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title: "SASS and Compass"
aliases:
- /en/master/tools/sass/
---
Cli container already contains `ruby`, `bundler`.
## Create Gemfile in Theme Folder
Example:
```ruby
source "https://rubygems.org"
gem 'sass', '~>3.4.0'
gem 'compass'
gem 'bootstrap-sass', '~>3.2.0'
```
## Install Tools
**Every developer runs `bundle install` to pull necessary gem dependencies**
This command should be run in the folder with Gemfile:
```bash
fin exec bundle install
```
Two folders (`.bundle`, `.bundler`) and a file (`Gemfile.lock`) will be created.
Please add these directories (`.bundle`/`.bundler`) to your `.gitignore` file.
## Compile SASS
Run in the theme folder:
```bash
fin exec bundle exec compass compile
```
This is important not run `compass compile` directly, but run it via `bundle exec` so that proper gem versions were used (defined in the Gemfile).
## Compass Watcher
You can run watcher to keep your CSS files up to date as changes are made:
```bash
fin exec bundle exec compass watch --poll
```
|
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"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}\label{s:intro}
It is estimated that some 285 million people in the world visually impaired, with the majority of cases potentially preventable or treatable if diagnosed promptly \cite{BlaiSandTuteWill+2016}. Unfortunately the burden of disease is greatest in low and middle income countries (East Asia, Latin America, and Africa) where access to diagnostic devices presents a major challenge \cite{GilbWormFielDeor+2015,BastHenn2012}. Observation of the retina by an eye care professional using an ophthalmoscope or through acquiring a video or photograph allows screening for key blinding diseases such as diabetic retinopathy, glaucoma and retinopathy of prematurity \cite{BlaiSandTuteWill+2016}.
\begin{figure}
\centering
\includegraphics[width=0.99\columnwidth]{highq-new.jpg}
\caption{ High quality images of the eye fundus acquired using a traditional, expensive, and bulky tabletop unit. }
\label{f:hq}
\end{figure}
However, one of the key barriers to fundal screening in low and middle income countries (LMICs) emerges from the lack of access to affordable retinal imaging equipment \cite{GilbWormFielDeor+2015}. The Arclight direct ophthalmoscope has been recently developed as a means of addressing this challenge \cite{BlaiSandTuteWill+2016}. The device, despite its simplified design and low cost, has been shown to be as effective as more expensive traditional devices \cite{LoweClelMgayFura+2015}. In addition, the Arclight can be attached to the camera of a mobile phone and acquire a digital recording of the fundus for the purposes of telemedicine. However, the captured images are intrinsically limited to a narrow field of view by the optics of direct ophthalmoscopy. Other limiting factors relating to the optics, sensor, and compression software of mobile phone cameras can further reduce the quality of the acquired data in comparison to more expensive dedicated traditional retinal cameras \cite{BlaiSandTuteWill+2016,GiarLiviJordBols+2014,BolsGiarBast2016}.
\section{Context and practical challenges}
Traditionally, retinal images (see Figure~\ref{f:hq}) are acquired using expensive and bulky tabletop units operated by trained technicians in a hospital setting \cite{RussMaphTuraCost+2016}. Consequently such devices are impractical for use in low income areas where resources to buy and maintain traditional cameras are limited and robustness and portability is key for allowing the sharing and movement between inhospitable sites \cite{BlaiSandTuteWill+2016,KandSmitWrigHart2013}.
However, the availability of low cost yet powerful smartphones in LMICs has led to the development of `m-health' \cite{QianYamaHausAltm+2011} whereby mobile phones can play several roles: contacting and reminding patients to take medicines and attend appointments, transferring money to pay for health care, storing guidelines and diagnostic algorithms, acquiring and archiving audio and visual information from patients, analysing data, utilizisng GPS tracking, and transferring information wirelessly. These approaches \cite{KandSmitWrigHart2013} have quickly been adopted in low resource health care settings where the portability and connectivity of smartphones \cite{Hull2008} has made them particularly valuable in telemedicine \cite{KandSmitWrigHart2013}, making it possible to reach patients and communities that currently receive suboptimal care due to geographical and financial barriers \cite{RussMaphTuraCost+2016}.
\subsection{The Arclight device}
The Arclight is a highly portable (dimensions: $110$mm $\times$ $26$mm $\times$ $9$mm, mass: $18$g) low-cost (\pounds 5 if purchased in high volume\footnote{See \url{https://iapb.standardlist.org/get-arclight/}}) ophthalmoscope that has been developed in response to the need for affordable diagnostic eye care tools in LMICs \cite{TuteYoun2017}. Uniquely the device is illuminated by an LED allowing it to be powered by a slim lithium battery that is charged by an integrated solar panel reducing the need for expensive and hard to find consumables. Employing a small LED allows simplification of the design of the device reducing manufacturing costs and its overall size, see Figure~\ref{f:arcl} \cite{BlaiSandTuteWill+2016}. The novel features of the Arclight make it ideal for use in mobile clinics and remote low resource settings with limited access to power. Consequently over 10000 devices have been distributed, enabling thousands of health care workers worldwide to perform fundal examinations for the first time \cite{BlaiSandTuteWill+2016,TuteYoun2017,LoweClelMgayFura+2015}.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{arcl.png}
\caption{Arclight: a miniature, low-cost ophthalmoscope-otoscope. }
\label{f:arcl}
\end{figure}
Studies have shown that the Arclight is just as accurate for screening for signs of diabetic retinopathy and glaucoma as more expensive traditional devices when used manually by an eye care professional \cite{BlaiSandTuteWill+2016}. However, one of the key inherent limitations of any direct ophthalmoscopy device is the narrow retinal field of view. Single still images are typically of limited utility but a sweeping video of a greater area of the fundus may offer greater clinical information, benefiting a telemedicine approach. Still, transferring video files is challenging in remote areas of a low income country where fast and reliable broadband internet may not be available. Synthesising a single wider field still image from multiple frames of a video fused together is a promising solution for overcoming many of the limitations of the optics of direct ophthalmoscopy as well as the data-heavy nature of video.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{input_example.png}
\caption{ Typical frames extracted from a video acquired using the Arclight device. Variable scale and brightness (which is also non-uniform), frequent occlusions of the region of interest, narrow field of view, and blur are just some of the notable challenges posed both to physicians and automated methods. }
\label{f:inExample}
\end{figure*}
\section{Proposed method}
In this section we describe a novel method for processing narrow field retinal video acquired using the Arclight ophthalmoscope while attached to an Apple iPhone 6s. By fusing the information contained in different frames of a video we synthesise a wider field image of higher quality, which is easier to interpret clinically and more compact to transmit and share. We start with an overview of the method and then follow up with detailed descriptions of each of the processing steps.
\subsection{Overview}
As we noted earlier, the quality and useful information content of different frames in a video varies greatly. Hence, in order to ensure robustness, our algorithm starts by automatically identifying the highest quality frame. Tracking, and thereafter incremental image stitching, commences both in forward and backward directions in time. Each new frame is processed in order to remove confounding appearance content, processed to extract salient information, registered with the growing synthetic image (initially the single frame which which tracking begun), and stitched with it.
\subsection{Detection of the region of interest}
A crucial step in the pipeline outlined in the previous section concerns the detection of the region of interest (ROI) i.e.\ the part of the image which corresponds to the fundus and contains salient appearance content: the blood vessels and the optic disc. The parts of the frame which are outside of it are of no use for us and correspond to the sclera (the white, outer part of the eyeball), eyelids, etc.
As the frames in Figure~\ref{f:inExample} illustrate, the region of the interest is elliptical in shape, and bright. However, colour in the image can not be used reliably as a discriminative feature for segmentation as significant variation can be observed across different individuals (e.g.\ there are differences between individuals of Caucasian and African descent). Variation in location, ellipse shape and orientation, as well as scale should also be noted as significant challenges.
Though the region of interest is elliptical in shape, observe that there is no sharp delineation, prohibiting the use of edge detection based approaches. Instead, we propose a two step method. In the first step, simple binary segmentation is performed using the well-known method first introduced by Otsu \cite{Otsu1979}. Here the frame is treated as a greyscale image, see Figure~\ref{f:otsu}. Then the contour of the largest region used to find the best elliptical fit. We achieve this using a genetic algorithm based approach. Our methodology is inspired by the work of Conn and Arandjelovi\'c who used ellipse fitting as the first stage in the task of localizing and segmenting ancient coins in real-world images~\cite{ConnAran2017}.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{otsu.png}
\caption{ Otsu's binary segmentation method based on automatic greyscale thresholding is used for the initial, rough localization of the region of interest. }
\label{f:otsu}
\end{figure*}
An ellipse embedded within a 2-dimensional Euclidean space is parameterized by five parameters. This can be readily seen from the implicit equation:
\begin{align}
x^2 + k_{xy} x y + k_{yy} y^2 + k_x x + k_y y + k = 0.
\label{e:ellipse}
\end{align}
The choice of the parameterization is crucial in making the most of our genetic search strategy. In particular, the parameterization should be such that when encoded as a `chromosome' (in the context of a genetic algorithm), operations such as crossover, mutations, and others, are likely to effect an improvement in the fitness of a hypothesis. With this goal in mind, we parameterize an ellipse (hypothesis) using five points on its circumference and, following Conn and Arandjelovi\'c~\cite{ConnAran2017}, use a short, non-binary chromosome comprising the coordinates of these points. As argued by Conn and Arandjelovi\'c, by enforcing the indivisibility of coordinate values we ensure that the constraint imposed by two circumference points is retained during evolutionary operations, thereby achieving a higher chance of greater generational fitness improvement.
The fitness of a specific hypothesis is evaluated in a straightforward fashion. From chromosomes formed by concatenating the coordinates of five points on an ellipse's circumference -- $(x_a,y_a)$, $(x_b,y_b)$, $(x_c,y_c)$, $(x_d,y_d)$, and $(x_e,y_e)$ -- the parameters in \eqref{e:ellipse} can be obtained by solving a simple linear equation:
\begin{align}
\left[
\begin{array}{cccccc}
x_a^2 & x_a y_a & y_a^2 & x_a & y_a & 1\\
x_b^2 & x_b y_b & y_b^2 & x_b & y_b & 1\\
x_c^2 & x_c y_c & y_c^2 & x_c & y_c & 1\\
x_d^2 & x_d y_d & y_d^2 & x_d & y_d & 1\\
x_e^2 & x_e y_e & y_e^2 & x_e & y_e & 1\\
\end{array}\right]
\left[\begin{array}{l}
1\\ k_{xy}\\ k_{yy}\\ k_x\\ k_y\\ k\\
\end{array}\right]
=\left[\begin{array}{l}
0\\0\\0\\0\\0\\
\end{array}\right].
\end{align}
From this parameterization, the fitness of a hypothesis can be readily quantified by (i) uniformly sampling the circumference of the corresponding ellipse, and (ii) computing the number of samples which are located on edge pixels of the edge image. Equispaced samples along the circumference of the corresponding ellipse are readily generated using the canonical form, i.e.\ using the major and minor radii ($a$ and $b$) of the ellipse, its centre $(x_0, y_0)$, and the rotation angle $\theta$ relative to the coordinate system:
\begin{align}
&a = \frac{-\sqrt{2 \psi \psi_1}}{k_{xy}^2 - 4k_{yy}} &\text{and}&& b = \frac{-\sqrt{2 \psi \psi_2}}{k_{xy}^2 - 4k_{yy}},\\
&x_0=\frac{2k_{yy} k_x - k_{xy} k_y}{k_{xy}^2 - 4k_{yy}} &\text{and}&& y_0=\frac{2 k_y - k_{xy} k_x}{k_{xy}^2 - 4k_{yy}},
\end{align}
where:
\begin{align}
\psi=& k_y^2+ k_{yy}k_x^2 - k_{xy}k_x k_y + (k_{xy}^2 - 4k_{yy})k,\\
\psi_1=& 1+k_{yy} + \sqrt{(1-k_{yy})^2+k_{xy}^2},\\
\psi_2=& 1+k_{yy} - \sqrt{(1-k_{yy})^2+k_{xy}^2}.
\end{align}
Then, if there are $n_s$ samples, and the samples are $\left\{ (x_1,y_1),(x_2,y_2),\ldots,(x_{n_s},y_{n_s})\right\}$, the fitness becomes:
\begin{align}
\phi = \frac{ \sum_{i=1}^{n_s} E(x_i,y_i) }{ n_s},
\label{e:fitness}
\end{align}
where $E(x,y)$ is the value of the pixel $(x,y)$ in the edge image $\mathbf{E}$.
\begin{figure*}
\centering
\includegraphics[width=2\columnwidth,height=1\columnwidth]{frangi.png}
\caption{ Examples of extracted vessel structures. Note that here only the raw output is shown: artefacts at the borders of the region of interest and the glare and removed thereafter. }
\label{f:frangi}
\end{figure*}
\subsection{Glare detection and removal}
As mentioned earlier, one of the challenges with the input videos is the presence of significant glare. It is of critical importance that the areas of the fundus occluded by glare are detected so as not to confound the tracking process, as well as so that the relevant regions can be filled-in by exploiting information from other frames in which they are not occluded.
Glare areas are for practical purposes of unpredictable shape, often but not universally present, and their location varies. All of these characteristics make their detection difficult. Another challenge which emerges in our specific application concerns the appearance of the optic disk which is usually brighter than the rest of the fundus, and could be mistaken for glare. Yet, the appearance of the optic disk is a crucial structure of interest to ophthalmologists, used to diagnose glaucoma and a number of other conditions \cite{KiagKherGichDamj+2013}.
Our approach to detecting glare is inspired by the work of Lange \cite{Lang2005} on images of the uterine cervix. Broadly speaking, we detect glare areas as those regions of the image which exhibit high contrast or saturation, and are approximately white. Formally, the criterion is based on the following local measure $g(x,y)$:
\begin{align}
g(x,y)=\min(R_{x,y},G_{x,y},B_{x,y}) - \frac{\min(R_{x,y},G_{x,y},B_{x,y})}{\max(R_{x,y},G_{x,y},B_{x,y})}
\end{align}
where $R_{x,y}$, $G_{x,y}$, and $B_{x,y}$ are respectively the red, green, and blue colour components of the pixel at the locus $(x,y)$. After computing $g(x,y)$ for all image loci the corresponding feature image is smoothed using the Alternating Sequential Filter (a sequence of morphological closing and opening operations), and the binary mask $\mathbf{M}$ computed with $M_{x,y} = 1$ iff $G_{x,y}$ exceeds a predefined threshold, and $M_{x,y} = 0$ otherwise. Examples are shown in Figure~\ref{f:glare}.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth,height=1.1\columnwidth]{glare.png}
\caption{ Glare detection result examples: original images (left), and the corresponding binary glare masks (centre) and images with glare removed (right). }
\label{f:glare}
\end{figure}
\subsection{Salient information extraction}
Having removed confounding image content (glare, background, etc.) the processed frames are ready for the main part of our algorithms with concerns the tracking of salient structures present in the fundus, and the subsequent stitching. However, notwithstanding the significant processing described thus far, these tasks are still far from straightforward. For example, note that neither simple appearance nor any of the commonly used local features can be used for tracking (or, equivalently, registration). Raw appearance changes greatly due to global illumination changes, as well as local changes caused by the inherent optics of the eye. Local features -- such as appearance based SIFT \cite{Lowe2004} and SURF \cite{BayEssTuytGool2008}, or edge based ones \cite{Aran2012f} -- can also not be reliably detected as the appearance of the fundus does not contain corner-like loci.
Our approach is specifically targeted at the type of appearance content of interest in fundal images, namely blood vessels. These are line like features which form complex and characteristic structures, and by virtue of this allow different frames to be mutually registered.
We extract characteristic vessel based features using the method introduced by Frangi \textit{et al}.\ \cite{FranNiesVincVier1998}. Their so-called vesselness filter, first proposed for the use on 3D MRI data but later successfully employed in a number of different applications \cite{GhiaAranBendMald2013a,GhiaAranBendMald2013,GhiaAranBendMald2014a}, extracts tubular structures from an image. For a 2D image consider the two eigenvalues $\lambda_1$ and $\lambda_2$ of the Hessian matrix computed at a certain image locus and at a particular scale. Without loss of generality let us also assume that $|\lambda_1| \leq |\lambda_2|$. The two key values used to quantify how tubular the local structure at this scale is are $\mathcal{R}_\mathcal{A} = |\lambda_1|/|\lambda_2|$ and $\mathcal{S} = \sqrt{\lambda_1^2 + \lambda_1^2}$. The former of these measures the degree of local 'blobiness'. If the local appearance is blob-like, the Hessian is approximately isotropic and $|\lambda_1|\approx|\lambda_2|$ making $\mathcal{R}_\mathcal{A}$ close to 1. For a tubular structure $\mathcal{R}_\mathcal{A}$ should be small. On the other hand, $\mathcal{S}$ ensures that there is sufficient local information content at all: in nearly uniform regions, both eigenvalues of the corresponding Hessian will have small values. For a particular scale of image analysis $s$, the two measures, $\mathcal{R}_\mathcal{A}$ and $\mathcal{S}$, are then unified into a single vesselness measure:
{\small\begin{align}
\mathcal{V}(s) =
\begin{cases}
0 &~ \text{if } \lambda_2 > 0\\
(1-e^{-\frac{\mathcal{R}_\mathcal{B}}{2\beta^2}}) \times (1-e^{-\frac{\mathcal{S}}{2c^2}}) &~ \text{otherwise},
\end{cases}
\end{align}}
where $\beta$ and $c$ are the parameters that control the sensitivity of the filter to $\mathcal{R}_\mathcal{A}$ and $\mathcal{S}$. Finally, if an image is analyzed across scales from $s_{min}$ to $s_{max}$, the vesselness of a particular image locus can be computed as the maximal vesselness across the range:
{\small\begin{align}
\mathcal{V}_0 = \max_{s_{min} \leq s \leq s_{max}} \mathcal{V}(s)
\end{align}}
We empirically set the parameter values to $\beta= 0.75$ and $c = 15$, and process frames at scales $s=3,4,5$. Examples of typical results are shown in Figure~\ref{f:frangi}.
\subsubsection{Algorithm initialization}
Recall that our algorithm starts the tracking (and thus the stitching process) from an automatically frame detected as the most reliable one i.e.\ one that contains sufficient salient information content to facilitate robust registration with the subsequent frames, and free of motion or out of focus blur. The vessel extraction method we just described is used to this end. In particular, following the computation of the vesselness image, the richness of salient information content is quantified by computing the entropy of this image. Input frames which do not contain many blood vessels, or which are corrupted by blur, will not produce many vessel detections, and thus result in low entropy vesselness images. Therefore, the frame with the highest entropy of the corresponding vesselness image is selected as the starting frame from which tracking commences, and as the initial synthetic image which is subsequently extended and improved though merging with newly processed frames.
\subsection{Image registration}
Recall that our algorithm incrementally expands the synthetic image of the fundus by expanding it by (stitching with) newly processed frames. The first step in the stitching process concerns the registration of a frame with the synthetic image. This is achieved by using the extracted vessel feature images. Specifically, we adopt the use of normalized cross-correlation as the hypothesis matching criterion \cite{AranPhamVenk2015c}, and exhaustively (though in a coarse to fine fashion, to speed up the process) explore the space of different scaling and translation parameters. The combination of the parameters which results in the highest normalized cross-correlation is accepted as the correct hypothesis but only if matching goodness exceeds that of the second best hypothesis (the second highest local normalized cross-correlation maximum) by a significant enough margin (herein we used a factor of 1.5). If this condition is not satisfied (which can happen, for example, when the newly acquired frame does not contain enough information to allow confident registration) the frame is discarded as unreliable.
\subsection{Incremental stitching}
Having registered a new frame with the growing synthetic image, the last step of our algorithm concerns the merging of the two images and with it the expansion of the synthetic result. There are several factors which make the process difficult. Firstly and as we noted before, there may be a significant illumination discrepancy between the two images. Moreover, a gradual darkening of the region of interest in the proximity of its boundary can be readily observed; note that this is a feature of the imaging process, rather than an artefact introduced by any of the steps of our algorithm. Lastly, the regions corresponding to glare areas, void of useful information, should be taken into account.
Ideally, the result of the merging process should not only produce a seamless output image, spatially expanded (in general), but also leverage information content from both input images to increase the quality of information in the overlapping areas too. To achieve this while at the same addressing the challenges summarized above, we adopt an adaptive weighting strategy whereby the relative contributions of the two input images are adjusted depending on the particular locus under the consideration. To motivate out approach intuitively, the idea is to use preferentially data from an image in which a particular location is more central (relative to the region of interest) i.e.\ where the optical setup of mini ophthalmoscopies and the eye itself, produce better quality data (higher signal to noise ratio). We formalize this through the use of the distance transform \cite{Borg1984,Aran2012b}. Specifically, after a linear illumination transform of brightness which normalizes global changes, for each pixel in an image we compute its approximate distance from the boundaries of the region of interest (the outer boundary and, if present, the boundary of the glare region). Then, when combining two corresponding pixels their relative contributions are computed by considering the inverse value of their distances from the boundaries in the original images, i.e.\ the output value of a RGB channel $c_{x,y}$ becomes:
\begin{align}
c_{x,y}= \frac{w_f c^f_{x,y} + w_s c^s_{x,y}}{w_f w_s}
\end{align}
where $c^f_{x,y}$ is the value of the pixel in a new frame and $c^s_{x,y}$ in the synthetic image, and:
\begin{align}
w_f = 1/d^f_{x,y} && w_s = 1/d^s_{x,y}
\end{align}
where $d^f_{x,y}$ and $d^s_{x,y}$ are the corresponding distance transform values (see Figure~\ref{f:dt}). The process can be therefore described as a form of adaptive linear fusion \cite{AranCipo2006a,AranHammCipo2010,Aran2016d}.
\begin{figure*}
\centering
\includegraphics[width=1.85\columnwidth,height=.57\columnwidth]{dt.png}
\caption{ Original frame (left), region of interest (after the removal of the background and the glare), and the corresponding distance transform used for adaptive pair-wise image merging. }
\label{f:dt}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.99\textwidth]{results_01.png}
\caption{ Examples of synthetically generated images. }
\label{f:res1}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.99\textwidth]{results_02.png}
\caption{ Examples of image fusion with the specific emphasis on the handling of glare. }
\label{f:res2}
\end{figure*}
\section{Evaluation}
The data corpus we used in this work was collected from healthy volunteers. The acquisition process was entirely unaffected and unguided by us, guaranteeing realistic video input as would be collected in the field. Thus, we found that the sequences were highly variable in length, lasting from approximately 20 up to 90 seconds. The majority of the sequences start with a distant view of the eye (and hence little useful fundal information), and progressively close in onto the areas of actual interest such as blood vessels and the optic disc. Major motion and out of focus blur can be observed in this early phase. Once a section of the retina is clear the acquisition becomes more controlled, exhibiting purposeful motion in an effort to capture as much of the fundus as possible. Only a small section can be seen at any given moment due to the narrow field of view of direct ophthalmoscopy devices. Throughout this process blur continues to pose problems, with the retina easily going out of focus, drastic changes in the direction of motion occurring, and the imaging stopping unexpectedly. Considering the nature of our data set, it was not possible to obtain objective ground truth high quality images using a traditional ophthalmoscope. Hence, at this stage we limited ourselves to a qualitative assessment of the results. Representative examples of synthetic images and the effects of specific challenges are shown in Figures~\ref{f:res1} and~\ref{f:res2}.
\section{Summary and conclusions}\label{s:summary}
The Arclight is a recently developed low-cost direct ophthalmoscope that offers great potential for the prevention of blindness amongst those living in low and middle income countries. However, currently this potential is limited by the inherent optical limitations of direct ophthalmoscopy, most notably the narrow field of view. In this paper we described the first method that uses this low quality data to create a synthetic, higher quality, wider view image comparable to one acquired using expensive and bulky traditional retinal cameras.
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Diluted earnings per share for the first quarter of fiscal increased Cover by James Harren. Especially not when that god is Thor. Thor is a American superhero film based on the Marvel Comics character of the same name, produced by Marvel Studios and distributed by Paramount Pictures. This is a really good launch to the new "Thor" series, a big issue with two cool stories. The artifacts of Asgard have been scattered across the earth, and to reclaim them, Thor …Thor Windsport 29m Model First Registration: THO has enjoyed a string of successful quarters -- most recently in the first quarter of fiscal three months ended Oct.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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//
// PPScanningViewController.h
// BarcodeFramework
//
// Created by Jurica Cerovec on 14/11/13.
// Copyright (c) 2015 MicroBlink Ltd. All rights reserved.
//
#import <Foundation/Foundation.h>
#import <UIKit/UIKit.h>
@protocol PPScanningDelegate;
NS_ASSUME_NONNULL_BEGIN
/**
* Protocol for View controllers which present camera and provide scanning features
*/
@protocol PPScanningViewController <NSObject>
@property (nonatomic, weak) id<PPScanningDelegate> scanningDelegate;
/**
* Scanning region
* Defines a portion of the screen in which the scanning will be performed.
* Given as a CGRect with unit coordinating system:
*
* @example CGRectMake(0.2f, 0.5f, 0.4f, 0.3f) defines a portion of the screen which starts at
* 20% from the left border
* 50% from the top
* covers 40% of screen width
* and 30% of screen heeight
*/
@property (nonatomic) CGRect scanningRegion;
/**
* ScanningViewController's shouldAutorotate will return this value.
*
* Default: NO.
*
* Set it to YES if you want scanning view controller to autorotate.
*/
@property (nonatomic) BOOL autorotate;
/**
* ScanningViewController's supportedInterfaceOrientations will return this value.
*
* Default: UIInterfaceOrientationMaskPortrait.
*/
@property (nonatomic) UIInterfaceOrientationMask supportedOrientations;
/**
* Pause scanning without dismissing the camera view.
*
* If there is camera frame being processed at a time, the processing will finish, but the results of processing
* will not be returned.
*
* @warning must be called from Main thread to ensure thread synchronization
*/
- (BOOL)pauseScanning;
/**
* Retrieve the current state of scanning.
*
* @return YES if scanning is paused. NO if it's in progress
*
* @warning must be called from Main thread to ensure thread synchronization
*/
- (BOOL)isScanningPaused;
/**
* Resumes scanning. Optionally, internal state of recognizers can be reset in the process.
*
* If you continue scanning the same object, for example, the same slip, or the same MRTD document, to get result
* with higher confidence, then pass NO to reset State.
*
* If you move to scan another object, for example, another barcode, or another payment slip, then pass YES to reset State.
*
* Internal state is used to use the fact that the same object exists on multiple consecutive frames, and using internal
* state provides better scanning results.
*
* @param resetState YES if state should be reset.
*
* @warning must be called from Main thread to ensure thread synchronization
*/
- (BOOL)resumeScanningAndResetState:(BOOL)resetState;
/**
* Resumes camera session. This method is automatically called in viewWillAppear when ScanningViewController enters screen.
*/
- (BOOL)resumeCamera;
/**
* Pauses camera session. This method is automatically called in viewDidDissapear when ScanningViewController exits screen.
*/
- (BOOL)pauseCamera;
/**
* Retrieve the current state of camera.
*
* @return YES if camera is paused. NO if camera is active
*/
- (BOOL)isCameraPaused;
/**
* Play scan sound.
*
* It uses default scan sound, you can change it by setting your own soundFilePath in PPUiSettings.
*/
- (void)playScanSuccesSound;
/**
* Call to turn on torch without camera overlay
*/
- (void)willSetTorchOn:(BOOL)torchOn;
@end
NS_ASSUME_NONNULL_END
|
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"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/doc.sagemath.org\/html\/en\/reference\/spkg\/meataxe.html","text":"# meataxe: Library for computing with modular representations\u00b6\n\n## Description\u00b6\n\nSharedMeatAxe 1.0 is an autotoolized shared library version of C MeatAxe 2.4.24, a set of programs for computing with modular representations. The package comprises a shared library \u201clibmtx\u201d, as well as several executables.\n\nSee http:\/\/users.minet.uni-jena.de\/~king\/SharedMeatAxe\/ for the package documentation.\n\n## Licence\u00b6\n\nThe Shared Meat-Axe is free software: you can redistribute it and\/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version. See the file COPYING.\n\noptional\n\n## Version Information\u00b6\n\npackage-version.txt:\n\n1.0.1\n\n\n## Equivalent System Packages\u00b6\n\narch:\n\n$sudo pacman -S shared-meataxe sharedmeataxe Fedora\/Redhat\/CentOS: $ sudo yum install shared-meataxe sharedmeataxe\n\n\nHowever, these system packages will not be used for building Sage because spkg-configure.m4 has not been written for this package; see https:\/\/trac.sagemath.org\/ticket\/27330","date":"2021-07-25 23:05:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.29292017221450806, \"perplexity\": 5932.9206080603735}, \"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-2021-31\/segments\/1627046151866.98\/warc\/CC-MAIN-20210725205752-20210725235752-00183.warc.gz\"}"}
| null | null |
Purpose of collecting information is to provide better services to all of our users. We use the information we collect from all of our services to provide, maintain, protect and improve them, to develop new ones, and to protect Okorana and our users.
cookies that may uniquely identify your browser or your Okorana Account.
