dash I assume). I have nothing to do with the programming though, so you might want to talk to someone else about this, especially if you would prefer it to be done another way. \u00a0:) fabiform | talk 07:27, 17 Feb 2004 (UTC)\n\nNot resolved. A few people involved in the field seem to find the use of the hypen by ordinary people for all of the purposes offensive. Those same ordinary people have no objecting to ignoring the attempt to prevent the language from evolving and ignore the variations in dash lengths now that normal people can easily write and publish and not follow the conventions which used to be used in the print world. Effectively, a small group is trying to enforce an undesired style rule on everyone else, when usage clearly indicates that the majority of contributors do not agree. Simply, the online style for almost everyone is to use - for everything. Since we're a wiki, we do have to accept that change in style expectations, because it's not practical for a few people to force everyone to do what they want. Just document the way most people do it - the simple hypehen - and document that it's accepted that those who are writing new text and object can do it the print way if they desire but are discouraged from changing the writing of others. Jamesday 02:46, 17 Feb 2004 (UTC)\n\nYou are contradicting yourself. You start out by arguing that the pro-dash people are trying to \u201cprevent the language from evolving,\u201d and then go on to say that we should definitely use hyphens everywhere because \u201cit\u2019s the online style.\u201d This begs the question: aren\u2019t you preventing the online style from evolving\/maturing?\nIn my not-so-humble opinion, it would be an insult to humanity\u2014and, specifically, the people who work with on-screen typography research, Unicode, the W3C, the Mozilla Project, and others who put in effort to enable to use of good typography on the World Wide Web\u2014to throw out the typographical lore accumulated over the course of centuries just because some PHP script can\u2019t do this or that, or because some people allegedly can read all text equally well no matter how badly formatted. As someone else noted, you\u2019re not proposing that we abandon other seemingly \u201cunnecessary\u201d and \u201ctroublesome\u201d English punctuation rules, such as italicising emphasized words or having a whole bunch of different punctuation marks\u2014e.g., comma, semicolon, period, colon, dash, parentheses\u2014that all basically mean \u201cshort pause\u201d\u2014or are you?\nFinally, yes, this is a wiki. This means that if you don\u2019t know or care about the difference between the variously sized line segments sprinkled about the text, that\u2019s okay, because I\u2014and probably a hundred other people who are willing to edit your text\u2014do. I completely fail to see the logic in arguing that the collective competence of thousands of editors of a wiki could somehow be less than that of, say, one or a few of a paper. No doubt, an article is read a lot more often than it is edited, so it makes sense to spare all the future readers some eye-strain. irritation, and confusion, at the editor\u2019s expense of a small one-time (or more likely few-times) typing cost.\nDaniel Brockman 08:50, Mar 7, 2004 (UTC)\nDaniel, please don't use HTML entities, it makes your text extremely hard to read in raw form. -- Tarquin 10:08, 7 Mar 2004 (UTC)\nSo who's going to want to read it in raw form when Wikipedia so nicely formats it for us? --Phil 10:10, Mar 9, 2004 (UTC)\nYour point seems to be based on the mistaken belief that the use of multiple types of dash is moving forward. It's not, it's moving back to conventions based on the limitations of newsprint and paper reproduction. The web needs to support the old print conventions, to support republication of old documents and to support those who want to apply print conventions online, because that's how they have been trained and what they are used to. We don't need to use things just because they can be used. Sparing readers and editors eyestrain, irritation and confusion is why I'm going with the usual practice here, which is the hyphen for everything. I've no problems at all with not using rules learned over centuries when rules learned over the last twenty years show them to be inappropriate. One good example of such abandonment of inappropriate conventions is the usual choice of sans- rather than serife fonts online. Since it's unlikely that those used to print will convert to other conventions - they are more likely to believe that they must be right - the leave it alone approach seems best here, so those used to print don't feel that their own writing is wrong. Jamesday 03:05, 9 Mar 2004 (UTC)\nSo you're saying the em and en dash are obsolete? Proper typography causes eyestrain? You're joking, right? Wikisux 00:20, 2 Jun 2004 (UTC)\nThe choice of sans-serif fonts rather than serif fonts has everything to do with the relatively low resolution of display monitors versus the printed page. Serifs simply don't render well at typical sizes on a 100dpi monitor. I think it is safe to assume when monitor resolution comes more into line with printed resolution, say around 300dpi, we will see a resurgence of the use of serif fonts which are far more readable than sans serif. I cite personal experience before I learned proper typography and references such as \"The PC is Not a Typewriter\" as evidence. Jason Michael Smithson 12:49, 2004 Sep 13 (UTC)\n\n#### Disputed paragraph\n\nEvolving language and the decreased reliance on print world conventions have led to the hyphen becoming an acceptable replacement for other dashes. Where hyphens have been used in place of other dashes, you are discouraged from changing these, in the same way that changing spelling forms is discouraged. (See #Usage and spelling).\n\nThe statement that hyphens are now acceptable substitutes for other types of dashes had been added to the main page. In an attempt to avoid an edit war, I added a notation that this is disputed rather than removing it again. I just do not believe that this is the case. I don't know of any style guides or professional online publishers that have said that punctuation rules have changed. And I don't buy that, just because lots of people do it, that that makes it correct. (By that rule, \"its\" and \"it's\" would be interchangeable, for example.) And I certainly object to having the statement put into the style guide that we're not supposed to correct somebody else's punctuation when we come across it. I could live with the inclusion of the observation that some people feel that hyphens are acceptable to use for other dashes. And I expect tht in Wikipedia, people will type what they're comfortable with. And then other people will come through and clean it up. I get the impression from all of the various preceding discussions on dashes and hyphens that \"hyphens-are-legitimate-for-anything\" is a minority opinion, and \"hyphen-once -typed-by-one-person-are-immutable\" is very much a minority opinion. Elf 05:48, 8 Mar 2004 (UTC)\n\nI couldn't agree more. I think that we should aim for Wikipedia articles to look clean and professional. This goes hand-in-hand with NPOV. There is plenty of Internet left for those who wish to experiment with new, \"online\" punctuation rules. \u2014Daniel Brockman 21:21, Mar 8, 2004 (UTC)\nI've commented this out for now and just noted that the hyphen is commonly used in place of other dashes. I disagree that it should be \"corrected\", and I believe the safest option is to go with the same policy we have for spelling to prevent edit wars over this. I personally regard pages containing text such as \"“”\" to be highly unreadable. Angela. 02:33, Mar 9, 2004 (UTC)\nIf you think it's a minority opinion, I suggest that you try the exercise of listing those who have on this talk page expressed opposition to or support for the use of different dash types. My quick count placed those opposing it in the majority, with those used to the print world being at least a high proportion of those who favor it. Jamesday 03:10, 9 Mar 2004 (UTC)\nLots of people will find the use of hyphens instead of dashes annoying, as you say, because they are used to the print world. After all, there's a pretty good amount of people who read books. On the other hand, I suspect that very few, if not no one, will find the use of proper dashes annoying to read. Granted, some might find it annoying to type, but that's been shown to be a non-problem, as other people will later come to clean it up.\nAs for being annoying to read in the edit box, well, let's face it: the source is and will always be harder to read than the rendered page. I think that in this case, we're talking about a rather minor decrease in source readability (em dashes are relatively rare) in exchange for a rather major enhancement in the appearance of the rendered page:\n\u2022 `Hey, what's—what?` \u21d2 Hey, what's\u2014what? vs.\n\u2022 `Hey, what's - what?` \u21d2 Hey, what's - what?\nCompare, for example, to Angela's example:\n\u2022 `He said, \"what's up?\"` \u21d2 He said, \"what's up?\" vs.\n\u2022 `He said, “what’s up?”` \u21d2 He said, \u201cwhat\u2019s up?\u201d.\nWhether the apostrophes are a little bent makes a minor difference, since people will know they're apostrophes anyway. The length of dashes, on the other hand, makes a major difference, since a long dash carries a completely different meaning than a hyphen.\nIt might be possible to render \" -- \" or \"---\" as an em dash automatically, just as two apostrophes are rendered as emphasis markup. I'm all for this, since it would give us the very best of both worlds. (The new table syntax is very low-priority compared to this, IMHO.) In fact, I don't see how anyone could object to it. So what's the status on this? `:-)` Is it in the process of being implemented? \u2014Daniel Brockman 09:38, Mar 9, 2004 (UTC)\nIn parallel with the automated conversion (which I think is a good idea: I reckon \"--\" should go to N-dash and \"---\" to M-dash, just like ''' makes things bold) maybe the toolbar could be extended to add in various types of dash. If people get used to clicking a button then you just need to alter the code behind the button. --Phil 10:14, Mar 9, 2004 (UTC)\nThe problem with having three dashes (---) go to emdash instead of two dashes (--) is that the latter has been taught as an emdash in typing classes probably since the advent of typewriters, and publishers who buy manuscripts from writers have used this as the standard for emdashes (and still do). I realize that TeK, which is a markup language, uses --- as the markup for an emdash, but that's a markup language, it's not the standard for typing. I'm going to reinsert the text that says use -- if you don't want to (or can't) use the others, because this is correct.\nWiki markup is just as much of a markup language as TeX is. Yes, representing em dashes by -- is an existing convention. But this was the case a few decades ago, too, when Donald Knuth decided to go aginst the convention, supposedly because it was too ambiguous. Today, both -- and --- are in widespread use. The former is still more common than the latter, but people who have used TeX are likely to stick to three hyphens, at least for text that is going to be parsed by a machine\u2014as in the case of wiki markup\u2014because they are aware of the ambiguities that would arise were both kinds of dashes represented by --. I feel confident that Knuth made the right choice, and I believe we are now facing the exact same choice. \u2014Daniel Brockman 22:41, Mar 9, 2004 (UTC)\nFurthermore, I don't really care whether people use dashes or the ampersand formats or whatever. What I do object to are (1) having info in the style guide that gives misinformation (hyphens just simply not the same as em dashes and just because people use single hyphens in their place doesn't make it correct; read some style guides) and (2) having info in the style guide that prohibits me from editing what other people have done. We made a real effort to specify what the different punctuations mean, to show the markup to use if you want to use it, and an alternative using regular dashes if that's what you want to use. I think that \"-\" no spaces for hyphen, \" - \" single with spaces for en dash, \"--\" double with or without spaces for em dash (I understand UK publishers sometimes prefer spaces) pretty much covers those options. Elf 20:05, 9 Mar 2004 (UTC)\nI appreciate your careful efforts here Elf, but the bottom line is that using -- to represent an em dash is doing just that: it represents an em dash, it does not pretend to be a replacement for it in anything bar a typed manuscript on its way to a publisher to be set properly. Bad typography is every bit as sloppy and unprofessional as bad spelling. I don't expect every Wiki contributor to get his or her punctuation right first time, but the Manual of Style certainly should not endorse bad punctuation as the standard. If you want to abandon correct language, please first demonstrate a consensus to do so. Tannin 20:41, 9 Mar 2004 (UTC)\nI expected to be stressed about this but instead you've made my day. I really am laughing--because I'm usually known as the harridan of correct typography and punctuation. It's so weird to be accused of being the opposite! Anyway, thanks for responding and I feel much better now. Maybe I'm repeating myself: I don't see that there is a disagreement that -- is never legit for em dashes, only whether to use markup of --- for emdashes (which see above) or whether single hyphens are acceptable substitutes, or whether markup should be required (which I think is your view) or shouldn't ever be used. I'm trying to be realistic. Double dashes have been taught in typing classes for so very long (I don't instinctively type — when I'm in the throes of writing text, I type --) that it can't be eradicated. And people who don't want to muck with markup are going to type something in place of em dashes, and it just seems to make more sense to identify the existing punctuation convention than to annoy people who hate typing markup. That's how we got into this whole discussion, because not everyone likes typing markup. Elf 20:57, 9 Mar 2004 (UTC)\n\nThis is perfectly fine with me. What I object to is reccomending that people use a single hyphen in place of an em dash, and forbid other people from correcting it (yes, I view this as changing incorrect langage to correct language). I do not expect people to use the correct punctuation all the time, just as I do not expect people to write perfect prose all the time. What I do expect is (1) the right to improve on other people's subperfect language, and (2) that someone will eventually come along and improve on my subperfect language. \u2014Daniel Brockman 22:41, Mar 9, 2004 (UTC)\n\n### Current status\n\nOK, we've tried to add NPOV description of the various dashes that does the following:\n\u2022 Briefly describes the standard usage for the various dash types. (For detailed descriptions, there's a reference to Dash (punctuation).)\n\u2022 Shows the special markup that's valid for each type of dash and also identifies how to represent each using the hyphen key on the keyboard. (Note that I don't have final info on whether someone implemented an automated tool that changes groups of hyphens to something else.)\n\u2022 Gives a nod to the fact that there might be technical issues involved in using the special markup.\n\nElf 17:17, 20 Feb 2004 (UTC)\n\nI agree with the above. But I must strongly oppose the use of (--) for en dashes or (---) for em dashes. -- for en dash is absolutely wrong as it is far too long, and -- for em dash is wrong because in all computer fonts I know of they do not connect, and thus instead of providing the \"long dash\" which it should (as the practice does on old-style typewriters) it gives an ugly series of characters. \u2014 Jor (Darkelf) 22:18, 16 Mar 2004 (UTC)\n\nWhat happened to the TeX-style autoexpansion of -- to \u2013 and --- to \u2014? This would be very helpful... \u2014Tkinias 21:23, 5 Apr 2004 (UTC)\n\nWhatever happened to this? I thought it was a good idea. It just doesn't work with table markup? Sounds easy to fix. - Omegatron 19:09, Jan 21, 2005 (UTC)\nI completely agree. I'd love the TeX behavior. If articles consistantly used en-dashes in the usual way and un-spaced em-dashes as a gramatical dash, then a \"dash style\" could turn that semantic dash into whatever format people like, just like the date style. (Personally, I like the look of this \u2013 as a dash over\u2014that, but semantic markup would allow us to all get along, even if we are using a cell phone and want - that as a dash.) \u2014 BenFrantzDale 06:30, 28 Jan 2005 (UTC)\nWhat's the status of this? Can someone explain the problem with dashes as they interact with tables? I'd relaly like to see a solution so that the wiki source isn't full of HTML, but so the typography looks good. \u2014BenFrantzDale 04:18, May 13, 2005 (UTC)\n\n### Minus signs vs figure dashes\n\nWow. This is an insane conversation. Anyway, I don't mean to start another feud, but I see in both the Dash (punctuation) article and in Manual of Style article (that goes with this page), figure dashes are equated with minus signs. The HTML entities for each are different, however, as I pointed out in the dash talk page.\n\nHyphen:\n+-=-=====\n-+-=-----\n\nMinus sign:\n+\u2212=\u2212=====\n\u2212+\u2212=\u2212\u2212\u2212\u2212\u2212\n\nFigure Dash:\n+\u2012=\u2012=====\n\u2012+\u2012=\u2012\u2012\u2012\u2012\u2012\n\nTeX:\n${\\displaystyle 1+2-3=}$\n\nHyphen:\n1 + 2 - 3 =\n\nMinus sign:\n1 + 2 \u2212 3 =\n\nFigure dash:\n1 + 2 \u2012 3 =\n\n${\\displaystyle 1+2-3=}$\n1+2-3=\n1+2\u22123=\n1+2\u20123=\n\nIt looks as if the TeX markup HTML rendition uses a plain old hyphen for a minus sign. I think the − works better in equations, since that is what it is designed for; to be the same width and height as the plus and equal signs. Hyphens and figure dashes look obviously bad in comparison. The figure dash doesn't even have space around it in my font. (8\u20128\u20120\u20120)\n\nSo:\n\n1. Should these be separated in the two articles?\n2. I have been using the − in my math articles. Is that ok?\n3. Should the TeX renderer use the minus sign too?\n\n- Omegatron 21:56, Mar 16, 2004 (UTC)\n\nFigure dashes are not minus signs, that is a mistake. To answer your questions:\n1. Yes. I'm doing so now.\n2. Using \u2212 is very good even, when used in mathematical operations.\n3. Probably. This is likely a font or encoding issue: the hyphen-minus is overused mainly because it is the only dash-like character known to exist on all platforms as it is in ASCII. If possible, Tex of course should use the real character.\n\u2014 Jor (Darkelf) 22:10, 16 Mar 2004 (UTC)\n2. Mostly, I was asking if it should be used, since it might not be supported by enough modern browsers. - Omegatron\nIt should be compatible with all modern browsers. I can't check archaic browsers like Netscape4, but any more recent browser should work. I know from personal experience the following support &minus:\n\u2022 Opera (3.5 and up)\n\u2022 Mozilla (any version), Netscape 6\/7 and up, and other Geckos,\n\u2022 Internet Explorer 4 and up.\n\u2022 The Lynx and Links terminal browsers (ASCII approximations)\n\u2014 Jor (Darkelf) 15:20, 17 Mar 2004 (UTC)\n\n## emdashes\n\nem dashes \u2014 dashes the width of an \"m\" character \u2014 are used in typesetting and \"upscale\" web page set off text, in a use similar to parentheses. where em dashes are unavailable, two \"regular\" dashes are often used: --\n\nWhile em dashes look much nicer, on certain displays and devices -- such as my handheld -- they aren't in the character set at all, and are displayed using a placeholder character, often a question mark or unfilled square (a \"box\").\n\nShould we use nice-looking em dashes, knowing they won't display properly on certain devices, or fall back on the less elegant but more portable double dash?\n\nThanks. orthogonal 04:22, 16 May 2004 (UTC)\n\nIs it that time already? Yeah, ok, we've had the mirror\/fork question and the case sensitivity question, I guess we must be back around to dashes. May I suggest we keep it civil this time? More like the time before last than last time. -- Tim Starling 04:28, May 16, 2004 (UTC)\n\nTim Starling: while it's certainly fine to note that this discussion has come up here before -- and I understand the annoyance of an old-timer seeing questions arise again and again -- I do it would be more helpful for you to let us know what the previous consensus was.\nSorry. There was a short discussion last August, here. Then the civil discussion I referred to last December: Wikipedia talk:Special characters#Unicode. Then there was the discussion on wikitech-l in January: [1]. And finally there was the February discussion here. The current situation as far as I'm concerned is that either TeX-style conversion or -- to — conversion will be implemented as soon as someone works out how to do it without breaking various things such as the table syntax. -- Tim Starling 04:59, May 16, 2004 (UTC)\nNo problem, and thanks Tim, that's good to know. (however, this doesn't answer Othagonal's original question -- what should be done about the incompatibility of em dashes with certian devices? Personally, I think it's the device's own damn fault.) Adam Conover 16:47, May 16, 2004 (UTC)\nIn principle we could change the rendering according to user agent, although that may make caching difficult. Obviously we have to think of everyone, not just those trying to view Wikipedia from a mobile phone. -- Tim Starling 01:23, May 17, 2004 (UTC)\nPersonally, though I use an emdash in my sig, I generally use double-dashes when writing, simply because that's what I'm used to doing -- most online communities don't support anything else, and '--' is the quickest way to write a dash in word -- and because they're must easier to type. I would guess that double-dashes will continue to be by far the most prevalent regardless of what we decide. Adam Conover 04:33, May 16, 2004 (UTC)\n\nYeah, I've been against them for some time now as they make editing difficult. Use \"--\" instead. RickK 04:43, 16 May 2004 (UTC)\n\n## Changing '-' to \"\u2014\"\n\nI don't see why they should all be changed to \"\u2014\". At large font sizes an \"\u2014\" is *huge* and looks very ugly. A '-' works just as well and doesn't stand out as much.\n\nDarrien 17:19, 2004 May 23 (UTC)\n\nThis perceived problem is a perceived font problem. Argue to have the Wikipedia font changed if you feel its m-dashes are too long. In the meanwhile, if a dash (\u2014) is meant, a dash should be used; if a hyphen (-) is meant, a hyphen should be used. Read any book from a major publisher \u2014 you won't find them using hyphens for dashes, either single, as you like them (-), or double (--), as most wikipedians seem to. Bad punctuation is ugly to educated people Chameleon 17:35, 23 May 2004 (UTC)\n\nIt is not a font problem at all. I'm surprised an \"educated\" person would expect a hyphen to have the same boldness as a dash. Perhaps you should concentrate on changing the all the 'U's and 'J's to 'V's and 'I's.\nDarrien 17:43, 2004 May 23 (UTC)\nI think it'd be better, more wikilike, to have '--' render as '\u2014'. (I agree that a distinction between hyphens and em-dashes should be made.) If there's no burning need to use HTML entities for something, they shouldn't be used. I vote for '--'. (Then again, we could go the TeX route and use '-' for the hyphen, '--' for the en-dash (used for ranges, like 10--25), and '---' for the em-dash (used for saparation---like this---of text). Higher-quality typesetting, fits well into the syntax ('----' is a horizontal rule, which is kinda sorta like the god-king of all dashes), and no more ugly HTML entities in the wikitext. Sounds like a win to me. grendelkhan|(blather) 18:45, 2004 May 23 (UTC)\nYes, I agree that there should be automatic conversion. That way, people could just type \"--\" or \"---\" and correct dashes would appear. That would be great. However, until this is implemented, we will have to use HTML, in the same way that we would have to use <i> <\/i> if we didn't have '' '' \u2014 Chameleon 19:12, 23 May 2004 (UTC)\nAutomated conversion is a good idea. But & m d a s h ; (rendered as \u2014) isn't supported in all browsers, as far as I know. And \"--\" seems to me a usenet style for m-dashes, not for n-dashes. (A related probleme is, btw, the difference between \"some text\u00a0? some extra text\u00a0? some text\" and \"some text?some extra text?some more text\" in different languages.) -- till we | Talk 18:54, 23 May 2004 (UTC)\n& mdash; displays correctly in all modern browsers these days. Let's work for forward, nor backward compatibility. Also bear in mind that \"-\", \"--\" or \"---\" for dashes do not display correctly (\u2014) in any browser. \u2014 Chameleon 19:12, 23 May 2004 (UTC)\nWe did have automatic conversion up and running for a couple of months ago: really cool, except that it broke the wikitable formatting commands and was deactivated in a matter of minutes. It'd be nice to have it back, but I'm afraid I'm not a programmer.\nRe the em-dashes and how they can look too long (a lot longer than a capital M here in Tahoma): the alternative is to use en-dashes (spaced), which is sanctioned by a fair number of schools of typography and punctuationalists. Hajor 01:14, 24 May 2004 (UTC)\nAh, that's a pity. Surely it can be programmed not to break wikitables. I hope the programmers are reading this. I've just been looking at a few fonts. It seems a lot have over-long m-dashes. Verdana, Arial Narrow, Book Antiqua, Comic Sans MS, Lucida Sans, High Tower Text, Trebuchet MS and others are okay though. Having said that, although dashes should not theoretically be any longer than an em, that fact that professional font designers make them a little longer shows that they agree with me in saying that it is more elegant for them to be too long than too short.\nIn any case, I think it's clear that those who believe it's okay to put just a hyphen are in a typographically-challenged minority. Darrien, please put the pages back how you found them, and stop turning my dashes into hyphens. \u2014\u00a0Chameleon 09:44, 24 May 2004 (UTC)\nI like the \"blank en-dash blank\" proposal (written \" -- \" or \" \u2013 \", possibly automatically transformed). Could we make a Wikipedia policy that this form should be used? \u2013 till we | Talk 16:16, 24 May 2004 (UTC)\nI prefer hyphens to any form of dash, but I'd be far less opposed to the spaced endashes idea than to emdashes. Angela. 17:09, May 24, 2004 (UTC)\nI would accept spaced endashes. That would seem to be the only compromise that can be reached. emdashes look horrible at large font sizes. endashes actually look slightly better than hyphens at large font sizes.\nDarrien 17:32, 2004 May 24 (UTC)\nSpaced en-dashes are a semi-acceptable approximation to the true em-dash. The most important thing is that dashes must never be turned into hyphens, whether single or double. Note that Darrien is still doing this.\nI know points should be argued, not proved by action... which is the only thing stopping me adding lol to every page and defending it on the basis that it is the internet standard for funny things. \u2014 Chameleon 03:41, 25 May 2004 (UTC)\nDoes this mean that you will accept the hyphens being changed to endashes?\nDarrien 04:14, 2004 May 25 (UTC)\nI'd wholeheartedly support making spaced endashes policy. How would we go about that? Hajor 04:18, 25 May 2004 (UTC)\nAdd it to the page. See if you get reverted. If you don't, it's policy.\u00a0;) Angela. 11:38, May 26, 2004 (UTC)\nHang on \u2014 this en-for-em hack only works if en-dashes are significantly longer than hyphens in this font, and they are not. We need to keep using correct em-dashes, and perhaps consider changing the font. This is what I see around Wikipedia:\n\u2022 some text-more text\n\u2022 some text - more text\n\u2022 some text -more text\n\u2022 some text- more text\n\u2022 some text--more text\n\u2022 some text -- more text\n\u2022 some text-- more text\n\u2022 some text --more text\nI'll carry on correcting all of these to:\n\u2022 some text \u2014 more text\n...though I'll leave en-dashes if I find them. Chameleon 09:12, 26 May 2004 (UTC)\nPlease stop \"correcting\" them. There is clearly no consensus that these are correct, and certainly no agreement that they should be used. Anyway, aren't spaced emdashes are always wrong? Angela. 11:38, May 26, 2004 (UTC)\nThe no-spaces thing is the old rule. Dashes have at the very least a hair-space these days, and it's a good thing too (because they are not hyphens).\nShould I stop \"correcting\" teh for the before I get specific consensus for it too? No. I should just carry on proofreading, whilst keeping an eye on any new policies that arise.\nIf I see old-fashioned spaceless em-dashes (1), or new-fangled spaced en-dashes (2), I'll leave them; but various hyphen hacks will be changed to (3).\n1. some text\u2014more text\n2. some text \u2013 more text\n3. some text \u2014 more text\nChameleon 15:52, 26 May 2004 (UTC)\nWhy change them to emdashes when there is far more agreement on spaced endashes? The compromise suggested above would be far more sensible than purposefully inserted characters which many users have stated look awful on their browsers. Angela. 22:56, May 26, 2004 (UTC)\nPlease, no. Using hyphens can be excused as ignorance, but purposefully using en dashes when you know an em dash is called for is a sin against typography. \u2014Steven G. Johnson 04:02, Jun 17, 2004 (UTC)\nI wouldn't say there's more agreement on spaced endashes. Spaced endashes are more modern (and recommended by Elements of Typographic Style) but entering them in wikitext looses their meaning. The problem is that there are three symbls (-, \u2013, and \u2014) and they convey several meanings (hyphen, en-length-hyphen, range delimiter (read \"to\"), dash, etc.). If there were a `‐` entity, the rendering could be a style preference as it is for dates. Then an unspaced `‐` could be The Right Way and you could have it render like\u2014that \u2014 that \u2013 that -- that - that or whatever. Personally, I use un-spaced em-dashes because they would be the safest to search and replace with a semantic dash (I can't think of a case where a character followed immediately by an em-dash followed immediately by a character would not be a semantic dash). \u2014BenFrantzDale 14:06, 4 October 2005 (UTC)\nUnspaced emdashes is\u2014pretty much\u2014an article of faith among those wikipedians who follow the Chicago Manual of Style. I suspect that on seeing your spaced ems, they'll feel a strong temptation to join 'em up. Faced with a spaced endash, the temptation would be less strong and the effort required greater. Chameleon, think about that one \u2014 which do you prefer, unspaced ems or spaced ens? (And while you're at it, take good note of Angela's comments there about the near-consensus we were approaching with spaced endashes: much closer than anything I've ever seen in the four or five times this issue has come up since I've been here.) Hajor 23:22, 26 May 2004 (UTC)\nThe consensus I see 'on the ground' is that those Wikipedians who know how to use these HTML entities use spaced em-dashes.\nAs for which I prefer, the main thing I want is hyphens not to be used. Chameleon 00:23, 27 May 2004 (UTC)\n\nThe question of em dashes vs. spaced en dashes is a matter of style. The latter has a couple of technical disadvantages, though. We should use the traditional em dash. Michael\u00a0Z.\u00a02005-10-4\u00a015:01\u00a0Z\n\n\u2022 A spaced dash is three characters instead of one. I'm not worried about database size, but it means text entry would not be as simple.\n\u2022 A spaced dash must be lead by a non-breaking space, or it may show up at the beginning of a line, which looks almost as ugly as using hyphens.\n\u2022 Unicode non-breaking spaces get destroyed in normal editing, because some browsers silently convert them to spaces.\n\u2022 Entering the entity or   makes entry too complicated and wikitext too ugly.\n\u2022 In some fonts the hyphen looks like an en dash (e.g. Lucida Grande, found on every Mac, and an excellent international font for a Wikipedia user's style sheet). this is not a serious visual problem, but it can make it impossible for an editor to tell which dash has been used.\n\n## UTF-8 can end this debate\n\nEver since I read the informative (and nit-picky, just my style) Dash (punctuation) article, I've been cheerfully using typographically correct dashes in my edits. And seeing as I can easily enter them as a single character, I've done so. Same goes for curly quotes, apostrophes, etc.: I've been typing them in directly to edit boxes.\n\nI've just discovered that in using these characters I am quite the shill for Microsoft [2] \u2014 the funny thing is, I'm editing with Firefox on a Mac! I suppose that technology has marched on, at least in some operating systems, allowing me to join in on the Windows party of using fancy punctuation marks that aren't part of the ISO-8859-1 standard. I recognize that this is wrong, since the English Wikipedia claims to use the ISO-8859-1 charset. Sorry.\n\nOnce you use edit this way, though, it's hard to go back. There's no need for ugly HTML code in the Wiki markup, just clean, correct typography. On a Mac you type in option key combos to get the symbols; on Windows you edit in Word and let it do its creepy thing, then copy and paste; on Linux, I don't know, you write a program or something. Anyway, it's better than the other options.\n\nOf course, the problem is that the ISO-8859 charset does not have these characters. The solution is to move the English wikipedia to UTF-8, like the French and other language Wikipedias have recently done. At that point we can use almost any single-character symbol we want.\n\nAs far as size goes, all ASCII characters still only take up one byte when represented in UTF-8. Characters that are not in ASCII but are in ISO-8859-1, like \u00e9, take up 2 bytes instead of 1, but I think that would be balanced by the savings from dashes being a 2 byte code instead of 8 bytes of HTML markup, or two ugly bytes when represented as --. Non-European characters (used in interwiki links) would take up less space in UTF-8, too. So in the end it might be an overall space savings, if the conversion is thorough.\n\nThoughts? This change would probably take some community support, as it can't be much fun changing character sets on the largest Wikipedia. Nathan 17:08, Jun 10, 2004 (UTC)\n\nI'll second your opinion. Unicode (UTF8 is the easiest) is the way to go. buckwad 03:38, 2004 Jun 11 (UTC)\nI used to enter special characters directly with my keyboard (Microsoft have released a free too that makes it possible to easily design your own keymap; I took the Spanish one and added a few extra characters), but people complained they couldn't view them, so I had to stop. It would be so good if I could start again. \u2014\u00a0Chameleon 12:39, 16 Jun 2004 (UTC)\nAFAIK only Mozilla based browsers like Firefox convert charts not present in the given encoding to Unicode Entities --Hhielscher 09:22, 27 Mar 2005 (UTC)\n\nPlease go to Wikipedia:Unicode and make sure your voice is heard. --Hhielscher 09:22, 27 Mar 2005 (UTC)\n\nWe have UTF-8 support now. Sweet! See: \"-\", \"\u2013\", \"\u2014\". \u2014 Bcat (talk | email) 28 June 2005 15:46 (UTC)\n\n## Should non-English words be italicized?\n\nShould non-English words be italicized? (Blackfoot music). Hyacinth 03:39, 14 Jun 2004 (UTC)\n\nYes, according to Wikipedia:Use other languages sparingly. Nathan 05:46, Jun 14, 2004 (UTC)\n\n## Proposed section for inclusion in the \"Dashes\" section\n\nOn repeated occasions we have been unable to reach agreement on what styles of dashes are acceptable in Wikipedia articles. In an attempt to avoid further petty dash reversion wars (in which I have readily \u2013 and pettily \u2013 participated in the past), I'd very much like to work for consensus towards including a paragraph along the following lines in the relevant section of the Manual of Style. It's basically the same as the approach proposed by User:Chameleon above.\n\n===Dash guidelines for Wikipedia editors===\nIn the interests of Wikipedia:Wikilove and pending a planned update of the Wikimedia software that will automatically convert groups of hyphens into the appropriate correct en- and em-dashes, editors are encouraged to be accepting of others' dash preferences and not to modify a chosen style arbitrarily, in the same way as they would refrain from arbitrarily changing \"artefact\" to \"artifact\" (or vice-versa). The following five dash styles are currently in use on Wikipedia. Please do not change them to reflect your preference except as indicated below.\n\u2022 Tight (unspaced) em-dashes\u2014like this. Entered by means of either — or —.\n\u2022 Spaced em-dashes \u2014 like this.\n\u2022 A very rare subset of this style separates the dash from the surrounding words using hair spaces; since many browsers cannot display hair spaces, these appear on the display as simple tight em-dashes.\n\u2022 Spaced en-dashes \u2013 like this. Entered by means of either &ndash or –. (Note: an unspaced en-dash may be used to indicate a range of numbers, but unspaced en-dashes should not be used for the parenthetical use under discussion in this Guideline.)\n\u2022 A pair of hyphens--either spaced or unspaced -- like that. These are ugly, but simple to type, and will be taken care of in the future by the automatic conversion feature; indeed, under that future version of the software, they are expected to become the default style. Editors who do not want the bother of keying in HTML entities are free to type their dashes in this fashion. However, subsequent editors are free to convert any double-hyphens they come across to any of the above three types, depending on:\n\u2022 personal preference between en-dashes and em-dashes, and\n\u2022 how the hyphens were initially entered -- i.e., a spaced double hyphen may only be converted into a spaced em-dash or a spaced en-dash. The original editor's spacing preference is respected.\n\u2022 A single spaced hyphen - actually, there's no real reason to flout the rules of good typesetting in this way. If you come across one of these, please feel free to convert it into your preferred dash style from the above list.\n\nIs there anything truly objectionable in this proposal? May I please include it in the Manual of Style? We've been through all the arguments before, several times; we all have our preferred ways of doing things and, until the software gets updated and we get automatic hyphen conversion, including this guideline in the Manual might help reduce the odd hot-spot of editing heat and, at the same time, improve the professional appearance of our articles. Please, comments, objections, amendments below. Hajor 22:48, 14 Jun 2004 (UTC)\n\nBeing a stickler for good spelling, grammar and punctuation, I obviously agree with this. \u2014\u00a0Chameleon 12:36, 16 Jun 2004 (UTC)\nThat seems quite reasonable to me, other than that it should be acceptable to substitute one's own preferred hyphen style along with one's own preferred spellings over that of the original editor when greatly expanding or greatly changing an article\u00a0\u2013 for example when expanding a stub or doing a complete reworking of an article. But not when making minor changes or additions.\nMy own dash style, however, is to use \" – \" to keep the spaced en-dash (or spaced em-dash) with the preceding word in case of a line break. jallan 23:58, 14 Jun 2004 (UTC)\nYes, I too often find it helpful to use a non-breaking space to keep the dash in the right place. \u2014\u00a0Chameleon 12:36, 16 Jun 2004 (UTC)\nI feel this is a good addition. I've had it with people not respecting my double hyphens. [[User:Meelar|Meelar (talk)]] 20:19, 15 Jun 2004 (UTC)\nAfter my perusual of the 2000 Chicago Manual of Style yesterday, I'm glad to see such a succinct, simplified, and inclusive policy suggested here. (I had forgotten how many kinds of dashes there were in typesetting!) I'm all for it, even more so with Jallan's amendment. One question: will the new software actually convert the text, or just display two hyphens as an m-dash? And (okay, two questions) will it (or can it be made to) incorporate Jallan's clever use of  ? \u2014 Jeff Q 20:34, 15 Jun 2004 (UTC)\nAs I understand it, this new software we are hoping for will interpret double hyphens as special wiki-markup for dashes. At that point, it will be possible for someone to write a script to change all the HTML entities (— and —) into double hyphens, since these double hyphens will in any case be converted into the appropriate entity (or indeed, directly insert correct dashes as Unicode if we go over to UTF-8) for viewing in the user's browser, and the use of entities will therefore be unnecessary and undesirable \u2014 just as we prefer the use of double apostrophes (''text'') to the use of HTML (<em>text<\/em> or <i>text<\/i>) when italics are required, now that the software supports this.\nN.B. Some argue that triple hyphens --- rather than double ones --- should be the markup for dashes, because this leaves \"--\" free to be the markup for those little en thingies (\"\u2013\"), in keeping with TE\u03a7 style. Personally, I don't mind either way. \u2014\u00a0Chameleon 12:36, 16 Jun 2004 (UTC)\nAn excellent proposal. I heartily endorse it. Tannin 14:56, 16 Jun 2004 (UTC)\nPS: As we all know by now, the ndash and the mdash are different things and are used for different purposes: i.e., the software should convert \"--\" and &ndash and – to , and convert \"---\" and &mdash and — to .\nBy the way, as one of the strongest supporters of professional standards in our presentation (i.e., correct punctuation), and in the interests of harmony, I some time ago voluntarily switched from the no-space mdash style to the spaced style, as this seems to incur less wrath from the typographically challenged ... er ... sorry ... I mean as this seems to incur less wrath from my good and helpful Wikibretheren. Tannin\nI think the convention of single, double and triple hyphens is a good convention that is easy and unambiguous to convert by software.\n--Ruhrjung 02:47, 2004 Jun 17 (UTC)\nAs a LaTeX user, I'm inclined towards -- and --- for en and em dashes myself as well. Regardless of the specific syntax, though, it's important that a simple markup style be present for both dash types, since both have common and important typographical uses. Of course, most users will probably not use either one correctly, but it should be easy to correct them and the resulting wikitext should be readable. (PS. Who thinks that a spaced en dash is an acceptable typographical replacement for an em dash, spaced or not???) \u2014Steven G. Johnson 03:48, Jun 17, 2004 (UTC)\nWell, I don't like it either but plenty of people here see it as the perfect compromise between us em-dashers and the hyphen mob. \u2014\u00a0Chameleon 11:13, 17 Jun 2004 (UTC)\nWho thinks that a spaced en dash is an acceptable typographical replacement for an em dash, spaced or not??? Germans, for one; their typographical convention is \"space, en-dash, space\" rather than (unspaced) \"em-dash\". A matter of style and tradition, perhaps.\nOh, and I also wouldn't mind seeing -- and --- being converted into en-dashes and em-dashes, respectively. -- pne 12:25, 17 Jun 2004 (UTC)\nYes. I've seen a number of dash recommendation specifications which suggest strongly that either em-dash or space,en-dash,space are equally acceptable. I normally use the second, but without any strong preference for it. Of course when used between quanities with the meaning 'to', an unspaced en-dash is always normal. But there are exceptions, for example \"\u22127\u2013\u22122\" looks far worse than \"\u22127\u00a0\u2013\u00a0\u22122\". Actually both look rather horrible. But in any case I agree that '--' for en-dash and '---' for em-dash is the best known easily readible convention that provides the most control. Presumably the new software would first run a pass and change all hard \"--\" to \"---\" and then all \" --- \" to \" -- \" before beginning the new interpretation. jallan 14:24, 17 Jun 2004 (UTC)\nWhile we're mentioning other languages, French uses spaced em-dashes. Spanish uses em-dashes spaced on one side, just like parentheses or brackets. \u2014\u00a0Chameleon 15:35, 17 Jun 2004 (UTC)\n\n## Dashes (moved from the village pump)\n\nOpinions are being sought regarding a proposal on this thorny issue for inclusion in the Wikipedia:Manual of Style. Please see the bottom of Wikipedia talk:Manual of Style\/Dashes. Thanks. Hajor 20:15, 15 Jun 2004 (UTC)\n\nThat huge and technical debate (still) seems to say that a software update will convert simple minus signs that everyone has one their keyboard into these funny entities that are not at all intuitive. One of the hugely disagreeable things about this debate is that usability (the essence of wikitext) seems to get repeatedly ignored, and we go round in circles about trivial details. Pcb21| Pete 08:07, 16 Jun 2004 (UTC)\nYou don't have to care about typography, but some editors do, and when they fix your typography it will be much easier for them, others, and you if they can write --- instead of — (\u2014) and -- instead of – (\u2013). \u2014Steven G. Johnson 06:07, Jun 17, 2004 (UTC)\nRight, that would be a software update extending the wiki-syntax. The discussion however seems to be about what combination of &mdash and &ndash to use right now. Pcb21| Pete 18:49, 17 Jun 2004 (UTC)\nNo, it is not about that. Except for one comment no-one has disagreed with Hajor's assertion that \"—\", \" – \" and even \" — \" are all acceptable. The discussion is on whether Hajor's modifications should replace the current statement which in any case already recommends use of — and –. There is discussion on what coding should be used which is another matter. Nor does using &emdash; instead of typing \"--\" or \" - \" in any way make any Wikipedia text less usable unless you mean usable in a strict ASCII text environment which barely exists any more. jallan 14:56, 18 Jun 2004 (UTC)\n\n## Can people be bothered?\n\nI know I'm asking to get lynched posting this here, but can most editors really be bothered to do anything special just to get a dash that looks the \"right\" length? I certainly can't. I'm all for correct spelling, punctuation and grammar, but I will stop short of fretting about whether my dash is the correct length. I don't perceive any difference when reading between hyphens, m and n dashes and whatever else you have (by that I mean it doesn't alter the way I would read a sentence, unlike other misplaced or misused punctuation). When I'm writing I will use just a single hyphen\/minus sign with a space either side - like this - when perhaps I \"should\" use a dash. It's easy to type and easy to read (both as source and when rendered on the page), and I frankly can't be bothered to go find the style guide every time I type a dash to check the correct way of doing it. Maybe you could blame my laziness on Microsoft Word for autocorrecting all my hyphens to dashes. Anyway, if you really have nothing more productive to do you're welcome to \"correct\" all my \"misused\" hyphens. But I'm sure Wikipedia could benefit from your time more if it were spent improving the content of articles rather than worrying about exactly what the dashes look like. Tjwood 17:34, 16 Jun 2004 (UTC)\n\nI would hope that most editors would be in favor of anything that improves the look and readibility of their text, including using dashes rather than hyphens when appropriate. Looking good is important. Your style is substandard in the technical sense. I don't believe you will find support for use of \" - \" as a dash in any normal style sheet. Instead you will find that use severely deprecated. Check also Google Search: publisher submission dash for specification after specification that in submissions to publishers a normal dash must be indicated by \"--\". If typing \"&emdash;\" is too difficult for you, then use \"--\" accordingly. That has been the standard keyboard substitute for dash for at least 60 years and is also more efficient to enter than \" - \". As you point out, even MS-Word corrects use of \" - \" as an error. That this is unimportant to you is irrevelant. It is important to others that Wikipedia text looks like it was written and edited by knowledgeable people. If you use non-standard style of any kind, whether punctuation, spelling, typographical style, grammar ... then you can expect to have your work edited to fit the standards. And you shouldn't expect those who know better to accept a generally deprecated style any more than they would accept a generally deprecated spelling. You seem to be suggesting that because you and some others don't know the rules and conventions and don't want to know the rules and conventions, that the rules and conventions don't matter. It doesn't work that way. jallan 01:33, 17 Jun 2004 (UTC)\nTJ, you are right that most editors cannot be bothered (nor should they) to understand the correct usages of the various dash types\u2014the most important thing is to get good writing and even better content. However, some editors will want to correct the typography, and it's important that they can do so (a) easily and (b) in a way that still produces readable wiki, so that editors like you are not too confused. \u2014Steven G. Johnson 03:56, Jun 17, 2004 (UTC)\n\u2026and (c) it is important that people must not convert (my!) correct dashes into hyphens, as has been happening. We need a policy in favour of correct punctuation to defend Wikipedia against silly edit wars. \u2014\u00a0Chameleon 11:13, 17 Jun 2004 (UTC)\nSorry, having re-read my first comment I don't think I made it quite clear. When I'm writing text that will be printed - in MS Word for instance - I will use an unspaced double hyphen which Word will automatically convert to an m-dash. Word will also convert a spaced hyphen to a spaced n-dash such as for date ranges etc. But for Web use where I'm limited to plain ASCII or otherwise using special entities, I use a spaced hyphen because it's too much hassle to look up the correct entities to use and the source looks a mess with entities all over the place. A double dash--like this--looks more ugly than a spaced hyphen.\nIf Wikipedia gets around to replacing double dashes with the correct entities when the pages are displayed (in a similar manner to Word), but leaving the source code readable, I will happily use them. But until that point I will stick to my \"incorrect\" spaced hyphens.\nAs for printed versions of Wikipedia, surely any possible printed version would need fairly widespread editing first to make things consistent anyway? I've seen different pages on Wikipedia written using completely different styles of layout etc, never mind just dashes. You can get away with it no problem on the web, but when it comes to paper it would look a mess without one fixed style decided by the publisher of the paper version.\nTjwood 13:14, 17 Jun 2004 (UTC)\nActually, you know, I've just looked at what I posted above, and I now think I was wrong when I said \"A double dash--like this--looks more ugly than a spaced hyphen.\" The double dash looks better. You may have just converted me.\u00a0:-) Tjwood 13:20, 17 Jun 2004 (UTC)\nIt doesn't really matter whether your hyphens are single or double, spaced or tight \u2014\u00a0just as long as you don't mind the rest of us correcting them to dashes. In the same way, it doesn't matter too much whether you misspell words when you make valuable factual contributions to the encyclopaedia \u2014\u00a0just as long as you don't mind us cleaning them up afterwards. \u2014\u00a0Chameleon 15:27, 17 Jun 2004 (UTC)\nIt may matter for ESL readers of Wikipedia. I am not sure, but I think the unspaced dashes, when used as parenthesis, are globally a relatively uncommon phenomena. Or at least, they introduce a resistance in the reading process for readers who aren't used to them. And unspaced dashes are not, let's be honest now, particularly common on the world wide web. I would favor a standard-rendering of spaces before and after m-wide dashes.\n--Ruhrjung 18:56, 2004 Jun 17 (UTC)\nAt this point I'd like to say that dashes are often poor punctuation anyway. Most of the time, the sentence is better with a comma, semi-colon, colon, full stop or parenthesis. \u2014\u00a0Chameleon 19:28, 17 Jun 2004 (UTC)\n\n## Consistency within articles\n\nHaving looked at the first ever printed version of Wikipedia (the German WikiReader Internet), I've noticed how awful mixing hyphens and varying length dashes within articles is. It's bad enough between articles, but can we please have a policy of not mixing them. I think consistency is more important than deciding which dashes are used. Angela. 12:43, 17 Jun 2004 (UTC)\n\nThat's fine, as long as it is not used as an excuse to justify bad punctuation and convert dashes into hyphens. This is really just an extension of the existing policy on spelling: correct mistakes, respect other people's variations within what is correct, try to keep usage consistent within articles even if usage differs between them... \u2014\u00a0Chameleon 15:27, 17 Jun 2004 (UTC)\nThat would be very good, and according to other wikipolicies I believe in (like consistent British or American spelling per article).\n--Ruhrjung 18:41, 2004 Jun 17 (UTC)\nIsn't consistency kind of a lost cause on Wikipedia? Which, if you ask me, is the way it probably ought to be?parts of articles suck, other parts are better, but overall it's getting pulled in the right direction. It'll never be complete. Wikisux 15:03, 23 Jun 2004 (UTC)\n\n## Going live with policy\n\nThe proposed text (above) appears to enjoy consensus support, so I'll be inserting it into the Manual of Style shortly. The outstanding issue to include, if anyone fancies adding in the extra language, is consistency within a given article: the guideline given for US\/UK spelling conventions is follow the standard set by the first major (non-stub) edit; the logical approach for dashes would be to follow the dash style used by the first editor to include a dash in the article.\n\nChanging to explicit — and – entities now (while admittedly not as transparent as we might wish wikitext to be) will also save us a lot of anguish and conflict when the promised automatic conversion comes on line, when presumably we'll switch to using two- and three-hyphen chains in the markup -- either manually or (as suggested above) by someone running a bot. (\"Conflict\" in the sense that a lot of people have been typing \"--\" in the expectation that they'll autoconvert into emdashes, when from what we saw a few months ago when the function was briefly activated, it'll be three hyphens for an emdash and two for an endash, similar to the TeX convention described above.)\n\nEverybody happy? Hajor 10:53, 28 Jun 2004 (UTC)\n\nThe trouble is that, if the software interprets double hyphens as en-dashes rather than em-dashes, and we have a policy of not changing other people's punctuation unless it's really bad (i.e. they use hyphens for dashes), then we'll have a lot of \"--\" that the author wanted to represent em-dashes that will now show up as en-dashes and our policy will specifically forbid us from manually turning these \"--\" into \"---\" (so that \u2014 rather than \u2013 appears).\nSo, the policy seems fair, but a technical glitch will unfairly favour the use of en-dashes. \u2014\u00a0Chameleon 11:32, 28 Jun 2004 (UTC)\n\nNo -- the third bullet says subsequent editors are free to change double hyphens to their choice of em-dashes or en-dashes (providing the original writer's spaced\/unspaced preference is respected). That was intended precisely to preempt that type of conflict when the auto-conversion comes on line: make them explicit now, and then change them back to either (1) \"---\" & \"--\" or (2) \"--\" and \" - \" following the upgrade. Hajor 12:51, 28 Jun 2004 (UTC)\n\nNo, Hajor, you're not getting me here. I understand that the policy says that now, in the short term I can turn hyphen hacks into real dashes. That wasn't my concern. I was concerned with how this policy would interact with future software that would automatically display \"--\" and \"---\" in the markup as \"\u2013\" and \"\u2014\" on the page.\nAt that point in time, the policy you are advocating would obviously be largely obsolete. The preferable way to put correct dashes into text will be, at that point, by using double and triple hyphens in the markup. Using \"—\" etc will be deprecated, in the same way that we prefer the use of double inverted commas instead of \"<i>\". That is to say, we only use HTML to do something when there is no wiki markup for it. It would be a good idea, at that point, to write a script that goes through Wikipedia changing all instances of \"—\" to \"---\". This would not affect the display of the pages in the slightest; it would just make the markup easier to read.\nMy point is that most of the policy would be obsolete, except for the part about not changing other people's punctuation (from one of the three acceptable forms of dashes to one of the others). I think people would argue that this part still stands. The problem with this is that, although those em-dashes originally entered into the text as \"—\" will now show up in the markup as \"---\" and on the page as \"\u2014\", there will also be many instances of dashes originally entered into the markup as \"--\" with the author's intention being to enter an em-dash, which would at that point show up on the page as \"\u2013\".\nThere would therefore be thousands of en-dashes that the original authors wanted to be longer. And policy would prevent me from changing the markup from \"--\" to \"---\" to make them display as \"\u2014\" instead of \"\u2013\". \u2014\u00a0Chameleon My page\/My talk 19:55, 30 Jun 2004 (UTC)\nSeems to me this whole mess could be avoided by turning \"--\" into em-dashes, as everyone expects, and \" - \" (hyphen surrounded by spaces) into en-dashes. Reverse compatibility, preserving authors' intent, and intuitive. (The three dashes proposal I find ridiculous.) Wikisux 21:19, 30 Jun 2004 (UTC)\n\nI've said this before, but I wonder if I am being understood: we should NOT use dashes that are created in source with characters that the average person cannot understand. The average person has NO IDEA what HTML entities are. Please put this ahead of nit-picking over correct and pretty typography. Agree on a way to represent the various typographical elements with \"-\", \"--\", \"---\" and spaces. -- Tarquin 22:34, 28 Jun 2004 (UTC)\n\nYou've said it before, you've been understood, but the consensus shown above indicates that you're simply not being agreed with. And I think you're insulting the intuitive abilities of the \"average person\", too. And why on Earth didn't you contribute your views in the two weeks since I posted the proposed text? Please -- could you try to live with the HTML entity solution until the automatic conversion function is activated (when we are sure to start using nice, transparent strings of hyphens)? Hajor 23:09, 28 Jun 2004 (UTC)\nI'm also suspicious of average person arguments, especially when almost all Wikipedians (and logically almost 50% of them are below the Wikipedian average) are able to master the other various stylistic features of Wikipedia source code. I think that even the average computer user today does know what HTML entities are. The forms — and – are hardly unituitive. But many Wikipedian editors find them ugly and annoying to type and some don't care about typography and some don't want to understand such things, balanced by those who very much care. But no-one wants to see \"---\" in the final presentation or \"--\" in a range, such as 1912--1914, in the final presentation. Accordingly we can't use such constructs before new interpretation software is applied. jallan 00:20, 29 Jun 2004 (UTC)\nAnd that is in fact what was mandated by the text of the Manual of Style as it stood when I made the suggestion two weeks ago: \"Historically a double hyphen (--) was used to represent an em dash because on a typewriter the hyphens tend to connect, creating a dash in appearance, but since this is almost never the case in ASCII do not use this.\" It's still there now, if you care to take a look. And no, it wasn't me who put it there. Hajor 04:21, 30 Jun 2004 (UTC)\n\nI apologize if this has already been brought up, but I'd suggest replacing double hyphens (--) with em dashes, and single hyphens surrounded by spaces ( - ) with en dashes. This is the Textile standard (if you've ever used Movable Type) and it makes more sense than any of the other ideas I've seen here. edit: Oh, I see it has been brought up already. I throw my weight behind this proposal, then. Wikisux 04:40, 30 Jun 2004 (UTC)\n\n## Quotation Dash \/ Horizontal bar\n\nThe Unicode Standard (v4.0) on page 155 in chap. 6 has the Quotation Dash (U+2015 \/ ―) to be called Horizontal Bar and gives Quotation Dash as an alternate name but officially refers to it as Horizontal Bar. Of course, calling it Horizontal Bar would conflict with the html tag <HR> which the resulting graphic people seem to want to call horizontal bar too, when it's really called a Horizontal Rule by the W3C standard. So yeah... Call U+2015 a quotation dash, a horizontal bar, or both? Keeping with the Unicode Standard for naming specific code characters would be best. Kevin Breitenstein 05:43, 3 Jul 2004 (UTC)\n\n## Software\n\nI\u2019ve come late into this debate, but I\u2019m pleased with the way it seems to have turned out. I\u2019d like to wish the people writing the conversion software the best of luck. It looks like a wicked problem, but there are some good precedents for a solution.\n\n\u2022 Textile, written in PHP, was already mentioned. It does dashes, smart quotes and lots of other character conversions. It even supports a table formatting scheme similar to wikitables. There\u2019s also a Textile plug-in for Movable Type, written in Perl.\n\u2022 SmartyPants, in Perl, is a simple processor which converts dashes, smart quotes, and ellipses. There's also a PHP version.\n\u2022 Markdown is a more full-featured text processor by the same author.\n\nIt was inspiring to see Wikipedians hash out a solution that respects everyone\u2019s wishes and maintains a high standard of typography.\n\n\u201cApostrophes and quotation marks, anyone?\u201d\n\nMichael Z. 06:04, 2004 Aug 27 (UTC)\n\n## Various ASCII to HTML conventions\n\nJust trying to list them to keep them straight. If the automatic conversion is ever implemented, this will probably become a vote.\n\n### TEX and SmartyPants convention\n\n\u2022 hyphen = \u201c`-`\u201d (one hyphen: \u201c`Ex-wife`\u201d)\n\u2022 en dash = \u201c`--`\u201d (two hyphens: \u201c`1995--2004`\u201d)\n\u2022 em dash = \u201c`---`\u201d (three hyphens: \u201c`em dashes---those beautiful things`\u201d)\n\u2022 spaced en dash = \u201c `--` \u201d (two hyphens surrounded by spaces: \u201c`November 1 -- December 26`\u201d)\n\u2022 spaced em dash = \u201c `---` \u201d (three hyphens surrounded by spaces: \u201c`em dashes --- those beautiful things`\u201d)\n\n### Textile convention\n\n\u2022 hyphen = \u201c`-`\u201d (one hyphen with no spaces: \u201c`Ex-wife`\u201d)\n\u2022 en dash = \u201c `-` \u201d (one hyphen surrounded by spaces: \u201c`1995 - 2004`\u201d)\n\u2022 em dash = \u201c`--`\u201d (two hyphens: \u201c`em dashes--those beautiful things`\u201d)\n\u2022 spaced en dash = ??? (can this be typeset at all using the Textile notation?)\n\u2022 spaced em dash = \u201c `--` \u201d (two hyphens surrounded by spaces: \u201c`em dashes -- those beautiful things`\u201d)\n\n### Nathan Hamblen's backwards convention\n\n\u2022 hyphen = \u201c`-`\u201d (one hyphen with no spaces: \u201c`Ex-wife`\u201d)\n\u2022 en dash = \u201c`---`\u201d (three hyphens: \u201c`1995---2004`\u201d)\n\u2022 em dash = \u201c`--`\u201d (two hyphens: \u201c`em dashes--those beautiful things`\u201d)\n\u2022 spaced en dash = \u201c `---` \u201d? (three hyphens surrounded by spaces: \u201c`November 1 --- December 26`\u201d)\n\u2022 spaced em dash = \u201c `--` \u201d? (two hyphens surrounded by spaces: \u201c`em dashes -- those beautiful things`\u201d)\n\nI believe that Textile and SmartyPants both allow you to reverse their behaviour by setting a preference. I\u2019m guessing TEX does too. The typist\u2019s convention going back many decades trumps the significance of these excellent software packages\u2019 behaviours. Michael\u00a0Z.\u00a02005-10-4\u00a015:04\u00a0Z\n\nMichael, I edited your nice list to (a) correct a small error in the example for \u201cen dash\u201d in Nathan Hamblen\u2019s backwards convention, (b) add entries for \u201cspaced en dash\u201d and \u201cspaced em dash,\u201d and (c) add some `<code>` tags to improve the formatting. I hope you don\u2019t mind. \u2013 Daniel Brockman 14:43, 26 December 2005 (UTC)\nThis is all moot since they can be entered directly into the markup now \u2014 like this. I've got a javascript button that converts a bunch of types automatically. \u2014 Omegatron 02:04, 1 December 2005 (UTC)\n\n## Contradiction on double hyphen use\n\nAccording to \"Dashes and hyphens used on Wikipedia\":\n\nHistorically a double hyphen (--) was used to represent an em dash because on a typewriter the hyphens tend to connect, creating a dash in appearance. Since this is almost never the case in digital type, do not use this technique.\n\nBut according to \"Dash guidelines for Wikipedia editors\":\n\nEditors who do not want the bother of keying in HTML entities or prefer to maintain the readability of the wikitext are free to type their dashes in this fashion.\n\n(my emphasis)\n\nIt seems odd to me that such a broad exception to the rule would be placed in the bottommost section. Also the two section titles seem redundant, no? Dforest 12:45, 8 November 2005 (UTC)\n\nYep. I like the second one better. They should be allowed, but can be changed to a proper dash by anyone who wants to. \u2014 Omegatron 14:41, 9 January 2006 (UTC)\nThere can be problems displaying em dashes (and curly quotes for that matter) with some OS and character sets -- that is a reason why perhaps there should be tolerance for the double hyphen (and straight quotes) option. RomaC 03:12, 21 January 2006 (UTC)\nBut allow it in an inconsistent manner? That doesn't make sense. Either allow them or don't. There's no point in letting people use double dashes in a few articles so that they can read them in their browser, while all the other articles use Unicode dashes. Those same browsers have a problem with foreign characters, too. Should we remove foreign language Wikipedias?\nIdeally, the Mediawiki would be smart enough to figure out where to use the real dashes, and we could use `<nowiki>` for the few cases where it would otherwise make a mistake. For instance, whenever it detects a dash between two dates, it renders it as an en dash. When it sees two dashes in a row, it renders it as an em, etc. Then you could turn it off in preferences if you had a broken browser. \u2014 Omegatron 19:40, 24 April 2006 (UTC)\n\n\"Allowed\" should be changed to \"tolerated\". That would mean that if a user types a double dash when creating or adding to an article, nobody should get bent out of shape, and there would also be the encouragement (more than allowing) of editors to change them to em dash. Thus it would read something like: Hu 22:49, 18 November 2006 (UTC)\n\nI agree 100% on this. There has been discussion and disagreement on this issue in the past, but I think it is worth revisting the topic. Especially now that the edit page has a link to insert the dashes without needing to use the HTML entity. \u2014Doug\u00a0Bell\u00a0talkcontrib 23:08, 18 November 2006 (UTC)\nI agree also, except that the wording should be: Double dashes may be tolerated, but editors are encouraged to convert any they might find to em dashes, or to spaced en dashes (depending on which of these two practices has already been established in the article). Remember that we allow spaced en dashes OR em dashes (spaced or not), as sentential punctuation. \u2013 Noetica 23:29, 18 November 2006 (UTC)\nThank you for re-emphasizing space em dashes. I think they are much preferred because the spaces enhance reading speed and comprehension since the words on either side are more easily recognized as individual cognates and don't have to be parsed away from the dash. Hu 10:32, 19 November 2006 (UTC)\n\n## Suggest cutting moot point at bottom\n\nMost of the bottom of this article (how to type em-dashes on various keyboards) is totally moot since anyone can just click the link below the edit box to add the em-dash. Any objections to me chopping it totally? It can always live at em dash or something. Stevage 22:19, 19 January 2006 (UTC)\n\nDon't. Some people will want to type them in by hand instead of the edit thingy, which is slower. \u2014 Omegatron 19:25, 26 January 2006 (UTC)\n\n## Should have a preferred dash style\u2014really\n\nI'm sure I can't say anything here that hasn't been said before, so please be patient with a new editor.\n\nI know about the many styles people use for dashes. I was hoping by coming to the manual of style I'd get some guidance about the preferred style on Wikipedia. My understanding\u2014generally, not specific to Wikipedia\u2014has been that the em dash without spaces (or with hair spaces) was the preferred style. After reading the style manual here it seems like the answer is \"whatever you want to type.\"\n\nAlso, having the guide tell editors not to change existing dash styles seems to go against WP:BB and will just result in ugly pages that use multiple dash styles. It's OK not to require a particular dash style, but not even stating a preference is, in my opinion, a mistake. \u2013 Doug Bell talkcontrib 04:07, 4 February 2006 (UTC)\n\n``` Sorry to rain on the parade, but I don't think 5 people is much of a consensus for the whole encyclopedia. I'd love to see an em-dash as preferred style for parenthetical clauses and similar breaking, but am very much against flush or hair spacing. Flush spacing, though common in print, is far from universal, and at Wikipedia seems to be in the minority of articles I've worked on (before tweaking them, that is). Hair spacing looks nice as displayed, but isn't worth the editing annoyance, especially when ordinary spaces look fine as well. \u2014 Jeff Q (talk) 02:18, 26 April 2006 (UTC)\n```\n``` Fine, let's ignore the spacing issue for now. Can we at least say that 1) it's okay to put in hyphens or em\/en dashes; 2) it's okay to replace hyphens with em or en dashes, as appropriate; and 3) it's not okay to replace em or en dashes with hyphens? \u2014Chowbok 04:03, 26 April 2006 (UTC)\n```\n``` * Support. Jeff Q (talk) 04:09, 26 April 2006 (UTC)\n* Support. (Seems like the sensible place to develop consensus on this is the dashes talk page, isn't it . . .) \u2014Simetrical (talk \u2022 contribs) 03:08, 17 May 2006 (UTC)\n* Support, preferably with the addtion that changes from en dash to em dash or em dash to en dash are discourgaed. Jeltz\n```\nNo one can agree on it, so it sits in limbo for now. \u00a0:-\\ \u2014 Omegatron 04:44, 4 February 2006 (UTC)\nI agree with Doug Bell. My ideal policy would be that we always use proper em and en dashes, but I think mandating hyphens would be a better solution than \"do what you like, but don't change anything\". Can't this just be voted on once and for all? Surely Wikipedia has settled thornier issues than this... \u2014Chowbok 22:51, 2 March 2006 (UTC)\nIt should be \"do what you like, but x is preferred. don't revert if people change to x.\" \u2014 Omegatron 03:10, 3 March 2006 (UTC)\nI agree. And as I've said elsewhere on this page, flush em dashes are the only style\u2014short of inventing a nonstandard ‐ entity\u2014with a prayer of automatically extracting semantic dashes if someone wanted to automatically convert Wikipedia to a different dash style. (PS: Is it a coincidence that this is the only Wikipedia discussion I've read in which all participants sign with a Unicode em dash?\u00a0:-) ) \u2014BenFrantzDale 04:10, 3 March 2006 (UTC)\nI second the agreement with Omegatron's recommended wording. Do we really need a vote to determine what the preferred x style is?\u2014it seems pretty obvious to me. (I've changed my signature from an en dash with spaces to an em dash with no spaces as a sign of support.\u00a0:-) \u2014Doug Bell talkcontrib 02:27, 18 March 2006 (UTC)\n\nI usually change double hyphens, spaced hyphens, and spaced dashes to flush em dashes when I see them, especially when they are inconsistently used in an article. I don't make it a crusade, just clean it up when I'm editing something else. Em dashes must look fine, because no one ever complains or reverts. Michael\u00a0Z.\u00a02006-03-03\u00a003:43\u00a0Z\n\nI think Omegatron's solution is exactly right. It won't effect anybody who doesn't want to bother with correct dashes, but we'll gradually move to proper usage. And I don't see why x should be so hard to resolve. These aren't actually controversial issues in the publishing world (unlike disputes like, say, the serial comma or the possessive of singular words ending in \"s\"); em dashes break up sentences, en dashes signify ranges and are used for compound hyphenation. \u2014Chowbok 02:42, 18 March 2006 (UTC)\nWhy are unspaced better? \u2014 Omegatron 03:46, 18 March 2006 (UTC)\nTo appeal to authority, unspaced em dashes are the style used by CMS and by Strunk & White. \u2014BenFrantzDale 05:07, 20 March 2006 (UTC)\n\nWe actually seem to have reached some sort of consensus here. Can we change the policy, then? \u2014Chowbok 01:51, 26 April 2006 (UTC)\n\nSorry to rain on the parade, but I don't think 5 people is much of a consensus for the whole encyclopedia. I'd love to see an em-dash as preferred style for parenthetical clauses and similar breaking, but am very much against flush or hair spacing. Flush spacing, though common in print, is far from universal, and at Wikipedia seems to be in the minority of articles I've worked on (before tweaking them, that is). Hair spacing looks nice as displayed, but isn't worth the editing annoyance, especially when ordinary spaces look fine as well. \u2014 Jeff Q\u00a0(talk) 02:18, 26 April 2006 (UTC)\nFine, let's ignore the spacing issue for now. Can we at least say that 1) it's okay to put in hyphens or em\/en dashes; 2) it's okay to replace hyphens with em or en dashes, as appropriate; and 3) it's not okay to replace em or en dashes with hyphens? \u2014Chowbok 04:03, 26 April 2006 (UTC)\n\u2022 Support. Jeff Q\u00a0(talk) 04:09, 26 April 2006 (UTC)\n\u2022 Support. (Seems like the sensible place to develop consensus on this is the dashes talk page, isn't it . . .) \u2014Simetrical (talk\u00a0\u2022\u00a0contribs) 03:08, 17 May 2006 (UTC)\n\u2022 Support, preferably with the addtion that changes from en dash to em dash or em dash to en dash are discourgaed. Jeltz talk 14:31, 21 August 2006 (UTC)\nHow come? The usage of those is pretty well-established (unlike with spacing). \u2014Chowbok 18:21, 21 August 2006 (UTC)\nWell established? I have seen both en dashes and em dashes used in serious typograhy (news papers, books). I think we should ignore en vs. em for now since there is no well-established rule there that all of the typograhic world agree on. Jeltz talk 11:22, 22 August 2006 (UTC)\nRight, both should be used. Whether it's an em or an en depends on the usage. Can you give me an example of a book that uses one or the other contrary to the standards? \u2014Chowbok 16:00, 22 August 2006 (UTC)\nHmmm, I feel like we might be talking about different things. What I'm talking about is that both en dashes and em dashes can be used for parentethical purposes, depnding on what you prefer. See Dash#En_dash_versus_em_dash for what I'm referring to. Jeltz talk 17:49, 22 August 2006 (UTC)\n\u2022 Support, of course. Ignore the spacing issue, and the en vs em issue for now. (Of course, Mediawiki should just replace -- with \u2014 automatically.) \u2014 Omegatron 15:13, 21 August 2006 (UTC)\nI personally prefer the LaTeX style where -- is replaced with \u2013 and --- is replaced with \u2014. That way you can write both. Jeltz talk 11:24, 22 August 2006 (UTC)\nI used to argue for that style, too. \u00a0:-) After creating the dash fixer, I realized that there is a vast number of double hyphens on Wikipedia already from people who learned it in typing class, and a few who just use a hyphen. Might as well just go with what they already use. (Besides, they'd just end up typing -- anyway, out of habit, getting an en dash, and leaving it that way; so we'd have ens everywhere we were supposed to have ems.) It could still be possible to create both dashes, if we were smart with the implementation. I tried to write up a Dash syntax summary, but never finished it. \u2014 Omegatron 11:45, 22 August 2006 (UTC)\n\u2022 Support, also hyphen-minus to minus (in math context) but not the reverse, and mediawiki should automatically do double -- (but leave more than two hyphens alone). This is the first I (who thought myself a bit of a typography geek) learned that hyphen and m-dash were not equivalent. In the long run, I'd say that type-anything is necessary, but any hyphen-minus outside a PRE tag or equivalent should be ripe for replacement \u2014 to minus if math; to en-dash when that's clear or, in parenthetical case, for consistency or when there's any indication that the original author prefers it so; to em-dash for the same reasons or for double hyphens; and to en-dash or em-dash as preferred (or possibly just em-dash) in the parenthetical case when there is no reason to infer authorial preference. --Homunq 14:25, 6 January 2007 (UTC) (or is it \u2014~~~~?)\n\u2022 Support Raif\u02bbh\u0101r Dorem\u00edtzwr 15:29, 6 January 2007 (UTC)\n\u2022 SupportDavid Eppstein 16:29, 6 January 2007 (UTC)\n\u2022 Support Strad 17:10, 6 January 2007 (UTC)\n\u2022 Support \u2014Ben FrantzDale 00:11, 8 January 2007 (UTC)\n\n## X11\n\nFollowing section was rather misleading, since e.g. most linux distributions have switched from xmodpad into using xkb, and the provided example does not work.\n\n```xmodmap - <<EOT\nkeysym m = m NoSymbol U2014 NoSymbol\nkeysym n = n NoSymbol U2013 NoSymbol\nkeysym KP_Subtract = KP_Subtract NoSymbol U2212 NoSymbol\nEOT\n```\n\nwill add the em dash (\u2014) to AltGr-M, the en dash (\u2013) to AltGr-N, and the minus sign (\u2212) can now be obtained by pressing the minus key on the numeric keypad while holding the AltGr key.\n\n## Ndashes are shorter than hyphens\n\nNotice:\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 twenty ndashes\n-------------------- twenty hyphens\n01234567890123456789 twenty digits\n\nAt least in my Lucida default font, the hyphen is at least a pixel longer than the ndash, contrary to this MoS section, and the hyphen matches the width of numbers, not the ndash.\n\nSo what I'm saying is, unless Lucida is unusual in this respect, please stop replacing hyphens in number ranges with ndashes. Thank you. --James S. 15:16, 16 February 2006 (UTC)\n\nFor the record: the character you describe as \"hyphen\" is the Unicode character U+002D called HYPHEN-MINUS. This is an old overloaded ASCII character that has in the past been used as both a HYPHEN (\u2010) and a MINUS (\u2212). Some font designers make HYPHEN-MINUS look more like a hyphen (short), others make it look more like a minus (longer, higher, matching +). As long as there is no easy way to enter the proper unambiguous Unicode HYPHEN on non-specialist keyboards, this problem will persist. Markus Kuhn 19:19, 24 March 2006 (UTC)\nIf that is ineed the case, then your copy of Lucida is quite unusual. The computer I am at right now has crappy fonts installed, but I am quite sure that at home the above would render with the first line longer than the second and with the line of numbers longer than the hyphens. Compare:\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 twenty ndashes\nNNNNNNNNNNNNNNNNNNNN twenty Ns\nThese should be very close to the same length.\nWhat operating system and browser are you using? \u2014BenFrantzDale 16:13, 16 February 2006 (UTC)\nLucida Grande has an unusually long hyphen; I assume other Lucidas do too. Normally a hyphen is very short, indeed. Michael\u00a0Z.\u00a02006-02-16\u00a016:27\u00a0Z\nOkay, that's just bizarre. My copy of Lucida Sans has en dashes longer than hyphens, but in Lucida Sans Unicode the hyphens are longer. Why would they do this? How dumb.\nBut yes, that particular Lucida is extremely unusual in that respect, so we shouldn't base policy on it. \u2014Chowbok 17:48, 20 March 2006 (UTC)\nIf you have Lucida Sans Unicode installed, this should demonstrate it:\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 twenty ndashes\n-------------------- twenty hyphens\n01234567890123456789 twenty digits\nChowbok 17:54, 20 March 2006 (UTC)\nWhereas plain Lucida Sans has\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 twenty n-dashes\n-------------------- twenty hyphens\n01234567890123456789 twenty digits\nnnnnnnnnnnnnnnnnnnnn twenty n's\nThe behavior of Lucida Sans Unicode is unusual and should be ignored. Arial, the default font, gives all elements their expected widths:\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 twenty n-dashes\n-------------------- twenty hyphens\n01234567890123456789 twenty digits\nnnnnnnnnnnnnnnnnnnnn twenty n's\nNotice how the n-dashes are precisely as long as the n's. \u2014Simetrical (talk\u00a0\u2022\u00a0contribs) 03:19, 17 May 2006 (UTC)\n\nThe Template:Style at the top of WP:MOSDASH made navigation more difficult than necessary. I've moved it to the bottom and filled out the missing shortcut in Template:Style-guideline. - Omniplex 18:02, 28 February 2006 (UTC)\n\n## Dashes not used on Wikipedia\n\nThe article states neither the figure dash (\"\u2012\") nor the quotation dash (\"\u2015\") should be used in Wikipedia articles because browser support for them is lacking. Is that still true? Under Mac OS X 10.3.9 (2003) they are rendered correctly: not only in the current version of Safari, but also in a 2003 version of Camino (a Mozilla-based browser) and in a 2001 version of Internet Explorer. Ian Spackman 08:13, 24 March 2006 (UTC)\n\nAs of 2005, the support of the UTF-8 end dash (\u2013), em dash (\u2014) and minus sign (\u2212) in web browsers has been excellent and there is no reason not to use these characters in Wikipedia. Even VT100-terminal based text-mode browsers such as w3m or lynx haven't had problems with them for many years. I don't know whether the same can already be said for figure dash and quotation dash, which are missing in most commonly-covered Unicode subsets, including WGL4. Markus Kuhn 19:10, 24 March 2006 (UTC)\n\n## Article text now outdated after UTF-8 switch\n\nThis policy page needs to be throoughly reviewed for technical inaccuracies after the Wikipedia UTF-8 switch. I found one specifically wrong passage, and I suspect there may be others.\n\nUse the HTML entity —, which the MediaWiki engine automatically converts into a numeric entity in the rendered HTML.\n\nFirst, this appears to contradict the up-front, emphasized statement about using UTF-8 characters. The context suggests its point was recommending the type of dash, not necessarily the preferred means to use it, but that's my point \u2014 it doesn't comprehend the availability of UTF-8. Second, it's most definitely wrong about rendering \"—\" as a numeric entity \u2014 it's rendered as a UTF-8 em-dash character, as it should be. I suspect that whoever added the bold-statement update to the top didn't examine the rest of the article for now-inaccurate statements or undesirable advice. I'd be bold and do an update myself, but I'm not confident enough of my knowledge about these topics. (I had to do some careful testing just to be sure of even the single case I'm mentioning here.) ~ Jeff Q\u00a0(talk) 11:45, 5 April 2006 (UTC)\n\nYes, the person who added the bold statement at the top (me) didn't alter the rest of the article very much. Originally it was more like a \"this page is invalid now\" notice. \u2014 Omegatron 14:08, 5 April 2006 (UTC)\nI do have a problem with encouraging the use of the UTF-8 characters in wiki source, because n-dash looks exactly like a hyphen in the edit box, at least in Windows Courier (m-dash, while a hair longer, also looks the same unless you compare them carefully side-by-side). It's fine for the original editor, but (a) it can lead to confusion for subsequent editors (\"hey, that's a date range, why did he put a hyphen there?\") (b) it can mislead newcomers who use existing articles for style guidance, when all they can see is a dash (c) it can lead to a pointless discussion such as the one I just had with User Quoth over what I thought was the replacement of a – with a hyphen in John Wilkes. It takes a true geek to look at an edit box and figure out whether one is looking at an ASCII hyphen or a typographically correct dash.\nCan we at least (1) say Entity and Unicode character are both valid; do not jump into an article and replace an HTML entity with a Unicode character just because you can (2) have the link in the Insert box create the entity instead of the Unicode character. David Brooks 00:09, 17 April 2006 (UTC)\nBut that makes the wikicode harder to read and more intimidating to the \"technically impaired\". Better to force the \"true geeks\" to deal with the complexity of unicode characters than to make it harder on all the regular people editing content by inserting all kinds of confusing HTML.\nIdeally, the software would recognize all the major cases in which each type of dash is used and generate them automatically (and curly quotes, too, while we're at it). The special cases could be handled with nowiki tags just like everything else. But no one seems to want to implement this. Textile does it. \u2014 Omegatron 02:26, 17 April 2006 (UTC)\n\n## Birth\u2013death dates\n\nA discussion at Wikipedia talk:Manual of Style (dates and numbers) has revealed an overwelming consensus (with, I think, only one editor disagreeing) for the use of en-rules for connecting birth and death dates. As this is the universal English standard, is there a reason for not making it the Wikipedia standard? Editors wouldn't of course (indeed, couldn't) be forced to use it, but I can't imagine why any editor would complain because a hyphen had been corrected to an en-rule. --Mel Etitis (\u039c\u03b5\u03bb \u0395\u03c4\u03b7\u03c4\u03b7\u03c2) 16:08, 15 May 2006 (UTC)\n\nProposing to keep Wikipedia talk:Manual of Style (dates and numbers)#n dash as central discussion place on this issue. --Francis Schonken 16:48, 15 May 2006 (UTC)\nThat certainly seem to be the wisest approach. --Mel Etitis (\u039c\u03b5\u03bb \u0395\u03c4\u03b7\u03c4\u03b7\u03c2) 22:17, 15 May 2006 (UTC)\n\n## Special character list\n\nThe character to the left of the emdash (\u2014) on the special character list under the edit box (\u2013) appears to be an ndash (\u2013). If this is correct, it would be helpful to have this fact added to Omegatron's bolded notice at the top of the article, for the benefit of casual skimmers of the page. With the broad lack of consensus over the styles discussed on this page, I don't feel so bold as to make the change myself. --Blainster 16:12, 18 May 2006 (UTC)\n\nI second that, and I do feel so bold. \u201471.208.125.132 22:38, 1 September 2006 (UTC)\n\n## Negative numbers\n\nI'd like to propose that negative numbers should use the traditional \"hyphen\/dash\" character \"-\" rather than \u2212. I believe this usage would make pages easier to maintain, and that pages using another convention would probably gradually migrate to this anyway just because of editor ignorance and common usage outside this wiki. I can see using \u2212 for subtraction in mathematical expressions (where length may be important for aesthetic reasons), but for plain negative numbers, I think the one or two pixel difference in length isn't worth the trouble. Thoughts? --Doradus 17:57, 23 July 2006 (UTC)\n\nI just type minus in in Unicode (0x2212) like this: \u2212. This shows up as correctly on the page and in the edit window. Anal typographers will certainly change dashes to minuses. I see no reason to avoid good typography when we have the tools to easily do it. \u2014Ben FrantzDale 05:23, 24 July 2006 (UTC)\nGenerally, \u201c+1-microsecond error\u201d and \u201c\u22121-microsecond error\u201d look okay, but \u201c-1-microsecond error\u201d doesn\u2019t look elegant, could be even confusing. Compare:\n\u2022 -1- to -5-microsecond error\n\u2022 \u22121- to \u22125-microsecond error\nU+2212 should be encouraged for more readable, non-ambiguous typesetting, although U+002D for the MINUS sign should not be disallowed. \u2014Gyopi 06:22, 24 July 2006 (UTC)\n\n## Dashes In Foreign Languages\n\ni'm a new editor to wikipedia, so i don't know if this issue has been raised already somewhere else, and i'm not sure this is the correct place to post it. if it isnt, pick my discussion up, and put it where it goes. tell me on my user page where u put it, or i'll just come back here for responses. here goes: hyphens are used frequently in french (Ile-de-France) and arabic (Burj al-Arab). i'm guessing the short guy (hyphen-minus) is the correct one to use? its the only one i've ever seen used in those languages. and in french, when do we put hyphens in phrases? i think the french department's name of \"Loir-et-Cher\" literally means \"Loir and Cher\". since english doesnt put hyphens there, does french really need them? i've seen people's and places's proper names with and without hyphens. are francophones arguing about whether to put or not put them, as much as anglophones are arguing about which type of dash to use? perhaps this is clarified in the french-language style manual, but im not a francophone. someone lemme know.4.230.174.207 01:57, 11 August 2006 (UTC)\n\nThere is actually a very specific hyphen character: Unicode U+2010 hex (8208 decimal)\u2014but it may not be widely supported in fonts or browsers.\nIn your browser\/font it looks like this: \u2018\u2010\u2019. (Looks about right in Safari.) Ian Spackman 00:49, 26 August 2006 (UTC)\nIt looks like a hyphen to me in Firefox on Ubuntu. Jeltz talk 10:07, 26 August 2006 (UTC)\n\n## Clarification re dashes separating surnames in page names\n\nGene Nygaard and I have been having a discussion on dashes in pagenames that could perhaps be clarified by a discussion here.\n\nSpecifically, it concerns dashes in the names of pages such as Erd\u0151s\u2013Straus conjecture that involve subjects named after multiple people. Clearly in text such phrases should use en-dashes but we are disagreeing on whether the same is true as part of a page name. MOSDASH says \"Hyphens and dashes are generally rather avoided in page names\" but it only specifically talks in that passage about numeric ranges, and also says that for greater precision dashes can be used (with appropriate redirects). I think that \"precision\" can be interpreted here to mean that dashes are appropriate as a signal that the page name refers to two people instead of to one person with a hyphenated name, while Gene thinks the preference for hyphens over dashes takes precedence. Also, the first example of a dash between names in MOSDASH is a seemingly-approving link to Poincar\u00e9\u2013Birkhoff\u2013Witt theorem; I think this example supports the position that dashes should be used in this context while Gene thinks that the dashed page name is a historical artifact related to the fact that the page name policy of MOSDASH was merged from elsewhere some time after that example was added.\n\nAlso, and I'm not sure how relevant this is, but in the specific case of Erd\u0151s\u2013Straus conjecture the url already has percent-encoded unicodes, so hyphens aren't needed to keep the url pretty.\n\nI'm happy to abide by whatever the consensus is on this issue, and I don't really want to turn this into an attempt to change policy, but I would like a broader sample of opinion on what the existing policy actually is. Anyone?\n\nIf anyone cares to see our existing discussion on the issue, it's here.\n\nDavid Eppstein 05:55, 8 October 2006 (UTC)\n\nNote that Wikipedia:Naming conventions main page now specifically defers to this MoS page for naming as well as use in articles, and the naming discussion on this MoS page is a recent addition here. I don't know where all the past discussions of this issue with respect to article names have taken place, but I do know that such discussions exist.\nErd\u0151s\u2013Woods number has a redirect from Erd\u0151s-Woods number, because it is a remnant that was left behind when Daniel Brockman moved the page.[3] If use of dashes in this context in the article names is encouraged, there will be a great many of them created without having that redirect from the hyphen form, just as we now have a huge number of article names with diacritics in the names which do not have the redirects from the English-alphabet spellings.\nLook at this list, then answer the following questions: Erd\u0151s-Woods, Erd\u0151s - Woods, Erd\u0151s\u2010Woods, Erd\u0151s\u2011Woods, Erd\u0151s\u2012Woods, Erd\u0151s\u2013Woods, Erd\u0151s\u2212Woods.\n1. How many different hyphens do you see as you view this page as you'd normally read it?\n2. How many different hyphens\/dashes do you see when you view it in edit mode (normally displayed in a different font)?\n3. How many of the seven in the above list above are true duplicates?\n4. How many other distinct possibilities could other editors come up with?\nNow, take a look at some of those possibilities contained in links to possible article names\/redirects:\nAs you can see, lots of missing redirects as I post this.\nEven more important, as far as the article naming question goes, if a link is made on a page as [[Erd\u0151s–Woods number]] (displays as [[Erd\u0151s–Woods number]]) or as [[Erd\u0151s–Woods number]] (displays as [[Erd\u0151s–Woods number]]) or as [[Erd\u0151s–Woods number]] (displays as Erd\u0151s\u2013Woods number), the latter has apparently been tweaked by the software to work (something we know to be true because when it gets there, it doesn't say \"Redirected from\"), but the first two remain redlinks as I type this, even though all three display as exactly the same en dash on the page, the same as if the character \u2013 were used in them (and we know from above the link works when that is used, and in fact that is the article rather than a redirect).\nAdditional missing redirects only peripherally related to the issue at hand in this discussion and not included in the list above: hyphen Erd\u00f6s-Woods number, en dash Erdos\u2013Woods number and Erd\u00f6s-Woods number. Note that encouraging use of dashes in article names would often mean that several new redirects should also be created (yet, in most cases, will not be done by the person creating or moving the article). Gene Nygaard 13:39, 8 October 2006 (UTC)\nHere is an additional problem related to the failure to make the situation clear, and a failure within this MoS project page to maintain a clear distinction to the naming-conventions role which has been passed here, and other uses. We can easily end up with copy-and-paste clone articles (which can then diverge from each other), something which has already happened at Bose\u2013Einstein condensate (page history) and Bose-Einstein condensate (page history). Gene Nygaard 13:56, 8 October 2006 (UTC)\nI feel we ought to keep ease for the reader as a top consideration, which would mean preferring hyphens (which exist on keyboards) to en-dashes (which generally don't). The name should be something the reader can easily type into the search box and find. Nareek 15:20, 8 October 2006 (UTC)\nJust make the hyphen version a redirect to the en dash version. \u2014 Omegatron 16:16, 8 October 2006 (UTC)\nAs you can see, that doesn't get done. And it is often not \"a\" redirect that is needed, but many redirects. People who see an en-dash version are likely to write it with – or other things that don't work in links, or use \u2012 instead of \u2013, which doesn't work, or use \u2212 instead of \u2013, which also doesn't work, as well as using the hyphen\/minus (-) instead of the en dash \u2013. That's much less of a problem if we have a rule to use the hyphens in the article names, and dashes only in redirects. People using \u2012 or \u2212 and getting a redlink are more likely to think of turning it blue by using a hyphen than they are to discover that \u2013 is the character they really want, rather than one of the others that is nearly identical to it.\nEncouraging use of dashes in article names will result in more inadvertent creation of duplicate articles (oblivious to the existence of the other) than continuing to recommend hyphens in article names will. Gene Nygaard 17:31, 8 October 2006 (UTC)\nThis isn't any different than any other naming convention. Many have special characters. Just make it very clear that redirects are necessary. \u2014 Omegatron 11:59, 14 October 2006 (UTC)\nAll those different variations will exist regardless of which one the policy tells us to use as the pagename. That's why the policy also says to put in the redirects. But if the search box can't find one of those names when people type in a different one, isn't that a flaw in the search box rather than a flaw in the naming convention? \u2014David Eppstein 16:36, 8 October 2006 (UTC)\nYep. \u2014 Omegatron 11:59, 14 October 2006 (UTC)\nThe different variations will exist; but it is more broken, more of a serious problem, for links using the characters we do have on our keyboards not to work, than it is for characters we do not have on our keyboards not to work.\nFurthermore, we need to work with what we have. We want to make our information accessible, now. If you want to claim that new features should be added to our software, go right ahead, but that isn't particularly relevant to what we have now. Gene Nygaard 17:31, 8 October 2006 (UTC)\nI don't know about yours, but the en-dash is definitely a character on my keyboard: option-minus. \u2014David Eppstein 17:46, 8 October 2006 (UTC)\nEven that isn't exactly \"on\" your keyboard, and is a feature unfamiliar to many users with computers like yours\u2014but as you already know, the computers which have an \"option\" key to even make that possible are a minority. Gene Nygaard 18:25, 8 October 2006 (UTC)\nSeems to me like a mediawiki problem. If the average user isn't going to know the difference between four or more different symbols that all look near-identical, why not just define them to be the same, like has already been done with the capitalization of the first letter? We'd need a way to solve conflicts that the initial change would cause, naturally, but in the end, I think we'd have a far more user-friendly system. Where there's an actual difference in meaning between two or more, we can use a disambiguation of some kind. -FunnyMan 01:49, 14 October 2006 (UTC)\nYou mean \"Why doesn't someone finish the {{DISPLAYTITLE|}} magic word and set rules for when it is appropriate?\"\nProbably, yes. Like one of the editors above I use a Mac and, since they are all easy to type, I am moderately scrupulous\/pedantic about using hyphens, en rules and em rules as I see fit at a given moment in time. However, while article titles should certainly be typographically correct in their canononical and display forms (the main ugliness is the near-universal use of\n'\nwhen what is meant is\n), it seems to be to be pretty barmy that a-1, a\u20131 and a\u20141 could all be distinct articles. Come to that I would certainly also like to see automatic case-insensitive redirects. \u2014Ian Spackman 12:20, 14 October 2006 (UTC)","date":"2018-02-18 22:45:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"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\": 0.6480520963668823, \"perplexity\": 2631.950004258326}, \"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-2018-09\/segments\/1518891812259.30\/warc\/CC-MAIN-20180218212626-20180218232626-00738.warc.gz\"}"} | null | null |
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<!--
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contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
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http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
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Category: Lucky Patcher
Everything you need to know about Lucky Patcher
Rajdeep Singh on Everything you need to know about Lucky Patcher
Science and technology are processing day by day. From one field to another, science has left no stone unturned to prove itself and keep on inventing something new. One of the best creation of science is a computer. The computer which was designed many years ago, many modifications are being made to improve its efficiency further. Computer technology is progressing day by day making the life of every individual easy and smooth. From the basic importance to the advance level improvements, everything is making life easier and smoother.
Mobile phones, yet another biggest and amazing invention of science has been stealing the heart of its user in each and every way. It is said that keeping a mobile phone in the pocket is equivalent to bringing the whole world in your pocket. From the calling facility to video calling, from camera facility to listening to music, everything is possible in a mobile phone. The internet facility is the biggest of all. Without the internet, one cannot survive. All the video calling and music and everything is only possible because of the internet. It is truly said, ' Necessity is the mother of invention.' and 'Man is the brilliant creature on the land.'
So now, after so many modifications, the applications were made that can be installed on the phone and used. From gaming application to chatting application to music application, everything is available on the Internet. But the most annoying thing about apps is boring advertisements. All of a sudden 30 seconds or 1 minute seems like an hour whenever the advertisement videos are played. Some ads also pop up on the screen and you can not skip it at least before 5 to 10 seconds. And this is the most unbearable part of all the apps. But nothing to worry about, as above mentioned phrase clears it all. Every problem has a solution and of course, man has created a good solution to this intolerable pain. The name of the solution or Application is Lucky Patcher. To know more about this amazing application, go through the article carefully.
What is Lucky Patcher?
Lucky Patcher is a hacking application created for the Android device. It is invented to patch apps to remove ads. It also helps in redirecting the billing or getting free subscriptions. Luckypatchers.com is the official website of the lucky patcher app.
Importance and Uses
Lucky Patcher application is known for its uses This app has multiple uses. Some of its usages are mentioned below.
It checks the list of apps in the mobile phone of the user and demonstrates the action the user can perform on the mobile phone.
It modifies the associated consent.
This app has the capacity to remove license verification.
It also extracts app data to perform backups.
It performs other illicit works like removing ads.
It also unlocks many paid apps so that they can be downloaded and installed on the device.
This app is popularly known for hacking online games or android games. Games are hacked so that items can be purchased totally free that is present inside the game app.
Sometimes android device apps require a huge amount of internal storage to save app data or cache. So if your device is running short of internal storage then you can easily shift an android app to the SD card storage and free up internal memory storage.
By using this app, you can convert any app to a system app.
Above mentioned uses looks interesting. Lucky patcher has been termed as Malware by Google Play Protect. This is so because it permits the user to make patches to android or delete it which is illegal.
Clearly, the above-mentioned features clarify that Lucky patcher is a mind-blowing app. It helps in many ways. This app directly gives you the control to manage the system. Doesn't it sound amazing?
Is Lucky Is patcher app safe?
The answer to this question is very simple. The safety depends on whether the devices are rooted or not rooted. There is no issue if the device is not rooted. But if your device is rooted then it is really a matter of risk and great concern.
Is Lucky Patcher app legal or illegal?
Lucky patcher app is not more than a tool. But the feature of this app that is removing license verification, free app purchases, are illegal or in other words, it is digital or technical theft. The other feature that is backup or restores apps, is totally not illegal. Rather, this feature is of great use. So, if you do not break any law or commit any crime like theft or anything, then everything is legal.
So, the above-mentioned app that is Lucky patcher is good for use. There are many advantages to this app which is already mentioned. Today, everything is possible with the help of technology. Science has been making continuous progress in its field day by day.
But there's a lot of misuse of technology also. People are misusing science to the peak level. Digital crimes are increasing day by day. While using any app, a particular rule should be followed. No crime should be committed by the user. Many laws have also been made against such kind of illegal activities. So it is good to be fair and enjoy the amazing features of the application.… | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/1775919\/construction-a-multiplicative-inverse-of-a-real-number-via-dedekind-cuts","text":"# Construction a multiplicative inverse of a real number via Dedekind cuts\n\nI'm reading Pugh's \"Real Mathematical Analysis\" where C.Pugh constructs real numbers system using Dedekind cuts. Unfortunately, he omits a construction of a multiplicative inverse of a real number $x = A|B$ where $A$ and $B$ are cuts in $\\mathbb{Q}$. First, some required definitions:\n\nA cut in $\\mathbb{Q}$ is a pair of subsets $A,B$ of $\\mathbb{R}$ such that\n\na)$A \\cup B = \\mathbb{Q}, A \\neq \\varnothing, B \\neq \\varnothing, A \\cap B = \\varnothing$\n\nb) If $a \\in A$ and $b \\in B$ then $a < b$\n\nc) A contains no largest element\n\nA real number is a cut in $\\mathbb{Q}$.\n\nThe additive inverse of a cut $x = A|B$ is defined as $(-x) = C|D$ where $C = \\{r \\in Q| \\exists b \\in B$ (not the smallest elements of $B$ ) $: r = -b \\}$ and $D = \\mathbb{Q} \\setminus C$\n\nMultiplication is defined for positive numbers $x > 0$. If $x = A|B, y = C|D$, then $xy = E|F$ where\n\n$E = \\{r \\in \\mathbb{Q}| r \\leq 0$ or $\\exists a \\in A \\ \\ \\exists c \\in C: a > 0, c > 0, r = ac \\}, F = \\mathbb{Q} \\setminus E$. If $x > 0, y < 0$ then we define $xy = -(x(-y))$\n\nNow I wonder how can we define a multiplicative inverse of a positive cut $x = A|B$. Obviously, it's a cut $\\frac{1}{x} = C|D$ such that $x \\frac{1}{x} = 1$\n\nSo, it's such a cut $C|D$ such that for $C$ we have $\\{r \\in \\mathbb{Q}| r \\leq 0$ or $\\exists a \\in A \\ \\ \\exists c \\in C : a > 0, b > 0, r = ac \\} = \\{r \\in \\mathbb{Q}| r < 1 \\}$\n\nAny ideas how we can define $C$(if we define $C$, then $D = \\mathbb{Q} \\setminus C$)?\n\nIf $x > 0$, $$C = \\{ r \\in \\Bbb Q \\mid r <= 0 \\text{ or } \\forall a \\in A,\\, ra < 1\\}$$\n\nIf $x < 0$, $$C = \\{ r \\in \\Bbb Q \\mid \\forall a \\in A, \\,ra > 1\\}$$","date":"2019-07-23 16:39:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9675557613372803, \"perplexity\": 145.52550141868542}, \"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-2019-30\/segments\/1563195529480.89\/warc\/CC-MAIN-20190723151547-20190723173547-00553.warc.gz\"}"} | null | null |
Tillmann K. Buttschardt, né en 1966 à Biberach an der Riß, est un géoécologue allemand.
Biographie
Il est enseignant chercheur à l'Institut de technologie de Karlsruhe et chargé de cours (Vertretungsprofessur/Dozent für Landschaftskunde) pour les sciences du paysages à la Fachhochschule Eberswalde. Depuis 2010, il est professeur de géographie et d'écologie du paysage à l'université de Münster.
Un des rares scientifiques allemands francophones, il est considéré comme l'un des rares spécialistes allemands du Bénin et du Burkina Faso. Tillmann Buttschardt est un élève de Manfred Meurer.
Notes et références
Lien externe
Homepage Universität Münster
Scientifique allemand
Naissance en 1966
Naissance à Biberach an der Riß | {
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Background: Obesity is a well-known cause of cardiovascular disease burden and premature death, but effects on psychological morbidity remain uncertain. This article reports findings following a systematic review of epidemiological studies to determine whether obesity causes depression. Methods: Multiple databases were searched for English-language studies of etiology of obesity (exposure variable, analyzed as an ordered category) on depression outcomes (dependent variables, continuous or categorical). Studies in children and in women during pregnancy or postpartum were excluded, as were nonrepresentative cross-sectional studies. Searches and identification of studies for inclusion were performed by EA, whereas a descriptive synthesis of important study characteristics was undertaken independently by us. Results: We reviewed 24 out of approximately 4500 potentially relevant studies; 4 were prospective cohort studies and 20 were cross-sectional studies (10 from the United States). Effect measures reported in all prospective cohort studies were consistent and suggested that obesity may increase the odds of future depression outcomes (symptoms or nonclinical diagnosis of depression). Effect measures reported in most cross-sectional studies from the United States supported the hypothesized association between obesity and prevalence of depression outcomes for women but not men, in contrast most cross-sectional studies from populations other than the United States consistently failed to find such associations. Conclusions: Overall, there is a weak level of evidence supporting the hypothesis that obesity increases the incidence of depression outcomes. Few high-quality prospective cohort studies exist, and cross-sectional studies account for the vast body of published evidence, and therefore firm conclusions for causality cannot yet be drawn. Our finding warrants additional high-quality etiological research on this topic. | {
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\section{Introduction}
We report on a custom analog front-end electronics system developed by the KEK-Kobe University joint project {\it LTARS} (Low-Temperature Analog Readout System).
The project goal was to establish an analog readout system for the {\it NEWAGE} directional dark matter experiment (NEw general Wimp search with an Advanced Gaseous tracker Experiment) \cite{nakamura}, and for a next-generation neutrino oscillation experiment \cite{nuexp}.
The {\it NEWAGE} detector is a $\mu$-TPC (micro-Time Projection Chamber), which uses an MPGD (Micro Patterned Gaseous Detector), \textit{e.g.}{}, $\mu$-PIC \cite{miuchi} or Micromegas (Micro-MEsh Gaseous Structure) \cite{giomataris} to read out the TPC. {\it NEWAGE} aims to detect the directional signature of galactic dark matter \cite{tanimori, mayet}. Typically, TPCs operate in electron-drift mode: a charged nuclear recoil ionizes the surrounding low-pressure gas to generate electron-ion pairs. An electric field then transports the electrons to a readout plane where they are amplified and collected by the MPGD. $\mu$-TPC{}s can therefore record a recoil track in three dimension. While three-dimensional tracking is possible with this approach, diffusion limits the drift length of the detector. To suppress diffusion and improve track reconstruction, a negative-ion drift gas can be used, in which ionization electrons attach to electronegative gas molecules to form negative ions \cite{martoff}. The negative ions are then transported to the readout plane. These TPCs are called negative-ion (NI) $\mu$-TPC{}s. The {\it DRIFT} experiment uses a NI TPC, and has demonstrated diffusion at the thermal limit \cite{DRIFT2}.
In addition to reduced diffusion, the use of a NI $\mu$-TPC{} also enables full-volume fiducialization through 3D position reconstruction of the event vertex \cite{minority}.
Because the main backgrounds of TPC-based dark matter searches are $\alpha$-particles and daughter nuclei from $^{238}$U/$^{232}$Th chain decays of isotopes on the MPGD and drift planes, fiducialization can enable background-free operation \cite{DRIFT2, DRIFT1}. Fiducialization along the drift direction is possible because NI gases can produce multiple species of charge carriers, each with different mass and therefore mobility. As a result, the difference of arrival times between species correlates with total drift distance. For example, nuclear recoils in SF$_6$ primarily produce SF$_6^-$, but also trace amounts of SF$_5^-$ (so-called a ``minority carrier''), whose mobility is $\approx 5\%$ larger \cite{phan}. Because the time-scale for events in a NI $\mu$-TPC{} is orders of magnitude longer than for an electron-drift TPC, new electronics, such as the \textit{LTARS}, are required to read out these detectors.
In addition to improving the sensitivity of directional dark matter searches, the \textit{LTARS} system can also benefit large-scale liquid argon (LAr) detectors. An $\mathcal{O}$(10~kt) LAr time projection chamber (LAr-TPC) will be utilized for next-generation neutrino oscillation studies, and also for studies of nucleon decay and astrophysical neutrinos.
Toward the realization of such a large detector,
a world-wide R\&D effort on kilo-ton-scale LAr-TPC demonstrators is underway \cite{WA105}.
One approach to reading out the ionization electron signal uses a dual-phase LAr-TPC \cite{LEM}.
In the dual-phase TPC, the ionization electrons are extracted from the liquid to the vapor phase, and extracted electrons are multiplied with a thick-GEM (Gas Electron Multiplier) and collected with a two-dimensional strip anode.
The main advantage of a dual-phase readout is the high signal-to-noise ratio afforded
by the gas multiplication. This enables longer drift lengths over which some primary electrons are lost to impurities in the LAr.
The high signal-to-noise ratio also benefits the
physics sensitivity for the
neutrino oscillation, nucleon decay, and astrophysical neutrino signals.
Both the directional dark matter search and LAr TPCs for neutrino experiments require innovations in readout electronics.
Since the drift speed of negative-ions is
orders of magnitude smaller than that of electrons in a low-pressure gas ($\approx 10^{-2}$ cm/$\mu$s \textit{vs.}{} $\approx 1$ cm/$\mu$s), existing electronics designed for high-rate TPC readout in collider experiments are too fast to collect charges from NI $\mu$-TPC{}. Additionally, since the minority carrier signal strength is typically only a few percent of the primary signal
in optimized gas conditions, a wide dynamic range is also required for the electronics to sense SF$_6^-$/SF$_5^-$ with high signal-to-noise ratio (\textit{e.g.}{}, typical signals are
3 fC for minority species and 80 fC for the main species).
Also, fine MPGD spatial resolution necessitates a high-channel count readout in a small form-factor (\textit{e.g.}{}, 1500 channels for a $30\times30$ cm$^2$ readout area with strip pitch of 400 $\mu$m).
The signal timescale for LAr-TPCs is similar to that of the NI $\mu$-TPC{} since the ionization electrons drift
in the liquid (the drift speed is typically $\approx 10^{-1}$ cm/$\mu$s).
The wide dynamic range requirement also applies to the LAr-TPC.
The typical ionization signal for a minimum ionization particle (MIP) track
is $\approx 10$ fC depending on the thick-GEM gain.
A single-channel signal for
events associated with electromagnetic or hadronic showers
is $\sim$50 times larger than for a MIP.
The number of readout channels is expected to be $\mathcal{O}(10^5)$ for
a $\mathcal{O}(10\,{\rm kt})$ detector.
Moreover, to minimize detector capacitance, the analog front-end electronics
must be located as close as possible to the strip readout.
As a consequence, the analog front-end must operate at $\sim$88 K.
In this article, we report on the recent development and characterization of a prototype analog front-end ASIC, which meets the needs of both NI $\mu$-TPC{} and LAr-TPC detectors.
We describe the ASIC design in section 2, report on the test results in section 3, discuss the design refinements in section 4, and give conclusions in section~\ref{sec:discussion}.
The characterization of 88~K operation is out of scope of the present paper. The full detail of the cryogenic operation is going to be described in a next paper.
\section{Specification of the ASIC}
\subsection{Overview of the ASIC}
The readout ASIC was implemented with Silterra 180-nm CMOS technology with a chip size of 5 $\times$ 5 mm$^2$. The objectives of the prototype were to establish operation of basic signal processing components such as a preamplifier and CR-RC shaper stage, and to demonstrate the successful operation of these elements on a detector.
For evaluation purposes, we implemented two different signal processing chains in the ASIC: (a) a pair of static gain stages to handle wide ($\pm$1600~fC) and narrow ($\pm$80~fC) dynamic-range signals (hereafter referred as ``static architecture''), and (b) a dynamic gain-switching architecture (hereafter ``dynamic architecture''). Each ASIC implements eight independent channels of each architecture.
Table \ref{tab1} lists the specification and performance requirements of the ASIC, and Figure~\ref{fig_foto} shows a photograph of the ASIC. Signals are processed from left to right on the chip. The main circuit core occupies the central quarter of the ASIC. The remainder of the chip houses bias nodes for the two architectures and decoupling capacitors.
\begin{table*}
\begin{center}
\caption{\label{tab1} Specification and requirements of the ASIC.}
\small
\begin{tabular}{|c|c|c|} \hline
Technology & \multicolumn{2}{|c|}{Silterra 180~nm CMOS} \\ \hline
Chip size & \multicolumn{2}{|c|}{5$\times$5~mm$^2$, 16 total channels} \\ \hline
Supply power & \multicolumn{2}{|c|}{1.8~V core/IO, $\pm0.9$~V operation, max. 2.4~mW/ch}\\ \hline
Fabrication options &\multicolumn{2}{|c|}{6 metals, deep N-well, high-value poly res., MIM cap.} \\ \hline \hline
Minimum signal charge & 3 fC (minority species) & 100 fC (main species)\\ \hline
ENC & 2000~e$^-$ (S/N=10) & $< 6.4 \times 10^4$ e$^-$ \\
& 4000~e$^-$ (S/N=5, see Section~\ref{sec:discussion}) & \\ \hline
Dynamic range & $\pm$80 fC (narrow range)& $\pm$1600 fC (wide range)\\ \hline
Voltage gain & 10~mV/fC & 0.5~mV/fC \\ \hline
Shaping time & \multicolumn{2}{|c|}{4--7~$\mu$s for NI $\mu$TPC / 1--4~$\mu$s for LAr-TPC} \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}[!t]
\centering
\includegraphics[width=2.4in, angle=180]{fig1.eps}
\caption{\label{fig_foto}Photograph of the readout ASIC. The chip size is $5\times5$~mm$^2$. A total of 16 readout channels are located in the central region, and the remainder of the chip houses decoupling capacitors to stabilize the bias and power supplies.}
\end{figure}
\subsection{Circuit Properties}
\subsubsection{Static gain architecture}
Figure~\ref{fig_MT} shows the block diagram of a single channel of the static architecture.
The detector is externally AC-coupled with electro-static discharge protection diodes to the charge-sensitive amplifier (CSA) input. Calibration test pulses can be injected via an on-chip capacitor of 100~fF.
The CSA implements a modified folded cascode topology with a large input NMOS transistor (W/L = 40~$\mu$m/300~nm, M = 100, g$_{\rm m}$/I$_{\rm d}$=7.5~mS/280~$\mu$A). The bias condition of input and load transistors are adjusted independently by acquiring constant currents from the VDD and VSS.
A transfer gate-type PMOS is employed for the CSA DC-feedback component.
The CSA is designed to handle signals of either polarity, and includes pole-zero cancellation.
The CSA drives a 2-stage shaping amplifier, whose output is shared by high- and low-gain amplifiers whose gains are optimized for narrow and wide dynamic-range signals (see Table \ref{tab1}). Although the time constants of each stage in Fig. \ref{fig_MT} seem to be set at 3~$\mu$s/1.5~$\mu$s ($C_{\rm sh1}R_{\rm sh1}$) and 3.6~$\mu$s/1.2~$\mu$s ($C_{\rm sh2}R_{\rm sh2}$), respectively, the overall response becomes slower since the PZC (pole-zero cancellation circuit) is directly connected to the CSA output instead of the source follower's output as a popular configuration. We thus carefully designed for a default peaking time of 7~$\mu$s/$3.5~\mu$s by SPICE simulation. In this work, we focus on the 7~$\mu$s performance.
All resistor components are implemented with high-value poly resistors (1.05~K$\Omega$/sq.).
\subsubsection{Dynamic gain-switching architecture}
The signal processing chain for the dynamic gain switching architecture is shown in Figure~\ref{fig_TK}. The circuit is based on Ref.~\cite{geronimo}.
The CSA uses a folded cascode configuration with an input PMOS transistor, and the CSA DC-feedback FET is implemented with NMOS.
The feedback capacitor is selectable via a CMOS switch (designated as \texttt{COMP\_FBIN}) between 340 fC and 8.58 pF for narrow-range and wide-range modes, respectively. This gain setting of either ``\texttt{L}'' or ``\texttt{H}'' (at first set as ``\texttt{H}'' with high-gain) is provided from a discriminator output after comparison with the CSA output and an external threshold voltage. An \texttt{RS latch} is inserted between the gain switch and the discriminator to prevent unexpected bit transition during the signal processing. After finishing each event process, the latched gain setting is reset from an external DAQ (or FPGA) board. The CSA output is fed into a band-pass filter circuit with pole-zero cancellation and a second-order low-pass filter. The latter utilizes two sets of high-resistance circuits, functioning as a selectable mega $\Omega$ resistor by implementing with a poly resistor of 66/330~k$\Omega$ and current mirrors \cite{kishishita}.
The transfer function of the band-pass filter (indicated in the dashed line in Figure~\ref{fig_TK}) is:
\[ T(s)=-\frac{R_1}{(s^2C_1C_2R_1R_2+sR_1C_1+1)}.
\]
By choosing $\displaystyle C_1R_1=4C_2R_2$, the transfer function becomes $\displaystyle T(s)=-\frac{R_1}{(2sC_2R_2+1)^2}$, resulting in the second order low-pass filter. The peaking times are selectable via the CMOS switch of the resistor circuits $R_1$ \& $R_2$.
Circuit configuration, such as test pulse enable and peaking time selection, is done by daisy chained shift registers.
\begin{figure*}[!t]
\centering
\includegraphics[width=5in]{fig2m.eps}
\caption{Block diagram of the static gain architecture. $R_{\rm f}$ and $R_{\rm pz}$ are within the range of several M$\Omega$ adjusted by the gate voltage $V_{\rm pre}$.}
\label{fig_MT}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5in]{fig3m.eps}
\caption{Signal processing chain of the dynamic gain switching architecture. Typical values of resistors and capacitors in the band-pass filter are $R_1=R_2=46.3$/9.3~M$\Omega$, $C_1=216$~fF (=$4\times C_2$), and $C_2=54$~fF.}
\label{fig_TK}
\end{figure*}
\section{Measurement Results}
\subsection{Experimental setup}
To evaluate the circuit properties, the ASIC was directly mounted on a test board before connecting to a detector. A computer interface was established using a commercially available digital board with a Xilinx FPGA (the SOY board from Bee Beans Technology \cite{bee}). All necessary bias voltages for the ASIC were provided from potentiometers and resistors on the PCB.
\subsection{Waveforms}
Figure~\ref{fig_waveform} shows the high-gain outputs for each architecture for an input charge of $-50$~fC. In the static architecture (left panel), the peak pulse height is measured to be $-512$~mV for a peaking time of $5.4~\mu$s. The simulation predicts a smaller pulse amplitude ($-426$~mV) and a slightly slower peaking time ($6.7~\mu$s).
As for the dynamic architecture, we needed bias optimization before injecting test pulses. This is because the default bias settings obtained from simulation did not provide appropriate conditions to the ASIC, and we observed the following phenomena: (1) the measured pulse height was 2--3 times smaller than the simulation for the default bias voltages, and (2) the effect of the detector capacitance $C_{\rm det}$ was worse than predicted, \textit{i.e.}{} the gain decreases rapidly as $C_{\rm det}$ increases. We describe these issues in detail in Section~\ref{sec:discussion}. After increasing the bias voltage on the \texttt{VM} node by 0.5~V (see Figure~\ref{fig_TK}), analog outputs were observed from the chip, as shown in the right panel of Figure~\ref{fig_waveform}. Even after bias optimization, however, the measured waveform peaking time ($\sim11$\,$\mu$s) does not match expectation, and the pulse height was still small by 10\%.
Next, we characterized the dynamic gain-switching response. Figure~\ref{fig_oscillo_dyn} shows screen shots for injected charges of 90~fC and 100~fC, respectively. The threshold level was set between those values. We can clearly see that the output of \texttt{COMP\_FBIN} (indicated in green at the bottom) in the left figure remains in the low state below the threshold, while the expected transition is seen in the right panel when the shaper output crosses threshold. A short spike in the shaper output of the right panel seems to coincide with a rising edge of \texttt{COMP\_FBIN}, however, this is an artifact caused by noise injection due to coupling between oscilloscope probes. The switching noise from the CMOS switch seems clearly removed in the band-pass filter, and the waveform shows the expected semi-gaussian shape.
\begin{figure*}[!t]
\centering
\begin{tabular}{c}
\begin{minipage}{0.48\hsize}
\includegraphics[keepaspectratio, scale=0.4, angle=0]
{fig4l.eps}
\end{minipage}
\begin{minipage}{0.02\hsize}
\hspace{2mm}
\end{minipage}
\begin{minipage}{0.48\hsize}
\includegraphics[keepaspectratio, scale=0.4, angle=0]
{fig4r.eps}
\end{minipage} \\
\end{tabular}
\caption{Output comparison of the static (left) and dynamic (right) architectures in the slow peaking time mode: dashed lines indicate SPICE simulation and solid lines show measurement results. Injected charges are both 50~fC with a pseudo-detector capacitance of 300~pF. The gain setting was fixed to high-gain mode for the dynamic architecture. }
\label{fig_waveform}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=5in]{fig5.eps}
\caption{Waveforms of the dynamic gain-switching response for input charges of 90~fC (left) and 100~fC (right). The signals from top to bottom are reset signal (magenta), test pulse from a function generator (yellow), shaper output (cyan), and \texttt{COMP\_FBIN} (green). }
\label{fig_oscillo_dyn}
\end{figure*}
\subsection{Voltage gain and Dynamic range}
Figure~\ref{fig_lin} shows measurements of the dynamic range for each architecture. The lower panels show the ratio of the measurements and a linear fit between $\pm 80$~fC for high-gain and $\pm 1000$~fC for low-gain, respectively. A $\pm 10$\% level is also presented in dashed lines. This level is defined as a negligible range, compared with an energy resolution of the gas detectors of typically $20$\%, including a position dependence of the MPGD \cite{nakamura2}.
For the static architecture, the measured gains are 10.05~mV/fC and 0.54~mV/fC, respectively, which meet the \textit{LTARS} design specification. For the fast peaking time mode (3.5\,$\mu$s), the measured gains are 10.29~mV/fC and 0.51~mV/fC, which also meet the design specification. For the dynamic architecture, the measured gains are 7.05~mV/fC and 0.49~mV/fC (7\,$\mu$s peaking time), and 7.33~mV/fC and 0.51~mV/fC (1\,$\mu$s peaking time).
The low-gain dynamic ranges for both architectures look narrower than the design specification of $\pm 1600$~fC. However, by adjusting the output baseline offset, the specification can be achieved. For standard operation, this is a satisfactory approach because the ASICs operate with a pre-determined polarity.
\begin{figure*}[!t]
\centering
\begin{tabular}{c}
\begin{minipage}{0.48\hsize}
\includegraphics[keepaspectratio, scale=0.5, angle=0]
{fig6l.eps}
\end{minipage}
\begin{minipage}{0.02\hsize}
\hspace{2mm}
\end{minipage}
\begin{minipage}{0.48\hsize}
\includegraphics[keepaspectratio, scale=0.5, angle=0]
{fig6r.eps}
\end{minipage} \\
\end{tabular}
\caption{Dynamic range and linearity curves of the high-gain (left) and low-gain (right) modes. Black and white points indicate dynamic and static architectures, respectively.}
\label{fig_lin}
\end{figure*}
\subsection{Noise characteristics}
The simulated and measured equivalent noise (rms) as a function of detector capacitance is shown in Figure~\ref{fig_noise}. The simulation includes parasitic resistance and capacitance of the core circuits. The noise slopes are roughly consistent with the simulation for both architectures, however, we can clearly see a noise offset of $\sim$1000~e$^-$.
The origin of that excess noise offset is under investigation, however the 2\,mm-long metal traces with a width of 3~$\mu$m between the ASIC pads and the CSA inputs are likely contributing and/or easily affected by external noises. On the other hand, the measured ENCs of the low-gain modes meet the design requirements for a detector capacitance of 300\,pF: 6950~e$^-$ for the static architecture and 3.5$\times10^4$~e$^-$ for the dynamic architecture. Although the noise measurement for the dynamic architecture was made without activating the gain-switching circuit, the noise contribution from CMOS switching is negligibly small ($\sim$3\%) compared with the ENC in the low-gain mode.
\begin{figure}[!t]
\centering
\includegraphics[width=3in]{fig7.eps}
\caption{ENC in the high-gain mode as a function of detector capacitance.}
\label{fig_noise}
\end{figure}
\section{Demonstration with a Micromegas detector}
\subsection{Experimental setup}
After characterizing the full chip functionality, we connected the \textit{LTARS} test boards to a NI $\mu$-TPC{} with Micromegas readout to assess the performance on a detector.
The detector was developed as a prototype directional dark matter detector at Wellesley College in the USA.
The experimental setup and the schematic readout diagram are shown in Figure~\ref{fig_setup1}.
The experiment was performed with an $\alpha$-ray source and the tracking capability for $\alpha$-rays was demonstrated.
The Micromegas was made by CERN with an amplification area and gap of 10 $\times$ 10 $\rm cm^2$ and 256 $\mu$m, respectively \cite{alex}.
The drift length was 1.5 cm and a radioactive source of $^{210}$Po was placed 5~mm away from the Micromegas.
SF$_6$ gas with a pressure of 60~Torr was used as a NI-drift gas. The high voltages of the drift plane and mesh were $-1525$\,V and $-625$\,V, which formed a drift electric field of 0.6~kV/cm. The gas gain was $\sim$100. A two-dimensional strip readout with a high-resistivity strip layer for spark protection on the top was used. The pitch of the readout and protection strips was 200~$\mu$m. The strip patterns were formed on a film with a thickness of 50~$\mu$m. Three films, namely Y-strips (bottom), X-strips (middle) and resistive-strips (top), were layered. The resistive strips were parallel to the Y-strips. The $\alpha$-source was mounted such that the mean $\alpha$-particle direction was parallel to the X-strips. A total of 32 strips (16 consecutive X-strips and 16 consecutive Y-strips) covering a detection area of $3.2\times3.2$~mm$^2$ were connected to four test boards, labeled as TOSHIZOU2 in Figure~\ref{fig_setup1}. The closest Y-strip to the $\alpha$-source was used as a trigger. The analog outputs from the test boards were digitized at 2.5~MHz by the FPGA board. The resulting digital data was recorded by a PC. All data were taken with the static gain architecture in the high-gain, slow peaking time mode to reduce electronic noise. The noise levels were 16~mV$_{\rm rms}$, corresponding to an ENC of 9937~e$^{-}$. The detector capacitance per strip (10 cm long) was measured to be 50 pF. This value is approximately four times larger than the estimation from Fig. 7. However, periodic noise is clearly visible in the waveforms of Fig. 10, indicating that the remaining noise originates elsewhere in the experimental setup.
\begin{figure*}[!t]
\centering
\begin{tabular}{c}
\begin{minipage}{0.4\hsize}
\includegraphics[keepaspectratio, scale=0.4, angle=0]
{fig8l.eps}
\end{minipage}
\begin{minipage}{0.43\hsize}
\includegraphics[keepaspectratio, scale=0.3, angle=0]
{fig8r.eps}
\end{minipage} \\
\end{tabular}
\caption{Experimental setup (left) and schematic readout diagram (right). Two of the four TOSHIZOU2 test boards (blue) mounted on the lateral sides of the setup are visible in the photograph.}
\label{fig_setup1}
\end{figure*}
\subsection{Track reconstruction}
Figures~\ref{fig_timingX} and \ref{fig_timing} show sample waveforms from each strip for a single $\alpha$-event. Some channels in the Y-strips were broken during the measurement. In this two-dimensional readout with resistive strips on the top, positive charges are induced when the amplified electrons are collected at the high-resistive strip. Thus, the polarities of observed waveforms from both strips are consistent with expectation. While we can see the TPC-like signal propagation of fast signals
clearly in the Y-strips (top waveforms first and bottom waveforms last), the
induced signals in X-strips are much longer in duration than Y-strips. The X-strip waveforms
seen in Figure 9 have long duration because charges propagate in the high-resistive strips
(parallel to the Y-strips) with a time constant of $\mathcal{O}$(10$\sim$100) $\mu$s and induce extra hits in the X-strips.
Although three-dimensional track reconstruction was hindered due to this effect, we successfully demonstrated that the \textit{LTARS} ASIC could successfully read out signal charges from the Micromegas with a very low gain of $\sim$100 and a strip pitch of 200~$\mu$m.
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{fig9.eps}
\caption{Waveforms from the 16 instrumented X-strips. The baseline for each channel is offset for clarity.}
\label{fig_timingX}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{fig10.eps}
\caption{Waveforms from 11 instrumented Y-strips (5 channels were not functioning). The baseline for each channel is offset for clarity. The upper strip is closest to the $\alpha$-source, and peaks earliest in time. This track thus pointed away from the Micromegas.}
\label{fig_timing}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3in]{fig11.eps}
\caption{$g_{\rm m}$ \textit{vs.}{} $I_{\rm d}$ of the input PMOS in the dynamic architecture.}
\label{fig11}
\end{figure}
\section{Discussion for performance improvements}
\label{sec:discussion}
In this section, we address the discrepancy between the measured and simulated performances for each architecture, and discuss design improvements for the next version of the \textit{LTARS} ASIC. The static architecture meets the design specification, but with slightly larger gain than expected (see Figure~\ref{fig_waveform}, left). This discrepancy cannot be explained by the transistor's corner parameters. We speculate instead that process variation of the poly resistor is the cause, \textit{i.e.}{}, three high-value poly resistors are placed in series with the minimum design width and dummy structures for a large resistance of 600~k$\Omega$ in the first shaping amplifier ($R_{\rm sh1}$ in Fig. \ref{fig_MT}). Since the poly resistors are formed by chemical etching on metal layers in this technology, the absolute resistance accuracy is typically 10\% even with dummy structures, when the poly width follows the minimum rule.
If we denote $R_{\rm sh1}$ as the feedback resistor in the first shaping amplifier (see Fig. \ref{fig_MT}), the voltage gain which is proportional to $R_{\rm sh1}$ becomes larger than expected under over-etching condition.
On the other hand, the performance discrepancies of the dynamic architecture cannot be explained only by the process variation since the circuit does not include poly resistors over 330~k$\Omega$ in the high-resistance circuit. The origin is likely due to the transconductance $g_{\rm m}$ of the input PMOS in the preamplifier ($W/L=11.2~\mu$m/ $L=200$~nm, $M=100$). The hypothesis that the dynamic architecture performance is affected by $g_m$ also explains other observations. Due to the Miller Effect, the effective input capacitance of the amplifier is $C_f (1+A)$, where $C_f$ is the feedback capacitance and $A$ is the open-loop gain of the amplifier. Hence, the fraction of charge generated in the detector and absorbed by $C_{\rm f}$ is written as $\displaystyle \frac{C_{\rm f}\cdot(1+A)}{C_{\rm det} + C_{\rm f}\cdot(1+A)}$, where $A=g_{\rm m}\times r_{\rm o}$, and $r_{\rm o}$ is the output impedance of the preamplifier. Ideally, $g_{\rm m}$ is sufficiently large, and the $C_{\rm det}$ dependence can be ignored. The $g_{\rm m}$ that is smaller than intended will degrade the S/N.
This is consistent with the observation in the $C_{\rm det}$ \textit{vs.}{} gain curve, which was worse than the design. A small $g_{\rm m}$ also affects the output peaking time. The rise time of the CSA is generally given as $t_{\rm r}=\frac{C_{\rm det}}{g_{\rm m}}+\frac{C_{\rm L}}{\mu_0 g_{\rm m}}$, where $\mu_0\equiv\frac{C_{\rm f}}{C_{\rm f}+C_{\rm det}}$, and $C_{\rm L}$ is a capacitive load at the output node of the preamplifier. With sufficiently large $g_{\rm m}$ (ideal case), the rise time of the preamplifier is fast, while the rise time becomes longer with decreasing $g_{\rm m}$ (current case). As a result, a slower waveform is observed at the final output stage (Figure~\ref{fig_waveform}, right). This is caused through multiple factors: (i) the expected $I_{\rm d}$ \textit{i.e.}{} designed as $I_{\rm}$=500~$\mu$A, is not supplied to the PMOS even under the maximum external bias, due to insufficient transistors' $W/L$ in bias current mirrors, and (ii) $g_{\rm m}$ is sensitive to the VM node since the node is not only connected to the source of the input FET, but also to the gate of the cascode transistor (Fig. \ref{fig_TK}). If we consider the IR drop in the VM node, the decrease of $g_{\rm m}$ is a reasonable conclusion. Figure \ref{fig11} shows the simulation result of $g_{\rm m}$ \textit{vs.}{} $I_{\rm d}$ of the input PMOS under various VM condition. After the optimization of the bias current mirrors, we separated the gate of cascode FET from the VM line, and also increased the finger number of the cascode FET. The improved $g_{\rm m}$ is overlaid in Fig. \ref{fig11}.
The remaining issue is the ENC. This should be considered from experimental viewpoints. As shown in Table~\ref{tab1} and Figure~\ref{fig_noise}, the targeted noise level of less than 2000~$e^{-}$ (rms) was not achieved in the current design.
If we relax the $S/N$ constraint to 5, fake events appear at 0.2 Hz in one strip, taking account of the band-width of $\sim$ 100~kHz. In the $\mu$-TPC{} experiment, we normally read
out multiple strips for each event, and the fake events can be easily recognized
if the noise rate is about 0.2 Hz. We thus reset the design specification of ENC to 4000~e$^-$ ($S/N$=5) and we will improve the layout to achieve the design specification in the next submission.
\section{Conclusion}
For the purpose of reading out NI $\mu$-TPC{} and LAr-TPCs, we have newly developed a dedicated analog ASIC in 180-nm CMOS technology. To meet the wide dynamic range of 1600~fC with a detector capacitance of 300~pF, the ASIC contains two different readout architectures: one was constructed with a static and separate gain-stages and another was implemented with a dynamic gain-switching topology. We mounted the ASIC on a test board and compared the ASIC performance to a SPICE simulation, in terms of shaping time, dynamic range, and ENC.
For the static architecture, the shaping time and dynamic range meet the design specifications. For the dynamic architecture, the gain switching works as designed, however, some design issues were found in the preamplifier circuit. By using a multi-finger cascode FET and separating the source of the input FET and gate of the cascode FET in the preamplifier, we expect to fix the problem, and achieve the required noise level in the next version of the ASIC. The ENCs of both architectures are still larger than desired, however, the excess noise is likely due to the long metal routing between a wirebonding pad and an analog input. This issue can be improved by optimizing the layout. In anticipation of cryogenic operation, we carefully chose the gate lengths of devices to reduce the impact ionization probability at 88 K. At low temperature, the CMOS transconductance increases due to the increased carrier mobility, which is expected to improve the noise performance. In order to assess such cryogenic effects, however, the overall noise performance must be improved. Therefore the cryogenic characterization of the LAr-TPC will be done with the next design.
We conclude that the static architecture reaches the requirements for an actual NI $\mu$-TPC{} dark-matter experiment. We also demonstrated the ASIC front-end on an NI $\mu$-TPC{} with Micromegas gas amplification and 200 $\mu$m-pitch strip electrodes.
\acknowledgments
This work was supported by the JSPS KAKENHI (Grant No. 25105008, 16H02189, 26104005, 15K21747, 17H04840), and the U.S.-Japan Science and Technology Cooperation Program in High Energy Physics (``Negative Ion Drift TPC Development for High-Resolution Tracking'' and ``Research and Development for Current and Future Long Baseline Neutrino Detectors''), as well as the National Science Foundation (EAGER PHY-1649966), and the Research Corporation Cottrell College Science Award (\#23325).
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Frank Hume (July 21, 1843 – July 17, 1906) was an American politician who served in the Virginia House of Delegates. The Hume School is named for him, and the Frank Hume Memorial Fountain, known as the "whispering wall", at the University of Virginia was built in his honor. In 2022, the memorial's engraved stones were replaced with plain stone.
During the Civil War, Hume fought with the Volunteer Southrons in the Confederate Army and received a thigh wound at the Battle of Gettysburg. Following the war, he had a clerkship in the Barruch Hall store in Washington, after which he and Richard Poole formed the Poole & Hume business. Hume took full control after Poole became ill, and he continued in charge until he died.
In 1870, Hume married Emma Phipps Norris. They had nine children.
References
External links
Democratic Party members of the Virginia House of Delegates
19th-century American politicians
20th-century American politicians
1843 births
1906 deaths | {
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Table of Contents
Title Page
Copyright Page
Dedication
Epigraph
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Author's Note
**The MiddLe Ages Come to Life . . .** to Bring Us Murder.
_**The Dame Frevisse Medieval Mystery Novels**_
**By Two-Time Edgar Award Nominee Margaret Frazer**
"An exceptionally strong series . . . full of the richness of the fifteenth century, handled with the care it deserves."
_—Minneapolis Star Tribune_
**Praise for**
_**The Hunter's Tale**_
"The book's charm lies in the author's meticulous research, notably on hounds and the changes in breeds of dogs over the centuries and on the intricacies of medieval wills and property rights. The plot moves at a stately pace appropriate to its time and setting." _—Publishers Weekly_
"With the author's usual attention to historical detail and keen psychological insight—but with more action than in previous entries—Frazer's latest will please both Frevisse aficionados and historical mystery readers new to the series."
— _Booklist_
"Beautifully written and captures the essence of the characters in a way that's bittersweet, as well as accurate."
_—The Best Reviews_
**Also in the series**
The Bastard's Tale
_Even in the charmed circle of medieval England's lavish royal court, no one is immune to murder . . ._
"Anyone who values high historical drama will feel amply rewarded by Edgar-nominee Frazer's latest Dame Frevisse mystery . . . History fans will relish every minute they spend with the characters in this powerfully created medieval world."— _Publishers Weekly_
"Frazer executes her . . . dramatic episode in fifteenth-century history with audacity and ingenuity."— _Kirkus Reviews_
The Clerk's Tale
_Dame Frevisse must find justice for the murder of an unjust man . . ._
"As usual, Frazer vividly recreates the medieval world through meticulous historical detail [and] remarkable scholarship . . . fans will rejoice that the devout yet human Dame Frevisse is back . . . a dramatic and surprising conclusion." — _Publishers Weekly_
The Novice's Tale
_Among the nuns at St. Frideswide's were piety, peace, and a little vial of poison . . ._
"Frazer uses her extensive knowledge of the period to create an unusual plot . . . appealing characters and crisp writing."
_—Los Angeles Times_
The Servant's Tale
_A troupe of actors at a nunnery is a harbinger of merriment—or murder . . ._
"Excellently drawn . . . very authentic . . . the essence of a truly historical story is that the people should feel and believe according to their times. Margaret Frazer has accomplished this extraordinarily well." _—Anne Perry_
The Outlaw's Tale
_Dame Frevisse meets a long-lost blood relative—but the blood may be on his hands . . ._
"A tale well told, filled with intrigue and spiced with romance and rogues."— _School Library Journal_
The Bishop's Tale
_The murder of a mourner means another funeral, and possibly more . . ._
"Some truly shocking scenes and psychological twists."
_—Mystery Loves Company_
The Boy's Tale
_Two young boys seek refuge at St. Frideswide's—but there is no sanctuary from murder . . ._
"Fast-paced . . . a surprise ending."— _Affaire de Coeur_
The Murderer's Tale
_Dame Frevisse's respite at Minster Lovell turns deadly when murder drops in_ . . .
"The period detail is lavish, and the characters are full-blooded." — _Minneapolis Star Tribune_
The Prioress' Tale
_When the prioress lets her family stay at St. Frideswide's, the consequences are deadly . . ._
"Will delight history buffs and mystery fans alike."
_—Murder Ink_
The Maiden's Tale
_In London for a visit, Frevisse finds that her wealthy cousin may have a deadly secret . . ._
"Great fun for all lovers of history with their mystery."
— _Minneapolis Star Tribune_
The Reeve's Tale
_Acting as village steward, Frevisse must tend to the sick—and track down a killer . . ._
"A brilliantly realized vision of a typical medieval English village . . . Suspenseful from start to surprising conclusion . . . another gem." _—Publishers Weekly_ (starred review)
The Squire's Tale
_Dame Frevisse learns that even love can spawn anger, greed, and murder . . ._
"Written with the graceful rhythms that thave garnered her two Edgar nominations . . . transports the reader to a medieval England made vivid and a world of emotions as familiar then as now." _—Publishers Weekly_ (starred review)
"Meticulous detail that speaks of trustworthy scholarship and a sympathetic imagination." _—The New York Times_
"As exquisitely woven as a medieval tapestry . . . dazzling."
_—The Cleveland Plain Dealer_
If you purchased this book without a cover, you should be aware that this book is stolen property. It was reported as "unsold and destroyed" to the publisher, and neither the author nor the publisher has received any payment for this "stripped book."
This is a work of fiction. Names, characters, places, and incidents either are
the product of the author's imagination or are used fictitiously, and any
resemblance to actual persons, living or dead, business establishments,
events, or locales is entirely coincidental.
THE HUNTER'S TALE
A Berkley Prime Crime Book / published by arrangement with the author
Copyright © 2004 by Gail Frazer.
The Edgar® name is a registered service mark of the
Mystery Writers of America, Inc.
All rights reserved.
No part of this book may be reproduced, scanned or distributed in any
printed or electronic form without permission. Please do not participate in or
encourage piracy of copyrighted materials in violation of the author's rights.
Purchase only authorized editions.
For information, address: The Berkley Publishing Group,
a division of Penguin Group (USA) Inc.,
375 Hudson Street, New York, New York 10014.
Visit our website at www.penguin.com
eISBN : 978-0-425-19942-8
Berkley Prime Crime Books are published by The Berkley Publishing Group,
a division of Penguin Group (USA) Inc.,
375 Hudson Street, New York, New York 10014.
The name BERKLEY PRIME CRIME and the BERKLEY PRIME CRIME
design are trademarks belonging to Penguin Group (USA) Inc.
<http://us.penguingroup.com>
_For Cheryl, because it's her fault._
_And for AKC Champion Stoneybrook Brighton, FCh, SC, TT,_
_PTF, a handsome lad who fulfills the Irish wolfhound's creed:_
_Gentle when stroked,_
_Fierce when provoked._
For in his huntyng hath he swich delit
That it is al his joye and appetit
To been hymself the grete hertes bane . . .
G. CHAUCER, _The Knight's Tale_
**Chapter 1**
Light trembled through the leaves, little flickers of sun and shade playing across Hugh's closed eyes where he sat with his head leaned back against the oak's deep-barked trunk, his face turned upward to the canopy of leaves and dark oak branches between him and the summer sky, listening to the whisper and sigh of leaves away into green distances, the soft sough of branches moving, the busy rustle of a squirrel passing through treetops so high the two wolfhounds—Bane's large, bony head resting on his chest and Brigand's against his hip—did not even flick an ear at it. With summer warmth all through him, Hugh let the quiet sink into him, too. Here, for a while, with manor, fields, and family—most especially family—left behind him, there was only the quiet and himself and the two hounds, with no need to speak when he'd rather be silent or feel anything when he'd rather feel nothing.
He wondered if the yelling at home had stopped yet.
His guess was that it hadn't. Sir Ralph had been at full cry and Tom ready to meet him yell for yell, a furious use of words the one thing he wholeheartedly shared with his father; and if they faltered, Miles was too often in black enough humour to start them up again. Today, when they had set to it at the midday dinner's end, Mother had risen from the table, called Lucy to her, bid Hugh fetch her spinning to her, and gone out to the arbor at the far end of her garden, not beyond sound of her husband and son's shouting in the hall but at least away from it.
Hugh had never understood whatever truce there was between his parents. Lady Anneys seemed to live somewhere beyond her husband's furious outbursts and he had never, so far as Hugh knew, ever lifted a hand against her. His sons and grandson and even Elyn once when she had back-talked him over something had, yes, all felt his fist, but Lady Anneys would simply rise and walk away, and he always let her go.
Today Hugh had escaped, too, because she had bid him come after her, and when he had fetched her distaff and basket of wool to the vine-shaded arbor, he had given them to her with a kiss on her cheek at the edge of the wimple that encircled her face framed by the stiff-starched wings of her white linen veil and said, "I'm away then, if anyone asks."
She had given him her small smile for answer but Lucy, being fourteen and eager for other than sitting the afternoon away, had started, "Can I—"
"No," her mother had said. "You can't."
Hugh had rubbed the top of his sister's head in the way he knew particularly irked her and left them by way of the garden's rear gate through the tall willow-woven fence, crossing over the two-plank bridge across the shallow-banked, narrow stream there to the packed earth of the cart-track running on its other side along the new-mown hayfield where the cut hay, laid out in its long windrows and turning to gold as it dried under the midsummer sun, smelled of all summer's sweetness. There had been half a dozen village women in the field this morning, turning the windrows over with their wooden rakes, the better for the hay to dry all through before it was haycocked, but they were gone to their dinners and Hugh had gone leftward along the cart-track without need to see or talk to anyone.
The way he went, the track curved behind the manor buildings, between them and the near fields, and then away to the farther fields. The other way it led to the church and village and more fields before it met the road that ran on the manor's other side and away. The manor of Woodrim had never been grand enough to warrant an enclosing wall; barn, byre, cartshed, ploughshed, dairy, poultry yard, and all else that went into making a manor's life were spread around and outward from the bare dirt yard in front of the hall in no great order. When Tom, Hugh, and Miles had all been small, the clutter of buildings had made for grand chase-and-hide games, and hiding was what Hugh had been at today, circling around to the kennel beyond the barn.
The kennel was the newest-built and best-kept place on the manor, a high, solid fence around a wide rectangle of grass with the hounds' low, square house in the middle made with close-planked walls, a thick-thatched roof, a fireplace for warming the hounds in winter or after wet hunting, and a loft for the hound-boy to sleep.
"The dogs live better than we do," Tom sometimes said bitterly, though not where their father would hear him.
It was Miles who said—and did not care at all if Sir Ralph heard him—that the hounds lived best of anyone on the manor because Sir Ralph cared for them more than he cared for anyone else; and Sir Ralph, hearing him, would roar, "You have the right of that, you useless bastard. Better than I care for you, that's sure."
Since Miles was neither useless nor a bastard, merely someone his grandfather wished had never been born, that was unfair as well as cruel, but fairness rarely mattered to Sir Ralph. Hugh doubted he knew how to be kind, except to his hounds. Hounds and the hunt were what mattered to Sir Ralph, and Hugh's good luck was that his own pleasure in hounds and hunting almost matched his father's. That had earned him something of his father's favor over the years, since the time he was ten years old and Sir Ralph had found him hiding in the kennel, crouched in the straw with his arms around Pensel's neck while Paliard licked at his wet face. His back had been aching from being beaten with Sir Ralph's leather belt for splashing wine on the cloth when serving at table during dinner and, hand to belt buckle again, Sir Ralph had stood glaring down at him, demanding, "What in hell are you doing here, messing with my dogs?"
Angry with pain and the shame of being found in tears and certain there was another beating coming, Hugh had said back, defiant with despair, "I'm not messing. I _like_ them. I like them better than I like _you_!"
That should have earned him another whipping but Sir Ralph had eyed Paliard licking now behind Hugh's ear, the strong stroke of her tongue up-ending his hair, and had said, "Do you, by God? Paliard, here."
Paliard had given Hugh a final swipe with her tongue, risen with her great, gaunt, wolfhound grace, and padded to Sir Ralph's side. Fondling her ears, his father had demanded, "What makes Paliard my best running hound?"
Hugh had promptly said, not needing to think about it, "Lean body, small flanks, long sides. Well-slanted shoulders, straight hocks—"
He could have gone on but Sir Ralph had interrupted, "How do you know that?"
"I've listened to you and Master Basing," who had been houndmaster and huntsman then, "talk about it all the time."
"Listen, do you? Come here, whelp."
Wary of pain but knowing there would be no escaping it if his father chose to give it, Hugh had obeyed. Sir Ralph had looked him up and down, felt of his shoulders (not bothering that Hugh flinched from his fingers rough on the belt-bruises) and his arms, his hands, his legs, then held him by the chin and turned his head from side to side, checking him over as if he were a hound himself, before he let him go with a shove that staggered Hugh to the side, and said, "You might do. Tell Master Basing you're to be his hound-boy until I say differently."
So hound-boy he had been, with Master Basing as well as his father to whip him when he failed to do things right. But he had learned quickly more for the love of it than because of the whippings, and as his skills grew, the whippings lessened; and when he had grown enough himself that, as Sir Ralph put it, "You're no longer so undersized that a hare could trample you into the ground, you skimp-boned whelp," the two men had begun to teach him a hunter's skills to go with his hound ones. And three years ago, when Master Basing had fallen, clutching his chest, during a midwinter deer hunt, and died before Father Leonel could be brought to him—"A cleaner death than a priest's mumbling could have given him anyway," Sir Ralph had said—Sir Ralph had slapped Hugh between the shoulders and told him he was houndmaster and huntsman now in Master Basing's place. "The great lords can afford houndmasters and a blast of hunters to their pleasure, but I'm a poor knight and nothing more, and by God's teeth, at least one of my brats can finally be worth his keep."