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Users have different privacy concerns. Our goal is to be clear about what information we collect, so that you can make meaningful choices about how it is used. We offer you choices regarding the collection, use and sharing of your privacy and we'll respect the choices you make.
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|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,683
|
Hans Ronald Mueck (ur. 1958 w Melbourne) – australijski rzeźbiarz hiperrealistyczny.
Mueck swoją karierę rozpoczął jako twórca modeli i kukiełkarz dla telewizji dziecięcej i filmów, pracował przy filmie Labirynt, gdzie również udzielił głosu postaci Ludo. Założył własną firmę w Londynie, przygotowującą fotorealistyczne rekwizyty i animatronics dla przemysłu reklamowego. Zwykle rekwizyty były projektowane bardzo szczegółowo, choć tak, aby były filmowane tylko pod pewnym kątem, ukrywając niedociągnięcia widoczne pod innym kątem. Mueck zapragnął tworzyć realistyczne rzeźby, które można by oglądać z każdej strony.
W 1996 Mueck rozpoczął współpracę z matką swojej żony, Paulą Rego tworząc niewielkie figurki będące częścią pracy jaką wystawiała w Hayward Gallery. Rego zapoznała go z Charlesem Saatchi, na którym jego prace zrobiły duże wrażenie i który zaczął je zamawiać i kolekcjonować. To doprowadziło do powstania pracy, która przyniosła mu rozgłos, "Dead Dad" (pol. Martwy Ojciec), praca ta została włączona do wystawy Sensation w Royal Academy w 1997. Do stworzenia "Dead Dad", Mueck wykorzystał silikon i inne materiały, rzeźba przedstawia zwłoki ojca Muecka, zredukowane do dwu trzecich realnej wielkości.
Hiperrealistycznie rzeźby Muecka wiernie odtwarzają najdrobniejszy szczegół ludzkiego ciała, jednocześnie przedstawiając je w zmienionej skali (nienaturalnie powiększone lub zmniejszone) co stwarza niepokojący wizualny efekt. Jego praca "Boy" (pol. Chłopiec), mająca ponad pięć metrów, wystawiana była w Millenium Dome, a następnie na biennale w Wenecji.
W roku 2002 rzeźba "Pregnant Woman" (pol. kobieta w ciąży) została zakupiona przez National Gallery of Australia za 800 tys. dolarów australijskich.
Wybrane prace
"Dead Dad" (1996-1997) – Martwy Ojciec, silikon, farba akrylowa ludzkie włosy – przedstawiająca nagie ciało ojca autora ułożone na plecach, 2/3 naturalnej wielkości – Saatchi Collection
"Boy" (1999) – Chłopiec, włókno szklane, żywica, silikon – 5-metrowa rzeźba chłopca. Na początku pokazywana w Millenium Dome. Obecnie jest w posiadaniu Muzeum Aros z Aarhaus, Dania
"Pregnant Woman" (2002) – Kobieta w ciąży, włókno szklane, żywica, silikon – 2,5-metrowa rzeźba nagiej ciężarnej kobiety zaciskającej dłonie nad głową
"Wild Man" (2005) – Dzikus, 3-metrowa postać nagiego brodatego mężczyzny ściskającego krawędzie krzesła na którym siedzi,
"Two Women Grim" (2005) – Dwie Ponure kobiety, dwie pomniejszone w skali figurki strasznych, ubranych kobiet.
Mueck, Ron
Hiperrealiści
Ludzie urodzeni w Melbourne
|
{
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| 4,949
|
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/41733\/circle-detection-using-invariance-kernels","text":"# Circle detection using invariance kernels\n\nI'm not sure, if this belongs more to the signal processing forum or if I can post it in here, but I want to solve it via mathematica and it's some kind of image-processing I guess.\n\nI have a noisy picture with a circle in it:\n\nNow I want to detect the circle, by using so called invariance Kernels like a Fourier-Mellin transform. In short, I want to replicate what the authors of this paper (see Section 4) did in mathematica. So what they did, was using Fourier-Mellin transform to detect a noisy image with a circle in it. I have some troubles grasping the mathematics behind it and I hope someone could help me out in understanding and implementing it in a mathematica code.\n\n\u2022 @holistic To get help for this kind of \"where to even begin\" type problem it should be original compared to already existing similar answers in this forum. If you little search this forum you will find many image feature detection related questions and plenty of them have excellent answers to help you with THE start. \u2013\u00a0PlatoManiac Feb 5 '14 at 13:10\n\u2022 I think we need to be careful here. The reason why there was a problem with the link is because the paper is copyrighted by the publisher and not available for free. Does the OP have the rights to provide free access to this paper through M.SE and his\/her dropbox account? \u2013\u00a0bobthechemist Feb 6 '14 at 0:06\n\u2022 @bobthechemist I'm sure that the authors of the paper wouldn't mind, and I think there are many of us who are quite unhappy about the behaviour of scientific publishers. \u2013\u00a0Szabolcs Feb 6 '14 at 0:52\n\u2022 @Szabolcs even the authors can't distribute on their own site. They transfer the publishing copyright to publisher. I think we are all unhappy except publishers. .. \u2013\u00a0s.s.o Feb 6 '14 at 1:10\n\u2022 @s.s.o actually, reading through Elsevier's guidelines, I disagree. They can, with caveats, post on a website, just not the final form. \u2013\u00a0rcollyer Feb 6 '14 at 15:03\n\nThis is based on a reading of that paper. It's a ways from my areas and I do not claim to get it correct, but the code might be of use in any case.\n\nImport the image.\n\nim = Import[\"http:\/\/i.stack.imgur.com\/POS2E.jpg\"]\n\n\nGet the basic data.\n\nimat = ImageData[im];\nidims = ImageDimensions[im];\n\n\nPrepare a kernel matrix to convolve with. I start with a disk.\n\nkmatinit = DiskMatrix[Min[idims]\/2];\nklen = Length[kmatinit];\nposns = Flatten[Outer[List, Range[klen], Range[klen]], 1];\nnf = Nearest[posns];\n\n\nNow empty the center. Then I'll put concentric rings at different radii into different phases, that is, multiply by the jth ring by Exp[2*I*Pi*j\/len] where len is the number of rings.\n\nmid = Ceiling[klen\/2];\nminr = 5;\nmaxr = mid;\nrrange = maxr - minr;\nrvals = Exp[Range[0, rrange - 1]*2.*Pi*I\/(rrange)];\n\nkmat = kmatinit;\npts = nf[{mid, mid}, {Infinity, minr}];\nMap[(kmat[[#[[1]], #[[2]]]] = 0) &, pts];\nDo[\nnewpts = nf[{mid, mid}, {Infinity, minr + j}];\nnewpts = Complement[newpts, pts];\nMap[(kmat[[#[[1]], #[[2]]]] *= rvals[[j]]) &, newpts];\npts = Join[pts, newpts];\n, {j, rrange}];\n\n\nHere is how it looks.\n\nMatrixPlot[kmat]\n\n\nNow convolve. If I understand correctly, the largest value should be in the vicinity of the circle's center (adjust due to convolution dimensions), and its phase should give us the radius.\n\nlc = Chop[ListConvolve[kmat, imat, {1, -1}, 0]];\nDimensions[lc]\nMax[Abs[lc]]\nabslc = Max[Abs[lc]] - Abs[lc];\npos = Position[abslc, 0.]\np1 = pos[[1, 1]];\np2 = pos[[1, 2]];\nlcpos = lc[[p1, p2]];\narg = lcpos\/Abs[lcpos]\n\n(* {200, 200}\n949.436832261\n{{97, 94}}\n-0.336662959549 - 0.941625218262 I *)\n\n\nSo the center should be in the whereabouts of {97,94}\/2. For the radius:\n\nIn[286]:= Nearest[rvals -> Range[Length[rvals]], arg]\n\nOut[286]= {33}\n\nIt is the 33rd value in our set. We began at radius of 6 so this indicates our radius is 6+33-1 or 38. From the picture and the Manipulate from @Stephen Luttrell I believe it is actually around 40-41 or so, so this is not a terrible outcome. That said, I confess I don't really know what I'm doing.\n\n--- edit ---\n\nHere are some additional remarks, partly in response to a question.\n\n(1) I make no claim to actually understanding that paper.\n\n(2) The multiplications are so that concentric rings in the kernel array will have different phases. The idea is that rings (circles) that are not of the right radius and\/or not centered with the circle we're seeking will tend to give noise that partly cancels. The ring of correct radius, when correctly centered, will all be in phase and not cancel with itself. We later use the phase information to figure out which radius ring this was.\n\nIt falls under the \"phase coded annulus\" description in the actual paper.\n\n(3) There are probably flaws insofar as the different rings will be weighted according to their radii, so there may be a bias toward larger circles. I was not able to adjust for that in any way that worked well.\n\n(4) If in doubt as to whether what I did makes sense, refer to remark (1). But rereading section 1.2.3 gives me the impression that at least I got the phase part correct.\n\n--- end edit ---\n\n\u2022 In your third code block, should kmatinit be kmat? kmat is not initialized. \u2013\u00a0Szabolcs Feb 7 '14 at 19:32\n\u2022 @Szabolcs Yes. Now fixed. \u2013\u00a0Daniel Lichtblau Feb 7 '14 at 19:46\n\u2022 Thanks @DanielLichtblau. Why did you multiply it with Exp[2*IPij\/len]? To which part of the paper did you refer? \u2013\u00a0holistic Feb 10 '14 at 18:56\n\u2022 I added some remarks that may help to explain.. \u2013\u00a0Daniel Lichtblau Feb 10 '14 at 19:13\n\u2022 @DanielLichtblau you're not happy until you've sneaked in some Do, are ye? \u2013\u00a0Yves Klett Feb 10 '14 at 20:19","date":"2021-06-24 00:39:06","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\": 0.3571324050426483, \"perplexity\": 1732.3098926546884}, \"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-2021-25\/segments\/1623488544264.91\/warc\/CC-MAIN-20210623225535-20210624015535-00523.warc.gz\"}"}
| null | null |
Marilyna - rodzaj ryb rozdymkokształtnych z rodziny rozdymkowatych.
Klasyfikacja
Gatunki zaliczane do tego rodzaju :
Marilyna darwinii
Marilyna meraukensis
Marilyna pleurosticta
Przypisy
Rozdymkowate
|
{
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|
Q: Finding the region in sample data that fits to a known distribution Given sample data that have a known probability distribution for large x and an unknown probability distribution for small x, what is the best way to find the value of x where the known probability distribution begins? I imagine trying a bunch of qq plots and then seeing where the best fits are would be how to go about this, but I'm wondering how to automate this / the best way of doing this, preferably in MATLAB. (I hope this makes sense. I'm not a mathematician / statistician.)
A: There several possible approaches. One is to optimize a goodness-of-fit criterion like the Kolmogorov-Smirnov (max CDF difference) or the Cramer-van-Mieses (L1 differences between CDFs) statistics. As you are interested, however, in the region $x>x_0$, you can just use the complementary CDF $\overline{F}(x) = P(X>x)=1-F(x)$. Then the Kolmogorov-Smirnov statistics, e.g., becomes
$$KS = \max_{x>x_0} | \overline{F}_{emp}(x) - \overline{F}_{fit}(x)|$$
where $\overline{F}_{emp}$ is the complementary CDF directly estimated from the data, and $\overline{F}_{fit}$ is the complementary CDF computed from the estimated distribution parameters.
Another approach, which could be combined with the above method, would be to demand that the probability density is continuous at the transition, i.e.
$$\hat{f}_{left}(x_0) = \hat{f}_{right}(x_0)$$
where $\hat{f}_{left}$ is estimated in a non-parametric way with a kernel-density estimator and $\hat{f}_{right}$ is estimated parametrically from your thereetical distribution.
For a use case, see sections 4.1 and 4.3 of the following study (the threshold is named $w_0$):
Dalitz: "Estimating Wealth Distribution: Top Tail and Inequality." Technical Report No. 2016-01, Hochschule Niederrhein, Fachbereich Elektrotechnik und Informatik, 2016
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,983
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Use Windows Server 2016 capabilities to meet operational and security challenges, freeing up IT resources to plan a strategy that uses the cloud for future applications and solutions.
Windows Server 2016 includes built-in breach resistance to help thwart attacks on your systems and meet compliance goals. Help prevent malicious attacks and detect suspicious activity where it matters: your operating system and workloads. Rely on security features built into Windows Server 2016 to control privileged access, protect virtual machines, and harden the platform against emerging threats.
The Azure-inspired, software-defined storage capabilities in Windows Server 2016 use policies and automation to reduce costs and add scale.
Bring flexibility and control to your datacentre while lowering costs with software-defined compute, storage, and networking. Run your datacentre with an automated and resilient operating system, and access many of the same cloud-efficient features found in Microsoft Azure datacentres.
Windows Server 2016 supports application innovation using container technology and microservices. Containers can help speed application deployment and streamline the way IT operations and development teams collaborate to deliver applications.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,149
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how to get a c1?
could anyone tell me waht i have to do to get the C1 needed to be able to drive a motorhome? my hubby can already drive one and i also want to be able to share the driving - just unsure what the test involves if i need lessons - like with a car?
If you passed your driving test on or after 1 January 1997 and you want to learn to drive a larger vehicle, you may be able to apply for the higher category. Once you have obtained provisional entitlement on your driving licence you can take the test in that category. For example, a driver must hold a full car licence (category B) before being able to apply for a provisional licence to drive a lorry, minibus or bus.
This is known as staging and will apply from the smallest to the largest vehicles.
Get yourself a provisional licence then go to a driving school for advice, pass your medical and eyesight test - (if you need one), pass the theory test, take some lessons and then take the driving test.
What the gross weight of your bus?
Surely over 3,500kgs needs an E license?
Hi the C1 test needs exactly the same medical and test standard as the C test which I took 2 years ago except you do it in an under 7.5ton vehicle.
then down to RTT and use their computers for the theory practice, then after a few weeks do the theory test, fairly easy as long as you have swotted up on the highway code.
The "E" is a supplement to a licence that allows you to tow various weight trailers.
Hi freakylady, so was I, hardly slept for the week I was doing the intensive driving, but after it was over, I wondered what I worried about, I mean it not as if my job depended on it.
Tester was very chatty and called Ian same as me, so I knew he was a good guy, Asked me if I wasn't a bit old for a change of career, cheeky sod.
If you fail you just take it again a week later after 1 more days training, that's what RTT (Rolling Transport Training) told me anyway, they block book tests so they can fit you in very quickly. I would imagine other training firms do the same.
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{
"redpajama_set_name": "RedPajamaC4"
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\section*{ }
It is shown that a given non-autonomous system of two first-order
ordinary differential equations can be expressed in Hamiltonian
form. The derivation presented here allow us to obtain previously
known results such as the infinite number of Hamiltonians in the
autonomous case and the Helmholtz condition for the existence
of a Lagrangian.\\[1ex]
{\it Keywords:} Non-autonomous systems; Hamilton equations;
Lagrangians \\[2ex]
Se muestra que un sistema dado, no aut\'onomo, de ecuaciones
diferenciales ordinarias de primer orden puede expresarse en forma
hamiltoniana. La deducci\'on presentada aqu\'{\i} nos permite
obtener resultados previamente conocidos tales como el n\'umero
infinito de hamiltonianas en el caso aut\'onomo y la
condici\'on de Helmholtz para la existencia de una lagrangiana.\\[1ex]
{\it Descriptores:} Sistemas no aut\'onomos; ecuaciones de Hamilton;
lagrangianas \\[2ex]
PACS: 45.05.+x; 45.20.-d
\section*{\tres 1. Introduction}
As is well known, it is very convenient to express a given system of
ordinary differential equations (not necessarily related to
classical mechanics) as the Euler--Lagrange equations associated
with some Lagrangian, $L$, or as the Hamilton equations associated
with some Hamiltonian, $H$ (see, {\em e.g.}, Ref.\ 1). One of the
advantages of such identifications is the possibility of finding
constants of motion, which are related to symmetries of $L$ or $H$.
Also, the Hamiltonian of a classical system is usually regarded as
an essential element to find a quantum version of the mechanical
system.
In the simple case of a mechanical system with forces derivable from
a potential (that may depend on the velocities), there is a
straightforward procedure to find a Lagrangian or a Hamiltonian.
However, in the case of non-conservative mechanical systems or of
systems not related to mechanics, the problem of finding a
Lagrangian or a Hamiltonian is more involved. A given system of $n$
second-order ordinary differential equations are equivalent to the
Euler--Lagrange equations for some Lagrangian if and only if a set
of conditions (known as the Helmholtz conditions) are fulfilled
(see, {\em e.g.}, Refs.\ 2, 3 and the references cited therein).
The aim of this paper is to give a straightforward procedure to find
a Hamiltonian for a given system of two first-order ordinary
differential equations (which may not be equivalent to a
second-order ordinary differential equation) that possibly involves
the time in an explicit form. The results derived here contain the
Helmholtz condition for $n = 1$ (in the case where the given system
is equivalent to a second-order equation). In Sec.\ 2 the main
results of this paper are established, demonstrating that a given
system of first-order ordinary differential equations can be
expressed in Hamiltonian form looking for an integrating factor of a
differential form made out of the functions contained in the system
and, in Sec.\ 3, several examples are presented. In Sec.\ 4 we show
that, in the appropriate case, our results lead to the Helmholtz
condition for the existence of a Lagrangian.
\section*{\tres 2. Hamiltonians and canonical variables}
We shall consider a system of first-order ordinary differential
equations of the form
\begin{equation}
\dot{x} = f(x,y,t), \qquad \dot{y} = g(x,y,t), \label{1}
\end{equation}
where $f$ and $g$ are two given functions. A system of this class
can be obtained from a second-order equation
\[
\ddot{x} = F(x, \dot{x}, t),
\]
by introducing the variable $y = \dot{x}$. We are initially
interested in finding a Hamiltonian function, $H$, and canonical
variables, $q$, $p$, such that the corresponding Hamilton's
equations be equivalent to the system (\ref{1}).
Assuming that there is an invertible relation between the variables
$x$, $y$ and a set of canonical coordinates $q$, $p$, $x =
x(q,p,t)$, $y = y(q,p,t)$, in such a way that Eqs.\ (\ref{1}) are
equivalent to the Hamilton equations for $q$ and $p$ with a
Hamiltonian $H$, making use of the chain rule, one finds that
\begin{equation}
-g {\rm d} x + f {\rm d} y = \frac{\partial(x,y)}{\partial(q,p)}
{\rm d} H - \frac{\partial y}{\partial t} {\rm d} x + \frac{\partial
x}{\partial t} {\rm d} y + {\rm terms\ proportional\ to\ } {\rm d}
t. \label{0}
\end{equation}
Therefore, given the system (\ref{1}) we start by considering the
differential form
\begin{equation}
- (g - \phi) {\rm d} x + (f - \psi) {\rm d} y, \label{2}
\end{equation}
where
\[
\phi(q,p,t) \equiv \frac{\partial y(q,p,t)}{\partial t}, \qquad
\psi(q,p,t) \equiv \frac{\partial x(q,p,t)}{\partial t}
\]
are functions unspecified by now (see Eq.\ (\ref{7.5}) below). For a
fixed value of $t$, the differential form (\ref{2}) is always
integrable (see any standard text on ordinary differential
equations, {\em e.g.}, Ref.\ 4); that is, there exist (locally)
functions $\sigma$ and $H$, which may depend parametrically on $t$,
such that
\begin{equation}
- (g - \phi) {\rm d} x + (f - \psi) {\rm d} y = \sigma {\rm d} H.
\label{3}
\end{equation}
Now, for simplicity, without any loss of generality (since, once we
have found a set of canonical coordinates, we have the liberty of
making any canonical transformation afterwards), we choose $q \equiv
x$ (hence, $\psi = 0$) and, therefore,
\[
\frac{\partial(x,y)}{\partial(q,p)} = \frac{\partial p}{\partial y}.
\]
Then, by comparing Eqs.\ (\ref{0}) and (\ref{3}), the canonical
momentum, $p$, must be such that
\begin{equation}
\frac{\partial p(x,y,t)}{\partial y} = \frac{1}{\sigma(x,y,t)}.
\label{5}
\end{equation}
Hence
\begin{equation}
{\rm d} p = \frac{\partial p}{\partial x} {\rm d} x +
\frac{1}{\sigma} {\rm d} y + \frac{\partial p}{\partial t} {\rm d} t
\label{6}
\end{equation}
or, equivalently,
\begin{equation}
{\rm d} y = - \sigma \frac{\partial p}{\partial x} {\rm d} x +
\sigma {\rm d} p - \sigma \frac{\partial p}{\partial t} {\rm d} t
\label{6.1}
\end{equation}
thus, recalling that $x = q$, this last expression shows that
\begin{equation}
\phi = - \sigma \frac{\partial p(x,y,t)}{\partial t} \label{6.5}
\end{equation}
and we can also write Eq.\ (\ref{6}) in the form
\begin{equation}
{\rm d} p = \frac{\partial p}{\partial x} {\rm d} x +
\frac{1}{\sigma} {\rm d} y - \frac{\phi}{\sigma} {\rm d} t.
\label{6.6}
\end{equation}
Since this is an exact differential, we have
\begin{equation}
\frac{\partial \sigma^{-1}}{\partial t} = \frac{\partial}{\partial
y} (- \sigma^{-1} \phi) = - \sigma^{-1} \frac{\partial
\phi}{\partial y} - \phi \frac{\partial \sigma^{-1}}{\partial y}.
\label{7.5}
\end{equation}
This equation establishes a relation between the integrating factor
and the function $\phi$ (see examples below).
From Eqs.\ (\ref{3}), with $\psi = 0$, and (\ref{6.6}) we have
\begin{eqnarray*}
{\rm d} H \!\!\! & = & \!\!\! - \frac{1}{\sigma} (g - \phi) {\rm d}
x + \frac{1}{\sigma} f {\rm d} y + \frac{\partial H}{\partial t} {\rm d} t \\
& = & \!\!\! - \frac{1}{\sigma} (g - \phi) {\rm d} x + f \left( {\rm
d} p - \frac{\partial p}{\partial x} {\rm d} x + \frac{\phi}{\sigma}
{\rm d} t \right) + \frac{\partial H}{\partial t} {\rm d} t \\
& = & \!\!\! - \left( \frac{g}{\sigma} - \frac{\phi}{\sigma} + f
\frac{\partial p}{\partial x} \right) {\rm d} q + f {\rm d} p +
\left( \frac{\partial H}{\partial t} + f \frac{\phi}{\sigma} \right)
{\rm d} t.
\end{eqnarray*}
Hence, considering $H$ as a function of $q$, $p$, and $t$,
\begin{equation}
\frac{\partial H}{\partial p} = f = \dot{q} \label{7}
\end{equation}
[see Eqs.\ (\ref{1})] and
\begin{equation}
- \frac{\partial H}{\partial q} = \frac{g}{\sigma} -
\frac{\phi}{\sigma} + f \frac{\partial p}{\partial x} = \dot{p},
\label{8}
\end{equation}
since, according to Eqs.\ (\ref{6.6}) and (\ref{1}),
\[
\dot{p} = \frac{\partial p}{\partial x} \dot{x} +
\frac{\dot{y}}{\sigma} - \frac{\phi}{\sigma} = \frac{\partial
p}{\partial x} f + \frac{g}{\sigma} - \frac{\phi}{\sigma}.
\]
Equations (\ref{7}) and (\ref{8}) are equivalent to the original
system (\ref{1}) and have the desired Hamiltonian form.
Summarizing, the system of equations (\ref{1}) can be written in the
form of the Hamilton equations, with the Hamiltonian determined by
Eq.\ (\ref{3}) and the canonical momentum defined by Eq.\
(\ref{6.6}).
The fact that the left-hand side of Eq.\ (\ref{3}) multiplied by
$\sigma^{-1}$ is an exact differential yields (when $\psi = 0$)
\[
\frac{\partial}{\partial y} [- \sigma^{-1}(g - \phi)] =
\frac{\partial}{\partial x} (\sigma^{-1}{f}),
\]
which amounts to
\begin{equation}
(g- \phi) \frac{\partial \sigma^{-1}}{\partial y} + \sigma^{-1}
\frac{\partial}{\partial y} (g - \phi) + f \frac{\partial
\sigma^{-1}}{\partial x} + \sigma^{-1} \frac{\partial f}{\partial x}
= 0. \label{7.6}
\end{equation}
Hence, making use of Eqs.\ (\ref{1}), (\ref{7.6}) and (\ref{7.5}),
we obtain
\begin{eqnarray}
\frac{{\rm d}}{{\rm d}t} \sigma^{-1} \!\!\! & = & \!\!\!
\frac{\partial \sigma^{-1}}{\partial x} \dot{x} + \frac{\partial
\sigma^{-1}}{\partial y} \dot{y} + \frac{\partial
\sigma^{-1}}{\partial t} \nonumber \\
& = & \!\!\! f \frac{\partial \sigma^{-1}}{\partial x} + g
\frac{\partial \sigma^{-1}}{\partial y} + \frac{\partial
\sigma^{-1}}{\partial t} \nonumber \\
& = & \!\!\! \phi \frac{\partial \sigma^{-1}}{\partial y} -
\sigma^{-1} \frac{\partial}{\partial y} (g - \phi) - \sigma^{-1}
\frac{\partial f}{\partial x} + \frac{\partial
\sigma^{-1}}{\partial t} \nonumber \\
& = & \!\!\! - \sigma^{-1} \left( \frac{\partial f}{\partial x}
+\frac{\partial g}{\partial y} \right). \label{pH}
\end{eqnarray}
(Note the cancelation of $\phi$.)
Equation (\ref{pH}) shows that the function $\sigma$ is determined
up to a factor that is a constant of motion and, therefore, there
exists an infinite number of Hamiltonians (and, correspondingly, of
expressions for $p$). It may be noticed that Eq.\ (\ref{pH}) is just
Liouville's theorem.
\section*{\tres 3. Examples}
A first example is provided by the equation
\[
\ddot{x} + \gamma \dot{x} + \omega_{0}^{2} x = \eta(t),
\]
where $\gamma$ and $\omega_{0}$ are constants, and $\eta(t)$ is an
arbitrary function, which corresponds to a forced damped harmonic
oscillator. Taking $y = \dot{x}$, we have $\dot{y} = - \gamma y -
\omega_{0}^{2} x + \eta(t)$, which is of the form (\ref{1}) with
$f(x,y,t) = y$, and $g(x,y,t) = - \gamma y - \omega_{0}^{2} x +
\eta(t)$. Then Eq.\ (\ref{pH}) reduces to
\[
\frac{{\rm d}}{{\rm d}t} \sigma^{-1} = \gamma \sigma^{-1}
\]
and we can take $\sigma = {\rm e}^{- \gamma t}$ (any other choice
would require the knowledge of the explicit form of $\eta$) then
from Eq.\ (\ref{7.5}) we see that
\[
\frac{\partial \phi}{\partial y} = - \gamma,
\]
which is satisfied with $\phi = - \gamma y$. Substituting all these
expressions into Eq.\ (\ref{3}) we have (with $t$ treated as a
constant)
\[
\big( \omega_{0}^{2} x - \eta(t) \big) {\rm d} x + y {\rm d} y =
{\rm e}^{- \gamma t} {\rm d} H
\]
and, therefore, we can take $H = {\rm e}^{\gamma t} (y^{2}/2 +
\omega_{0}^{2} x^{2}/2 - \eta(t) x)$. Finally, from Eq.\ (\ref{6.6})
we find that $p$ can be chosen as $p = {\rm e}^{\gamma t} y$. The
corresponding Lagrangian can be calculated in the usual way, by
means of the Legendre transformation.
The results of the previous section allow us to readily derive those
of Ref.\ 5, corresponding to the autonomous case. In fact, when the
functions $f$ and $g$, appearing in Eqs.\ (1), do not depend
explicitly on the time, from Eqs.\ (\ref{3}) and (\ref{1}), taking
$\phi = 0 = \psi$, we have $\sigma \dot{H} = - g \dot{x} + f \dot{y}
= -gf + fg = 0$. This means that $H$ is {\em some}\/ constant of
motion, which is not unique; we can replace it by $H' = G(H)$, with
$G$ being an arbitrary function. $H'$ is also a constant of motion
and $\sigma$ will not depend explicitly on $t$ [see Eq.\
(\ref{7.5})], no matter what (time-independent) Hamiltonian we
choose.
The expressions given above allow us to find $H$, which need not be
related to the total energy. In the example considered in the
appendix of Ref.\ 5, $f(x,y) = y$, $g(x,y) = - ky$, where $k$ is a
constant ({\em i.e.}, $\ddot{x} = - k \dot{x}$). Then, $-g {\rm d} x
+ f {\rm d} y = ky {\rm d} x + y {\rm d} y = y {\rm d}(kx + y)$ and,
therefore, we can take $\sigma = y$ and $H = kx + y$.
We end this section by considering the problem studied in Ref.\ 6
(which corresponds approximately to a relativistic particle
subjected to a constant force, $\lambda$, and a force of friction
proportional to the square of the velocity), namely (with the
appropriate changes in notation)
\[
m \dot{y} = (\lambda - \gamma y^{2}) (1 - \alpha^{2} y^{2}),
\]
where $m$ represents a mass, $\lambda$, $\gamma$, and $\alpha$ are
constants. Thus, $f(x,y) = y$, and $g(x,y) = (\lambda - \gamma
y^{2}) (1 - \alpha^{2} y^{2})/m$. Thus,
\begin{eqnarray*}
-g {\rm d} x + f {\rm d} y \!\!\! & = & \!\!\! - \frac{1}{m}
(\lambda - \gamma y^{2}) (1 - \alpha^{2} y^{2}) {\rm d} x + y {\rm
d} y \\
& = & \!\!\! (\lambda - \gamma y^{2}) (1 - \alpha^{2} y^{2}) \left[
- \frac{{\rm d} x}{m} + \frac{y {\rm d} y}{(\lambda - \gamma y^{2})
(1 - \alpha^{2} y^{2})} \right].
\end{eqnarray*}
Comparing with Eq.\ (\ref{3}) (with $\phi = 0 = \psi$) we
immediately see that we can take
\[
\sigma = (\lambda - \gamma y^{2}) (1 - \alpha^{2} y^{2})
\]
and
\begin{eqnarray*}
H \!\!\! & = & \!\!\! - \frac{x}{m} + \int \frac{y {\rm d}
y}{(\lambda - \gamma y^{2}) (1 - \alpha^{2} y^{2})} \\
& = & \!\!\! - \frac{x}{m} + \frac{1}{2(\lambda \alpha^{2} -
\gamma)} \ln \left| \frac{\lambda - \gamma y^{2}}{1- \alpha^{2}
y^{2}} \right|.
\end{eqnarray*}
According to Eq.\ (\ref{6.6}), the canonical momentum $p$ can be
taken as
\[
p = \int \frac{{\rm d} y}{(\lambda - \gamma y^{2}) (1 - \alpha^{2}
y^{2})}.
\]
Despite the huge difference with the expressions given in Ref.\ 6,
one can show that the Hamiltonian obtained in that reference is
essentially the exponential of our $H$. (See, Eqs.\ (23) and (26) of
Ref.\ 6.)
\section*{\tres 4. The Helmholtz condition}
The case where one starts with a second-order equation of the form
\begin{equation}
\ddot{x} = F(x, \dot{x}, t) \label{11}
\end{equation}
(considered in Refs.\ 2, 3), is a particular case of the treatment
above if one defines, {\em e.g.}, $y \equiv \dot{x}$, that
transforms Eq.\ (\ref{11}) into the system
\[
\dot{x} = y, \qquad \dot{y} = F(x,y,t),
\]
which is of the form (\ref{1}) with $f(x,y,t) = y$ and $g(x,y,t) =
F(x,y,t)$. Then Eq.\ (\ref{pH}) reduces to
\begin{equation}
\frac{{\rm d}}{{\rm d}t} \sigma^{-1} = - \sigma^{-1} \frac{\partial
F}{\partial y},
\end{equation}
which is the Helmholtz condition when there is one degree of freedom
(see, {\em e.g.}, Ref.\ 2 and the references cited therein; note
that $\sigma^{-1} = \partial p/ \partial y = \partial p/
\partial \dot{x} = \partial^{2} L / \partial \dot{x}^{2}$ is the
integrating factor $w_{11}$ employed in these references).
On the other hand, not every system of equations of the form
(\ref{1}) comes from a second-order equation $\ddot{x} = F(x,
\dot{x}, t)$. An example is given by
\[
\dot{x} = f(x,t), \qquad \dot{y} = g(y,t),
\]
where there is no coupling between the variables $x$, $y$. Here
(choosing $\phi = 0 = \psi$)
\[
-g {\rm d} x + f {\rm d} y = fg \left( - \frac{{\rm d} x}{f} +
\frac{{\rm d} y}{g} \right).
\]
Therefore, if we assume that $\sigma = fg$ does not depend
explicitly on of $t$ [see Eq.\ (\ref{7.5})], we can take
\[
H = - \int \frac{{\rm d} x}{f} + \int \frac{{\rm d} y}{g}
\]
and, from Eq.\ (\ref{5}),
\[
p = \int \frac{{\rm d} y}{\sigma} = \frac{1}{f} \int \frac{{\rm d}
y}{g}.
\]
Thus, $H = pf - \int f^{-1} {\rm d} x$ and with the Hamiltonian
being a linear function of $p$, the Legendre transformation is not
defined nor the Lagrangian.
\section*{\tres 5. Concluding remarks}
As we have shown, at least in the case of a system of two
first-order ordinary differential equations, finding a Hamiltonian
is essentially equivalent to finding an integrating factor for a
linear differential form in two variables. The integrating factor
also determines the expression for the canonical momentum. Equation
(\ref{pH}) is analogous to the Helmholtz condition, but, in the
present approach, it leads directly to the Hamiltonian (in the
standard approach, finding a solution to the Helmholtz conditions,
only gives the second partial derivatives $\partial^{2} L/\partial
\dot{x}_{i} \partial \dot{x}_{j}$). When the system is
non-autonomous, it is convenient to find the integrating factor
using Eq.\ (\ref{pH}), while in the autonomous case, it may be more
simply obtained from the linear differential form itself. Finally,
as shown in Sec.\ 4, there are systems of equations for which a
Lagrangian does not exist, but a Hamiltonian description can be
given.
\section*{\tres Acknowledgment}
The authors would like to thank Dr.\ M.\ Montesinos for enlightening
discussions.
\section*{References}
\newcounter{ref} \begin{list}{\hspace{1.3ex}\arabic{ref}.\hfill}
{\usecounter{ref} \setlength{\leftmargin}{2em}
\setlength{\itemsep}{-.98ex}}
\item H.\ Goldstein, {\it Classical Mechanics}, 2nd ed., (Addison-Wesley,
Reading, Mass., 1980).
\item S.A.\ Hojman and L.C.\ Shepley, {\it J.\ Math.\ Phys.}\ {\bf
32} (1991) 142.
\item S.K.\ Soni and M.\ Kumar, {\it Europhys.\ Lett.}\ {\bf 68}
(2004) 501.
\item G.F.\ Simmons, {\it Differential Equations with
Applications and Historical Notes}, 2nd ed., (McGraw-Hill, New York,
1991).
\item G.F.\ Torres del Castillo, {\it Rev.\ Mex.\ F\'{\i}s.}\ {\bf 50} (2004) 379.
\item G.\ Gonz\'alez, {\it Int.\ J.\ Theor.\ Phys.}\ {\bf 43} (2004)
1885.
\end{list}
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,957
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Q: bootstrap 5 navbar offcanvas I got the code below from the official bootstrap 5 demos and for the life of me, can't figure out how to move the off-canvas menu from Left-to-Right. The documentation code to place the offcanvas-start and the demo code is completely different.
I am creating a landing page in Bootstrap 5, where the default menu should be replaced by an off-canvas menu with a close icon.
<!doctype html>
<html lang="fa" dir="rtl">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="css/bootstrap.rtl.min.css">
<title>Hello, world!</title>
</head>
<body>
<div class="container">
<!-- Static navbar -->
<nav class="navbar navbar-expand-lg fixed-top navbar-dark bg-dark" aria-label="Main navigation">
<div class="container-fluid">
<a class="navbar-brand" href="#">Offcanvas navbar</a>
<button class="navbar-toggler p-0 border-0" type="button" id="navbarSideCollapse" aria-label="Toggle navigation">
<span class="navbar-toggler-icon"></span>
</button>
<div class="navbar-collapse offcanvas-collapse" id="navbarsExampleDefault">
<ul class="navbar-nav me-auto mb-2 mb-lg-0">
<li class="nav-item">
<a class="nav-link active" aria-current="page" href="#">Dashboard</a>
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<a class="nav-link" href="#">Notifications</a>
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<form class="d-flex">
<input class="form-control me-2" type="search" placeholder="Search" aria-label="Search">
<button class="btn btn-outline-success" type="submit">Search</button>
</form>
</div>
</div>
</nav>
<!-- Main component for a primary marketing message or call to action -->
<div class="jumbotron">
<h1>Offcanvas Navbar example</h1>
<p>This example demonstrates using the offcanvas plugin with the navbar.</p>
</div>
<p>
By default the navbar is show on the right side of the screen. You can show it on the left side instead by
adding <code>.navmenu-fixed-left</code> to the <code>.navbar-offcanvas</code>.
</p>
</div> <!-- /container -->
<script src="js/bootstrap.bundle.min.js"></script>
</body>
</html>
<!doctype html>
<html lang="fa" dir="rtl">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
<title>Hello, world!</title>
</head>
<body>
<div class="container">
<!-- Static navbar -->
<nav class="navbar navbar-expand-lg fixed-top navbar-dark bg-dark" aria-label="Main navigation">
<div class="container-fluid">
<a class="navbar-brand" href="#">Offcanvas navbar</a>
<button class="navbar-toggler p-0 border-0" type="button" id="navbarSideCollapse" aria-label="Toggle navigation">
<span class="navbar-toggler-icon"></span>
</button>
<div class="navbar-collapse offcanvas-collapse" id="navbarsExampleDefault">
<ul class="navbar-nav me-auto mb-2 mb-lg-0">
<li class="nav-item">
<a class="nav-link active" aria-current="page" href="#">Dashboard</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Notifications</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Profile</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Switch account</a>
</li>
<li class="nav-item dropdown">
<a class="nav-link dropdown-toggle" href="#" id="dropdown01" data-bs-toggle="dropdown" aria-expanded="false">Settings</a>
<ul class="dropdown-menu" aria-labelledby="dropdown01">
<li><a class="dropdown-item" href="#">Action</a></li>
<li><a class="dropdown-item" href="#">Another action</a></li>
<li><a class="dropdown-item" href="#">Something else here</a></li>
</ul>
</li>
</ul>
<form class="d-flex">
<input class="form-control me-2" type="search" placeholder="Search" aria-label="Search">
<button class="btn btn-outline-success" type="submit">Search</button>
</form>
</div>
</div>
</nav>
<!-- Main component for a primary marketing message or call to action -->
<div class="jumbotron">
<h1>Offcanvas Navbar example</h1>
<p>This example demonstrates using the offcanvas plugin with the navbar.</p>
</div>
<p>
By default the navbar is show on the right side of the screen. You can show it on the left side instead by adding <code>.navmenu-fixed-left</code> to the <code>.navbar-offcanvas</code>.
</p>
</div>
<!-- /container -->
<script src="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/js/bootstrap.bundle.min.js" integrity="sha384-MrcW6ZMFYlzcLA8Nl+NtUVF0sA7MsXsP1UyJoMp4YLEuNSfAP+JcXn/tWtIaxVXM" crossorigin="anonymous"></script>
</body>
A: You do not need extra CSS and JS, just a tiny bit of inline styling. So all off canvas features and from left to right are in bootstrap itself. Here you go.
<!doctype html>
<html lang="zxx">
<head>
<link rel="icon" type="image/png" href="img/favicon.png">
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="">
<meta name="J. Stulemeijer" content="Mark Otto, Jacob Thornton, and Bootstrap contributors">
<title>Offcanvas Component Navbar Example · Bootstrap v5.2.0 Beta1</title>
<link rel="canonical" href="https://getbootstrap.com/docs/5.0/examples/starter-template/">
<link href="https://cdn.jsdelivr.net/npm/bootstrap@5.2.0-beta1/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-0evHe/X+R7YkIZDRvuzKMRqM+OrBnVFBL6DOitfPri4tjfHxaWutUpFmBp4vmVor" crossorigin="anonymous">
</head>
<body>
<nav class="navbar navbar-expand-lg navbar-dark bg-dark mb-4 fixed-top" id="navbar">
<div class="container-fluid">
<a class="navbar-brand" href="index.html">Bootstrap 5</a>
<button class="navbar-toggler" type="button" data-bs-toggle="offcanvas" data-bs-target="#offcanvasNavbarExample-expand-lg" aria-controls="offcanvasNavbarExample-expand-lg">
<span class="navbar-toggler-icon" data-bs-target="#offcanvasNavbarExample-expand-lg"></span>
</button>
<div class="offcanvas offcanvas-start bg-dark" data-bs-hideresize="true" tabindex="-1" id="offcanvasNavbarExample-expand-lg" aria-labelledby="offcanvasNavbarExample-expand-lg">
<div class="offcanvas-header">
<h5 class="offcanvas-title" id="offcanvasLabel" style="color:white;">Bootstrap 5</h5>
<button type="button" class="btn-close btn-close-white text-reset" data-bs-dismiss="offcanvas" aria-label="Close"></button>
</div>
<div class="offcanvas-body bg-dark">
<ul class="navbar-nav justify-content-end flex-grow-1 pe-3">
<li class="nav-item">
<a class="nav-link active" aria-current="page" href="index.html">Home</a>
</li>
<li class="nav-item">
<a class="nav-link" href="index.html">Link</a>
</li>
<li class="nav-item dropdown">
<a class="nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
Dropdown
</a>
<ul class="dropdown-menu" aria-labelledby="navbarDropdown">
<li><a class="dropdown-item" href="#">Action</a></li>
<li><a class="dropdown-item" href="#">Another action</a></li>
<li><hr class="dropdown-divider"></li>
<li><a class="dropdown-item" href="#">Something else here</a></li>
</ul>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Disabled</a>
</li>
</ul>
<form class="d-flex">
<input class="form-control me-2" type="search" placeholder="Search" aria-label="Search">
<button class="btn btn-outline-success" type="submit">Search</button>
</form>
</div>
</div>
</div>
</nav>
<div style="padding-top:5em">
<section class="container mb-4">
<h2>Text to test body scrolling</h2>
<p>Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.</p>
<p>Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat.</p>
<p>Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi.</p>
<p>Nam liber tempor cum soluta nobis eleifend option congue nihil imperdiet doming id quod mazim placerat facer possim assum. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat.</p>
<p>Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis.</p>
<p>At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, At accusam aliquyam diam diam dolore dolores duo eirmod eos erat, et nonumy sed tempor et et invidunt justo labore Stet clita ea et gubergren, kasd magna no rebum. sanctus sea sed takimata ut vero voluptua. est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur</p>
<p>Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.</p>
<p>Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat.</p>
<p>Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi.</p>
<p>Nam liber tempor cum soluta nobis eleifend option congue nihil imperdiet doming id quod mazim placerat facer possim assum. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat.</p>
<p>Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis.</p>
<p>At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, At accusam aliquyam diam diam dolore dolores duo eirmod eos erat, et nonumy sed tempor et et invidunt justo labore Stet clita ea et gubergren, kasd magna no rebum. sanctus sea sed takimata ut vero voluptua. est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur</p>
</section>
</div>
<!-- JavaScript Bundle with Popper -->
<script src="https://cdn.jsdelivr.net/npm/bootstrap@5.2.0-beta1/dist/js/bootstrap.bundle.min.js" integrity="sha384-pprn3073KE6tl6bjs2QrFaJGz5/SUsLqktiwsUTF55Jfv3qYSDhgCecCxMW52nD2" crossorigin="anonymous"></script>
</body>
</html>
A: I made a guess on the source of your example.
It has CSS specific to it and JavaScript specific to it.
Here I added the CSS in the snippet CSS block and the JavaScript in that block
Perhaps download instead of copy from the example site? FWIW I just used the browser to look into the actual source CSS in play here.
(function () {
'use strict'
document.querySelector('#navbarSideCollapse').addEventListener('click', function () {
document.querySelector('.offcanvas-collapse').classList.toggle('open')
})
})()
html,
body {
overflow-x: hidden; /* Prevent scroll on narrow devices */
}
body {
padding-top: 56px;
}
@media (max-width: 991.98px) {
.offcanvas-collapse {
position: fixed;
top: 56px; /* Height of navbar */
bottom: 0;
left: 100%;
width: 100%;
padding-right: 1rem;
padding-left: 1rem;
overflow-y: auto;
visibility: hidden;
background-color: #343a40;
transition: transform .3s ease-in-out, visibility .3s ease-in-out;
}
.offcanvas-collapse.open {
visibility: visible;
transform: translateX(-100%);
}
}
.nav-scroller {
position: relative;
z-index: 2;
height: 2.75rem;
overflow-y: hidden;
}
.nav-scroller .nav {
display: flex;
flex-wrap: nowrap;
padding-bottom: 1rem;
margin-top: -1px;
overflow-x: auto;
color: rgba(255, 255, 255, .75);
text-align: center;
white-space: nowrap;
-webkit-overflow-scrolling: touch;
}
.nav-underline .nav-link {
padding-top: .75rem;
padding-bottom: .75rem;
font-size: .875rem;
color: #6c757d;
}
.nav-underline .nav-link:hover {
color: #007bff;
}
.nav-underline .active {
font-weight: 500;
color: #343a40;
}
.text-white-50 { color: rgba(255, 255, 255, .5); }
.bg-purple { background-color: #6f42c1; }
<!doctype html>
<html lang="fa" dir="rtl">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
<title>Hello, world!</title>
</head>
<body>
<div class="container">
<!-- Static navbar -->
<nav class="navbar navbar-expand-lg fixed-top navbar-dark bg-dark" aria-label="Main navigation">
<div class="container-fluid">
<a class="navbar-brand" href="#">Offcanvas navbar</a>
<button class="navbar-toggler p-0 border-0" type="button" id="navbarSideCollapse" aria-label="Toggle navigation">
<span class="navbar-toggler-icon"></span>
</button>
<div class="navbar-collapse offcanvas-collapse" id="navbarsExampleDefault">
<ul class="navbar-nav me-auto mb-2 mb-lg-0">
<li class="nav-item">
<a class="nav-link active" aria-current="page" href="#">Dashboard</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Notifications</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Profile</a>
</li>
<li class="nav-item">
<a class="nav-link" href="#">Switch account</a>
</li>
<li class="nav-item dropdown">
<a class="nav-link dropdown-toggle" href="#" id="dropdown01" data-bs-toggle="dropdown" aria-expanded="false">Settings</a>
<ul class="dropdown-menu" aria-labelledby="dropdown01">
<li><a class="dropdown-item" href="#">Action</a></li>
<li><a class="dropdown-item" href="#">Another action</a></li>
<li><a class="dropdown-item" href="#">Something else here</a></li>
</ul>
</li>
</ul>
<form class="d-flex">
<input class="form-control me-2" type="search" placeholder="Search" aria-label="Search">
<button class="btn btn-outline-success" type="submit">Search</button>
</form>
</div>
</div>
</nav>
<!-- Main component for a primary marketing message or call to action -->
<div class="jumbotron">
<h1>Offcanvas Navbar example</h1>
<p>This example demonstrates using the offcanvas plugin with the navbar.</p>
</div>
<p>
By default the navbar is show on the right side of the screen. You can show it on the left side instead by adding <code>.navmenu-fixed-left</code> to the <code>.navbar-offcanvas</code>.
</p>
</div>
<!-- /container -->
<script src="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/js/bootstrap.bundle.min.js" integrity="sha384-MrcW6ZMFYlzcLA8Nl+NtUVF0sA7MsXsP1UyJoMp4YLEuNSfAP+JcXn/tWtIaxVXM" crossorigin="anonymous"></script>
</body>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,835
|
{"url":"https:\/\/asvabtestpro.com\/quiz\/if-the-handles-of-a-wheelbarrow-are-4-ft-from-the-wheel-627234dbdb5cd741a80b4d59\/","text":"# If the handles of a wheelbarrow are 4 ft. from the wheel axle, what force must you exert to lift the handles if it's carrying a 340 lb. load concentrated at a point 0.5 ft. from the axle?\n\n42.5 lbs.\n\nExplanation\n\nThis problem describes a second class lever and, for a second class lever, the effort force multiplied by the effort distance equals the resistance force multipied by the resistance distance:\n\n$$P_{e} d_{e}=F_{r} d_{r}$$\n\nPlugging in the variables from this problem yields:\n\n$$F_{e} \\times 4 f t .=340 \\mathrm{lbs} . \\times 0.5 \\mathrm{ft.}$$\n$$F_{e}=170 \\mathrm{ft} . \\mathrm{lbs} . \\div 4 \\mathrm{ft} .=42.5 \\mathrm{lbs}$$","date":"2022-12-03 06:05:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6630547642707825, \"perplexity\": 1840.5575315757649}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710924.83\/warc\/CC-MAIN-20221203043643-20221203073643-00521.warc.gz\"}"}
| null | null |
Q: SHL instruction with AND instruction to isolate each bit and jump on carry I am trying display 01011011B AND 11000111b as an ASCII string of binary digits. I am having real trouble jumping and getting anything to print. I am new at this so any help would be great. Going on 7 hours with minimal progress. Thank you
.stack 100h
.model small
.386
.data
str1 db 20 dup(?)
lstring EQU 9
.code
main:
mov ax, @data ; initialize DS
mov ds, ax
mov cx, lstring
L1:
mov al,01011011b
and al,11000111b
shl al, 1
loop L1
mov str1, al
mov ax, 8
int 21h
mov ax, 9 ; dos service to display...
mov bx, 1 ; to screen
mov cx, lstring ; number of bytes
mov dx, OFFSET str1 ; where to get data
int 21h
MOV AH, 4CH ; return control to DOS
INT 21H
end main
A: *
*Set result to ascii '0'
*Shift test register left once
*Add with carry 0 to result (will still be '0' if carry clear, '1' if set)
*Append that character to the output.
*Repeat 8 times, once per bit.
Edit:
Bear in mind that it's been a great many years since my (brief) dalliance with x86 assembler and I don't have a DOS setup to test it.
mov al,01011011b
and al,11000111b ; Only need to do this once
; now al is the intermediate result
mov cx, 8 ; Do this 8 times, cx is the loop ctr
mov bx, OFFSET str1 ; Destination for resulting chars - start at beginning
L1: ; This is the loop ; do {
mov dl, '0' ; Ascii character zero
shl al, 1 ; Upper bit now in carry flag
adc dl, 0 ; Adds carry flag - '0' or '1'
mov [bx], dl ; Save ASCII digit to current position
inc bx ; Next position
loop L1 ; }while(--cx != 0) ; Counts down cx
mov byte ptr [bx], '$' ; Terminate DOS string
mov ah, 9 ; dos service to display $-terminated string
mov dx, OFFSET str1 ; where to get data
int 21h
MOV AH, 4CH ; return control to DOS
INT 21H
DOS system-call documentation: http://spike.scu.edu.au/~barry/interrupts.html#ah09
Try that but also learn to use a debugger to see where it's going wrong.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,155
|
\section{Introduction}\label{sec:intro}
The advent of wide-field time domain surveys has revolutionized the field of transient astrophysics. Coverage on short timescales in the optical/NIR bands, in particular, has facilitated rapid strides in our understanding of both supernova (SN) explosions and peculiar transients. This work focuses on short-lived transients, whose light curves evolution happens in $\lesssim$10 days, as well as slower transients (evolving in months time scales) that show fast evolving features at some phases.
Rapidly evolving transients are generally poorly understood, and may be associated with a variety of phenomena, including accretion-induced white dwarf collapse \citep{Metzger2009}, underluminous and fallback SN~\citep{Moriya2010}, ultra-stripped SN \citep{Drout2013,Kasliwal2010,Tauris2015,De2018}, compact-object mergers \citep{Kasen2015,Metzger2010}, orphan gamma-ray burst afterglows \citep{Totani2002}, and common-envelope ejections \citep{Blagorodnova2017}. ``Infant'' supernovae (SNe)---hours to days after explosion---evolve quickly and their observations provide vital constraints on their explosion mechanisms and progenitor systems \citep{Nakar2010,Rabinak2011,Nugent2011}, potential non-spherical behavior \citep{Matzner2013,Salbi2014}, and shock collision with a binary companion \citep{Kasen2010}.
Despite progress, the detection rate for both rapid transients and rapidly evolving \emph{phases} in SN explosions has remained low due to a combination of survey efficiency and intrinsic event rates. The volume surveyed by the new and upcoming surveys Zwicky Transient Factory (ZTF; \citealt{ztf}) and the Large Synoptic Survey Telescope (LSST; \citealt{lsst}\footnote{as accessed in its \url{arxiv.org} version, which is a living document, on January 30, 2019}) bring the promise of detecting many more intrinsically rare events. However, recognizing and using these events to probe the science questions described herein (\autoref{sec:transients}) requires adequate time- and filter-sampling of relatively short-lived features.
In this work, we explore the minimal cadence requirements that allow a multi-band imaging survey to effectively \emph{recognize} young and rapidly evolving transients. We focus on an implementation for the LSST main survey (also known as Wide-Fast-Deep, WFD). The LSST main survey is designed with four science goals in mind: understanding dark matter and dark energy, cataloging the orbits of millions of moving objects in the solar system, understanding the structure of the Milky Way via resolved stellar population, and exploring the transient sky \citep{lsst}. To deliver on these science goals, the LSST is planning a 6-filter (\emph{ugrizy}), $\sim18,000$~square degree survey of the southern hemisphere to a single-image (coadded) depth of $r\gtrsim24.5$ mag ($27.5$ mag) for a signal-to-noise ratio of $5$ for point sources, with subsecond spatial resolution \citep{lsst}\footnote{For more details, see also the LSST Science Requirements Document \url{https://ls.st/srd}.}. However, the LSST main survey's baseline cadence results in a typical intra-night revisit rate of $\sim$30 minutes \citep{COSEP}, which is too rapid to recognize effectively fast transients and fast transient features from ``normal'' transients. Furthermore, the current baseline planned cadence repeats observations in the same filter with a median gap of 2 weeks: too long to capture rapidly evolving transients with enough epochs for characterization, and in extreme, but also extremely interesting cases, \emph{e.g.} kilonovae (KNe), to even capture them with anything more than a single epoch. We are most interested in optimizing the cadence of LSST because, compared to other ongoing and upcoming all sky transient surveys, LSST will have the greatest depth. This provides better uncertainties on the flux compared to other surveys, enabling a more accurate color or light curve evolution determination from a small number of observations, and could be critical in the determination of these quantities from a relatively small number of observations. Second, the higher depth allows LSST to probe a larger volume for the same sky area, which is better for building a sample of intrinsically rare transients and features --- but a key constraint is the potential availability of spectroscopic follow-up facilities for the fainter targets.
The optimization of the LSST survey strategy is, at the time of writing, open to community input, and the LSST Project Science Team issued a call for white papers suggesting LSST cadences details in 2018 November, which prompted this work and many other strategy implementation ideas\footnote{https://www.lsst.org/submitted-whitepaper-2018}.
We explored the \emph{minimal} requirements necessary to achieve the goal of detecting and recognizing fast transients and fast transient features, and design a strategy for the LSST main survey (WFD) that accommodates the core LSST science goals (and other science goals identified by the scientific community) but which, unlike many other WFD proposals (including the current baseline strategy\footnote{the {\tt baseline2018a} simulation described in https://ls.st/Document-28453}), allows for the identification of young and rapidly evolving transients from the millions of LSST alerts every night. This cadence will allow for the prompt triggering of external follow-up resources (e.g., spectroscopy, non-optical facilities) that are necessary to confirm the discoveries and study the temperature, composition, and explosion mechanisms of explosive phenomena.