The main lie in that was that Sir Ralph was only a poor knight. He had started life plainly enough—though not poor—as the heir to a small manor in Leicestershire but taken himself to London, become a lawyer, married a rich merchant's daughter, and earned a goodly fortune and his knighthood there, then taken his wealth away to the country, giving up the law for hounds and hunting, his true passions. He had bought the Oxfordshire manor of Woodrim not for what profits it might make him but because of the hunting rights in the king's neighboring forest of Charlbrook Chase that went with it; and when his first wife died, he had married Lady Anneys, another merchant's daughter, for the dower of town properties that came with her and the sons she might birth for him. The properties' rents were to help to pay for his hounds and hunting. The sons were to replace the disappointing one his first wife had given him. For that, to Sir Ralph's mind, Hugh with his love of hounds and hunting had proved the best; and if Hugh had not friendship or affection from his father, at least he had less of Sir Ralph's wrath than did Tom or Miles.
Tom, being the eldest living son and therefore likely his heir, drew Sir Ralph's wrath the way a tall tree in an open field drew lightning, but—possessing much of their father's hot blood and love of yelling himself—he usually gave as good as he was given whenever he and Sir Ralph set to it, which was often enough because Tom served as his father's steward, responsible both for Woodrim and all his other properties. Today's dinnertime quarreling had come out of Tom's wanting, because of the overhot weather, to give more than the usually allotted ale to the haymakers presently working Sir Ralph's fields instead of their own, and Sir Ralph's refusing him with oaths.
With his own escape made before he was drawn into their argument, Hugh had called Bane and Brigand to him from among the twenty other hounds in the kennel yard and set out by the nearest field path across the rough pasture to the wide-cut greenway into the forest that so closely curved around two sides of the manor. If he had followed the greenway far enough, it would have taken him through Woodrim's thick-grown forest to the more open Charlbrook Chase, which was as much meadow and rough heath as woods, for all it was called a royal forest, because dense woods did not make for good galloping after deer. But neither the manor's forest nor the king's was only for hunting. Wood was taken from them for building and for fuel and for sale, and in autumns the lord's and villagers' pigs were herded in among the trees to fatten on fallen nuts before slaughter-time at Martinmas.
The grassy greenway served for passage into and through the forest for all of that, but Hugh had followed it only a short way before turning from it into a narrow deer-path leading away among the trees. Less by thought than by long usage, he had shifted into a more careful-footed going, quiet in his soft-soled boots, barely stirring the forest's midday stillness; and Bane and Brigand, knowing they were on no hunt, had moved easily at his heels, padding as silently as he did. Before long they had come out of the green shadows into this small clearing around the tree Hugh called, to himself, the grandfather-oak because of its age and grand-spreading size. It was a companionable tree, as good company in its way as were the hounds, nor was anyone likely to happen on him here by chance.
But on purpose was another matter.
Hugh opened his eyes. The silence had changed, and the two hounds lifted their heads to listen with him. Someone almost as silent-footed as he had been was coming along the deer-path. Then Bane stood up, tail swaying in a slow wag, and Hugh stayed where he was, unsurprised when Miles came into the clearing, tugged onward by Skyre, the lean, young, not-much-trained lymer-hound that was halfway to throttling herself on the leash with eagerness.
Brigand rose to join Bane in greeting her and Miles, and Hugh asked, "How did she do?" as Miles, still holding Skyre's leash, folded his long legs and sat down in the grass beside him.
"She followed your scent as easily as if you were dragging a musk bag."
"Me and two hounds. I should hope she could follow us."
"No leash manners, though."
"Did you bother trying to teach her some on the way?"
"Not my business," Miles said.
"Not your business to have her out at all."
"She got to pacing and whining in the hall. Sir Ralph said to take her out so I did."
Eyes closed again, Hugh flopped one hand sideways from where it lay on his chest to hit Miles idly on the knee. "He didn't mean this far, I warrant, you idiot. Are they still at it?"
"If there was a wind, you could probably hear them loud and clear." Miles shifted to lean back against the oak's trunk beside Hugh. "You'd think they'd get tired of listening to each other."
"Don't you know the only sound sweeter to my father's ears than his hounds baying in the hunt is his own voice baying at one of us?"
"How good for him. How unfortunate for us." Skyre flopped down with a sigh beside him. Miles likewise sighed and added, "Wouldn't it be good if he'd choke on his own bile and be done with it?"
Hugh said nothing; no answer was needed. The wolfhounds settled beside him again and the forest's quiet came back, with only the hum of midges in the clearing's sunlight and Brigand sometime snuffling in his sleep. For Hugh, Miles' ability to be silent was one of his virtues. That they were able to be silent together was part of their friendship, more than the fact that, close in age though they were, they were uncle and nephew because Sir Ralph—disappointed in the son he had had by his late first wife—had married Lady Anneys much about the time that disappointing son had come back from the French war with a French wife. Lady Anneys had borne Tom a year later and had Hugh a year after that, a few months before Miles was born to their half-brother, who had shortly thereafter died, leaving his wife and infant son to his father's and stepmother's care.
Miles' mother, much loathed by Sir Ralph and loathing him in return, had lived for another three years, then died, too; and Lady Anneys, with Tom and Hugh and by then a daughter of her own, had taken Miles to herself as if he were another son rather than her stepgrandson. Hugh did not know if she had quarreled with Sir Ralph over that; but then, he had never in his life seen his parents quarrel over anything. What he had equally never seen was Sir Ralph show anything like affection toward Miles. Instead, all the disgust Sir Ralph must have had against his firstborn son and the French wife had turned on Miles. And in return Miles as fully and openly loathed him.
Brigand stood up, turned around and around, and lay down again, this time dropping his head heavily onto Hugh's stomach. Jarred out of his almost-doze, Hugh ran a hand down a gray, rough-coated shoulder and said at Miles, "You're going to be in trouble for wandering off, you know."
"I didn't just wander off," Miles said lazily. His hands clasped behind his head now, he was gazing idly away at nothing across the clearing. "If anyone asks, I've been to see how the charcoal making is going at the far end of the woods."
" _Is_ there charcoal making going on at the far end of the woods?"
"No."
"Miles," Hugh said with feigned deep weariness, "that's careless." That it was also a lie was not the matter. Lying to Sir Ralph was simply the safest way to go through life.
"Old Roaring Ralph won't find out. He hardly takes enough interest in the manor to know when it's haying time, let be if there's charcoal making or not. He won't know the difference or care what I'm doing, so long as I'm 'earning my keep' and not anywhere near him."
That was all too true. Once, when they were half-grown and hiding in the hayloft nursing beaten backs for some forgotten offense, Miles had said, "He beats you to make sure you obey him. He beats me because he wants me to break." Had said it flatly, not so much in complaint as merely telling a truth that Hugh had not seen until then.
That they all three—he and Miles and Tom—were now grown too big to be beaten anymore only meant Sir Ralph had other ways of making sure they understood his hand was upper here, and one way was to make use of all of them. Hugh had his houndmaster and huntsman's duties, seeing to it that hounds and game were ready when Sir Ralph wanted to hunt, while Tom had had writing and arithmetic driven into him so that he could be Sir Ralph's steward. "And any year you don't show a profit, you'd better be able to show me the reason why," Sir Ralph had said. "You're not so big I can't take it out of your hide if I want to."
"You'll get what's coming to you," Tom had snarled. "Don't fear that."
Sir Ralph had risen from his chair, snarling back, "Don't loose your lip at me, boy! You haven't inherited yet. By God's teeth, you cross me and I'll leave it all to the Church. I swear I will!"
"Leave away! The day you do, you lose all hold on me and any of us and we'll be gone so fast you'll choke on the dust!"
"Good riddance to the lot of you then! Maybe I'd have a little peace around here!"
When the two of them set to one of their shouting-matches, like today, everyone who could tried simply to get out of hearing, and one good thing about Tom being steward was that he had to ride out and away to see to Sir Ralph's other properties, taking him out of shouting range for a few weeks at a time. Miles was not so fortunate. As Woodrim's forester, overseeing the manor's forest and Charlbrook Chase for the sake of the hunting and for Sir Ralph's profit, his duties took him away from the manor, yes, but never far and never for long. It was his business to see to it that the villagers took no more deadwood than they were allowed by right and custom; that timbering and coppicing were done where and when they should be to keep the trees not only from overgrowing but constantly fit for harvesting; and that new plantings were steadily made, to keep both forests fit for all their uses. It was ceaseless work, year-around in all weathers, and Miles did it well, Hugh knew. What he did not know was whether Miles had any actual care for the work or only for the excuse it gave him to be away from the manor and Sir Ralph for hours and sometimes whole days at a time. It was something Hugh had never asked, but he sometimes thought—and shied away from the thought when it came—how they all—he and Miles and Tom and everyone else—despite how much they lived together, lived so much alone, careful behind walls of silence that kept them a little safe from Sir Ralph but also shut away from each other.
So quietly that for a moment Hugh did not take in what he had said, Miles murmured, "The wonder is that one of us hasn't killed him yet." And after another moment's silence, "Or that someone hasn't, if not one of us. He's crossed enough men in his life that you'd think somebody would have done for him by now."
"Kill him?" Hugh repeated, still watching the leaves above him.
"Or beaten him bloody," Miles went on, in the same mild voice.
Hugh pushed Brigand's head aside and propped himself up on one elbow to look at Miles, still gazing away across the clearing. "Miles," he said. "Don't."
Miles rolled his head sideways to meet his gaze. "Don't say it or don't do it?"
"Don't do either one."
Miles turned his face away, back to looking across the clearing. "The pity is that I won't. I just wish someone would. His death is the only way we'll ever be cut loose from him. You have to know that. His death or ours."
Hugh lay back down and closed his eyes. Away from Sir Ralph, he did not want even to think about him, let alone pointlessly talk about him; but feigning mildness as deliberately as Miles had, he asked, "What thing worse than usual did he say today?"
He waited through the silence, knowing Miles would answer, and finally Miles did, still low-voiced but now openly bitter. "In the middle of yelling at Tom over the cost of rope—you missed them moving on from ale to rope—he of a sudden pointed at me and said, 'The only thing around here, Tomkin, more useless than you, is that.'"
Hugh, trying for lightness, said, "Which doubtless made Tom no happier than it made you?"
"I'd guess so, judging by how hard he threw his cup against the wall."
"Meaning it was a good thing it was the wooden cups at dinner today?"
"A very good thing. One of the pewter ones would have holed the wall. The wooden one only broke. While Tom was yelling that Sir Ralph was wrong, the most useless thing on the manor was _him_ and Sir Ralph was yelling back, I left. You don't suppose today is the day they'll finally kill each other, do you?"
Hugh doubted it would ever come to killing. They used yelling instead of blows these days. But he looked at Miles again and said, "Just mind _you_ don't go killing him, that's all. You'd be caught for it and hanged and he'd rejoice from the grave to know he'd brought you down. You don't want to make him that happy, do you?"
Miles caught Hugh's mockery and tossed it back. "Right. I can see me standing on the scaffold with the noose around my neck and him sniggering up at me from Hell. Damn. I'll have to let him live."
"Besides that, if you killed him, you'd go to Hell for it and have to spend eternity with him."
Miles threw up his hands in surrender. "Then he's surely safe from me. I'm not about to risk that."
"Unless," Hugh went on, "he's left money in his will for Masses enough to send him to Heaven."
"Ha! There aren't enough priests in the world to say that many Masses before Judgment Day."
Hugh moved Brigand from his stomach and sat up. "Of course, after you killed him, you could repent of it, and then if you were hanged, you'd save your soul and be safe away to Heaven, leaving him to Hell on his own."
"Repent?" Miles said with huge indignation. He signed himself with a large cross. "God forgive me if I'm ever _that_ much of a hypocrite!" And he and Hugh both burst into laughter. He put his hands behind his head again and, still softly chuckling, shifted to stare up into leaves in his turn while Hugh began checking the dogs' paws for thorns or cuts. A hunting dog's fate was in their paws, Master Basing had told him often enough, and yesterday he had heard himself saying to young Degory, the present hound-boy, "A dog that can't run is good for nothing," sounding so much like Master Basing he had almost laughed at himself.
The forest's stillness and their own companionable silence folded around him and Miles again, until the sunlight's slant through the trees told them they had been gone as long as they safely could be. There was no need to say it. When Hugh stood up, so did Miles. The three dogs rose, too, and mightily shook themselves; and in silence they all started homeward, out of the clearing into the forest path's thick shade.
**Chapter 2**
Dame Frevisse awoke in the dark to the whisper of rain along the roof of the nuns' dorter and smiled to herself with pleasure at the sound. This summer in the year of God's grace 1448 had thus far been all warm days and clear skies and just rain enough to keep the pastures thick-grassed for the cattle's grazing and bring the green grain tall in the village fields around St. Frideswide's nunnery in northern Oxfordshire. In the wider world all rumors of trouble roiling around the king and among his lords were, since spring, faded away into the distance like a thunderstorm disappearing along the horizon, too far off to matter aught to anyone here. That rain was come again to the fields after almost a week without was a greater matter than the doings or not-doings of king and lords.
The rain's whispering kept Frevisse and St. Frideswide's eight other nuns company in the thinning darkness toward dawn as they went soft-footed down from the dorter and along the roofed cloister walk to the church for Prime's prayers and psalms; and the rain was still falling when they finished the Office and passed along the walk again, around the cloister garth to breakfast in the refectory; but by the time they returned to the church for Mass, only a light pattering was left, and by midmorning, after Sext's prayers, only a dripping off the eaves under a clearing sky.
Frevisse smiled up at the cloud-streaked blue above the cloister roof as she came along the cloister walk from the infirmary with a small pot of freshly made ink. Her hope this morning was for time at her desk before Nones to go a little farther with copying out the book of Breton stories promised to a clothier's wife in Banbury by Michaelmas in return for enough black woolen cloth to make two nuns' winter gowns. If she could copy through to the end of Sir Degare's troubles today . . .
She was passing the passageway to the cloister's outer door when a knocking at it brought her to an abrupt stop. St. Frideswide's was too small a nunnery to need a constant doorkeeper. Whoever was nearest was expected to answer a knock when it came and there was no denying that not only was she nearest but that there was no one else in sight—either nun or servant—and reluctantly she turned and went along the passageway to open the shutter of the door's small, grilled window and look out, asking, "Yes?"
Old Ela from the guesthall across the courtyard made a short curtsy, bobbing briefly out of view and back again before saying, "My lady, there's someone come to see Domina Elisabeth. Master Hugh Woderove, if you please."
"Kin to Ursula?" Frevisse asked.
"Her brother, he says, my lady."
Careful of the inkpot, Frevisse unlatched the door one-handed and opened it. The cloister was a place men rarely and only briefly came but it was allowed when necessary, and without bothering to take clear look at the young man standing beside Ela, she bowed her head and stood aside to let him enter, thanked Ela, shut the door again, said to him, "This way, please," and led him into the cloister walk, where she paused to set the inkpot on the low wall between the walk and the cloister garth at its center before leading him away to the stairs up to Domina Elisabeth's parlor door. There she asked him to wait while she scratched at the door and at Domina Elisabeth's _"Benedicite"_ went in to say he was here.
Both for courtesy and in matters that could not be done in chapter meetings, the prioress often received important guests or others here, and therefore her parlor was more comfortably provided than any other place in the cloister, having its own fireplace, two chairs, brightly embroidered cushions on the window seat, and a woven Spanish carpet on the table. Besides that, because she shared her nuns' copying work, Domina Elisabeth had her own slant-topped desk set near the window that overlooked the courtyard and guest-halls, where the light fell well for most of the day. She was there now but looked up from wiping a quill pen's tip clean to ask as Frevisse entered, "Who is he? Has he said why he wants to see me?" Thereby proving that her copywork had not kept her from looking out the window to see that someone had arrived.
"He's Master Hugh Woderove, Ursula's brother," Frevisse answered. "I don't know why he's here."
"To visit her, belike." Domina Elisabeth rose from the desk's tall-legged stool and shook out her skirts. "Bring him in, please, and stay." Since no nun should be private with a man.
Frevisse saw Hugh Woderove in and stood aside beside the door, hands folded in front of her and head a little bowed, but not so low she could not take her first clear look at him as he crossed the room. He was a wide-shouldered, brown-haired young man in a short, dark gray houppelande slit at the sides for riding and knee-tall leather boots, well made but with many miles and probably years of use to them. His bow and his "My lady" to Domina Elisabeth were good enough but constrained, as if he were unused to using his manners, and when Domina Elisabeth sat in her own high-backed, carven chair and bade him to the room's other, he sat uncomfortably forward on the chair's edge, hands clasped tightly around his black leather riding gloves as he said without other greeting, "By your leave, my lady, I won't keep you long. I've only come to take my sister home."
"Home?" Domina Elisabeth permitted quiet surprise into her voice despite she was undoubtedly—as Frevisse was—rapidly wondering why. "So suddenly?"
Twisting the gloves in his hands, he started, "Our father—" but stopped, interrupted by a soft scratching at the door frame.
A stranger's arrival in the cloister never went unknown for long. Frevisse's only question had been whether Dame Emma or Sister Amicia would be first, claiming to wonder if Domina Elisabeth wished food or drink brought for him. As it happened, it was Sister Amicia, panting a little from the haste she had made, who came in at Domina Elisabeth's rather sharp _"Benedicite,"_ and curtsyed, taking an open, questioning look at their guest while asking, "Does my lady wish I bring wine and something to eat?"
"Please, no," Hugh said to Domina Elisabeth before she could answer. "We can be home early tomorrow if we leave soon."
Accepting his pressing need without more question, Domina Elisabeth said, "Sister Amicia, please bring Ursula Woderove here and give order for her things to be packed." She turned a questioning look to Ursula's brother. "All of her things?"
"What? No. She'll be back. I hope to bring her back in a few weeks."
"Enough of her things for a visit," Domina Elisabeth said, and added crisply, "Now, please," when it seemed Sister Amicia might linger in hope of hearing more. At that, Sister Amicia made a reluctant curtsy and went and Domina Elisabeth turned her heed back to Hugh, saying, "Your sister will be here soon. Can you tell me what the trouble is? Might our prayers help?"
"Yes. No. I mean, yes." Hugh paused, having lost himself among answers, drew breath, and said, smoothing his gloves over his knee, "It's our father. Sir Ralph. He's dead."
Frevisse and Domina Elisabeth both made the sign of the cross on themselves, Frevisse murmuring a brief prayer while Domina Elisabeth asked, "We knew nothing of him being ill. It came suddenly?"
"Yes. Suddenly," Hugh agreed awkwardly. "It was . . . he was killed. Someone killed him. We found him dead two days ago."
"Not . . . an accident?" Domina Elisabeth asked.
"Not an accident, no."
"But you don't know who . . ."
Domina Elisabeth stopped, caught between a wish to know and uncertainty of how much to ask, but Hugh answered, "We don't know who, no. It was in the woods. He'd been hunting. Whoever did it was well away before we even knew it had happened. We made search with men and dogs but to no use."
All of which explained the young man's unsteadiness, Frevisse thought. He had hardly had time to find his own balance and was here to tell his sister in her turn. At least he did not have to wait long; small, light footsteps were already running up the stairs, with Sister Amicia calling, "There's no need to run, Ursula," but too late. Ursula burst into the room. Wearing her favorite red gown, she was always startling among the nuns forever in their Benedictine black, but as Dame Claire had said last time the question had come up of dressing her in something more seemly, its boldness suited her, and besides, no one was truly willing to spoil her open pleasure in its bright color or forgo their own pleasure in seeing hers. But like her gown, she was sudden, and forgetful of manners and the courtesy due Domina Elisabeth, she cried out, "Hugh!," and flung herself across the room at him.
Rising from his chair, he caught her and lifted her from the floor into a great hug that she returned with her arms around his neck and hearty kissing of his cheek. He kissed her as heartily back, set her to the floor, and said the inevitable, "You've grown!"
Ursula flashed a happy smile up at him. "You haven't seen me since Christmastide and I turned eleven the morrow of St. Mark's. Of course I've grown." She poked at his belt buckle accusingly. "You said you'd come at Easter but you didn't. Nobody did."
"I only said might and it happened I couldn't. But I sent you a red ribbon for your birthday, didn't I? I did do that."
Ursula flung her arms around his waist, as high as she could reach, and hugged him again. "You did, you did. Just like I asked."
"But, Ursa," Hugh said, and the brightness was gone from his voice, warning her there was something else.
Dame Perpetua was always saying that Ursula was a quickly clever child and now she immediately read her brother's voice and pulled back from him as far as she could without letting go of him and asked, frightened, "Has something happened to Mother?"
"Not to Mother, no. She's well." Hugh moved to sit again, drawing Ursula to stand in front of him so they were face to face. "It's Father. He's dead. He was killed while he was out hunting. Two days ago."
Ursula's face went blank and her color drained away. With widened eyes, she echoed faintly, "He's dead? He's really dead?"
"Two days ago. I'm here to take you home for the funeral."
Belatedly Domina Elisabeth rose to her feet. "We'll leave you together for now. Take as long as need be. Your things will be waiting for you when you're ready to leave, Ursula."
Ursula vaguely thanked her without looking away from her brother or loosing her hold on him. Frevisse, following Domina Elisabeth out of the room, heard her ask again, "He's really dead?" with what sounded oddly more like unbelieving hope than grief, but Frevisse rid herself of that thought by the time she reached the stairfoot, where Domina Elisabeth was sending Sister Amicia to ask Father Henry, the nunnery's priest, to meet her in the church.
"Sir Ralph Woderove was not overgenerous to us but he did send that haunch of venison when Ursula returned to us at Christmastide, and he was her father. For her sake he shall have a Mass and prayers from us," she said.
Frevisse agreed that would be well and, her part in it finished, retrieved her inkpot and was at her desk on the cloister walk's north side, bent over her work, when Domina Elisabeth saw Ursula and her brother out. Keeping her eyes on her copying, Frevisse did not see them actually leave but heard, when they were gone and Domina Elisabeth was come back into the cloister, the soft rush of skirts and feet along the walk and Dame Emma eagerly asking Domina Elisabeth where Ursula was going and would she be back and who was that who'd come for her and what had happened. Domina Elisabeth told her, briefly, and sent her back to whatever she should have been doing.
Frevisse missed whatever talk came after that among the other nuns, keeping at her copying through the rest of the day save for the Offices and dinner when no talking was allowed. Only in the hour's recreation allowed each day before Compline's final prayers and bed did she hear more, when the nuns went after supper to the high-walled garden behind the cloister. The fading day was warm under a sky soft with westering sunlight, and in the usual way of things, they would have sat on the benches or strolled alone or a few together, talking and at ease, along the graveled paths between the carefully kept beds of herbs and flowers. This evening, though, while Frevisse walked slowly in company with Dame Claire, and Sister Thomasine went her usual way to the church to pray, the other nuns hurried ahead along the narrow slype that led out of the cloister walk, crowding on each other's heels into the garden and immediately clustering just inside the gateway with heads together and tongues going. There was no way for Frevisse not to hear, as she and Dame Claire made to go past them, Dame Juliana saying excitedly, "Yes! I asked the servant who came with him." Presently hosteler, with the duty of seeing to the nunnery's guests, Dame Juliana had plainly used her duty today as a chance to learn what she could from Hugh Woderove's servant. "He told me everything."
"It truly was murder?" Sister Johane asked. "Someone did kill him?"
"They did. Very much killed him." Dame Juliana was as eager to tell as her listeners were to hear. "Someone bashed his head in with a stone. They'd been hunting and he went into the woods after a dog, I think, and when he didn't come back, they went looking for him and found him _dead._ "
"They don't know who did it?" Dame Emma asked.
"Well, they think it must have been a poacher and Sir Ralph surprised him and the man killed him. Whoever it was, he escaped clean away and that's why they think a poacher, because a poacher would know the woods well enough to get away and how to lose the hounds when they tried to trail him."
Frevisse's own thought was that it must have been a singularly stupid poacher if he chose to be there in the woods when a hunt was going on, but Dame Claire asked as they moved away, neither of them interested in spending their hour in fervent talk over a murder that had nothing to do with them, "How was it with Ursula when her brother told her, poor child?"
Cautious with her own uncertainty, Frevisse said, "I was there only at first, before she'd had time to fully feel it, I think."
Dame Claire bent to run her fingers through a tall clump of lavender. "We'll have a goodly harvest of this, it seems."
Frevisse agreed, and leaving the dead man forgotten, they strolled on.
**Chapter 3**
The trouble with summer days was how long they lasted, Hugh thought wearily, watching almost the last of the funeral guests ride out of sight around the far curve of the road along the woodshore beyond the church. Today had stretched out forever from sunrise until finally now when dusk had set the last guests homeward. Master and Mistress Drayton had a four-mile ride to go but twilight this near past Midsummer went on forever; they would be home before full dark. Hugh lowered his hand from a last wave after them and turned back through the gateway into the manor's foreyard, trying to hold his shoulders straight against his weariness. Long summer days and their drawn-out twilight had always been pleasure for him, but today he had found himself wanting a brief winter's day that would be done with and over, with a long night to follow when he could go to bed and not be anything except—with luck—asleep for hours upon hours and no need to say or do any of all the things he had had to do and say these past five days.
As it was, having no need for haste to be home before dark, the guests had lingered over the funeral feast. Not that there had been that many guests. "And most of them are here simply for the pleasure of seeing him dead," Miles had said low-voiced to Hugh as they came out of the uncrowded church after the funeral Mass. Nor was anyone there who had to take much trouble over coming except Master Wyck from Banbury and that was more because he had been Sir Ralph's attorney than for respect. Certainly there had been no great grieving from anyone, unless Elyn and Lucy's tears meant much, which Hugh doubted. That his sisters' weeping seemed more from duty—they owed it to themselves to weep—than from feeling was among the thoughts weighing on Hugh all day. Another heavy thought was that despite Sir Ralph had lived a goodly number of years, been married, had children, known a fair number of people along the way, at the end of it all he was buried with no one much caring and only duty-tears over his grave.
If anyone is going to miss him, Hugh thought as he crossed the foreyard, back toward the hall, his shadow stretching out black ahead of him, it will be me and the hounds. And I don't think we do. Except maybe Bevis. The brindled wolfhound had been Sir Ralph's present favorite, taken with him almost everywhere, even into church, where Father Leonel had frowned but not dared say anything. These days since Sir Ralph's death, Bevis had limped restlessly through the hall and foreyard or else lain beside the hall hearth, long muzzle on paws, eyes fixed on the outer door as if awaiting Sir Ralph's return. Yet when Tom had brought him to lie beside the bier, he would not, and today was tied in the kitchen yard, out of the way. Hugh had wondered before now how things would have gone that last day if Bevis had been with Sir Ralph. Assuredly not the way they had.
But Bevis had cut a forepaw on a stone the day before and the morning of the hunt Sir Ralph had rumpled his ears and said, "Best you lie up today, old fellow. It's only hare-hunting anyway." But a very good hare-hunting, as it happened; the best there had been that summer. Only fallow and roedeer bucks and hare were allowable to the hunt through the summer months and early autumn, between St. John's Day and Holy Rood, and since Sir Ralph had hunted roedeer a few days earlier, he had been in the humour for hare-hunting, and so Hugh had been out in the green-gold dawn that day, a-foot and without dogs, to quarter the rough pastureland beyond the village fields, looking for the best place to bring the hunt. He had been glad the signs for likely best hunting looked to be in the farthest of the pastures, well away from the fields where the grain was ripening toward harvest. Sir Ralph had no care if his hounds coursed through the standing grain, but it set the villagers to fury to see their work and hope of winter bread trampled by hounds and hunters for the sake of sport. Sir Ralph's answer to their protests when they came into the manor court about it was always, "I have to live with your poaching my game out of my forest whenever you've a mind to it and stealing my wood whenever my back is turned. You can live with my trampling a little of your fields in return. Now get out."
A year ago Hary Gefori, the hayward's grown son and shaping well to take his father's place when the time came, had dared say angrily back, "Aye, we poach, and when we're caught at it, we're beaten and fined for it. So, in like, when you and your hounds and horses have robbed us of our grain, why shouldn't you pay?"
Sir Ralph had half-risen from his chair, his hands gripping its arms so his knuckles stood out white, his face purpled with fury and his words almost throttled by his anger. "Pay for what's mine? I own all this manor and everything on it, including you and every stalk of wheat and rye and barley and plain pasture grass. If anyone's going to pay, it'll be you—with half the teeth in your head and the skin off your back. Tom and you there, Duff, take him. Hugh, fetch my dog whip. I'll show . . ."
Hary had not waited to be taken or whipped but had spun on his heel and shoved his way among the men gathered to the court, with no one—including Tom—trying to stop him before he was out the door. That had earned Tom a yelling-at and every man there the fine of a penny each, including Tom, though Hugh doubted Tom ever paid it, since Tom and Father Leonel between them kept the manor accounts and Sir Ralph "never cares what the accounts say," Tom had raged once to their mother. "So long as the hounds are healthy and the roof isn't falling in, he doesn't care. I could be stealing him blind and he'd never know."
"Are you stealing him blind?" Lady Anneys had asked.
"No. The more fool me," Tom had said bitterly.
Now everything was Tom's, and if the villagers had warily held back from outright celebration of Sir Ralph's death, Hugh did not doubt there was nonetheless hidden joy among them because Tom, for all that his anger could flare like Sir Ralph's, was far more even-handed in his dealings. He had been even more pleased than Hugh the morning of that last hunt to hear the chase would likely keep well away from the grain fields.
"It will be closer to the gathering place, too," he had said. "Farther for the servants but closer for us, and Mother and the girls won't mind the walk." Which they would have to make, whether they minded or not, because last night Sir Ralph had pointed at Lady Anneys across the parlor and ordered, "See to it there's food laid out at the spring after the hunt. We might as well make a day of it, since Sir William is bringing both Elyn and his girl." Sir William being their near neighbor and as passionate to the hunt as Sir Ralph.
That night Miles said, while he and Hugh and Tom had been readying to bed in the chamber they shared over the kitchen at the hall's other end from Sir Ralph and Lady Anneys' own room, "So we're to hunt in the morning, guzzle through midday, and return to the slaughter in the afternoon. I wonder if I feel a sickness coming on and must keep to my bed for the day?"
Tom had thrown a wadded shirt at him. "If I have to be there, so do you."
"I hate hare-hunting."
"You hate all hunting."
Miles threw the shirt back at him. "Hare-hunting is worse. You can hunt the fool things twice in a day. Everything else you hunt and then go home. Red deer, roedeer, fallow deer, otter, badger, fox, boar, bear, wolf . . ."
"Boar? Bear? Wolf?" Tom had repeated cuttingly. "When have any of us ever hunted boar or bear or wolf?"
"Never, thank St. Eustace. It's been bad enough listening to Sir Ralph moaning on about lacking them. Years and years of him moaning there's no wild boar or bear or wolf left for him to slaughter. Moaning on and on . . ."
"Nephew," Tom warned, "if you don't shut yourself up . . ."
". . . and on and on and . . ."
Tom and Hugh together had shoved him backward onto the bed and made to smother him with a pillow until laughter broke up their wrestling and they had all settled to sleep in the cheerfulness they so often had together when away from Sir Ralph.
In the morning Hugh had been first up and dressed and away, leaving them pulling on their heavy hunting hosen and debating whether hare was better baked in gravy in a crusted pie or roasted crisp on a spit. He had returned, more than ready for his breakfast, to find Sir William had arrived. His pair of scent hounds were in the foreyard, held on leash by Sir William's steward, Master Selenger.
Master Selenger was a man as lean and ready to the hunt as the hounds he held, and Hugh traded a few words with him before going in to tell what he had found and snatch some bread and cheese while his father and Sir William discussed which hounds they meant to use today. Sir William was not quite Sir Ralph's age nor given to anything like Sir Ralph's rages but their shared passion for hunting had made them "as near to friends as Sir Ralph is ever likely to come," Miles once said. Friends enough that Sir Ralph had married Elyn, his eldest daughter, to him and lately begun to talk with him of marrying Tom to Sir William's daughter, Philippa.
For a wonder, given how readily Tom quarreled with Sir Ralph over anything and everything else, he had made no protest against that. Not that there was much to protest. Besides being Sir William's only child and therefore his heir, Philippa was a pleasant-featured, pleasant-mannered girl, friends with Elyn and Lucy and so often at Woodrim that Lady Anneys, fond of her, said she was already more than halfway to belonging there. The only present complication was Sir William's marriage to Elyn two years ago. Besides that it made him Tom and Hugh's brother-in-law, it raised the likelihood he would father more children, lessening Philippa's inheritance or, if there were a son, replacing Philippa altogether. Any marriage agreement made now would have to be most carefully made to ensure she stayed worth Tom's marrying and as yet Sir Ralph and Sir William had not settled down to the task and Tom knew better than to push the matter. And since Elyn wasn't bearing yet, everything was mayhap and maybe anyway and more important that morning was the hare-hunt.
They had gone on foot to the far pasture, the hounds knowing what was coming and as eager to the business as the men. At the pasture's edge Sir Ralph had blown three glad notes on his hunting horn, and Hugh, Degory, and Master Selenger had uncoupled the six lymers—the scent hounds—who had the first work. Set forward with Hugh's cry of "Avaunt, sire, avaunt!," they had surged away into the pasture with Hugh's following call of, "So howe, so howe, so howe!" to urge them onward. Not that urging was needed. Hares were cunning. As if ever-aware they might be hunted, one hare never, for choice, traveled straightforwardly but rather went one way, then back on its trail for a ways before going another way, over and over again, ten times or more and crisscrossing its own trails while it did, with sometimes a sideways leap to start a different way all over again. The lymers' challenge was to sort out the trails and thereby track a hare to its form—its resting place—and rout it out, and Somer, Sudden, Sendal, and young Skyre, along with Sir William's lymers, set to it joyfully, questing rapidly back and forth through the long grass, heads down and tails madly wagging. As always for Hugh, their intensity became his as he watched them searching, spreading apart, sweeping this way and that across the pasture to untangle the scents, while beside him and Tom, Sir Ralph, Sir William, and Master Selenger, the coursing hounds waited their chance with quivering eagerness. Miles, as always, stood a little apart, there because he had to be, but even so he was watching, smiling, the hounds' joy at their work impossible not to share.
Hugh saw Sendal start to swing too far apart from the others and called out, "Howse, Sendal, howse!" to bring him back to the others, now closing in on a hare it seemed, to judge by how they were rushing forward, crowding together, then spreading apart and crowding forward again, all a-quiver and their tails wild. As Sendal rejoined them, the hare burst up from the grass and into a run hardly three hounds' length ahead of Sudden in the lead.
"She goes!" Sir William cried. "Ears up! She's a good one!" Because a hare that waited until the hounds saw it and ran with its ears up was confident of its strength and chance of escape. This one was as cunning in its running as it had been in laying its trail, swift in its turns and twice cutting sideways and away under the very muzzles of the hounds before Sir Ralph said, "That's enough. Now," and set Bertrand loose with the cry, "Venes!" The rest of them loosed the other hounds with him and—white, and brown-spotted, and brindled gray—Makarie, Melador, Bane, Brigand, and Sir William's Chandos streaked away to join the others.
The end came quickly then, and hunters trotted out to the kill-site where Melador was standing over the dead hare while the other hounds seethed around with high-waving tails, panting and pleased with themselves. Sir Ralph held the corpse high and blared the death on his horn, they went through the various ceremonies demanded at a kill, and then the hare was dismembered, bits of it mixed with blood-soaked bread and cheese and given to the hounds, but the better parts handed to a servant to carry back to the manor. Then they began again.
Two more hares were started, both of them escaping after good runs, before another one was taken. By then the morning was far enough along that Sir Ralph declared the hunting done for the while and they headed across the pastureland and along the woodshore to the greenway, men and hounds all glad of the shade under the tall-arched trees and gladder when they turned aside from the way into the wide, smooth-grassed clearing of the gathering place. At its upper end a spring bubbled out of the slope, its cold water filling a stone-built basin before flowing over and away in a shallow, broadening stream the length of the clearing and out of sight among the trees. Along the near side of the clearing wooden tabletops on trestle legs had been put up and cold chicken, new cheese, bread, and ale set out by servants waiting now to serve, while beyond the little stream Lady Anneys, Elyn, Philippa, and Lucy were seated on cushions on the grass, Philippa with her small lute on her lap, the others with their embroidery.
As usual, Hugh, Master Selenger, and Degory saw to watering the hounds, then Hugh and Master Selenger left them to Degory to tie in the shade and feed while they joined the others at the tables to eat and drink and talk over the morning's hunt. Afterward the women returned to their cushions, and the men, fed and tired and ready to rest, dropped down around them on what Lady Anneys called the hunt-cushions—large, old, not very clean cushions kept for this use. Only Miles went to sit on the wide rim of the spring's basin, dabbling his fingers in the water. The midday warmth and well-fed bellies slowed the talk and even the women sat idle, their sewing in their laps, except Philippa took up her lute again and began to stroke small, silver-sounding notes from its strings, simple as the sound of the water flowing and hardly more noticeable. Sir Ralph tried to pick a quarrel with Tom over a pasture that had been grazed last month instead of left rough for the summer's hunting but they were both too full and tired for it, and giving it up, Sir Ralph joined Sir William in dozing, stretched out with their heads laid on cushions and hands clasped on stomachs. Tom shifted to sit beside Philippa, watching her play. The family jest was that he had no more ear than a post for music, and when Elyn asked Philippa to sing, it was to Master Selenger that Philippa turned, asking, "Join me?"
Lying on his side, propped up on one elbow on one of the cushions, he smilingly said, "Gladly," and sat up. He was Philippa's uncle, brother to Sir William's first wife, dead these dozen years, and their singing together was always everyone's pleasure. Their voices—his dark, hers light—blended around each other, wending through "At sometime merry, at sometime sad, At sometimes well, at sometimes woe" as gracefully as dancers winding a maypole until at the end they sang exactly together, "He is not wise, he is but mad, That after worldly wealth does go," and then laughed at their own delight in their shared pleasure while Lady Anneys, Elyn, Tom, Hugh, and Miles lightly applauded them.
Looking back from afterward, Hugh could only think how usual it all had been. A midday gathering like uncounted others, with no warning of what would come. But what warning could there have been, Hugh wondered now, crossing the foreyard back to the hall. Almost a week was past, with time enough to remember, and he did not see what . . .
"Hugh," his mother said quietly from the hall doorway.
He had been looking down, making no haste back to the hall, but at her call he looked up and lengthened his pace. He was tired and knew she was and he had not expected her to wait for him there after seeing the guests away. She had taken these past days quietly, the way she took everything. Not even in the first horrible surprise of Sir Ralph's death had she cried. Elyn and Lucy had shed more than enough tears and Hugh's thought was that Lady Anneys' own were dried up with comforting them, but his fear might have been that with the funeral done and the guests gone, her grieving would come over her in a storm. Except, in a carefully buried part of his mind, he doubted that she grieved at all.
But then, did any of them?
He smiled at her as he joined her at the doorway and asked, "Will you rest now, Mother? Can I send Elyn and Lucy off to a far corner of a far field and let you be at peace for a while?"
She smiled back at him. "Not just yet." Against the black of her mourning gown and veil, her always fair-skinned face was even paler, the gray shadows under her eyes almost its only color. "Master Wyck wants to speak with us all."
"Now? Does it have to be now?"
"To have it done with," Lady Anneys said. "He's to spend the night with Sir William, to be that much further on his road to home come the morning with no need to come back here."
"You're tired." So was he, come to that. Since Sir Ralph's death there had been no pause in things to be done, beginning with the useless hunt through the woods for his murderer, looking for track of someone and not finding it, casting the hounds wide and wide again, trying to flush someone out but failing. Then there had been sending for the crowner—the king's officer charged with seeing into unexpected deaths—and when he came, his questions and demands to be answered. And then fetching Ursula from the nunnery and all the preparations for the funeral and finally today the funeral itself and the funeral feast and all that went with that, and now the lawyer wanted them.
Hugh knew he himself was faltering, doubted how much more his mother could endure, and protested more strongly, "You're too tired for this."
She laid her hand on his arm, briefly smiled at him, said, "It's no matter," and turned away, back into the hall.
Hugh, needing more time than that to gather himself, paused a moment longer on the threshold between the day's sunlight and the hall's shadows. Woodrim had never been more than a minor knight's manor and then for forty years an aged and childless widow's dower land until, after her death, Sir Ralph had bought it from a distant heir. But he had bought it for the hunting and never much bothered with anything else about it, keeping the hall as he had found it—plain, bare-raftered, not overlarge, with an open hearth in the middle and the smoke meant to escape through a penticed louver in the roof. There was not even a screen-wall here at its lower end to block the draughts when the outside door was opened and from where he stood in the doorway Hugh could see its length, past the servants clearing away the remains of the funeral feast from the long table set up along one side, in a hurry to have their own feasting in the kitchen, to the far end where the dais raised the master's table a step above the rest of the hall for whoever sat there to see and be seen by the rest of the household.
The hall's only window was there, looking out from one end of the dais onto the foreyard, tall and narrow and glassed at the top above the shutters so that even in winter or ill weather when the heavy wooden shutters were closed, there could be some light in the hall besides through the doorway or from candles or rushlights. Today in the warm afternoon the shutters stood open, letting the westering sunlight fall the dais' length across Tom, Sir William, and Master Wyck standing together behind the table there, next to Sir Ralph's tall-backed chair. Tom's chair now, Hugh reminded himself; but neither Tom nor anyone else had sat there since Sir Ralph's death. Tom would have to, sooner or later, there being only their mother's smaller chair and the benches otherwise, but presently the men were all standing, talking, a cluster of blackness in their mourning gowns and doublets. At the dais' farther end Elyn, Lucy, Ursula, and Philippa were gathered close together, Miles standing near them but somehow not with them, still in the silence that he had kept heavily around him this while since Sir Ralph's death.
"Since I can't mourn and shouldn't openly rejoice, best I just keep quiet," he'd said when Hugh had asked how he was.
Still wishing that Master Wyck would wait with whatever he wanted to say, Hugh followed his mother up the hall. Sir William and Master Wyck bowed to her as she joined them and Tom took her by the hand to bring her to her chair beside Sir Ralph's. She sat and Sir William leaned over her, laying a hand on hers on the chair arm, saying something too low for Hugh to hear. She shook her head and said something back. At the dais' other end Miles opened the door to the parlor and stepped aside for the girls and Elyn to go in. Lucy and Ursula did, but as Lady Anneys answered Sir William, Elyn and Philippa both paused and, with Miles, looked back toward them. Then Miles said something to Elyn and she nodded and went on, but Philippa paused a moment longer, looking from Miles to her father and back to Miles until from the parlor Elyn ordered loudly, sharply, "Philippa!"
Philippa winced. Miles ruefully shrugged at her and she ruefully shrugged back with a slight, uneven smile and followed Elyn into the parlor, Miles closing the door behind her. Hugh had sometimes wondered, in the two years since Elyn had married Sir William, how Philippa, only two years younger and often with Elyn while they were growing up, felt at having Elyn for a stepmother, wielding a stepmother's authority over her. Hugh doubted Elyn troubled herself with wondering. Elyn had welcomed marriage, been glad to become Lady Elyn and free of anyone telling her what to do except her husband, nor had she yet shown any regrets; and since there was a great deal of their father in Elyn, Hugh well supposed she probably did not care what Philippa thought or felt about any of it so long as Philippa did what she was told and made no quarrel about it.
Sir William was still speaking to Lady Anneys, now sitting with her head bowed and her hands folded on her lap, seemingly making no answer. It was Tom, standing behind her with a hand on her shoulder, who interrupted whatever Sir William was saying, saying instead to Master Wyck, "I agree with Sir William. Why isn't this something that can wait?"
Hugh joined Miles at the end of the table. Low-voiced he asked, "What's the trouble?"
"Master Wyck wants to talk of the will. Nobody else does."
"Maybe it's better to have it over with?" Hugh said.
"Or better to put off knowing the worst until later," Miles returned. There was laughter under his low-kept voice. Miles too often found laughter where no one else did, was sometimes reckless with it, and had brought Sir Ralph's wrath down on himself more than once that way. These past few days Hugh had caught glints of it behind Miles' few words and long silences that warned his outward seemliness was very thin and barely holding; but this wasn't the time to give way and Hugh punched him just hard enough in the small of his back to remind him and said, keeping his voice low, "Easy enough for you, anyway. You already know the worst for you."
"True," Miles returned. "One badly neglected and diminished manor, complete with Sir Ralph's curse because he couldn't find a way to keep it from me."
And Sir Ralph would have kept it from him if he could. About that, Sir Ralph had always been very clear. All else that he held was his to dispose of as he chose and he had more than once goaded Tom and Hugh with, "I can leave you no more than the clothes you stand in. You cross me once too often and that's all I will leave you. I swear it." But the manor of Goscote in Leicestershire had come to him by right of blood and was entailed by law to pass to the eldest male heir of the blood, meaning Miles, only son of his loathed eldest son. Bitterly grudging that, Sir Ralph had taken pains over the years to take as much from the manor as he could, do as little for it as possible, and make certain Miles knew it. "Still," said Miles cheerfully now, "better a broken manor without Sir Ralph than Paradise with him. Not that any place with Sir Ralph would be Paradise."
But the rest of them—except Lady Anneys with the dower land of her marriage agreement—were not assured of anything. Despite all his talking and threats, Sir Ralph had never told them for certain what was in his will; for all any of them knew, he might have left everything to the Church or a cousin they had never heard of or "a long-discarded mistress," Miles had once speculated, "who's become a nun and will pray forever for his soul."
"I doubt it," Tom had said. "If he was going to do a thing like that, he'd tell us, to watch us writhe."
With that Hugh fully agreed, so it was not to avoid something he feared to hear that Tom was delaying as he turned from Sir William and Master Wyck and said, "Hugh, help me here. Mother doesn't need more today. She—"
Lady Anneys straightened, lifted her head, and quietly, firmly interrupted him. "I've said that I'm ready." She looked at Master Wyck, waiting in front of her with papers in his hand. "If you think this time is good, then let's be done with it. Go on."
"Lady Anneys," Sir William said kindly, "this may not be wise."
Lady Anneys looked past Master Wyck and Tom to Hugh and Miles. "Hugh. Miles. What do you say?"
Hugh hesitated. It was Miles who answered, "If you say now, then now it should be."
Lady Anneys returned her quiet gaze to Master Wyck. "Now, sir, if you please."
He made her a small bow and said to the rest of them, "If you would care to be seated, gentlemen?"
Miles promptly hitched a hip onto the edge of the table. Hugh and Sir William sat on the benches. Tom hesitated, looked at his mother, who understood what he was silently asking and nodded at Sir Ralph's chair. "Now, for you, too," she said with a smile. And he sat down in it.
"Well done," Miles said, lightly mocking. "You fit."
Tom shifted a little uneasily, settling himself more firmly, and turned his heed to Master Wyck.
The attorney had stayed standing, his papers at the ready. With them settled, he briefly bowed his head to them all and said, "What I have here is Sir Ralph's will, with his sign and seal upon it, witnessed by my clerk and Master Carrow. I think you are acquainted with Master Carrow, my lady?"
"The saddler in Banbury. Yes."
"Another copy, likewise signed, sealed, and witnessed, is in my keeping in Banbury. The provisions of the will are much as you probably expect. If there's no objection, I will summarize, rather than read them out at length?" He paused and, when no one objected, went on, "Of course to Master Miles Woderove goes the manor of Goscote in Leicestershire, as entailed."
"And be damned to Sir Ralph," Miles muttered, so low only Hugh heard him. Miles' hands, clenched one around the other, eased. He had been waiting, Hugh realized, for one final bitterness from his grandfather.
"This manor of Woodrim," Master Wyck went on, "goes to Master Thomas Woderove with all appurtenances and rights and so forth." He made a general gesture with one hand. "You are already well acquainted with what those are, I'm sure, Master Woderove. Also all other of his properties except as are otherwise given elsewhere in this will. Master Hugh Woderove is to have the hounds, all that goes with them, and his horse and his father's recommendation as a worthy and skilled huntsman and master of hounds, should he need or choose to seek other hire than with his brother." Master Wyck paused and looked at Hugh. "I have that recommendation in my possession, written out in his own hand, signed and sealed, against such time as you may desire it."
Hugh made an acknowledging nod, caught between surprised pleasure and complete relief. Until now he had not dared let himself think about what might happen with him now Sir Ralph was dead. Here and the hounds were all he knew and if he had lost them . . .
"You are likewise to have such property as Sir Ralph held in Banbury and Northampton," Master Wyck went on. "Also ten marks in coin or else property to that amount upon such time as you marry, so long as you marry with Lady Anneys' approval and before you are thirty years of age."
Miles insufficiently smothered a laugh. Hugh slapped the back of a hand against his leg and muttered, "Shut up."
Tom was trying but failing to hold in a grin. Their mother was a little frowning but whether with displeasure or disquiet Hugh could not tell.
Master Wyck continued, "Lady Elyn, having already been provided with her dowry upon her marriage, has no further provision made for her. As is usual and as I think you knew?" he inquired of Lady Anneys and Sir William together. They both nodded that they did. Unless a daughter were also an heiress, her inheritance was usually considered complete upon her marriage. "For his other daughters, Sir Ralph has provided as much for their marriages as was given with Lady Elyn, so long as they marry with your approval, Lady Anneys, though if Ursula chooses to become a nun she shall have five marks more than otherwise." Master Wyck cleared his throat. "There are a variety of other provisions made, mostly concerning Masses for his soul . . ."
Hugh kicked Miles' foot and Miles choked his laughter into a smothered cough.
"I'll leave those to Master Woderove"—he nodded respectfully to Tom—"and the executors to read in detail. It is the matter of the executors, however, that I should like to directly address."
Tom leaned forward. "Is this where the bastard twists us over?"
"Tom," Lady Anneys said.
"Three executors are appointed," Master Wyck said. "Lady Anneys. Sir William Trensal. Master Hugh Woderove."
Tom looked at Hugh, his surprise matching Hugh's own, and Hugh let his discomfort and discomfiture show with a frown. He had neither expected nor wanted that duty. Tom shook his head, telling him that he didn't care, and said to Master Wyck, "Sir William is to be main executor, I suppose?"
Master Wyck hesitated before admitting, somewhat uncomfortably, "In truth . . . no."
"No?" Again Tom's surprise matched Hugh's. "Then who is?"
Master Wyck bowed his head to their mother. "Lady Anneys."
She drew in her breath and said, sharp with protest, "He would never have given me that!"
"He did, my lady," Master Wyck assured her. He cleared his throat. "With conditions, however."
"Ah. Conditions," Miles said. " _This_ is where the bastard twists us over."
**Chapter 4**
They are not so bad as that," Master Wyck said stiffly. "There are, however, certain provisions that I thought best to make known as soon as possible, particularly concerning you, my lady."
"Please, then," Lady Anneys said. Her face was calm, her voice was even, but her hands were clasped tightly together in her lap. "Continue."
Master Wyck cleared his throat again. "It might be best if I read them out to you as they stand, so there can be no mistaking."
"We're in for it," Miles said, for only Hugh to hear.
Hugh jerked a fist sideways against his thigh to shut him up.
"What Sir Ralph willed is this," Master Wyck said."' To my wife Lady Anneys Woderove, besides her dower lands I leave ten marks yearly, such sum to be paid from such lands as our son Thomas inherits in his name and from such heirs as follow him, for the term of her life or until she fails in such provision as here follows, namely that she live chastely, virtuously, and unmarried. Likewise, she is to have full say and rule concerning our unmarried children's marriages as given above but her rights therein to be utterly lost should she marry or prove, by the determination of her fellow executors, to have been unchaste or lived unvirtuously. Should such happen, the right and control of such of our children as are yet unmarried and the rights and profits of their marriages shall fall to my other executors, with Sir William Trensal to become chief executor with final say in all matters and the ten marks yearly that were hers to go to him in her stead. Likewise, should any child go against her wishes in the matter of their marrying or against Sir William Trensal's wishes in like matter, should she fail in her duty as above stated and her rights fall to him, their inheritance is to be utterly and finally lost, to pass to their heir as hereafter given in this will.'"
Master Wyck ceased to read, looked up from his paper, and said, "There is a clause then detailing which of his children inherits from whom in the event of one or another's death. Master Hugh is to inherit should Master Thomas die before him, but should Master Hugh die first and without heirs of his body, his inherited properties are to revert to Master Thomas. One daughter's dowry is to be increased by the other daughter's should one or the other of them die before marriage and all to be shared equally between their brothers should both daughters die unmarried."
He finished. Around him the silence stretched out, becoming too long before Miles said very quietly, "What it comes to, then, is that Lady Anneys has full and final decision over not only Lucy and Ursula's marriages but Tom's and Hugh's."
"Yes." Master Wyck's face was blank and his voice level.
"She chooses or agrees to whom Tom and Hugh and Lucy and Ursula marry," Miles went on, "or they lose every part of their inheritance. Even Tom."
"Yes," Master Wyck agreed again.
"Unless _she_ marries or lives unchaste or . . . what was the other word?"
"Unvirtuously," Master Wyck supplied.
"Unvirtuously," Miles said. "If she does, then Sir William is to become chief executor in her place."
"Yes."
"Meaning," said Miles, "that Sir Ralph has bound at least her and Tom almost as completely as when he lived. Neither of them has any freedom outside the narrow limits he's made for them."
"I would hardly say the limits are so narrow as all that," Master Wyck said stiffly.
"Only because you don't have to live inside them!" Miles snapped.
"Miles," Lady Anneys said quietly. But Tom, having sorted everything into place, burst out, "He's right, though! The thing is written to keep a stranglehold on both of us. Even dead, he won't let go!"
"That is commonly the way with wills," Master Wyck pointed out. "The testator determines what his heirs—"
"Most testators don't try to keep their dead hands clenched around their heirs' throats," Miles snarled. "Do they?"
"No. No, not usually to the extent herein. No. But that does not render it less legal . . ."
"What it comes down to," said Tom, "is that either Mother lives exactly the way Father has willed her to or else she gets nothing and has no say in anything."
"One can hardly say she gets nothing. Her dower land is hardly nothing," Master Wyck protested.
"It's nothing compared to ten marks yearly. Come to that, how the hell am I going to find that much money out of what's left to me without beggaring myself anyway? And who's he to say whether she marries again or not now she's finally quit of him? How—"
"Tom," Lady Anneys said gently. "I have no need of those ten marks."
"My lady," Master Wyck said, "you can hardly refuse—"
"I can 'loan' them back to my son before he's given them to me," Lady Annneys said, unexpectedly firm. "There's nothing in the will against that. As for marrying or not marrying, my living chastely or unchastely, I have no intention toward either marriage or unchastity so there is no problem there either." She looked, smiling, at Hugh and reached to touch Tom's hand gripping the arm of his chair; he turned his hand over to hold hers, and she went on, "Nor do I think either of my sons is such a fool that I'm likely to refuse who they choose to marry." She moved her smile to Miles. The angers that had been writhing up around them all were suddenly, simply gone. Even Miles' and even before she said, "The most I might ask is that they help their nephew, whose inheritance has been so badly handled by their father. Is there anything else from my husband's will that we should know?"
Master Wyck fumbled, "No. There's nothing else, I think. The minor bequests. The provision for Masses. Those are, of course, the executors' concern . . ."
Lady Anneys rose to her feet. "We'll see to them, surely." She looked toward the window, where the sunlight had begun to fade, the sun gone behind the forest. "You and Sir William have a ways to ride. You'll wish to be going?"
It was a gracious dismissal and Master Wyck took it graciously, though as everyone stood up, Miles muttered in Hugh's ear, "Probably glad to escape so easily," before offering aloud to fetch Elyn and Philippa.
"If you would," Lady Anneys said.
While Miles went parlorward, she moved with Sir William and Master Wyck toward the hall's outer door, making light talk and no haste, with Tom and Hugh following behind them. In the yard, while they waited for Elyn and Philippa and for their horses to be brought, Sir William took Lady Anneys by the hand and promised, "I'll come back in a few days, my lady. We can go over matters then, to see where we stand and all when you've had time to rest."
She thanked him gravely, thanked Master Wyck, prompted Tom to do the same, kissed both Elyn and Philippa when they came out, and stood waving to them all as they rode away, with Tom on her right and Hugh and Miles on her left, each with a hand raised in vague farewell. But when the riders were well gone, Lady Anneys' hand dropped to her side, her shoulders slumped, and her weariness was suddenly, nakedly plain. Tom put an arm around her, saying, "It's over with, Mother. It's done. We can all have rest now."
She leaned against him, her eyes shut, her head sideways against his shoulder. "Rest," she agreed on a long sigh. "Yes. Let's all do with that."
What surprised Hugh in the days that followed was how easily they learned to live without Sir Ralph. A few times in the first week after the funeral Lucy and Ursula went together to the churchyard to pray at his grave. No one else did and Lucy soon stopped and Ursula, so far as Hugh knew, went only once more after that and then bothered no more either.
The household remained, as always, Lady Anneys' concern, with Lucy and Ursula learning beside her as she not only gave orders to the servants but worked at such things as pastry-making in the kitchen, weeding among the cabbages in the greens garden, and sometimes scrubbing clothes in the laundry because, as she often said to Lucy's protests, "There's no better way to know a thing than by learning to do it. Even when you have servants to do it for you, you have to know it yourself to understand whether they do it well or ill, and always there are things better done yourself than by anyone else."
Tom went on much as he had been, ordering and overseeing things about the manor, but all went more evenly and easily without Sir Ralph there to make trouble where none had to be. Tom could even be heard whistling as he went striding from one task to another, and in the evenings he often told about his day in ways that made them all laugh aloud with him. Laughter, Hugh realized, had been scant here while Sir Ralph lived.
For Hugh there were still the hounds. If they missed Sir Ralph, they did not show it. Even Bevis had given up his restlessness and unexpectedly attached himself to Miles, following him everywhere—not entirely to Miles' pleasure—except up the steep stairs to the bedchamber. Besides that, nothing much was changed except the hounds were now all Hugh's own, any choices about them all his to make and no one else's. Not that he was making many decisions yet, willing to let things go on as they presently were. One of the smooth-coated coursing hounds, Baude, was in whelp from a breeding with Makarie and that was as much of the future as Hugh was ready to look at. For now it was enough to ride out every day, not to hunt but simply with the hounds loping beside his horse to keep them fit. His lovely hounds. _His._
And then there was Miles. Without his anger at Sir Ralph, he sometimes seemed almost uncertain what to feel, as if thrown off balance by the void; but he joined in the laughter when it came and seemed, like Tom, to go easier about his work. It was maybe awareness that he'd be losing his work soon that made it harder for him than the rest of them to find his balance. The vague plan was that he would stay at Woodrim until the escheat of Sir Ralph's lands was done—the determining by a royal officer of what the deceased had lawfully held, to whom it lawfully went, and what fees could be had for the king from it all. Once that was settled, hopefully by Michaelmas at September's end, he would be away to his own manor and what they would do for a woodward here was undecided yet. In the meanwhile he went on as usual, often away to the woods—but never by himself these days, always with a servant or at least the wolfhound Bevis because the day after Sir Ralph's funeral Lady Anneys had asked it of him. "Please," she had said. "For safety's sake, don't go alone."
That was the nearest anyone spoke of Sir Ralph's death. Or of Sir Ralph at all. Except that Lady Anneys continued to dress in widow-black, he might have died long ago and in an ordinary way, instead of less than a month ago and murdered, with his murderer yet unfound. Hugh wished he could leave off thinking about that day as completely as everyone had left off talking about it, but all too often his thoughts circled back to it, trying to see something, anything, that might have told how it was going to end. Something . . .
But there was nothing, even now. Philippa and her uncle had sung together. Sir Ralph and Sir William had sat up from their dozes and some chance remark by Sir William about someone they knew lately buying a hawk had set Sir Ralph to one of his favorite rages against hawking—how he would never take it up, not to save himself, no, by God's teeth he wouldn't. All that fiddling with hoods and jesses and staying up nights on end and the rot about who can have what kind. "Gerfalcons for a king, peregrines for earls, merlins for ladies, hobys for boys. God's teeth, give me hounds and a long chase with a death at the end of it and venison on the table or a good hare pie. If I want a damned duck for my supper, I'll send to the poultry yard for one."
The talk had gone off then to the morning's hare-hunt and somehow come around to Sir Ralph saying suddenly at Tom, "They're pushing their plough-land out again, those peasants of yours. I swear I've lost an acre of pasture."
From where he still sat beside Philippa, Tom had answered even-voiced, apparently not in the humour for quarreling, "I'll see to it."
"You damnably well will. I don't keep this manor for the peasants' pleasure. You're not to let them graze that pasture again this summer either. Cattle-cropped grass is no good for hare-hunting and you know it."
"They have to graze somewhere."
"They want to keep too many cattle. That's where the trouble lies."
"They—"
"Don't quarrel with me, boy! This is my manor. I'll do as I please about it. If any of your peasants don't like it, let him pay to go free and get out of here and be damned to him."
"It's said Clement is going to do that come Michaelmas quarter day," Tom said bitterly.
"He's one of the ones who's been nothing but trouble at almost every manor court, isn't he? Good riddance to him."
"He's one of the best men in the village," Tom had said hotly. "It's shame to lose him."
"It's not shame to be quit of his whining over 'his rights' any longer. It's bad enough I have to listen to you whine for him."
"Then don't!" Tom had sprung to his feet. "But don't _you_ whine at me when the place goes to ruin, because when it does, it will be your doing, not mine!"
"Like hell it will!"
But Tom, red-faced and hands in fists, was striding out of the clearing toward the greenway and neither looked back nor answered. Sir Ralph spat into the grass after him and then, as if satisfied by a good day's work, lay back onto the cushions, smiling. The wary silence that always followed one of his angers ended with Lady Anneys gently pointing out a flaw in Lucy's embroidery. Elyn said something about how well her own was coming on and Miles moved from the spring's edge to sit where Tom had been, beside Philippa, asking her and Master Selenger, "Sing something else?"
Master Selenger obligingly started the lilting, "The Lady Fortune is friend and foe . . ." As he reached the second line, "The poor she makes rich, the rich poor also . . ." Philippa began it over again with, "The Lady Fortune is friend and foe . . ." and as Master Selenger reached, "Turning woe to good and good to woe," Miles joined in with, "The Lady Fortune is friend and foe . . ." so that the song turned and turned around on itself as, following behind him, Master Selenger started over again.
Not about to spoil their singing by adding his own, Hugh rose and went to where Degory and the hounds were sprawled asleep in the shade beyond the spring. He stirred Degory awake with a friendly foot in the ribs and they settled together to check the hounds' paws for cuts or thorns. That Miles could sing—and sing well—for some reason always surprised Hugh, probably because Miles so rarely sang at all, but he and Philippa and Master Selenger were happily winding and warring their way through the Lady Fortune song, repeating its one verse faster and faster to see who would lose their way first, while Sir William and Elyn clapped to the beat and Lady Anneys propped up Lucy, who had fallen over with laughter.
Never given to singing or to noticeable pleasure in listening to it, Sir Ralph stood up and came over to join Hugh presently crouched beside young Skyre, seeing to a slight scrape on her ear. Despite she was nearly full-grown, she was still as scatter-brained as a puppy, with more eagerness than good sense when it came to the chase and at least twice this morning had been knocked over by Somer for being too bold among her elders and betters. But to Sir Ralph's "How is she?," Hugh answered, "Taken no hurt that I can find besides this scratch and it's not much."
"Let me see."
Hugh had shifted a little aside without getting up, and Sir Ralph had squatted down on his heels to take hold of the young hound's head in his usual rough way, holding her hard by the muzzle while pulling her ear up for a better look.
Sir William strolled toward them, asking, "Trouble?"
"Nothing much," Sir Ralph had said, had let her go and stood up, turning away from her.
It was just then that something among the trees must have caught Skyre's eye—a squirrel maybe, or the bright flit of a bird among the bushes. No one ever knew what. All that Hugh—crouched beside her and rummaging in a leather bag for an ointment for the scratch—saw was her head snap up, suddenly alert. Knowing what a fool she could be, he had dropped the bag and grabbed for her collar but too late; she was away in a single long bound and gone among the trees and he was left sprawled stomach-down on the grass while above him, Sir Ralph roared out, "Skyre!" Snarling, "Get up, you fool," he kicked at Hugh's hip, grabbed up a leash lying there, and whipped it across Degory's bare legs with, "Idiot! Get after her!" Swore, "Damned idiot!" at Hugh just scrambling up, hit him across the back with the leash for good measure, and went furiously away into the woods himself, slashing the leash at the underbrush as he went.
Within the hour he was dead.
When the body was found, they had made the hue and cry for his murderer. Law required that and fear made certain of it. The search had spread through and beyond the woods. They had tried to find a track to set the dogs on but maybe they had trammeled too much in the first horror of finding the body or there was too much blood; even Somer, best of the lymers, failed to take up a scent. Nor did Hugh with his huntsman's skills find any track to follow nor had anyone at all been seen. They had been left with nothing more to do but carry the body home and send a man to bring the crowner.
It sometimes took days for a crowner to come but Master Hampden had ridden in with his men late the next day, while Hugh was gone to fetch Ursula. He had viewed the body and where it had been found, asked questions, but received few answers because no one had many to give. By the time Hugh had returned from St. Frideswide's, he was holding his inquest, where it was officially found that Sir Ralph's death was indeed murder by person or persons unknown. "And that," Master Hampden had apologized afterward to Lady Anneys, "is the best I can presently give you."
He had ridden away before the funeral, with promise that a search would be made and questions asked about likely strangers seen anywhere around there, adding a warning to keep watch themselves for anyone and anything—and as easily as that it had all been settled, tidied away into the crowner's records as tidily as Sir Ralph into his grave. Hugh wished his thoughts could be as tidily done with and put away; but aside from them—and time was dulling their edge, he found—things on the whole were very good. The summer was coming on to Lammastide with promise of a fine harvest if the weather held, and Tom had asked him what the chances were of having venison for a start-of-harvest feast he was minded to give the villagers.
Hugh had warned, "You do this, you risk making a new custom they'll want every year," and Tom had answered, "Father made enough bad customs here over the years that a good new one will likely get us more than it loses."
Therefore, tomorrow Hugh would ride out with some of the hounds to find where best to hunt a roedeer stag in a day or two; and though neither he nor Tom had said it, they both knew the fact that the hunt would be the first one since Sir Ralph's death made it all the sweeter.
But this afternoon Hugh had spent helping Degory clean out the kennel and kennel yard and spread clean straw, and he was satisfyingly tired and hardly thinking of anything at all as he bent over the washbasin set on the bench outside the hall door to scrub his face and hands before going in to supper. The late afternoon sun was warm on his back through his shirt, and when he dried his face and hands, the linen towel smelled of the rosemary bush over which it must have been draped after laundering. Inside the hall there was the thud of tabletops going onto trestles as the servants set up for supper and his stomach growled with timely hunger; but the soft thud of a horse's hoofs behind him turned him around to see Gib of the stable leading a saddled and bridled gray horse toward him across the yard.
Frowning with puzzlement rather than displeasure, Hugh said, "That's Master Selenger's horse, isn't it?" Knowing it was.
"Aye," Gib answered. "The man is back again. I make that three days running he's been here."
"It is," Hugh agreed.
Sir William, a few days after his promised visit to see what help or comfort he might give Lady Anneys, had sent Master Selenger to ask if Tom needed help with anything and promise that he had only to ask if he did. Tom had thanked Master Selenger but said, "It's what I've been doing for years here. The only thing that's changed is that it's all easier done without Father to tell me what's wrong with everything I do."
Master Selenger had laughed at that, said he was likewise charged with asking after Lady Anneys, and had ended by sitting with her in her garden, talking for a somewhat while. When he came back yesterday, he had brought Elyn with him and not seen Tom at all but kept company with Lady Anneys, Elyn, Lucy, and Ursula in the garden for much of the afternoon.
And here he was back again. Without Elyn. And not to see Tom, who had gone past the kennel two hours ago on his way to the east field and not yet come back.
Hugh, frowning, turned back toward the hall and immediately smoothed the frown away to greet Master Selenger coming out.
"Hugh," Master Selenger returned cheerfully. "Good day. I hear there's to be a hunt."
"We mean so," Hugh said. "Would you and Sir William be minded to join us for it when the time comes, do you think?"
"Surely," Master Selenger answered and they talked hounds a little before he made his farewell, thanked Gib, and gave him a small coin for waiting with his horse.
Watching him ride away, Gib gave the coin a toss and said, "He's a gentleman, is that one."
Hugh agreed to that, but while Gib tucked the coin into his belt pouch and headed back for the stable, he stood watching Master Selenger out of sight. A man much about Lady Anneys' own age. Well-featured, well-kept, pleasant-mannered.
All things Sir Ralph had not been or bothered to be.
Hugh went in search of his mother and found her in her garden, alone, standing at the gate toward the cart-track, looking outward across the field where the last of the hay, dried and carefully gathered into haycocks, was waiting to be stacked or else carted away to byres and the stable. In the westering sunlight they looked like heaps of gold, and in their way they were—food through the winter for horses and cattle. Lady Anneys turned as Hugh neared her and said, smiling, "I was thinking I might get a rosebush next year. When I was a girl, a neighbor had one in her garden. The flowers were more red than anything I've ever seen and smelled so beautifully. I've always wanted one of my own."
Hugh had not known that. He had never heard her want anything at all, he realized; and wondered, with a twitch of what felt like guilt for never having wondered it before, what else she had wanted and never had. Her silence about anything she felt or wanted had been a way of hiding from Sir Ralph, he suddenly thought. They had all hidden from him in whatever ways they could. Tom had used his anger, Miles his mockery, Hugh the talk of hounds and hunting. Lady Anneys had had her silence.
But in keeping Sir Ralph shut out, they had kept shut away from each other, too. For safety's sake you left as few gaps in your wall as possible. Even banded together the way he had been with Miles and Tom against Sir Ralph, Hugh knew how much he had never said. And Lady Anneys had had no one at all. No friends because around Sir Ralph friends had been impossible to have. Not her children. She could give them her love and what comfort there was in that but not her protection and assuredly not her thoughts.
But none of that was anything he could say to her and he said instead, "I'll find you a rosebush, come the spring. I'll ride to Northampton, Oxford, or even London to do it, if I have to."
Lady Anneys smiled at him and said, "That would be lovely." But not as if she believed it would truly happen. Which goaded Hugh to ask, a little more abruptly than he might have, "Where are Lucy and Ursula?" Had she been here alone with Master Selenger?
"I sent them in with my sewing when Master Selenger left. I was ready to be alone awhile."
Hugh stepped back, ready to leave her, but she held out her hand and said, "But I'd welcome your company. You're not a chattering young girl."
Hugh returned to her side. She tucked her hand into the crook of his arm and they stood together in companionable silence, Hugh looking out at the hayfield, Lady Anneys toying with the purple flowers of whatever it was growing tall beside the gate, until in what he thought might be a safe while, Hugh asked easily, as if merely making talk, "Was Master Selenger good company this afternoon?"
"Very good," Lady Anneys said.
Hugh waited but she said nothing more and in another while he said, "I wish nothing would change from how it is now."
Lady Anneys let that lie for a few moments before she said, "That would be good. But Tom will likely marry soon."
"Philippa?" Hugh asked, despite he knew the answer.
Lady Anneys a little nodded. "And Miles will go away to his manor before then. It's time and past for him to take up his own life."
"This is his life," Hugh said and could not keep an edge from his voice. "Here. With us."
Lady Anneys slightly shook her head against that. "Miles hasn't had a life here. He's had Hell. He needs to be free of here. He needs the chance to heal as best he can from everything Sir Ralph did to ruin him."
"Sir Ralph is gone," Hugh said stubbornly.
"In body," Lady Anneys answered.
And though his soul was surely gone to Hell, he lingered in other ways, Hugh bitterly, silently admitted.
"And you," his mother went on. "You're free to go, too, if you want."
"There's nowhere else I want to go." Why should there be, when everything he wanted was here? "You, too," he said. "You're free, too. To stay or go."
"I'll likely go," she said. She must have felt him tense under her hand because she squeezed his arm and added mildly, "Once Tom is married, Philippa should be mistress here without a mother-in-law at her back, watching her every move. I have my dower lands to go to and I will." Her smile deepened. "I can find you a wife and husbands for Lucy and Ursula from there as well as from here, probably. Unless you want to find your own?"
Hugh made a sound that admitted to nothing.
Lady Anneys laughed at him, squeezed his arm again, and let him go. "There's no hurry, though. And after all, Ursula may choose the nunnery."
"Do you think she will?"
"I don't know."
Hugh tried to think of Ursula grown up and shut away into a nunnery but couldn't. Not that she'd be any more lost to them in the nunnery than married, he supposed. And a nunnery might be easier to visit than a brother-in-law, he half-jestingly supposed to himself, ready to let go of thought about what might come and be simply at peace in the summer-warm quiet, waiting to be called to supper.
But quietly, hardly louder than the bees humming in the beebalm in the garden bed behind her, Lady Anneys said, "I think, when you return Ursula to St. Frideswide's, I'll go with her and stay, too, for a time."
Startled, Hugh demanded, "Why?" More roughly than he might have if she had not taken him so much by surprise.
For a long moment she did not answer; then said only, still quietly, "It would be best, I think," in a way that somehow stopped him asking more.
**Chapter 5**
Although dawn's cobweb-gray shadows were barely gone from the cloister walk, the day was already warm and promised to be warmer and Frevisse made no more haste than the other nuns as they left the cool inside of the church after Mass to go the short way along the roofed walk to their morning chapter meeting.
St. Frideswide's was neither a large nor wealthy nunnery. It maintained itself but barely more and the room used in the mornings for the daily chapter meeting, where a chapter of St. Benedict's Rule was read and matters of business and discipline were discussed, was a plain place, like nearly everywhere in the nunnery, with plastered but unpainted walls, a chair for Domina Elisabeth, stools for her nuns, a small wooden worktable, and nothing more. In wet or cold weather it served for the nuns' evening hour of recreation before Compline's prayers and bed, and in winter it was their warming room, having the nunnery's only fireplace save for those in the kitchen and the prioress' parlor.
Presently, though, the hour of recreation was a long summer's day away and there was most definitely no need for warming. Instead, the door stood open and someone had already lowered the shutter from the window, letting in the soft-scented morning air and a long-slanted shaft of richly golden light from the newly risen sun. Nuns whose joint stools were in its way shifted aside and turned their backs to it with a scrape of wooden legs on stone, except Sister Thomasine went to stand directly in its brightness, her eyes shut, her face held up to the light. Sister Thomasine had always lived her nun's life more intently than most did. Given her choice, she would have been in the church praying on her knees at the altar more hours of the day and night than not. There was even sometimes whispered hope among some of the nuns that she might prove to be a saint, and Frevisse—who only slowly over the years had come to accept her as other than merely annoying—granted to herself that for Sister Thomasine the touch of the sunlight was probably like the touch of God's hand in blessing on her.
But then it _was_ God's blessing, Frevisse thought. All of life was God's blessing, forget it though mankind might and ill-use it as mankind surely did. Sister Thomasine's skill—or gift—was that she did not forget but lived her life in certainty of the blessing.
It made her very hard to endure sometimes.
"Sister Thomasine, sit, please," Domina Elisabeth said. Already seated herself in the tall-backed chair that had served all of St. Frideswide's prioresses through the hundred years since the priory was founded, she did not wait to be obeyed but leaned forward to say something to Dame Claire, the priory's infirmarian, about an ache she had in her back. Sister Thomasine, with the same quietness she had given to the sunlight, sat down on the remaining stool, clasped her hands on her lap, and bowed her head to pray through the wait for Father Henry to put off his vestments and join them. Around her, the other nuns went on in steady talk. The rule of silence—that there be only necessary words within the cloister save for each evening's recreation—had slackened in the years since Frevisse had entered St. Frideswide's. She missed the quiet it had enforced but saw no sign that anyone else did. Dame Emma was explaining to Sister Margrett the value of cutting the kitchen garden's green onions fine when for a salad while Sister Amicia tried to convince Dame Juliana there was no need to weed any herb bed today and Dame Perpetua and Sister Johane discussed some copying work they meant to begin.
Content to keep her own silence, Frevisse followed Sister Thomasine into prayer for the little while until Father Henry hurried in, rumpled and flushed with heat and hurry, his fair hair in unruly curls around his tonsure. In his time as the priory's priest he had grown from young manhood into middle age and a certain stoutness of girth that came with the aging of a burly body rather than from sloth or self-indulgence. He never slacked his priestly duties to the souls in his keeping but he was not a deep-minded man; Frevisse had never found any spiritual challenge in him, only the challenge of putting up with the unfailingly simple goodness he brought to everything he did, until finally experience had taught her how deeply difficult "simple" goodness could be.
She rose to her feet with the other nuns and bowed her head willingly to receive his blessing for the day and found herself smiling to remember how she had struggled against that lesson. Humility, she too well knew, was a virtue to which she was coming only very slowly. Her smile, kept to herself by bowed head and the fall of her veil to either side of her face, went wry as she considered how much easier everything would have been if she could have started out wise and been done with it, instead of having to learn by effort and errors how far she still had to grow.
He finished the blessing.
Domina Elisabeth said, _"Dominus vobiscum."_ The Lord be with you.
They answered, _"Et cum spiritu tuo."_ And with your spirit.
They sat again, the opening prayer was made, and Father Henry read the day's chapter from St. Benedict's Rule, first in Latin, then in English, followed it with a short homily easily listened to—or not, in Frevisse's case, despite her best intent. Then he blessed them and left and from there the chapter meeting went its usual way, with such complaints as any nun deemed necessary and the officers' reports and confession of faults and giving of warnings or penance as Domina Elisabeth saw fit. Sister Amicia, presently Cellarer and therefore bound to worry over food, said _someone_ had eaten so carelessly at dinner yesterday that they had wastefully left bread crumbs on the refectory floor. Domina Elisabeth gave warning that _no one_ should eat so carelessly again. Sister Thomasine as Sacrist murmured that the silver polish was running low; Domina Elisabeth gave leave for Dame Claire to make more. Sister Margrett confessed to nodding to sleep during Lauds last night; Domina Elisabeth bade her say twenty-five _mea culpa_ s on her knees in front of the altar as soon as chapter was done.
When there seemed no other business to report or deal with, Domina Elisabeth looked around at them all and asked, "What of Lady Anneys and Ursula then? They've been with us a week and, so far as I know, have given no trouble. Is that so?"
Everyone looked at everyone else and there was a general shaking of heads that, no, neither Ursula nor her mother had been any trouble. "In truth," Dame Perpetua said, "Lady Anneys has eased my work. She gives Ursula some of her lessons and has helped with the mending." She frowned a little. "Although Ursula's sewing has not improved."
"She keeps to herself," Dame Emma said. "Lady Anneys, I mean. She comes to the Offices, of course—all but Matins and Lauds and Compline, of course, and that's understandable—and she brings Ursula with her, which is good, it spares one of the servants the task. But she doesn't talk. I've tried with her more than once but she 'saves her breath to cool her fingers,' as the saying goes. I don't think I've had more than ten words with her at a time . . ."
And if anyone could get away from Dame Emma with less than ten words, they had accomplished something indeed, Frevisse thought.
Domina Elisabeth raised a hand, stopping Dame Emma's present outpouring, and smiled on them all. "I shall take it, then, that all is well there. There's nothing else? Then it's time to tell you that because today is St. Swithin's holy day and because we well deserve it, too, we will have holiday this afternoon. Not merely holiday from duties either. I've provided for something altogether different for us."
Sister Margrett forgot herself so far as to clap her hands and exclaim, "Oh! What, my lady?" with such delight that rather than rebuke her excess, Domina Elisabeth smiled and said, "You'll see when the time comes," but that was all she would say.
Morning tasks were not so well attended to as they might have been and at each Office of prayer—Tierce, Sext, Nones—only Domina Elisabeth's sternest looks stopped the whispers running among Dame Emma, Sister Amicia, and Sister Margrett before the Office could begin, and when finally at their midday dinner's end Domina Elisabeth bade them gather in the cloister walk, there was an unseemly hurry of scraping benches and fluster of skirts. Dame Emma's stiffening joints kept her behind the younger nuns' rush out the refectory door, but even among the older nuns who chose to put on a front of more dignity, no one lingered. Most days in the nunnery were much like other days. The most constant change was in the Offices themselves as their prayers circled through the seasons of the Church—Whitsuntide just past, then the summer and autumn holy days, on to Advent and Christmastide, Lent and Lady Day and Easter, and around to Whitsun again. The promise of something other than the ordinary was welcomed by nearly everyone, save maybe Sister Thomasine, who had to be almost shooed ahead by Sister Johane to have her out the door quickly enough.
They found Lady Anneys and Ursula waiting in the cloister walk, Ursula bouncing a little on her toes with impatient delight. Frevisse had expected a solemn little girl to return from her father's funeral but she had not; nor had Lady Anneys shown any signs of deep grief, only a grave willingness to keep to her own and Ursula's company. Today, though, they plainly both knew something of what Domina Elisabeth purposed because they were dressed for some kind of work, their gowns plain, Lady Anneys with simply a veil pinned over her hair, and Ursula's long hair fastened up around her head instead of hanging down her back. But whatever Domina Elisabeth had in mind for them she did not yet say, merely nodded to Lady Anneys to walk beside her and, taking Ursula's hand, led the rest of them along the cloister walk and through the slype, the narrow passage leading out of the cloister toward the nuns' high-walled garden. Coming out at its far end, she turned not toward the garden's gate but leftward along the garden wall to the usually locked back-gate into the orchard. Enclosed by a steep earthen bank, the orchard was nearly as shut away from the world as the cloister, and sometimes the nuns were allowed to have their recreation among the apple, pear, and cherry trees and the peaceful unmarked graves of former nuns under the long grass; but today Domina Elisabeth led them through the fruit-burdened trees to the always-locked gate in the short length of board-made fence closing the gap between the church's north wall and the orchard's earthen bank. There, as Domina Elisabeth brought out a key, even Dame Emma's chatter stopped. Whatever else of nunnery life had eased under Domina Elisabeth's rule, she still held her nuns to strict enclosure. To go outside the cloister walls was a rare adventure for most of them, and in silence they watched her unlock the gate. Only when she started to open it did Dame Perpetua say, faint-voiced, "We're going out, my lady?"
"We're going out," Domina Elisabeth said and set the gate wide open.
Sister Thomasine started to drift backward and away. The times Sister Thomasine had been outside the nunnery since she had taken her vows could probably be counted on less than one hand, and given her own choice, she would never go at all; but Domina Elisabeth pointed at her and said firmly, "This includes you, Sister Thomasine."
Ursula slipped away from her mother and around Dame Juliana to take hold of Sister Thomasine's sleeve, looking up at her and saying with earnest assurance, "You can walk with me."
One way and another, Frevisse had learned that Sister Thomasine did not lack courage to go out, merely inclination, but to Ursula it must have seemed like fear and her offer was a kindness that Sister Thomasine solemnly accepted by taking hold of her hand and following with the rest, the more eager nuns crowding to follow Domina Elisabeth and Lady Anneys through the gate into the board-fenced alley, the back way for the going and coming of carts and workers between the priory's foreyard, with its byres and barns and all the business needed to sustain the nunnery's life of prayer, and the nearer fields outside the nunnery's walls. Domina Elisabeth went right, away from the foreyard and toward altogether outside, and the other nuns' laughter and talk began to rise with excitation. Even Thomasine, drawn on by Ursula, was not last out the gate. Frevisse and Dame Claire were.
But they caught up to the others at the alley's outer end where almost everyone's eagerness faltered and they slowed and bunched together, some of them even stopping, discomfited after months of the closeness of cloister walls by the sudden distance of low-grown green fields of beans and peas stretching away to far-off hedgerows, with more sky all at once than could ever be seen above cloister roof or garden walls. But their uncertainty was only momentary. As Domina Elisabeth and Lady Anneys went on unbothered and Ursula pulled Sister Thomasine forward, everyone else's pause turned to a rush to follow them along the cart-track running there, Sister Margrett asking delightedly where they were going, Dame Emma and Sister Amicia making guesses, and Domina Elisabeth smilingly refusing any answer.
It was Ursula who could not bear her own excitement. "Fishing!" she exclaimed. "We're going fishing!"
"Fishing?" Sister Johane exclaimed with almost disbelief, and Domina Elisabeth said, laughing, "Yes. Fishing."
To meet the nunnery's constant need for fish for fast days, feast days, and every day, two square ponds had been made beyond near fields, with a stream diverted to feed them and alder planted around their banks for shade. Because their upkeep and expenses were matters discussed and dealt with in chapter meetings, all the nuns knew of them in detail, whether they had ever seen them or not, and because fish-ponds were part of almost every manor and therefore part of most of the nuns' lives before they entered St. Frideswide's, they knew about fishing, too. More, perhaps, than Domina Elisabeth did, Frevisse thought, because early afternoon under a high summer sun was hardly the best time for catching fish.
But actually catching fish was hardly the point, she soon decided. Village boys were waiting with rods, lines, hooks, and bait in the shade among the alder trees along the first pond, and the first squealing and protests over worms on hooks from some of the nuns and laughter from most of the others led on to elbowing and nudging each other toward the water, until finally shoes and short hosen came off and skirts were hitched above ankles and soon thereafter the inevitable happened and Sister Amicia was standing in the water, grimacing at the mud between her toes and laughing at the water's coolness. Ursula, Sister Johane, Sister Margrett, and finally Dame Emma followed her, while those determined to fish went farther along the bank where their chances were hardly bettered by their flailing rods and jerking lines.
Faced with all of that, the village boys' first stiff respect crumbled, and when Sister Johane and Sister Margrett began a splashing battle against each other and anyone else in reach, Colyn, the reeve's younger son, gave up to laughter, rolling on the grass and holding his sides. So they splashed him, too.
Then Lady Anneys drew her skirts up through her belt and waded in, too, only barely avoiding her daughter's fate when Ursula, leaning to splash water at Sister Johane, overbalanced and sat down with a great splash. Lady Anneys, backing away from her, stumbled and grabbed hold of one of the boys to keep from falling, both of them laughing as Ursula rose dripping and muddied to the waist, laughing, too.
Not long after that, Sister Thomasine, finally persuaded to cast a line since she would not wade, somehow and against all likelihood hooked a fish and even—with help from the boys—landed it, a large carp. Domina Elisabeth, paying one of the boys a farthing to run it to the nunnery kitchen, said,"'Tis not our Lord's miracle of the loaves and fishes but assuredly a miracle nonetheless." Which brought on more laughter.
The alders' afternoon shadows were stretching long across the water and weariness was overtaking merriment when Domina Elisabeth called an end. While the boys set to gathering the fishing gear for going home, the nuns sat on the grassy bank to put on stockings and shoes again before beginning a slow walk back toward the nunnery, a very bedraggled Ursula holding to her mother's hand and no one's tired legs able to make haste despite the time for Vespers was nearing. The church's gray-lead roof showed dully above the orchard's trees, reminder that the cloister was waiting to close them in again, and as they reached the outer gateway, Frevisse saw Sister Margrett look back across the green fields toward the deeper green of distant trees under the richly blue, high-arching sky, her young face showing a mixture of longing and puzzlement that Frevisse understood. They had taken pleasure in that world today but now were going to shut themselves away from it again, away from all its possibilities—away from the places they would never have chance to go and people they would never have chance to meet.
That was a thing a girl or woman understood when she chose to become a nun, and by the time she came to take her final vows, she understood it even better; but no one ever fully understood it until she had lived in it, year around into year after year, knowing it was for all the rest of her life. For a very few, like Sister Thomasine with her desire for nothing except God, the life absorbed them utterly. Others were content enough, accepting where they were and willing to be satisfied with it. Most, alas, were never so easily one way or the other and sometimes old longings would return, whether wanted or not. Frevisse was, mostly, past them herself, but she understood Sister Margrett's momentary longing. And then Lady Anneys, probably to help Ursula's dragging feet along, began to sing, swinging her daughter's hand, "Hand and hand we shall us take, And joy and bliss we shall us make . . ." And in ones and twos the nuns joined in until they were all singing, some more tunefully than others but with no one's feet quite so heavy as they had been and Sister Margrett as happily as everyone else, all sign of other longing gone.
They were to the orchard gate when a boy came running along the alleyway from the priory's foreyard and Dame Juliana said, "It's Sim from the guesthall. Sim, is there trouble?"
Flushed with hurry and the importance of bearing a message, he said, "There's a man been waiting this while at the guesthall, my lady. My lady," he added with a bow to Domina Elisabeth. "Ela said to tell you he's someone come to see Lady Anneys. Ela said . . ."
Lady Anneys took a quick step forward, anxiously asking, "One of my sons?"
"Mistress Ela said to say he's John Selenger?" the boy said questioningly.
The mingled hope and worry in Lady Anneys' face changed to something less easily read. "Yes," she agreed. "John Selenger. He's our neighbor's steward."
"You'll see him?" Domina Elisabeth asked.
"I'd best." Though by her look she would rather not. Then she added with deliberately lightness, "He's probably brought some word from home. That's all."
"Bid Ela see him to the guest parlor," Domina Elisabeth told Sim, adding as he made to dart away, "There's no need for haste. We'll be making none."
Nor did they. Indeed, Domina Elisabeth, having seen everyone through the gate and on their way across the orchard, followed only slowly, somewhat behind them all when Lady Anneys gave Ursula's hand to Dame Perpetua and fell back to her side to say, "By your leave, my lady?" in a quick, low voice that Frevisse heard only because she was side by side with Dame Claire a little ahead of them.
"Yes?" Domina Elisabeth said.
"May one of your nuns be with me in the parlor while I talk with Master Selenger?"
Domina Elisabeth was quiet a moment, then began, "If you're afraid of this man . . ."
"No," Lady Anneys said quickly. "Not that. I'm not afraid of him. Only I'd rather not . . . see him alone."
Frevisse had only time to begin to wonder why not when Domina Elisabeth said, "Dame Frevisse, come here, please you."
Thus, simply because she had been near when Domina Elisabeth had need of someone, Frevisse went with Lady Anneys—once they were inside again and had washed hands and faces and straightened veils—around the cloister walk with Lady Anneys to the small, bare room near the outer door where nuns met any visitors they might have; and while they went, Lady Anneys said, "This man. This Master Selenger. I've nothing against him. But . . . since my husband's death he's . . . shown interest in me. I don't want his interest. That's . . . why I want someone with me."
"Of course," Frevisse murmured, readily able to suppose that in all likelihood this Master Selenger was too old or too young or too ill-favored or too obviously intent on Lady Anneys' dower properties, and that Lady Anneys had yet to find a way to turn him away without giving offense; but when she followed Lady Anneys into the parlor, the man standing there in the middle of the room was neither aged nor ill-favored and his deep bow to them was both graceful and gracious.
That left only the likelihood that he was ambitious rather than amorous, Frevisse thought dryly, as Lady Anneys answered his questioning look with, "It's hardly suitable I talk alone with a man inside the cloister, Master Selenger."
Since Lady Anneys was under no vows, that was not true, nor had there been any reason except her own choice for not seeing him in the guesthall, and by his slight frown Frevisse guessed Master Selenger knew as much. But then, very likely, Lady Anneys had known he knew it and this was simply her quiet way of saying she did not want to see him alone while leaving him no choice but to accept that or else to argue with her. He chose to accept it, slightly bowing his head to Frevisse, who bowed hers in return while Lady Anneys asked, "Is everyone well at home? You've brought no ill news?"
"Everyone's well. There's no ill news, my lady. Will you sit?"
The room had only a small square table, a bench, and three stools. Master Selenger gestured toward the bench. Lady Anneys refused with a curt shake of her head and insisted, "Everything's well?"
Her curtness was just barely short of unmannerly but Master Selenger kept his smile and assured her, "Very well, save that Lady Elyn and Lucy don't go on as well together as they might."
Lady Anneys gave a tight laugh. "They haven't gone on well together since Lucy was born. I only insisted she stay with Elyn because she'd trouble her brothers even worse. Has there been any word from the crowner?"
Her change from Elyn and Lucy left Master Selenger behind her. "The crowner?" he repeated blankly, then caught up with, "No, no word. Nothing has been found out."
"Nor anyone?" Lady Anneys said.
"Nor anyone."
"Nor any word when the escheater will come?"
"No word of that either. There won't be any trouble over it, whenever it's held, though."
"I'm not supposing there will be. Haven't you brought me any messages from anyone?"
Master Selenger paused at his answer before finally saying, "I didn't tell anyone I was coming here."
Lady Anneys stared at him in surprise and coldly, and said nothing.
Master Selenger ended the uncomfortable silence with, "You could ask me how I come to be here."
"I could," Lady Anneys answered, cold as her look. "I presumed you had business this way."
Master Selenger hesitated, his eyes flickering toward Frevisse still standing in the doorway. She had deliberately taken her "I'm not here" stand beside the doorway, her hands tucked out of sight into her opposite sleeves and her head bowed—though not so far that she could not see, with a little upward look through her lashes, everything happening there. She saw Master Selenger make up his mind and return his gaze to Lady Anneys to say, "My only business this way was with you."
His directness gave Lady Anneys pause. She might even have wavered, between one heartbeat and another, in her sharpness at him; but if she did, it was for no longer than that and she said, still sharply, "My thanks for letting me know that all's well and may you fare well on your ride home. Dame Frevisse and I have to ready for Vespers now. God go with you."
She was drawing back from him, turning toward the door as she spoke. Master Selenger put out a hand to stop her, protesting, "My lady . . ."
Lady Anneys kept going, repeating more firmly, without looking back, "God go with you, Master Selenger."
As taken aback by the suddenness as Master Selenger was, Frevisse stepped aside, out of her way. Master Selenger moved as if to follow her out the door but with a quick gathering of her wits Frevisse stepped back into his way and said, "I'll see you out, sir, if you will."
He stopped, looking past her with confusion and an edge of understandable anger, but Lady Anneys was already out of sight, heading along the cloister walk toward the room she shared with Ursula; and he gathered himself and answered Frevisse with at least outward good grace over whatever else he was thinking, "If you would be so good, my lady, yes, I'll go now."
Frevisse led him in silence to the door to the guesthall yard. Only as she opened it for him did she ask, very mildly, "You'll stay the night and leave in the morning?"
Gone somewhere in his own thoughts, her question seemed to take him by surprise and he answered with something of Lady Anneys' sharpness. "What? Yes." He recovered and said more evenly, "Yes. I suppose so. In the morning. Thank you, my lady," and went out.
And yet the next morning at Tierce, when he might have been supposed to be well on his way homeward, he was still there.
Frevisse, intent on readying her mind for the Office as the nuns settled into their choir stalls, half to either side, facing each other across the choir, would not have known it except Sister Johane whispered rather too loudly to Sister Amicia, "He's there again. I told you he would be. Look. He's watching her."
A slight clearing of Domina Elisabeth's throat stopped anything Sister Amicia might have answered, but Frevisse slightly turned her already bowed head and slid her eyes sideways to look past the edge of her veil down the length of the church. The nuns had mentioned among themselves at chapter meeting that neither Lady Anneys nor Ursula had been at Prime or Mass this morning. Frevisse, after supposing to herself that Lady Anneys had probably chosen not to chance meeting Master Selenger again before he left, had forgotten it. Now her careful look told her both Lady Anneys and Ursula were there in the nave, standing not far beyond the choir with the few of the nunnery servants that came now and again to Offices. And that Master Selenger was standing not far behind them.
Domina Elisabeth set the Office firmly on its way by saying in her clear, determined voice, _"Pater noster, qui es in caelis . . ."_ —Our father, you who are in heaven . . .—and her nuns obediently followed her into the prayer.
Domina Elisabeth took a workmanlike approach to the Offices and every other service due to God and his saints, believing—as nearly as Frevisse could tell—that God and his saints would in return see to the priory's well-being—fair pay for fair work, as it were—and indeed St. Frideswide's was prospering compared to what it had been; but there were still times when Frevisse greatly missed Domina Edith, prioress when she first entered St. Frideswide's. For Domina Edith, the Offices' beauty and passion had been a way to deepen her own and her nuns' devotion, a way to bring them nearer to God. There had been none of the "I give you thus and you give me so in return" that seemed to be Domina Elisabeth's way. Instead Domina Edith had tried for as full a giving of herself as was possible, in the hope of growing to be worthy of God's great love.
Much of what Frevisse understood of nunhood had come from her, but Domina Edith had been dead these twelve years, and at least it could be said that under Domina Elisabeth's careful governance no one was allowed to scant their prayers. The chant rose, _"Nunc, Sancte, nobis, Spiritus, Unum Patri cum Filio . . ."_ —Now, Holy Spirit, one with the Father, with the Son . . .—and Frevisse gave her full heed to that and for the while forgot all else, until at the ending _Amen_ Domina Elisabeth promptly closed her breviary and rose to her feet.
Her nuns did likewise and, two by two, made procession from the choir and out the church's side door into the cloister walk in a busy bustle. There Domina Elisabeth briefly blessed them and left them to scatter to whatever tasks they might do in the while before Sext. The day was bright and dry and warm again, and minded to work at her copying, Frevisse went the little way along the walk to the desk she used among the five set endwise there to the church's wall for the sake of the best light. The half-filled page she had been working on before Tierce was waiting for her, but rather than sitting down, she stood looking back to the church door, waiting to see Lady Anneys come out.
The servants came. And Ursula, led by Malde from the kitchen, whose grip on her hand was just short of an open struggle as Ursula twisted to pull free.
"Malde," Frevisse said. "What's the matter?"
Not letting go of Ursula, Malde stopped and started to say something. But Ursula said fiercely past her, still pulling to be loose, "He gave her money and told her to take me away so he could talk to Mother. Mother doesn't want to talk to him. Let me _go._ "
She wrenched hard and it might have turned to tussling but Frevisse said, moving toward them, "I'll see to it, Ursula. Go on with Malde. Malde, I'll talk with you later about this."
Malde looked suddenly uncertain she had been paid enough for that, but Frevisse went past her and Ursula to the door still standing open into the church. Yesterday Lady Anneys had made it plain to Master Selenger she had no wish to talk with him anymore. If she had changed her mind today, then Frevisse would discreetly withdraw but . . .
She was not even a single pace inside the door before she knew Lady Anneys had not changed her mind. Standing much where she had been during Tierce, she was turning away from whatever Master Selenger had said to keep her there, saying at him angrily, "There's no use to this. Leave me alone."
He reached out and caught her by the arm. "Lady Anneys, listen . . ."
"Sir?" Frevisse said, bland as if blind to what was happening. "My lady?"
Master Selenger let go his hold so suddenly that Lady Anneys, still pulling back from him, stumbled. He caught her arm again to steady her, but as fierce as Ursula had been against Malde, she jerked free again, turned her back on him, and said with open anger to Frevisse, "Master Selenger was just making his good-byes. He's leaving now."
"I'm certain he is," Frevisse agreed, staring coldly at him as she moved forward to Lady Anneys' side. "The Lord's blessing on your going, sir."
She made that more of a command than a blessing, and Master Selenger's face was flamingly red and stiff with things unsaid as he bowed rigidly first to her, then to Lady Anneys' back still turned firmly to him. He opened his mouth as if to speak, decided against it, swung around, and left, going at a swift walk down the nave and out the church's west door to the guesthall yard without looking back.
Frevisse and Lady Anneys stayed where they were. Only when the heavy door had thudded shut behind him and the church was safely empty did Lady Anneys turn to Frevisse. A tear was sliding down one cheek and her voice shook a little as she said, "This mustn't be talked about. I pray you, Dame Frevisse, say nothing about it to anyone. If anyone— _anyone_ —asks if anything passed between Master Selenger and me, tell them what you saw. That he wanted to keep me here against my will and I was trying to leave. That yesterday I encouraged him to nothing."
"I will," Frevisse promised, since all that was nothing more than the truth.
Lady Anneys wiped at the tear, seeming angry it was there; shut her eyes and pressed her fingers to them to stop more from coming.
Carefully, Frevisse said, "Cry if need be. There's no one here to mind."
Lady Anneys dropped her hands and opened her dry eyes, refusing both the offer and more tears. " _I_ would mind."
"It's sometimes best to cry and be done with it."
"If once I started," Lady Anneys said, "I might never be done. There are too many in me that I never shed. My late, damned husband would have treasured every one he ever forced from me, so even now I won't give them to him."
Frevisse had long since judged that Lady Anneys was hardly in mourning for her husband, but the hatred naked in Lady Anneys' words and voice surprised her enough she showed her surprise, and Lady Anneys said bitterly, "Oh, yes, I hate him. I didn't dare while he was alive, but now that he's dead and I'm free of him, I hate him very, very much. That's one of the reasons I needed to be here, away from everything. I need this time to pray and purge myself of him."
She had said "one of the reasons," Frevisse noted but did not ask the others, only offered, "Do you want I should leave to your prayers now?"
Lady Anneys looked uneasily the way Master Selenger had gone.
"I'll see that he's left or that it's understood he's to go before I do anything else," Frevisse said. Under St. Benedict's Rule, the priory was required to receive such guests as God might send them. That did not mean they had to put up with those who spoiled their own welcome by making trouble.
"Yes," Lady Anneys said. "Yes. Thank you. If I know he's gone . . . yes, I 'd like to pray for a while. But . . ." She hesitated, then asked, "No questions? No wanting to know anything else?"
"I may want to know," Frevisse said in all honesty, "but I don't think you want to tell me."
That surprised a half-unwilling laugh from Lady Anneys. "I don't, no. Thank you."
"But if sometime you do, I'll listen. And still not ask questions if you don't want them."
Lady Anneys regarded her in searching silence for a moment, then slightly bowed her head in thanks again. "If the time comes, I'll remember." She drew back a step, looking toward the altar. "For now, though, I think prayer will suffice. By your leave, my lady."
**Chapter 6**
The bright, late summer days had turned from warm to hot, and Tom, Hugh, and Miles were lingering in the shadowy hall after midday dinner, none of them in haste to be about their afternoon work. Hugh had spent the early morning riding out to exercise the dogs while the day was still somewhat cool and then he and Degory had rather uselessly worked with Skyre, who was seeming less and less likely to ever be a usable scent-hound after all. In the alarm after Sir Ralph was found, no one had remembered her save Degory, and when they carried Sir Ralph's body back to the manor, he had stayed behind, searched for her, found her, "Cowering under a bush close by where _he_ was," he told Hugh when he went to ask his help. "She maybe saw it or something. It's like she's witless. I've brought her in. She won't stop shaking." Nor did she until Hugh had wrapped her in a blanket, tightly swaddling her almost the way a nurse would do a howling baby to quiet it. But Skyre had not howled, only trembled, and even now still trembled and cringed at any sudden noise or movement. Still, she had been too promising a hound to let go without he tried to save her, but it had made for a discouraging morning and because he had nothing particularly planned for this afternoon, he was simply sitting on the dais step with Baude between his knees, stroking a brush down her back and sides not so much because she needed the grooming as for the pleasure it gave them both.
Tom, with seemingly no more ambition toward the rest of the day than Hugh had, was leaned back in his chair behind the table, legs stretched out in front of him, his eyes shut although—if anyone had asked him—he would have said he was not sleeping, only not ready to move. He had spent the morning walking the fields with Lucas, the reeve, overseeing the start of the barley harvest, and once he bestirred himself he'd be out again all the afternoon.
Miles had his head down on his crossed arms on the table, and under the table Bevis was stretched out with his chin resting on Miles' foot. When Hugh had laughingly goaded him at first about the hound's unwanted devotion, Miles had grumbled, "Can't you kennel him with the rest of the hounds?"
"He's too used to being with Sir Ralph. I doubt he'd do anything but make trouble if put with the other hounds."
"Instead, he's making trouble for _me,_ " Miles had muttered. But lately, except when he remembered to complain, he had begun to seem as content in Bevis' company as the hound was in his; he had not, Hugh noted, moved his foot since Bevis' chin came to rest on it.
Miles and Bevis had been out about woodward duties from not much after dawn today, away to Skippitt Coppice to see where best to start the autumn cutting. Most times lately when Miles went out, George from the village went with him and, "I think he might do well enough in my place the while until you find someone else," Miles had told Tom over dinner. "He knows what he's seeing when he looks at it."
That would be something to tell Mother when she came home, Hugh thought. What they would do for a woodward was one of the things she had talked of when he was taking her and Ursula to St. Frideswide's. He had taken the chance to ask her then when she thought to come home and been relieved when she said, "Before Miles leaves us, surely."
He had been half-afraid she was thinking of nunhood for herself. It would be the most straightforward way to fulfill Sir Ralph's order that she live a virtuous, unmarried life, and almost Hugh could want it to be that simple for her. Almost but not quite.
Nor did he want it for Ursula, he had admitted to himself. She had been quiet out of the ordinary on the ride to St. Frideswide's. Once, to Hugh's pleasure, she had started to whistle back to a linnet in a hedgerow as they rode past, but Lady Anneys had reminded her that whistling was unladylike and Ursula had fallen quiet again, until—outside the cloister door in St. Frideswide's guesthall courtyard when Hugh had lifted her up for a better embrace good-bye—she had clung to him and whispered in his ear, "I wish I was going home with you."
He had whispered back, "I wish you were, too," but had set her down and let the waiting nun take her hand as he turned to embrace his mother, too.
She had said nothing, only when she drew back from him had looked deeply into his eyes as if searching for something before briefly touching his cheek and turning away. He had stood there in the courtyard, watching as she and Ursula went inside, until the door was closed behind them, shutting them in and him out, and then he had ridden away, as oddly bereft as if that door would never open to let either of them out again.
To the good, he was not bereft by Lucy being gone to stay with Elyn. That was merely pleasant; but along with missing Lady Anneys and Ursula, he missed Tom sharing the loft with him and Miles. Before she had left, Lady Anneys had told Tom he should move into the lord of the manor's bedchamber above the parlor when she was gone. "It's yours," she had said. "You may as well grow used to it."
"But where will you sleep?" Tom had protested.
"Not there again, if I can help it," she had said. "In the girls' chamber with Lucy, probably."
But it was odd to have Tom gone from their own chamber. "It's not that we miss your snoring," Miles had told Tom the first morning after he had shifted. "It's just hard getting used to the quiet." They had been standing in the hall, breaking their morning fast, and Tom had thrown a heavy-crusted chunk of bread at him. Miles had ducked out of the way and Bevis had seized the crust for his own.
Now Miles gave a wide yawn and sat up.
"Don't," said Tom without opening his eyes. "If you start moving, I'll have to."
"All right," Miles said and put his head down again.
Hugh, still stroking the brush down Baude's back, said, "I'm thinking to hare-hunt tomorrow morning. Helinor," the manor's cook, "says she wouldn't mind some fresh meat for the pot. Either of you interested in coming?"
Tom stretched and said, "Yes. Good. I'll come."
Miles mumbled into the crook of his arm, "Not me."
They sat in silence a little longer, until Miles said, still without lifting his head, "Did you know there's a copy of Sir William's will in Sir Ralph's strong-chest, Tom?"
"Um-hum," Tom murmured. His eyes were closed again. "There's Sir William's will, and deeds, and rent statements, and papers that look left over from Sir Ralph's lawyer-days. I'm still reading through them. It's killing me."
Idly, Hugh asked, "What were you doing in the strong-chest, Miles?" The iron-bound, padlocked box was kept under Sir Ralph's . . . under Tom's bed, to hold what ready money there was and whatever deeds, charters, and suchlike as were worth safe-guarding.
"I had him fetch me some pence yesterday," Tom answered, rather than Miles. "To pay old Wat for turfing that place where the stream bank had started to slide. He made good work of it. Wat, I mean."
"Sir William's will was lying right on top of everything else," Miles said, "and I was curious."
"You were snooping," Tom said lazily and not as if he minded.
Miles sat up, stretched his back and crooked his neck as if they were stiff, and asked, "What did you think of the will?"
"I thought it read pretty much like Father's." Tom sounded not in the least interested. "He'll have to make a new one, now Sir Ralph is dead and can't be his executor. But if Philippa and I are married soon enough, he won't have to bother with deciding who's to oversee her marriage in his place."
"You're thinking you'll marry soon?" Hugh asked.
Tom rolled his shoulders in a lazy shrug. "Might as well, since we're going to do it sooner or later anyway and none of us are much interested in a year's mourning for Sir Ralph anyway, are we?"
"If Sir Ralph had a copy of Sir William's will in his safe-keeping," Miles said, "don't you suppose Sir William has a copy of Sir Ralph's?"
"I suppose." Tom did not sound like he was supposing it very hard.
"Then he must have known everything that was in it long before Master Wyck told us. Just as Sir Ralph must have known what was in his will, too," Miles persisted.
Still lazily, Tom granted again, "I suppose."
Hugh looked up, frowning warily. Miles was going somewhere with this.
Miles reached over and shook Tom's knee. "But remember how Sir William tried to put off Master Wyck telling us about it?"
"He thought it was too soon to be burdening Mother with it," Tom said. "That's all."
"Was it?" Miles demanded. "What about Selenger?"
Tom finally drew himself up straight in his chair. "What about him?"
"Mind how he was here so often before she went away? Always coming to see her? Keeping her company for half an afternoon at a time?"
"Yes," Tom granted.
"I half-thought . . ." Hugh started, thought better of saying it but knew there was no going back, and finished slowly, "I've half-thought she went with Ursula to be away from him."
Tom was deeply frowning now. "She never said that, did she?"
"No. It was just something I . . . felt."
Carefully, as if he had considered the words for quite a while, Miles said, "He's been to the nunnery to see her."
"To see Mother? Don't be witless," Tom protested. "Why would he go without telling us he was? Or else, when he came back, tell us how she is?"
Miles held silent, waiting for them to figure it out for themselves. Tom took hardly longer than Hugh did to see where Miles was going; but while Hugh said nothing, Tom protested more strongly, "Miles, don't try that one. He's not angling for her, if that's what you're going around the bush to say."
"Then why did he go?" Miles said.
"I don't know." Tom was impatient about it. "A letter from Elyn. Complaints from Lucy. Sir William wanting to ask her something. I don't know."
"Why not ask if we had any word to send her?" Miles demanded. "That would be courtesy."
Tom made an impatient gesture at him. "Give it over, Miles. He's not fool enough to be after her if that's what you're on about. She won't marry again. Why would she? She's too old to want another husband, anyway." He stood up and stretched his arms wide to the sides. "It's too hot for this much thinking. I'm away to see how far they're likely to get with Pollard Field today. Time you shifted yourself, too."
Miles waved a lazy hand at him, saying nothing. Nor did Hugh. Tom left, and when he was gone, Miles stood up, still saying nothing, and left, too, with Bevis at his heels. Hugh stayed a little longer where he was, stroking Baude with his hand now, considering everything Miles had said and some of the things he had not, before finally rising to his feet. Baude, who was beginning to feel the weight of her whelping, rose more grudgingly, and he told her, "You stay here." He pointed at the hall's cool stone floor. "Stay."
With a whoof that Hugh took for gratitude, Baude lay down again, and he went in search of Miles.
He found him in the stable, sitting on a manger's edge feeding carrot bits to Lanval, the squat roan gelding with carthorse in its ancestry that Sir Ralph had said was good enough for him. Bevis, lying flat on his side under the manger, opened an eye at Hugh, decided he was no problem, and shut it again. Hugh leaned a shoulder against the post outside the stall to show he meant to stay awhile and asked, "Where did you get the carrots? And why's Lanval in stable rather than pasture?"
"I'm riding out again this afternoon. There's a stretch of timber over Ashstock way that's ready for thinning, I think, but it's been a while since I've had a look at it. As for the carrots, they're from the kitchen garden, of course."
Raiding the kitchen garden had been a favored pastime with them and Tom when they were small. Helinor would have let them have what they wanted for the asking but the skulking and "stealing" had been better sport, and Helinor had obliged by calling dire threats after them whenever she saw them at it. She never complained of them to Lady Anneys, though, and always had some treat for them when they ventured into the kitchen itself.
Hugh grinned. "Nobody saw you?"
"Well, yes, but only Alson." The kitchen maid. "I gave her a kiss to keep her quiet."
"You would," said Hugh. For years he and Miles and Tom had jested at each other's alleged passion for Alson, who was somewhat old enough to be their grandmother. But this time Miles let the jest lie and Hugh did not keep it up. Lanval crunched carrots and swept his tail at flies whose lazy buzzing was the only other sound in the stable. The other horses were in pasture or at work, and Gib, like everyone else on the manor who could be spared, was out to the barley harvest. There was no one there, no one to overhear, and in a while Hugh asked, "How did you know Master Selenger went to St. Frideswide's of late?"
Miles slowly drew his hand down Lanval's long face before saying, seemingly to the gelding's forelock, "Philippa told me."
Carefully, because so far as he knew none of them had been near Sir William's manor in a week, Hugh asked, "When did you talk with Philippa?"
Miles looked at him, saying nothing at all.
Hugh looked back at him and did not ask again but said instead, very, very quietly, "She's meant for Tom."
Miles turned away, ducking under Lanval's neck to the other side of the stall. "We know," he said.
"If you know . . ."
"We _know._ " Miles pulled his saddle from the stall wall and slung it over Lanval's back.
Hugh tried again. "Miles . . ."
Miles began fastening up the girths. "How long have you known about us?"
"I haven't known." Nor could he say how long he had been working hard not to know it. "I've only thought it . . . possible. Miles, it's no good."
Miles paused in drawing up the rear girth and looked at him across Lanval's back. "No," he agreed. "It isn't any good. Nor is it well-witted of us. And no, it isn't safe either. We know all that. And no, I'm not going to take Philippa from Tom. We're neither of us fools, Philippa and I. It will be years, if ever, before I can afford a wife, before my ruin of a manor—a manor I've never even seen, mind you—is worth anything again."
"But you and Philippa . . ."
"Know we have no hope of each other. That she's all but promised elsewhere. That I can't give her anything worth having. We sometimes see each other, yes. That's all. We see each other and take what comfort there is in that, for this little while until we won't be able to see each other anymore."
Miles' sharp bitterness warned Hugh to let that be the end of it, but more than the bitterness, Hugh heard the pain behind the words—he had always been too able to hear Miles' pain—and found himself saying against good sense, "Even if Elyn bears Sir William children, Philippa will still have a goodly dowry. Enough to make a good start toward bringing your manor back."
Miles finished with the girth, heaved a sigh heavy with patience, and hands clasped, leaned his arms across Lanval's back. "Do you really think," he said, "that Sir William will give her much of a dowry—if any at all—if she marries against his wish?"
No answer was needed to that, and Miles turned away from Hugh's silence to take down his bridle from the peg where it hung at the end of the stall. But as he turned back, Hugh asked, "About Master Selenger. You really think there's something to worry about him?"
Slipping the bit into Lanval's mouth, Miles said evenly, "I think his interest in Lady Anneys is very sudden."
"It might not be. He might have . . . been interested for a long time. He'd hardly have let it be known while Sir Ralph was alive."
"I'd be willing to have a go at believing that, save for how hard Sir William tried to delay her knowing Sir Ralph's will."
"He feared Mother was too tired for it just then. That's all."
Miles paused in buckling the bridle's throat lash to look at him. "Hugh, you can think better than that. You just don't bother."
Hugh crossed his arms and glared at him, refusing to be drawn.
That did not stop Miles, who went on, "In truth, most of the time, you'd rather not think further than your hounds if you could help it."
"I think enough to know you're trying to goad me into an argument."
Miles grinned. "See? You can think when you're forced to it."
"You want an argument so I'll be forced to think about the possibility that Sir William maybe wanted Mother not to know how the will bound her because he wanted Master Selenger to have better chance of . . . something . . . with her."
"Or ruining her," Miles said curtly. "She doesn't even have to marry again, remember. She only has to be 'unvirtuous' for her to lose her control of everything and Sir William gain it. And what's 'unvirtuous' can be almost anything, depending on who's deciding. Which means Sir William."
"And me. I'm executor with him."
"Making you an executor was a jest and you know it. Sir Ralph never expected you'd do aught but leave everything to Lady Anneys and Sir William."
Yes, Hugh knew that. Had known it from the first.
It had not bothered him until now.
But his bother was not the point at present and he said, "Miles, we don't even know that Sir William knows Master Selenger is giving her such heed, let alone that he's behind it."
"How does Selenger get away from Denhill so often and easily, if not with Sir William's leave?"
"How does Philippa get away to see you?"
"She doesn't. We meet at the far end of the orchard, beyond the hedge and very rarely. Leave that. Listen. If Selenger brings Lady Anneys to marriage or anything else or even the seeming of anything else, it's you who are going to be in as much trouble as Tom. It won't be Lady Anneys but Sir William who has the say over who you and Tom and Lucy and Ursula marry. It will be Sir William who'll have the profit from selling you all to the highest bidders, whoever they are, and never mind how any of you feel about it. If any of you refuse his choice, you lose your inheritance."
"I know all that, but—"
"Think about the possibilities. Because I'll warrant Sir William has."
"Tom is to marry Philippa . . ."
"There's no betrothal yet. Nothing that seals and sets it to happen. Let's suppose Sir William decides instead to find someone else for Tom. Someone who'll pay Sir William a goodly amount to marry a daughter to the well-propertied young man that Tom now is."
"Then Tom will have her for a wife instead of Philippa."
"And Sir William will still have Philippa to marry off profitably to someone else altogether. That's double profit for him. Well and good and all very reasonable, as such things go. But suppose whoever he finds for Tom is so ghastly that Tom can only refuse the marriage? Then everything—this manor and everything else that are Tom's—goes to you. Whose marriage Sir William likewise controls. You're now in possession of Tom's inheritance as well as your own and very well worth marrying to Philippa, who thereby gains more than she would have if Sir William had settled for merely marrying her to Tom. Or he may play the same game on you he played Tom, finds you a bride you have to refuse. Then everything goes to Lucy, and can you imagine what someone is likely to pay him to marry a son to her at that point? He can—"
"Miles!" Hugh said harshly. "Stop it!"
Bevis stood up uneasily under the manger and Lanval tossed his head. Miles caught the gelding by the bridle and stroked his face, speaking to him soothingly. Hugh, regaining control of his anger, said, trying to be reasonable, "None of that's happened yet and nothing says it's going to."
"It hasn't happened yet," Miles agreed very quietly. "And it may not. But it's possible."
"Not very," Hugh insisted.
Miles straightened and began to back Lanval from the stall.
"Wait," Hugh said. "Is George going with you or only Bevis?"
"Only Bevis. George is gone to the harvesting."
"Then I'll come with you, if you'll wait while I fetch Foix and saddle him." His tall-legged bay palfrey. Because he suddenly did not want Miles to ride out alone with himself this afternoon. Not alone with his dark thoughts anyway. "I haven't been toward Ashstock for a while either and ought to see how the hunting seems that way."
Miles did not frown against the thought, only looked momentarily surprised before—surprisingly—he smiled and said, "Hurry up, then. The day won't last forever."
Despite of that, they made no haste on their ride, let the horses set their own easy, head-bobbing walk along one of the lesser trails through the forest's green quiet, and did not talk at all themselves, the only sounds around them the soft fall of the horses' hoofs on the trail's bare earth and the softer whispering of the trees among the sunlight-flecked green shadows. And when they reached the Ashstock woods and talked a little, it was only about the five well-grown ash trees and an oak that Miles judged ready for cutting and how many trees could be planted to replace them. Hugh found out enough signs of deer to show there would likely be good hunting this way, and then in companionable silence they started homeward, still making no haste, so that by the time they rode out of the woods above the manor, the sun had slid from sight, leaving the green twilight sky feathered by a few high clouds turned gold from the vanished sun, and smoke from supper fires was rising lazily above the hall and village roofs. Home and supper, and afterward a quiet evening, and then bed. That, Hugh thought, was the way every day should end.
He remembered that day afterward, because, in its way, it was the last of the good ones.
**Chapter 7**
The morrow started as fair a summer's day as could be hoped for, warm and clear-skied and the hounds ready to run. The hare-hunt went well. The hares gave fine sport, both the ones that were caught and the ones that escaped—sometimes especially the ones that escaped. At the hunt's end Tom sent the smallest hare away to old Goditha in the village who was ailing. "It's old age and she won't get over that," he said blithely, "but meat in the cooking pot will make her daughter-in-law better resigned to having care of her."
The rest of the dead hares were handed over to Degory to take to Helinor in the kitchen. "Tell her to make as many hare pies as she can from them. Two for the hall at supper tonight, the rest for the harvest-folk at their noontide break tomorrow," he said and Degory trotted away happy at the likelihood of hare pie tonight and maybe tomorrow, too, because when he was not needed for the hounds, he had been at the harvest.
Not ready to end the morning, Hugh and Tom saw the hounds back to the kennel together and helped Degory, back from the kitchen, to settle the hounds to a well-earned, well-fed rest. They made easy talk while they did and while they headed to the hall to wash before dinner, beginning to feel their tiredness now but laughing as they went, reminding each other how one of the young hounds had leaned too far over in turning after a hare and gone down in a flail of legs and humiliation. By the time he had untangled himself and stood up, Bane had been standing with the dead hare in her mouth and a distainful look at him.
"Baseot's young," Hugh said in his defense. "He hasn't learned to match eagerness to balance yet."
"I'll warrant he remembers after today!" Tom laughed.
They came from between the stable and byre into the sideyard where the wide-trunked elm that must have been young when the manor was spread welcome shade and found Miles there, sitting on the bench beneath it, doing nothing.
"Very grand," Tom said. "Wish I had that kind of lazying time."
Standing up, ignoring the jibe, Miles said, "One of Sir William's men is here to see you. I thought you'd want to know before you saw him."
"What's he want?" Tom asked.
"To see you. That's all I know."
"And thought to warn me so I'd be ready, in case there's an ambush in it?" Tom scorned. "Give it over, Miles."
They rounded the corner of the hall into the foreyard and Sir William's man rose from the bench beside the door where the late morning shadow still lay and it was cooler than inside the hall. Someone had properly given him a cup of ale and he had probably been happy enough in his waiting, but leaving the cup on the bench, he came to meet them, bowed, and said to Tom, "Sir, Sir William asks you come to see him."
"When?" Tom asked.
"Now, please you."
"Now? For what?"
"I don't know, sir."
"Well, I'm not going now. I want to wash, sit down awhile, and have my dinner. When all that's done, then I'll go to Sir William."
He moved to go on, into the hall. The man said, "Please, sir, I think he meant you to come directly back with me and I've been waiting awhile."
Tom stopped, turned back to the man, and smiled on him, not unkindly but not yielding either. "I don't care what Sir William meant. I'm dirty and tired and starting to be hungry. When I've taken care of all that, then I'll go to Sir William. You're welcome to ride back and tell him so, or you can wait, eat with us, and ride with me."
Looking as if neither choice made him comfortable, the man bowed. "I think I'd best go back. To let him know you're coming."
"As you choose," Tom said and went past him into the hall, leaving the man to bow to his back and tossing over his shoulder as he went, "If he'd had the courtesy to say what he wanted me for, I might be more willing to oblige, but since all he did was order me to come, he can damn all wait until I'm ready."
The man was staring, startled, at Tom's back as Hugh and Miles went past him, too. Hugh could only hope the man had wit enough not to pass that along to Sir William with the rest of the message and that dinner would take the edge off Tom's sudden ill-humour. He had sounded exactly the way he had when readying to do battle with Sir Ralph, and Miles must have thought so, too, because he said, low-voiced for Tom not to hear, "You better go with him, Hugh."
"Hah," said Hugh. "You think I can stop him if he gets hot-humoured?"
"If worse comes to worse, you can throw yourself between him and Sir William."
"If you're keen on throwing, you go."
"Have you ever known me to be a soothing presence when there's trouble?"
"A point well-made," Hugh said. "I'll go."
Washed and in clean hosen and shirts, they fed well on a plain dinner of new-baked bread, fresh cheese, cold pork tarts, and garden greens—lettuce, borage, cress, parsley, fennel, and young onions mixed and dressed with vinegar and salt. Afterward, leaving Miles still comfortably at the table, Tom and Hugh grudgingly shrugged into their good doublets that were unfortunately—considering the day was now moving into its midday heat—of wool. Deep crimson for Tom and forest green for Hugh. Tom, who had begun to mellow over dinner, began to unmellow, grumbling while fastening the stiffly standing upright collar high under his chin, "I've probably already irked Sir William enough by not being there before now that I doubt making pretty for him is going to help."
Not any happier about it himself, Hugh said, "It can't hurt."
"It's hurting me. I'm roasting."
"I like the way your face matches your doublet," Miles offered.
"Remind me to wrap you in a blanket and set you by the kitchen fire when we get home," Tom returned.
They rode at an easy walk, making no haste. Sir William's manor was barely two miles away and the day too warm for hurry. Once past Woodrim's church, the road to Denhill curved between this year's fallow field sloping green up to the woodshore on one side and a grassy verge, thick hedge, and golden-standing wheatfield ready to the harvest when its turn came. Because presently the work was farther off, in Pollard Field, here was strangely unpeopled, and there was not even the usual distant calling back and forth because all the harvest-folk were still at their midday rest, stretched out in the shade of the hedge and trees along Pollard Field and the stream there. The day was drowsing. Gay blue chicory, bright yellow yarrow, and bold red poppies still bloomed in the tall summer-growing grasses along the road; and if the grasses' greens were beginning to dull with the year's turn toward autumn, the sky was clear, high-arched, and deeply blue, and contentedly Tom said, "It's going to be a good year for the harvest. Everything is the best I've ever seen it."
"There looks to be deer in plenty, too," Hugh said. "The hunting should be good all winter."
"Praise St. Eustace. Not so much salt meat." Tom had never cared much for salt-kept meat, no matter how it was cooked.
They rode beyond their own fields and for a while were quiet, easy-swaying in their saddles to their horses' even stride, until Tom said out of wherever his thoughts had been, "You don't mean to leave, do you, Hugh? I mean, you can if you want, but all in all I'd rather you stayed."
Hugh had been gazing aside at the green roll of country-side, the land sloping away just here to a long view of fields and woods familiar to him all his life. They were as part of him as his own arms or legs—or heart—and no, he did not want to go away. Not now, not ever, and he turned his head toward Tom and answered as quietly as Tom had asked it, "No. I'm not wanting to leave."
"Good." Tom suddenly held out his hand. Hugh without hesitation reached for him in return, but better than only a handshake, they gripped each other's arms near the elbow, forearm to forearm in promise that their friendship would go on. Then, just as suddenly, they both felt foolish for showing so much feeling and let each other go and Hugh said, for the sake of saying something, "Of course your wife may want me out from underfoot when the time comes."
"Philippa? Not likely. She's as used to you being underfoot as the rest of us are."
"Do you think it's about your marriage Sir William wants to see you?"
"Probably."
"Are you ready to talk about it?"
"Why not? We're getting no younger, Philippa and I. If Sir William and I can make our agreement soon enough, then the banns can be said, we can be wed by maybe Michaelmas, and I'll have someone warm in my bed come winter."
"And Philippa?" Hugh asked despite himself.
"She'll have someone warm in bed, too, you dolt."
That had not been what Hugh meant but he let it go, because after all, Tom probably had the right of it. There was small use in wondering what Philippa thought of the marriage when she was going to have no choice about it.
They were riding through Sir William's manor now, the land that would someday be Tom's if he married Philippa. And if Elyn birthed no sons, Hugh silently added. Ahead, as the road rolled over another lift of low hill, the manor hall and its surrounding buildings came into sight, settled tightly in a curve of land below a hill, gardens and orchard flanking it before the fields began, with an arm of forest stretched behind it, hedged from the rest. That made it easy for Miles to come unseen to the orchard's end to meet with Philippa, Hugh thought; and wished he had not.
"Father kept too much of Woodrim for hunting," Tom said. "Sir William has more in fields. If I can put Woodrim and Denhill together someday, it will make a manor well worth the having."
"Do you know," he added as they neared the manor yard, "I think I'd prefer to live here when the time comes that Sir William is no more. You can have Woodrim for a hunting lodge and all the hounds you can fit into it."
"I have been thinking . . ." Hugh started, but stopped, unsure how to say what had only lately begun to shape itself in his mind.
"Go on," Tom encouraged.
Hesitantly, Hugh said, "I've thought of maybe setting to breed and train hounds for more than my own use. To sell, you know. We have two good lines already. If I worked at it . . ."
He stopped because it sounded overbold when said aloud; but Tom reached over to punch him lightly in the arm, saying, "Good. Good for you. Do it. I'll breed sons and you can breed hounds. And sons of your own, too, when the time comes. Once I'm married, we'll find you a wife."
Two servants came out from the hall to meet them as they rode into the manor foreyard, one to lead them inside, the other to take their horses away to the stable. Sir William greeted them in the great hall, not with the—at best—simmering rage Sir Ralph would have had for Tom not coming immediately, but merely courtesy as he led them into the parlor. It was larger and more comfortable than anything at Woodrim. There was a fireplace, for one thing, with a long, cushioned settle in front of it, and everything—polished wooden table, short-backed chairs, close-woven reed matting—was better-made and better-kept. It was more comfortably lived in, too. At Woodrim there was a barrenness still there from all the years of no one daring to show much of themselves to Sir Ralph, hiding from him by having very little of their own for him to attack, either by word or, in his angers, by deed. Here, there were Elyn's tapestry frame beside a chair with the wall-hanging she had been working at for a year and more; an untidy sewing basket at one end of the settle that was surely Lucy's; Philippa's lute lying on the end of the seat below the window that looked out into the garden.
Besides that, there was a wooden chest standing open along one wall, scrolls and papers inside it. More writing-covered papers, an inkpot, several clean quills, and a penknife with a knotwork pattern carved around its wooden hilt were on the table where Sir William's chair waited at one side and a tall-legged joint stool at the end. Gesturing Tom to the stool and seating himself in the chair, Sir William said, continuing his jest, "It's a wife you need, Tom, to stir you up in the morning and set you going."
Left to sit where he chose, Hugh went to the window bench. A brief look into the garden showed him Elyn, Philippa, and Lucy sitting together on the turf-topped seat in the slight shade cast by a trellised vine at the garden's far end before he turned back to the room as Tom answered Sir William with an easy smile, "That's what we're to talk about, isn't it? Marriage?"
Sir William, shuffling the papers on the table in front of him, said, "That's why you're here, yes. Your marriage to Philippa."
A maidservant came in with a pitcher and three pewter goblets. Sir William bade her set them at the table's other end and waved her out. "Hugh, if you would?" he asked, and Hugh rose, went, and poured the dark wine for them and for himself while Tom said, "What sort of agreement were you and Father shaping before he died?"
"Well." Sir William settled comfortably back in his chair. "Things have changed since then. You're no longer your father's heir but lord of the manor. That puts a stronger front on things for you."
Hugh returned to the window seat with his wine and sat down. Tom had already set his goblet aside and was leaning toward Sir William, saying, "It doesn't change that you're married to a young wife who'll likely give you other heirs. For all I know, Elyn is bearing right now."
Sir William slightly flushed. "She's not."
"That's not to say she won't," Tom insisted. "If she does, there goes much if not most of Philippa's inheritance. A daughter would mean Philippa's inheritance is cut by half. A son would mean Philippa loses almost everything. Any marriage settlement we make has to allow for both those chances."
"You're the son of my best friend. He and I meant from the very first for our children to marry. It was his hope." Sir William spread his hands in a gesture of open-heartedness. "It's still mine."
"My father's main hope was to make everyone closest to him as miserable as possible," Tom said curtly. "Now that he's dead, I don't have to oblige him anymore. We know what Philippa will get if she remains your heir. The question, then, is what do you have to offer in compensation for everything she loses if you have a son to replace her?"
Sir William rapped the tabletop with his knuckles. "Tom. Let's not take a harsh line with this. We're here to talk it out . . ."
"That doesn't mean I lie down and roll over and take what I'm given. Because Sir Ralph wanted me to marry Philippa doesn't mean I _have_ to marry Philippa and no more than gamble that I'll gain more than I lose by it."
Sir William took a deep drink of his wine, maybe hoping to wash down the anger tightening his face. Hugh guessed he had not counted on argument from Tom, while Tom was suddenly feeling the freedom Sir Ralph's death had given him; and Hugh said, hoping to forestall trouble, "Wouldn't this better wait until Mother is here to have say in it?"
Sir William dismissed that with a flick of one hand. "We don't need a woman's dealing in this. Besides, she's put herself away into that nunnery."
"Not forever," Hugh said. "She—"
"She'll accept what I decide," Tom cut in impatiently.
Sir William looked back and forth between them and—to give himself somewhere else to look, Hugh guessed—restlessly picked up the small-bladed penknife and one of the untrimmed quills lying on the table and began to work a point onto it, saying while he did, "Besides, it's doubtful she'll much longer have any say at all in these matters, what with this uncommon interest she's taking in John Selenger. Not that he isn't worth her interest but—"
"It's Selenger who's shown uncommon interest in her," Hugh said. "She went into the nunnery to get away from him."
Tom gave him a surprised look. He clearly had never connected the two, despite what Miles had said yesterday.
With a slight and maybe scornful smile at one corner of his mouth, Sir William said, trimming more intently at the quill, "It's odd, then, that she sent him a letter asking for him to come see her there." He looked up, first at Hugh, now as surprised as Tom, and then at Tom. "Or didn't you know that? Sent for him and was not at all ungiving of her . . . company, shall we say?"
Tom and Hugh traded looks. Hugh shook his head, not believing it. Nor did Tom, who said angrily at Sir William, "I doubt both the letter and that she welcomed him. I think Miles is right. I think you've set him on to her."
"John needs no 'setting on' to have interest in a comely woman. And why shouldn't she have an interest in him? Their marrying would give her protection and the pleasures of a husband and—"
"And give him her property to use as he chooses," Tom snapped, "while she loses all say in my marriage and in Hugh's and the girls'. _You'd_ have say over all our marriages. You'd make them to suit yourself and take the profits of them for good measure. You knew what was in Sir Ralph's will. You wanted to keep it from us long enough for Selenger to have a clear run at her." Tom swung around to Hugh. "Miles was right!"
Red-faced with open anger, still clutching the quill in one hand, Sir William pointed at Tom with the penknife and said furiously, "Miles has nothing to do with this. Now listen, Tom . . ."
Tom stood up so sharply he knocked over the stool behind him. "If Selenger wants a wife, he can have Lucy. She's ready for marriage and it'll save the bother of searching out someone else for her. But if all he wants is to make trouble for Mother so you can have the profit of it, then be damned to you both!"
The quill bent and broke in Sir William's grasp. He drew in breath for an angry answer but Tom stepped around the corner of the table and leaned over him, one hand on the table, the other braced on the back of Sir William's chair. Thrusting his face close at Sir William, he said, "As for Philippa, the next time you want to talk marriage with me . . ."
Sir William, crowded back into his chair, swept a hand at him like warding off a fly too close to his face. Tom jerked away, enough that Sir William was able to shove himself out of his chair and swing around to put it between them, saying angrily, "Listen, you young fool . . ."
But by then Hugh was across the room and gripping Tom by the arm, pleading, "Tom. Let it go. Let's go home. You're both too angry for this. Let's go." Tom shook him off. Hugh moved in front of him, between him and Sir William, insisting, "It's wrong, Tom. Let it go for now. The day's too hot for talking. This isn't the time. Let's go home for now. We can all talk later." Spilling words the way he would have spilled water on a fire to stop it.
And Tom fell back a step and then another, threw up his hands in surrender, and said, "Yes. Fine. Good enough." Forcing himself back from his anger, he slapped the side of Hugh's shoulder. "You're right. Let's go." And before Hugh could answer that, he swung around and stalked from the room.
Still churned with the suddenness of it all, Hugh turned around to Sir William, trying desperately to think of some apology that was not betrayal of Tom, saying, "He'll settle. I'm sorry. We'll be back. Or you can come to Woodrim next time."
"Supposing there is a next time," Sir William snapped; but like Tom, he was forcing his anger back into control and said a little less tersely, "Yes. Next time I'll come to Woodrim maybe. I'm sorry, too."
He made to lay the penknife still clutched in his fist onto the table beside him, then went rigid, staring at it so that Hugh looked, too. And saw the short, sharp blade was bloodied. He and Sir William raised their eyes from it together, looking at each other with mutual unbelief before Sir William half-whispered, "I forgot I was even holding it."
"You must have . . . when you swung at him . . . you must have . . ."
"I didn't feel it. I didn't feel anything. I didn't know . . . I didn't mean . . ."
"He didn't know either." Hugh was already away from the table, leaving. "It's so sharp he must not even have felt it." And with the high, crimson collar of his doublet and no one looking for blood . . .
By the time he was out of the parlor, Hugh was running. Running, he passed through the great hall without seeing it, ran out of it into the yard where no one was, and crossed it, still at a run, into the stableyard, where a stableman leading Hugh's unsaddled and unbridled Foix away from the watering trough startled to a stop, staring.
"My brother," Hugh demanded. "Where is he?"
"Gone," the man said. "Stormed in and rode out. He's gone."
"You couldn't have saddled his horse in the time."
"I hadn't unsaddled it yet. I was tending to this one. I'd just finished wiping him down and was coming out for the other one. It was tied there." He nodded to a ring in the stable wall. "Sir William said you'd be here a longish while so I was making no hurry about it . . ."
Hugh grabbed the lead rope from him. "My bridle. Get it. Fast."
The man went—not fast enough—into the stable and came out with both saddle and bridle. Hugh grabbed the bridle from him, flung a rein around Foix's neck to hold him where he was, stripped off the halter, and bridled Foix as fast as his suddenly clumsy fingers would let him. Foix shook his head and tried to back away and Hugh cursed him. The stableman, still not fast enough, made to throw the saddle on Foix but Hugh said, "No, I haven't time," and flung himself up and astride Foix's bare back, gathered the reins, swung him around, and set him into a gallop out of the yard.
Tom couldn't be far ahead. He'd know by now he was bleeding. He'd have stopped to deal with it. Over the first rise of the road he'd surely be in sight.
He wasn't. And beyond the rise the road curved and the forest came down to its edge, cutting off longer view, and Hugh, leaned low over Foix's neck, dug his heels harder into the horse's sides to set him faster. But Tom must have ridden with an anger that more than equaled Hugh's desperation because it was beyond there, after another rise of the road, that Hugh finally saw . . .
. . . not Tom. His horse. A bay, like Hugh's. Riderless. Grazing on the grassy verge beside Woodrim's wheatfield near a spreading oak tree that in harvest served the workers for shade when they broke for their dinner or rest times. But it was barley harvest now. There was no one in sight here. Not even Tom. Only his bay horse.
Confused, Hugh drew rein, bringing Foix to a halt beside the other horse. And only then saw Tom. Lying on his stomach in the long grass at the edge of the oak's deep shade, his head pillowed on one arm as if he might be sleeping, his face turned away from Hugh, who suddenly wanted to go no nearer.
But he slid from Foix's back and went toward him, saying unsteadily, "Tom? What are you doing?"
He knelt and touched Tom's shoulder; said, "Tom?" again; and then—all unwillingly—took hold of his shoulder and started to turn him over.
And knew by the body's utter slackness, even before he saw the dulled, empty-staring eyes, that Tom was no longer there.
**Chapter 8**
In the warm, clear-skied evening the nuns were come into the walled garden for their hour of recreation before Compline and bed. The sun was not yet set, Frevisse supposed, but it was gone below the cloister's roofs, leaving the garden in gentle evening shadows, the day's warmth lingering with the mingled scents of the summer fields beyond the nunnery walls and the garden's herbs and flowers.
Most of the younger nuns and Dame Emma were clustered on the turf benches near one another, heads together in busy talk. Dame Juliana, ever in love with flowers, was drifting from bed to bed, humming happily under her breath, touching her more treasured blooms, stooping now and again to smell one or another, sometimes plucking out a daring, doomed weed.
Dame Claire as infirmarian cared more for the herbs from which she made the nunnery's medicines than for merely flowers but this evening she was simply walking slowly, alone, back and forth along the path beside their beds, a sprig of some plant between her fingers and a thoughtful look on her face. She had been in talk with Margery, the village herbwife, today, Frevisse knew, and was maybe considering something newly learned from her. The powdered cinnamon and pepper mixed with honey they had tried last year against Dame Emma's toothache had not been a success but a vervain poultice they had devised two winters ago had worked well against an ulcerating sore on a villager's leg.
Tired from a day spent mostly at her copying work, Frevisse was pacing quietly back and forth along a different path, at the garden's far end and well away from the talking, Dame Juliana, and even Dame Claire. Oddly, Sister Thomasine was there, too, pacing the same path rather than at her prayers in the church, her pace and Frevisse's so nearly the same that they passed each other at almost the same point every time, never speaking or ever looking at one another. Over the years, Frevisse had come to trust Sister Thomasine's silences. Like Frevisse, she had no need for constant talk to reassure herself that she was real. Her intense piety was more than Frevisse could match, but for Frevisse the quiet of her own thoughts was company enough for now. Some people withdrew from the world into nunnery or monastery because either they had no other choices or they could not face what choices they had. Frevisse had become a nun not so much in withdrawal from the world as in a glad going toward God.
What had surprised her was how much, even in the nunnery, a day's necessities and duties got in the way of that going. She found herself, sometimes, regretting she did not have Sister Thomasine's gift for making a prayer of almost everything she did, but at least in little whiles like this, when she was alone and silent, she could go a small way toward where she wanted to be in her mind and heart.
The evening's quiet was stirred—not disturbed—by Ursula's laughter from the orchard beyond the garden's wall. Domina Elisabeth only sometimes spent the recreation hour in her nuns' company, preferring to be in solitude for the one hour of the day when no one was likely to need her for anything; but this evening, as often of late, she was with Lady Anneys and Ursula, and whatever their pastime in the orchard was, Ursula was very happy with it. Indeed, as the days had drawn on since Master Selenger's visit, Lady Anneys seemed happier, and a few times her own laughter rose from the orchard.