We argue that three observations per night are sufficient to disambiguate fast transients and trigger prompt follow-up, as long as the three images are collected in two filters with appropriate time gaps between the filters in the sequence. We find that this cadence should put minimal strain on an observing strategy, and still allow for further strategy details to be optimized for different science cases (e.g. covering large areas of the sky within a night, or the specific pointing sequence, or obtaining long observation sequences in a single filter). Specifically, the strategy we envision has two requirements:
\begin{enumerate}
\item Observations in {\bf two filters} obtained in quick succession so that the transient's {\bf color} can be measured. The color is critical to both allow us to distinguish different classes of transients and as a probe of the physics operating during rapid transient evolution phases.
\item A {\bf same-filter} revisit separated by hours (before or after the filter pair) so that the light curve {\bf evolution (slope)} can be constrained and distinguished from slower-evolving transients.
\end{enumerate}
Since this cadence is designed to return both a transient's color and rate of brightness change in a single night, we call it the {\em Presto-Color} (rapid-color) cadence.
The exact form of the {\em Presto-Color} cadence is provided with more detail in \autoref{sec:implementation}.
In order to demonstrate the ability of the {\em Presto-Color} survey cadence to achieve our goals, we have selected four exemplary types of extragalactic fast transients, and fast features of longer duration transients, for our discussion in \autoref{sec:transients}. Representative light curves for each type of transient are shown in \autoref{fig:examples}, and graphical representations of where they separate from normal SN in color and intra-night rate-of-change are shown in \autoref{fig:phasespace} and \autoref{fig:classifier}.
\section{Scientific Motivation}\label{sec:transients}
The main science cases that motivate the design of the {\em Presto-Color} cadence are discussed below, and simplified light curves are shown in \autoref{fig:examples}.
\begin{center}
\begin{figure*}[!t]
\includegraphics[height=4.4cm]{figures/fasttrans2.png}
\includegraphics[height=4.4cm]{figures/Tanvir_fig2_remake_small.png}
\includegraphics[height=4.4cm]{figures/2016gkg2_small.png}
\includegraphics[height=4.4cm]{figures/bluebump2.png}
\caption{{Light curves of exemplary fast transients and fast features. From left to right: (\emph{a}) fast transient PS1-10bjp \citep{Drout2014}; (\emph{b}) the kilonova associated to GW170817 \citep{Tanvir2017}; (\emph{c}) the shock breakout model fits for SN\,IIb 2016gkg's stellar radius \citep{Bersten2018}; and (\emph{d}) Type Ia SN 2017cbv's ``blue bump'' compared to the ``normal'' Type Ia SN 2011fe~\citep{Graham2015, Hosseinzadeh2017}.} In some cases, these plots have been adapted from their original form for clarity in the context of this paper.}\label{fig:examples}
\end{figure*}
\end{center}
\subsection{The Nature of Rapidly Evolving Luminous Transients}
``Rapidly evolving transients'' are defined as extragalactic events that reach SN luminosity but have timescales an order of magnitude faster. To date, only a small number have been identified, but recent sample studies \citep{Drout2014} have shown that they are not \emph{intrinsically} rare: few have been detected simply because current surveys are not designed to be efficient at short timescales. They are a significant fraction ($\sim$5\%--10\%) of the core-collapse events, which we must understand to have a complete picture of massive star death. Known events have rise times spanning 1--3 days and blue colors at maximum \citep{Drout2014,Pirsiainen2018,Rest2018}; an example of a fast transient light curve is shown in \autoref{fig:examples} panel (a). While their true nature is unknown, leading theoretical models include black hole formation in failed SN and the birth of binary neutron star systems, with recent observations of AT2018cow showing evidence for a central engine \citep{Kashiyama2015,Margutti2018,Prentice2018}.
It is clear that larger samples, as well as more detailed and complete observations of individual event are required to understand their true nature and diversity.
\subsection{KNe and the Origin of Heavy Elements}
KNe are powered by the radioactive decay of \emph{r}-process nuclei synthesized in the ejecta of neutron star mergers \citep{Li1998,MetzgerKN}. These objects have been described by theories and models, and candidate KNe have been observed in conjunction with Gamma Ray Bursts \citep{Berger13, Tanvir13, Jin15}. The first detection of an optical counterpart associated with a gravitational wave (GW) event came in 2017 in a search triggered by the \emph{Advanced LIGO} \citep{Aasi15} and \emph{Advanced Virgo} \citep{Acernese15} detection of
GW170817 \citep{Abbott2017}. Observations of the KN associated with GW170817 revealed thermal emission that rose in $<$1 day and cooled from a temperature of $>$10,000 K to 3,000 K over 5 days \citep{Drout2017}. The initially blue optical light faded at a rate of $>$1 mag $\mathrm{day}^{-1}$, and was followed by a longer-lived red transient consistent with the production of a significant quantity of \emph{r}-process elements of \emph{multiple} compositions \citep[][and references therein]{Cowp+17,Drout2017,Kasliwal2017,Smartt2017,Tanvir2017,Villar2017}. An example of this $\sim$10 day long red light curve is shown in \autoref{fig:examples} panel (b). Additional examples --- with or without associated LIGO triggers --- are required to ascertain whether GW170817 was typical. Sample studies will provide constraints on the ejecta composition, mass, and velocity that strongly influences the resulting \emph{color}, \emph{magnitude}, and \emph{timescale} of the emission. Once the effect of the ejecta on the color is understood, the frequency of the early blue emission will set critical constraints on the ratio of light and heavy elements formed, and the total contribution of NS mergers to cosmic nucleosynthesis \citep{Metzger2018,Piro2018,Rosswog2018}.
Unfortunately, the current baseline cadence planned for the LSST WFD Survey repeats observations in each filter with a median gap of 2 weeks. This is longer than the anticipated time scale of KNe, as well as many other putative systems that produce fast transients. The call for LSST white papers recognized that this need is largely unmet, and several cadence proposals are aimed at improving LSST's ability to probe short ($< 1$~week) duration time-domain phenomena. Notably, responding to the same call for white papers, \citet{Andreoni2018} advocates for an LSST ``rolling'' cadence with colors for the purpose of increasing the chance detection rate of KNe. This strategy is similar to ours advocating for 2-filter (\emph{g} and \emph{i}) observations on consecutive days.
\subsection{Progenitors and Pre-explosion Mass Loss of Core-collapse SN}
Early observations of core-collapse SN (CCSN) provide critical constraints on the progenitor radius and envelope structure through the detection of either shock breakout ($\sim$1 day) or cooling envelope ($\sim$1-5 day) emission \citep{Modjaz2009,Nakar2010,Arcavi2011,Bersten2018}. An example of a CCSN that exhibited both types of fast features, and a hydrodynamical model fit to the data, is shown in \autoref{fig:examples} panel (c). Indeed, there has been growing evidence that many CCSN either explode in ``non-standard'' evolutionary states or undergo enhanced pre-SN mass-loss and outbursts in their terminal years \citep{Nakar2014,Khazov2016}. Theoretical studies have pointed to a range of potential explanations to accommodate the observations, such as pulsation-driven superwinds \citep{Yoon2010}, wave heating outbursts \citep{Fuller2017}, and inflated progenitor envelope \citep{Grafener2012}. However, the nature of this mass loss and the types of SN experiencing it remain uncertain.
\subsection{Progenitors and Explosion Mechanisms of Thermonuclear SN}
SN Ia result from the thermonuclear disruption of a carbon-oxygen white dwarf star which has either accreted mass from or merged with a binary companion \citep{Hillebrandt2000}. However, questions remain regarding the nature of the companion star (e.g., a red giant, main sequence, or another white dwarf star). Answering these questions are not only important for a complete and accurate picture of stellar evolution and death, but also for understanding how the diversity of progenitor systems and differences in the SN evolutionary pathways over cosmic time may induce systematic differences in standardization, and thus systematic errors in precision cosmological inference (e.g., \citealt{2011PhDT........37D,2013PhDT.......326H}).
Observable signatures of a stellar companion star in the progenitor system of a SN\,Ia is, for the above reasons, an active pursuit of the SN community. For example, \citet{Kasen2010} predicted that a red giant or main sequence companion star could shock the SN\,Ia ejecta at early times, and cause an observable ``blue bump" in the first few days of a SN\,Ia light curve. In this model the distance and size of the companion star relative to the white dwarf drives the color and duration of this fast blue light curve feature, while the viewing angle of the observer relative affects the intensity of the observed light curve ``bump.'' Sample studies of type Ia SN lead to constraints on the progenitor fractions \citep{Bianco11, Hayden2010}. Singular examples of ``blue bumps" have been observed: for example, observations within $\sim$1 day of explosion revealed a rapidly-rising blue ``bump'' for SN\,Ia 2017cbv (panel (d) of \autoref{fig:examples}), which has been interpreted by some as a collision with a non-degenerate companion star \citep{Hosseinzadeh2017}.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.9\textwidth]{figures/cadence_highlevel.png}
\caption{A schematic example of the {\em Presto-Color} cadence with two alternating filters covering four regions of sky over 5.5 hours to obtain three observations per region with appropriate time gaps to measure light curve color and shape. In this implementation the time gaps are: $\mathrm{30~min} \lesssim \ensuremath{\Delta T_1}\lesssim \mathrm{1~hour}$ and $\mathrm{2~hour} \lesssim \ensuremath{\Delta T_2}\lesssim \mathrm{3~hour}$ . Preliminary results for such an implementation of {\em Presto-Color} for LSST (performed with {\tt OpSim}, see \autoref{sec:implementation} and Appendix) indicate a $\sim1\%$ efficiency loss due to the increased number of filter changes.}\label{fig:implementation}
\end{center}
\end{figure*}
However, larger sample studies are required to resolve the degeneracy between the nature of the companion and the geometric alignment, and to furthermore help distinguish between the effects of a companion star and the influence of the explosion itself on the early light curve. For example, preliminary population studies suggest the possibility of an as-yet-unexplained red/blue color dichotomy in early ($<$ 5 days) rapidly rising light curves of SN\,Ia \citep{Stritzinger2018}, and implications for outwardly mixed radioactive material are predicted by the double detonation explosion model \citep{Piro2016,Polin2018}. The latter physical quality may also be related to dust formation models from SNe, and also important for sub-Chandrasekhar mass models for SN\,Ia progenitors \citep{Polin2018}. The {\em Presto-Color} cadence for LSST will not only help to identify such fast features early enough for follow-up, but lead to a significantly higher fraction of LSST's discovered SN\,Ia having the requisite early time sampling for large statistical analyses.
\subsection{Additional Science Cases}\label{sec:additional}
While we have focused on extragalactic fast transients here, a cadence that allows measurement of both color and rate-of-change on the timescale of $\sim$hours will have general applicability across many areas --- from variable stars and microlensing to characterization of solar system objects. Requests for multiple observations in the same and in different filters, in fact, were made in several other white papers submitted to the LSST Project in November including \citet{streetWP}, which focuses on microlensing, and \citet{Bricman2018} and \citet{Gezari2018}, both of which focus on tidal disruption events.
\section{Implementation}\label{sec:implementation}
The {\em Presto-Color} cadence aims to obtain both the color and light curve shape of a transient in a single night and it is originally designed as an LSST WFD strategy. It requires observations in two different filters, $f_1$ and $f_2$, within a short interval of time \ensuremath{\Delta T_1}, and then to return to the same field with either of those filters at a later time \ensuremath{\Delta T_2}. The minimal technical constraints in this observing strategy are:
\begin{enumerate}
\item {\tt max(\ensuremath{\Delta T_1})}, an upper limit on the time between the two visits that provide color,
\item {\tt min(\ensuremath{\Delta T_2})}, a lower limit on the time between the two visits that provide light curve shape, and
\item The filter pair $f_1$ and $f_2$.
\end{enumerate}
As we envision the implementation of {\em Presto-Color} for a synoptic survey, we have to keep in mind efficiency is generally a priority for such surveys, and for LSST in particular: the implementation of {\em Presto-Color} must not interfere with the achievement of the four main science goals described in \autoref{sec:intro}. We tested a simple implementation of a strategy that delivers on our requirements by using the LSST {\tt OpSim}\footnote{https://www.lsst.org/node/656} software \citep{opsim}: an application that simulates the field selection and image acquisition process of the LSST survey over the lifetime of the survey, balancing strategic requirements on different time scales (e.g. minimum number of observations per filter per field over the survey lifetime \emph{vs} target filter gaps and filter sequences), simulating realistic weather and seeing conditions, scheduled engineering downtime, and current telescope and camera parameters (a more detailed discussion of our simulations is presented in the Appendix). As illustrated in \autoref{fig:implementation}, the {\em Presto-Color} cadence can be implemented by alternating pairs of visits on a field and single visits on the previous field, thus minimizing slew time. The single visits separated by \ensuremath{\Delta T_2}\ could alternate between the two filters, thereby reducing the number of filter changes (which incurs a $2$ minute overhead with LSST). In the example implementation of \autoref{fig:implementation}, half of the transients would have their light curve slope measured in ``blue" filter, the other half in ``orange" filter, and all would have a data point in ``blue-minus-orange" or ``orange-minus-blue'' color.
The main science goals of {\em Presto-Color} might be reachable if the proposed cadence is limited to extra-galactic LSST fields, although we noted in \autoref{sec:additional} that some galactic transients (e.g., microlensing) would also benefit from it. Obtaining two observations within a short time interval, comparable to our \ensuremath{\Delta T_1}, is necessary to distinguish moving objects (\emph{i.e.} Solar System objects) from transients and variables, so a repeat observation within a short interval of time is built in almost all LSST cadence implementations. However, collecting these observations in two different filters $f_1$ and $f_2$ limits the detectability of Solar System objects to the sensitivity of the shallowest filter in the $f_1-f_2$ pair. The {\em Presto-Color} cadence could be avoided in regions of the extended ecliptic plane (i.e., the region of interest for solar system studies). Essentially, given the incredible increase in the LSST's accessible volume compared to past surveys, implementing the {\em Presto-Color} cadence in just some sky regions, some of the time, should yield more (and better) observations of fast transients and fast features than ever before.
\section{Recognizing fast transients and fast features}
We explore our ability to recognize fast transients and fast features in transients, distinguishing them from ``normal'' transients, assuming an observing strategy that collects three images within a single observing night. We parameterize transients in a phase space of observed color versus observed intra-night magnitude-change to determine which region of this space each transient occupies.
\subsection{Sample Light Curves}
We first compile a sample of light curves for both fast transients and ``normal'' supernova from the literature, which will be used to test the effectiveness of different observing strategies in distinguishing targets of interest.
For our sample of fast transients/features, we include example light curves for each of the classes of events described in \autoref{sec:transients}.
Our sample of rapidly evolving transients includes the gold sample of events from \cite{Drout2014} as well as the rapidly declining transients SN\,2005ek \citep{Drout2013} and AT2018cow \citep{Margutti2018}. For KNe, we consider the fiducial case of the best fit models to the emission of GW170817 from \citet{Villar2017}. We include models both with and without the early blue component (labeled as 'Blue Kilonova' and 'Red Kilonova'). For ``infant'' SNe we consider observations of supernova obtained within the first 5 days after explosion. We include both empirical data from the core-collapse SN\,2016gkg (supplemented by a hydrodynamical model from \citealt{Piro2016} to obtain finer time steps) and the Type Ia SN2011fe and SN2017cbv \citep{Graham2015,Hosseinzadeh2017} as well as simulated observations of SNe Ia with and without binary companions based on the models of \citet{Kasen2010}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.5\textwidth]{figures/LSST_PhaseSpace.png}
\caption{Fiducial phase space plot showing separation between classes of transients in observed color and intra-night magnitude change. We show the location of transients for observations obtained in $g$-and-$i$ band filters within 30 minutes (\ensuremath{\Delta T_1}) and a second $g-$band observation obtained 4 hr (\ensuremath{\Delta T_2}) after the first. All fast transients/features are shown in colors, SNe Ia, as representatives of ``normal transients,'' are plotted in gray, clustering around 0 on the $x$-axis due to their slower evolution (see legend and text for details). Each transient is shown at a range of epochs in its evolution. Rising light curves correspond to positive magnitude changes (positive $x$-axis). }\label{fig:phasespace}
\end{center}
\end{figure}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/FTclassifier_g.png}
\includegraphics[width=0.49\textwidth]{figures/FTclassifier_r.png}
\caption{{The result of a binary classification (``normal'' \emph{vs.} ``fast'') performed with a Gaussian Processes (GP) Probabilistic Classifier for two fiducial strategies: (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0.5 hr, 0.5 hr), shown at the \emph{top}, and (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0, 4 hr), at the \emph{bottom}. The \emph{left} panel shows the result for observations in $g$-and-$i$, the \emph{right} panel in $r$-and-$i$, with the $g$ and $r$ filter repeated twice, respectively, leading to a light curve shape constraint $\Delta g$ and $\Delta r$. The red dots indicate transients of interest (same sample as the colored data points in \autoref{fig:phasespace}) and blue and gold dots denote SN Ia and II populations respectively. Gray circles indicate the points included in the training set. The heat map in the background represents the probability of a transient being ``not-normal'' from three observations collected according to the specific implementation of our \emph{Presto-Color} strategy. The majority of the phase space has a probability $p~\sim~0.5$ because it is far from any known transients: the region densely occupied by known transients is relatively compact, except when observed in $g$-and-$i$ at (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0, 4 hr). The error bars at the bottom indicate the photometric uncertainty in $\Delta mag$ based on the typical rate of change for a normal SN\,Ia (0.3 mag/day) for a transient initially observed by LSST at mag~20, 22, and 24 in the $g$ (\emph{left}) or $r$ (\emph{right}) band: thick lines represent a $1\sigma$, thin lines a $3\sigma$ uncertainty. Transients at a horizontal separation smaller than an error bar have indistinguishable light curve evolution within that confidence level. When observed with gaps (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0.5 hr, 0.5 hr) nearly all transients $g/r\gtrsim22$ at nearly all epochs are indistinguishable at a $1\sigma$ level in both filter combinations. Even at $g/r\sim20$ transients are indistinguishable at a high ($3\sigma$) confidence.
With time gaps (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0, 4 hr), in the $r$-and-$i$ observations case (\emph{right}),
nearly all transients $r\gtrsim22$ are consistent at a $3\sigma$ level. Furthermore, the distribution is more compact along the $y$-axis, leading to less insightful clues into the physical phenomena. Among these strategies, the $g$-and-$i$ observations with \ensuremath{\Delta T_1}= 0 and \ensuremath{\Delta T_2}= 4 hours time gaps is optimal for recognizing fast transients, allowing us to distinguish most transients $g~\gtrsim~22$ from SNe Ia and II at $\gtrsim1\sigma$. High resolution PDF versions of this images are available on the project repository \url{https://github.com/fedhere/prestocolor/tree/master/FastTrWP/PhaseSpace/Figures}}}\label{fig:classifier}
\end{center}\end{figure*}
We also include a population of normal Type Ia and Type II/P/L SNe. Together, these represent slower evolving transients from which we wish to \emph{distinguish} our transients of interest. For the SN Ia we include a population of events between $-$7 and $+15$ days of maximum light at a range of redshifts. The SNe Ia SNe are generated with the {\tt SNcosmo} python library \citep{sncosmo}. LSST-like $g$, $r$, and $i$ light curves are generated at redshifts $0.3~\leq~z~\leq~0.7$ using the Spectral Adaptive Lightcurve Template framework {\tt SALT2} \citep[][a light curve fitting model]{salt2}. The three parameters given to {\tt SALT2} are: an observer flux normalization for the spectral templates $x_0$, a light curve shape-luminosity parameter (related to stretch) $x_1$, and a color parameter $c$ related to the $B - V$ color at peak. Following \citealt{Kessler2010} (figures 2 and 10 therein) we select {\tt SALT2} parameters randomly from Gaussian distributions as $x_1~\sim~N(0,1)$, and set $c~\sim~N(0,0.05)$, and let {\tt SNcosmo} set $x_0$ to produce realistic lightcurves. We generate 100 SNe Ia, constraining the sampling to $-7~\mathrm{days}~<~\mathrm{phase}~<~15~\mathrm{days}$ to peak $B$ brightness. Our rationale to set an upper limit of +15 days is that older transients would have already been discovered by a survey like LSST in previous observations of the same field, so that more data would be available to constraint their characteristics. Younger SNe, on the other hand, would be ``fast transients.''
For SNe II, we consider a few representative examples: SN~2006bc, SN~2008M SN~2009N, SN~2007od, SN~2008aw~\citep[personal communication]{sncsp}.
Finally, we consider intermediate luminosity transients (ILOTs) and Calcium rich transients (CARTs) and events from sub-luminous SN Iax. The first two of these models have significantly faster time evolution from type Ia and core collapse SNe, while the Iax subclass exhibit similar properties to SNe Ia using our methodology. These events were simulated for the recently concluded Photometric LSST Astronomical Time-series Classification Challenge~\citep[PLAsTiCC,][]{plasticc}. Light curves were generated assuming the \texttt{minion1016} cadence (see \citealt{COSEP} for details on this strategy) in the LSST Deep Drilling Fields. We selected these objects over a range of redshifts and Milky Way reddening values, as well as intrinsic parameters of each of these models.
In order to use these sample light curves to test the effectiveness of various observing strategies we uniformly sample the light curves to a cadence of 30 minutes. Model light curves were generated at this cadence while the light curves from observed transients were fit with low order polynomials and interpolated to a fixed 30 minute cadence within the observed range (no extrapolation). In general, we restrict these template light curves to relatively young objects, interpolating from the \texttt{minion1016} cadence to a 30 minute cadence from $-21$~days to $+15$~days post-maximum.
Each individual transient leads to a family of data-points, corresponding to observations at different epochs in its lifetime.
In \autoref{fig:phasespace}, we plot the location of these sample light curves (excluding the PLAsTiCC transients) in a phase space of observed color (vertical axis) versus observed magnitude change (horizontal axis). In creating this plot, we adopt an observing strategy in which observations are obtained in $g$-and-$i$ band within 30 minutes of each other and a second $g-$band observation is obtained 4 hr later: (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0.5 hr, 4 hr).
For each individual event the family of data-points generated by observing the transient at different epochs is visible as a `track.'
Red transients are located at the top of the plot and blue at the bottom. Declining transients are found on the left and rising transients on the right. The infant core-collapse SN found in the declining portion of the plot is the rapidly fading component of the cooling envelope emission observed in SN2016gkg (see \autoref{fig:examples}). With this survey strategy, our transients of interest (colored points) are easily distinguishable from ``normal'' SNe Ia and II (which are clustered near zero magnitude change). At the same time, the observed color provides vital context on the \emph{type} of transient of interest.
In the section below, we investigate how details of the adopted observing strategy influences our ability to identify transients of interest within this phase space.
\subsection{The sample in the color versus magnitude-change space}\label{sec:classification}
With this data set in hand, we can explore how the color versus magnitude-change space is occupied by normal and unusual transients when two observations are performed in $g$-and-$i$ band or $r$-and-$i$ band at a time interval \ensuremath{\Delta T_1}\ (in \autoref{fig:phasespace}, \ensuremath{\Delta T_1} = 30 minutes), and with one further observation in $g-$ or $r-$band \ensuremath{\Delta T_2}\ after the first image (in \autoref{fig:phasespace}, \ensuremath{\Delta T_1} = 4 hr).
We train a machine learning classifier to recognize fast transients in this phase space. We use a Gaussian Processes (GP) Probabilistic Classifier \citep{Rasmussen06gaussianprocesses} with a radial basis function (RBF) kernel as implemented in the {\tt scikit-learn} python package~\citep{scikit-learn}, training it to disambiguate fast transients from SNe Ia and II in $g$-and-$i$ and $r$-and-$i$ observations at (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0, 0.5 hr) and (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0.5 hr, 4 hr). The goal of this algorithm is separating ``normal'' transients from objects that evolve ``fast'': we are approaching this as a binary classification problem (\autoref{fig:classifier} and \autoref{fig:PLAsTiCC}).
We balance these two classes by selecting $N_\mathrm{fast}$ random observations among the ``normal'' light curves' data points, with $N_\mathrm{fast}$ equal to the number of data points available for all fast transients combined. The ``normal'' sample is split as two-third SN Ia and one-third SN II. We note that these classes will be significantly unbalanced in real observations, with orders of magnitudes more ``normal'' than unusual objects, but our goal here is to develop a classifier that is \emph{sensitive} or has a high true-positive rate for unusual objects, rather than optimizing for overall accuracy, which would lead to good classification performance on the most common objects, but poor performance on rare events. Additionally, a realistic survey can expect to have GW triggers for KNe, or detections from other wide-field observatories. These further limit the search area on the sky, and constrain the evolution, which further justifies our focus on sensitivity over sample completeness or contamination. Building an optimal photometric classifier that serves the needs of the entire LSST community is a research problem of great interest, and there has been much recent activity on this front \citep[e.g.][]{plasticc}.
The classifier generates a probability for the transient being ``normal'' or ``fast'' anywhere in our phase space. The goal of using a classifier here is to leverage it as a data-driven way to identify which set of parameters provides the most solid and useful ``not-normal'' label for a newly discovered object. Note that the classifier will default to a 0.5 probability in regions of the space that are not occupied by any of our transients. This does not imply that these regions are uninteresting, and indeed it is in these regions that we might expect to find true outliers: unexpected and unusual, as-of-yet unknown objects!
We train the classifiers with all possible combination of the following time gaps:
$\ensuremath{\Delta T_1}~=$~[0.5, ~1.5, ~3.5, ~4, ~4.5, ~6.5]~hr and
$\ensuremath{\Delta T_2}~=$~[0, ~0.5, ~1, ~2]~hr. We do not consider time scales longer than 6.5 hr, due to general consideration on the visibility of a field within a night (see \autoref{sec:implementation}).
The results of our classification exercise are shown in \autoref{fig:classifier}, where red dots are all transients of interest, blue and gold are SNe~Ia and II, gray circles are data points included in the training set, and the classification probability for each point in the phase space is indicated by the background color. The (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0.5 hr, 0.5 hr) strategy classification results are shown in the top panel, and (\ensuremath{\Delta T_1},\ensuremath{\Delta T_2}) = (0, 4 hr) in the bottom.
On the left-hand side is the result for the filter combination $g$-and-$i$, and on the right for $r$-and-$i$. In all cases we obtain high ($\gtrsim$95\%) accuracy in cross-validation.
To test the robustness of this classification result, we separately implement a Random Forest Decision Tree Classifier \citep[implemented using scikit-learn,][]{scikit-learn} using the same sample data set of objects as described above on a fiducial set of data using the ($g,i$) filters and (\ensuremath{\Delta T_1}, \ensuremath{\Delta T_2}) = (0.5, 4.5 hr) time gaps which approximate the median case. Starting with a sparse feature space that only uses the observed color and magnitude difference we find that, as with the GP classifier above, we have a somewhat optimistic classification accuracy of $\gtrsim 97\%$ when using 70\% train and 30\% test data and aggregating across n = 1000 individual decision trees.
One benefit of tree classification algorithms is that they allow us to straightforwardly interpret the relative importance of our features in determining the assigned class by asking in what fraction of trees is a particular feature at a higher rank in the tree than the others. When evaluating the relative strengths assigned by the ensemble of trees, we find that the change in magnitude is roughly twice as important as the observed color when assigning a coarse classification ($\sim 0.68$ versus $\sim 0.32$).
These results lead us to the following considerations:
{\tt { max(\ensuremath{\Delta T_1}) ---}} The two observations in different filters, separated by \ensuremath{\Delta T_1}, will be used to assess the color of the transient.
The fast transients are distinguished mostly due to their $\Delta$mag, but their high rate of change in luminosity implies that only for small \ensuremath{\Delta T_1}\ we can obtain true color information that would direct our follow-up strategy choices.
Smaller values of \ensuremath{\Delta T_1}\ --- $\sim$30 minutes --- are better to provide a true estimate of the intrinsic color (see \autoref{sec:classification}). However, values of \ensuremath{\Delta T_1}\ of up to a few hours, while no longer sensitive to the true ``instantaneous" color of the transient, can still provide some diagnostic power (e.g., of whether a given transient is red or blue), and thus insight into the ongoing physical processes.
\vspace{1.5mm}
{\tt { min(\ensuremath{\Delta T_2}) ---}} The two observations in a matching filter, separated by \ensuremath{\Delta T_2}, will be used to assess the rate of brightness change of the transient (the light curve slope). While our classifier successfully separated SNe Ia and II from fast transients in all \ensuremath{\Delta T_1}-\ensuremath{\Delta T_2}\ combinations, with accuracy exceeding 95\%, disambiguation is obviously harder for smaller \ensuremath{\Delta T_2}. This is intuitively obvious looking at \autoref{fig:classifier}.