But in the garden, at its other end from Frevisse, the nuns' talk stopped and heads turned at sound of someone coming at a hurry along the slype from the cloister. A servant surely, but this far toward the day's end, with all business done, they were almost surely bringing trouble, and Sister Amicia and Sister Margrett sprang up and ran to the gate to meet whoever was there. By the uneven pad of the footsteps Frevisse had guessed it was lame Ela from the guesthall and was not surprised to see her at the gateway; but she only briefly asked something of Sister Amicia and Sister Margrett and immediately disappeared again. The rap a moment later on the orchard's gate told where she had gone, and Sister Amicia turned back to the other nuns to say, disappointed, "Just old Ela wanting Domina Elisabeth. Probably somebody's arrived she ought to know about." But no one likely to divert them, her disappointment said.
Not needing to be diverted, Frevisse continued her slow pacing, content with her own thoughts about nothing in particular until the bell rang to Compline.
It was at next morning's chapter meeting they learned that, after all, it had been trouble that brought Ela into the cloister. After Father Henry had given his blessing and left them, Domina Elisabeth said with no other beginning, "There was ill news came to Lady Anneys yesterday evening. Her older son was killed two days ago."
While making the sign of the cross and murmuring a quick prayer for his soul, Frevisse tried to remember if the son she had met had been the older or younger one. Or had she ever known which he was?
"Her younger son brought word of it," Domina Elisabeth was going on. "He's here to take her and Ursula home again today."
"How was he killed?" Dame Emma asked, more eagerly than seemly.
Domina Elisabeth fixed her with a look that warned her to silence, said crisply, "That is something for later, dame," and returned to the rest of them. "They would be gone already except Lady Anneys has asked for one of us to go with her. That being the only comfort besides our prayers that we can offer her, I've agreed. Dame Frevisse, it was you she asked for. Sister Johane will go with you." Because no nun should go from the nunnery uncompanioned by another nun. "You are both excused to ready what you'll need and meet Lady Anneys in the guesthall yard in as few minutes as possible. Sister Johane, be guided by Dame Frevisse in this and on your journeying."
Sister Johane, already eagerly on her feet, bobbed a deep, quick curtsy, ready to be out the door and not minding the hard looks turned her way by Sister Amicia and Sister Margrett, who probably felt that if Domina Elisabeth wanted a younger nun to go, they might equally well—nay, better—have been chosen. Frevisse, nothing like so ready to go out into the world for whatever reason, rose more slowly, her eyes down to keep her prioress from reading her unwillingness, and curtsied as deeply.
Domina Elisabeth made the sign of the cross toward them and said, "My blessing on you both. Father Henry and I will be in the yard to see you away."
Despite bishops' best efforts that nuns be kept strictly enclosed, they were not. Now and again some family matter or other reason—even simply fishing—would take them out of their cloister. As lately as last Martinmas Sister Margrett had gone, companioned with Dame Emma, to her sister's lying-in with a fourth child, stayed for the christening because she was to be godmother, and returned to St. Frideswide's barely in time for Advent. Now, safely out of Domina Elisabeth's sight, Sister Johane all but bounced up the stairs to the dorter ahead of Frevisse. Only the sad reason for their going out probably kept her from outright singing, Frevisse suspected, while they gathered a change of clothing and what little else they would need; and she remembered to be at least outwardly subdued when she followed Frevisse out the cloister door into the guesthall's cobbled yard and with lowered eyes gave her bag to a servant to be strapped behind the saddle of one of the nunnery horses waiting for them.
Lady Anneys and Ursula were already on theirs, ready to leave, and Frevisse, giving over her bag in turn, saw with relief that the young man waiting with them was the same son who had come for Ursula those few weeks ago. The death of either of Lady Anneys' sons was a sorrow, but Frevisse knew this one a little while the dead man was no one to her, without even a face she could put to him in her prayers unless the chance came to see him unshrouded in his coffin. With no particular feelings of her own about him, no burden of personal sorrow, she would be more free to pay heed to curbing Sister Johane—should it come to that—and in giving what comfort she could to Lady Anneys and Ursula, she supposed. Just now, though, they both looked beyond comforting, and Hugh—that was his name, Frevisse remembered—looked no better. Ursula was mounted behind him, leaning against his back, the side of her face pressed to him, her arms tightly around his waist. Lady Anneys had her own horse but was close enough to her son that they were reached out to each other, holding hands as if that were their last hold on life. They had both been crying, that was plain, and Ursula still was, her eyes red and swollen, her cheeks shiny with tears. All the grief that Frevisse had not seen in Lady Anneys or her for their husband and father was terribly there now. That death had not hurt. This one did, and Frevisse foresaw that despite the golden sunshine of yet another perfect summer's day, today's riding was going to be dark with their pain.
She and Sister Johane had just swung astride their horses and were settling their skirts, neither of them interested in fashionable side-riding in box saddles, when Domina Elisabeth and Father Henry came from the cloister into the yard. The priest went to say something to Lady Anneys and the others. Domina Elisabeth came instead to Sister Johane, spoke too low to her for Frevisse to hear, then handed her a fair-sized, cloth-wrapped parcel that Sister Johane, nodding agreement to something, turned to tuck into the bag tied behind her saddle. Coming to Frevisse, Domina Elisabeth said, still in a low voice, "I've given some herbs to Sister Johane that Dame Claire thought might be useful if Lady Anneys or anyone is too uncalm to sleep or rest."
Domina Elisabeth's choice of Sister Johane to come with her suddenly made sense to Frevisse. All of the nuns helped Dame Claire, turn and turn again, at her infirmarian tasks, but of them all, Sister Johane had so far proved the most apt to the work. Very probably the comfort she could offer by way of soothing herbs would be worth as much or more than Frevisse's these following days, and Frevisse nodded to Domina Elisabeth with brisk understanding.
Domina Elisabeth gave a brisk nod in return and stepped back as Father Henry raised his hand to bless their journeying.
It proved to be a long, hard day's journeying. Frevisse remembered that when Hugh had come for Ursula, he had said that by rights it should take a day and a little more of reasonable riding to reach their manor. Today they did not ride reasonably. Rather than an easy, steady pace of no great haste, they went at trot or even canter more than half the time, easing the horses only when necessary and themselves hardly at all.
The choice did not seem to be Hugh's. More than a few times Frevisse saw him speak to Lady Anneys with concern on his face, but each time Lady Anneys shook her head against whatever he was asking her. Even when they stopped for a midday meal of sorts at some village's alehouse, Frevisse thought Lady Anneys would have stayed in the saddle to eat, except her son dismounted, lifted Ursula down, and came to help her so firmly that, although she hesitated, she did not refuse and, when she was dismounted, let him lead her to a bench in the shade by the alehouse door.
But if Lady Anneys did not want the respite, Frevisse most definitely did. She rode well but not often enough to be ready for this kind of riding and was willing to admit that her years were telling on her. It was only pride that kept her from groaning even more than Sister Johane did at the effort of pulling themselves back into their saddles when time came to ride on.
They rode through the afternoon and into the long, pale twilight, paused for a slight supper at another alehouse, then rode on into the darkening blue of evening. A rising moon gave light enough they hardly slowed their pace, but the servant who had come with Hugh rode on ahead to warn of their coming, and when they finally rode into the manor yard, there were people waiting and lighted torches flaring in the darkness. Hugh had long since moved Ursula to ride in front of him, cradled in the curve of his arm, and she must have been asleep because as the torchlight fell on her face, she startled upright with a small cry. Hugh said something to her and she answered, "Home?," then leaned against him, softly crying again.
A young man, who had gone first to Lady Anneys, turned to them and took Ursula from Hugh into his own arms. In the torchlight and Frevisse's tiredness, he looked so much like Hugh that he could have been his brother. But Lady Anneys had only two sons and one of them was dead, she thought a little confusedly. But whoever he was, Ursula clung to him as readily as she had to Hugh as he carried her away, into the house, leaving Hugh to dismount and help Lady Anneys from her saddle while two girls—as dressed in black as Lady Anneys and crying—hovered close, waiting only until Hugh stepped aside before they flung themselves at her, crying harder.
More daughters, Frevisse supposed.
Wearily, she dragged herself from her own saddle, lowering herself carefully to the ground and not letting go of the horse until she had convinced her legs that they not only had to hold her up but were going to walk, too. But now, please God, this day was nearly ended and soon there would be somewhere she could lie down and sleep.
As she followed Sister Johane following a servant toward the hall, she heard Lady Anneys say to someone in a tired and aching voice, "Where's Tom? I want to see him."
**Chapter 9**
In the morning Frevisse remembered, as she awoke, where she was and wished herself asleep again. She and Sister Johane had shared a truckle bed rolled from under Lady Anneys' own in an upper bedchamber, and although she had stayed awake and upright long enough to undress down to her undergown and fold her clothing onto a nearby stool, she had noticed nothing beyond that, simply lain down and fallen to sleep. She had awakened when Lady Anneys came in with her daughters but only enough to realize that beyond the bedchamber there was a room that must be theirs. Then she had slept again. Now it was morning, with nothing she looked forward to about the day.
Wary of her aches and stiffness, she sat up. Lady Anneys' bed had been slept in but she was not there now, and to judge by the open door and the quiet from the farther room, the girls were gone, too. That she had not heard them at all, as well as slept through her usual hours of prayer, told Frevisse how tired and deeply sleeping she must have been. Beside her, Sister Johane was still deeply sleeping and Frevisse took the chance to see better where they were. Someone had unshuttered the bedchamber's one, unglassed window before they left, letting in the gray light of an overcast day, letting her see not only the wide bed that nearly filled the room but its faded, plain green curtains and the two large, flat-topped chests set along one wall with a hunting dagger in its sheath lying on one of them. There was a small table beside the door with a pottery pitcher and basin and a white towel on it. A man's brown doublet and white shirt hung somewhat carelessly over the single wall-pole. Her own and Sister Johane's travel bags were leaning against the wall beside the truckle bed. That was all.
It was not so much a bare room, Frevisse thought, as a barren room. As if someone had been here but not lived in it. Except for the man's clothing and the dagger—the dead son's, she realized; he would have slept here as the manor's new lord, when his mother was gone—it was a room curiously empty of anyone. Even empty, a room usually carried some sense of who lived in it. This room was no one's. Admittedly the young man had maybe had too little time to make it fully his own, but even though Lady Anneys must have lived and slept here for years and very probably birthed her children here, there was nothing of her either and there should be. Embroidered cushions on the chests for softer sitting. A plant on the windowsill. Bright painted patterns on the plaster walls or on the roof beams. A woven mat on the floor. Something that said someone belonged here. But there was nothing. As if she had never been here at all.
Frevisse made to crawl out of the bed, deliberately clumsy at it so that Sister Johane awoke, mumbled, rolled over, awoke a little more, enough to open her eyes and say, pleased, "We slept in. Wonderful. If we haven't missed breakfast."
"We're late for Prime," Frevisse answered.
"This morning?" Sister Johane protested. "Now?"
"Now," Frevisse said.
Sister Johane sighed heavily but made no other protest. They dressed and took their breviaries from their bags and, kneeling on either side of the truckle bed, set to the shortened Office that was allowed to nuns when traveling. Sister Johane, despite Frevisse's attempt to hold to a reasonable pace, rushed at the prayers and psalms, shortening the Offices more, reached the end with, _"Et fidelium animae per misericordiam Dei requiescant in pace. Amen"_ —And the souls of the faithful through the mercy of God rest in peace—in a burst of speed, slapped her breviary closed, and was climbing stiffly to her feet before Frevisse had finished saying that final _Amen._ Another time Frevisse would have been irked into snapping at her for her haste—or, better, gently rebuked her—but just now the effort was too much and Frevisse let it go. Working her way to her own feet was trouble enough; but when Sister Johane headed for the stairs down to the hall while Frevisse was still slipping on her shoes, Frevisse said, "Sister," just quellingly enough that Sister Johane stopped, abashed, and waited, as was proper, for Frevisse to lead.
The overcast sky made judging the time difficult but there had been sounds enough, both from outside and downstairs, for Frevisse to think the morning was well begun. She found, upon opening the door at the stairfoot, that indeed breakfast had already happened for some, but a pitcher and the ample remains of a cold meat pie still waited at one end of the high table, and Lady Anneys, Ursula, and the two girls Frevisse presumed were her other daughters were standing nearby it. The gray day, the hall in its shadows, and their mourning dresses—that they had begun to wear for one death and now would wear for two—made for more gloom. But presently, mercifully, no one was crying.
Frevisse admitted the ungraciousness of that thought even as she had it. She was still weary from yesterday's ride, did not want to be here, did not know what was expected of her, and was altogether far from happy about anything. Was she supposed to give comfort to Lady Anneys? The woman had a son and three daughters who were surely better suited to that than a nun she barely knew. On yesterday's long ride, Frevisse had considered that she was maybe meant to be a guard for Lady Anneys against that man who had troubled her at the nunnery. But surely he wouldn't be so much a fool as to plague her with his attentions for this while.
There were no servants in sight but Ursula moved immediately to pour ale from the pitcher into two waiting cups while Lady Anneys said, weary-voiced but with attempted graciousness, "My ladies, good morrow." Though "good" was probably the last thing the day seemed to her. She was pale and holding herself in the way of someone determined to go on despite a wound whose pain was almost overwhelming them. "Would you please to meet my other daughters?" She gestured to the older of the girls. "Lady Elyn." Who was not so young as Frevisse had thought her by torchlight last night. She was not a girl but a young woman, and as she briefly curtsied, her mother said, "She's wed to our neighbor Sir William Trensal."
At mention of her husband, Lady Elyn turned away, stifling a sob.
"And this is Lucy," said Lady Anneys.
A round-faced, half-grown girl who might have dimples when she smiled, Lucy made a curtsy and sniffed on tears that were not far from being shed.
Ursula, bringing the ale to Frevisse and Sister Johane, asked, "Would you like some of the hare pie, my ladies?"
When they said they would, she returned to the table to cut it, and Frevisse asked Lady Anneys, "How does it go with you this morning, my lady?"
"Not very well," Lady Anneys answered quietly. "And it will go worse. I have to warn you the crowner will be here shortly. He arrived late yesterday and stayed the night with our priest in the village. He'll hold his inquest here this morning, when he's viewed Tom's . . ." Her steadiness faltered. She took hold on it again and went on, ". . . when he's viewed Tom's body and the jurors have come."
It was something that could not be avoided. A crowner's inquiry must always come after any unexpected or violent death, to determine where guilt lay or if there was guilt at all; and who, if anyone, should be arrested; and whether the sheriff must needs be called in; and what fines were due to the king. All that would be far too familiar here already this summer, Frevisse thought. To go through it again was surely nightmare added to nightmare for Lady Anneys. But she also thought, as she thanked Ursula for the thick wedge of pie the girl now handed to her, that by hearing the inquest she would learn everything about this Tom's death without need to ask questions of her own and that would be to the good, both for her curiosity's sake and as help in giving better comfort to Lady Anneys afterward.
Lucy was rubbing at her eyes, murmuring that they hurt. They were red and swollen and probably salt-scalded with tears and Lady Elyn pressed her fingers to the outer corners of her own eyes, which looked no better, and said, "So do mine. We'll look dreadful for the inquest." Trembling toward a new siege of weeping, Lucy nodded agreement with that.
The unkind thought crossed Frevisse's mind that besides their very real grief, they both were feeling very sorry for themselves at being so unhappy.
She was instantly sorry for that thought. Each person had to grieve in their own way, as best or worst, greatest or least they could; and very possibly their way was better than Lady Anneys' stiff, braced quiet, as if she hardly dared move for fear of the hurt in her despite it would come no matter what she did or did not do.
Sister Johane, putting down her emptied ale cup and her partly eaten piece of pie, said, "Yes, your poor eyes. I have something that might help. An ointment that I brought. It's meant exactly for soothing sore eyes. I have it with my things. If you'll come to the bedchamber?"
Lucy nodded readily and Lady Elyn said, "Oh, please, yes," and Frevisse felt even more contrite. Sister Johane offered needed help, while all she had were unkind thoughts that were no use to anyone.
"Mother, you'll come, too?" Lady Elyn said as she and Lucy started toward the stairs with Sister Johane.
"Miles is here," Lady Anneys said. "He probably has something to tell me. You go on."
They did; and with a word of apology to Frevisse, Lady Anneys went the other way, Ursula following her like a small shadow afraid of being lost, down the hall toward a man who had just come in, the one who looked so much like Hugh, with a tall, brindled wolfhound at his side. Frevisse, quite abruptly left to herself, took her first chance to look freely around her while finishing her breakfast. The hall was a plain one, bare-raftered and small, paved with plain stone and without even a screens passage at the far end, just the door to the yard at one side and another door opposite it, presumably to the kitchen and whatever rest of the house there might be. Just as in the bedchamber, everything was plain and well-worn, including the aged and ordinary high-backed chair meant for the lord of the manor here at the high table. The only thing beyond bare necessity was a rather poorly painted tapestry on the wall behind the table, with hunters and hounds striding stiff-legged across a green field strewn with flowers after an oddly proportioned deer leaping away from them toward a grove of scrawny trees.
Servants had come into the hall now. One of them was heading toward the high table, probably to clear the breakfast things away. The others were starting to shift benches from along the walls into rows across the middle of the hall, facing the table. Finished eating and not minded either to sit or return upstairs or join Lady Anneys still in talk at the hall's far end, Frevisse set down her cup for the servant to take and moved away toward the tall window at one end of the dais, out of everyone's way.
The shutters were open, letting in air soft with coming rain. It seemed they had been fortunate in fair weather for their traveling yesterday. She stood looking out at the manor's foreyard, still wishing she were elsewhere, until behind her someone said, "My lady?"
She turned around to find it was the young man who had been in talk with Lady Anneys, his hound still with him. Seen close to and by daylight, he was leaner than Hugh and perhaps a little taller but with the same brown hair and eyes and general look about him, she thought, as he bowed and said, "I'm Miles Woderove, Lady Anneys' stepgrandson. She's gone to be sure all's going well in the kitchen. She asked me to ask if there's anything you need or I could do for you."
She looked at him, trying to decide if she dared ask what she wanted to know. He seemed the most calm, least grieved, of any of the family she had so far met, but as he looked steadily back at her, she took in the tight-drawn lines around his eyes, the thin line of his mouth, and judged that his calm was a shield he was barely keeping between himself and the world. So he was someone both controlled and deeply caring, and she said, though she had not thought it out beforehand, "Tell me what you can of how things are here, if you please."
His eyes flickered. Stiffly he asked, "About Tom's death, you mean?"
"No. I'll hear enough and more than enough about that at the inquest. What I need to know . . . If I'm to help Lady Anneys at all, I need to have some thought of how things are, how they've been for her here."
Either puzzled over what she meant or considering whether he should answer, Miles looked down at the dog beside him. From what little Frevisse knew of dogs, this was a very fine wolfhound, lean and long-legged like his master, alertly looking at her with dark, intelligent eyes, and so tall that Miles' hand rested on his head without effort.
"I've gathered," Frevisse said carefully, feeling her way, "that Lady Anneys doesn't much grieve for her husband's death."
Miles' head snapped up. The calm was gone. "Nobody grieves for Sir Ralph's death," he said curtly. "Even his own dog doesn't grieve for him. You want to help Lady Anneys? Help her forget my late, unlamented grandfather ever existed."
Frevisse had not expected that much of an answer. Falling back on the obvious, she said, "He wasn't a good man?"
Miles made a sound too harsh to be laughter. "He was the human equivalent of something you'd scrape off the bottom of your shoe."
Frevisse was so startled that she said, "He's dead, and you still hate him that badly?"
"That badly and three times worse, my lady."
If he could be so open, so could she. "Why?"
"Why? Because he had pleasure in only two things. Hunting and being cruel to everyone around him. Now that he's gone, _I_ enjoy being cruel about _him_."
Frevisse had recovered enough balance by now to ask without showing particular feeling about it, "You've come back here on visit, now that he's dead?"
Bitter laughter bent the corners of Miles' mouth upward into what could not be called a smile. "I've never left, my lady. I tried. Twice. Once, when I was ten or so, I took off into the woods. He tracked me with hounds and beat me and brought me back. I tried again when I was fifteen and made a better go of it. That time he took three days to find me but brought men with him to hold me while he beat me. He said I was his heir and, by God's teeth, I was going to stay where he could see to me." He shrugged, maybe to shrug off the memory and the anger, looked down at the hound, and said, his smile all bitterly twisted to one side, "You shouldn't have set me on to this."
She maybe should not have, but now that she had, she said, "I thought Hugh's brother Tom was the heir."
"To everything Sir Ralph had purchased with his ill-taken fortune from his lawyer days in London, yes. All of that except what Sir Ralph left to Hugh and the girls went to Tom. But there's a Leicestershire manor entailed from eldest son to eldest son to eldest son. That's me, the only son of Sir Ralph's only son by his first wife. Sir Ralph hated him, too."
"Your mother is dead?"
"Long ago. Lady Anneys raised me." Miles' voice and smile lost their bitterness when he named her.
It was good to know his hatred did not reach to everyone around him and Frevisse asked, "Was Sir Ralph cruel to her, too?"
Miles slightly frowned, almost as if considering a riddle. "Sir Ralph would hit anyone else in his reach, even the girls—though rarely them and not Tom or Hugh or I since we grew big enough to hit back hard enough to make it count. But none of us ever saw him raise a hand against her. No, he never struck her. He never had much to do with her at all. She left him alone and he left her."
But there were other ways than blows to hurt and ways of "leaving alone" that could cut to the heart, Frevisse thought. Aloud, she only asked, "There's been no word of who killed him?"
"None," Miles said sharply. "Nor no more searching either, I think. I hope they never catch who did it."
"What about Tom? Was he like his father?"
"No." Miles harshly refused that. "Tom hated him, too. We all did. But Tom . . ." Miles broke off. There seemed easily enough hatred in him to include the man who had inherited so much that might have been his if Sir Ralph had been other than he was, but rather than anger, it was grief that twisted Mile's voice as he recovered and said, "After Hugh, Tom was my best friend. I could strangle Sir William with my own hands for killing him."
"Sir William?" She had just heard of a Sir William here, she thought.
Miles' spasm of grief was gone, the bitterness and anger back. "Our near neighbor. Lady Elyn's husband."
Yesterday, on the long ride to here, there had been no talk about Tom's death. Frevisse had supposed everything had been said the night before between Lady Anneys and Hugh and to Ursula, leaving them with only the grim need to reach home. Faced with that, neither she nor Sister Johane had asked anything. Now Frevisse wished they had, because there seemed to be far too many things she ought to know—and no chance to ask them because Miles was turning away from her to look out the window into the yard where harness-jangle and the thudding of hoofs warned riders had arrived. Three men and a girl, Frevisse saw when she looked out, too, as Miles drew in a hissing breath and stepped back, saying, "St. Anne help us. He's brought Philippa. My lady, _that_ is Sir William and I doubt he has the good grace to wait in the yard until the crowner comes. Pray, excuse me."
With a curt bow and not waiting for Frevisse's answer, he went away toward the outer door, the hound beside him. Left on her own, Frevisse watched the newcomers, more accepting of Lady Elyn's tears if her husband was indeed her brother's murderer but wondering why, if he was known to be Tom Woderove's murderer, this Sir William was riding free.
At least she could easily tell which of the three men he was. If his richer clothing—a long, high-collared, black houppelande and dark blue, brimless hat with a glinting, silver-set jewel pinned to it—and better horse—a black palfrey—had not told her, the speed with which one of the men with him, servant-dressed in simple doublet and hosen, dismounted and went to hold his bridle would have. And the girl, who must be Philippa, would be his daughter, Frevisse guessed. Not someone's wife, anyway, because her long, fair hair was bound back but covered by only a black veil pinned to a small hat's padded roll. A married woman or a widow like Lady Anneys would have worn a wimple that circled her face and hid her hair and been finished with a starched veil.
But the next moment Frevisse's heed went from the girl to the man now lifting her down from her side-saddle. The other man who had ridden in with Sir William.
Master Selenger.
What did Master Selenger have to do with Sir William? He looked to be in attendance on Philippa but was too well dressed to be only servant.
Frevisse was suddenly deeply annoyed to be so ignorant of everything and everyone here.
The newcomers' horses were being led away toward the stables and Master Selenger had led Philippa to Sir William, who was saying something to her. Though Frevisse could not see the doorway from where she stood, she guessed Miles was standing there, in their way, instead of going out to them, but if he meant some challenge to Sir William, he was forestalled by a half dozen more horsemen riding into the yard. Frevisse immediately judged them likely to be the crowner with his clerk and men, all dressed in a plain business way with loose surcoats over doublets and hosen and riding boots, and was disappointed that none of them was the one Oxfordshire crowner she knew. It would have been good to have a friend here.
Sir William and Master Selenger moved to meet them as they dismounted; or rather, they moved to meet the man who dismounted and stepped forward from among them, leaving his horse for someone else to hold. He wore authority as openly as he wore his fullsomely cut surcoat of deep-dyed dark green and seemed to greet both Sir William and Master Selenger familiarly.
Philippa had stayed where she was, standing alone; but now Miles came into sight, likewise crossing the yard toward the crowner and his men, and he paused by her to say something. His anger toward Sir William seemed not to include her. Whatever he said, she answered with a nod, and when he reached to touch her shoulder briefly, she briefly raised her own hand to touch his before Miles went onward to the men and Philippa toward the hall.
Having seen how much anger was in Miles, his gentleness toward the girl surprised Frevisse, and surprised her the more because it was toward someone linked to the loathed Sir William. But Hugh had appeared from somewhere beyond the hall, crossing the yard to join the crowner and Sir William, reaching them at the same time as Miles and putting himself with what looked like purpose between Miles and Sir William as they all greeted the crowner. It might have been by chance but Frevisse thought it was deliberate. From where she stood she could not tell if Hugh had the same anger toward Sir William that Miles did, though Miles' rigid back was clear enough even from here.
The men spoke together for a few moments, then walked on toward the hall. Frevisse moved away from the window. Sister Johane, Lady Elyn, and Lucy were just returning to the hall. At its far end Lady Anneys, with Ursula still beside her, was in talk with a maidservant but finished and dismissed her as her daughters and Sister Johane approached her, Frevisse following behind them. But it was to Frevisse and Sister Johane she said, "My ladies, you'd do well to take a place on one of the benches before more people come, I think. Besides our own folk there'll be neighbors surely, and whoever Sir William has brought, and the crowner's men and jurors. There are to be ten jurors, I'm told. Five men of ours and five from Sir William's manor. But . . ." Her flood of words suddenly stopped. She paused, bewildered, seemingly trying to remember why she had been saying any of that and why it mattered.
Frevisse, understanding the need to flee from overwhelming grief by clinging to practical things and how easy it was to stumble in that flight and be overtaken, hoped there was a strong quieting draught included in whatever Dame Claire had provided Sister Johane and that Lady Anneys could be persuaded to take it when this day was done; but for now she only said, "There's no need to think about us, my lady. We'll do well," and went away with Sister Johane to the last of the three bench-rows facing the dais. For the ease of whoever came later, she and Sister Johane sat in the middle of the bench they chose so it could fill in to either side of them. Two other long benches—for the jurors, Frevisse supposed—had been set at an angle between the dais with its high table and the bench-rows, and presently two manservants were wrestling a long settle through a door behind the dais that Frevisse presumed led to the parlor since the other door there led to the stairs to the bedchambers. Once through the door, the men brought the settle down from the dais, lurching a little with its weight, and set it behind the jurors' benches, facing the high table where the crowner would sit. That would be for Lady Anneys and her daughters, Frevisse guessed.
Not interested in benches, Sister Johane had twisted around to watch the outer door and said excitedly, "Someone's arrived."
"Sir William and the crowner," Frevisse said. "I saw them from the window."
Sister Johane twisted back to say low-voiced and close to her, "He's Lady Elyn's husband, did you know? Sir William, I mean. They've been married about two years and his daughter Philippa was going to marry this Tom who's dead. That's why he was at Sir William's. To talk about the marriage. Only they quarreled and Sir William killed him by mistake, Lady Elyn says."
"Is that why Lady Elyn is here instead of with her husband? Because he killed her brother?"
"Oh, no, not at all. She says nobody blames Sir William. It was a mistake. She's here because, well, she was needed here more than there, with her mother being gone and all." Sister Johane broke off and twisted around again to see what was happening.
Frevisse somewhat turned, too. The crowner, companioned with Miles, was greeting Lady Anneys where she stood with her daughters beside her not far inside the hall door. She gave him her hand and he bowed over it and spoke briefly to her before going on toward the dais with Miles, followed by a man who was probably his clerk, carrying a leather bundle of probably papers.
Sir William had come in behind them and now stepped forward, his hands held out to Lady Anneys, his voice carrying as he said, "Lady Anneys, I'm sorry beyond words for this. It was all . . ."
Lady Anneys, rigid, snatched her hands away from him and pressed them together over her heart. Sir William, his own hands still out, pleaded, "Lady Anneys, please. For Elyn's sake if nothing else, don't turn this to a quarrel between us. I never meant—"
She interrupted, her voice carrying as clearly as his. "It isn't a quarrel between us, Sir William. All the anger in this world won't bring Tom back to me. But I can't . . . I won't take the hand that held the knife that killed my son."
Sir William looked down at his out-held right hand with the surprise of a man seeing a thing in a way he never had until now. Then he withdrew it, took a step back from her, and stiffly bowed. Lady Anneys as stiffly bowed her head to him in return and stood staring over his shoulder at nothing, waiting for him to go away. He looked at Lady Elyn and held out his hand to her. After a bare moment of hesitation and a flicker of her eyes toward her mother, she took it and he led her up the hall to the front bench.
In Frevisse's ear, Sister Johane whispered, "Lady Elyn said her mother told her she must go with Sir William when the time came. She said Lady Anneys said there wasn't choice to make between her husband and family. That they were wed and that was what she had to do."
To Frevisse that spoke well of Lady Anneys, but now she was faced by Master Selenger and the girl Philippa, come in together behind Sir William. Master Selenger looked ready to speak to Lady Anneys but Philippa was hesitating, uncertain whether to she should do that or go immediately after her father. Lady Anneys, paying Master Selenger no heed at all, held out both her hands to the girl and said tenderly and smiling, "Philippa," and with a grateful gasp Philippa moved quickly to take her hands, holding tightly to them and saying something too low to be heard. Lady Anneys answered quietly, too, then let her go, with a crisp nod and nothing else at Master Selenger, letting him know to keep his distance. He bowed silently and moved off with Philippa, taking her to sit on her father's other side from her stepmother before seating himself on the bench directly behind them.
Since Sister Johane seemed to have learned a great deal in the while she was tending to Lady Elyn and Lucy, Frevisse whispered to her, "Who is Master Selenger?"
"Who?"
"The man who came in with Philippa just now?"
"Oh. I don't know."
"What does Lady Elyn say about having a stepdaughter not much younger than she is?"
"Not anything to me. They all grew up together, I think. Philippa and Lady Elyn and the others. So they know each other and all."
Frevisse wondered whether that would make it harder or easier among them. And how difficult was it for Philippa to sit there beside her father, who had killed the man she was supposed to marry?
"The two families were close until now?" she asked.
"I gather so. Sir William and Sir Ralph were friends, from what Lucy was saying. They both loved hunting. Sir William and Lady Anneys are both executors of her husband's will and that makes all this even worse because how are they ever going to deal together now?" She broke off as Hugh entered beside a shuffling old man in a priest's black gown, then said, "That will be Father Leonel. Lady Elyn says he's useless as a priest but that was what her father wanted, not somebody who'd try to change him. Lucy said she shouldn't talk like that about their father or priests but Elyn said it's the truth and she'll say it if she wants to."
Miles returned to Lady Anneys from seeing the crowner to the dais just as Hugh and the priest joined her. They all spoke low-voiced together and seemed to agree on something before Hugh went back outside and Miles, with Lady Anneys, Lucy, and Ursula following, led Father Leonel to the end of a bench near the settle. While the priest seated himself with the carefulness of stiff joints, Lady Anneys took her place on the settle with Lucy on one side of her and Ursula on the other and Miles took his place standing at the settle's end, looking ready to do whatever Lady Anneys might need but also able from there to see whatever else went on in the hall.
Hugh and one of the crowner's men came in with ten men. The inquest's jurors. Five men from each manor, Lady Anneys said. They were all solemn and well-scrubbed, well-combed, and wearing their best clothes—plain, belted tunics in dull reds, blues, greens, browns, and loose hosen. They bowed to the crowner seated behind the high table where his clerk was laying out papers in front of him, and took their places on the benches set for them. Watching them, Frevisse saw there was no mistaking which five went together against the other five. Cold shoulders and wary looks told more than enough and for the first time Frevisse thought beyond the grief of the young man's family to the raw possibility of open enmity between one manor's folk and the other's because of it. It was not usual for lesser folk to serve on a jury against someone like Sir William but here it might well be best because a decision reached among them would more likely be accepted on both manors.
Once they were sitting, however grudgingly with each other, the crowner looked down the hall and nodded to someone and a moment later people in quantity began to enter. They must have been gathering in the yard ever since Frevisse left the window. Neighbors, she judged—other gentry like Sir William and the Woderoves; mostly men, a few women. They filled the benches quickly. Two men with polite bows took the place to Frevisse's right; a husband and wife sat to Sister Johane's left. Later comers were directed by some of the crowner's men to places along the side of the hall across from Lady Anneys and her daughters, and then the common folk were let in. They crowded quietly in, filling the hall's end behind the benches. Manor folk, Frevisse decided with a quick look back at them. There were both sorrow and in-held anger in their faces. They were here to know for certain what had happened to their young lord and ready to make trouble over his death if things came to that. And that told Frevisse much more about Tom Woderove than the little she knew, because while it was one thing for his family to grieve his death, it was something more for people whose lives had been under his rule for ill or good to grieve for him, too.
From what she had so far heard, she doubted they had grieved for his father.
The crowner's clerk, a solid, middle-aged man, stood up at the end of the high table and announced the inquest would now begin, bowed to the crowner with "Master Hampden," sat down, took up a quill pen, and waited with it poised over an inkpot, paper ready in front of him.
The inquest went the usual way, with statements as to where it was and why and whose death was in question. It was established that Sir William Trensal and Hugh Woderove—now Master Woderove since his brother's death—were the only witnesses, besides the deceased, to the actual occasion of the death. Master Hampden averred that he had viewed the body and that the only wound on it was a small cut to the right side of the neck that had been sufficient to sever a blood vessel. The deceased had then bled to death.
Lady Anneys, her head already bowed, shuddered. Ursula buried her face against her mother's shoulder and Lucy openly sobbed. Lady Anneys put her arms around them.
There were no other marks or wounds upon the body, Master Hampden said, and Hugh was called forward to describe what had happened.
Frevisse, listening while Hugh told what had passed between his brother and Sir William, watched Lady Anneys and her daughters listening, too, and ached with how pointless the death had been. Pointless and, it would seem, never intended.
"You believe, from what you saw," Master Hampden asked when Hugh had told of seeing the blood on Sir William's penknife, "that Sir William was surprised to see the blood? That he had had no knowledge until then that he had stabbed your brother?"
"I believe that, yes," Hugh said. "He looked surprised to find he was still holding the knife. Nor did he stab Tom. He . . ." Hugh made a gesture. "He just swept a hand at him like that. To make him back off."
"Which Master Woderove did."
"Yes."
"Did he show any sign of knowing he was hurt?"
"No."
"You saw no blood on him at the time?"
"No. He was wearing a dark red doublet. It must have hid that he was bleeding, and I was standing on the other side of him anyway. And he wasn't there long. He said an angry thing or two more at Sir William and stormed out."
"What happened then?"
"Sir William and I said a few things at each other. Then he saw the blood on his penknife and I knew Tom was hurt and went after him."
"You overtook him before he reached home?"
"I saw his horse grazing at the roadside maybe a quarter mile from the yard here."
"And your brother? When did you see him?"
"Not until I was nearly to him. He was stretched out in the shade of a tree there. He was lying down, face down, his head resting on one crooked arm. I thought he was sleeping. I thought . . ." For the first time the young man's stiff attempt to say what he knew but feel nothing while he said it broke down. "I don't know . . . what I thought," he fumbled. "Then I turned him over and saw he was . . . dead."
Lady Elyn broke into open sobs. Lucy in the circle of her mother's arm and Ursula huddled against Lady Anneys' other side could cry no harder than they were, but Lady Anneys sat with lifted head, looking dry-eyed at her son. Only the pleading in her haunted eyes begged for it to be over soon. Frevisse saw Master Hampden glance toward her before going on to ask what Hugh had done next.
"I cried out. I stood up and I shouted for help."
"Who were you shouting to?"
"No one. Anyone who could hear me."
"There was no one there who might have seen your brother fall?"
"I didn't see anybody."
"But somebody came?"
"Finally. From one of the further fields where they were barley-harvesting. Somebody heard me and some of the men came and we carried Tom home."
"But they hadn't heard or seen anything before then?"
"No."
"Are those men here?"
Hugh moved one hand toward the jurors. "There. In front."
The five men on the nearest bench acknowledged that with nods.
"So you were first-finder of the body but they came immediately afterwards."
"Yes."
"What did you do then?"
The questions and answering went on, through sending one of the men running for Father Leonel while the others carried the body to the hall, to sending one of the manor men to Sir William to tell him Tom was dead and to bring Lucy and Lady Elyn back to Woodrim to sending another manor man away to find out and tell the crowner he was needed.
"You didn't send or go yourself to seize Sir William," Master Hampden said. "Why?"
Hugh had to have known that question would come. By law and under penalty of fine, the first-finder of a body, when the murderer was known, had to raise the hue and cry and, joined by everyone who heard him, pursue the murderer. If taken, the murderer was then to be held until the crowner came and claimed him into custody. But firmly Hugh said, "I saw no likelihood Sir William would seek escape. He knew and I knew that he never meant harm to Tom. It was chance and nothing else that Tom . . . died." He choked on the word and for the first time bent his head, tears thick in his voice.
Master Hampden drew a penknife from under the papers in front of him, laid it on the open palm of his hand, and held it out for Hugh to see. "Is this the penknife you saw in Sir William's hand when your brother was wounded?"
Hugh raised his head and looked. "Yes."
Master Hampden asked the jurors if they had any questions of their own for him. They did not and Master Hampden thanked him and bade him sit down.
Sir William was next. He was solemn, as well he should be, but he carried himself assuredly. To the crowner's questions he said much the same things as Hugh had. That he had asked Master Woderove to come to him in order to discuss the planned marriage. That they had somehow fallen into a quarrel, Sir William was not sure how. "One nothing thing leading to another. That was all. He had his father's hot humour and Sir Ralph could go into a fury over things most men would not."
There was general head-nodding agreement from the Woodrim manor men of the jury and among the onlookers. Master Hampden noted it. It was his task to learn as much as he could about what had happened and where blame should be laid, if blame there should be, and because he had the right of inquest, he had the power to ask all the questions he could think to ask and to expect answers for them under oath; but he was also expected to make use of anyone who best knew the accused and victim, to better understand what might lie behind and around the actual crime itself. So the jury was made of local men and a competent crowner gave them heed.
Frevisse had known a crowner who, unless forced to it, never bothered with more than his own opinion of who the guilty should be. He had been a dangerous man in his narrow way and she was thankful to find that Master Hampden was a different sort.
He was come now to the moment Sir William had struck at Tom. "He leaned toward you and you swept your hand with the penknife in it at him. Is that what you say?"
"And that I'd forgotten I was holding the knife, yes."
"Were you frightened at that moment? Did you fear he was going to attack you?"
"No, sir, I was not frightened. He was angry, he was leaning too close, I wanted him to stand back. That was all. I never meant even to touch him. I didn't know I had."
"But you admit that you did?"
"I must have but I was unaware of it at the time."
Outside the day had grown grayer and the rain begun to fall, soft as weary weeping. The questioning went on. Sir William readily admitted the penknife was his own, the one he had had that day, and agreed that, yes, it was very sharp; penknives had to be able to trim tough quills to usable points.
"What did you do after Master Woderove and then his brother left you?" Master Hampden asked.
"I called one of my servants to saddle my horse and set out after them both, to find out how badly Tom . . . Master Woderove was hurt. I meant to apologize, before the matter could fester into something worse."
"You were not angry?"
"I was irked it had come to this. A petty quarreling for no good reason except he was young and hotheaded. But I was not angry. I simply wanted to end it before it worsened."
"In regard to Master Woderove's death, what would you say you're guilty of?"
"Of misadventure," Sir William replied firmly. "I didn't strike at him with intent to hurt, much less to kill. I didn't mean to strike him at all."
Master Hampden gave the jurors their turn to ask whatever questions they might have, but they had few and Sir William was permitted to sit again. They asked to see and handle the penknife and then talked among themselves, twisted around on the benches, heads close together, and Frevisse noted the stiff wariness among them was gone. When they had finished and all faced forward again, Master Hampden asked if they had come to a conclusion and one of the Woodrim men stood up to say that to all their minds Master Woderove's death had been by misadventure. The other jurors all nodded their agreement and after that everything was nearly done. Master Hampden asked if there were three men here who would stand bond for Sir William to appear at the next county court. Five men among the onlookers stood up. Master Hampden accepted all five, told them to speak to his clerk, and declared the court ended.
People who had been standing began shifting from where they stood. Those who had been sitting began to stand up, Sir William among them. He turned to say something to the men behind him. He would almost surely be found guilty of lesser manslaughter at the county court and somewhat fined for it, but from everything that had been said here, he had no great guilt in Master Woderove's death. So far as the law was concerned, the matter was all but done. All that was left, Frevisse thought as she stood up along with everyone else, was for the two families to come somehow to peace with each other and to terms with their grief. If there had been outright murder done, there would have been a harsh sundering of one kind or another—outright rage at Sir William and maybe crude satisfaction for his punishment—but because it had been mischance, not murder, somehow, for the sake of everyone, there would be healing.
At least Frevisse, watching Lady Elyn rise and go to her mother, prayed there would be.
**Chapter 10**
The crowd thinned as the manor folk shifted out the door. The rain was still lightly falling, barely enough to dampen anyone or harm the harvest-ready crops, but there would have been no work in the fields today even without the inquest and the manor folk were not heading out to work again, merely removing themselves now that they were satisfied about their lord's death. Some of the neighbors and others who had come were in talk with Sir William; others were going to Hugh; the five men who had offered to stand bond for Sir William were with the crowner's clerk at the high table; most of the women were gathering toward Lady Anneys.
Beside Frevisse, Sister Johane said, "At least it's over."
The inquest was, Frevisse thought; but nothing else. "When will the funeral be?" she asked.
"I think we're meant to go from here to the church for it." Sister Johane lowered her voice and leaned close to say, "It's been four days." As if it were a secret that burial now would be best since no body would keep for long in this warm weather.
Frevisse made no answer. She was watching Sir William excuse himself from the men with whom he had been in talk and went toward Lady Anneys. Miles, seeing him, too, moved to Lady Anneys' side. Frevisse, finding herself very interested in what would be said among the three of them, went, with Sister Johane following her, to join the cluster of people drawing away to either side to let Sir William reach Lady Anneys, who stood stiffly looking at him, saying nothing when he stopped in front of her.
"Lady Anneys," he said, holding out his hand as he had before, "let it end here?"
"Let what end here?" she said back at him. "My son's death?"
Sir William did not withdraw his hand. "I'm acquitted of guilt in that."
"You're acquitted of willful guilt," Lady Anneys said quietly. "You never purposed Tom's death. I accept and believe that. I hope in return you'll accept that my pain is still too new for me to be willing toward peace with you." She looked down at his out-held hand. "And that is still the hand that killed my son."
Sir William had begun to redden above his collar. He drew his hand back. "We can't ignore each other. By Sir Ralph's will you and I must—"
"Work together. I know. But not for this while. Not soon."
Sir William's color deepened. Frevisse suspected he was not used to being interrupted or refused, and Lady Anneys had just done both.
At Lady Anneys' side, Miles said—as quietly as Lady Anneys but with warning below the words, "Leave it, Sir William."
Sir William's eyes narrowed with the anger he had been holding back from Lady Anneys, but Hugh slipped between people to his side and said calmly, respectfully, "Master Hampden asks leave to speak with you, if you please, Sir William."
Sir William turned his head, looking as ready to snap at Hugh as at Miles, except that Hugh added, "And Philippa would like to speak with my mother, if she may."
As Lady Anneys said with warm affection, "Of course she may speak with me," Sir William's angry look went to Philippa standing just behind Hugh, then shifted quickly back and forth between them, his anger giving way to some other thought, and with a curt bow of his head to Lady Anneys, he stepped from Philippa's way and went toward the crowner at the high table. A look passed between Hugh and Miles, agreeing on something before Hugh followed Sir William and Miles turned to talk with several men nearby. Lady Anneys, keeping one arm around Ursula, took Philippa by the hand with her other hand and drew her close so they could talk with their heads close together, leaving Lady Elyn and Lucy to the other women come to offer their comfort and regrets. Sister Johane shifted to join them, but Frevisse drifted away in Sir William and Hugh's wake with some vague purpose of speaking at least briefly to the crowner.
That proved easier than she had supposed it might. Whatever Master Hampden had to say to Sir William was short. Well before Frevisse was near, Sir William had gone on along the table to where the five men who were to stand his bond were signing papers for the clerk and pressing their seals to wax. Hugh in his turn said something to Master Hampden, who answered him and then Hugh went away, too, toward the hall's outer door where Father Leonel was waiting for him.
Taking her chance, Frevisse stepped onto the dais across the table from Master Hampden, who was now gathering together his papers. He looked up, gave a slight bow in answer to the respectful bending of her head to him, and asked, "May I help you, my lady?"
"I'm Dame Frevisse of St. Frideswide's priory."
"I know of it. A worthy house. You're here as comforter to Lady Anneys, I believe."
She slightly bent her head to him again, admitting that but saying, "I met one of your fellow crowners a few years ago. Master Christopher Montfort. I wondered if you could tell me how he does?"
Master Hampden smiled. "Young Christopher. Yes. I saw him two weeks or so ago. He was in fine health and doing well. He's to be married close on to St. Edward's Day, I hear."
Frevisse expressed her pleasure at that and said, before Master Hampden could excuse himself or else his clerk come to rescue him, "I'm here because Lady Anneys was visiting her youngest daughter at the priory when word came of her son's death. As you said, Sister Johane and I hope to give what help and comfort we can. Our trouble is that we hardly know the family and fear we're as likely to say the wrong thing as the right in trying to comfort her. We don't want to pry among the servants to learn more and know no one else to ask. Could you, as someone outside it all, tell me something about Lady Anneys' husband's death and how things seem to you here, so we'd have a better chance of giving help instead of hurt?"
Master Hampden considered that with a slight, not unfriendly frown. Farther along the table Sir William and the other men had finished their business with the clerk and were going away. The clerk, gathering up his papers, pens, and ink, was watching with a careful eye to see if Master Hampden wanted rescuing or not, and with a small lift of one hand, Master Hampden let him know no rescue was needed yet before saying to Frevisse, "I can willingly tell you everything that's generally known. Come aside and we'll talk."
They went along their opposite sides of the table, Master Hampden pausing to say a few words to his clerk before meeting Frevisse at the window, much where she had stood in talk with Miles. By then Master Hampden had a question of his own and asked, "How much do you know, or understand, about the Woderove family?"
"I've gathered that Sir Ralph was its head. Miles is his grandson and heir by a first son long dead, by a first wife also long dead. Lady Anneys is Sir Ralph's second wife and by her he had two sons and three daughters that I know of. The eldest daughter is married to Sir William. Sir Ralph was murdered earlier this summer, no one knows by whom. All I've heard is that he was killed while hunting. What I know about Tom Woderove's death I heard here, just now, but he seems mourned by everyone while no one seems grieved for Sir Ralph at all."
She stopped and waited and Master Hampden said, "You have all of that right, especially about Sir Ralph. The inquest on his death was very different from this one. You can see the grief there is for Master Woderove. I don't think anyone at all minded Sir Ralph was dead. The suddenness of his death had them thrown off balance but there was no grief."
"Why?" Although, after talk with Miles, she had some thought about that.
"He seems to have been a brutal man. As part of finding out who might have wanted him dead I asked about him elsewhere as well as here. There was no good report to be had of him anywhere except somewhat from Sir William, who was as near to a friend as he had, I gather. They had both been lawyers and shared a love for hunting. Aside from that, the only other person Sir Ralph even somewhat got on with was his son Hugh—Master Woderove as he is now that his brother is dead. He served as his father's huntsman and saw to his hounds. He inherited them at his father's death, I believe. For the most part, though, Sir Ralph was careless about people. Or maybe he was deliberately cruel. I think 'cruel' is the correct word in his case. I only ever met him in passing and have had mostly to go by what I gather from other people. Either way, there was no mourning for him here. In helping Lady Anneys, it's only her son's death you need worry for and about that you know as much as everyone."
Her son's death, yes, Frevisse silently agreed; but there were surely also years-old wounds left by a man careless of everybody but himself. Lady Anneys had hardly had chance to begin to heal from those before this new wound came. And not only Lady Anneys. From even the little Frevisse had so far seen and heard here, she could guess that still-raw hurts ran deep in probably everyone ever in Sir Ralph's reach. It wasn't only her own pain with which Lady Anneys had to deal, but her broken family's, worse broken now with Tom Woderove's death just when there had been hope of healing.
Quickly, because Master Hampden was readying to make his excuse to be done with her, Frevisse asked, "Sir Ralph, then, was well disliked outside his family as well as in?"
"He was disliked by almost anyone who ever had to deal with him, yes."
"And outright hated by some?"
Master Hampden hesitated before answering carefully, "I'd say so, yes."
"But you've no one to suspect more than another for his death?"
"Not yet, no. There was a man driven off the manor by Sir Ralph a year ago. I was told he was in no danger of Sir Ralph bothering to find him and force him back, but we found him out anyway, on the chance he'd come back for revenge."
"Had he?"
"He's found work in an inn's stable off Kettering way, looks likely to marry the owner's daughter, and blesses the day he left here. Nor could my man find that he'd been away for even a day this summer, let be the time he'd need to come back here and lie in wait on the chance of finding Sir Ralph alone. Now, if you'll pardon me, my clerk wants me for something."
He parted from her graciously but firmly nor did she try to keep him. She gave him thanks and let him go and stood alone, looking out the window and thinking. Lady Anneys' husband had been disliked to the point of hatred. By his grandson certainly and by others undoubtedly, so many that the crowner had no one in particular to suspect of his death. What did she remember hearing about how he had died? That he had been found dead in the woods, his head smashed in? Was that what someone had said at the nunnery? If that was all, it told her almost nothing. He had been hunting, been separated from others who had been hunting with him, and found dead. Had he happened to be alone or had he been on purpose, maybe secretly to meet someone who had then killed him? Or had someone been lurking in the woods, hoping for the chance to kill him and taking the chance when it came their way? Or had it been pure happenstance—someone had happened on him when he was alone and killed him because the chance was suddenly there? Surely the crowner had tried to learn of any recent quarrel that could have brought someone to murder. Or did Sir Ralph have so many quarrels there was no way to point at one before another as possibly the cause?
And what business was it of hers?
It wasn't. Except that she was here and she doubted, knowing herself, that she would let go of it so long as there was chance of having an answer.
She looked down the hall at the gathering of family, manor folk, and neighbors and knew it was not simply curiosity in her. Very probably a wound or wounds given by Sir Ralph had festered in someone's mind into murder and as with any wound that festered and went bad there would be no healing until the wound was cleansed, which in this would be by finding out the murderer.
Even if, as was all too likely, the murderer was someone here.
The rain fell lightly, steadily, all through the funeral and the afternoon, and Hugh felt much like the weather—gray and weeping, despite he had no outward tears. Partly his lack of outward grieving was simply that he was tired almost past thinking. Tom's death, followed by two days of hard riding to bring Mother and Ursula back from the priory, then keeping the vigil beside Tom's body all last night because Miles had kept it the night before and one of them at least should sleep . . . With all of that, the best he could do with his howling grief was keep it buried until there was time and place to give way to it. But time and place wouldn't come today, and all through today, from somewhere aside from it all, he had watched himself doing what needed to be done, heard himself being a comfort to his mother and sisters and grateful to everyone who offered their consolations; had listened to himself making needed decisions and saw himself move with apparent purpose through the hours of a day five times longer than any day he could remember.
But finally it was done. Everyone who was not staying was gone and he was standing alone at the hall's tall window, watching a watery yellow sun slide below thinning gray clouds toward a damp sunset. It was all over.
Except it would never be over.
Tom was never coming back.
By the soft footfall he knew it was Ursula behind him and he did not turn around but waited until she slipped her small hand into his before he looked down at her and tried a smile. It was a smile as thin and damp as the sunset, but so was hers and neither of them said anything as she leaned against him, clinging to his hand with both of hers and rubbing her face on his sleeve, the silence maybe as comforting to her as it was to him.
It could not last. Comforts so rarely did, and much too soon Ursula said, "Mother wants to see you, Hugh. She's in her garden."
"Now?" With evening coming on at the end of a wet day, a dripping garden was hardly the best of places, surely. But even as he questioned it, Hugh knew why she was there. Her garden was her place of comforting. He could not remember his father going into it even once, just as she had never, in Hugh's memory, ever gone on any hunt.
But it was still no place to be when night vapors would soon be rising and he kissed the top of Ursula's head and said, "I'll bring her in. Where are you going to be? Where's Lucy?"
"In the parlor with the nuns. We're going to have a cold supper there."
"That's good. I'll bring Mother."
He found Lady Anneys standing at the garden's rear gate, looking out across the cart-track and the stubbled hayfield where the cattle would be turned to graze come frost time. He joined her without a word, and much the way Ursula had done, she took his hand. She was not crying nor did she lean against him like Ursula but stayed standing straight and gazing outward. Dark was coming quickly on, bringing chill with it, but despite what he had told Ursula, Hugh did not try to draw her inside. He guessed that, like his little while in the hall, this was her pause before taking up life again in its new, unwanted shape, and he was willing to wait as long as she needed until she was ready—or able—to say why she had sent for him.
But what she said when finally she spoke was, quietly, "At this hour three days ago I had just lately finished praying at Vespers for my children's safety and good health. That day and all the next I was thinking of Tom as alive and he was already dead."
Not knowing what to say, not trusting his voice, Hugh held silent.
His mother looked at him with a smile both tender and bleak. "I haven't been able to pray. Nothing beyond 'God keep his soul' and a rather desperate thanks that I still have you and Miles."
"And the girls."
"And the girls. But I'm thinking of sons just now."
Her eyes were fixed on his face, as if to be certain of it past ever forgetting. He looked away, out into the gray-blue mist and evening shadows now hiding the far end of the hayfield. "I can't pray at all," he said. "Not even as much as you've done."
She squeezed his hand. "It will come back. Even feeling will come back, though I think I'd rather it didn't. I don't think I want to feel again."
Hugh made a small, assenting sound. The fierceness of his first grief had torn him into shreds with pain. After that, dealing with necessities had brought on a numbness he feared to lose despite he knew it would not last.
Lady Anneys was still looking at him. He went on gazing into the gathering darkness, willing her to say nothing else, to leave them where they were for just a little longer.
"You're Master Woderove now," she said.
Hugh hoped his shudder—as if someone had walked on his grave—did not show. "People kept calling me that all day. I kept thinking they were somehow talking to Tom."
"Tom's place is yours now."
"I _know._ " Hugh kept the words muted but not the anguish in them. He snatched his hand from hers, put both his hands to his face, and rubbed at the pain behind his eyes. "I know, and I would to God it wasn't."
His mother's only reply was to lay a hand on his shoulder. For a time they stood in silence, his face hidden behind his hands. Only finally did he drop one to his side and put the other on his mother's still resting on his shoulder. Keeping his voice even but fooling neither of them, he said, "The rain today wasn't enough to hurt the fields. Tomorrow looks to be fair. Maybe we can get on with the harvest by afternoon."
**Chapter 11**
Through the next few days Frevisse found herself oddly more alone in the midst of Woodrim's household bustle than ever she was able to be in St. Frideswide's cloister. At St. Frideswide's she had place and duties. Here she had neither, except to keep Lady Anneys company, and even that seemed purposeless because Lady Anneys, with a household to bring back into balance and order, hardly seemed in need of company besides her own people here and her children. She had no idle time to fill nor did she seem in want of comforting talk. If she had, Father Leonel could probably have given it as well as anyone. He made daily visits to the hall, and from what she saw of him, Frevisse judged that, elderly and slow a-foot though he certainly was, he was not feeble of mind. But Lady Anneys never made more than ordinary talk with him. The while he was there each evening was spent with Hugh, showing him how the manor's accounts were ordered.
It was hard going for both of them. Frevisse gathered from what she heard around her that the dead Sir Ralph had given all his time and heed to hunting, while his elder son had had the running of the manor and his younger son had seen to the hounds and huntsman's duties. Now Father Leonel was trying to show Hugh in short order all that his brother had learned and known over years, and Hugh did not seem happy at it.
Things were easier, Frevisse supposed, for Miles. As woodward, overseeing the forest both for the hunting there and for what profits could be had from it, his duties were not changed at all and he and his hound were away into the forest early and for most of every day, so that Frevisse had seen very little of him since the inquest. He sometimes returned to the hall for midday dinner but usually was only there in time for supper and afterward stayed near Hugh and Father Leonel while they worked in a corner of the parlor and Lady Anneys made Lucy leave them alone.
"Master Woderove really doesn't want to be lord here, does he?" Sister Johane said as she and Frevisse walked back from Mass in the village church their fourth morning at the manor. "Lucy says Lady Anneys says he's well-witted and that Father Leonel worked the accounts for years with Tom and will show him all he needs to know. But he doesn't even like to be called Master Woderove. Have you noted that?"
Frevisse had. In truth, Hugh all but flinched when called Master Woderove, and Father Leonel, from forgetfulness or kindness, mostly said "Master Hugh" when they worked together.
But Sister Johane, pleased to have so much new to talk about after the small, same things of the nunnery, was going on, "Lucy hopes her mother will make a marriage for her soon. She says it's impossible having Lady Elyn for a sister now Lady Elyn is married and all. She has all the power to settle her children's marriages. Lady Anneys, I mean. Her husband gave it to her in his will. Even her sons' marriages. Well, just Master Woderove's now, of course."
"Not Miles'?" Frevisse asked, no more than mildly curious.
"Not Miles'. Lucy says she thinks Sir Ralph just hoped Miles would rot. She says . . ."
Lucy seemed to say a great deal, but despite the haste of Sister Johane's tongue, Frevisse did not hurry their walk. At least the weather, after the funeral day's rain, was turned dry and late-summer warm, as perfect as could be wished for the harvest, now moved on to the wheatfield, Frevisse understood. They were nearly to the foreyard and she was wondering rather longingly how the harvest was going at St. Frideswide's when Sister Johane chatted brightly, "Everything is much better here since Sir Ralph died, you know. Everyone is sorry about Tom being dead but everybody's glad about Sir Ralph, even if he was murdered."
Frevisse stopped short and faced her. "Did Lucy say that about her father? That plainly?"
Sister Johane had the grace to turn pink within the white circle of her wimple around her face and she dropped her gaze to the dusty road at her feet. "One of the maidservants did," she murmured; then added with unwilling honesty, "Two of the maidservants."
"When were you talking with the servants?"
Still toward the road, Sister Johane said, "When I was in the kitchen yesterday, making the eyebright and clary poultice for Lucy's eyes. There's a lovely lot of clary in Lady Anneys' garden."
There were so many objections to falling into light talk with servants that Frevisse did not know where to start, nor did she mean to be turned from her disapproval by a surfeit of herbs; but a second's more thought told her there was small point to saying any of the objections since Sister Johane surely knew them already. Instead, surprising herself, Frevisse said, "Dame Claire would be pleased at how much you've helped Lucy with her sore eyes. Whatever you've given Lady Anneys to help her sleep has surely helped her, too, these past two nights."
Sister Johane looked up, openly startled at praise when she had expected rebuke. "It's only . . . it's all very simple," she fumbled.
"When something makes the difference between suffering and not-suffering, it's more than 'only.'"
Sister Johane brightened with pleasure and said, "Thank you," in a way that made Frevisse think that maybe she should find more things for which to praise Sister Johane, both because she deserved it and in the hope that encouraging her skills might serve to draw her away from talk with servants and a silly girl.
The morning passed in what were become usual ways, spent in the garden when Lady Anneys had finished giving to the servants what orders were needed for the day. Sometimes she worked among her flowers and herbs, teaching Lucy and Ursula and asking Sister Johane for all she knew about such medicinal things as she had growing here. Other times, like today, they sat in the arbor, Lady Anneys spinning thread from this season's flax, her spindle's whorl twisting as she worked the thin strands into fine thread, while the girls and Sister Johane sewed at the white linen shirts they had begun yesterday for Miles and Hugh, and Frevisse—because her sewing skill reached no further than hemming—read aloud to them of John Mandeville's travels to the far reaches of the world. It was the only book on the manor besides Lady Anneys' prayer book, unless Father Leonel had some others and Frevisse had not yet asked him, putting off her disappointment if he did not.
When the times came for each of the Offices, she and Sister Johane went inside and up to Lady Anneys' bedchamber to say them. At first Frevisse had half-expected Lady Anneys would ask them to stay in the garden for the Offices and join in with her daughters, the way she and Ursula had done at St. Frideswide's, but Lady Anneys never did. Her interest in prayers—even in going to morning Mass—had gone, nor did anyone else in her family show inclination that way. Frevisse had considered speaking with Father Leonel, to find if there were anything she might do to help, but so far had put it off. There were too many hurts here, both old and new, for her to begin carelessly probing at them, she thought.
Because most of the house servants were gone to the fields to help with the harvest instead of here for the cooking, the midday meal was no more than herb fritters, a cold cheese and onion tart, and fruit. Miles did not appear, but Hugh was there, sitting in the lord's chair where he looked so ill at ease. Lady Anneys was on his left, with Lucy and Ursula beyond her, while Frevisse and Sister Johane were on his right and the hound bitch Baude, round-bellied in whelp, lay on the floor behind the chair.
Talk was small, merely about Hugh's morning spent seeing how the harvest work was going. "Gefori says the weather will hold a few more days for sure before there's chance of rain again," he said, wiping the rim of the goblet he shared with his mother and passing it to her.
"Then it will," Lady Anneys said. "He always knows."
"How?" Ursula asked.
"Reads the signs, he says," answered Hugh.
"What signs?" Ursula persisted.
"I've never asked him."
Ursula gave her brother a disgusted look. He made a face back at her and said, "Ask him yourself if you're so interested."
"I will."
"After harvest," Lady Anneys said. "He doesn't need distracting now."
As they finished the pears baked with spices, Hugh said, "I'm away to Charlbrook Chase this afternoon. We're closing fast on Holy Rood Day." When the mercy-time on game ended. "I want to see how the hunting looks likely to be that way."
"You're not taking Baude, are you?" Lady Anneys asked.
Hearing her name, the bitch raised her head. Hugh reached back and scratched between her ears, saying, "She's too close to whelping. That's another reason to go today rather than some other. I want to be here when she does." He stood up to leave.
"You're going alone?" Lady Anneys asked, her worry ill-concealed.
Hugh bent and kissed her cheek. "I'm not going alone. I'm taking Bane and Brigand. I might meet up with Miles, too."
He bade them all good afternoon; and Frevisse, watching Lady Anneys watch him stride away toward the hall door, Baude following him, saw her lips move, silently bidding him, "Be careful." At the door he stopped to pet the hound, ordered her to stay, then left. Baude lingered in the doorway, still hoping, before giving in to disappointment and going to lie beside the cold hearth.
Lady Anneys rose and led her daughters, Frevisse, and Sister Johane out to the garden again and the tasks they had left in the arbor's shade when called to dinner. One shirt was ready for hemming, and Frevisse worked at that rather than any more reading aloud because Lady Anneys had brought out one of the household accounts scrolls to lesson Lucy on how the accounts could be used to plan the autumn buying of what would be needed to see the manor through the winter.
"You set how much salt we have on hand now against how much we bought a year ago, on the chance we overbought last year and can do with buying less this year," Lady Anneys was saying. "But against that you must needs allow for how well the calving went this spring, and the summer's haying, to judge whether there will be less or more beef to salt down at this autumn's slaughtering. If it's been a poor year for cattle, you'll need less salt because there'll be less meat to cure. Or if it was a good year for cattle but a poor one for hay and there'll have to be a greater slaughtering of cattle we can't over winter, you'll need much more salt for the curing. You see?"
Lucy nodded but looked closer to napping than thinking in the afternoon's drowsy warmth until Ursula raised her head from her sewing and said, "Someone's coming." Her younger ears must have heard footfall or something because the arbor was set so that, from inside of it, the house was out of sight. Lucy instantly said, "I'll see who!" and sprang up to lean out of the arbor.
Probably no one but Frevisse saw Lady Anneys' hands clutch the account roll on her lap and her eyes widen with the unthinking fear of someone always afraid that anything sudden meant news of trouble; or saw her relief when Lucy said, "It's Elyn!," so that she was smiling when her daughter came into the arbor, dressed in a plain blue gown for riding and smiling, too. She exchanged quick kisses with her sisters and mother, bent her head with a respectful, "Good day, my ladies," to Frevisse and Sister Johane, and sat down beside Lady Anneys where Lucy had been, saying as she pulled off her riding gloves, "My, isn't it hot today?"
They all agreed on that before Lady Anneys asked, "Isn't Philippa with you?"
"I left her home. She's my stepdaughter, not my dog. She doesn't have to go everywhere with me," Lady Elyn laughed.
"You didn't ride alone?"
"Sawnder came with me. I left him in the yard with the horses. I can't stay long."
"What's the matter?" Lady Anneys asked.
It was a reasonable question. Lady Elyn was sitting on the edge of the seat, looking ready to spring up again, holding her riding gloves in one hand and drawing them again and again through the other. But she was discomposed to be so readily found out, looked quickly around at everyone, and said as if she could not hold it in a moment longer, "Mother, can I talk with you alone a little?"
Lady Anneys stiffened but said with seeming ease, "Of course. Sister Johane, would you be so good as to take Lucy and Ursula for a walk? Perhaps along the stream toward the village." Well away from hearing anything that might be said in the arbor.
Sister Johane stood up readily, Lucy and Ursula less readily and looking the protest they did not make, but when Frevisse stood up, too, Lady Anneys said, "I'd have you stay, please, Dame Frevisse."
Frevisse sat down again, as unwilling to stay as Lucy and Ursula were to go, and Lady Elyn started to protest, "Oh, Mother . . ." but Lady Anneys silenced her with a look, but waiting until Sister Johane had led the others away toward the garden's back gate before she said, "Dame Frevisse is here to advise and help me. You needn't worry about what you say for her to hear." While Frevisse kept her surprise at that to herself, Lady Anneys laid a quieting hand on her daughter's restless ones and said, "Now, what is it? Do you think you're with child?"
"Oh, Mother. No." Lady Elyn shook her head impatiently. "That isn't it. It's Hugh. No. It's Sir William. It's what Sir William is saying about Hugh."
Lady Anneys' hand tightened on her daughter's and her voice was strained, for all that she kept it low as she asked, "What's Sir William saying?"
"And to whom," Frevisse said.
Lady Anneys cast her a sharp, agreeing glance and added, "And to whom?"
"To Master Wyck today. I don't know if he's ever said it to anyone else."
"To Master Wyck?" Lady Anneys repeated. "Today? Why was he at Denhill?"
"I don't know. Sir William doesn't tell me things. About his will maybe, or some property. I don't know. They were in the parlor. I was going to go in to see if they needed more wine or wanted aught to eat. I supposed they were talking business and it would be better if I went in than a servant. The way you taught me, Mother."
"And Sir William said something while you were there."
"Not while I was really there. It was just as I was about to knock. I heard Tom's name and tears came up in my eyes and I stopped to dry them. Sir William doesn't like tears." New tears welled in her eyes as she said it, but with more indignation than grief she said, "He doesn't care how I feel. I swear it. All he wants is not to be bothered."
"Men are cowards that way," Lady Anneys said evenly. "It doesn't matter if you're in pain, whether of mind or body. What matters is that you don't trouble them with it. I warned you of that when you married."
Frevisse had now and again known men with courage enough to care and to show their care, but she could easily guess Sir Ralph had not been one and Sir William must be no better; but Lady Anneys was taking Lady Elyn back to the point, saying, "It was while you were drying your eyes that you heard something."
"I heard Sir William say . . ." Lady Elyn wiped more tears away, drew a deep breath, and said it all at a rush. "He said it was as well Hugh didn't try to make trouble at the inquest, because if Hugh had, he would have pointed out that when Tom left Denhill, he was alive and that when he was next seen by anyone besides Hugh, he was dead and maybe someone should take closer look at that."
Lady Anneys drew in a sharp breath. "He said that?"
Lady Elyn now could not tumble the words out fast enough. "He said he'd point out that maybe Hugh took advantage of the quarrel to make it seem Tom's death was Sir William's fault but wasn't it more likely Hugh killed him for the sake of having everything for himself?"
Angrily, Lady Anneys said, "There was nothing that showed anything like that. There was no sign Tom's death was anything but accident and ill chance. How could he even think to blame Hugh?"
"Master Wyck said that, too. That there was no proof that way at all. But Sir William said proof didn't matter and he didn't want to have Hugh into true trouble. He would have said it just to draw trouble off from himself if Hugh had tried to make it. I was so angry I didn't even dare go in. I just went away and . . . and came here because I had to tell someone!" Lady Elyn sniffed on her dried tears. Having shifted some of her upset's burden onto someone else, she was beginning to calm. "At least it never came to him really saying it to anyone, so I suppose it's all right. But I thought it was a vile thing for him to think of."
"It was," Lady Anneys said, the words flat and hard.
"Do you think Hugh should know?"
"He should not." Lady Anneys was sharply certain of that. "No one should know it. And you must never, ever, speak of it again to anyone. Even me. _Ever,_ Elyn. Do you understand?"
Startled by her mother's vehemence, Lady Elyn fumbled, "Well . . . yes." She looked aside at Frevisse. "But . . ."
"Dame Frevisse will never speak of it either. None of us will. Even the slightest whisper of something as ugly as that can fly into full-blown rumor clear across the county before you can turn around. Do you understand?"
"Yes." Lady Elyn was impatient at having to say it again. "But what am _I_ going to do? How am I even going to look at Sir William, knowing he thought such a thing about Hugh?"
Lady Anneys' fierceness was suddenly gone. Or . . . not gone, Frevisse thought, but drawn back out of sight, into wherever she kept it. Her face was returned to its usual smooth quietness as she let go Lady Elyn's hands, patted them gently, and said, "You'll find a way. We all have to find our ways." She took her hands back into her own lap, looking at them rather than Lady Elyn as she added, sounding just as she had when lessoning Lucy over the account roll, "Accept him as he is. Submit to what he asks of you. Be dutiful. Demand nothing."
Lady Elyn gave an impatient sigh. "But it makes me so _angry_ he even thought of it."
"Feel whatever you need to feel," Lady Anneys said, "but never let Sir William know it."
Lady Elyn gave another sigh, heavy this time, and stood up. "I have to go back."
Lady Anneys stood up, too, embraced her gently and, as they drew apart touched her cheek lovingly and said, "St. Anne be with you."
"And with you, Mother," Lady Elyn said.
Frevisse watched Lady Anneys watch her daughter leave and did not try to hide what she was thinking when Lady Anneys turned around and looked at her.
"You don't agree with what I told her," Lady Anneys said.
"You know her husband better than I do. You know whether your advice was good."
Lady Anneys sat down, took up the account roll, put it down again, took up Ursula's sewing, searched out the needle thrust into the cloth, and began to stitch the gathering of a sleeve into a cuff, before she said, wearily defiant, "It was at least necessary advice, if she's to live in anything like peace. Sir William won't be changed." She laid the sewing on her lap and met Frevisse's gaze. "You understand that it was a lie? What Elyn said Sir William threatened to say about Hugh? Hugh never harmed his brother."
Slowly, measuring her thoughts and words, Frevisse said, "I think, from what I heard at the crowner's inquest, that the right verdict was given. Your son's death was by mischance." She paused, thinking to leave it there, but after all went on. "What I don't see is why Sir William would think Hugh might try to use his brother's death against him."
She made that a statement, not a question, and waited. Lady Anneys took up the sewing again, stitched a single, jerky stitch, put the shirt down, and said to her lap, "In his will, my late husband made me his chief executor and left me considerable property beyond simply my dower lands. Enough that I could live very well rather than merely in bare comfort. He also gave me control of all our children's marriages and of our daughters until they marry, with disinheritance for any of them who refuses my choice."
"He must have thought well of you to trust you that far," said Frevisse carefully.
"It wasn't trust. It was bribe." Anger and bitterness sharpened Lady Anneys' voice. "In order to have that property and to keep control of my children's marriages, I have to live chastely and unmarried. If I'm unchaste or if I marry again, I lose it all, save for my dower land. The property is no matter. If that were all that stood between me and being finally, fully free of Sir Ralph, I'd spit on his will and let it go. But if I forfeit my say in my children's lives, Sir William takes my place. If I fail Sir Ralph's strictures against me, the girls' marriages and Hugh's all become Sir William's to control and profit from as he pleases."
"And you don't trust him?" Frevisse asked, still very carefully choosing her words.
"I trust him no more," Lady Anneys said coldly and precisely, "than I would have trusted my cur of a husband."
She held Frevisse's eyes in a long look that Frevisse met, letting Lady Anneys see that she understood. And Lady Anneys drew a shaken breath and said, "It helps to say it." She looked at her hands lying on the shirt across her lap and went on, "And while I'm saying so much, I have to warn you that any day now Master Selenger will begin to visit me here."
"The man who came to see you at the nunnery."
"Him. Yes."
"He was at your son's inquest and funeral but kept his distance. He was with Sir William, I thought."
"He's Sir William's steward and Philippa's uncle." Lady Anneys lowered her gaze and began to smooth the shirt across her lap. "He began coming to see me after my husband's death. He says . . . he said once, when he had the chance to do out of anyone else's hearing, that he's long loved and wanted me."
"Wanted you," Frevisse repeated, keeping the words as empty of meaning as she could.
"For his wife. He claims."
"You don't believe him?"
"He's Sir William's brother-in-law and his steward. How likely do you think it is that he doesn't know about the will and how much power would come into Sir William's hands if I marry again or am unchaste?"
"It isn't open knowledge?"
"We agreed among ourselves—Tom, Hugh, Miles, Sir William, and I—that no one else needed to know."
"Your daughters don't know?"
"Not even Lady Elyn. Of that I'm sure because if she knew she'd talk of it. Discretion is not her better part, unless she's frightened. The way she was frightened by what she overheard today because she has wit enough to know what trouble that kind of talk could make."
"But you think Sir William has told Master Selenger?"
"I don't know. But if Master Selenger does know and hasn't said, then I have to fear he's working to Sir William's purpose, to bring me to forfeit my place in the will, giving everything over to Sir William."
"Is it the kind of thing Sir William would undertake? Is Master Selenger the kind of man who would do it?"
"I haven't enough trust left toward men to say Master Selenger wouldn't do it. As for Sir William, look what he would have done to Hugh if he'd felt threatened at the inquest. With rumor creeping about that Hugh might have killed his brother, who would be eager to marry their daughter to him? Only Sir William. He and Sir Ralph were set on Philippa marrying Tom . . ."
"Were Tom and Philippa set on it?"
"There was no reason for them not to be. They knew each other and there was nothing against either one."
"But she'll marry Hugh instead?"
"The marriage is a good one for all the reasons there were before and no reasons against it. It's what Sir William would do with the girls' marriages that I worry over."
Frevisse sat silent, considering, and finally asked, "What if Master Selenger _isn't_ working to Sir William's purpose?"
"You mean what if he does truly desire me?" Lady Anneys shoved the shirt aside, onto Lucy's sewing basket again. "Then I have only myself to fear. No." She made a sharp dismissing movement of one hand. "That's wrong. I doubt there's enough womanhood left in me to rouse to any desire. What I'm afraid is that somehow some _seeming_ might happen that could be used against me." She dropped her voice to a strained whisper, sounding almost ashamed to say it. "What I fear is that if he can't seduce me to marriage, Master Selenger may simply claim we've done . . . wrong together. That would be enough to serve Sir William's purpose."
Frevisse briefly wondered whether Lady Anneys were grown so cold as she said. At a guess, she was hardly to forty years and the body's fires were rarely burned out by then, however weary the heart and mind might be.
But that same weariness could lessen the guard against the body's lusts and maybe, whether she admitted it to herself or not, Lady Anneys did well to fear, if only her own body's possible treachery. But if she were right in her suspicions against Sir William and Master Selenger . . . _if_ she were right . . . then she had much more than that to fear, and slowly Frevisse said, "So you wanted me here to stand guard of you against Master Selenger, the way I did at St. Frideswide's. But you have family and servants here who could do that."
"I don't dare let Hugh or Miles know of this! If ever they suspected such a thing . . ." She made a taut gesture of helpless fear of their anger and what might come of it. "And the girls are too young and any servant's word not enough. Your oath that I'd done nothing wrong would have weight, if it comes to that, but yes, I want you to stand guard between me and Master Selenger."
Rather than immediately answering, Frevisse looked away from her, stared out of the arbor to the bright sunlight on the garden path while inwardly seeing the layers of Lady Anneys' fears—of Sir William and Master Selenger, who possibly intended ill against her; for what Hugh and Miles might do in anger; for her daughters if she lost the care of them. If Master Selenger and Sir William were indeed sporting with Lady Anneys' life the way she feared, then she needed whatever help Frevisse could give. But . . .
Frevisse returned her gaze to Lady Anneys and said, "We can't stay here forever, Sister Johane and I."
"I know. I just need . . ." Lady Anneys shook her head, impatient at being unable to find the words. "When Hugh came for me, to tell me Tom was dead, I didn't have time to think anything through. Tom . . ." Tears flooded her eyes. She fought them, steadied her voice, and said, "I couldn't think of anything but that he was dead and I had to be back here and I was afraid. I needed . . . needed . . ."
"Someone to somewhat guard you until you're certain of things again. Until you have chance to find your balance," Frevisse offered.
"Yes." Lady Anneys put a world of heartsick weariness into the word. "Yes. That." Relieved to have someone say it for her.
Voices warned that Sister Johane, Lucy, and Ursula were returning, and Lady Anneys straightened her back, wiped her eyes free of tears, and smoothed her face to calm again, so that to outward seeming there might have been nothing but the day's quietness in her. But her gaze sought Frevisse's, asking, and quietly to her quietness Frevisse said, "I'll do all that I can."
**Chapter 12**
Hugh made no haste homeward despite the day was wearing out and suppertime was near. The small excuse he would give if asked what was wearing out and suppertime was near. The small excuse he would give if asked what kept him so long was that in the afternoon's heat neither he nor the hounds had felt like hurrying. That was somewhat true enough; Bane and Brigand, pacing patiently beside Foix, were lightly panting and he had taken off his doublet and untied his shirt at the throat. But the true reason he was making no haste—the reason he hardly admitted to himself, let be to anyone else—was that he did not want to be home. Away from the manor, he could almost believe things were as they had been those few weeks after his father's death. Could almost believe that when he reached the manor he would see to the hounds, talk with Degory awhile, wander to the hall, and wash off the day's dirt while listening to Tom complain over some village disagreement he had probably enjoyed settling. Miles would be leaning easily on an end of the high table, teasing Lucy or Ursula. Mother would be coming from the kitchen, having seen that supper was nearly ready and tucking a stray strand of hair back under her wimple. Hugh could not remember her hair ever straying loose while Sir Ralph was alive.
Those had been good days, with Tom happy to have all his own way with the manor, Miles not constantly ready to be angry, Mother content and smiling, no one wary with waiting for Sir Ralph's next anger. Even when Mother and Ursula were gone to the nunnery and Lucy to stay with Elyn, the days had gone on being good.
And now they were not.
He did not mean to turn aside from the greenway as he came through the woods. He had not been to the gathering place since the day Sir Ralph had died and had no thought to go there now, but as they reached the side-trail that went to it, Brigand lagged, looked up at him, and whined, letting Hugh know he was thirsty and saw no reason he should wait until the kennel when water was close here. Hugh hesitated but suddenly wanted a drink, too, and why not of fresh, cold spring water? It was hardly as if Sir Ralph had died there. And without admitting to his thought that he would now be even later reaching home, he turned Foix into the side-trail.
Bane and Brigand went eagerly ahead of him. It was their tails stirring into a pleased wagging that told him someone was in the clearing, someone they knew, and so he was only half-surprised to see Miles sitting sideways on the wall around the spring, staring down into water, so far into whatever he was seeing or thinking that, with the water's soft burbling over the pool's edge to cover any small sounds of their coming, his first warning he was not alone was Bevis standing up from the grass beside him. He put a hand on the hound's shoulder and twisted a little around to quirk half a smile at Hugh dismounting at the clearing's edge. "Caught me doing nothing," he said.
"There's two of us then." Hugh wrapped his reins over a branch low enough that Foix could reach the grass and went to sit on the low wall across the spring from Miles. The hounds, having circled with Bevis a few times and decided they were all still friends, were lapping busily from the stream, and Hugh leaned over to splash water up into his face before drinking from his cupped hands.
When he straightened up, Miles was watching him with a shut, unreadable look.
"Hot, isn't it?" Hugh said for the sake of saying something.
"Hot enough." Miles dropped his gaze and plucked up a handful of grass. The hounds, having finished their lapping, collapsed on the grass with sighs of pleasure, and Miles began to drop the grass blades one by one into the pool, watching one swirl away over the edge before dropping in the next. The forest's late afternoon quiet closed around them, and Hugh, lulled by the stream's burble and chuckle, slipped into a small measure of welcome unthinking until Miles said, "You'll agree soon with Sir William over marrying Philippa?"
"What? Philippa? She . . ." Almost Hugh blurted out "She's meant for Tom" but stopped himself and said instead, even-voiced, "I can't marry her. You want her. She wants you."
Miles shrugged.
"Doesn't she?" Hugh persisted.
Miles shrugged again. "It doesn't matter, does it?" He dropped another grass blade, with great care, into the very middle of the pool. "With Tom dead, Sir William will be looking for you to marry her."
"I don't want to marry her."
He said it maybe more strongly than need be but he hated when Miles went distant like this. Hated it worse when Miles finally looked at him blank-faced and said with no feeling at all in his voice, "You had better. Because I won't."
Hugh glared at him across the pool and demanded, "Why not?"
Miles went back to dropping grass blades into the water. "She'd lose everything and gain nearly nothing."
"She'd gain you."
"As I said. Nearly nothing."
There was no point in trying to reason with Miles in his black humour but Hugh said, setting his stubbornness against Miles', "You might as well marry her because, come what may, I'm not going to."
Miles did not answer that, only went on dropping grass blades. One. Two. Three. Then said quietly, barely above the sound of the flowing water, "Remember how we used to plot that we'd run off if ever we saw the chance? Join a cry of players or else cattle drovers, maybe? Anything to get away from here forever."
"I remember. You even tried to go. Twice."
"Only to be hauled back both times, beaten and bloody."
"At least you tried." Guilt stirred in Hugh. "I never did."
"Because you had the sense to know he'd never let us go. Not even me."
"It wasn't more sense. I wasn't as desperate."
"Desperate, or else mad to think there was ever escape."
There was a bleakness in Miles' voice that Hugh could not answer and silence came back between them, Hugh watching the water flow, hoping they were done with talk, until Miles said, even more quietly, "Marry Philippa."
Hugh jerked up his head too sharply, hurting his neck. Sharp with the pain, he said, "No! Not if you—"
"Marry her," Miles said, still quietly, not looking up, watching his last grass blade swirl away out of the pool. "If you don't, who knows what kind of man Sir William will marry her to. Marry her so she'll be safe."
Staring at the top of Miles' head, Hugh looked for some answer but found none, and said finally, bitterly, "I'm glad Sir Ralph is dead."
"The pity," Miles said, "is that it didn't happen sooner." He sprang to his feet, startling the hounds, who all sat up, looking around for whatever the trouble was. "Come on. Let's go home. I'm hungry."
He seemed to have shaken off whatever humour had been on him, but they said little as they walked together, Hugh leading Foix and the hounds ranging around them. They left Bane and Brigand at the kennel with Degory, who was just back from helping with the harvest and said all was going well, and took Foix to the stable, where Gib was just back, too. With Bevis still at Miles' side, they headed for the hall then, their shadows long across the foreyard ahead of them, and were met by Lucas, the village reeve, coming out of the hall doorway. He gave Hugh a deep bow and Miles a brief one and said, "I was come to find you, sir. This was something you needed to know as soon as might be." He held out a sickle in two pieces, the metal blade broken off from the wooden handle. "It's the metal broke, not the handle," Lucas said. "I'd not say it was Hal's fault it broke but it's the manor's sickle, not his, and he's all tied up in fits that you'll make him pay for it or have it out of his hide."
Hugh's insides turned sickly over with the thought that with Tom dead he now had the kind of power that could make people afraid of him that way, and maybe too quickly he said, "I won't do that. Let me see it."
Lucas gave the pieces to him and he easily saw for himself how worn thin the haft was where it had gone into the handle. He remembered Tom had been at Sir Ralph early in the summer about the need to remake some of the sickles before harvest-time. Sir Ralph had refused, grudging the cost because the manor had no smith and the work would have had to be taken elsewhere. Once he was lord, Tom had probably meant to see to it but had not had time before . . .
"So we're a sickle short," Hugh said. And therefore a worker short. How much would that set back the harvest, he wondered.
"Nay," Lucas said. "As it happens, Master Woderove bought a new one that time he went to Banbury about Whitsuntide. So all's well there. But this one needs remaking and there's always chance another one may go, so I thought you'd best know."
"A new one?" Hugh echoed. Where had Tom found the money for that? But aloud he only said, "Yes. Thank you. I'll see to having it done."
Lucas bowed, took a step toward leaving, but stopped and said, "Begging your pardon if I speak out of turn, but you might want to talk a bit to Father Leonel. He might be able, like, to tell you things."
"Tell me things?"
"About sickle blades and such like."
Before Hugh had sorted himself to ask another question, Lucas quickly made another bow and left. Hugh watched him go, looked down at the broken scythe in his hands, looked after the now-vanished Lucas, and finally looked at Miles in hope of an answer from him.
Miles shrugged and said, "You'd best talk to Father Leonel, I guess."
"What am I supposed to say? 'Father, have you anything to confess to me?'"
"That's as good a way as any." Miles grinned, the task not being his, and went inside.
Hugh stared broodingly at the white-plastered wall beside the doorway, knowing yet again how little he liked being lord of the manor. But there was no use putting off what was probably better done sooner than later and he went, first, to find Lady Anneys, to ask if it would be no trouble to have Father Leonel to supper, then sent Lucy to ask if he would come.
The priest returned with Lucy as they were readying to sit down. He took his place on Hugh's other side from Lady Anneys, Lucy, and Ursula, with the nuns sitting beyond him and Miles sitting between them, being courteous to both. Uncourteous though Sir Ralph had been, he had forced courtesy at table on everyone else. "I'll not have you disgracing me in front of others," he had said. A sure way to goad at least Miles into rampant discourtesy; but because Lady Anneys had asked it, too, and taught them, he had learned and this evening gracefully served the nuns the roast chicken in black sauce and a barley frumenty set before them and asked about their day. Lady Anneys talked with Father Leonel about a village woman who had badly cut her foot on a stone just before harvest started while Hugh, passing a small piece of chicken to Baude lying behind his chair, wondered what he could ask Father Leonel.
The day's last sunlight, slanted across the far end of the hall's east wall and banded with shadow from the tall, unglassed window's wooden mullions, was slipping upward as the sun sank downward. Dusk thickened in the hall but there was light enough by which to finish the meal and talk, with no need for candles. Afterward, for the while until bedtime, they would probably go into Lady Anneys' garden or else to the parlor, and Hugh wished that was all he need think about this evening; but when the fish in a green tart and the carrots roasted in herbs, oil, and vinegar had been served, Lady Anneys claimed Frevisse's attention with a question about what flax was grown at St. Frideswide's, Sister Johane was talking to Miles, and Hugh took the chance to ask Father Leonel, "Ivetta's foot is healing then?"
"The poultice Lady Anneys recommended seems to be drawing the poison out," the priest said. "The cut is so deep, though, she likely won't be walking well before Martinmas. She's worried less about her foot, though, than that she can't do her boon work this harvest. She fears you'll either demand rent money she can't pay or else force her out."
"Force her out?" Hugh repeated, bewildered. "Because she's hurt?"
"Because she can't do her share of the fieldwork she owes you and can't pay you for either," Father Leonel said patiently. "You'd be within your rights, as lord of the manor."
Hugh gave intent heed to spearing a piece of carrot on his knife point, giving himself time to stop his insides' turning before he finally said, "Weren't you saying with Mother that she could help here in the kitchen until her foot is healed? In place of her fieldwork?"
"That would need your agreement," Father Leonel said in the careful way he would have said it to Sir Ralph.
Hugh held back an urge to smash his knife down on the table and yell, "Don't talk to me like that!" Instead, because lord of the manor was a thing he was, whether he liked it or not, he laid his knife down quietly, spread his hands flat on the tabletop to either side of his bread trencher, and said, "Tell me about her."
Father Leonel did—how Ivetta had never given trouble or failed in her duty before now, nor had her late husband, and that their son kept his holding well and never failed of his rent or boon work. "A good family. First and last, a good family. It would be shame to—"
He broke off abruptly. Hugh knew well enough that he was thinking how it had always been ill to tell Sir Ralph what he should or should not do—and said, hating anyone could think he might be like his father that way, "It would be shame to repay them with unkindness in her need. She's welcome to work in the kitchen this while for her boon work. Would it help if I sent a cart for her every day, so she doesn't have to walk here?"
The relief that swept over Father Leonel's face was even better payment than his quick thanks; and on Hugh's other side Lady Anneys briefly laid a hand over his own, telling Hugh that despite she was still in talk with Dame Frevisse, she had heard and approved his answer, too. On the warmth of that, Hugh waited while Alson from the kitchen served out the meal's final dish—apple pudding sprinkled with nutmeg—in wooden bowls to each of them, and said when Alson was gone, "This afternoon Lucas said I should maybe talk to you about some things, Father."
"Anything in particular?" Father Leonel asked lightly, openly much eased now the matter of Ivetta was settled.
"Sickle blades and suchlike, he said." From the side of his eye Hugh saw the priest's hand, about to dip his spoon into the cream, cease to move. Trying to seem he had not seen that, Hugh went on, "He was here to tell me that an old sickle blade had broken and that it was a good thing Tom had bought a new one at Whitsuntide. Then, as he was going, he said you might be able to 'tell me things.'"
Father Leonel set down his spoon, drew back his hands, and folded them together on the edge of the table. "Now?" he said, very softly.
Hugh put down his own spoon, knowing food would not go past his throat's sudden tightness even if he did pretend to eat. He turned in his chair toward the priest and said, "Now would likely be good." Wishing he had not chosen now to ask, here with everyone to hear. He had thought it would be easier, done friendliwise over a meal. Instead he was suddenly afraid it was something he was going to wish he had never done at all. "Unless we should go apart," he said hurriedly and too late. The change in their voices and in Father Leonel had drawn Lady Anneys' heed toward them and now everyone else along the table was looking, too.
"No." Father Leonel pushed back his shoulders, straightening his bent back as much as might be, and faced Hugh squarely—an old man whose body was failing him but not his courage. And that startled Hugh because "courage" was not a word he had ever put to Father Leonel shambling about the manor in his old priest's gown and worn shoes, always worried about one person or another. He had come to the manor when Hugh was small, had never had interest in hunting and had therefore been despised by Sir Ralph; but he had gently taught all the boys and the girls their reading, writing, and numbers and had kept the manor accounts for Sir Ralph until Tom was old enough to have a hand in them and after that he and Tom had kept them together. A kindly, useful man and that was all, Hugh would have said if there had been need to say anything about him at all, until now he faced Hugh and said firmly, "There's no need to go apart. It's not a thing of shame. What Lucas meant was that Tom and I had been deceiving Sir Ralph with the manor accounts for years. I did it when I kept them by myself. When Sir Ralph set Tom to be steward and I had to show him the accounts, Tom saw almost at the first, without my telling him, what I had been doing. From then on we did it together."
The silence along the table was complete and stunned, until Lady Anneys breathed, "Sir Ralph would have all but killed you if he'd found out."
The priest smiled at her. "I trusted to God's mercy that he'd not find out." He sobered. "But likewise I was willing to pay the price if he did."
"But why?" Hugh asked. Then answered for himself. "So there would be money for such things as spare sickle blades."
Father Leonel smiled the way he had when Hugh had been especially apt at some lesson. "For such things as that, yes, and sometimes we'd write that someone had paid a fine when they had not, if Tom and I thought the fine unjust or someone was unable to pay it for good reason. Things like that. Never much. We only did it because there were always needs but Sir Ralph never cared for anything but himself and . . ."
Father Leonel stopped, not because his courage failed him there, Hugh saw, but out of pity. That pity hurt worse than anger could have and Hugh finished for him, not able to keep bitterness out of the words, "For anything but himself and his hounds and hunting. It was always everything for his hounds and hunting."
"And to hell with the rest of us," said Miles.
Hugh shoved his chair roughly back from the table—careful not to hit Baude lying there—and stood up. "And you're afraid I'll be the same," he said. It was a struggle to speak evenly but somehow he did. "Tomorrow morning you can show me what you and Tom have done. After that we'll make it so there's no more need for deceiving anymore. No," he said as everyone started to rise with him, and ordered Baude struggling to pull herself to her feet, "Stay," before swinging around, away from his chair and through the nearest doorway, into the parlor, shutting the door hard behind him.
But that was not far enough away and he crossed to the room's one window, its shutters standing open to the warm evening, and swung himself over the sill and out. If he had been younger, there were places enough where he could have gone to hide, but he was too old for that and only went to his usual refuge, the kennel. Degory was scrubbing out the dogs' feeding dishes after their supper and welcomed him much the same way the hounds did—without surprise or need to talk. He went on with his work and Hugh squatted on his heels just inside the gate, welcoming one hound after another as they ambled over to snuffle at him and be briefly petted. Only the lymer Somer stayed with him, flopping down with a hearty sigh in front of him, and Hugh was absently fondling her ears when Miles appeared, leaned on the gate with deliberant ease, and said nothing.
Neither did Hugh. Degory finished with the dishes, judged their silence with a wary look, and slipped out through the kennel door, away on some business of his own. The silence drew out until Hugh gave way and said, half-bitterly, half-bewildered, "If it's been secret all this long, why did Lucas set me on to Father Leonel like that?"
Miles did not answer for an uncomfortably long time. Though the west was still ablaze with orange from the vanished sun, the shadows were gathered deeply blue in the kennel-yard and Hugh could read Miles' face no better than Miles could probably read his; he had to wait for answer until finally Miles said with the gentleness that—coming from him—was always surprising, "Maybe the folk are as tired of walking wary as the rest of us are. Maybe Lucas wanted it settled what kind of lord you're going to be."
Hugh stood up, startling Somer. "Walking wary?" he protested fiercely. "You know me. You know I'm not Sir Ralph."
"You're not Tom either," Miles pointed out kindly. "What you've been is Sir Ralph's huntsman. Hounds, hunting, Sir Ralph, and you. That's what the folk here have known. What are Lucas and everyone else to think but that they'll matter less to you than the hounds and hunting? Just like with Sir Ralph."
"What do _you_ think?" Hugh asked harshly.
Slowly, seeming to make sure of the words as he went, Miles said, "Sir Ralph used every spare penny—and sometimes pennies that weren't to spare—for his hounds and hunting and be-damned to the rest of us. I don't see you ever be-damning."
Somer had left, offended. Bounder, one of the younger hounds, wandered to Hugh, tail swaying behind him, and Hugh absently took his great head between his hands, stroking the broad forehead while saying slowly to Miles, "No. I don't think I'm any good at be-damning."
"And pleased the manor folk will be to learn it," Miles said. "But they'll have to learn it. You're going to have to show them."
"Can I show them?" Hugh had never said it aloud, never even let himself clearly think it until now. "How good am I going to be at this? I won't be-damn like Sir Ralph, but I'm not Tom. There's nothing to say I won't be damned bad no matter how I try."
Offensively cheerful, Miles said, "We'll find out, won't we?"
Even knowing that the cheerfulness was meant deliberately to goad him into cheerfulness himself, Hugh wanted to throw something at him. Unfortunately, a clean-kept kennel-yard offered little to throw besides straw and dogs, and before Hugh came up with at least some words to throw, Miles shifted aside for Degory to lean on the gate, too, carrying a slice of the fish tart in one hand and a thick piece of bread folded around other things for the rest of his supper in the other. The hounds gathered to the fence, too well trained to grab and snatch but assuring him of their coming gratitude if he cared to share with them.
"I don't eat your suppers," Degory told them, then said to Hugh over their backs and waving tails, "That Master Selenger is here again. He'll be going home by moonlight, won't he?"
**Chapter 13**
It was that evening Frevisse could no longer hide from knowing how wrong things were at Woodrim. After Hugh's leaving, supper finished in a stiff silence and eyes kept to bowls and tabletop until Lady Anneys asked Father Leonel to give the final grace. That done, Ursula hurriedly slid off the bench and went to open the parlor door too quickly for her mother to say more than "Leave him be . . ." before Ursula said from the doorway, "He's gone," all disappointment and worry.
"He's gone to the kennel then," said Lady Anneys. "Miles."
Miles rose quickly, gave her a slight smile, a brief bow, and left.
Calmly, Lady Anneys invited Father Leonel to join her in the garden for a while but he declined. "Not because I don't want to talk about . . ." He made a vague gesture of distress, to include all there might be to talk—or not to talk—about.
Lady Anneys caught his hand and held it, saying affectionately, "I wasn't going to ask you anything. I only wanted your company."
He clasped her hand in both of his. "Bless you, my lady. But I'm promised this evening to Roberd and Mariote. He's Lucas' younger son, you remember? They're planning to marry just after Michaelmas. We're to set when the banns will be and decide other things tonight."
"That I'll not keep you from," Lady Anneys said. "Be sure to let me know with what I can best gift them when the time comes."
"I will, my lady."
Father Leonel blessed her with a quickly sketched cross in the air between them. He was making another in a general way at Frevisse, Sister Johane, Lucy, and Ursula scattered along the table, when Lucy exclaimed, "Someone is coming," and dashed to the window, looked out, and swung around to say with a glowing look at her mother, "It's Master Selenger!" She dashed back to catch Ursula by the hand. "Come on. Let's walk Father Leonel to the village. Are you coming, Father? Sister Johane, Dame Frevisse, you'll come with us, too?"
Even if she had wanted a different choice, Ursula was given no chance for it. Lucy was already dragging her toward the outer door, Father Leonel following, smiling, and after the barest hesitation and a look at Frevisse, who refused with a small shake of her head, Sister Johane went, too.
"Dame Frevisse!" Lucy insisted from the hall's far end.
"No," Lady Anneys said, too low for anyone but Frevisse to hear, all her smiling ease of a few moments ago gone.
Frevisse waved the others onward. They met Master Selenger at the door, their brief exchange of greetings among them all giving her chance to say to Lady Anneys, "It might go easier if your daughters knew you didn't want to be left alone with him."
"They're the least burdened by everything that's happened," Lady Anneys said. "I'd like to keep them that way as long as possible. Ursula is still so young."
Remembering how Ursula had taken the news of her father's death with relief rather than grief, Frevisse doubted Ursula was so young as her mother thought her. Grief had come only with her brother's death and surely left her even less young than she had been before it. But this was hardly the time to take that up with Lady Anneys, who was moving from behind the table, down from the dais to meet Master Selenger coming up the hall toward her. Frevisse, tucking her hands into her opposite sleeves, followed her, eyes lowered in the seeming of quiet nunhood but not so far she could not see Master Selenger meet Lady Anneys with, "Good evening, my lady," and a bow and a hand held out to take one of hers.
Seeming not to see his hand, keeping her own folded at her waist, one over the other, Lady Anneys said with no particular feeling to the words, "Good evening to you, too, Master Selenger. What brings you here?"
By then Frevisse was beside her, and Master Selenger, his hand returned to his side, made her a bow while answering, "Hope for the pleasure of your company and to ask about Lady Elyn."
Alarm sharpened Lady Anneys' voice. "She left here hours ago. Isn't she home yet?"
"Yes! Oh, yes," Master Selenger said with instant, matching alarm. "I'm sorry. I didn't mean to fright you. She's home and safe. But . . ."
He cast a meaningful look at the servants now coming to clear supper's remains from the table. "We'll be cooler in the garden, don't you think?" he asked.
Lady Anneys murmured unwilling agreement but added, "Dame Frevisse, you'll come, too?"
Frevisse accepted with a slight bow of her head.
What Master Selenger thought he kept to himself nor, when they were in the garden, did he offer his hand to Lady Anneys again even while they walked side by side along the path. He could take an unspoken suggestion when it was given, Frevisse thought, following behind them. Except he as yet refused to understand that Lady Anneys did not want his company.
The garden was in deepening twilight but not much cooled by the slight wind beginning to stir the evening air. The day's warm scents of flowers and herbs still lingered and a last few bees were bumbling in the bee-flowers, late at going hive-ward. Lady Anneys paused near the door to break off stems of fern-leaved tansy for herself and Frevisse to keep off whatever evening midges might seek them out, but left Master Selenger to pluck his own, which he did. Since leaving the hall, none of them had spoken. Lady Anneys led the way to the long, wood-sided, turf-topped bench along one side of the garden's wall and sat down, leaving room for Frevisse between herself and one end of the bench and nodding Master Selenger toward the bench's far end, well away from her.
He somewhat took the hint, sitting not altogether to the bench's end but an arm's length and a little more away as Lady Anneys demanded, "What about Lady Elyn, then?"
"She came to see you this afternoon."
"She did." Lady Anneys said the words flatly and let them lie there, leaving it to Master Selenger to make of them what he would. She was looking not at him but across the path and down at a cluster of red gillyflowers so that the soft fall of her veil on either side of her face served to hide her from him and Frevisse both.
"Why?" he asked.
"To see me. Isn't that reason enough?"
"She seemed unhappy when she returned."
"Sir William sent you to find out why from me, rather than ask her himself?"
"He asked. She said she wanted to see her mother."
"Why wasn't that enough?"
Master Selenger very slightly smiled. "Because of the way she sniffled while she did it and wouldn't look at him. He's worried there's something wrong that he should know." Master Selenger paused, then added, subdued and apologizing, "Besides what's obviously wrong, of course. He would have come himself but won't until he's asked. Besides, he knew I"—Master Selenger's voice was very low—"would not mind the chance."
Lady Anneys still had not looked at him, nor did her voice give anything away as she said mildly, "I'm grateful for Sir William's concern and consideration, but since she was unhappy when she came here, the reason for it would be better sought there than here."
"Save that she might have spoken to you more freely than she would to Sir William and you could advise him of what best he might do."
Lady Anneys drew a long breath and sat up straighter, still without looking at him. "That would be somewhat betraying my daughter's trust."
"If, truly, it's something Sir William can't help, then of course you won't tell it," Master Selenger said.
"Such as 'women's problems,'" Lady Anneys said, still mildly but probably fully aware that Master Selenger instantly shifted uneasily. There were few things so sure as "women's problems" to set a man back.
"Um," he said. "Yes."
"Or," said Lady Anneys, still mildly and toward the gillyflowers across the path, "Sir William may be worried that I'm trying to turn her against him because of her brother's death."
Master Selenger went very still and in the blue gathering dusk Frevisse could see his eyes searching for some clue in Lady Anneys' faceless stillness to her seriousness in that.
She answered his unasked question, "I'm not."
Master Selenger went on looking at her with a worry Frevisse could not quite read. Then he reached a hand toward Lady Anneys' lying quietly together in her lap; but her shoulder nearest him jerked forward, warning him away, and he took his hand back and after a moment tried, even more gently than he had been speaking, "Lady Anneys—"
Her own voice suddenly crisp, she interrupted him. "I'll tell you this much." She raised her head and finally looked at him; Frevisse wished she could see her face. "Lady Elyn came to me with a woman's worry. I advised her just as my mother advised me before ever I married—that a wife's duty is to submit and accept. So I advised her and so I hope she will do. Will that satisfy Sir William?"
Uncertainly Master Selenger said, "It should."
Lady Anneys nodded once and looked away again, letting him see she was waiting for him to leave; but he went on sitting there, still looking at her and silent as she was. In the quiet, small birds were twittering and settling for the night among the vine leaves over the arbor and from somewhere, faintly carried on the small wind, came distant laughter. From the village, Frevisse supposed; she did not think she had heard anyone here at the hall laugh since she had come, save maybe Miles, and darkness ran under his laughter.
But then darkness seemed to run under and behind everything and everyone here.
And from more than Tom Woderove's death.
Why that thought should come to her so clearly here in the garden's quiet, Frevisse did not know, nor did she fully understand what it meant, but she followed it, to find where it would go. Sir Ralph was the core of it, she thought. Assuredly he seemed to have created darkness enough in his life that it still lay over everyone here. Tom Woderove's death had only added to it. But there was some shadow more than that. Something sharper, newer. There was . . . fear?
Frevisse moved carefully around the edge of that thought, looking more closely at it. Lady Anneys was afraid, surely, and admitted as much. She was afraid of Master Selenger because of the trouble he could bring on her; and of Sir William because if she was right about him, he was the cause of that trouble; and for her children—afraid not only of losing more of them but of how amiss their lives might go. But those were fears every parent had, and her worry over Sir William and Master Selenger was reasonable, too; but both those fears had shape and boundaries. It was her fear beyond that that Frevisse did not understand. There was a ceaseless wariness in Lady Anneys, a ceaseless waiting for something _more_ to happen, as if some other, deeper, secret fear were feeding her more open, reasonable ones.
And though to say Hugh or Miles or Ursula were afraid might be to say too much, still, there was something . . . There was a wariness in them, too. The sense of a guard being kept. Against what?
Master Selenger stirred and said slowly, as if he regretted the need, "There's something else, my lady."
For the first time Lady Anneys let impatience into her voice. "What?"
Both looking and sounding on the edge of apology, Master Selenger said, "Sir William asked me to speak to you about Hugh's marriage to Philippa."
Lady Anneys lifted her head and faced him again, said nothing, then said, with each word carefully separated from the others, "Hugh's marriage to Philippa?"
"My lady—" Master Selenger began.
Lady Anneys interrupted him with cold anger. "You can tell Sir William that he'll be told when anyone here is ready to discuss marriage and that until then I do not want to hear about it from him or anyone else."
"You can tell him, too," Hugh said over the rear gate, "that when and _if_ the time comes to talk marriage, _I_ will be the one to talk to." He shoved open the gate and came in with Miles behind him. "Not my mother. She has grief and troubles enough without having to deal about that or with Sir William. Especially now."
Master Selenger stood up. "Hugh. Miles. Lady Anneys and I were—"
"We heard," Hugh said. The arbor and the evening's shadows had hidden his and Miles' coming along the cart-track until they were at the gate. There was no knowing how much they had heard, but like his mother, Hugh was angry. Unlike her, he was not cold and sounded ready to include Master Selenger in his anger.
Lady Anneys stood up, too, saying quickly, "Master Selenger was only asking because Sir William told him to."
"Then Master Selenger can take our message back to him that no one here is ready to talk marriage yet," Hugh answered, his look fixed on Master Selenger.
Master Selenger accepted that with a small bow and started, "Sir William only thought that—"
"Sir William can keep his thoughts to himself," Hugh said tersely.
"—that the marriage would reassure everyone that there's peace between the families."
"They can be reassured by the fact that we're not openly fighting."
" _Is_ there peace?" Master Selenger persisted.
"Yes. There's peace," Hugh snapped.
"Simply not ease," Miles put in quietly. "Ease will take longer than peace."
"Tell Sir William we need time," Lady Anneys said. "That's all. He's simply too soon with it."
Frevisse saw Master Selenger's swift look from her face to Hugh's and Miles' before he bowed to her and said, "I'll tell him so, my lady. He'll understand."
He held out his hand for hers. It was a reasonable courtesy but unreasonable here and now since she had so lately refused it. But after only the barest pause and probably for the sake of keeping up a seeming of courtesy in front of Hugh and Miles, she gave her hand to him in return. He bowed over it, did not make so bold as to kiss it but, when he straightened, held it a moment overlong, taking the chance to gaze into her eyes. Only the instant before she would have snatched her hand away did he let her go, smoothly turning to bow to Frevisse and bend his head to Hugh and then to Miles, who said, "I'll keep you company to the yard," in a way that warned he would see him out of the yard and well away, too.
While they left, silence except for the settling rustle of the birds in the arbor vine filled in the garden. Not until they were gone did Hugh ask, "Is that all he came for? To talk about Philippa's marriage?"
Again Lady Anneys paused, barely, before answering, "He asked first why Elyn came here today."
Hugh frowned, puzzled. "Why shouldn't Elyn come here?"
"She was upset about something. Sir William was worried for her because she wouldn't say why. I told Master Selenger that we talked and that I reassured her."
"Is she with child?"
"No. It was a woman trouble."
That put Hugh off that track as thoroughly as it had Master Selenger, but he went to, "What about this marriage business?"
"Nothing beyond the ordinary. Philippa's marriage to Tom had been purposed for years. You know that and why."
"Because adding Sir William's manor to ours is sensible. But not as sensible as it was before Father married Elyn to Sir William."
"Sir William was thinking to marry again. Sir Ralph offered Elyn to him as a way to double-guard our interest. Nor was Elyn unwilling."
"Of course Elyn wasn't unwilling," Hugh said, sounding impatient at being told what he knew too well. "And even if Elyn does bear a child or children, Philippa has enough inheritance from her mother that her marriage to Tom would still have been a profitable thing. I know all that. I suppose what Sir William wants is assurance I'll marry her in Tom's place. The trouble with that is—"
Either catching his impatience or else giving way to the anger she had not dared show Master Selenger, Lady Anneys said sharply, "The trouble is that no one ever seems to wonder what Philippa might feel or want in the way of marriage. That's the trouble and the pity. You've never thought about what she might want in the matter, have you? Of course you haven't. No man ever does. All that's ever asked of women is to submit and accept. That's what I told Elyn today. Submit and accept. Just like every woman has to do."
Frevisse had not found, over the years, that simple submission and plain acceptance were all that common among women, whether maidens, wives, widows, or nuns. For some women, yes, for some women, no, with most women living somewhere between, sometimes yes and very often no. Just as most men lived in between what was expected from them in their lives and what they truly wanted. Most men but not men like Sir Ralph—and Frevisse had known women like him, too—who demanded and forced and took what they wanted and left the pain and scars of it on others. The way the pain and scars were left on Lady Anneys, who said now with deep-scored bitterness, "That's how I survived your father. By submitting and accepting, accepting and submitting." She spat the words. "And wished him dead so many times, so badly wanted him dead . . ."
Frevisse was unable to see her face, but Hugh's was raw with pain. He tried to speak but Lady Anneys cut him off again with, "Do you know how I stopped him from hitting me? You never saw him hit me, did you? You know how I did that? I was six months married when I found I was with child. With Tom." Her voice threatened to break on his name but she pulled back into her anger and went on, "I was so afraid Sir Ralph would do something to me and I'd lose my baby that I told him I'd see to it I'd never bear him any child at all if he ever, _ever_ laid hand, fist, or whip on me again."
"Again?" Hugh said hoarsely. "He'd already hit you?"
"Already? Better to ask how often. I spent that first half-year of my marriage being hit whenever he felt like it. After all, I wasn't one of his hounds, to be gently handled and cherished. I was his brood mare, to give him sons to replace Miles' father, his disappointing firstborn. So I told him that if he ever struck me again, there'd be no sons at all, only a long, barren marriage. I made him believe there'd be no children if he ever hit me. Later, when we'd had children enough, he'd ceased noting me enough for me to be worth hitting. But the threat to never bear him children at all was how I won free of him."
"Could you have done it?" Hugh asked, sounding half-frightened of the answer.
"Yes." Her voice was flat with certainty but bitterly weary as she went on, "Only I never found a way, after you all were born, to save any of you from him. I could only save myself." She sank down on the bench beside Frevisse, all the anger and strength gone suddenly out of her. "And now I'm worn out. Childbirths and all those years of never showing weakness to Sir Ralph—of never showing _anything_ to Sir Ralph—have left nothing of me except tiredness."
She made to stand up but might have failed if Frevisse had not hurriedly stood up, too, to help her. Hugh, too, took her by one arm and helped to steady her when she was on her feet; but it was on Frevisse she leaned and to Frevisse she said, hardly above a whisper and on the edge of tears, "I want to go inside now. I want to go to bed and never get up again. Please."
**Chapter 14**
Hugh let his mother and Dame Frevisse leave him standing there beside the bench in the gathering darkness. He was not needed by them nor did he know where to go or what to do, and when they were gone, he sat down on the bench again, leaned forward with his elbows on his knees and his head in his hands, and waited to feel something else besides sick with the weariness of worry. Too many things were too wrong in too many ways, with no way he saw to make anything better.
When Degory had said Master Selenger was here again, he had been angry—and been angrier when he saw how badly troubled by him Lady Anneys was. Did she guess the same thing Miles guessed about Selenger's attention to her? Or should she be warned? He didn't know. There were so many things he didn't know—didn't know what to think, didn't know what to do, was beginning not to know even what to feel.
And now there was what his mother had just said.
That she could have kept him, Tom, the girls from ever being at all.
He had never even thought much about being or not being until lately, and that was strange, because death was such a close part of life. There were always, in the usual way of things, the deaths of manor folk or neighbors and every autumn the slaughter of whatever cattle and pigs could not be kept through the winter and sometimes the need to put down hounds that were too hurt or sick to live, let be that he had hunted enough animals to the death for sport and food.
But Sir Ralph's death . . .
That had been different.
. . . his head all splintered bone and blood and gray brain matter . . .
And then Tom's death.
Tom lying there so . . . empty.
Gone past ever having back.
And now their mother said she could have made it so they never were at all . . .
Hugh gripped his head more tightly, wishing he could stop his thoughts. Any and all of his thoughts. Everything was become so tangled out of sense and shape, the whole world unraveling around him, with nothing left to his life but rough, torn, unmendable edges . . .
"Hugh?" Ursula asked softly.
He lifted his head. He had not realized it was grown so late; she was only a shadow-shape among the other garden shadows, and he said, "Come looking for me? I'm sorry," and made to rise.
But she asked, "Could we just stay awhile? It's quiet here," sat down beside him when he nodded, took his hand between her own small ones, and leaned against him. Her head came just to his shoulder.
Content to stay if she was, Hugh asked with feigned lightness, "Is someone fighting again? Is that why you came for me?"
"They're just talking. Lucy is. She always does and then everybody else has to talk, too."
"Words get to be heavy after a while, don't they? They wear you down."
She nodded against his shoulder but did not say anything and neither did he, only slipped his near hand free of hers and put his arm around her waist, cuddling her to him the way he might have a lonely hound. He guessed they were both in need, himself and her, of someone to hold to and be held by; so he held to her and she held to him and they sat there in the garden quiet, neither of them saying anything while the last afterglow of the sunset faded behind the hall's roof and the stars thickened across the sky.
The green-gold morning was already warm when Frevisse and Sister Johane came out of the church after Mass next day. Such manor servants as had been there headed back to their morning duties at the hall and the villager folk went away to whatever work they could do about their own places while they waited for the dew to dry so harvest could go on. Lady Anneys, as was usual, had not come, nor Miles, but Frevisse had seen him at Mass only the one Sunday since she had come to Woodrim, and Hugh had been no better until this morning. Today he had accompanied his sisters, which made it easier at the Mass' end, when Frevisse said that she and Sister Johane were going to stay at the church awhile, for them to go happily away with him, back to the hall.
She had told Sister Johane on their way to church that she wanted to stay ". . . to talk a little with Father Leonel. He knows everyone here far better than we yet can. If he can help us understand more, perhaps we can help more."
Sister Johane had readily accepted that. Only while they waited in the churchyard for him to come out did Frevisse suggest she should talk with him alone. "He might be easier with only one of us, rather than two," she said.
"He might be, yes," Sister Johane agreed, possibly a little doubtfully, but she sat down, seemingly content to wait as long as need be, on the mounting block at the churchyard gate while Frevisse went back to the church door. Until last night she had not considered consulting the priest; he had seemed only an old man enduring a thankless place by being and seeing as little as he could. The exchange at supper last night had changed her mind about that and now she wanted to know not only what he knew but what he thought, and that, indeed, he would probably better say to one person than to two.
Father Leonel seemed surprised but not unwelcoming to find her there when he came out, but when she asked if she might speak with him, he only asked, "Inside or out here, my lady?"
"Out here, if it please you. The day is so fair."
"It is indeed. But I pray you pardon my aged body and let us sit."
There was a bench made of a roughly flattened log pegged a-top two sawn-off stumps of wood beside the low churchyard wall well away from the gateway. Father Leonel led her there across the sheep-cropped grass of the churchyard's grave mounds and hollows, sat down with stiff carefulness, and said, "Oddly enough, hot, damp days bother my aches more than cold ones do. It's a sorry thing when one has to be grateful for winter." He settled himself, rested his gnarl-knuckled hands on his knees, and asked as Frevisse sat down beside him, "Is it spiritual counsel you want or help to deal with the Woderoves?"
Frevisse gave a small, surprised laugh. "The Woderoves," she said, returning the favor of his directness. "Sister Johane and I want to give comfort and be a help but there's so much pain here that we don't understand."
"I can give so little of the so much comfort that's needed that any comfort you can give will be most welcome. But where to start." Father Leonel drew and released a deep breath. "To begin, let it be said straight out that we're all the better for being rid of Sir Ralph."
"I've gathered that," Frevisse agreed. Though she had hardly expected him to say it out so bluntly.
"You've probably gathered enough that I need say nothing else about him?" He looked at her questioningly and she nodded that that was true. "What have you determined about Lady Anneys in the while you've known her?" he asked.
"That she's endured by burying herself so deeply it will be a wonder if she ever finds herself again."
Father Leonel's gaze at her became considering. "That's well seen," he said slowly. "Very well seen." He thought about something for a moment, then said, "My hope is that enough of the strength she used to bury herself remains for her to bring herself back to life. Her children are going to need her."
"And Miles."
"Miles is a son to her in all but blood, but yes, perhaps Miles most especially. And Ursula."
"Ursula? Was Sir Ralph particularly cruel to her?"
"It was Miles he was particularly cruel to, having come to hate the boy's father so greatly and despising his mother."
"Why?"
"Sir Ralph hated Miles' father, his son, because the young man went against his wishes in everything. He hated Miles' mother because, as Sir Ralph often said—often enough that I remember the words—'The woman is French. What in the devil's name is there to like about her?' Besides, like his son, she fought Sir Ralph almost every day she knew him. Then, like his son, she died, and that left only their son for Sir Ralph to pay back for the both of them."
Frevisse considered the ugliness of soul a man had to have to keep up that cruelty for year after year against a boy who had done nothing but be born.
"As for Ursula," Father Leonel sighed, "her trouble has been that she was her father's favorite. Can you imagine being the favored, petted, well-beloved child of a man who was a monster to everyone else around you?"
Barely, Frevisse could and said with muted horror, "Blessed St. Nicholas." The patron saint of children.
Father Leonel nodded in dark agreement. "Yes. A person might well turn either into a monster or, if their heart is good enough to resist that, live constantly trying not to cringe from him. Happily, Ursula's heart is good."
"But if she was his favorite, how did he come to send her away to the nunnery?"
Father Leonel rubbed at one hand with the other. "I think he saw it as a way to keep her for himself forever. Rather than give her up to a worldly husband, he intended she should become a nun and spend her life in prayer for him."
"And Lady Elyn's marriage was because Sir William was his friend?"
"Partly, but mostly as a way to keep up hope of his properties and Sir William's being someday joined, if not in this generation, then perhaps a later. Sir William's only child, Philippa, had always been intended to marry Tom as Sir Ralph's heir. When Sir William began to think of marrying again, meaning he might have a son to replace Philippa as his heir, Sir Ralph reckoned to bring Sir Ralph's lands his way by marrying Elyn to Sir William. That way any more children Sir William had would be his grandchildren."
"But now that Tom is dead?"
"I expect the expectation is for Master Hugh to marry her."
It very surely was, but, "Will he?"
"Very probably. He favors no one else so far as I know. They get on well enough together and all the reasons for marriage between Sir William's and Sir Ralph's heirs remain."
"I gather Hugh got on better with his father than anyone else did?"
"He got on best with Sir Ralph, yes. Hugh is quiet-spirited enough he would rather oblige than quarrel, and their shared love of hounds and hunting meant he had some use and value to his father."
"And Tom?"
"Tom," Father Leonel said with a smile and softened voice that showed who had been his favorite. "Tom had something of his father's quick angers but a better heart. A far better heart." The old man's voice twisted with grief. "He's a loss beyond measure."
Frevisse paused for his grief, then said gently, "It must have been more than only hard for you to see all this and be able to do so little. At least I suppose, from what I hear of Sir Ralph, there was little to be done with him?"
Father Leonel's aged face and voice hardened. "There was nothing anybody could do with Sir Ralph except endure him, and yes, it was very bitter being priest to a man whom nothing could touch. Not guilt or pity. And never love."
"But you stayed. Was there never chance to go? Plead your age and be given an easier parish?"
"I was needed here," Father Leonel said simply. "By everyone else if not by Sir Ralph. At least I could give comfort to some, shelter to others, and keep from quarreling with the man. Another priest might have given way to quarreling and, believe me, that would have done far more harm than good. To cross Sir Ralph in anything was merely to make the matter worse."
"How did he come to die? I know he was killed in the forest but nothing else."
Father Leonel sighed from far down inside himself and turned his head away to stare toward the forest's edge dark along the rising ground beyond the manor; stared for a long while before he said at last, his gaze still away, "There had been a hunt that day. Only hare-coursing since it was nigh to high summer. There's a place not far inside the wood where they often gather before or during or after a day's hunting to rest and eat. A clearing with a spring and a small stream. They were there that day for the midday meal and resting afterwards, meaning to hunt again in the afternoon. I understand what happened was that one of the young hounds ran off into the woods and Sir Ralph and some of the others went after it. A while later, still searching for the dog, Hugh found Sir Ralph instead. Very dead. He had been savagely attacked, struck on the head with a rock. Struck many times. No one let Lady Anneys see the body. Nor his daughters. All was bad enough as it was. By the time I was brought, Lady Elyn was screaming and crying so that her mother had to be comforting her when it should have been the other way around."
"Lady Elyn was there?"
"What? Oh, yes. They all were. Sir William, Lady Elyn, Philippa, Master Selenger. Lucy, too."
"Sir William's steward was there?"
"He's Philippa's uncle, too. Brother to Sir William's first wife. As much family as steward, you see."
"Was that usual, for them all to be here?"
"Oh, yes. Sir Ralph and Sir William shared a passion for hunting. They often hunted together."
Frevisse hesitated over what she wanted most to ask: Did he have any thought of who might have killed Sir Ralph? She had claimed her questions were for the sake of understanding more so she could comfort better, and she had already ranged somewhat far from that—far enough that another straying question would make small difference, she decided and asked, "There's no talk about who might have killed Sir Ralph?"
"Talk?" Father Leonel pulled himself straight as if his back ached. "Alas, there's always talk." He braced his hands on the bench to either side of himself and began the slow work of pushing himself to his feet. Frevisse quickly helped with a careful hold on his arm. On his feet at last, Father Leonel thanked her and began a stiff shuffle back toward the gate. She matched her steps to his but asked nothing more. They went in silence until, almost to the gate, he added, "The only surety is that whoever did it must be far away by now, escaped from the law's judgment maybe, if not from God's."
There was a second surety, too, Frevisse thought suddenly—that everyone _wanted_ Sir Ralph's murderer to be someone long gone.
And that, unwillingly, made her wonder if he was.
At the gate she thanked Father Leonel for his time and help. He blessed both her and Sister Johane and was shuffling back toward his church as they went their way away toward the hall. Taken up with her own thoughts, Frevisse did not notice Sister Johane's silence until, almost to the manor yard, she asked, "Was he able to help?"
Nearly, Frevisse said, "Very little," before she remembered the reason she had given for wanting to talk with him and said instead, "Lady Anneys' husband was something of a monster. You've spent most time with the girls. Have they talked about him?"
"So very little that I've been wondering about him. They talk about their brother Tom but almost never about their father."
"He seems to have cared for nothing much beyond himself and his dogs. Everyone lived afraid of his anger. Even Father Leonel, I think."
"Nobody seems to mind he's dead, that's certain," Sister Johane said. "It's their brother that Lucy and Ursula are mourning, not him at all."
"Everyone seems to mourn Tom. Everyone seems to have liked him and thought things would go well with him as lord of the manor."
"Well, Master Hugh seems a good young man, too. He's kind to his sisters and his mother, certainly."
"What do you think of Miles?"
Sister Johane was silent a time before she answered, "He makes the girls laugh and I think he's Master Hugh's good friend, but he always seems like he's about to be angry at something."
"By what Father Leonel says, Sir Ralph was worse to him than to anybody else."
"Well, I'm just glad he can make Lucy laugh. He was the first one to give her something else to do besides cry." Sister Johane wrinkled her forehead thoughtfully. "Though she would have stopped sooner or later. You can only cry so long before you begin to bore even yourself."
Surprised, because she had not thought Sister Johane saw things that sharply, Frevisse said, "True," so warmly that Sister Johane looked at her with answering surprise, making Frevisse wonder why. But something more had come into her mind while she and Sister Johane talked: they had both said "seems" again and again. Now that she thought of it, it was a word she had been using often, if only to herself, since coming to Woodrim, and she thought on it harder as she and Sister Johane went on across the foreyard. So many things "seemed" here at Woodrim. Why? And if so much "seemed," what truly was?
If so much was seeming here—if so many were holding up a mask of themselves between their truth and what they wanted others to see of them—then what were they hiding? Or hiding from? Not from Sir Ralph anymore. That only left that they were hiding from each other.
Or hiding something they didn't want known.
Or else hiding from each other what they knew and did not want to know.
Or else were hiding to keep from knowing something more than they did. Because if you buried yourself deeply enough, you could keep from knowing almost anything.
Let them alone, she told herself. Leave things as they are. Let these people piece their lives back together and heal as best they could. It was no business of hers nor did she want to make it her business.
But buried things had a tendency to rot.
And what else had her questions to Father Leonel been except the beginning of making it her business? Something was deeply wrong here. She knew it, and she knew herself well enough to know that, once begun, she would not stop her seeking to know what.
They were nearly to the hall door but she said, "Shall we go see the kennel and dogs we keep hearing about?"
"Yes!" Sister Johane said instantly, then added more hesitantly, "Well, yes, but wouldn't it be better to ask Master Hugh to go with us?"
Because it was not to Hugh she wanted to talk, Frevisse said, "Do we really want to hear that much about dogs?"
Sister Johane smiled at that, then a little frowned and looked vaguely around. "Do we know where the kennel is?"
Frevisse pointed vaguely away to their left. "I've seen Master Hugh go that way sometimes," she said.
They found their way well enough, past a large elm tree beyond the hall and by way of the stableyard to finally the kennels, where the dog-boy was leaning at ease on the kennel-yard's gate, scratching under the chin of a young wolfhound standing on its hind legs, its forepaws on the gate top, its head towering above him. About a dozen other hounds of various sizes and kinds—rough-coated, smooth-coated, brindled, plain, and spotted brown and black—lay in the early morning sunlight or paced around the yard, and if the dog-boy was idle now, it was because he had already done his morning work; the kennel-yard was clean and the water in the well-scrubbed wooden trough unslobbered yet. He straightened from the fence and bowed low as Frevisse and Sister Johane approached, and Frevisse saw his eyes shift past them, expecting Hugh to be there.
"We thought we'd like to see the dogs without troubling Master Woderove," she said before he could say anything. "We thought surely you could tell us enough about them."
The boy showed doubtful but willing. "That I can, probably." He looked past them again, maybe still hoping Hugh would somehow be there after all. "You don't know where Master Hugh is, do you? He usually comes of a morning to see things."
"He walked his sisters home from Mass," Frevisse said. "I don't know where he is now."
The boy grimaced. "Likely he's shut up with papers and ink again. There's too much of that when you're lord, looks to me."
"I doubt he likes it any better," Sister Johane said, kindly. "What's your name?"
"Nay, I don't suppose he does," the boy sadly agreed. "Degory, my lady."
"You keep a fine kennel, Degory. It reminds me of my father's. He keeps about thirty hounds. Or did when I was last at home."
"Does he?" Degory's deference slid into eagerness. "We've but fifteen. But they're good ones."
"And beautifully kept. You do all the work yourself?" Sister Johane asked.
"Master Hugh and me, we did it between us. It's mostly me now, he's so taken up with other things. But I don't mind," he added hurriedly. "He comes when he can."
"You're breeding here, too, aren't you? How many lines do you have?" Sister Johane asked.
Frevisse had been wondering how to set about putting the boy at ease enough to answer the questions she wanted to ask him. Sister Johane with her unexpected interest was solving that and Frevisse left her to it until in a while, when Sister Johane and Degory were talking of the training of hounds, Sister Johane said, "My father won't have whip or stick used on his hounds. How is it with Master Woderove?"
"He's the same. He never hits at all. It was Sir Ralph used to hit all the time."
"The dogs?" Frevisse put quickly in.
"Me mostly. He liked the hounds best, see, and Master Hugh and I, we train'em well. There's never cause for hitting them."
"With dogs this big, you want them well-trained, I suppose," Frevisse said.
"Aye, well, they're quiet-minded anyway, by nature," Degory said. "You wouldn't want hounds that big wanting to fight you all the time on everything. They're quiet-natured but then we train them, too."
"But one of them ran off the day Sir Ralph was killed," Frevisse said, with a carefully concerned frown. "That's what I've heard, anyway."
"Oh, aye. That was Skyre, the silly bitch. She's young and hadn't learned better. That's her there." Degory pointed to a smooth-coated, yellow hound lying alone in a corner of the yard, muzzle on paws but round, dark eyes fixed on the unknown women at the fence as if worried they were dangerous. "She was shaping to be a good lymer but likely she's ruined, Master Woderove says. Whatever happened with Sir Ralph there in the woods, by the time I found her she was frighted silly out of her wits and doesn't look to be getting over it. Twitchy all the time, see. So maybe we'll keep her for a litter or two, if she breeds well, and then see what can be done with her."
"What did happen in the woods with Sir Ralph?" Frevisse said, meaning to sound no more curious than anyone might be who lacked the courtesy not to ask at all. "No one ever quite says and we don't like to ask Master Hugh. He found his father, didn't he?"
Degory's shudder looked a little practiced, as if maybe he had done it too often and hardly meant it anymore; and he answered readily enough, "It was terrible. There's none of us will forget it."
"Skyre had run off and Sir Ralph had gone after her?" Frevisse prompted.
"Aye. Silly bitch," Degory said again, looking at the disgraced dog with pity and disgust. "Saw a squirrel on the ground or something and took off after it. Sir Ralph yelled and that didn't help. Then he hit me and he hit Master Hugh and said to get after her and took off himself after her, too. That was the last we saw him. Until he was dead."
"But you and Master Hugh went after her, too."
"Oh, aye. Wouldn't dare otherwise once Sir Ralph said to."
"Together?"
"No. Could cover more going separate, see."
"And everyone else stayed where they were."
"Master Tom wasn't there anymore. He'd fought with Sir Ralph already and gone home. I guess Sir William after a while tired of waiting and told Master Selenger they could look, too. Master Selenger was there at Sir Ralph's body even afore me when Master Hugh started yelling for help after he'd found Sir Ralph. Daft that was—all of us out looking for her. More likely to scare her off than not." Degory lowered his voice but did not keep his satisfaction out of it. "Sir William isn't so good with hounds as he likes to think he is. They're just things to hunt with, that's all, to him."
"I've known dogs with finer feelings than any person," said Sister Johane. "There was one my father had . . ."
Frevisse let her and the boy talk on awhile longer before drawing Sister Johane off by suggesting they would be missed by now. Sister Johane gave way unwillingly, told Degory again that he kept a fine kennel, and talked happily all the way back to the hall about dogs she had known. Grateful for the unwitting help she had been, Frevisse let her.
**Chapter 15**
Since coming to Woodrim, Frevisse had tried to keep the Offices of prayer at something like their proper times. This morning they were late about it because of her time with Father Leonel and the dog-boy, and while they were going back to the hall, Sister Johane said a little hopefully, "We've missed Tierce. We could simply forgo it and do only Sext."
"Or we could do both," Frevisse said in a way that said they _were_ going to do both.
Sister Johane sighed but made no protest. There were servant-sounds from kitchenward as they crossed the hall toward the stairs but the hall was empty, and the bedchamber, too. Breviaries in hand, they sat together on one of the chests and began. The familiar web of prayers and psalms— _Et posuit in ore meo canticum novum, carmen Deo nostro . . . Beatus vir, qui posuit in Domino spem suam._ And he put in my mouth a new song, a song to our God . . . Happy man, who puts in the Lord his hope—quickly drew Frevisse away from all the ways her thoughts had been twisting since yesterday. That was the pleasure and much of the blessing of the Offices: they were reminder that there was more than only here and this brief now; that there were other passions than the passing ones of the body; that there was Love beyond love and Joy beyond the world's so easily lost happinesses.
There were surely lost happinesses enough here at Woodrim. Nor had Sir Ralph's death purged the ugliness he had made of his life. _Exstingue flammas litium._ Put out the flames of quarrel. _Aufer calorem noxium, Confer salutem corporum, Veramque pacem cordium._ Take away guilty love, Give health to the body, And true peace to the soul.
But there wasn't peace here. There was a shadow through everything, like blight through a field of grain, sickening and blackening what should have been well and fine.
From what did the shadow come? And how many people knew of it?
Most here might well be living in it without knowing that they were, or else, like herself, they knew there was a darkness without knowing what it was. But it was here, subtly eating at hearts and minds.
Or maybe not so subtly, for some.
Was it suspicion that cast the shadow, she suddenly wondered. She had her own suspicion, surely, and it was a dark one—that Sir Ralph's murderer was not someone long gone; that he was still here. And very probably she was not the only one who suspected that. Suspicion without certainty—that was a darkness very hard to live in. Or—worse—not suspicion but certainty the murderer was still here without knowing who he was. That would cast a darkness deep enough to make the shadow she felt here.
But worse yet was her guess that no one wanted Sir Ralph's murderer caught. That they would rather, given the choice, live in the shadow.
When the Office ended, she would have sat quietly awhile longer, but Sister Johane closed her prayer book with a satisfied sigh and stood up, ready for whatever came next. Frevisse set aside a stir of impatience, looked up at her, and said, kindly rather than accusing, "You're enjoying being here."
With open pleasure, Sister Johane said, "I am. It makes such a change, being around people who aren't nuns."
With effort Frevisse kept hidden her worry at that, but Sister Johane said next, with a small, thoughtful frown, "It helps me remember how good it is to be a nun. It's not as if being a nun is easy." She was frowning harder, staring at the wall above Frevisse's head with concentration, her breviary held to her breast. "But so many of the problems are inward. They can be made all right if I look at them hard enough, grow enough so I can understand them and change myself. Not like poor Lady Anneys, who's had so many things happen _to_ her she can't find her balance at all. Or like Lucy, who doesn't even try to think about things, just feels them as hard as she can."
Sister Johane suddenly realized she was saying all that aloud and maybe read the surprise open on Frevisse's face as disapproval because she ducked her head and said hurriedly, "It's not that I don't like Lucy. I do. She's just so . . . so . . ."
"So very young?" Frevisse supplied, with a silent laugh at herself because, to her, Sister Johane was very young; but Sister Johane had taken her vows when she was hardly older than Lucy was and had been a nun almost ten years.
"Yes," Sister Johane said, encouraged. "She's very young."
"And Ursula?" Frevisse asked, finding for the first time that she was interested in what Sister Johane thought about something.
Sister Johane a little frowned again. "I don't know about Ursula. She reminds me of her brother, the way she keeps herself to herself. Or do I mean she reminds me of Miles? No, she doesn't have Miles' anger. She just has Hugh's quietness. Is she going to be a nun, do you think?"
"I don't know," Frevisse answered, thoughtful in her turn because Sister Johane was right: Ursula did not show herself.
She considered that as she and Sister Johane put away their breviaries and went downstairs. Despite Ursula seemed an open child and Hugh an open young man, Frevisse's talk with Father Leonel had shown her there were very likely deeply hidden places in them both, places where they had hidden from their father. Nor were they probably the only ones. Lady Anneys, assuredly. Tom very probably. Even Father Leonel, elderly and crippled, had kept hidden his secret reworking of the manor accounts. Oddly enough, Miles in his way was probably the most open of any of them. His hatred for Sir Ralph had never been hidden, she had gathered, and his pleasure that Sir Ralph was dead was completely open.
Or did he have other things hidden so deep there was no hint of them for anyone to see?
When her thinking turned that far around on its own track, with nothing to feed on but itself, Frevisse knew to be done with it for a time and she was glad to find Lady Anneys, Lucy, and Ursula in the garden, weeding one of the herb beds. All three of them were lightly dressed in simple linen gowns over their underdresses, with sleeves pushed up and only Lady Anneys bothering with shoes though they all wore broad-brimmed straw hats against the sun riding high now, the day bright and warm.
They all straightened to greet Frevisse and Sister Johane, with Lucy asking, "Aren't you too hot, wearing all that and your wimples and veils and all? I'm simply roasting. Can't you take some of it off?"
"We're used to it," said Sister Johane. She sounded faintly surprised, though perhaps not at Lucy so much as at herself, maybe only just realizing how completely her nun's clothing was part of her. Frevisse well remembered the moment, well into her own nunhood, when she had realized how used she was to her nun's clothing, how unnatural she would have felt wearing anything else or even less of it.
"Besides," said Ursula at her sister, scornfully, "they _have_ to wear it."
Lucy wiped at her forehead. "Then there's another reason I'll never be a nun."
Lady Anneys wiped at her forehead, too, then pressed her hand to it, her eyes closed. For coolness' sake, she wore only a loose veil over her hair and no wimple at all, but her face was an odd shade of white and Frevisse asked, thinking the day's warmth was too much for her, "My lady, are you ill?"
"One of my headaches," Lady Anneys said.
"It isn't better?" Ursula asked, then said to Sister Johane, "She took fennel and that usually helps."
"It hasn't this time," Lady Anneys said. "It's that the day's so warm, I think." Her voice wavered, as if thinking were difficult. "It's worsening. Do you have something stronger that might help, Sister?"
"Yes, but you'd best come inside, out of the sun, and lie down while I ready it." Sister Johane sounded very much like Dame Claire in one of her brisker moments. "Ursula, please pick me some balm and bring it after us. I'll be in the kitchen after I've seen your mother to her bed."
Lady Anneys and Sister Johane went away together, Ursula picked a goodly handful of the balm and hurried out of the garden after them, and Frevisse asked Lucy, "What can I do? If you point me to what a weed looks like, I can try to pull out nothing else."
"I'm tired of weeds," Lucy said. "Let's head the basil for a while."
Frevisse was willing to that. Whereas one green plant looked much like another to her, making her a peril in the garden when it came to weeding, she could tell a plain leaf from a flower and plucking off the flower stems from basil plants, to keep the herb usable for longer, was well within her ability. Besides, the spicey scent of the basil in the warm sunshine and on her fingers was pleasant and soothing and she was willing to be soothed just now. Even Lucy's talk was undisturbing, since so little thought needed be given to it, Lucy giving so little to it herself. She started with mourning her mother's headache, went on to say she liked the smell of basil, and, "Sister Johane is so clever about herbs. She made my eyes stop being red after I'd cried so much for Tom. She's given me something to bathe them with when I've been crying again. I don't know what it is, though."
"Eyebright," Frevisse murmured.
"That might be it. There can hardly be anything more useful than knowing how to do things like that, don't you think? I mean to have her teach me things like that while she's here. When there's less going on."
Frevisse kept to herself that thought that Lucy was probably someone who often meant to do things, only to be satisfied that the intent was as good as the deed and did nothing. Working far more slowly from plant to plant than Frevisse was—despite Frevisse was making no haste—Lucy had talked her way around to how she hoped Ursula didn't go back to the nunnery, she was better at reading aloud than Lucy was and so Lucy didn't have to do it if Ursula was home—"Besides, I like listening better than having to read"—by the time Sister Johane came back into the garden, her own sleeves rolled up now and worry on her face. Interrupting Lucy as if she knew that were the best way to get a word in, she said, "Dame Frevisse, I know it's almost time for Nones but I'm in the midst of mixing the drink for Lady Anneys and I'd like to sit with her after she's taken it. Might I miss prayers this time?"
"There are more ways than one to pray," Frevisse said. "What you're doing can be prayer in its way. And it's certainly a mercy. Of course go on with it."
Lucy watched Sister Johane hurry out of the garden and said as if the two had no business together, "She's quite pretty. Why did she ever become a nun?"
Frevisse settled for, "Her aunt was our prioress at the time," and that was sufficient to satisfy Lucy, who shifted to talking happily of her hope that Hugh would see to arranging her marriage soon.
"Though he'll probably marry Philippa before he sees to mine, and that will have to wait until we've done the mourning time for Tom."
Frevisse had been about to excuse herself to say Nones but asked instead, "Does Philippa mind she's to marry Hugh, not Tom?"
"Oh, I don't know she liked Tom better than she likes Hugh so it's much the same to her, I should think." Lucy gave another sigh, this one just short of tears. "It's Tom I feel sorry for. I do so miss Tom. Everything is so wrong without him."
"Wrong?"
"Not the same. I was already used to Father being gone, and Master Selenger had been coming to see Mother all the time and that was so sweet—except she went away and I don't understand why, except it was too soon after Father died for her to be interested in Master Selenger, I suppose—and Tom was going to marry Philippa. Everything was _good_ and now it isn't. I hardly get to see Elyn or Philippa, and Master Selenger isn't welcome here anymore and Hugh is unhappy . . ."
"More unhappy than after your father's death?"
"Oh, much more unhappy. He loved Tom." She lowered her voice. "He doesn't want to be lord of the manor. He just wants to be with his dogs and go hunting." She tossed her head. "But that's just too bad for him. He's just going to have to make the best of it and that's all there is to it."
Frevisse had long since ceased to be surprised that "that's all there is to it" always applied to other people and rarely to the person saying it. She knew she should go aside to her prayers now, but since Lucy was so ready to talk and another chance with her alone might not come so easily, Frevisse quelled her conscience and said, "If Hugh's badly missing his father and brother, that makes it all harder for him."
"If anyone is missing Father, it's Hugh," Lucy granted. "They both liked the hounds and hunting. Now there's just Hugh likes the hunting. Until we can be friends with Sir William again, I suppose."
"Don't you like the hunting?"
Lucy shrugged a shoulder. They had finished with the basil but she was too happy with talking to move on to something else. "Not much. Noise and blood. That's all it is."
"But you were there on the hunt the day your father died, weren't you?"
"There for the hare-hunting? Never." Lucy found the thought displeasing. "You can't imagine how stupid that is. No, Mother and Philippa and I were at the gathering place with the food and servants, that's all. That's the only good part about the hunting. We eat and sit on cushions by the stream and talk. That day we sang, too. Philippa and her uncle sing lovely together. And Miles," she added, as if surprised. "He can sing, too, when he wants to. But he hardly ever wants to." She giggled. "He did that day, though, because Philippa was." She leaned toward Frevisse and lowered her voice, as if imparting a choice secret. "I think he's fond of her. You know?"
"Ah," Frevisse said, trying to sound only lightly interested. "And is she fond of him?"
Lucy frowned. "She _likes_ him but I can't get her to say it's more than that, no matter how I push her to."
Frevisse held back from pointing out that if Philippa was to marry Hugh, trying to make her admit to caring for Miles was hardly kind. Instead, while she looked for a way to lead on to anything about Sir Ralph's murder, Lucy went on, "They went off together that day. That's all I know. And they didn't hurry back even when the shouting started. When Hugh found Father's body, there was so much shouting, you know."
"They went off together? Miles and Philippa? Looking for the dog that ran off, you mean?"
Lucy laughed. "Hardly! Father went, and Hugh and Degory, and they were no more than gone when Sir William said he might as well look, too, as sit there, and he and Master Selenger went off, too. I asked Miles if he wasn't going to go. He said he wasn't, but Philippa said she was tired of sitting and wanted him to walk with her."
"So the forest is fairly open there?" Most forests—especially those kept for hunting—were less trees than open ground, with large clearings and little undergrowth. It was something Frevisse only now considered about Sir Ralph's death. How had he been so away from the others that he could be killed and no one either see or hear it happen or catch sight of his murderer?
But Lucy said, "Oh, no. It's not open at all. The king's forest—the royal chase that Father has right . . . had right to—that's clear and open, of course, but Father always kept our woods thicker. For coverts, you know. If the deer had places to hide, they'd stay nearer, that's what he said. He only let the villagers use it a little, for fattening the pigs on acorns in the autumn and hardly at all to gather wood. He and Tom used to quarrel over that. Tom said it wasn't fair people had to go so far for wood—there's another bit of woodland at a far end of the manor—and Father said he didn't care. I remember once Tom yelled at him, 'If you didn't like pork, you probably wouldn't even let the pigs in to fatten,' and Father yelled back, 'You're damnably right I wouldn't!'" Lucy laughed, then lowered her voice again, though Frevisse didn't know who else was likely to hear her. "But I think the villagers took more wood from there than Father knew about and I think Tom let them. He just kept quiet and never fined anyone unless Father caught them at it himself."
"But Philippa wanted to walk there?"
Lucy shrugged her shoulder again. "There are paths. They went off along one of those." She lowered her voice to almost a whisper and leaned toward Frevisse. "I don't think Mother or Elyn liked it but I asked Elyn something and kept her talking to me until they were gone."
And they had not come quickly back even when the shouting over Sir Ralph's body began. Frevisse, not worried Lucy would make too much of any questions—the girl enjoyed talking too much to pause for thinking—asked, "So you and your mother and Lady Elyn were alone, still at the gathering place, when Sir Ralph's body was found? What did you do?"
Lucy's eyes grew big. "We didn't know _what_ to do. We didn't even know what the shouting was about." Whatever horror there had been that day had been, for her, replaced by the pleasure of being horrified by it all over again. "Nobody else knew either. Sir William came crashing back into the clearing, wanting to know what was happening, were we all right, told us to stay there, then ran off toward the shouting. The dogs all started barking and we tried to quiet them and then Degory came, with a great, bleeding scratch down one side of his face, and he was crying and said Sir Ralph was dead and he had to fetch Tom and Father Leonel. After that Mother and Elyn and I just stood there, holding on to each other, not even crying, just waiting to find out what had happened, not believing it could be that bad. It was forever before Tom and Father Leonel came and of course they didn't know anything either, but then Master Selenger came back to us and he knew. Sir William had sent him and he told us all that someone had killed Sir Ralph and that Father Leonel and Tom had better come see the body because they couldn't leave it there for the crowner, but Mother and Elyn and Philippa and I had better go back to the manor. It was terrible."
"When did Philippa and Miles come back to you?"
"Oh, sometime then, while all that was happening. We told them what Degory had said and Miles went off to join the rest of them." Lucy lowered her voice again. "We were never even let to see Father's body when they brought it back. Did you know that? Not even Mother. Isn't that awful?"
Frevisse agreed it was, tried to think what else she might get out of Lucy, but decided to excuse herself at last to say Nones. "I'll just go into the arbor if that won't bother you."
"Oh no, of course it won't. But don't you need your book?"
"It's in the bedchamber and I won't chance disturbing Lady Anneys. I know it mostly by heart."
"By heart," Lucy said with awe. "Oh, my. I mean to learn some of my prayers that way sometime. I mean, besides the paternoster and ave."
Frevisse granted that would be a good thing and went to settle to Nones in the arbor, leaving the girl poking among the feverfew for weeds.
**Chapter 16**
Lady Anneys was not at dinner but, "I gave her something to help her to sleep as well as ease her headache," Sister Johane said to Hugh's question.
Nor was Miles there, and when Ursula asked about him, Hugh answered, "He said at breakfast he was going off to Beech Heath. That's too far for being back by now."
"He's always gone somewhere," Ursula complained. "Even more than before."
"He'll be gone altogether after Michaelmas," said Lucy. She leaned forward to say past Hugh to Frevisse and Sister Johane, "That's when he means to go to his own manor in Leicestershire." She sat back and added to Ursula, goading older sister to younger, "We'll probably never see him again after that."
"Of course we'll see him," said Hugh impatiently, shoving the cheese tart to her instead of politely serving her.
Lucy stared indignantly first at the tart, then at him, but Ursula reached for it and willingly helped herself while insisting, "He doesn't have to go. Father hardly ever went. He could just stay here like always."
"He'll want to marry, you goose," Lucy said. "His wife won't want to be here with all of us. Whoever she is," she added, trying to sound meaningful about it.
No one chose to take a meaning from it; or if they did, they kept it to themselves in a thick silence that made Frevisse wonder if some besides Lucy had thoughts about Miles and Philippa. Did Hugh suspect he might have a rival for his proposed wife? The day of the inquest, when everyone had been here and she might have noted something, Frevisse had known too little about anyone to be much judging what she saw. Had she seen anything particular between Miles and Philippa then? Or between Hugh and Philippa, come to that. She remembered nothing, but it had hardly been the time or place for something that way, anyway. But what if Lucy was not merely making trouble when she talked of Miles and Philippa? What would happen if that was real and they were foolish?
In the while she had been here, it had become plain to her that there was friendship between Hugh and Miles. That made her think better of Miles than she might have. With his bitter edge and sharp tongue, friendship with him could not have come easily; that it was there between him and Hugh meant there must be more to him than she had so far seen. And judging by the tender way Lady Anneys had leaned from her chair and stroked Miles' hair back from his hot forehead the other evening, when he had been sitting on the parlor floor playing at jackstraws with Ursula, and how he had paused to smile up at her, he was—as Father Leonel had said—a son in all but blood, as dear as Hugh and her daughters openly were. Frevisse guessed that despite all that Sir Ralph had done to ruin these people—when they might have chosen to be as savage to each other as he was to all of them—they had all banded together instead, made a hard knot of family against him and guarded each other as best they could. And that, Frevisse suspected, was Lady Anneys' doing more than anyone's. She had outwardly submitted to Sir Ralph because she had to and must have kept much of herself buried beyond his reach through the years of her marriage; but she had never surrendered. She had loved and sheltered her sons and Miles and her daughters as best she could and now—if she could have peace for a while and a chance to heal—maybe she could become herself, instead of Sir Ralph's wife, just as they all had chance to live as themselves now Sir Ralph was dead.
And yet . . .
Listening without heed to Lucy trying to persuade Hugh that they all needed to go to Banbury market next week—"For something different to do!" she pleaded—and him telling her, "Not while harvest is on," and eating without noting the pottage of broad beans fried with onions and sage set before her, Frevisse probed at the uneasy edge of her thoughts. And yet . . . what?
Behind the affection among them all and beyond the relief of Sir Ralph's death and the still-raw grief for Tom there was something else.
She put a word to it.
Wariness.
Not that wariness was unreasonable, given the hell Sir Ralph seemed to have kept them in while he lived.
But he was dead now.
Was it simply that it was too soon after his death to leave behind all the old ways of feeling? Lucy seemed to have done so; Frevisse felt no undercurrents or shut doors in her. But from the others . . . even Ursula . . . there was that sense of wariness, of being guarded. Against what?
The meal was ended. As they rose from their places, Ursula asked, "Can you keep us company awhile, Hugh?"
He tweaked one of her long, braided plaits. "I fear not. I'm going out with Father Leonel to see how the harvest is coming on."
"The way Tom used to," Lucy said, all mournful.
Hugh sent her a glance of dislike. "Yes. Like Tom used to," he snapped, then said to Ursula, more kindly, "I want to see how much longer it's likely to be before we need to hunt a deer for the harvest-home feast."
"Kill two this year," said Lucy. "So there'll be enough for us afterwards."
"Greedy," he said, not meaning it, and bowed to Frevisse and Sister Johane. "By your leave, my ladies."
They bent their heads to him in return and he left, leaving them with all the afternoon ahead of them. Sister Johane went to be sure Lady Anneys was sleeping quietly while Ursula and Lucy fetched their sewing and Mandeville's _Travels_ from the parlor, and when Sister Johane came down, the four of them went out to the arbor's shade. This warm middle of the day, the quiet there was as thick as the sunlight save for the hum of bees. Frevisse would have willingly joined the quiet or else read, but when the girls had opened their sewing baskets and set to work on Hugh's and Miles' shirts again, Lucy began to talk of how she meant to persuade her mother to buy her some red cloth to make her a new gown this autumn.
"Something very, very bright, but I don't know yet whether I want crimson or scarlet," she said.
"You won't be able to wear it for almost a year," Ursula pointed out.
"I won't have to hurry at making it, then, will I? And I'll have it waiting for me when I don't have to be in mourning anymore, while you'll have to go back to wearing your same old gowns then."
"By then Ursula will probably have outgrown anything that fits her now and need a new gown, too," Sister Johane said, maybe trying to head off trouble.
But well able to see to her sister herself, Ursula retorted at Lucy, " _You'll_ probably be too fat by this time next year to fit into anything you make now."
"Fat?" Lucy protested. "I won't be!"
"Fat _ter,_ " Ursula said with great and insulting precision.
"Shall I read?" said Frevisse, taking up the book at the same moment that Sister Johane said, "Would you like me to stitch this neckband together for you, Lucy?," picking up the pieces of cloth from the sewing basket between them.
The distraction worked. While Lucy showed how wide a seam the neckband should have, Frevisse opened the book at the ribbon marking her place and began to read aloud about the court of the Great Khan in Cathay, her voice pitched low to match the garden's quiet. She was reading of the Tartars' round houses when Sister Johane cried out, "Oh!" with such sharp dismay that Frevisse, Lucy, and Ursula all looked at her. She stared back, wide-eyed with alarm. "I forgot! Lady Anneys asked if I'd make certain Helinor in the kitchen had sent the boon-ale to the field." The lord of the manor's expected daily gift to his people during harvest.
"The workers won't like it if she hasn't," Lucy said. "Even Father never dared stop their boon-ale. He wanted to, though."
"Helinor wouldn't forget," Ursula said scornfully.
Sister Johane started to lay her sewing aside. "But I told Lady Anneys I'd make certain."
"I'll go," said Frevisse. She placed the ribbon where she had left off, closed the book, and stood up before Sister Johane could. "You go on with your sewing."
Sister Johane accepted that with willing thanks that gave some ease to Frevisse's slight guilt, because she had offered not to be helpful but because she saw suddenly a chance to talk unsuspiciously with whoever was at work in the kitchen this afternoon.
When she had gone inside and through the pantry into the kitchen, she found it a square and high-roofed room with a deep stone fireplace along one wall, flanked with racks of pans, kettles, grill, and skillet and hanging long-handled spoons, ladles, and forks. A long, narrow table ran along the wall beside the door from the pantry, for setting out readied dishes to be carried into the hall during a meal, while the middle of the room was filled by the solid bulk of a worktable where three people could work side by side without touching elbows and broad enough a person would have to stretch to set a bowl across it to the far edge. To Frevisse's regret, though, there was only one person presently at work there, an aproned woman of late middle-years briskly slicing a young onion's long green leaves to small pieces on a well-scrubbed cutting board at the table's far end.
She stopped when Frevisse entered and, with the knife poised over her work, gave her a quick curtsy while asking, "May I help you, my lady?"
The Woodrove household was not large but Frevisse knew more than a lone woman was needed in the kitchen and asked in return, surprised, "Where's everyone else?" Then immediately answered her own question. "Gone to fields for the harvest, of course."
"That's right," the woman agreed.
She still stood with the knife ready over the onion and Frevisse said, "Go on, please," looked at the basket sitting beside the cutting board, heaped with more onions and clean, slender carrots, and said, "You've more than enough for one woman to be doing alone, if that's to be for supper."
"Aye." The woman went back to slicing, deft and quick about it. She was not unfriendly, just very busy. "Alson will be back in time to help finish it all off, though."
"May I help?"
The woman gave her a quick, doubting glance.
"I often do it at the nunnery," Frevisse said. St. Frideswide's was not so prosperous it could afford many servants nor so large that any nun could be allowed to stand on her dignity; they all had turns at what work needed to be done, including in the kitchen.
"If you like, then, it would be a help, aye," the woman said without pausing at her work. "There's more knives there."
She nodded to the rack fastened to the table's end and Frevisse took one, saying while reaching for a carrot, "I came to ask if Helinor had seen to the boon-ale going to the field. Are you Helinor?"
"I'm Helinor and the ale has gone. I'd not be allowed to forget for long, that's sure. Somebody would be at the kitchen door asking."
The kitchen door stood open to a garden far less well-kept than Lady Anneys' but flourishing with all the different greens of herbs and summer vegetables useful to cooking. Frevisse complimented it as a way to starting talk with Helinor, who said cheerfully, "Oh, aye. It's been a good-growing summer."
"The harvest looks to be fine, from all I've heard."
"Very fine. It's going to be a happy harvest-home when it comes."
"Will it be much of a harvest-home, what with the mourning and all?"
"Aye, there's that," Helinor said soberly. "Master Tom is going to be missed. He loved harvest-home, he did. Every year he worked hard as anyone to make it happen. He would have been a good lord to have. He'll be kindly remembered. There's been more than a few candles lighted in the church for him."
"But not for Sir Ralph?"
Helinor hacked with sudden, unnecessary savagery at a defenseless onion she had just put on the cutting board. "Any candles lighted will be in thanks to whoever did for the old bastard. Begging your pardon for speaking out," she added.
"No pardon needed," Frevisse said easily, to show she was willing to hear more. "From what I've heard, he needed killing."
"He did that." The onion, having suffered enough, was swept aside with the knife blade to join the growing pile of its predecessors. "At birth, if you ask me." Helinor began on another onion with no more mercy than she had shown the last one. "The wonder is he lasted so long as he did."
Working less viciously at a carrot, Frevisse ventured, "I don't think anyone even cares who did it."
"They don't that. Not in the least. Except maybe we'd like to thank him. Only there's so many likely to have done it, we'll never know. My own guess is it was one of the men he'd driven off from here, come back to do for him and long gone again."
"He'd forced a great many men away?"
"More than his fair share." The pieces of the onion were swept aside and another took its place. "He didn't care about people at all, or their rights. The manor and its folk were here long before he came, but he acted like nobody mattered aught but him and what he wanted. We weren't even supposed to keep the deer out of the fields. They come out of the woods into the fields to graze and we were supposed to let them, because then they'd be near to hand for hunting. 'Eaten by the deer' is what we say about someone when they can't bear it anymore and leave. Half a dozen men in the past ten years. And women, too. It's been bad."
"What sort of lord do you think Master Hugh will be?"
Helinor paused to consider that, then set at the onion again. "Not so bad as Sir Ralph, that's sure. He has a kinder heart."
"No one's afraid he'll be like his father about the deer?"
"He's let Master Tom's order stand that we could keep them out of the fields this year. That's enough to satisfy everyone for now."
Finishing one carrot and starting another, Frevisse said thoughtfully, "What I've found odd is that no one ever says how Sir Ralph died. They say he was found dead in the forest but not how he was killed. It must have been terrible?"
"Terrible enough that nobody will ever talk of it around Lady Anneys or her daughters if they've any kindness toward them at all."
"You saw his body?" Frevisse prompted.
"Ha. I was one of them that had to clean it and ready it for burial. Nobody was going to let Lady Anneys see it, let alone do any of that. His head was all smashed in. Someone took a rock to it and smashed it to bits."
Because it was expected of her, Frevisse made a wordless sound of horror.
"Aye, it was bad." Memory of it subdued even Helinor. She paused, staring down at the half-chopped onion on the cutting board. "His head was broken so bad I don't even know if all the pieces were there. We had to wrap it in waxed cloth to hold together what there was, then pad it around with more cloth to give it a head-shape before we wrapped it in the shroud. So it would look right." She shook her head and started cutting again. "It was bad, aye."
"There weren't any other wounds on the body? Just his head all smashed?"
"That's all. He'd not been stabbed or beaten or anything else. Just his head smashed to bone-bits and bloody pulp. That's why I say it was someone he'd wronged and they'd come back to do for him. It was revenge, not simple killing, if you see what I mean."
Frevisse saw and went back to carefully slicing the carrot she had in hand before she said, "Well, Lady Anneys is better off, from all I hear."
"She is that. But Master Tom's death is a grief she didn't need. Nor none of us. Still, she's rid of Sir Ralph and that's more than a little to the good. I just hope she doesn't make haste over marrying that Master Selenger."
"Is she likely to? Are people saying it's likely?"
"Well, he's here as often as not, isn't he? Started a few days after Sir Ralph was buried and it's not for the sake of anyone else's company he comes." Helinor smiled at her present onion, going at it less brutally with her knife. "He's a good-looking man and well-mannered. He could do with a wife and she could do worse. Besides, what's she to do with herself after Lady Philippa marries Master Hugh? She's too good a lady to think there can be two mistresses in one house or want to keep Master Hugh's wife from her rights."
So the provisions of Sir Ralph's will concerning Lady Anneys' remarrying were still as secret as Lady Anneys supposed, Frevisse thought, and there seemed to be no suspicions about Hugh marrying Philippa. Knowing she must go back to the garden soon but hoping for more, she said mildly, "Things will be different around here when all that has happened. Master Miles will be leaving soon, too, I gather."
"So we hear. Poor lad. It's time he had a life of his own but he's going to be lonely at it, I fear."
"He could well marry, now he had his own manor."
"Not from what I hear about that manor. Run-down and neglected. He'd need a wife who'd bring money with her, and what woman with money would marry someone with no more to offer than a ruined manor?" Helinor had finished the last onion and began on the carrots, their pile already much lessened by Frevisse. "But you have the right of it about things here. They'll be different and God be praised for that. The sorry thing is that it was hardest on Master Miles, maybe, when that old devil Sir Ralph was alive, but now that everything's changed, it's still Master Miles who's going to have it hardest. Those three boys were good at guarding each other's backs. They worked together and now Master Hugh has the manor and will have a wife, while Master Miles must go off on his own to somewhere he's never been before and make a whole new life for himself. That's going to be hard for him. Hard, too, for Master Hugh," she added after a thoughtful moment. "Being left alone when for so long there's been the three of them."
Frevisse thought the same, now she came to think of it at all. But she was also thinking of what else Helinor had said. Hugh, Miles, and Tom had worked together; but Hugh hadn't known about Tom's and Father Leonel's twisting of the accounts. Had Miles? Who had been keeping secrets from whom? And what other secrets might there be?
And then there were the provisions against Lady Anneys remarrying. Both Hugh and Miles knew about those but neither her daughters nor the servants did. Come to that, Lady Anneys did not know whether Master Selenger did or not, and on that hung the question of whether or not Sir William was playing some sort of double game of marriage—his daughter's and Lady Anneys'—to his own advantage.
How many layers of secrets were here? Frevisse wondered. When she had first come to Woodrim, she had assumed that with a family torn by the grief of two deaths, there would be sorrow in plenty but a straightforward sorrow, straightly dealt with. Instead, she was finding almost nothing was straightforward here at all.
She worked awhile longer, leading Helinor to talk about Lucy, Ursula, and Lady Elyn but learning nothing she did not already know except that, "Oh, aye, Elyn was ready to be married, she was. Couldn't wait to be Lady Elyn and away from here. Took to that marriage from the moment it was offered her and who could blame her?" Helinor said. "Anything to be away from here. Another one on the run from Sir Ralph."
"She wanted it more to be away from Sir Ralph than for love of Sir William, you mean?"
"Very much more." About that Helinor had no doubts.
"But now Sir Ralph is dead and no one was counting on that."
"She took her chance when it came," Helinor said, unconcerned. "She made her bed and must go on lying in it, just like everyone else does."
**Chapter 17**
As Frevisse passed through the garden, returning to the arbor, she saw over the garden's back gate a manservant sitting on the grassy edge of the far side of the cart-track, holding the reins of two grazing horses. She knew neither the man nor the horses and for a moment did not know the girl, either, seated in the arbor between Lucy and Ursula, turning the pages of Mandeville's _Travels,_ then realized she was Sir William's daughter, the talked-of Philippa, as the girl stood up and curtsied to her, saying, "My lady."
Frevisse bent her head in return and they both sat, Frevisse beside Sister Johane on the bench facing her and taking the chance for a long look as Lucy went on saying, ". . . didn't come with you only because it's too hot? She's feeling well, isn't she? She's not, um, not . . ."
"Breeding?" Ursula asked brightly, looking up from her sewing.
_"Ursula,"_ Lucy said, sending quick looks toward Sister Johane and Frevisse, probably to warn against talking about such things in front of nuns.
But Ursula had spent enough time among nuns not to think their living out of the world meant they were unworldly, and said, still brightly, "But that's what we all want to know, isn't it? Is Elyn going to have a baby and spoil Philippa's chance to marry Hugh?"
"That won't spoil her chance to marry Hugh. She just won't bring as much to the marriage," Lucy snapped.
"How far have you gone with Mandeville?" Philippa asked somewhat quickly, opening the book again. She was an even-featured girl, nothing particular about her and her hair simply brown, but she knew when and how to change the course of a conversation. "I like the part about the dog-headed people the best. What about you, Lucy?"
"There are too many dog-heads around here as it is," said Lucy. "I like to hear about all the riches in Cathay. When is Master Selenger coming to visit Mother again?"
"I don't know," Philippa said with a lightness that failed to ring completely true to Frevisse. "He probably wouldn't be good company if he did. He had angry words with Father this morning."
"About what?" Lucy asked eagerly.
_"Lucy,"_ Ursula said.
But Philippa answered, "I don't know. I could hear that they were angry, but not about what."
"It's because of Mother," said Lucy certainly. "He's pining and angry because she won't say she'll marry him."
"Lucy, why would that make my father angry at him?" Philippa said patiently. "Besides, Uncle John isn't that unsensible, to expect Lady Anneys to be ready to marry again so soon."
"How is your father?" Sister Johane asked, taking the talk a different way. Deliberately, Frevisse thought, but this time regretted it. She would have liked to hear more about Selenger. And if Philippa had come visiting to forget troubles at home, no one was helping her do it.
"He's still unsettled by . . . all of it," Philippa answered. "And that Lady Anneys won't have anything to do with him."
"It's early days yet," Sister Johane said comfortingly. "Lady Anneys only needs time until she can face him again."
"Besides," Lucy said cheerily, "you and Hugh can't be married for probably a year anyway. There's plenty of time yet."
"Lucy, you don't care who marries who," Ursula said impatiently, "just so long as somebody is marrying somebody."
At the opening of the arbor Miles said, "If Hugh is wise, he'll find a husband for you, Lucy, before he does anything else and get you out of here."
"Miles!" Ursula exclaimed happily while Lucy complained, "I wish he would," and Miles laughed at her. He was still in the rough, dark green tunic, hosen, and soft-soled leather boots he wore to the forest and his hound was beside him.
"Oh, Miles, not Bevis," Lucy protested. "Not here."
Ignoring her, he made a half-bow to Philippa and said, "I saw your horse and man and thought I'd find you here. You look well, my lady."
"As do you, good sir," she answered as lightly.
"Where's Lady Anneys?"
"Gone to bed with a headache," said Lucy, "and Hugh is gone off with Father Leonel, so you might as well join us. Here. But do leave Bevis there." She started to shift sideways to make room between her and Philippa.
"By your leave," Miles said, coming only a single pace into the arbor's shade before he folded his legs and sank down cross-legged on the ground. "Here will suit me better, where I can look on all your lovely faces." The hound Bevis lay down beside him with a heavy huff. "How go things at home, Philippa?"
"Well," she said, then amended, "Well enough."
"What you mean is that Sir William is still glooming because Lady Anneys doesn't want to talk him just now, and that your Uncle John is glooming because she doesn't want him wooing her. And you're glooming because you have to wait a year to marry Hugh?" Just barely he turned the last into a question and put a stinging edge to it.
Lucy protested, "Oh, Miles, don't be mean!"
But Philippa laughed and tossed his challenge back at him with, "And Elyn is glooming because everyone else is. I came here so I could gloom somewhere else for a change, yes."
Miles laughed, too, and with mocking regret said, "It seems Father Leonel will have to give us a sermon on the virtue of patience some Sunday soon."
"And on how blessed it is to have a cheerful heart in the face of adversity," Philippa returned.
"And about the sin of gloom," Miles said.
"And the sin of sloth," Philippa retorted.
"Sloth?" Miles protested. "Where does sloth come into it?"
"Sloth in pursuing the virtue of patience. Sloth in avoiding the sin of gloom." Philippa paused in counting them off on her fingers to say, "Though I've never heard gloom listed among the sins. And _your_ sloth in not giving up tormenting your sisters and me."
Miles and Ursula laughed, while Lucy looked back and forth between him and Philippa and said, "Oh, you two. You're so strange."
"Strange and stranger," Miles agreed. "So, as long as we're talking of gloom and marrying, who do you want to marry, Lucy?"
Frevisse immediately guessed it was a question to which they all knew the answer, because Ursula rolled her eyes upward and took up her sewing again and Philippa sat back with smothered laughter while Lucy unhesitatingly launched into telling how she wanted a husband who was wealthy, not too old, and certainly not ugly, and lived in a town because she was so tired of living in the country and never seeing anything, and for her wedding dress she wanted . . .
Frevisse watched the others listening to her and wondered what angry things Sir William and Master Selenger had been saying to each other. Philippa had seemed unsure her uncle's anger this morning was from disappointment over Lady Anneys, but even if it had been, why would that bring him to angry words with Sir William? Because Sir William disapproved of his interest? Or because Lady Anneys was right—Sir William had set Master Selenger onto her for Sir William's own ends? If that was it, then both Sir William's and Master Selenger's anger could have been not at each other but at Lady Anneys for forestalling them. Or Sir William might be angry at Master Selenger for, thus far, failing in his purpose. Or just possibly Master Selenger was unwilling to the work and angry at having to do it.
Or this morning might have had nothing to do with Lady Anneys at all.
And possibly Master Selenger was only what he outwardly seemed—a man honestly drawn to a woman.
But what was the likelihood that Sir William had indeed ordered Master Selenger at her, the way Lady Anneys feared? Uncomfortably, Frevisse had to admit that, from what she had been told, Lady Anneys was the one hindrance between him and the profitable control of Hugh's, Lucy's, and Ursula's marriages. If he had ambitions that way, Lady Anneys was the obstacle he needed to remove.
Once Sir Ralph was out of the way.
Which brought her back to the question at the heart of everything here. Who had killed Sir Ralph?
Miles was telling Lucy that however rich a husband she married, she would undoubtedly use up all his money within a year. Lucy was telling him he was mean and Philippa was saying that they simply needed to marry Lucy to someone so rich she couldn't possibly use up all his money in a mere year.
No one more than barely noticed when Frevisse stood up and, with a murmur that she would walk awhile, left the arbor.
The garden was drowsy in the afternoon's warmth and sunshine. Hands quietly clasped in front of her, Frevisse moved along the paths. The beds enclosed in their low wattle fencing were full as could be with the end-of-summer flourish of flowers and herbs. Blue scabious and borage, feverfew with its flood of yellow and white flowers, tall and golden St. John's Wort, towering scarlet hollyhocks, thick-growing low thyme and marjorem, others that Frevisse did not know by name. Pleasures for the eye, ease for the mind, healing for the body. But her thoughts came with her and the garden's loveliness did not keep her from turning them over and over with the care she would have given to coals she feared were hot enough to burn.
Everyone said Sir Ralph must have been murdered by someone now long gone and never likely to be found out. They all said it . . . but how many of them believed it?
Some of them might. Others might doubt but were willing, Frevisse thought, to ignore their doubt. What she feared was that someone here knew for certain it was a lie. Because if the murderer was not long gone away, he was still here.
And there had to be those among the doubters who feared it, too.
Feared . . . but refused to face it: would rather live with the murderer still among them than find out who he was.
Whoever had killed Sir Ralph that savagely, beating his head in even after it had to be clear he was dead, had to have hated him. Unhappily, that hardly limited who might have done it. To guess from all she had heard, there were surely villagers in plenty who hated him; and certainly no one in his family loved him. And all of his family had been in the woods that day when he was killed, except for Ursula.
And Tom, Frevisse amended. He had quarreled with his father and gone back to the manor before the dog ran off.
But had he gone back to the manor? He wouldn't have purposefully lain in wait for Sir Ralph since they were hare-hunting that day and Sir Ralph unlikely to be alone in the forest. But what if he had stayed in the woods to walk off his anger, rather than going back to the manor, and had happened on his father and, still in a fury at him, killed him?
That was possible. Someone had told her it had taken a long time to find and bring Father Leonel and Tom after Sir Ralph's body was found. Although how long "a long time" was to someone frantically looking for someone would be difficult to determine.
And besides his immediate anger, Tom could have been afraid his cheating on the accounts with Father Leonel was about to be found out. Had Sir Ralph lately been growing suspicious? If so, Tom had better reason than a quarrel to want him dead and better reason to kill him if he suddenly had the chance.
Who else had immediate reason or need to have Sir Ralph dead? Hugh? Had he stood to gain anything from his father's death? Unless he had simply been unable to bear his father any longer, he had already had all he seemed to want—the hounds and hunting. Or was there more he wanted that she had yet to see? Tom had stood between him and inheriting the manor and now Tom was dead, too; but nothing about Hugh told Frevisse he had wanted the manor at all, let alone wanted it badly enough to kill for it. The only other thing that would come to him because of Tom's death was Philippa. Did he want her that badly? Frevisse had no way to know.
For what it was worth, Lucy claimed it was Miles, not Hugh, who was something more than only friendly with Philippa. From what Frevisse had just seen of them together, they did accord well together; but accord and love were two different things and neither one was enough for murder. Or rather, love could be used as a reason to kill but only by the most desperate or foolish, and Miles did not seem to be either. Not for the kind of murder that had been done on Sir Ralph. Hatred, though, was another matter, and Miles did not in the slightest hide his hatred of Sir Ralph or his pleasure that he was dead. Besides, with Sir Ralph's death he gained his freedom. Lucy had said he and Philippa were together when Sir Ralph was killed, but if there was love between them, she might well lie for him. But could she lie well enough? And keep up the lie? And if she could, then what sort of person was she, to see murder like that done and not only hold quiet about it afterward, but stay a laughing friend with the murderer?
That was, of course, supposing she had not killed Sir Ralph herself. But the objection to that was the same Frevisse already had against Lady Anneys or Lady Elyn or Lucy doing it: a woman repeatedly smashing a man's skull could not have avoided such a splashing and spattering of blood onto her skirts as would have been afterward seen and questioned.
For that matter, a man would not have escaped being bloodied either but after a morning spent hunting there had likely been blood on more than one of them already. If no one noted Sir Ralph's when it was fresh, later it would simply be dried blood along with other dried blood and unremarkable.
Her pacing of the garden had brought her past the arbor again and Sister Johane said, "You won't come back into the shade and sit?"
Frevisse made a smile whose worth she doubted and said, "No," and walked on, wishing she had not seen that Miles had shifted onto the bench facing Philippa and was leaning toward her with one hand out to hold a fold of her skirt while she leaned toward him, a hand resting on his knee, the two of them laughing together over something. But she had seen it and she put it with the rest.
The rest of what? What, altogether, did she have? There was Tom, who might have killed his father to protect himself. And Hugh, who maybe wanted either the manor or Philippa or both badly enough to kill for them not once but twice, meaning Tom's death was not by chance after all. And Miles, who openly hated Sir Ralph and had gained freedom by his death but not Philippa. To have Philippa, if he did want her, which was not certain, he needed both Tom and Hugh out of his way and, true, Tom was dead, but not by Miles' doing. So far as anyone had yet said, Miles had been altogether somewhere else when it happened.
But they weren't the only possibilities in Sir Ralph's death. There were Sir William and Master Selenger, too. Master Selenger could have wanted him dead, to clear his way to Lady Anneys if his interest in her was real rather than a thing made up between him and Sir William for their profit. And for Sir William himself, of course, there was the obvious reason that Sir Ralph's death put him in reach of profitable control of the Woderove marriages. That did not explain the savagery of the killing, though.
Of course, Sir William might have had an entire other reason for needing or wanting Sir Ralph dead, and the control of the marriages be only an afterthought. Come to that, Master Selenger might have other reason of his own, too. Some old anger or wrong only finally avenged.
If that were it, Frevisse had no thought on how she might find it out, but what did she know about them that day at least? Lucy had said they left the clearing together but that Sir William had come back alone. If he and Master Selenger had separated in the woods, one of them could have come on Sir Ralph and killed him. Or they could have killed him together and then separated. Either way, Master Selenger was Sir William's man. If Lady Anneys was right in believing his wooing of her was at Sir William's bidding, it could be supposed he might equally well lie to keep Sir William clear of murder.
Or Sir William could be lying for him.
About what had they been angry and arguing today?
She swiped an impatient hand at a tall borage plant as she passed. All she had were guesses and questions and no thought on how to find answers. Someone had murdered Sir Ralph. That was one certainty. Another was that he had been well-hated by many people, while some of them and too many others stood to profit by his death. Set the question of "who" to the side, then, since it was so wide. What about "why then" and "why there"? Why that time and place for his death?
Those questions told her something anyway—that his killing hadn't been planned. There had been no way to know the dog would run off and Sir Ralph go after it, no reason to expect he would be alone at all that day. That meant the chance to kill him had merely happened and someone had taken it, probably without thought of afterward. And very likely only blind luck, rather than forethought, had kept whoever it was from being caught at it or found out soon afterward.
Frevisse paused to rub thyme leaves between her fingers, releasing the scent into the warm air. A chance murder, yes. But had it been someone taking the chance when it came or someone forced to it then and there? There was still the possibility that it had been nothing more than Sir Ralph catching someone where they should not have been—someone poaching or a peasant taking wood. Though how could someone have been fool enough to be anywhere near where Sir Ralph was hunting when it was known he favored that place in the woods for his midday rest she did not know.
Hunting. The dog-boy Degory. She had forgotten him but he had been there. He had even said that Sir Ralph had hit him when the dog ran off. Had it been one blow too many?
He had also said Sir Ralph had hit Hugh then, too. Had _that_ been one blow too many?
Frevisse found she had stopped and was standing over the same cluster of red gillyflowers that Lady Anneys had looked at so long last evening. They seemed such simple flowers until one not only looked but truly _saw_ them; then, with their finely veined, delicately fringed petals, their careful stems and leaves, their rich and subtle coloring, they were not simple at all. Beauty, at its heart, was rarely simple, and yet the world held so much of it, and what Frevisse found forever hard to understand was why mankind so often chose ugliness when beauty was so readily, amply there.
Was it because the ugliness was easier? Because it let a person feel powerful without the cost of being anything but selfish?
She walked on. Sir Ralph, by all that she had so far heard of him, had been that kind of petty, small-hearted man, his life a blight on everyone around him. The pity was that the blight had not been cleansed by his death. It was still here, a blood-tainted shadowing in minds and hearts.
She came to the garden's rear gate and stopped there to gaze out across the field stubbled yellow from the summer-harvested hay. Come autumn, the cattle would be turned out there to feed and in the spring it would be ploughed and planted, to be harvested when another autumn came and after that left fallow, to be hay again another year.
Year went around into year, the pattern of them repeating and yet never the same. This year had brought murder here. She wished she could forget that. Wished she would let the matter lie, for someone else to take up or leave, as they would. She could do that, she told herself—could just leave it all alone. Guilt and justice were the crowner's and sheriff's business, not hers. If the crowner was satisfied and everyone here content, shouldn't she be, too?
But she couldn't be. Someone here was a murderer, and even though no one might know who he was, enough of them knew he was here that their denial, their willed unknowing, was a rot at the heart of things—and rot, left to itself, only spread, rotting everything around it.
**Chapter 18**
The afternoon had drawn on while Frevisse walked and the others talked. She was thinking she should rejoin them when Sister Johane joined her at the gate, leaned on it beside her, and said, "In spite of Dame Emma and Sister Amicia, I'm not used to this much talking. I'm going to see how Lady Anneys does. Should we do Vespers after that?"
Pleased that for once she need not remind Sister Johane about an Office, Frevisse said, "I'll come with you and save you coming back."
As they crossed the silent, empty hall, the hound Baude—stretched out on the cool floor-tiles near the empty hearth—opened one eye as they passed but did not bother to raise her head. Upstairs, they found Lady Anneys just awakened, sitting on the side of her bed trying to pin up her hair. Still slow with sleep, she was not doing well and Sister Johane went to take comb and hairpins from her, asking, "How do you feel, my lady?"
Lady Anneys gave a soft half-laugh. "I'm not sure I'm awake enough yet to know. But my headache is gone. Thank you for that."
"You can have the sleeping draught again tonight if you like. It's a mild one. Having it twice in one day this once will do no harm."
"I should like that, I think." Lady Anneys sighed and moved her neck from side to side as if she were stiff. "It's so tiring when even going to sleep takes effort."
Still eased from her sleep and maybe from the quieting aftereffects of whatever Sister Johane had given her for the headache, Lady Anneys unprotestingly let Sister Johane finish her hair and then pin on her veil for her. "But not my wimple," she said. "I know as a properly grieving widow I should go wimpled, veiled, and all but invisible, but the day is hot and I'm not properly grieving and we have no guests to be offended. So just the veil."
"Lady Philippa is here," Frevisse said.
"Philippa is family, not guest." Lady Anneys made to rise, found herself a little unsteady at it, and thanked Sister Johane for a steadying hand to her back. On her feet, she straightened her shoulders with a visible effort and asked, "Is Elyn here, too?"
"Only Lady Philippa," Frevisse said.
"Where is everyone?"
"In the garden, except Master Hugh, who went to see how the harvest does."
Lady Anneys was both fully awake and had her balance now and said as she started for the door, "I'll go to the parlor then. I don't want all Lucy's chatter just yet. But could you ask Philippa to come to me, Dame Frevisse?"
Frevisse took the errand willingly. By the mingle of voices from the kitchen as she passed, she guessed someone was back from the field and readying supper, and in the garden she found Hugh had returned, too, and was sitting beside Ursula, pretending to find fault with her sewing. Philippa had moved to the other bench, while Miles had shifted back to the ground and was leaning with his back against the wooden wall of the bench near her, his hound stretched out beside him, chin resting on his knee, his hand on its shoulder. Frevisse gave her message. Philippa said she would come in a few moments, and after returning to tell that to Lady Anneys in the parlor, Frevisse went thankfully up to the bedchamber with Sister Johane again.
Vespers was supposed to draw the day's work to an end in prayer that left the heart and mind free of whatever burdens the day had held. In the nunnery there would be supper afterward, the nuns silent while someone read aloud from a suitable book, then the hour of recreation followed by Compline's brief prayers and bed in good time for a few hours' sleep before Matins and Lauds in the middle of the night. But today Frevisse could not settle her mind to the Office. Vespers' comfort did not come and she was merely grateful when at last she and Sister Johane ended, _"Fidelium animae per misericordiam Dei requiescant in pace. Amen"_ —The souls of the faithful through the mercy of God rest in peace—and could cross themselves and stand up.
She made herself wait, though, until they had put their breviaries away before she said, "Shall we see if Lady Anneys wants our company?" Because this might be her chance to talk with Philippa apart from the others.
Her restlessness must have showed despite herself or else she had betrayed herself in some other way, because rather than immediately answering, Sister Johane gave her a long look and finally said, "You're doing it again, aren't you?"
Frevisse was surprised into asking, "I'm doing what again?"
"We talk about it sometimes. What you do. Finding out murderers. You're trying to find out who murdered Sir Ralph, aren't you?"
Frevisse opened her mouth, closed it again, and at last answered, "Yes."
"Is it wise?" Sister Johane asked seriously.
Again Frevisse hesitated before saying with matching seriousness, "I don't know."
"You can't just let it go?"
That at least she could answer without hesitation. "I can't, no."
Now Sister Johane paused, before finally saying, simply, "If you need any help, I'll give it."
The simple steadiness of that took Frevisse by more surprise. She had to will herself to remember that, after all, Dame Claire trusted this girl—this woman—and that said much for her, before she could bring herself to say, "Just listen for anyone saying anything about the day Sir Ralph was killed. Or anything that's said that's . . . not right to your mind."
Sister Johane gave a small nod. "I understand."
"But don't ask anything. Don't let anyone know you've any interest. I'd have you promise me that."
Steadily, Sister Johane said, "I promise."
Frevisse stared at her a moment longer. Sister Johane levelly returned her gaze in a way that satisfied Frevisse, she understood the danger there could be, the odd thing being that until now Frevisse had not clearly considered the danger that someone who had killed once might well kill again to protect himself. But there was no going back from it now and she only said, "I'd like to see Lady Anneys and Philippa alone. Would you go back to the garden without me?"
Sister Johane agreed to that with a nod. They went down to the hall and parted and outside the partly open parlor door Frevisse forwent the temptation to pause and listen, instead scratched lightly at the doorframe and, at Lady Anneys' bidding, went in. Lady Anneys and Philippa were seated on the long chest under the open window to catch whatever breeze might come that way, both of them leaning on the windowsill, Philippa with her head bowed and resting in one elbow-propped hand, Lady Anneys with her arm lying along the sill, her hand touching the girl's arm. By the look of it, Philippa had been crying, and although she was recovering from it, she turned her face away without lifting her head, hiding behind her hand, as Frevisse entered. Contrariwise, Lady Anneys was calm-faced and asked in an even voice, "Where's Sister Johane?"
"Gone back to the others in the garden."
"Please, sit, if you like." Lady Anneys nodded toward a carved joint stool near them and went on as Frevisse crossed to it, "Philippa has been warning me of exactly what I feared."
Philippa raised her head, began a protest. Lady Anneys briefly, tenderly squeezed her arm and said, "I've already talked about the will and Sir William and your uncle with Dame Frevisse. She knows enough there's no harm in her knowing more. Please, tell her what you told me."
Philippa let out her breath on an unsteady half-sob but obediently straightened up, folded her hands into her lap, and faced Frevisse. Dry-throated from her crying and her voice not so even as Lady Anneys', she said, "You heard me saying my father and uncle were angry at each other this morning. What I didn't tell was why." She looked down at her hands in her lap and said, pacing the words like something learned by rote, "Father wants Lady Anneys married or unchaste so he can take over Hugh and Lucy and Ursula's marriages. He's set my uncle on to do it and is impatient because it hasn't happened yet. This morning he was saying he wants it done sooner rather than later, before she does anything on her own about the marriages. About Hugh's marriage especially. Father is afraid she won't agree to me marrying him."
Frevisse looked at Lady Anneys. "You're thinking against it now?"
"My lady," Lady Anneys said wearily, "Sir William is ahead of me on this by a long ways. I'm too tired and hurting, heart and mind both, to think about any marriages. Left to myself, I'd leave at least theirs to Hugh and Philippa's choice anyway."
"That's one of the things that worries Father," said Philippa, her head still bowed. "He doesn't want it left to our choice."
"Because he thinks you or Hugh will refuse the marriage?" Frevisse quickly asked.
Philippa finally looked up. "He thinks Hugh is too cool toward it. He's afraid Lady Anneys will influence Hugh against it. Because of Tom."
"And you? Is he afraid you're cool toward it?"
Philippa looked down at her lap again. "He expects me to do what I'm told to do."
"And will you?" Frevisse asked.
"Yes." Flat and without feeling.
"What about Miles?"
Philippa jerked up her head and stared at Frevisse, eyes frightened wide and not answering.
It was Lady Anneys who asked, puzzled, "Philippa?"
Philippa turned her head to stare at her, still making no answer.
Carefully, making no mention of Lucy's tattling talk, Frevisse said to her in a meaningful way, "I watched you together today."
She did not add that she had seen nothing that truly betrayed them, but Philippa gave a small gasp and covered her face with both hands.
Beginning to be alarmed, Lady Anneys asked, "Philippa?"
Philippa dropped her hands into her lap again and lifted her head. She was flushed but not crying, and said, somewhere toward defiant, "Miles and I. Yes. If I had my choice, I'd choose Miles. Only he won't let me. Because we know there's no hope in it. Father will never let it happen." She gave a brief, unhappy laugh. "Nor will Miles. He says I'd lose too much by marrying him. He says he won't do that to me."
Gently Frevisse asked, "Does Sir William know what's between you?"
"If he did, I doubt I'd be let out of the house again until safely married to Hugh."
"Does Elyn know of it?" Lady Anneys asked.
"No. Nor even suspect. She'd surely tell Father if she did."
"But when you went alone with Miles, didn't that make her wonder?" Frevisse asked.
"How do you know I've been alone with Miles?" Philippa asked with sudden fear. "Who else knows we've been meeting?"
"Oh, Philippa," Lady Annys said, sharp with worry. "You've not been meeting secretly?"
"A few times. That's all. Just a poor, few times. But no one else is supposed to know about it at all. Who's seen us?" she demanded of Frevisse, half-angry as well as frightened. "Who told you?"
Quickly, deliberately mildly, to soothe her, Frevisse said, "All I know is that someone said that the day of Sir Ralph's death you and Miles went walking together while the others were searching for the dog."
Philippa put a hand over her mouth, realizing she had said things she could have left unsaid; but an instant later she dropped her hand and said, "Miles and I went off on our own that day, yes. There was nothing wrong about it. Lady Anneys was there. She saw us and said nothing."
"Because I thought it was simply that you and Miles couldn't bear Lucy's and Elyn's nattering any longer. There seemed less harm in your walking together than in Miles knocking their heads together. I never thought . . ." Lady Anneys shook her head. "I never thought you and Miles . . ."
Or maybe she had unknowingly chosen not to think it, Frevisse thought. Among the things Lady Anneys had not needed in her life then, any more than now, was one more trouble. She might have unknowingly chosen not to see that one.
"It doesn't matter anyway," said Philippa. "Nothing can come of what we want. It's Father is the trouble. And Uncle John." Whose treachery—to judge by her voice—hurt her worse than her father's did.
"He's completely willing to do Sir William's bidding in this?" Lady Anneys asked bitterly.
"I . . . I gather so. What he seemed angry at was being pushed too hard at it. Father wants it done now and he says it can't be."
"Is he supposed to go so far as to marry me? Or will ruining my reputation be enough?"
"I don't know," whispered Philippa.
"Or why not do both?" Miles asked angrily from the doorway.
He and Hugh stood together there, both of them dark-faced with matching anger, and again Frevisse was struck by how much like brothers they looked. But it was Miles who came forward ahead of Hugh, saying at Lady Anneys, "Why shouldn't Selenger ruin your reputation for Sir William's behalf, then marry you for his own? That way both of them will have something out of it."
Behind him Hugh said warningly, "Miles."
Miles, halfway across the room, swung around and said at him, "For years, because of Sir Ralph, we've always talked around things, never straight at them. Where has it put us? Anywhere we want to be?"
"In that, you're right," Lady Anneys said before Hugh could answer. "So let's talk straight at them now. But let's not have anger in place of silence. Sit down. You and Hugh both."
It was an order more than a request, and Hugh came forward to sit on a joint stool. Miles held back, on the edge of rebellion for a moment, then went to lean his hips against the table edge, arms folded across his breast.
"First," said Lady Anneys to both of them, "Philippa came here today to warn me about Sir William and Master Selenger. Dame Frevisse and Hugh both know I already feared it. Miles, did you?"
"Yes. From almost the first day Selenger came here after Sir Ralph's death."
"Well, now there's no more need to fear because now that we know for certain and I don't have to worry you'll find out, I won't let him near me again. It's finished. That leaves us with you, Miles. You and Philippa."
She must have seen as clearly as Frevisse did the look that flashed between Miles and Hugh because she asked quickly, "Hugh knows?"
For once Miles was less than assured, his glance at Philippa uncertain as he answered, "We've talked some of Philippa and me, yes, but this isn't the time to—"
"You said you were tired of talking carefully around things," Lady Anneys pointed out. "So am I. Philippa has already given away more of what's between you than she meant to, so Dame Frevisse and I know. I take it Hugh knows, too?"
Grudging and uncomfortable, looking aside rather than at any of them, Miles said, "I've told him, yes."
"Hugh? What have you thought of it?"
Hugh looked as if he were thinking he'd like very much to be somewhere else just then but met his mother's gaze and said, "I favor it."
Philippa stood quickly up and went to Miles. Laying her hands on his folded arms, she said gladly, "You see?"
"Except," Hugh said with a wary look from his mother to them and back again, "he keeps saying I should marry her."
Philippa looked unbelievingly from Miles to him to Miles again, stepped back, and said fiercely, "You idiot!"
Miles tried to catch her hands but she did not let him and he spread out his own, half-entreating, half-insisting, "We've talked of the hopelessness of it often enough. What can I give you except trouble? It's better that you're safe with Hugh than ruined with me."
"Idiot!" she said again, still fiercely.
Lady Anneys laughed, startling them, maybe startling herself; but she pointed a finger at Miles and said, "I don't know much of love, but when a woman calls you 'idiot' in that way, my guess is she's much too much in love with you to reason about it."
Stubbornly Miles said, "It doesn't solve anything." But even as he said it, one of his hands reached toward Philippa, who moved close to him and took it.
"No," Lady Anneys agreed. "But with it in the open among us, it makes one less secret."
For her part, Frevisse was watching Hugh. He had been tensely silent until, when Miles reached out to Philippa, his face lightened into a smile of relief and pleasure, telling her he had meant it when he said he favored Philippa for Miles rather than for himself. The friendship between them was that true. But because she was unlikely to have better chance than this, she asked, "Miles, when you and Philippa were away together the day Sir Ralph was killed, did you hear anything—any angry voices, a quarrel, someone moving in the woods who shouldn't have been there . . ."
At mention of Sir Ralph, Philippa pressed close to Miles and hid her face against his shoulder. Putting his arm around her, Miles answered, "If we had, we would have said so."
"Except maybe you were afraid of Sir William learning you'd been away together?" Frevisse asked.
"I don't want to hear or ever think about that day again," Philippa said, her face still to Miles' shoulder. Lady Anneys, her own head bowed over her hands now clenched together in her lap, nodded sharp agreement. Hugh made no answer at all except to turn his head away from all of them. Only Miles met Frevisse's gaze and said steadily, "It was murder. We would have said if we'd heard or seen anything, Sir William or no."
Hugh turned back. "That's one thing. We none of us heard anything. There was no shouting."
"And Sir Ralph was a famous shouter," Miles said dryly.
"The crowner asked the same thing," Hugh said, "but whoever it was must have taken him by surprise. They must have been lying in wait and taken him by surprise."
Philippa stepped back from Miles. "I'd best go home now."
Miles moved as if to see her out, but Lady Anneys said, "Let Hugh go with her, Miles. He'd be expected to and for now we might do best not to bother giving anyone other thoughts."
Hugh and Miles exchanged looks and matching wry smiles before Miles nodded. Philippa, missing none of that, smiled much the same, curtsied to Frevisse and Lady Anneys while Hugh bowed, and then left easily with him, except as they went out the door she pushed his arm and said, "He told you about us?"
Miles, as if abruptly weary, sank down on the joint stool Hugh had left. Lady Anneys rose and went to him, leaned down, and kissed his forehead before laying a hand on his shoulder and saying quietly, "Heed me in this, Miles. Don't lose or ruin what there is between you and Philippa. You can only guess at how much hurt there'll be for her if she marries someone she doesn't want, but I know, and I say that whatever she loses from Sir William is nothing when set against it."
Miles stood abruptly up and without a word or even a bow went out of the room.
Lady Anneys watched him go, then turned back to the window, sat again on the chest, and said, "I would never be so young again for all the world."
"Nor I," Frevisse agreed. She went to sit at the chest's other end. There were heartaches of one kind and another all through life but those of one's young years, even looked back on, seemed to cut with a sharper edge than almost any later ones, simply—or not so simply—because one had not yet learned any defense against them. And that, in time, like all things, they passed. But slowly, feeling her way through the thought as it came to her, Frevisse said, "I would never have thought, seeing Philippa in the garden this afternoon, that she was as deeply upset as she showed here about what she'd overheard between her father and uncle."
Lady Anneys looked out the window into the dusty sideyard where the broad-trunked, thickly green-leaved elm tree spread its branches to half-hide the back of the stable and byre and slowly said, "We've all learned not to show our thoughts or what we feel. We hide almost everything from one another. Around men like Sir Ralph and Sir William, that's safest, but sometimes we even hide ourselves from ourselves, I think."
She was calm rather than bitter about it. But she was calm about most things, Frevisse had found. Except now Frevisse was coming to think it was not so much calm as the smoothness of old, old scars over deep wounds. The kind of scars that ache when the weather is wrong and sometimes hurt with shadow of their old pain. Gently, and again because she could not be sure she would have such another chance, she said, "Neither you nor anyone else cares at all that your husband is dead, do they?"
Lady Anneys paused before answering, then said, "If by 'cares,' you mean 'grieves,' no. No one grieves for him. His death lessened, not increased, our grief here."
"No one even cares that his murderer goes unknown."
Again the pause. Then, "Whether we care or not makes no difference. His murderer is long gone from here and there's an end of it."
"And you're content with that?"
"I'm content _because_ of that."
From the hall came the sounds of the tables being set up for supper and servants' voices. The girls and Sister Johane would come in from the garden soon, or Hugh come back from seeing Philippa away, and Frevisse considered quickly asking who else had left the clearing the day Sir Ralph died and what Lady Anneys might have noticed about anyone when they came back there after his death. But those questions were too open; Lady Anneys would understand too readily why she asked; and unable to think of anything else to say, Frevisse let the silence lie between them, looking out the window at the elm tree's leaves hanging unmoving in the still, hot air.
It was Lady Anneys who stirred and spoke first, answering something that had not been asked. "My husband was murdered. My son died by mischance. But that mischance came because of all the wrongs and ugliness my husband made while he lived. Those are deaths and griefs enough. I only want it all to be done with and forgotten and the rest of us left free to go on with our lives. I want to remember Tom and forget my husband as completely as if he had never lived."
And could there be worse epitaph than that for anyone, Frevisse thought. To be so hated that someone wished to forget that you had ever been at all.
**Chapter 19**
Hugh awoke too early, when only the first bird was twittering outside the loft's small window, and lay in the darkness too long with only his thoughts and the even sound of Miles' breathing for company. When finally the black square of the window lightened to the dark blue of dawn's beginning and there were the sounds of the servants who slept in the kitchen below him getting up, he could no longer bear it and rolled off his bed, gathered up his clothes from on and around the stool where he had tossed them as he undressed last night, and groped his way to the ladder-steep stairs.
Miles mumbled a question. Hugh answered, "Out," and creaked down the steps into the passage between hall and kitchen. Bevis, lying on an old blanket put along the wall for him there, raised his great head but did not bother to rise. Hugh touched him briefly, silently letting him know everything was well, and went along the passage to let himself out the rear door into his mother's garden. He stopped there to dress. The birds were in full choir now and the world was brightening from blue shadows into colors. By the time he went, fully dressed, out the garden gate in the already warm half-light of dawn, he could hear Helinor in the kitchen threatening Alson over something and, from above, at the far end of the hall, Lucy complaining to Ursula in their room that Ursula's bad dreams had kept her awake in the night.
With no particular purpose except to be away from all that household busyness beginning the day, he went to the kennel where the hounds, strewn in long-legged sprawls around the yard, lifted their heads to look at him, but only Bane rose and ambled to the fence. Degory, burrowed into sleep with an arm around Skyre's neck on a heap of straw in a corner of the yard, stirred less readily and asked, not fully awake, "What is it?"
"Nothing. I just came to see the hounds." He stroked Bane's long head. "I don't see them enough anymore."
Yawning, Degory sat up with straw in his hair. Skyre sat up, too, and licked his face.
"Is she doing better?" Hugh asked.
Degory climbed to his feet, scratching. "Better, but I doubt she'll ever be right." He ambled to the fence much the way Bane had. "We're going to be short on lymers without her."
"Not if she breeds true. I'm thinking to try her with Makarie when the time comes."
"Aye, that might do. How is it with Baude? Are you going to bring her back here to whelp?"
"I mean to, yes."
"I've her place ready for her. She can't be long off it now."
They made comfortable dog-talk for a time, daylight growing around them and the manor coming well awake. The hounds roused and wandered around their yard, came to be petted, wandered off again. "What about old Bevis, then?" Degory asked. "Still following Master Miles everywhere?"
"Except into the loft to sleep with us."
"Odd, that," Degory said on a yawn. "Him being so much Sir Ralph's dog but taking to Master Miles like this."
Hugh kept back from saying that it showed Sir Ralph had not been worth even a dog's loyalty, but offered, "Come to that, it's just as odd that Miles has taken to him."
"That's true enough. Master Miles never much cared for dogs that I ever saw."
"Maybe he'll be one of us yet. What do you say? Are you and the pack up to some hunting tomorrow?"
"That we are!" Degory exclaimed, then added hopefully, "A hart maybe?"
"I can't take any men off the harvest to make a hart-hunt. It will have to be hare again, but I'm thinking we'll go all the way off to Beech Heath. We haven't coursed there since spring and Miles said he saw signs of hare in plenty of late."
"They'll be fat and lazy by this time," Degory said happily. "Ready for the hounds to tickle them up they'll be. We'll have to be off early though, to be ahead of the heat. Is it staying this hot?"
"Gefori says there'll be rain in a day or two."
"Too much for the harvest?"
"He doesn't know."
"St. Peter bless us," Degory said. He slipped the latch on the gate and came out of the yard, pushed several hopeful muzzles back inside, and fastened the gate. "I'm to breakfast. You?"
"No. I'll linger here a time. You go on."
Degory left and Hugh stayed where he was, a hand out to stroke heads and receive licks. The sun edged into sight over the horizon, sharpening the day to long-thrown shadows and molten gold. Hugh wished he felt that bright about the day. His heart had lightened with talk of hounds and hunting and the thought that tomorrow he'd not be bound to the manor by one duty and another but away and rid of everything save the needs of the hunt. But that was tomorrow. There was still today and then the days after tomorrow and him tethered here at the manor for most of them. Tom had had everything in hand and ready for the harvest, and the reeve saw to most of the rest, so that presently there was too little to keep himself fully busy here, just duties enough that he wasn't free to go about such hound and hunting work as he could have otherwise.
Dissatisfied with almost everything, he left the kennel and, vaguely minded to go to Mass, circled the stable and saw, when he reached the road toward the church, that the two nuns were ahead of him, bound the same way, their black gowns and veils dull against the morning's late summer green-and-gold. Hugh wondered, not for the first time, if Ursula would choose that life. He hoped not. He couldn't imagine her a nun, shut up for the rest of her life. But then he could not imagine her married, either. She was just his little sister.
On the other hand, the sooner a husband was found for Lucy, the better. Tom had said something once about asking Master Wyck if he knew anyone in Banbury who might be suitable. That was likely the best way to go about it and he would, once the harvest was done and he had talked with Mother about it.
By then this trouble with Master Selenger and Sir William might be settled, too—another thing off his mind.
Ahead of him the nuns went into the church. A few village folk were going in, too, and Father Leonel would be making ready to begin the Mass, and in his mind Hugh could see it all, familiar all of his life. The short, windowless nave; the chancel hardly big enough to hold the altar; the small sheen of candlelight on the silver altar goods given by the widow who had held Woodrim before Sir Ralph bought it; the green altar cloth that Hugh could remember his mother embroidering with vines and wheatsheaves when he was small; the small, plain-glassed window high in the east gable end of the roof letting in so very little light that gray shadows filled all the rest of the church. And suddenly Hugh knew he could not go in there, to the shadows and the prayers, and already in the churchyard, halfway between gate and door, he turned aside, onto the little dirt path that curved through the churchyard and around the church.
He would sit awhile with Tom, he thought.
They had made Tom's grave on the other side of the church and just outside the chancel, as near to the altar as could be without being buried inside, and the heaped earth of his grave-mound was still raw under the turfs of green grass laid over it at his burial's end.
Sir Ralph was buried across the churchyard, almost against the churchyard wall, the hump of his grave already well-settled, its green turfs melded together by the summer's rains. He had made no provision in his will for where he should be buried. It had been Lady Anneys' choice and she had pointed there, a place that would be easily lost among the other graves and grass if no one bothered to mark it with more than the plain wooden crosses that marked the village folk's graves. Those, being wood, decayed and the graves were lost along with them as folk who might remember died in their turn. No one had known whose bones were dug up when Sir Ralph's grave was made; their bones were simply added to the other bones piled in the small charnel house screened by yew trees at the far corner of the yard, where someday Sir Ralph's bones would go and maybe sooner than some because no one was minded to mark his grave with even a wooden cross, let alone visit or remember it.
Hugh meant somehow to find the money to lay a stone grave-slab over Tom, though. Even if it were only flat to the ground and carved with no more than a cross, it would be something, would mean Tom would maybe sleep undisturbed in his grave until Judgment Day.
Hugh had no clear thought why he was going to Tom's grave. To pray for him maybe, or just to be somewhere no one else was; but as he came around the church's corner, he saw Miles at the churchyard's far end, standing near the yews around the charnel house. Miles had his back to him and Hugh paused. He knew Miles came here sometimes—possibly more often than Hugh knew—to where his mother was buried.
"There's not much I remember of her," Miles had said once. "She had black hair, very long. She'd let it down at bedtime and hold me on her lap and wrap it around me and tickle my ear until I curled up laughing. I remember walking somewhere with her, too. On a sunny day. Picking flowers out of the wayside grass to make a flower crown. Not much else."
Not much at all but enough that he still came to her grave, and Hugh's half-thought to leave before Miles knew he was there; but Bevis lying beside Miles raised his head to look at Hugh, and Miles turned around like someone not wanting to be taken unaware from behind.
But seeing Hugh, he waved and started toward him and Hugh waited where he was, beside Tom's grave. Father Leonel had begun the Mass, his old voice faint from inside the church, and Miles, joining Hugh, cocked his head, listening, and asked, "You came too late to go in?"
"Found I didn't want to, once I was here."
"You were out and about early, gone when I awoke. Where've you been?"
"To see the hounds. I'm thinking to have a hare-hunt at Beech Heath tomorrow, to give them a run and bring in some fresh meat. Bevis ought to come. Will you?" Before Miles could answer, Hugh felt bound to add, "I'm going to ask Sir William if he wants to join us."
"Why?" Miles asked, more rude than curious.
"Because we can't afford to have him angry at us."
"What about our being angry at him?"
"We can't afford that either. Not when there's no point to it."
"There wasn't any point to Tom's death either."
"That wasn't meant, Miles. I was there. I know."
"Tom is dead, whether it was meant or not," Miles said bitterly.
Hugh abruptly sank down on an old grave's low green mound across the path from Tom's unweathered one, drew up his legs, wrapped his arms around his knees, bent forward to press his forehead onto them, and said miserably, his eyes tightly shut to close out the world, "God's mercy. I'm so tired." He didn't mean "tired." He meant "in pain" but he didn't know how to talk about pain, or about the fear or the wariness or the darkness in him. When Sir Ralph was alive, he had known what the pain was for, had known from where all of it—the pain, the fear, the wariness, the darkness—came. Now Sir Ralph was gone and all of those things should have gone, too. But they hadn't. They were all still with him.
The difference was that he no longer knew the reason for them.
And he was afraid to find out.
Miles sat down beside him on the grave-mound, briefly touched his shoulder, and said with the rare gentleness that was always a surprise from him, "I'm sorry. I shouldn't have done that to you."
Without lifting his head, Hugh rolled it from side to side, refusing Miles' guilt. "You didn't do it. It's me. Everything is wrong since Tom died and I doubt I can ever make it better."
"You're doing well at it," Miles said. "Give it time."
"I'd rather give it away. All of it."
"Not until you've married Lucy and Ursula off, and by then your own wife, whoever she is, won't let you."
Hugh raised his head and looked at Miles."'Whoever she is'?" he echoed, and was pleased he matched Miles' light touch on the words. "You mean you've decided not to make me marry Philippa?"
"I don't think I'd like what she'd do to either one of us if we tried."
"Then you'll marry her before you leave for Leicestershire?" Hugh pressed.
"If I marry her, I'll _have_ to leave and it won't have anything to do with Leicestershire. It will be because Sir William is after me."
Hugh let go the lightness, said seriously, "Have you thought that once Philippa is married, Sir William loses one of his reasons for trying to take over Sir Ralph's will? Then I'll find a husband for Lucy as soon as may be and someone for myself, and with only Ursula left, none of it will be worth Sir William's bother anymore and Master Selenger will give over troubling Mother."
"Selenger had better give over troubling her before then," Miles said. "And how are you going to protect her in the meanwhile?"
"She's doing that, by seeing to it she's never alone. That's why she brought the nuns back with her, I think."
"You can't keep them forever."
"When they leave, I'll do something else. Meanwhile, the thing is to get you married to Philippa."
"Leave it, Hugh. For now, just leave it," Miles said amiably. But firmly. He slid down to lie on his back, his head pillowed on the grave. Bevis, who had been sitting patiently this while, promptly lay down and nudged his muzzle into Miles' side to remind him he was there. Miles obediently began to play with his ears.
Hugh held silent. Silence was better than making a quarrel where he did not want one, and although the sun was well up now and the dew already dried from the grass, warning the day would be hot later on, just now it was good simply to sit here in the warm, heavy sunshine. He pulled at the grass beside him until he had a handful, then let it fall in a small scatter, listening to Father Leonel without making out the words, while the quiet drew out between him and Miles, until he said, "You still mean to go to Leicestershire come Michaelmas?"
Miles made a wordless, assenting sound.
Hugh plucked more grass and let it fall. "I've thought of another reason you should marry Philippa."
"Hugh . . ." Miles started warningly.
"Just think how angry it would have made Sir Ralph."
That took Miles enough by surprise that he laughed, choked, began to cough, and had to sit up. Hugh cheerfully beat on his back, helping not at all. Bevis, looking confused about whether he should worry or not, sat up, too. Miles, fending off Hugh's blows with one hand, put his other arm around Bevis' neck and croaked, "Let be, you dolt."
Hugh let be and stood up out of reach before Miles recovered enough to repay him. "Mass seems done," he said. "I'm going in to talk to Father Leonel before going back to the hall. Will you come?"
"May as well," Miles said, and together they circled back to the other side of the church. The village folk were already out, spreading homeward down the village or to work, but as Hugh and Miles reached the broad, round-topped door of thick planks and heavy ironwork standing open into the nave, Hugh heard Father Leonel, inside, saying to someone, "Go on then," not sounding pleased about it.
"The day Sir Ralph was killed . . ." Dame Frevisse said.
Hugh stopped in midstride. So did Miles beside him. ". . . why did whoever went looking for Tom Woderove have such trouble finding him? I understand he'd come back to the manor well before his father's body was found, but he couldn't be readily found when he was looked for."
Crisply, in a way that made Hugh think she had been asking other questions and the priest was not happy about it, Father Leonel answered, "Tom took so long to be found because no one thinks to look first thing in a church for a young man. That's where he was. He'd come to me to talk off his anger and make confession of it. Why is knowing that of any help to Lady Anneys?"
Hugh and Miles looked at each other, each silently asking if the other knew what this was about, both of them shaking their heads that they did not.
And Dame Frevisse was ignoring Father Leonel's question to ask, "Was he worried that Sir Ralph was going to find out what you both had been doing with the accounts?"
"There was nothing to worry about that way. We always made certain there was money enough for his hounds and Sir Ralph cared nothing about the rest. Dame, be advised—let all of this go. Let it rest with the dead. It's better there than being raised up to plague the living."
Hugh could not make out her murmured answer to that. Maybe she had bowed her head and was accepting the priest's order, because the next that Hugh heard was Father Leonel blessing her in farewell. Beside him, Miles drew back a step, making to leave before they came out. Hugh was ready to retreat with him but they had waited too long. Nor was it Father Leonel who came out but both the nuns, and there was nothing either Hugh or Miles could do, caught flat-footed and in the open, except bow, wish them good-day, and move to go past them into the church. But Dame Frevisse, quicker than they were, said, "Master Woderove, we were just talking to Father Leonel, trying to understand matters better, so we can maybe better help your mother."
"Help her how?" Miles demanded.
"She's in deep grief," Dame Frevisse said.
"For Tom," Hugh said stiffly. "We all are."
"For Tom, yes," Dame Frevisse agreed. "But she's in grief for her own life, too."
"In grief for her life?" Miles echoed, sounding as if he understood no better than Hugh did.
"For her life," Dame Frevisse repeated. "We gather that Sir Ralph was not . . . kind . . . to her."
Miles gave a short, harsh laugh. Hugh said only, "No. He wasn't."
"Or to any of you," Dame Frevisse persisted.
"Or to anyone," Miles snapped.
"And to you most especially," Dame Frevisse said to him evenly; but she returned to Hugh with, "You got on the best of anyone with him, didn't you?"
"For what that was worth," Hugh said, "and only because of the hounds and the hunting. He probably got on best with Sir William."
"Since they were both of a kind," Miles said, not hiding his raw dislike of that "kind."
"Does Lady Anneys ever talk about Sir Ralph's murder?"
"We've none of us talked about it," Hugh answered sharply. "Ever."
"Not at all? Not even when it happened?"
"Not then and not now," Hugh said almost harshly.
"We were too glad that he was gone," said Miles.
Dame Frevisse fixed a suddenly narrowed look on him. "Somebody wanted him gone badly enough that they killed him."
Miles met her look with his own and answered, "Yes. And blessings on them for it."
"Don't you care who?"
"Not greatly, no."
"Whoever did it," Hugh put in, "is long gone and not likely to be found. So there's no point in caring."
"What about Master Selenger?" she asked them both.
Hugh traded quick, questioning looks with Miles, who did not look as if he understood the question either, before Hugh answered, "Master Selenger didn't kill him, no more than any of us did."
"How do you know he didn't? He was there that day. And he's made plain his interest in Lady Anneys."
"He's only interested because Sir William told him to be," said Miles.
"Or was Master Selenger interested before," Dame Frevisse asked, "and it's Sir William who's being led, rather than the other way on?"
Hugh and Miles traded looks again, and this time it was Miles who answered, somewhat slowly, thinking it out as he went, "Because if one of them murdered Sir Ralph, it would be the one who first thought of how they could gain by his death. Master Selenger because he wanted Lady Anneys. Or Sir William through the will."
"Yes," Dame Frevisse said.
He and Hugh looked at each other again.
"That," said Miles, still slowly, "could be worth the finding out."
But Hugh said, "My lady, please leave it alone. Please."
"The day Sir Ralph died . . ." she started.
Worried and puzzled, Sister Johane said, "Dame . . ." at the same moment Miles burst out, "Hugh said let it lie, my lady! So let it!"
Hugh, still pleading more than demanding, held out a hand to silence Miles and said, with more quiet than he felt, "Dame Frevisse, please believe this—that we're all far the better off with Sir Ralph dead than we ever were with him alive. His death is neither a trouble nor grief to anybody here. If there's any grief over Sir Ralph, it's for the wasted years he was alive."
And there was the terrible truth. That was what Sir Ralph's life had been and it was what he had made of all their lives while he lived. A waste. His life had been a waste, and Tom's death was a waste, and the grief of both those truths suddenly choked Hugh. "So let his death go, my lady," he forced out. "We've grief enough and don't need more." And because he could not trust himself beyond that, he turned away from her and the other nun and Miles. Turned away from them and everyone and everything except his grief and tangled thoughts. Turned and went out of the churchyard and across the road, blindly headed toward the forest's edge across the pasture there, wanting the sanctuary the forest always gave him. Sanctuary and time to think. Sir Ralph's life had been a waste of all of their lives and none of them talked of his death because no one wanted to know . . . who among them had done it.
Because, very surely, one of them had.
And Dame Frevisse knew it as well as he did.
**Chapter 20**
With Hugh almost to the woods, Miles broke the startled silence left behind him, saying, "If you'll pardon me, too, my ladies," and went the other way, across the churchyard, not even to the gate but bracing one hand on the low wall when he came to it and swinging over. His hound cleared it in an easy leap after him and they disappeared together into the village.
When Frevisse looked back toward the woods, Hugh was out of sight. Beside her, Sister Johane said softly, "Oh my," and when Frevisse looked at her, her eyes were large with pity and unease.
"Oh my indeed," Frevisse agreed.
"Do you think one of them did it?" Sister Johane almost whispered, though there was no one to overhear them.
Slowly Frevisse answered, "There's nothing that says so."
"Nor anything that says not," Sister Johane said, her gaze fixed on Frevisse's face.
"No. There's nothing to say that either. In truth"—and the truth came hard—"I've not yet learned anything that tells against anyone more than another. Anyone at all."
Hopefully, Sister Johane asked, "Then you're going to let it go?"
Staring downward at the grassy edge of the graveled path, Frevisse said slowly, "I don't know what else I can ask or where else I can look for answers. But to leave it like this . . ."
"If it's the only place you can leave it, you have to," Sister Johane said.
Frevisse lifted her head with a sigh somewhere between accepting that and impatient at herself, her heart and mind heavy with more than the day's growing heat.
Hugh was afraid and Miles was angry, and she understood Miles' anger. Sir Ralph had done enough to him to fuel a lifetime's anger. But what was Hugh afraid of?
Of being found out for his father's murder?
Or of finding out who _had_ done it?
Or did he know who had done it and was afraid for them?
Or afraid _of_ them.
She stood staring at the woodshore where Hugh had disappeared. It was the weather, she told herself. It was too hot for her to think clearly. But the woods' rich greens of high summer were already dulled toward the dusty beginnings of autumn. The year was on the turn.
"This weather can't hold," she said. "There'll be a storm before long."
"Shall we stay here for Tierce and say it in the church?" asked Sister Johane.
"No," Frevisse said, finding she did not want to meet Father Leonel again just yet, with his burden of knowledge about the Woderoves. Had Sir Ralph's murderer confessed to him yet? Did he know who it was? Or maybe, with his deeper knowledge of everyone here, did he at least have too true a suspicion? "No," she said again to Sister Johane's question. "Let's go back to the manor for it."
The day passed somehow. The heat grew worse, weighing on everyone, stilling even Lucy's chatter. Neither Hugh nor Miles came in to midday dinner and for once Lady Anneys was impatient at them, saying, "They could at least say when they're not going to be here."
At her order, a double share of ale was sent out to the fields for the harvesters' midafternoon rest-time. Later, Helinor came into the garden to tell her, "Alson brought back word Master Hugh and Master Miles are both in the field, helping to harvest. Thought you'd want to know," and afterward Lady Anneys was a little farther away from the edge of her ill-humour. She even had supper delayed until nearly dark, waiting for their return, and buckets of water set on the bench outside the hall door to the foreyard to warm in the afternoon sun, with towels and a bowl of soap and clean tunics laid beside them, so that when Hugh and Miles finally walked wearily into the yard, they were able to wash there, stripped to the waist and scrubbing each other's backs, Ursula reported, hanging out the tall window to watch them.
"Well, tell them to hurry. I'm starved," Lucy said irritably from the table.
"Not so starved as they surely are," Lady Anneys said curtly. "You've done nothing all day except moan about the heat while they've been out working in it. Ursula, come here and sit down. They don't need your help to wash themselves."
They came in, in their clean tunics and with their hair slick to their heads from dunking in a bucket. Baude, wide with her whelps, heaved up from beside the hearth and waddled to meet them, she and Bevis circling each other with waving tails. Talk through the meal was of how the harvest went and whether the weather would break in a storm sooner rather than later.
"Gefori says sooner," Hugh said. "Late tomorrow maybe. If it holds off to late afternoon, we'll have most of the wheat safe."
"What of the beans and peas?" Lady Anneys asked. A storm that battered them into the ground when they should be drying on their plants could ruin the crop and mean much of the manor's food for the next year was gone.
"Father Leonel is praying," Hugh said.
"Do you still mean to hunt tomorrow?" Lady Anneys asked.
"The hounds need it, if nothing else," Hugh said. "But I'll make it short and be done early." Taking great care at spreading butter on a piece of bread, probably so he did not have to look at her, he added, "I asked Sir William if he'd join me."
Suddenly no one except Lucy was looking at anyone else, until after a moment Lady Anneys said with careful quiet, "And is he going to?"
"He sent back his thanks but said he'd not."
"Another time, then."
Lady Anneys spoke as if hardly interested one way or the other; and Hugh with relief said, "Yes. Likely later."
The message had been passed between them that, when the time came, she would accept it.
Because they had dined so late, there was little time for staying up and by the last fade of twilight they all went to their beds, before there was need to light candles to see their way. Shut into Lady Anneys' bedchamber, Frevisse freed her head from veil and wimple with a relief matched by Sister Johane's sigh of pleasure as she rubbed at her bared neck. While Sister Johane settled onto their truckle bed, Frevisse moved to close the window shutter but Lady Anneys said, "Pray, leave it open." She was sitting on one of the chests with her hair already loosed and falling to her waist, Ursula combing it with long, slow strokes in which both she and her mother seemed to be taking pleasure. "I'd rather risk the night vapors," Lady Anneys said, smiling, "than smother the way we surely will with the window shut."
So would Frevisse and she willingly left it open, but it made small difference. The hoped-for evening coolness did not come and sleep was hard to reach, no matter how much it was wanted. And Frevisse wanted it very much, because otherwise she lay thinking when there was far too much she did not want to think about because there was far too little she knew.
But without sleep, she found herself considering, in the quiet darkness after Lady Anneys had gone to her bed and Lucy and Ursula to theirs, how comforting it would be to believe Tom Woderove had killed his father.
Found herself likewise thinking how unfortunate it was that she believed Father Leonel when he said Tom had been with him.
If she believed that and refused to believe there had been an unknown someone there or that one of the women or girls could have done it without being bloodstained, she was down to the four men and Degory for the murderer. Or three men if she accepted that Miles and Philippa had been together—and hadn't themselves killed Sir Ralph. Because it might have been planned between them, to go off together but Philippa wait somewhere alone while Miles stalked Sir Ralph and killed him, with her to claim afterward that they had never been apart. It would have been a very quickly made plan, though, and if they were going to kill someone, wouldn't Sir William have been the better choice? His death would have made Philippa sure of her inheritance, setting her and Miles free to marry in despite of whatever rage Sir Ralph might have against it.
Perhaps Philippa had balked at plotting her father's murder. But killing Sir Ralph was purposeless for them. Or maybe not so very purposeless, because Sir Ralph's death freed everyone from him. Which was the trouble. Everyone had that reason to want him dead.
But if Tom was left out of it, who had had best hope of profit from Sir Ralph's death? Sir William if he thought he could take control of her children's marriages from Lady Anneys. Master Selenger if he wanted Lady Anneys and thought he could win her once she was widowed. Hugh if it mattered enough to him to have the hounds all for himself . . .
Sleep was finally starting to come, her mind drifting loose of connected thought . . . and into the thought that maybe they were all lying. That they had all killed him and were all lying. That Hugh and Miles and Sir William and Master Selenger and even Degory had all planned his death and done it, and everything any of them said was lies for the sake of protecting each other . . .
Frevisse found she was stark awake again, staring angrily at her thoughts.
She was like a hen scratching in the dust, throwing bits of everything around at random in hopes of finding a stray fact to feed on, she told herself. And if that was all she could do, she would be better putting her efforts into prayer for peace for the souls of everyone here, that they might come to God's mercy, since it looked unlikely anyone would come to Man's judgment.
The familiar ways of prayer brought her slowly toward sleep; but when it finally came, she found herself in a troublous dream where a man that she knew—in the way one knew things in dreams—was Sir Ralph was struck down and killed and stood up to be killed again, first by Tom—again, in the way of dreams, Frevisse knew who he was, though she had never seen him—and then by Hugh, and then by Miles, and then by Sir William and Master Selenger together, and finally by Lady Anneys, who—unlike the others—beat and beat and beat on him after he was down, making sure of his death and that he did not rise again.
Frevisse awoke from that with a start that wrenched her upright in bed, gasping for breath, certain something terrible was happening. And heard, not in any dream, the high-pitched scream of someone in mortal agony or fear.
Beside her, wrenched equally awake, Sister Johane sobbed, "Holy Mary, Mother of God, protect us now and in the hour of our death!" But from the bed above them Lady Anneys said, "It's Baude. She's begun to whelp and is frighted by it. She always does this."
"Are you sure?" Sister Johane gasped.
"I'm sure," Lady Anneys said with a calm that could only come from complete assurance. A plaintive voice called from the girls' room and Lady Anneys called back, "It's only Baude. Go back to sleep."
Reassured along with Lucy and Ursula, Sister Johane lay down with a relieved sigh. Frevisse lay down less quickly, her mind still tainted by her dream. Baude gave one more agonized howl before Hugh must have reached her and begun to soothe her because the night's quiet came back and in it Frevisse tried to say some of the prayers she should have awakened to do at midnight. A slight stir of air had cooled the room a little and she soon slept again, but not deeply. The ugly tendrils of her dream still strayed through her mind and she was fully awake sometime later when Lady Anneys rose restlessly from her bed.
The darkness beyond the window had thinned to the blue-gray of coming dawn. In the dimness Lady Anneys groped for her bedgown where she had laid it aside before going to bed, found it, and slipped it on before—so quiet-footed that she must have had every squeaking floorboard held in her memory—she went out and down the stairs. Frevisse, having pretended to be as asleep as Sister Johane was, settled herself determinedly for a last small sleep before the household stirred awake and the day began, but there seemed to be no more sleep in her and impatience took over, and rather than resort to pillow-pummeling in the hope of finding a shape both cool and sleep-inducing, she rolled warily off the bed. The ropes stretched under the mattress squeaked slightly the way ropes stretched under mattresses always did, but Sister Johane did not stir, and Frevisse dressed by feel and pinned on her wimple and veil blindly but with practiced fingers before going down the stairs. Lady Anneys had probably gone to the garden for the early morning's coolness. So would she.
Faint lamplight shining around the shut door at the stairfoot somewhat surprised her, and she was more surprised when she opened it to find Hugh and Miles and Degory all there in the hall, crouched near the hearth around a heap of straw where Baude lay on her side, looking far flatter than she had last evening. Hugh and Degory barely glanced up as Frevisse circled them, careful not to come too near. It was Miles who looked up, a wholehearted smile on his tired face, to explain, "She was too frantic. We didn't want to move her to the whelping shed. So Degory brought straw and we've done it here."
"All's well?" Frevisse asked softly.
"Five so far and we don't think she's done."
Craning her neck to see without coming too close, Frevisse saw the little bodies lined along Baude's belly, suckling mightily. Baude looked the least pleased of anyone about the whole business, lying still but with her eye rolled sideways and fixed rather desperately on Hugh, who was crooning to her, telling what a brave, good girl she was.
"You won't be hunting today," Frevisse half-whispered to Miles.
"We won't, no. The hares will live to another day," he agreed.
The long, hard surge of a contraction rolled down Baude's length. She tried to flounder to her feet in protest against it, setting Hugh to talking more earnestly to her and Miles to handing the whelps to Degory to put tenderly into a wide, waiting basket nested with rags. Frevisse quickly departed, most definitely not needed there.
At the hall's far end she found Bevis stretched out across the doorway to the rear passage, his great head resting on his outstretched forepaws and such a desolate look on his face that she stopped to tell him comfortingly, "It will be done soon and you can have Miles back." He regarded her solemnly but not as if he believed her, then pushed himself up on his pony-long forelegs, clearing the doorway for her. She thanked him because his dignity seemed to require it and, when she was past him, heard him lie down again behind her with a heavy, patient sigh and slight thud.
She likewise heard people early-morning mumbling among themselves in the kitchen as she passed and the rustle of straw-stuffed mattresses being put away for the day. Ahead of her the door to the garden stood open, the morning's coolness flowing in, and she stopped on the threshold to breathe it in with relief. The morning birds were chirruping welcome to the new day, though the sunrise sky was a sullen red along the horizon that warned the threat of storm to come was real.
But overhead the sky was still clear and brightening to rich blue, the last stars washed away by the swelling daylight that showed Lady Anneys standing at the gate, facing outward toward the red sunrise, her arms wrapped around herself as if her bedgown was not warm enough in the morning's cool. From where Frevisse stood, able to see only her back and the long fall of her hair, she looked more a young girl than a husband-wearied wife and mourning mother. There was such peace in her standing there that Frevisse held where she was in the doorway's shadow with a sudden ache for the moment's perfection: Lady Anneys quiet here in her garden; Hugh and Miles intent and content in their work together over Baude and her newborn whelps; Lucy and Ursula safely sleeping; the household servants setting about their everyday early work. This was how lives were supposed to be lived—with pleasure and work and rest in fair amounts all around. Not in the tangled ugliness of angers and fears Sir Ralph had made of it for everyone here, nor the torn, painful mess of secrets, broken hopes, distrusts, and doubts there had been through these past weeks.
Slowly, sadly, Frevisse crossed herself, praying better would soon come.
At the gate Lady Anneys startled and her head flinched sideways, to look leftward along the cart-track toward the stable and other manor buildings. Frevisse's view that way was blocked by the garden's fence and the arbor but the next moment she heard the soft hoof-fall of a slowly ridden horse and had just time to wonder who it could be at that hour and place before Lady Anneys opened the gate and stepped out onto the small bridge across the stream there, saying, wary and worried together, "Master Selenger. Is there something wrong? Has something happened to Elyn?"
He rode into Frevisse's view. Beyond him the sun had just begun to rise, throwing the shadow of him and his horse sharp-edged and black across Lady Anneys still at the gate as he said, "Everything's well. There's nothing's wrong."
"Then why are you here?" She put a hand behind her, to the gate, ready to retreat. "Sir William said he wasn't going to hunt today."
Master Selenger swung down from his horse, stood holding his reins but very near her now. "Lady Elyn has said you always rise to see her brothers and Miles off on a hunt. I thought this would be a better time than most to see you alone. To hunt Beech Heath they'd have to leave before full light and they have. I came around by the kennels to be sure and all's quiet there. But you see." He gestured at his high riding boots and old, forest-green hunting doublet. "I came ready with my excuse. If they'd been still here, I would have simply joined them on the hunt. But I've won my chance and here you are alone."
"They're still here. They didn't go on the hunt," Lady Anneys said. She was fumbling to clear the folds of her bedrobe from around her feet, trying to back away but hampered by them and kept by the bridge's narrowness from turning around. "Baude's whelping. They . . ."
Selenger dropped his reins and moved toward her with abrupt purpose, caught her by the arm, and drew her to him, off the bridge and away from the gate.
Lady Anneys tried to pull free, protesting, "Let me go!," and Frevisse stepped out of the doorway's shadow, demanding in a clear, carrying voice, "Master Selenger! Have done!"
He sent her a single, swift, dismissing look, pulled Lady Anneys—now outright struggling to be loosed—against him, put an arm around her waist, and took hold of her other arm, pinioning her to him. She was struggling desperately now, beginning fully to believe what was happening, and Frevisse—her skirts caught up, away from her feet—began to run toward them, but thinking even as she did that oddly Selenger was neither forcing Lady Anneys back into the garden or toward his horse but only out into the middle of the cart-track. He had even let his horse go and it was drifting away to crop grass along the track's far side. What . . .
Sir William's voice cracked whip-sharp into the morning air. "Lady Anneys!"
Both Selenger and Lady Anneys froze and Frevisse came to a stop in the gateway. Scarcely fifteen yards away, from the same way Selenger had come, Sir William sat on his black palfray, frowning at Selenger and Lady Anneys. He cast Frevisse only a short look that dismissed her as completely as Selenger had done before he said with dark displeasure, "What have I caught you at, Lady Anneys?"
"Nothing!" Lady Anneys said shrilly. "Tell him to let me go!"
"Master Selenger?" he demanded. "What was happening here?"
"As you see," Selenger said. He pulled Lady Anneys more tightly to him.
It was wrong. Where he should been defiant, even angry, or coarse with laughter and satisfaction, he sounded sullen, the words and gesture made almost more by rote than will; and then he let Lady Anneys go except for a hold on her near wrist.
An ugly suspicion awoke in Frevisse, made more ugly by Sir William saying with a flicker of what could only be pleasure across his face, "What I see is that you've been at the man-woman sport together."
Lady Anneys started a strangled refusal of that. Over it Frevisse said strongly, "I've been with Lady Anneys since before Master Selenger came. Nothing has passed between them except her refusal of him and his seizing of her against her will."
"It's good of you to say so, Dame," Sir William said coldly at her. "But this has openly gone past your lies being any use. I saw them embracing and how willing she was to it. And in her bedgown for worse measure."
Lady Anneys cried out in wordless protest and wrenched her wrist free of Master Selenger's hold, but Sir William pointed an accusing finger at her. "You've violated your husband's will, my lady. You are unchaste and thereby have lost the right to control your children's marriages. I fear that—"
" _I_ fear," Miles said, his voice barely recognizable with fury, "that you're wrong, Sir William." He pushed past Frevisse and crossed the small bridge to the cart-track and Lady Anneys' side, Bevis stalking beside him, both of them undoubtedly brought by the angry voices. Raw with anger, he said, "No woman affairs with a man in the presence of her grandson and a nun, and that grandson and nun will testify we saw nothing between her and Selenger save his ill manners."
"Don't waste your time with perjury, Miles," Sir William said scornfully back. "It'll do no good against my word and Selenger's."
"My word won't be perjury," Frevisse said and found she was as angry as Miles.
Sir William's look at her was cold. "One woman defending another. Worthless." He urged his horse forward at Lady Anneys. Caught now between outrage and tears, she pulled loose from Selenger's slight hold and tried to retreat, but Sir William rode nearer, looming over her, saying with thick satisfaction, "No, my lady. Even Selenger will say you've been willingly his. His word and mine against yours. You—"
_"You,"_ Miles said. " _You_ set him on to this. He's to perjure himself to ruin her for your gain, you miserable . . ." He sprang forward and seized Sir William by belt and sleeve. _"_. . . _cur!"_ And hauled him from his saddle.
Sir William's horse shied violently away, adding to the force of Sir William's fall; he hit the ground hard enough to jar every bone in his body and Miles, his grip torn loose, staggered backward, momentarily off balance. Selenger, who might have come to Sir William's aid, instead caught Lady Anneys by the shoulders from behind and drew her backward, away from the two men and toward Frevisse. Miles caught hold of Bevis' rough-coated back and regained his balance while Sir William rolled over and crawled to his hands and knees, then lurched to his feet, gasping for breath and red-faced with fury.
"You damned whelp," he panted. "I'll hang you with my own hands for that. You tried to kill me!"
"If I'd tried," Miles snarled back, "you wouldn't be getting up."
Bevis moved forward, putting himself in front of Miles, trying to push him away from Sir William with a wolfhound's instinct to keep his master out of danger.
"Miles," said Hugh, suddenly at Frevisse's side. "He's armed."
Sir William was, only with the kind of belt-hung dagger that men wore most of the time but he was drawing it and neither Miles nor Hugh had anything at all, dressed only in the loose shirts and hosen they had pulled on while seeing to Baude. Bevis was still pushing against Miles but his dark eyes were fixed on Sir William and he began to growl, the hackles rising the length of his back as Sir William, too blind with fury to heed him, moved toward Miles.
Miles, as able as Bevis to read Sir William's fury, stepped back as Sir William thrust at him. Sir William was just far enough away that the thrust was probably meant more in threat than actually to stab—but Miles stumbled sideways as if his foot had caught in a trackway rut, throwing him off balance, unable to defend himself, and that made Sir William's move too much a threat for Bevis, who—so swiftly there was no chance to stop it or guard against it—twisted sideways between the two men and reared on his hind legs to his full wolfhound height, forequarters and head towering over Sir William for the bare instant before his forepaws hit Sir William on chest and shoulder, driving him backward, snapping his head back to expose his throat that Bevis seized from the side in his jaws with all the intent to death that he had ever seized on a stag in the hunt. Sir William stabbed at him once but then was being shaken and flung from side to side like a barn-killed rat. There was a gurgling that must have been his last attempt to cry out and then blood was spurting from his flopping body and Hugh and Miles both yelled, "Drop it!" at Bevis, who—well-trained hound that he was—immediately did, letting blood from Sir William's ripped-out throat fountain red into the red morning light, spraying wide and over everything.
**Chapter 21**
In the instant that Bevis seized Sir William, Selenger swung himself in front of Lady Anneys, between her and sight of what was happening but not enough blocking Frevisse's own view, everything happening so fast that she had time to see but do no more than throw her hand up against the spraying blood, and then Lady Anneys was screaming and Selenger was pushing her at Frevisse, saying, "Get her away!"
"Carry her!" Frevisse ordered back at him, that being the surest way to have Lady Anneys away, turning from them back into the garden herself as she said it. And Selenger obeyed, caught Lady Anneys into his arms and followed Frevisse as Lady Anneys broke off her screaming and began to struggle against Selenger's hold, crying, "Hugh! Miles! I can't leave . . ."
"Ursula," Frevisse returned sharply. "Lucy. You can't let them come out to this. And the servants. They shouldn't see it."
They were to the door but, "Put me down," Lady Anneys ordered at Selenger with such sudden angry certainty that he stopped and did but kept hold on her, which was as well because she swayed, looking either about to faint or be ill; but she steadied, straightened in Selenger's hold, and shoved his hands away, saying, "The girls. Yes. And Father Leonel. We need him," as if air were hard to find but rigidly in control of herself again. "And the servants." Who were only now—it had all happened so fast—coming out of the kitchen in answer to Lady Anneys' screaming, both women with a knife or heavy pan in their hand, looking uncertain whether they should be angry or afraid.
Lady Anneys, her bedgown caught up and gathered to her in both hands—a way to hide their shaking as well as clear her feet—went forward to turn them back from going out, assuring them that, yes, something terrible had happened but that Hugh and Miles were seeing to it, no, they were unhurt but it was better that everyone else stay inside.
"Go with her," Frevisse said at Selenger. "See to it she has something strong to drink. Wine, if there is any. And you, too." Because his face was the same ashen gray as Lady Anneys'.
And so was her own, Frevisse supposed as Selenger nodded with full understanding and followed after Lady Anneys shepherding her servants into the hall ahead of her. Frevisse, wishing she were going with them, turned back because Lady Anneys was right—Hugh and Miles should not be left alone. Though going back to the cart-track was among the last things she wanted ever to do. Part of her was even trying to believe it had not happened, but her hand was sticky with blood and there must be blood on her and when she went through the garden gate into the golden-slanted morning light of the newly rising sun, scarlet droplets of Sir William's blood were falling from the leaves where the arbor's vine overhung the fence; and there was red brightness on the fence and in the grass, and dark red pools and streams across the beaten-smooth earth of the cart-track, spreading out from the ruin that had been a man bare moments ago.
Both Sir William's and Selenger's horses had startled away along the track. Near Sir William's body were only Hugh kneeling and Miles crouched, both soaked with blood, beside the equally bloodied Bevis, who was standing very still, panting, his neck stretched out, his head lowered, Sir William's dagger hanging out of a red, blood-welling gash along his side.
The removed, assessing part of Frevisse's mind judged that the dagger must have raked shallow over bones from the hound's shoulder to almost his flank rather than been thrust deep, and now it was stuck into a rib, not buried in the hound's side. Otherwise he would be dead instead of standing there.
But even as she thought that, Miles jerked into movement, grabbed out the dagger with a cry of pain and rage, and spun around, his arm rising with his clear intent to stab down into Sir William's sprawled body. But Hugh cried out, "Miles!" and lurched forward in time to catch his arm and force him around, putting them face to face as he cried again, "Miles!" and let go his arm to take hold of his face with both hands, saying desperately, "Don't! Don't make it worse!"
And Miles shut his eyes, shuddered, let his arm fall and the dagger drop, then bent over, to slide forward onto his knees and cover his face with both hands.
Hugh, with the wide eyes and stark paleness of someone hit too hard but not yet crumpled to the blow, wrapped an arm around his shoulders, looked up at Frevisse, and asked hoarsely, "What . . . do we do now?"
By morning's end the sky had clouded to gray and in the early afternoon the rain began.
At least it wasn't the battering storm they all had feared, Hugh thought, watching it through the parlor window. The light, steady pattering would likely hold up the harvest no more than a day. But maybe it would be enough to wash away Sir William's blood, dried and darkened now on the cart-track, the grass, the fence, the leaves. Enough to wash away the sight of it, the smell of it. Wash away everything except Hugh's memory of it.
He could still hear his own hoarse voice asking Dame Frevisse what they should do and her crisp answer, "Tend to the dog. Miles, get up and go find men and canvas to cover and move the body."
Miles had pulled himself to his feet and gone. Dame Frevisse had helped Hugh tear his shirt and bind it around Bevis, the wound covered by the time Miles came back with Gib and Duff and a long piece of canvas from the stable. There was pause while Gib threw up in the ditch and then they spread out the canvas where the track was unbloodied, and Hugh and Miles—already bloodied past more blood mattering—carefully lifted and shifted Sir William's body onto it, troubled most by the head. It had so little still holding it on . . .
Not until the body was safely wrapped and out of sight did Miles go aside and throw up in his turn.
Father Leonel came through the garden then, disheveled and with more haste than a man his age should have made. Clutched to his chest he had his box of things needed to give the last rite to the dying but the canvas-wrapped bundle and all the blood told him they were past use and he only said, "Best bring him to my house."
Hugh moved to help Gib and Duff with the burden but Father Leonel said, "Not you or Miles. You need to clean yourselves. And you, my lady," he had added to Dame Frevisse, who was still standing beside Bevis, a quieting hand on his head.
Only then did Hugh realize she was bloodied, too—nothing like so badly as he and Miles, soaked as they were, and her black nun's gown hiding more than it showed, but bloodied enough, her white wimple spattered and her face smeared where she had wiped at it. But as Father Leonel led Gib and Duff bearing Sir William's body away, she only said, "I'll order water heated in the laundry shed and someone to bring you clothing. Clean yourselves before you come in," to him and Miles before she went away toward the house, leaving them to take Bevis to the kennel, to the corner kept for hurt or sick hounds.
Together they cleaned his wound, medicined and bandaged it, and in all that while Hugh said nothing and Miles no more than, "It's a shallow cut. He should heal well."
It wasn't much to bring out of the wreck of the morning, and against a man's death, a wounded dog maybe should not count for much; but there were a good heart and faithfulness in Bevis. Sir William had had neither.
"I'll stay awhile with him," Miles said when they were finished, and Hugh had left him and gone around to the far side of the kitchen yard to the laundry shed, where both Helinor and the two large laundry kettles full of heated water were waiting for him. Helinor's stark stare at him told him he looked as ghastly as he felt but she said nothing, only bustled at having him out of his hosen—nearly all he was still wearing after ripping up his shirt for Bevis—threw them into one kettle to soak, and left him to wash as he would, saying as she went out, "Don't forget your hair, too."
He had scrubbed thoroughly, beginning with his blood-matted hair, then dressed, and all the while felt that everything was being done by someone else. Most of him was not here, was hiding behind what needed to be done, rather than facing everything it was not yet safe to feel or think.
He found the hall empty save for Selenger sitting on the end of a bench behind the high table, head bowed and hands hanging between his knees, and Degory still with Baude beside the hearth. Hugh paused to ask how she did, counted the heads nursing heartily along her belly, and said with surprise, "Eight? There's eight of them?"
Degory nodded happily. "They came right quick after you left. Strong ones, all of them."
Very briefly Hugh thought that Degory was probably the only happy person here this morning. The hounds were the center and circumference of his world, and if all was well with them, all was well with him. Hugh, remembering when his own life had been almost that simple, dropped a hand on the boy's shoulder in acknowledgment that he had done well and moved on to Selenger, who looked up, haggard-faced, and said, "Your mother is upstairs with the nun to help her. She was spared most of the blood and seeing too much. She . . ." His voice shook and he stopped.
"That was your doing," Hugh said, remembering what he had hardly seen at the time—Selenger putting himself between her and . . . the death. "Thank you." Then added, hearing the words come in the strange, removed voice that did not sound like his, "And you? How are you?"
"I don't know." Selenger seemed truly bewildered by the question. "He's dead, isn't he?"
"He's dead. Should I send word to . . . to Lady Elyn? Or do you want to take it?"
"I'll take it. In a little while. I'd like . . . I'd like to see Lady Anneys before I go."
Hugh did not have it in himself to say that Selenger had seen enough of Lady Anneys. Instead, the words a little thick in his throat, he said, "Your back, it's all soaked with blood."
Selenger shifted his shoulders inside his shirt and doublet, stiff and sticking to him with dried blood. He must not have noticed it until then because sickened horror began to spread across his face and Hugh said quickly, "Go and wash. Someone in the kitchen will show you where. I'll find you something else to wear." And added, when Selenger started to refuse, "You can't go back to Lady Elyn and Philippa like that. Leave your clothes in the laundry. We'll see to washing them."
Practical things. Small things to set against the enormity of death but no less needing to be thought of, needing to be done, and small things were what saw Hugh through the rest of the morning. Small things and Dame Frevisse. It was by her doing that, by the time Master Selenger returned to the hall, clean and in some of Tom's old clothing Hugh had found for him, Lady Anneys was waiting there, dressed in her black mourning clothes, with Ursula on one side of her, pressed close to her and clinging to her hand, and Lucy on her other with tear-blotched face and a packed bag at her feet, Sister Johane standing behind her in a quelling sort of way and Dame Frevisse to one side of them all as if making certain they stayed where she had put them. With some part of his mind determined to note things without feeling them, Hugh saw she, too, was no longer bloodied.
Without other greeting, Lady Anneys said to Selenger as he and Hugh approached her, "Lucy is going back with you. She can be a comfort to Lady Elyn." Not mentioning the comfort it would be to have Lucy away from here. "Sister Johane is going with her, to help them all."
Selenger accepted that with, "Yes, my lady," and went down on one knee in front of her, his head bent low, his voice choked but clear enough as he said, "I have to beg your pardon, my lady, for the wrong I tried to do you. It was not willingly done, I swear to you and by any saint you ask of me."
For a startled moment no one moved or said anything, until Lady Anneys asked uncertainly, "Not willingly done?"
Head still bowed, Selenger said, "Sir William told me what was in Sir Ralph's will and that he wanted you out of his way. He set me on to do it. If you could be brought to marry me, fine and well and good because that would be enough; but if not marriage, then whatever else I could manage that would ruin you."
"So everything you did was at his orders?" Lady Anneys said coldly.
Selenger raised his head to look at her and let her see his face. "By his orders but by my choice, too, because he meant to ruin you. If not by me, then by a rougher way."
"A . . . rougher way," Lady Anneys echoed.
"He would have hired someone to . . . do whatever was necessary. To take you by force if need be, so you could be declared unchaste and no longer fit to be executor of Sir Ralph's will. Rather that let it come to that, I agreed to . . . what I did."
"And this morning?" Dame Frevisse asked. "What was this morning supposed to be?"
"It was a trap," Selenger said. "Sir William was impatient to have the matter settled. You weren't giving way to me, my lady, and I wouldn't . . . force you the way Sir William wanted me to. With Hugh and Miles gone hunting this morning, I was to find a way to be alone with you and Sir William would come on us and accuse you. Just as he did."
Selenger bowed his head again, away from Lady Anneys' stare into his face. Lady Anneys went on staring, now at the back of his bent head, until finally she said, "And if you hadn't done this, he would have done something worse to me."
"Yes, my lady."
Lady Anneys bent forward and touched his shoulder. "Then you have my thanks, John Selenger, and . . ." She paused, then brought herself to say, "And my forgiveness."
Selenger raised his head. For a moment their eyes held, before he stood up and bowed low to her.
Not until Hugh turned then to see him and Lucy and Sister Johane to the yard where their horses waited—Dame Frevisse had given order for that, too—did he see Miles standing in the doorway, the look on his face telling he had heard it all.
Selenger, Lucy, and Sister Johane left. Duff was sent with word to the crowner. Lucas the reeve was summoned and told what had happened so he could report it to the manor's folk and slow whatever rumors were surely starting. Baude and her whelps were moved to the whelping kennel for Degory to keep watch on them and Bevis. One thing and then another was seen to, and somewhere along the way Hugh had the sudden discovery that he had had no breakfast and was starving. If he had thought about it beforehand, he would have thought he would never have desire to eat again, sick as he was inside himself, but hunger came on him and there was bread and cheese and—at Lady Anneys' order—wine for him and Miles that she said Miles must drink, whether he ate anything or not.
That had been somewhere around midday. The clouds closing across the sky by then had made it hard to know the time, but soon afterward Philippa alone rode into the foreyard. She said she had been at Father Leonel's to pray beside her father's body and she did not want to go home where Elyn and Lucy were wailing pointlessly on each other's shoulders. Could she stay here a time before she faced them again?
She had been crying herself but was calm then. Too calm, Lady Anneys maybe thought, because she gathered Philippa to her and took her into the parlor alone and a while later, when Hugh went in, Philippa had been crying more. "It cleanses," Lady Anneys said to him, though he'd said nothing. "Crying cleanses the weight of grief and makes it easier to bear."
And he remembered how she had cried not at all after Sir Ralph's death.
Soon after that, when somehow there was nothing else to be done for the while and the rain had started, bringing coolness with it and the autumn smell of rain-settled dust, Dame Frevisse and Miles came into the parlor, too. Ursula was gone off to be with Baude and her whelps and that was good, because she should not be here in their silence, so many things going unsaid and all of them sitting apart from each other. Lady Anneys was on one end of the chest below the window, Dame Frevisse at the other end, both of them with their hands folded into their laps, their eyes down. Philippa, sitting very straightly in one of the chairs, her hands gripping its arms, was staring out the window, the marks of her tears still on her cheeks though she was dry-eyed now. Miles was not with her but on one of the joint stools, leaning forward with his arms resting on his knees and his hands tightly clasped, his gaze toward the floor. Hugh, able neither to sit nor pace, leaned against the wall near the door, watching the softly falling rain, its light pattering the only sound in the room.
Afterward Hugh thought how that should have been enough. To listen to the rain and be simply quiet for a while before having to do whatever came next.
But into the quiet Miles said, "It had to have been Sir William who killed Sir Ralph."
Heads lifted and turned toward him, no one saying anything and Hugh silently begging him not to start this.
But, "It makes sense," Miles said, looking around at them all. "He killed Sir Ralph and set Selenger on to ruin Lady Anneys. Maybe Tom's death was an accident or maybe it wasn't, but he surely murdered Sir Ralph."
"Miles," Dame Frevisse said softly, barely louder than the rain.
Miles looked toward her.
She met his look and said, still softly, "Not once since I came here have I heard anyone care about Sir Ralph's death. No one has cared who did it or why. Now, suddenly, you're accusing Sir William. Why? Why now this need to accuse when there was no need before?"
Miles straightened. "Because I didn't know before. Now it's all of a whole. He killed Sir Ralph and then tried to ruin Lady Anneys, all as a way to have control over the marriages so he could profit from them."
"It was a savage murder done on Sir Ralph," Dame Frevisse said, her voice still soft and careful, her gaze still fixed on Miles. "He wasn't hit only once. He wasn't simply killed. His head was smashed in, struck, and crushed again and again and again. That wasn't murder merely for profit's sake. It was murder done from hatred, and I've heard no one say Sir William hated Sir Ralph. But you did. You still do."
Hugh straightened away from the wall and said, "We all hated Sir Ralph. That makes it likely for any of us to have done it. Me as much as Miles."
"Yes," Dame Frevisse agreed without looking away from Miles. "Except you didn't deliberately murder Sir William. Miles did."
Lady Anneys stood up, protesting, "He didn't!" as Hugh exclaimed, "You're wrong," and Philippa said, _"No!"_
Only Miles said nothing and Dame Frevisse looked away from him to Hugh to ask, "You know what happened to make Bevis attack Sir William?"
"Miles stumbled and while he was off guard Sir William went for him," Hugh answered without hesitation. "Went for him with the dagger. That's enough to set any wolfhound from defense to attack when they're defending someone the way Bevis was defending Miles then."
"I know," Dame Frevisse said. "I've had to do with wolfhounds before this."
Hugh went wary, hoping his face was as blank as he meant it to be. Most nuns were of gentry if not noble families; any of them might well have had to do with hounds before entering the nunnery. But the way Dame Frevisse said it warned him there was something more than that.
"Wolfhounds," she said, "except when on the hunt, are the gentlest of dogs. They're bred to be."
"Except when on the hunt or defending someone," Hugh said.
"Or when they _think_ they're defending someone," Dame Frevisse said. "You could see what happened even more clearly than I did. Miles didn't truly stumble, did he?"
A swift denial of that would have been best; but Hugh, fatally, froze and Dame Frevisse returned her look to Miles. "You _seemed_ to stumble as Sir William came at you with his dagger. Bevis was defending you, and because you seemed thrown off your balance and off guard, he did what he had been warning he would do. He attacked."
"He was defending Miles," Hugh said fiercely—the more fiercely because Miles was saying nothing.
"Sir William knew wolfhounds, too," Dame Frevisse said. "He must have known how far he could push what he was doing, His dagger thrust wasn't meant to come anywhere near to Miles and it didn't. It was because Miles seemed to stumble just at that moment . . ."
"Did stumble," Hugh said sharply. "Not seemed. _Did._ " Said it too sharply, too desperately, willing Miles to say nothing, to let things lie where they were, leave the denying to him.
Maybe Miles would have, but Philippa stood up and said, "It wasn't my father killed Sir Ralph. I did."
**Chapter 22**
In weary, protesting grief Lady Anneys said, "Oh, Philippa, no," at the same moment that Miles stood up, saying, _"Philippa!"_
Ignoring them both, Philippa said at Dame Frevisse, "Anyone can tell you I'd left the gathering place and wasn't there when Sir Ralph was killed."
"You left the gathering place with Miles and you came back with him," Dame Frevisse said.
"He . . . we didn't stay together. He went off one way and I met Sir Ralph and . . . killed him."
"That's feeble," Miles said at her angrily.
"It's the truth," Philippa returned, refusing to look away from Dame Frevisse.
Miles caught her by the arm and turned her roughly toward him. "Stop it!"
Philippa looked down at his hand on her. He looked, too, then jerked his hand away from her as if burned and would have turned away except Philippa now caught him by his arm, holding him where he was before she moved closer to him, took his face between her hands, and said up at him, intensely tender, "Never think it, my heart. He never cared what he did. You do. You're not him in any way."
Hugh held back from any word or movement, until after a long moment of staring down at Philippa, Miles closed his eyes and let out a shaken breath, accepting what she had said. She put her arms around him and his arms went around her and they clung together. Hugh breathed again. If they could hold like that—together and silent—then there was chance . . .
"My guess would be," Dame Frevisse said evenly, "that Sir Ralph did indeed come on you in the wood, Philippa, but you were with Miles, and Sir Ralph went into a rage. He did something or said something or both, and Miles killed him."
Philippa turned around, out of Miles' arms, to face her. "No. It happened like I said."
"That can't have been the way of it, Philippa," Lady Anneys said. "You had no blood on you when you came back to the gathering place. I know. I looked for it on everyone and there was none on anyone. Not on you or Miles or anyone. You don't have to tell this lie."
"It's not a lie. It's no more a lie than saying Miles set Bevis to kill my father!"
And Miles did the one thing Hugh had hoped he would not do. He turned his head and looked at him—a long meeting of their eyes—and Hugh knew that everything he was feeling was naked on his face for Miles to read there—beginning with the sickened certainty that Miles had indeed deliberately set Bevis to kill Sir William.
Hugh had had that dark knowing with him all day, since he had wrenched his eyes away from Sir William's torn corpse and seen, instead, Miles' face triumphant with raw pleasure.
Then Bevis had staggered and they had realized he was hurt and had both gone to him, and when Miles saw how bad the wound was, he had grabbed the dagger out of the rib-bone where it had stuck, and made at Sir William's corpse. Only when Hugh had grabbed him and they had stared into each other's faces did Miles seem suddenly to understand what he had been about to do—maybe saw Hugh's horror at him, too—because his raw savagery had turned to something close to that same horror and he had dropped the dagger and hidden his face.
Since then Hugh had taken care not to meet Miles' gaze. Until now. When very surely Miles could see his certainty that it wasn't only Sir William he had killed.
And still Dame Frevisse went on, now at Miles, saying, "Philippa will lie for you for as long as she has any hope it will save you. She'll lie and you'll lie and others will believe your lies or pretend to believe them and lie for you, too, until neither you nor anyone else knows when you're telling the truth and when you're not. You won't know what the truth is anymore, even between each other. You know too well how long-lived anger can corrupt and darken lives. Do you think living in lies won't do the same?"
Miles was heeding her. He had turned his head away from Hugh and was listening to her, and Hugh searched desperately for something to say, anything he could say to keep Miles or Philippa from answering her.
But it was Lady Anneys who said with quiet force, "Leave it, Dame Frevisse. All of it. Sir Ralph's death. Sir William's death. Leave them as they are. And you, Miles, you will be quiet. You'll say nothing else. Nor you, Philippa. Leave it, Dame Frevisse. Here and now it's done."
"It can't be left where it is," Dame Frevisse insisted. "You can't—"
"We can." Lady Anneys did not raise her voice. "We're broken pieces of what should have been a family. Sir Ralph's cruelty kept us from ever being whole and we've had too little chance to mend since he's been dead. We need that chance. We'll never be as we might have been if we'd been always whole, but mended is better than broken, healed wounds better than unhealed wounds. If we start searching for 'justice' now, everything will only be made worse, broken into pieces so small we'll never mend. Leave it as it is. That Sir Ralph was killed by someone unknown who will never be caught. That Sir William brought his death on himself. Leave it and leave us to make what we can of what we have left. Please."
That last was as much command as plea.
"The truth—" Dame Frevisse started to answer.
Lady Anneys cut her off. "The truth is that between them Sir Ralph and Sir William tore us all into pieces and broke us, and if ever we're going to mend, the truth of things here is no one's business but our own." Her voice hardened into plain command. "So leave it."
Dame Frevisse stared at her. She stared back. No one moved. No one spoke.
Then Miles drew breath as if to say something, and more sharply than Hugh had ever heard her speak, Lady Anneys ordered at him, _"No."_
And Miles held silent.
Dame Frevisse rose abruptly to her feet and went toward the door. Hugh barely gathered his mind enough to open it and stand aside before she reached it, and without even glance at him, she left, leaving silence behind her save for the soft fall of the rain outside the window.
It was a long silence. Then finally Lady Anneys drew an unsteady breath and was no longer rigid but only sitting, her shoulders a little bowed, and that released the rest of them. Miles gently seated Philippa in her chair again and stayed standing beside her while Hugh crossed to sit on the chest under the window where Dame Frevisse had been, and reached a hand toward Lady Anneys, who gave him one of her own, holding to him tightly as she said softly, "It would have maybe been better, Philippa, if you'd said nothing."
"I was afraid," Philippa said. "I was afraid that if she went on saying . . ." She stopped but the unsaid words were there, no need to say them. If Dame Frevisse had gone on saying what she had been saying about Sir William's death, Miles might have admitted what none of them wanted to hear. "I was afraid," Philippa repeated and left it there.
"Unfortunately," Lady Anneys said, her voice oddly empty of feeling, "what she said about lies and their corruption is true. And it's a pointless corruption here among us, because, lie how we may, we know the truth. That one of us did kill Sir Ralph." She looked at Miles, who looked back, his face as empty as her level voice going on, "We've all known from the very first that it was one of us who killed him. It's only been desperate pretense that it wasn't."
In silence again, she and Miles looked at each other. Then he said, "I killed him. Yes." He straightened. Weight seeming to slip from his shoulders. A weight that had been crippling him, Hugh realized. And now that those few words were said, Miles said the rest, pouring it out with a hatred-tainted anger oddly mixed with raw grief. "Sir Ralph came on Philippa and I alone together. It was too apparent why we were together and alone. He flew into one of his rages. He was so angered he choked on it. He couldn't even yell he was so furious. He backhanded me in the face and swore he'd see Philippa locked up until she was married to Tom and that he'd geld me if I tried to see her again. He grabbed Philippa by the arm. He hurt her and I hit him." Miles stopped and looked at Hugh. "The way one of us should have done years ago, because all the years when I hadn't hit him were in that blow. I knocked him down and before he could get up I kicked him in the head and then I took up a rock that was there and . . . finished it."
Hugh had dropped his gaze to the floor before Miles finished, unable to look at him, remembering what Sir Ralph's head had been like. As if all the beatings he had ever given had been returned to him in one. And maybe they were all thinking that because for a long moment no one said anything. Only finally, with no particular feeling except curiosity, Lady Anneys asked, "How did you do that and stay unbloodied?"
Perhaps because she asked it so evenly, Hugh was able to look up and watch Miles as he answered evenly back, "I had my doublet off when Sir Ralph found us together. It was my shirt and upper hosen that took the blood. I washed the shirt . . ."
"We washed it," Philippa said, claiming her part in it.
"We washed it," Miles agreed. "We went away from Sir Ralph's body, back along the path we'd taken from the gathering place to where there's a small stream. We were there when you found him, Hugh." Miles looked at him and this time Hugh met his look, accepting it because his mother and Philippa were accepting it. "That's why we were so long coming back to everyone. I had to put the shirt back on wet, but the doublet is thick. By the time it was soaked through, it passed for sweat if anyone noticed at all. I don't think anyone did. Not with all else that was going on. And the doublet is long enough it hid most of the blood-spattered part of my hosen and they were dark enough the blood didn't much show after I'd rubbed some dirt over them. By the time I gave them over to be washed here, the blood was so long dried there was no telling it wasn't from the hunt."
"Then all we had to do was keep quiet," said Philippa. "Pretend to know nothing more than anyone else did."
She said it simply but then shuddered, her hands suddenly gripping each other in her lap, and she began to cry. Miles dropped to his knees in front of her, taking her hands in his own and looking up into her face, saying, "I'm sorry, Philippa. I'm sorry it came to this. I'm sorry."
Philippa fell into greater sobs and bent forward to put her arms around him and her face against his shoulder. He put his arms around her waist, his face pressed against her hair; but behind him Lady Anneys said, "Now admit the rest," and Miles jerked back from Philippa.
Without rising, he pivoted to face Lady Anneys and said, barely above a whisper, "The rest?"
"The thing Philippa kept you from saying to Dame Frevisse."
"No."
Lady Anneys leaned toward him. "It has to be said." Her voice was low and steady, her look at him unwavering. "It's leaving it to rot in the dark will poison your life and Philippa's."
"He doesn't have to say it," Philippa said, laying her hands on Miles' shoulders from behind. "There's no need!"
But there was need. If Hugh had not seen Miles' face after Sir William's death, he might well have failed to understand that, but what he had seen in Miles then should not be left in silence where the rot of it would only spread. The only hope Hugh had came from the horror that had swept though Miles afterward, because it meant he was not dead inside himself to what he'd done. But it had to be said. Whatever damage the truth would do among them, lies—or even a silence that was the same as lies—would do worse.
"Miles," Lady Anneys said. "Philippa already knows. We all know. It's for you that you have to say it. Admit to it. Accept it. Leave no room to lie to yourself about it. It's how I survived those years with Sir Ralph. By never lying to myself about what I felt or what I thought. It's by truth and whatever penance goes with it that we stay strong and grow. Not by darkness and lies. You have to say it."
"No," Philippa sobbed.
"Philippa, you have to accept it as fully as he does," Lady Anneys said relentlessly. As she must have been relentless with herself all of her years with Sir Ralph. "You have to accept it or else reject him because of it, but don't think you can ignore it or deny it. It can't be left to twist and go foul in the dark between you. Miles."
Miles stood up, turned to Philippa, took her by the hands and drew her to her feet, then let go her hands and said, "I killed your father. I seemed to stumble because I knew that would set Bevis at him. I meant for Bevis to kill him."
And now it was Philippa who had to choose but her choice had been made before ever Miles said anything and she drew close to him, whispered something in his ear that no one else heard, and then, as Miles sank to his knees in front of her, his body shaken with sobs, she sat quickly down, and took his head on her lap, her own tears falling on him as she leaned over him, holding him.
"How often I wished," Lady Anneys said softly, maybe for only Hugh beside her to hear, "through the years, that I could have cried. Instead of simply hating."
But she had not simply hated, Hugh thought. She had loved, too. Loved strongly enough that they had not been destroyed by Sir Ralph's dark-heartedness. They had all been tainted by it, yes, but not so destroyed that, now, they couldn't face the darkness and fight it instead of giving up to it.
Not so destroyed that they couldn't love.
The rain had stopped and the cool, gray afternoon was drawing on when Sister Johane found Frevisse in Lady Anneys' arbor. The thick leaves had kept off most of the rain; the benches were dry enough and in the while that Frevisse had been there only Alson from the kitchen had come out once, to ask if she wanted anything. Frevisse had not and otherwise she had been left alone; had tried to pray but not much succeeded; had tried to make peace with her thoughts and utterly failed. So she welcomed the distraction Sister Johane might be.
But Sister Johane sat on the facing bench, still a little unsure of her welcome, and explained, "Lady Elyn has come to cry on her mother for a while. Lucy and I came back with her."
Frevisse nodded to that, would have listened to more if Sister Johane had offered it, but she did not, and Frevisse had nothing to say either and for a while the raindrops' drip from the leaves was all there was, before Sister Johane made a small movement of her head toward the house and said, "It's very strange in there. Among all of them. It felt like something other than grief."
"Yes," Frevisse said. "It would."
And maybe she would have been the one to say more but a footfall on the path gave brief warning that someone was coming before Hugh appeared outside the arbor. His look went uneasily from Frevisse to Sister Johane and returned to Frevisse before he asked, "My lady, if I might speak with you?"
"Alone?"
His eyes flickered back and forth again. "Unless she knows, unless you've told her . . ."
Sister Johane immediately stood up. "I'll walk the far side of the garden for the while," she offered.
"If you would," Frevisse said. "Thank you."
Hugh stood aside to let Sister Johane leave and Frevisse moved one hand, letting him know he should sit where Sister Johane had been. He did, and Frevisse, her hands tucked into her opposite sleeves and resting quietly on her lap, waited for whatever he had come to say.
He very obviously would rather have not been there to say anything at all and said, to have it done, "My mother sent me. She wants you to know what passed after you left."
That was not what Frevisse had expected. A plea for her silence, or assurance she had misunderstood what she had heard, or a veiled—or even unveiled—threat against her if she ever spoke of what she more than suspected. Any of those had seemed possible. But to tell her of what had happened after she left? Unless, of course, he had come to lie to her, in hopes of deceiving her into silence.
"Why didn't she come herself?"
"She said it would be too hard for her. She said it would be better for me to do it, who doesn't know you so well."
Frevisse slightly bent her head, accepting that, and waited, regarding him with a steady gaze while he readied himself and finally said, "We talked after you left. Miles admitted to Sir Ralph's death. Philippa was there when it happened and knew why and helped him hide the signs he'd done it."
"And Sir William's death?"
"He admitted that, too."
"With Philippa there to hear him?"
"Yes."
"And?"
"She loves him."
"He's twice committed murder."
"Any of us could have killed Sir Ralph at almost any time. The time happened to come to Miles first."
"And Sir William's death?"
She heard the coldness in her voice and so must have Hugh and he held back his answer, before saying at last, slowly, "Sir William's death was a mistake. Miles was angry at him. Rightly angry. I think Miles thought that because killing Sir Ralph came so . . . easily, that killing Sir William wouldn't matter to him either. Afterwards . . . he found out it did."
"And you? Does or doesn't it matter to you?"
Her question startled him. "Matter to me?"
"That he killed your father. That he killed Sir William."
Hugh paused over his answer, then said slowly, "I'd rather have him free than paying for those men's deaths with his own."
"So you'll lie for him and go on lying."
"Yes." Hugh did not hesitate over that but then paused again before adding, his eyes locked to hers, "We'll all lie. But not to each other anymore. And none of us, anymore, to ourselves."
They stayed staring at one another for a long moment more before, slowly, still meeting his look, Frevisse nodded, accepting that as something better than nothing.
Hugh dropped his gaze, stood up, bowed to her. "Lady Anneys simply wanted you to know. That's all."
He made to leave but Frevisse asked, "Where's Miles now?"
"He's gone to Father Leonel. To confess and take whatever penance he's given."
That was something, too. But there was one more thing and she asked it before he could leave as he so openly wanted to do. "Are you still Miles' friend?"
She had startled him again. He stopped, eyes widened with surprise, and said, "Yes," plainly never having thought _not_ to be Miles' friend. Then he left.
Sister Johane did not come immediately back to the arbor when he was gone and Frevisse stayed where she was, watching a last few rain droplets fall from leaves. Over the hall's roof a waterish gray and yellow sunset trying to happen through the westward clouds gave some hope that tomorrow would be clear. A hope but not a certainty.
Frevisse had been haunted after she left the parlor by the thought of the reckoning that had come—a reckoning as vile as everything that had gone before it, however long it took to come—if Lady Anneys, Miles, Hugh, and Philippa refused to face the depth of wrongs there had been done here. If they tried to ignore what they all knew, the corruption of it would destroy them, heart and soul. Of that she had been sure, and that fear, at least, Hugh had taken from her. They had ended their lies to themselves and to each other.
For the world's authorities to know—crowner and sheriff and all—Sir Ralph's death would go unsolved and Sir William's be seen only as a fearsome mistake. For Miles there was the hope that penance might finally cleanse his heart and spirit of hatred's ugly dross and bring, in God's eyes, absolution for his sin of murder. And Frevisse found she could live with the law's justice not being done if a deeper justice was being answered, if payment was being made—payment of maybe a deeper and more healing kind than the law's justice would have exacted. Penance and love might well save Miles: the others' love for him and, as important, his love for them.
Sister Johane came hesitantly back and sat again, watching her for a silent while before saying, "Something is better?"
"Something is better."
"But not well?"
"Not well. Not yet. But now there's hope." Where, before, there had been none.
"It has to do with the deaths, doesn't it?"
"Yes."
"But you can't tell me?"
"No."
Sister Johane thought on that, then said, "But I should pray for all of them because of it, shouldn't I? For Hugh and Miles and Lady Anneys."
"And Philippa," said Frevisse. "Yes. For all of them."
Sister Johane accepted that in silence, before saying, sad with longing, "May we go home soon?"
Something tight-coiled around Frevisse's heart began to ease at simply the thought of that. "Yes," she said. "Soon we'll go home. For now, though, shall we say vespers?"
Author's Note
As always, so many sources studied over so many years have been drawn on for this book that there is no way to detail most of them, but very specifically used this time were John Cummins' marvelous _The Hound and the Hawk_ —a gathering in and explaining of a number of medieval hunting treatises—and _The Master of Game_ by Edward, second duke of York (died 1415 in the Battle of Agincourt)—a Middle English rendering and extension of Gaston Phoebus, count of Foix's famous _The Book of Hunting_ from the 1300s—and _The Hunting Book_ by Gabriel Bise for its reproductions of full-color pictures from a 1400s manuscript of Gaston Phoebus' work. For a discourse on modern open-field coursing, _Gazehounds_ _and Coursing_ by M.H. Dutch Salmon was invaluable.
Medieval breeds of dogs were not necessarily the same as modern ones, and terms from then may have different meanings now or be totally unfamiliar. Rather than explain the differences and specifics of lymers, raches, alauntes, kenets, harriers, spaniels, mastiffs, greyhounds, and more, I have kept, for the most part, simply to "hound," lest I end up writing my own treatise on medieval hounds instead of a novel.
The time of grace—when some animals could be hunted but not others—mentioned here differs from dates given in some sources because times of grace differ from one medieval source to another, dependent on author and place. I used what seemed most likely for where the story is set.
The "picnic" in Chapter 3 is not an anachronism but so much a part of medieval hunting that Chapter XXXIII of _The Master of Game_ is given over to describing how it should be done, including, "And the place where the gathering shall be made should be in a fair mead . . . beside some running brook."
The story of the herdboy shifting the cows with a slingshot was my father's story from his own boyhood—though, almost needless to say, the type of slingshot differed.
Specific and particular thanks must go to Cheryl Tregillis of Wyndfal Irish Wolfhounds for urging me to write a book with hounds and hunting in it and then providing me not only with information on wolfhounds but several chances to spend time with some of her own gentle, beautiful, charming Irish wolfhounds, successful in the show ring and fleet of foot in the field.
Thanks are likewise due Dr. Carol Manning, who advised on how a small wound in the throat could be sufficient to kill. An author has ideas but needs authorities to tell her what's possible.
_Now I pray unto every creature that hath heard or read this little treatise . . . that where there is too little of good language that of their benignity and grace they will add more, and there where there is too much superfluity they will also abridge it as may seem best by their good and wise discretion._
[from the end of _The Master of Game_ ]
| {
"redpajama_set_name": "RedPajamaBook"
} | 1,199 |
Q: Laravel pivot table test I've got a test where I check if records in a pivot table are being removed. The test looks like this:
/** @test */
public function when_a_relation_is_destroyed_linked_products_will_go_away()
{
$products = create(Product::class, [], 3)->toArray();
$relation = create(Relation::class)->syncProducts($products);
$this->actingAs($this->user)->deleteJson('/relaties/'.$relation->id);
$this->assertDatabaseMissing('product_has_relations', [
"product_id" => "1", "relation_id" => "1",
"product_id" => "2", "relation_id" => "1",
"product_id" => "3", "relation_id" => "1"
]);
}
When I test this in my real application the records in the pivot table are removed correctly. But my test keeps failing:
Failed asserting that a row in the table [product_has_relations] does not match the attributes {
"product_id": "3",
"relation_id": "1"
}.
Found: [
{
"product_id": "1",
"relation_id": "1"
},
{
"product_id": "2",
"relation_id": "1"
},
{
"product_id": "3",
"relation_id": "1"
}
].
This is not working aswel:
$this->assertDatabaseMissing('product_has_relations', [
["product_id" => "1", "relation_id" => "1"],
["product_id" => "2", "relation_id" => "1"],
["product_id" => "3", "relation_id" => "1"]
]);
Any idea what could be wrong here?
A: Your problem is here $this->actingAs($this->user)->deleteJson('/relaties/'.$relation->id);.
Records are not deleted hence the test fails.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
\section{Introduction}
Power-law tails in various branches of natural science have been
intensively investigated during last several decades. These
studies deal with experimental results on existing of power-law
tails as well as attempts of theoretical explanation of this
phenomenon. Some experimental material have been collected in a
popular review \citep{FCP}, other examples important for
astrophysics see, for example, in \cite{a1, a2}. As an example of
theoretical description it is possible to note a progress in
modern statistical physics which predicts power-law tails by using
non-classical definitions of entropy (this approach starts from
works of \cite{R} and \cite{T}, for a modern review see, for
example, \cite{Rudoi}). For other approaches to possible source of
this phenomenon see, for example, \cite{Montroll}.
In hamiltonian dynamics power-law tails appear in two different
situations: for systems with escape we can study the number of
trajectories $N$ which survive up to time $t$ (or, alternatively,
the number of trajectory $dN$ experienced the escape in the time
interval from $t$ to $t+dt$ we will study this differential form
of the time distribution of escapes in the present paper), for
systems with a compact phase space it is possible to construct a
function which describes Poincare recurrence time depending on the
initial position in the phase space. Many empirical results show
that the appearance of power-law tails in such systems is
connected with islands of a regular motion in the phase space and
``stickness'' of trajectories to the boundaries of these islands
\citep{Zaslavsky}, though strict mathematical results about this
kind of systems are still rather poor.
Three-body problem is an example of combined dynamics -- it is
known that stable periodic orbits surrounded by islands of regular
dynamics exist in a chaotic "sea" representing the bulk of phase
space. For equal-mass problem (when all three masses have the same
value) there exists stable configuration called a ``figure-eight''
orbit \citep{fe}, as well as several other orbits \citep{Orlov}.
It is also known that the "figure-eight" orbit becomes unstable
when relative differenses in masses reach approximately $10^{-2}$
\citep{Simo}, so this very interesting orbit is hardly important
for any real astrophysical problem. The Schubart orbit \citep{s}
remaines stable in a much broader interval of masses
\citep{Shuborb}
On the other hand, famous triangle Lagrange
points become stable in the situation of suffitiently different masses, and we can imagine that
set of stable orbits have rather nontrivial dependence on mass differences.
The phenomenon of trajectory ``stickness'' to boundary of a
regular region for the three-body problem has already been
remarked by \cite{st} and \cite{Orlov}. Keeping this in mind we
can expect existence of power-law tails in $N(t)$ distribution.
For other theoretical arguments supporting this suggestion see
\cite{Agekian}. Earlier attempts to describe this function have
been done by \cite{Mikkola} where exponential ansatz for $N(t)$
have been used. However, Orlov et al. indicated that this
modelling works only for a limited time interval, and for large
enough time $t$ power-law tails become clearly detectable
\cite{Orlnew}. Both these works deal with the equal-mass problem.
This construction may be too symmetric for making general
conclusions (keeping in mind loss of stability for "figure-eight"
solution" with a small mass inequality), so it is reasonable to
provide similar analysis in a general situation with non-equal
masses. In the present paper we have chosen several different mass
ratios for investigation, starting from near-equal mass system and
concluding for the system with masses $(0.1, 1, 10)$ mass units,
which means that Schubart orbit is already unstable, and Lagrange
triangle points are still unstable for this mass ratio.
\section{System and initial data}
\begin{figure}
\includegraphics[width=8cm]{test5.eps}
\caption{The layout of the initial conditions (coordinates). The $m_1$ body is always in the (0, 0)
point, $x$-coordinate of $m_2$ is varying within certain range. As for $m_3$, its both coordinates
are varying --- see text for details.}
\end{figure}
In the present paper we cover wide range of mass ratios from
equal-mass problem to near hierarchical system. We use Aarseth
code \citep{Aarseth} and start from zero-velocity initial
conditions. Time variable is chosen so that $G=1$.
The layout of the initial conditions we used in our study is given in
Fig. 1. The first body with mass $m_1$ is initially located in the
(0, 0) point, the second body $m_2$ initially lies on the ($a$, 0)
segment with $a\in (n+\delta, 2n+\delta)$, where $n$ is a space scale
and $\delta$ is some small separation introduced to avoid
the situation when two bodies have totally coinciding initial coordinates. Finally, the third body
$m_3$ initially lies within a square $(b, c)$ with $b\in (0, 2n)$ and
$c\in (0+\delta, 2n+\delta)$.
This choice cover rather wide area in the initial condition space for our results to
be representent, on the other hand, different permutations of non-equal masses
results in different initial condition sets allowing us to study possible dependence
of our results upon initial conditions chosen.
Using the described above layout we evenly distribute about 4
millions initial configurations and calculate their evolution
until the decay (for exact meaning of the decay in Aarseth code
see, for example, \cite{Orlnew}).
\begin{figure}
\includegraphics[angle=-90, width=9cm]{test1.eps}
\caption{The typical plot of $N(t)$ in the logarithmic scale.}
\end{figure}
\section{Decay spectra}
A typical plot of the differential distribution $N(t)$ in the
logarithmic scale is shown in Fig. 2. Power-law tail following a
steeper function after $t$ reaches the value of about one thousand
is clearly seen. The distribution of escaping time for small $t$
is clearly different from power-law, and to obtain correct values
of power index in the modelling
$$
N(t)= {\rm const} \times t^{-\gamma}
$$
we should exclude this range of $t$ from
the analysis. As an illustration we plot in Fig. 3 power indices calculated in the time range from
$t_{in}$ to $T_f$ where the maximal time of integration $T_f=7000$. We
can see that the power index typically does not change if $t_{in}$
is bigger than $2000$. In what follows we will ignore first $2000$
time units in calculation of power index $\gamma$. In the table
below we present minimal and maximal values of $\gamma$ for
several mass combinations calculated for different sets of initial
conditions. Error bars, also presented in the table indicate that
in some situations the difference in power indices calculated for
the same system using different sets of initial conditions can be
real (though not so big). We can see also that despite mass ratio
of the systems under investigation cover a broad interval from
near-equal to near-hierarchical systems, the value of $\gamma$
does not change significantly with the slightly developed tendency
of growing $\gamma$ with the increasing hierarchicity of the
triple system. We should, however, note that our values of
$\gamma$ for equal mass system is bigger than the value found in
the paper \cite{Orlnew}. Can different sets of initial conditions
be the only cause of this difference remaines unclear and requires
further investigations.
\begin{figure*}
\includegraphics[angle=-90, width=14cm]{fig1m2_3k.eps}
\caption{An examples of dependence of the slope $\gamma$ on the cutoff time
$t_{in}$. Black lines corresponds to (1, 2, 3)$_{n=3}$ mass combination, grey -- to
(1, 3, 5)$_{n=5}$. Different line styles correspond to different distribution of masses between
$m_1$, $m_2$ and $m_3$.}
\end{figure*}
\begin{table}
\centering
\begin{minipage}{70mm}
\caption{Maximal and minimum values for the slope $\gamma$, calculated at $t_{in}=3000$ for different mass
combinations. First column gives absolute masses, second -- masses, normalized
to $m_1 + m_2 + m_3 = 1$, third -- the slope $\gamma$ with an error, minimal (upper row) and maximal (lower row) values.}
\begin{tabular}{@{}cccc@{}}
\hline
\multicolumn{2}{c}{Masses} & \multicolumn{2}{c}{$\gamma \pm \delta\gamma$} \\
\hline
(0.1, 1, 10) & (0.009, 0.09, 0.901) & 1.762 & 0.011 \\
& & 1.863 & 0.016 \\
\hline
(1, 3, 10) & (0.071, 0.214, 0.715) & 1.845 & 0.013 \\
& & 1.862 & 0.012 \\
\hline
(1, 3, 5) & (0.111, 0.333, 0.556) & -1.828 & 0.013 \\
& & 1.848 & 0.014 \\
\hline
(1, 2, 3) & (0.167, 0.333, 0.5) & 1.857 & 0.014 \\
& & 1.905 & 0.015 \\
\hline
(0.9, 1, 1.1) & (0.3, 1/3, 11/30) & 2.149 & 0.009 \\
& & 2.181 & 0.010 \\
\hline
(1, 1, 1) & (1/3, 1/3, 1/3) & 2.125 & 0.014 \\
& & 2.178 & 0.012 \\
\hline
\end{tabular}
\end{minipage}
\end{table}
\section{Conclusions}
We studied the escaping rate statistics in a general three-body
problem with non-equal masses. We argue that for large enough time
$t$ this statistics can be modelled by power-law functions with a
good accuracy. For all cases studied (with normalized mass ratios
from to) the power index is located within a rather narrow
interval (from $1.76$ to $2.19$).
\section*{Acknowledgments}
Authors are grateful to Artur Chernin, Viktor Orlov and Sergey
Prants for helpful discussions. As well as we'd like to thank
Sergey Karpov (SAO RAS) for useful discussions and various
technical help.
\onecolumngrid
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,991 |
Q: How to filter data in table (crosstab) by date I need some help in creating report using iReport for this layout. I am able to create records using crosstab, but its creating duplicate records.