For \ensuremath{\Delta T_2}\ smaller than $\sim4$ hr it is effectively impossible to distinguish fast transients from ``normal'' ones due to the photometric uncertainties, since the classification is mostly driven by the rate of change in luminosity. The uncertainty in $\Delta$mag is plotted for a typical SN Ia (0.3 magnitudes $\mathrm{day}^{-1}$) as observed by LSST with the first measurement at $g/r~=~$20, 22, 24 in the $g$ or $r$ band ($1\sigma$ and $3\sigma$ are shown). At \ensuremath{\Delta T_2}\ smaller than $\sim4$ hr nearly all fast transient observations fall within the $1\sigma$ region of type Ia SNe for transients $g~\gtrsim~22$.
\vspace{1.5mm}
{\tt { max(\ensuremath{\Delta T_2})}} --- We advocate for three observations \emph{within the same night} to trigger a follow-up campaign and catch the early, extremely informative stages of a transient's evolution. In addition, there are general consideration on the visibility of a field within a night to make: $\ensuremath{\Delta T_2}\sim4$~hours means we are now observing a field that moved $60\degr$ across the sky: if the first observations happened near zenith the field would now be at a prohibitively high airmass.
\vspace{1.5mm}
{\bf Filter sequence ---} The repeat observation in the same filter can be obtained either \emph{before or after} the filter pair (while the illustration in \autoref{fig:implementation} always shows it as occurring after). However, given that the main motivation of the {\em Presto-Color} cadence is to identify new objects with rising light curves, we note that beginning the sequence with the image pair is a strategy that is more robust to saturation of very rapidly rising objects: it still provides color and a \emph{slope constraint} even if the final image is saturated. This aspect of the implementation is especially important to consider in light of the relatively faint saturation limit of LSST ($r\sim16$ mag; \citealt{lsst}).
On the other end, beginning the observing sequence with the single visit would lead to a color determination and \emph{lower limits} to the slope even with an initial non-detection.
\vspace{1.5mm}
{\bf Filter pair $f_1$-$f_2$ ---}
\vspace{-3mm}
\begin{enumerate}
\item
The types of energetic, rapidly rising events which the {\em Presto-Color} cadence is designed to find, such as infant SNe, are very blue in color at their earliest phases. This favors the inclusion of LSST filters $g$ or $r$ in the pair (LSST's $u$-band filter's throughput is low, \citealt{oliver2008}).
\item
The magnitude evolution is generally faster at bluer wavelengths, due to rapid cooling which is observed in many of the transients considered here. Thus the transients are better separated along the $\Delta$mag axis when $g-$band observations are repeated.
\item
Non-adjacent filter pairs provide a larger lever arm to determine the slope of the spectral energy density (SED).
Specifically, the $g$ and $i$ filter choice separates the transients much better than $r$ and $i$ (\autoref{fig:classifier}). This allows far more insight into the transient from the initial observation triplet.
\end{enumerate}
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\textwidth]{figures/GPclassifierPLASTICC_40_00.png}
\caption{The phase space of color vs. magnitude-change is mapped with the probability of a data point belonging to a fast transient/fast feature, or to a ``normal'' transient. We overlay to this map transients from the PLAsTiCC challenge~\citep{plasticc}: ILOTs and CARTs (\emph{left} panel) and SN Iax (\emph{right} panel). The data points are plotted in red if our simple GP classifier identifies them as ``fast,'' in gray if they are classified as ``normal'' transients.}
\end{center}
\label{fig:PLAsTiCC}
\end{figure*}
\subsection{Generalization to the PLAsTiCC intermediate luminosity objects}
The simple segregation of the phase space we obtained by training a GP classifier on a few exemplary fast transients/features is not expected to be particularly robust. Nonetheless, we place additional fast transients in this phase space and test the performance of the classifier on these objects, which were not in the training set, to see how generalizable our conclusions are.
As described in \autoref{sec:transients}, from the PLAsTiCC challenge dataset~\citep{plasticc} we obtained a set of ILOTs, CARTs, and SNe Iax \citep[][and references therein]{Jha2017}. In this limited sample, ILOTs and CARTs display rapid luminosity evolution, while SNe Iax's behavior is not very different from standard type Ia's. The ILOTs and CARTs are shown in the left panel of \autoref{fig:PLAsTiCC}. The data points are plotted in red if they are classified as ``unusual',' in gray otherwise. ILOTs and CARTs are in part recognized by our classifier, mostly in phases in which their evolution is rapid and the ``tracks'' depart from the middle of the plot along the \emph{x}-axis, but also when their color is blue (bottom of the plot), enforcing the importance of obtaining color information for classification of fast transients. Note that some of the rapid evolution tracks that stray away from the region inhabited by the ``normal'' transients extending in the \emph{red} and \emph{rising} portion of the plot (top right) are not classified as unusual by our classifier simply because we had no points in this region. However, they are in a region of the phase space where the probability is $p~\sim~0.5$, and thus they should be considered interesting. The evolution of Iax SNe, however, is similar to that of normal type Ia's, and they are mostly missed by our classifier. This indicates that our simplistic classification scheme is not sufficient to distinguish SN Iax, or that more observations on longer time scales or contextual data (e.g. proximity to a galaxy center) are required; we remind the reader that photometric classification of transients is a field in rapid evolution \citep[\emph{e.g.}][]{plasticc}.
\section{Conclusions}
We identified minimal requirements that a photometric survey should meet to detect and distinguish unusual, fast-evolving astrophysical transients associated with explosive physics from other transients. We refer to our proposed strategy as {\em Presto-Color} and focus our design on the upcoming LSST.
The detection and prompt identification of fast transients, such as KNe and rapidly evolving features in type Ia and core-collapse SN, is critical to future synoptic surveys. These transients are poorly observed due to the general inefficiency of synoptic surveys at detecting and recognizing them, and they are poorly understood. Some of these transients are intrinsically rare (KN). Others, while not intrinsically rare, are seldom detected \citep{Drout2014}, as their evolution happens on time-scales shorter than probed by the cadence of ongoing wide-field surveys. They are characterized by rapid luminosity evolution and unusual colors, relative to more common sources. We thus identify the minimal requirements to be three observations in two filters within a night, with a maximum separation for the observations in different filters of $\ensuremath{\Delta T_1}~\lesssim~ 30$~minutes, which yields an accurate color determination in the presence of a rapidly evolving luminosity, and an optimal repeat of $\ensuremath{\Delta T_2}~\gtrsim~ 4$~hr, which provides sufficient leverage on the lightcurve shape to detect the rapid evolution.
The preferred implementation options for LSST is to use $g$-and-$i$ and repeat the $g$ observation, but alternating between $g$ and $i$ for the repeat filter maximizes observing efficiency (minimizing slew and filter changes). However, we note that LSST is planning to collect nearly twice as many images in $r$ than in $g$\footnote{\url{https://ls.st/srd}} to achieve its four main science goals (\autoref{sec:intro}). If this design feature is preserved in the final LSST observing strategy, {\em Presto-color} could be implemented in $r$-and-$z$.
We find that a minimum separation between observations in the same filter of $\gtrsim 1.5$~hr is sufficient to capture a significant change in luminosity only for transients $g\gtrsim22$.
These two requirements, while not sufficient for accurate studies of individual transients, are sufficient to separate fast transients/features with contemporary machine learning methods. This, in turn, can be used to trigger timely photometric or spectroscopic follow-up, and will lead to valuable rate estimates, as well as observational constraints on the nature of the progenitor systems of these sources. These requirements are only minimal constraints, and still permit significant survey strategy optimization: field selection, sky coverage, and filter selection (provided our constraints) can all be optimized for other science goals.
This research is reproducible. The code is accessible on GitHub\footnote{\url{https://github.com/fedhere/prestocolor}}
\software{scikit-earn \citep{scikit-learn},
sncosmo \citep{sncosmo},
Opsim \citep{opsim},
MAF \citep{maf},
astropy \citep{astropy2013, astropy2018}}
\section{Acknowledgments}
This work was developed within the Transients and Variable Stars Science Collaboration (TVS) and the author acknowledges the support of TVS in the preparation of this paper.
\noindent
The authors acknowledge support from the Flatiron Institute, Heising-Simons Foundation, and LSST Corporation for the development of this paper.
\noindent
Research support to IA is provided by the GROWTH (Global Relay of Observatories Watching Transients Happen) project funded by the National Science Foundation Partnership in International Research Program under NSF PIRE grant number 1545949.
\noindent
P.S.C. is grateful for support provided by NASA through the NASA Hubble Fellowship grant \#HST-HF2-51404.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.
\noindent
G.N. is supported by the Lasker Fellowship at the Space Telescope Science Institute.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=6.5cm,height=4.5cm]{figures/3visits_pontus_2591_gi.png}
\includegraphics[width=6.5cm,height=4.5cm]{figures/3visits_baseline2018a_gi.png}
\includegraphics[width=6.91cm,height=4.5cm]{figures/3visits_pontus_2591_rz.png}
\includegraphics[width=6.5cm,height=4.5cm]{figures/3visits_baseline2018a_rz.png}
\includegraphics[width=6.5cm,height=4.5cm]{figures/3visits_pontus_2591_grgzriiz.png}
\includegraphics[width=6.5cm,height=4.5cm]{figures/3visits_baseline2018a_grgzriiz.png}
\caption{{The results of our {\em diagnostic} metric, {\tt threeVisitsWColorMetric}, that checks for fields in {\tt OpSim} LSST strategy simulations that were observed three times in a night with two filters (as labeled) satisfying constraints on \ensuremath{\Delta T_1}\ and \ensuremath{\Delta T_2}. We explore year-one of the {\tt pontus\_2591} (our {\em Presto-Color} test run; \emph{left} panels), and year-one of the {\tt baseline2018a} LSST {\tt OpSim} runs (\emph{right} panels). We use a {\tt HEALPix} \citep{Gorski05} sky pixelization at a resolution of $nside=128$. All {\tt HEALPix128} ``pixels" that met the conditions of our metric are shown as points, colored by the filter pair $f_1$-and-$f_2$.
In {\tt pontus\_2591}, $\sim20$ thousand observations in year-one satisfy our constraints strictly ($\ensuremath{\Delta T_1}<0.5$ and $\ensuremath{\Delta T_2}>1.5$) in $g$-and-$i$ and over 40 thousand in $r$-and-$z$.
Very few {\tt HEALPix128} pixels in the {\tt baseline2018a} {\tt OpSim} have observations in triplets in two filters and none that satisfy our constraints.}}\label{fig:metricresult}
\end{center}
\end{figure*}
|
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{"url":"https:\/\/greprepclub.com\/forum\/the-quantity-will-end-in-how-many-zeros-5882.html?sort_by_oldest=true","text":"It is currently 19 Sep 2019, 22:12\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# The quantity will end in how many zeros\n\n Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nTAGS:\nFounder\nJoined: 18 Apr 2015\nPosts: 8139\nFollowers: 157\n\nKudos [?]: 1710 [0], given: 7485\n\nThe quantity will end in how many zeros\u00a0[#permalink] \u00a009 Aug 2017, 15:41\nExpert's post\n00:00\n\nQuestion Stats:\n\n67% (02:04) correct 32% (03:02) wrong based on 52 sessions\n\nThe quantity $$3^3 4^4 5^5 6^6$$ - $$3^6 4^5 5^4 6^3$$ will end in how many zeros ?\n\nA. 3\n\nB. 4\n\nC. 5\n\nD. 6\n\nE. 9\n[Reveal] Spoiler: OA\n\n_________________\nIntern\nJoined: 08 Dec 2017\nPosts: 40\nFollowers: 1\n\nKudos [?]: 44 [1] , given: 70\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a007 Feb 2018, 01:18\n1\nKUDOS\n3^3 4^4 5^5 6^6 - 3^6 4^5 5^4 6^3\n=...(2^14)(5^5) - ....(2^13)(5^4)\nThere should be 4 zeroes ending up this quantity.\nThe answer is B.\nDirector\nJoined: 07 Jan 2018\nPosts: 684\nFollowers: 10\n\nKudos [?]: 657 [3] , given: 88\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a008 Feb 2018, 22:51\n3\nKUDOS\nrealize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$\n\nsimplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$\nSimilarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$\nSubtracting 2nd term from 1st term:take the common term which is whole of the 2nd term\n\n$$3^9*2^1^3*5^4$$$$(10-1)$$\nnow we have to find out the number of zeros in the common term because non common term is 9\n2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros\n_________________\n\nThis is my response to the question and may be incorrect. Feel free to rectify any mistakes\n\nManager\nJoined: 15 Feb 2018\nPosts: 53\nFollowers: 1\n\nKudos [?]: 18 [0], given: 33\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a023 Feb 2018, 05:21\nHi, I don't quite understand your last sentence. What do you mean by the limiting number?\n\namorphous wrote:\nrealize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$\n\nsimplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$\nSimilarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$\nSubtracting 2nd term from 1st term:take the common term which is whole of the 2nd term\n\n$$3^9*2^1^3*5^4$$$$(10-1)$$\nnow we have to find out the number of zeros in the common term because non common term is 9\n2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros\nDirector\nJoined: 07 Jan 2018\nPosts: 684\nFollowers: 10\n\nKudos [?]: 657 [2] , given: 88\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a026 Feb 2018, 09:34\n2\nKUDOS\ngremather wrote:\nHi, I don't quite understand your last sentence. What do you mean by the limiting number?\n\namorphous wrote:\nrealize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$\n\nsimplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$\nSimilarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$\nSubtracting 2nd term from 1st term:take the common term which is whole of the 2nd term\n\n$$3^9*2^1^3*5^4$$$$(10-1)$$\nnow we have to find out the number of zeros in the common term because non common term is 9\n2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros\n\nsince for a '$$0$$' to occur $$5$$ has to be multiplied by $$2$$. The number of zeros will depend on the minimum power raised of either of the two numbers\n\nFor eg.\n$$100 = 5^2 * 2^2 = 2$$ zeros at the end (because both the terms have power raised to 2)\n$$125 = 5^3 * 2^0 = 0$$ zeros at the end (because 2 is raised to a power of 0 hence 2 becomes the limiting number)\n_________________\n\nThis is my response to the question and may be incorrect. Feel free to rectify any mistakes\n\nManager\nJoined: 18 Jun 2019\nPosts: 86\nFollowers: 0\n\nKudos [?]: 6 [0], given: 41\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a006 Sep 2019, 07:34\nHi! looking for another solution to this question.\n\nI've simplified upto 3^9*2^13*5^4(9) and don't know how to proceed to fnd the number of zero's.\n\nFounder\nJoined: 18 Apr 2015\nPosts: 8139\nFollowers: 157\n\nKudos [?]: 1710 [1] , given: 7485\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a006 Sep 2019, 08:04\n1\nKUDOS\nExpert's post\nThe trick is how many \"five you do have in the quantity ??\n\nwe do have nine five numbers.\n\nNow, you will have a zero whenever you do have $$2 \\times 5 = 10$$\n\nIn our quantity we do have 4 couples of 5 plus one 5 disparaged\n\n5*5\n5*5\n5*5\n5*5\n5\n\nThat means two couples of five are 4 zeros.\n\nHope this helps\n_________________\nManager\nJoined: 18 Jun 2019\nPosts: 86\nFollowers: 0\n\nKudos [?]: 6 [0], given: 41\n\nRe: The quantity will end in how many zeros\u00a0[#permalink] \u00a006 Sep 2019, 08:07\nCarcass wrote:\nThe trick is how many \"five you do have in the quantity ??\n\nwe do have nine five numbers.\n\nNow, you will have a zero whenever you do have 2 \\times 5 = 10\n\nIn our quantity we do have 4 couples of 5 plus one 5 disparaged\n\n5*5\n5*5\n5*5\n5*5\n5\n\nThat means two couples of five are 4 zeros.\n\nHope this helps\n\nMuch clearer!! Thank you!\nRe: The quantity will end in how many zeros \u00a0 [#permalink] 06 Sep 2019, 08:07\nDisplay posts from previous: Sort by\n\n# The quantity will end in how many zeros\n\n Question banks Downloads My Bookmarks Reviews Important topics\n\n Powered by phpBB \u00a9 phpBB Group Kindly note that the GRE\u00ae test is a registered trademark of the Educational Testing Service\u00ae, and this site has neither been reviewed nor endorsed by ETS\u00ae.","date":"2019-09-20 06:12:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.5572390556335449, \"perplexity\": 2129.494832725206}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573832.23\/warc\/CC-MAIN-20190920050858-20190920072858-00179.warc.gz\"}"}
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Q: Interop Structure: Should Unsigned Short be Mapped to byte[]? I have such a C++ structure:
typedef struct _FILE_OP_BLOCK
{
unsigned short fid; // objective file ID
unsigned short offset; // operating offset
unsigned char len; // buffer length(update)
// read length(read)
unsigned char buff[240];
} FILE_OP_BLOCK;
And now I want to map it in .Net. The tricky thing is that the I should pass a 2 byte array for fid, and integer for len, even though in C# fid is an unsigned short and len is an unsigned char
I wonder whether my structure ( in C#) below is correct?
[StructLayout(LayoutKind.Sequential, Pack = 1, CharSet = CharSet.Auto)]
public struct File_OP_Block
{
[MarshalAs(UnmanagedType.ByValArray, SizeConst = 2)]
public byte[] fid;
public ushort offset;
public byte length;
[MarshalAs(UnmanagedType.ByValArray, SizeConst = 240)]
public char[] buff;
}
A: Your CharSet property on the [DllImport] attribute is definitely wrong, you need CharSet.Ansi to get the P/Invoke marshaller to convert it to a char[]. Declare the buff member as a string for easier usage. While declaring the fid member as a byte[] isn't wrong, I really don't see the point of it. That the unmanaged code copies a char[] into it is an implementation detail. Thus:
[StructLayout(LayoutKind.Sequential, Pack = 1, CharSet = CharSet.Ansi)]
public struct File_OP_Block
{
public ushort fid;
public ushort offset;
public byte length;
[MarshalAs(UnmanagedType.ByValTStr, SizeConst = 240)]
public string buff;
}
A: Given that the C++ short data type is actually two bytes, a two byte array should work. The integer sizes in C/C++ are not strictly defined, so the standard only says that a short is at least two bytes.
The C# char data type is a 16 bit unicode character, so that doesn't match the C++ char data type which is an 8 bit data type. You either need an attribute to specify how the characters are encoded into bytes, or use a byte array.
You might need an attribute to specify the packing, so that there is no padding between the members.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
We are interested in unsupervised learning of data with temporal dependencies.
The aim of such learning is to extract the statistical structure from ``trajectories'' or sequences,
after which it will be possible, for example, to predict the next frame in the sequence, to make optimal inferences about the dynamical state of the underlying system being observed (``filter''), or to generate novel sequences with the same statistical properties.
Ideally, we would make use of some of the sophisticated unsupervised-learning techniques developed for static data.
If, however, a single training vector consists of the observations at a single time step, the temporal dependencies obviously will not be learned.
The opposite extreme, feasible in some contexts, is to assign to a single training vector an entire trajectory of observations; but in general, trajectories may be very long, vary in length, or required to be learned on-line.
A middle course is to try to construct, at each time step, \emph{augmented} observations that are shorter than an entire trajectory of observations, but that form, unlike the raw observations, a Markov chain.
When the \emph{direct} temporal dependencies are not, indeed, the length of the trajectory itself, past observations' bearing on future observations can be summarized compactly.
Data consisting of the current observation, augmented by such a summary, form a Markov chain across time, which facilitates the application of recursive procedures for filtering and prediction.
In the most familiar example of this strategy, the Kalman filter, the past noisy observations of a discrete-time, linear-Gaussian dynamical system are summarized by the posterior cumulants over a hidden dynamical state.
These cumulants, which are also the sufficient statistics for the hidden state (conceived as a parameter), are the state-estimate vector, whose length is proportional to the length of the direct temporal dependencies, and a covariance matrix that encodes the (inverse) reliability of this state estimate.
The recursive filtering procedure in turn makes parameter learning possible---in this case, with an expectation-maximization (EM) algorithm \citep[see, e.g.,][]{Ghahramani1996a}.
Our goal is to extend this ``middle'' strategy to more general temporal sequences, for which we are unable, for example, to compute analytically the posterior distribution over the hidden state, or even to formulate equations of state or an observation model.
The compact summaries, consequently, we shall require the algorithm itself to formulate, rather than deriving them analytically.
In particular, our basic building block will be the exponential-family harmonium (EFH) \citep{Welling2004}, a two-layer Markov random field with full interlayer connections and no intralayer connections, in which the conditional distributions over the hidden layer given the visible layer and vice versa are products over exponential-family distributions.
(The restricted Boltzmann machine corresponds to the special case of Bernoulli conditional distributions.)
For the summaries, with which we augment the observations at each time step, we choose the hidden-unit means from the previous time step.
We have recently \citep{Makin2015b} introduced this temporal extension to the EFH, the ``recurrent EFH'' (rEFH); investigated its predictions for receptive-field formation in primate cerebral cortex; and compared its performance as a filter for simple dynamical systems against Kalman filters learned with EM.
Here, we investigate the analytical properties of the rEFH, supplying sufficient conditions and a proof for when the training procedure results in a model that provides optimal inference.
The central idea behind the proof is to abandon the assumption of model correctness that underlies use of, e.g., linear-Gaussian dynamical systems, in favor of a weaker assumption about model identifiability.
This assumption, which holds for the rEFH, guarantees that, at each time step, the sufficient statistics for the hidden units of the trained model are also sufficient statistics for the current \emph{true} hidden state.
That is, the model's hidden units learn to encode the compact summary corresponding to the filter cumulants.
The rEFH is very similar to, but importantly different from, two recent temporal extensions to the EFH, the TRBM \citep{Sutskever2007} and the RTRBM \citep{Sutskever2009}.
We show why the proof does not apply to these models, and then compare all three empirically by training and testing on data where the optimal posterior estimate of the true dynamical state is computable analytically.
We also compare model performance on the synthetic data set used in those papers:\ the rEFH is superior to the TRBM and competitive with the RTRBM, despite a simpler learning algorithm.
Finally, we show that generating sample trajectories---\emph{in the reverse direction}---is also simple in the rEFH, in contrast to the TRBM and RTRBM, for which generation is hard.
\section{Theory}\label{sec:theory}
We begin by describing our strategy for extending the EFH to temporal data (\sctn{proofstrategy}) before justifying it more rigorously (\sctn{proofformal}), and finally describing the training procedure in detail (\sctn{rEFHtraining}).
Our aim is a training procedure and architecture that can be guaranteed to provide \emph{optimal inference}.
That is, after training, our model should allow us to compute the posterior distribution, $p(\worldstates{t}|\stateobsvstillnow)$, over the dynamical state, $\Worldstates{t}$, of the underlying causes of our observations, $\stateobsvs{t}$, conditioned on the preceding sequence of such observations.
In particular, we require that the (mean) hidden-unit activities of the model at time $t$ encode the cumulants of this distribution.
\subsection{Strategy for extending the EFH to temporal data}\label{sec:proofstrategy}
We consider temporal data $\Stateobsvsalltime$ that have been generated by a very general discrete-time dynamical process, anything consistent with the independence statements asserted by the hidden Markov model in the top two rows of \subfig{HMMaugmented}, without any other assumptions about the distributions parameterizing it.
(The Markov assumption is weak in the sense that longer temporal dependencies can be accommodated under it simply by lengthening the state vector.)
We shall, however, introduce below some constraints relating to a technical condition called ``identifiability.''
Our goal is to make optimal inferences about the underlying ``causes,'' $\Worldstatesalltime$, for the observations, $\Stateobsvsalltime$, with joint distribution
\begin{equation}\nonumber
p(\worldstatesalltime,\stateobsvsalltime)
= \prod_{t=0}^T p(\stateobsvs{t}|\worldstates{t}) p(\worldstates{t}|\worldstates{t-1}),
\end{equation}
corresponding to \subfig{HMMaugmented}.
(Here as elsewhere, for notational compactness, $p(\worldstates{0}|\worldstates{-1}) := p(\worldstates{0})$, and likewise for other expressions that reference negative times.)
Given nothing more than samples from a ``data distribution'' of this general form, we construct our own latent-variable model, $q$, and attempt to bring the probabilities it assigns to observation sequences close to the probabilities with which they occur in the data; i.e., we seek parameters $\params$ such that $q(\stateobsvsalltime;\params) = p(\stateobsvsalltime)$ (``density estimation'').
\FigTikzHMMs
Here we introduce an important distinction.
We do not assume that the latent variables of the model have the same form as those ``in the world,'' $\Worldstates{}$, so in fact we use a different letter for them, $\Modelstates{}$.
In training latent-variable density estimators, the correspondence between $\Modelstates{}$ and $\Worldstates{}$ is not always straightforward, even after the model has become a good one for the observed data.
Nevertheless, our recent result about the character of this correspondence \citep{Makin2015a} can be used to show how to achieve our goal, optimal inference of the true causes, $\Worldstates{}$.
Consider in particular observed (static) random variables $\Augobsvs{}$.
In many cases one can say that the sufficient statistics\footnote{``T,'' the usual symbol for sufficient statistics, is already used for time, so we use ``U'' instead.} for the model's latent variables, $\SSformodelstates{}$, are also sufficient statistics for the true latent variables, $\Worldstates{}$.
This is intuitive:\ the same aspects of $\Augobsvs{}$ are in need of explanation, whether via the world or via the model.
The result does not hold for all latent-variable density estimators, but for a class of such models that obeys a set of rather weak technical conditions \citep[see][and \thm{maintheorem} below]{Makin2015a}, it is the case that
\begin{equation}\label{eqn:sufficiencyForLatents}
\forall \augobsvs{}\hspace{0.1in} q(\augobsvs{};\params) = p(\augobsvs{})
\implies \MI{\SSformodelstates{}}{\Worldstates{}} = \MI{\Augobsvs{}}{\Worldstates{}},
\end{equation}
with $\MI{\cdot}{\cdot}$ the mutual information.
That is, after a good model has been learned (left-hand equation), transforming $\Augobsvs{}$ into any set of sufficient statistics for $\Modelstates{}$ discards no information about the underlying cause, $\Worldstates{}$.
When information about $\Worldstates{}$ reaches $\Modelstates{}$ \emph{only} via $\Augobsvs{}$, \eqn{sufficiencyForLatents} is equivalent \citep[see][]{Makin2015a} to
\begin{equation}\label{eqn:posteriorMatchingResult}
\forall \augobsvs{}\hspace{0.1in} q(\augobsvs{};\params)
= p(\augobsvs{})
\implies \forall \augobsvs{},\worldstates{}\hspace{0.1in}
p(\worldstates{}|\ssformodelstates{})
= p(\worldstates{}|\augobsvs{}).
\end{equation}
In words, transforming $\Augobsvs{}$ into sufficient statistics for $\Modelstates{}$ does not change the posterior distribution over the underlying cause, $\Worldstates{}$.
For EFHs, the sufficient statistics $\SSformodelstates{}$ are the posterior mean, $\xpct{q}{\Modelstates{}|\Augobsvs{}}$; hence, in a trained model, the mean activity of the hidden units captures all the information about the true cause, $\Worldstates{}$, as was contained in the vector of observed responses, $\Augobsvs{}$.
We construct a procedure for training the EFH on data with temporal dependencies that exploits this result.
In particular, at every time step, we train the EFH on an ``augmented'' input vector $\augobsvs{t}$ that consists of the current observations, $\stateobsvs{t}$, concatenated together with the sufficient statistics for $\Modelstates{t-1}$ from the previous time step, $\ssformodelstates{t-1}$---i.e., the previous posterior mean, $\xpct{q}{\Modelstates{t-1}|\augobsvs{t-1}}$.
The central intuition for why this procedure will produce models providing optimal inference is as follows.
Training the model at the first time step ($t=0$) forces the hidden units---or, more precisely, the sufficient statistics for them, $\xpct{q}{\Modelstates{0}|\stateobsvs{0}}$---to extract from observations $\stateobsvs{0}$ a good summary of the underlying state, $\Worldstates{0}$, no information about it being lost.
The sufficient statistics at the second time step ($t=1$), then, are forced to become a good ``meta-summary'' of both the information in $\Stateobsvs{1}$ about $\Worldstates{1}$ \emph{and} the previous summary.
Continuing in this fashion builds up in these sufficient statistics---the means of the hidden units---a good running summary about the true state and its history.
We call the resulting model the recurrent, exponential-family harmonium (rEFH).
\subsection{Rigorous justification of the training procedure}\label{sec:proofformal}
We now formalize this intuitive argument.
We begin with the result from our previous work:
\begin{theorem}
Let $p(\augobsvs{}) = \sum_\worldstates{} p(\augobsvs{}|\worldstates{})p(\worldstates{})$ and
$q(\augobsvs{};\params) = \sum_\modelstates{} p(\augobsvs{}|\modelstates{};\params)p(\modelstates{};\params)$
be expressible as \emph{identifiable} finite mixtures of the same conditional distribution $f(\augobsvs{}|\proxcauses{})$,
$\sum_\proxcauses{} f(\augobsvs{}|\proxcauses{})p(\proxcauses{})$ and
$\sum_\proxcauses{} f(\augobsvs{}|\proxcauses{})q(\proxcauses{})$, respectively.
That is, the map from prior distributions over ``parameters'' $\Proxcauses{}$ to mixtures (marginal distributions over $\Augobsvs{}$) is injective.
Then \eqn{sufficiencyForLatents} holds.
Furthermore, if $\Worldstates{}$ and $\Modelstates{}$ are independent conditioned on $\Augobsvs{}$, then \eqn{posteriorMatchingResult} holds as well.
\end{theorem}
For the proof, see the work of \citet{Makin2015a}.
Here we prove by induction that the training procedure just proposed generalizes this theorem to temporal data.
\begin{theorem}\label{thm:maintheorem}
For all $t$, let the marginal distributions
\begin{equation}\nonumber
\begin{split}
p(\augobsvs{t};\params)
&:= \sum_{\worldstatestillnow;\params}p(\augobsvs{t}|\worldstatestillnow)p(\worldstatestillnow)\\
q(\augobsvs{t};\params)
&:= \sum_\modelstates{t} q(\augobsvs{t}|\modelstates{t};\params)q(\modelstates{t};\params)
\end{split}
\end{equation}
be expressible as \emph{identifiable} finite mixtures of the same conditional distribution $f(\augobsvs{t}|\proxcauses{t})$,
$\sum_\proxcauses{t} f(\augobsvs{t}|\proxcauses{t})p(\proxcauses{t})$ and
$\sum_\proxcauses{t} f(\augobsvs{t}|\proxcauses{t})q(\proxcauses{t})$, respectively.
Furthermore, let the distribution of observed data $\Augobsvs{t}$ be consistent with the independence statements asserted by \subfig{HMMaugmented}, in which in particular each observation consists of the emissions of a hidden Markov model, concatenated together with the sufficient statistics, $\SSformodelstates{t-1}$, for the latent variables $\Modelstates{t-1}$ of $q$ at the preceding time step:
\begin{equation}\label{eqn:augmentedData}
\Augobsvs{t} := [\SSformodelstates{t-1},\Stateobsvs{t}].
\end{equation}
Then
\begin{equation}\label{eqn:sufficiencyForTemporalLatents}
\forall t, \forall \augobsvs{t}\hspace{0.1in} q(\augobsvs{t};\params) = p(\augobsvs{t};\params)
\implies \MI{\SSformodelstates{t}}{\Worldstates{t}} = \MI{\Stateobsvstillnow}{\Worldstates{t}}.
\end{equation}
If, moreover, $\Worldstates{t}$ and $\Modelstates{t}$ are independent conditioned on $\Augobsvs{t}$, then
\begin{equation}\label{eqn:posteriorGivenHiddensIsFilteringDistribution}
\forall t, \forall \augobsvs{t}\hspace{0.1in} q(\augobsvs{t};\params) = p(\augobsvs{t};\params)
\implies p(\worldstates{t}|\ssformodelstates{t}) = p(\worldstates{t}|\stateobsvstillnow).
\end{equation}
\end{theorem}
Note that the variables $\ssformodelstates{t-1}$ can be calculated deterministically from $\stateobsvstillprev$, so the data distribution $p$ induces a distribution over these sufficient statistics as well as the observations.
This is the joint distribution we have called $p(\augobsvs{t};\params)$, whose generative model is depicted in \subfig{HMMaugmented}.
The equality on the left of \eqn{sufficiencyForTemporalLatents} says that the EFH is a good density estimator for its inputs at time $t$, $\Augobsvs{t}$ (cf.\ \subfig{REFHgraphicalmodel}).
The information equality on the right says that all the information about the current state, $\Worldstates{t}$, that was available in all the observations up to the present, $\stateobsvstillnow$, is also available in a particular function, $\suffstats{\Modelstates{t}}(\cdot)$, of just the current augmented input vector, $\augobsvs{t}$.
Just as \eqn{sufficiencyForLatents} can be re-expressed as \eqn{posteriorMatchingResult}, this information equality can be alternatively expressed as \eqn{posteriorGivenHiddensIsFilteringDistribution}.
The critical technical condition guaranteeing \eqn{sufficiencyForLatents}, on which the proof depends, is that $p(\augobsvs{t})$ and $q(\augobsvs{t})$ be expressible in terms of the same finite-mixture model, and that the map from distributions over parameters of this mixture to the marginal distribution over $\Augobsvs{t}$ be injective, except possibly on sets of measure zero.
Precise characterization being rather technical, we defer it to the \app{technicalconditions}.
We now prove theorem \thm{maintheorem} by induction, under the assumption that these technical conditions hold.
\begin{proof}
It is easily verified that $\Worldstates{t}$ and $\Modelstates{t}$ are independent conditioned on $\Augobsvs{t}$, for all time, so we shall work throughout with \eqns{posteriorMatchingResult}{posteriorGivenHiddensIsFilteringDistribution} rather than \eqns{sufficiencyForLatents}{sufficiencyForTemporalLatents}.
Now, the base case of the induction is simply an application of \eqn{posteriorMatchingResult}.
Assume that $q(\augobsvs{0};\params) = p(\augobsvs{0})$, or equivalently that $q(\stateobsvs{0};\params) = p(\stateobsvs{0})$ (since at $t=0$ there is no recurrent vector to concatenate with the current observation $\stateobsvs{0}$; see \subfig{REFHgraphicalmodel}); that is, that the EFH has become a good density estimator for the initial data.
Then \eqn{posteriorMatchingResult} implies that
\begin{equation}\label{eqn:filteringInductionBaseCase}
p(\worldstates{0}|\ssformodelstates{0})
= p(\worldstates{0}|\suffstats{\Modelstates{0}}(\stateobsvs{0}))
= p(\worldstates{0}|\stateobsvs{0}).
\end{equation}
Now for the induction step, assume that
\begin{equation}\label{eqn:filteringInductionIf}
p(\worldstates{t-1}|\ssformodelstates{t-1}) = p(\worldstates{t-1}|\stateobsvstillprev).
\end{equation}
We want to show that under this assumption, the EFH having become a good density estimator for its inputs at time $t$, $q(\augobsvs{t};\params) = p(\augobsvs{t};\params)$, implies
\begin{equation}\label{eqn:filteringInductionThen}
p(\worldstates{t}|\ssformodelstates{t}) = p(\worldstates{t}|\stateobsvstillnow).
\end{equation}
Together with the base case, this will show that the implication holds for all time---\eqn{posteriorGivenHiddensIsFilteringDistribution}.
To save space we abbreviate $\Rcrnts{t} := \SSformodelstates{t}$ and omit references to the parameters.
Now, since $q(\augobsvs{t}) = p(\augobsvs{t})$, \eqn{posteriorMatchingResult} implies that
\begin{subequations}
\begin{align}
p(\worldstates{t}|\rcrnts{t})
&= p(\worldstates{t}|\augobsvs{t}) \nonumber\\
&= p(\worldstates{t}|\stateobsvs{t},\rcrnts{t-1}) \nonumber\\
\implies p(\worldstates{t}|\rcrnts{t}) p(\stateobsvs{t}|\rcrnts{t-1})
&= p(\worldstates{t}|\stateobsvs{t},\rcrnts{t-1})
p(\stateobsvs{t}|\rcrnts{t-1}) \label{eqn:rEFHproofSequenceA}\\
&= p(\stateobsvs{t}|\worldstates{t},\rcrnts{t-1})
p(\worldstates{t}|\rcrnts{t-1}) \nonumber\\
&= p(\stateobsvs{t}|\worldstates{t}) \sum_{\worldstates{t-1}}
p(\worldstates{t},\worldstates{t-1}|\rcrnts{t-1}) \label{eqn:rEFHproofSequenceB}\\
&= p(\stateobsvs{t}|\worldstates{t}) \sum_{\worldstates{t-1}}
p(\worldstates{t}|\worldstates{t-1},\rcrnts{t-1})
p(\worldstates{t-1}|\rcrnts{t-1}) \nonumber\\
&= p(\stateobsvs{t}|\worldstates{t}) \sum_{\worldstates{t-1}}
p(\worldstates{t}|\worldstates{t-1})
p(\worldstates{t-1}|\rcrnts{t-1}) \label{eqn:rEFHproofSequenceC}\\
&= p(\stateobsvs{t}|\worldstates{t}) \sum_{\worldstates{t-1}}
p(\worldstates{t}|\worldstates{t-1},\stateobsvstillprev)
p(\worldstates{t-1}|\stateobsvstillprev) \label{eqn:rEFHproofSequenceD}\\
&= p(\stateobsvs{t}|\worldstates{t})
p(\worldstates{t}|\stateobsvstillprev) \nonumber\\
&= p(\stateobsvs{t}|\worldstates{t},\stateobsvstillprev)
p(\worldstates{t}|\stateobsvstillprev) \label{eqn:rEFHproofSequenceE}\\
&= p(\worldstates{t}|\stateobsvstillnow)
p(\stateobsvs{t}|\stateobsvstillprev). \label{eqn:rEFHproofSequenceF}
\end{align}
\end{subequations}
The independence conditions used in the transitions to
\eqnsss{rEFHproofSequenceB}{rEFHproofSequenceC}{rEFHproofSequenceD}{rEFHproofSequenceE}
are licensed by the graph in \subfig{HMMaugmented}, in particular by the ``explaining away'' property of the unobserved $\Rcrnts{t}$.
Marginalizing out $\worldstates{t}$ from the left-hand side of \eqn{rEFHproofSequenceA} and from \eqn{rEFHproofSequenceF} shows that
that $p(\stateobsvs{t}|\rcrnts{t-1}) = p(\stateobsvs{t}|\stateobsvstillprev)$,
and therefore $p(\worldstates{t}|\rcrnts{t}) = p(\worldstates{t}|\stateobsvstillnow)$, which is \eqn{filteringInductionThen}.
This completes the induction and therefore the proof of the theorem.
\end{proof}
\subsection{Training the rEFH}\label{sec:rEFHtraining}
The proof shows that optimal inference---what is expressed by the information equality on the right side of \eqn{sufficiencyForTemporalLatents}---can be achieved, in density estimators that can be expressed as identifiable mixture models, by establishing the equality on the left side, $q(\augobsvs{t};\params) = p(\augobsvs{t};\params)$,
where $\augobsvs{t}$ is constructed according to \eqn{augmentedData}.
To establish that equality, we train an exponential-family harmonium \citep[EFH, depicted in \subfig{EFH}, and introduced by][]{Welling2004} on these ``augmented'' data vectors.
The joint distribution then takes the form of a Boltzmann distribution.
For simplicity, we consider only exponential families whose sufficient statistics are the identity function.
In accordance with \eqn{augmentedData}, the visible layer consists of the current observations, $\Stateobsvs{t}$, and the previous hidden-layer sufficient statistics, $\Rcrnts{t-1}$; see \subfig{EFHinrEFH}.
Thus the joint distribution is
\begin{equation}\label{eqn:REFHjointdstrb}
q(\modelstates{t},\rcrnts{t-1},\stateobsvs{t};\params)
\propto h(\rcrnts{t-1})h(\stateobsvs{t})h(\modelstates{t})\exp\big\{
\modelstates{t}\tr (\vishidwts \stateobsvs{t} + \rcrnthidwts\rcrnts{t-1} + \hidbiases)
+ \visbiases\tr\stateobsvs{t} + \rcrntbiases\tr\rcrnts{t-1}\big\}.
\end{equation}
Here the functions $h$ are the base measures, determined by the choices of exponential families.
The intractable normalizer for $q$ has been omitted.
The corresponding conditional distributions are%
\begin{subequations}\label{eqn:REFHconditionals}%
\begin{align}
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t};\params)
&= \frac{1}{\mathcal{Z}_\modelstatevar(\stateobsvs{t},\rcrnts{t-1})}h(\modelstates{t})
\exp\big\{ (\vishidwts\stateobsvs{t} + \rcrnthidwts\rcrnts{t-1} + \hidbiases)\tr\modelstates{t} \big\},
\label{eqn:REFHhiddencond}\\
q(\stateobsvs{t}|\modelstates{t};\params)
&= \frac{1}{\mathcal{Z}_\stateobsvvar(\modelstates{t})}h(\stateobsvs{t})
\exp\big\{ (\vishidwts\tr\modelstates{t} + \visbiases)\tr\stateobsvs{t} \big\},
\label{eqn:REFHstateobsvcond}\\
q(\rcrnts{t-1}|\modelstates{t};\params)
&= \frac{1}{\mathcal{Z}_\rcrntvar(\modelstates{t})}h(\rcrnts{t-1})
\exp\big\{ (\rcrnthidwts\tr\modelstates{t} + \rcrntbiases)\tr\rcrnts{t-1} \big\}.
\label{eqn:REFHrcrntcond}
\end{align}
\end{subequations}
The conditional distribution for the visible units has been separated into two conditionals, \eqns{REFHstateobsvcond}{REFHrcrntcond}, corresponding to the observations and the previous hidden-layer means, respectively.
From the graphical model, \subfig{EFHinrEFH}, their product forms the conditional distribution over the visible units:\
$q(\rcrnts{t},\stateobsvs{t}|\modelstates{t};\params) = q(\rcrnts{t}|\modelstates{t};\params) q(\stateobsvs{t}|\modelstates{t};\params)$.
The data distribution over $\Stateobsvsalltime$ is assumed to be unknown, but the conditional distribution it induces over the hidden-layer sufficient statistics is known and deterministic:
\begin{equation}\label{eqn:rcrntdstrb}
p(\rcrnts{t}|\rcrnts{t-1},\stateobsvs{t};\params)
:= \delta(\rcrnts{t} - \xpct{q}{\Modelstates{t}|\rcrnts{t-1},\stateobsvs{t};\params}).
\end{equation}
In words, the distribution over the current hidden-layer sufficient statistics is a delta function located at the posterior mean.
In the EFH, this is computed as a deterministic ``upward'' (visible-to-hidden) pass through the weight matrix, followed by the pointwise nonlinearity appropriate for the exponential family to which the posterior distribution belongs \citep[see][]{Welling2004}.
We emphasize that, despite being parameterized (and by parameters of the model, no less), we treat the conditional distribution in \eqn{rcrntdstrb} as part of the data distribution ($p$) because it provides part of the training data for the EFH.
\FigTikzEFHs
Exact density estimation in an EFH---that is, gradient descent in the KL divergence between $p(\augobsvs{t};\params)$ and $q(\augobsvs{t};\params)$---requires prolonged Gibbs sampling, which introduces variance into the learning procedure as well as slowing it down \citep{Hinton2002}.
We therefore proceed by stochastic gradient descent of the approximate ``$n$-step contrastive divergence'' (CD$_n$) objective function, which has $q = p$ at its minimum \citep{Hinton2002,Hinton2006a}.
At the cost of this approximation, contrastive-divergence learning buys fast, low-variance learning, since it requires only $n$ (which can be small) full steps of Gibbs sampling from the model before the weights can be updated.
We also ignore the dependence of the data distribution, $p(\augobsvs{t};\params)$, on the parameters, introduced by \eqn{rcrntdstrb},
an approximation that we discuss further in \sctn{conclusions}, below.
The learning rules are therefore the standard rules for the EFH under one-step contrastive divergence \citep[see][]{Welling2004}, applied to our data:
\begin{align}\label{eqn:rEFHparameterUpdates}
\Delta\vishidwts &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}\Stateobsvs{t}\tr
-\smplavg{
q(\updtmodelstates{t}|\updtrcrnts{t-1},\updtstateobsvs{t})
q(\updtrcrnts{t-1},\updtstateobsvs{t}|\modelstates{t})}
{\Updtmodelstates{t}\Updtstateobsvs{t}\tr
}
},\nonumber \\
\Delta\rcrnthidwts &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}\Rcrnts{t-1}\tr
-\smplavg{
q(\updtmodelstates{t}|\updtrcrnts{t-1},\updtstateobsvs{t})
q(\updtrcrnts{t-1},\updtstateobsvs{t}|\modelstates{t})}
{\Updtmodelstates{t}\Updtrcrnts{t-1}\tr}
},\nonumber \\
\Delta\hidbiases &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}
-\smplavg{
q(\updtmodelstates{t}|\updtrcrnts{t-1},\updtstateobsvs{t})
q(\updtrcrnts{t-1},\updtstateobsvs{t}|\modelstates{t})}
{\Updtmodelstates{t}}
},\nonumber \\
\Delta\visbiases &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Stateobsvs{t}
-\smplavg{
q(\updtstateobsvs{t}|\modelstates{t})}
{\Updtstateobsvs{t}}
},\nonumber \\
\Delta\rcrntbiases &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Rcrnts{t-1}
-\smplavg{
q(\updtrcrnts{t-1}|\modelstates{t})}
{\Updtrcrnts{t-1}}
},
\end{align}
where the angle brackets indicate sample averages rather than true expectations.
\section{Alternative extensions of the EFH to temporal data.}
Sutskever and colleagues introduced an extension of the restricted Boltzmann machine (RBM) to temporal data, and proposed two different methods for training it \citep{Sutskever2007,Sutskever2009}.
We re-present them here to contrast them with the rEFH, and to show that the proof of the preceding section does not apply to these methods.
For simplicity, we confine ourselves to the special case of the TRBM where the only direct dependence of an RBM on any of its predecessors in the chain is on the \emph{hidden} units of its \emph{immediate} predecessor (\subfig{TRBMgraphicalmodel}).
(We consider the more general model in the discussion below.)
We derive both models in terms of an EFH, rather than an RBM as in the original papers, because the EFH is more general and can be accommodated within the framework of the cited works with minor adjustments only.
Finally, our development of the models is slightly different, although the final result is the same.
\citet{Sutskever2007} define the following joint distribution:
\begin{equation}\nonumber
q(\modelstates{0},\stateobsvs{0},\ldots,\modelstates{T},\stateobsvs{T};\params)
= \prod_{t=0}^T q(\modelstates{t},\stateobsvs{t}|\modelstates{t-1};\params),
\end{equation}
reflecting the independence properties of the graph in \subfig{TRBMgraphicalmodel}.
The graph also asserts the independence of $\Stateobsvs{t}$ and $\Modelstates{t-1}$, given $\Modelstates{t}$.
This follows, straightforwardly, from the definition of the central component of the model,
\begin{equation}\nonumber
q(\modelstates{t},\stateobsvs{t}|\modelstates{t-1};\params)
:= \frac{1}{\mathcal{Z}(\modelstates{t-1})}h(\stateobsvs{t})h(\modelstates{t})\exp\big\{
\modelstates{t}\tr \vishidwts \stateobsvs{t}
+ \visbiases\tr\stateobsvs{t}
+ (\rcrnthidwts\modelstates{t-1}
+ \hidbiases)\tr\modelstates{t}
\big\},
\end{equation}
since the components of the right-hand side that reference $\modelstates{t-1}$ and $\stateobsvs{t}$ can be factored into separate pieces.
This equation has the form of a harmonium; that is, it can be produced by two conditional distributions of the form
\begin{subequations}\nonumber
\begin{align}
q(\modelstates{t}|\modelstates{t-1},\stateobsvs{t})
&= \frac{1}{\mathcal{Z}_\modelstatevar(\stateobsvs{t},\modelstates{t-1})}h(\modelstates{t})
\exp\big\{ (\vishidwts\stateobsvs{t} + \rcrnthidwts\modelstates{t-1} + \hidbiases)\tr\modelstates{t} \big\},\\
q(\stateobsvs{t}|\modelstates{t},\modelstates{t-1})
&= \frac{1}{\mathcal{Z}_\stateobsvvar(\modelstates{t})}h(\stateobsvs{t})
\exp\big\{ (\vishidwts\tr\modelstates{t} + \visbiases)\tr\stateobsvs{t} \big\}
= q(\stateobsvs{t}|\modelstates{t}).
\end{align}
\end{subequations}
As it stands, learning in this model is intractable, since inference does not admit of a recursive procedure (like the Kalman filter).
In order apply such a procedure, the authors decouple the EFHs through time with a gross approximation.
They define a recursion $\rcrnts{t} = \xpct{q}{\Modelstates{t}|\rcrnts{t-1},\stateobsvs{t}}$, where the expectation is taken under the first conditional.
Then they simply substitute $\rcrnts{t-1}$ for $\modelstates{t-1}$.
Thus, the approximate model is defined by the joint distribution
\begin{equation}\label{eqn:RTRBMjointdstrb}
q(\modelstates{t},\stateobsvs{t}|\rcrnts{t-1};\params)
:= \frac{1}{\mathcal{Z}(\rcrnts{t-1})}h(\stateobsvs{t})h(\modelstates{t})\exp\big\{
\modelstates{t}\tr \vishidwts \stateobsvs{t}
+ \visbiases\tr\stateobsvs{t}
+ (\rcrnthidwts\rcrnts{t-1}
+ \hidbiases)\tr\modelstates{t}\big\},
\end{equation}
corresponding to the graphical model in \subfig{RTRBMgraphicalmodel};
and the augmented data distribution is, as for the rEFH, defined by \eqn{rcrntdstrb}.
This is the defining equation for the TRBM and the RTRBM.
Applying the substitution ($\rcrnts{t-1}$ for $\modelstates{t-1}$) to the conditional distributions yields precisely \eqn{REFHhiddencond} and \eqn{REFHstateobsvcond}.
This makes the deterministic recursion the same for both models, to wit, \eqn{rcrntdstrb}.
Thus, the only difference between the (R)TRBM and the rEFH is that the latter additionally defines a conditional distribution over the previous hidden means, \eqn{REFHrcrntcond}.
This can also be seen in the joint distributions:\ the EFH defined in \eqn{RTRBMjointdstrb} is \emph{conditioned} on $\rcrnts{t-1}$, rather than defining a (joint) distribution over it, as in \eqn{REFHjointdstrb}.
\emph{The (R)TRBM does not define a distribution ($q$) over} $\Rcrnts{t}$, although the recursion extends the \emph{data} distribution ($p$) to $\Rcrnts{t}$.
The consequence can be seen in the subtle but important difference in the graphical models:\ the connections from $\Rcrnts{t-1}$ to $\Modelstates{t}$ are directed in the (R)TRBM (\subfig{RTRBMgraphicalmodel}), rather than undirected as in the rEFH (\subfig{REFHgraphicalmodel}).
Essentially, the vector $\Rcrnts{t-1}$ is treated in the (R)TRBM like a (dynamic) bias rather than data, as in the rEFH; it is not allowed to ``fantasize'' about the past, only the present.
This leads to different learning rules, which we exhibit below.
Treating the past hidden-unit activities as data is, however, required for our proof that the rEFH provides optimal inference (\sctn{proofformal}).
Recall the basic idea:
The hidden-unit activities of a(n identifiable) generative model become good summaries for the data they are required to generate; if the past hidden-unit activities are among those data, the summaries accumulate recursively.
It is not obvious, therefore, that such a proof can be extended to the (R)TRBM (the left-hand equality in \eqn{sufficiencyForTemporalLatents} will not be satisfied).
Nevertheless, the RTRBM does appear to achieve nearly optimal inference, as we show in our experiments below.
But choosing to treat the previous hidden-unit activities as data (the rEFH) does turn out to have a very important consequence:\ the dependence of these ``data'' on the rEFH's parameters, introduced by \eqn{rcrntdstrb}, can be safely ignored.
When the recurrent activities are treated as a dynamic bias, on the other hand, ignoring their dependence on the parameters (the TRBM) produces suboptimal inference; while including the dependence in the learning rules (the RTRBM) introduces
backpropagation-through-time.
We show this in the following sections and argue why it is so in \sctn{conclusions}.
\FigTemporalEFHs
\subsection{The TRBM}
There are two training procedures for the graphical model of \subfig{RTRBMgraphicalmodel}.
The first employs the approximation we have made use of for the rEFH.
In minimizing KL divergence between data, $p(\stateobsvs{t};\params)$, and model, $q(\stateobsvs{t}|\rcrnts{t-1};\params)$, the dependence $\rcrnts{t-1}$ on the parameters (\eqn{rcrntdstrb}) is ignored.
Applying the method of one-step contrastive divergence to the distribution in \eqn{RTRBMjointdstrb} then yields
\begin{equation}\label{eqn:TRBMparameterUpdates}
\begin{split}
\Delta\vishidwts &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}\Stateobsvs{t}\tr
-\smplavg{
q(\updtmodelstates{t}|\rcrnts{t-1},\updtstateobsvs{t})
q(\updtstateobsvs{t}|\rcrnts{t-1},\modelstates{t})}
{\Updtmodelstates{t}\Updtstateobsvs{t}\tr
}
},\\
\Delta\rcrnthidwts &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}\Rcrnts{t-1}\tr
-\smplavg{
q(\updtmodelstates{t}|\rcrnts{t-1},\updtstateobsvs{t})
q(\updtstateobsvs{t}|\rcrnts{t-1},\modelstates{t})}
{\Updtmodelstates{t}\Rcrnts{t-1}\tr}
},\\
\Delta\hidbiases &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Modelstates{t}
-\smplavg{
q(\updtmodelstates{t}|\rcrnts{t-1},\updtstateobsvs{t})
q(\updtstateobsvs{t}|\rcrnts{t-1},\modelstates{t})}
{\Updtmodelstates{t}}
},\\
\Delta\visbiases &\propto \sum_{t=0}^T
\smplavg{
q(\modelstates{t}|\rcrnts{t-1},\stateobsvs{t})
p(\rcrnts{t-1},\stateobsvs{t})}
{\Stateobsvs{t}
-\smplavg{
q(\updtstateobsvs{t}|\rcrnts{t-1},\modelstates{t})}
{\Updtstateobsvs{t}}
}.
\end{split}
\end{equation}
We emphasize the subtle difference between \eqn{rEFHparameterUpdates} and \eqn{TRBMparameterUpdates}; in particular, no ``updated'' version of the posterior means, $\updtrcrnts{t-1}$, ever appears in the latter.
\subsection{The RTRBM}
If the dependence of the KL divergence on the parameters via the data distribution, introduced by \eqn{rcrntdstrb}, is \emph{not} ignored \citep[][]{Sutskever2009}, the learning rules accrue an extra term, which must be computed with backpropagation-through-time (BPTT).
Thus the RTRBM learning rules are
\begin{align*}
\Delta\vishidwts
&= (\Delta\vishidwts)_\text{TRBM} + \eta\sum_{t=0}^T \BPTTvec{t}\stateobsvs{t}\tr,
&
\Delta\rcrnthidwts
&= (\Delta\rcrnthidwts)_\text{TRBM} + \eta\sum_{t=0}^T \BPTTvec{t}\rcrnts{t-1}\tr\\
\Delta\hidbiases
&= (\Delta\hidbiases)_\text{TRBM} + \eta \sum_{t=0}^T \BPTTvec{t},
&
\Delta\visbiases
&= (\Delta\visbiases)_\text{TRBM};\\
\BPTTvec{t}
&= \transitionjacobian{t}\tr\rcrnthidwts\tr \bigg(
\colgradient{\log q(\stateobsvs{t+1}|\rcrnts{t};\params)}{\hidbiases}
+ \BPTTvec{t+1}\bigg).
\end{align*}
where the last line defines the backwards recursion for BPTT (for a derivation, see \app{BPTT}), $\eta$ is the learning rate, and $\transitionjacobian{t}$ is the Jacobian of the nonlinearity (e.g., logistic function for Bernoulli units).\footnote{Of course, the nonlinearity acts element-wise, so the Jacobian is diagonal, and in practice is implemented as a Hadamard product rather than a matrix multiplication.}
\section{Experiments}\label{sec:experiments}
We compare performance of the rEFH, TRBM, and RTRBM on two different kinds of data:\ a linear dynamical system (LDS) that is observed via a ``probabilistic population code'' (PPC) \citep{Ma2006}; and the bouncing-ball data used by \citet{Sutskever2007} and \citet{Sutskever2009}.
For the LDS, we have reported previously a subset of the results for the rEFH \citep{Makin2015b}.