Male Female
Date
Total Count
Query
SELECT table1.`gender` AS table1_gender,
registration_details.`date_reg` AS registration_details_date_reg
FROM
`table1` table1 INNER JOIN `date_reg` date_reg ON table1.`candidate_id` = registration_details.`candidate_id`
where
gender is present in table1 (candidate_id : pk)
date_reg is present in registration_details
The jrxml:
<?xml version="1.0" encoding="UTF-8"?>
<jasperReport xmlns="http://jasperreports.sourceforge.net/jasperreports" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://jasperreports.sourceforge.net/jasperreports http://jasperreports.sourceforge.net/xsd/jasperreport.xsd" name="report12" language="groovy" pageWidth="595" pageHeight="842" columnWidth="555" leftMargin="20" rightMargin="20" topMargin="20" bottomMargin="20">
<style name="Crosstab Data Text" hAlign="Center"/>
<style name="table">
<box>
<pen lineWidth="1.0" lineColor="#000000"/>
</box>
</style>
<style name="table_TH" mode="Opaque" backcolor="#F0F8FF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<style name="table_CH" mode="Opaque" backcolor="#BFE1FF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<style name="table_TD" mode="Opaque" backcolor="#FFFFFF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<style name="table 1">
<box>
<pen lineWidth="1.0" lineColor="#000000"/>
</box>
</style>
<style name="table 1_TH" mode="Opaque" backcolor="#F0F8FF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<style name="table 1_CH" mode="Opaque" backcolor="#BFE1FF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<style name="table 1_TD" mode="Opaque" backcolor="#FFFFFF">
<box>
<pen lineWidth="0.5" lineColor="#000000"/>
</box>
</style>
<subDataset name="New Dataset 1">
<queryString language="SQL">
<![CDATA[SELECT
table1.`gender` AS table1_gender,
registration_details.`date_reg` AS registration_details_date_reg
FROM
`table1` table1 INNER JOIN `date_reg` date_reg ON table1.`candidate_id` = registration_details.`candidate_id`]]>
</queryString>
<field name="table1_gender" class="java.lang.String"/>
<field name="registration_details_date_reg" class="java.sql.Date"/>
</subDataset>
<subDataset name="Table Dataset 1"/>
<queryString language="SQL">
<![CDATA[SELECT
table1.`gender` AS table1_gender,
registration_details.`date_reg` AS registration_details_date_reg
FROM
`table1` table1 INNER JOIN `date_reg` date_reg ON table1.`candidate_id` = registration_details.`candidate_id`]]>
</queryString>
<field name="table1_gender" class="java.lang.String"/>
<field name="date_reg" class="java.sql.Date"/>
<detail>
<band height="125" splitType="Stretch">
<crosstab>
<reportElement x="0" y="0" width="555" height="125"/>
<crosstabDataset>
<dataset>
<datasetRun subDataset="New Dataset 1">
<connectionExpression><![CDATA[$P{REPORT_CONNECTION}]]></connectionExpression>
</datasetRun>
</dataset>
</crosstabDataset>
<rowGroup name="date_reg" width="70" totalPosition="End">
<bucket>
<bucketExpression class="java.lang.String"><![CDATA[(new SimpleDateFormat("yyyy-MM-dd")).format($F{date_reg})]]></bucketExpression>
</bucket>
<crosstabRowHeader>
<cellContents backcolor="#F0F8FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="70" height="25"/>
<textElement/>
<textFieldExpression class="java.lang.String"><![CDATA[$V{date_reg}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabRowHeader>
<crosstabTotalRowHeader>
<cellContents backcolor="#BFE1FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<staticText>
<reportElement x="0" y="0" width="70" height="25"/>
<textElement textAlignment="Center" verticalAlignment="Middle"/>
<text><![CDATA[Total date_reg]]></text>
</staticText>
</cellContents>
</crosstabTotalRowHeader>
</rowGroup>
<columnGroup name="table1_gender" height="30" totalPosition="End">
<bucket>
<bucketExpression class="java.lang.String"><![CDATA[$F{table1_gender}]]></bucketExpression>
</bucket>
<crosstabColumnHeader>
<cellContents backcolor="#F0F8FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="50" height="30"/>
<textElement/>
<textFieldExpression class="java.lang.String"><![CDATA[$V{student_gender}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabColumnHeader>
<crosstabTotalColumnHeader>
<cellContents backcolor="#BFE1FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<staticText>
<reportElement x="0" y="0" width="50" height="30"/>
<textElement textAlignment="Center" verticalAlignment="Middle"/>
<text><![CDATA[Total student_gender]]></text>
</staticText>
</cellContents>
</crosstabTotalColumnHeader>
</columnGroup>
<measure name="table1_genderMeasure" class="java.lang.Integer" calculation="Count">
<measureExpression><![CDATA[$F{table1_gender}]]></measureExpression>
</measure>
<crosstabCell width="50" height="25">
<cellContents>
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="50" height="25"/>
<textElement/>
<textFieldExpression class="java.lang.Integer"><![CDATA[$V{table1_genderMeasure}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabCell>
<crosstabCell height="25" rowTotalGroup="date_reg">
<cellContents backcolor="#BFE1FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="50" height="25"/>
<textElement/>
<textFieldExpression class="java.lang.Integer"><![CDATA[$V{table1_genderMeasure}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabCell>
<crosstabCell width="50" columnTotalGroup="student_gender">
<cellContents backcolor="#BFE1FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="50" height="25"/>
<textElement/>
<textFieldExpression class="java.lang.Integer"><![CDATA[$V{table1_genderMeasure}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabCell>
<crosstabCell rowTotalGroup="date_reg" columnTotalGroup="table1_gender">
<cellContents backcolor="#BFE1FF" mode="Opaque">
<box>
<pen lineWidth="0.5" lineStyle="Solid" lineColor="#000000"/>
</box>
<textField>
<reportElement style="Crosstab Data Text" x="0" y="0" width="50" height="25"/>
<textElement/>
<textFieldExpression class="java.lang.Integer"><![CDATA[$V{table1_genderMeasure}]]></textFieldExpression>
</textField>
</cellContents>
</crosstabCell>
</crosstab>
</band>
</detail>
</jasperReport>
Currently its showing all the records , Need suggestions/help in creating report
Where a User will be able to fetch records by providing date.
also its printing single records multiple time.
I would appreciate any help.
A: For filtering data by date you can add paramater (java.lang.String or java.util.Date) type and use it in the query.
The example:
<parameter name="dateParam" isForPrompting="false" class="java.lang.String"/>
...
<parameter name="dqlDateFilter" isForPrompting="false" class="java.lang.String">
<defaultValueExpression ><![CDATA[$P{dateParam} == null ? "1=1" : "registration_details.date_reg=$P{dateParam}"]]></defaultValueExpression>
</parameter>
...
<queryString language="SQL">SELECT.... FROM.... WHERE $P!{dqlDateFilter}</queryString>
You should write the correct date filter for your case.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,723 |
Пу́шкинская у́лица — улица в городе Воронеже, которая берёт своё начало от Никитинской площади и заканчивается у улицы 20-летия Октября. Появилась согласно регулярному плану Воронежа XVIII века.
История
В XVIII веке после своего появления улица являлась дорогой на Острогожск. Поэтому она называлась Острогожской. Ко второй половине XIX века сформировалась улица, которая получила название 2-й Острогожской (современная улица Кирова). Острогожская улица была переименована в 1-ю Острогожскую.
Улица считалась одной из основных торговых. У Соборной, Щепной, Хлебной и Конной площадей располагались магазины, лавки, мастерские.
На углу Щепной площади находилась открытая Д. А. Кашкиным в начале 1830-х гг первая в Воронеже книжная лавка, при которой в ежедневно открытой библиотеке можно было за плату брать книги на дом, подписываться на журналы и газеты. В этом книжном магазине часто бывали А. В. Кольцов, И. С. Никитин.
Знаменитая на весь город аптека Л. И. Мюфке занимала двухэтажное каменное здание рядом с современной Никитинской площадью, в начале улицы Пушкинской, по четной стороне. На месте здания Пушкинская, 2. Помимо аптеки, здесь находились парфюмерный магазин и заводы искусственных вод и одеколона.
В этом доме с 1870-х по 1884 год снимал квартиру и вёл уроки в своей экспериментальной школе известный воронежский педагог Н. Ф. Бунаков.
В 1884 г. здесь жил и давал нотные и фортепианные уроки В. Г. Ростропович (1858—1913).
С 1884 года здесь жил врач, редактор-издатель журнала «Медицинская беседа» А. Х. Сабинин.
Аптека просуществовала до начала Великой Отечественной войны, когда здание было разрушено.
В 1922 году улица была переименована в Советскую, а потом в 1-ю Советскую. В 1936 году в честь 100-летия со дня смерти А. С. Пушкина улица получила современное название.
Во время Великой Отечественной войны многие здания пострадали столь сильно, что в дальнейшем многие постройки по западной (нечётной) стороне были снесены.
Сейчас основная застройка в начале улицы — 1950—1960 годов, в середине, по нечетной стороне, — около 1970—1980.
С юга улица замыкается зданием цирка, построенного на месте Новостроящегося кладбища.
По чётной стороне улицы имеются одно-двухэтажные каменные жилые дома XIX — начала XX века.
Ценные особняки первой половины XIX в. (№ 12,14,26) обрамляют перекрёстки с улицами Куцыгина и 9 Января. Построенное из красного кирпича здание мужского приходского училища им. А. С. Пушкина начала XX в. — дом № 16. Жилой дом в стиле конструктивизма — № 22.
Архитектура
№ 10 — построен в 1911 году, мемориальная доска Л. Л. Семаго
№ 12 — дом эпохи классицизма, построен в начале XIX века. В третьей четверти века принадлежал семье купца А. А. Протасова. Украшен декоративными лепными вазами, установленными в нишах.
№ 14 — построен в 1810-е годы в стиле классицизма. Известен, как дом городского головы А. Р. Михайлова
№ 16 — построен в 1908—1909 годы
№ 20 — дом городского головы А. Н. Безрукова. Дом с мезонином сохранился с середины XIX века.
№ 26 — дом Клочкова, построен в середине XIX века
Литература
Источники
Улица Пушкинская, Воронеж по источникам:
Акиньшин А. К. Храмы Воронежа // Воронеж, 1994. С. 11-16;
Акиньшин А.К, Ласунский О. Г. Записки старого пешехода // Воронеж, 1995. С.295-298;
Ласунский О. Г. Литературно-общественное движение в русской провинции. Воронеж, 1985. С. 157;
Попов П. Алфавит из наших улиц: улица Пушкинская//Воронежский курьер. 1994. 16 июля;
Попов П. Путешествие по прошлому и настоящему // Коммуна. 1989. 1 мая;
Улицы Центрального района Воронежа
Улицы Ленинского района Воронежа
Воронеж | {
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{"url":"http:\/\/brianschiller.com\/blog\/2013\/12\/12\/pthreads-and-prngs-oh-my.html","text":"#### Brian Schiller\n\nProgrammer, Mathematician, and Perpetual Student\n\nEmail\n\nGithub\n\n# pthreads and PRNGs, oh my!\n\nWe learn in concurrency class that if you have a race condition in your code, the results are unpredictable. Certainly, the results are not usually what you wanted, but does that make them unpredictable?\n\nFor those who haven't taken a concurrent programming class, race conditions most often occur when two or more pieces of code (usually called threads) are running at the same time, and the behaviour of the program will change depending on which thread gets to a milestone first.\n\nFor example, you might have two threads, hp, and ij. They're both frantically setting the value of a variable winner to be different values (note that this is not real code in any language I know of):\n\nwinner = \"\"\n\nwhile winner_is_unannounced:\nwinner = \"Harry Potter\"\n\nthreap ij:\nwhile winner_is_unannounced:\nwinner = \"Indiana Jones\"\n\n...wait a moment (let them fight over it)...\nprint \"The winner is:\", winner\n\n\nDepending on which thread most recently set the value of winner, a different name is printed out. Either one is possible.\n\nYou might be thinking that whether Harry or Dr. Jones ends up winning would be random. Well, you wouldn't be alone. Austin Amoruso (who I know from WWU and who is currently a developer at SnapBi.com) posted the following on facebook:\n\nConcurrency teacher and textbooks will tell us that programs with race conditions are unpredictable. So if we make a program with a race condition, can we use it as a pseudorandom number generator (PRNG). Before we can answer that, I think we need a better definition for unpredictable.\n\n## What is unpredictable?\n\nOn wikipedia, you can read about a whole host of criteria for deciding if a PRNG is 'good enough'. For PRNGs that are used for cryptography, there is a requirement that \"an adversary has only negligible advantage in distinguishing the generator's output sequence from a random sequence\".\n\nThat means that no adversary -- a program that, given the earlier bits of a sequence, tries to predict the next one -- should do better than a random guesser.\n\nWe'll use a more relaxed criteria today. I want to see if the adversary from this page can do better than expected.\n\nThat website, Man vs. Machine, or, why Man is not a Particularly Good Source of Entropy, pits your own \"random\" bit generation against a fairly simple algorithm that tries to guess your next move.\n\nThe website gives the machine the benefit that it can decide to pass, and forgo guessing for a turn without penalty. That doesn't seem fair to me, so I've made the machine instead flip a (metaphorical) coin when it would otherwise pass, effectively forcing it to make a guess.\n\n## Methods\n\nIn order to give the PRNG a fighting chance, I added a sleep of 0.01 seconds between subsequent drawings of the bits. That is, the PRNG code examines the contested value, prints it out and then waits for 0.01 seconds before starting again.\n\nMeanwhile, the adversary code is reading each bit and making a prediction about the next one. After reading the last bit, it reports the proportion that it guessed correctly.\n\nI performed 100 trials of 100 bits each. Each trial consisted of setting up the PRNG to produce 100 bits, allowing the adversary code to make its guesses, and then storing only the proportion correct.\n\nFinally, to determine whether the adversary has \"a non-negligible advantage\", we use a one-sample t-test against the hypothesis that the distribution of those 100 proportions has a true mean of 0.5.\n\nIf I lost you in that last paragraph, the idea is just that we are going to do some stats to decide if the proportions correct are enough higher than 0.5 to say that the adversary has an advantage over a random guesser.\n\n## Results\n\nAs you may have guessed, we have statistical significance! In fact, $$p \\approx 0.000489$$, which is way lower than the usual 0.05 needed to qualify as significant. This $$p$$ measures the likelihood that, in fact, the adversay has no advantage and it only did better by chance. Since we have such a low $$p$$-value, it's very unlikely that the adversary has no advantage.\n\nHowever, the advantage is pretty small. The mean of the proportions comes out to only $$0.5213$$, just $$0.0213$$ better than a random guesser.\n\nThis doesn't mean that our PRNG is close to random -- just that maybe we didn't choose the best adversary for the job.\n\n## Challenger Approaching\n\nAfter looking at the output of the PRNG, I came up with a much simpler adversary. Here's how it goes: always guess whatever bit was most recently seen.\n\nThere are a lot of 'runs' in the bit sequences from this PRNG. For example, here is a sequence of 40 bits:\n\n0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1\n\n\nJust from looking, there is more repetition than we would expect. This new adversary takes advantage of this fact, to pretty good results.\n\nOur $$p$$-value for this adversary is embarrassingly low at $$0.00000000000000000000006208$$. Very significant, then. On average, this new adversary gets 58.77% of its guesses correct, 8.77% better than a random guesser.\n\nThe code is available as a GitHub Gist. Please give it a run if you'd like. Maybe the PRNG performs better on a computer with lots of cores (I only have two). Maybe it does better if you increase the sleep time. I'm excited to hear what you find out!","date":"2015-01-29 00:10:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5706692934036255, \"perplexity\": 815.4198133971877}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-06\/segments\/1422122233086.24\/warc\/CC-MAIN-20150124175713-00089-ip-10-180-212-252.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.clutchprep.com\/physics\/practice-problems\/144989\/a-car-is-traveling-at-10-m-s-a-how-fast-would-the-car-need-to-go-to-double-its-k","text":"Net Work & Kinetic Energy Video Lessons\n\nConcept\n\n# Problem: A car is traveling at 10 m\/s.How fast would the car need to go to double its kinetic energy?\n\n###### FREE Expert Solution\n\nKinetic energy:\n\n$\\overline{){\\mathbf{K}}{\\mathbf{=}}\\frac{\\mathbf{1}}{\\mathbf{2}}{\\mathbf{m}}{{\\mathbf{v}}}^{{\\mathbf{2}}}}$\n\nK = 2K0\n\n###### Problem Details\n\nA car is traveling at 10 m\/s.\nHow fast would the car need to go to double its kinetic energy?","date":"2020-10-24 23:31:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.23579519987106323, \"perplexity\": 3131.164946767419}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107885059.50\/warc\/CC-MAIN-20201024223210-20201025013210-00336.warc.gz\"}"} | null | null |
mhall
=====
Monty Hall Problem simulator.
| {
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Q: CSS - incorrect using of Selectors? This is the code I have right now with jQuery. 1.fiddle. What I want is to create the same scenario without jQuery. This is where I ended up. 2. fiddle.
Can anyone point out what I have done wrong here?? (As far as I can think of, my problem is with the CSS Selectors)
Thank you.
A: The is no .hide() method in HTML. To hide an element, set the display style to none.
setTimeout(function()
{
document.getElementById("blocker").style.display = "none";
}, 3000);
Also, the original CSS can work without jQuery.
A: Try below. Don't use hide() in native javascript.
setTimeout(function()
{
document.getElementById("blockid").style.display = "none";
}, 1000);
A: There is no function hide() in native javascript. Replace the hide() function with following line.
document.getElementById("blocker").style.display = 'none';
| {
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Echo das Damas foi um jornal brasileiro de divulgação feminista do século XIX. Foi fundado por Amélia Carolina de Silva Couto no Rio de Janeiro, tendo circulado por oito anos entre 1879 e 1888. Os artigos abordavam temas como valorização da mulher na família, defesa da educação para as mulheres e emancipação financeira da mulher.
Características
A palavra "eco" no nome sugere que os textos publicados no periódico deveriam ser propagados na sociedade. O primeiro número do jornal foi publicado em 4 de julho de 1879 e tinha o subtítulo "Órgão dedicado aos interesses da mulher – Crítico, recreativo, científico, literário e noticioso", que foi alterado para "Órgão dos interesses, da mulher, científico e literário" a partir do quarto número. É provável que a remoção do termo "recreativo" seja uma tentativa de garantir maior legitimidade e seriedade ao periódico, que sofria diversos ataques de opositores à sua causa.
Possuía quatro páginas por edição, sendo a última dedicada a publicidade, e com colunas que variavam a cada publicação. Seus artigos criticavam os costumes da época e os papéis de gênero, além de reivindicar o voto e a educação para mulheres. Também contou com divulgação de poesias escritas por mulheres, reportagens sobre feitos femininos em áreas consideradas masculinas e dicas domésticas, de moda e de beleza.
Ainda que o jornal pretendesse ser diário, sua periodicidade variou muito ao longo do tempo que circulou, provavelmente por questões financeiras, uma vez que tanto a tipografia quanto o escritório da sede mudaram de lugar diversas vezes. Tentou-se contornar o problema ao abrir assinaturas do jornal e em 1888 existiam 272 assinantes, que continha o nome da Princesa Isabel. O último número publicado pelo jornal foi em 28 de agosto de 1888.
Ligações Externas
Digitalização do Echo das Damas - Hemeroteca Digital (Biblioteca Nacional)
Jornais extintos do Rio de Janeiro
Jornais extintos do Brasil | {
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\section{Introduction}
Quantum entanglement is
used as a resource in quantum information processes such as teleportation \cite{teleportation1}, dense coding \cite{densecoding} and quantum-key distribution \cite{key-dist}. The success probability in fulfilling quantum information tasks depends on the properties of the entangled state which is used as a resource \cite{agrawal,pati}. This requires a deep understanding of the transformation properties of entangled states under local operations and classical communication (LOCC). One line of research is the interconvertability of multi-qubit states, and the optimal \cite{Optimal_GHZ_distil,Yildiz,optimal1,optimal2,sheng1,deng,Torun} and deterministic \cite{turgut, tajima,zhao} transformations of some classes of multi-qubit states have been widely studied. Another line of research is the interconversion of bipartite entangled states \cite{sheng2,sheng3}. In particular, Bennett et al. \cite{bennett} showed that the entanglement in any pure state of
a bipartite system can be concentrated by LOCC into maximally entangled states, and conversely, an arbitrary partly entangled state of a bipartite system can be prepared by LOCC using maximally entangled states as the only source of entanglement. Vidal \cite{Vidal1} obtained the maximum transformation probability of $n\otimes n$ pure states in terms of the Schmidt coefficients and explicitly constructed a local protocol. Lo and Popescu \cite{Lo} showed that any general transformation between bipartite pure states using LOCC can be performed with
one-way classical communications only, and one way communication is more powerful than those without communications. Chau et al. \cite{chau} presented necessary and sufficient conditions for the probabilistic transformations of quantum states using local operations (without classical communication) only. Jonathan and Plenio \cite{jonathan} used a minimal set of entanglement monotones, and presented an optimal local strategy for entanglement concentration.
Deterministic transformation of a state by LOCC is of fundamental importance in quantum information theory, because if a state $\ket{\psi}$ can be transformed into another state $\ket{\phi}$ with unit probability by LOCC, then the information tasks that can be performed by using the state $\ket{\psi}$ can also be performed by using the state $\ket{\phi}$. He and Bergou \cite{He} showed that classical communication is necessary in realizing the deterministic transformations of a single bipartite entangled state.
Nielsen \cite{nielsen} used the algebraic theory of majorization and obtained the necessary and sufficient condition for the deterministic transformations of bipartite pure states.
The same condition was derived by using the method of areas \cite{hardy}. Roa et al. \cite{roa} proposed a method for the probabilistic transformation of bipartite pure states based on local overlap modification and they obtained the deterministic transformation as a special case for states satisfying the majorization condition.
Majorization condition states that a state in Schmidt form
\begin{eqnarray}\label{111}
\ket{\psi}=\sum_{j=1}^{n}\psi_j \ket{j}\ket{j}, \quad (\psi_1 \geq \psi_2 \geq ... \geq \psi_{n} > 0)
\end{eqnarray}
can be transformed into another state
\begin{eqnarray}\label{222}
\ket{\phi}=\sum_{j=1}^{n} \phi_j \ket{j}\ket{j}, \quad (\phi_1 \geq \phi_2 \geq ... \geq \phi_{n} \geq 0)
\end{eqnarray}
by LOCC with unit probability if and only if $\lambda (\psi)$ is majorized by $\lambda (\phi)$, written $\lambda (\psi)\prec \lambda (\phi)$, where $\lambda (\psi)$ and $\lambda (\phi)$ denote the vectors of decreasingly ordered eigenvalues of the reduced density matrices $\rho_{A}^{\psi}$ and $\rho_{A}^{\phi}$, respectively. Majorization condition implies that the transformation $\ket{\psi}\rightarrow \ket{\phi}$ can be obtained with unit probability if and only if the inequality
\begin{eqnarray}\label{majorization}
\sum_{j=k}^{n}\phi_j^2 \leq \sum_{j=k}^{n}\psi_j^2
\end{eqnarray}
is satisfied for any $k$ $(1\leq k\leq n)$, with equality holding when $k=1$.
However, deterministic transformations of states by a single measurement get more complicated as the dimension increases, since the construction of the doubly stochastic matrices, measurement operators and unitary operators gets more complicated \cite{nielsen-chuang}. Hence simple and explicit protocols for the deterministic transformations of quantum states are of paramount importance. If the explicit solutions for the deterministic transformations of states in higher-dimensional space are known, then deterministic transformations of states in lower-dimensional space can be obtained as the special cases of higher-dimensional solutions.
However, the protocols for explicit transformations of states in higher-dimensional space using the lower-dimensional solutions still need to be developed.
In this Letter, we propose an explicit protocol for deterministic transformations of $n\otimes n$ states using the solutions for the deterministic transformations of lower-dimensional quantum states $m\otimes m$ ($m<n$) . As an example, we find the explicit solutions for the deterministic transformations of $3\otimes 3$ pure states by a single three-outcome measurement followed by unitary transformations, and use them to construct a protocol for the deterministic transformation of $n\otimes n$ states in $\lfloor n/2 \rfloor$ steps.
\section{Deterministic transformations of $n\otimes n$ pure states}
In this section, we are going to present a protocol for the deterministic transformations of the source state $\ket{\psi}$ to the target state $\ket{\phi}$, satisfying the majorization condition, using the solutions for the deterministic transformations of lower-dimensional quantum states $m\otimes m$ ($m<n$). Suppose that the solutions for the deterministic transformation of
a state $\ket{\chi}=\sum_{j=1}^{m}\chi_j \ket{j}\ket{j}$ to any state $\ket{\omega}=\sum_{j=1}^{m}\omega_j \ket{j}\ket{j}$ satisfying the majorization condition, $\lambda (\chi)\prec \lambda (\omega)$, are known. Our protocol consists of deterministic transformations of $m$-dimensional subspace of the source state in each step. We now describe the first step: The source state $\ket{\psi}$ can be written as
\begin{equation}
\ket{\psi}=\sum_{j=1}^{n-m}\psi_j \ket{j}\ket{j}+C\ket{\chi}
\end{equation}
where \begin{equation}
\ket{\chi}=\sum_{j=n-m+1}^{n}\chi_j \ket{j}\ket{j},\quad C=\sqrt{\sum_{j=n-m+1}^{n}\psi_j^2},\quad \chi_j=\frac{\psi_j}{C}.
\end{equation}
The state $\ket{\psi}$ can be deterministically transformed to any state $\ket{\phi^{(1)}}$
\begin{equation}
\ket{\phi^{(1)}}=\sum_{j=1}^{n-m}\psi_j \ket{j}\ket{j}+C\ket{\omega},\quad \ket{\omega}=\sum_{j=n-m+1}^{n}\omega_j \ket{j}\ket{j}
\end{equation}
satisfying the majorization condition, $\lambda (\chi)\prec \lambda (\omega)$. We choose the state $\ket{\omega}$ such that maximum number of the Schmidt coefficients of the source state is transformed to the Schmidt coefficients of the target state. In the most general case, at least $m-1$ smallest Schmidt coefficients of the target state $\ket{\phi}$ can be obtained by choosing
\begin{equation}
\omega_j=\phi_j /C,\quad n-m+2\leq j\leq n, \quad \omega_{n-m+1}=\sqrt{1-\sum_{j=n-m+2}^{n}\omega_j^2}.
\end{equation}
We note that only in the special case where the Schmidt coefficients of the source state $\ket{\psi}$ and target state $\ket{\phi}$ satisfy $\sum_{j=n-m+1}^{n}\psi_j^2=\sum_{j=n-m+1}^{n} \phi_j^2$, then $\omega_{n-m+1}=\phi_{n-m+1}/C$, and $m$ smallest Schmidt coefficients of the source state can be transformed to $m$ smallest Schmidt coefficients of the target state.
One may continue with $m$-dimensional deterministic transformations until the target state is obtained. The transformation $\ket{\psi}\rightarrow \ket{\phi}$ can be obtained in $l$ steps
\begin{equation}\label{tf7}
\ket{\psi}\rightarrow \ket{\phi^{(1)}}\rightarrow \ket{\phi^{(2)}}\rightarrow...\rightarrow \ket{\phi^{(l-1) }}\rightarrow \ket{\phi}
\end{equation}
where the intermediate states are given by
\begin{eqnarray}\label{fk7}
\ket{\phi^{(k)}}&=&\sum_{j=1}^{n-k(m-1)-1} \psi_j \ket{j}\ket{j}
+\tilde{\phi}_{n-k(m-1)}\ket{n-k(m-1)}\ket{n-k(m-1)} \nonumber \\
&& +\sum_{j=n-k(m-1)+1}^{n} \phi_j \ket{j}\ket{j},
\end{eqnarray}
\begin{equation}\label{n17}
\tilde{\phi}_{n-k(m-1)}=\sqrt{\psi_{n-k(m-1)}^2+\sum_{j=n-k(m-1)+1}^{n} \left(\psi_j^2-\phi_j^2\right)},\quad 1\leq k\leq l-1.
\end{equation}
By using eqs. \eqref{fk7}-\eqref{n17} it can be shown that the majorization conditions $\lambda (\phi^{(k)}) \prec \lambda (\phi^{(k+1)})$ are satisfied and hence the transformations $\ket{\phi^{(k)}}\rightarrow \ket{\phi^{(k+1)}}$ can be obtained with unit probability.
In each step, at least $m-1$ more coefficients of the target state $\ket{\phi}$ are obtained, {\it i.e.}, the smallest $k(m-1)$ coefficients of $\ket{\phi^{(k)}}$ and $\ket{\phi}$ are equal. The states $\ket{\phi^{(k)}}$ and $\ket{\phi^{(k+1)}}$ have $m$ nonequal Schmidt coefficients and the transformation $\ket{\phi^{(k)}}\rightarrow \ket{\phi^{(k+1)}}$ is effectively an $m$-dimensional transformation. In the final transformation, $\ket{\phi^{(l-1)}}\rightarrow \ket{\phi}$, the states $\ket{\phi^{(l-1)}}$ and $\ket{\phi}$ may have $q$ ($2\leq q\leq m$) nonequal Schmidt coefficients which can be transformed by a single transformation.
In summary, using successive $m$-dimensional deterministic transformations $l$ times, it is possible to obtain deterministic transformations of $n\otimes n$ ($(l-1)(m-1)+2 \leq n \leq (l-1)(m-1)+m$) bipartite pure states.
We now discuss the possibility of obtaining the target state, instead of transforming the smallest nonequal Schmidt coefficients, by transforming the greatest Schmidt coefficients in each step. In this case, the target state is again obtained, if possible, by successive deterministic transformations
\begin{equation}\label{tfm2}
\ket{\psi}\rightarrow \ket{\psi^{(1)}}\rightarrow \ket{\psi^{(2)}}\rightarrow...\rightarrow \ket{\psi^{(l-1)}}\rightarrow \ket{\phi}.
\end{equation}
In that case, the intermediate states turn out to be
\begin{eqnarray}\label{fk8}
\ket{\psi^{(k)}}=\sum_{j=1}^{k(m-1)} \phi_j \ket{j}\ket{j}
+\tilde{\psi}_{k(m-1)+1}\ket{k(m-1)+1}\ket{k(m-1)+1}
+\sum_{j=k(m-1)+2}^{n} \psi_j \ket{j}\ket{j},
\end{eqnarray}
\begin{equation}\label{n18}
\tilde{\psi}_{k(m-1)+1}=\sqrt{\phi_{k(m-1)+1}^2+\sum_{j=k(m-1)+2}^{n} \left(\phi_j^2-\psi_j^2\right)},\quad 1\leq k\leq l-1.
\end{equation}
Using the majorization conditions it can be shown that $\tilde{\psi}_{k(m-1)+1}$ can be equal to zero, and the number of nonzero Schmidt coefficients is reduced by one. In this case, the probability of obtaining the target state $\ket{\phi}$ from $\ket{\psi^{(k)}}$ will be zero, since the number of nonzero Schmidt coefficients cannot be increased by LOCC \cite{Lo}.
This rules out the possibility of any deterministic transformation other than the transformation of the smallest Schmidt coefficients in each step, as given by eq. \eqref{tf7}.
\section{Deterministic transformations of $3 \otimes 3$ pure states by a single measurement}
In this section, we are going to present a protocol for the deterministic transformation of the state
\begin{eqnarray}\label{3.32}
\ket{\psi}=a_1\ket{1}\ket{1}+b_1\ket{2}\ket{2}+c_1\ket{3}\ket{3},\quad (a_1\geq b_1 \geq c_1> 0)
\end{eqnarray}
to the state
\begin{eqnarray}\label{3.33}
\ket{\phi}= a_2\ket{1}\ket{1}+b_2\ket{2}\ket{2}+c_2\ket{3}\ket{3},\quad (a_2\geq b_2 \geq c_2\geq 0)
\end{eqnarray}
satisfying the majorization condition by a single measurement. A generalized three-outcome measurement with the POVM elements leaving the Schmidt form invariant,
\begin{eqnarray}\label{op3}
M_i=\sqrt{\alpha_i}\ket{1}\bra{1}+\sqrt{\beta_i}\ket{2}\bra{2}+\sqrt{\gamma_i}\ket{3}\bra{3},\ \ \sum_{i=1}^{3} M_i^\dag M_i =I,
\end{eqnarray}
is performed on one of the particles. The state after the measurement turns out to be one of the states
\begin{eqnarray}\label{3.2}
\ket{\psi_i}&=&\frac{M_i \ket{\psi}}{\sqrt{p_i}}
=\frac{1}{\sqrt{p_i}}\Big(\sqrt{\alpha_i} a_1\ket{1}\ket{1}+\sqrt{\beta_i} b_1\ket{2}\ket{2}+\sqrt{\gamma_i}c_1\ket{3}\ket{3}\Big)
\end{eqnarray}
with probabilities $p_i=a_1^2\alpha_i+b_1^2 \beta_i+c_1^2 \gamma_i$. We impose the condition that all three states, $\ket{\psi_i}$, can be transformed to the state $\ket{\phi}$ by local unitary transformations, then the transformation $\ket{\psi}\rightarrow\ket{\phi}$ is a deterministic transformation. Although, due to the majorization condition, $a_1\leq a_2$ and $c_{1}\geq c_{2}$ there is not an unique relation between the other Schmidt coefficients, $\it {i. e.}$, both $b_1\geq b_2$ and $b_2\geq b_1$ are possible. We find that the local operations for the transformations turn out to be different for cases $b_1\geq b_2$ and $b_2\geq b_1$. We present the solutions for both cases:
\textit{$(i)$ The case $b_1\geq b_2$;}
The three-outcome generalized measurement with POVM elements
\begin{eqnarray}\label{3.76}
M_1&=&\sqrt{p_1}\Big(\frac{a_2}{a_1}\ket{1}\bra{1}+\frac{b_2}{b_1}\ket{2}\bra{2}+\frac{c_2}{c_1}\ket{3}\bra{3}\Big), \nonumber \\
M_2&=&\sqrt{p_2}\Big(\frac{b_2}{a_1}\ket{1}\bra{1}+\frac{a_2}{b_1}\ket{2}\bra{2}+\frac{c_2}{c_1}\ket{3}\bra{3}\Big), \\ M_3&=&\sqrt{p_3}\Big(\frac{c_2}{a_1}\ket{1}\bra{1}+\frac{b_2}{b_1}\ket{2}\bra{2}+\frac{a_2}{c_1}\ket{3}\bra{3}\Big)\nonumber
\end{eqnarray}
transforms the state $\ket{\psi}$, given by eq. \eqref{3.32}, to one of the states
\begin{eqnarray}\label{3.73}
\ket{\psi_1}&=& a_2\ket{1}\ket{1}+b_2\ket{2}\ket{2}+c_2\ket{3}\ket{3},\nonumber \\
\ket{\psi_2}&=& b_2\ket{1}\ket{1}+a_2\ket{2}\ket{2}+c_2\ket{3}\ket{3},\\
\ket{\psi_3}&=& c_2\ket{1}\ket{1}+b_2\ket{2}\ket{2}+a_2\ket{3}\ket{3}\nonumber
\end{eqnarray}
with probabilities
\begin{equation}\label{3.81}
p_1=\frac{a_1^2}{a_2^2}-\frac{b_2^2}{a_2^2}\frac{(b_1^2 - b_2^2)}{(a_2^2 - b_2^2)}- \frac{c_2^2}{a_2^2}\frac{(c_1^2 - c_2^2)}{(a_2^2 - c_2^2)},\ \
p_2=\frac{b_1^2 - b_2^2}{a_2^2 - b_2^2},\ \
p_3=\frac{c_1^2 - c_2^2}{a_2^2 - c_2^2},
\end{equation}
respectively. The state $\ket{\psi_1}$ is already the target state $\ket{\phi}$, and the states $\ket{\psi_2}$ and $\ket{\psi_3}$ can be transformed to the target state by local unitary transformations $\ket{1}\leftrightarrow\ket{2}$ and $\ket{1}\leftrightarrow\ket{3}$, respectively. Since all states obtained after the measurement are transformed to the target state by local unitary transformations and total probability of success is unity ($p_1+p_2+p_3=1$), we may conclude that the deterministic transformation $\ket{\psi}\rightarrow\ket{\phi}$ can be obtained by LOCC described above. The majorization condition and the condition $b_1\geq b_2$ imply the ordering of the parameters $a_2\geq a_1\geq b_1\geq b_2 \geq c_2$, and hence for $a_2=b_2$ the source and target states are already equal.
\textit{$(ii)$ The case $b_2\geq b_1$;}
The three-outcome generalized measurement with POVM elements
\begin{eqnarray}\label{3.77}
M_1&=&\sqrt{p_1}\Big(\frac{a_2}{a_1}\ket{1}\bra{1}+\frac{b_2}{b_1}\ket{2}\bra{2}+\frac{c_2}{c_1}\ket{3}\bra{3}\Big), \nonumber \\
M_2&=&\sqrt{p_2}\Big(\frac{c_2}{a_1}\ket{1}\bra{1}+\frac{b_2}{b_1}\ket{2}\bra{2}+\frac{a_2}{c_1}\ket{3}\bra{3}\Big), \\ M_3&=&\sqrt{p_3}\Big(\frac{a_2}{a_1}\ket{1}\bra{1}+\frac{c_2}{b_1}\ket{2}\bra{2}+\frac{b_2}{c_1}\ket{3}\bra{3}\Big)\nonumber
\end{eqnarray}
transforms the state $\ket{\psi}$ to one of the states
\begin{eqnarray}\label{3.74}
\ket{\psi_1}&=& a_2\ket{1}\ket{1}+b_2\ket{2}\ket{2}+c_2\ket{3}\ket{3},\nonumber \\
\ket{\psi_2}&=& c_2\ket{1}\ket{1}+b_2\ket{2}\ket{2}+a_2\ket{3}\ket{3},\\
\ket{\psi_3}&=& a_2\ket{1}\ket{1}+c_2\ket{2}\ket{2}+b_2\ket{3}\ket{3}\nonumber
\end{eqnarray}
with probabilities
\begin{equation}\label{3.82}
p_1=\frac{a_1^2}{a_2^2}-\frac{c_2^2}{a_2^2}\frac{(a_2^2 - a_1^2)}{(a_2^2 - c_2^2)}- \frac{(b_2^2 - b_1^2)}{(b_2^2 - c_2^2)},\ \
p_2=\frac{a_2^2 - a_1^2}{a_2^2 - c_2^2},\ \
p_3=\frac{b_2^2 - b_1^2}{b_2^2 - c_2^2},
\end{equation}
respectively. The state $\ket{\psi_1}$ is already the target state $\ket{\phi}$, and the states $\ket{\psi_2}$ and $\ket{\psi_3}$ can be transformed to the target state by local unitary transformations $\ket{1}\leftrightarrow\ket{3}$ and $\ket{2}\leftrightarrow\ket{3}$, respectively. Since all states obtained after the measurement are transformed to the target state by local unitary transformations and total probability of success is unity ($p_1+p_2+p_3=1$), we may conclude that the deterministic transformation $\ket{\psi}\rightarrow\ket{\phi}$ can be obtained by LOCC described above. The majorization conditions, given by eq. \eqref{majorization}, and the condition $b_2\geq b_1$ imply the ordering of the parameters $a_2\geq b_2\geq b_1\geq c_1 \geq c_2$, and hence for $b_2=c_2$, the source and target states are equal.
We note that for $b_1=b_2$, the three-dimensional problem reduces to the two-dimensional problem, and both solutions, given by eqs. \eqref{3.76} and \eqref{3.77}, reduce to the solution of the two-dimensional problem.
Motivated by the solutions for the deterministic transformations of $3\otimes3$ states presented above, one may consider any possible generalization of the method to any dimension, \textit{i.e.}, deterministic transformation of \eqref{111} into \eqref{222} by a single measurement followed by local unitary transformations. The solution of the state transformation problem depends on the ordering of the Schmidt coefficients of the source and target states. There are only two different cases ($b_1\geq b_2$ and $b_2\geq b_1$) for transformations of $3\otimes3$ states as discussed above. In the $n\otimes n$-dimensional problem, there are at least $2^{n-2}$ inequivalent cases, each of which requires the solutions for highly nontrivial problem of finding local measurement operators and unitary transformations in high dimensions. The complexity of the problem increases exponentially as the dimension increases, and hence there is no simple generalization of deterministic state transformation by a single measurement to higher-dimensional spaces. However, it is possible to use the solutions obtained for transformation of states in lower dimensions to obtain transformation of states in higher dimensions.
\section{Deterministic transformations of $ n\otimes n$ pure states by three-outcome measurements}
In this section, we are going to present a protocol for the deterministic transformations of the state given by eq. \eqref{111}, to the state given by eq. \eqref{222}, satisfying the majorization condition, using the results obtained for the deterministic transformations of $3\otimes 3$ pure states. Our protocol consists of $\lfloor n/2 \rfloor$ deterministic transformations,
\begin{equation}\label{tf}
\ket{\psi}\rightarrow \ket{\phi^{(1)}}\rightarrow \ket{\phi^{(2)}}\rightarrow...\rightarrow \ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}\rightarrow \ket{\phi},
\end{equation}
where the intermediate states are given by
\begin{equation}\label{fk}
\ket{\phi^{(k)}}=\sum_{j=1}^{n-2k-1} \psi_j \ket{j}\ket{j}+\tilde{\phi}_{n-2k}\ket{n-2k}\ket{n-2k}+\sum_{j=n-2k+1}^{n} \phi_j \ket{j}\ket{j},
\end{equation}
\begin{equation}\label{n1}
\tilde{\phi}_{n-2k}=\sqrt{\psi_{n-2k}^2+\sum_{j=n-2k+1}^{n} \left(\psi_j^2-\phi_j^2\right)},\quad 1\leq k\leq \lfloor (n-2)/2 \rfloor.
\end{equation}
In each step, two more coefficients of the target state $\ket{\phi}$ are obtained, {\it i.e.}, the smallest $2k$ coefficients of $\ket{\phi^{(k)}}$ and $\ket{\phi}$ are equal.
Deterministic transformations require that the majorization conditions, $\lambda (\phi^{(k)}) \prec \lambda (\phi^{(k+1)})$, which lead to the inequalities given by
\begin{eqnarray}\label{c1}
\tilde{\phi}_{n-2k-2}^2+\phi_{n-2k-1}^2+\phi_{n-2k}^2 &\leq& \psi_{n-2k-2}^2+\psi_{n-2k-1}^2+\tilde{\phi}_{n-2k}^2 , \nonumber \\
\phi_{n-2k-1}^2+\phi_{n-2k}^2&\leq& \psi_{n-2k-1}^2+\tilde{\phi}_{n-2k}^2 , \\
\phi_{n-2k}&\leq& \tilde{\phi}_{n-2k} \nonumber
\end{eqnarray}
should be satisfied. By substituting $\tilde{\phi}_{n-2k}^2$, given by eq. \eqref{n1}, and using the majorization condition it can be shown that the inequalities given by eq. \eqref{c1} are satisfied.
We give the explicit solutions for the first transformation, $\ket{\psi}\rightarrow \ket{\phi^{(1)}}$, and the last transformation, $\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}\rightarrow \ket{\phi}$, for illustrative purposes, and other transformations can be done similarly.
For the first transformation, we write the states $\ket{\psi}$ and $\ket{\phi^{(1)}}$ as
\begin{eqnarray}\label{t1}
\ket{\psi}&=&\sum_{j=1}^{n-3}\psi_j \ket{j}\ket{j}+\mu \Big(a_1 \ket{n-2}\ket{n-2}+b_1 \ket{n-1}\ket{n-1}+c_1 \ket{n}\ket{n}\Big),\\
\label{t21}\ket{\phi^{(1)}}&=&\sum_{j=1}^{n-3}\psi_j \ket{j}\ket{j}+\mu \Big(a_2 \ket{n-2}\ket{n-2}+b_2 \ket{n-1}\ket{n-1}+c_2 \ket{n}\ket{n}\Big)
\end{eqnarray}
where
\begin{equation}\label{t13}
\mu=\sqrt{\psi_{n-2}^2+\psi_{n-1}^2+\psi_{n}^2},\quad a_1= \frac{\psi_{n-2}}{\mu}, \quad b_1= \frac{\psi_{n-1}}{\mu}, \quad c_1=\frac{\psi_{n}}{\mu},
\end{equation}
\begin{equation}
a_2=\frac{\tilde{\phi}_{n-2}}{\mu}= \sqrt{1-b_{2}^{2}-c_{2}^{2}}, \quad b_2= \frac{\phi_{n-1}}{\mu}, \quad c_2=\frac{\phi_{n}}{\mu}.\nonumber
\end{equation}
A three-outcome measurement with POVM elements
\begin{eqnarray}
M_{i}&=&\sum_{j=1}^{n-3} \sqrt{p_i}\ket{j}\bra{j}+\sqrt{\alpha_i}\ket{n-2}\bra{n-2}+\sqrt{\beta_i}\ket{n-1}\bra{n-1} \nonumber \\
&&+\sqrt{\gamma_i}\ket{n}\bra{n}, \quad p_i=a_1^2\alpha_i+b_1^2 \beta_i+c_1^2 \gamma_i,\quad i=1,2,3
\end{eqnarray}
should be performed on one of the particles. The solutions for $\alpha_i$, $\beta_i$, $\gamma_i$ and the local unitary transformations which should be done after the measurement depend on the parameters $\phi_{n-1}$ and $\psi_{n-1}$.
\textit{$(i)$ The case $\psi_{n-1}\geq \phi_{n-1}$}; A three-outcome measurement with POVM elements
\begin{eqnarray}
&M_1=\sqrt{p_1}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{a_2}{a_1}\ket{n-2}\bra{n-2}+\frac{b_2}{b_1}\ket{n-1}\bra{n-1}+\frac{c_2}{c_1}\ket{n}\bra{n}\Big),& \nonumber\\
&M_2=\sqrt{p_2}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{b_2}{a_1}\ket{n-2}\bra{n-2}+\frac{a_2}{b_1}\ket{n-1}\bra{n-1}+\frac{c_2}{c_1}\ket{n}\bra{n}\Big), & \\ &M_3=\sqrt{p_3}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{c_2}{a_1}\ket{n-2}\bra{n-2}+\frac{b_2}{b_1}\ket{n-1}\bra{n-1}+\frac{a_2}{c_1}\ket{n}\bra{n}\Big)&\nonumber
\end{eqnarray}
transforms the state $\ket{\psi}$, given by eq. \eqref{111}, to one of the states
\begin{eqnarray}
&\ket{\psi_1}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+\tilde{\phi}_{n-2}\ket{n-2}\ket{n-2}+\phi_{n-1}\ket{n-1}\ket{n-1}+\phi_{n}\ket{n}\ket{n},&\nonumber \\
&\ket{\psi_2}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+ \phi_{n-1}\ket{n-2}\ket{n-2}+\tilde{\phi}_{n-2}\ket{n-1}\ket{n-1}+\phi_{n}\ket{n}\ket{n},&\\
&\ket{\psi_3}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+\phi_{n}\ket{n-2}\ket{n-2}+\phi_{n-1}\ket{n-1}\ket{n-1}+\tilde{\phi}_{n-2}\ket{n}\ket{n}&\nonumber
\end{eqnarray}
with probabilities $p_1$, $p_2$ and $p_3$ given by eq. \eqref{3.81}.
The state $\ket{\psi_1}$ is already the state $\ket{\phi^{(1)}}$, and the states $\ket{\psi_2}$ and $\ket{\psi_3}$ can be transformed to the state $\ket{\phi^{(1)}}$ by local unitary transformations $\ket{n-2}\leftrightarrow\ket{n-1}$ and $\ket{n-2}\leftrightarrow\ket{n}$ respectively.
\textit{$(ii)$ The case $\phi_{n-1}\geq \psi_{n-1}$};
\begin{eqnarray}
&M_1=\sqrt{p_1}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{a_2}{a_1}\ket{n-2}\bra{n-2}+\frac{b_2}{b_1}\ket{n-1}\bra{n-1}+\frac{c_2}{c_1}\ket{n}\bra{n}\Big),& \nonumber \\
&M_2=\sqrt{p_2}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{c_2}{a_1}\ket{n-2}\bra{n-2}+\frac{b_2}{b_1}\ket{n-1}\bra{n-1}+\frac{a_2}{c_1}\ket{n}\bra{n}\Big), & \\ &M_3=\sqrt{p_3}\Big(\sum_{j=1}^{n-3}\ket{j}\bra{j}+\frac{a_2}{a_1}\ket{n-2}\bra{n-2}+\frac{c_2}{b_1}\ket{n-1}\bra{n-1}+\frac{b_2}{c_1}\ket{n}\bra{n}\Big) &\nonumber
\end{eqnarray}
transforms the state $\ket{\psi}$ to one of the states
\begin{eqnarray}
&\ket{\psi_1}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+\tilde{\phi}_{n-2}\ket{n-2}\ket{n-2}+\phi_{n-1}\ket{n-1}\ket{n-1}+\phi_{n}\ket{n}\ket{n}, \nonumber \\
&\ket{\psi_2}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+ \phi_{n}\ket{n-2}\ket{n-2}+\phi_{n-1}\ket{n-1}\ket{n-1}+\tilde{\phi}_{n-2}\ket{n}\ket{n}, &\\
&\ket{\psi_3}= \sum_{j=1}^{n-3} \psi_j \ket{j}\ket{j}+\tilde{\phi}_{n-2}\ket{n-2}\ket{n-2}+\phi_{n}\ket{n-1}\ket{n-1}+\phi_{n-1}\ket{n}\ket{n}&\nonumber
\end{eqnarray}
with probabilities $p_1$, $p_2$ and $p_3$ given by eq. \eqref{3.82}.
The state $\ket{\psi_1}$ is already the state $\ket{\phi^{(1)}}$, and the states $\ket{\psi_2}$ and $\ket{\psi_3}$ can be transformed to the state $\ket{\phi^{(1)}}$ by local unitary transformations $\ket{n-2}\leftrightarrow\ket{n}$ and $\ket{n-1}\leftrightarrow\ket{n}$ respectively.
The last transformation, $\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}\rightarrow \ket{\phi}$, is effectively a two-dimensional transformation if $n$ is an even number, or a three-dimensional transformation if $n$ is an odd number.
For $n$ is an even number, the states $\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}$ and $\ket{\phi}$ can be written as
\begin{eqnarray}\label{4.13}
\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}&=&\psi_1 \ket{1}\ket{1}+\tilde{\phi}_{2}\ket{2}\ket{2}+\sum_{j=3}^{n} \phi_j \ket{j}\ket{j},\nonumber \\
&=& \nu \left( a'_1 \ket{1}\ket{1}+b'_1 \ket{2}\ket{2}\right)+\sum_{j=3}^{n} \phi_j \ket{j}\ket{j},\\
\ket{\phi}&=& \nu \left( a'_2 \ket{1}\ket{1}+b'_2 \ket{2}\ket{2}\right)+\sum_{j=3}^{n} \phi_j \ket{j}\ket{j} \nonumber
\end{eqnarray}
where
\begin{equation}\label{t27}
a'_{1}= \frac{\psi_{1}}{\nu}, \quad b'_{1}= \frac{\tilde{\phi}_{2}}{\nu} =\frac{\sqrt{\phi_{1}^2+\phi_{2}^2-\psi_{1}^2}}{\nu}, \quad
a'_{2}=\frac{\phi_{1}}{\nu}, \quad b'_{2}= \frac{\phi_{2}}{\nu},\quad \nu=\sqrt{\phi_{1}^2+\phi_{2}^2}.
\end{equation}
A two-outcome measurement on one of the particles with the POVM elements
\begin{eqnarray}\begin{aligned}\label{4.18}
M'_1&=\sqrt{p'_1}\Big(\frac{a'_2}{a'_1}\ket{1}\bra{1}
+\frac{b'_2}{b'_1}\ket{2}\bra{2}+\sum_{j=3}^{n}\ket{j}\bra{j}\Big),\quad p'_1=\frac{(a'_{1})^2-(b'_2)^2}{(a'_2)^2-(b'_2)^2}, \\
M'_2&=\sqrt{p'_2}\Big(\frac{b'_2}{a'_1}\ket{1}\bra{1}
+\frac{a'_2}{b'_1}\ket{2}\bra{2}+\sum_{j=3}^{n}\ket{j}\bra{j}\Big), \quad p'_2=\frac{(a'_2)^2-(a'_1)^2}{(a'_2)^2-(b'_2)^2}
\end{aligned}\end{eqnarray}
can be used to obtain the deterministic transformation. If the measurement gives the outcome corresponding to the operator $M'_1$, with probability $p'_1$, then the resulting state is $\ket{\phi}$. But if the measurement gives the outcome corresponding to operator $M'_2$ , with probability $p'_2$, then both parties perform unitary operations $\ket{1}\leftrightarrow \ket{2}$ on their particles to obtain the state $\ket{\phi}$.
For $n$ is an odd number, the states $\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}$ and $\ket{\phi}$ can be written as
\begin{eqnarray}\label{4.111}
\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}&=&\psi_1 \ket{1}\ket{1}+\psi_2 \ket{2}\ket{2}+\tilde{\phi}_{3}\ket{3}\ket{3}+\sum_{j=4}^{n} \phi_j \ket{j}\ket{j},\nonumber \\
&=& \kappa \left( a''_1 \ket{1}\ket{1}+b''_1 \ket{2}\ket{2}+c''_1 \ket{3}\ket{3}\right)+\sum_{j=4}^{n} \phi_j \ket{j}\ket{j},\\
\ket{\phi}&=& \kappa \left( a''_2 \ket{1}\ket{1}+b''_2 \ket{2}\ket{2}+c''_2 \ket{3}\ket{3}\right)+\sum_{j=4}^{n} \phi_j \ket{j}\ket{j} \nonumber
\end{eqnarray}
where
\begin{equation}
a''_{1}= \frac{\psi_{1}}{\kappa}, \quad b''_{1}=\frac{\psi_{2}}{\kappa},\quad c''_{1}= \frac{\tilde{\phi}_{3}}{\kappa}=\frac{\sqrt{\phi_{1}^2+\phi_{2}^2+\phi_{3}^2-\psi_{1}^2-\psi_{2}^2}}{\kappa},\nonumber
\end{equation}
\begin{equation}
a''_{2}= \frac{\phi_{1}}{\kappa},\quad b''_{2}= \frac{\phi_{2}}{\kappa},\quad c''_{2}= \frac{\phi_{3}}{\kappa},\quad \kappa=\sqrt{\phi_{1}^2+\phi_{2}^2+\phi_{3}^2}.
\end{equation}
The deterministic transformation of states, $\ket{\phi^{(\lfloor (n-2)/2 \rfloor)}}\rightarrow \ket{\phi}$, is effectively a three-dimensional transformation problem. The solutions for the three-outcome measurement of one the particles with POVM elements
\begin{eqnarray}\label{op4}
M''_i=\sqrt{\alpha''_i}\ket{1}\bra{1}+\sqrt{\beta''_i}\ket{2}\bra{2}+
\sqrt{\gamma''_i}\ket{3}\bra{3}+\sqrt{p''_i}\sum_{j=4}^{n}\ket{j}\bra{j},\ \ \sum_{i=1}^{3} (M''_i)^\dag (M''_i) =I
\end{eqnarray}
and the unitary transformations can easily be found using the solutions for the three-dimensional case given by Eqs. \eqref{3.76}-\eqref{3.82}.
\section{Conclusion}
We have presented an explicit protocol for the deterministic transformation of any bipartite pure state of higher dimensions using the solutions of that of lower dimensions. We have divided the higher-dimensional transformation problem into some number of effectively smaller dimensional transformation problems. In each step, some number of Schmidt coefficients of the source state have been transformed to the Schmidt coefficients of the target state. The constraint on each transformation is that the intermediate states obtained by lower-dimensional transformations should be deterministically transformable to the target state, $\it {i. e.}$, each intermediate state should satisfy the necessary majorization conditions. This constraint also implies that the number of nonzero Schmidt coefficients of the intermediate states cannot be less than that of the target state. We have shown that the transformation of greater nonequal Schmidt coefficients of the source state could reduce the number of nonzero Schmidt coefficients of intermediate states, and obtaining the target state would be impossible. This leaves the transformations of the smallest nonequal Schmidt coefficients of the intermediate states to the smallest nonequal Schmidt coefficients of the target state as the only option. We have also shown that the intermediate states in our protocol satisfy the necessary majorization conditions.
As an example of the proposed protocol, we first obtain the deterministic transformations of $3 \otimes 3$ pure states which consist of a single three-outcome measurement, one-way classical communication and local unitary transformations. Then we use the results of the deterministic transformations of $3 \otimes 3$ states, and construct the explicit operations for the deterministic transformations $n \otimes n$ states in $\lfloor n/2 \rfloor$ steps. We think that the proposed
protocol sheds some light on the transformations of pure states.
\section{Acknowledgments}
This work has been partially supported by the Scientific and
Technological Research Council of Turkey (TUBITAK) under Grant
113F256.
\bibliographystyle{elsarticle-num}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,723 |
Q: problemsn with xcode and 64 bit app compatability Hi despite following all the steps indicated in this thread
I am still not able to get passed the 64 bit validation on iTunes wondering if anyone can assist please
I have developed the app based on cordova platform yet I use Xcode as the developer interface not sure if this may be a contributing factor ???
A: Have you connected your devices when archiving the app? If connected, please unplug them, and archive again.
BTW, the common reason is you set "build active architecture only"(in the Building Settings Section) release mode to "yes". So just check that, and change it to "no".
Hope this help you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,475 |
Corporate Finance & Accounting Financial Analysis
CAPM vs. Arbitrage Pricing Theory: What's the Difference?
Steven Nickolas
Steven Nickolas is a freelance writer and has 10+ years of experience working as a consultant to retail and institutional investors.
Margaret James
Reviewed by Margaret James
Peggy James is a CPA with over 9 years of experience in accounting and finance, including corporate, nonprofit, and personal finance environments. She most recently worked at Duke University and is the owner of Peggy James, CPA, PLLC, serving small businesses, nonprofits, solopreneurs, freelancers, and individuals.
How to Value a Company
CAPM vs. Arbitrage Pricing Theory: An Overview
In the 1960s, Jack Treynor, William F. Sharpe, John Lintner, and Jan Mossin developed the capital asset pricing model (CAPM) to determine the theoretical appropriate rate that an asset should return given the level of risk assumed. Thereafter, in 1976, economist Stephen Ross developed the arbitrage pricing theory (APT) as an alternative to the CAPM. The APT introduced a framework that explains the expected theoretical rate of return of an asset, or portfolio, in equilibrium as a linear function of the risk of the asset, or portfolio, with respect to a set of factors capturing systematic risk.
Capital Asset Pricing Model
The CAPM allows investors to quantify the expected return on an investment given the investment risk, risk-free rate of return, expected market return, and the beta of an asset or portfolio. The risk-free rate of return that is used is typically the federal funds rate or the 10-year government bond yield.
An asset's or portfolio's beta measures the theoretical volatility in relation to the overall market. For example, if a portfolio has a beta of 1.25 in relation to the Standard & Poor's 500 Index (S&P 500), it is theoretically 25 percent more volatile than the S&P 500 Index. Therefore, if the index rises by 10 percent, the portfolio rises by 12.5 percent. If the index falls by 10 percent, the portfolio falls by 12.5 percent.
CAPM Formula
The formula used in CAPM is: E(ri) = rf + βi * (E(rM) - rf), where rf is the risk-free rate of return, βi is the asset's or portfolio's beta in relation to a benchmark index, E(rM) is the expected benchmark index's returns over a specified period, and E(ri) is the theoretical appropriate rate that an asset should return given the inputs.
Arbitrage Pricing Theory
The APT serves as an alternative to the CAPM, and it uses fewer assumptions and may be harder to implement than the CAPM. Ross developed the APT on the basis that the prices of securities are driven by multiple factors, which could be grouped into macroeconomic or company-specific factors. Unlike the CAPM, the APT does not indicate the identity or even the number of risk factors. Instead, for any multifactor model assumed to generate returns, which follows a return-generating process, the theory gives the associated expression for the asset's expected return. While the CAPM formula requires the input of the expected market return, the APT formula uses an asset's expected rate of return and the risk premium of multiple macroeconomic factors.
Arbitrage Pricing Theory Formula
In the APT model, an asset's or a portfolio's returns follow a factor intensity structure if the returns could be expressed using this formula: ri = ai + βi1 * F1 + βi2 * F2 + ... + βkn * Fn + εi, where ai is a constant for the asset; F is a systematic factor, such as a macroeconomic or company-specific factor; β is the sensitivity of the asset or portfolio in relation to the specified factor; and εi is the asset's idiosyncratic random shock with an expected mean of zero, also known as the error term.
The APT formula is E(ri) = rf + βi1 * RP1 + βi2 * RP2 + ... + βkn * RPn, where rf is the risk-free rate of return, β is the sensitivity of the asset or portfolio in relation to the specified factor and RP is the risk premium of the specified factor.
Key Differences
At first glance, the CAPM and APT formulas look identical, but the CAPM has only one factor and one beta. Conversely, the APT formula has multiple factors that include non-company factors, which requires the asset's beta in relation to each separate factor. However, the APT does not provide insight into what these factors could be, so users of the APT model must analytically determine relevant factors that might affect the asset's returns. On the other hand, the factor used in the CAPM is the difference between the expected market rate of return and the risk-free rate of return.
Since the CAPM is a one-factor model and simpler to use, investors may want to use it to determine the expected theoretical appropriate rate of return rather than using APT, which requires users to quantify multiple factors.
The CAPM lets investors quantify the expected return on investment given the risk, risk-free rate of return, expected market return, and the beta of an asset or portfolio.
The arbitrage pricing theory is an alternative to the CAPM that uses fewer assumptions and can be harder to implement than the CAPM.
While both are useful, many investors prefer to use the CAPM, a one-factor model, over the more complicated APT, which requires users to quantify multiple factors.
Journal of Economic Perspectives. "The Capital Asset Pricing Model." Accessed April 9, 2020.
Knowledge@Wharton. "How the Father of Arbitrage Pricing Theory Influenced Wall Street." Accessed April 9, 2020.
How is the capital asset pricing model (CAPM) represented in the security market line (SML)?
CAPM Model: Advantages and Disadvantages
Tools for Fundamental Analysis
Is Apple's Stock Over Valued Or Undervalued?
Catch on to the CCAPM
Alpha vs. Beta: What's the Difference?
Understanding Arbitrage Pricing Theory
Arbitrage pricing theory is a pricing model that predicts a return using the relationship between an expected return and macroeconomic factors.
Consumption Capital Asset Pricing Model (CCAPM)
The consumption capital asset pricing model (CCAPM) is an extension of the capital asset pricing model but one that uses consumption beta instead of market beta.
International Beta Definition
International beta (often known as "global beta") is a measure of the systematic risk of a stock or portfolio in relation to a global market.
Excess Returns Definition
Excess returns are returns achieved above and beyond the return of a proxy. Excess returns will depend on a designated investment return comparison for analysis.
International Capital Asset Pricing Model (CAPM)
The international capital asset pricing model (CAPM) is a financial model that extends the concept of the CAPM to international investments.
What Is the Fama and French Three Factor Model?
The Fama and French Three-Factor model expanded the CAPM to include size risk and value risk to explain differences in diversified portfolio returns. | {
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Qene Games Partners with Ethio Telecom
by vbassey in Business, Home, News, Press Releases
2 min read – April 13, 2022
Qene Games, in partnership with Ethio Telecom, will make high-quality mobile games available to the Ethiopian market by utilizing the service providers fintech solutions.
The partnership between Qene Games and Ethio telecom will be a prominent achievement for the Ethiopian gaming sector. The partnership ultimately aims to solve the gaming sector's primary challenge – payment options. Using Ethio telecom's fintech solutions, Qene will make its games available through various models, including subscriptions and in-app purchases.
"We believe that Africa has a great potential to become a major games exporter and compete in the global creative and entertainment industry. However, the first step we need to take to make this a reality is to give African creators easy access to sell their content in the African market. We are happy to have taken a step forward in this direction with our recent partnership with Ethio Telecom." says Dawit Abraham, CEO of Qene Games.
The partnership will kick off using Qene's first-ever mobile game, Kukulu. The game is being localized into more than four Ethiopian languages and exhibits culturally revelant characters and settings that Ethiopian players find relevant. Following this launch, Qene will work to make its games library and more locally and continentally sourced high-quality games available to the Ethiopian market.
This collaboration's long-term vision is to boost the number of African games published in the continent and grow the industry by providing easy access to market for talented game creators. "Through this and coming partnerships, we want to provide African countries with enjoyable and relatable content, while at the same time showing young people in Africa that the gaming industry is a promising and financially viable sector to get into." says Samuel Sisay, CTO and lead developer at Qene Games.
Qene Games also aims to utilize its membership in the Pan-African Gaming Group (PAGG) to realize this vision. Launched in February 2022, the Pan African Gaming Group brought ten top African gaming studios as one to unify the continent's gaming sector. This group currently has access to fifty mobile games from its games library. | {
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\section{Introduction}
A line junction (LJ) \cite{renn,oreg,kane1,mitra,kollar,kim,zulicke,papa,das}
separating two edges of fractional quantum Hall (QH) states
allows the realization of one-dimensional systems of interacting electrons
for which the Luttinger parameter can be tuned \cite{gogolin,giamarchi1,rao}.
A LJ is formed by using a gate voltage to create a narrow barrier which
divides a fractional QH state such that there are two chiral edges flowing in
opposite directions (counter propagating) on the two sides of the barrier
\cite{kang,yang1,yang2,roddaro1,roddaro2}. For a QH system corresponding to
a filling fraction which is the inverse of an odd integer such as $1, 3, 5,
\cdots$, the edge consists of a single mode which can be described by a
chiral bosonic theory \cite{wen1}. In a system with two QH states separated
by a line junction, the edges on the two sides of the LJ generally
interact with each other through a short-range density-density interaction
(screened Coulomb repulsion); such an interaction can be treated exactly in
the bosonic language. The physical separation between the two edges and,
therefore, the strength of the interaction can be controlled by a gate
voltage. In general, a LJ also allows tunneling between the two edges; if the
LJ is disordered, the tunneling amplitude is taken to be a random variable.
The tunneling amplitude is also dependent on the separation between the edges.
Recently, QH systems with a sharp bend of $90^0$ have been
fabricated \cite{grayson1,grayson2}. An application of an appropriately
tilted magnetic field in such a system can produce QH states on the two
faces which have different filling fractions $\nu_1$ and $\nu_2$, since
the filling fractions are governed by the components of the magnetic field
perpendicular to the faces. The two perpendicular components can even have
opposite signs if the magnetic field is sufficiently tilted. Depending on
whether $\nu_1$ and $\nu_2$ have the same sign or opposite signs, the edge
states on the two sides of the line separating the two QH states may
propagate in opposite directions or in the same direction; we call these
counter or co-propagating edges respectively. A QH system with a bend
therefore provides a new kind of LJ in which the filling fractions can be
different on the two sides of the LJ, and the two edges can be co-propagating.
In an earlier paper \cite{sen}, we developed a microscopic model for the
direct current (DC) conductivity of a finite length LJ with either counter
or co-propagating edges. The conductivity is expressed by a current
splitting matrix $S_{dc}$ which depends on the filling fractions $\nu_1$ and
$\nu_2$, the choice of current splitting matrices which provide boundary
conditions for the bosonic fields at the two ends of the LJ, the
tunneling conductance per unit length $\sigma$, and the length
$L$ of the LJ. The Coulomb interaction between the edges was ignored in the
calculation of the DC conductivity, but the effect of the interaction was
then taken into account to study the renormalization group flow of the
tunneling conductance and therefore the conductivity.
In this paper, we will generalize the results of Ref. \cite{sen} to
find the alternating current (AC) conductivity along a LJ; the inter-edge
interactions will be taken into account in this calculation. In the limits
in which the AC frequency $\omega$ goes to zero, we will recover the
results obtained in Ref. \cite{sen}.
The paper is organized as follows. In Sec. II, we introduce a microscopic
model for the LJ and discuss the general form of the current splitting matrix
$S$ for both DC and AC. We obtain the condition for zero power dissipation.
We also discuss the possible boundary conditions which can be imposed
at the ends of the line junction; if we require that the commutation
relations of the incoming and outgoing bosonic fields be preserved,
we find that the boundary conditions must take one of two forms, which are
described by matrices $S_0$ and $S_1$. In Sec. III, we introduce short-range
interactions (whose strength is given by a parameter $\lambda$) and a
local tunneling conductance (denoted by $\sigma$) between
the edges of the line junction. We then discuss the case of counter
propagating edges and present the frequency dependent matrix $S_{ac}$
for some simple choices of the filling fractions and velocities of the edge
modes. In Sec. IV, we discuss the case of co-propagating edges and present
the matrix $S_{ac}$, again for some simple cases. We present some plots of
the elements of $S_{ac}$ as functions of the AC frequency $\omega$. In Sec. V,
we discuss the implications of a renormalization group analysis for the low
temperature behavior of $S_{ac}$, and how this may be checked experimentally.
Section VI summarizes our results and discusses possible extensions of our
work. In the Appendix we present the details of the calculations for the
general case of arbitrary filling fractions and velocities of the edge modes.
\section{Model for the line junction}
We consider a LJ with two different QH liquids on the two sides. The edges
of the QH liquids on the two sides of the LJ are assumed to be spatially
close to each other; hence there are density-density interactions
between the two edges, and electrons can also tunnel between the edges.
For simplicity, we will assume that the interaction strength and the
tunneling conductance have the same magnitude at all points along the LJ.
We will also assume that the incoming and outgoing fields connect
continuously to the corresponding fields at each end of the LJ.
Consider the counter propagating (co-propagating) LJ systems shown in Figs.
1 (a) and (b) respectively.
The currents (voltages) in the two incoming edges are denoted as $I_1$ ($V_1$)
and $I_2$ ($V_2$), and in the two outgoing edges as $I_3$ ($V_3$) and $I_4$
($V_4$). Here edges $1$ and $3$ correspond to a QH system with filling
fraction $\nu_1$, while edges $2$ and $4$ correspond to a system with filling
fraction $\nu_2$. We also assume that the QH edge modes are locally
equilibrated; discussions of equilibration at zero frequency have been
presented in Refs. \cite{kane2,buttiker}. Namely, at each point $x$, which may
lie either on one of the outer edges 1-4 or inside the line junction (where
$x$ goes from 0 to $L$), we assume, for small bias, that
\begin{equation} I_{i} (x,t) ~=~ \frac{e^2}{h} ~\nu_i ~V_{i} (x,t). \label{iv} \end{equation}
In the linear response regime (when the applied voltage bias is small),
we expect the outgoing currents to be
related to the incoming ones in a linear way. Let us denote the alternating
current on edge $i$ by $I_i = \alpha_i e^{i(k_i x - \omega t)} + c.c.$, where
$\alpha_i$ is a complex number in general. The numbers $\alpha_i$
are related by a current splitting matrix $S_{ac} (\omega)$ as
\begin{eqnarray} \left( \begin{array}{c} \alpha_3 \\
\alpha_4 \end{array} \right) &=& S_{ac} ~\left( \begin{array}{c} \alpha_1 \\
\alpha_2 \end{array} \right), \nonumber \\
{\rm where}~~~~ S_{ac} &=& \left( \begin{array}{cc} r(\omega) & \bar{t}(\omega) \\
t(\omega) & \bar{r}(\omega) \end{array} \right). \label{sac} \end{eqnarray}
When all the edge states are in equilibrium, the power dissipated is given by
the difference of the incoming and outgoing powers, namely,
\begin{eqnarray} \label{P1}
P &=& \frac{1}{2} ~[I_1 V_1 ~+~ I_2 V_2 ~-~ I_3 V_3 ~-~ I_4 V_4]. \end{eqnarray}
If there is no power dissipation in the system (we will see below that this is
true if the tunneling conductance is zero all along the LJ), then the average
over one oscillation cycle of the incoming energy must be equal to the
outgoing energy. This imposes the following condition
\begin{equation} S_{ac}^\dagger ~\left( \begin{array}{cc} 1/\nu_1 & 0 \\
0 & 1/\nu_2 \end{array} \right) ~S_{ac} ~=~ \left( \begin{array}{cc}
1/\nu_1 & 0 \\
0 & 1/\nu_2 \end{array} \right), \label{cons1} \end{equation}
or, more explicitly,
\begin{eqnarray} \frac{|r(\omega)|^2}{\nu_1} ~+~ \frac{|t(\omega)|^2}{\nu_2} &=&
\frac{1}{\nu_1}, \nonumber \\
\frac{|\bar{r}(\omega)|^2}{\nu_2} ~+~ \frac{|\bar{t}(\omega)|^2}{\nu_1} &=&
\frac{1}{\nu_2}, \nonumber \\
{\rm and} ~~~~\frac{r^* (\omega) \bar{t} (\omega)}{\nu_1} ~+~ \frac{t^* (\omega)
\bar{r}(\omega)}{\nu_2} &=& 0. \end{eqnarray}
When the incoming currents are DC in nature, the currents satisfy a linear
relation and are related by a current splitting matrix $S_{dc}$. This is a
real matrix which can be characterized by a single parameter $\gamma$ (called
the scattering coefficient); it has the general form \cite{wen2,halperin,sen}
\begin{equation} \left( \begin{array}{c} I_3 \\ I_4 \end{array} \right) ~=~
\left( \begin{array}{cc} 1 - \frac{2 \gamma \nu_2}{\nu_1+\nu_2} &
\frac{2 \gamma \nu_1}{\nu_1+\nu_2} \\ \frac{2 \gamma \nu_2}{\nu_1+\nu_2} &
1-\frac{2 \gamma \nu_1}{\nu_1+\nu_2} \end{array}
\right) \left( \begin{array}{c} I_1 \\
I_2 \end{array} \right). \label{sdc} \end{equation}
The power dissipated is the difference of the incoming and outgoing energy
flux (\ref{P1}), and it is given by
\begin{equation} P ~=~ \frac{e^2}{h} ~\frac{2\nu_1 \nu_2}{\nu_1 + \nu_2} ~\gamma (1 - \gamma)~
(V_1 - V_2)^2. \end{equation}
The condition that $P \ge 0$ requires that $0 \le \gamma \le 1$. No power is
dissipated if $\gamma = 0$ or $1$, and maximum power dissipation occurs when
$\gamma = 1/2$.
\begin{figure}[t] \begin{center}
\includegraphics[width=1.0 \linewidth]{fig1.ps} \end{center}
\caption{(Color online) Schematic picture of a line junction with (a) counter
propagating and (b) co-propagating edges, with two incoming and two outgoing
edges.} \end{figure}
The end points of the LJ shown in Fig. 1 lie at $x=0$ and $L$. At each of
these ends, we have two incoming edges and two outgoing edges; two of these
correspond to the outer edges marked $I_1$, $I_2$, $I_3$, and $I_4$, while
the other two edges are internal to the LJ and are marked $J_1$ and $J_2$.
An important ingredient of the model for such a system is the
boundary condition which should be imposed at the end points. In Ref.
\cite{sen}, it was shown that there are two possible boundary conditions
which can be imposed at each end; both of these allow us to smoothly connect
the bosonic fields which may be used to calculate the currents in the system.
The two possible boundary conditions correspond to using one of two matrices
$S_0$ or $S_1$ to related the incoming and outgoing modes at each end, where
\begin{eqnarray} S_0 &=& \left( \begin{array}{cc} 1 & 0 \\
0 & 1 \end{array} \right), \nonumber \\
{\rm and} ~~~ S_1 &=& \frac{1}{\nu_1 + \nu_2} ~\left( \begin{array}{cc}
\nu_1 - \nu_2 & 2 \nu_1 \\
2 \nu_2 & \nu_2 - \nu_1 \end{array} \right). \label{s01} \end{eqnarray}
In this paper, we will only consider the boundary condition corresponding to
the matrix $S_0$; this is physically the more plausible boundary condition
since it just connects the incoming edge to the outgoing one for each
QH liquid separately.
\section{Counter propagating case}
We will now present a microscopic model of a LJ for the case of counter
propagating edges. For simple filling fractions $\nu_i$ given by the inverse
of an odd integer, each edge is associated with a single chiral boson mode.
For the counter propagating case, shown in Fig. 1 (a), the mode on one edge
propagates from $x=0$ to $x=L$, while the mode on the other edge propagates
in the opposite direction; let us call the corresponding bosonic fields as
$\phi_1$ (right mover) and $\phi_2$ (left mover) respectively. In the absence
of density-density interactions between these edges, the Lagrangian is given by
\begin{eqnarray} {\cal L} &=& \frac{1}{4\pi \nu_1} ~\int_0^L dx ~\partial_x \phi_1~ (- \partial_t -
v_1 \partial_x) ~\phi_1 \nonumber \\
& & + ~\frac{1}{4\pi \nu_2} ~\int_0^L dx ~\partial_x \phi_2 ~(\partial_t - v_2 \partial_x) ~
\phi_2 , \label{lag1} \end{eqnarray}
where $v_i$ denotes the velocity of mode $i$. The density and current fields
are defined as
\begin{eqnarray} \begin{array}{cccc}
\rho_1 ~=& \partial_x \phi_1/(2\pi), ~~& J_1 ~=& ~ -\partial_t \phi_1/(2\pi), \\
\rho_2 ~=& - ~\partial_x \phi_2/(2\pi), ~~& J_2 ~=& \partial_t \phi_2/(2\pi).
\end{array} \end{eqnarray}
For a short-range density-density interaction between the two edges, the term
in the Lagrangian is of the form
\begin{eqnarray} {\cal L}_{int} &=& \frac{\lambda}{4\pi \sqrt{\nu_1 \nu_2}} ~\int_0^L
dx ~\partial_x \phi_1~ \partial_x \phi_2, \label{lag2} \end{eqnarray}
where $\lambda$ is the interaction strength (positive for repulsive interactions)
with the dimensions of velocity.
The equations of motion for the Lagrangian given in Eqs. (\ref{lag1}) and
(\ref{lag2}), written in terms of the density and the current fields, are
\begin{eqnarray} J_1 ~-~ v_1 \rho_1 ~-~ \frac{\lambda \nu_1}{2 \sqrt{\nu_1 \nu_2}}~
\rho_2 ~=~ 0, \nonumber \\
J_2 ~+~ v_2 \rho_2 ~+~ \frac{\lambda \nu_2}{2 \sqrt{\nu_1 \nu_2}}~ \rho_1 ~=~
0. \label{eqm1} \end{eqnarray}
A model of tunneling at zero frequency between different edges or point
contacts in a QH system has been developed in Ref. \cite{kane2}. By adding
a time derivative term in their expressions, we can model tunneling at finite
frequencies between the two edges along the LJ using the following equations
\begin{eqnarray} \partial_t \rho_1 ~+~ \partial_x J_1 &=& \frac{\sigma h}{e^2}\left(\frac{J_2}{\nu_2}
-\frac{J_1}{\nu_1} \right), \nonumber \\
\partial_t \rho_2 ~+~ \partial_x J_2 &=& \frac{\sigma h}{e^2}\left(\frac{J_2}{\nu_2}
-\frac{J_1}{\nu_1} \right), \label{eqm2} \end{eqnarray}
where $\sigma$ is the conductance per unit length across the LJ. Physically, Eq.
(\ref{eqm2}) is the continuity equations for the edge states with a source
term \cite{kane2}, the source term being the current tunneling into the system
because of the voltage difference between the corresponding points on the line
junction, $I_{source}=\sigma (V_2-V_1)=(\sigma h/e^2) (J_2 /\nu_2 - J_1 /\nu_1)$.
We will assume $\sigma$ to be constant along the LJ. Unlike Eq. (\ref{eqm1}),
the model of tunneling given in Eq. (\ref{eqm2}) cannot be derived from any
Lagrangian since it is non-unitary, and a non-zero value of $\sigma$ implies
that there is dissipation in the system.
For the DC case, in a non-interacting system, the current splitting matrix
is given by Eq. (\ref{sdc}), and $\gamma$ is given by \cite{sen}
\begin{equation} \label{tdc1}
\gamma = \frac{\nu_1+\nu_2}{2} \frac{1-e^{-L/l_c}}{\nu_2 -\nu_1e^{-L/l_c}} ~~~
{\rm and} ~~~ l_c^{-1} = \frac{\sigma h}{e^2} \left( \frac{1}{\nu_1} -
\frac{1}{\nu_2} \right) \end{equation}
when $\nu_1 \neq \nu_2$. For the special case \cite{kane1} of $\nu_1 = \nu_2
= \nu$, we obtain $\gamma = [1 + \nu e^2/(\sigma L h)]^{-1}$.
Now we solve the problem for the general case with interactions and for an
arbitrary value of $\omega$. We can combine Eqs. (\ref{eqm1}) and (\ref{eqm2})
to obtain
\begin{eqnarray} \label{eqm3}
\left(\partial_t+v_1 \partial_x + \frac{\alpha}{\nu_1}\right)J_1 &+& \left( \frac{\lambda
\nu_1}{2 \sqrt{\nu_1\nu_2}} \partial_x - \frac{\alpha}{\nu_2} \right) J_2=0, \nonumber \\
\left(\partial_t-v_2 \partial_x + \frac{\beta}{\nu_2}\right)J_2 &-& \left( \frac{\lambda
\nu_2}{2 \sqrt{\nu_1\nu_2}} \partial_x + \frac{\beta}{\nu_1}\right)J_1=0, \nonumber \\
\end{eqnarray}
where
\begin{eqnarray} \alpha &=& \frac{\sigma h}{e^2} ~(v_1+\frac{\lambda \nu_1}{2 \sqrt{\nu_1\nu_2}}),
\nonumber \\
\beta &=& \frac{\sigma h}{e^2} ~(v_2+\frac{\lambda \nu_2}{2 \sqrt{\nu_1\nu_2}}). \end{eqnarray}
Solving these equations with appropriate boundary conditions gives us the
current splitting matrix $S_{ac}$. We will work with the boundary condition
that connects the fields along the LJ continuously to the corresponding
incoming and outgoing fields, i.e., the incoming field $I_{1/2}=J_{1/2}(0)$
at $x=0$ and $J_{1/2}(L) = I_{3/4}$ at $x=L$. The most general case will
have $\nu_1 \neq \nu_2 $ and $v_1 \neq v_2$. We solve Eq. (\ref{eqm3}) in its
most general form in the Appendix and present the matrix $S_{ac}$.
In this section we present results for some relatively simple cases.
For the case of LJ with interactions but no tunneling ($\sigma=0$), same filling
fraction ($\nu_1=\nu_2=\nu$) and same velocity ($v_1=v_2=v$), we find that
\begin{eqnarray} t(\omega) &=& \bar{t}(\omega) ~=~ - ~\frac{\lambda \sin (k L)}{2 \left[
i \tilde{v} \cos (k L) + v \sin (k L) \right]}, \nonumber \\
r(\omega) &=& \bar{r}(\omega) ~=~ \frac{i \tilde{v}}{i\tilde{v} \cos (k L) + v
\sin (k L)}, \label{trom} \end{eqnarray}
where $k=\omega/{\tilde v}$ and ${\tilde v}=\sqrt{v^2-\lambda^2/4}$.
Note that there is no dissipation in this case, and the AC current
splitting matrix satisfies Eq. (\ref{cons1}). We also note that this
solution is consistent with Eq. (11) of Ref. \cite{oreg} in the limit
of $\lambda << v$, (our $r(\omega)$ corresponds to their $t(\omega)$). Similar
expressions have also appeared in Refs. \cite{safi2,safi3}. Also note that
in the DC limit ($\omega \to 0$), $|r(\omega)|=1$. This implies that conductance
across the LJ, $G=\nu e^2/h$. This is expected and is consistent with
Ref. \cite{maslov,ponomarenko,safi1,safi4}.
For the case of same $\nu$'s and same velocities but with the tunneling $\sigma$
switched on, we have
\begin{eqnarray} t(\omega) =& \frac{2 \left(k^2 v^2 \nu ^2+(\alpha -i \nu \omega )^2
\right)}{k \nu (2 v \alpha +\lambda (\alpha -i \nu \omega )) \cot (k L) +
\left(-k^2 v \lambda \nu^2 +2 \alpha (\alpha -i \nu \omega ) \right)}, \nonumber \\
r(\omega) =& \frac{1}{\cos (k L)+\frac{\left(-k^2 v \lambda \nu^2 + 2 \alpha
(\alpha -i \nu \omega) \right) \sin (k L)}{k \nu ( 2 v \alpha +\lambda
(\alpha -i \nu \omega ))}}, \end{eqnarray}
where $\alpha = (\sigma h /e^2) \left (v+\lambda/2 \right)$, and
\begin{equation} k ~=~ \frac{\omega}{\sqrt{v^2-\lambda^2/4}} ~\left(1+\frac{2 i \sigma h}
{\nu e^2 \omega}(v+\lambda/2)\right)^{1/2}. \label{kom} \end{equation}
The expression for the most general case is given in the Appendix in Eq.
(\ref{KA3}) and (\ref{TRA5}). In the DC limit $\omega \to 0$, Eq. (\ref{KA3})
gives $k_1 \to il_c^{-1} = i(\sigma h /e^2) (\nu_1^{-1} - \nu_2^{-1})$ and
$k_2 \to 0$, while Eq. (\ref{TRA5}) gives
\begin{equation} \label{r1}
r(\omega \to 0) ~=~ \frac{\nu_1 - \nu_2}{\nu_1 - \nu_2 ~ e^{L/l_c}}. \end{equation}
Comparing with the expression in Eq. (\ref{sdc}), we get the same value of
$\gamma$ as in Eq.(\ref{tdc1}).
In Fig. 2, we show the absolute values of the various reflection and
transmission amplitudes as functions of the frequency $\omega$ (in units of
$v_1/L$ which has been set equal to unity) for various choices of the filling
fractions $\nu_i$, velocities $v_i$, length $L$, interaction $\lambda$ (in units
of $v_1 = 1$), and tunneling conductance per unit length $\sigma$ (in units of
$e^2/(hL)$). Figs. 2 (a) and (b) show the cases of
$\sigma = 0$ (zero tunneling) for equal filling fractions and different filling
fractions respectively; in Fig. 2 (a), $|r| = |\bar r|$ and $|t| = |\bar t|$
by symmetry. In both figures, we see prominent oscillations as a function of
$\omega$. This is clear from Eq. (\ref{trom}) where we see that $k$ is real and
the different amplitudes oscillate with a wavelength $2\pi /k$. In the $\omega
\to 0$ limit, $|r| = |\bar{r}| = 1$ for both cases, as is evident from Eq.
(\ref{r1}). Both these cases are dissipationless and the amplitudes satisfy
Eq. (\ref{cons1}). In Fig. 2 (b), the curves for $|r|$ and $|\bar r|$ coincide
for all $\omega$; this can be shown to hold if $\sigma =0$, no matter what the
filling fractions and velocities are. Figs. 2 (c) and (d) show the cases of
$\sigma \ne 0$ for equal filling fractions and different filling fractions
respectively; we have assumed for simplicity that $\sigma$ itself does not
depend on $\omega$. In this case, $k$ is complex as shown in Eq. (\ref{kom}) and
(\ref{KA3}); the imaginary part of $k$ remains finite for large $\omega$. Hence
the different amplitudes show oscillations but they also decay as $\omega$
increases. In the $\omega \to 0$ limit, $r$ is given by Eq. (\ref{r1}).
Note that we have treated the fully interacting problem in this section,
but in the DC limit $\omega \to 0$, we recover the results for the
non-interacting case given in Ref. \cite{sen}. This is because in that
limit, the terms involving $\partial_t \rho$ vanish in Eq. (\ref{eqm2}). The
currents $J_i (x)$ can then be found from Eq. (\ref{eqm2}) alone, and Eq.
(\ref{eqm1}) becomes unnecessary. The values of the currents therefore
do not depend on the interaction parameter $\lambda$ and the velocities $v_i$
appearing in Eq. (\ref{eqm1}). However, for the AC case, Eq. (\ref{eqm1})
and (\ref{eqm2}) are both required to find the currents, and the results
are different for the interacting and non-interacting cases in general. It
is also interesting to note that if there is no tunneling ($\sigma = 0$), the
corrections to lowest order in the AC frequency for the reflection and
transmission amplitudes in Eq. (\ref{trom}) are of order $w$, but if there is
tunneling ($\sigma \ne 0$), the lowest order corrections are of order
$\omega^{1/2}$. This follows from Eq. (\ref{kom}) which shows that for small
$\omega$, $k \sim \omega$ if $\sigma = 0$, but $k \sim \omega^{1/2}$ if $\sigma \ne 0$.
\begin{figure}[t] \begin{center}
\includegraphics[width=1.0 \linewidth]{fig2.eps} \end{center}
\caption{(Color online) Absolute values of AC current splitting amplitudes as
functions of frequency $\omega$ for a counter propagating line junction, for
interaction $\lambda = 0.5$. Figures (a) and (b) correspond to tunneling $\sigma =
0$, while (c) and (d) are for $\sigma = 0.5$. The units of $\omega$, $\lambda$ and $\sigma$
are explained in the text.} \end{figure}
\section{Co-propagating case}
In the absence of density-density interactions between the co-propagating
modes, the Lagrangian is given by
\begin{eqnarray} {\cal L} &=& \frac{1}{4\pi \nu_1} ~\int_0^L dx ~\partial_x \phi_1~ (- \partial_t -
v_1 \partial_x) ~\phi_1 \nonumber \\
& & + ~\frac{1}{4\pi \nu_2} ~\int_0^L dx ~\partial_x \phi_2 ~(-\partial_t - v_2 \partial_x) ~
\phi_2. \label{lag4} \end{eqnarray}
Here both the edge modes are taken to be propagating from $x=0$ to $L$. The
corresponding density and current fields are defined as $\rho_{1/2}= \partial_x
\phi_{1/2}/(2\pi)$ and $J_{1/2}= ~ -\partial_t \phi_{1/2}/(2\pi)$. The short-range
repulsive density-density interaction between the two edges takes the form
\begin{eqnarray} {\cal L}_{int} &=& - ~\frac{\lambda}{4\pi \sqrt{\nu_1 \nu_2}} ~\int_0^L
dx ~\partial_x \phi_1~ \partial_x \phi_2, \label{lag5} \end{eqnarray}
where $\lambda$ is the interaction strength (positive for repulsive interactions)
with the dimensions of velocity.
The equations of motion for the Lagrangian corresponding to Eq. (\ref{lag4})
and (\ref{lag5}), written in terms of the density and the current fields, are
\begin{eqnarray} J_1 ~-~ v_1 \rho_1 ~-~ \frac{\lambda \nu_1}{2 \sqrt{\nu_1 \nu_2}}~
\rho_2 &=& 0, \nonumber \\
J_2 ~-~ v_2 \rho_2 ~-~ \frac{\lambda \nu_2}{2 \sqrt{\nu_1 \nu_2}}~ \rho_1 &=&
0. \label{eqm4} \end{eqnarray}
The tunneling along the LJ will be modeled using the following equations
\begin{eqnarray} \partial_t \rho_1 ~+~ \partial_x J_1 &=& \frac{\sigma h}{e^2}\left(\frac{J_2}{\nu_2}
-\frac{J_1}{\nu_1} \right), \nonumber \\
\partial_t \rho_2 ~+~ \partial_x J_2 &=& - \frac{\sigma h}{e^2}\left(\frac{J_2}{\nu_2}
-\frac{J_1}{\nu_1} \right). \label{eqm5} \end{eqnarray}
We can combine Eq. (\ref{eqm4}) and (\ref{eqm5}) to give
\begin{eqnarray} \label{eqm6}
\left(\partial_t+v_1 \partial_x + \frac{\alpha}{\nu_1}\right) J_1 &+& \left( \frac{\lambda
\nu_1}{2 \sqrt{\nu_1\nu_2}} \partial_x - \frac{\alpha}{\nu_2} \right) J_2 ~=~0, \nonumber \\
\left( \partial_t +v_2 \partial_x + \frac{\beta}{\nu_2} \right) J_2 &+&
\left( \frac{\lambda \nu_2}{2 \sqrt{\nu_1\nu_2}} \partial_x - \frac{\beta}{\nu_1}
\right) J_1 ~=~ 0, \nonumber \\
& & \end{eqnarray}
where
\begin{eqnarray} \alpha &=& \frac{\sigma h}{e^2} ~(v_1-\frac{\lambda \nu_1}{2 \sqrt{\nu_1\nu_2}}),
\nonumber \\
\beta &=& \frac{\sigma h}{e^2} ~(v_2-\frac{\lambda \nu_2}{2 \sqrt{\nu_1\nu_2}}). \end{eqnarray}
Solving these equations with the appropriate boundary conditions gives us the
current splitting matrix.
For the DC case, in a non-interacting system, the current splitting matrix
is given by Eq. (\ref{sdc}), and $\gamma$ is given by \cite{sen}
\begin{equation} \gamma = \frac{1 - e^{-L/l_c}}{2} ~~~ {\rm and} ~~~ l_c^{-1} = \frac{\sigma h}{
e^2} \left( \frac{1}{\nu_1}+\frac{1}{\nu_2} \right). \label{tdc2} \end{equation}
We now turn to the AC case. For the simplest case of same filling fraction
($\nu_1=\nu_2=\nu$), same velocity ($v_1=v_2=v$)and and no tunneling ($\sigma=0$),
we obtain
\begin{eqnarray} t(\omega) =& \frac{2 \left(e^{i L k_1}-e^{i L k_2} \right) \left(\omega - v
k_1\right) \left(\omega -v k_2\right)}{\lambda \omega \left(k_1-k_2 \right)}, \nonumber \\
r(\omega) =& \frac{\left(\omega -v k_2\right) k_1 e^{-i L k_1} - \left(\omega -v k_1
\right)k_2 e^{-i L k_2} }{\omega \left(k_1-k_2\right)}, \end{eqnarray}
where $k_1=\omega/(v-\lambda/2)$ and $k_2=\omega/(v+\lambda/2)$. Note that this is the
dissipationless case and the amplitudes satisfy Eq. (\ref{cons1}), and in the
DC limit, $r \to 1$.
For the case of same filing fractions ($\nu_1=\nu_2=\nu$), same velocities
($v_1=v_2=v$), but $\sigma\neq 0$, we get
\begin{eqnarray} t(\omega)&=& [2 i(e^{i L k_1}-e^{iLk_2})(\alpha -i \nu (\omega -v k_1))(\alpha -i\nu
\nonumber \\
&\quad& (\omega -v k_2)) ] / [(2 v \alpha \nu +\lambda \nu(\alpha -i \nu \omega ))(k_1
-k_2)] \nonumber \\
r(\omega) &=& [(e^{i L k_2}-e^{iLk_1})\left( 2\alpha (\alpha -i\nu\omega ) + \lambda v \nu^2
k_1 k_2 \right) \nonumber \\
&\quad& + i \lambda (\alpha-i\nu\omega)(k_1 e^{i L k_1}-k_2 e^{i L k_2})+ 2 i v \alpha \nu
\nonumber \\
&\quad& (k_1 e^{i L k_2}-k_2 e^{i L k_1})]/ [ i \nu (2 \alpha v +\lambda (\alpha -
i \nu \omega )) \nonumber \\
&\quad& (k_1-k_2)]l, \end{eqnarray}
where the values of $k_{1/2}$ are given by Eq. (\ref{KA7}).
\begin{figure}[t] \begin{center}
\includegraphics[width=1.0 \linewidth]{fig3.eps} \end{center}
\caption{(Color online) Absolute values of AC current splitting amplitudes as
functions of frequency $\omega$ for a co-propagating line junction, for
interaction $\lambda = 0.5$. Figures (a) and (b) correspond to tunneling $\sigma =
0$, while (c) and (d) are for $\sigma = 0.5$. The units of $\omega$, $\lambda$ and $\sigma$
are explained in the text.} \end{figure}
The expression for the most general case is given in the Appendix in Eq.
(\ref{KA7}) and (\ref{TRA9}). In the DC limit $\omega \to 0$, Eq. (\ref{KA7})
gives $k_1 \to il_c^{-1} = i(\sigma h /e^2) (\nu_1^{-1} + \nu_2^{-1})$ and
$k_2 \to 0$, while Eq. (\ref{TRA9}) gives
\begin{equation} r(\omega \to 0) ~=~ \frac{\nu_1+\nu_2~e^{- L/ l_c}}{\nu_1+\nu_2}. \end{equation}
Comparing with the expression in Eq. (\ref{sdc}), we find the same value of
$\gamma$ as in Eq. (\ref{tdc2}).
In Fig. 3, we show the absolute values of the various reflection and
transmission amplitudes as functions of the frequency $\omega$ (in units
of $v_1/L = 1$) for various choices of $\nu_i$, $v_i$, $L$, $\lambda$ (in units
of $v_1 = 1$), and $\sigma$ (in units of $e^2/(hL)$). Figs. 3 (a)
and (b) show the case of $\sigma = 0$ (zero tunneling) for equal filling
fractions and different filling fractions respectively, while Figs. 2 (c) and
(d) show the case of $\sigma \ne 0$, again assuming that $\sigma$ does not
depend on $\omega$. As in Fig. 2, we see prominent oscillations
as a function of $\omega$ when $\sigma = 0$, and both oscillations and decay when
$\sigma \ne 0$. Just as in Fig. 2 (b), we note that the curves for $|r|$ and
$|\bar r|$ coincide for all $\omega$ in Fig. 3 (b) also.
Once again we note that we have treated the fully interacting problem here,
but in the DC limit we recover the non-interacting results given in Ref.
\cite{sen}. The reasons for this are the same as those explained at the
end of Sec. III. For the AC case, the expression for the current splitting
matrix is different for the interacting and the non-interacting cases.
\section{Renormalization group and experimental implications}
Before discussing how measurements of the AC reflection and transmission
amplitudes can provide information about the parameters of the system such
as the tunneling conductance $\sigma$, the interaction strength $\lambda$ and the
edge mode velocities $v_i$, we have to consider the renormalization group (RG)
flow of $\sigma$. For the DC case, this has been discussed in Ref. \cite{sen}.
Briefly, the tunneling amplitude is given by a term in the Hamiltonian density
\begin{equation} {\cal H}_{tun} ~=~ \xi (x) ~\psi_1^\dag (x) \psi_2 (x) ~+~ h.c.,
\label{htun} \end{equation}
where $\psi_i (x)$ denotes the electron annihilation operator at point $x$ on
edge $i$ of the LJ. The tunneling conductance $\sigma$ is proportional to
$|\xi|^2$ when $|\xi|$ is small. The presence of impurities near the LJ makes
$\xi (x)$ a random complex variable; let us assume it to be a Gaussian
variable with a variance $W$. Then $W$ satisfies an RG equation; to lowest
order (i.e., for small $\xi$), this is given by \cite{giamarchi2,kane1}
\begin{equation} \frac{dW}{d \ln l} ~=~ (3 - 2d_t) ~W, \label{rg} \end{equation}
where $l$ denotes the length scale, and $d_t$ is the scaling dimension of the
tunneling operator $\psi_1^\dag \psi_2$ appearing in Eq. (\ref{htun}). We
will present expressions for $d_t$ below for both counter and co-propagating
cases. There is also an RG equation for the interaction strength $\lambda$, but
that can be ignored if $W$ is small.
Next, let us assume that the phase de-coherence length $L_T = \hbar v/(k_B T)$
(the length beyond which electrons lose phase coherence due to thermal
smearing) is much smaller than both the length $L$ and the scattering mean
free path $L_m$ of the LJ. Successive backscattering events are then
incoherent, and quantum interference effects of disorder are absent. One can
then show \cite{kane1} that $\sigma$ scales with the temperature $T$ as
$T^{2d_t - 2}$. It therefore seems that $\sigma L \to 0$ as $T \to 0$ if
$d_t > 1$. However, it turns out that this is true only if $d_t > 3/2$, i.e.,
if $W$ is an irrelevant variable according to Eq. (\ref{rg}). If $d_t > 3/2$
(called the metallic phase), one can simultaneously have $L \gg L_T$ (which
justifies cutting off the RG flow at $L_T$ rather than at $L$), and $\sigma L
\to 0$, i.e., $LT \gg 1$ and $L T^{2d_t - 2} \to 0$, for some range of
temperatures. Further, $L_m$ scales with temperature \cite{kane1} as
$T^{2-2d_t}$ and $L_T \sim T^{-1}$. Thus throughout the metallic phase, $L_m
\gg L_T$ as $T \to 0$. But if $d_t < 3/2$ (i.e., $W$ is a relevant variable),
we have $L/L_T \sim LT \gg 1$ and $T^{2d_t - 3} \to \infty$; hence $\sigma L
\sim L T^{2d_t - 2} \to \infty$ (we call this the insulating phase).
The above analysis breaks down if one goes to very low temperatures where $L_T
\gtrsim L$ or $L_m$. In that case, the RG flow of $W$ has to be cut off at the
length scale $L$ or $L_m$, rather than $L_T$; hence $\sigma$ and therefore the
scattering coefficient $\gamma$ become independent of the temperature $T$.
The scaling dimension $d_t$ can be computed using bosonization \cite{sen}.
For the counter propagating case, we find that
\begin{eqnarray} d_t &=& \frac{1}{4K} \left[ (1+K^2) \left( \frac{1}{\nu_1} +
\frac{1}{\nu_2} \right) - \frac{2(1-K^2)}{\sqrt{\nu_1 \nu_2}} \right], \nonumber \\
{\rm where} ~~K &=& \sqrt{\frac{v_1+v_2-\lambda}{v_1+v_2+\lambda}}. \label{dtk} \end{eqnarray}
Thus $d_t$ depends on the interaction strength $\lambda$ and the velocities $v_i$.
(The stability of the system requires that $4 v_1 v_2 > \lambda^2$). For the
co-propagating case, we have
\begin{equation} d_t ~=~ \frac{1}{2 \nu_1} ~+~ \frac{1}{2 \nu_2}, \end{equation}
which is independent of $\lambda$ and the $v_i$.
We can now discuss the experimental implications of our results for the
various reflection and transmission amplitudes as a function of the
temperature. As mentioned earlier, a gate voltage can be used to control the
distance between the two edges of the LJ. Making the gate voltage less
repulsive for electrons is expected to reduce the distance between the edges;
this should increase both the strength of the density-density interactions as
well as the tunneling conductance \cite{kane1}. In this way, one may be able
to vary the scaling dimension $d_t$ across the value $3/2$ for the case of
counter propagating edges in the LJ. We then see that quite different things
should occur depending on whether the system is in the metallic phase ($d_t
> 3/2$) or in the insulating phase ($d_t < 3/2$). In the metallic phase, $\sigma
L \to 0$ as $T \to 0$; based on Figs. 2 and 3, we expect that oscillations
versus $\omega$ of the various amplitudes should become more prominent at
low temperatures. In the insulating phases, $\sigma L \to \infty$ as $T \to 0$;
Figs. 2 and 3 then indicate that the oscillations versus $\omega$ of the
different amplitudes should become less prominent at low temperatures. It
would be interesting to check this qualitative prediction experimentally.
\section{Discussion}
In this paper, we have discussed the response of a LJ separating two QH
states to an AC voltage in the linear response (or small bias) regime.
Depending on the filling fractions on the two sides of the LJ, the edges
of the LJ may be counter propagating or co-propagating. We have presented
a microscopic model for the system which includes short-range density-density
interactions and electron tunneling between the two edges. The AC response
can be described by a current splitting matrix $S_{ac}$; we have presented
expressions for this matrix in terms of the AC frequency $\omega$,
the length of the LJ, the tunneling conductance, the strength of the
interaction between the edges of the LJ, the filling fractions, and the
velocities of the modes on the two sides of the LJ. In general, the elements
of $S_{ac}$ oscillate with the frequency $\omega$; the amplitude of oscillations
depends on $\omega$ and the tunneling conductance $\sigma$ across the LJ. We
find the interesting result that the matrix $S_{ac}$ does not depend on the
interaction strength and the velocities in the DC limit $\omega \to 0$, but
does depend on those parameters for non-zero frequencies. [The fact that
the DC conductivity is independent of the interaction strength has been
observed earlier in the context of quantum wires modeled as non-chiral
Tomonaga-Luttinger liquids
\cite{safi2,safi3,maslov,ponomarenko,safi1,safi4,furusaki,rech}.]
The low temperature behaviors of the elements of $S_{ac}$ can then be
predicted based on a renormalization group analysis. In the case of counter
propagating edges, we find that depending on the interaction strength, the
system can be in either a metallic phase or an insulating phase. The two
phases exhibit quite different behaviors of $S_{ac}$ as we go to low
temperatures.
We emphasize that in the absence of any tunneling between the two edges,
our calculation is valid in the linear response (small AC amplitude) regime
and only for frequencies $\omega$ which lie within the linearization
regime for each LL wire (i.e., $\omega < v/\alpha$).
In the presence of inter-edge tunneling also, we
have assumed that the current is proportional to the potential (voltage)
at every point with no phase difference between the two (Eq. (\ref{iv})).
This is only true if $\omega$ is less than the inverse of the relaxation time
$\tau$ (for equilibration after each tunneling event).
However, in the absence of a detailed theory of equilibration at finite
frequencies, we do not know the precise form of $\tau$.
Another limitation of our calculation is that the intrinsic frequency
dependence of $\sigma$ is not known, although the RG analysis discussed in
Sec. V gives an idea of the length scale dependence of $\sigma$ arising due
to the interactions. We note that our various expressions for $t(\omega)$
and $r(\omega)$ will remain valid even if we take $\sigma$ to be frequency
dependent. However, Figs. 2 (c-d) and 3 (c-d) have been made under the
assumption that $\sigma$ does not depend on $\omega$.
Before ending, we would like to mention some other studies of QH systems
with multiple filling fractions \cite{sandler,lal,rosenow1}. It may be
interesting to extend our studies of AC response to these systems. Finally,
we note that the AC response of non-chiral Tomonaga-Luttinger liquids with
disorder has been studied earlier in some papers
\cite{maura,fogler,rosenow2,safibena}.
\section*{Acknowledgments}
A.A. thanks Yuval Oreg for suggesting this problem and CSIR, India for
financial support. We thank Sumathi Rao for comments and DST, India for
financial support under Project No. SR/S2/CMP-27/2006.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,337 |
using Meridian.Enum;
using Meridian.Utils.Helpers;
using System;
using Windows.UI.Xaml.Data;
namespace Meridian.Converters
{
public class SortTypeToStringConverter : IValueConverter
{
public object Convert(object value, Type targetType, object parameter, string language)
{
var sortType = (TracksSortType)value;
switch (sortType)
{
case TracksSortType.DateAdded:
return Resources.GetStringByKey("Toolbar_SortByDateAdded");
case TracksSortType.Title:
return Resources.GetStringByKey("Toolbar_SortByTitle");
case TracksSortType.Artist:
return Resources.GetStringByKey("Toolbar_SortByArtist");
}
return null;
}
public object ConvertBack(object value, Type targetType, object parameter, string language)
{
throw new NotImplementedException();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,479 |
from rctkdemos.demos import serve_demo, standalone
from rctk.widgets import Panel, Button, StaticText
from rctk.layouts import HBox, VBox
class Demo(object):
title = "Panel"
description = "Demonstrates the Panel control"
def build(self, tk, parent):
parent.setLayout(HBox())
p1 = Panel(tk,width=200, height=200)
p2 = Panel(tk, scrolling=True, width=200, height=200)
p2.setLayout(VBox())
p1.append(StaticText(tk, "Non-scrolling panel"))
p2.append(StaticText(tk, "Scrolling panel"))
b = Button(tk, "Click me")
p1.append(b)
def click(e):
p2.append(StaticText(tk, "Line added"))
parent.layout()
p2.scrollbottom()
b.click = click
parent.append(p1)
parent.append(p2)
Standalone = standalone(Demo)
if __name__ == '__main__':
serve_demo(Demo)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,030 |
ATLANTA (AP) — Atlanta police are reminding the public that no drones are allowed around sites hosting events related to the Super Bowl.
Police spokesman Carlos Campos said in an email Sunday that there would be a zero-tolerance policy for drones flying in areas that include Mercedes-Benz Stadium, the Georgia World Congress Center, State Farm Arena, Centennial Olympic Park and the Fox Theatre.
Campos said hundreds of local, state and federal law enforcement officers will be watching for illegal drone use in the prohibited areas.
Anyone who violates the no-drone rules may be subject to criminal penalties. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,808 |
\section*{Abstract}
We study motion of a relativistic charged particle in a plane slow electromagnetic wave and background uniform magnetic field. The wave propagates normally to the background field. Under certain conditions, the resonance between the wave and the Larmor motion of the particle is possible. Capture into this resonance results in acceleration of the particle along the wave front (surfatron acceleration). We analyse the phenomenon of capture and show that a captured particle never leaves the resonance and its energy infinitely grows. Scattering on the resonance is also studied. We find that this scattering results in diffusive growth of the particle energy. Finally, we estimate energy losses due to radiation by an accelerated particle.
\section{ Introduction}\label{intro}
Motion of a charged particle in an inhomogeneous electromagnetic field can be described by a Hamiltonian nonlinear system, which cannot be solved analytically in a general case \cite{LL:field}. In such a system various resonant effects take a place \cite{Chirikov:1959, AKN}, and some of them can by described using the adiabatic approach. In particular, a charged particle in the field of an electromagnetic (or electrostatic) wave and a background magnetic field can be trapped in the potential well of the wave and accelerated along the wave front. This phenomenon is called a surfatron acceleration. The mechanism of the surfatron acceleration of charged particles is often considered for description of various plasma-physics phenomena \cite{Sagdeev66,KatsouleasDawson:1983}. Originally this mechanism was suggested for description of charged particles acceleration along the front of a shock wave \cite{Sagdeev66} and this application is still actual \cite{Kichigin:shock}. On the other hand, there are various astrophysical applications of the surfatron acceleration mechanism to problems of generation of high energy particles \cite{Erokhin89, Eliasson, Wang07, Dieckmann05} and consequent radiation \cite{Zaslavskii86, Bulanov86, Bulanov00}. Surfatron acceleration of relativistic particles was considered, for example, in \cite{Chernikov:1992,Itin:2000}. In all these papers authors consider a particle interaction with an electrostatic wave.
Surfatron acceleration of a particle by an electromagnetic wave is less studied. The analytical theory was constructed only for nonrelativistic \cite{AVNZ:2009} or ultrarelativistic \cite{Chernikov:1992,Itin:2002} particles. The effect of large particle velocity was estimated numerically in \cite{Takeuchi87,Itin:2002}. Also, numerical calculations were carried out for the case when the wave amplitude is small compared to the background magnetic field \cite{Ginet90, Karimabadi90}; in this case, the particle is accelerated by multiple scatterings on the wave. In addition, several laboratory experiments with relativistic particles and large wave amplitudes were carried out for the investigation of surfatron acceleration of charged particles by electromagnetic waves (see \cite{Yugami96} and references therein). Therefore, a complete analytic theory of relativistic charged particle captures by electromagnetic waves and the resulting acceleration is important.
Particle capture and surfatron acceleration is possible if phase velocity of the wave is smaller than the absolute value of the particle velocity (and hence, smaller than the speed of light). In this case the projection of particle velocity onto the wave vector direction can become equal to value of the phase velocity of the wave and the resonance takes a place. There are several plasma modes that can support a wave with needed properties: the magnetosonic wave with frequency close to lower-hybrid \cite{Neishtadt09}, the plasma wave with frequency close to higher-hybrid \cite{Erokhin89} or various drift modes of plasma instability \cite{Zelenyi09}. In addition, a relatively small population of trapped particles can decrease the phase velocity of a wave \cite{Krasovsky10} and establish the condition of resonance interaction.
A secondary effect of surfatron acceleration is the radiation of accelerated particles due to the oscillatory component of their motion \cite{Zaslavskii86, Zaslavskii86:jetp}. A captured particle accelerates along the wave front, and at the same time it oscillates near the minimum of the wave potential well. Due to these oscillations the particle can radiate. Estimates of this radiation were carried out in the case of an electrostatic wave. The question of particle radiation in the system with electromagnetic wave is discussed in our paper.
We study the problem of interaction of a charged particle in a uniform magnetic field with an electromagnetic wave using the theory of resonant phenomena. The study of slow passages of a Hamiltonian system through a nonlinear resonance was started in \cite{Chirikov:1959}. In the present paper we use the theory of resonant processes in Hamiltonian systems with slow and fast motions in the form developed in \cite{Neishtadt:1997,Neishtadt:1999,NV:2006} (see \cite{Neishtadt:1997} for references to preceding works). The description of scattering on resonances and captures into resonances plays a central role in this theory. Resonant phenomena arise in a variety of problems of physics, including hydrodynamics, celestial mechanics, and plasma physics. For several examples of resonant phenomena, see, e.g., \cite{Shklyar:1986,FeingoldandKadanoffandPiro:1988,Gendelman:2001,Mezic:2001,GrosfeldFriedland:2002,VWG:2008,RomKedar:2009}.
\section{ Main equations}\label{main}
We consider motion of a relativistic charged particle of mass $m$ and charge $e$ in a uniform magnetic field ${\bf B}_0 = (0,0,B_0)$ and the field of a plane linearly polarized electromagnetic wave propagating in perpendicular direction to ${\bf B}_0 $. Thus, in the Cartesian coordinates $(\hat q_1,\hat q_2,\hat q_3)$ the resulting magnetic field components are $B_1 = B_2 = 0, \, B_3 = -B_w \sin (\hat k\hat q_1 - \hat \omega \hat t) + B_0$, where $B_w$ is the amplitude of the magnetic field of the wave, $\hat k$ is the magnitude of the wave vector directed along the $\hat q_1$-axis, and $\hat \omega$ is the wave frequency. The corresponding vector potential can be chosen as
\begin{equation}
{\bf A} = \left(0, B_0 \hat q_1 + \frac{B_w}{\hat k}\cos(\hat k\hat q_1 - \hat \omega \hat t) ,0 \right).
\label{2.1}
\end{equation}
Let $\hat p_i , \, i=1,2,3$ be components of the particle's momentum. Introduce
\begin{equation}
{\cal P}_2 = \hat p_2 +\frac{e}{c}B_0 \hat q_1 + \frac{eB_w}{c\hat k}\cos(\hat k \hat q_1 - \hat \omega \hat t).
\label{2.1a}
\end{equation}
The Hamiltonian of the system is:
\begin{equation}
\hat H=\sqrt{m^2 c^4 + c^2\hat p_1^2 + \left(c{\cal P}_2 - eB_0\hat q_1 - \frac{eB_w}{\hat k}\cos(\hat k\hat
q_1 - \hat \omega \hat t)\right)^2 + c^2\hat p_3^2},
\label{2.2}
\end{equation}
and pairs of canonically conjugate variables are $(\hat p_1,\hat q_1),({\cal P}_2,\hat q_2),(\hat p_3,\hat q_3)$. The Hamiltonian does not contain variables $\hat q_2$ and $\hat q_3$. Thus canonically conjugate momenta ${\cal P}_2$ and $\hat p_3$ are constants of motion. We can put $\hat p_3 = 0$ (it always can be done by redefining the particle's mass) ; one can also make ${\cal P}_2 = 0$ choosing properly the origin in $\hat q_1$. Introduce Larmor frequency $\omega_L = eB_0/(mc)$ and dimensionless parameter $\varepsilon = eB_w/(m\hat kc^2)$. We assume that $\varepsilon$ is small: $0<\varepsilon\ll 1$. Use the following rescaling to make the system dimensionless:
\begin{eqnarray}
p_i = \frac{\hat p_i}{mc}, \;\; q_i = \frac{\omega_L \hat q_i}{\varepsilon c}, \nonumber \\
t = \frac{\omega_L \hat t}{\varepsilon}, \;\; H = \frac{\hat H}{mc^2}, \nonumber \\
k = \frac{\hat k c\varepsilon}{\omega_L}, \;\; \omega = \frac{\hat \omega \varepsilon}{\omega_L}. \nonumber
\end{eqnarray}
The Hamiltonian in the new variables is:
\begin{equation}
H = \sqrt{1+ p_1^2 + (\varepsilon q_1 + \varepsilon \cos(kq_1 - \omega t))^2}.
\label{2.3}
\end{equation}
Introduce new variable $U = \omega t$. Let $P_U$ be the variable, canonically conjugate to $U$. Thus we obtain a 2 d.o.f. Hamiltonian system. The Hamiltonian takes the form:
\begin{equation}
{\cal H} = \omega P_U + \sqrt{1+ p_1^2 + (\varepsilon q_1 + \varepsilon \cos(kq_1 - U))^2}.
\label{2.4a}
\end{equation}
Now we introduce the wave phase $\varphi = kq_1 - U$ as a new variable. To this end, we make canonical transform with generating function $W = pq_1 + I(kq_1 - U)$, where $I$ is a new variable canonically conjugate to $\varphi$. For the new variables $p,q,I$ we have $q = q_1, \, p = p_1 - Ik, I = -P_U$. Denote $\varepsilon q = \tilde q$. Omitting the tilde, we find for the Hamiltonian in the new variables:
\begin{equation}
{\cal H} = -\omega I + \sqrt{1 + (p+Ik)^2 + (q+\varepsilon \cos \varphi)^2},
\label{2.4}
\end{equation}
where the pairs of canonically conjugate variables are $(I,\varphi)$ and $(p,\varepsilon^{-1} q)$. This Hamiltonian can be represented in the form ${\cal H} = H_0 + \varepsilon H_1$, where
\begin{eqnarray}
H_0 = -\omega I + \sqrt{1 + (p+Ik)^2 + q^2}, \nonumber \\
\varepsilon H_1 = \varepsilon \frac{q\cos\varphi}{\sqrt{1 + (p+Ik)^2 + q^2}} + O(\varepsilon^2).
\label{2.5}
\end{eqnarray}
In the main approximation, the equations of motion are
\begin{eqnarray}
\dot I = -\frac{\partial {\cal H}}{\partial \varphi} = \varepsilon \frac{q\sin\varphi}{\sqrt{1 + (p+Ik)^2 + q^2}}
\label{2.6} \\
\dot \varphi = \frac{\partial {\cal H}}{\partial I} = -\omega + \frac{k(p+Ik)}{\sqrt{1 + (p+Ik)^2 + q^2}}
\nonumber \\
\dot p = -\varepsilon \frac{\partial {\cal H}}{\partial q} = -\varepsilon \frac{q}{\sqrt{1 + (p+Ik)^2 + q^2}}
\label{2.7} \\
\dot q = \varepsilon \frac{\partial {\cal H}}{\partial p} = \varepsilon \frac{p+Ik}{\sqrt{1 + (p+Ik)^2 + q^2}}.
\nonumber
\end{eqnarray}
In this system, variable $\varphi$ is fast (its time derivative is a value of order $\varepsilon$), and the other variables are slow (their time derivatives are of order $1$). Thus, one can average over fast phase $\varphi$ and obtain the {\it averaged system}. Motion in this system is just the Larmor rotation in the uniform magnetic field $B_0$. The averaging, however, does not describe the motion adequately near the resonance $\dot\varphi = 0$. At the resonance, projection of particle's velocity onto the $q_1$-axis equals the phase velocity of the wave. Resonance condition $\partial H_0/\partial I = 0$ defines a {\it resonance surface} in the $(p,q,I)$-space:
\begin{equation}
(p+Ik)^2(k^2 - \omega^2) = (1+q^2)\omega^2.
\label{2.8}
\end{equation}
We denote the value of $I$ on this surface as $I_r$. This is a function of variables $p,q$:
\begin{equation}
I_r (p,q) = \frac1{k} \left( \omega \sqrt{\frac{1+q^2}{k^2-\omega^2}} - p \right).
\label{2.8a}
\end{equation}
Intersection of surface (\ref{2.8}) and isoenergetic surface $H_0 = h$ defines the {\it resonance curve}. Its projection onto the $(p,q)$-plane is a hyperbola given by equation
\begin{equation}
(h-v_{\phi}p)^2 = (1+q^2)(1-v_{\phi}^2),
\label{2.9}
\end{equation}
where we introduced the dimensionless phase velocity of the wave $v_{\phi} = \omega/k$. Note that $v_{\phi}$ is always smaller than 1.
Variable $I$ is an integral of motion of the averaged system (see (\ref{2.6})) and thus an adiabatic invariant of the exact system. Far from the resonance the value of $I$ is preserved with the accuracy of order $\varepsilon$ on time intervals of order $1/\varepsilon$ (see, e.g., \cite{AKN}). The adiabatic invariance of $I$ breaks down near the resonance, where the averaging does not work properly. In a neighborhood of the resonance phenomena of {\it capture} and {\it scattering} can take place. We study dynamics of the system near the resonance in the following sections.
\section{ Motion near the resonance}\label{resonance}
To study the system near the resonance, we apply the approach formulated in \cite{Neishtadt:1999} (see also \cite{NV:2006}). Close to the surface (\ref{2.8}) the Hamiltonian can be expanded into series in $(I-I_r)$:
\begin{equation}
{\cal H} =\Lambda(p,q) +\frac12 g(p,q)(I-I_r(p,q))^2 + \varepsilon H_1|_{I=I_r} + O(|I-I_r|^3) + O(\varepsilon(I-I_r(p,q))^2).
\label{3.1}
\end{equation}
Here $\Lambda(p,q) = H_0|_{I=I_r}$ is the unperturbed Hamiltonian $H_0$ restricted onto the resonant surface (\ref{2.8}). Function $g(p,q)$ in (\ref{3.1}) is $\partial^2 H_0/\partial I^2$ restricted onto the resonant surface. It is straightforward to find
\begin{equation}
\Lambda(p,q) = pv_{\phi} + \sqrt{(1+q^2)(1-v_{\phi}^2)}, \;\; g(p,q) = k^2
\frac{(1-v_{\phi}^2)^{3/2}}{\sqrt{1+q^2}}.
\label{3.2}
\end{equation}
Introduce notation $d = q\sqrt{1-v_{\phi}^2}/\sqrt{1+q^2}$. Then
\begin{equation}
\varepsilon H_1|_{I=I_r} = \varepsilon d \cos\varphi.
\label{3.3}
\end{equation}
Now we introduce new momentum $K=I - I_r(p,q) + O(\varepsilon)$. To this end, we make a canonical transformation of variables $(I,\varphi,p,q)\mapsto (K,\bar\varphi,\bar p,\bar q)$ with generating function $W_1 = \bar p \varepsilon^{-1} q +(K + I_r(\bar p,q))\varphi$. Omitting bars, we find for the Hamiltonian in the new variables (we neglect terms of higher orders):
\begin{equation}
{\cal H} = \Lambda(p,q) + \frac12 g(p,q)K^2 +\varepsilon d \cos\varphi +\varepsilon b(p,q)\varphi,
\label{3.4}
\end{equation}
where
\begin{equation}
b(p,q) = \{I_r,\Lambda\} = \frac{\partial I_r}{\partial q}\frac{\partial \Lambda}{\partial p}-\frac{\partial I_r}{\partial p}\frac{\partial \Lambda}{\partial
q}= \frac{q}{k\sqrt{1+q^2}}\frac1{\sqrt{1-v_{\phi}^2}}.
\label{3.5}
\end{equation}
Introduce $P = K/\sqrt\varepsilon$, $\theta = t\sqrt\varepsilon$, and rescaled Hamiltonian $F = {\cal H}/\varepsilon$. The rescaled system is Hamiltonian, and pairs of canonically conjugate variables are $(P,\varphi)$ and $(p,\varepsilon^{-3/2}q)$. In the main approximation, the Hamiltonian is
\begin{equation}
F=\varepsilon^{-1}\Lambda(p,q) + F_0(P,\varphi,p,q), \;\; F_0 = \frac12 gP^2 + d\cos\varphi +b\varphi,
\label{3.6}
\end{equation}
and the equations of motion are
\begin{eqnarray}
p^\prime = -\sqrt\varepsilon \frac{\partial \Lambda}{\partial q}, \;\; q^\prime = \sqrt\varepsilon \frac{\partial \Lambda}{\partial p},
\nonumber\\
P^\prime = - \frac{\partial F_0}{\partial \varphi}, \;\; \varphi^\prime = \frac{\partial F_0}{\partial P},
\label{3.7}
\end{eqnarray}
where prime denotes derivative over $\theta$. One can see that variables $(P,\varphi)$ are fast, and variables $(p,q)$ are slow. Thus, as the first step to study this system, one can consider variation of the fast variables at fixed values of $p$ and $q$. Dynamics of the fast variables is defined by Hamiltonian $F_0$, which contains $q$ as a parameter. This is a Hamiltonian of a pendulum under the action of external torque. Consider phase portrait of this system (fast subsystem). If $d/b > 1$, there is a separatrix on the phase portrait, and, correspondingly, domain of oscillatory motion (see Fig.1). In the opposite case $d/b < 1$, there is no separatrix. Note that ratio $d/b = k(1-v_\phi^2)$ is independent of $q$. In original dimensional units $d/b = (B_w/B_0)(1-\hat v_\phi^2/c^2)$, where $\hat v_{\phi}=\hat \omega/\hat k$. We consider first the case $d/b>1$.
\begin{figure}[t]
\center\epsfig{file=schem.eps, width=0.8\textwidth}
\caption{\label{schem} Schematic view of the phase portrait of the system (\ref{3.6}) for $d/b>1$ (a) and
for $d/b<1$ (b)}
\end{figure}
It is straightforward to obtain for the area $S$ inside the separatrix on the plane $(\varphi,P)$
\begin{equation}
S = 2\int_{\varphi_s}^{\varphi_m}\left(2\frac{d}{g}\left[\cos\varphi_s + \frac{b}{d}\varphi_s - \cos \varphi -
\frac{b}{d}\varphi\right]\right)^{1/2} \mbox{d} \varphi,
\label{3.8}
\end{equation}
where $\varphi_s = \arcsin(b/d)$ and $\varphi_m$ is the root of equation $\cos\varphi +(b/d)\varphi = \cos\varphi_s +(b/d)\varphi_s$ different from $\varphi_s$. One can see that both integration limits and the expression in square brackets in (\ref{3.8}) do not depend on $q$. Thus,
\begin{equation}
S=\sqrt{\frac{d}{g}}A = \sqrt{\frac{q}{k^2 - \omega^2}}A,
\label{3.9}
\end{equation}
where $A$ is a constant independent of $q$ (and $p$).
Now take into account slow variation of $p$ and $q$ according to the first two equations in (\ref{3.7}). It follows from the expression (\ref{3.2}) for $\Lambda(p,q)$ that if $v_\phi>0$, variable $q$ grows with time. This means that area $S$ also grows in the process of motion. Therefore, phase points on the phase portrait of the fast subsystem can cross the separatrix and enter the domain of oscillations. This corresponds to a capture into resonance. The area $J$ encircled by a trajectory in the domain of oscillations on this portrait is an adiabatic invariant (it is called the {\it inner adiabatic invariant}). Thus, as $S$ monotonously grows with time, a captured particle goes deeper and deeper inside the oscillation domain and cannot leave it. This means that a particle captured into the resonance is captured forever.
\begin{figure}[t]
\center\begin{tabular}{cc} \hspace*{-10mm}\epsfig{file=fig01.eps,width=2.8in}&
\epsfig{file=fig02.eps,width=2.8in}
\end{tabular}
\caption{\label{Fig1}Left panel: trajectory of the particle in momentum space $(p_1,p_2)$, Right panel:
the same situation as in left panel with a longer time range. }
\end{figure}
Motion of a captured particle can be described as follows. In the main approximation, it moves with the {\it resonant flow} defined by Hamiltonian $\Lambda(p,q)$. The corresponding equations of motion are:
\begin{eqnarray}
\dot p = -\varepsilon \frac{\partial \Lambda}{\partial q}= \varepsilon q \,\sqrt{\frac{1-v_{\phi}^2}{1+q^2}},
\nonumber\\
\dot q = \varepsilon \frac{\partial \Lambda}{\partial p}= \varepsilon v_{\phi}.
\label{3.10}
\end{eqnarray}
It means that a captured particle moves in $\hat q_1$-direction with the wave at a speed of the wave's phase velocity. It follows from (\ref{2.1a}) and the fact that ${\cal P}_2 = 0$ that $p_2 \approx -q$, and, hence $p_2 \sim -\varepsilon v_{\phi} t$. Therefore $\hat p_2$-component of the particle's momentum varies (on average) linearly in time:
$$
\hat p_2 \sim -\frac{e}{c} B_0 \hat v_{\phi}\hat t \sim mc(\omega_L \hat v_{\phi} \hat t/c)
$$
Thus, the particle is accelerated along the wave front. This acceleration is called surfatron one. To find variation of $p_1$ in this motion, we use that $p_1 = p +Ik$ and expression (\ref{2.8a}) for $I$ on the resonant surface. Thus we obtain $p_1 = v_{\phi}\sqrt{(1+q^2)/(1-v_{\phi}^2)}$ and
\begin{equation}
\dot p_1 = \frac{\varepsilon v_{\phi}^2}{\sqrt{1-v_{\phi}^2}} \frac{q}{\sqrt{1+ q^2}},
\label{3.11}
\end{equation}
where we used the second equation in (\ref{3.10}). At $q \gg 1$ we find that $p_1$ grows with time as $\varepsilon v_{\phi}^2 t/\sqrt{1-v_{\phi}^2}$. In dimensional variables, we find that
$$
\hat p_1 \sim \frac{eB_0 \hat v_{\phi}^2}{c^2 \sqrt{1 - \hat v_{\phi}^2/c^2}} \hat t \sim mc \frac{ \hat
v_{\phi}/c}{\sqrt{1 - \hat v_{\phi}^2/c^2}} (\omega_L \hat v_{\phi} \hat t/c).
$$
For the energy of a captured particle $E= \sqrt{1 + p_1^2 +p_2^2}$ we find that it also grows linearly with time at large enough values of $q$. Namely, we have $E \sim \varepsilon v_{\phi} t/ \sqrt{1 - v_{\phi}^2}$ and, in dimensional variables, $\hat E = mc^2 E \sim mc \hat v_{\phi} \omega_L \hat t/ \sqrt{1 - v_{\phi}^2/c^2}$.
\begin{figure}[t]
\center\epsfig{file=fig03.eps, width=0.7\textwidth}
\caption{\label{Fig3} Particle's energy as a function of time. The inner panel shows the short time interval before
capture.}
\end{figure}
The captured particle also oscillates in the potential well of the wave. These oscillations correspond to motion in the oscillatory domain in Fig. 1. One can evaluate the amplitude and the frequency of the oscillations using conservation of the inner adiabatic invariant $J$. For a captured particle $J$ equals the area inside the separatrix {\it at the time when the particle crossed the separatrix}. It follows from the expression for $F_0$ in (\ref{3.6}) that for a captured particle
\begin{equation}
J = \frac{2^{3/2}}{k} \,\sqrt{\frac{q}{1-v_{\phi}^2}} \int_{\varphi_1}^{\varphi_2}\sqrt{\tilde f_0 - \cos\varphi -
\frac{\varphi}{k(1-v_{\phi}^2)}}\, \mbox{d}\varphi,
\label{3.12}
\end{equation}
where $\tilde f_0 = F_0/d$ does not depend on $\varphi$; $\varphi_1$ and $\varphi_2$ are the minimal and the maximal values of $\varphi$ on a trajectory with fixed values of $F_0$ and $q$. Thus, growth of $q$ results in decreasing of the amplitude of the $\varphi$-oscillations. When the amplitude of these oscillations is sufficiently small, one can expand the Hamiltonian $F_0$ to obtain $2(F_0 - \bar F_0) \approx gP^2 + d |\cos \varphi_0| \cdot (\varphi -\varphi_0)^2$, where $\bar F_0$ and $\varphi_0$ are values of $F_0$ and $\varphi$ at the bottom of the potential well inside the separatrix (see Fig. 1a). The frequency of oscillations (in terms of rescaled time $\theta$) is approximately $\sqrt{gd|\cos \varphi_0| }$, and in this approximation $2\pi J \sim (F_0 - \bar F_0)/\sqrt{gd}$. When the particle is captured, $J = J_0$. Using conservation of $J$ along the trajectory of the captured particle, one obtains the scalings $\Delta\varphi \sim q^{-1/4} (k^2-\omega^2)^{1/4} $ and $\Delta P \sim q^{1/4}(k^2-\omega^2)^{-1/4}$, where $\Delta\varphi$ and $\Delta P$ are amplitudes of $\varphi$-oscillations and $P$-oscillations accordingly. Thus, the amplitude of oscillations in $q$ decreases with time proportionally to $t^{-1/4}$, while amplitude of oscillations in $p_1$ grows with time proportionally to $t^{1/4}$. Accordingly, amplitude of oscillations in $p_2$ decreases with time as $t^{-1/4}$. In dimensional variables we have $\Delta\hat q_1 \sim \varepsilon ( \hat v_{\phi} \hat t)^{-1/4}$ and $\Delta \hat p_1 \sim \sqrt{\varepsilon} ( \hat v_{\phi} \hat t)^{-1/4}$. The frequency of these oscillations is $\omega_0 \approx \sqrt{\varepsilon gd}$ (we recall that we made time rescaling to obtain (\ref{3.6})). Thus,
\begin{equation}
\omega_0 \approx k(1-v_{\phi}^2) \sqrt{\frac{\varepsilon q}{1+q^2}} .
\label{3.13}
\end{equation}
At $q \gg 1$ we find that this frequency decreases as $\sqrt{\varepsilon/q}$. In dimensional variables we find that $\hat \omega_0 \sim (\varepsilon \hat t)^{-1/2}$.
\begin{figure}[t]
\epsfig{file=fig04.eps, width=5.0in}
\caption{\label{Fig4} Oscillation frequency of $\dot \varphi$ as a function of time for two particles with different
values of initial momentum. Grey dashed lines show the theoretical dependence $\omega_0 \sim t^{-1/2}$}
\end{figure}
Here are more complete formulas in dimensional variables for the amplitudes of the oscillations at large enough values of $\hat q_1$, such that we can put $\hat q_1 = \hat v_{\phi} \hat t$. One obtains:
$$
\Delta \hat q_1 \sim \varepsilon c/\omega_L \sqrt{B_0/B_w} \left(1 - \frac{ \hat v_{\phi}^2}{c^2}\right)^{1/4}(
\omega_L \hat v_{\phi} \hat t/c)^{-1/4},
$$
$$
\Delta \hat p_1 \sim \varepsilon^{1/2} \sqrt{B_w/B_0} \left(1 - \frac{ \hat v_{\phi}^2}{c^2}\right)^{1/4} (
\omega_L \hat v_{\phi} \hat t/c)^{1/4} ,
$$
$$
\hat\omega_0 \sim \omega_L \varepsilon^{-1/2} \sqrt{B_w/B_0} \left(1 - \frac{ \hat v_{\phi}^2}{c^2}\right) (
\omega_L \hat v_{\phi} \hat t/c)^{-1/2}.
$$
Capture into the resonance is a probabilistic phenomenon (see, e.g., \cite{Neishtadt:1999,NV:2006}). Its probability is a small value of order $\sqrt\varepsilon$. However, the geometry of the system makes the particle pass through the resonance repeatedly, at each Larmor turn. The probability of capture after $\sim\varepsilon^{-1/2}$ Larmor turns is a value of order one (provided that $d/b > 1$).
For comparison with theoretical results we present the numerical solution of the system with Hamiltonian (\ref{2.3}), parameters $\varepsilon=0.1$, $\hat \omega/(\hat kc)=0.25$ and initial value of the $p_1 = 0.1$. The particle trajectory in momentum space $(p_1, p_2)$ is shown in Fig. \ref{Fig1}. At the first stage of modelling the particle rotates in the constant background magnetic field (Larmor rotation). This motion is slightly perturbed by influence of the wave: the Larmor circles in $(p_1, p_2)$ plane are ``scattered''. Then after certain time interval the particle is captured by the wave and the magnitude of momentum $p_2$ grows with time while momentum $p_1$ oscillates around the resonant value (it is also increasing, yet much slower). To compare the scale of growth of momenta $p_1$ and $p_2$ we plot the same picture on a longer time range (see Fig. \ref{Fig1}). The relation between $p_1$ and $p_2$ after initial time interval is $1/40 \sim \varepsilon\hat \omega/(\hat kc)$, in agreement with the theory (see equation (\ref{3.11})). The particle energy $E = \sqrt{1+p_1^2+p_2^2}$ is shown in Fig. \ref{Fig3}. The energy is almost constant before capture (if we neglect small scatterings due to wave impacts) and after the capture the energy grows linearly with time. In addition we examine the theoretical equation for frequency of oscillation of the captured particle - equation (\ref{3.13}). For this purpose we plot the oscillation frequency of $\dot \varphi$ around the null value - Fig.~\ref{Fig4}.
\section{Scatterings on the resonance}
Capture into the resonance is impossible in the case $d/b \le 1$, when there is no oscillatory domain on the phase portrait of the pendulum-like system (see Fig. 1b). However, in this case the particle energy also changes at the resonance crossing. This happens due to scatterings on the resonance. If $d/b \le 1$, the average value of the jump in the energy is zero (see \cite{Neishtadt:1999}), but the dispersion is non-zero, and thus diffusive variation of the particle energy may be possible. Here we study this topic more attentively.
When the particle is far from the resonance, its energy is approximately constant, because the impact of the wave can be averaged. Thus, to study variation of the particle energy we find its time derivative according to equations of motion (\ref{3.7}) and integrate it near the resonance. We have
\begin{equation}
E=\sqrt{1+p_1^2 + q^2} = \sqrt{1+(p+kI)^2 + q^2}.
\label{4.1}
\end{equation}
Using (\ref{2.4}) we can write
\begin{equation}
\dot E = \frac{\mbox{d} }{\mbox{d} t} ({\cal H} + \omega I) = \omega \dot I = -\omega \frac{\partial {\cal H}}{\partial \varphi}.
\label{4.2}
\end{equation}
From (\ref{3.1}) and (\ref{3.3}) we find that near the resonance
\begin{equation}
\dot E = \varepsilon \omega \, d \sin \varphi ,
\label{4.3}
\end{equation}
To integrate (\ref{4.3}) we change the integration variable from time $t$ to phase $\varphi$ according to $\dot \varphi = \sqrt{\varepsilon} \partial F_0/\partial P$. Thus we find for variation (jump) of the particle energy on one resonance crossing
\begin{equation}
j_E = 2\sqrt{\varepsilon} \frac{v_\phi}{\sqrt{1-v_\phi^2}} \sqrt{q} \int_{-\infty}^{\varphi_*} \frac{\sin \varphi \,
\mbox{d}\varphi}{\sqrt{2\left[\cos\varphi_* + \frac{b}{d}\varphi_* - \cos \varphi - \frac{b}{d}\varphi\right]}},
\label{4.4}
\end{equation}
where $\varphi_*$ is the wave's phase at the resonance crossing, and $q$ is taken also at the crossing of the unperturbed trajectory with energy $E$ and the resonant surface. The value of $\varphi_*$ strongly depends on initial conditions and should be treated as random. Therefore, change in the particle's energy on the resonance is also a random variable. If there is no separatrix on the phase portrait in Fig. 1 (i.e., if $d/b \le 1$), the average value of this latter random value is zero. An important question is whether these values at successive crossings are statistically independent. Expressing $q$ in (\ref{4.4}) on the resonance via $E$ (we use that at the resonance $E = \sqrt{(1+q^2)/(1-v_\phi^2)}$) we find that at $E \gg 1$ the variation of energy at the resonance scattering is a value of order
\begin{equation}
j_E \sim \sqrt{\varepsilon E}.
\label{4.5}
\end{equation}
On the plane $(q,p_1)$ unperturbed motion of the particle is rotation along the circle $p_1^2 + q^2 = E^2 - 1$. Using Hamiltonian equations of the unperturbed motion, one immediately finds that the frequency of this rotation is $\omega_0 = \varepsilon/E $. The resonant curve on the plane $(q,p_1)$ is a branch of hyperbola $p_1^2 (1-v_\phi^2) - q^2 v_\phi^2= v_\phi^2$ with $p_1>0$. At large enough values of $E$ the trajectory of the unperturbed motion crosses the resonant curve at two points. It is straightforward to find that the time of motion between these two points is a value of order $E/\varepsilon$. Consider two successive resonance crossings. Let the values of $\varphi$ at the first and the second crossings be $\varphi_*^{(1)}$ and $\varphi_*^{(2)}$ accordingly. To find $\varphi_*^{(2)} - \varphi_*^{(1)}$ one can integrate equation of motion for $\varphi$ in (\ref{2.6}). Thus, one obtains $\varphi_*^{(2)} - \varphi_*^{(1)} \sim E/\varepsilon$. A small variation $\delta\varphi_*^{(1)}$ of the phase $\varphi_*^{(1)}$ results in variation of the energy jump $\delta j_E \sim \sqrt{\varepsilon E}\delta\varphi_*^{(1)}$ at the first resonance crossing. The resulting variation $\delta \varphi_*^{(2)}$ of the phase at the second crossing can be found as $\delta j_E\, \partial(\varphi_*^{(2)} - \varphi_*^{(1)})/\partial E \sim \delta j_E/\varepsilon \sim \delta\varphi_*^{(1)}\sqrt{E}/\sqrt{\varepsilon}$. Thus, the resulting variation in the phase at the second resonance crossing is much larger than $\delta\varphi_*^{(1)}$. Therefore, the values of phase at successive resonance crossings are statistically independent. Hence, the jumps in the particle's energy at the resonance produce diffusive variation of the energy and its unlimited stochastic growth. Note, that in \cite{AVNZ:2009} the diffusive growth of energy was studied in nonrelativistic case. It was found that, unlike in the relativistic case, for a nonrelativistic particle the energy diffusion slows down and finally stops at large enough energies.
\begin{figure}[t]
\center\epsfig{file=dif.eps, width=0.95\textwidth}
\caption{\label{dif} The Poincar\'e sections for several cases with different initial energy ($\varepsilon
= 0.1$). Number of resonance crossings in both cases is $10^6$. The energy diffusion is clearly seen.}
\end{figure}
One can estimate the rate of the energy diffusion as follows. Consider a long trajectory that crosses the resonance $N \gg 1$ times. Introduce new variable $\kappa = \sqrt E$. It follows from (\ref{4.5}) that at every resonant crossing $\kappa$ changes by a value of order $\sqrt\varepsilon$. If successive jumps in energy are not correlated, it follows from (\ref{4.5}) that typical displacement of $\kappa$ after $N$ resonance crossings is $\Delta\kappa \sim \sqrt{\varepsilon N}$. Hence, typical value of energy after $N$ jumps is proportional to $N$:
\begin{equation}
E \sim \varepsilon N.
\label{4.6}
\end{equation}
Time interval between successive jumps is a value of order of the Larmor period. Hence, the time interval corresponding to $N$ resonance crossings is $t \sim N E/\varepsilon$. Combining this with (\ref{4.6}), we find that the energy typically grows with time as
\begin{equation}
E \sim \varepsilon \sqrt{t}.
\label{4.7}
\end{equation}
We also obtain that the number of jumps (resonance crossings) grows with time as $N \sim \sqrt{t}$.