Here we compare the filtering performance of the rEFH with that of the TRBM and RTRBM on these data, and investigate the failure modes of the latter, especially in connection with the order of the dynamics that can be learned.
For the bouncing-ball data, \citet{Sutskever2013} characterized the performance of the TRBM and RTRBM in terms of prediction of the next frame.
Here, we characterize the rEFH's performance in the same way on the same data.
\subsection{Details of the training procedures}
With the exception of the rEFHs trained on LDSs (\sctn{LDSwithPPCs}), all models and experiments used the same training scheme, which was matched to that reported by \citet{Sutskever2013}.
Weight changes were made according to 25-step contrastive-divergence with a fixed momentum of 0.9 and a learning rate declining linearly from 1/1500 (LDS) or 1/100 (bouncing balls)\footnote{The differing learning rates reflect the size of the inputs:\ the LDS data are Poisson distributed with maximum mean firing rates of about 10, whereas the bouncing-ball ``pixels'' are Bernoulli distributed.}, on ``minibatches'' consisting of 100 time steps of a single trajectory.
New batches of 400 trajectories (i.e., 400*100 = 40000 sample vectors) were created every 5 epochs, for 250 epochs.
These models were initialized with 30 batches of static pre-training (i.e., setting recurrent activities to zero) at CD-5.
The rEFHs of \sctn{LDSwithPPCs}, on the other hand, could be trained much more quickly.
No static pre-training was required, only CD-1 was used throughout, and training terminated after only 90 epochs.
Minibatches consisted of 40 vectors, each corresponding to the same time step of 40 different trajectories.\footnote{This minibatch scheme is precluded for the RTRBM by the use of BPTT, which requires many steps of the \emph{same} trajectory.}
Batches, renewed every 5 epochs, consisted of 1000 time steps (i.e., 40*1000 = 40000 sample vectors).
Learning rates declined \emph{exponentially}, according to $1.1^{-\text{epoch}}$, from 1/500 (Poisson-Bernoulli weights), 1/50 (Bernoulli-Bernoulli weights) or 1/120 (biases).
Momentum started at about 0.5 and exponentially approached 0.98 ($\rho = 0.98 - 0.5*1.1^{-\text{epoch}}$).
A small weight decay of 0.001 was also used.
\subsection{Linear dynamical system with PPC observations}\label{sec:LDSwithPPCs}
We aimed to quantify model performance precisely, so we constructed a generative model for the data (consistent with \subfig{HMMaugmented}) for which the posterior distribution can be computed in closed form---despite nonlinear, non-Gaussian emissions.
The ``backbone'' of the model is a second-order linear dynamical system; that is, position and velocity depend on previous position and velocity.
The observations at each time step are a product over fifteen conditionally independent Poisson ``neurons'' with fixed-width Gaussian-shaped tuning to the position---a ``probabilistic population code'' \citep{Ma2006}.
This could be thought of as a crude model of sensory responses to a moving object.
Since the tuning curves evenly tile the space, the posterior distribution over the hidden dynamical state is in fact (approximately) a normal distribution \citep{Ma2006}.
This makes the filtering distribution (the RHS of \eqn{posteriorGivenHiddensIsFilteringDistribution}) computable in closed form---the Kalman filter (KF)---where, in place of the observation and the emission variance, respectively, one uses the center of mass of the population code and the tuning-curve width scaled by the total number of spikes \citep{Ma2006}.
This model can be learned with an EM algorithm \citep{Ghahramani1996a}, which allows us to benchmark the rEFH, TRBM, and RTRBM against easily interpreted quantities.
In particular, we can consider Kalman filters applied to models learned by EM under the assumption of first- (\KFone) or second-order (\KFtwo) dynamics.
We also considered a zeroth-order model (\KFnaught), which assumes no dynamics at all; and a KF that has access to the parameters of the true dynamical system (\KFopt---no learning required).
We chose these parameters so that any first-order approximation will be poor, allowing us to distinguish these cases from each other.
Fresh data were generated for testing:\ for each model the posterior mean, $\xpct{q}{\Modelstates{t}|\augobsvs{t}}$, was computed at each time step.
From each vector of hidden activities, the position of the underlying ``stimulus'' was decoded (using the technique we have introduced for similar, static models; see \citealt{Makin2013b} or \app{experimentdetails}) and used to compute an error with respect to the actual position.
We did the same using the posterior mean from Kalman filters learned using \KFone, \KFtwo; the zeroth-order model; and the ``optimal'' (no-learning) model.
For each, we report the mean, across time steps (1000) and trajectories (40), of the square of these errors (MSE).
In order to characterize susceptibility of training to local minima, 20 of each of the EM-based models (\KFone, \KFtwo) were trained from scratch.
Likewise, 20 of each of the rEFH, TRBM, and RTRBM were trained from scratch for each of 20 different hidden-layer sizes.
The distribution of the resulting MSEs are shown in the box-and-whisker plot in \fig{ErrorStatsVsNumHiddens}.
Also indicated with lines are the performances of the \emph{best} of each of the \KFone\ and \KFtwo\ models, and the performances of the models requiring no training, \KFnaught\ and \KFopt.
(Since a single set of testing data was used for all models, there is no variation in performance---or, therefore, box plots---for the no-training models.)
Evidently, the TRBM learns only first-order dynamics, as suggested by \citet{Sutskever2013}, who reported that it tends to generate data that move in a random walk.
The best RTRBMs approach the performance of the best second-order Kalman filter (\KFtwo, the order of the true underlying dynamics).
The rEFHs are on average slightly worse, but given at least 195 hidden units, the best are very close to the best RTRBMs, which do not appear to improve after about 165 hidden units.
The rEFHs are certainly able to learn second-order dynamics, unlike the TRBM; and they do not require BPTT, unlike the RTRBM.
In fine, the performance deficit of the rEFH relative to the RTRBM appears not to be insuperable, as with the TRBM, but rather to be remediable---at least on these data---with an increase in hidden units.
This cost may be offset to some extent by the savings accrued from not needing BPTT; we explore this below on a more difficult task, the bouncing-ball data set.
Finally, we note that only outliers among the \KFtwo\ models actually achieve performance superior to that of the median rEFH or RTRBM, as long as these neural networks have at least 60 hidden units.
\FigErrorStatsVsNumHiddens
\subsection{Bouncing balls}
Each data vector consists of a 30$\times$30 image patch containing three ``balls'' (circles) that bounce off each other and the walls with complete energy conservation.
The nonlinear dynamics greatly complicate computation of the posterior distribution over the hidden state---the positions and velocities of the balls---so we compute from the hidden units not the optimal posterior means, but the best predictions of the next frame, as proposed by \citet{Sutskever2013}.
We note first of all that the \emph{current frame} predicts the next frame with a per-pixel, mean square error (MSE) of 0.015.
Our implementation of the TRBM, with 400 hidden units, achieves a MSE of 0.046, close to the 0.04 reported by \citet{Sutskever2013}.
Interestingly, this is higher (worse) than the current-frame prediction, which suggests that this TRBM does not learn any dynamical model at all.
In fact, this is consistent with the results above:
The TRBM can learn only first-order dynamics; and for the bouncing balls, the optimal first-order model is the zeroth-order model, since the velocities are indeed constant (until collisions), and the \emph{average}, over all trajectories, of the next position is the current position (velocities are on average zero).
That the TRBM fails even to learn even the zeroth-order model (the current-frame predictions) suggests, perhaps, a limitation imposed by the nonlinearities (collisions) in the dynamics.
Our implementation of the RTRBM, with 400 hidden units, achieves a MSE of 0.008, again close to the results reported by \citet{Sutskever2013}, in this case 0.007.
The rEFH with 400 hidden units achieves a per-pixel MSE of 0.014:\ superior to the TRBM, as well as the zeroth- and first-order predictions, but inferior to the RTRBM.
However, training time is also much shorter (see \tbl{bbresults}), since BPTT is not required.\footnote{The comparison is between our own implementations of the RTRBM and the rEFH.
Our highly vectorized, GPU-optimized {\sc Matlab} implementations take on the order of hours; on the same machine, Sutskever's original {\sc python} implementation of the RTRBM takes a few days.
All of our code is publicly available at \tt{https://github.com/jgmakin}.}
Increasing the number of hidden units to 625 increases the rEFH's training time to that of the RTRBM, about 2.5 hours, and brings its MSE closer, down to 0.010 MSE.
Allowing a little under five hours accommodates 1000 hidden units in the rEFH and approximately matches the RTRBM's MSE at 0.008.
\begin{table}[ht]
\begin{minipage}{\textwidth}
\begin{center}
\begin{tabular}{c|c|c|c}
model & $\#$ hidden & MSE & training time (hrs)\\
\hline
\hline
0$^\text{th}$-order & --- & 0.015 & ---\\
TRBM & 400 & 0.046 & 1.43\\
TRBM & 625 & 0.040 & 2.53\\
RTRBM & 400 & 0.008 & 2.56\\
rEFH & 400 & 0.014 & 1.42\\
rEFH & 625 & 0.010 & 2.56\\
rEFH & 1000 & 0.008 & 4.76\\
\end{tabular}
\caption{For the bouncing-ball data, a comparison of next-frame per-pixel MSE, and training times.}
\label{tbl:bbresults}
\end{center}
\end{minipage}
\end{table}
\subsection{Generating trajectories}\label{sec:generation}
The rEFH enjoys a different advantage over the (R)TRBM:\ generating sequences is orders of magnitude less costly.
Because the rEFH defines a distribution over the previous hidden units, generation in the reverse direction is particularly simple:\ from each hidden vector, a ``previous'' hidden vector as well as the current observation can be drawn immediately, and the process iterated.
In the (R)TRBM, by contrast, generating sequences (in either direction) requires at each time step repeated Gibbs sampling of the hidden and visible units, conditioned on the previous hidden means, before the draws converge to the stationary distribution.
In practice, this appears to be about 50 steps---i.e., 50 upward and 50 downward passes \citep{Sutskever2013}.
Examples of bouncing-ball trajectories generated by the rEFH can be found in the supplemental material.
\section{Conclusions}\label{sec:conclusions}
The ``recurrent exponential-family harmonium'' (rEFH) generalizes EFHs to time-series data.
Its central motivation is to provide optimal inference to the hidden state of ``the world'' that produces those data, even when the world's true generative model is intractable to Bayesian inversion, hard to express in closed-form equations, or even unknown.
Thus, in contrast to directed graphs like dynamic Bayes Nets \citep{Murphy2002}, where one attempts to formulate the true generative model and then derive an inference algorithm for it, we relaxed the assumption that the generative model of the world and the rEFH correspond, in favor of a weaker one about identifiability:
The rEFH's and the world's marginal distributions over observations must be expressible in terms of the same identifiable finite mixture model (see \app{technicalconditions} for technical details).
Consequently, the sufficient statistics for the true hidden state are not derived in closed form (like, e.g., the posterior cumulants recursively computed in a Kalman filter), but are nevertheless guaranteed to be encoded in the hidden units of the trained rEFH.
The idea behind the proof of this guarantee is simple:\ good density estimators make their hidden units---or rather, the sufficient statistics for them---good summaries of their input data.
When such summaries are also included among the inputs---at the next time step (\fig{tikzHMMs})---the succeeding hidden vector is forced to summarize both the new inputs and the old summaries.
This recursive procedure builds up a master summary that takes into account all relevant past information.
The other conspicuous consideration regarding the choice of directed and undirected (or at least EFH-like) models is the trade-off between inference and generation.
In directed models, generation is easy, but inference---in all but a few special cases---is hard, and approximate techniques are necessary; whereas in the EFH-like undirected models, inference (to its own hidden states) is easy, but generation is hard, because it requires Gibbs sampling.
But in the rEFH, generating sample trajectories---in reverse time order---is easy, requiring only a single pass through the model.
This is not the case in the alternative architecture proposed by \citet{Sutskever2013}, the TRBM and RTRBM (\subfig{RTRBMgraphicalmodel}), where prolonged Gibbs sampling is required.
In practice, some of the assumptions of the proof are mildly violated:\ e.g., that the model has become a perfect model for the training data.
To show that the proof is robust against such violations, we tested the model experimentally on data from a generative model that has been contrived to yield closed-form solutions to the inference problem, a second-order system whose dynamics are, and whose emissions can be re-formulated to be, linear-Gaussian.
This enabled us to demonstrate conclusively that the rEFH learns second-order dynamics from reports of position only---even though its recurrent connections span only one time step.
But the most interesting approximation is the use of a sub-optimal learning rule.
During learning, changes to the model's parameters ramify into the data, since they consist in part of the model's own hidden-unit means (\eqn{rcrntdstrb}).
We ignored this, which amounts to assuming that whatever improvements to the model are effected by a parameter update are not (entirely) undone by the ensuing changes to the data.
This obviates the need for backprop through time (BPTT).
In Sutskever's architecture (\subfig{RTRBMgraphicalmodel}), the same approximation turns the RTRBM into a TRBM, and loses the ability to learn dynamics beyond first-order, at least without increasing the span of its recurrent connections.
In contrast, the only advantage the RTRBM appears to enjoy over the rEFH is more economical use of hidden units (\fig{ErrorStatsVsNumHiddens}, \tbl{bbresults}), which comes at the price of slower training times (BTPP being required), and the inability to generate trajectories in one pass.
Some of the rEFH's training-time advantage can be bartered for performance by increasing the size of its hidden layer ---although to eliminate completely the performance gap on Sutskever's bouncing-ball data set ultimately requires about a factor of 2 \emph{more} time for the rEFH (\tbl{bbresults}).
But for data with trajectories longer than 100 time steps, the advantage of neglecting BPTT is even greater, and the relative merits of the rEFH increase.
Why can BPTT be neglected for the rEFH but not the (R)TRBM?
The answer does not follow obviously from application of BPTT to the rEFH architecture, which yields non-zero terms.
We consider this the most important theoretical question for future work.
Second, empirically, the accumulation of master summaries described above is ultimately limited, obviously, by the capacity of the hidden layer, so not all past history can be ``summarized'' in the way just described.
(This is consistent with the proof because the capacity limits would likewise limit the generative fidelity of the model to the data, compromising one of the proof's requirements.)
Finally, it also seems unlikely that the model, without modification, could learn ``pathologically'' long dependencies.
Thus, although the toy data sets we picked allowed us to benchmark the models, it remains to determine how the model performs on more challenging time-series data.
\acks{Funding was provided by DARPA (N66001-10-C-2010, W911NF-14-2-0043), NIH Vision Training Grant (5T32EY007120-22), the UCSF Wheeler Center for the Neurobiology of Addiction, and (BKD) the National Science Foundation Graduate Research Fellowship under Grant No.\ 1144247.
Some of the EFHs were trained using Tesla K40 GPUs, the generous donation of the Nvidia Corporation.}
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Italian Government Approves Marijuana Home Growing
By: Maria Loreto
CannabisMarijuana LegislationNewsPolitics
Photo by Duncan McCulloch/Getty Images
Italy's Supreme Court now protects the cultivation of small amounts of cannabis grown in people's homes.
Over the holidays, the Italian Supreme Court ruled that it was legal to grow small amounts of marijuana at home for personal use. The court declared that narcotics laws should exclude "small amounts grown domestically for the exclusive use of the grower."
According to the Independent, the ruling was approved on December 19, but it wasn't announced until after Christmas. It has predictably caused outrage from conservative political parties and support from cannabis advocates.
"Drugs cause harm, forget about growing them or buying them in shops," said Matteo Salvini, leader of the far-right League Party. Before leaving office in August, Salvini had been one of the most vocal opponents of cannabis, pushing for the closure of legal weed shops and looking to ban "light cannabis" products, which contain levels of THC below 0.6%.
RELATED: Italy Hopes To Nearly Triple Domestic Medical Marijuana Production
Photo by Stocksnap via Pexels
Luca Fiorentino, founder of cannabis supply company Cannabidiol Distribution, expressed his relief and support of the Supreme Court's ruling in favor of cannabis users. "It's the end of a nightmare," he said. "After Salvini's witch hunt I had to fire 10 people and I lost 68% of my revenues."
RELATED: Which Is Better, Indoor Or Outdoor Weed?
In Italy, the use of medicinal and industrial marijuana was legalized three years ago, but growing and cultivating the plant remained illegal. The Supreme Court's ruling is credited to a recent case where a man was prosecuted due to the cultivation of two cannabis plants.
Although the Supreme Court's ruling doesn't specify what "small-scale cultivation" of cannabis entails, we can assume that growing a plant or two at home should fall under the protection of the law.
home grow
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Residents Of Legal Marijuana States Have Lower Rates Of This Disorder, New Study Finds
Maria Loreto - January 13, 2023
The study found that while alcohol consumption between states didn't vary greatly, twins that lived in legal states were less likely to face harm when under the influence.
Legal Marijuana Leads To More Jobs And Economic Improvements — Here's How
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Cannabis January 9, 2023
Here's How Many Kids Have Accidentally Gotten High From Eating Edibles
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{sec:intro}
Automatic speech recognition (ASR) has achieved high accuracy by adopting a range of neural network architectures \cite{chan2016listen, li2021recent, DBLP:journals/corr/abs-1211-3711, he2019streaming, guo2020efficient, zhang2020transformer,watanabe2018espnet}. One remaining challenge is that ASR performance can vary substantially by speaker~\cite{tatman2017effects, koenecke2020racial,martin2020understanding,tatman2017gender, gao2022seamless, sari2021counterfactually,liu2021model}. The problem is associated with a variety of factors, including geographic location, age, and accent. In this research, we focus on reducing disparity in ASR performance between different geographic regions.
We can enhance the performance for an underperforming region by collecting more training data from that region. In general, there are two mechanisms to employ this additional data: retraining the ASR model from scratch with both current data and the new data from high-error regions, or adapting the already-trained model with data from the high-error regions. The first mechanism can be computationally expensive and requires access to past data, which is not always possible. However, it might lead to better performance than the second approach. The second mechanism can save training time, and is suitable for many real-world scenarios, such as adapting the ASR model on edge devices \cite{peinl2020open} with local data, where access to the cloud-based training data is impractical. Here we focus on the second mechanism (adaptation) and make use of the first mechanism (retraining) as an empirical upper bound on performance.
In this paper, we employ elastic weight consolidation (EWC)~\cite{kirkpatrick2017overcoming} to adapt the pretrained ASR model. This method is designed to keep the model from forgetting important parameters learned in the initial training while acquiring new knowledge. More specifically, EWC adds a regularization term to the ASR loss to make the model parameters of the new task (recognition of regions having high WER) to be close to the best parameters for the previous task (recognition of all regions), along the dimensions that matter the most to the previous task. We compare the EWC method with several transfer learning \cite{pan2009survey} techniques.
EWC belongs to the continual-learning \cite{delange2021continual} family of methods, referring to the ability of a machine learning model to learn continually from new data being ingested over time \cite{parisi2019continual}. Regularization and memorization-based techniques are two popular directions to deal with the forgetting problem arising in continual learning.
In the regularization-based approach, there are ``importance scores'' assigned to the parameter weights, so the model has the flexibility to move along the dimensions that are not important for the old task. Generally, the regularization-based approach has a loss of the form
\begin{equation}
\mathcal{L}(\theta)= \mathcal{L}_{new}(\theta) + \lambda \sum_{k} I_k (\theta_{k} - \theta_{old,k}^*)^2
\end{equation}
where $\mathcal{L}$ is the loss, $\theta_{k}$ is the $k$th parameter for the new task, $\theta_{old,k}^*$ is the same parameter for the old task at convergence, $I_k$ is the importance score for the $k$th parameter, and the scalar $\lambda$ controls the balance between learning the new task and remembering the old task. There are several ways to assign the importance scores. While EWC utilizes an off-line Fisher information matrix, synaptic intelligence (SI) \cite{zenke2017continual} computes the importance scores online, and memory-aware synapse (MAS) \cite{aljundi2018memory} uses the squared (L2) norm of the gradient of the output with respect to the parameters.
In other related work, \cite{riemer2018learning} proposes the meta-experience relay (MER) method, which utilizes a fixed-size memory buffer holding random samples from both old and new tasks, and optimizes the network within the meta-learning framework. Additionally, \cite{nguyen2018variational} presents the variational continual learning (VCL) method, which uses a Bayesian core set (similar to memory) sampled by K-center \cite{gonzalez1985clustering} or randomly. In \cite{nguyen2018variational}, the old task is the prior and the new task is the posterior in the Bayesian framework, and Bayesian variational inference is applied to approximate the posterior distribution.
In this paper, we propose a loss function to which an additional EWC regularization term is added to make the model perform better in high-WER regions, without forgetting knowledge about the general user population. In Section~\ref{sec:methods} we describe our method in detail. Section~\ref{sec:experiments} describes the data, experiments and training parameters. Results are presented in Section~\ref{sec:results}.
\section{Methods}
\label{sec:methods}
\subsection{Problem statement}
As input to our task, we are given an ASR model pretrained on a dataset of speech from the general user population. This pretrained model has good performance on average, but has high error rate for speakers from some geographic areas. Our task is to eliminate (or reduce) the performance gap \cite{mehrabi2021survey} against these regions, without access to the data of the pretraining stage and without degrading average performance for all regions.
\subsection{ASR Model}
Our ASR system is an end-to-end model utilizing the recurrent neural network transducer (RNN-T) architecture \cite{DBLP:journals/corr/abs-1211-3711, he2019streaming}, which is suitable for streaming. More specifically, the system includes a long short-term memory (LSTM) \cite{hochreiter1997long} encoder, an LSTM prediction network, and a joint network. At each time frame, the encoder output is a vector $\mathbf{h}_t^{enc}$ that summarizes input information over time:
\begin{equation}
\mathbf{h}_t^{enc} = \mathrm{LSTM}(\mathbf{x}_{t})
\end{equation}
where $\mathbf{x}_{t}$ is the vector of acoustic features at time $t$.
The predictor takes the previous predicted labels $\mathbf{\hat{y}}_{m-1}$ as the input and returns $\mathbf{h}_m^{pred}$ as the output:
\begin{equation}
\mathbf{h}_m^{pred} = \mathrm{LSTM}(\mathbf{\hat{y}}_{m-1}) \\
\end{equation}
The outputs of the encoder and predictor are combined by a joint network, which is a simple feed-forward network (FFN) \cite{li2019improving}:
\begin{equation}
F^{joint}(\mathbf{h}_t^{enc},\mathbf{h}_m^{pred}) = \mathrm{tanh}(\mathbf{W}\mathbf{h}_t^{enc} + \mathbf{V}\mathbf{h}_m^{pred})
\end{equation}
where $\mathbf{W}$ and $\mathbf{V}$ are weight matrices.
Finally, a softmax layer is applied to the output of the joint network to make the final prediction:
\begin{equation}
p(\mathbf{\hat{y}}_{m}|\mathbf{x}_{1:t},\mathbf{\hat{y}}_{1:m-1}) = \mathrm{softmax}(F^{joint}(\mathbf{h}_t^{enc},\mathbf{h}_m^{pred}))
\end{equation}
\subsection{Elastic weight consolidation}
Elastic weight consolidation (EWC) \cite{kirkpatrick2017overcoming} can be applied to make the model acquire new knowledge while retaining information already learned. We denote model parameters by $\mathbf{\theta}$, and old and new data by $D_A$ and $D_B$, respectively.
Applying Bayes' theorem
\begin{align} \label{eq:1}
\log p(\mathbf{\theta}|D_A, D_B) &= \log p(D_B|\mathbf{\theta}) + \log p(\mathbf{\theta} | D_A) \nonumber \\
&- \log p(D_B | D_A)
\end{align}
EWC approximates $\log p(\mathbf{\theta} |D_A)$ with the second Taylor expansion at $\theta_A^*$. As $\log p'(\mathbf{\theta}_A^*|D_A) = 0$ at $\mathbf{\theta}_A^*$,
\begin{align}
\log p(\theta|D_A) &\approx \log p(\theta_A^*|D_A) + \frac{1}{2} \log p'' (\theta_A^* | D_A)(\theta - \theta_A^*)^2 \nonumber \\
& \approx const + \frac{1}{2} \log p'' (\theta_A^* | D_A)(\theta - \theta_A^*)^2
\end{align}
We observe that a normal distribution $\mathcal{N}(\mu,\sigma^2)$ has log probability density in the form $constant - \frac{1}{2\sigma^2}(x-\mu)^2$, then
\begin{equation}
\log p(\theta|D_A) \approx \mathcal{N}(\theta_A^*,F^{-1})
\end{equation}
where $F = \frac{1}{2}\log p'' (\mathbf{\theta}_A^*|D_A)$ is the Fisher information matrix.
EWC assumes the Fisher information matrix is diagonal, giving
\begin{equation} \label{eq:2}
\log p(\mathbf{\theta}|D_A) \approx constant + \frac{1}{2} F_i (\mathbf{\theta} - \mathbf{\theta}_A^*)^2
\end{equation}
Because $\log p(D_B | D_A)$ is a constant, substituting $\log p(\mathbf{\theta}|D_A)$ from Eq.~(\ref{eq:2}) in Eq.~(\ref{eq:1}), we obtain
\begin{align} \label{eq:3}
\log p(\mathbf{\theta}|D_AD_B) \approx \log p(D_B|\mathbf{\theta}) + \frac{1}{2} F_i (\mathbf{\theta} - \mathbf{\theta}_A^*)^2
\end{align}
As we can see, there is a regularization term $\frac{1}{2} F_i (\mathbf{\theta} - \mathbf{\theta}_A^*)^2$ added to the model's loss on the new task $\log p(D_B|\mathbf{\theta})$, to keep the parameters on the new task $\mathbf{\theta}_B$ close to the best parameters of the old task $\mathbf{\theta}_A^*$ along the dimensions that matter to task A. $F_i$ can be interpreted as the importance scores that EWC assigns to the $i$th parameter, where $F_i$ is the element at position $i$ on the main diagonal of the Fisher information matrix.
\subsection{Loss function for adaptation}
Here, we formulate the problem of eliminating geographic disparities in a continual learning setup for ASR, where we adapt a pretrained model (trained on data from all regions) on new data that contains only speech from regions having high WER, without access to previous data. We propose a new loss function that is a combination of the EWC loss and the RNN-T loss. This loss function can mitigate the problem of distribution-shift between the pretraining data and fine-tuning (adaptation) data.
Our loss function is defined as follows:
\begin{equation}
\mathcal{L}(\mathbf{\theta}) = \mathcal{L}_{ASR}(\mathbf{\theta}) + \frac{\lambda}{2}\sum_{i} F_i (\mathbf{\theta}_{i} - \mathbf{\theta}_{p,i}^*)^2
\label{eq:4}
\end{equation}
As before, the additional EWC regularization term forces the parameters of the ASR model trained on high-WER regions $\mathbf{\theta}$ to be close to the best parameters of the model trained on all regions $\mathbf{\theta}_p^*$, along the dimensions that are important to the pretrained task. More specifically, every parameter of the ASR model has a different penalty when it moves away from the optimal pretrained value. The parameters that are important to the pretraining task have high penalties, so they cannot change much from the prior optimal values. On the other hand, the parameters that are less essential to the pretraining task are allowed to vary more freely. Thus, the ASR model can still improve performance during fine-tuning for regions with high WER.
Our work follows \cite{kunstner2019limitations} in approximating the Fisher information matrix, where the empirical $F_i$ is derived as:
\begin{equation} \label{eq:fim}
F_i = \sum_{j \in D} \mathbb{E}(\frac{\partial \mathcal{L}_{ASR}(y_j,\hat{y_j})}{\partial \theta_i})^2
\end{equation}
where $y_j$, $\hat{y}_j$ are the label and ASR output respectively, $D$ is the dataset used to extract the Fisher information matrix, $\mathcal{L}_{ASR}$ is the ASR loss, and $\theta_i$ is a parameter of the ASR model. $F_i$ is the $i$th element on the main diagonal of the Fisher information matrix, which acts as an importance score for parameter $\theta_i$ with respect to the previous task, in our case reflecting how important parameter $\theta_i$ is to ASR performance on the general user population.
\subsection{Identifying regions with high word error rate by geo-clustering}
\label{sec:tree}
To improve ASR performance for regions with high WER, we aim to adapt the model with additional data from those areas. Thus, we built a clustering tree to select training samples belonging to those regions. The clustering tree is trained and tested on a dataset separate from the ASR training set. The tree training data is split into different regions by using an algorithm that maximizes the WER differences between regions. The geographic clustering tree keeps splitting the data into left and right branches by either approximate longitude or latitude while the number of devices (as a proxy for users) in each leaf is larger than or equal to a predefined threshold. The decision to split by longitude or latitude depends on which coordinate yields the larger WER difference between the two branches. Pseudo-code for geographic clustering is given in Algorithms~\ref{alg:alg1} and~\ref{alg:alg2}.