These results on the energy diffusion of a relativistic particle can be examined numerically. For this purpose we construct the Poincar\'e section of a particle trajectory in $(p_1, p_2)$ plane. Points on this plane are plotted with time period $2\pi/\omega$. One can see that the diffusion in $(p_1, p_2)$ space become stronger as the initial energy of particles grows (Fig. \ref{dif}). In Fig. \ref{dif2}, we present results of numerics illustrating estimates (\ref{4.6}) and (\ref{4.7}).
\begin{figure}[t]
\center\epsfig{file=dif2.eps, width=0.5\textwidth}
\caption{\label{dif2} The energy of particle ensemble (number of particles is $10^3$) as a function of
number of resonance crossings in the system with $\varepsilon = 0.1$. In subpanel: the number of resonance
crossings and averaged energy as a function of time.}
\end{figure}
\section{Energy loss due to radiation}
The oscillations of the captured particle across the wave front result in energy loss due to radiation. The energy loss has the following rate (see \cite{LL:field}):
\begin{equation}
\frac{\mbox{d} \hat E_-}{\mbox{d} \hat t} = \frac{2e^2 }{3c^3 }\left(\frac{\mbox{d}^2\hat q_1}{\mbox{d} \hat t^2}\right)^2.
\label{5.1}
\end{equation}
In the dimensionless variables this expression takes the form:
\begin{equation}
\frac{\mbox{d} E_-}{\mbox{d} t} = \frac{2r_0 \omega_L}{3 \varepsilon^3 c}\left(\frac{\mbox{d}^2 q}{\mbox{d} t^2}\right)^2,
\label{5.2}
\end{equation}
where we introduced notation $r_0 =e^2/mc^2$. For a captured particle moving deep inside the domain of oscillations on phase portrait in Fig.~1b, we find
\begin{equation}
\frac{\mbox{d}^2 q}{\mbox{d} t^2} = -\varepsilon k^{-1} \Delta\varphi \omega_0^2 \sin\omega_0 t = K_1 \frac{\varepsilon^2 q^{3/4}}{1+q^2} k^{3/2}(1-v_{\phi}^2)^{9/4} \sin\omega_0 t.
\label{5.3}
\end{equation}
Here $\Delta\varphi$ and $\omega_0$ are amplitude and frequency of small oscillations of the captured particle, and $K_1$ is a value of order one; $K_1$ depends on the value of $q$ at the instance of capture into the resonance (see Section 3). Thus, we find
\begin{equation}
\frac{\mbox{d} E_-}{\mbox{d} t} = \varepsilon K_2 \frac{r_0 \omega_L}{ c} k^3 \frac{q^{3/2}}{(1+q^2)^2} (1-v_{\phi}^2)^{9/2}.
\label{5.4}
\end{equation}
Here $K_2$ is also a value of order one. In the latter expression, the fraction containing $q$ reaches its maximum value $f_{max} \approx 0.266$ at $q = \sqrt{3/5}$. Hence, loss of energy due to radiation is maximal at this value of $q$.
On the other hand, the captured particle is accelerated along the wave front, and thus it gains energy at the rate (see Section 3)
\begin{equation}
\frac{\mbox{d} E_+}{\mbox{d} t} = \varepsilon v_{\phi}/\sqrt{1-v_{\phi}^2}.
\label{5.5}
\end{equation}
Comparing expressions (\ref{5.4}) and (\ref{5.5}) we find that if
\begin{equation}
K_2 f_{max} \frac{r_0}{cv_{\phi}} \omega_L k^3(1-v_{\phi}^2)^{5} < 1,
\label{5.6}
\end{equation}
the radiation cannot stop the particle acceleration. In the opposite case,
\begin{equation}
K_2 f_{max} \frac{r_0}{cv_{\phi}} \omega_L k^3(1-v_{\phi}^2)^{5} > 1,
\label{5.7}
\end{equation}
the acceleration can be stopped under the additional condition that the capture took place at a value of $q$ smaller than the largest of the two roots of equation $\mbox{d} E_+/\mbox{d} t =\mbox{d} E_-/\mbox{d} t$.
Note that inequality (\ref{5.7}) can be written in dimensional form as $ r_0 \omega _L \left( B_w /B_0 \right)^3 \left( {1 - \hat v_\phi ^2 /c^2 } \right)^5 > \hat v_\phi $. To be valid, this inequality needs either $B_w$ very large or $v_{\phi}$ very small. Neither so large value of magnetic field, nor so small value of phase velocity can be found in physically realistic situations, and hence (\ref{5.7}) cannot be valid in such situations.
\section{Conclusions}
In this paper we considered dynamics of a relativistic charged particle in the field of an electromagnetic wave in the presence of a background magnetic field. We have described the particle capture into resonance with the wave and consequent acceleration using approach of the adiabatic theory of motion. During the acceleration the particle's momentum in the direction of the wave vector $\hat p_1$ and along the wave front $\hat p_2$ change with time linearly ($\hat p_1 \sim mc \frac{ \hat v_{\phi}/c}{\sqrt{1 - \hat v_{\phi}^2/c^2}} (\omega_L \hat v_{\phi} \hat t/c) $, $\hat p_2 \sim -mc(\omega_L \hat v_{\phi} \hat t/c)$ ). As a result the particle energy $\hat E \sim \sqrt{m^2c^4+\hat p_1^2+\hat p_2^2}$ grows with time as $\hat E \sim mc^2(\omega_L \hat v_{\phi} \hat t/c)$. The estimates of energy loss due to radiation of the accelerated particle show that it is not sufficient to stop the acceleration and the particle energy grows infinitely. If the condition of capture into the resonance is not satisfied (the magnitude of the wave is less than a certain value), particle can nevertheless gain energy by scatterings on the resonance. In this case $\hat E \sim N$, where $N$ is a number of resonance crossings.
\section*{Acknowledgements}
The work was supported in part by the Russian Foundation for Basic Research (project nos. 09-01-00333, 08-02-00201), and the Council of the Russian Federation Presidential Grants for State Support of Leading Scientific Schools (project no. NSh-8784.2010.1).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,902 |
Turkiet deltog i olympiska vinterspelen 2002. Turkiets trupp bestod av tre idrottare varav två var män och en var kvinna.
Trupp
Alpin skidåkning
Atakan Alaftargil
Längdskidåkning
Kelime Aydın
Sabahattin Oglago
Resultat
Alpin skidåkning
Slalom
Atakan Alaftargil - 32
Längdskidåkning
Sprint herrar
Sabahattin Oglago - 60
30 km herrar
Sabahattin Oglago - ?
Sprint damer
Kelime Aydın - 56
5+5 km damer
Kelime Aydın - 67
Källor
SR/Olympic Sports
2002 i Turkiet
Nationer i olympiska vinterspelen 2002
2002 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,794 |
sudo su -c 'rpm -Uvh http://dl.fedoraproject.org/pub/epel/6/x86_64/epel-release-6-8.noarch.rpm'
sudo yum -y install nginx
rm -rf /var/www
sudo ln -fs /vargrant /var/www
sudo chkconfig --levels 235 nginx on
sudo rm /etc/nginx/conf.d/default.conf
sudo cp /vagrant/playing_with_vagrant.conf /etc/nginx/conf.d/default.conf
sudo usermod -a -G vagrant nginx
sudo service nginx restart | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,363 |
\section{Introduction}
We let $O(n)$ denote the orthogonal group of the Euclidean $n$-space $\mathbb{R}^{n}$ and $\theta_n$ its Haar probability measure. We metrize $O(n)$ with the usual operator norm. Let also $\mathcal L^n$ stand for the Lebesgue measure on $\mathbb{R}^{n}$ and let $\dim$ stand for the Hausdorff dimension and $\mathcal H^s$ for $s$-dimensional Hausdorff measure. We shall prove the following theorem:
\begin{thm}\label{theo1}
Let $s$ and $t$ be positive numbers with $s+t > n+1$. Let $A$ and $B$ be Borel subsets of $\mathbb{R}^{n}$ with $\mathcal H^s(A)>0$ and $\mathcal H^t(B)>0$. Then
there is $E\subset O(n)$ such that
$$\dim E\leq 2n-s-t+(n-1)(n-2)/2=n(n-1)/2-(s+t-(n+1))$$
and for $g\in O(n)\setminus E$,
\begin{equation}\label{eq4}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z)\geq s+t-n\})>0.
\end{equation}
\end{thm}
The version stated in the abstract concerning the case $\dim A + \dim B > n+1$ is slightly weaker than Theorem \ref{theo1}. Notice that the above upper bound for the dimension of $E$ is at least $(n-1)(n-2)/2$ which is the dimension of $O(n-1)$. In Section \ref{Examples} we show that it is needed in the estimates. The assumption $s+t>n+1$ only comes from the fact that the statement is trivial if $s+t\leq n+1$: then the above upper for $\dim E$ is at least $n(n-1)/2=\dim O(n)$ and we could take $E=O(n)$.
This is an exceptional set estimate related to the following result of
\cite{M2}: (\ref{eq4}) holds for $\theta_n$ almost all $g\in O(n)$ if one of the sets has dimension bigger than $(n+1)/2$, see also Chapter 13 in \cite{M4} and Chapter 7 in \cite{M5}. This of course is satisfied when $s+t>n+1$, as in the theorem. It is expected that this generic result with respect to $\theta_n$ should hold whenever $\dim A + \dim B > n$. Under this condition it was proved (without exceptional set estimates) in \cite{K} and \cite{M1} provided the orthogonal group is replaced by a larger transformation group, for example by similarity maps as in \cite{M1}, or, more generally by Kahane in \cite{K}, by any closed subgroup of the general linear group acting transitively outside the origin. For the orthogonal group no dimensional restrictions are needed provided one of the sets satisfies some extra condition, for example if it is rectifiable, see \cite{M1}, or a Salem set, see \cite{M3}.
It is easy to see, cf. the remark at the end of \cite{M2}, that in Theorem \ref{theo1} the positivity of the Hausdorff measures cannot be relaxed to $\dim A = s$ and $\dim B = t$.
If one of the sets supports a measure with sufficiently fast average decay over spheres for the Fourier transform, we can improve the estimate of Theorem \ref{theo1}. Then the results even hold for the sum sets provided the dimensions are big enough. This is given in Theorem \ref{theo3}. It yields immediately the following result in case one of the sets is a Salem set. By definition, $A$ is a Salem set if for every $0<s<\dim A$ there is $\mu\in\mathcal M(A)$ such that $|\mu(x)|^2\lesssim |x|^{-s}$. A discussion on Salem sets can be found, for example, in \cite{M5}, Section 3.6.
\begin{thm}\label{theo4}
Let $A$ and $B$ be Borel subsets of $\mathbb{R}^{n}$ and suppose that $A$ is a Salem set. Suppose that $0<u<\dim A + \dim B - n$.
(a) If $\dim A+\dim B>2n-1$, then
\begin{equation}\label{eq5}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (B+z)\geq u\})>0.
\end{equation}
(b)If $\dim A+\dim B\leq 2n-1$, then there is $E\subset O(n)$ with
$$\dim E\leq n(n-1)/2-u$$
such that for $g\in O(n)\setminus E$,
\begin{equation}\label{eq7}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z)\geq u\})>0.
\end{equation}
\end{thm}
Another consequence of Theorem \ref{theo3} is the following improvement of Theorem \ref{theo1} in the case where one of the sets has small dimension:
\begin{thm}\label{theo6}
Let $A$ and $B$ be Borel subsets of $\mathbb{R}^{n}$ and suppose that $\dim A\leq (n-1)/2$. If $0<u<\dim A + \dim B - n$, then there is $E\subset O(n)$ with
$$\dim E\leq n(n-1)/2-u$$
such that for $g\in O(n)\setminus E$,
\begin{equation}\label{eq18}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z)\geq u\})>0.
\end{equation}
\end{thm}
The method used to prove Theorem \ref{theo1} can easily be modified to other subgroups of the general linear group $GL(n)$ in place of the orthogonal group. For example, let $S(n)$ be the group of similarities, the compositions of orthogonal maps and dilations. Then $\dim S(n)=n(n-1)/2+1$ and for any $x,z\in\mathbb{R}^{n}\setminus\{0\}$, the dimension of $\{g\in S(n): g(z)=x\}$ is the same as the dimension of $O(n-1)$, that is, $(n-1)(n-2)/2$. With small changes in the proof of Theorem \ref{theo1} this leads to
\begin{thm}\label{theo2}
Let $s$ and $t$ be numbers with $0<s,t<n$ and $s+t > n$. Let $A$ and $B$ be Borel subsets of $\mathbb{R}^{n}$ such that $\mathcal H^s(A)>0$ and $\mathcal H^t(B)>0$. Then
there is $E\subset S(n)$ with
$$\dim E\leq 2n-s-t+(n-1)(n-2)/2$$
and for $g\in S(n)\setminus E$,
\begin{equation}\label{eq6}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z)\geq s+t-n\})>0.
\end{equation}
\end{thm}
\section{Peliminaries}
The proof of Theorem \ref{theo1} is based on the relationship of the Hausdorff dimension to the energies of measures and their relations to the Fourier transform. For $A\subset\mathbb{R}^{n}$ (or $A\subset O(n)$) we denote by $\mathcal M(A)$ the set of non-zero Radon measures $\mu$ on $\mathbb{R}^{n}$ with compact support $\spt\mu\subset A$. The Fourier transform of $\mu$ is defined by
$$\widehat{\mu}(x)=\int e^{-2\pi ix\cdot y}\,d\mu y,~ x\in\mathbb{R}^{n}.$$
For $0<s<n$ the $s$-energy of $\mu$ is
\begin{equation}\label{eq10}
I_s(\mu)=\iint|x-y|^{-s}\,d\mu x\,d\mu y=c(n,s)\int|\widehat{\mu}(x)|^2|x|^{s-n}\,dx.
\end{equation}
The second equality is a consequence of Parseval's formula and the fact that the distributional Fourier transform of the Riesz kernel $k_s, k_s(x)=|x|^{-s}$, is a constant multiple of $k_{n-s}$, see, for example, \cite{M4}, Lemma 12.12, or \cite{M5}, Theorem 3.10. These books contain most of the back-ground material needed in this paper.
We then have for any Borel set $A\subset\mathbb{R}^{n}$, cf. Theorem 8.9 in \cite{M4},
\begin{equation}\label{eq3}
\dim A=\sup\{s:\exists \mu\in\mathcal M(A)\ \text{such that}\ I_s(\mu)<\infty\}.
\end{equation}
Let $\nu\in\mathcal M(\mathbb{R}^{n})$ and let $\psi_{\varepsilon}$ be an approximate identity: $\psi_{\varepsilon}(x)=\varepsilon^{-n}\psi(x/\varepsilon)$ where $\psi$ is a non-negative $C^{\infty}$-function with support in the unit ball and with integral $1$. Let $\nu_{\varepsilon}=\psi_{\varepsilon}\ast\nu$. Then the $\nu_{\varepsilon}$ converge weakly to $\nu$ when $\varepsilon\to 0$. Notice that $\widehat{\nu_{\varepsilon}}(x)=\widehat{\psi}(\varepsilon x)\widehat{\nu}(x)$.
By the notation $M\lesssim N$ we mean that $M\leq CN$ for some constant $C$. The dependence of $C$ should be clear from the context. By $C(a)$ and $c(a)$ we mean positive constants depending on the parameter $a$. The closed ball with centre $x$ and radius $r$ will be denoted by $B(x,r)$.
\begin{lm}\label{lemma2} Let $\theta\in\mathcal M(O(n))$ and $\a>(n-1)(n-2)/2$. If $\theta(B(g,r))\leq r^{\a}$ for all $g\in O(n)$ and $r>0$, then for $x,z\in\mathbb{R}^{n}\setminus\{0\}, r>0$,
\begin{equation}\label{eq11}
\theta(\{g:|x-g(z)|< r\})\lesssim (r/|z|)^{\a-(n-1)(n-2)/2}.
\end{equation}
\end{lm}
\begin{proof}
First we may clearly assume that $|z|=1$, and then also that $|x|=1$, because $|x-g(z)|< r$ implies $|x/|x|-g(z)|< 2r$. Then $O_{x,z}:=\{g\in O(n): g(z)=x\}$ can be identified with $O(n-1)$. Hence it is a smooth compact $(n-1)(n-2)/2$-dimensional submanifold of $O(n)$ which implies that it can be covered with roughly $r^{-(n-1)(n-2)/2}$ balls of radius $r$. If $g\in G$ satisfies $|x-g(z)|< r$, then $g$ belongs to the $r$-neighbourhood of $O_{x,z}$. The lemma follows from this.
\end{proof}
\section{Proof of Theorem \ref{theo1}}
The key to the proof of Theorem \ref{theo1} is the following energy estimate.
For $\mu, \nu\in\mathcal M(\mathbb{R}^{n}), g\in O(n)$ and $z\in\mathbb{R}^{n}$, let $\nu_{\varepsilon}=\psi_{\varepsilon}\ast\nu$ as above and set
\begin{equation}\label{eq0}
\nu_{g,z,\varepsilon}(x)=\nu_{\varepsilon}(g^{-1}(x)-z),\ x\in\mathbb{R}^{n}.
\end{equation}
\begin{lm}\label{lemma1} Let $\beta>0$ and $\theta\in\mathcal M(O(n))$ be such that $\theta(O(n))\leq 1$ and for $x,z\in\mathbb{R}^{n}\setminus\{0\}, r>0,$
\begin{equation}\label{eq9}
\theta(\{g\in O(n):|x-g(z)|<r\})\leq (r/|z|)^{\beta}.
\end{equation}
Let $0<s,t<n, 0<u=s+t-n$ and $u>n-\beta$. Let $\mu, \nu \in \mathcal M(\mathbb{R}^{n})$. Then
\begin{equation}\label{eq15}
\iint I_u(\nu_{g,z,\varepsilon}\mu)\,d\mathcal L^nz\,d\theta g \leq C(n,s,t)I_s(\mu)I_t(\nu).
\end{equation}
\end{lm}
\begin{proof}
We may assume that $I_s(\mu)$ and $I_t(\nu)$ are finite.
Define
$$\tilde{\nu}_{g,x,\varepsilon}(z)=\nu_{\varepsilon}(g^{-1}(x)-z),\ z\in\mathbb{R}^{n}.$$
Then
$$\widehat{\tilde{\nu}_{g,x,\varepsilon}}(z)=e^{-2\pi ig^{-1}(x)\cdot z}\widehat{\nu_{\varepsilon}}(-z).$$
Hence by Parseval's formula for $x,y\in\mathbb{R}^{n}, x\neq y,$
\begin{align*}
&\int\nu_{\varepsilon}(g^{-1}(x)-z)\nu_{\varepsilon}(g^{-1}(y)-z)\,d\mathcal L^nz\\
&=\int\widehat{\tilde{\nu}_{g,x,\varepsilon}}(z)\overline{\widehat{\tilde{\nu}_{g,y,\varepsilon}}(z)}\,d\mathcal L^nz\\
&=\int\widehat{\nu_{\varepsilon}}(-z)\overline{\widehat{\nu_{\varepsilon}}(-z)}e^{-2\pi ig^{-1}(x-y)\cdot z}d\mathcal L^nz\\
\end{align*}
It follows by Fubini's theorem that
\begin{align*}
&I:=\iint I_u(\nu_{g,z,\varepsilon}\mu)\,d\mathcal L^nz\,d\theta g\\
&=\iiiint k_u(x-y)\nu_{\varepsilon}(g^{-1}(x)-z)\nu_{\varepsilon}(g^{-1}(y)-z)\,d\mu x\,d\mu y\,d\mathcal L^nz\,d\theta g\\
&=\iiint k_u(x-y)\left(\int\nu_{\varepsilon}(g^{-1}(x)-z)\nu_{\varepsilon}(g^{-1}(y)-z)\,d\mathcal L^nz\right)\,d\mu x\,d\mu y\,d\theta g\\
&=\iiint k_u(x-y)\left(\int|\widehat{\nu_{\varepsilon}}(z)|^2 e^{2\pi ig^{-1}(x-y)\cdot z}\,d\mathcal L^nz\right)\,d\mu x\,d\mu y\,d\theta g\\
&=\iiint k_{u,g,z}\ast\mu(x)\,d\mu x|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\,d\theta g,
\end{align*}
where
\begin{equation*}
k_{u,g,z}(x)=|x|^{-u}e^{2\pi ig^{-1}(x)\cdot z}=|x|^{-u}e^{2\pi ix\cdot g(z)}
\end{equation*}
One checks by direct computation that the Fourier transform of $k_{u,g,z}$, in the sense of distributions, is given by
\begin{equation*}
\widehat{k_{u,g,z}}(x)=
c(n,u)|x-g(z)|^{u-n}.
\end{equation*}
It follows that
$$\iint k_{u,g,z}\ast\mu\,d\mu=\int\widehat{k_{u,g,z}}|\widehat{\mu}|^2\,d\mathcal L^n=c(n,u)\int|x-g(z)|^{u-n}|\widehat{\mu}(x)|^2\,d\mathcal L^nx.$$
As $I_u(\mu)<\infty$, this is easily checked approximating $\mu$ with $\psi_{\varepsilon}\ast\mu$ and using the Lebesgue dominated convergence theorem.
Thus
\begin{equation}\label{eq8}
I=c(n,u)\iiint |x-g(z)|^{u-n}\,d\theta g|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz.
\end{equation}
We first observe that if $|x|\geq 2|z|$, then
$$\int |x-g(z)|^{u-n}\,d\theta g\leq \theta(O(n))2^{n-u}|x|^{u-n}\leq 2^{2n}|x|^{s-n}|z|^{t-n}.$$
Similarly if $|z|\geq 2|x|$, then
$$\int |x-g(z)|^{u-n}\,d\theta g\leq \theta(O(n))2^{n-u}|z|^{u-n}\leq 2^{2n}|x|^{s-n}|z|^{t-n}.$$
Suppose then that $|z|/2\leq|x|\leq2|z|.$ Then by the assumption
\begin{align*}
&\int|x-g(z)|^{u-n}\,d\theta g = \int_0^{\infty}\theta(\{g:|x-g(z)|^{u-n}>\lambda\})\,d\lambda\\
&=(n-u)\int_0^{\infty}\theta(\{g:|x-g(z)|<r\})r^{u-n-1}\,dr\\
&\lesssim \int_{0}^{|z|}(r/|z|)^{\beta}r^{u-n-1}\,dr+\int_{|z|}^{\infty}r^{u-n-1}\,dr\\
&\approx |z|^{u-n}\approx |x|^{s-n}|z|^{t-n},
\end{align*}
since $\beta+u-n>0$.
It follows that
\begin{equation}\label{eq13}
I\lesssim \iint |x|^{s-n}|z|^{t-n}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(y)|^2\,dx\,dy\lesssim I_s(\mu)I_t(\nu),
\end{equation}
as required.
\end{proof}
Next we show that, with $\theta$ as above, for $\theta\times\mathcal L^n$ almost all $(g,z)$ the measures $\nu_{g,z,\varepsilon}\mu$ converge weakly as $\varepsilon\to 0$. It is immediate that for almost all $(g,z)$ this takes place through some sequence $(\varepsilon_j)$, depending on $(g,z)$, but we would at least need one sequence which is good for almost all $(g,z)$. The proof of the following theorem was inspired by an argument of Kahane in \cite{K}.
\begin{thm}\label{theo5}
Let $s,t$ and $u$ be positive numbers with $u=s+t-n>0$ and let $\mu, \nu\in\mathcal M(\mathbb{R}^{n})$ with $I_s(\mu)<\infty$ and $I_t(\nu)<\infty$. Let $\psi_{\varepsilon}$ be an approximate identity and $\nu_{\varepsilon}=\psi_{\varepsilon}\ast\nu$. For $g\in O(n)$ and $z\in\mathbb{R}^{n}$, let $\nu_{g,z,\varepsilon}$ be as in (\ref{eq0}). Finally, let $\theta\in\mathcal M(O(n))$ be as in Lemma \ref{lemma1}. Then for $\theta\times\mathcal L^n$ almost all $(g,z)$, as $\varepsilon\to 0$, the measures
$\nu_{g,g^{-1}(z),\varepsilon}\mu$ converge weakly to a measure $\lambda_{g,z}$ with the properties
\begin{itemize}
\item[(a)] $$\spt\lambda_{g,z}\subset\spt\mu\cap(g(\spt\nu)+z),$$
\item[(b)]$$\int\lambda_{g,z}(\mathbb{R}^{n})\,d\mathcal L^nz = \mu(\mathbb{R}^{n})\nu(\mathbb{R}^{n})\ \text{for}\ \theta\ \text{almost all}\ g\in O(n),$$
\item[(c)]
$$\iint I_u(\lambda_{g,z})\,d\mathcal L^nz\,d\theta g\leq C(n,s,t)I_s(\mu)I_t(\nu).$$
\end{itemize}
\end{thm}
\begin{proof}
If the convergence takes place, the support property (a) is clear. Using the change of variable from $z$ to $g^{-1}(z)$ in the appropriate places, it is then sufficient to show that for $\theta\times\mathcal L^n$ almost all $(g,z)$, as $\varepsilon\to 0$, the measures
$\nu_{g,z,\varepsilon}\mu$ converge weakly to a measure $\tilde{\lambda}_{g,z}$ such that (b) and (c) hold with $\lambda_{g,z}$ replaced by $\tilde{\lambda}_{g,z}$.
Let $\phi\in C^+_0(\mathbb{R}^{n})$. Then by Lemma \ref{lemma1},
$$\iint (\int\nu_{g,z,\varepsilon}\phi\,d\mu)^2\,d\mathcal L^nz\,d\theta g\lesssim I_s(\mu)I_t(\nu)<\infty.$$
Hence by Fatou's lemma
$$\int \left(\liminf_{\varepsilon\to 0}\int(\int\nu_{g,z,\varepsilon}\phi\,d\mu)^2\,d\mathcal L^nz\right)\,d\theta g\lesssim I_s(\mu)I_t(\nu)<\infty.$$
Thus for $\theta$ almost all $g$ there is a sequence $(\varepsilon_j)$ tending to $0$ such that
$$\sup_j\int(\int\nu_{g,z,\varepsilon_j}\phi\,d\mu)^2\,d\mathcal L^nz<\infty.$$
On the other hand, defining the measure $\mu_{\phi,g}$ by $\int h\,d\mu_{\phi,g}=\int h(-g^{-1}(x))\phi(x)\,d\mu x$, we have
$$\int\nu_{g,z,\varepsilon}\phi\,d\mu=\int\nu_{\varepsilon}(g^{-1}(x)-z)\phi(x)\,d\mu x=\mu_{\phi,g}\ast\nu\ast\psi_{\varepsilon}(-z),$$
and the measures $\mu_{\phi,g}\ast\nu\ast\psi_{\varepsilon}$ converge weakly to $\mu_{\phi,g}\ast\nu$. Consequently, $\mu_{\phi,g}\ast\nu$ is an $L^2$ function on $\mathbb{R}^{n}$ and the convergence takes place almost everywhere. It follows now that for $\theta$ almost all $g\in O(n)$ and for every $\phi\in C^+_0(\mathbb{R}^{n})$ the finite limit
\begin{equation}\label{eq2}
L_{g,z}\phi:=\lim_{\varepsilon\to 0}\int\nu_{g,z,\varepsilon}\phi\,d\mu=\lim_{\varepsilon\to 0}\mu_{\phi,g}\ast\nu\ast\psi_{\varepsilon}(-z)=\mu_{\phi,g}\ast\nu(-z)
\end{equation}
exists for almost all $z\in\mathbb{R}^{n}$. Let $\mathcal D$ be a countable dense subset of $C^+_0(\mathbb{R}^{n})$ containing a function $\phi_0$ which is $1$ on the support of $\mu$. Then there is a set $E$ of measure zero such that (\ref{eq2}) holds for all $z\in\mathbb{R}^{n}\setminus E$ for all $\phi\in \mathcal D$, the exceptional set is indenpendent of $\phi$. Applying (\ref{eq2}) to $\phi_0$ we see that
$$\sup_{\varepsilon>0}\int\nu_{g,z,\varepsilon}\,d\mu<\infty.$$
Then by the Cauchy criterion the denseness of $\mathcal D$ yields that whenever $z\in\mathbb{R}^{n}\setminus E$, there is the finite limit $L_{g,z}\phi:=\lim_{\varepsilon\to 0}\int\nu_{g,z,\varepsilon}\phi\,d\mu$ for all $\phi\in C^+_0(\mathbb{R}^{n})$. Hence by the Riesz representation theorem the positive linear functional $L_{g,z}$ corresponds to a Radon measure $\tilde{\lambda}_{g,z}$ to which the measures $\nu_{g,z,\varepsilon}\mu$ converge weakly.
The claim (b) follows from (\ref{eq2}):
\begin{align*}
\int\lambda_{g,z}(\mathbb{R}^{n})\,d\mathcal L^nz = &\int L_{g,z}\phi_0\,d\mathcal L^nz
= \int\mu_{\phi_0,g}\ast\nu(-z)\,d\mathcal L^n\\
&=\mu_{\phi_0,g}(\mathbb{R}^{n})\nu(\mathbb{R}^{n})=\mu(\mathbb{R}^{n})\nu(\mathbb{R}^{n}).
\end{align*}
The claim (c) follows from Lemma \ref{lemma1}, Fatou's lemma and the lower semicontinuity of the energy-integrals under the weak convergence.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theo1}]
Theorem \ref{theo1} follows from Lemma \ref{lemma1} and Theorem \ref{theo5}: Let
$$G=\{g\in O(n): \mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z))\geq s+t-n\})=0\}.$$
Then $G$ is a Borel set. We leave checking this to the reader. It is a bit easier when $A$ and $B$ are compact. We may assume the compactness since $A$ and $B$ as in the theorem contain compact subsets with positive measure, cf \cite{Fe}, 2.10.48. Suppose, contrary to what is claimed, that $\dim G > 2n-s-t+(n-1)(n-2)/2.$ Let
$\dim G > \a > 2n-s-t + (n-1)(n-2)/2$. Then by Frostman's lemma, cf. \cite{M4}, Theorem 8.8, and Lemma \ref{lemma2} there is $\theta\in\mathcal M(G)$ satisfying (\ref{eq9}) with $\beta=\a-(n-1)(n-2)/2>2n-s-t$.
By Frostman's lemma there are $\mu\in\mathcal M(A)$ and $\nu\in\mathcal M(B)$ such that $\mu(B(x,r))\leq r^s$ and $\nu(B(x,r))\leq r^t$ for all balls $B(x,r)$. Then by easy estimation, for example, as in the beginning of Chapter 8 in \cite{M4}, $I_{s'}(\mu)<\infty$ and $I_{t'}(\nu)<\infty$ for $0<s'<s$ and $0<t'<t$. By Theorem \ref{theo5}(b) the set $E_g=\{z:\lambda_{g,z}(\mathbb{R}^{n})>0\}$ has positive Lebesgue measure for $\theta$ almost all $g$. It then follows from Theorem \ref{theo5}(a) and (c) and (\ref{eq3}) that for $\theta$ almost all $g$,~ $\dim A\cap(g(B)+z)\geq s+t-n$ for almost all $z\in E_g$. This contradicts the definition of $G$ and the fact that $\theta$ has support in $G$.
\end{proof}
\section{Intersections and the decay of spherical averages}\label{decay}
For $\mu\in\mathcal M(\mathbb{R}^{n})$ define the spherical averages
$$\sigma(\mu)(r)=r^{1-n}\int_{S(r)}|\widehat{\mu}(x)|^2\,d\sigma_r^{n-1}x, r>0,$$
where $\sigma_r^{n-1}$ is the surface measure on the sphere $S(r)=\{x\in\mathbb{R}^{n}:|x|=r\}$.
Notice that if $\sigma(\mu)(r)\lesssim r^{-\gamma}$ for $r>0$ and for some $\gamma>0$, then $I_s(\mu)<\infty$ for $0<s<\gamma$. We now prove that under such decay condition we can improve Theorem \ref{theo1}:
\begin{thm}\label{theo3}
Let $t,\gamma$ and $\Gamma$ be positive numbers with $t,\gamma<n$. Let $\mu, \nu\in\mathcal M(\mathbb{R}^{n})$ with $\sigma(\mu)(r)\leq \Gamma r^{-\gamma}$ for $r>0, I_{\gamma}(\mu)<\infty$ and $I_t(\nu)<\infty$.
(a) If $\gamma+t>2n-1$, then
\begin{equation}\label{eq16}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim \spt\mu\cap (\spt\nu+z)\geq \gamma+t-n\})>0.
\end{equation}
(b) If $\gamma+t\leq 2n-1$, then there is $E\subset O(n)$ with
$$\dim E\leq 2n-1-\gamma-t+(n-1)(n-2)/2$$
such that for $g\in O(n)\setminus E$,
\begin{equation}\label{eq17}
\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim \spt\mu\cap (g(\spt\nu)+z))\geq \gamma+t-n\})>0.
\end{equation}
\end{thm}
\begin{proof} Let $u=\gamma+t-n$.
As above, we only need to show that the the conclusion of Lemma \ref{lemma1} holds under the present assumptions, but now the upper bound in (\ref{eq15}) will be a constant involving $\Gamma, I_t(\nu),I_{\gamma}(\mu), I_u(\mu),\mu(\mathbb{R}^{n})$ and $\nu(\mathbb{R}^{n})$.
For the statement (a) there is no $\theta$ integration (or $\theta$ is the Dirac measure at the identity map) and $n-1<u<n$, and for the statement (b) $\theta$ satisfies (\ref{eq9}) with some $\beta$ with $n-1-u<\beta<n-u$.
We again have (\ref{eq8}). The integration over $|z|\leq 2$ is easily controlled for example by
\begin{align*}
&\iiint_{|z|\leq 2} |x-g(z)|^{u-n}\,d\theta g|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\\
&\lesssim \mu(\mathbb{R}^{n})^2\nu(\mathbb{R}^{n})^2\iint_{|z|\leq 2, |x|\leq 3} |x-g(z)|^{u-n}\,d\mathcal L^nx\,d\mathcal L^nz+
\nu(\mathbb{R}^{n})^2\int |x|^{u-n}|\widehat{\mu}(x)|^2\,d\mathcal L^nx\\
&\lesssim (\mu(\mathbb{R}^{n})^2+I_u(\mu))\nu(\mathbb{R}^{n})^2.
\end{align*}
For the part where $|x|>2|z|$ or $|z|>2|x|$ we can argue as before.
To prove (a), suppose $n-1<u<n$. Then it suffices to show
$$\iint_{|z|/2\leq |x|\leq 2|z|, |z|>2} |x-z|^{u-n}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\lesssim \Gamma I_t(\nu).$$
Since $-1<u-n<0, n-1-\gamma\leq t-n$ and $|x-z|\geq||x|-|z||$, the integral over the part $||x|-|z||\leq 1, |z|>1,$ can be estimated by
\begin{align*}
&\iint_{||x|-|z||\leq 1, |z|>2} |x-z|^{u-n}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^n\,z\\
&\leq \int_{|z|>2}\int_{|z|-1}^{|z|+1}|r-|z||^{u-n}\int_{S(r)}|\widehat{\mu}(x)|^2\,d\sigma_r^{n-1}x\,dr|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma\int_{|z|>2} |z|^{n-1-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\leq \Gamma\int |z|^{t-n}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\lesssim \Gamma I_t(\nu).
\end{align*}
For the remaining part we have
\begin{align*}
&\iint_{||x|-|z|| > 1, 1<|z|/2\leq|x|\leq 2|z|} |x-z|^{u-n}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\\
&\leq\int\sum_{1\leq2^j\leq3|z|} \int_{2^j\leq||x|-|z||\leq 2^{j+1},|z|/2\leq|x|\leq 2|z|}||x|-|z||^{u-n}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\\
&\lesssim \int\sum_{1\leq2^j\leq3|z|} 2^{j(u-n)}\int_{2^j\leq||x|-|z||\leq 2^{j+1},|z|/2\leq|x|\leq 2|z|}|\widehat{\mu}(x)|^2\,d\mathcal L^nx|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&= \int\sum_{1\leq2^j\leq3|z|} 2^{j(u-n)}\int_{2^j\leq|r-|z||\leq 2^{j+1},|z|/2\leq r\leq 2|z|}\int_{S(r)}|\widehat{\mu}(x)|^2\,d\sigma_r^{n-1}x\,dr|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma \int\sum_{1\leq2^j\leq3|z|} 2^{j(u-n)}2^j|z|^{n-1-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma\int |z|^{u-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\lesssim \Gamma I_t(\nu).
\end{align*}
We establish the statement (b) by showing that
$$\iiint_{|z|/2\leq |x|\leq 2|z|,|z|>2} |x-g(z)|^{u-n}\,d\theta g|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\lesssim \Gamma I_t(\nu),$$
where $\theta\in\mathcal M(O(n))$ is as in Lemma \ref{lemma1} with $n-1-u<\beta<n-u$.
We first have as in the proof of Lemma \ref{lemma1}
\begin{align*}
&\int|x-g(z)|^{u-n}\,d\theta g =(n-u)\int_0^{\infty}\theta(\{g:|x-g(z)|<r\})r^{u-n-1}\,dr\\
&\lesssim \int_{||x|-|z||}^{\infty}(r/|z|)^{\beta}r^{u-n-1}\,dr=(n-\beta-u)||x|-|z||^{\beta+u-n}|z|^{-\beta},
\end{align*}
because $\theta(\{g:|x-g(z)|<r\})=0$ if $r<||x|-|z||$.
Using $-\beta+n-1-\gamma<t-n$ the integral over the part $||x|-|z||\leq 1$ can be estimated by
\begin{align*}
&\iiint_{||x|-|z||\leq 1,|z|>2} |x-g(z)|^{u-n}\,d\theta g|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(x)|^2\,d\mathcal L^nx\,d\mathcal L^n\,z\\
&\lesssim\int_{|z|>2} \int_{|z|-1}^{|z|+1}|r-|z||^{\beta+u-n}|z|^{-\beta}\int_{S(r)}|\widehat{\mu}(x)|^2\,d\sigma_r^{n-1}x\,dr|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma\int_{|z|>2} |z|^{-\beta+n-1-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\leq \Gamma\int |z|^{t-n}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz
\lesssim \Gamma I_t(\nu).
\end{align*}
For the remaining part we have
\begin{align*}
&\iiint_{||x|-|z|| > 1, 1<|z|/2\leq|x|\leq 2|z|} |x-g(z)|^{u-n}\,d\theta g|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\\
&\leq\int\sum_{1\leq2^j\leq3|z|} \int_{2^j\leq||x|-|z||\leq 2^{j+1},|z|/2\leq|x|\leq 2|z|}||x|-|z||^{\beta+u-n}|z|^{-\beta}|\widehat{\mu}(x)|^2|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nx\,d\mathcal L^nz\\
&\lesssim \int\sum_{1\leq2^j\leq3|z|} 2^{j(\beta+u-n)}|z|^{-\beta}\int_{2^j\leq||x|-|z||\leq 2^{j+1},|z|/2\leq|x|\leq 2|z|}|\widehat{\mu}(x)|^2\,d\mathcal L^nx|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&= \int\sum_{1\leq2^j\leq3|z|} 2^{j(\beta+u-n)}|z|^{-\beta}\int_{2^j\leq|r-|z||\leq 2^{j+1},|z|/2\leq r\leq 2|z|}\int_{S(r)}|\widehat{\mu}(x)|^2\,d\sigma_r^{n-1}x\,dr|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma\int\sum_{1\leq2^j\leq3|z|} 2^{j(\beta+u-n)}|z|^{-\beta}2^j|z|^{n-1-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\\
&\lesssim \Gamma\int |z|^{u-\gamma}|\widehat{\nu_{\varepsilon}}(z)|^2\,d\mathcal L^nz\lesssim \Gamma I_t(\nu).
\end{align*}
\end{proof}
For $0<s<n$ denote by $\gamma_n(s)$ the supremum of the numbers $\gamma$ such that
\begin{equation}\label{eq19}
\sigma(\mu)(r)\lesssim I_s(\mu)r^{-\gamma}\ \text{for}\ r>0
\end{equation}
holds for all $\mu\in\mathcal M(\mathbb{R}^{n})$ with support in the unit ball. Estimates for $\gamma(s)$ are discussed in \cite{M5}, Chapter 15. For $s\leq (n-1)/2$ the optimal, rather easy, result $\gamma(s)=s$ is valid, see \cite{M5}, Lemma 3.15. This together with Theorem \ref{theo3} yields immediately Theorem \ref{theo6}. For $s>(n-1)/2$ the optimal estimate fails and Theorem \ref{theo3} only gives a lower bound for the dimension of intersections which stays below and bounded away from $\dim A + \dim B - n$. The deepest estimates are due to Wolff \cite{W} in the plane and to Erdo\u gan \cite{E} in higher dimensions. They give $\gamma_n(s)\geq (n+2s-2)4$ for $n/2\leq s \leq (n+2)/2$. Theorem \ref{theo3} combined with this leads to the result that if $\dim A + \dim B/2 - (3n+2)/4>u>0$, then
$$\mathcal L^n(\{z\in\mathbb{R}^{n}: \dim A\cap (g(B)+z)) > u\})>0$$
for $g\in O(n)$ outside an exceptional set $E$ with $\dim E \leq n(n-1)/2 - u$. Plugging into Theorem \ref{theo3} other known estimates for $\gamma_n(s)$ gives similar rather weak intersection results.
\section{Examples}\label{Examples}
The first example here shows that in Theorem \ref{theo1} the bound $(n-1)(n-2)/2$ is sharp in the case where both sets have the maximal dimension $n$. This of course does not tell us anything in the plane but it explains the appearance of the dimension of $O(n-1)$.
In the following we identify $O(n-1)$ with a subset of $O(n)$ letting $g\in O(n-1)$ mean the map $(x_1,\dots,x_n)\mapsto (g(x_1\dots,x_{n-1}),x_n)$.
\begin{ex}\label{ex1}
Let $n\geq 3$. There are compact sets $A, B\subset\mathbb{R}^n$ such that $\dim A=\dim B=n$ and for every $g\in O(n-1)$,~ $\dim A\cap (g(B)+z)\leq n-1$ for all $z\in\mathbb{R}^n$.
\end{ex}
\begin{proof}
Let $C, D\subset\mathbb{R}$ be compact sets such that $\dim C = \dim D = 1$ and for every $z\in\mathbb{R}$ the intersection $C\cap(D+z)$ contains at most one point. Such sets were constructed in \cite{M1}, the construction is explained also in \cite{M4}, Example 13.18. Let $F$ be the closed unit ball in $\mathbb{R}^{n-1}$ and take $A=F\times C$ and $B=F\times D$. These sets have the required properties.
\end{proof}
The following example shows that we need some additional assumptions, for example as in Theorem \ref{theo3}, to get any result using only translations:
\begin{ex}\label{ex2}
There are compact sets $A, B\subset\mathbb{R}^n$ such that $\dim A = \dim B =n$ and for every $z\in\mathbb{R}^n$ the intersection $A\cap(B+z)$ contains at most one point.
\end{ex}
\begin{proof}
Let $C, D\subset\mathbb{R}$ be the compact sets of the previous example. Take $A=C^n$ and $B=D^n$. These sets have the required properties.
\end{proof}
I do not know what are the sharp bounds for the dimension of exceptional sets of Theorem \ref{theo1}. For simplicity, let us look at this question in the plane. Let $d(s,t)\in[0,1], 0<s,t\leq 2, s+t>2,$ be the infimum of the numbers $d>0$ with the property that for all Borel sets $A,B\subset\mathbb{R}^2$ with $\dim A = s, \dim B = t$ and for all $0<u<s+t-2$,
$$\dim\{g\in O(2): \mathcal L^2(\{z\in\mathbb{R}^2:\dim A\cap(g(B)+z)>u\})=0\}\leq d.$$
The problem is to determine $d(s,t)$. We know from Theorem \ref{theo1} that if $s+t>3$, then $d(s,t)\leq 4-s-t$. In particular $d(2,2)=0$. This suggests that $d(s,t)$ might be $4-s-t$ when $s+t>3$. However we know from Theorem \ref{theo6} that $d(s,t)\leq 3-s-t$ whenever $s\leq 1/2$. I would be happy to see some examples sheding light into this question.
\section{Concluding remarks}
As mentioned in the Introduction, intersection problems of this type for general sets were first studied by Kahane in \cite{K} and by the author in \cite{M1}, involving transformations such as similarities. For the orthogonal group the result in \cite{M2} concerned the case where one of the sets has dimension bigger than $(n+1)/2$. In \cite{M3} a general method was developed to get dimension estimates for the distance sets and intersections once suitable spherical average estimates (\ref{eq19}) for measures with finite energy are available. Such deep estimates were proved by Wolff in \cite{W} and Erdo\u gan in \cite{E}. They gave the best known results for the distance sets, see \cite{M5}, Chapters 15 and 16, but only minor progress for the intersections, as mentioned in Section \ref{decay}. The known estimates for $\sigma(\mu)$ are discussed in \cite{M5}, Chapter 15, see also \cite{LR} for a recent one.
The reverse inequality in Theorem \ref{theo1} fails: for any $0\leq s\leq n$ there exists a Borel set $A\subset \mathbb{R}^{n}$ such that
$\dim A\cap f(A)=s$ for all similarity maps $f$ of $\mathbb{R}^{n}$. This follows from \cite{F2}, see also Example 13.19 in \cite{M4} and the further references given there. The reverse inequality holds if one of the sets is a reasonably nice integral dimensional set, for example rectifiable, or if $\dim A\times B = \dim A + \dim B$, see \cite{M1}. This latter condition is valid if, for example, one of the sets is Ahlfors-David regular, see \cite{M4}, pp. 115-116. For such reverse inequalities no rotations $g$ are needed (or, equivalently, they hold for every $g$).
Exceptional set estimates in the spirit of this paper were first proved for projections by Kaufman in \cite{Ka}, then continued by Kaufman and the author \cite{KM} and by Falconer \cite{F1}. Peres and Schlag \cite{PS} proved such estimates for large classes of generalized projections. Exceptional set estimates for intersections with planes were first proved by Orponen \cite{O1} and continued by Orponen and the author \cite{MO}. In \cite{O2} Orponen derived estimates for radial projections. All these estimates expect those in \cite{MO} and some in \cite{PS} are known to be sharp. Some of these and other related results are also discussed in \cite{M5}.
Recently Donoven and Falconer \cite{DF} investigated Hausdorff dimension of intersections for subsets of certain Cantor sets and Shmerkin and Suomala \cite{SS} for large classes of random sets.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,183 |
{"url":"https:\/\/bryantravissmith.com\/2016\/01\/09\/implementing-logistic-regression-from-scratch-part-3-regularization\/","text":"This is a third part in a series of posts where I am implementing logistic regression from scratch. In the first post I discussed the theory of logistic regression, and in the second post I implemented it in python providing comparison to sklearn. In this post I will be showing how I implemented regularization from scratch.\n\nIn my previous post I also discussed the theory of regularization in the context of logistic regression where I show how different prior beliefs on the weights in the model lead to different update equations. This post will take advantage of the weight updates showed in that post.\n\n### Update for LASSO Regularization (Laplace Prior)\n\n$w^i = w^i + \\alpha \\ \\Bigg ( \\sum_{k} \\big(y_k - p(y_k = 1 |X,w)\\big) w^i - \\lambda_1 \\thinspace sign(w^i) \\Bigg )$\n\n### Update for Ridge Regularization (Gaussian Prior)\n\n$w^i = w^i + \\alpha \\ \\Bigg ( \\sum_{k} \\big(y_k - p(y_k = 1 |X,w)\\big) w^i - \\lambda_2 \\thinspace w^i \\Bigg )$\n\nThe simplest solution is to take the code from the previous post on implementing the code and just change the gradient function to include the regularized terms.\n\n# Starter Code\n\nThis is the template code of logistic regression similar to what I implemented in the previous post\n\nimport numpy as np\nclass LogisticRegression:\n\ndef __init__(self,X,y,tolerance=1e-5,l1=0.,l2=0.):\n"""Initializes Class for Logistic Regression\n\nParameters\n----------\nX : ndarray(n-rows,m-features)\nNumerical training data.\n\ny: ndarray(n-rows,)\nInterger training labels.\n\ntolerance : float (default 1e-5)\nStopping threshold difference in the loglikelihood between iterations.\n\nl1: float (default 0)\nLASSO regularization value\n\nl2: float (default 0)\nRidge regularization value\n\n"""\nself.tolerance = tolerance\nself.labels = y.reshape(y.size,1)\nself.w = np.zeros((X.shape[1]+1,1))\nself.features = np.ones((X.shape[0],X.shape[1]+1))\nself.features[:,1:] = X\nself.shuffled_features = self.features\nself.shuffled_labels = self.labels\nself.l1=l1\nself.l2=l2\nself.likelihood_history = []\n\ndef log_likelihood(self):\n"""Calculate the loglikelihood for the current set of weights and features.\n\nReturns\n-------\nout : float\n"""\n#Get Probablities\np = self.probability()\n#Get Log Likelihood For Each Row of Dataset\nloglikelihood = self.labels*np.log(p+1e-24) + (1-self.labels)*np.log(1-p+1e-24)\nloglikelihood = -1*loglikelihood.sum()\n#Return Sum\nreturn loglikelihood\n\ndef probability(self):\n"""Computes the logistic probability of being a positive example\n\nReturns\n-------\nout : ndarray (1,)\nProbablity of being a positive example\n"""\nreturn 1\/(1+np.exp(-self.features.dot(self.w)))\n\n"""Calculate the loglikelihood gradient for the current set of weights and features.\n\nReturns\n-------\nout : ndarray(n features, 1)\n"""\nerror = self.labels-self.probability()\nproduct = error*self.features\n\nParameters\n----------\nalpha : float\nThe learning rate for the algorithm\n\nmax_iterations : int\nThe maximum number of iterations allowed to run before the algorithm terminates\n\n"""\nprevious_likelihood = self.log_likelihood()\ndifference = self.tolerance+1\niteration = 0\nself.likelihood_history = [previous_likelihood]\nwhile (difference > self.tolerance) and (iteration < max_iterations):\ntemp = self.log_likelihood()\ndifference = np.abs(temp-previous_likelihood)\nprevious_likelihood = temp\nself.likelihood_history.append(previous_likelihood)\niteration += 1\n\ndef predict_probabilty(self,X):\n"""Computes the logistic probability of being a positive example\n\nParameters\n----------\nX : ndarray (n-rows,n-features)\nTest data to score using the current weights\n\nReturns\n-------\nout : ndarray (1,)\nProbablity of being a positive example\n"""\nfeatures = np.ones((X.shape[0],X.shape[1]+1))\nfeatures[:,1:] = (X-self.mean_x)\/self.std_x\nreturn 1\/(1+np.exp(-features.dot(self.w)))\n\ndef get_coefficients(self):\nreturn self.w.T[0]\n\nI will be \u00a0using the same data from the the regularization post:\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy as sp\n%matplotlib inline\ndef generate_1D_data(loc_positive=1,loc_negative=-1,size=100):\nX = np.hstack((np.random.normal(loc=loc_positive,size=size),np.random.normal(loc=loc_negative,size=size)))\nX = X.reshape(X.size,1)\ny = np.hstack((np.ones(size),np.zeros(size)))\nreturn X,y\nX,y = generate_1D_data()\nplt.figure(figsize=(16,8))\nplt.hist(X[y==1],alpha=0.5,color='steelblue',label='Positive Example')\nplt.hist(X[y==0],alpha=0.5,color='darkred',label='Negative Example')\nplt.xlabel('Variable Value')\nplt.ylabel('Count')\nplt.title('Histogram of Simulated 1D Data')\nplt.show()\n\nUsing the data generated from \u00a0above, my implementation generates the following results:\n\nlog = LogisticRegression(X,y,tolerance=1e-6,l1=0,l2=0)\nprint 'Cost Before Fit:', log.log_likelihood()\ncoef = log.get_coefficients()\nprint 'Cost After Fit:', log.log_likelihood()\nprint 'Intercept: ',coef[0]\nprint 'Slope: ',coef[1]\n\nCost Before Fit: 138.629436112\nCost After Fit: 62.6436146605\nIntercept: -0.0574759097926\nSlope: 2.31981998386\n\nWe can compare this to sklearn\u2019s implementation of logistic regression to make sure the results still match.\n\nfrom sklearn.linear_model import LogisticRegression as SKLogisticRegression\nlogSK = SKLogisticRegression(tol=1e-6,C=1e35)\nlogSK.fit(X.reshape(X.size,1),y)\nprint "Intercept: ",logSK.intercept_[0]\nprint "Slope: ",logSK.coef_[0,0]\n\nIntercept: -0.0576359266481\nSlope: 2.32334711964\n\nThe results are very close. Lets update the code to include regularization.\n\n# Update Code\n\nI have to update two functions to implement regularization. One is the log_likelihood, and the other is the log_likelihood_gradient.\n\n## Log Likelihood Code\n\ndef log_likelihood(self):\n"""Calculate the loglikelihood for the current set of weights and features.\n\nReturns\n-------\nout : float\n"""\n#Get Probablities\np = self.probability()\n#Get Log Likelihood For Each Row of Dataset\nloglikelihood = self.labels*np.log(p+1e-24) + (1-self.labels)*np.log(1-p+1e-24)\nloglikelihood = -1*loglikelihood.sum()\nloglikelihood += self.l1*np.abs(self.w).sum()\nloglikelihood += self.l2*np.power(self.w, 2).sum()\/2\n#Return Sum\nreturn loglikelihood\n\nIdeal we should probably check that only l1 or l2 is set, but this is only toy code. The first term implements the addition of the LASSO regularization to the cost function.\n\n$\\textrm{LASSO Cost} = \\lambda_1 \\thinspace \\sum_k |w^k|$\n\nThe second term implements the Ridge regularization to the cost function.\n\n$\\textrm{Ridge Cost} = \\frac{1}{2} \\thinspace \\lambda_2 \\thinspace \\sum_k |w^k|^2$\n\nThe next code update is to update the gradient equations as shown at the beginning of the post.\n\n"""Calculate the loglikelihood gradient for the current set of weights and features.\n\nReturns\n-------\nout : ndarray(n features, 1)\n"""\nerror = self.labels-self.probability()\nproduct = error*self.features\n\nThe first term implements subtracting the gradient for the LASSO regularization of the gradient.\n\n$\\frac{\\partial}{\\partial w^k} \\textrm{LASSO Cost} = \\lambda_1 \\thinspace sign(w^k)$\n\nThe second term implements subtracting the Ridge regularization of the gradient.\n\n$\\frac{\\partial}{\\partial w^k} \\textrm{Ridge Cost} = \\thinspace \\lambda_2 \\thinspace w^k$\n\n# Checking the implementation\n\nI will compare a couple of toy examples in this implementation compared to sklearn.\n\n## Ridge (Gaussian Prior)\n\n### Lambda = 100, C = 0.01\n\nfrom LogisticRegressionRegularization import LogisticRegression as BTSLogisticRegression\nlog = BTSLogisticRegression(X,y,tolerance=1e-9,l1=0,l2=100)\ncoef = log.get_coefficients()\nlogSK = SKLogisticRegression(tol=1e-9,C=0.01)\nlogSK.fit(X.reshape(X.size,1),y)\nprint "Sklearn Intercept, Bryan Intercept: ",logSK.intercept_[0], coef[0]\nprint "Sklearn Slope, Bryan Slope: ",logSK.coef_[0,0], coef[1]\n\nSklearn Intercept, Bryan Intercept: -0.00217832261688 -0.00217735345805\nSklearn Slope, Bryan Slope: 0.548644577533 0.54864036825\n\n### Lambda = 10, C = 0.1\n\nlog = BTSLogisticRegression(X,y,tolerance=1e-9,l1=0,l2=10)\ncoef = log.get_coefficients()\nlogSK = SKLogisticRegression(tol=1e-9,C=0.1)\nlogSK.fit(X.reshape(X.size,1),y)\nprint "Sklearn Intercept, Bryan Intercept: ",logSK.intercept_[0], coef[0]\nprint "Sklearn Slope, Bryan Slope: ",logSK.coef_[0,0], coef[1]\n\nSklearn Intercept, Bryan Intercept: -0.021489688722 -0.0214850033177\nSklearn Slope, Bryan Slope: 1.40733200454 1.40730512612\n\n## LASSO (Laplace Prior)\n\n### Lambda = 100, C = 0.01\n\nfrom LogisticRegressionRegularization import LogisticRegression as BTSLogisticRegression\nlog = BTSLogisticRegression(X,y,tolerance=1e-9,l1=100,l2=0)\ncoef = log.get_coefficients()\nlogSK = SKLogisticRegression(tol=1e-9,C=0.01,penalty='l1')\nlogSK.fit(X.reshape(X.size,1),y)\nprint "Sklearn Intercept, Bryan Intercept: ",logSK.intercept_[0], coef[0]\nprint "Sklearn Slope, Bryan Slope: ",logSK.coef_[0,0], coef[1]\n\nSklearn Intercept, Bryan Intercept: 0.0 -0.0524505475518\nSklearn Slope, Bryan Slope: 0.0892273998403 0.089274906091\n\n### Lambda = 10, C = 0.1\n\nfrom LogisticRegressionRegularization import LogisticRegression as BTSLogisticRegression\nlog = BTSLogisticRegression(X,y,tolerance=1e-9,l1=10,l2=0)\ncoef = log.get_coefficients()\nlogSK = SKLogisticRegression(tol=1e-9,C=0.1,penalty='l1')\nlogSK.fit(X.reshape(X.size,1),y)\nprint "Sklearn Intercept, Bryan Intercept: ",logSK.intercept_[0], coef[0]\nprint "Sklearn Slope, Bryan Slope: ",logSK.coef_[0,0], coef[1]\n\nSklearn Intercept, Bryan Intercept: 0.0 -0.00436123552719\nSklearn Slope, Bryan Slope: 1.58593771765 1.58595857714\n\n# Review\n\nIn this post I made minimal changes to the logistic regression code to implement Ridge and LASSO regularization. I also showed that the results between this implementation and sklearn on a simple toy model are consistent. The simple gradient decent algorithm, with its manual tuning of the learning rate, keeps this method from being ready for production. Attemping to use my code on larger and more nuanced data will lead to difficulties with training time and finding the \u2018optimal\u2019 learning rate.\n\nIn the next and last post in this series, I am going to alter the cost functions\/regularization to choose set priors that reflect prior experience, data, or conclusions that are not reflected in the dataset.\n\n# Thank You\n\nI appreciate you taking the time to read this post. I hope you gained something from it. \u00a0Feel free to reach out with any questions in the comments or directly through the email link in the menu.\n\n1 Comment","date":"2019-09-19 19:15:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 12, \"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\": 0.30358099937438965, \"perplexity\": 10017.532699590805}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573570.6\/warc\/CC-MAIN-20190919183843-20190919205843-00452.warc.gz\"}"} | null | null |
\section{Introduction}
\par
Thermal convection has been of major interest to the scientific community owing to its ubiquitous presence in many natural flows and engineering applications. Rayleigh-B\'{e}nard convection (RBC), in which a thin fluid layer is confined between a hot isothermal plate at the bottom and cold isothermal plate at the top, is an ideal prototype model to study convective turbulence. It is one of the few systems in which the flow approaches to the turbulent state upon increase in the parameter quite systematically, i.e., static region is followed by the time dependent region and then chaos which is followed by the spatio-temporal chaos and then turbulence \cite{Krishnamurthi_Howard_1981}. Flow characteristics of RBC depend on
the Rayleigh number $Ra=g\beta \Delta T H^{3}/\nu \alpha$, the Prandtl number $Pr = \nu/\alpha$ and the aspect ratio $~\Gamma=D/H$, the ratio of the horizontal to the vertical dimension of the container.
Where $\Delta T$ is the temperature difference between the bottom and top plates, $g$ the acceleration due to gravity, and $\alpha$, $\beta$ and $\nu$ are respectively the thermal diffusivity, thermal expansion coefficient and kinematic viscosity of the fluid.
The other parameters which are important are the Nusselt number $Nu$, defined as the ratio of the total heat flux to conductive heat flux and the Rossby number $Ro=V/2\Omega H$ (in case of rotating RBC), where $V$ is the free-fall velocity and $\Omega$ is the rotation rate.
\smallskip
\par
In turbulent state of RBC above a threshold Rayleigh number statistical coherence of the flow is restored with the appearance of large-scale circulation also known as ``mean-wind'' with turbulent background which is a self-organized structure having length scale of order of size of the container and time-scale of the order of eddy turn-over time.\cite{Krishnamurthi_Howard_1981,Casting_etal_1989,Sano_Wu_1989,Ciliberto_etal_1996,Qiu_Tong_2001a,Sun_etal_2005a,Sun_etal_2005b,Tsuji_etal_2005,Brown_Ahlers_PRL_2007}
On the course of time, the plane containing LSC exhibits change in its orientation also known as reorientations of LSC, which can occur in two ways in cylindrical domain: (a) by an azimuthal rotation of the plane containing LSC without considerable change in the circulation strength \cite{Cioni_etal_1997,Brown_Nikolaenko_Ahlers_PRL_2005}, called as rotation-led reorientation. (b) by momentary vanishing of the circulation strength accompanied by arbitrary change in the orientation of flow, termed as cessation-led reorientation \cite{Hansen_etal_1992,Brown_Nikolaenko_Ahlers_PRL_2005}.
Reorientations in which the large-scale flow completely changes its direction of circulation are termed as flow reversals \cite{Cioni_etal_1997}. Azimuthal changes in the orientation of LSC has been previously observed by many researchers \cite{Keller_1966, Welander_1967, Creveling_etal_1975, Gorman_etal_1984, Brown_Ahlers_JFM, Sreenivasan_etal_2002, Xi_Xia_PRE_2008}. Sudden changes in the orientation of LSC are not only interesting phenomenon, but are extremely relevant in many of the geophysical and astrophysical flows. For example, field reversals are common in the natural convection of Earth's atmosphere \cite{Van_Doorn}, magnetic field reversals in geodynamo \cite{Mishra_etal_EPL_2013}, etc.
Understanding the genesis of LSC and the nature of the instabilities that lead to the reorientations of LSC is one of the classical problems in RBC \cite{Ahlers_Grossmann_Lohse}.
\par
Experiments have played a key role in understanding the characteristics and dynamics of LSC in turbulent RBC.
Cioni \emph{et al} \cite{Cioni_etal_1997} identified LSC from dipolar temperature distribution obtained from probes placed along the azimuth of the container. Reversals in the flow were also obtained as the fluctuations in temperature switched sign randomly. They computed the Fourier transform of the temperature field along the azimuthal direction and observed that the amplitude of the first Fourier mode fluctuates about a finite value. However, the phase changes by $\pi$-radians time to time indicating the flow reversals. Niemela \emph{et al.} \cite{Niemela_etal_2001} and Sreenivasan \emph{et al.} \cite{Sreenivasan_etal_2002} also performed experiments in turbulent RBC with helium and reported reversals of LSC. Similar experiments were carried out by Brown \emph{et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005} and Brown and Ahlers \cite{Brown_Ahlers_JFM} with water as the working fluid. The probes were placed at three different horizontal planes ($z=1/4H$, $1/2H$ and $3/4H$) to get better idea on the dynamics of LSC. They identified reorientations of LSC using the phase of the first Fourier mode $\Phi_1$, and quantified them as rotation-led and cessation-led based on the amplitude of the modes. Brown and Ahlers \cite{Brown_Ahlers_JFM} observed that rotation-led reorientations were more frequent than cessation-led reorientations.
The rotation-led reorientations exhibited power law distribution, while cessation-led reorientations followed uniform distribution.
Further, both rotations and cessations showed Poisson distribution in time.
They observed that the probability density function (PDF) of time interval between successive reorientations follows an exponential function. Similar observations were made by Xi and Xia \cite{Xi_Xia_PRE_2008}. However, Sreenivasan \emph{et al.} \cite{Sreenivasan_etal_2002} showed that although the PDF of time interval between successive reorientations shows exponential nature, for lower time intervals it could be well fitted with power law distribution.
Xi and Xia \cite{Xi_Xia_PRE_2008} experimentally studied the aspect ratio dependency of reorientations in turbulent RBC. They observed that in contrast to the power law distribution in unit aspect ratio container, reorientations in $\Gamma=0.5$ geometry follows an exponential distribution.
In another study \cite{Xi_Xia_PRE_2007} they found that the reorientations occur with an order-of-magnitude more frequently in $\Gamma=0.5$ than these happen in $\Gamma=1$. Funfschilling and Ahlers \cite{Funfschilling_Ahlers_2004} observed that in addition to the azimuthal meandering of direction of circulation, the upper and lower halves of LSC are also found to undergo twisting oscillations.
\par
The numerical studies on the reorientations or reversals of LSC in RBC are rare compared to the experimental ones. Verzicco and Camussi \cite{Verzicco_Camussi_1999} performed a detailed numerical study on the effect of Prandtl number on the dynamics of convective turbulent flow inside a cylindrical cell. They noticed that there is significant drop in Nusselt number when the strength of LSC decreases for lower $Pr$ compared to higher $Pr$.
Later Stringano and Verzicco \cite{Stringano_Verzicco_JFM_2006} studied the mean flow structure in RBC in a slender cylindrical cell and observed a single convection roll breaking into two counter-rotating rolls stacked vertically one above the other. Benzi and Verzicco \cite{Benzi_Verzicco_2008} numerically investigated the statistical properties of the large scale flow in RBC at $Pr=0.7$ and $Ra=6\times10^5$, by using an external random perturbation on the temperature field.
A detailed numerical study on the dynamics of reorientations of LSC in RBC was carried out by Mishra \emph{et al.} \cite{MishraDe}, where they reproduced the experimentally observed rotation-led reorientations, cessation-led reorientations and double-cessations for a Rayleigh number range $6\times10^5\leq Ra\leq 3\times10^7$ which is considerably lower than the experimental counterparts.
\par
Over the years many works on LSC dynamics in 2D RBC have reported that corner-rolls play an important role in flow reversals.
Here the LSC is confined to a single plane and the complicated three-dimensional dynamics are absent \cite{Breuer_Hansen_2009,Paul_2010_Pramana,Hansen_etal_1990,Hansen_etal_1992}.
Sugiyama \emph{et al.} \cite{Sugiyama_Q2D_2010} combined both numerical and experimental analysis to evaluate the LSC dynamics in 2D/quasi-2D systems for a range of $Ra$ and $Pr$. The kinetic energy and size of the corner-rolls grow with time as a result of plume detachments from
the boundary layers, and finally they take over the main large-scale diagonal flow, thus resulting in reversal.
Similar relevance of corner-rolls in flow reversals has been reported in many numerical and experimental studies in 2D/quasi-2D systems \cite{Chandra_Verma_2011, chandra_Verma_2013,Vasilev_etal_2011,Yanagisawa_etal_2011,Wagner_Shishkina_2013,Ni_Huang_Xia_2015}.
However in 3-dimensional flow these phenomena are much complex. It is established that corner-rolls do exist in 3-D systems also \cite{Sun_etal_2005a,Foroozani_etal_2017}, but unlike the 2D/quasi-2D counterparts, these rolls are not strictly confined in the LSC plane (for 3-D systems). Resultantly, when fed with energy, the corner rolls need not essentially grow in diameter at the expense of shrinking the main LSC roll; rather, they can move or grow outside the LSC plane. De and Mishra \cite{De_Mishra_2018_IHTC} investigated the dynamics of the LSC in the turbulent RBC ($2\times10^6 \leq Ra \leq 2\times10^7$) inside a cubic box and reported that the plane containing LSC is aligned along one of the diagonal direction of the box. Further, they observed that over the course of time this plane containing LSC undergoes intermittent and chaotic switching between the two diagonals. Recent simulations by De \emph{et al.}\cite{De_EJMB_2018} have shown that in case of periodic boundary conditions, the thermal plumes drift in the lateral direction that leads to the flow reversals.
\par
Although, there are many existing literatures on the reorientation dynamics of LSC in turbulent RBC, the effect of rotation on these dynamics is less explored.
Kunnen \emph{et al.} \cite{kunnen2} investigated RRBC both numerically and
experimentally at $Ra=2\times10^9$ and $Pr=6.4$. At low rotation rates a domain-filling LSC was observed ($Ro^{-1}\lesssim 0.83$), while it breaks down at higher rotation rates. At higher rotation rates the Ekman-vortex structures diminishes the strength of the LSC, and as a result for $Ro^{-1}$ significantly above unity, the LSC structure breaks down \cite{Zhong_Ahlers_JFM_2010,Hart_etal_pof_2002}. They observed a retrograde precession of the LSC structure which depends on the rotation rate. However, prograde precession of the LSC at modest rotation rates ($Ro^{-1}\lesssim 1$) was observed by Weiss and Ahlers \cite{Weiss_Ahlers_JFM_2011_LSF} in their experimental investigations on RRBC in cylinder with aspect ratio $\Gamma=0.5$ for Rayleigh number range $2.3\times10^9\lesssim Ra\lesssim 7.2\times10^{10}$.
Further, they observed that for taller aspect ratio containers LSC can exist as single-roll or double-roll systems and modest rotation has an effect of stabilizing the single-roll and destabilizing the double roll. This was later numerically observed by Stevens \emph{et al.} \cite{stevens7} for the same parameter range.
\par
The questions which revolve around the research of the dynamics of LSC are: how the coherent structure appears amidst the turbulent background; how the heat transfer and boundary layer dynamics get affected due to its presence; what are the different characteristics of the reorientations and the corresponding distributions of occurrence of these events, etc. As discussed above there are many experimental works which have taken lead in these directions, but still comprehensive numerical and theoretical studies which may give some quantitative picture to them are lacking. In this paper, using the direct numerical simulations of RRBC we have been able to distinguish the different flow regimes (LSC dominated or not) by calculating relative energy contained in the Fourier mode corresponding to the LSC. We have identified a threshold value of the rotation rate below which the flow is mainly LSC dominated. Further, we have computed the variation in the boundary layer width in the plane in which LSC is confined and quantitatively showed its overall effect on the heat transfer rate. We have also identified the presence of different kind of reorientations, like, rotation-led and cessation-led, same as RBC and have extensively showed the effect of rotation on these events. Interestingly through our numerical work we have been able to consider the statistics of reorientations, which have not been previously attempted numerically and obtain the power law characteristics of them which are very close to the earlier experimental results.
\par
The paper is organized as follows. Section \ref{Sec:NUM_DET} details the mathematical formulations and numerical methods used along with grid independence study. The results of the present study with detailed discussions on the dynamics of reorientations of LSC and flow statistics are summarized in section \ref{Sec:Results}. Finally, the present work is concluded by section \ref{Sec:Conclusion} by outlining the main findings.
\section{Numerical details}\label{Sec:NUM_DET}
In this section, we briefly describe the mathematical formulations of the problem followed by grid independence study. A schematic diagram of the flow domain with relevant boundary conditions used is shown in Fig. \ref{flowdomain}(a). We consider a cylindrical cell of unit aspect ratio $\Gamma=D/H$ with bottom heated ($T_H$) and top cooled ($T_C$) configuration.
Rotation is applied about the vertical axis of the cylinder. Mathematically the problem can be described using the momentum, mass and energy conservation equations written in the rotating frame of reference using the Boussinesq approximation as
\small
\begin{eqnarray}
\frac{\partial \bm{u}}{\partial t}+\bm{u}\cdotp \nabla \bm{u} &=& -\nabla P + \sqrt{\frac{Pr}{Ra}} \nabla^{2} \bm{u} + \theta \bm{\hat{k}}-\frac{1}{Ro}(\bm{\hat{k}} \times \bm{u}) \label{gov_eqn1} \\
\nabla \cdotp \bm{u} &=& 0 \label{gov_eqn2}\\
\frac{\partial \theta}{\partial t} + \bm{u}\cdotp \nabla \theta &=& \frac{1}{\sqrt{Pr Ra}} \nabla^{2}\theta \label{gov_eqn3}
\end{eqnarray}
\normalsize
\smallskip
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.30]{CylinderD.pdf}\
\caption{(a) Schematic diagram of the flow domain with relevant boundary conditions. (b) Schematic representation of the location of numerical probes at mid-vertical cross section of the cylinder used for signal analysis.}
\label{flowdomain}
\end{figure}
\noindent
where $\bm{u}$ is the velocity in the rotating frame, $P$ is the modified pressure, $\hat{k}$ is the unit vector in the vertical direction and $\theta$ ($=(T-T_{C})/\Delta T$) is the non-dimensional temperature. We use $H$, $V=\sqrt{g\beta \Delta T H }$ and $\Delta T = (T_H - T_C)$ as the scales for normalization of length, velocity and temperature, respectively.
\smallskip
\par
The governing Eqs. \ref{gov_eqn1}- \ref{gov_eqn3} are solved using finite volume formulation with collocated arrangement of variables.
No-slip boundary conditions are implemented on all the surfaces. For temperature, horizontal surfaces are iso-thermal and the lateral wall is adiabatic.
The convective term is approximated using the $2$nd-order Adams-Bashforth scheme while the buoyancy and diffusive terms are approximated by the Crank-Nicholson scheme. Time increments of $\Delta t=10^{-3}$ and $5\times10^{-4}$ are used for $Ra=2\times10^6$ and $2\times10^7$, respectively. To adequately address the issue of velocity-pressure coupling, a predictor-corrector based momentum interpolation method \cite{Peter_De_Ocean} is used as the solution technique. Bi-Conjugate
Gradient Stabilized (BiCGSTAB) technique preconditioned by Stone's Strongly Implicit Procedure (SIP) \cite{Ferziger_Peric} is used to solve all the resulting sparse
linear systems. For details of the numerical procedures Peter and De \cite{Peter_De_Ocean} can be referred.
\begin{table*}[t!]
\caption{Different grids with corresponding number of grid points within the thermal boundary layer $N_{BL}$, the critical grid parameters $l_{max}/\eta_{k}$, $\Delta z_{max}/\eta_{k}$ and $\Delta_{max}/\eta_{k}$ followed by volume averaged Nusselt number $Nu_{vol}$, Nusselt number at the bottom wall $ Nu$ and ratio of numerical to analytical dissipation rates ($\langle\makebox{\large\ensuremath{\epsilon}}^{num}_u\rangle / \langle\makebox{\large\ensuremath{\epsilon}}^{th}_u\rangle $ and $\langle\makebox{\large\ensuremath{\epsilon}}^{num}_{\theta}\rangle / \langle\makebox{\large\ensuremath{\epsilon}}^{th}_{\theta}\rangle$) at $Ra=2\times10^6$ and $2\times10^7$ for non-rotating and high rotation case. }
\centering
\begin{tabular}{cc cc cc cc cc cc cc cc cc }
\hline \hline
\vspace {0.5em}
$Ra$&$Ro^{-1}$ &$N_{r} \times N_{\phi} \times N_{z}~ $&$N_{BL}$&\large{$\frac{l_{max}}{\eta_{k}}$}&\large{$\frac{\Delta z_{max}}{\eta_{k}}$}&\large{$\frac{\Delta_{max}}{\eta_{k}}$}&$~ Nu_{vol}~$&$~Nu~$& \large{$~\frac{\langle\makebox{\large\ensuremath{\epsilon}}^{num}_u\rangle}{\langle\makebox{\large\ensuremath{\epsilon}}^{th}_u\rangle }~$}&\large{$~\frac{\langle\makebox{\large\ensuremath{\epsilon}}^{num}_{\theta}\rangle}{\langle\makebox{\large\ensuremath{\epsilon}}^{th}_{\theta}\rangle }~$}&\\[.10ex]
\hline
\hline
\small \multirow{2}{*}{$2\times10^6$}& 0 & $43\times 101 \times 85$&$9$&$0.804$ & $0.515$&$0.455$&$10.75$&$10.71$&$0.981$& $0.987$&\\
\small & 10 & $43\times 101 \times 85$&$13$&$0.715$ & $0.457$&$0.404$&$3.68$&$3.69$&$0.978$& $0.990$&\\
\hline
\small \multirow{2}{*}{$2\times10^6$}& 0&$50\times 101 \times 101$&$11$&$0.804$ & $0.432$&$0.409$ &$10.68$&$10.66$&$0.990$& $0.983$&\\
\small & 10&$50\times 101 \times 101$&$15$&$0.715$ & $0.384$&$0.363$ &$3.68$&$3.69$& $0.990$& $0.982$&\\
\hline
\small \multirow{2}{*}{$2\times10^6$}& 0&$60\times 101 \times 115$&$13$&$0.804$ & $0.380$&$0.367$ &$10.65$&$10.60$&$0.991$& $0.984$&\\
\small & 10&$60\times 101 \times 115$&$19$&$0.715$ & $0.338$&$0.326$ &$3.67$&$3.68$&$0.985$&$0.984$&\\
\hline
\hline
\small \multirow{ 2}{*}{ $2\times10^7$}& 0 &$93\times 175 \times 185$&$12$&$0.968$& $0.492$&$0.492$ &$20.47$&$20.42$&$0.978$&$0.986$&\\
\small & 10 &$93\times 175 \times 185$&$27$&$0.724$& $0.368$&$0.350$ &$6.49$&$6.50$&$0.984$&$0.989$&\\
\hline
\small\multirow{ 2}{*}{ $2\times10^7$}& 0&$101\times 175 \times 201$&$13$&$0.968$& $0.453$&$0.453$ &$20.46$&$20.46$&$0.980$&$0.988$&\\
\small & 10&$101\times 175 \times 201$&$30$&$0.724$& $0.339$&$0.331$ &$6.37$&$6.38$&$0.986$&$0.992$&\\
\hline
\small \multirow{ 2}{*}{ $2\times10^7$}& 0 &$115\times 175 \times 229$&$15$&$0.968$& $0.391$&$0.391$ &$20.42$&$20.39$&$0.982$&$0.988$&\\
\small & 10 &$115\times 175 \times 229$&$32$&$0.724$& $0.293$&$0.304$ &$6.39$&$6.40$&$0.989$&$0.993$&\\
\hline \hline
\end{tabular}
\label{tab:grid_stats_full}
\smallskip
\end{table*}
\smallskip
\par
While reporting direct numerical simulations an important requirement is that the grid is resolved adequately to capture the flow physics correctly at the same time it consumes less computational resources.
Two criteria are imposed for calculating the grid sizes. Firstly, the boundary layers are aptly resolved, for which we necessitate that at least 10 grid points are present within the boundary layers (both near horizontal and lateral walls). As a result $\Delta z_{min} \approx \delta_{\theta} /10 $ and $\Delta z_{min} \approx \Delta r_{min}$. Secondly, the mesh size must be of the order less than or equal to the Kolmogorov scale. We implement the conditions $\Delta_{max}\lesssim \eta_K$, $l_{max}(=2\pi r/N\theta) \lesssim \eta_K$ , $\Delta z_{max} < \eta_K$ and $\Delta r_{max} \approx \Delta z_{max}$. Here $\Delta_{max}$ indicates the maximum of mean grid size calculated as $(r\Delta \phi \Delta r \Delta z)^{1/3}$ and $\eta_K$ is the Kolmogorov scale estimated using the analytical formula $\eta_K \approx \pi (Pr^2/RaNu)^{1/4}$ given by Gr\"{o}tzbach \cite{grotzbach}. Table \ref{tab:grid_stats_full} shows the different computational grids with corresponding number of grid points within the boundary layer followed by the maximum grid size in the azimuthal and axial direction, and the maximum mean grid size, normalized with the Kolmogorov scale for both the Rayleigh numbers at $Ro^{-1}=0$ and $10$.
\par
The Nusselt number at the bottom wall is computed as $Nu = \langle \partial \theta / \partial z \rangle_{A,t}$, where $\langle .. \rangle_{A,t}$ represents averaging over horizontal plane and time. Further, the volume averaged Nusselt number over the entire domain $ Nu_{vol}=1+\sqrt{RaPr} \langle w\theta \rangle$ ($\langle ..\rangle$ denotes averaging over volume and time) is also calculated. Both values agree well and the difference between the $ Nu_{vol}$ (or $Nu$ ) values computed at the coarsest and finest grid for both $Ra$ is less than $2\%$, which indicates a good grid convergence. To further confirm the spacial resolution, the ratio of the numerical to the theoretical estimate of the viscous ($\langle\makebox{\large\ensuremath{\epsilon}}^{num}_u\rangle/\langle\makebox{\large\ensuremath{\epsilon}}^{th}_u\rangle$) and thermal dissipation rates ($\langle\makebox{\large\ensuremath{\epsilon}}^{num}_{\theta}\rangle/\langle\makebox{\large\ensuremath{\epsilon}}^{th}_{\theta}\rangle$) are calculated for both non-rotating ($Ro^{-1}=0$) and sufficiently high rotation rate $Ro^{-1}=10$.
Here $\langle\makebox{\large\ensuremath{\epsilon}}^{num}_u\rangle =\nu V^{2}H^{-2}\langle\lvert\nabla \bm{u}\rvert^{2}\rangle$~and~$\langle\makebox{\large\ensuremath{\epsilon}}^{num}_{\theta}\rangle=\alpha(\Delta T)^{2}H^{-2}\langle\lvert\nabla\theta \rvert^{2}\rangle$~are the numerically calculated values of viscous and thermal dissipation rates, respectively, while $\langle\makebox{\large\ensuremath{\epsilon}}^{th}_u\rangle=\nu^{3}(Nu-1)RaPr^{-2}H^{-4} $ and $\langle\makebox{\large\ensuremath{\epsilon}}^{th}_{\theta}\rangle =\alpha (\Delta T)^{2}NuH^{-2}$ are their analytical counterparts\cite{Shraiman_Siggia_1990}.
Note that the theoretical estimates of the dissipation rates are functions of Nusselt number, for which we substitute the corresponding numerically obtained value \cite{stevens5,MishraDe}.
The ratios are always close to unity (minimum $\approx 0.978$), suggesting that the present grid is well resolved to perform the proposed numerical calculations.
Considering the computational cost and the accuracy of the above mentioned parameters, a grid size of $50\times 101 \times 101$ and $101\times 175 \times 201$ are chosen for $Ra=2\times10^6$ and $2\times10^7$, respectively.
\vspace{3 mm}
\section{Results and Discussions}\label{Sec:Results}
In this section we discuss the principal findings of the present work.
First we establish the presence of LSC in our simulations of RBC with and without rotation.
Then we study the characteristics of LSC and its relation with heat transfer, boundary layer dynamics, and the thermal and viscous dissipation rates.
This is followed by a detailed discussion on the dynamics of LSC, quantified by the Fourier mode analysis. Finally, we investigate the statistics of reorientations of LSC.
\subsection{Identification of LSC}
Large-scale circulation is a coherent structure in RBC where the thermal plumes organize to form a ``large-scale'' convection roll with hot and cold fluid rising and dipping, respectively, along opposite sides of the lateral wall.
Over the years a number of experimental \cite{Krishnamurthi_Howard_1981,Funfschilling_Ahlers_2004,Sun_etal_2005a,Brown_Ahlers_JFM} and numerical \cite{Stringano_Verzicco_JFM_2006,Verzicco_Camussi_1999,MishraDe} studies have reported presence of LSC and its associated dynamics for non-rotating RBC. However, such attempts on the dynamics of LSC for flows modulated by rotation are few. Here we focus on the identification of LSC in rotating and non-rotating RBC using flow structures and time signals of vertical velocity.
\par
An instantaneous snapshot of the temperature iso-surfaces is shown in Fig. \ref{LSC_Diss1}(a) for $Ra=2\times10^7$ and $Ro^{-1}=0.2$. Xi \emph{et al.} \cite{Xi_Lam_Xia_2004} experimentally studied RBC and observed that LSC is a result of organization of the thermal plumes. On a similar note, we observe that hot and cold plumes are clustered separately along opposite side of the cylinder, thus indicating an LSC structure. The dotted line indicates an approximate azimuthal orientation of LSC.
The temperature contours at the vertical mid-plane along which LSC is aligned [indicated in Fig. \ref{LSC_Diss1}(a)] is shown in Fig. \ref{LSC_Diss1}(b). For clarity, we have shown a dashed line which gives the directional overview of LSC in that plane. The hot (cold) plumes rise (fall) from the bottom (top) surface, proceed along the cylinder bulk and plunge into the boundary layer near the opposite wall. A similar flow structure is observed at higher rotation rate ($Ro^{-1}=0.5$) shown in Fig. \ref{LSC_Diss1}(c), with a change in the sense of circulation. We observe similar large-scale structure at $Ro^{-1}<1$.
As we further increase the rotation, the coherence at large scale is lost. For example at $Ro^{-1}=2$ shown in in Fig. \ref{LSC_Diss1}(d), the temperature contours indicate a distorted structure.
\begin{figure}
\centering
\includegraphics[scale=0.110]{LSC_Ra7Ro5.jpg}\\
\includegraphics[scale=0.110]{LSC_Ra7Ro2_05.jpg}
\caption{(Colour online) (a) Temperature iso-surfaces at $\theta=0.4$ (blue) and $\theta =0.6$ (red) indicating LSC for $Ra=2\times10^7$ and $Ro^{-1}=0.2$. The dotted line indicates the azimuthal orientation of LSC. (b) Temperature contour at the vertical plane where the LSC is oriented. The dotted line indicates the direction of circulation. (c) Temperature contour at LSC plane for $Ro^{-1}=0.5$. (d) Temperature contour at $\phi=0$ plane for $Ro^{-1}=2$.}
\label{LSC_Diss1}
\end{figure}
\par
In order to characterize the LSC, time signals of vertical velocity are recorded at $36$ equispaced azimuthal stations that are located at the mid-vertical plane ($z=0.5$) near the lateral wall ($r=0.4R$).
Figure \ref{Ra7_Sig2} shows time signals of vertical velocity at either ends of two diametrical planes separated by an angle $\pi/2$. The LSC is primarily identified by the occurrence of non-zero mean vertical velocities that are anti-correlated, shown in rectangular windows (red). Note while the LSC persists along a diameter, high-variance fluctuations are observed on the other and they switch between each other.
The time over which the LSC persists along a diameter depends strongly on the rotation rate. Figure \ref{Ra7_Sig1} shows that as rotation rate is increased the switching frequency of the signals increases. These are partly associated with reorientations of LSC along the azimuthal plane, discussed in subsequent sections.
The time average vertical velocity along the above-mentioned azimuthally equispaced numerical probes is shown in Fig. \ref{fig:Wazi}, where the averaging is carried out within a time over which LSC remains in a particular plane. For the non-rotating case (and low rotation rates) the profile shows a cosine function with a single cycle spanning the entire ($2\pi$) domain [see Fig. \ref{fig:Wazi}(a)] which confirms the presence of LSC structure, and is consistent with previous studies \cite{Cioni_etal_1997,StevensPoF2011,Xi_etal_PRL_2009,Brown_Ahlers_JFM}. This particular nature of the flow in the azimuthal direction can be associated with the dipolar structure.
As rotation rate increases to $Ro^{-1}=2$ we find a distortion from the cosine fit, as shown in Fig. \ref{fig:Wazi}(b).
Further increase in rotation leads to deviation from the dipolar structure and flow appears to be more close to the quadrupolar structure, indicated by two pairs of $+$ve and $-$ve peaks in the velocity profile [refer Fig. \ref{fig:Wazi}(c)].
At very high rotation rates ($Ro^{-1}=20$), the vertical velocity profile appears to be more close to a sextupolar behaviour with $+$ve (up-flow) and $-$ve (down-flow) velocity separated azimuthally by $\pi/3$. Note that at lower and higher rotation rates we observe almost perfect cosine fits for the vertical velocity profile, while at intermediate rotation rates it deviates from cosine fit.
\begin{figure}[h!]
\centering
\includegraphics[scale=1.150]{Ra7_Sig_pofA3_Ro.pdf}
\caption{(Colour online) Time traces of vertical velocity for $Ra=2\times10^7$ and $Ro^{-1}=0.1$. The red and blue boxed regions indicate the anti-correlated signals and high-variance data, respectively, at diameters which are $\pi/2$ apart.}
\label{Ra7_Sig2}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=1.30]{Ra7_Sig_pofB3_Ro.pdf}
\caption{Time traces of vertical velocity from numerical probes which are $\pi$ apart in the azimuthal direction at $Ra=2\times10^7$ for different $Ro^{-1}$. The switching frequency of the signals increases with the increase in rotation rate.}
\label{Ra7_Sig1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.55]{WziF_A1.pdf}\\
\includegraphics[scale=0.55]{WziF_B1.pdf}
\caption{The temporal averaged vertical velocity profile along the azimuthal direction for $Ra=2\times10^6$ at (a)$Ro^{-1}=0$, (b)$Ro^{-1}=2$, (c)$Ro^{-1}=7.5$ and (d)$Ro^{-1}=20$. The solid red line indicates a pure cosine fit.}
\label{fig:Wazi}
\end{figure}
\par
Following the works of Cioni \emph{et al} \cite{Cioni_etal_1997}, Brown and Ahlers \cite{Brown_Ahlers_JFM} and Mishra \emph{et al.}\cite{MishraDe},
to get further insights into LSC and its dynamics, we carry out Fourier transform of the vertical velocity recorded along the azimuthal direction at mid-vertical plane.
Note that $u_j$ is the velocity signal from $N~(=36)$ data points along the cylinder azimuth [refer Fig. \ref{flowdomain}(b)] and its Fourier transform is written as
\begin{eqnarray}
\hat{u}_{k}= \sum_{j=1}^{N} u_{j} e^{-i2\pi k j/N} \label{gov_eqn4}
\end{eqnarray}
\noindent
where $\hat{u}_{k}$ represents the $k-$th Fourier mode with $\Phi_k$ being its phase.
The first ($\hat{u}_{1}$), second and third modes are associated with the dipolar, quadrupolar and sextupolar structure of the flow, respectively \cite{Xi_Zhang_JFM_2016}.
Based on the formalism proposed by Kunnen \emph{et al.} \cite{kunnen2}, Weiss and Ahlers \cite{Weiss_Ahlers_JFM_2011_LSF} and Xi \emph{et al.} \cite{Xi_Zhang_JFM_2016}, we have characterized the LSC using the fraction of energy contained in the $k-$th mode $E_{k}/E_{tot}$, where $E_{k}=|\hat{u}_k|^2$ and $E_{tot}= \sum_{k=1}^{N/2} E_{k}$. The evidence of LSC is often ascertained from the energy fraction of first Fourier mode ($E_{1}/E_{tot}$).
Figure \ref{fig:Fmodes} shows the variation of the energy fraction of first three Fourier modes with rotation rate.
We identify different flow regimes based on the dominance of specific modes. In regime \Romannum{1} ($0 \leq Ro^{-1} \lesssim 1$), the first Fourier mode is clearly dominant with $E_{1}/E_{tot} > 0.5$ and is identified as the LSC regime. Consequently, the average vertical velocity profile along the azimuthal direction clearly shows a perfect cosine fit [see Fig. \ref{fig:Wazi}(a)] as discussed before. A decrease in the energy fraction of the first Fourier mode accompanied by increase in that of second ($E_{2}/E_{tot}$) with the increase in rotation rate is observed here. At high rotation rates regime \Romannum{2} is identified ($1 \lesssim Ro^{-1} \lesssim 10$) where all the three modes share almost equal energy. As a result the vertical velocity profiles in this regime are far from perfect cosine fit, as shown in Figs. \ref{fig:Wazi}(b) and (c).
For $Ra=2\times10^6$ although the second mode marginally dominates at moderately high rotation rates ($5 < Ro^{-1} \lesssim 10$), it declines at extremely high rotation rates. Here, we even observe a third Fourier mode dominated flow for $Ro^{-1} \gtrsim 10$. This is identified as regime \Romannum{3}, where the higher mode (second or third) shares more than $50\%$ of the total energy.
For $Ra=2\times10^7$ the second Fourier mode dominates over the other modes in regime \Romannum{3}.
\begin{figure}
\centering
\includegraphics[scale=0.40]{F_modesB.pdf}
\caption{(Colour online) The variation of energy fraction of the first three Fourier modes with rotation rate for $Ra=2\times10^6$ and $2\times10^7$.}
\label{fig:Fmodes}
\end{figure}
\par
In order to make it more quantitative we also compute the strength of LSC ${S}_{k1}$ \cite{StevensPoF2011}, defined as
\begin{eqnarray}
{S}_{k1} = Max \Big[ \Big( \frac{ E_{1}}{ E_{tot}} -\frac{1}{N} \Big) / \Big(1-\frac{1}{N}\Big), 0 \Big]
\end{eqnarray}
\noindent
The value of ${S}_{k1}$ lies between $0$ and $1$. The strength close to unity indicates that most of the energy is contained in the first Fourier mode, i.e., $E_{1}/E_{tot}\approx1$ and the azimuthal vertical velocity profile is close to a cosine fit, thus indicating LSC. On the other hand, its value near to zero implies that the energy fraction of first Fourier mode is smaller ($E_{1}/E_{tot}<< 1$), in effect most of the energy is contained in the higher modes and the flow behaviour is far from LSC structure.
In Fig. \ref{fig:Sk} we plot the circulation strength against the rotation rate. At low rotation rates the strength is as high as $0.7$, while it drops exponentially with the increase in rotation rate. We observe a perfect exponential fit as $S_{k1}\propto\exp (-0.5 Ro^{-1} )$ and $S_{k1}\propto\exp (-0.8 Ro^{-1} )$ for $Ra=2\times10^6$ and $2\times10^7$, respectively. The decay of $S_{k1}$ is at a higher rate for the higher Rayleigh number. Similar observation is made from the energy fraction of the first Fourier mode (refer Fig. \ref{fig:Fmodes}), where it declines steeply for $Ra=2\times10^7$.
We identify the LSC regime as $Ro^{-1}\lesssim1$ where the circulation strength is greater than $0.5$.
This is consistent with the observations made by Kunnen \emph{et al.} \cite{kunnen2} and Stevens \emph{et al.}\cite{stevens7}, where they reported the break-down of LSC for $Ro^{-1}\gtrsim 0.86$.
\begin{figure}
\centering
\includegraphics[scale=0.290]{SkB2.pdf}
\caption{Exponential decay of the strength of the first Fourier mode with the increase in rotation rate for $Ra=2\times10^6$ and $2\times10^7$.}
\label{fig:Sk}
\end{figure}
\subsection{Effect of LSC on heat transfer and boundary layer dynamics}
In the previous subsection we identified several flow regimes based on the energy contained in the Fourier modes related to the large scale circulation. In this subsection we investigate the nature of heat transfer, dissipation rates and dynamics of boundary layer in those regimes. In addition we also explore
how the presence of LSC overall affects the features of the boundary layer.
\begin{figure}
\centering
\includegraphics[scale=0.2750]{Bl_theory_pof.pdf}
\caption{(a) Schematic representation of the time averaged temperature profile showing the intersection of linear fit near the bottom plate with the bulk temperature. Inset shows the geometric construction of the thermal boundary layer thickness.}
\label{BLT_plot1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.1300]{LSC_TblA_Ra7Ro5_colA.jpg}\\
\includegraphics[scale=0.330]{Bl_LSC_B.pdf}
\caption{(Colour online) (a) Instantaneous ($t=4400$) temperature iso-surfaces at $\theta=0.4$ (blue) and $\theta =0.6$ (red) indicating LSC. The dotted line represents the azimuthal orientation of LSC. (b) Time averaged temperature iso-surfaces at $\theta=0.4$ and $\theta =0.6$, obtained from a time span ($4200-4400$) at which LSC persists. (c) Contours of thermal boundary layer thickness near the top and bottom plate. (d) Variation of boundary layer thickness along and perpendicular to the direction of LSC, at the bottom. }
\label{BLT_plot_LSC}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.250]{TblNu_Avg_RmsB.pdf}\includegraphics[scale=0.270]{Ra6_blazi.pdf}\includegraphics[scale=0.270]{Ra7_blazi.pdf}
\caption{(Colour online) (a) Thermal boundary layer thickness computed by different methods, as a function of inverse $Ro$ for $Ra=2\times10^7$. $\langle \delta_{\theta}^{sl} \rangle_A$ indicates the area averaged boundary layer thickness calculated from the slope method. (b) and (c) Variation azimuthal averaged thermal boundary layer thickness along the radial direction for $Ra=2\times10^6$ and $2\times10^7$, respectively, at different rotation rates.}
\label{Bl}
\end{figure*}
\par
In order to characterize the boundary layer we compute the thermal boundary layer thickness using three techniques. Firstly, $\delta_{\theta}^{Nu}$ which indicates the analytical thermal boundary layer thickness obtained from the relation $\delta_{\theta}^{Nu} \approx 1/2 Nu $ \cite{Belmonte_Tilgner_Libchaber,Verzicco_Camussi_1999}.
Secondly, $ \delta_{\theta}^{\sigma}$ which corresponds to the boundary layer thickness computed as the vertical distance of the peak rms of temperature from the bottom plate \cite{Verzicco_Camussi_2003}.
Finally, $ \delta_{\theta}^{sl}$ which is the boundary layer thickness calculated as the vertical distance of the point from the bottom plate where the linear fit of the time-averaged temperature profile approaches the bulk temperature \cite{Zhou_n5_PoF_2011,Lui_Xia}. A sample temperature profile is shown in Fig. \ref{BLT_plot1} where the geometric construction of the thermal boundary layer thickness is shown in the inset. Similar technique is adopted to compute the boundary layer thickness for all the spatial locations ($r,\phi$).
\par
At this point we are interested to see how the boundary layer thickness varies along and perpendicular to the plane containing LSC.
Figure \ref{BLT_plot_LSC}(a) shows the instantaneous temperature iso-surfaces indicating LSC structure. The time averaged flow structure for the same is shown in Fig. \ref{BLT_plot_LSC}(b), which is used to compute the thermal boundary layer thickness. Here averaging is performed within a time-span at which LSC persists in a particular plane.
The planar contours thus obtained near the top and bottom plates are shown in Fig. \ref{BLT_plot_LSC}(c), where the orientation of LSC is shown by dashed line.
Close to the bottom plate, thermal boundary layer is thicker near the region where hot plumes rise and is thinner near the opposite side of the lateral wall where the cold plumes plunge into the boundary layer. Similar observation is made near the top plate. Further, we plot $\delta_{\theta}^{sl}$ along and perpendicular to the direction of LSC in Fig. \ref{BLT_plot_LSC}(d). Note that perpendicular to LSC the boundary layer thickness varies almost symmetrically. However, along the plane containing LSC, $\delta_{\theta}^{sl}$ shows an asymmetric trend, as it is thicker at one side and thinner near the opposite side of the lateral wall. This is consistent with the time traces of vertical velocity discussed in the previous subsection (refer Fig. \ref{Ra7_Sig2}), wherein, across LSC the signals show high-variance, zero-mean fluctuations and along LSC they exhibit non-zero mean with negative correlation near opposite sides of the lateral wall.
\par
Figure \ref{Bl}(a) shows the comparison between the thermal boundary layer thickness computed by different methods for $Ra=2\times10^7$. The boundary layer thickness remains constant at low rotation rate and increases at higher rotation rates.
We observe that at low rotation rates, boundary layer thickness obtained from all three methods are consistent, while at high rotation rates ($Ro^{-1}\gtrsim10$) the one computed from the rms profile underpredicts the other two.
Similar observation was made by Zhou \emph{et al.} \cite{Zhou_n5_PoF_2011} in two-dimensional RBC, where the thickness from the rms method was found about $20\%$ below that obtained from the slope method.
The variation of azimuthal averaged boundary layer thickness $\langle \delta_{\theta}^{sl} \rangle_{\phi}$ along the radial direction is shown in Figs. \ref{Bl} (b) and (c).
We find that, the boundary layer is thinner (and almost same as the non-rotating case) at low rotation rates and increases considerably at higher rotation rates.
The effect of rotation is evident near the core region of the cylindrical domain.
However, it is less apparent near the lateral wall due to the near wall viscous effects. Further, we observe that for low rotation rates, particularly in the LSC regime [red lines in Figs. \ref{Bl}(b) and (c)], the boundary layer is thicker near the wall and thinner near the core region, while at high rotation rates the trend is reversed.
As rotation rate increases, the flow stabilizes as per Taylor Proudman theorem \cite{taylorp,proudmant} and the temperature field approaches a conduction-like profile.
Here we identify that flow stabilization is reflected noticeably near the core region as $\langle \delta_{\theta}^{sl} \rangle_{\phi}$ is much thicker along the core compared to that near the lateral wall.
\par
Two quantities which play an important role in local and global heat transport process are the thermal and viscous dissipation rates.
The global average of dissipation rates are connected to the global heat transport through the analytical relations $\langle\makebox{\large\ensuremath{\epsilon}}^{th}_u\rangle = \nu^{3}H^{-4} (Nu-1)RaPr^{-2}$ and $\langle \makebox{\large\ensuremath{\epsilon}}^{th}_{\theta}\rangle=\alpha (\Delta T)^{2}NuH^{-2}$ \cite{Shraiman_Siggia_1990}.
Dissipation rates are often used to identify different flow regimes \cite{Grossmann_Lohse_JFM_2000, Grossmann_Lohse_2004}, establish scaling laws and quantify different flow characteristics. For instance, Shishkina and Wagner \cite{Shishkina_Wagner_2006,Shishkina_Wagner_2008} investigated the formation and interactions of thermal plumes by evaluating the dissipation rates.
\par
Here we analyze the dissipation rates and their association with the orientation of LSC.
The contours of thermal and viscous dissipation rates near the top and bottom plates [at the same instant as in Fig. \ref{BLT_plot_LSC}(a)] are shown in Figs. \ref{LSC_Diss}(a) and (b), where the azimuthal orientation of LSC is represented by the dashed line. We observe that the dissipation rates vary significantly along the direction of LSC. The thermal and viscous dissipations are maximum in the regions where the plumes collide with the boundary layer. Similar high-amplitude dissipation events associated with the collision of thermal plumes were observed by Schumacher and Scheel \cite{Schumacher_Scheel_2016}. Further, the contours of Nusselt number at the top and bottom plates are shown in Fig. \ref{LSC_Diss}(c), which also suggests maximum heat transfer occurs at locations where the plume collide with the boundary layer.
\begin{figure}
\centering
\includegraphics[scale=0.370]{LDC_Diss_Nu_ColB1.jpg}
\caption{(a) and (b) Contours of thermal and viscous dissipation near the bottom (left panel) and top (right panel) plate. The maximum dissipation is observed in the region where the plumes splash into boundary layers near the isothermal plates. (d) Contours of Nusselt number at the bottom (left panel) and top (right panel) plate. The dotted line indicates the azimuthal orientation of LSC.}
\label{LSC_Diss}
\end{figure}
\par
Next we discuss the behaviour of the Nusselt number which is a measure of heat transport, and dissipation rates in different flow regimes.
Figure \ref{Nu_scaling} shows the variation of average Nusselt number with rotation rate at different Rayleigh numbers.
In regime \Romannum{1} ($0 \leq Ro^{-1} \lesssim 1$) or LSC regime, heat transfer remains almost constant. As rotation rate increases Nusselt number drops linearly as $\langle Nu \rangle \propto 1/Ro$ in regime \Romannum{2} given by $1 \lesssim Ro^{-1} \lesssim 10$. However, at high rotation rates ($Ro^{-1} \gtrsim 10$, regime \Romannum{3}) the Nusselt number shows a power law behaviour, $\langle Nu \rangle \propto (1/Ro)^{b}$, with the exponent $b=-0.93$ and $-0.53$ for $Ra=2\times10^6$ and $2\times10^7$, respectively. The solid red and blue lines in Fig. \ref{Nu_scaling} represent the corresponding linear and power law fits at regimes \Romannum{2} and \Romannum{3}, respectively.
\begin{figure}
\centering
\includegraphics[scale=0.3]{Nufull_pof_scalingsC.pdf}
\caption{(Colour online) Variation of average Nusselt number with rotation rate at $Ra=2\times10^6$ and $2\times10^7$. The solid red and blue lines indicate linear and power law fits, respectively}
\label{Nu_scaling}
\end{figure}
\par
Figure \ref{Diss_scale} shows the ensemble averaged non-dimensional dissipation rates, $\langle\makebox{\large\ensuremath{\epsilon}}_u\rangle =\sqrt{Pr/Ra} \langle\lvert\nabla \mathbf{u} \rvert ^{2}\rangle$ and $\langle\makebox{\large\ensuremath{\epsilon}}_{\theta}\rangle=\sqrt{1/RaPr} \langle\lvert\nabla \theta \rvert^{2}\rangle$, normalized by the corresponding non-rotating dissipation rates $\langle\makebox{\large\ensuremath{\epsilon}}_u^{o}\rangle$ and $\langle\makebox{\large\ensuremath{\epsilon}}_{\theta}^{o}\rangle$, respectively, for different rotation rates. In regime \Romannum{1} where LSC is observed both dissipation rates remain almost constant and close to the non-rotating value.
However, at higher rotation rates a significant departure from the non-rotating case is noticed.
At regime \Romannum{2} the dissipation rates decrease linearly with rotation rate, as observed for the Nusselt number. This is followed by rotation dominated regime \Romannum{3} where they follow a power law behaviour as $\langle\makebox{\large\ensuremath{\epsilon}}\rangle \propto (1/Ro)^{b}$, with exponent $b=-1.95$ and $-0.68$ for $Ra=2\times10^6$ and $2\times10^7$, respectively, for $\langle\makebox{\large\ensuremath{\epsilon}}_u\rangle$, while $\langle\makebox{\large\ensuremath{\epsilon}}_{\theta}\rangle$ follows the same power law as that of $\langle Nu \rangle$.
\par
In RBC, dissipation fields are often expected to follow a log-normal distribution \cite{Kolmogorov_1962}. However, considerable deviations due to the highly intermittent nature of the local dissipation has also been reported \cite{Zhang_Zhou_SunJFM2017,Emran_Schumacher_2008,Schumacher_Sreenivasan_2005,Kaczorowski_Wagner_2009}. In Fig. \ref{DissEt_PDF}, PDF of thermal dissipation rate is represented in log-normal coordinates, where $\mu$ and $\sigma$ indicate the mean and standard deviation of log $\makebox{\large\ensuremath{\epsilon}}_{\theta}$.
In the LSC regime, both $\makebox{\large\ensuremath{\epsilon}}_{u}$ and $\makebox{\large\ensuremath{\epsilon}}_{\theta}$ follow the log-normal behaviour. However at regime \Romannum{3}, a clear departure from log-normality is noticed ($Ro^{-1} = 10$).
Further, the tails of PDF become shorter at regime \Romannum{3}, implying a reduced probability of high-amplitude dissipation events.
\begin{figure}
\centering
\includegraphics[scale=0.3]{Diss0full_pof.pdf}
\caption{(Colour online) Variation of $ \langle\makebox{\large\ensuremath{\epsilon}}_{u}\rangle / \langle\makebox{\large\ensuremath{\epsilon}}_u^{o}\rangle$ and $\langle\makebox{\large\ensuremath{\epsilon}}_{\theta}\rangle /\langle\makebox{\large\ensuremath{\epsilon}}_{\theta}^{o}\rangle$ with rotation rate at $Ra=2\times10^6$ and $2\times10^7$.}
\label{Diss_scale}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.330]{Ra7PDF_LogNormal_ET.pdf}
\caption{(Colour online) PDF of log$\epsilon_{\theta}$ at $Ra=2\times10^7$ for different $Ro^{-1}$ in log-normal coordinates.}
\label{DissEt_PDF}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.375]{DissT_2RoiA.pdf}\\
\includegraphics[scale=0.375]{DissU_logA.pdf}
\caption{(Colour online) (a) Enlarged view (near bottom plate) of the variation of thermal dissipation along the vertical direction at $Ra=2\times10^6$ for different $Ro^{-1}$. The inset shows the complete profile. (b) Variation of average viscous dissipation rate along the vertical direction for different rotation rates in semi-log scale. The inset shows the zoomed view of the same near the bottom plate. Arrows indicate the direction of increase in rotation rate.}
\label{Diss_Axial}
\end{figure}
\par
We now examine the spatial distribution of the viscous and thermal dissipation rates in turbulent RRBC.
The variation of the horizontal plane and time averaged dissipation rates along the vertical direction are shown in Fig. \ref{Diss_Axial}, where the complete profile of $\langle \makebox{\large\ensuremath{\epsilon}}_{\theta} \rangle_{A,t}$ is shown in the inset (a).
We observe that maximum value of the dissipation rates occur at the top and bottom plates which is consistent with the observations made by Zhang \emph{et al.} \cite{Zhang_Zhou_SunJFM2017} and Emran and Schumacher \cite{Emran_Schumacher_2012}.
In the LSC regime, the bulk region shows negligible thermal dissipation (in comparison to boundary layers), indicating most of the thermal energy is dissipated near the thermal boundary layers. As rotation rate increases (regime \Romannum{3}) the magnitude of thermal dissipation decreases near the top and bottom surfaces. The straight vertical lines in regime \Romannum{3} suggest that the energy is dissipated more uniformly throughout the domain.
On a similar note, the viscous dissipation also follows a different profile in the LSC regime compared to that at high rotation rates. In the LSC regime two local peaks are identified in the $\langle \makebox{\large\ensuremath{\epsilon}}_{u} \rangle_{A,t}$ profile before it becomes stable near the bulk region. However at regime \Romannum{3}, only a single peak is identified. As rotation rate increases the average dissipation rates decrease and, $\langle \makebox{\large\ensuremath{\epsilon}}_u \rangle_{A,t}$ and $\langle \makebox{\large\ensuremath{\epsilon}}_{\theta} \rangle_{A,t}$ approaches steady profiles, as all the velocity and temperature fluctuations in the system are damped out. The stabilizing effect of rotation on RBC is evident from these profiles. In the following subsection we discuss the dynamics of LSC, mainly the changes in the azimuthal direction and further present their statistics.
\subsection{Dynamics of reorientations}
The vertical plane containing LSC is known
to show spontaneous and erratic drifts or directional changes with time \cite{Keller_1966,Welander_1967,Creveling_etal_1975,Gorman_etal_1984,Hansen_etal_1992}.
These sudden and significant changes in the orientation of LSC are referred as reorientations of LSC \cite{Brown_Ahlers_JFM}.
During some reorientations the amplitude of the first Fourier mode almost drops to zero and such events are termed as cessation-led reorientations. While in others, the LSC rotates azimuthally without considerable change in the amplitude of the first Fourier mode \cite{Cioni_etal_1997,Brown_Nikolaenko_Ahlers_PRL_2005}. These are referred as rotation-led reorientations.
Although these events are reported experimentally by Brown \emph{et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005}, Brown and Ahlers \cite{Brown_Ahlers_JFM} and Xi \emph{et al.} \cite{Xi_Zhou_Xia_2006}, confirmation by numerical studies is rare.
Here we probe the dynamics and statistics of reorientations of LSC in RRBC using direct numerical simulations.
\par
Reorientations are generally quantified using the amplitude $A_{k}=|\hat{u}(k)|$ and phase $\Phi_{k}= tan^{-1} (\mathbf{Im~} \hat{u}(k) / \mathbf{Re~}\hat{u}(k))$ of the Fourier modes (refer to Eqn. \ref{gov_eqn4}). The characterization of reorientations of LSC is not a well established concept, but in literature we find few works in this direction. For example, Brown \emph{et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005} and Brown and Ahlers \cite{Brown_Ahlers_JFM} identified the reorientations based on the phase of the first Fourier mode $\Phi_1$ using two criteria:
(a) The net angular change $|\Delta \Phi_1|$ has to be greater than $\pi/4$, and (b) magnitude of the net azimuthal rotation rate $|\Delta \Phi_1/\Delta t|$ has to be greater than $\pi/5T_{eddy}$, where $T_{eddy}$ is the eddy turn over time calculated as $T_{eddy}=2H/w^{rms}$. Here $w^{rms}$ is the average value of the rms of vertical velocity time signals from the previously mentioned $36$ numerical probes placed at the mid-vertical plane.
The above procedure results in multiple overlapping subsets of reorientations and thus requires further conditioning using quality factors to pick the appropriate reorientation. To avoid this complexity, in the present study we ensure that the azimuthal rotation rate between any two adjacent points is greater than the specified cutoff ($\pi/5T_{eddy}$).
Further, we identify these reorientations as
complete reversal if the phase change $|\Delta \Phi_1| = \pi$ and as partial reversal if $|\Delta \Phi_1|\neq \pi$, as proposed by Mishra \emph{et al.} \cite{MishraDe}.
\begin{figure}
\centering
\includegraphics[scale=0.38]{FMERa6RoiE.pdf}
\caption{(Colour online) (a) Time signals of vertical velocity from different numerical probes at mid-plane of the cylinder and (b) the phase of first Fourier mode showing complete (blue box) and partial reversals (green box) for $Ra=2\times10^6$ and $Ro^{-1}=0$. (c) Schematic representation of the location of the numerical probes.}
\label{Rot_Re_sig}
\end{figure}
\par
The time evolution of the phase of the first Fourier mode along with the vertical velocity signals from different numerical probes are shown in Fig. \ref{Rot_Re_sig}.
Here we identify reversals by the sudden and considerable change in the phase $\Phi_1$, along with the switching of mean vertical velocity.
At $t\approx 1380$, a complete reversal is observed, where the phase change $|\Delta \Phi_1| \approx \pi$. This is accompanied by the switching of mean flow across all the probes. However, for the reversal at $t\approx 1810$, the mean value of $w$ does not switch at all probes. Clearly, at probes $S_{4}~(\phi=\pi/6),~ S_{22}~(\phi=7\pi/6),~S_{16}~(\phi=5\pi/6)$ and $S_{34}~(\phi=11\pi/6)$ the mean vertical velocity remains unaltered, while it switches sign at $S_{10}~(\phi=\pi/2)$ and $S_{28}~(\phi=3\pi/2)$. This is a partial reversal where the phase change is less than $\pi$. We observe several partial reversals in our simulations while the number of complete reversals are very less.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.350]{Ra6Roi_CP_reversalsC.pdf}
\caption{(Colour online) The time series of first and second Fourier modes, their corresponding amplitude ratio $A_{2}/A_{1}$ and phase of the first Fourier mode during complete (blue box) and partial (green box) reversals at $Ra=2\times10^6$ and $Ro^{-1}=0$.}
\label{CP_reveRa6i}
\end{figure}
\par
To get better insights into the nature of these reversals we analyze the time evolution of the first and second Fourier modes. We observe that for RBC and RRBC at low rotation rates (where LSC persists) the first Fourier mode dominates over the other modes. In Fig. \ref{CP_reveRa6i}, we observe that during the reorientations amplitude of the first Fourier mode $A_1=|\hat{u}_{1}|$ drops below its mean value. On closer inspection, although $A_1$ shows a small dip in magnitude near $t\approx1380$, the rise in the amplitude of the second mode $A_2=|\hat{u}_{2}|$ is marginal and thus the amplitude fraction $A_{2}/A_{1}$ remains negligibly small. Thus we identify these as rotation-led reorientation, where the LSC structure rotates azimuthally to reorient itself to a new direction without much change in the circulation strength.
\par
Rotation-led reorientation observed for rotating RBC is shown in Fig. \ref{P_Rot} for $Ra=2\times10^6$ and $Ro^{-1}=0.1$.
At $t\approx 2765$ the phase of the first Fourier mode shows an angular change $|\Delta\Phi_1|\approx 2\pi/3$. Although, $A_1$ shows a local minimum at this instant, $A_2$ is also minimal and resultantly the amplitude fraction is close to zero as shown in frame (b).
However, during a different reorientation (refer Fig. \ref{C_Rot}) the amplitude of the first Fourier mode drops considerably ($A_{1} \rightarrow 0$) and that of the second rises. These are identified as cessation-led reorientations. Here the strength of LSC diminishes at the cessation point (peak of $A_{2}/A_{1}$) and the flow reorients itself at any arbitrarily chosen direction. Due to the fluctuations in the $A_1$ and $A_2$ time signals, we use the amplitude fraction $A_{2}/A_{1}$ to clearly recognize the cessations. In the present study cessations are identified as sharp peaks in the amplitude fraction such that $A_{2}/A_{1}\gtrsim10$.
Here we observe a a partial cessation-led reorientation near $t\approx 4250$ with an angular change $|\Delta \Phi_1| \approx 3\pi/4$.
The experimental studies by Brown \emph{et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005}, Brown and Ahlers \cite{Brown_Ahlers_JFM} and Xi \emph{et al.} \cite{Xi_Zhou_Xia_2006} have reported similar reorientations.
\begin{figure}
\centering
\includegraphics[scale=0.40]{RotRa6Ro10.pdf}
\caption{The time series of first and second Fourier modes, their corresponding amplitude ratio $A_{2}/A_{1}$ and phase of the first Fourier mode during rotation-led partial reversal obtained for $Ra=2\times10^6$ and $Ro^{-1}=0.1$. The boxed region identifies the reorientation. }
\label{P_Rot}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.40]{CessRa6Ro5.pdf}
\caption{The time series of first and second Fourier modes, their corresponding amplitude ratio $A_{2}/A_{1}$ and phase of the first Fourier mode during cessation-led reorientation obtained for $Ra=2\times10^6$ and $Ro^{-1}=0.2$. The dotted line in the amplitude fraction plot indicates the cutoff value $A_{2}/A_{1}=10$, used to identify cessations. }
\label{C_Rot}
\end{figure}
\par
Occasionally, two cessations occur in quick succession, i.e., within an eddy turnover time. These are referred as double-cessations.
Figure \ref{Cess_D} shows a double-cessation observed for $Ra=2\times10^6$ and $Ro^{-1}=0.3$. The boxed region bounds an eddy turnover time ($T_{eddy}\approx 11$ free-fall time units), within which the double-cessation occurs. The amplitude of the first Fourier mode drops ($A_{1} \rightarrow 0$) at $t\approx1687$ and $t\approx1692$, accompanied by sharp rise in the amplitude fraction $A_{2}/A_{1}$. Here we observe that the phase of first Fourier mode $\Phi_{1}$ changes successively, first by $|\Delta \Phi_1| \approx \pi$ and second by $|\Delta \Phi_1 |\approx \pi/2$. The net angular change as a result of the double-cessation is accounted as the sum of these two successive events. Similar double-cessations were observed experimentally by Xi \emph{et al.} \cite{Xi_Zhou_Xia_2006}, and were later numerically reported by Mishra \emph{et al.} \cite{MishraDe}. The number of double-cessations observed is much fewer compared to regular (or single) cessations.
\par
Interestingly, in our simulations, we observe more than two cessations occur within an eddy turnover time. This has not been reported before and we term this phenomenon as multiple-cessation. Figure \ref{Cess_M} shows the multiple-cessation observed for $Ra=2\times10^7$ at $Ro^{-1}=0.2$. The boxed region bounds an eddy turnover time within which the multiple-cessation occurs. We notice that $A_1$ drops close to zero successively at $t\approx2199, 2201$, $2205$ and $2210$, and the amplitude fraction $A_{2}/A_{1}$ shows sharp peaks at these locations. The discontinuities (sudden fluctuations) in the phase are jitters near $-\pi$ and $\pi$.
We observe only a single instance of multiple-cessation in our entire set of simulation.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.39]{Ra6Ro333_DCessationsD.pdf}\\
\caption{The time series of first and second Fourier modes, their corresponding amplitude ratio $A_{2}/A_{1}$ and phase of the first Fourier mode during double cessation observed for $Ra=2\times10^6$ at $Ro^{-1}= 0.3$. The dotted line in the amplitude fraction plot indicates the cutoff value $A_{2}/A_{1}=10$, used to identify cessations. The boxed region bounds an eddy turnover time within which the double-cessation prevails. }
\label{Cess_D}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.39]{Ra7Ro5_MCessationsC.pdf}
\caption{The time series of first and second Fourier modes, their corresponding amplitude ratio $A_{2}/A_{1}$ and phase of the first Fourier mode during multiple cessations observed for $Ra=2\times10^7$ at $Ro^{-1}=0.2$. The dotted line in the amplitude fraction plot indicates the cutoff value $A_{2}/A_{1}=10$, used to identify cessations. The boxed region bounds an eddy turnover time within which the multiple-cessation prevails.}
\label{Cess_M}
\end{figure}
\par
Next we study the relation between cessations and the dynamics of flow structure. We observe that during cessations, as the amplitude of first Fourier mode diminishes the flow structure corresponding to the first mode (dipolar structure) becomes weak, and the quadrupolar structure corresponding to the second Fourier mode become dominant. However, after the cessation, the dipolar structure reappears accompanied by a change in the azimuthal orientation of LSC. This is consistent with the observations made by Mishra \emph{et al.}\cite{MishraDe} for non-rotating RBC. Here we substantiate this by illustrating the temperature contours at mid-plane ($z=0.5$) before, during and after the cessation, as shown in Fig. \ref{Cess_Contours}. Before the cessation, the contour at $t\approx1865$ follows an almost dipolar structure, where hot and cold fluid appear at opposite sides ($\pi$ apart) of the lateral wall. However, during cessations, i.e., at $t\approx2201$, $2205$ and $2210$ (refer Fig. \ref{Cess_M} also) the temperature contours indicate a quadrupolar behaviour. Here the hot and cold fluid are observed alternately and separated azimuthally by about $ \pi/2$. After the cessations the flow goes back to the dipolar structure as shown in Figs. \ref{Cess_Contours} (e) and (f).
\begin{figure}
\centering
\includegraphics[scale=0.150]{Cess_Cont_Ra7Ro5C.jpg}
\caption{Temperature contours at mid-vertical plane showing quadrupolar and dipolar structures observed before ($t=1865$), during ($t=2201,2205$ and $2210$) and after ($t=2240$ and $2267$) cessation, for $Ra=2\times10^7$ at $Ro^{-1}=0.2$. }
\label{Cess_Contours}
\end{figure}
\subsection{Statistics of reorientations}
\par
After investigating the dynamics of reorientations of LSC, we turn our focus on its statistics.
The number of reorientations ($N_R$) and cessations ($N_C$) per eddy turnover time at different rotation rates are shown in Fig. \ref{Reore_stats} for the two Rayleigh numbers.
On an average we consider about $550$ and $300$ eddy turnover times ($T_{eddy}$) for $Ra=2\times10^6$ and $2\times10^7$, respectively.
With the increase in rotation rate we observe a considerable increase in the number of reorientations.
Further, the number of reorientations are higher for lower $Ra$ for both rotating and non-rotating RBC. Although, we observe comparatively more cessations for lower $Ra$, the number of cessations are quite low to arrive at any conclusion.
As reported by previous studies cessations are rare phenomenons and double-cessations are much rarer. Brown and Ahlers \cite{Brown_Ahlers_JFM} observed that cessations account for about $5\%$ of the total reorientations. In our simulations we observe that for $Ra=2\times10^6$ nearly $4$-$7\%$ reorientations are cessation-led. Likewise, for $Ra=2\times10^7$ the percentage of cessations is about $4\%$. However, for the non-rotation case at $Ra=2\times10^7$ cessations account for about $13\%$ for the total reorientations.
Since our criterion for identification of cessations depends on the amplitude fraction of the Fourier modes ($A_{2}/A_{1}$), and that of reorientations depends on the angular change $|\Delta \Phi_1|$ and angular rotation rate $|\Delta \Phi_1/\Delta t|$, which are independent of each other, some events appear in both the categories while some are exclusive to either cessations or reorientations. Note that the number of reorientations observed in the present study are much lesser compared to the experimental counterparts \cite{Brown_Ahlers_JFM,Brown_Nikolaenko_Ahlers_PRL_2005,Xi_Zhou_Xia_2006} due to the limitations in the length of time histories.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.19]{Reore_statsC.pdf}\includegraphics[scale=0.19]{Cess_statsC.pdf}
\caption{Number of reorientations and cessations per eddy turnover time observed for $Ra=2\times10^6$ and $2\times10^7$. }
\label{Reore_stats}
\end{figure}
\par
Next, we quantify the possibility of occurrence of the reorientations based on the angular change $|\Delta \Phi_1|$.
Figure \ref{PDF_Reore1} shows the probability distribution $P(|\Delta \Phi_1|)$ for different rotation rates.
We observe that the statistics of reorientation follow a power law distribution as $P(|\Delta \Phi_1|) \propto |\Delta \Phi_1|^{-m} $, where $m =3.7\pm 0.5$. The power law fit is carried out by maximum likelihood estimation (MLE) \cite{Powerlaw_Newman_2005,Bevington_book}, to prevent the errors arising due to the binning of data. MLE is preferred over least squares method which can produce substantially inaccurate estimates of parameters for the distribution due to smaller size of the data sample \cite{Bevington_book}.
In MLE the parameter values of the distribution are calculated such that they maximize the likelihood function, which is the product of probability densities of the individual events.
The exponent of the power law distribution is calculated as
\begin{eqnarray}
m=1+n \left[ \sum_{i=1}^{n}\mathrm{ln}\frac{|\Delta\Phi_{1}^{i}|}{|\Delta\Phi_{1}^{min}|} \right]^{-1}
\end{eqnarray}
where $|\Delta\Phi_{1}^{i}|$, $i=1, ...n$ are the angular changes from $n$ number of observations and $|\Delta\Phi_{1}^{min}|$ is the minimum angular change. Similar power law fits with negative exponents are observed for all cases (rotating and non-rotating RBC), indicating a monotonically decreasing distribution of $|\Delta \Phi_1|$. This suggests that smaller reorientations are more frequent events, while the larger ones are rare occurrences.
Note that the exponent is well within the range of $-3.25$ to $-4.45$ experimentally observed for different $Ra$ by Brown and Ahlers \cite{Brown_Ahlers_JFM} and Brown \emph {et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005}. The works of Xi and Xia \cite{Xi_Xia_PRE_2008} also found the slope of the power law fit to be of the order of $-3.3$ for unit aspect ratio container. However, all these experiments were on non-rotating RBC. In the present study, we infer that even for rotating RBC, the reorientations follow a power law distribution.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.79]{Pdf_Reore_1.pdf}
\caption{PDF of the angular change $\Delta \Phi_1$ for $Ra=2\times10^6$ and $Ra=2\times10^7$ at $Ro^{-1}=0.1$ and $0.5$. Solid line indicates the power law fit by MLE. }
\label{PDF_Reore1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.80]{FullTau_pof.pdf}
\caption{PDF of time interval between successive reorientations for $Ra=2\times10^6$ at $Ro^{-1}=0.2$ and $0.5$. The solid line indicates the exponential function $P(\uptau/\mu_{\uptau})$~=$~\exp(\uptau/\mu_{\uptau})$. }
\label{PDF_Tau}
\end{figure}
\par
We now evaluate the occurrence of reorientations with time. For this we consider the time interval between two successive reorientations ($\tau$). Experimental studies by Xi and Xia \cite{Xi_Xia_PRE_2008}, Brown and Ahlers \cite{Brown_Ahlers_JFM} and Brown \emph {et al.} \cite{Brown_Nikolaenko_Ahlers_PRL_2005} have reported that the PDF of time interval of reorientations can be well fitted with an exponential function, which suggests that the occurrence of reorientations follows a Poisson process.
Figure \ref{PDF_Tau} shows the probability distribution $P(\uptau/\mu_{\uptau})$ for different rotation rates ($Ro^{-1}=0.2$ and $0.5$), where the solid line shows the exponential function $P(\uptau/\mu_{\uptau})~$=$~\exp(\uptau/\mu_{\uptau})$ with $\mu_{\uptau}$ the average time interval of reorientations.
Note that for $Ra=2\times10^6$ the time interval is in good agreement with the exponential function, while deviations are observed in $Ra=2\times10^7$, which might be due to the lesser number of observations. Poisson distribution of the time interval suggests that reorientations are independent of each other and they do not have any memory of the previous event \cite{Xi_Xia_PRE_2008}.
\section{Conclusions}\label{Sec:Conclusion}
We have presented a detailed three-dimensional numerical investigation on the dynamics and statistics of reorientations of LSC in turbulent RRBC in a unit aspect ratio cylindrical cell with $Pr=0.7$ and two different Rayleigh numbers $Ra=2\times10^6$ and $2\times10^7$, for a wide range of rotation rate ($0\leq Ro^{-1}\leq 30$). We have been able to identify different flow regimes based on the energy contained in the Fourier modes computed for the field recorded in the azimuthal direction of the mid-plane of the container. We find LSC dominated flow at the low rotation rate ( $Ro^{-1}\lesssim1$) while presence of multiple roll structures like quadrupole and sextupole have been observed at higher rotation rates. Presence of LSC significantly affects the overall heat transfer and the boundary layer thickness. Along the direction of LSC, the thermal boundary layer width $\delta_{\theta}$ shows an asymmetric trend, as it is thicker at one side and thinner near the opposite side of the lateral wall.
However, perpendicular to LSC, the boundary layer thickness varies almost symmetrically.
\par
Further, the reorientations of LSC have been characterized as rotation-led and cessation-led based on their nature of occurrence, and as partial and complete reversal depending on the degree of phase change. In addition to the previously reported rare events like cessations and double-cessation, an interesting event of multiple-cessation have been observed, where more than two cessations occur in quick succession. Cessation-led reorientations are rare events, which account for about $4$-$7\%$ of the total number of reorientations. Double-cessations are even rarer, and multiple-cessations are the rarest of all reorientations.
At moderate rotation rates, significant increase in the number of rotation-led reorientations are observed, while cessation-led ones are unaffected by rotation. Similar to non-rotating RBC, we find that the probability distribution of the reorientations exhibits a power law distribution with the exponent nearly equal to $-3.7$. In addition the PDF of time interval between two successive reorientations follows a Poisson distribution, which indicates that reorientations are independent of each other.
\section*{Acknowledgement}
All simulations have been carried out in the `Param-Ishan' computing facility at IIT Guwahati.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,910 |
Победа — опустевший поселок в Рогнединском районе Брянской области в составе Фёдоровского сельского поселения.
География
Находится в северной части Брянской области на расстоянии приблизительно 16 км на север по прямой от районного центра поселка Рогнедино.
История
Известен с 1930-х годов. На карте 1941 года отмечен как поселение с 16 дворами.
Население
Численность населения: 13 человек (русские 100 %) в 2002 году.
Примечания
Населённые пункты Рогнединского района | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,104 |
\section{Introduction}
This paper contains a version of a generic identification theorem for simple groups of finite Morley rank adapted from the main result of \cite{BB04} by weakening its assumptions. It is published because it is being used in the authors' study of actions of groups of finite Morley rank \cite{BB11}.
A general discussion of the subject can be found in the books \cite{bn} and \cite{abc}.
A group of finite Morley rank is said to be of {\em $p'$-type}, if
it contains no infinite abelian subgroup of exponent $p$. Notice
that a simple algebraic group over an \acf $K$ is of $p'$-type if
and only if ${\rm char}\, K \ne p$. Other definitions can be found in
the next section.
The aim of this work is to prove the following.
\begin{theorem}
Let\/ $G$\/ be a simple group of finite
Morley rank and\/ $D$ a maximal\/ $p$-torus in $G$
of Pr\"{u}fer rank at least\/ $3$. Assume that
\begin{itemize}
\item[{\rm (A)}] Every proper connected definable subgroup of $G$ which contains $D$ is a $K$-group.
\item[{\rm (B)}] For every element\/ $x$ of order\/
$p$ in $D$, the group $C^\circ_G(x)$ is of\/ $p'$-type and
$C^\circ_G(x)=F^\circ(C^\circ_G(x))E(C^\circ_G(x)) $.
\item[{\rm (C)}] $\langle C^\circ_G(x) \mid x \in D, \;|x|=p\rangle = G$.
\end{itemize}
Then $G$ is a Chevalley group over an \acf of characteristic distinct
from
$p$.
\label{sis-kebap}
\end{theorem}
Notice that, under assumption (B) of Theorem~\ref{sis-kebap},
$C^\circ_G(x)$ is a central product of $F^\circ(C^\circ_G(x))$
and $E(C^\circ_G(x))$.
The predecessor of this theorem, Theorem 1.2 of \cite{BB04}, was based on a stronger assumption than our assumption (A), namely, that \emph{every} proper definable subgroup of $G$ is a $K$-group, a condition that was difficult to check in its actual applications.
The proof of Theorem~\ref{sis-kebap} is given in
Section~\ref{sec:sis-kebap}.
\subsection{Definitions}
All definitions related to groups of finite Morley rank in general
can be found in \cite{bn} and \cite{abc}.
{}From now on $G$ is a group of finite Morley rank. The group $G$
is called
a {\it $K$-group}, if every
infinite simple definable and connected section of the group is an
algebraic group over an algebraically closed field.
A {\em $p$-torus} $S$ is a direct product of finitely many copies
of the quasicyclic group ${{\Bbb Z}}_{p^\infty}$. The number of
copies is called the {\em Pr\"{u}fer $p$-rank} of $S$ and is
denoted by ${\rm pr}(S)$. For a definable group $H$, ${\rm
pr}(H)$ is the maximum of the Pr\"{u}fer ranks of $p$-subgroups
in $H$. It is easy to see that the Pr\"{u}fer $p$-rank of any
subgroup of a group of finite Morley rank is finite.
A group $H$ is called {\it quasi-simple} if $H'=H$ and $H/Z(H)$
is simple and non-abelian. A quasi-simple subnormal subgroup of
$G$ is called a {\it component} of $G$. The product of all
components of $G$ is called the {\it layer} of $G$ and denoted by
$L(G)$, and $E(G)$ stands for $L^\circ(G)$. It is known (see
Lemmas 7.6 and 7.10 in \cite{bn}) that $G$ has finitely many
components and that they are definable and are normal in $E(G)$.
$F(H)$ is the
{\em Fitting subgroup} of $H$, that is, the maximal normal definable
nilpotent subgroup.
If $H$ is a group of finite Morley rank then $B(H)$ is the
subgroup generated by all definable connected $2$-subgroups of
bounded exponent in $H$. Note that $B(H)$ is connected by
Assertion~\ref{zilb}.
\section{Background Material}
\subsection{Algebraic Groups}
For a discussion of the model theory of algebraic groups, the reader
might like to see Section 3.1 in \cite{berkman}.
The basic structural facts and definitions related to algebraic groups
can be easily found in the standard references
such as \cite{cartk,hump}.
First note that a connected algebraic group $G$ is called {\em
simple} if it has no proper normal connected and closed
subgroups. Such a group turns out to have a finite center, the
quotient group being simple as an abstract group. The classical
classification theorem for simple algebraic groups states that
simple algebraic groups over algebraically closed fields are
Chevalley groups, that is, groups constructed from Chevalley
bases in simple complex Lie algebras as described, for example, in
\cite{cartk}.
Now fix a maximal torus $T$ in a connected algebraic group
$G$ and denote the corresponding
root system by $\Phi$, and for each $\alpha\in \Phi$, denote the
corresponding root subgroup by $X_\alpha$. The subgroup $\langle
X_\alpha,X_{-\alpha}\rangle$ is known to be isomorphic to $SL_2$
or $PSL_2$ and is called a {\em root $SL_2$-subgroup}.
If $G$ is simple,
the roots can have at most two different lengths, and the terms `short
root $SL_2$-subgroup' and `long root $SL_2$-subgroup'
have the obvious meanings.
A simple algebraic group is generated by its root $SL_2$-subgroups. In a
simple algebraic group, all long root
$SL_2$-subgroups are conjugate to each other, and similarly all short
root $SL_2$-subgroups are conjugate to each other.
\begin{fact}
\label{prop:class-in-algebraic} Suppose that\/ $G$ is a simple algebraic
group over an algebraically closed field. Let\/ $T$ be a
maximal torus in $G$ and $K$, $L$ closed subgroups of $G$
that are isomorphic to $SL_2$ or\/ $PSL_2$ and are
normalised by $T$. Then the following hold.
\begin{enumerate}
\item Either\/ $[K,L]=1$ or\/ $\langle
K,L\rangle$ is a simple algebraic group of rank $2$;
that is of type $A_2$, $B_2$ or\/ $G_2$.
\item The subgroups $K$ and $L$
are embedded in $\langle K,L\rangle$ as root $SL_2$-subgroups.
\item If $\langle K,L\rangle$ is of type $G_2$, then $G=\langle K,L\rangle$.
\end{enumerate}
\end{fact}
\paragraph{Proof.} The proof follows from the description of closed
subgroups in simple algebraic groups normalised by a maximal torus
\cite[2.5]{seitz}; see also \cite[Section~3.1]{seitz2}. \hfill $\Box$
\begin{fact}
Let\/ $G$ be a simple algebraic group over an algebraically
closed field of characteristic $\neq p$,
and let $D$ be a maximal $p$-torus in $G$. Then
$C_G(D)$ is a maximal torus in $G$. \label{fact:p-torus}
\end{fact}
\begin{proof} The proof follows from the description of
centralisers of subgroups of commuting semisimple elements in
simple algebraic groups \cite[Theorem~5.5.8]{ste1}.
\end{proof}
\subsection{Groups of Finite Morley Rank}
\begin{fact}
\label{zilb} {\rm (Zil'ber's Indecomposability Theorem)}
A subgroup of a
group of finite Morley rank which is generated by a family
of definable connected subgroups is also definable
and connected.
\end{fact}
\begin{proof} See \cite{zilber} or Corollary~5.28 in \cite{bn}.
\end{proof}
\begin{fact} {\rm \cite[Theorem~8.4]{bn}}
Let\/ $G\rtimes H$ be a group of finite Morley rank, where $G$
and\/ $H$ are definable, $G$ is an infinite simple algebraic group
over an \acf and\/ $C_H(G)=1$. Then $H$ can be viewed as a
subgroup of the group of automorphisms of\/ $G$, and moreover\/
$H$ lies in the product of the group of inner automorphisms and
the group of graph automorphisms of $G$ (which preserve root
lengths). In particular, when $H$ is connected, then $H$ consists
of inner automorphisms only. \label{defaut}
\end{fact}
\begin{fact}
{\rm \cite{ac}} \label{centext}
Suppose $G$ is a group of finite
Morley rank, $G=G'$, and\/ $G/Z(G)$ is a simple algebraic group
over an \acfd, and is of finite Morley rank, then $Z(G)$ is
finite and\/ $G$ is also algebraic.
\end{fact}
\begin{lemma}
Let\/ $G$ be a connected $K$-group of\/ $p'$-type and\/ $D$ a
maximal\/ $p$-torus in $G$. If\/ $L \lhd G$ is a component in $G$,
then $D\cap L$ is a maximal\/ $p$-torus in $L$ and\/ $D=C_D(L)(D\cap L)$.
\label{lm:torus-in-component}
\end{lemma}
\begin{proof} As $G$ is connected, $L \triangleleft G$. Now the lemma
immediately follows from Assertions~\ref{centext}, \ref{defaut} and
\ref{fact:p-torus}. \end{proof}
\begin{lemma}
\label{algcomps} Under the assumptions of Theorem~{\rm
\ref{sis-kebap}}, we have, for every $p$-element\/ $t\in D$,
\[C^\circ_G(t)=F\cdot L_1\cdots L_r,\]
where $F=F^\circ(C_G^\circ(t))$ and each\/ $L_i$ is a simple
algebraic group over an algebraically closed field of
characteristic $ \ne p$.
\end{lemma}
\begin{proof} For every $p$-element $t$ in $G$, $C^\circ_G(t)=F\cdot
E(C^\circ_G(t))$. And assumption (C) ensures that
$C^\circ_G(t)$ is a $K$-group, hence its components are algebraic
groups by Assertion~\ref{centext}. \end{proof}
\subsection{Lyons's Theorem}
A detailed discussion of this particular version
of Lyons's theorem can be found in \cite{berkman}.
\begin{fact}[(Lyons \cite{gls,gls3})]
\label{Lyons} Suppose that\/ ${\Bbb F}$ is an algebraically closed
field, $I$ is one of the connected Dynkin diagrams of the simple
algebraic groups of Tits rank at least\/ $3$ and $\tilde{G}$ is
the simply connected simple algebraic group of type $I$ over
${\Bbb F}$. Let\/ $G$ be an arbitrary group and for each $i\in I$,
$K_i$ stand for a subgroup of $G$ which is centrally isomorphic to
$PSL_2({\Bbb F})$, and\/ $T_i < K_i$ denote a maximal torus in
$K_i$. Also assume that the following statements hold.
\begin{enumerate}
\item The group $G$ is generated by $K_i$ where $i\in I$.
\item For all\/ $i,j \in I$, $[T_i,T_j]=1$.
\item If\/ $i\ne j$ and $(i,j)$ is not an edge
in $I$, then $[K_i,K_j]=1$.
\item If\/ $(i,j)$ is a single edge in $I$, then
$G_{ij} =\langle K_i,K_j\rangle$ is isomorphic to
$(P)SL_3({\Bbb F})$.
\item If\/ $(i,j)$ is a double edge in $I$, then
$G_{ij}=\langle K_i,K_j\rangle$ is isomorphic to
$(P)Sp_4({\Bbb F})$.
Moreover, in that case, if\/ $r_i\in N_{K_i}(T_iT_j)$ and
$r_j\in N_{K_j}(T_iT_j)$ are involutions, then
the order of\/ $r_ir_j$ in $N_{G_{ij}}(T_iT_j)/T_iT_j$
is $4$.
\item For all $i,j\in I$, $K_i$ and $K_j$
are root\/ $SL_2$-subgroups of\/ $G_{ij}$
corresponding to the maximal torus $T_iT_j$ of\/ $G_{ij}$.
\end{enumerate}
Then there is a homomorphism from $\tilde{G}$ onto
$G$, under which the root $SL_2$-subgroups of $\tilde{G}$
{\rm (}for some simple root system{\rm )} correspond to the subgroups $K_i$.
\end{fact}
\subsection{Reflection Groups}
A linear semisimple transformation of finite order is called a
{\it reflection} if it has exactly one eigenvalue which is not 1.
\begin{theorem}
Let\/ $W$ be a finite group and assume that the following
statements hold.
\begin{enumerate}
\item There is a normal subset $S\subseteq W$ consisting of involutions
and generating $W$.
\item Over $\Bbb C$, $W$ has a faithful irreducible
representation of dimension $n\geqslant 3$ in which involutions from $S$ act as reflections.
\item For almost all prime numbers $q$, $W$ has faithful irreducible
representations {\rm(}possibly of different dimensions{\rm )} over
fields\/ ${\Bbb F}_q$. Moreover, for every such representation,
involutions in $S$ act as reflections.
\end{enumerate}
Then $W$ is one of the
groups $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ for
$n\geqslant 3$.
\label{reflection}
\end{theorem}
\begin{proof} A proof, based on the classification of irreducible
complex reflection groups \cite{shephard-todd}, can be found in
\cite{berkman}. \end{proof}
\section{Proof of Theorem~\ref{sis-kebap}}
\label{sec:sis-kebap}
The strategy is to construct the Weyl group and the root system
of $G$, and then to apply Lyons's Theorem. So from now on $G$ is
a simple group of \fmr and $D$ is a maximal $p$-torus in $G$
of Pr\"{u}fer rank $\geqslant 3$ and such that every proper definable connected subgroup contatining $D$ is a $K$-group. We also assume that
$C^\circ_G(x)$ is of $p'$-type for every element $x \in D$ of
order $p$, $C^\circ_G(x)=F^\circ(C^\circ_G(x))E(C^\circ_G(x)) $
and
\[G = \langle C^\circ_G(x) \mid x \in D, \; |x|=p\rangle.\]
Notice that if $M$ is a proper definable subgroup in $G$ normalised by $D$ then $MD$ is a proper definable subgroup of $G$, for otherwise $M$ would be normal in $G$, which contradicts simplicity of $G$. Therefore, $MD$ and hence $M$ are $K$-groups by assumption (A). This observation will be used throughout the proof.
We also shall systematically use the following observation.
\begin{lemma}
$F^\circ(C^\circ_G(x))$ centralises
$D$ for every element $x \in D$ of order $p$. \label{lm:F}
\end{lemma}
\begin{proof} The result immediately follows from the fact that a
$p$-torus in a definable nilpotent group belongs to the center of
this group \cite[Theorem~6.9]{bn}. \end{proof}
\subsection{Root Subgroups}
{}From now on, $SL_2$ will be used instead of $SL_2({\Bbb F})$, etc.
Denote by $\Sigma$ the set of all definable subgroups isomorphic
to $(P)SL_2$ and normalised by $D$. These are our future root
$SL_2$-subgroups.
If $N$ is a subgroup of $G$ which is normalised by $D$, then set
$H_N:=C_N(D\cap N)$. Note that if $K\in\Sigma$, then $H_K$ is
an algebraic torus in $K$.
\begin{lemma}
The set\/ $\Sigma$ is non-empty.
\end{lemma}
\begin{proof} Assume the contrary. If $L=E(C^\circ_G(x)) \ne 1$ for
some element $x$ of order $p$ from $D$, then, $L$ being a central
product of simple algebraic groups, contains a definable
$SL_2$-subgroup normalised by $D$. Therefore $C^\circ_G(x) =
F^\circ(C^\circ_G(x))$ centralises $D$ by Lemma~\ref{lm:F}. But
then \[G = \langle C^\circ_G(x) \mid x \in D, \; |x|=p\rangle\]
centralises $D$ which contradicts the assumption that $G$ is
simple. \end{proof}
\begin{lemma}
\label{pairs} Let $K,L\in \Sigma$ be distinct and set
$M = \langle K, L\rangle$. Then the following statements hold.
\begin{itemize}
\item[{\rm (1)}] The subgroup
$C_D(K) \cap C_D(L) \ne 1$ and $M$ is a $K$-group.
\item[{\rm (2)}] Either\/ $K$ and\/ $L$ commute or\/
$M$ is an algebraic group of type $A_2$, $B_2$ or $G_2$.
\item[{\rm (3)}] $D\cap M = (D\cap K)(D\cap L)$ is a maximal\/ $p$-torus in
$M$.
\item[{\rm (4)}] If\/ $K$ and\/ $L$
do not commute then $H_M$ is a
maximal algebraic torus of the algebraic group $M$, and
$K$ and\/ $L$ are root\/ $SL_2$-subgroups of the algebraic
group $M$ with respect to the maximal torus $H_M$.
\item[{\rm (5)}] For all $K, L\in\Sigma$, we have $[H_K,H_L]=1$.
\item[{\rm (6)}] For any $K,L\in\Sigma$, if the $p$-tori
$D\cap K$ and\/ $D\cap L$ have
intersection of order $>2$, then $K=L$.
\end{itemize}
\label{lm:tori}
\end{lemma}
\begin{proof} For the proof of $C_D(K) \cap C_D(L) \ne 1$ we refer the reader to the proof of \cite[Lemma~9.3]{berkman}. After that $M \leqslant C_G(C_D(K) \cap C_D(L))$ is a proper definable subgroup of $G$ and is a $K$-group since $D$ normalizes $M$.
(2)-(3) For $L\in \Sigma$ set $R_L = C^\circ_D(L)$. If $n =
{\rm pr}(D)$, then the Pr\"{u}fer $p$-rank of $R_L$ is $n-1$.
Note that since $D$ is maximal and $D$ centralises a $p$-torus in
$L$, $D \cap L$ is a maximal $p$-torus in $L$. Now let $x$ be an
element of order $p$ in $R_K\cap R_L$. Then $K,L\leqslant
E(C_G(x))$ by the assumptions of the theorem. Set $E =
E(C_G(x))$. It follows from Lemma~\ref{lm:torus-in-component}
that the subgroup $D \cap E$ is a maximal $p$-torus of $E$, and
the subgroups $K$ and $L$, being $D$-invariant, lies in
components of $E$. If $K$ and $L$ belong to different components
of $E$, then they commute. Otherwise the component $A$ that
contains both $K$ and $L$ is a simple algebraic group, and
moreover $D\cap A$ is a maximal $p$-torus in $A$. Hence
the results follow from Assertion~\ref{prop:class-in-algebraic}.\\
(4)-(6) These follow by inspecting case by case and Assertion~\ref{prop:class-in-algebraic}. \end{proof}
\begin{lemma}
The subgroups in $\Sigma$ generate $G$.
\label{lm:sigmagenerates}
\end{lemma}
\begin{proof} Let $x\in D$ be of order $p$, then by assumption (B) of
the theorem
\[C^\circ_G(x) = F \cdot L_1\cdots L_n,\]
where $F = F^\circ(C^\circ_G(x))$ and $L_i \lhd C^\circ_G(x)$ is
a simple algebraic group, for each $i=1,\ldots,n$.
The first step is to prove that $L_1\cdots
L_n\leqslant\langle\Sigma\rangle$. Note that $D\leqslant
C^\circ_G(x)$ and $D\cap L_i$ is a maximal $p$-torus in $L_i$ by
Lemma~\ref{lm:torus-in-component}. Let $H_i$ stand for the
maximal algebraic torus in $L_i$ containing $D\cap L_i$ and
$\Gamma_i$ be the collection of root $SL_2$-subgroups in $L_i$
normalised by $H_i$. Since $D\cap L_i\leqslant H_i$, $D\cap L_i$
normalises the subgroups in $\Gamma_i$. By Lemma~\ref
{lm:torus-in-component}, we have $D=C_D(L_i)(D\cap L_i)$, hence
$D$ normalises $\langle\Gamma_i\rangle=L_i$; that is
$\Gamma_i\subseteq \Sigma$ for each $i=1,\ldots,n$. This proves
the first step.
Hence for each $x\in D$ of order $p$, $C^\circ_G(x)=F\cdot
E(C^\circ_G(x))\leqslant F\langle\Sigma\rangle \leqslant
C_G(D)\langle\Sigma\rangle$. Therefore
\[G=\langle C^\circ_G(x)\mid x\in D, |x|=p\rangle\leqslant C_G(D)
\langle\Sigma\rangle.\] Since $C_G(D)$ normalises
$\langle\Sigma\rangle$, we have $\langle\Sigma\rangle\unlhd G$.
Now the result follows, since $G$ is simple. \end{proof}
We make $\Sigma$ into a graph by taking $SL_2$-subgroups $L \in
\Sigma$ for vertices and connecting two vertices $K$ and $L$ by
an edge if $K$ and $L$ do not commute.
\begin{lemma}
The graph $\Sigma$ is connected. \label{lm:connected}
\end{lemma}
\begin{proof} Otherwise consider a decomposition $\Sigma = \Sigma'
\cup \Sigma''$ of $\Sigma$ into the union of two non-empty sets
such that no vertex in $\Sigma'$ is connected to a vertex in
$\Sigma''$. Then we have
\[G = \langle \Sigma \rangle = \langle \Sigma' \rangle \times \langle \Sigma'' \rangle,
\]
which contradicts the assumption that $G$ is simple. \end{proof}
\begin{lemma}
If\/ $L \in \Sigma$ then $L = E(C_G(C_D(L)))$. \label{lm:L-unique}
\end{lemma}
\proof Let ${\rm pr}(D) = n$, then ${\rm pr}(C_G(C_D(L)))=n$
and ${\rm pr}(E(C_G(C_D(L)))) =1$. Since $L \leqslant
E(C_G(C_D(L)))$ and $E(C_G(C_D(L)))$ is a central product of
simple algebraic groups over algebraically closed fields of
characteristic $\ne p$, we immediately conclude that $L =
E(C_G(C_D(L)))$. \hfill $\Box$
\subsection{Weyl Group}
Recall that when $L\in\Sigma$, $H_L$ stands for the maximal
algebraic torus $H_L:=C_L(D\cap L)$ in $L\cong SL_2$. Now set $H =
\langle H_L \mid L \in \Sigma \rangle$ and call it the {\em
natural torus associated with $D$}. It easily follows from
Lemma~\ref{lm:tori}(5) that $H$ is a divisible abelian group.
For any $L \in \Sigma$, $W(L):=N_L(H)H/H=N_L(H_L)/H_L$ is the
Weyl group of $SL_2$ and has order $2$; hence $W(L)$ contains a
single involution, which will be denoted by $r_L$. Note that the
subgroup $L$ is uniquely determined by $r_L$, since $C_D(L) =
C^\circ_D(r_L)$ and $L = E(C_G(C_D(L)))$ by
Lemma~\ref{lm:L-unique}.
\begin{lemma}
\label{refgp} Consider a graph\/ $\Delta$ with the set of vertices
$\Sigma$, in which two vertices $K$ and\/ $L$ are connected by an edge
if\/ $[r_K, r_L] \ne 1$. If\/ $K$ and\/ $L$ belong to different
connected components of\/ $\Delta$, then $[K,L] =1$.
\end{lemma}
\proof It suffices to check this statement in the subgroup
$M=\langle K,L\rangle$, where it is obvious by Lemma~\ref{pairs}(2).
\hfill $\Box$
\bigskip
Notice that $D$ is a $p$-torus, the subgroups $N_G(D)$ and
$C_G(D)$ are definable and the factor group $N_G(D)/C_G(D)$ is
finite. Set $W:=N_G(D)/C_G(D)$. The images of involutions $r_L$,
for $L\in \Sigma$, in $W$ generate a subgroup which we denote by
$W_0$. Since, by their construction, involutions $r_L$ normalise
$D$, there is a natural action of $W_0$ on $D$.
\begin{lemma} The $p$-torus $D$ lies in the natural torus $H$.
In particular, $D$ is the Sylow $p$-subgroup of\/ $H$. \label{lm:D<H}
\end{lemma}
\proof Set $D' = \langle D\cap L \mid L \in \Sigma \rangle$. It suffices
to prove that $D'=D$. First note that $D'\leqslant D \cap H$.
If $D'<D$ then, since $[D, r_L] = D\cap L$,
all involutions $r_L$ act trivially on the factor group $D/D'$
which is divisible.
Let us take an element $d \in D$ which has sufficiently big order
so that the image of $d^{|W_0|}$ in $D/D'$ has order at least
$p^2$. Then the element $$z=\prod_{w\in W_0} d^w$$ has the same
image in $D/D'$ as $d^{|W_0|}$ and thus $z$ has order at least
$p^2$. Since $D = C_D(L)(D\cap L)$ and $|C_D(L) \cap (D\cap L)|
\leqslant |Z(L)| \leqslant 2$, we see that $|C_D(r_L):
C_D(L)|\leqslant 2$. Of course, the equality is possible only if
$p=2$. In any case, since $z \in C_D(r_L)$, $z^p \in C_D(L)$ for
all $L \in \Sigma$ and $z^p \ne 1$. Hence $z^p \in C_G(\langle
\Sigma \rangle) = Z(G)$. This contradiction shows that $D = D'$
and $D \leqslant H$. \hfill $\Box$
\begin{lemma}
$N_G(D)= N_G(H)$. \label{lm:N=N}
\end{lemma}
\proof The embedding $N_G(H) \leqslant N_G(D)$ follows from
Lemma~\ref{lm:D<H}. Vice versa, if $x \in N_G(D)$ then the action
of $x$ by conjugation leaves the set $\Sigma$ invariant, hence it
leaves invariant the set of algebraic tori $\{H_L \mid L \in
\Sigma\}$ which generates $H$. Therefore $x \in N_G(H)$. \hfill
$\Box$
\begin{lemma}
$C_G(D)= C_G(H)$. \label{lm:C=C}
\end{lemma}
\proof Let $x \in C_G(D)$, then, for every $L \in \Sigma$, $x$
centralises $C_D(L)$ and thus, by Lemma~\ref{lm:L-unique},
normalises $L = E(C_G(C_D(L)))$. Since $x$ centralises a maximal
$p$-torus $D\cap L$ of $L$, it centralises the maximal torus $H_L
= C_L(D\cap L)$. Hence $x\in C_G(H)$. This proves $C_G(D)
\leqslant C_G(H)$. The reverse inclusion follows from
Lemma~\ref{lm:D<H}. \hfill $\Box$
\bigskip
In view of Lemmata~\ref{lm:N=N} and \ref{lm:C=C}, we can refer to
$W_0$ either as the subgroup generated by the images of
involutions $r_L$ in the factor group $N_G(D)/C_G(D)$ or as the
subgroup generated by the images of involutions $r_L$ in
$N_G(H)/C_G(H)$. Also, we now know that the group $W_0$ acts on
$D$ faithfully.
\subsection{Tate Module}
Now the aim is to construct a $\Bbb Z$-lattice on which $W_0$
acts as a crystallographic reflection group. For that purpose we
shall associate with $D$ the {\em Tate module} $T_p$. It is
constructed in the following way.
Let $E_{p^k}$ be the subgroup of $D$ generated by elements of order
$p^k$. Notice that every $r_L$ acts on $E_{p^k}$ as a reflection,
that is, $E_{p^k} = C_{E_{p^k}}(r_L)\times [E_{p^k}, r_L]$, $[E_{p^k},
r_L]\leqslant D \cap L$ is a cyclic group and $r_L$
inverts every element in $[E_{p^k}, r_L]$.
Consider the sequence of subgroups
\[E_p \longleftarrow E_{p^2} \longleftarrow E_{p^3} \longleftarrow
\cdots\] linked by the homomorphisms $x \mapsto x^p$. The
projective limit of this sequence is the free module $T_p$ over
the ring ${\Bbb Z}_p$ of $p$-adic integers. The action of $W_0$
on $D$ can be lifted to $T_p$, where it is still an irreducible
reflection group. By construction, $T_p /pT_p$ is isomorphic to
$E_p$ as a $W_0$-module. Notice also that $W_0$ acts on the
tensor product $T_p \otimes_{{\Bbb Z}_p} {\Bbb C}$ as a (complex)
reflection group, and that the dimension of $T_p
\otimes_{{\Bbb Z}_p} {\Bbb C}$ over $\Bbb C$
coincides with the Pr\"{u}fer
$p$-rank of $D$, hence is at least $3$.
\subsection{More Reflection Representations for ${W_0}$}
Now let us focus on odd primes $q\ne p$.
Consider the elementary abelian $q$-subgroups $E_q$ generated in
$H$ by all elements of the fixed prime order $q$. For the sake of
complete reducibility of the action of $W$ on $E_q$, one can
consider only $q > |W|$.
Lemmas~\ref{kerdcc}, \ref{long} and \ref{equiv} below are
similar to some results in \cite{berkman}.
We include the proofs here for the sake of completeness of exposition.
\begin{lemma} {\rm \cite[Lemma 9.7]{berkman}}
\label{kerdcc} Let\/ $N=N_G(H)$, then $C_N(E_q) = C_G(H)$.
\end{lemma}
\proof It is clear that $C_G(H)\leqslant C_N(E_{q})$. To see the
converse, let $x\in C_N(E_{q})$. Since $x\in N$,
it acts on the elements of $\Sigma$ by conjugation.
First let us prove that $x$ normalises each subgroup in $\Sigma$.
To get a contradiction, assume that there is some subgroup
$L\in\Sigma$ such that $L^x\neq L$. But then by
Lemma~\ref{pairs}, $L$ and $L^x$ either commute or generate a
semisimple group as root $SL_2$-subgroups. Hence $|L\cap
L^x|\leqslant 2$. But this gives a contradiction since $q$ is an
odd prime and $L\cap E_{q}=L^x\cap E_{q} \leqslant L\cap L^x$.
Hence for each $L\in \Sigma$, $L^x=L$ and $x$ acts on $H\cap L$ as
an element from $N_L(H\cap L)$, since $SL_2$ does not have any
definable outer automorphisms. Note that the Weyl group of $SL_2$
is generated by an involution which inverts the torus $H\cap L$.
Since $x$ centralises $E_{q}\cap H$, $x$ centralises $H\cap L$ for
each $L\in\Sigma$, and hence $x$ centralises $H=\langle H\cap
L\mid L\in \Sigma\rangle$ and $x\in C_G(H)$. This proves the
equality. \hfill $\Box$
\bigskip
Now notice that $[E_q,r_L]$ is generated by a $q$-element in $H_L$
and thus has order $q$. Hence $E_q$ is a finite dimensional
vector space over ${\Bbb F}_q$ on which $W_0$ acts as a group
generated by reflections.
\begin{lemma}
\label{wirr} The group $W_0$ acts irreducibly on $E_q$.
\end{lemma}
\proof Note that $W_0$ acts on $E_q$ faithfully by
Lemma~\ref{kerdcc}. Since $q > |W|$, the action of $W_0$ on $E_q$
is completely reducible. So if the action is reducible, then we can write
$E_q = E' \oplus E''$ for two proper $W_0$-invariant subspaces.
Assume that $W_0$ acts trivially on one of the subspaces $E'$ or
$E''$, say on $E'$. If $L\in \Sigma$, then $E_q = C_{E_q}(L)
\times (E_q \cap L)$, and, obviously, $C_{E_q}(L) = C_{E_q}(r_L)$.
Hence all $L \in \Sigma$ centralise $E'$ and $E' \leqslant
C_G(\langle \Sigma\rangle) = Z(G) =1$. Therefore $W_0$ acts
nontrivially on both $E'$ and $E''$.
For $L \in \Sigma$, the
$-1$-eigenspace $[E_q, r_L]$ of $r_L$ belongs to one of the
subspaces $E'$ or $E''$ and hence $r_L$ acts as a reflection on
one of the subspaces $E'$ or $E''$ and centralises the other.
Set $\Sigma'=\{\,L\in\Sigma \mid [E_q,r_L] \leqslant E'\,\}$
and $\Sigma'' = \{\,L\in\Sigma \mid [E_q,r_L] \leqslant E''\,\}$. It
is easy to see that $[r_K,r_L] =1$ for $K \in \Sigma'$ and $L \in
\Sigma''$. By Lemma~\ref{refgp}, $K$ and $L$ commute for all $K
\in \Sigma'$ and $L \in \Sigma''$, which contradicts
Lemma~\ref{lm:connected}. Hence $W_0$ is irreducible on $E_q$.
\hfill $\Box$
\begin{lemma}
The group $W_0$ acts irreducibly on $T_p \otimes_{{\mathbb{Z}}_p}
\mathbb{C}$.
\end{lemma}
\proof The proof is analogous to that of the previous lemma.
\hfill $\Box$
\subsection{Root System}
The aim of this subsection is to construct a root system on which $W_0$
acts as a crystallographic reflection group. The existence of such a
root system is guaranteed by the following lemma.
\medskip
\begin{lemma}
\label{rootsys} There exists an irreducible root system on which\/
$W_0$ acts as a crystallographic reflection group.
\end{lemma}
\proof Recall that $n \geqslant 3$. By Theorem~\ref{reflection},
the quotient group $W_0$ is one of the crystallographic reflection
groups $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4$ and acts on the
corresponding root system. \hfill $\Box$
\bigskip
Let now $R= \{\, {r}_i \mid i \in I \,\}$ be a simple
system of reflections in $W_0$. We shall identify $I$ with the
set of nodes of the Dynkin diagram for $W_0$. It is well known
that every reflection in an irreducible reflection group $W_0$ is
conjugate to a reflection in $R$.
\begin{lemma} {\rm \cite[Lemma~9.9]{berkman}}
\label{long} Every reflection\/ $r \in W_0$ has the form $r_K$ for
some $SL_2$-subgroup $K \in \Sigma$.
\end{lemma}
\proof Let $ r\in W_0$ be a reflection. Working our way back
through the construction of the module $T_p$, one can easily see
that the Pr\"{u}fer $p$-rank of $[D,r]$ is $1$.
Let $r_L\in W_0$ be a reflection which corresponds to a
$SL_2$-subgroup $L\in \Sigma$. By comparing the Pr\"{u}fer
$p$-ranks of the groups $C_D(r_L)$ and $C_D(r)$, we see that
$Z=(C_H(r_L) \cap C_H(r))^\circ$ has Pr\"{u}fer $p$-rank at least $1$.
Hence the subgroup $\langle L,H, r \rangle$ contains a non-trivial
central $p$-torus; also note that $\langle L,H, r \rangle$ is a
is a $K$-group, since $D$ lies in $H$. It is well known that a finite irreducible reflection
group contains at most two conjugate classes of reflections.
Therefore, after replacing $r$ and $r_L$ by their appropriate
conjugates in $W_0$, we can assume without loss of generality that
the images of $r_L$ and $r$ in $W_0$ correspond to adjacent nodes
of the Dynkin diagram. Now we can easily see that $\langle L,H, r
\rangle = Y * Z$ for some simple algebraic group $Y$ of Lie rank
$2$, and that $r=r_K$ for some root $SL_2$-subgroup $K$ of $Y$
such that $K\in \Sigma$. \hfill $\Box$
\subsection{Final Step}
The next task is to prove that the conditions of Lyons's Theorem
(Theorem~\ref{Lyons}) are satisfied.
\begin{lemma} {\rm \cite[Lemma~9.10]{berkman}}
\label{equiv} Attach an $SL_2$-subgroup $L_i \in \Sigma$ to each
vertex of the Dynkin diagram $I$ in such a way that the simple
reflection ${r}_i$ corresponding to this vertex is $r_{L_i}$ in
$W_0$. Then the following statements hold.
{\rm (1)} $[L_i,L_j]=1$
if and only if\/ $| {r}_i {r}_j|=2$.
{\rm (2)} $\langle
L_i,L_j\rangle$ is isomorphic to $(P)SL_3$ if and only if $| {r}_i
{r}_j|=3$.
{\rm (3)} $\langle L_i,L_j\rangle$ is isomorphic to
$(P)Sp_4$ if and only if\/ $| {r}_i {r}_j|=4$.
{\rm (4)} $L_i$
and $L_j$ are embedded in $\langle L_i,L_j\rangle$ as root\/
$SL_2$-subgroups.
\end{lemma}
\begin{proof} It is well known that for each $i, j\in I$, the order
$| {r}_i {r}_j|$ takes the values $2$, $3$ or $4$, in a Dynkin
diagram of type $A_n, B_n, C_n, D_n, E_6, E_7, E_8$ or $F_4$. By
Lemma~\ref{pairs}, $L_i$ and $L_j$ either commute or generate
$(P)SL_3$, $(P)Sp_4$ or $G_2$. However $\langle
L_i,L_j\rangle\cong G_2$ is not possible in our case since $|
{r}_i {r}_j|=6$ does not occur in $I$.
The `only if' parts of (1) and (2) are easy to see.
In the case of part (3), that is when $L_i$ and $L_j$
generate $(P)Sp_4$, we have to show that $| {r_i} {r_j}|\neq 2$.
To get a contradiction, assume that $L_i$ and $L_j$
generate $(P)Sp_4$ and $| {r_i} {r_j}|=2$.
But then
$L_i$ and $L_j$ are both
short root $SL_2$-subgroups. Note that ${r_i}$ and ${r_j}$ are
simple reflections, and it can be checked by inspection that one
of them must be a long reflection. This proves the `only if' part
of (3). Now parts (1), (2) and (3) follow from Lemma~\ref{pairs}
and the previous discussion. Part (4) is a direct consequence of
Lemma~\ref{pairs}. \end{proof}
\begin{lemma}
Each subgroup in $\Sigma$ is isomorphic to $(P)SL_2({\Bbb F})$ for
the same field $\Bbb F$.
\end{lemma}
\begin{proof} By Lemma~\ref{lm:connected} any two subgroups of
$\Sigma$ are connected by a sequence of edges. Note that each pair
which is connected by a single edge generates a simple group of
Lie rank 2 by Lemma~\ref{pairs}(2), hence their underlying fields
coincide. Thus the underlying fields of any two subgroups in
$\Sigma$ coincide. \end{proof}
Finally, we are in a position to apply Lyons's Theorem. Set $G_0$
to be the subgroup of $G$ generated by the subgroups $L_i$ for $ i
\in I$. By Lyons's Theorem, $G_0$ is a simple algebraic group over
$\Bbb F$ with the Dynkin diagram $I$. Its Weyl group, with respect
to the torus $T$, is $W_0$, hence $G_0$ contains all subgroups
from $\Sigma$. Therefore by Lemma~\ref{lm:sigmagenerates}, $G_0=G$
is a Chevalley group over $\Bbb F$. Since $G$ is of $p'$-type,
$\Bbb F$ is of characteristic different from $p$. \hfill $\Box$
$\Box$
\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,658 |
My wife and I split up 4 years ago and she has remained in the house since. I received a charging order but no CCJ on the house around 4 years ago. We have not officially divorced as yet, but she is looking to either move or remortgage the house without me on it, which she lives in with our daughter.
We have read in places that the charge can be avoided for her, but then we cannot find how to do this.
The mortgage deposit came from her bank account as have all of the payments so if the worse comes to the worst could we state that I have little or no equity in the house?
Any help would be great, plus also if there is a solicitor who can assist then this would be amazing! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,197 |
package main
import (
"context"
"encoding/json"
"errors"
"flag"
"fmt"
"log"
"net/http"
"os"
"path/filepath"
"strings"
"text/template"
"time"
"github.com/thrasher-corp/gocryptotrader/common"
"github.com/thrasher-corp/gocryptotrader/common/file"
"github.com/thrasher-corp/gocryptotrader/core"
)
const (
// DefaultRepo is the main example repository
DefaultRepo = "https://api.github.com/repos/thrasher-corp/gocryptotrader"
// GithubAPIEndpoint allows the program to query your repository
// contributor list
GithubAPIEndpoint = "/contributors"
// LicenseFile defines a license file
LicenseFile = "LICENSE"
// ContributorFile defines contributor file
ContributorFile = "CONTRIBUTORS"
)
var (
// DefaultExcludedDirectories defines the basic directory exclusion list for GCT
DefaultExcludedDirectories = []string{".github",
".git",
"node_modules",
".vscode",
".idea",
"cmd_templates",
"common_templates",
"communications_templates",
"config_templates",
"currency_templates",
"events_templates",
"exchanges_templates",
"portfolio_templates",
"root_templates",
"sub_templates",
"testdata_templates",
"tools_templates",
"web_templates",
}
// global flag for verbosity
verbose bool
// current tool directory to specify working templates
toolDir string
// exposes root directory if outside of document tool directory
repoDir string
// is a broken down version of the documentation tool dir for cross platform
// checking
ref = []string{"gocryptotrader", "cmd", "documentation"}
engineFolder = "engine"
)
// Contributor defines an account associated with this code base by doing
// contributions
type Contributor struct {
Login string `json:"login"`
URL string `json:"html_url"`
Contributions int `json:"contributions"`
}
// Config defines the running config to deploy documentation across a github
// repository including exclusion lists for files and directories
type Config struct {
GithubRepo string `json:"githubRepo"`
Exclusions Exclusions `json:"exclusionList"`
RootReadme bool `json:"rootReadmeActive"`
LicenseFile bool `json:"licenseFileActive"`
ContributorFile bool `json:"contributorFileActive"`
}
// Exclusions defines the exclusion list so documents are not generated
type Exclusions struct {
Files []string `json:"Files"`
Directories []string `json:"Directories"`
}
// DocumentationDetails defines parameters to update documentation
type DocumentationDetails struct {
Directories []string
Tmpl *template.Template
Contributors []Contributor
Config *Config
}
// Attributes defines specific documentation attributes when a template is
// executed
type Attributes struct {
Name string
Contributors []Contributor
NameURL string
Year int
CapitalName string
DonationAddress string
}
func main() {
flag.BoolVar(&verbose, "v", false, "Verbose output")
flag.StringVar(&toolDir, "tooldir", "", "Pass in the documentation tool directory if outside tool folder")
flag.Parse()
wd, err := os.Getwd()
if err != nil {
fmt.Println("Documentation tool error cannot get working dir:", err)
os.Exit(1)
}
if strings.Contains(wd, filepath.Join(ref...)) {
rootDir := filepath.Dir(filepath.Dir(wd))
repoDir = rootDir
toolDir = wd
} else {
if toolDir == "" {
fmt.Println("Please set documentation tool directory via the tooldir flag if working outside of tool directory")
os.Exit(1)
}
repoDir = wd
}
fmt.Print(core.Banner)
fmt.Println("This will update and regenerate documentation for the different packages in your repo.")
fmt.Println()
if verbose {
fmt.Println("Fetching configuration...")
}
config, err := GetConfiguration()
if err != nil {
log.Fatalf("Documentation Generation Tool - GetConfiguration error %s",
err)
}
if verbose {
fmt.Println("Fetching project directory tree...")
}
dirList, err := GetProjectDirectoryTree(&config)
if err != nil {
log.Fatalf("Documentation Generation Tool - GetProjectDirectoryTree error %s",
err)
}
var contributors []Contributor
if config.ContributorFile {
if verbose {
fmt.Println("Fetching repository contributor list...")
}
contributors, err = GetContributorList(config.GithubRepo, verbose)
if err != nil {
log.Fatalf("Documentation Generation Tool - GetContributorList error %s",
err)
}
// Github API missing contributors
contributors = append(contributors, []Contributor{
{
Login: "herenow",
URL: "https://github.com/herenow",
Contributions: 2,
},
{
Login: "mshogin",
URL: "https://github.com/mshogin",
Contributions: 2,
},
{
Login: "soxipy",
URL: "https://github.com/soxipy",
Contributions: 2,
},
{
Login: "tk42",
URL: "https://github.com/tk42",
Contributions: 2,
},
{
Login: "daniel-cohen",
URL: "https://github.com/daniel-cohen",
Contributions: 1,
},
{
Login: "DirectX",
URL: "https://github.com/DirectX",
Contributions: 1,
},
{
Login: "frankzougc",
URL: "https://github.com/frankzougc",
Contributions: 1,
},
// idoall's contributors were forked and merged, so his contributions
// aren't automatically retrievable
{
Login: "idoall",
URL: "https://github.com/idoall",
Contributions: 1,
},
{
Login: "mattkanwisher",
URL: "https://github.com/mattkanwisher",
Contributions: 1,
},
{
Login: "mKurrels",
URL: "https://github.com/mKurrels",
Contributions: 1,
},
{
Login: "m1kola",
URL: "https://github.com/m1kola",
Contributions: 1,
},
{
Login: "cavapoo2",
URL: "https://github.com/cavapoo2",
Contributions: 1,
},
{
Login: "zeldrinn",
URL: "https://github.com/zeldrinn",
Contributions: 1,
},
{
Login: "starit",
URL: "https://github.com/starit",
Contributions: 1,
},
{
Login: "Jimexist",
URL: "https://github.com/Jimexist",
Contributions: 1,
},
{
Login: "lookfirst",
URL: "https://github.com/lookfirst",
Contributions: 1,
},
{
Login: "merkeld",
URL: "https://github.com/merkeld",
Contributions: 1,
},
{
Login: "CodeLingoTeam",
URL: "https://github.com/CodeLingoTeam",
Contributions: 1,
},
{
Login: "Daanikus",
URL: "https://github.com/Daanikus",
Contributions: 1,
}, {
Login: "CodeLingoBot",
URL: "https://github.com/CodeLingoBot",
Contributions: 1,
},
{
Login: "blombard",
URL: "https://github.com/blombard",
Contributions: 1,
},
{
Login: "soxipy",
URL: "https://github.com/soxipy",
Contributions: 2,
},
{
Login: "lozdog245",
URL: "https://github.com/lozdog245",
Contributions: 2,
},
}...)
if verbose {
fmt.Println("Contributor List Fetched")
for i := range contributors {
fmt.Println(contributors[i].Login)
}
}
} else {
fmt.Println("Contributor list file disabled skipping fetching details")
}
if verbose {
fmt.Println("Fetching template files...")
}
tmpl, err := GetTemplateFiles()
if err != nil {
log.Fatalf("Documentation Generation Tool - GetTemplateFiles error %s",
err)
}
if verbose {
fmt.Println("All core systems fetched, updating documentation...")
}
UpdateDocumentation(DocumentationDetails{
dirList,
tmpl,
contributors,
&config})
fmt.Println("\nDocumentation Generation Tool - Finished")
}
// GetConfiguration retrieves the documentation configuration
func GetConfiguration() (Config, error) {
var c Config
configFilePath := filepath.Join(toolDir, "config.json")
if file.Exists(configFilePath) {
config, err := os.ReadFile(configFilePath)
if err != nil {
return c, err
}
err = json.Unmarshal(config, &c)
if err != nil {
return c, err
}
if c.GithubRepo == "" {
return c, errors.New("repository not set in config.json file, please change")
}
return c, nil
}
fmt.Println("Creating configuration file, please check to add a different github repository path and change preferences")
// Set default params for configuration
c.GithubRepo = DefaultRepo
c.ContributorFile = true
c.LicenseFile = true
c.RootReadme = true
c.Exclusions.Directories = DefaultExcludedDirectories
data, err := json.MarshalIndent(c, "", " ")
if err != nil {
return c, err
}
if err := os.WriteFile(configFilePath, data, file.DefaultPermissionOctal); err != nil {
return c, err
}
return c, nil
}
// IsExcluded returns if the file path is included in the exclusion list
func IsExcluded(path string, exclusion []string) bool {
for i := range exclusion {
if path == exclusion[i] {
return true
}
}
return false
}
// GetProjectDirectoryTree uses filepath walk functions to get each individual
// directory name and path to match templates with
func GetProjectDirectoryTree(c *Config) ([]string, error) {
var directoryData []string
if c.RootReadme { // Projects root README.md
directoryData = append(directoryData, repoDir)
}
if c.LicenseFile { // Standard license file
directoryData = append(directoryData, filepath.Join(repoDir, LicenseFile))
}
if c.ContributorFile { // Standard contributor file
directoryData = append(directoryData, filepath.Join(repoDir, ContributorFile))
}
walkFn := func(path string, info os.FileInfo, err error) error {
if err != nil {
return err
}
if info.IsDir() {
// Bypass what is contained in config.json directory exclusion
if IsExcluded(info.Name(), c.Exclusions.Directories) {
if verbose {
fmt.Println("Excluding Directory:", info.Name())
}
return filepath.SkipDir
}
// Don't append parent directory
if strings.EqualFold(info.Name(), "..") {
return nil
}
directoryData = append(directoryData, path)
}
return nil
}
return directoryData, filepath.Walk(repoDir, walkFn)
}
// GetTemplateFiles parses and returns all template files in the documentation
// tree
func GetTemplateFiles() (*template.Template, error) {
tmpl := template.New("")
walkFn := func(path string, info os.FileInfo, err error) error {
if err != nil {
return err
}
if info.IsDir() {
if path == "." || path == ".." {
return nil
}
var tmplExt *template.Template
tmplExt, err = tmpl.ParseGlob(filepath.Join(path, "*.tmpl"))
if err != nil {
fmt.Println(err)
if strings.Contains(err.Error(), "pattern matches no files") {
return nil
}
return err
}
tmpl = tmplExt
return filepath.SkipDir
}
return nil
}
return tmpl, filepath.Walk(toolDir, walkFn)
}
// GetContributorList fetches a list of contributors from the github api
// endpoint
func GetContributorList(repo string, verbose bool) ([]Contributor, error) {
contents, err := common.SendHTTPRequest(context.TODO(),
http.MethodGet,
repo+GithubAPIEndpoint,
nil,
nil,
verbose)
if err != nil {
return nil, err
}
var resp []Contributor
return resp, json.Unmarshal(contents, &resp)
}
// GetDocumentationAttributes returns specific attributes for a file template
func GetDocumentationAttributes(packageName string, contributors []Contributor) Attributes {
return Attributes{
Name: GetPackageName(packageName, false),
Contributors: contributors,
NameURL: GetGoDocURL(packageName),
Year: time.Now().Year(),
CapitalName: GetPackageName(packageName, true),
DonationAddress: core.BitcoinDonationAddress,
}
}
// GetPackageName returns the package name after cleaning path as a string
func GetPackageName(name string, capital bool) string {
newStrings := strings.Split(name, " ")
var i int
if len(newStrings) > 1 {
// retrieve the latest spacing to define the most childish package name
i = len(newStrings) - 1
}
if capital {
return strings.Replace(strings.Title(newStrings[i]), "_", " ", -1) // nolint // ignore staticcheck strings.Title warning
}
return newStrings[i]
}
// GetGoDocURL returns a string for godoc package names
func GetGoDocURL(name string) string {
if strings.Contains(name, " ") {
return strings.Join(strings.Split(name, " "), "/")
}
if name == "testdata" ||
name == "tools" ||
name == ContributorFile ||
name == LicenseFile {
return ""
}
return name
}
// UpdateDocumentation generates or updates readme/documentation files across
// the codebase
func UpdateDocumentation(details DocumentationDetails) {
for i := range details.Directories {
cutSet := details.Directories[i][len(repoDir):]
if cutSet != "" && cutSet[0] == os.PathSeparator {
cutSet = cutSet[1:]
}
data := strings.Split(cutSet, string(os.PathSeparator))
var temp []string
for x := range data {
if data[x] == ".." {
continue
}
if data[x] == "" {
break
}
temp = append(temp, data[x])
}
var name string
if len(temp) == 0 {
name = "root"
} else {
name = strings.Join(temp, " ")
}
if IsExcluded(name, details.Config.Exclusions.Files) {
if verbose {
fmt.Println("Excluding file:", name)
}
continue
}
if name == engineFolder {
d, err := os.ReadDir(details.Directories[i])
if err != nil {
fmt.Println("Excluding file:", err)
}
for x := range d {
nameSplit := strings.Split(d[x].Name(), ".go")
engineTemplateName := engineFolder + " " + nameSplit[0]
if details.Tmpl.Lookup(engineTemplateName) == nil {
fmt.Printf("Template not found for path %s create new template with {{define \"%s\" -}} TEMPLATE HERE {{end}}\n",
details.Directories[i],
name)
continue
}
err = runTemplate(details, filepath.Join(details.Directories[i], nameSplit[0]+".md"), engineTemplateName)
if err != nil {
fmt.Println(err)
}
}
continue
}
if details.Tmpl.Lookup(name) == nil {
fmt.Printf("Template not found for path %s create new template with {{define \"%s\" -}} TEMPLATE HERE {{end}}\n",
details.Directories[i],
name)
continue
}
var mainPath string
switch {
case name == LicenseFile || name == ContributorFile:
mainPath = details.Directories[i]
default:
mainPath = filepath.Join(details.Directories[i], "README.md")
}
if err := runTemplate(details, mainPath, name); err != nil {
log.Println(err)
continue
}
}
}
func runTemplate(details DocumentationDetails, mainPath, name string) error {
err := os.Remove(mainPath)
if err != nil && !(strings.Contains(err.Error(), "no such file or directory") ||
strings.Contains(err.Error(), "The system cannot find the file specified.")) {
return err
}
f, err := os.Create(mainPath)
if err != nil {
return err
}
defer func(f *os.File) {
err := f.Close()
if err != nil {
log.Printf("could not close file %s: %v", mainPath, err)
}
}(f)
attr := GetDocumentationAttributes(name, details.Contributors)
return details.Tmpl.ExecuteTemplate(f, name, attr)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 856 |
The National Head Start Association and the Early Head Start program at Carey Services will be celebrating the 50th anniversary of the national program with a Red, White and Blue day on Wednesday, Sept. 17.
"Education is not a problem," President Lyndon B. Johnson said 50 years ago. "Education is an opportunity" Head Start has opened that window of opportunity for the nation's most at-risk children and families since Johnson signed legislation creating the program.
On Wednesday, Head Start students, teachers, staff members, parents and supporters will be wearing red, white and blue to help celebrate the anniversary.
Carey Services' Early Head Start program provides services for families in Grant and Blackford counties. The program provides services for expectant mothers through children up to age 3.
For information about Carey Services' Early Head Start program, call (765) 668-8961.
For information about the 50th anniversary of the national Head Start program, visit their website. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,963 |
Q: Azure Time Trigger java function - not working locally Azure time trigger function java - is not scheduling while running locally in Intellij ,any lead would help...
sample code generated by Intellij is as follows
public class TimerTriggerFunction {
/**
* This function will be invoked periodically according to the specified schedule.
*/
@FunctionName("TimerTrigger-Java")
public void run(
@TimerTrigger(name = "timerInfo", schedule = "0 */5 * * * *") String timerInfo,
final ExecutionContext context) {
context.getLogger().info("Java Timer trigger function executed at: " + LocalDateTime.now());
}
}
logs:
The listener for function 'Functions.TimerTrigger-Java' was unable to start. Azure.Core: Retry failed after 6 tries.
Retry settings can be adjusted in ClientOptions.
Retry.
(No connection could be made because the target machine actively refused it. (127.0.0.1:10000)) (No connection could be made because the target machine actively refused it. (127.0.0.1:10000))
A: The error that you are getting because you have not installed Azurite in your local as,
The Azurite open-source emulator allows you to test your Azure blob,
queue, and table storage apps locally for free. Switch to using an
Azure Storage account in the cloud if you're satisfied with how your
application works locally.
For more information please refer to this Microsoft document: Create your first Java function in Azure using IntelliJ & this blog
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,384 |
\section{Introduction}
In a previous study \cite{Plath2019}, the acoustics, timbre, and perception of a piano have been
investigated at two stages, firstly right after manufacture (further denoted as
\textit{stage 1}) and secondly after one year of performance in a concert hall
(\textit{stage 2}).
In an ABX double-blind comparison test \cite{Clark1982} all 88 keys of both
stages were presented to subjects. Additionally, subjects were asked to
verbalize how they where able to distinguish between stages (regardless of whether they could tell correctly). The text input was cleaned and filtered utilizing natural language processing (NLP) methods and further categorized
as affiliated to the domains \textit{timbre}, \textit{pitch}, \textit{temporal}, \textit{spatial}, and \textit{loudness}.
Although listeners were significantly able to distinguish between the two stages, data analysis
was not able to clearly identify the reason(s). Analyzed acoustical features
were driving point mobility, which did not change considerably,
the crown bearing, and the action speed. Although the latter two showed
differences, like a clear homogenization of the action speed over the year, they could not be
associated with perception.
Signal processing of the data sets showed deviations between the two stages:
The spectral centroid was considerably lower for the mid- and high-frequency
range one year after performance. Still this did not correlate with the distribution
of verbalizations in the timbre category. Regarding pitch, the majority of
verbalizations was given in frequency-ranges where no measurable pitch
differences were present. Also, although verbalizations of temporal differences
were reported, no considerable differences of attack time or decay time (RT60) could
be observed. Regarding loudness domain features, Sound Pressure Level (SPL) and
Interlevel Differences (ILD) decreased in the mid- and high-frequency range one
year after playing, still verbalizations were too few to find correlations.
In this paper, a new approach is pursued to distinguish
the two piano stages using Music Information Retrieval (MIR) parameters and
Machine Learning (ML), namely a Self-Organizing Kohonen Map (SOM).
\section{Method}
The \textit{apollon} and \textit{COMSAR} frameworks \cite{apollon,COMSAR},
developed at the Institute of Systematic Musicology over the last years, have
been used within the project of \textit{Computational Phonogram Archiving }
\cite{Bader2019,
Bader2021a, Bader2021b, Bader2021c}. The system uses the following timbre features to generate a feature vector:
spectral centroid, spectral spread, spectral flux, roughness,
sharpness, SPL, and fractal correlation dimension. The fractal correlation
dimension counts the amount of inharmonic partials and strong amplitude
fluctuations which appear during the initial transient of musical
sounds \cite{Bader2013}. Therefore it is a measure of the chaoticity of initial
transients.
All 2$\times$88 single played tones are analyzed with respect to these features
in frames of 50 ms over the cause of each sound and the arithmetic mean for each timbre
feature is calculated.
The aim is to cluster the two piano stages on a two-dimensional Kohonen map.
Therefore, different feature combinations are used. At first, all features are
trained by different maps as single features. Therefore the feature vector on
each nodal point has only one entry. Secondly, different combinations of
feature vectors are used and the resulting trained maps are selected according
to clustering of piano stages as well as other sub-clusters within the two
piano stages. It turns out that only single and double-features succeed in
finding such clusters.
\section{Results}
\subsection{Single feature SOMs}
In Fig. \ref{SOM_Single} four single timbre features are used to train the SOM.
For the two-dimensional maps tones of all 88 keys at two stages are displayed as
dots, where the hue is changed from lowest to highest keys. Stage 1 keys are
red (lowest) to yellow (highest), stage 2 are blue (lowest) to violet
(highest). Additionally, the key number is shown next to the dot. Sometimes
more than one key is placed on one neuron. The background color shows the
u-matrix of the SOM, which is the distance between neighboring neurons. Darker
colors imply high similarity for neighboring neurons, lighter colors
like yellow mean that the neighboring neurons are very different.
Therefore, yellow ridges indicate boundaries between more homogeneous regions
displayed in blue.
Spectral flux (upper left) is able to perfectly cluster the two piano stages, where stage 1 is on the upper right and stage 2 on the lower left side of the map. Both stages are clearly separated by a strong ridge in the u-matrix (yellow line). The SOM orders spectral flux as a continuous decrease from a maximum at the lower right corner to a minimum at the upper left corner. Therefore, stage 1 has considerably more flux than stage 2. This might be the main reason for the successful separation of the two stages by listeners found in \cite{Plath2019}. Furthermore, stage 2 shows a continues decay of flux from lower keys to higher ones, where stage 1 is not at all so clearly sorted.
SPL (upper right) is not able to cluster the stages, but clearly puts the low keys on the upper left corner where SPL is high and the high keys in the lower right corner where SPL is low. Additionally, next to a main ridge separating the lower right from the upper left corner, two ridges separate a middle region. Therefore, in terms of SPL, four clusters are found, where the two middle clusters sort keys in the medium-frequency range.
Roughness (lower left) clusters lower keys of both stages at the lower left corner and distributes all other keys in the rest of the map. This region of strong roughness, expected from lower keys is strong in the u-matrix (yellow region) meaning that additionally to strong roughness, the low keys are also very diverse with respect to roughness.
The map for fractal correlation dimension (lower right) behaves similarly to the one for roughness, pointing to a correlation between the two parameters with respect to piano sounds. Training a SOM with only these two timbre features does not result in a perfect orthonormal relation, but also not in a perfect correlation.
Therefore, spectral flux is assumed to be the main cause of listeners being able to distinguish the two stages aurally. The other timbre parameters go into details which is further investigated using timbre combinations below. Other timbre features did not result in any clear clusters.
\begin{figure}[!]
\begin{center}
\includegraphics[width=1\columnwidth]{PianoKI_SOM_Single}
\includegraphics[width=.7\columnwidth]{viridis_legend}
\caption{SOM for four single timbre features: spectral flux (upper left), SPL (upper right), roughness (lower left), and fractal correlation dimension (lower right). Stage 1 pianos keys: red(lower keys)-yellow(upper keys), stage 2 piano keys: blue(lower keys)-violet(upper keys). Background: u-matrix displaying similarity between neighboring neurons. Spectral flux clusters the pianos perfectly. SPL, roughness, and fractal correlation cluster according to keys.}
\end{center}
\label{SOM_Single}
\end{figure}
\subsection{Double-feature SOMs}
As shown with multi-dimensional scaling (MDS) listening tests over the last decades, timbre perception is multi-dimensional (for a review see \cite{Bader2013}). Therefore it is likely that listeners use a combination of timbre features for distinguishing between tones. Almost all MDS investigations resulted in a maximum of three timbre features, where the third one is most often very weakly present, hereinafter combinations of two features are presented. Since spectral flux is the only feature being able to distinguish between stage 1 and 2 and as listeners were asked for performing a difference discrimination, spectral flux was used in all three combinations discussed below. The respective second features used in combination with spectral flux are SPL, roughness, and fractal correlation dimension, as again only these features did result in reasonable clusters as single features (see above).
The combination of spectral flux vs. SPL in Fig. \ref{SOM_Double} (upper left) maintains a perfect cluster of the two stages, where stage 1 is in the upper right and stage 2 in the lower left corner. Additionally, both stages are clearly ordered in terms of their keys, where lower keys are in the lower right and upper keys in the upper left corner. The distribution of the two features over the map are almost perfectly orthonormal, as shown in Fig. \ref{fig:pianokicomponentleftfluxrightspl}. Again, the spectral flux is stronger for stage 1, and SPL is stronger for lower keys. Additionally, a distinct ridge splits the lower keys from the upper ones, going from the lower left to the upper right corner. Another strong ridge can be seen in the lower right side which is considerably stronger for stage 1 than for stage 2. Another smaller ridge is found in the upper left corner, again a bit stronger for stage 1 than for stage 2. These ridges clearly separate stage 1 into four distinct regions, ordered in terms of low keys, lower middle keys, upper middle keys, and high keys. Such a separation can also slightly be seen for stage 2, although it is less clearly developed. This points to a homogenization of SPL from stage 1 to stage 2, without influence on spectral flux.
The combination of spectral flux vs. roughness, shown in Fig. \ref{SOM_Double} (upper right) is no longer able to perfectly cluster the stages. The feature distribution, shown in Fig. \ref{fig:pianokicomponentleftfluxrightroughness} is also not perfectly orthonormal, flux is strongest at the upper right side, as expected, is weakened towards to upper left side only to get stronger again at the upper left corner. This is reflected in the distribution of stage 1 keys of both pianos at the upper left corner. The roughness distribution is clearer, with a maximum at the upper left corner, where the low keys are located, smoothly decreasing towards the lower right corner. Due to the decrease in clustering of stage 1 and stage 2 this combination of timbre features seems unlikely to be used by listeners.
The last combination shown is spectral flux vs. fractal correlation dimension in Fig. \ref{SOM_Double}, lower left. The feature distributions shown in Fig. \ref{fig:pianokicomponentleftfluxrightfractal} are now again nearly perfectly orthonormal to one another, with flux being strongest in the upper right corner, where stage 1 is located, and fractal dimension being strongest in the upper left corner, where low keys are distributed. Interestingly, stage 2 pianos have a sub-cluster at the lower end of the plot, separated from upper mid-frequency keys by a ridge. Another sub-cluster is found at the left side roughly between neurons 11-16. Stage 1 sounds are sub-clustered in a different way, where two sub-clusters are separated by a ridge from the upper right corner to the center of the plot. Additionally, stage 1 keys 87 and 83 (yellow) are found around neurons (10-12, 17-20), so in the range of low keys. Overall, stage 2 pianos have a much clearer sub-clustering and smoother key distribution, pointing to a more systematic fractal dimension distribution. As the fractal dimension displays the chaoticity in the initial transient of the signal, the transient of stage 2 seems to be clearer organized.
\begin{figure}[!]
\begin{center}
\includegraphics[width=1\columnwidth]{PianoKI_SOM_Double}\\
\includegraphics[width=.7\columnwidth]{viridis_legend}
\caption{SOM for three double-feature correlations: spectral flux/SPL (upper left), spectral flux/roughness (upper right), and spectral flux/fractal correlation dimension (lower left). Coloring like in \ref{SOM_Single}. While the combinations with SPL and fractal correlation maintain perfect clustering of stage 1 and 2, combining with roughness mixes the clusters. Also additional sub-clustering is observable.}
\end{center}
\label{SOM_Double}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{PianoKI_Component_left_Flux_right_SPL}
\caption{Feature distribution on SOM for the combination spectral flux/SPL of Fig. \ref{SOM_Double} (upper left). Left: spectral flux, right: SPL. Yellow: large values, blue: low values. The two features are nearly perfectly orthonormal on to another.}
\label{fig:pianokicomponentleftfluxrightspl}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{PianoKI_Component_left_Flux_right_Roughness_}
\caption{Distribution of features spectral flux (left) and roughness (right) for SOM shown in Fig. \ref{SOM_Double}, upper right. Color as in Fig. \ref{fig:pianokicomponentleftfluxrightspl}.}
\label{fig:pianokicomponentleftfluxrightroughness}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{PianoKI_Component_left_Flux_right_Fractal}
\caption{Distribution of features spectral flux (left) and fractal correlation dimension (right) for SOM shown in Fig. \ref{SOM_Double}, lower left. Color as in Fig. \ref{fig:pianokicomponentleftfluxrightspl}.}
\label{fig:pianokicomponentleftfluxrightfractal}
\end{figure}
\section{Conclusions}
Spectral flux clusters the two piano stages perfectly and therefore is assumed to be the main cause of the clear separation of the two stages by listeners found in \cite{Plath2019}. Furthermore, stage 2 has a much smoother distribution of SPL and a clearer sub-clustering structure of the chaoticity of initial transients. This is expected due to the one-year treatment of piano tuners of stage 2 compared to state 1, which seem to have these features as their aim. Assuming at least two timbre parameters to be used by listeners to separate the two stages, two candidates appear from the present analysis, being spectral flux with SPL and spectral flux with fractal correlation dimension.
\newpage
\addtocounter{page}{2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,315 |
Q: How do I pass props to the onSubmitSuccess callback in Redux Form? I am buildng a react native app and am using Redux-form for my registration form. I currently have a form and on success, I would like to navigate to the next page only on success though. In my code, I am using redux-form's onSubmitSuccess function and I need to use props to navigate to the next page, but when I try using props in onSubmitSuccess it tells me that "props is not defined". In redux-form v6.5.0, you can pass props to onSubmitSuccess but I'm unsure of exactly how to do this. This is the documentation for it: https://github.com/erikras/redux-form/blob/master/docs/api/ReduxForm.md#onsubmitsuccess--function-optional
Below is a snippet of my code:
render() {
const {handleSubmit, touched, error} = this.props
const onSubmit = (values, dispatch) => {
this.props.addFirstName(values.firstName)
this.props.addLastName(values.lastName)
this.props.addEmailAddress(values.emailAddress)
this.props.addPassword(values.password)
}
return (
<Container>
<View theme={theme}>
<Image source={require('../../../images/BG-signUp.png')} style={styles.background} >
<Content padder scrollEnabled={false}>
<Text style={styles.signupHeader}>
CREATE ACCOUNT
</Text>
<View style={styles.signupContainer}>
<Field name = "firstName" component = {renderFirstName} />
<Field name = "lastName" component = {renderLastName} />
<Field name = "emailAddress" component = {renderEmailAddress} />
<Field name = "password" component = {renderPassword} />
<Field name = "confirmPassword" component = {renderConfirmPassword} />
<Button
rounded transparent block
onPress={handleSubmit(onSubmit)}
style={styles.signupBtn}
>
Continue
</Button>
<TouchableOpacity>
<Text style={styles.termsText}>Terms & Conditions</Text>
</TouchableOpacity>
</View>
</Content>
</Image>
</View>
</Container>
)
}
function bindAction(dispatch) {
return {
addFirstName: (firstName) => dispatch(addFirstName(firstName)),
addLastName: (lastName) => dispatch(addLastName(lastName)),
addEmailAddress: (emailAddress) => dispatch(addEmailAddress(emailAddress)),
addPassword: (password) => dispatch(addPassword(password)),
reset: key => dispatch(reset([{ key: 'orgPage' }], key, 0)),
};
}
const mapStateToProps = state => ({
navigation: state.cardNavigation,
credentials: state.credentials
});
/*export default connect(mapStateToProps, bindAction)(SignUp);*/
SignUp = reduxForm({
form: 'signup',
validate,
onSubmitSuccess: () => {
this.props.reset(this.props.navigation.key)
}
})(SignUp);
SignUp = connect(mapStateToProps, bindAction)(SignUp);
A: SignUp = reduxForm({
form: 'signup',
validate,
onSubmitSuccess: () => {
this.props.reset(this.props.navigation.key)
}
})(SignUp);
this has no meaning inside of this fat-arrow function. The props are given to onSubmitSuccess as the third parameter.
SignUp = reduxForm({
form: 'signup',
validate,
onSubmitSuccess: (result, dispatch, props) => {
props.reset(props.navigation.key) // <---------
}
})(SignUp);
Hope that helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,485 |
Olaus Olai, latinisering av Olof Olofsson, kan avse:
Olaus Olai Wexionius
Olof Wexionius
Olaus Olai Nauclerus
Olaus Olai (kyrkoherde i Fellingsbro) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,107 |
{"url":"https:\/\/www.physicsforums.com\/threads\/why-can-a-bird-survive-on-a-high-voltage-power-line-without-getting-fried.772578\/","text":"# Homework Help: Why can a bird survive on a high voltage power line without getting fried?\n\n1. Sep 23, 2014\n\n### Hereformore\n\n1. The problem statement, all variables and given\/known data\nBirds can be seen perched on a high voltage power line yet seem unaffected by the high voltage. This is due to...\n\nA) Low resistance of the birds compared to the wire, minimizing current through the birds\nB) birds keeping only one foot on the wire, thus not completing a circuit\nC) low capacitance of the birds (?)\nD) Minimal potential difference across the birds\n\n2. Relevant equations\nV=IR\n\nR = p(L\/A)\n\n3. The attempt at a solution\n\nAssuming that current is constant across the entire transmisison line (it is right?), since the distance between the birds feet is very small, and since resistance is determined by length, the resulting resistence across that portion of wire would be very small.\n\nSince current is constant across the wire, and the resistence is very small, then the resulting Voltage across those two points must be very small as well. The bird is effectively in parallel with the wire portion so it feels the same voltage,. Since the voltage is so small, it is essentially unaffected.\n\nSo that is why D is the correct answer.\n\nWhat i dont understand is why a bird keeping only one foot on the wire wouldn't be a legitate reason either. There is no complete circuit so charge cant flow.\n\nI was thrown off by the \"low capacitance of the birds\" statement, and since q=CV , I knew that would be false. But does capacitance even apply here? I thought it only applied when you had two parallel plates?\n\nFinally, in this scenario, we know the voltage difference between the beginning and end of the wire (thats what we refer to when we say it's \"high voltage\" right?), current is constant throughout the wire right? I ask because it seemed odd to base my calculations of V=IR upon the constant current. Usually you use voltage to find the curent.\n\nThanks!\n\n2. Sep 23, 2014\n\n### Staff: Mentor\n\nBecause most birds don't stand on the wire on one leg.\n\nHowever, it is not true that birds survive on a high voltage power line without getting fried. Sometimes they don't. There is a 15 kV three phases line not far from here, often used by stock doves. Once when they started I saw a flash and heard an arc. Apparently one unlucky dove short circuited two phases.\n\n3. Sep 23, 2014\n\n### KL7AJ\n\nIn Alaska we have ravens with enough wingspan to bridge between the two legs of a 250Kv liine. Every once in a while one manages to do it, and there is a rather impressive mushroom cloud of feathers and beaks. \")\n\n4. Sep 23, 2014\n\n### cwasdqwe\n\nI do agree on D as the right answer. As you say, across the wire the potential is really small in the wire portion a bird can occupy, so the electrons will rather travel through the metal wire (where conducting electrons move freely) than through the bird's body. If this bird was to touch another wire which had a different potential, then the electrons would prefer, this time, to take the shortcut that's the bird... And the electrons move, and our friend... would shine electric. So it has to do with potential and resistance (exactly the opposite of answer A).\n\n5. Sep 23, 2014\n\n### Staff: Mentor\n\nLow capacitance is an important point as well. You can view the birds as one capacitor \"plate\", with the whole space around them as \"gap\" (there is no need for a second plate, but the other lines, the ground and so on work as one). As (nearly all) HV transmission lines operate with AC, each cycle charges and discharges the birds a bit. As their capacitance is tiny, this is not relevant for birds. In some countries, there are workers on live wires, where you can see this effect - the approaching helicopter does have a significant capacitance (example, starting at 1:00).\n\n6. Sep 23, 2014\n\n### Orodruin\n\nStaff Emeritus\nJust to clarify. The \"high-voltage\" refers to the potential difference between the wires, not between the beginning and end of a wire. Generally you want to drive a load which takes some given power and you do not want significant losses in the wires. Power lost in the wires is $P = R I^2$ and so the less current you need to use the less power you will lose in the wires. Thus you get more power left for the load if you can have a high voltage difference in the load (load power is UI where U is the potential across it - more U, less current). Typically the \"load\" would be a transformer station which then distributes electricity for local consumption at a lower voltage.\n\n7. Sep 24, 2014\n\n### rcgldr\n\nThere's also a voltage differential along a high voltage wire if the length doesn't correspond to the effective wave length of the AC voltage wave. There are line \"inspector\" vehicles that hang from a transmission line via two wheels about 1 meter apart and use transformer effect to extract energy from a single line.\n\nI also had the impression that birds could get zapped (arc or antenna effect) by the really high voltage wires, 500 kv to 750kv used in power tranmssion lines. These usually have a ground wire positioned at the highest point (this wire is grounded locally, and not connected to the ground of the generating source), and hopefully birds will choose to land on that ground wire. The humming and or sensation of the voltage on the transmission lines may scare off the birds. The guys that service these lines wear a faraday cage suit, approach the lines from helicopter based platform, and during transition on and off the line, they have to connect a wire to synchronize the helicopter with the voltage potential of the transmission wire. Youtube video:","date":"2018-07-21 23:58:09","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\": 0.5701060891151428, \"perplexity\": 965.7185909140733}, \"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-2018-30\/segments\/1531676592861.86\/warc\/CC-MAIN-20180721223206-20180722003206-00299.warc.gz\"}"} | null | null |
\section{Introduction}
Realization of synthetic spin-orbit coupling (SOC) for cold atoms has enabled studies of many exotic quantum phenomena in a highly controllable environment of atomic Bose-Einstein condensates (BEC)
\cite{Goldman_2014,Zhai2015,ZHANG201975}.
A peculiar feature of spin-orbit-coupled systems is their energy-momentum dispersion, which contains two minima. The interplay between collisional interactions and SOC then leads to a formation of two different phases: single-momentum phase, when atoms occupy one of the two minima, and stripe phase, when both minima are occupied. In the latter case, interference of the two momentum states leads to characteristic spatially periodic density modulations, thus the name ``stripe phase''. Due to a broken translational symmetry and an emergent long-range order this phase is also often associated with the supersolidity phenomenon \cite{Li2017,RevModPhys.84.759}.
The pioneering realizations of synthetic SOC in bosonic \cite{Lin2011} and fermionic \cite{PhysRevLett.109.095301} quantum gases were based on a two-photon Raman coupling between atomic hyperfine states.
If the Raman transition is accompanied by a change in the center-of-mass momentum of the atom, the pseudo-spin and momentum degrees of freedom become coupled in the condensate.
We will refer to such coupling as the linear SOC.
Subsequently
a number of other techniques have been developed to achieve similar effects \cite{luo2016tunable,PhysRevLett.117.185301}.
However, flexibility of the Raman coupling scheme together with a wide variety of accessible light fields allowed to propose and generate many different types of SOC in cold-atom setups \cite{Wu2016,PhysRevLett.118.190401}.
One specific type of Raman-induced SOC relevant for the present work is the spin-orbital-angular-momentum coupling (SOAMC), which is achieved when Raman laser beams carry a non-zero orbital angular momentum, e.g. Laguerre-Gaussian (LG) beams \cite{PhysRevLett.121.113204,PhysRevLett.122.110402,PhysRevA.93.013629,PhysRevA.94.033627,PhysRevA.91.063627,PhysRevResearch.2.033152,Chiu_2020}.
Interestingly, most of the previous studies focus on the SOAM coupling in harmonically trapped condensates. In this case the phase singularity of the LG Raman beam is located inside the condensate, which makes the system considerably different from the one with linear SOC. Most notably, such a singularity may lead to a formation of various vortex states in the ground state \cite{PhysRevA.91.033630,PhysRevA.92.033615,PhysRevA.102.063328}.
In the present work we investigate ground-state phases of a toroidal SOAM-coupled atomic condensate.
If the trap potential has a toroidal shape, then the phase singularity is excluded from the condensate region.
This leads to absence of vortices in the ground state and makes analogies with linear SOC more straightforward \cite{PhysRevA.91.063627}.
Using an accurate numerical minimization of the mean-field energy as well as several simplified variational approximations we reveal and analyze a peculiar internal structure of the stripe phase.
The predicted sub-phases produce distinctive density distributions of the ground state and can be detected through certain physical observables, such as the average angular momentum of the system.
The paper is organized as follows.
In Sec. II we introduce our model physical system and several commonly used approximations, which are necessary to reduce the Hamiltonian to a convenient one-dimensional form.
In Sec. III we analyze the spectrum of the single-particle Hamiltonian and establish the phase diagram of the noninteracting system.
In Sec. IV we present several variational models extending on the single-particle solutions to account for the effects of interactions.
We derive a number of analytical relations to estimate the region in parameter space, where the stripe phase may exist.
In Sec. V we discuss the phase diagram of the interacting system and structure of the ground state within different phases. We reveal and analyze the internal fine structure of the stripe phase and highlight the apparent limitations of the variational approximations.
In Sec. VI we show example solutions for a realistic three-dimensional system and confirm that previously identified features of the stripe phase can be observed in a realistic scenario.
Finally, Sec. VII contains a short summary and conclusions of the work.
\section{Model equations}\label{sec:model}
We consider a zero-temperature atomic BEC in a toroidal trap.
Spin-orbit coupling in the condensate is achieved by applying a continuous coherent Raman coupling between two different hyperfine substates from the ground state manifold \cite{Lin2011}.
In this case we can effectively use spin-1/2 algebra and characterize the condensate by a (pseudo)spinor field
\begin{equation}\label{eq:psi2c}
\Psi = \begin{bmatrix}
\psi_a(\mathbf{r}) \\ \psi_b(\mathbf{r})
\end{bmatrix}.
\end{equation}
The single-particle Hamiltonian is then represented as a $2\times2$ matrix
\begin{equation}\label{eq:ham0}
\mathcal{H}_0 = \frac{\hbar^2\nabla^2}{2M}\mathbb{I}_2 + V (\mathbf{r}) \mathbb{I}_2 + \mathcal{H}_\mathrm{c},
\end{equation}
consisting of (spin-independent) kinetic energy and trap potential, as well as the spin-coupling term $\mathcal{H}_\mathrm{c}$.
In cylindrical coordinates $\mathbf{r} = (r,\varphi,z)$ toroidal trap can be approximated by harmonic potentials in radial and longitudinal directions
\begin{equation}\label{eq:torpot}
V(\mathbf{r}) = \frac{M}{2}\left[\omega_r^2 (r-r_0)^2 + \omega_z^2 z^2 \right],
\end{equation}
where $\omega_z$ and $\omega_r$ are longitudinal and radial trap frequencies, respectively.
The Raman coupling term has the following form within the rotating wave approximation:
\begin{equation}\label{eq:2praman}
\mathcal{H}_c = \frac{\hbar}{2}\begin{bmatrix} -\tilde\delta & \tilde\Omega f(r) e^{-2im_0\varphi} \\ \tilde\Omega f(r) e^{2im_0\varphi} & \tilde\delta \end{bmatrix}.
\end{equation}
Such form of the coupling Hamiltonian arises when Raman transition is induced by two copropagating Laguerre-Gaussian (LG) beams with orbital angular momenta $m_0$ and $-m_0$ and with their beam axes aligned along the $z$-axis \cite{PhysRevLett.122.110402,PhysRevA.91.063627}.
In this case the Raman absorption and stimulated emission sequence transfers $2m_0$ angular momentum quanta to the condensate. Previous experiments with SOAM coupling in BEC were conducted mostly with $m_0=1$. However, it is also possible to generate and manipulate LG laser modes with much higher OAM \cite{Campbell:12,Fickler13642}. Throughout the present work we use the value $m_0=10$, which is large enough to see a more general picture, valid for any OAM transfer implemented within the Raman setup.
Other quantities that enter in the coupling Hamiltonian (\ref{eq:2praman}) are two-photon frequency detuning $\tilde\delta$ and the Raman coupling with the amplitude $\tilde\Omega$ and the radial distribution $f(r)$ reflecting the intensity profile of the LG laser beam. If the BEC ring is thin and its radius $r_0$ coincides with the intensity maximum of the Raman coupling, then the specific shape of $f(r)$ is of minor importance \cite{PhysRevA.91.063627,PhysRevA.96.011603}.
In order to make the system accessible for analytical investigation we can use several commonly applied transformations and approximations.
First, we consider the parametric regime when characteristic energy scales for atoms movement in $r$ and $z$ dimensions are much higher then those of the Raman coupling, i.e. $\omega_r,\omega_z \gg \tilde \Omega,\tilde \delta$.
This means that movement in $r$ and $z$ directions can be considered frozen and these dimensions can be factorized in the wave function.
We also introduce a unitary transformation of the wave function in the angular dimension, such that
\begin{equation}\label{eq:utrans}
\begin{array}{l}
\displaystyle \psi_a(\mathbf{r}) \rightarrow \psi_a(\varphi) \chi(r,z) e^{-im_0\varphi},\\[2mm]
\displaystyle \psi_b(\mathbf{r}) \rightarrow \psi_b(\varphi) \chi(r,z) e^{im_0\varphi},
\end{array}
\end{equation}
with normalization conditions
\begin{equation}\label{eq:rz-norm}
\int dr dz |\chi(r,z)|^2=1,\quad \int dr dz f(r)|\chi(r,z)|^2=1.
\end{equation}
The above transformation removes explicit dependence of the Hamiltonian on the angular coordinate $\varphi$ and allows to integrate out $r$ and $z$ dimensions,
leaving us with the following one-dimensional Hamiltonian
\begin{equation}\label{eq:h0dim}
\mathcal{\tilde H}_{\varphi} = \frac{ \hbar^2}{2Mr_0^2} \left(i\partial_{\varphi}+m_0 \sigma_z\right)^2 + \frac{\hbar\tilde \Omega}2 \sigma_x - \frac{\hbar\tilde \delta}2 \sigma_z,
\end{equation}
where $\sigma_z$ and $\sigma_x$ are the standard $2 \times 2$ Pauli matrices.
The Hamiltonian can be further rewritten in the dimensionless form using the energy unit $\epsilon=\hbar^2/(Mr_0^2)$, which is a characteristic energy scale of rotation in the system \cite{PhysRevA.87.013619}. We finally get the following dimensionless Hamiltonian:
\begin{equation}\label{eq:h0}
\mathcal{H}_{\varphi} = \frac{ (L_z-m_0 \sigma_z)^2}{2} + \frac{\Omega}2 \sigma_x - \frac{\delta}2 \sigma_z,
\end{equation}
where $\Omega=\hbar\tilde\Omega/\epsilon$, $\delta=\hbar\tilde\delta/\epsilon$, $L_z=-i \partial_\varphi$ are dimensionless Raman coupling strength, detuning and the angular momentum operator respectively. A single-particle Hamiltonian of this general form is commonly used to study spin-orbital coupling in two-component spinor BECs \cite{Lin2011,PhysRevLett.108.225301,PhysRevA.91.063627}.
The ground state of a weakly interacting spinor condensate is defined through a minimization of the Gross-Pitaevskii energy functional
\begin{multline}\label{eq:gpe_energy1d}
\!\!\!E = \int\limits_0^{2\pi} d\varphi \left[
\begin{pmatrix}
\psi_a^* &
\psi_b^*
\end{pmatrix}
\mathcal{H}_{\varphi}
\begin{pmatrix}
\psi_a\\
\psi_b
\end{pmatrix} + \frac{g_{aa}}2 |\psi_a|^4 + \frac{g_{bb}}2 |\psi_b|^4 \right. \\ \left. + g_{ab} |\psi_a|^2 |\psi_b|^2
\right].
\end{multline}
where $g_{ij}$ are (one-dimensional) nonlinear interaction constants between atoms in states $i$ and $j$, also expressed in the units of $\epsilon$. We assume here for simplicity that interactions are spin-symmetric ($g_{aa}=g_{bb}=g$) and set the normalization condition as
\[
\int\limits_0^{2\pi} d\varphi \left(|\psi_a|^2 + |\psi_b|^2\right) = 1,
\]
so that $g$ and $g_{ab}$ are proportional to the total number of atoms in the condensate. Additionally, we limit the present study to the situations when interactions are repulsive and two components of the condensate are fully miscible, i.e. $g>g_{ab}>0$.
Before we continue with analyzing the spectrum and eigenstates of the Hamiltonian (\ref{eq:h0}) it is worth briefly discussing its symmetry properties.
The most important is the rotational symmetry, stemming from commutation of the Hamiltonian (\ref{eq:h0}) with the angular momentum operator $L_z$. This symmetry plays here the same role as the translational symmetry in the systems with linear SOC. Spontaneous breaking of rotational symmetry leads to formation of a stripe phase which is one of the most stunning features of the spin-orbit coupled systems.
One crucial difference between linear SOC and SOAMC, however, arises from the periodic boundary conditions producing a quantization of (angular) momentum, which is absent in systems with linear SOC. As we will see in the next sections, this difference leads to a considerably richer phase diagram of the SOAM-coupled ring system in comparison to a thoroughly studied uniform BEC with linear-momentum SOC.
Another important symmetry of the Hamiltonian (\ref{eq:h0}), which exists only with $\delta=0$, is a $\mathbb{Z}_2$ symmetry often associated with the time reversal. The corresponding symmetry transformation combines the complex conjugation of the wave function and spin inversion:
\begin{equation}\label{eq:t-op}
\mathcal{T}\begin{bmatrix}
\psi_a \\ \psi_b
\end{bmatrix} = \begin{bmatrix}
\psi_b^* \\ \psi_a^*
\end{bmatrix}.
\end{equation}
Time reversal symmetry also can be spontaneously broken in the ground state leading to formation of polarized phases \cite{PhysRevA.102.063328}. Any non-zero detuning $\delta$ leads to the absence of such symmetry in the Hamiltonian. Still, the symmetry transformation (\ref{eq:t-op}) plays an important role for the eigenstates of the system, as will be seen in the next section.
\section{Noninteracting system}
As a result of the rotational symmetry of the Hamiltonian (\ref{eq:h0}), its
eigenstates can be characterized by a well defined angular momentum projection.
The corresponding energy spectrum can then be written in terms of the angular quantum number $m$ as follows
\begin{equation}\label{eq:sp_spectr}
E_{\pm} (m) = \frac12 \left[m_0^2+m^2 \pm \sqrt{\Omega^2+(2m_0m+\delta)^2}\right].
\end{equation}
The energy spectrum consists of two branches (see Fig.~\ref{fig:sp_spectr}). The lower branch $E_-$ may contain two minima. Depending on $\Omega$ and $\delta$ these minima may be located at different values of $m$ within the range $[-m_0,m_0]$ and either one or both of them correspond to the ground state of the system. Understanding which value of $m$ represents the ground state will be the main goal of the present section.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig1}
\caption{Single-particle energy spectrum defined by Eq.~(\ref{eq:sp_spectr}) with $m_0=10$.
Left panel corresponds to $\delta=0$ and various values of $\Omega$.
Right panel corresponds to $\Omega=100$ and various values of $\delta$.
Lines are guides for the eye marking the lower (solid lines) and upper (dotted lines) branches of the spectrum.}
\label{fig:sp_spectr}
\end{figure}
If $m$ was a continuous variable, we would find the ground state from the extremum condition $\partial E_-/\partial m = 0$. This condition, when solved with respect to $\Omega$, yields the following expression:
\begin{equation}\label{eq:omega_sp_cont}
\Omega= \left(2m_0 +\frac{\delta}{m}\right) \sqrt{m_0^2-m^2}.
\end{equation}
However, as $m$ is a discrete quantum number, we have to use a different approach. We can find regions in $(\Omega,\delta)$ space where specific value $m$ correspond to the ground state by solving the equation $E_-(m)=E_-(m+1)$.
This equation can be solved for $\Omega$ analytically, providing a critical coupling, where the ground states switches from $m$ to $m+1$:
\begin{equation}\label{eq:omega_sp}
\Omega_{m} = \sqrt{\left[\left(2m_0 +\frac{\delta}{m+1/2}\right)^2 - 1 \right]\left[m_0^2- \left(m+\frac12\right)^2\right]}.
\end{equation}
This expression allows us to define the regions in $(\Omega,\delta)$-space where angular quantum number $m$ represents the ground state, providing effectively a single-particle ``phase diagram'' of the system, which is shown in Fig.~\ref{fig:sp_pd}. Similar expression can be found in Ref.~\cite{PhysRevA.91.063627}.
Interestingly, if we consider $(2m_0)^2 \gg 1$ then Eq.~(\ref{eq:omega_sp}) simplifies to
\begin{equation}\label{eq:omega_sp_cont2}
\Omega_{m}\approx \left(2m_0 +\frac{\delta}{m+1/2}\right) \sqrt{m_0^2-\left(m+\frac12\right)^2},
\end{equation}
which is equivalent to Eq.~(\ref{eq:omega_sp_cont}) evaluated at the half-integer value $m+1/2$. Such equivalence between the continuous and discrete extremum conditions will be especially useful in the next section for the analysis of the interacting system.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig2.eps}
\caption{Single-particle ``phase diagram'' for $m_0=10$. Blue lines and numbers between $-10$ and $10$ show the regions where corresponding value of $m$ represents the ground state of the system. Red dotted lines show the same for the second minimum. Black dashed line shows the region where two minima exist in the energy spectrum and is defined by Eq.~(\ref{eq:cusp})}
\label{fig:sp_pd}
\end{figure}
If the energy $E_-(m)$ contain two minima, then the extremum conditions derived above will be satisfied not only for the ground state, but also for the second energy minimum and the maximum in between.
Therefore, we can also find regions in parameter space where certain quantum number $m$ corresponds to a metastable excited state as well as the region where two energy minima exist (see Fig.~\ref{fig:sp_pd}).
This region, where the energy spectrum has two minima, is localized at small values of $\Omega$ and $\delta$ and bounded by the astroid curve
\begin{equation}\label{eq:cusp}
\Omega^{2/3} + |\delta|^{2/3} = \Big(2m_0^2\Big)^{2/3}.
\end{equation}
It will be useful for the discussions in the next section to also
introduce the eigenstates corresponding to the energy spectrum (\ref{eq:sp_spectr}). These normalized eigenstates read
\begin{equation}\label{eq:sp_wavefunc}
\Psi_m = \begin{bmatrix}
\psi_a(\varphi)\\
\psi_b(\varphi)
\end{bmatrix}
=
\frac{1}{\sqrt{2\pi}}
\begin{bmatrix}
\cos(\theta/2)\\
-\sin(\theta/2)
\end{bmatrix} e^{i m \varphi},
\end{equation}
with
\[
\theta = \arctan\left(\frac{\Omega}{2m_0m+\delta}\right).
\]
Eigenstates in such a form describe both branches of energy spectrum. The values $0<\theta<\pi$ correspond to the lower energy branch $E_-$, while $\pi<\theta<2\pi$ corresponds to $E_+$.
Eigenstates with the opposite angular quantum numbers $m$ and $-m$ are related through a time-reversal transformation (\ref{eq:t-op}):
\begin{equation}\label{eq:trsym}
\Psi_{-m} = \mathcal{T} \Psi_m = \frac{1}{\sqrt{2\pi}}
\begin{bmatrix}
-\sin(\theta/2)\\
\cos(\theta/2)
\end{bmatrix} e^{-i m \varphi}.
\end{equation}
If $\delta = 0$ then the states $\Psi_m$ and $\Psi_{-m}$ have the same energy due to a time-reversal symmetry of the Hamiltonian.
On a final note, the energy spectrum (\ref{eq:sp_spectr}) is symmetric with respect to $\Omega$, and also symmetric with respect to $\delta$ with a replacement $m\rightarrow-m$. We can therefore restrict all further analysis to $\Omega>0$ and $\delta>0$ without a loss of generality.
\section{Variational treatment of the interacting system}
We now extend the above one-dimensional analysis to the case of interacting system.
In the presence of interactions minimization of the energy functional (\ref{eq:gpe_energy1d}) can not be performed analytically.
Nevertheless, some analytical treatment is possible within the variational approach.
But first, for the convenience of further analysis we introduce instead of nonlinear parameters $g$ and $g_{ab}$ new parameters $G_1 = (g+g_{ab})/8\pi$ and $G_2 = (g-g_{ab})/8\pi$. The energy functional (\ref{eq:gpe_energy1d}) is then rewritten in an equivalent form
\begin{multline}\label{eq:gpe_energy2}
\!\!\!E = \int\limits_0^{2\pi} d\varphi \left[
\begin{pmatrix}
\psi_a^* &
\psi_b^*
\end{pmatrix}
\mathcal{H}_{\varphi}
\begin{pmatrix}
\psi_a\\
\psi_b
\end{pmatrix} + 2\pi G_1 \left(|\psi_a|^2 + |\psi_b|^2\right)^2 \right. \\ \left. +
2\pi G_2 \left(|\psi_a|^2 - |\psi_b|^2\right)^2
\right].
\end{multline}
It is worth mentioning here that for the most experimental realizations of spin-orbit-coupled BECs with alkali atoms the nonlinear interaction coefficients satisfy the inequality $0<G_2\ll G_1$ \cite{PhysRevLett.116.160402}.
\subsection{Single-mode ansatz}
The simplest, yet quite instructive approach is to take a single noninteracting solution $\Psi_m$ from Eq.~(\ref{eq:sp_wavefunc}) as the variational trial function and consider $m$ and $\theta$ as variational parameters. After inserting (\ref{eq:sp_wavefunc}) into the energy functional (\ref{eq:gpe_energy2}) we get
\begin{multline}
E(m,\theta) = \frac{m_0^2+m^2}{2} - \left(m_0 m + \frac\delta2\right)\cos\theta \\ - \frac\Omega2\sin\theta + G_1 + G_2\cos^2\theta.
\end{multline}
This expression can be minimized analytically with respect to $m$ and $\theta$ only if $m$ is considered as a continuous variable. We therefore follow the analogy with the noninteracting case and express the critical coupling $\Omega_m$ for the transition between $m$ and $m+1$ ground states by inserting $m+1/2$ into a solution of the system of equations $\partial E/\partial m=0$ and $\partial E/\partial \theta=0$. We obtain the following expression very similar to Eq.~(\ref{eq:omega_sp_cont2}):
\begin{equation}\label{eq:omega_sp1}
\Omega_{m} = \left[ 2 m_0 +\frac{\delta}{m+1/2} - \frac{4G_2}{m_0} \right] \sqrt{m_0^2- \left(m+\frac12\right)^2}.
\end{equation}
We see that within the single mode ansatz interactions lead only to shifts of boundaries on the single-particle phase diagram.
Qualitatively, the picture remains the same as in Fig.~\ref{fig:sp_pd}.
Validity of the above expression will be evaluated numerically in the next section.
It is important to note that Eq.~(\ref{eq:omega_sp1}) does not depend on $G_1$ and in the case of spin-independent collisional interactions, i.e. $g=g_{ab}$, and consequently $G_2=0$, it becomes fully identical to the result of noninteracting system (\ref{eq:omega_sp_cont2}).
\subsection{Two-mode ansatz}
As already mentioned above, the stripe phase is associated with simultaneous occupation of two minima of the energy spectrum.
Therefore, the most commonly used variational ansatz that is able to describe the stripe phase is based on a linear combination of two wave functions with opposite angular momenta $m$ and $-m$ (see e.g. \cite{PhysRevLett.108.225301,PhysRevA.91.063627,PhysRevResearch.2.033152,Chiu_2020}):
\begin{equation}\label{eq:2wave}
\Psi = C_+ \Psi_m + C_- \Psi_{-m}.
\end{equation}
The functions $\Psi_m$ and $\Psi_{-m}$ are defined by Eq.~(\ref{eq:sp_wavefunc}) and related through a time-reversal transformation (\ref{eq:trsym}). The coefficients $C_+$ and $C_-$ are constrained by a normalization condition $|C_+|^2+|C_-|^2=1$. Consequently, we have a variational problem with three independent variational parameters: $m$, $\theta$ and $\beta = |C_+|^2-|C_-|^2$.
Without a loss of generality we may consider $|C_+| \geq |C_-|$ and restrict $\beta$ within the range $[0,1]$.
After inserting (\ref{eq:2wave}) into (\ref{eq:gpe_energy2}) we get the following expression of energy as a function of variational parameters:
\begin{multline}\label{eq:en-2wave}
E(m,\theta,\beta) = \frac{m_0^2+m^2}{2} - \left(m_0 m + \frac{\beta \delta}2\right)\cos\theta \\ - \frac\Omega2\sin\theta + G_1 + G_1\frac{1-\beta^2}{2}\sin^2\theta + G_2 \beta^2 \cos^2\theta.
\end{multline}
Many other useful physical quantities can also be readily expressed in terms of our variational parameters. These include spin polarizations
\begin{equation}
\langle \sigma_z \rangle = \beta \cos \theta, \quad \langle \sigma_x \rangle = - \sin \theta,
\end{equation}
expectation value of angular momentum
\begin{equation}\label{eq:lz-an}
\langle L_z \rangle = m \beta,
\end{equation}
and its standard deviation
\begin{equation}
\Delta L_z=\sqrt{\langle L_z^2 \rangle-\langle L_z \rangle^2} = m \sqrt{1-\beta^2},
\end{equation}
as well as the particle density distribution
\begin{multline}\label{eq:dens-2m}
|\Psi(\varphi)|^2 = |\psi_a|^2+|\psi_b|^2 \\ = \frac{1}{2\pi} \left[1+\sqrt{1-\beta^2} \sin \theta \cos (2m\varphi)\right].
\end{multline}
The variational parameter $\beta$ describes the degree of mixing between the two angular momentum modes. With $\beta=1$ we get a state with well defined angular momentum and recover the above single-mode result. It corresponds to the single-momentum (SM) phase of the system. With $\beta < 1$ two angular momentum components are populated. Such states contain $2m$ oscillations in the particle density and represent the stripe phase of the system.
In general, the energy functional (\ref{eq:en-2wave}) can be minimized only numerically.
There are, however, several important particular cases that can be studied analytically in order to estimate the region in $(\Omega,\delta)$ space where the stripe phase may exist.
First, Eq.~(\ref{eq:en-2wave}) can be relatively easily analyzed in the limit $\Omega \rightarrow 0$. In this case the energy minimum is achieved with
\begin{equation}\label{eq:small-delta}
m=m_0,\quad \theta=0, \quad \beta=\frac{\delta}{4G_2}
\end{equation}
if $\delta<\delta_\mathrm{c} = 4 G_2$.
This region corresponds to the stripe phase. If $\delta>\delta_\mathrm{c}$ then the energy is minimized by
\begin{equation}
m=m_0,\quad \theta=0, \quad \beta=1,
\end{equation}
representing the SM phase with the momentum $m_0$.
Another important limit case is zero detuning $\delta=0$.
With $\delta=0$ the energy (\ref{eq:en-2wave}) becomes a linear function of $\beta^2$.
Therefore, energy minimum with respect to $\beta$ can be achieved only at one of the limit values.
The value $\beta=1$ represents the SM phase, while $\beta=0$ corresponds to the stripe phase. The boundary between the two phases can be found by minimizing separately $E(m,\theta,0)$ and $E(m,\theta,1)$ and finding the crossing between the two. Minimization of the single-momentum energy $E(m,\theta,1)$ was already discussed above. Minimization of the stripe-phase energy $E(m,\theta,0)$ is more complicated, since in this case application of continuous extremum condition on $m$ is not justified. Relatively simple solutions are possible to obtain in strongly-interacting regime, when $G_1 \gg \Omega$. In this case the energy minimum is achieved with the following values of variational parameters
\begin{equation}
m=m_0,\quad \sin\theta \approx \frac{\Omega}{2(G_1+m_0^2)}, \quad \beta=0.
\end{equation}
For the transition between the stripe phase and the SM phase there are two possible distinct cases. If
\begin{equation}\label{eq:g2-cond}
G_2>\frac{1}{2}\left[ \sqrt{G_1(m_0^2 + G_1)}-G_1\right],
\end{equation}
then the transition is observed at the critical coupling strength
\begin{equation}\label{eq:omc-2mode1}
\Omega_{\mathrm{c}} = 2\left[ m_0^2 + G_1 - \sqrt{G_1(m_0^2 + G_1)} \right].
\end{equation}
Below this critical coupling ($\Omega<\Omega_{\mathrm{c}}$) the energy minimum corresponds to the stripe phase and if $\Omega>\Omega_{\mathrm{c}}$, the energy is minimized with $m=0$ and $\beta=1$ corresponding to a SM phase with zero angular momentum.
If the condition (\ref{eq:g2-cond}) is not satisfied, then the transition occurs to the state with non-zero angular momentum. In this case, analogously to the above single-mode discussion, the SM energy $E(m,\theta,1)$ needs to be minimized approximating $m$ as a continuous variable. The resulting critical coupling reads
\begin{equation}\label{eq:omc-2mode2}
\Omega_{\mathrm{c}} = 2\sqrt{\frac{2G_2(m_0^2 + G_1)(m_0^2 - 2G_2)}{G_1+2G_2}}.
\end{equation}
Eqs. (\ref{eq:g2-cond}) -- (\ref{eq:omc-2mode2}) allow us to estimate the range of Raman couplings $\Omega$ where the stripe phase may exist.
These results are consistent with previous findings for systems with linear SOC \cite{PhysRevLett.108.225301}.
\subsection{Multi-mode ansatz}
In order to develop a more accurate representation for possible ground states, which would generalize the above two-mode ansatz, we start with writing the wave function as a following superposition of angular modes:
\begin{equation}\label{eq:nwave}
\Psi = \sum\limits_{n=-\infty}^{\infty}
\begin{pmatrix}
A_n \\ B_n
\end{pmatrix}
e^{i(m-n k)\phi},
\end{equation}
with $k > m \geq 0$, so that $m$ represents the lowest non-negative angular momentum component and $m-k$ is the lowest negative one.
The idea behind this representation is that two central components $n=0$ and $n=1$ with angular momenta $m$ and $m-k$ represent simultaneous population of two minima in the energy spectrum. All other higher order components appear due to nonlinear mixing caused by interactions.
Similar representations were used previously to describe the stripe phase in a system with linear Raman-induced SOC \cite{PhysRevLett.110.235302} and SOAMC \cite{Chiu_2020}.
Importantly, the infinite sum (\ref{eq:nwave}) is still an exact representation of both stripe and SM states of the system.
In the latter case the sum contains only one pair of non-zero coefficients $A_0$ and $B_0$.
To use the representation (\ref{eq:nwave}) as a variational trial function, the sum must be truncated to a finite number of terms. Truncating the the sum to only two central components $n=0,1$ can be considered as a generalization of the two-wave ansatz (\ref{eq:2wave}).
The variational problem then contains in total five independent variational parameters. Those are $m$, $k$, and three out of the four coefficients $A_0$, $B_0$, $A_1$, $B_1$, which are constrained by the normalization condition.
In order to see the role of higher-order angular modes
for the structure of the stripe phase
we will also use four central components of Eq.~(\ref{eq:nwave})
as a trial function, such that
\begin{multline}\label{eq:4mode}
\Psi =
\begin{pmatrix}
A_{-1} \\ B_{-1}
\end{pmatrix}
e^{i (m+k) \phi}+
\begin{pmatrix}
A_0 \\ B_0
\end{pmatrix}
e^{i m \phi} \\ +
\begin{pmatrix}
A_1 \\ B_1
\end{pmatrix}
e^{i (m-k) \phi}+
\begin{pmatrix}
A_{2} \\ B_{2}
\end{pmatrix}
e^{i (m-2k) \phi}.
\end{multline}
The coefficients $A_n$ and $B_n$ and characteristic wave numbers $m$ and $k$ are considered as variational parameters.
Accounting for the normalization condition, we have then in total 9 independent variational parameters to be optimized.
A similar variational approximation was previously used in Ref.~\cite{Chiu_2020} showing a considerable improvement over a two-mode atsatz for the determination of a stripe contrast.
\section{Phase diagram}
We now use the variational ans\"atze described above as well as the full numerical minimization of the energy functional (\ref{eq:gpe_energy2}) to build and analyze a phase diagram of the system.
We have in total four external parameters that can be used to drive the phase transitions. Those are Raman coupling strength $\Omega$, detuning $\delta$, and nonliear interaction coefficients $G_1$ and $G_2$. Phase diagrams can be built by fixing two of these parameters and tracing the ground state of the system through a range of values for other two.
Fig.~\ref{fig:int_pd} shows the calculated phase diagram in $(\Omega,\delta)$-plane for fixed values of $G_1=500$ and $G_2=50$.
The phase diagram consists of two large regions representing two phases of the system. The stripe phase is observed in the the region of small values of $\Omega$ and $\delta$ and single-momentum (SM) phase covers the rest of the parameter space.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig3}
\caption{Phase diagram of SOAM-coupled ring system with $m_0=10$, $G_1=500$, $G_2=50$ calculated with full numerical minimization (a), two-mode variational ansatz (b) and four-mode atsatz (c).
Thick black line shows the boundary between the stripe phase and the single-momentum phase. Dashed black lines show boundaries between ground states with different well-defined angular momentum [marked also with numbers on panel (b)]. Blue lines mark boundaries between different types of stripe states.
Thin red lines on panel (b) represent the result of Eq.~(\ref{eq:omega_sp1}).
}
\label{fig:int_pd}
\end{figure}
Both phases exhibit certain internal structure.
Inside the SM phase we observe regions of different well-defined angular momentum. Boundaries of these regions are similar to the single-particle phase diagram and can be very accurately approximated by Eq.~(\ref{eq:omega_sp1}). The observed internal structure of the stripe phase is much more peculiar, since it does not directly follow from any of the above analytical predictions and was not observed in previous studies.
The stripe phase consists of a number of sub-phases, which we label as S$_k$. The index $k$, introduced here to distinguish these sub-phases, represents the number of periodic density modulations (stripes) observed in the particle density of the ground state. This number is always integer due to the periodicity of the wave function.
In the region of small $\delta$ and close to the transition to the SM phase we observe an increasing number of stripe sub-phases, which become difficult to resolve numerically.
Examples of ground-states representing two different stripe sub-phases are shown in Figs.~\ref{fig:sol1} and \ref{fig:sol2}. The density distributions show a clear periodic structure with $k$ periods, though it is clearly different from the cosine-like shape predicted by Eq.~(\ref{eq:dens-2m}).
Similar structures were recently predicted theoretically for ultracold Fermi gases with SOAM coupling \cite{PhysRevLett.126.193401}.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig4a}
\includegraphics[width=\linewidth]{fig4b}
\caption{
Numerically calculated particle density (top panel) and power spectrum (bottom panel) of the ground state at $\Omega=60$, $\delta=0$.
The state corresponds to the sub-phase $S_{20}$, as marked in Fig.~\ref{fig:int_pd}, and contains 20 density modulations. Four central peaks in $|\widetilde \Psi|^2$ are located at $ m= \{\pm 10, \pm 30 \}$. Note the logarithmic scale of the vertical axis on the lower panel.}
\label{fig:sol1}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig5a}
\includegraphics[width=\linewidth]{fig5b}
\caption{Same as Fig.~\ref{fig:sol1} but for $\Omega=118$, $\delta=40$ representing the sub-phase $S_{13}$. Four central peaks in $|\widetilde \Psi|^2$ are located at $ m= \{7,-6,20,-19\}$.}
\label{fig:sol2}
\end{figure}
In order to find the mode composition of the ground states and relate it to Eq.~(\ref{eq:nwave}) we also show in Figs.~\ref{fig:sol1} and \ref{fig:sol2} the power spectra, which can be defined for a spinor state as $|\widetilde{\Psi}(m)|^2 = |\widetilde \psi_a(m)|^2 + |\widetilde \psi_b(m)|^2$, with $\widetilde \psi_a(m)$ and $\widetilde \psi_b(m)$ being Fourier transforms of the corresponding wave functions.
By calculating such power spectra for different sub-phases we confirm that they always follow the Eq.~(\ref{eq:nwave}) with $k$ having the same meaning as in the label of our sub-phases and $m=k/2$ for even-$k$ states, and $m=(k+1)/2$ for odd-$k$ states. One may notice from Fig.~\ref{fig:int_pd}, that with zero detuning we observe only even-$k$ states, while odd-$k$ states appear only with finite $\delta$. This is quite natural, since the superposition (\ref{eq:nwave}) can fulfill the time reversal symmetry only with even values of $k$ and with $m=k/2$.
In Fig.~\ref{fig:int_pd}(b) and (c) we also show the phase diagrams calculated within the variational approximations described in the previous section.
We see that the two-mode variational ansatz provides the phase diagram with only one stripe phase $S_{20}$ and fails to describe any of the sub-phases observed in the full calculation.
This effectively means that if we consider the variational function as a sum of two angular modes, then these two modes are always identified with $m=\pm m_0$ \cite{PhysRevA.91.063627}.
We should emphasize that parameter space of the two-mode atsatz is not restricted to $m=m_0$. Still, other values did not appear within the stripe phase during energy minimization.
The four-mode variational ansatz shows qualitatively some of the fine structure of the stripe phase. However, it still deviates from the correct result, especially for low-$k$ sub-phases. Importantly, the difference in the results obtained with the two-mode and the four-mode variational approximations clearly shows that the fine structure of the stripe phase is the result of admixing of higher-order angular-momentum modes in the wave function.
All the phases and phase transitions, including internal structure of the SM and stripe phases, can be identified from the behavior of physical observables, such as previously introduced expectation value of angular momentum $\langle L_z \rangle$ and its standard deviation $\Delta L_z$. In Fig.~\ref{fig:surf} we show these quantities calculated within the same parametric region as Fig.~\ref{fig:int_pd}.
Well defined angular momentum in the SM phase produces a characteristic
staircase-like pattern in $\langle L_z \rangle$ with zero values of $\Delta L_z$.
Within the stripe phase we observe mostly linear growth of the angular momentum with the detuning $\delta$. In the limit of small Raman coupling $\Omega$ this behavior can be deduced from Eqs.~(\ref{eq:lz-an}) and (\ref{eq:small-delta}) giving the expression $\langle L_z \rangle = m_0\delta/4G_2$, which correlates nicely with our numerical results. Interestingly, while the angular momentum is not a well-defined quantity in the stripe phase, it still shows a discontinuous behavior, indicating first-order phase transitions between the stripe sub-phases S$_k$.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig6}
\caption{Expectation value of the angular momentum $\langle L_z \rangle$ and its standard deviation $\Delta L_z$ (shown with color) as functions of Raman coupling $\Omega$ and detuning $\delta$. All parameters are the same as in Fig.~\ref{fig:int_pd}}
\label{fig:surf}
\end{figure}
Next, we analyze the dependencies of phase boundaries on the nonlinear parameters $G_1$ and $G_2$. To this end Fig.~\ref{fig:g2dep} shows phase diagrams calculated within $(\Omega,G_2)$ and $(\Omega,G_1)$ planes. We restrict these phase diagrams to the region $G_1 \gg G_2$.
Comparing the obtained results with the two-mode variational atsatz we see that the phase boundary of the stripe phase can be only qualitatively reproduced by Eqs.~(\ref{eq:g2-cond}) -- (\ref{eq:omc-2mode2}). The quantitative agreement is, however, rather poor. Throughout the whole parametric range shown in the figure the stripe phase exist in considerably wider region then it is predicted by the two-mode variational approximation.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig7a}
\includegraphics[width=\linewidth]{fig7b}
\caption{Top panel shows the phase diagram on the $(\Omega,G_1)$--plane with other parameters fixed as $\delta=0$ and $G_2=20$.
Lower panel shows the phase diagram on the $(\Omega,G_2)$--plane with $\delta=0$ and $G_1=1000$. On both panels thick black line shows the boundary between the stripe phase and the single-momentum (SM) phase, blue solid lines represent boundaries of different stripe states S$_k$ (only first two are labeled) and dashed black line shows the analytical prediction of the two-mode ansatz [Eqs.~(\ref{eq:g2-cond}) -- (\ref{eq:omc-2mode2})]. Red dashed lines on the lower panel correspond to transitions between different single-momentum states. }
\label{fig:g2dep}
\end{figure}
The most important observation from Fig.~\ref{fig:g2dep} is that different sub-phases S$_k$ persist within the stripe phase through a wide range of parameters $G_1$ and $G_2$.
One may also notice that the boundaries of different stripe sub-phases S$_k$ are completely independent on $G_2$. This behavior is understandable, since with zero detuning $\delta=0$ the stripe states possess a time-reversal symmetry, implying the relation between the wave function components $|\psi_a|=|\psi_b|$, and consequently, the energy functional (\ref{eq:gpe_energy2}) becomes independent on $G_2$. Somewhat more surprising observation is that the boundaries of stripe sub-phases also become almost independent on $G_1$ in the region of large values of this interaction parameter.
It is therefore expected that the internal fine structure of the stripe phase is quite robust with respect to the nonlinear coupling parameters and should be manifested in experimentally accessible scenarios.
\section{Application to a realistic ring system}
So far we have analyzed the internal structure of the stripe phase and have shown that this structure persists within a wide range of parameters of the system.
In this section we test these predictions for a realistic system with experimentally realizable parameters.
As an example we consider a BEC of $^{23}$Na atoms in two hyperfine substates $|F=1,m_F=\pm1\rangle$ from the ground-state manifold. The total number of atoms in the condensate is $N=5\times10^5$ and the ring trap is defined by the potential (\ref{eq:torpot})
with $\omega_r=2\pi\times400\,\mathrm{Hz}$, $\omega_z=2\pi\times542\,\mathrm{Hz}$, $r_0=22\,\mathrm{\mu m}$, which is close to the existing experiments with sodium condensates \cite{Wang_2015}.
We now minimize a full three-dimensional Gross-Pitaevskii energy functional
\begin{multline}\label{eq:gpe_energy3d}
E = \int d\mathbf{r} \left[
\begin{pmatrix}
\psi_a^* &
\psi_b^*
\end{pmatrix}
\mathcal{H}_0
\begin{pmatrix}
\psi_a\\
\psi_b
\end{pmatrix} + \frac{g^{(\mathrm{3D})}}2 |\psi_a|^4 \right. \\ \left. + \frac{g^{(\mathrm{3D})}}2 |\psi_b|^4 + g^{(\mathrm{3D})}_{ab} |\psi_a|^2 |\psi_b|^2
\right],
\end{multline}
with the single particle Hamiltonian from Eq.~(\ref{eq:ham0}) and the nonlinear interaction constants defined as
\[
g^{(\mathrm{3D})} = \frac{4N\pi\hbar^2}{M}a, \qquad g^{(\mathrm{3D})}_{ab} = \frac{4N\pi\hbar^2}{M}a_{ab},
\]
where $M$ is the atom mass, $a=54.54a_B$ and $a_{ab}=50.78a_B$ are s-wave scattering lengths between the same and different spin states of $^{23}$Na, with $a_B$ being the Bohr radius \cite{PhysRevA.97.063615,PhysRevA.83.042704}.
The Raman coupling is considered here with its radial distribution $f(r)$, which is of a Laguerre-Gaussian shape with an intensity maximum at $r_0$, so that
\[
f(r) = A \left(r/r_0\right)^{2m_0} e^{-m_0(r/r_0)^2},
\]
where $A$ is a normalization coefficient, defined according to Eq.~(\ref{eq:rz-norm}).
The above physical parameters correspond to the reduced one-dimensional model described in Sec.~\ref{sec:model} with the energy scale $\epsilon = 2\pi/\hbar \times 0.91\,\mathrm{Hz}$ and the dimensionless nonlinear interaction coefficients $G_1=1053$ and $G_2=37.6$.
From these values we can estimate the stripe phase region using the approximate expressions derived above using the two-mode variational model. We find, that stripe phase should be expected for $\tilde\delta<2\pi\times 136.6\,\mathrm{Hz}$ and $\tilde\Omega < 2\pi\times 93\,\mathrm{Hz}$.
The ground state of the three-dimensional system is obtained by direct numerical minimization of Eq.~(\ref{eq:gpe_energy3d}) using the gradient flow method with discrete normalization and with backward Euler pseudospectral discretization scheme \cite{doi:10.1137/140979241}.
In Fig.~\ref{fig:na1} we show three examples of ground states obtained with $\tilde\delta=0$ and varying $\tilde\Omega$.
The values of the Raman coupling $\tilde\Omega$ are chosen such that the ground state represents different stripe sub-phases S$_{20}$, S$_{16}$ and S$_{12}$.
It is worth noticing, that the value $\tilde \Omega = 2\pi\times 112\,\mathrm{Hz}$ lies already beyond the region of stripe phase predicted by the two-mode model. This is in line with the results shown in figs.~\ref{fig:int_pd} and \ref{fig:g2dep} and confirms that the two-mode prediction is only a rough estimate of the phase boundary.
By integrating the particle density distribution over the radial and longitudinal directions we obtain a one-dimensional angular density, which show a nearly perfect agreement with the results of the one-dimensional model [see Fig.~\ref{fig:na1}(d)--(f)].
Shown examples confirm the validity of approximations used for the one-dimensional model in Sec.~\ref{sec:model} and also confirm the general claim about multiple types of stripe states in the system.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig8}
\caption{
Results of the ground state calculations for the example three-dimensional system
with $\tilde \delta=0$ and $\tilde \Omega = 2\pi\times 54\,\mathrm{Hz}$ [panels (a) and (d)], $\tilde \Omega = 2\pi\times 93\,\mathrm{Hz}$ [panels (b) and (e)], $\tilde \Omega = 2\pi\times 112\,\mathrm{Hz}$ [panels (c) and (f)]. Panels (a)--(c) show two-dimensional column density (in arb. units) and panels (d)--(f) show one-dimensional angular density (dimensionless). Blue lines are obtained from three-dimensional ground states by integrating out the longitudinal and radial dimensions, red lines are the results of one-dimensional model.
}
\label{fig:na1}
\end{figure}
Next, we consider a slightly different example with the ring radius reduced to $r_0=10\,\mathrm{\mu m}$, the total number of atoms increased to $N=2\times10^6$ and the other physical parameter same as above. In this case the one-dimensional model is not justified any more. Additionally, the LG profile of the Raman coupling, which we also change to have a maximum at the new $r_0$, is now considerably inhomogeneous within the condensate [see Fig.~\ref{fig:na2}(a)].
By calculating the ground states of this system with different values of $\tilde \Omega$ we can again identify stripe states with different number of density modulations [see Fig.~\ref{fig:na2}(b--d)], which shows that such features can be observed also beyond purely one-dimensional regimes.
More detailed analysis of stripe sub-phases in realistic scenarios and the effects of the trap potential will be presented in a forthcoming publication.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{fig9}
\caption{Results of the ground state calculations for the second example three-dimensional system. Upper panel (a) shows the radial profile of the Raman coupling (blue line and left axis) and of the condensate particle density (red line and right axis). Lower panels show calculated ground-state column densities for $\tilde \delta=0$ and $\tilde \Omega = 2\pi\times 114\,\mathrm{Hz}$ (b), $\tilde \Omega = 2\pi\times 185\,\mathrm{Hz}$ (c), $\tilde \Omega = 2\pi\times 242\,\mathrm{Hz}$ (d).
}
\label{fig:na2}
\end{figure}
\section{Conclusions}
In the present work we have analyzed the structure of the ground-state phases in a toroidal two-component Bose-Einstein condensate with a spin-orbital-angular-momentum coupling between the components. In particular, a series of sub-phases is revealed inside the stripe phase of the system. These sub-phases are distinguished by the number of spatial density modulations and their characteristic power spectra. The existence of such fine structure of the stripe phase is a natural result of the wave function periodicity.
It is worth noticing that all observed phase transitions are of the first order, which is due to quantization of the angular momentum. While this result seems natural within the SM phase, which obeys the rotational symmetry, it is somewhat surprising withing the stripe phase, where the rotational symmetry is spontaneously broken.
The obtained results are analyzed and verified through a comparison with several well established variational ans\"atze.
We observe that within a wide region of physical parameters the commonly used two-mode approximation gives only a rough description of the stripe phase of the SOAM-coupled system. Predictions of this approximation for the phase boundaries show only limited qualitative agreement with full numerical calculations. The two-mode model also completely fails to predict the internal fine structure of the stripe phase.
These problems can be partially rectified by extending the variational model to four angular modes.
Therefore, while the two-mode approximation can provide useful analytical estimates for SOAM-coupled condensate, one should be careful with drawing physical conclusions from this approximation.
Finally, we have demonstrated that the predicted fine structure of the stripe phase can be observed within experimentally accessible parametric regimes. To this end we have performed a full three-dimensional modeling of a realistic two-component condensate of $^{23}$Na atoms.
We were able to observe different types of stripe states and their spatial structure agrees with predictions of the one-dimensional model.
\begin{acknowledgments}
Y.B. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- Projektnummer 445408588.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,610 |
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HvA/Amsterdam University of Applied Sciences | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 103 |
Q: How to read a directory in html5 from a folder picker input? I want to have a input field that lets you select a folder. And then use html5 directory reader api to recursively read all files and folders in it.
Does anyone know any tutorials doing this specifically? I have code here, but it doesn't work, it always lists empty array when reading.
<input type="file" id="filepicker" name="fileList" webkitdirectory directory />
<button id="ff">Read</button>
<ul id="filelist"></ul>
js
function toArray(list) {
return Array.prototype.slice.call(list || [], 0);
}
function listResults(entries) {
// Document fragments can improve performance since they're only appended
// to the DOM once. Only one browser reflow occurs.
var fragment = document.createDocumentFragment();
entries.forEach(function(entry, i) {
var img = entry.isDirectory ? '<img src="folder-icon.gif">' :
'<img src="file-icon.gif">';
var li = document.createElement('li');
li.innerHTML = [img, '<span>', entry.name, '</span>'].join('');
fragment.appendChild(li);
});
document.querySelector('#filelist').appendChild(fragment);
}
function onInitFs(fs) {
var dirReader = fs.root.createReader();
var entries = [];
// Call the reader.readEntries() until no more results are returned.
var readEntries = function() {
dirReader.readEntries (function(results) {
if (!results.length) {
listResults(entries.sort());
} else {
entries = entries.concat(toArray(results));
readEntries();
}
}, errorHandler);
};
readEntries(); // Start reading dirs.
}
function errorHandler(e) {
var msg = '';
switch (e.code) {
case DOMError.QUOTA_EXCEEDED_ERR:
msg = 'QUOTA_EXCEEDED_ERR';
break;
case DOMError.NOT_FOUND_ERR:
msg = 'NOT_FOUND_ERR';
break;
case DOMError.SECURITY_ERR:
msg = 'SECURITY_ERR';
break;
case DOMError.INVALID_MODIFICATION_ERR:
msg = 'INVALID_MODIFICATION_ERR';
break;
case DOMError.INVALID_STATE_ERR:
msg = 'INVALID_STATE_ERR';
break;
default:
msg = 'Unknown Error';
break;
};
console.log('Error: ' + msg);
}
document.getElementById("ff").addEventListener("click", function() {
var selectedFiledf = $("#filepicker").val();
window.webkitRequestFileSystem(selectedFiledf, 1024*1024, onInitFs, errorHandler);
});
A: You could have a function attached to the onchange event of the file input.
When this function is called, you will be able to access the file list by accessing event.target.files.
Then, when the user presses the Read button, you can process the files.
var files;
document.getElementById("filepicker").addEventListener("change", function(event) {
files = event.target.files;
}, false);
This guide might be useful: https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/webkitdirectory
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 286 |
package com.google.security.zynamics.binnavi.Gui.Debug.MemorySelectionPanel;
import java.awt.BorderLayout;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.border.TitledBorder;
import com.google.common.base.Preconditions;
import com.google.security.zynamics.binnavi.Gui.Debug.MemoryPanel.CMemoryPanel;
import com.google.security.zynamics.binnavi.Gui.Debug.MemoryRefreshButton.CMemoryRefreshButtonPanel;
import com.google.security.zynamics.binnavi.Gui.Debug.MemorySectionPanel.CMemorySectionPanel;
import com.google.security.zynamics.binnavi.Gui.GraphWindows.Panels.CDebugPerspectiveModel;
/**
* Encapsulates the memory viewer, the memory section combobox, and the memory refresh button.
*/
public final class CMemorySelectionPanel extends JPanel {
/**
* Used for serialization.
*/
private static final long serialVersionUID = 2787600129006804398L;
/**
* Panel where the target process memory is shown.
*/
private final CMemoryPanel m_memoryPanel;
/**
* Used to select the currently visible memory region.
*/
private final CMemorySectionPanel m_memorySectionPanel;
/**
* Used to refresh the visible memory.
*/
private final CMemoryRefreshButtonPanel m_refreshPanel;
/**
* Creates a new memory selection panel.
*
* @param parent Parent window used for dialogs.
* @param debugPerspectiveModel Describes a debug GUI perspective.
* @param refreshPanel Used to refresh the visible memory.
*/
public CMemorySelectionPanel(final JFrame parent,
final CDebugPerspectiveModel debugPerspectiveModel,
final CMemoryRefreshButtonPanel refreshPanel) {
super(new BorderLayout());
Preconditions.checkNotNull(parent, "IE01456: Parent argument can not be null");
Preconditions.checkNotNull(debugPerspectiveModel, "IE01457: Debugger argument can not be null");
m_memorySectionPanel = new CMemorySectionPanel(debugPerspectiveModel);
m_refreshPanel = refreshPanel;
m_memoryPanel = new CMemoryPanel(parent, debugPerspectiveModel);
setBorder(new TitledBorder("Target Memory"));
// There is the memory panel which contains the memory layout combobox
// and the refresh button.
final JPanel headerPanel = new JPanel(new BorderLayout());
headerPanel.add(m_memorySectionPanel, BorderLayout.CENTER);
headerPanel.add(m_refreshPanel, BorderLayout.EAST);
add(headerPanel, BorderLayout.NORTH);
add(m_memoryPanel, BorderLayout.CENTER);
}
/**
* Frees allocated resources.
*/
public void dispose() {
m_memoryPanel.dispose();
m_memorySectionPanel.dispose();
m_refreshPanel.dispose();
}
/**
* Returns the memory panel.
*
* @return The memory panel.
*/
public CMemoryPanel getMemoryPanel() {
return m_memoryPanel;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,522 |
package com.paradoxbomb.inkcannon.common.misc;
import com.paradoxbomb.inkcannon.LogHelper;
import com.paradoxbomb.inkcannon.common.blocks.ModBlocks;
import com.paradoxbomb.inkcannon.common.items.ModItems;
import net.minecraft.init.Items;
import net.minecraft.item.ItemStack;
import net.minecraftforge.fml.common.registry.GameRegistry;
public class Crafting
{
//initalizes crafting recipes, called during init phase
public static void initCrafting()
{
LogHelper.info("Regostering mod recipes...");
//shaped
GameRegistry.addRecipe(new ItemStack(ModBlocks.testBlock), new Object[] {"##", "##", '#', ModItems.testItem});
GameRegistry.addRecipe(new ItemStack(ModItems.inkBow), new Object [] {"#i|", "iU|", "#i|", 'i', Items.stick, '|', Items.string, 'U', Items.bucket, '#', Items.iron_ingot});
//shapeless
GameRegistry.addShapelessRecipe(new ItemStack(ModItems.testItem), new Object [] {Items.redstone, new ItemStack(Items.dye, 1, 4)});
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 245 |
#include "yajl.h"
#include <errno.h>
#include <stdio.h>
#include <string.h>
#include <kdbease.h>
#include <kdberrors.h>
#include <kdbmacros.h>
#include <yajl/yajl_parse.h>
static void elektraYajlSetArrayLength (KeySet * ks, Key * current)
{
// Update array length in array key
cursor_t cursor = ksGetCursor (ks);
Key * arrayKey = keyNew (keyName (current), KEY_END);
keySetBaseName (arrayKey, 0);
Key * foundKey = ksLookup (ks, arrayKey, 0);
keySetMeta (foundKey, "array", keyBaseName (current));
keyDel (arrayKey);
ksSetCursor (ks, cursor);
}
/**
@retval 0 if ksCurrent does not hold an array entry
@retval 1 if the array entry will be used because its the first
@retval 2 if a new array entry was created
@retval -1 error in snprintf
*/
static int elektraYajlIncrementArrayEntry (KeySet * ks)
{
Key * current = ksCurrent (ks);
const char * baseName = keyBaseName (current);
const char * meta = keyString (keyGetMeta (current, "array"));
if (!strcmp (meta, "empty"))
{
current = keyNew (keyName (current), KEY_END);
keyAddName (current, "#0");
ksAppendKey (ks, current);
elektraYajlSetArrayLength (ks, current);
return 1;
}
else if (baseName && *baseName == '#')
{
// we are in an array
current = keyNew (keyName (current), KEY_END);
elektraArrayIncName (current);
ksAppendKey (ks, current);
elektraYajlSetArrayLength (ks, current);
return 2;
}
else
{
// previous entry indicates this is not an array
return 0;
}
}
static int elektraYajlParseNull (void * ctx)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * current = ksCurrent (ks);
keySetBinary (current, NULL, 0);
ELEKTRA_LOG_DEBUG ("parse null");
return 1;
}
static int elektraYajlParseBoolean (void * ctx, int boolean)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * current = ksCurrent (ks);
if (boolean == 1)
{
keySetString (current, "1");
}
else
{
keySetString (current, "0");
}
keySetMeta (current, "type", "boolean");
ELEKTRA_LOG_DEBUG ("%d", boolean);
return 1;
}
static int elektraYajlParseNumber (void * ctx, const char * stringVal, yajl_size_type stringLen)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * current = ksCurrent (ks);
unsigned char delim = stringVal[stringLen];
char * stringValue = (char *) stringVal;
stringValue[stringLen] = '\0';
ELEKTRA_LOG_DEBUG ("%s %zu", stringVal, stringLen);
keySetString (current, stringVal);
keySetMeta (current, "type", "double");
// restore old character in buffer
stringValue[stringLen] = delim;
return 1;
}
static int elektraYajlParseString (void * ctx, const unsigned char * stringVal, yajl_size_type stringLen)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * current = ksCurrent (ks);
unsigned char delim = stringVal[stringLen];
char * stringValue = (char *) stringVal;
stringValue[stringLen] = '\0';
ELEKTRA_LOG_DEBUG ("%s %zu", stringVal, stringLen);
keySetString (current, stringValue);
// restore old character in buffer
stringValue[stringLen] = delim;
return 1;
}
static int elektraYajlParseMapKey (void * ctx, const unsigned char * stringVal, yajl_size_type stringLen)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * currentKey = keyNew (keyName (ksCurrent (ks)), KEY_END);
keySetString (currentKey, 0);
unsigned char delim = stringVal[stringLen];
char * stringValue = (char *) stringVal;
stringValue[stringLen] = '\0';
ELEKTRA_LOG_DEBUG ("stringValue: %s currentKey: %s", stringValue, keyName (currentKey));
if (currentKey && !strcmp (keyBaseName (currentKey), "___empty_map"))
{
// remove old key
keyDel (ksLookup (ks, currentKey, KDB_O_POP));
// now we know the name of the object
keySetBaseName (currentKey, stringValue);
}
else
{
// we entered a new pair (inside the previous object)
keySetBaseName (currentKey, stringValue);
}
ksAppendKey (ks, currentKey);
// restore old character in buffer
stringValue[stringLen] = delim;
return 1;
}
static int elektraYajlParseStartMap (void * ctx)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * currentKey = ksCurrent (ks);
Key * newKey = keyNew (keyName (currentKey), KEY_END);
// add a pseudo element for empty map
keyAddBaseName (newKey, "___empty_map");
ksAppendKey (ks, newKey);
ELEKTRA_LOG_DEBUG ("with new key %s", keyName (newKey));
return 1;
}
static int elektraYajlParseEnd (void * ctx)
{
KeySet * ks = (KeySet *) ctx;
Key * currentKey = ksCurrent (ks);
const char * meta = keyString (keyGetMeta (currentKey, "array"));
// If array is still empty by the time we reach the end, replace with ""
if (!strcmp (meta, "empty"))
{
keySetMeta (currentKey, "array", "");
return 1;
}
Key * lookupKey = keyNew (keyName (currentKey), KEY_END);
keySetBaseName (lookupKey, 0); // remove current baseName
// lets point current to the correct place
Key * foundKey = ksLookup (ks, lookupKey, 0);
#ifdef HAVE_LOGGER
if (foundKey)
{
ELEKTRA_LOG_DEBUG ("%s", keyName (foundKey));
}
else
{
ELEKTRA_LOG_DEBUG ("did not find key %s", keyName (lookupKey));
}
#else
(void) foundKey; // foundKey is not used, but lookup is needed
#endif
keyDel (lookupKey);
return 1;
}
static int elektraYajlParseStartArray (void * ctx)
{
KeySet * ks = (KeySet *) ctx;
elektraYajlIncrementArrayEntry (ks);
Key * currentKey = ksCurrent (ks);
Key * newKey = keyNew (keyName (currentKey), KEY_END);
keySetMeta (newKey, "array", "empty");
ksAppendKey (ks, newKey);
ELEKTRA_LOG_DEBUG ("with new key %s", keyName (newKey));
return 1;
}
/**
* @brief Remove all non-leaf keys except for arrays
*
* @param returned to remove the keys from
*/
static void elektraYajlParseSuppressNonLeafKeys (KeySet * returned)
{
ksRewind (returned);
Key * cur = ksNext (returned);
while (cur != NULL)
{
cursor_t cursor = ksGetCursor (returned);
if (ksNext (returned) == NULL) break;
Key * peekDup = keyDup (ksCurrent (returned));
keySetBaseName (peekDup, 0);
if (!strcmp (keyName (peekDup), keyName (cur)))
{
const char * baseName = keyBaseName (ksCurrent (returned));
// TODO: Add test for empty array check
if (strcmp (baseName, "#0"))
{
ELEKTRA_LOG_DEBUG ("Removing non-leaf key %s", keyName (cur));
keyDel (ksLookup (returned, cur, KDB_O_POP));
ksSetCursor (returned, cursor);
}
else
{
// Set array key to NULL to avoid empty ___dirdata entries
keySetBinary (cur, NULL, 0);
}
}
keyDel (peekDup);
cur = ksCurrent (returned);
}
}
/**
* @brief Remove ___empty_map if thats the only thing which would be
* returned.
*
* @param returned to remove the key from
*/
static void elektraYajlParseSuppressEmptyMap (KeySet * returned, Key * parentKey)
{
if (ksGetSize (returned) == 2)
{
Key * lookupKey = keyDup (parentKey);
keyAddBaseName (lookupKey, "___empty_map");
Key * toRemove = ksLookup (returned, lookupKey, KDB_O_POP);
#ifdef HAVE_LOGGER
if (toRemove)
{
ELEKTRA_LOG_DEBUG ("remove %s", keyName (toRemove));
}
else
{
ksRewind (returned);
Key * cur;
while ((cur = ksNext (returned)) != 0)
{
ELEKTRA_LOG_DEBUG ("key %s has value %s", keyName (cur), keyString (cur));
}
ELEKTRA_LOG_DEBUG ("did not find %s", keyName (lookupKey));
ksRewind (returned);
}
#endif
if (toRemove)
{
keyDel (toRemove);
}
keyDel (lookupKey);
}
}
static inline KeySet * elektraGetModuleConfig (void)
{
return ksNew (30, keyNew ("system/elektra/modules/yajl", KEY_VALUE, "yajl plugin waits for your orders", KEY_END),
keyNew ("system/elektra/modules/yajl/exports", KEY_END),
keyNew ("system/elektra/modules/yajl/exports/get", KEY_FUNC, elektraYajlGet, KEY_END),
keyNew ("system/elektra/modules/yajl/exports/set", KEY_FUNC, elektraYajlSet, KEY_END),
#include "readme_yajl.c"
keyNew ("system/elektra/modules/yajl/infos/version", KEY_VALUE, PLUGINVERSION, KEY_END),
keyNew ("system/elektra/modules/yajl/config", KEY_END),
keyNew ("system/elektra/modules/yajl/config/", KEY_VALUE, "system", KEY_END),
keyNew ("system/elektra/modules/yajl/config/below", KEY_VALUE, "user", KEY_END),
keyNew ("system/elektra/modules/yajl/config/needs/boolean/restoreas", KEY_VALUE, "none", KEY_END), KS_END);
}
int elektraYajlGet (Plugin * handle ELEKTRA_UNUSED, KeySet * returned, Key * parentKey)
{
if (!strcmp (keyName (parentKey), "system/elektra/modules/yajl"))
{
KeySet * moduleConfig = elektraGetModuleConfig ();
ksAppend (returned, moduleConfig);
ksDel (moduleConfig);
return 1;
}
yajl_callbacks callbacks = { elektraYajlParseNull,
elektraYajlParseBoolean,
NULL,
NULL,
elektraYajlParseNumber,
elektraYajlParseString,
elektraYajlParseStartMap,
elektraYajlParseMapKey,
elektraYajlParseEnd,
elektraYajlParseStartArray,
elektraYajlParseEnd };
ksAppendKey (returned, keyNew (keyName ((parentKey)), KEY_END));
#if YAJL_MAJOR == 1
yajl_parser_config cfg = { 1, 1 };
yajl_handle hand = yajl_alloc (&callbacks, &cfg, NULL, returned);
#else
yajl_handle hand = yajl_alloc (&callbacks, NULL, returned);
yajl_config (hand, yajl_allow_comments, 1);
#endif
int errnosave = errno;
unsigned char fileData[65536];
int done = 0;
FILE * fileHandle = fopen (keyString (parentKey), "r");
if (!fileHandle)
{
yajl_free (hand);
ELEKTRA_SET_ERROR_GET (parentKey);
errno = errnosave;
return -1;
}
while (!done)
{
yajl_size_type rd = fread ((void *) fileData, 1, sizeof (fileData) - 1, fileHandle);
if (rd == 0)
{
if (!feof (fileHandle))
{
ELEKTRA_SET_RESOURCE_ERRORF (parentKey, "Error while reading file: %s", keyString (parentKey));
fclose (fileHandle);
yajl_free (hand);
return -1;
}
done = 1;
}
fileData[rd] = 0;
yajl_status stat;
if (done)
{
#if YAJL_MAJOR == 1
stat = yajl_parse_complete (hand);
#else
stat = yajl_complete_parse (hand);
#endif
}
else
{
stat = yajl_parse (hand, fileData, rd);
}
int test_status = (stat != yajl_status_ok);
#if YAJL_MAJOR == 1
test_status = test_status && (stat != yajl_status_insufficient_data);
#endif
if (test_status)
{
unsigned char * str = yajl_get_error (hand, 1, fileData, rd);
ELEKTRA_SET_VALIDATION_SYNTACTIC_ERRORF (parentKey, "Yajl parse error happened. Reason: %s", (char *) str);
yajl_free_error (hand, str);
yajl_free (hand);
fclose (fileHandle);
return -1;
}
}
yajl_free (hand);
fclose (fileHandle);
elektraYajlParseSuppressNonLeafKeys (returned);
elektraYajlParseSuppressEmptyMap (returned, parentKey);
return 1; /* success */
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,489 |
\section{ Introduction.}
Group classification of differential equations consists in the
specification of non-equivalent classes of such equations which
possess the same symmetry groups. It is a rather attractive research
field which has both fundamental and application values.
A perfect example of group classification of fundamental equations
of mathematical physics was presented by Boyer~\cite{Boy} who had
specified all inequivalent Schr\"odinger equations with time
independent potentials admitting symmetries with respect to Lie
groups, see also papers \cite{Hag,Nied,And} where
particular important symmetries were discussed, and paper \cite{NB} were the Boyer results are corrected. These old results
have a big impact since include a priori information about all
symmetry groups which can be admitted by the fundamental equation of
quantum mechanics. Let us mention also that the nonlinear
Schr\"odinger equation as well as the generalized Ginsburg--Landau
quasilinear equations have been classified also \cite{pop,N1} as well as symmetries of more general systems of reaction-diffusion equations \cite{NW1,NW2}. For general discussion of {\it supersymmetries} of Schr\"odinger equation see, e.g., papers \cite{bek1,bek2,AK,AN1}.
In contrary, the group classification of Schr\"odinger equations
with position dependent mass (PDM) was waited for a very long time.
There were many papers devoted to PDM Schr\"odinger equations with
particular symmetries, see, e.g., \cite{11,rac,Koch,Cru}. But the complete group classification of
these equations appears only recently in papers \cite{NZ} and \cite{NZ2,NN} for the stationary and time dependent
equations correspondingly. A systematic search for the higher order symmetries if the PDM systems started in paper \cite{N2}. So late making of such important job
have to cause the blame for experts in group analysis of
differential equations, taking into account the fundamental role
played by such equations in modern theoretical physics!
Let us remind that the PDM Schr\"odinger equations are requested for
description of various condensed-matter systems such as
semiconductors, quantum liquids, and metal clusters, quantum wells,
wires and dots, super-lattice band structures, etc., etc.
It happens that the number of PDM systems with different Lie
sym-metries is rather extended. Namely, in~\cite{NN} seventy classes
of such systems are specified. Twenty of them are defined up to
arbitrary parameters, the remaining fifty systems include arbitrary
functions.
The knowledge of all Lie groups which can be admitted by the PDM
Schr\"odinger equations has both fundamental and application values.
In particular, when construct the models with a priory requested
symmetries we can use the complete lists of inequivalent PDM systems
presented in~\cite{NZ2} for $d=2$ and~\cite{NN} for $d=3$. Moreover, in
many cases a~sufficiently extended symmetry induces integrability or
exact solvability of the system, and just this aspect will be
discussed in the present paper.
It will be shown that all PDM systems admitting six parametric Lie
groups of symmetries or more extended symmetries are exactly
solvable. Moreover, the complete sets of solutions of the
corresponding stationary PDM Schr\"odinger equations will be
presented explicitly.
There exist a tight connection between the complete solvability and
various types of higher symmetries and supersymmetries. We will see
that extended Lie symmetries also can cause the exact solvability.
More-over, the systems admitting extended Lie symmetries in many
cases are supersymmetric and superintegrable.
\section{ PDM Schr\"odinger equations with extended Lie sym\-met\-ries.}
In paper \cite{NN} we present the group classification of PDM
Schr\"odinger equations
\begin{gather}\label{seq}L\psi\equiv\left(\ri\frac{\p}{\p
t}-H\right)\psi=0,\end{gather} where $H$ is the PDM Hamiltonian of
the following generic form
\begin{gather}\label{B1} H=\tfrac14\big(m^\alpha p_a m^{\beta}p_am^\gamma+
m^\gamma p_a m^{\beta}p_am^\alpha\big)+ \hat V,\quad
p_a=-\ri\frac{\p}{\p x_a}.\end{gather} Here $m=m({\bf x})$ and $\hat
V=\hat V({\bf x})$ are the mass and potential depending on spatial
variables ${\bf x}=(x_1,x_2,x_3)$, and summation w.r.t.\ the
repeating indices $a$ is imposed over the values $a=1,2,3$. In
addition, $\alpha$, $\beta$ and $\gamma$ are the ambiguity parameters
satisfying the condition $\alpha+\beta+\gamma=-1$.
The choice of values of the ambiguity parameters can be motivated by
physical reasons, see a short discussion of this point in~\cite{NN}.
Hamiltonian (\ref{B1}) can be rewritten in the following more compact form
\begin{gather}\label{B2}H= \tfrac12p_a fp_a+ V,\end{gather}
where
\begin{gather}\label{B3} V=\hat V+\tfrac14(\alpha+\gamma)f_{aa}+
\alpha\gamma\frac{f_a f_a}{2f}\end{gather} with $f=\frac1m,\
f_a=\frac{\p f}{\p x_a}$ and $f_{aa}=\Delta f=\frac{\p f_a}{\p
x_a}$.
In the following text just representation (\ref{B3}) will be used.
In accordance with \cite{NN} there is a big variety of Hamiltonians
(\ref{B3}) ge\-ne\-rating non-equivalent continuous point symmetries of
equation~(\ref{B1}). The corresponding potential and mass terms are
defined up to arbitrary parameters or even up to arbitrary
functions.
In the present paper we consider the PDM systems defined up to
arbitrary parameters. Just such systems admit the most extended Lie
symmetries. Using the classification results presented in \cite{NZ} and~\cite{NN}
we enumerate these systems in the following Table~1, where $\varphi=\arctan\frac{x_2}{x_1}$ and the other Greek letters denote arbitrary constants parameters, which are supposed not to be zero simultaneously. Moreover, $\lambda$ and $\omega$ are either real or imaginary, the remaining parameters are real.
\vspace{1mm}
\begin{center} Table 1.
PDM systems with extended Lie symmetries.\end{center}
\vspace{1mm}
\begin{tabular}{ccll}
\hline
No&$\begin{array}{c}\text{Inverse} \ \text{mass} \ f
\end{array}$&Potential $V$&Symmetries\\
\hline
\vspace{2mm}
1&$\big(r^2+1\big)^2
$&\vspace{1mm}$-3 r^2$&$\begin{array}{l}M_{41}, M_{42}, M_{43},\\ M_{21}, M_{31}, M_{32}\end{array}$ \\
2&\vspace{1mm}$\big(r^2-1\big)^2$&$-3 r^2$&$ \begin{array}{l}M_{01}, M_{02}, M_{03},\\ M_{21}, M_{31},
M_{32}\end{array}$ \\
3&\vspace{1mm}$x_3^2$&$\nu\ln(x_3)$&$\begin{array}{l}P_1,\ P_2,\ M_{12},\ D+\nu
t\end{array}$
\\
4&\vspace{1mm}$\tilde r^3$&$ \kappa x_3+\lambda\tilde r$&$
P_3+\kappa t,
\ D+\ri t\p_t, \ M_{12}$\\
5&\vspace{1mm}$x_1^3$&$\lambda x_1+\kappa x_3$&$P_3+\kappa t, \ P_2,
\
D+\ri t\p_t$\\
6&\vspace{1mm}$x_3^{\sigma+2}$&$\kappa
x_3^\sigma$&$\begin{array}{l}P_1,\ P_2,\ M_{12}, \ D+\ri\sigma t\p_t, \\
\sigma\neq
0,1,-2\end{array}$\\
7&\vspace{1mm}$\tilde r^{\sigma+2}{\rm e}^{\lambda\varphi}$&$\kappa \tilde
r^{\sigma}{\rm e}^{\lambda\varphi}$&$\begin{array}{l}M_{12}+\ri\lambda t\p_t,\
P_3,\\
D+\ri\sigma t\p_t, \ \sigma\neq0\end{array}$\\
8& \vspace{1mm} $\tilde
r^2$&
\frac{\lambda^2}2\varphi^2+\mu\varphi+\nu \ln(\tilde
r)
$& $ \begin{array}{l}B^1_1, B^1_2,\\D+\nu t, \ P_3 \end{array}
$\\
9&\vspace{1mm}$\tilde r^2{\rm e}^{\sigma \varphi}$&$\kappa
{\rm e}^{\sigma\varphi}+\frac{\omega^2}{2} {\rm e}^{-\sigma\varphi}$&$\begin{array}{l}N^1_1, \ N^1_2, P_3, \ D,\ K_3\end{array}$\\
10&\vspace{1mm}$ r^2$&$\nu\ln(r)+
{\frac{\lambda^2}2}\ln(r)^2$&$\begin{array}{l}B^2_1,\ B^2_2, \
L_1, \ L_2, \ L_3 \end{array}$\\
11&\vspace{1mm}$r^{2+\sigma}$&$\kappa r^{\sigma}+
\frac{\omega^2}{2}r^{-\sigma} $&$\begin{array}{l}N^2_1,\
N^2_2,\
L_1, \ L_2, \ L_3 \end{array}$\\
\hline\hline
\end{tabular}
\vspace{2mm}
The symmetry operators presented in column 4 of the table are given by the following formulae:
\begin{gather}
P_{i}=p_{i}=-i\frac{\partial}{\partial x_{i}},\quad D=x_n p_n-\tfrac{3\ri}2,\nonumber\\
M_{ij}=x_ip_j-x_jp_j,\quad M_{0i}=\tfrac12\big(K^i+P_i\big), \quad M_{4i}=\tfrac12\big(K^i+P_i\big),\nonumber \\
B^1_1=\lambda\sin (\lambda t) M_{12}
\big(\lambda^2\varphi+\nu\big)\cos (\lambda t), \quad B^1_2=\frac{\p}{\p t}B^1_1,\nonumber\\
B^2_1= \sin(\lambda t)D- \cos(\lambda
t)\left({\lambda}\ln(r)+\tfrac\nu\lambda\right), \quad B^2_2=\frac{\p}{\p t}B^2_1,\nonumber\\
N^1_1=\omega\cos
(\omega\sigma t) L_3-\sin(\omega\sigma t)\big(\ri
\p_t- \omega^2{\rm e}^{-\sigma\Theta}\big) ,\quad N^1_2=\frac{\p}{\p t}N^1_1,\nonumber \\
N^2_1=\omega\cos(\omega\sigma
t)D+\sin(\omega\sigma t)\big(\ri \p_t-{\omega^2} r^{-\sigma}\big)
,\quad N^2_2=\frac{\p}{\p t}N^1_1,\label{SO}
\end{gather}
where $K_{i}=x_nx_n p_i -2x_iD$ and indices $i$, $j$, $k$, $n$ take the values $1$, $2$, $3$. In addition, all the presented systems admit symmetry operators $P_0=\ri\frac{\p}{ \p t}$ and the unit operator, the latter is requested to obtain the closed symmetry algebras.
Rather surprisingly, all systems
presented in Table 1 (except ones given in items 4 and 5 with $\kappa\neq0$) are exactly solvable. In the following sections we
present their exact solutions. To obtain these solutions we use some
nice properties of the considered systems like superintegrability and supersymmetry
with shape invariance. Let us remind that the quantum
mechanical system is called superintegrable if it admits more integrals
of motion than its number of degrees of freedom.
In accordance with Table 1 we can indicate 11 inequivalent PDM
systems which are defined up to arbitrary parameters and admit Lie
symmetry algebras of dimension five or higher. Notice that the systems
fixed in items 4 and 5 admit five dimension symmetry algebras while the
remaining systems admit more extended symmetries.
\section{ Systems with fixed mass and potentials.}
First we consider systems whose mass and potential terms are fixed, i.e., do not include arbitrary parameters.
These systems are presented in items 1, 2 of Table~1 and others provided the mass does not depends on parameters and parameters of the potential are trivial.
\subsection{ System invariant w.r.t.\ algebra $\boldsymbol{\mathfrak{so}(4)}$.}
Consider Hamiltonian (\ref{B2}) with functions $f$ and $V$ presented
in item~1 of Table~1:
\begin{gather}\label{H2}H=\tfrac12p_a\big(1+r^2\big)^2p_a-3 r^2.\end{gather}
The eigenvalue problem for this Hamiltonian can be written in the
following form:
\begin{gather}\label{ep}H\psi= 2E\psi,\end{gather} where
$E$ are yet unknown eigenvalues.
Equation (\ref{ep}) admits six integrals of motion $M_{AB}$, $A,B=1,2,3,4$, presented in equation~(\ref{SO}). Let us write them explicitly
\begin{gather} M^{ab}=x^ap^b-x^bp^a,\quad
M^{4a}=\tfrac12\big(r^2-1\big)p^a
-x^ax^bp^b+\tfrac{3\ri}2x^a.\label{im} \end{gather}
Operators (\ref{im}) form a
basis of algebra $\mathfrak{so}(4)$. Moreover, the first Casimir operator of this algebra is proportional to Hamiltonian (\ref{H2}) up to the constant shift
\begin{gather*
C_1=\tfrac12M_{AB}M_{AB}=\tfrac12 ( H-9),\end{gather*}
while the second Casimir operator $C_2=\varepsilon_{ABCD}M_{AB}M_{CD}$ appears to be zero.
Thus like the Hydrogen atom system (\ref{ep}) admits six integrals of motion belonging to algebra $\mathfrak{so}(4)$ and is maximally superintegrable.
Using our knowledge of unitary representations of algebra $\mathfrak{so}(4)$ is possible to find eigenvalues~$E$ algebraically:
\begin{gather}\label{ev2} E=4 n^2+5,\end{gather}
where $n=0,1,2,\dots $ are natural numbers.
To find the eigenvectors of Hamiltonian (\ref{H2}) corresponding to eigenvalues (\ref{ev2}) we use the rotation invariance of (\ref{ep}) and separate variables. Introducing spherical variables and expanding solutions via spherical functions
\begin{gather}\label{rv}\psi=\frac1r\sum_{l,m}\phi_{lm}(r)Y^l_m\end{gather}
we come to the following equations for radial functions
\begin{gather*} \label{re}\left(-\big(r^2+1\big)^2\left(\frac{\p^2}{\p r^2}-\frac{l(l+1)}{r^2}\right)-4r\big(r^2+1\big)\frac{\p}{\p r}-2r^2\right)\varphi_{lm}\\
\quad{} =\left(4 n^2+1\right)\varphi_{lm},
\end{gather*} where $l=0,1,2,\dots$ are parameters numerating eigenvalues of the squared orbital momentum.
The square integrable solutions of these equations are
\begin{gather}\label{soll}\varphi_{lm}=C_{lm}^n\big(r^2+1\big)^{-n-\frac12}r^{l+1}
{\cal F}\left([A,B],[C] -r^2\right),\end{gather}
where
\begin{gather*}A=-n+l+1, \quad B=-n+\tfrac12,\quad C= l+\tfrac32.\end{gather*}
${\cal F}(\cdots)$ is the hypergeometric function and $C_{lm}^n$ are integration constants. Solutions~(\ref{soll}) tend to zero at infinity provided $n$ is a natural number and $l\leq n-1$.
Thus the system (\ref{ep}) is maximally superintegrable and exactly sol\-vable.
\subsection{ System invariant w.r.t.\ algebra $\boldsymbol{\mathfrak{so}(1,3)}$.}
The next Hamiltonian we consider corresponds to functions~$f$ and~$V$
presented in item~2 of Table~1.
The related eigenvalue problem includes the following equation
\begin{gather}\label{ep1} H\psi\equiv-\tfrac12\big(\p_a\big(1-r^2\big)^2\p_a+6 r^2\big)\psi=E\psi.\end{gather}
Equation (\ref{ep1}) admits six
integrals of motion $M_{\mu\nu}$, $\mu, \nu=0, 1,2,3,$ given by equation (\ref{SO}), which can be written explicitly in the following form
\begin{gather} M_{ab}=x^ap^b-x^bp^a, \nonumber\\
M_{0a}=\tfrac12\big(r^2+1\big)p^a -x^ax^bp^b+\tfrac{3\ri}2x^a, \quad a,b=1,2,3.\label{im1} \end{gather}
These operators form a basis of algebra~$\mathfrak{so}(1,3)$, i.e., the Lie algebra of Lorentz group.
As in the previous section, the corresponding first Casimir operator is expressed via the Hamiltonian, namely
\begin{gather}\label{uh} C_1=\tfrac12M^{ab}M^{ab}-M^{0a}M^{0a}=\tfrac12 ( H+9),\end{gather} while the second one appears to be zero.
Using our knowledge of irreducible unitary representations of Lorentz group we find eigenvalues of $C_1$ and $C_2$ in the form
\cite{naimark,IM}:
\begin{gather*} c_1=1-j_0^2-j_1^2, \quad c_2=2\ri j_0j_1,\end{gather*} where $j_0$ and $j_1$ are quantum numbers labeling irreducible representations. Since the second Casimir operator $C_2$ is trivial, we have $c_1=j_0=0$. So there are two possibilities~\cite{naimark}: either~$j_1$ is an arbitrary imaginary number, and the corresponding representation belongs to the principal series, or $j_1$ is a real number satisfying $|j_1|\leq1 $, and we come to the subsidiary series of IRs. So
\begin{gather}\label{j1}j_1={\rm i}\lambda, \quad c_1=1-j_1^2=\lambda^2+1, \end{gather}
where $\lambda$ is an arbitrary real number, or, alternatively,
\begin{gather}\label{j2}0\leq j_1\leq 1, \quad c_1=1-j_1^2.\end{gather}
In accordance with (\ref{uh}) the related eigenvalues $ E$ in~(\ref{ep1}) are
\begin{gather}E=-5-j_1^2.\label{EE}\end{gather}
In view of the
rotational invariance of equation (\ref{ep1}) it is convenient to
represent solutions
in form~(\ref{rv}). As a result we obtain the following radial
equations
\begin{gather} \left(-\big(r^2-1\big)^2\left(\frac{\p^2}{\p
r^2}-\frac{l(l+1)}{r^2}\right)-4r\big(r^2-1\big)\frac{\p}{\p
r}-2r^2\right)\varphi_{lm}\nonumber\\
\quad {} =(\tilde E+4)\varphi_{lm}. \label{re1}\end{gather}
The general solution of (\ref{re1}) is
\begin{gather} \varphi_{lm}=\big(1-r^2\big)^{-
\frac12-k}\big(C_{lm}^k r^{l+1}{\cal F}\big([A,B],[C],r^2\big) \nonumber \\
\hphantom{\varphi_{lm}=}{} +\tilde C_{lm}^k r^{-l}{\cal F}\big([\tilde A,\tilde B],[\tilde C], r^2\big)\big),\label{soll1}
\end{gather}
where
\begin{gather*} A=-k+l+1, \quad B= -k+\tfrac12,\quad C= l+\tfrac32,\\
\tilde A = -k-l, \quad \tilde B=-k+\tfrac12,\quad \tilde C=\tfrac12-l, \quad k=\tfrac12\sqrt{-\tilde E-5}
\end{gather*} and is singular at $r=1$. However, for $\tilde C_{lm}^k=0$ and $k= j_1$ the solutions are normalizable in some specific metric~\cite{NZ}.
Thus the system presented in item 7 of Table~1 is exactly solvable too. The corresponding eigenvalues and eigenvectors are given by equations (\ref{j1}), (\ref{j2}), (\ref{EE}) and (\ref{soll1}) correspondingly.
\subsection{ Scale invariant systems.}
Consider one more PDM system which is presented in item~3 of the
table and includes the following Hamiltonian:
Let us note that the free fall effective potential appears also one more system specified in Table~1. Thus, considering the inverse mass and potential specified in item~3 we come to the following Hamiltonian
\begin{gather}\label{oj}\begin{split}&
H=-\frac12\left(x_3\frac{\p}{\p x_3}x_3\frac{\p}{\p x_3}+x_3\frac{\p}{\p x_3}+x_3^2\left(\frac{\p^2}{\p x_1^2}+\frac{\p^2}{\p x_2^2}\right)\right)\\&+\nu\ln(x_3)\end{split}.\end{gather}
Equation (\ref{ep1}) with Hamiltonian given in (\ref{oj}) can be easily solved by separation of variables in Cartesian coordinates. Expanding the wave function $\psi$ via eigenfunctions of integrals of motion $P_1$ and $P_2$:
\begin{gather}\label{buh}\psi=\exp(-\ri(k_1 x_1+k_2x_2))\Phi(k_1,k_2,x_3)\end{gather}
and introducing new variable $y=\ln(x_3)$ we come to the following equation for $\Phi=\Phi(k_1,k_2,x_3)$:
\begin{gather}\label{uaa}
-\frac{\p^2 \Phi}{\p{y^2 }}+\left(\big(k_1^2+k_2^2\big)\exp(2y)+2\nu y\right)\Phi=\tilde E\Phi \end{gather}
where $\tilde E=2E-\frac14$.
Here we consider the simplest version of equation (\ref{uaa}) when parameter $\nu$ is trivial:
\begin{gather}\label{ua}
-\frac{\p^2 \Phi}{\p{y^2 }}+\big(k_1^2+k_2^2\big)\exp(2y)\Phi=\tilde E\Phi.\end{gather}
This equation is scale invariant and can be easily solved. Its square integrable solutions are given by Bessel functions
\begin{gather*
\Psi=C^E_{k_1k_2}K_{\ri \sqrt{\tilde E}}\Big(\sqrt{k_1^2+k_2^2}\ln(x_3)\Big),\end{gather*}
where $C^E_{k_1k_2}$ are integration constants and $\tilde E$ are arbitrary real parameters.
It is interesting to note that there are rather non-trivial relations between the results given in the present and previous sections. Equation (\ref{ua}) admits six integrals of motion which are nothing but the following operators{\samepage
\begin{gather}\label{onem}P_1,\ P_2,\ K_1,\ K_2, \ M_{12},\ D,\end{gather}
which are presented in equations (\ref{SO}).}
Like operators (\ref{im1}) integrals of motion (\ref{onem}) form a basis of the Lie algebra of Lorentz group, and we again can find the eigenvalues of Hamiltonian (\ref{ua}) algebraically by direct analogy with the above. We will not present this routine procedure since there exist strong equivalence relations between Hamiltonians (\ref{ua}) with zero $\nu$ and (\ref{H2}). To find them we note that basis (\ref{onem}) is equivalent to the following linear combinations of the basis elements:
\begin{gather} M_{01}, \ M_{02}, M_{04}, \ M_{41}\ M_{42}, \ M_{12},\label{mecem}\end{gather} whose expressions via operators~(\ref{onem}) are given by equation~(\ref{SO}). To reduce~(\ref{mecem}) to the set~(\ref{im1}) it is sufficient to change subindices 4 to 3, i.e., to make the rotation in the plane 43. The infinitesimal operator for such rotation is given by the following operator
\begin{gather*}M_{43}=\tfrac12(K_3+P_3)=\tfrac12\big(r^2-1\big)p_3
-x_3x_bp_b+\tfrac{3\ri}2x_3
\end{gather*}
which belongs to the equivalence group of equations.
Solving the corresponding Lie equations and choosing the group parameter be equal $\frac{\pi}{2}$ we easily find the requested equivalence transformations.
One more scale invariant system is presented in item~8 where all parameters of potential are zero. The relation Hamiltonian looks as follows:
\begin{gather} \label{12} H=-\tilde r\frac{\p}{\p{x_\alpha}}\tilde r\frac{\p}{\p
{x_\alpha}} -x_\alpha\frac{\p}{\p{x_\alpha}}-\tilde r^2\frac{\p^2}{\p {x_3^2}},\quad \alpha=1,2.\end{gather}
Considering the eigenvalue problem for (\ref{12}) it is convenient to use the cylindrical variables
\begin{gather}\label{CV}\tilde r=\sqrt {x_1^2+x_2^2}, \ \varphi=\arctan\frac{x_2}{x_1},\ x_3=z\end{gather}
and expand solutions via eigenfunctions of $M^{12}$ and $P_3=-\ri\frac{\p}{\p z}$:
\begin{gather}\label{CV1} \Psi=\exp[\ri(\kappa\varphi +\omega z)]\Phi_{\kappa\omega}(\tilde r),\quad \kappa=0,\pm1, \pm2,..., -\infty<\omega<\infty.\end{gather}
As a result we come to the following equations for radial functions $\Phi=\Phi_{\kappa\omega}(\tilde r)$:
\begin{gather*}-\left(\tilde r\frac{\p} {\p \tilde r}\tilde r\frac{\p}{\p \tilde r}+\tilde r\frac{\p} {\p \tilde r}+\omega^2\right)\Phi=(\tilde E-\kappa^2.)\Phi\end{gather*}
Square integrable (with the weight $\tilde r$) solutions of this equation are:
\begin{gather}\label{soso}\Phi_{\kappa\omega}=\frac1{\tilde r}J_\alpha(\omega \tilde r),\quad \alpha=\kappa^2+1-\tilde E \end{gather}
where $J_\alpha(\omega \tilde r)$ is Bessel function of the first kind.
Functions (\ref{soso}) are normalizable and disappear at $\tilde r=0$ provided $\alpha\leq0$. The rescaled energies $\tilde E$ continuously take the values $\kappa^2\leq\tilde E\leq\infty$.
The last scale invariant system which we have to consider is fixed in item~10 where $\nu=\lambda=0$ We will do it later in Section
\section{ Systems defined up to arbitrary parameters.}
In previous section we present exact solutions for systems with
fixed potential and mass terms. In the following we deal with
the systems defined up to arbitrary parameters.
\subsection{ The system with oscillator effective potential.}
Let us consider equation (\ref{seq}) with $f$ and $V$ are
functions fixed in item~10 of Table~1, i.e.,
\begin{gather*
\ri\frac{\p \psi}{\p t}=
\left(-\frac12\frac{\p}{\p{x_a}} r^{2}\frac{\p}{\p{x_a}}+\nu\ln(r)+
{\frac{\lambda^2}2}\ln(r)^2\right)\psi.\end{gather*}
These equations admit extended symmetries Lie symmetries (whose generators are indicated in the table) being
invariant w.r.t.\ six-pa\-ra\-met\-ri\-cal Lie group. Let us show that they also admit hidden supersymmetries.
In view of the rotational invariance and
symmetry of the considered equations with respect to shifts of time variable, it is reasonable to
to search for their solutions in spherical variables, i.e., in the
following form
\begin{gather}\label{psi}\Psi={\rm e}^{-iEt}R_{lm}( r)Y_{lm}(\varphi,\theta),\end{gather}
where $\varphi$ and $\theta$ are angular variables and
$Y_{lm}(\varphi,\varphi)$ are spherical functions, i.e.,
eigenvectors of $L^2=L_1^2+L_2^2+M_{12}^2$ and $ M_{12}$. As a result we
come to the following radial equations
\begin{gather} \left(-r\frac{\p R_{lm}}{\p r}r\frac{\p R_{lm}}{\p r}-r\frac{\p R_{lm}}{\p r}\right.\nonumber\\\left.
\quad {} +{l(l+1)}+\nu\ln(r)+ {\frac{\lambda^2}2}\ln(r)^2\right)R_{lm}
=2ER_{lm}.\label{equq7} \end{gather}
Introducing new variable $y=\sqrt{2}\ln (r)$ we can rewrite equation (\ref{equq7}) in the following form:
\begin{gather}\label{ree}\left(-\frac{\p^2}{\p y^2}+l(l+1)+\nu y+\frac{\lambda^2}2y^2\right)R_{lm}(y)=\tilde ER_{lm}(y),\end{gather}
where $\tilde E=E-\frac14$.
Let $\lambda\neq0$ then equation (\ref{ree}) is reduced to the 1D harmonic oscillator up to the additional term $l(l+1)$ The admissible eigenvalues $\tilde E$ are given by the following formula
\begin{gather*}\tilde E =n+l(l+1).\end{gather*}
where $n$ is a natural number.
The corresponding eigenfunctions are well known and we will not presented them here. The same is true for supersymmetric aspects of the considered system.
If parameter $\lambda$ is equal to zero then (\ref{ree}) reduces to equation with free fall potential slightly modified by the term $l(l+1)$. The corresponding solutions can be found in textbooks devoted to quantum mechanics.
\subsection{ The systems with potentials equivalent to 3d oscillator.}
Consider now the system represented in item~11 of the table. The corresponding equation (\ref{seq}) takes the following form:
\begin{gather}\label{equ5}\ri\frac{\p \psi}{\p t}=
\left(-\frac12\p_a r^{\sigma+2}\p_a+\kappa r^{2\sigma}+
\frac{\omega^2}{r^{2\sigma}}\right)\psi.\end{gather}
Like in previous section we represent the wave function in the form given in (\ref{psi}) and came to the following radial equation
\begin{gather} -r^{2\sigma+2}\frac{\p^2R_{lm}}{\p r^2}-(2\sigma+4)r^{2\sigma+1}\frac{\p R_{lm}}{\p r}\nonumber\\
\quad {} +
\big(r^{2\sigma}({l(l+1)+\kappa})+\omega^2r^{-2\sigma}\big)R_{lm}
=2ER_{lm}.\label{equ7} \end{gather}
Using the Liouville transform
\begin{gather*
r\to z=r^{-\sigma},\quad R_{lm}\to\tilde R_{lm}=z^{\frac{\sigma+3}{2\sigma}}R_{lm},\end{gather*} we reduce (\ref{equ7}) to the following form
\begin{gather}\label{equ9}-\sigma^2\frac{\p^2\tilde R_{lm}}{\p z^2}+\left(\frac{l(l+1)+\delta}{z^2}+\omega^2z^2\right)\tilde R_{lm}
=2E\tilde R_{lm},\end{gather} where
$ \delta=\frac3{4}(\sigma+1)(\sigma+3)+
2\kappa$.
Equation (\ref{equ9}) describes a deformed 3d harmonic oscillator
including two deformation parameters, namely, $\sigma$ and $\kappa$.
Let
\begin{gather}\label{con}2\kappa=-\sigma^2-3\sigma-2,\end{gather}
then equation (\ref{equ9}) is reduced to the following form
\begin{gather}\label{eg1}H_l\tilde R_{lm}\equiv\left(-\sigma^2\frac{\p^2}{\p z^2}+\frac{(2l+1)^2
{-\sigma^2}}{4z^2}+\omega^2z^2\right)\tilde R_{lm}
=2E\tilde R_{lm}.\end{gather}
Equation (\ref{eg1}) is shape invariant. Hamiltonian~$H_r$ can be factorized
\begin{gather}H_l=a_l^+a_l-C_l,\label{eg2}\end{gather}
where
\begin{gather*} a=-\sigma\frac{\p}{\p z}+W,\quad a^+=\sigma\frac{\p}{\p z}+W,\nonumber\\
W=\frac{2l+1+\sigma}{2z}+\omega z,\quad C_l=\omega(2l+2\sigma+1)
\end{gather*}
The superpartner $\hat H_l $ of Hamiltonian~(\ref{eg2}) has the following property
\begin{gather*}\hat H_l\equiv a_la_l^++C_l=H_{l+\sigma}+C_l
\end{gather*}
Thus our Hamiltonian is shape invariant.
Thus to solve equation (\ref{eg1}) we can use the standard tools of SUSY quantum mechanics and find the admissible eigenvalues in the following form{\samepage
\begin{gather}\label{eg6}E_n=\omega\left(2n\sigma+l+
\sigma+\tfrac12\right)=\omega\left(2n+l+\tfrac32\right)+\delta\omega(2n+1),\end{gather}
where $\delta=\sigma-1$.}
Equation (\ref{eg6}) represents the spectrum of 3d isotropic harmonic oscillator deformed by the term proportional to $\delta$.
For equation (\ref{equ9}) we obtain in the analogous way
\begin{gather}\label{spect}E_n=\frac\omega2\big(\sigma(2n+1)+
\sqrt{(2l+1)^2+\tilde \kappa}\big),\end{gather} where $\tilde
\kappa=8(\kappa+1)+\sigma(\sigma+3).$
The related eigenvectors are
expressed via the confluent hypergeometric functions $\cal F$:
\begin{gather*
R_n={\rm e}^{-\frac{\omega r^{\sigma}}{2\sigma}}r^{\sigma n-\frac{E_n}{\omega}}{\cal F}\left(-n, \frac{E_n}{\sigma\omega}-n,\frac{\omega}{\sigma} r^{-\sigma}\right),
\end{gather*}
where $n$ is integer and $E_n$ is eigenvalue (\ref{spect}).
\subsection{ System with angular oscillator potential.}
The next system which we consider is specified by the inverse mass and potential presented in item~8 of the table. The corresponding Hamiltonian is:
\begin{gather*
H=p_a r^2p_a +\frac{\lambda^2}2\varphi^2+\sigma\varphi+\nu \ln(\tilde r).\end{gather*}
The corresponding eigenvalue equation is separable in cylindrical variables, thus it is reasonable to represent the wave function as follows
\begin{gather}\label{cv} \psi=\Psi(\tilde r)\Phi(\varphi)\exp(-{\rm i}kx_3).\end{gather}
As a result we obtain the following equations for radial and angular variables
\begin{gather}\label{ep6}\left(-\tilde r\p_{\tilde r}\tilde r\p_{\tilde r}-\tilde r\p_{\tilde r}+\nu\ln(\tilde r)+k^2\tilde r^2-\mu\right)\Psi(\tilde r)=0\end{gather}
and
\begin{gather}\label{ep7}\left(-\frac{\p^2}{\p \varphi^2}+\frac{\lambda^2}{2}\varphi^2+\sigma\varphi-
\mu\right)\Phi(\varphi)=0,
\end{gather}
where $\mu$ is a separation constant.
For $\lambda$ nonzero equation (\ref{ep7}) is equivalent to the Harmonic oscillator. The specificity of this system is that, in contrast with (\ref{ree}), it includes angular variable $\varphi$ whose origin is \begin{gather}\label{uhuhuh}0\leq\varphi\leq2\pi.\end{gather}
For trivial $\lambda$ our equation (\ref{ep7}) is reduced to equation with free fall potential, but again for the angular variable satisfying (\ref{uhuhuh}).
The radial equation (\ref{ep6}) is simple solvable too. In the case $k=0$ we again come to the free fall potential.
\subsection{ Systems with Morse effective potential.}
The next system we consider is specified by the inverse mass and potentials represented in item~9 of Table~1. The corresponding Hamiltonian is
\begin{gather*
H= -\frac{\p}{\p x_a}\tilde r^2{\rm e}^{\sigma \varphi}\frac{\p}{\p x_a}+\kappa
{\rm e}^{\sigma\varphi}+\frac{\omega^2}{2} {\rm e}^{-\sigma\varphi}.\end{gather*}
Introducing again the cylindric variables and representing the wave function in the form (\ref{cv}) we come to the following equations for the radial and angular variables
\begin{gather*
\left(-\left(\frac{\p^2}{ \p y^2}+\frac{\p}{ \p y}\right)+\mu+k^2{\rm e}^{2y}\right)\Psi(\tilde r)=\mu\Psi(\tilde r)\end{gather*}
and
\begin{gather}\label{olala}
\left(-{\rm e}^{\sigma\varphi}\left(\frac{\p^2}{ \p \varphi^2}+\kappa-\mu\right)
+\frac{\omega^2}{2} {\rm e}^{-\sigma\varphi}\right)\Phi(\varphi)=\tilde E\Phi(\varphi ).\end{gather}
Dividing all terms in (\ref{olala}) by $\exp({\sigma\varphi})$ we obtain the following equation:
\begin{gather*}
\left(-\left(\frac{\p^2}{ \p \varphi^2}+\kappa-\mu\right)
+\frac{\omega^2}{2} {\rm e}^{-2\sigma\varphi} \right)\Phi(\varphi)={\rm e}^{-\sigma\varphi} \tilde E\Phi(\varphi ).\end{gather*}
or
\begin{gather}\label{olalala}
\left(-\left(\frac{\p^2}{ \p \varphi^2}\right)
+\frac{\omega^2}{2} {\rm e}^{-2\sigma\varphi}-\tilde E{\rm e}^{-\sigma\varphi}\right)\Phi(\varphi )=\hat E\Phi(\varphi ),\end{gather}
where we denote $\hat E=\mu-\kappa$.
Formula (\ref{olalala}) represents the Schr\"odinger equation with Morse potential. This equation is shape invariant and also can be solved using tools of SUSY quantum mechanics. We demonstrate this procedure using another system.
Considering the mass and potential presented in item~6 of Table 1 we come to the following Hamiltonian
\begin{gather*}H=\frac12p_ax_3^{\sigma+2}p_a+\kappa x_3^\sigma.\end{gather*}
Equation (\ref{ep1}) with Hamiltonian (\ref{oj}) can be solved by separation of variables in Cartesian coordinates. Expanding the wave function $\psi$ via eigenfunctions of integrals of motion $P_1$ and $P_2$ in the form (\ref{buh}) and introducing new variable $y=\ln(x_3)$
we reduce the problem to the following equation for $\Phi(k_1,k_2,x_3)$:
\begin{gather}\label{lll}\left(-\frac{\p}{\p{x_3}}x_3^{\sigma+2}\frac{\p}{\p{x_3}}+
x_3^{\sigma+2}k^2+2\kappa x_3^\sigma\right)\Phi=2E\Phi\end{gather}
were $k^2=k_1^2+k_2^2$.
Dividing all terms in (\ref{lll}) by $x_3^\sigma$ we can rewrite it in the following form:
\begin{gather*}\left(-\frac{\p^2}{\p y^2}-(\sigma+1)\frac{\p}{\p y}-2E\exp(-\sigma y)+k^2\exp(2 y)+2\kappa\right)\Phi=0 \end{gather*}
In the particular case $\sigma=2$ we again come to the equation with Morse effective potential.
One more system which can be related to Morse potential is represented in item~7 and include the following Hamiltonian:
\begin{gather*}H=\frac12p_a\exp(\lambda\varphi)\tilde r^{\sigma+2}p_a+\nu\exp(\lambda\varphi)\tilde r^{\sigma}.\end{gather*}
The corresponding equation (\ref{ep1}) is separable in the cylindrical variables (\ref{CV}) provided $\sigma\cdot\lambda=0$ and again includes the Morse effective potential.
Let us return to equation (\ref{equ7}) and solve it using approach analogous to the presented above. In other words, we will change the roles of eigenvalues and coupling constants.
First we divide all terms in by $r^{2\sigma}$
and obtain
\begin{gather}-r^{2}\frac{\p^2R_{lm}}{\p r^2}-(2\sigma+4)r\frac{\p R_{lm}}{\p r} \nonumber\\
\quad {} +
\big(\omega^2r^{-4\sigma} +\mu r^{-2\sigma}\big)R_{lm}=\varepsilon R_{lm},\label{equq1}\end{gather} where
\begin{gather}
\label{B}\varepsilon={-l(l+1)-2\kappa}, \quad \mu=-2E.\end{gather}
Applying the Liouville transform
\begin{gather*
r\to \rho=\ln(r),\quad R_{lm}\to\tilde R_{lm}={\rm e}^{-\frac{\sigma+3}2}R_{lm}\end{gather*}
we reduce (\ref{equq1}) to a more compact form
\begin{gather}\label{equq3} H_\nu\tilde R_{lm}\equiv\left(\!-\frac{\p^2}{\p \rho^2}+\omega^2{\rm e}^{-2\sigma\rho} +
(2\omega\nu+\omega{\sigma}){\rm e}^{-\sigma\rho}\!\right) \tilde
R_{lm}=\hat\varepsilon\tilde R_{lm},\!\!\! \end{gather} where
\begin{gather}\label{A}\hat\varepsilon=\varepsilon-
\left(\frac{\sigma+3}2\right)^2 ,\quad
\nu=\frac{\mu}{2\omega}-\frac{\sigma}2.
\end{gather}
Like (\ref{olalala}) equation (\ref{equq3}) includes the familiar Morse potential and so
is shape invariant. Indeed, denoting $\mu=2\omega(\nu+\frac\sigma2)$
we can factorize hamiltonian $H_\nu$ like it was done in (\ref{eg2})
where index $l$ should be changed to $\nu$ and{\samepage
\begin{gather*}\label{w}
W=\nu-\omega{\rm e}^{-a\rho},\quad C_\nu=\nu^2
\end{gather*}
and the shape invariance is easy recognized.}
To find the admissible eigenvalues $\varepsilon$ and the
corresponding eigenvectors we can directly use the results presented
in paper~\cite{Khare}, see item~4 of Table~4.1 there
\begin{gather*
\hat\varepsilon =\hat\varepsilon_n=-(\nu-n\sigma)^2,\quad
\big(\tilde R_{lm}\big)_n=y^{\frac\nu\sigma-
n}{\rm e}^{-\frac{y}2}L_n^{2(\frac\nu\sigma-n)}
(y), \end{gather*} where $y=\frac{2\omega}\sigma
r^{-\sigma}$.
Thus we find the admissible values of $\hat\varepsilon_n$. Using
definitions (\ref{B}) and~(\ref{A}) we can find the corresponding
values of $E$ which are in perfect accordance with~(\ref{spect}).
\section{ Discussion.}
The results presented above in Section 2 include the
complete list of continuous symmetries which can be admitted by PDM
Schr\"odinger equations, provided these equations are defined up to arbitrary
parameters.
It is important to note that the list of symmetries presented in the
fourth column of the table is valid only for the case of nonzero parameters
defining the potential and mass terms. If some (or all) of these parameters
are trivial, the corresponding PDM Schr\"odinger equation can have more
extended set of symmetries. For example, it is the case for the potential
and PDM presented in item 3 of the table, compare the list of symmetries
presented in column 4 with (24). The completed list of non-equivalent
symmetries can be found in [13] which generalizes the Boyer results [3] to
the case of PDM Schr\"odinger equations. As other extensions of results
of [3] we can mention the group classification of the nonlinear Schr\"odinger
equations [15] and the analysis of its conditional symmetries [6].
Thanks to their extended symmetries the majority of the presented
systems is exactly solvable. In Sections 3 and 4 we present the corresponding
solutions explicitly and discuss supersymmetric aspects of
some of them. However, two of the presented systems (whose mass and
potential are presented in items 4 and 5 of Table 1) are not separable, if
arbitrary parameter $\kappa$ is nonzero. And just these systems
have "small" symmetry, admitting five parametrical Lie groups. For $\kappa$ equal to zero these systems are reduced to particular cases presented in items 6 and 11.
On the other hand, all systems admitting
six- or higher-dimensional Lie symmetry algebras are separable and
exactly solvable.
In addition to the symmetry under the six parameter Lie group, equation
(32) (which we call deformed 3d isotropic harmonic oscillator) possesses
a hidden dynamical symmetry w.r.t. group SO(1, 2). The effective
radial Hamiltonian is shape invariant, and its eigenvalues can be found
algebraically. In spite on the qualitative difference of its spectra (37)
and (38) of the standard 3d oscillator, it keeps the main supersymmetric
properties of the latter. We note that the shape invariance of PDM
problems usually attends their extended symmetries.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,226 |
Krvavi meridijan ili Večernja rumen na zapadu () je epski vestern( ili anti-vestern<ref>Hage, Erik. Cormac McCarthy: A Literary Companion. North Carolina: 2010. p. 45</ref>) roman Cormaca McCarthya objavljen 1985. godine. Roman je utemeljen na stvarnim događajima iz 19. stoljeća, i prati grupu lovaca na skalpove predvođenu Sucem Holdenom tijekom njihovih masakra Indijanaca i ostalih ljudi na području granice Meksika i SAD-a. Dok u početku to rade zbog nagrada od tjeralica, kako se radnja razvija motiv njihove okrutnosti sve je teže razumjeti.
Iako su prve recenzije bile osrednje, Krvavi meridijan'' se danas smatra McCarthyevim remek-djelom. Časopis Time ga je stavio na popis najboljih romana na engleskom jeziku iz razdoblja 1923. – 2005.
Izvori
Američka književnost | {
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} | 1,199 |
'use strict';
describe('buildMonitor', function () {
describe('stats', function () {
var stats,
googleAnalyticsBackend,
$timeout,
$window,
sandbox,
interval = seconds(10);
beforeEach(function() {
sandbox = sinon.sandbox.create();
module('buildMonitor.stats', function(statsProvider) {
statsProvider.configure({ flushIntervalInSeconds: interval });
});
inject(function (_stats_, _googleAnalyticsBackend_, _$timeout_, _$window_) {
stats = _stats_;
googleAnalyticsBackend = _googleAnalyticsBackend_;
$timeout = _$timeout_;
$window = _$window_;
});
});
afterEach(function () { sandbox.restore(); });
it ('does not flush the metrics if there are none recorded', function () {
googleAnalyticsBackend.flush = sandbox.spy();
$timeout.flush();
expect(googleAnalyticsBackend.flush).not.toHaveBeenCalled();
});
it ('flushes the timers after a "flushInterval" seconds', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
$timeout.flush(seconds(10));
expect(googleAnalyticsBackend.flush).toHaveBeenCalledWith({
'category name': {
'timer name': {
upper: 250,
lower: 250,
mean: 250
}
}
});
});
it ('removes recorded metrics after the flush', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
$timeout.flush(interval);
$timeout.flush(interval);
expect(googleAnalyticsBackend.flush).toHaveBeenCalledOnce();
});
it ('starts fresh with every interval', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
$timeout.flush(interval);
stats.timer('category name', 'timer name', 475);
$timeout.flush(interval);
expect(googleAnalyticsBackend.flush).toHaveBeenCalledWith({
'category name': {
'timer name': {
lower: 475,
upper: 475,
mean: 475
}
}
});
});
it ('accumulates metrics within an interval', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
stats.timer('category name', 'timer name', 350);
stats.timer('category name', 'timer name', 425);
stats.timer('category name', 'timer name', 167);
$timeout.flush(interval);
expect(googleAnalyticsBackend.flush).toHaveBeenCalledWith({
'category name': {
'timer name': {
lower: 167,
upper: 425,
mean: 298
}
}
});
});
it ('accumulates metrics within a category', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
stats.timer('second category name', 'timer name', 350);
stats.timer('second category name', 'timer name', 425);
stats.timer('category name', 'timer name', 167);
$timeout.flush(interval);
expect(googleAnalyticsBackend.flush).toHaveBeenCalledWith({
'category name': {
'timer name': {
lower: 167,
upper: 250,
mean: 209
}
},
'second category name': {
'timer name': {
lower: 350,
upper: 425,
mean: 388
}
}
});
});
it ('accumulates metrics per parameter', function () {
googleAnalyticsBackend.flush = sandbox.spy();
stats.timer('category name', 'timer name', 250);
stats.timer('category name', 'second timer name', 350);
stats.timer('category name', 'timer name', 425);
stats.timer('category name', 'second timer name', 167);
$timeout.flush(interval);
expect(googleAnalyticsBackend.flush).toHaveBeenCalledWith({
'category name': {
'timer name': {
lower: 250,
upper: 425,
mean: 338
},
'second timer name': {
lower: 167,
upper: 350,
mean: 259
}
}
});
});
describe('integration with google analytics script', function () {
it ('works for a single timer', function () {
$window.ga = sandbox.spy(); // let's pretend we've got google analytics script loaded
stats.timer('my timing category', 'timer name', 315);
$timeout.flush(interval);
expect(ga).toHaveBeenCalledWith('send', 'timing', 'my timing category', 'timer name lower', 315);
expect(ga).toHaveBeenCalledWith('send', 'timing', 'my timing category', 'timer name upper', 315);
expect(ga).toHaveBeenCalledWith('send', 'timing', 'my timing category', 'timer name mean', 315);
});
it ('works for a timers from different categories', function () {
$window.ga = sandbox.spy(); // let's pretend we've got the google analytics script loaded
stats.timer('my timing category 1', 'timer', 250);
stats.timer('my timing category 1', 'timer', 251);
stats.timer('my timing category 1', 'timer', 254);
stats.timer('my timing category 1', 'timer', 252);
stats.timer('my timing category 1', 'timer', 267);
stats.timer('my timing category 1', 'timer', 221);
stats.timer('my timing category 1', 'timer', 260);
stats.timer('my timing category 1', 'timer', 243);
stats.timer('my timing category 1', 'timer', 215);
stats.timer('my timing category 1', 'timer', 237);
stats.timer('my timing category 1', 'timer', 280);
stats.timer('my timing category 2', 'second timer', 350);
stats.timer('my timing category 3', 'third timer', 167);
$timeout.flush(interval);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 1', 'timer lower', 215);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 1', 'timer upper', 280);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 1', 'timer mean', 248);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 2', 'second timer lower', 350);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 2', 'second timer upper', 350);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 2', 'second timer mean', 350);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 3', 'third timer lower', 167);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 3', 'third timer upper', 167);
expect($window.ga).toHaveBeenCalledWith('send', 'timing', 'my timing category 3', 'third timer mean', 167);
});
});
function seconds(number) {
return number * 1000;
}
});
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,235 |
Demon's Mark (Hell Unleashed Series)
FREE on Kindle Unlimited.
DEMON'S MARK
As a half-demon, Cary has survived a lot... but this love thing might be her undoing.
Levi Walker, master demon hunter, is on a mission to find the demon bastard who killed his best friend. As he finally tracks down the beast, he's suddenly distracted by the newest Argos recruit - the sexy vixen who also broke his heart - and the battle takes a deadly turn. Now with the recruit in the beast's sights, both she and Levi are instantly marked for death.
If Cary Stone's secret is revealed (she's a demon half-breed, after all), she'll be thrown straight into Hell. After a lifetime following her father's advice - never let anyone get close, never stay too long in one city, and ALWAYS work alone - in the past year she's broken all the rules.
Not only did she get a job hunting demons, she also fell in love with the most badass hunter of them all. Oh and, if things weren't interesting enough, she managed to grab the top spot on the hit list of the biggest mercenary in Hell. Yeah, shit just got real.
Please note: Demon's Fever has been previously published in the boxed set - Dark Legends.
On sale at Amazon. CLICK TO GET YOUR COPY.
WANT MORE SEXY PARANORMAL STORIES?
SEE MY COMPLETE BOOKLIST HERE: T.F. WALSH'S BOOKS
Sign up to my newsletter for latest releases in the series: T.F. Walsh Newsletter | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 92 |
from django.contrib import admin
from core.models import EventPageContent
from sponsor.models import Sponsor
class SponsorInline(admin.TabularInline):
model = EventPageContent.sponsors.through
extra = 1
verbose_name_plural = 'Sponsors'
class SponsorAdmin(admin.ModelAdmin):
list_display = ('id', 'name', 'logo_display_for_admin', 'url')
list_per_page = 50
search_fields = ('name', )
def get_queryset(self, request):
qs = super(SponsorAdmin, self).get_queryset(request)
if request.user.is_superuser:
return qs
return qs.filter(eventpagecontent__event__team=request.user).distinct()
def get_form(self, request, obj=None, **kwargs):
form = super(SponsorAdmin, self).get_form(request, obj, **kwargs)
if not request.user.is_superuser:
if 'eventpagecontent' in form.base_fields:
qs = EventPageContent.objects.filter(event__team=request.user)
form.base_fields['eventpagecontent'].queryset = qs
return form
admin.site.register(Sponsor, SponsorAdmin)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,640 |
Report: Georgia assistant coaches could be entertaining moving on
Stephen Samra•3 days
Article written by: Stephen Samra
Steve_S13
Jeffrey Vest/Icon Sportswire via Getty Images
Georgia finally ended their national championship drought on Monday, and Kirby Smart and his staff will forever be known as the coaches who led the charge. Now, a pair of Bulldogs assistants could have their eyes on moving elsewhere.
According to FOX's Bruce Feldman, Georgia could lose offensive coordinator Todd Monken to the NFL. Additionally, defensive backs coach Jahmile Addae could also be leaving the program.
"A couple of potential staff moves to keep an eye on at Georgia: OC Todd Monken may return to the NFL and DBs coach Jahmile Addae also could be moving on, per sources," revealed Feldman.
First, Monken has a bevy of NFL experience under his belt already. Prior to joining Smart at Georgia, the offensive coordinator coached offense for the Jacksonville Jaguars, Tampa Bay Buccaneers and Cleveland Browns. With many head coaching positions open this offseason in the league, a newly-hired coach could look toward Monken to call their offense.
Additionally, Addae has a wealth of experience coaching defensive backs. The former West Virginia star began his coaching career as a graduate assistant with the Mountaineers, then made stops at Michigan, Cincinnati, Arizona and Minnesota. Finally, he returned to the Mountaineers as their defensive backs coach in 2019, and joined Smart's staff in 2021.
As you can see, both coaches will be in-demand. In addition to the pair of coaches, Smart is already losing his defensive coordinator, as Dan Lanning is becoming the head coach of Oregon.
The Bulldogs will enjoy their championship, but Kirby Smart has to rework his coaching staff if the program is looking to win back to back titles.
Smart on replacing players, coaches
As you can see, Kirby Smart has a lot on his plate this off-season. While he's enjoying Georgia's first title in forty years, the head coach talked about roster turnover not only with players turning professional, but also the transfer portal and the Bulldogs' incoming recruiting class.
"It's right now. We'll have multiple kids. Probably we'll have a few kids go into the portal that we've had discussions with that held that off, out of respect for the team. I thought that was tremendous. We'll have that," Smart said Tuesday morning. "Everybody will think that's a terrible thing. But some cases it's actually a good opportunity for kids. We've got 16 or 17 mid-years that have enrolled. We've got to finish off this class. We don't know how many spots we have because we don't know what some guys are doing, like Lewis and other guys. There's a lot of work to be done."
Kirby Smart went on to talk about how every other team in the country, besides Alabama, already has a leg up on Georgia going into next year. Most teams saw their season end by Jan. 1 with just a one-off bowl game. However, he credited Nick Saban, saying he learned how to deal with the turnover from him.
"And I know people don't understand it, but other teams are 10 days up on you because they finished January 1st, your competitors. And you're going, wait, they're doing what? We're trying to win a national championship," Smart continued. "Fortunately, I had a really good mentor, teacher of dealing with that because he's in it every year. But you fall behind because you're trying to manage your roster. My biggest concern in terms of complacency is the guys coming back. We'll lose some really good players on defense. We've got a lot of guys coming back on offense. But we'll have to replace some good talented backs as well." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,863 |
Man pleads guilty in case connected to gun theft
Published 6:19 pm Tuesday, June 14, 2022
By Eric Johnson
A man who was arrested and who allegedly was involved in a case of stolen guns has made his pleas in the case in Mower County District Court Monday.
Bradley Blaine Beckstead, 31, of Blooming Prairie, has pleaded not guilty to three counts of felony violent felon in possession of a firearm and one misdemeanor for driving after cancellation.
A jury trial has been scheduled for Nov. 14, 2022.
Beckstead was arrested earlier this year after he was found in possession of gun from a case involving April Taylor Arends, 27, of Austin, who has been charged with 12 felony counts of burglary and theft and one gross misdemeanor count of theft and who Beckstead had been dating.
One of the guns was found in the center console of a rented vehicle driven by Beckstead. Beckstead allegedly admitted to an Austin Police Detective that he drove the car, but was unaware of the burglary Arends was charged in.
During a search of Arends' apartment, her phone and tablet were seized. An instant message conversation had been discovered between Arends and Beckstead on the afternoon of March 2, 2022, after the rental car was seized by law enforcement.
In the exchange Beckstead tells Arends, "the ar's in the car, I'm walking and I'm just hoping they don't look." Later in the conversation Beckstead messaged Arends and said "So if they figure out your story was a lie you gotta come clean cry and tell them it was your and you didn't know."
Beckstead has a lengthy criminal history including convictions for felony threats of violence in July of 2021, fifth degree drug possession in September of 2020, fifth degree assault in April of 2015 and domestic assault in August of 2012. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,306 |
{"url":"https:\/\/gamedev.stackexchange.com\/questions\/23652\/what-alternatives-to-glut-exist\/23660","text":"# What alternatives to GLUT exist?\n\nI am trying to learn OpenGL, and I just found out that GLUT is obsolete. I already know SDL, and it seems it is a good alternative. Should I use SDL to develop games with OpenGL, or are there any better alternatives. I am new to game development, so I don't know much about the state of the art.\n\nI just found out that GLUT is obsolete.\n\nThat's what happens when you accept the first answer you come across; you get bad information. GLUT 3.7 shouldn't be used, but FreeGLUT is completely backwards compatible with it.\n\nHowever, your question goes elsewhere. For making an actual game, GLUT of * form is inappropriate. It doesn't give you control of the main loop, and you really need that in a game (FreeGLUT does have a way to control the main loop, but it's still odd). GLUT is for writing graphics demos. That's very important when testing new graphical effects, so it should be there in your toolbox when needed. But your main game code should never use it.\n\nGLFW is a game-centric lightweight alternative to GLUT. It provides basic support for things beyond creating an OpenGL window. It lets you get input, load images as textures, and a couple of other things.\n\nSDL, SFML, and Allegro 5 are all multimedia toolkits. They can create OpenGL windows, but they do a lot more than that. They provide full support for input, audio, and various other stuff you need to make a game. If you're making a game, any of these are a good starting point. I personally like Allegro 5, but that's just my bias towards good documentation and a clean API.\n\n\u2022 What does the AAA titles(both pc only and multi-platform) uses? Do they use these(GLFW, Allegro) that you've mentioned, for handling opengl window? or they write their own 'everything' from the scratch? \u2013\u00a0Quazi Irfan Feb 10 '12 at 17:45\n\u2022 @iamcreasy: \"AAA titles\" (I really hate that term) don't use any of these. In general, they don't use anything open-source. They could, but if they're going to use a library, they'd much rather use a library that is supported. There are some exceptions like ODL or Bullet. But those are exceptions rather than the rule. \u2013\u00a0Nicol Bolas Feb 10 '12 at 17:55\n\u2022 Since when does GLFW 'load images as textures'? Definitely isn't in the feature list; nor any functions that remotely serve that purpose in my glfw header. \u2013\u00a0deceleratedcaviar Feb 10 '12 at 18:05\n\u2022 The other big downside to GLUT is that its input model isn't particularly well-suited to games; there's no way to get raw keycodes or to get press\/release events on the modifier keys themselves (e.g. using ctrl as a fire button), nor is there a way to get the underlying GL context (which is necessary to enable vsync on X11, for example). \u2013\u00a0fluffy Feb 10 '12 at 18:56\n\u2022 @NicolBolas ah, I forgot about 2.7, they [threads, images] have been declared 'out of context' for the GLFW 3.0 release. \u2013\u00a0deceleratedcaviar Feb 10 '12 at 21:38\n\nhttp:\/\/www.opengl.org\/resources\/libraries\/windowtoolkits\/\n\nTo that i would also add SFML:\n\nhttp:\/\/www.sfml-dev.org\/index.php\n\n### If your goal is to develop a game,\n\nIt depends on whether you want your game to be cross-platform or not. For portable (Windows, Mac, Linux) games SDL seems to be the only choice.\n\nIf you only target Windows and\/or XBox, there are much better alternatives, e. g. XNA which frees you from writing a lot of code manually.\n\n### If your goal is to learn OpenGL,\n\nJust program in raw OpenGL, but be prepared to face a lot of unintuitive concepts, math and huge manuals.\n\n\u2022 As Takumi said, there's also SFML as a portable library. Better than SDL imho (it's more up-to-date.) \u2013\u00a0kaoD Feb 10 '12 at 15:26\n\u2022 -1: \"For portable (Windows, Mac, Linux) games SDL seems to be the only choice.\" This is not even remotely true. SFML and Allegro 5 both exist as perfectly valid alternatives, and both have wider platform bases. \u2013\u00a0Nicol Bolas Feb 10 '12 at 16:22\n\nGLUT is an API as well as a library. The original library is indeed obsolete. The API is a different thing, it was pretty good to begin, it just needs revisions for overcoming limitations and addressing new needs. The primary current implementation is FreeGLUT, which is supported and not obsolete. That means that a good alternative to GLUT (as in the library GLUT) is GLUT (as in FreeGLUT).\n\nThe problem noted above with \"control of the main loop\" is incorrect. With newer GLUT versions you can have better control over your game loop (if that is important to you). The limitation was fixed by glutMainLoopEvent(). This and other limitations are addressed here:\n\nhttp:\/\/en.wikipedia.org\/wiki\/OpenGL_Utility_Toolkit\n\n\u2022 \"With newer GLUT versions you can have better control over your game loop (if that is important to you).\" It's a game, so yes, that's important to you. Also, glutMainLoopEvent is hardly a good way to go about managing the main loop. It still uses an event-driven model for feeding you data, rather than a polling model where you get the data you want when you want it. \u2013\u00a0Nicol Bolas Apr 1 '13 at 10:39\n\u2022 I must argue that the event driven model is more efficient than polling, unless you are polling into data stored from the events (which is just fine). Polling system resources should be done carefully. \u2013\u00a0Ingemar Ragnemalm Jul 30 '17 at 12:58\n\nQt does support OpenGL. Take a look to Open Inventor and consider if make it sense for you to go straight to an higher level 3D API.","date":"2020-01-27 10:53:23","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\": 0.21960461139678955, \"perplexity\": 2110.0828627618557}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251696046.73\/warc\/CC-MAIN-20200127081933-20200127111933-00359.warc.gz\"}"} | null | null |
Good Day Philadelphia at 6A
Pat Toomey
Women Vote!
Good Day Philadelphia at 6A : WTXF : July 5, 2016 6:00am-7:00am EDT
not be sitting here. so i'm outraged that pat toomey voted to defund planned parenthood... ...which thousands of pennsylvania women depend on for cancer screenings. pat toomey was even willing to shut down the federal government to eliminate funding for planned parenthood. shut down the government over planned parenthood? i think we ought to shut down pat toomey. women vote is responsible for the content of this advertising. (male off-screen announcer) what's it feel like to win the mustang instant game,
not be sitting here. so i'm outraged that pat toomey voted to defund planned parenthood... ...which thousands of pennsylvania women depend on for cancer screenings. pat toomey was even willing to shut down the federal government to eliminate funding for planned parenthood. shut down the government over planned parenthood? i think we ought to shut down pat toomey. women vote is responsible for the content of this advertising. (male off-screen announcer) what's it feel like to win the mustang... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 248 |
85' - Corner - Watford. Conceded by Marcus Bettinelli.
83' - Offside - Watford. José Holebas with a pass, however Roberto Pereyra is in offside.
80' - Offside - Watford. Ben Foster with a pass, however Isaac Success is in offside.
80' - Substitution - Watford. Isaac Success for Andre Gray.
79' - Craig Cathcart - Watford - won a free kick in defence.
78' - GOAL!!! Fulham 1, Watford 1. Aleksandar Mitrovic - Fulham - shot with right foot inside of six yard box - left side to the centre of the goal. Assist - Luciano Vietto.
78' - Shot blocked. André Schürrle - Fulham - shot with right foot from outside the box is blocked. Assist - Jean Michael Seri.
75' - Delay in match Ryan Sessegnon - Fulham - - injury.
74' - Corner - Watford. Conceded by Timothy Fosu-Mensah.
73' - Shot blocked. Calum Chambers - Fulham - shot with right foot from the left side of the box is blocked. Assist - André-Frank Zambo Anguissa.
73' - Missed chance. André-Frank Zambo Anguissa - Fulham - shot with right foot from outside the box missed to the left after corner.
72' - Corner - Fulham. Conceded by Kiko Femenía.
72' - Shot blocked. André Schürrle - Fulham - shot with right foot from outside the box is blocked. Assist - Jean Michael Seri.
70' - Substitution - Watford. Kiko Femenía for Will Hughes.
69' - Troy Deeney - Watford - won a free kick in defence.
67' - Offside - Fulham. Timothy Fosu-Mensah with a pass, however Aleksandar Mitrovic is in offside.
66' - Missed chance. Aleksandar Mitrovic - Fulham - shot with the head from the centre of the box missed. Assist - Jean Michael Seri with a cross after corner.
66' - Corner - Fulham. Conceded by Christian Kabasele.
65' - Missed chance. Troy Deeney - Watford - shot with right foot from the centre of the box missed to the left. Assist - Will Hughes.
64' - Substitution - Fulham. André-Frank Zambo Anguissa for Stefan Johansen.
63' - Shot blocked. Aleksandar Mitrovic - Fulham - shot with right foot from outside the box is blocked. Assist - André Schürrle.
63' - Missed chance. José Holebas - Watford - shot with right foot from outside the box missed to the left.
61' - Corner - Fulham. Conceded by Daryl Janmaat.
61' - Shot blocked. Aleksandar Mitrovic - Fulham - shot with the head from the centre of the box is blocked. Assist - Timothy Fosu-Mensah with a cross.
59' - New attacking attempt. Andre Gray - Watford - shot with right foot from the right side of the box is saved by goalkeeper in the centre of the goal. Assist - Abdoulaye Doucouré.
58' - José Holebas - Watford - receive yellow card for a foul.
58' - Luciano Vietto - Fulham - won a free kick in defence.
57' - Will Hughes - Watford - won a free kick in attack.
56' - Ryan Sessegnon - Fulham - won a free kick in defence.
53' - Aleksandar Mitrovic - Fulham - won a free kick in defence.
52' - Missed chance. Luciano Vietto - Fulham - shot with right foot from outside the box missed. Assist - Aleksandar Mitrovic.
51' - Floyd Ayité - Fulham - won a free kick in defence.
50' - Floyd Ayité - Fulham - won a free kick in defence.
49' - Missed chance. Aleksandar Mitrovic - Fulham - shot with the head from the centre of the box goes high. Assist - André Schürrle with a cross.
48' - Offside - Fulham. Marcus Bettinelli with a pass, however Floyd Ayité is in offside.
45' - Substitution - Fulham. Floyd Ayité for Kevin McDonald.
45' - Substitution - Fulham. Denis Odoi for Alfie Mawson.
45' - Second Half starts Fulham 0, Watford 1.
45'+2' - First Half ended - Fulham 0, Watford 1.
45'+2' - Offside - Watford. Christian Kabasele with a pass, however Andre Gray is in offside.
45'+1' - Andre Gray - Watford - won a free kick on the right wing.
45' - Missed chance. Roberto Pereyra - Watford - shot with right foot from the left side of the box is close, but missed. Assist - Ben Foster.
44' - Corner - Fulham. Conceded by Daryl Janmaat.
44' - Luciano Vietto - Fulham - won a free kick in attack.
42' - Troy Deeney - Watford - won a free kick on the left wing.
40' - Corner - Fulham. Conceded by Craig Cathcart.
39' - Offside - Watford. Will Hughes with a pass, however Andre Gray is in offside.
37' - Corner - Watford. Conceded by Calum Chambers.
34' - New attacking attempt. Abdoulaye Doucouré - Watford - shot with left foot from outside the box is saved by goalkeeper in the centre of the goal.
33' - Corner - Watford. Conceded by Timothy Fosu-Mensah.
32' - Timothy Fosu-Mensah - Fulham - receive yellow card for a foul.
32' - Troy Deeney - Watford - won a free kick in attack.
31' - Offside - Fulham. Kevin McDonald with a pass, however Aleksandar Mitrovic is in offside.
31' - Jean Michael Seri - Fulham - won a free kick in defence.
30' - Alfie Mawson - Fulham - receive yellow card for a foul.
29' - Roberto Pereyra - Watford - won a free kick on the left wing.
28' - Will Hughes - Watford - won a free kick on the right wing.
25' - Shot blocked. Aleksandar Mitrovic - Fulham - shot with right foot from the centre of the box is blocked. Assist - Timothy Fosu-Mensah with a through ball.
22' - Corner - Fulham. Conceded by Craig Cathcart.
22' - Ryan Sessegnon - Fulham - won a free kick on the left wing.
18' - Offside - Watford. Will Hughes with a pass, however Andre Gray is in offside.
17' - Corner - Watford. Conceded by Marcus Bettinelli.
17' - New attacking attempt. Andre Gray - Watford - shot with right foot from the centre of the box is saved by goalkeeper in the centre of the goal.
16' - José Holebas - Watford - won a free kick on the left wing.
15' - Offside - Watford. Roberto Pereyra with a pass, however Andre Gray is in offside.
14' - Shot blocked. Christian Kabasele - Watford - shot with the head from the centre of the box is blocked. Assist - José Holebas with a cross.
13' - Corner - Watford. Conceded by Marcus Bettinelli.
13' - New attacking attempt. Troy Deeney - Watford - shot with right foot from the right side of the six yard box is saved by goalkeeper in the centre of the goal.
12' - Missed chance. Luciano Vietto - Fulham - shot with right foot from the left side of the box missed following a fast break.
12' - New attacking attempt. Luciano Vietto - Fulham - shot with left foot from the left side of the box is saved by goalkeeper in the centre of the goal. Assist - André Schürrle.
7' - Corner - Watford. Conceded by Stefan Johansen.
7' - Corner - Watford. Conceded by Alfie Mawson.
6' - Roberto Pereyra - Watford - won a free kick on the left wing.
3' - André Schürrle - Fulham - won a free kick in defence.
2' - GOAL!!! Fulham 0, Watford 1. Andre Gray - Watford - shot with right foot from the centre of the box to the left corner. Assist - Will Hughes. | {
"redpajama_set_name": "RedPajamaC4"
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Sign up for New Product News!
Excitement is building around the Traxxas booth at SEMA, the largest aftermarket automotive trade show in the world, taking place in Las Vegas. This week, Traxxas is unveiling several new product releases to the attendees of the show. Take a look at some of the products coming soon to a hobby store near you.
Justin Lofton's #41 Fox trophy truck will be featured this year in the Traxxas booth for show-goers to see first-hand. With two consecutive Mint 400 championships in 2015 and 2016, Lofton has made an impressive mark on the competitive world of desert racing. Some notable specs on the #41 Fox truck are: 890-horsepower, Fox Factory shocks, front/rear wheel travel 24"/32", BFGoodrich 40" KR'3 tires, and a 150+mph top speed.
Traxxas has replicated Lofton's #41 Fox trophy truck in stunning Pro-Scale realism with the Traxxas Unlimited Desert Racer. This innovative high performance radio controlled racer mirrors the huge suspension travel; plush, multi-shock damping, and extreme horsepower with intense accuracy. The Traxxas Unlimited Desert Racer will be on display alongside the Fox Shox Trophy Truck this year at SEMA, October 30 to November 2, 2018 in Las Vegas, NV.
Ford fans already have the amazing Ford GT and Mustang GT bodies for the 4-Tec 2.0 racing chassis. Now it's time for Chevy fans to get excited. Traxxas debuts these stunning Cadillac CTS-V and Corvette Z06 bodies for your 4-Tec 2.0 chassis. CTS-V bodies will be available in blue or silver. Corvette Z06 bodies will come in red or blue. Both bodies are available fully painted with highly detailed injection molded mirrors and spoilers. For those wanting custom creations, clear bodies will follow shortly.
2019 will be a big year for crawling fans. The TRX-4 Sport shows off some highly detailed scale accessories at SEMA. From roof racks to light kits, the TRX-4 Sport makes a statement out on the trail. Eagle-eyed viewers will also spot a number of performance parts for all TRX-4 scale and trail crawlers. Aluminum portal housings, wheel weights, and other high-performance accessories will be popular choice for dedicated RC crawling fans.
Already the most recognized name in monster trucks, BIGFOOT® No. 1 debuts a new flame-paint body at SEMA this year. The RTR truck includes all the scale details of the original such as chrome bumpers and grille, high-rise carburetor, roll bars, deep-dish wheels, and replica lights. This wild new body will be a hit whether you're catching air, crushing cars, or ripping roost in the backyard.
The recently-released Rustler 4X4 has already rewritten the rulebook on stadium truck performance. Soon, a full line of color options will allow you to customize your Rustler 4X4 to make the ultimate off-road statement. Look for these high-performance accessories on the parts wall of your nearest hobby shop in the coming months.
Literally the biggest thing ever to come from Traxxas, X-Maxx defies physics with its jaw-dropping acceleration and 50+mph speed. The new orange and black X-Maxx adds another eye-catching option to the powerful X-Maxx lineup.
Subscribe now for the latest news and updates from Traxxas.
"BIGFOOT"® and "The Original Monster Truck"® are registered trademarks of BIGFOOT 4X4, Inc. FIRESTONE and the FIRESTONE logo are trademarks of Bridgestone Brands, LLC and Bridgestone Licensing Services, Inc.
©2017 BIGFOOT 4X4, Inc., 2286 Rose Lane, Pacific, MO 63069 USA Bigfoot4x4.com. All rights reserved. | {
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Hearts & Minds Books
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MY FAVORITE 10 BOOKS of 2022 — ALL 20% OFF
Posted on January 12, 2023 by Byron Borger
It often happens this time of year. My head is nearly spinning, giddy with the thought of sharing with you our picks for the Best Books of 2022. Christmas is always hectic here in retail-land and then the new year is often (thank goodness) a bit more hectic than we expect or remember. We're so glad folks are sending us interesting requests for us to weigh in on, asking for book ideas for their winter programing, for their reconvened book clubs, for their upcoming preaching season, for their classes, for their own personal reading plans. It's just a busy time of year to be creating this momentous post.
Which then makes me ponder about the point of all the list-making and honorable mentioning. As a struggling bookseller, I'll admit: our goal is to persuade you to buy books. From us. Obviously.
And yet, there is some altruistic motivation, too, an educational offer for the common good. We really do want to inform the reading public and (insofar as they notice or care) honor publishers and their authors who do good work. There's no real awards show or prize money, of course, so our little hoopla goes mostly unnoticed.
Yet, our fans and friends and followers want to know what we think. I am so honored by that. We thank you for caring.
So, here is a righteous shout-out to a handful of really good books. My top ten.
Here's my caveat, offered as bluntly as I can put it. I am not insisting these are the "best" books, whatever that may mean. Heaven knows, I am not a judge of that.
But they are books I loved. Books I think others should read. Books we want to honor.
Soon, we'll do a second list of more titles and authors that we consider the cream of the crop of 2022; the best of the best, in the literary world that I know, at least. I can't wait to list those for you.
But here, today, I want to celebrate my choices for my own favorite books of the year. These are the ones I most enjoyed and that I think are worthy of being on a year's end recommended reading list. These achieved that sweet spot of being delightfully written, artful and entertaining, and important, with something vital to say. These are my own choices for my favorite (nonfiction) books of 2022. I commend them all, strongly so. Each is a masterpiece that gave me many meaningful hours with these good volumes in my lap. I hope you order something from this list today.
The ORDER LINK is at the very bottom of this column… don't forget to scroll down the whole way.
MY VERY FAVORITE BOOKS OF 2022
Wastelands: The True Story of Farm Country on Trial Corban Addison (Knopf) $30.00 OUR SALE PRICE = $24.00
This is one of the most dramatic and well written books I have ever read. Like a novel (and I've read this author's fiction) it has such lush description and well crafted sentences — it's a beauty to behold. I have to say this is the book I enjoyed the most all year and I highly recommend it to anyone who likes vivid, energetic prose and one heckuva story.
Yet, it is hard to say I "liked" it because it is horrendous, deeply so. It is about a major, years long, lawsuit against the CAFO (Concentrated Animal Feeding Operation) hog operations in North Carolina and the extraordinary lengths the fat cats at Smithfield (now owned by a communist Chinese business tycoon) and their acolytes in the North Carolina legislature went to fight this suit.
Along the way you meet the folks — mostly Black — who inherited land generations previous and their hopes for a quiet, agrarian life. Most are church-goers and since the author himself is a Christian (I know from his other work) he seems particularly aware of the sustenance and value simple country folks get from their small churches. Drawing on faith and hope and a good bit of love they take their stand against the sickening odors and disease from these shit-filled facilities.
(One learns quite a bit about the extraordinary amount of refuse that comes from packing tens of thousands of hogs in a small space — more waste than a small city would handle, but devoid of regulation. The pollution is gross and offensive with foul lagoons overflowing, trucks that carry out dead hogs disrupting the quiet nights, the spray method — spraying the excrement into the air — beyond ridiculous; cruel. Other proven methods less harmful to public health would cut into Smithfield's profits a bit, so were obviously tabled time and again. The public officials tasked with protecting people simply ignored the issues. The EPA was nowhere to be found, in any event, useless against the war chests of big meat.)
Faithful followers of Christ or not — and many seem to be — the heroes of this story (besides the dignified homeowners who protested, and are forever hated by many of their neighbors) are the valiant lawyers from a small practice in North Carolina (who recruited a passionate public interest attorney from Colorado) and their teams who are on the side of the angels. Over and over and over they face obstacles, including threats of intimidations and violence, by those who start up these huge industrial hog facilities (I will not call them farms; one does not have to be Wendell Berry to see the utter disregard for and even disdain of traditional agriculture from these CAFO billionaires.) This is the wildest David and Goliath story I have ever read.
The great foreword is by Baptist bestseller, the one and only John Grisham. He lamented that he hadn't made up a story this thrilling or told one so well as this. It is remarkable praise and after reading Grisham's foreword I was hooked. If you like his legal fiction (or movies like The Pelican Brief or the iconic Erin Brockovich) you have to get this book immediately. You won't be able to stop turning pages, believe me. Who knows, you may end up wanting to go to law school to take up public interest advocacy.
The history of black property owners is explored. The rise of the meat industry and the big slaughterhouses are explained. There's some fascinating studies of the science of aroma (and the bogus study of smells that the high-priced, pseudo-scientist Smithfield put on the stand.) From pondering legal jurisprudence to the inside look at a mid-size law firm (these are not the big shots from "The Good Fight or other such big city practices) to the toll stressful cases take on mental health and relationships — again the case took years and years with millions of dollars of expenses and multi-millions at stake — Wastelands is so informative. Anybody interested in lawyering or legal practice has to read it. The details of jury selection and court process and how an opening argument is crafted (and rehearsed) and the complexities of cross examination and — yes — appeals are all fascinating, written with expert detail but colorfully textured that, again, is like an unfolding novel. Beside the turn-paging plot and the struggle for justice (one could hardly make up such corrupt bad guys), the telling is gripping. I've said before that Corbin Addison is one heck of a storyteller and an artful writer.
As Grisham himself puts it, Wastelands is:
Beautifully written, impeccably researched, and told with the air of suspense that few writers can handle…
This absorbing book that evokes thrills and emotions and makes you think about so very much will, in the words of the remarkable nonfiction master Wilbur Smith, "hold you spellbound with his elegant prose from his first word to his last."
Jonathan Harr, author of the best-seller A Civil Action, says:
In this book, Addison turns a novelist's eye to the thorny complexities of a real legal case. The prose is lyrical, the cast of characters jump to life on the page, and the result is a captivating account of how a small group of citizens bring a huge corporation to justice.
A few more things to be aware of, things that make this even more page-turning and so very important, given how our democracy is these days.
There is evidence here — explicitly documenting and powerfully exposing — a propaganda campaign on the part of the meatpackers at Smithfield. They spent millions airing sweet TV footage (and creating billboards) of family farms with bucolic scenes in lovely rural villages (and their smiling children) all the while implying that the litigants — who have hog excrement (from the CAFO's spraying methods) on their laundry lines and cannot stand having a picnic or worship service from the affront of the odor and presence of the CAFO's waste — hate farmers, hate meat, hate bacon. What a batch of lies, these industrial hog-facilities portraying themselves as quaint rural farmers with traditional (Christian) folkways. And people fell for it, believing the litigants were liberal leftists who are trying to stop farmers and American business and small farmers. That the lawyers were just in it for the money. This PR campaign was insidious and despicable but it framed the lawsuits in a certain (utterly untrue) light. It is an ugly part of the book even though Addison doesn't dwell on it.
Secondly, the ungodly relationship between government and business was so odd that even super strict conservative constitutional scholars opposed the machinations of the North Carolina House, if to no effect. We've seen shenanigans in our own State House and we all know what goes on in DC sometimes. This is the most egregious move to pass legislation that would prop up an industry under fire of which I know, and I've studied this sort of injustice a bit. Man. Read it and weep, especially if you live in North Carolina. Those who voted to protect Smithfield from litigation should be ashamed of themselves.
Thirdly, although it isn't directly an overt part of this story, there is hovering around this plot the question of how we eat, what sort of farming practices we want to encourage, and how to reform large scale agriculture. (For another expose of the meat industry as such, see Raw Deal: Hidden Corruption, Corporate Greed, and the Fight for the Future of Meat by Chloe Sorvino.) Insofar as the Smithfield-led CAFO industry was forced to grapple with their irresponsible business practices which clearly harm the air and Earth and neighborhoods (not to mention abusive of the animals) there is some modicum of reform. But the bigger questions this raises are themselves huge.
Wastelands brilliantly in captivating detail offers an important investigation, creating a truly great story, making this a rare, exceptional book. I am happy to name it as one of the Very Best of 2022.
South to America: A Journey Below the Mason-Dixon to Understand the Soul of a Nation Imani Perry (Ecco) $28.99 OUR SALE PRICE = $23.19
I have reviewed this, briefly, before, and I have pondered my comments last summer when I read this the first time. I admittedly skimmed some in the beginning as I just knew it was so important I had to describe it for you. It is so rich, so vast, covering so much — more on that in a moment — and is so wondrously creative that I was glad to commend it. It subsequently won the prestigious National Book Award and, so, I was proven right. South to America is extraordinary, nothing short of brilliant. As the great Isabel Wilkerson notes, describing its elegance, it is "by an esteemed daughter of the South and one of the great intellectuals of our time."
I was correct to explain that this is part African American history, part cultural studies, in the format of a travelogue or memoir. I noted the book's light touches — she talks about candy or sweets that she likes, the outfits she wears, shoes, pop music, memories of her girlhood, gentle conversations she has along the way.
Let it also be said that there are reports here, consistently, of the savage ways Black people have been abused, from the Middle Passage to the slave blocks, from the Plantations of certain parts of the South to the Jim Crow lynchings and the onerous daily indignities faced by Black people everywhere in America, almost always. Isabel Wilkerson is right to say that this is a meditation on "the complexities of the American South — and thus of America." I believe this is one of the most educational, informative, inspiring books on America I have ever read.
To remind you, the structure of the book is splendid. In each chapter Perry visits a certain location, usually a city or region, ranging from Virginia, Annapolis, Baltimore, and West Virginia through Louisville, Memphis and Nashville, into a chapter on North Carolina called 'Tobacco Road in the Bible Belt" and to the deeper South (Birmingham and Mobile, Atlanta and the famous "Black Belt.") From Baton Rouge and New Orleans (what a chapter) to Florida (and its important survey of indigenous people and Spanish colonization) to (yes, and it's important) the Bahamas, Cuba, and Haiti, she explores so very, very much.
If you like travelogues, this should appeal to you. I had little idea about any of these places, even though I've visited a few and read about others. South to America, though, filled in the color, the details, the local history, good and bad. If you want to learn some fine details through the eyes of an expert in Black history and Black literature, Perry is the best possible guide you could find. Read South to American, please.
You should know this; Perry is from the South, but now lives around Philly while she teaches at Princeton. (And, yes, there is a chapter on that most Southern of the Ivies.) Her parents were Black civil rights and movement activists and they introduced her to everybody. She is young, but knows so much about so much. The book shifts and moves, almost stream of consciousness-like, at times. One minutes she is describing something about the colors of historic cloth used for certain things in the colonial era or wild Savannah legends or Haitian slave revolts or the racial history of Beale Street and Elvis or riffing on mobile homes in Mobile or the rise of black Catholic nuns in New Orleans or offering great details about entertainers or artists. She knows quite a bit about African history and she knows tons of details about characters she introduces us to in the US cities she visits. She gets around.
Dr. Perry is a scholar, a writer, an artist, and a historian. (She wrote a previous book that was seriously awarded on the life of Raisin in the Sun playwright Lorraine Hansberry called Looking for Lorraine: The Radiant and Radical Life of Lorraine Hansberry that illustrates her wide knowledge.) I admire her greatly. There are moments, though — I'll admit this but I hope it doesn't scare you off — I had no idea what she was talking about. There are sentences that are gloriously poetic but, frankly, made no sense. Over and over I've re-read certain paragraphs and, not unlike the hippest of contemporary creative writing, some of it leaves me scratching my head. I had to look up a few words — and oh, what a wordsmith she is. Some of it, I suspect, is my lack of familiarity with deep Black culture. In any case, this is mostly wonderfully written, deep, thoughtful, poetic, mystical, inspiring. It's an amazing book, one I have read twice and will surely find time to read again. We eagerly add our voices to the many who have named it one of the great books of our time.
I do not want to overstate this but I have read a number of books about Black history, African- American culture, racial justice and multi-ethnic ministry. We have a lot of good ones. There is something about South To America that was so rich it demanded more than one read and there is so very much happening, so many turns of events and so many places she talks about that I think it is one of the most important books in this field that I've ever read. I cannot imagine a white person, at least, and probably most others, who will not learn something new.
South to America marks time like Beloved did. Similarly, we will talk not solely of books about the south, but books generally as before or after South to America. I have known and loved the South for four decades and Imani Perry has shown me that there is so much more in our region's fleshy folds to know, explore and love. It is simply the most finely crafted and rigorously conceived book about our region, and nation, I have ever read. — Kiese Laymon, author of Heavy
The Pastor's Bookshelf: Why Reading Matters for Ministry Austin Carty (Eerdmans) $19.99 OUR SALE PRICE = $15.99
If there is one recent book that I wish every pastor and Christian leader would read, it is this one. My, my, it is a blast, fun, even. The author was on the national TV show Survivor, for crying out loud — how can it not be? It is nicely written, moving at times, serious minded without being fussy. His unique combo of clarity and sophistication and teacherly explanation reminds me of Richard Mouw, and that is saying a lot. He's a very good writer, not overly flamboyant, but he's obviously got some colorful writing chops.
But more to the point, this chronicles Carty's increasing awareness that to be a good pastor in the classic sense, he simply must be a reader. He must make it a point to read daily, and to read widely, including lots of fiction. He is, I dare say, in this regard, at least, the closest thing we've got to Eugene Peterson whose love of reading and convictions about that hover around the entire book. If you appreciated Peterson's almost grumpy resistance to trends about mega- leadership and speedy techniques for growth and formulas for hipper churches, you'll love Austin Carty. He isn't quite so old as Eugene but there he is, citing Peterson's stories about reading well as a pastoral duty.
I so appreciated his calm guidance, his stories of meeting with others to invite them to read more, his suggestions on how to make it happen. This is exciting for any of us who are book lovers and great for those who need an extra push in the right direction. He's got both a deep and big perspective but he's also a good teacher about this stuff, practical and helpful.
Naturally, he uses the trope of reading for "formation, not information" which if you've heard me on this topic, you know I recite routinely. He shows how reading can nurture wisdom and help us learn to love. This whole section about formation is rich and inspiring and, I might add, good for anyone, not just pastors.
The next portion covers specific practices and aspects of the reading life as it works out in a pastors life. I am not a clergy person, but this still was fabulously inspiring for me as he explores how reading well can help with sermons and pastoral care, vision casting and leadership. He is right that reading is "not a luxury."
The final part includes six great chapters on reading with a good attitude, on the spiritual discipline of study, of how to choose what to read. (He even has a section on marking and filing, which was awe-inspiring, but I'm not there. Whoa.) The final chapter, which is fantastic, is about reading the Scriptures — obviously. It's a great way to end the book and wise on any number of levels.
The Pastor's Bookshelf: Why Reading Matters is truly one of my favorite books of the year. I hope you get a few and share them with readers and maybe those who are not as given to the reading life. Pastors, certainly, and others.
As the great pastoral leader and author Thomas Long puts it in his fabulous introduction, "One remarkable feature of Carty's writing in this volume is how much of it is done in conversation with others, particularly parishioners and others who are on the receiving end of ministry. Carty hopes to encourage pastors who read, but not merely as a form of gratuitous self-improvement, but reading done among, with, and for the people of God." Nice, huh?
Christians are a people of the Word, yet we are formed more and more today by wanton, careless words. Those who will lead the church well will be those who are formed by good words — those who know the power words have over our hearts and minds. Those who read good books well will be such leaders. Pastors who read and live by the wisdom in this book will be changed, as will their ministries and the people to whom they minister. This book belongs on every pastor's shelf. — Karen Swallow Prior, author of On Reading Well: Finding the Good Life through Great Books
Reading is crucial for ministry, not as a mine for anecdotes and illustrations, but as an apprenticeship of the imagination. In this warm and wise book, Austin Carty invites pastors to develop capacious reading habits, as wide and curious and wonderful as the world in which they serve. I hope this book is an occasion for many pastors to build new shelves of poetry and fiction, biography and memoir, all of them adventures in understanding humanity. — James K. A. Smith, editor of Image journal, author of You Are What You Love
I am gobsmacked by this book's threefold beauty: its writing, its erudition, and the author's deep commitment to what true reading can give not only pastors, but us all. — Maryanne Wolf, author of Reader, Come Home: The Reading Brain in a Digital World
Surrender: Forty Songs, One Story Bono (Knopf ) $34.00 OUR SALE PRICE = $27.20
I have written in my mind about five different reviews of this, sharing this or that feature, recalling this or that story, saying which parts I was most moved by. I think to honor this as one of my very favorite books that I read this year, nearly devouring it, I'll just share what I wrote back when it first came out before Christmas. Here goes:
Surrender: 40 Songs, One Story is one of the very best books I've read all year and it will certainly be in my personal favorites list coming up next month. In a way, it is a book of a lifetime for me. As a fan of U2, as a music-lover, as a uniquely Christian music-lover, this book resonated with me so very, very much. It brought stunning insight and joy; lots of joy. And, man, does Bono know his stuff. He knows so much stuff. Sure, he's got the swagger, and he (as one reviewer noted, here "embraces his contradictions.") But he really is smart. This book is an education in the popular culture of the last 40 years.
Let me just say four quick things about Surrender. I could, and surely should, wax more eloquently about it (it's over 550 pages, after all) but I want to keep this relatively succinct. I want you to know (if you don't already) whether this book is for you.
Uno, the book is not exactly linear and chronological (would you expect it to be?) but it mostly is. And there are song titles for chapter headings; naturally the first chapters are entitled from songs from their earliest recordings. (And the last few are, naturally, from their last albums; the important penultimate chapter is called "The Moment of Surrender" which you know from the No Line on the Horizon album.)
We learn from Mr. Paul Hewson in his own words a lot about his boyhood, the rough and rowdy ways of the religiously-conflicted Northern Ireland during the years of the troubles. With famous songs about "Sunday Bloody Sunday" outspoken pacifist tirades by the socially aware frontman of the social aware band, with nuanced lyrics recalling about how they cut down the few trees in their neighborhood and used them against their enemies (from "Peace on Earth" on All That You Can't Leave Behind) I would have expected a bit more of the Troubles. Instead we hear about his love of bands, his school experiences, the impact of books he read, like Lord of the Flies, and — a theme throughout the whole book — the sudden death of his mother, Iris, when he was a young teen. So many of the lyrics of his long career, we come to find out, are veiled (or not so veiled) references to his mother and father. (As he sings in "Iris (Hold Me Close)" on Songs of Experience, "The ache in my heart is so much a part of who I am…")
He's a hurting punk and wanted to be a punk rocker, and man, I grew to love him more, learning a bit, in impressionistic style, about his youth and his longing for a more stable family.
He met his best friend, Ali, in his teen years in Dublin, Ali who became a girlfriend, who became his wife, early on. Again, this bit of his past is exceedingly important to him, enduring for him. My hunch is that many celebrities and certainly many rock stars are less connected to their youth, their past, their families. Or at least they think it isn't cool to share that sort of sentimental family stuff. I loved that Bono has such affection for his dad (even if there was a lot of brokenness) and it was fun learning about Ali. It was fun learning about how he met the other three guys in the band and the importance of their friendships. His loyalty to these men is remarkable and in a way Surrender is a memoir of the trusting loyalty of these friendships.
I am a serious fan of the music, a real fan of Bono's political action, and have admired his sly art as it transfigured and changed over the years. I really enjoy all of the albums and admire them all. (As we suspected, by the way, the changes were often very intentional; the Zoo-TV era antics of the Fly and the sensory overload of the shows were almost fully satire, some of it literally informed by C.S. Lewis's Screwtape Letters, messing with the devil, their dangerously materialistic lifestyles mostly an embodied prophetic experiment.) So I know their work. But I have not read fan bios and knew very little about Bono's family life. Maybe other fans knew about Ali and his children, but I think this is the most forthcoming he has been about them. There are beautiful pages, lovely episodes shared with many passages about the hard conflicts and honest struggles. Bono knows he has a very good woman by his side and he knows if he isn't careful (as one of the most famous rock stars in the world) he could blow it. He almost did. But, man, his candor and poetic insight was some of the most romantic stuff I have read, ever. My hat is off and my heart is warmed.
Dos, did I mention the music? I could quote pages and pages about this (and have sticky notes throughout the book in case I want to do a serious study.) He goes on tangents — not really tangents, just colorful side-journeys, into his friendships with other artists. From punk guys to Frank Sinatra, soul singers to new wave artists, from Prince to black gospel choirs, he tells endearing and sometimes heartbreaking tales of the many people he admires and loves. It is very obvious — he never speaks badly of anyone (except himself) and even when talking honestly about the horrors of drug or alcohol abuse (even Adam's) he is not judgmental or mean-spirited. His generosity is lovely and his Irish storytelling — often of drinking late at night — is captivating. As a celebrity he knows he has been given quite remarkable opportunities, but he is also a gregarious bridge-builder and he knows more artists, working in different genres, than you could imagine.
He has encouraged many rising artists to apply their craft to anti-poverty and other justice measures; he tells of fashion designers, models, film-makers, poets, novelists, painters, dancers. Wow. Not bad from a kid from the Northside.
His story of how Pavarotti got him involved in relief work in Sarajevo is, by the way, hilarious. Annoying as it was, he applauded Pavarotti's tenacity in pursuing him. "Miss Sarajevo" (from the pseudonymical "Passengers" album) remains one of Bono's favorite pieces of his career. His moving reflections on Sinatra were powerful; his tribute to Michael Hutchence (of INXS) and his suicide was very tender.
Do you recall when a hard rock band was playing in Paris (in 2015) and a mass shooting killed dozens of audience members? U2 were doing a series of stadium shows also in Paris that week and their show was shut down — it wasn't the only time Bono had experienced a mass shooting, by the way. When they rescheduled the cancelled show they brought the smaller bar band — Eagles of Death Metal — onto their stage so they could finish their show that was so horrifically interrupted. These small stories of bands and stages and colleagues in the music biz were a blast to read and often inspiring.
And the recordings! I have read lots of books about rock music. Serious music lovers who read this sort of stuff may know Greil Marcus's magisterial work Mystery Train: Images of American in Rock 'n Roll Music or his book on Dylan and the Band's "Basement Tapes" sessions (Old, Weird America.) And there are some really cool books on the details of certain recording sessions. Bono doesn't give us that much of that sonic and technical detail, but there is plenty for even the most geeky fans of recording studios. Not to mention the small revelations of the band's work with lighting artists and staging designers creating what have been some of the most outlandish, brilliant, and expensive stage shows in the rock touring world. This is all so interesting but it never turns self-indulgent, naming the obscure brands of tubes or speakers or the sorts of electronics in the amps. (Although it might be said that it is self-indulgent in a different way as he talks much about the personal stuff going on in the midst of these urgent sessions, squeezing in so much global activist between tours and recordings, struggles with his voice, and the constant guidance of producers like Brian Eno and Daniel Lanois.)
I love hearing bits about so many songs — his reflections nearer the end about writing songs about friendship ("Bad" for instance) or linking the famous "I Still Haven't Found What I'm Looking For" to Bunyan's Pilgrim's Progress or how one song was co-written by Salman Rushdie. I was glad to hear about them holding their ground on changing the plans for a nice, spared-down, acoustic rendition of "Ordinary Love" (from the Mandela: Long Walk to Freedom Movie) for a live, Oscar show performance.
So there's family stuff, friendships, relationships, old ones and new ones. There's music, U2s and Bono's numerous friendships with so many other rock artists. There's great stuff on performing, on singing, or writing, or recording. If you like rock music (and especially if you like U2) this book is going to be a true joy.
Tres — and this is huge —there is a whole lot on politics. I found these portions hard to put down and as one who has engaged in a tiny, tiny bit of lobbying and protesting and building civic coalitions, I found this insider's look to be a blast. Early on, Bono learned (from a story about Dr. King told to him by Harry Belefonte) to build bridges even with those one might not want to work with. There were times when Bono was deeply lobbying the Bush administration — with the Jubilee campaign to cancel the third world debt, with ONE and then with his DATA and (RED) to fund life-saving drugs against AIDS in Africa — and was becoming friends with those who others on his team (and in his band) found unsavory. Bush was bombing Iraq, of course, torturing Muslims in off-the-grid black sites, and cutting budgets for the poor in the US. Yet, as he endured, learning from all sides, he came to be convinced of some of the value of conservative economic theory and in his famous office visits to right-winger Jesse Helms even found a friendly prayer partner. I was on the edge of my seat as Bono had to make some decisions regarding the leader of the free world and consequential choices about aid and trade, war and peace. Interestingly, though not surprisingly, friends like the late Mike Gerson are named. What a thrill, knowing how it finally turned out.
From his meetings with Nelson Mandela (and other lesser known African leaders) to his off-the-record opening of his home to Mikhail Gorbachev (despite Ali's outspoken work with anti-nuclear power activists resisting Russian malfeasance at Chernobyl) to his palling around with (and fallings asleep at) the Obama's, it is very entertaining, although none of it feels like name-dropping. To listen in on one with such amazing global connections who was actually nervous about it all — imposter syndrome, don't ya know — and his bits of candor about, say, fretting about what to wear when one is a rock star visiting the Oval Office, made for a great read. If you care at all about how the world works and how change happens, if you've donated money to ONE or (RED) or other similar anti-poverty groups, listening to Bono will be as inspiring as listening to the likes of Gary Haugen or Melinda Gates. He knows a lot about the facts of economic development and global politics and he weaves it into magical stories, often with stories of his on-the-ground, real-life volunteerism in poor villages. You'll learn a lot.
Catorce? I sort of hate to mention this final element as a discreet point since it is interwoven so naturally throughout the book, but it should be noted that Bono's Christian faith — unorthodox and uneasy as it may seem to some — is central to the whole story. It is not just cited a little, it is not just mentioned briefly. There are Bible allusions and explications, basic theology, Christian authors mentioned, and spiritual realities talked about in significant ways during every portion of his life, so throughout the 40 chapters. (And you know, of course, that one of their most famous songs ("40") is a nearly verbatim rendition of Psalm 40. Fans used to leave the stadium singing over and over "How long…")
There is even a moving telling of the family's deeply affecting religious tour of the Holy Land, which, for a glitzy rock star seems such a conventional, churchy practice. This is from the guy who says he "has never left Jesus out of the most banal or profane actions of my life."
Most know how Bono's father was a not terribly active Catholic and his mother was a good Protestant and how three of the band members came to a lively faith in a charismatic, Jesus-movement sort of evangelical ministry in their young adult years.They remained in touch with some of that crowd even after their faith moved to more ecumenical and liberationist ways and Bono continues to be haunted by that robust sense of the Spirit and that strong teaching of Biblical truth. For many of us, his casual, humorous, but serious-minded love/hate relationship with the church, is an inspiration. His honest lament and plea, of the sort found in "Wake Up Dead Man", (from 1997's Pop) means more than any number of happy-clappy CCM ditties.
Through his fame and tenacity and righteous commitments Bono has had contact with world-class Christian leaders, from Desmond Tutu to Eugene Peterson to a hilarious episode that he writes about with Pope John Paul II. When he is visiting dignitaries he mentions that he sometimes gives away books— often a volume of Yeats or other Irish poetry. But I happen to know he's given away his share of The Message, too. I admit to getting teary-eyed when I read his brief acknowledgment of Eugene Peterson.
Relationships, music, politics, faith. Stories galore, goodness and failure, temptation and joy, meaning and vision, art and wealth, compromise, justice, romance, sex, life and death. There is so much in this marvelous, stimulating book.
One final word: Surrender is creatively and colorfully written. Bono can really write; it is whimsical, a bit stream-of-consciousness, and, man, can he turn a phrase. There are witty lines on every page, brilliant sentences, wondrous prose. His clever honesty has him say things like about his ego being "far taller than my self-esteem." Ha.
As the flyleaf of this well designed volume puts it,
A remarkable book by a combative artist, who finds he's at his best when he learns how to surrender.
The Invisible Kingdom: Reimagining Chronic Illness Meghan O'Rourke (Riverhead Books) $28.00 OUR SALE PRICE = $22.40
Memoirs of those with chronic pain or the drama of trying to find a diagnosis are important to me. So many of us suffer with unusual ailments and for some, the pain (or the quest for answers) can nearly lead to madness. For others, it generates remarkable insight into the broken human condition and allows for a sense of grace, despite all. From practical guides for life lived while in chronic pain like the very good Hurting Yet Whole: Reconciling Body and Spirit in Chronic Pain and Illness by Liuan Huska (IVP; $17.00) to powerful stories like the brilliant The Deep Places: A Memoir of Illness and Discovery about Ross Douthat's battle with Lyme disease (Convergent; $26.00) that carries a fine endorsement by Kate Bowler, there are many to commend.
The Invisible Kingdom is one of the best books I've read this year because it is so full of pathos, yes, but also because it is so elegantly written, so very thoughtful, so honest and real.
O'Rourke is an academic who teaches in New York City and makes her living also as a writer and poet. Besides contributing intelligent pieces to The Atlantic, The New Yorker, and the like she has done several volumes of poetry and, a decade or so ago, a moving memoir about the early death her mother called A Long Goodbye.
The Invisible Kingdom was a finalist for the American Book Award and that, of course, carries its own weight and recommendation. It is notable because it is so well written (and in that sense, I say, carefully, that it is in a sense entertaining, a good read) but also because it covers not only her own story, her marriage, her work, her aspirations and fears and (yes, of course) her many encounters with medical practitioners (And, like Ross Douthat's with alternative medicine folks who she might have been averse to in another life, but, now, desperate, is willing to try nearly anything in the search for answers and relief.)
If this were all this book did — a moving memoir offering a writer's candid glimpse into the life of one with eccentric disorders and unclear diagnoses — it would be more than enough. I honor Ms O'Rourke for her candor and courage in sharing this stuff so very, very well. Her coping with a compromised autoimmune disorder is not uncommon but her writing is extraordinary.
But there is, as the telemarketers say, still more. Besides this eloquent and life-giving memoir there are excursions into the history and philosophy of science, of medicine, of women's medicine. About the common accusation that "it is all in your head." The trends in medical care and the backstory value systems that are in the air — Freudian assumptions, views about hysteria, and more — are vital to understand and O'Rourke carefully dissects many of the prevalent motifs that entangle even caring physicians.
Not to mention those who are not so caring. And those who wish to be but may not buck the managed care strictures of the insurance companies demanding that they spend less time with patients, not more. Anybody who cares about humane and effective health care simply must keep up with some of the debates that this book evokes, and The Invisible Kingdom is a good way into those discussions. At times searing, and at times tender, I recommend it to doctors and nurses of all sorts.
Allow me to be blunt: if you opposed President Obama's reform of health care, popularly known as Obamacare, a few years ago, you need to read this book, hearing from the ground up what a middle class professional goes through in managing her appointments and less than efficient doctors, all time strapped due to financial considerations of the management teams to supervisor most hospitals in America. If you worked for passage of some kind of health care reform, this will be a refresher course — in the first person — of some of what is at stake. This is not, I'll remind you, a book of politics or policy but a narrative of illness and struggle, a search for hope. But some of the health care reforms (starting with doctors not dismissing their patients and their stories of discomfort) that are so very needed are insinuated into this well-crafted story.
Of the many rave reviews this book has garnered here are just two:
In this elegant fusion of memoir, reporting, and cultural history, O'Rourke traces the development of modern Western medicine and takes aim at its limitations, advocating for a community-centric healthcare model that treats patients as people, not parts. At once a rigorous work of scholarship and a radical act of empathy, The Invisible Kingdom has the power to move mountains. —Esquire
Listen to this amazing quote by Esme Eijun Wang, author of The Collected Schizophrenias, a quote saying it is "the best book on the subject." Wow.
I've gone through much literature about being sick, hoping to better understand the tangle of circumstances relevant to chronic illness. In The Invisible Kingdom, O'Rourke brilliantly unpicks the threads, creating the best book on the subject that I've read yet.
Wildland: The Making of America's Fury Evan Osnos (Farrar Straus & Giroux) $30.00 OUR SALE PRICE = $24.00 // now in paperback $20.00 OUR SALE PRICE – $16.00
I think I am not going to try to say much about this other than to assure you that if you are one (like so many in America) are trying to figure out how it all happened — people falling for Trump's election lies, the storming of the Capitol, the right of the alt-right, the wild weirdness of our civic life these days — if you are among those grasping at ways to discern how we got here, this, then, is a must read. It is more detailed than some may want but it, in my view, top notch reporting and expose, narrative nonfiction that rivals classics such as Dopesick or Soul Full of Coal Dust (among my favorite books in this genre.) Not quite as eloquent and dramatic as Wastelands it is none-the-less a page turner. I couldn't put it down and didn't stop reading (as I often do) until I finished the acknowledgements on page 433. What a book.
The short version is this: Wildland is a study of three pockets of American culture, including the very rich in the financial sector, including dubious hedge funds (in Greenwich CT — I had no idea that place was that wealthy) and the financially hard off in small town West Virginia, and in quintessentially mid-western Chicagoland. The author (a master craftsman of investigative writing who won a National Book Award for his book The Age of Ambition) had lived in the Middle East and in China for years and upon returning home could hardly recognize the culture he left more than a decade previous. He went to three towns in which he had lived, which he thought he understood, and began to explore the stories of his former neighbors, telling about how things changed in the fast-paced era leading up to the Trump era. I like nonfiction narrative that is colorfully written with a personal touch and as Esnos tells of his connections to these three very different places, we learn to trust his instincts and become eager for his analysis, his evaluations. We are caught up in his own story, the search for answers, or at least some clues.
The stories switch back and forth from one place to the other, so much that it has been called "sprawling" and a "reportorial tour de force." There are the super rich who came to favor President Trump (even though some despised him, personally) since his policies favored their Wall Street investments. (There are shades in this portion of the great literature and film that came out of the years of the Great Recession, with works about the subprime scandal like The Big Short.) There is the declining local newspaper in Clarksburg, West Virginia (and, yes, some shades of Dopesick and other studies of the complexities of Appalachian towns and governance.) The politics of Chicago and the black neighborhoods? Oh man — there's much to be said but you can imagine.
This is not the study that is typical these days of disaffected poor white folks hitching their hopes to the straight talking Trump, duped into thinking he was on their side. That's a story that has been well told, often. This, or so it seems to me, went after a different nuance, the shift in worldviews among the rich and the powerless, blacks and whites, from here and there, but often with a realization of how the greedy left such an impact. The scope is broad, if focused, and what he comes up with is nuanced and insightful. It is less about the wild rebellion of the Tea Party movement inspired by Bannon, segueing into the Proud Boys and Oath Keepers. It's maybe less dramatic than that kind of wild story, but, at the end, it pays off with greater insight into what makes this sad country tick. I really recommend it.
Diligent and deeply researched . . . Osnos offers intimate portraits of the men and women in the three communities on his radar . . . Wildland is written in first person, which often gives the book a satisfying immediacy . . . Osnos himself seems too driven, too idealistic to give up on the America that he once promoted on his travels abroad. But as he makes painfully clear in Wildland, the underbrush is still parched, and a mere ember could set it ablaze. — James S. Hirsch, Boston Globe
Evan Osnos' Wildland is a reportorial tour de force, describing the kaleidoscopic changes that threaten to cause America to come apart at the seams. He deftly connects the dots between the hedge-fund billionaires of Greenwich, Connecticut, the opioid-soaked towns of Appalachia, and the gun-heavy gangs of Chicago. By turning his trained eye as a former foreign correspondent on his own country, Osnos paints an indelible picture that is heart-rending, appalling and hard to put down. — Jane Mayer, chief Washington correspondent for The New Yorker and author of Dark Money: The Hidden History of the Billionaires Behind the Rise of the Radical Right
The Life We're Looking For: Reclaiming Relationship in a Technological World Andy Crouch (Convergent) $25.00 OUR SALE PRICE = $20.00
From a few pages in I knew I would love this book and early on I said to myself (and anyone else listening) that it surely will be among my favorite books of the last year. Yes, indeed. To honor it as a Best Book of 2022, I'll reprise some of what I wrote back in April.
Andy is an elegant and careful writer, always offering interesting insights that set him above most authors of what I might call profoundly Christian cultural studies. But the genre of "cultural studies" doesn't exactly capture the nature of his work which is notably personal as well. For instance, his book (one of my all time favorites) Culture-Making is about the human vocation to make things, to shape culture, to better the world —"recovering our creative calling" as the subtitle puts — so while it is in some ways about culture, informed by remarkable Bible study and a uniquely Christian world and life viewpoint, it isn't essentially cultural criticism or social analysis; it is an invitation to a better way of life in and for the world.
It's the same with his exceptionally important book on using power, Playing God: Redeeming the Gift of Power where he makes an impeccable case that, like other things in God's good but fallen world, institutions and organizations and influential leadership (that is to say authority and power) can be misused but, with great intentionality and care and prayerful innovation, can be exercised with grace and justice…
The Life We're Looking For is new and after writing a small and practical book about using smartphones and computers well in a family setting, Crouch has returned to offer another major work. It comes to us in a slightly trim-sized hardback so even though it is 226 pages (including the intriguingly exquisite footnotes) it isn't too long or hefty. It is mature and stimulating in his most gracefully crafted style yet. Tech scholar Sherry Turtle of MIT raves about it, calling it a "personal meditation" written with "warmth and erudition."
Already in the pocket-sized Tech-Wise Family Crouch explored what it does to us when we are shaped — our habits, our inclinations, our desires, our sense of how the world is to work — by our devices. (He doesn't note James K.A. Smith's notion of "secular liturgies" in the new one but he surely could have.) The Life We're Looking For goes a lot deeper and in different directions than Tech-Wise, inviting us to reflect on the meaning of being human, the importance of being made in God's image, and how to exercise our capacities (even drawing on self-help neurological studies about "flow" and the like.) He is trying to consider what we really want — what Tish Warren in her back cover blurb calls "the most vulnerable longings of the human heart." He knows that we have a primal need to be seen, to be "embedded in rich relationships."
As we all know, one-click ordering from faceless corporations is designed to be seamlessly smooth and very efficient. And, as we should know, these algorithms and bureaucracies are damaging us, eroding community fabric and molding our attitudes. His brief rumination on not knowing the UPS and other delivery drivers in his neighborhood anymore, since Amazon has gig workers is quite observant and telling.
Do we really want this shallow connection — quantity over quality? Do we want devices to make everything easy (even if they could) as we become consumers, but less adept at using our bodies, playing instruments, cooking our own food? Do we want to be tethered to multitasking and regularly speed-speed-speed through increasingly impersonal environments? Do we want rugged individualism or do we long for communities of care, extended households, even? Andy's chapters on the formation of such households — a countercultural move to subvert the individualism and even loneliness brought on by technocracy — were simple and radical and moved me to tears.
In an otherwise very positive review in Publisher's Weekly they worried that The Life We're Looking For was uneven, and I sort of understand why — Crouch covers a lot of ground. There are sections which include a bit on the history of technology, a look at alchemy (with a nod to Harry Potter), a good bit on artificial intelligence and "boring robots", a powerful reflection on money/Mammon and nature of capitalism, his wife's use of instruments in her scientific research, and an admitted excursion in what calls an "intermission" on a very moving story from the New Testament as a remarkable reminder of the early church's presence within first-century Roman Empire.
All of that, and more, could be seen as tangents, but, trust me: they are not. The chapters weave together and reward the patient reader as connections are made and insights circle back and layer on one another, bit by bit. We all know these are confusing and contradictory times ("we love it, we hate it, ain't that life?" the late, great Mark Heard sung so powerfully in "Nod Over Coffee.") Andy's book helps us navigate in healthy and even beautiful ways, the tensions and trade-offs of these days.
To glean the evocative style and deep wisdom of this book, ponder (and be enticed) by just a few of twelve chapter titles, paying close attention to the helpful subtitles:
The Superpower Zone: How We Trade Personhood for Effortless Power
Modern Magic: The Ancient Roots of Our Tech Obsessions
From Devices to Instruments: Truly Personal Technology
None of this is bombastic or heavy-handed (as passionately concerned as he obviously is.) The book is not overly polemical nor alarmist. It is often gentle, even a bit quiet, in a way that seems proper for the human-scale ecology into which he is inviting us. In a chapter called "From Charmed to Blessed" he tells a story that you will long remember as he calls us to "the community of the unuseful." In the telling about our New Testament friend Gaius and the odd, diverse community under Roman rule of which he was a part, he describes them as "fragile." He follows that with reflections on own ancestry and I found it very tender.
This lovely, thoughtful book offers plenty of well-researched information and teaches us much, even a bit, as we've noted, on AI and computer science, which is enlivened when he mentions his Roomba and how he sneakily alludes to a dishwasher as a computer. He observes a bit about the human-scale texture of our lives when we give ourselves over to automation. (Which is, by the way, one of the reasons we always reply with a personal note acknowledging orders here on line, trying in our small way to redeem the online buying experience, inefficient as that may seem.) Naturally, the book includes poignant stories (and some fun ones, for instance about his own bike riding habits and his driving habits as well.) In the second portion, Crouch offers "redemptive moves" — new postures and habits to "help us begin, right now, to live more fully human lives." I am convinced that it just might. I am sure reading it will help us cope with — you'll have to read it to get the full irony — "the loneliness of a personalized world." For starters, he reminds us that we "do not have to accept our technology's default settings." They can be adapted "to serve a new and better set of purposes."
As Andy so nicely invites us to move away from "ever-increasing isolation" and create homes "that become creative centers far more consequential than the refuges of consumption and leisure have let them become" he also pushes us to include the outcast, the unwell, the unproductive, the overlooked.
And he is hopeful:
The great news is that there are already examples of these redemptive moves — some seedlings, some saplings, some beginning to bear widespread fruit — and we all have a part to play in helping them grow.
Quiet and reflective as it may be, finally, The Life We're Looking For: Reclaiming Relationships in a Technological World is not a gentleman's armchair treatise. It is a reformational manifesto, calling us to renewal, change, redemptive efforts, to become agents of the shalom God wills for us all. Or at least a bit more sane in this fast-paced, digital world. It is a wise and wonderful book.
Alan Jacobs is one of the prominent essayists of our time, and knows a thing or two about the implications of our shift to machines in the middle of modernity. (Read his spectacular Year of Our Lord: 1943 for a particularly in-depth study of what five important Christian thinkers were thinking about these very things.) Here he says this is Andy's best book yet:
The Life We're Looking For is, and this is saying something, Andy Crouch's best book: a deeply moving meditation on the human need to find true personhood, which means, among other things, to know as we are known. Strong and cogent critiques of Mammon's empire–which, as Crouch shows, is where we live–are not unheard of, but a book that goes this deeply into the heart of things, into the heart of God, is a pearl of great price. — Alan Jacobs, author of How to Think and Breaking Bread with the Dead
Listen to Tish Harrison Warren, who writes so well about so much:
As I read this breathtaking book, I was surprised to find myself tearing up often, not because it is a book about tragedy or loss, but because Andy Crouch, perhaps more than any other writer of our day, perceives and names the deepest and most vulnerable longings of the human heart. The Life We're Looking For describes the confusion and contradictions of our cultural moment in clear and resonant ways and, more important, offers hope that we might find a beautiful way of living amidst them. – Tish Harrison Warren, author of Liturgy of the Ordinary and Prayer in the Night
The Flag, The Cross, and the Station Wagon: A Graying American Looks Back at His Suburban Boyhood and Wonders What the Hell Happened Bill McKibben (Henry Holt & Company) $27.99 OUR SALE PRICE = $22.39
I raved about this in the June 23, 2022 BookNotes, a column I was proud of. Here are a few key excerpts, sharing why I think this book was so very good, such a great read, and one of my favorites of 2022.
The format of this memoir is, obviously, in three big parts. But first, why do I say "quasi-memoir"? I'll leave aside if that is even a word, but the point is made clear in the beginning when McKibben wryly notes that the best memoirs have drama and adventure and pathos and since he grew up in a pretty ordinary, middle-class, stable family, was dealt a good hand and has had a fairly uninteresting life, his memoir, such as it is, is going to be more social history than gut-wrenching autobiography.
McKibben is a prolific author of great renown, actually, and he not only is a professor of note at Middlebury College in Vermont but has founded several environmental organizations. He has travelled the world. His campaigns to do MLK-type mass civil disobedience to try to stop dangerously polluting pipelines and his mobilizing even against the Obama White House is, frankly, quite thrilling and would be, for most, drama enough for several lifetimes, so he's being a bit demure in suggesting that his life doesn't have enough angst for a full-on memoir. Maybe that will come later.
For now, though, he uses this storytelling format of recalling moments and eras of his life as a window to see other, bigger things. He has an agenda and it is to illuminate much about the last fifty years of US history and how patriotism, religion, and our consumerist way of life (rooted as we are in suburbs and automobiles) have shaped our culture and the world's climate, how these things have themselves changed in recent decades, and need now to be reimagined and refined if we are going to rise to the occasion of being faithful in this day and age. That he is finally getting at how to more urgently and effectively mobilize to lower our carbon output and mitigate the disastrous climate change (of which he is an expert) should not surprise anyone who knows him. That he would do so with antidote and charm and a lovely survey of his own patriotism and faith, while not exactly surprising, is a writerly delight and makes for one tremendous book. I am not alone in suggesting this may be his best book yet. (And I've got The Bill McKibben Reader by my bedside!)
"Bill McKibben has written a great American memoir, using the prism of his own life to reflect on the most important dynamics in our society. Bill McKibben's writing is poignant, engrossing and revealing. His message is a clarion call for a generation to understand what happened to their American Dream, and to fight for our common future." — Heather McGhee, author of The Sum of Us: How Racism Costs Everyone and How We Can Prosper Together
This is a fabulous couple of breezy chapters – not long — where McKibben shares his own passion for US history, especially colonial history. You see, he grew up in Lexington, Massachusetts, and worked as a teen as one of those New England historical interpreters, wearing his tricorn hat and waxing eloquent about the shot heard round the world and the nearby Battle of Bunker Hill and Paul Revere's famous ride. He knows that stuff cold and his retelling of it is vital, and with enough local color and backstory to make it captivating. In memoir fashion, he is telling us of his telling of it (a standard colonial-era joke, playing the board game Risk while waiting for the next crowd to arrive, his "half hour spiel" to "maybe one family with a couple of bored teenagers, maybe an entire bus of Japanese tourists" and holding out that hat" to collect tips, which was his summer job pay.) This respected scholar and activist-leader and New Yorker writer who is known around the world was a pimply faced kid wearing that three-sided cap and earning tips by sharing his passion for the great US revolution. I loved it, hearing of his love for our country and its founding story.
As McKibben puts it, "I came by my patriotism honestly." He writes about the account the guides would deliver and the lasting influence of the basic importance of the story,
It was a clean and brave story, and, as I say, it has informed me ever since. The valor of standing up to unjust and arbitrary power seemed to me its clear and obvious moral. Indeed in the years that followed, as I read more deeply in American history, the importance of that stand sank further in.
I want to talk about that – to tell how and why the Revolution came to seem so important to me. I want to draw the picture in as bold lines as possible. Because soon enough the picture is going to get much more shaded, much less noble. But not quite yet.
Here it gets even more interesting and provides good instruction for many of us. He tells briefly about the important work of the 1619 Project and what new insights it brings to our understanding of our nation's early history. He is embarrassed that his teenage job holding forth about the revolutionary years in Lexington didn't have him telling the truth about the indigenous people nor the role of enslaved people (or black freemen like Crispus Attucks, say.) He ponders which is worse, that they knowingly ignored the unpleasant fact or if they just didn't think to include them – an example of the generous but candid self-awareness that gives this book much of its appeal. That is, it is neither a white wash or a diatribe. It's just a good man trying to say what he's learned to be true and ponder its significance for us all today in our own cultural moment.
It is, as I've implied, earnest and fair and wise. Terry Tempest Williams (whose most recent, luminous writings are collected in Erosion: Essays of Undoing) described McKibben as an "everyday hero" and says the book is plainspoken, direct, and conversational. His candid and well-informed critique of the right-wing pushback against the 1619 Project is worth the price of the book; it is not overly zealous and it is not unfair. But on just a few pages and with a few key examples he shows why we need the insights of black and native peoples and why their stories need to be part of our national story. I've read a bit on the controversy and think McKibben is sensible and right; I'm surprised that some writers I respect have fretted about the Project – I just don't get it, and so appreciated McKibben's sensible generosity. It is interesting how he gets a bit passionate and names what needs to be named, but comes back to the memories of his own early patriotism formed there in the Lexington Green.
(There is one paragraph unlocking a racially-consequential line in the famous poem about Paul Revere that will take your breath away if you do not know about it; I did not, and McKibben's discovery is stunning.)
There is a pivotal event that happened in the town when he was a kid and I won't spoil the show by saying anything about it, but I want to say for the record that I so admire his parents and was very glad for this fascinating glimpse into small town New England politics in the late 1960s. Kudos to local historians and small town storytellers who write booklets and make tapes and keep records and oral accounts alive in local libraries and historical societies. McKibben comes back to this episode throughout the book, but I don't want to ruin it by saying more.
There is a part that explains, too, about the economic realities that emerged from our troubling history of white privilege. Books like Richard Rothstein's must-read The Color of Law, Dorothy Brown's scholarly treatise The Whiteness of Wealth, Randall Robinson's The Debt and Ta- Nehisiha Coates's stunning 2014 call for reparations are mentioned and it becomes clear that McKibben's commitments to the flag, seen in his telling of his pride in raising Old Glory with his Boy Scout troop – a lovely paragraph that made me smile — are now deeply tied to true truths about economic injustice stemming from a history of institutional racism. What the hell happened? This book explains it as clearly and succinctly as any I've read.
I needn't say much more about his early formation as a proud, if now sobered, US citizen, but I will note this: I'm inclined to protest, or, these days, at least compliment those who do, when things go haywire. But the sort of honest lament McKibben names about our sinfulness doesn't mean we cannot affirm the good ideas and good things that emerged from our founding as a nation. McKibben's reflections on the flag and proper patriotism are solid, balanced, and, I think, very important.
Beth and I were thrilled even by the first page or so of this section where McKibben describes the character and tone of his youth group (often held in "fellowship hall") and church camp and mission/service trip and endlessly singing songs like "Kum Ba Yah" and "Day by Day" from Godspell. (Does anybody out there remember "Pass It On"?) These were the early and mid-1970s and kids didn't sing "praise and worship" songs like they do today. His testimony of the value of his UCC church was as wonderful to read as, well, some of the scenes in Jonathan Franzen's Crossroads novel that I devoured last summer. His earnest mainline faith, his reading of the gospels, his telling of his own faith journey is simply delightful.
Those that have followed the nature writer, environmentalist, anti-global warming activist, and social critic, have known of his faith. He writes for Sojo and had a book published years ago (on Job, actually) by the prominent religious publisher Eerdmans out of Grand Rapids. But to hear him talk about his Sunday school teachers and his spiritual concerns as a young adult is terrific and encouraging. Importantly, his description is not offered only for the purpose of literary memoir but to make an observation, to testify, about the positive formative nature of much mainline Protestantism and the social ethic that emerged from this broad, non-fundamentalist youth ministry which so influenced him. In fact, this piece is, in many ways, a eulogy for a certain sort of healthy civil religion that allowed mainline Protestant public intellectuals (from Reinhold Niebuhr, say, to Martin Luther King to Dorothy Day) to have influence over the discourse and values of American culture.
I might want to push back in conversation about his take on mainline Protestantism although, given his framing of it – in the 60s we had Tillich and Barth and King and the brilliant William Sloan Coffin as public representatives of Jesus and in more recent times we have had the Jerry Falwells, Franklin Graham, and Trump sycophants that seem to care little for the Bible or Jesus – it is hard to argue. Hipster evangelicals mock "Kum Ba Yah" (as did Donald Trump, for that matter) but if singing that around the church camp campfire gave us the likes of Bill McKibben, I'll take it.
McKibben is earnest, also, about his college years and it is a great grace that he sought out thoughtful Christian leaders while a student at Harvard. He is never proud or smug about this but it is clear that he was mentored, in part, by the black, Republican (and gay) preacher there, Peter Gomes. McKibben is nearly evangelistic when he wishes others would read Rev. Gomes's book The Scandalous Gospel of Jesus: What's So Good About the Good News?
There is little doubt that Christianity has been a hugely important influence, for better or worse, in both forming and fraying our social fabric in the last few decades. No social commentator can ignore the role of faith communities, or what we sometimes called "the religious landscape." It is helpful that McKibben here shares both personally and more broadly, about his sense of how the Christian faith ought to be an influence for the common good. It is all so very interesting, informative and at times beautiful.
As in the previous section, he refrains from academic footnotes, but there is a fabulously interesting essay about sources and book recommendations in a final epilogue. His passion for early US history is evident and his suggestions there offer a year's worth of reading, at least, starting, not least, with the important work of Gordon Wood (for instance, his early The Radicalism of the American Revolution.)
For the section "The Cross" he thanks his friend Diana Butler Bass (a fine church historian and contemporary writer who I mention often in BookNotes) and he commends her on-line newsletter "The Cottage." He names the magisterial collection, The Future of Mainline Protestantism in American, edited by James Hudnut-Beumler and Mark Silk, the fabulous edited IVP volume by Mark Labberton called Still Evangelical? and he highly recommends Jemar Tisby's The Color of Compromise. I was glad to see that he pointed readers to Kristin Kobes Du Mez's must-read Jesus and John Wayne — again, a book that we very, very highly recommend. One would almost think he's reading BookNotes. Ha.
THE STATION WAGON
I suppose it makes sense that McKibben uses the station wagon – indeed, one that his family owned and for which he has great affection to this day – as a symbol of the consumerism and social inequity caused by the rise of the American suburbs. I mentioned the rowdy critique of the ugliness and ecological harms of suburbia described with such wit and zeal by James Howard Kuenstler and McKibben stands in his tradition, I suppose, without any of his cynicism or rudeness. (I kept wishing for a quote from The Geography of Nowhere or Home from Nowhere.) Happily suburban bred and raised, McKibben realizes that the rise of individual homes, and the station wagons that transported the (mostly white) kids across this land in those glory days of the American Dream, became detrimental to the planet and here his passion about climate change begins to appear.
We knew it would, of course. He's been writing about this since his 1989 breakout bestseller The End of Nature — inspired, as I recall, by his colleague Jonathan Schell's The Fate of the Earth — and the influential Eaarth that came out in 2010. In the previous sections of The Flag, the Cross, and the Station Wagon he exposed how the lovely boyhood and middle class, churched upbringing contributed to a distorted understanding of our society and how things work in the world, but here – oh my. His reporting continues to shine; his prose riveting and his insight brilliant. The relationship of the flag and the cross are coming into focus and much of it is about, well, not exactly the station wagon, but the money accrued from the homes where those station wagons were parked. I know housing bubbles and interest rates and zoning battles may not seem like the sexiest topics for an entertaining nonfiction read, but trust me.
McKibben has lived this stuff, but he has also researched it well, drawing on the definitive and the most fascinating works, such as Meg Jacobs, who he thanks, for her Panic at the Pump: The Energy Crisis and the Transformation of American Politics in the 1970s.
Here, particularly, is where McKibben's writing style makes this complicated stuff accessible – he is able to tell a story or two, bringing in a big picture analysis, and name the need for some kind of repair – all while keeping the prose light, the information interesting, the story compelling.
You see, he is showing the ways in which owning one's home naturally creates wealth; generational wealth. Yes, certain suburban cul de sac lifestyles can cause increased pollution and alienation from place and creation, and yes, ice caps are melting due to our materialistic extravagance. But that is only the most calamitous of the implications. Along the way we got racist policies (like redlining and the gross injustice in disallowing black World War II vets access to the benefits of the GI Bill, educational opportunity, lines of credit, jobs.) Deeper wealth discrepancies developed and the economic injustices based on the rise of the US suburbs (and subsequent home ownership and banking) is damning. That he isn't even more emphatic and prophetic in his denunciation is admirable. The social evils are so obvious in his telling, the book could have gone off the rails with screeds and anger and extremist proposals. He verges on it, but he returns to his town, the ups and downs, the good and bad, rooted in a good family and good faith and decent folks who mostly want to make a difference. The "Station Wagon" section, like the others, is a fair-minded, honest critique. It is the kind of analysis that, if widely heard – that is, if this book sells well and is discussed widely – could become a compelling game-changer. We hope you consider it and order a few. It is, no doubt, one of my favorite books of 2022.
The Flag, The Cross, and the Station Wagon is a great read, and much of it is really quite entertaining. For instance, what a blast to hear his own story of the famous solar panels installed on the White House by Jimmy Carter and famously taken down, out of spite or ideology stupidity, by Ronald Reagan. (I recall that Ed Meese called them "a joke.") McKibben actually knew a bit about those panels as they were rescued from some Washington warehouse and ended up being used effectively at a small college in Maine. Bill tells the rest of the story, including a large Chinese business startup making more of these panels, inspired by one they got from the college in Maine, and how he got some students to create some holy trouble when they brought some of the remaining panels — still working good as new — as a gift to Obama whose people refused to meet with them, let alone put them up on the White House roof.
In retrospect, McKibben writes, sharing his disillusionment, "it was pretty clear why Obama wanted nothing to do with those solar panels: they were tainted by their association with Carter. The 1980 election, thirty years later, still dominated our politics." Yup.
There is a final piece to The Flag, The Cross, and the Station Wagon which I will only mention briefly, but it is his big altar call / patriotic ending. It is hinted at in the story about taking college kids on the road trip – stopping for PR events along the way – to the White House to give the historic Carter-era solar panels back to the Obamas. He mentions that he truly felt bad causing the disillusionment of the youth he had brought along. It was a tell-tale line in passing, but McKibben – as the subtitle notes – is graying. And this is how he ends the book, with some remarkable stuff about how older people can become mentors for younger ones, can encourage and fund them, how our more experienced citizens can mobilize alongside the idealistic younger leaders. He has (of course he has) started a great organization to help facilitate that and it is already going strong. Check out his group, Third Act.
Mr. McKibben's last, short chapter is inspiring, entitled "People of a Certain Age." He writes about caring for kids and grandkids and is encouraged that many older people are ready to act differently in their public duties, as well. He notes that "many of us are now emerging into our latter years with skills, with more than our share of resources, and with grandchildren. Surely that might give us the capacity and the reason to help."
Evangelical Anxiety: A Memoir Charles Marsh (HarperOne) $27.99 OUR SALE PRICE = $22.39
I started out my BookNotes review last June with this:
I want to start this BookNotes edition telling about a book that blew me away and which I am now reading for the second time. It is rare when I make time for an immediate repeat but this book was so engaging – that is, it was very well-written in a way that was artful and richly-crafted but was equally inviting and enjoyable. Not every book that is written with literary verve and gorgeous prose is, frankly, still that interesting. This, though, a memoir of a journey in and coming out of a southern sort of fundamentalism, and finding a way through the hurts and hang-ups of that milieu, is a page-turner. My mind is reeling thinking of a dozen things to say about it as there is so much going on in this breathtaking story.
The least I can do to honor this extraordinarily honest and fascinating story is to name it as a Best Book of 2022. It was certainly one of my favorite reads, even if some of it left me pondering. I suppose that is the sign of a good book, eh? Kudos!
Evangelical Anxiety by Charles Marsh is some of what I might have expected in a memoir from him, knowing a bit about Dr. Marsh's journey and scholarly interests from his previous books. I am not sure I can describe simply his current, lively, Episcopalian kind of mere Christianity, but his conservative, Southern evangelical past is the swamp he has slogged through. As a Bonhoeffer expert (his second Bonhoeffer book, Strange Glory, is an essential one), a scholar of and advocate for racial justice (and author of several excellent books on these exact themes, including one co-written with John Perkins called Welcoming Justice) and Director at the Project for Lived Theology at Charlottesville's UVA, I assumed his story surely included some shift away from evangelicalism and distancing himself from the ugly compromises made by many white evangelicals in the last decades as they've drifted from gospel clarity and focused increasingly on right wing politics. When former evangelical leaders like Jerry Falwell, Jr say that Jesus doesn't have anything to do with their values voting, what is a somewhat socially progressive evangelical Christian to do? Needing to disavow the weirdness of a shallow God and Country sort of so-called evangelicalism has produced a number of memoirs and a number of serious studies about the value and wisdom – for the sake of the gospel! – to no longer use the phrase evangelical. Other books in this recent genre are less about the current state of the evangelical brand, and are artful tellings of the tales of living through what was often a toxic sort of legalism. There's a lot of deconstruction type narratives with a lot of anxiety.
We have commended, for instance, the much-discussed Where the Light Fell by Philip Yancey as a gripping look at the boyhood and young adulthood and ongoing faith journey set in this complicated and sometimes religiously harmful subculture. As any good memoir, Yancey's book allows us to look over his shoulder and into his life as he navigated his broken family and harsh faith – it's an entertaining if intense read; I often say that well-written memoirs provide a reading experience akin to reading great fiction. In some cases, one could hardly make up such astonishing stuff. Let's face it, regardless of what one thinks about or what relationship one has with a given religious subculture, it makes for great literature.
This is a long way to get to a major point about the exquisitely written story, Evangelical Anxiety. It is, in fact, mostly not about the somewhat predictable question of how a smart young scholar and person of conscience with devout commitments to Christ can abide being a evangelical, given how grubby that phrase has become these days. I was wrong to assume he was talking about that anxiety. (Although not as a memoir, Marsh did explore this already in 2007 with an acclaimed Oxford University Press book, Wayward Christian Soldiers: Freeing the Gospel from Political Captivity.) With Charles's masterful writing chops and his extraordinary mind and learning, it would be fabulous to have a book about his worries, struggles, disagreements, cognitive dissonance, and theological ruminations, about his evangelical past. However, I was wrong about this being mostly that kind of a story, really. It is, in fact, about his real anxiety disorder. In this stunning report that reveals more than I expected, we learn about Marsh's years of psychotherapy to cope with his nearly debilitating panic attacks and something akin to depression.
Within the opening paragraphs we realize how very well written Evangelical Anxiety is and what an artful reading experience it will be. Wow. Soon enough, we realize that even as some of the themes are what we might first expect – a strict religious background giving way to a more expansive faith, the struggle to understand for oneself the spiritual life in the college and young adult years, the not uncommon journey from sparred down fundamentalist preaching services to a more liturgical (Episcopalian) worship — we soon realize that coping with real anxiety is much of what this book is about. And, well. What a story it is!
You should know this: Mr. Marsh wrote an earlier autobiographical account of his growing up years and it focused on an exceptional episode in his young life, a life-changing season at his father's church, and while Evangelical Anxiety is not a sequel or second part of his life story, the remarkable stuff told in that previous one, does inform this new one. He explains those years briefly since it comes up over and again.
In The Last Days: A Son's Story Of Sin And Segregation At The Dawn Of A New South, written in 2001, Marsh tells about how his father, a good preacher and thoughtful Baptist pastor, realized that there were violent KKK guys in his church, even on his leadership council. The Last Days tells of their horrible crimes, his father coming to terms with it. Like many Protestant pastors in those hard years in the South, Reverend Marsh was not an active anti-segregationist nor grossly bigoted. He was, perhaps, the sort of leader who would have realized that King's famous Letter from a Birmingham Jail was, in part, addressed to him. He was a good man and a good father, if conventional in that Southern Baptist setting and slow to come around to the courage needed to confront the likes of Sam Bowers, the Imperial Wizard of the White Knights of the Mississippi KKK who lived near the Marsh's in Laurel, Mississippi.
The violence in the town, aimed at Blacks, of course, was in the air, and Marsh was not unaware of this fearful texture to daily life. But when his father removed the men of the KKK (and their families) from his church, the violence was aimed at his family as well. Young Marsh — athletic, popular, strong — couldn't sleep at nights.
In a nutshell – and Marsh describes it with considerable beauty and pathos and understanding and keen insight based on his own decades of studying the civil rights movement in the American south – this consequence, this violence and fear of violence, is the origin of Marsh's own crippling anxiety attacks.
Yes, he reads literature and theology by authors outside of his evangelical world in high-school and then college; yes he ends up at Harvard Divinity School after Gordon College with some nearly anti-Christian teachers, or so it sounds. Yes, there is this refining and reframing of his faith and church life (perhaps akin to what some today call deconstruction) but all of that, or so Evangelical Anxiety suggests, is colored by the trauma of growing up in a repressive fundamentalist subculture, and of coming of age in a time and way that might suggest his fears are a fallout from his father's fidelity to the gospel. Charles does not cheaply pat his father and mother on their backs and does not portray himself as collateral damage from their small, if belated, part in the civil rights struggle. But he knows the cost of discipleship in his bones. It drives him to seek help.
And this – oh my – is where the book gets even more captivating. He ends up (to make a colorfully long story of his circuitous path a bit shorter) in Freudian psychotherapy.
Think what you will about the appropriateness of a Biblically-trained evangelical young man heading to the couch to talk about his sexual desires and his mother and such, it is at the heart of this story. In a few spots, the book admits to Marsh's own awareness of the irony of this (he knows all about Jay Adams and the anti-Freudian teaching from conservative evangelical thinkers who propose more overtly Biblical counseling; golly, he had a meeting with Francis Schaeffer when he was a teen as he sought guidance and direction. The story is made that much more fascinating knowing just how unlikely it was for this young man to take up such therapy.
As a broke young scholar with a new wife, feeling drawn into therapy, he ends up in a rare situation of doing analysis with a new shrink in a nearly free program in Baltimore. For those who know anything about this, you won't be surprised that it goes on, almost daily, for years. Some of his breakthroughs and insights are disclosed here although the book never devolves into a mere account of his id and superego. It's a great memoir, not a document of his therapy. Nonetheless, working on the couch has been a significant part of his life in coping with his disorders, and, well, there it is, written about with candor and wit. It is sharp and at times funny.
As Jemar Tisby puts it:
Marsh probes the realms of piety and mental health with engrossing prose and naked honesty, showing us how the sacred can be found in literature and on the therapist's couch. Anyone curious about a better way to navigate mental health and belief will find hope and inspiration in this.
I do not think I have ever read a book like this. The glimpse into a professional and religious life in which debilitating panic attacks and gripping depression and unusual ticks and so many concerns are described in such detail (without being overly dramatic or maudlin or self-pitying) is rare and so very interesting. I was stunned when I read early on in the book about his first attack. (Sorry for the spoiler alert — I hadn't seen it coming. It is, no matter, an amazing piece of writing.) If you care about how some people cope with psychological disorder and their subsequent physiological consequences, this book will be illuminating. (Granted, he is from an educated class, a world-famous scholar, and award-winning author and the book is a memoir, not a guidebook, of his particular experience as a professor, academic, writer, and theologian.) I do not think it is a bad thing to say Evangelical Anxiety will be entertaining, a good read, as they say. There were descriptions and well-crafted sentences that just made me shake my head in wonder and there were episodes described that made me laugh out loud. Publisher's Weekly called it an "endearing and rewardingly unusual account of mental illness and faith."
Patricia Hampl, author of The Art of the Wasted Day says,
"A harrowing book but, weirdly and wonderfully, also a hoot. I kept laughing aloud – and then sighing. A remarkable achievement."
But, again: Mr. Marsh's story unfolds against the backdrop of considerable anxiety around the religious questions of leaving behind a strict version of faith; it is, as more than one reviewer observed, connected to the questions about the relationships of the so-called secular and sacred; the split between body and soul, desire and duty. As in the Yancey memoir, moving away from the faith and very worldview of one's youth, especially if it was a demanding subculture defined as over-and-against all others, can be painful and can create relational ruptures. Fortunately, Marsh's parents were not toxic or harmful and some of his faith experiences (and the webs and networks of relationships he experiences) were perhaps less caustic that the caricature of this harsh setting might conjure; still, getting severely paddled by high school coaches and terribly shamed by youth group leaders was part of what was considered ordinary in that time in that place.
As Marsh comes of age in the 1970s there is cultural change in the wind, not unrelated to the seismic shifts begun in the 1960s. "The Times They are a Changin'" Dylan sang and the words were prescient. I feel it in my gut as I type it, knowing how I myself snarled out the words with my own cheap guitar in the '70s. Marsh reports well how one person and family – including his beloved wife K – negotiated these changes in these times as they moved into their early married years of the 1980s and on. Again, I could not put this book down and was very deeply moved by it all; I can at least say that anyone who is aware of the nature of Protestant life during the end of the 20th century and into the new millennium will find it fascinating.
We need you, dear and gentle reader, to know something else about this stunning memoir. It is honest. Marsh is exceedingly candid about his fears and his failures. Do I need to issue a trigger warning? Perhaps. He is candid, particularly, about his sexuality and, given the way purity culture was made into a fetish in some evangelical circles and how the Biblical teachings not to have sex before marriage were made exceptionally clear and linked to the looming threat of hell in his subculture, it is no wonder, I suppose, that he, uh, had issues.
A scene in Evangelical Anxiety of Charles and his then girlfriend reading wildly together while house-sitting in the home of Elizabeth Elliot of all people (look her up if you don't know) is so erotically charged I don't know how they remained chaste. In any case, there is some very frank talk in parts of the book. As a reader with a pretty wide palate for "language" in stories and who doesn't think that human sexuality needs to be off limits for writers telling about their life story, I still have to say that some of this felt gratuitous. I think an editor should have put her foot down a time or two, even if Publisher's Weekly enjoyed the "bawdy" parts.
Nonetheless, the book really does need to explore this stuff: it is an integral part of the story. It was the heavy-handed sexual repression combined with the ubiquitous racial violence that helped shape the psyche of a man who realized he could not manage a life in these modern times, as a faithful person, without unpacking it. And, so, he goes there, sharing without shame some intimate details of his life and not so unusual desires.
The very discerning James K.A. Smith called it "at once transgressive and faithful." Perhaps that's it — both transgressive and faithful.
Other early readers also have raved about this long-awaited memoir by Mr. Marsh. We know it may not be for everyone but it is a major book by an important voice, and it was very difficult to put down. I'm happy to tell you about it and hope you'll send us an order.
There is vivid storytelling, there are remarkable recollections of important stuff, and there is some broad-brush cultural analysis, placing his own journey in the context of the fundamentalist and evangelical world of the past generation, up to and including his own worship experiences today.
The opening page describing in smooth detail the crisp khaki trousers and brand name shirt of the Anglican worship leader presiding at worship, the tasteful praise songs, and the shockingly weird sermon, was so well written and deftly designed, shifting to a line at the end of the page that made tears well up in my eyes, alerted me that this was going to be one great read.
There is also great tenderness in Evangelical Anxiety. Marsh writes about taking his kids to a Christian camp. He describes his love for his mother, including the solace she offered during his fearful nights as a boy. He is deeply remorseful when he has hurt his wife. He struggles with how to relate his own scholarship – he writes about Bonhoeffer, after all – with his own practice of lived discipleship. He holds what he knows to be true about the world, its racism and violence, and is learning how to carry on as a sane and happy person. In a simple passage about finding joy in good things in God's creation, the spirituality of the ordinary, so to speak, he mentions how his friend, the evangelical, black leader John Perkins likes blue berries. I got a lump in my throat, just such a lovely little line about a man who has suffered much and experienced great fame, Charles's friend. Many who pick up this book and enter this story will also be struck by Marsh's great love for literature and the often beautiful way he mentions novels and authors, his intimate relationship with their truths and artful pleasures. I so enjoyed reading about a man I respect and the books he loves and the authors who have informed him.
This is a bold, beautiful memoir, at once transgressive and faithful. Marsh embodies a theology with the courage to tackle the taboo, including depression and desire, in prose that is evocative and seductive. In the end, we learn that the most astounding grace is found in the God we can tell our secrets. — James K.A. Smith, Calvin University, editor in chief, Image, author of You Are What You Love and On the Road with Saint Augustine
Stories of My Life Katherine Paterson (Westminster/John Knox) $22.00 OUR SALE PRICE = $17.60
This. Yes — one of my favorite books of 2022.
I will never forget the spectacular talk the esteemed YA novelist gave at the 1998 Calvin College Festival of Faith and Writing on the role of the imagination, the best of the keynotes that year, sandwiched between Elie Weisel and John Updike. In Stories of My Life, Patterson tells about that lecture, and staying in the Grand Rapids home of the legendary writer (and Calvin prof) Gary Schmidt for that weekend, when she got the phone call saying she won the prestigious Hans Christian Andersen Award. It is just one of the great, tender moments in her new book of memories, Stories of My Life. It is utterly magnificent, touching, and I found it captivating — a gentle but absolute page-turner!
I was very moved by this sprawling set of memories from her life, including what she has learned about parents and grandparents on her side of the family. We learn a bit about the colorful stories of her late husband — he was a Paterson ("with one t" she noticed, quickly) whose own father had very colorful years of, among other things, serving in World War One, being gassed, losing a leg, and being treated for TB. It is very nicely written with a calm and no-nonsense style. It was truly lovely, without being luminous, engaging without pretension. She says firmly in the beginning that it is not a memoir. It is, as the title sensibly proclaims, a set of stories from her life.
And what a life she has led. I can hardly say enough about this wonderful read about a wonderful Christian woman whose contribution to (and fame in) the world of contemporary children's literature is nearly unsurmountable. She is, certainly, one of the great children's writers of these times.
How can I persuade you to order this handsome volume full of entertaining and edifying stories of a life well lived? If you do not know that she was born in China (her parents were medical missionaries) who fled as a little one during the Japanese invasion, and who returned, only to be exiled again (mostly due to tensions with the communists that time) and you do not know that she herself was a Presbyterian missionary in Japan, if you do not know her moving stories like Bridge to Terabithia and her several nonfiction books about the role of the imagination and of faithful but not overtly religious storytelling, I hardly know where to begin. Any good library would have her many out of print children's titles and now you can easily learn about her life and times.
Here are just a few fascinating and enjoyable moments you will encounter if you order Stories From My Life. I'm only scratching the surface — it is such a great read.
Firstly, you know you are in for a treat (and will be walking among the gods of stories) when you open the book to find a fabulous short intro by none other than Kate DiCamillo. She highlights a key moment of vulnerability in the narrative when she alerts us to Katherine's story of being a child (home from the Chinese mission field, wearing second-hand clothing, a bit shy, and seemingly not welcomed into her new school) and not receiving any Valentine Day cards in school. Kate notes that Katherine writes that she told her mother many years later about this and she was, of course, aghast. Mother wondered why Katherine never wrote about the hurtful incident. As DiCamillo recalls, Katherine "answered her by saying, "All my books are about the day I didn't get any valentines."
And then, in her own great gift, the great Kate DiCamillo says:
This book is a valentine.
It is Katherine's Valentine to her parents and to her children. It is her Valentine to life and to stories.
It is her valentine to us.
Kate Dicamillo also has badgered Katherine to include the story of Maude, a relative of her grandfather's who was the last person to kiss Robert E. Lee, who, in turn kissed little Katherine. DiCamillo loved the story so, she threatened to write it up herself if Katherine didn't write it down. So, yes, here is the bit about Lee, although, personally, I enjoyed the episode about her brother and the bones of Lee's famous horse, Traveller. You will have to read it to discover that yourself.
The second foreword is so endearing and masterfully written and insightful that I've read it twice — it is by writer Nancy Price Graff and she tells of the twosome's weekly lunch at a diner in their town in Vermont. For over twenty years the women have grown old together in their regular Naugahyde booth. Paterson has written 40-some books in fifty years, performing what she calls "the fragile magic" of spinning stores for children and young adults. But she doesn't talk about her writing or much about her fame.
"Week after week," Graff writes,
…one of the great storytellers in the world has told me the story of her exceptional life. Diners no more than three feet away, deep into their meatloaf, are oblivious to the presence of the former National Ambassador for Young People's Literature, the winner of the Astrid Lindgren Memorial Award, and the Hans Christian Andersen Award. It would never cross their minds that the gray-haired woman sitting two booths over, wearing a turtleneck and a pink sweater, might have had dinner last week with the librarian of Congress or the empress of Japan.
The stories are not exactly chronological and in fact, starts with a good piece responding to "Three Frequently Asked Questions." I was hooked. In one of these early pages she tells of being at the Presbyterian School of Christian Education in Richmond, Virginia. (Now known as Union, it was started when there was a need for theological education for women headed to the mission field or into educational tasks since the Presbyterian churches were not yet ordaining women so women were not as likely to attend a seminary like Princeton, say.) One of her favorite professors stopped her in the hall — he has been reading her exam —and said in made her wonder if she ever thought of being a writer.
Now I, the lifelong reader, the summa cum laude graduate in English literature, knew what great writing was, so how could Dr. Little imagine, on the basis of an essay on an exam, that I could be a writer? 'No,' I said primly. I had no intention of being a writer because I wouldn't want to add another mediocre writer to the world.
Well, the prof pushed back, wondering if perhaps that was exactly what God was calling her to do. Katherine tell us simply:
It was hard to imagine that God needed a lot more mediocre writers in the world, so I didn't become a writer or movie star. I became a missionary.
Her first piece of writing, by the way, was a great Sunday school book for middle school age kids published in 1966, Who Am I?, which is still in print from Eerdmans. She was by then home from her Japanese mission experience (1957 – 1961), had married John whom she had met at Union Theological Seminary in New York, and was teaching in Pennington NJ, while John attended Princeton. By '66 they had moved to Tacoma Park, MD.
One of the opening questions in that long opening chapter was "How does it feel to be famous?" Children and others ask this at book readings and interviews and it is a question she is not fully comfortable with. She tends to be shy, although has learned to be brave. She tells of being at very fancy dinners at a head table and being ignored. She has been shunted here and there on book tours and speaking engagements and sometimes mistreated. She reports that she'd come home whining to her husband that she is not treated "like a human being." This reveals both her insecurities, it seems, and her life's overarching principle — that people, made in God's image, should be treated with understanding and kindness. This matter of being uncared for comes up over and over and I found it quite gripping. Near the end of Stories… she admits that she wrote The Great Gilly Hopkins after pondering a question of how it would feel to be considered disposable.
The Paterson's have adopted two children, one from Hong Kong and one an indigenous Native baby. She and her husband were great parents, it seems. They have been foster parents, too, and it was painfully difficult. "After The Great Gilly Hopkins was published, I realized, belatedly, that I had put two foster children in the story. I might not have been Gilly. I might well have been William Ernest."
She was an honorable child, usually, it seems, but playful and adventurous (and a good reader, bored with early school book readers.) There are stories of family in China, and of being back in the US, a "home" she did not know, of course. (Today we call this phenomenon being "third culture kids.") Her parents loved her dearly, even though there were harrowing times of her dad being on Chinese riverboats trying to smuggle life-saving medicines and supplies to Christian hospitals for the Chinese people. There were times when he'd to travel undetected for remarkable distances, keeping away from the Japanese invaders and the young communists and certain military officials. What a story!
(Her parent's backgrounds were fascinating themselves. She is somehow distantly related to Mark Twain. After WW I her father was cared for by a Mrs. Lathrop Brown, whose husband was a special assistant to the Department of Interior, high up in the Wilson administration. She had been a New York debutante and her husband had been Franklin Roosevelt's roommate at Harvard. As a disabled veteran, he was fortunate to have her as a caregiver and she stayed in touch with her parents until she died. In fact, she sent boxes of children's books to little Katherine in China. When they were exiled from China and spent an awful time in 1938 as refugees, she had arranged for a chauffeur to meet them at the boat in New York harbor.)
Katherine's time in Japan is explained and there are a few memorable stories. It doesn't take much —she's working that 'fragile magic' — and I was in tears at a going away party which had a Japanese pastor reading Ephesians 2:14 (a personal favorite, about the dividing walls being broken down in Christ) and Paul's revolutionary words in Galatians 3:28. It is especially powerful knowing that Katherine had admitted that she had trained to return to her native China. Going there on mission was not to be and when she was assigned to Japan — the feared and despised enemy that had attacked China (and perpetrated atrocities such as the Rape of Nanking in 1937) — it caused turmoil in her soul. Of course she went and then, knowing the language and caring for the people, a Japanese pastor says,
Katherine is young, I am old. She is a woman, I am a man. She is American. I am Japanese. When she was the child of missionaries in China, I was a colonel in the occupying army in Manchuria. She comes from the Presbyterian tradition, I come from the Pentecostal. The world would think it is impossible that she and I should love each other. But Christ has broken down the barriers that should divide us. We are one in Christ Jesus.
After her own sermonizing just a bit, she notes how the influence of Japan is evident in all her work. "My first three novels are set there, as well as the beautiful picture book The Tale of the Mandarin Ducks, whose illustrations by Leo and Diane Dillard garnered a Boston-Globe-Horn Book Award." She has translated some Japanese folk tales, as well, illustrated by the award-winning Suekichi Akab. She quietly notes that she wouldn't be the writer she is if it were not for her four years in ministry there. "To be loved by people you thought hated you is an experience I wish everyone could have."
She loved her job as a teacher, then, first in 1955 reading aloud to poor rural kids in a one-room school in Virginia. Oooh, was she irritated that these kids were all said to be dumb because they supposedly had done poorly on IQ tests. These kids were not dumb! (And she henceforth distrusted standardized tests.) She doesn't think she was much of a teacher but she gave them good experiences (including a trip to the National Zoo in DC that, trust me, was a great episode to read about.)
I choked back tears when she tells of going to visit the little school in Lovettsville years later while passing through the region. It was now a community center so she found the newer building. School had just closed but teachers were there, cleaning up as they do on their last day. Katherine marveled at the well-stocked library. Nobody had time, really, to chat until it became evident that she used to teach there and some old-timer had some recollections of people she had known decades beforehand they realized who she was. Oh, were they pleased, confident that Bridge to Terabithia's Lark Creek was based on Lovettsville. The current sixth grade teacher said that he tells their students that each year and they never believe him. She assured him that he was right.
She also taught for a while in a Methodist boys school, teaching the Bible. There's a great story there about a boy complaining about how all the kings of the Old Testament seemed to be getting killed and how irrelevant that all was. Before she could even answer, the classroom door was thrown open. "The history master was standing in the doorway, ashen faced. 'The president has been shot,' he said." She comments,
Without a word, we filed out into the common area where there was a large television set and watched in horror until Walter Cronkite finally announced the news that Kennedy was dead. The boys didn't try to argue about the stupidity of the ancient Hebrew ever again.
This is a typical passage — casually reported yet full of pathos, poignant, even, and sort of sly. There are some fun laughs in the book —her young married life was hard and she had four young children (two of two different races) and yet she and her husband made do and did well. It's a glorious part of the book, hearing about their married life and her efforts as a parent.
One of the most moving stories comes at about page 270 as she tells of her son, David, finally getting a good friend; Katherine had been diagnosed with cancer and worried about her children, but David, especially, needed a good pal. And then one day he met Lisa. Who — to cut a tender story short — was suddenly killed, struck by lightning at Bethany Beach. This was, of course, the genesis of the tragic story of a boy/girl friendship and the way youth cope with death that became Bridge to Terabithia. (You may recall that in that story they read The Chronicles of Narnia together.) She hardly wanted to finish writing the story and tells of putting off doing the chapter when Leslie Burke would die.
And to think Bridge to Terabithia has been maligned and banned! To think we have been criticized for carrying it!
It is fascinating how Katherine Paterson has often written about serious things. Her story of struggling with her first novel, The Sign of the Chrysanthemum, comes to mind. It is nicely told and she tells us much, but she offhandedly observes that she was doing a juvenile novel set in 12th century Japan (with a storyline of ancient civil strife, poverty, and which included trafficking and a brothel) at the same time that the nation's number one best seller of adult fiction was a calming, almost silly narrative about a seagull named Jonathan. If you are a baby boomer, you know what she means.
There are fun things to learn while reading Stories of My Life. She has a whole chapter called "Pets" and it involves more than their beloved dogs. Yes, the great Gilly Hopkins is named after Gerard Manley Hopkins. Jacob Have I Loved came out of thorny ground and a difficult time — her editor, the famous Virginia Buckley, had to push and pull to get her to develop it suitably. She missed the first Newbery Award press conference when a plane couldn't land —Peter Spier was the only one who made it and he "single-handedly charmed the press and the American Library Association, melting the heart of blizzard-bound Chicago." The story of what she allowed her kids to do with the first Newbery Award check —one thousand dollars was the prize amount and it was the most disposable income they ever had —is cute and made me chuckle.
Early on Paterson notes that there were stories she heard growing up as the family did the dishes together. She wondered why many of these stories were not passed on to her own now grown children and grandchildren. You never told us that, they'd exclaim. (The answer is easy — they had a dishwashing machine which eroded such family time.) In a way, this book was written for her own loved ones. She set about finding diaries and letters and researched things in far away courthouses and museums to get more information behind the anecdotal stories she grew up with. She added much from her own life, her writing career, her travels around the world as an internationally known figure promoting children's literacy and the imaginative arts.
I will not spoil the last two chapters but they are tender and touching. The very last is short but she ponders that one famous reviewer said, looking back over her work, said that she is a writer of hope. Indeed. But there is something behind that, she insists, and it is the Biblical doctrine of grace. She cites the words to the hymn Come Thou Font of Every Blessing. She is now 90, doing well, and active at the First Presbyterian Church in Barre, VT, where, as she puts it, she has "experienced the true communion of the saints." It is a lovely ending to a marvelously entertaining book. It is surely one of my favorite books of 2022.
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