\textbf
\SetAlgoNoLine
\DontPrintSemicolon
\LinesNumbered
{
\begin{algorithm}[t]
\scriptsize
\SetKwFunction{bestsplit}{Best split}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\bestsplit{data X, threshold t}}{
Best coordinate = None, WER difference = 0 \;
\For{coordinate in \{longitude,latitude\}}
{
$X_{left}$ = \{$x \in X|x <$ median coordinate \} \;
$X_{right}$ = \{$x \in X|x \geq$ median coordinate\} \;
\If{$|X_{left}| < t$ or $|X_{right}| < t$ devices } {
go to line 14 \;
}
\# $X_{left}$ WER is the WER of a dataset contains all \;
\# utterances on the left branch \;
d = ($X_{left}$ WER - $X_{right}$ WER)$^2$ \;
\If{d $>$ difference}{
Best coordinate = coordinate \;
WER difference = d \;
}
\KwRet Best coordinate, WER difference}
}
\caption{Best split}
\label{alg:alg1}
\end{algorithm}}%
}
\SetAlgoNoLine
\DontPrintSemicolon
\LinesNumbered
{
\begin{algorithm}[t]
\scriptsize
\SetKwFunction{bestsplit}{Best split}
\SetKwProg{Fn}{Function}{:}{}
\SetKwFunction{buildtree}{Build tree}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\buildtree{data X, threshold t}}{
Best-coordinate, WER-diff = \bestsplit{X, t} \;
\If{WER-diff = 0 or (WER-diff $\neq$ 0 and Best-coord is None)}{
create new region; compute WER for new region \; }
$X_{left}$ = \{$x \in X|x <$ median coordinate \} \;
$X_{right}$ = \{$x \in X|x \geq$ median coordinate\} \;
$tree_{left}$ = {\buildtree{$X_{left}$, t}} \;
$tree_{right}$={\buildtree{$X_{right}$, t}} \;
\# tree is an object containing information about left, right tree \;
\# branches and the coordinate \;
\KwRet tree \;
}
\caption{Geo-clustering tree}
\label{alg:alg2}
\end{algorithm}}%
\begin{table*}[tb]
\footnotesize
\caption{Relative WER and variance reduction on the test set. Region WER for a dataset containing only utterances from a specific geographic region. Overall WER is over the whole test set. For the variance column, negative numbers indicate a smaller variance; for the other columns, a negative number indicates a relative WER improvement. $D_p$ is the pretraining dataset. $D_c$ and $D_r$ are two fine-tuning datasets, where $D_c$ contains utterances from the regions with high WER while $D_r$ has equally many randomly selected utterances. Each of $D_p$, $D_c$ and $D_r$ contain 10k hours of speech. WERR is WER reduction, lower is better.}
\label{tbl:result1}
\centering
\begin{tabular}{llcrrrrr}
\toprule
\textbf{Experiment} & \textbf{Description} & \textbf{Data} & \multicolumn{4}{c}{\textbf{Region WERR}} & \textbf{Overall} \\
\cline{4-7}
& & & \textbf{variance} & \textbf{mean} & \textbf{max} & \textbf{min} & \textbf{WERR} \\
\midrule
Experiment 1 & Baseline & $D_p$ & 0 & 0 & 0 & 0 & 0 \\
Experiment 2 & No freeze & $D_c$ & -5.3 & -0.9 & -2.9 & -4.6 & -1.1 \\
Experiment 3 & Freeze Encoder & $D_c$ & -1.8 & 0.0 & -1.4 & -5.4 & -0.1 \\
Experiment 4 & Freeze Predictor & $D_c$ & 1.8 & -0.3& -1.3 & -8.5 & -0.4 \\
Experiment 5 & Freeze 3 lowest encoder \newline layers and 1 predictor layer & $D_c$ & -0.9 & -0.5 & -2.5 & -2.7 & -0.4 \\
Experiment 6 & Proposed method & $D_c$ & \textbf{-7.9} & \textbf{-1.1} & \textbf{-3.2} & \textbf{-5.8} & \textbf{-1.3} \\
\midrule
Experiment 7 & Empirical bound & $D_p$ +$D_c$& -5.3 & -1.2 & -2.3 & 0.2 & -1.0 \\
Experiment 8 & & $D_p$ +$D_r$& -12.3 & -2.3 & -0.9 & -7.3 & -2.1 \\
\bottomrule
\end{tabular}
\end{table*}
\section{Data and Experiments}
\label{sec:experiments}
\subsection{Identifying geographic regions}
We utilize the clustering tree algorithm in Section~\ref{sec:tree} to split the dataset into different subsets, where each subset corresponds to a specific geographic region. Note that the data is not split by state, city or zip code, but by approximate longitude and latitude. The clustering tree is trained with 5-fold cross-validation on de-identified user data from a commercial voice-enabled artificial intelligence assistant, consisting of 1.1k hours of audio. Thus for every fold, the training and test sets contain 0.9k and 0.2k hours, respectively. After 5-fold cross-validation, we obtain five different clustering trees. We select the tree with the lowest L1-distance between the predicted and true WERs per region, over all regions from the test set. The tree keeps splitting the dataset into smaller subsets, with the condition that each subset have at least $t$ devices. We evaluated different values of $t$ (1500, 2000 and 2500) and chose $t = 1500$, as this resulted in the largest WER disparity between highest and lowest region-specific WERs. The resulting tree split the dataset into 126 subsets, corresponding to 126 longitude/latitude-bounded geographic regions, as illustrated in Figure \ref{fig:fig1}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figure3_continuous_mix.png}
\caption{126 regions identified by the clustering tree. The color does not indicate specific WER, however regions with the same color have the same WER.}
\label{fig:fig1}
\end{figure}
\subsection{Datasets}
In addition to the set of 1.1k hours for training the clustering tree, we used another de-identified dataset from the same system, comprising 47k hours of audio, to train the ASR system. In the transfer learning setup, there is a pretraining and a fine-tuning stage. As every stage requires a different dataset, we split the 47k-hour corpus ($D_w$) into several subsets. The ASR pretraining set ($D_p$) is created by drawing randomly 10k hours of speech from $D_w$. After removing the pretraining set, we use 37k hours of speech to form two fine-tuning sets. The first set $D_c$ is created by taking utterances from the regions in order of decreasing WER until $D_c$ contains 10k hours of speech. As a result, we ended up with 35 geographic regions in $D_c$. As a control, we sample 10k hours randomly from the 37k-hour set to form another fine-tuning set $D_r$. We found that $D_r$ and $D_c$ have 2.8k hours in common. The development and test sets comprise 25 and 125 hours, respectively.
\subsection{Training parameters}
The encoder is a five-layer LSTM network with 1024 hidden units per layer. The predictor is a two-layer LSTM with 1024 units per layer. The joint network is a single-layer feedforward network with 512 hidden units. All these models are trained with Adam \cite{Adam} optimizer. The baseline is trained with a learning rate of $6.25\times 10^{-5}$. Other models are fine-tuned with an initial learning rate of $6.25\times 10^{-5}$, and a smaller learning rate of $1\times 10^{-5}$
after 100k steps (every step uses 5 hours of speech). The $\lambda$ value in Eq.~(\ref{eq:4}) that defines the weight between the EWC regularization loss and the ASR loss is set to 1, giving ASR loss and EWC regularization equal influence. Given limited time and the large size of our training set, we did not optimize $\lambda$. The speech feature consists of 64-dimensional log-filterbank energies \cite{raeesy2018lstm}; sampling rate is 16\,kHz. Training uses Tensorflow.
\subsection{Experiments}
We compare our proposed method against other transfer learning techniques. In Exp.~1 we train a baseline ASR model on the $D_p$ subset of 10k hours of random speakers. This baseline model after pretraining is used to initialize the weights of all other transfer learning methods (Exps.~2-5) and our proposed technique (Exp.~6). In Exp.~2, we adapt the pretrained model on the dataset $D_c$ without freezing any parameters. The encoder parameters are frozen in Exp.~3 while the predictor parameters are frozen in Exp.~4. In Exp.~5, the parameters of the first three layers of the encoders and the first layer of the predictor are frozen. The first layers are frozen because the representations at these early stages of the network often capture basic patterns of speech, which might transfer better to a new dataset than the representations at later layers, which we expect to capture more abstract information.
In Exp.~6, we implement our proposed method with EWC regularization for RNN-Ts. In this experiment, the Fisher matrix is derived with the converged pretrained model from Exp.~1 following Eq.~(\ref{eq:fim}).
We then carry out Exp.~7 by training an ASR model with both pretraining $D_p$ and fine-tuning $D_c$ datasets from scratch, rather than adapting a pretrained model on $D_c$ as in Exps.~2-6. In Exp.~7, there is no distribution shift because the training utterances are randomly selected from the union of $D_p$ and $D_c$. Exp.~7 represents the ideal case where we have all the data available for training, compared to the case of no access to pretraining data. In Exp.~8, we train the ASR from scratch with a dataset that is a combination of $D_p$ and $D_r$.
\section{Results and Discussion}
\label{sec:results}
Results are shown in Table \ref{tbl:result1}. We report the relative test set error rate reductions compared to the baseline (which thus has relative WER 0). We define a region WER as the WER of a dataset that contains only utterances belonging to a specific geographic region. In this way, we have 126 WER values for 126 geographic regions. We report the mean, variance, minimum and maximum of those 126 WER values. The WER over the whole test set (column 'Overall WER') is also reported.
The results show that among the conventional transfer learning approaches (Exps.~2-5), the best result overall is obtained when the ASR parameters are not frozen (Exp.~2). In that case, the region WER max (from the region with the highest WER) is reduced by 2.9\% compared to the baseline, while freezing the encoder, or only its first three layers and the first predictor layer, reduces the region WER max by 1.4\% and 2.5\%, respectively.
Freezing the predictor (Exp.~4) reduces the region WER max by 1.3\%. Since the predictor predicts the next character given previous characters, it acts as a lightweight language model for the RNN-T. Therefore, freezing the predictor prevents the model from learning new linguistic information. Freezing the encoder (Exp.~3), on the other hand, restricts the ASR system from capturing new acoustic knowledge. The variance is reduced by 1.8\% in Exp.~3, while increasing by 1.8\% in Exp.~4; adapting the predictor (Exp.~3) is especially important for reducing variance between regions, indicating that language usage (words, grammar) contributes much to the performance disparity.
Moreover, our proposed method improves the geographic WER disparity the most, reducing both the WER of the region with highest WER (by 3.2\%) and the variance across regions (by 7.9\%), more than any other adaptation method investigated here. Also, the proposed approach reduces the WER on the whole test set by 1.3\%, versus the empirical bound of 1.0\% given by Exp.~7. Exp.~7 can be considered a notional optimum for adaptation because the model is trained on both pretraining dataset $D_p$ and fine-tuning dataset $D_c$ jointly, while in the continual learning setup, only fine-tuning data may be used. In Exp.~8, the WER of the region with highest WER only decreases by $0.9\%$, as a result of using random data from $D_r$ compared to $3.2\%$ in Exp.~6 ($D_r$ has 2.8k hours in common with $D_c$).
Our proposed method improves performance on the whole user population by 1.3\%, so is effective at preserving knowledge learned by the model from past data in the pretraining stage. More specifically, the proposed method can enhance performance in regions with the highest WER without sacrificing performance in other regions. Also, other conventional transfer-learning methods forget little past knowledge; e.g., Exp.~4 has the same overall WER performance as the baseline. We can credit the small learning rate in fine-tuning for this result. However, while a small learning rate mitigates the forgetting problem, it also limits their model's ability to learn new knowledge. Furthermore, the pretraining data already contains some of the regions with high WER, further alleviating forgetting.
Finally, although we have not explored the hyperparameter $\lambda$ in Eq.~(\ref{eq:4}), it could be used to balance the learning of new knowledge against the forgetting of old knowledge in an operational setting.
\section{Conclusion}
We have investigated a method to address geographic disparities and fairness in ASR \cite{mehrabi2021survey}. We propose an RNN-T loss function combining the standard ASR loss with EWC regularization loss to encourage the ASR model to find parameters that have good performance for the user population overall, while reducing the performance degradation for speakers from regions with high WER. Our proposed method reduces the WER in the region with highest WER by 3.2\% relative and reduces the overall WER by 1.3\% relative. Moreover, our results suggest that at least in our setting, adapting the language modeling component of the system is important for reducing the performance gap. Our empirical results focus on reducing geographic differences in ASR performance, but our method is equally applicable to other scenarios with a need to adapt a model to a specific dataset without degrading overall performance.
\vfill
\pagebreak
\balance
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Winter 2021: Other Significant Articles
| March 29, 2021 | no comments
David Livermore, "The Group Excluded from Diversity Programs" Center for Cultural Intelligence (November 11, 2020).
PneumaReview.com author David Livermore challenges DEI advocates not to exclude the white working class from their efforts. "I'm not suggesting we should shy away from calling racism what it is. Nor am I suggesting that polite conversations that treat all viewpoints as equally valid are the answer. But we have to stop and consider why so many working-class individuals feel like diversity programs teach people to tolerate and include everyone except them."
J. Lee Grady, "Don't Put a Lid on the Holy Spirit" Fire In My Bones (November 18, 2020).
The former editor of Charisma magazine writes: "People need the reality of God's power. I wish we would stop being so worried about how the Holy Spirit might show up in church. Here are seven practical things we can do to encourage the freedom of the Spirit."
Paul King, "Tribute to My Most Intensive Mentor—Dr. Chuck Farah" The King's Round Table (November 28, 2020).
Paul King writes of Chuck Farah, who wrote the critique of the Word Faith movement, From the Pinnacle of the Temple: "He was a man of the Spirit—who combined the academic and the supernatural, a Ph.D. who experienced speaking in tongues."
"We Prayed for Healing. God Brought a Pandemic: A coronavirus outbreak at France's biggest Pentecostal megachurch changed their view of providence, judgment, and fellowship" Christianity Today (November 23, 2020).
In this (translated) interview with Jean-Paul Rempp, he describes how his church has taken a stance of forgiveness for those that accused them of becoming vectors for COVID-19 and how they have learned how to be a renewed and more welcoming community.
"Stanford E. Linzey, Jr. Collection Deposited at Flower Pentecostal Heritage Center" iFPHC.com (December 2, 2020).
For more on Stanford Linzey, Jr., see John Miller's review of his book, The Holy Spirit in the Third Millennium: Handbook on the Holy Spirit: A Guide to the Spirit within.
Image: Le Duc
Frank Macchia, "How Flawed Is Our Gratitude? Learning from Jonah" FrankDMacchia.wixsite.com (December 7, 2020).
Roger E. Olson, "Critical Race Theory Examined and Analyzed" Patheos (December 21, 2020).
Brian Blount, "How Vineyard History Can Give Practical Insight to Discerning Prophetic Words" CrestwoodVineyard.org (December 12, 2020).
Pastor Brian Blount reflects on Bill Jackson's book, The Quest for the Radical Middle and on what going through the prophetic movement taught him and other leaders in the Vineyard association. "We need to remember all three parts of discerning prophecy. We should weigh not only the revelation, but also the interpretation and the application. Correct revelation with incorrect interpretation or application can be very damaging."
Thanks to William De Arteaga for pointing out this article.
Roger E. Olson, "About 'Socialism' and Christianity" Patheos (December 26, 2020).
Historian of religion and theology, Roger Olson, discusses the many definitions of socialism, including the long-forgotten calls from Christians leaders of the past to urge the State to help the poor.
Michael Brown, "Don't Let Anyone Threaten You With Prophetic Manipulation" YouTube.com (January 5, 2021).
John Lathrop writes, "Here is a new video, about 10 minutes long, in which Dr. Michael Brown speaks about manipulative prophecies." Brown describes how "Touch not the Lord's anointed" is often abused and says, "As leaders we are not just accountable to God, but we are accountable to other leaders and ultimately accountable, on a certain level, to whom we minister."
Dave Johnson, "Interview with Lora Timenia about her book Third Wave of Pentecostalism in the Philippines: Understanding Toronto Blessing Revivalism's Signs and Wonders Theology in the Philippines" (December 14, 2020).
In this interview from APTS Press, PneumaReview.com author Dave Johnson interviews Lora Timenia about her book.
David Livermore, "What I've Learned From 10 Years Leading The CQ Center" Cultural Intelligence Center (January 6, 2021).
Pneuma Review author David Livermore reflects on his work bringing Cultural Intelligence to the world. What is CQ and why is it important? Quick introduction: https://vimeo.com/468569477
"Revival Timelines" Revival-Library.org
The Revival Library has announced the publication of easy-to-read timelines of all major revival movements that have occurred in biblical history as well as prior and since the Great Reformation.
Médine Keener, "Lessons from Susie King Taylor" Asbury University (February 24, 2021).
A chapel message given by PneumaReview.com author Médine Keener has been converted into a podcast. From the introduction: "Dr. Medine Keener shares what we learn about racial and ethnic reconciliation through the life of Susie King Taylor—an invitation to embrace all God's people."
Paul J. Palma, The Antioch Legacy, by John P. Lathrop PJPalma.net (February 27, 2021).
Pentecostal scholar Paul Palma reviews Pastor John Lathrop's book, The Antioch Legacy. Roger E. Olson, "American Evangelical Christianity: A Failed Movement?" Patheos (February 27, 2021).
Anthony Bradley, "Critical Race Theory Isn't a Threat for Presbyterians" MereOrthodoxy.com (February 3, 2021).
While written to Presbyterians, the principles apply to all followers of Jesus: "CRT is not a real threat to the work of the Presbyterian Church in America. It does not tell us any more than what the Bible and the Christian tradition already acknowledge as a point of fact—namely, that people can be evil and that the parasite kingdom is wreaking havoc as far as the curse is found. There is racism in America and, at times, that racism can take on structural forms. It is proper to the work of the church to actively seek to bring solidarity and peace where there is racial conflict because of the hope of Resurrection. Presbyterians, then, can eat the meat and spit the bones of CRT, or any other secular social theory that does not presuppose the Triune God because the real war is against any manifestations of the principalities and powers (Eph 6:12) at work parasitically undermining the goodness of God's creation."
Thanks to Mark Galli for recommending this article.
Tags: 2021, articles, significant, winter
Category: Winter 2021
About the Author: The PneumaReview.com editors are Raul Mock, Mike Dies, Joe Joslin, and Jim Dettmann with significant input from other writers including John Lathrop, Amos Yong, Tony Richie, and Kevin Williams.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,407
|
"""
SleekXMPP: The Sleek XMPP Library
Copyright (C) 2010 Nathanael C. Fritz, Lance J.T. Stout
This file is part of SleekXMPP.
See the file LICENSE for copying permission.
"""
import unittest
import sleekxmpp
from sleekxmpp import ClientXMPP, ComponentXMPP
from sleekxmpp.stanza import Message, Iq, Presence
from sleekxmpp.test import TestSocket, TestLiveSocket
from sleekxmpp.xmlstream import StanzaBase, ET, register_stanza_plugin
from sleekxmpp.xmlstream.tostring import tostring
class SleekTest(unittest.TestCase):
"""
A SleekXMPP specific TestCase class that provides
methods for comparing message, iq, and presence stanzas.
Methods:
Message -- Create a Message stanza object.
Iq -- Create an Iq stanza object.
Presence -- Create a Presence stanza object.
check_jid -- Check a JID and its component parts.
check -- Compare a stanza against an XML string.
stream_start -- Initialize a dummy XMPP client.
stream_close -- Disconnect the XMPP client.
make_header -- Create a stream header.
send_header -- Check that the given header has been sent.
send_feature -- Send a raw XML element.
send -- Check that the XMPP client sent the given
generic stanza.
recv -- Queue data for XMPP client to receive, or
verify the data that was received from a
live connection.
recv_header -- Check that a given stream header
was received.
recv_feature -- Check that a given, raw XML element
was recveived.
fix_namespaces -- Add top-level namespace to an XML object.
compare -- Compare XML objects against each other.
"""
def runTest(self):
pass
def parse_xml(self, xml_string):
try:
xml = ET.fromstring(xml_string)
return xml
except SyntaxError as e:
if 'unbound' in e.msg:
known_prefixes = {
'stream': 'http://etherx.jabber.org/streams'}
prefix = xml_string.split('<')[1].split(':')[0]
if prefix in known_prefixes:
xml_string = '<fixns xmlns:%s="%s">%s</fixns>' % (
prefix,
known_prefixes[prefix],
xml_string)
xml = self.parse_xml(xml_string)
xml = xml.getchildren()[0]
return xml
# ------------------------------------------------------------------
# Shortcut methods for creating stanza objects
def Message(self, *args, **kwargs):
"""
Create a Message stanza.
Uses same arguments as StanzaBase.__init__
Arguments:
xml -- An XML object to use for the Message's values.
"""
return Message(None, *args, **kwargs)
def Iq(self, *args, **kwargs):
"""
Create an Iq stanza.
Uses same arguments as StanzaBase.__init__
Arguments:
xml -- An XML object to use for the Iq's values.
"""
return Iq(None, *args, **kwargs)
def Presence(self, *args, **kwargs):
"""
Create a Presence stanza.
Uses same arguments as StanzaBase.__init__
Arguments:
xml -- An XML object to use for the Iq's values.
"""
return Presence(None, *args, **kwargs)
def check_jid(self, jid, user=None, domain=None, resource=None,
bare=None, full=None, string=None):
"""
Verify the components of a JID.
Arguments:
jid -- The JID object to test.
user -- Optional. The user name portion of the JID.
domain -- Optional. The domain name portion of the JID.
resource -- Optional. The resource portion of the JID.
bare -- Optional. The bare JID.
full -- Optional. The full JID.
string -- Optional. The string version of the JID.
"""
if user is not None:
self.assertEqual(jid.user, user,
"User does not match: %s" % jid.user)
if domain is not None:
self.assertEqual(jid.domain, domain,
"Domain does not match: %s" % jid.domain)
if resource is not None:
self.assertEqual(jid.resource, resource,
"Resource does not match: %s" % jid.resource)
if bare is not None:
self.assertEqual(jid.bare, bare,
"Bare JID does not match: %s" % jid.bare)
if full is not None:
self.assertEqual(jid.full, full,
"Full JID does not match: %s" % jid.full)
if string is not None:
self.assertEqual(str(jid), string,
"String does not match: %s" % str(jid))
# ------------------------------------------------------------------
# Methods for comparing stanza objects to XML strings
def check(self, stanza, xml_string,
defaults=None, use_values=True):
"""
Create and compare several stanza objects to a correct XML string.
If use_values is False, test using getStanzaValues() and
setStanzaValues() will not be used.
Some stanzas provide default values for some interfaces, but
these defaults can be problematic for testing since they can easily
be forgotten when supplying the XML string. A list of interfaces that
use defaults may be provided and the generated stanzas will use the
default values for those interfaces if needed.
However, correcting the supplied XML is not possible for interfaces
that add or remove XML elements. Only interfaces that map to XML
attributes may be set using the defaults parameter. The supplied XML
must take into account any extra elements that are included by default.
Arguments:
stanza -- The stanza object to test.
xml_string -- A string version of the correct XML expected.
defaults -- A list of stanza interfaces that have default
values. These interfaces will be set to their
defaults for the given and generated stanzas to
prevent unexpected test failures.
use_values -- Indicates if testing using getStanzaValues() and
setStanzaValues() should be used. Defaults to
True.
"""
stanza_class = stanza.__class__
xml = self.parse_xml(xml_string)
# Ensure that top level namespaces are used, even if they
# were not provided.
self.fix_namespaces(stanza.xml, 'jabber:client')
self.fix_namespaces(xml, 'jabber:client')
stanza2 = stanza_class(xml=xml)
if use_values:
# Using getStanzaValues() and setStanzaValues() will add
# XML for any interface that has a default value. We need
# to set those defaults on the existing stanzas and XML
# so that they will compare correctly.
default_stanza = stanza_class()
if defaults is None:
known_defaults = {
Message: ['type'],
Presence: ['priority']
}
defaults = known_defaults.get(stanza_class, [])
for interface in defaults:
stanza[interface] = stanza[interface]
stanza2[interface] = stanza2[interface]
# Can really only automatically add defaults for top
# level attribute values. Anything else must be accounted
# for in the provided XML string.
if interface not in xml.attrib:
if interface in default_stanza.xml.attrib:
value = default_stanza.xml.attrib[interface]
xml.attrib[interface] = value
values = stanza2.getStanzaValues()
stanza3 = stanza_class()
stanza3.setStanzaValues(values)
debug = "Three methods for creating stanzas do not match.\n"
debug += "Given XML:\n%s\n" % tostring(xml)
debug += "Given stanza:\n%s\n" % tostring(stanza.xml)
debug += "Generated stanza:\n%s\n" % tostring(stanza2.xml)
debug += "Second generated stanza:\n%s\n" % tostring(stanza3.xml)
result = self.compare(xml, stanza.xml, stanza2.xml, stanza3.xml)
else:
debug = "Two methods for creating stanzas do not match.\n"
debug += "Given XML:\n%s\n" % tostring(xml)
debug += "Given stanza:\n%s\n" % tostring(stanza.xml)
debug += "Generated stanza:\n%s\n" % tostring(stanza2.xml)
result = self.compare(xml, stanza.xml, stanza2.xml)
self.failUnless(result, debug)
# ------------------------------------------------------------------
# Methods for simulating stanza streams.
def stream_start(self, mode='client', skip=True, header=None,
socket='mock', jid='tester@localhost',
password='test', server='localhost',
port=5222):
"""
Initialize an XMPP client or component using a dummy XML stream.
Arguments:
mode -- Either 'client' or 'component'. Defaults to 'client'.
skip -- Indicates if the first item in the sent queue (the
stream header) should be removed. Tests that wish
to test initializing the stream should set this to
False. Otherwise, the default of True should be used.
socket -- Either 'mock' or 'live' to indicate if the socket
should be a dummy, mock socket or a live, functioning
socket. Defaults to 'mock'.
jid -- The JID to use for the connection.
Defaults to 'tester@localhost'.
password -- The password to use for the connection.
Defaults to 'test'.
server -- The name of the XMPP server. Defaults to 'localhost'.
port -- The port to use when connecting to the server.
Defaults to 5222.
"""
if mode == 'client':
self.xmpp = ClientXMPP(jid, password)
elif mode == 'component':
self.xmpp = ComponentXMPP(jid, password,
server, port)
else:
raise ValueError("Unknown XMPP connection mode.")
if socket == 'mock':
self.xmpp.set_socket(TestSocket())
# Simulate connecting for mock sockets.
self.xmpp.auto_reconnect = False
self.xmpp.is_client = True
self.xmpp.state._set_state('connected')
# Must have the stream header ready for xmpp.process() to work.
if not header:
header = self.xmpp.stream_header
self.xmpp.socket.recv_data(header)
elif socket == 'live':
self.xmpp.socket_class = TestLiveSocket
self.xmpp.connect()
else:
raise ValueError("Unknown socket type.")
self.xmpp.register_plugins()
self.xmpp.process(threaded=True)
if skip:
# Clear startup stanzas
self.xmpp.socket.next_sent(timeout=1)
if mode == 'component':
self.xmpp.socket.next_sent(timeout=1)
def make_header(self, sto='',
sfrom='',
sid='',
stream_ns="http://etherx.jabber.org/streams",
default_ns="jabber:client",
version="1.0",
xml_header=True):
"""
Create a stream header to be received by the test XMPP agent.
The header must be saved and passed to stream_start.
Arguments:
sto -- The recipient of the stream header.
sfrom -- The agent sending the stream header.
sid -- The stream's id.
stream_ns -- The namespace of the stream's root element.
default_ns -- The default stanza namespace.
version -- The stream version.
xml_header -- Indicates if the XML version header should be
appended before the stream header.
"""
header = '<stream:stream %s>'
parts = []
if xml_header:
header = '<?xml version="1.0"?>' + header
if sto:
parts.append('to="%s"' % sto)
if sfrom:
parts.append('from="%s"' % sfrom)
if sid:
parts.append('id="%s"' % sid)
parts.append('version="%s"' % version)
parts.append('xmlns:stream="%s"' % stream_ns)
parts.append('xmlns="%s"' % default_ns)
return header % ' '.join(parts)
def recv(self, data, stanza_class=StanzaBase, defaults=[],
use_values=True, timeout=1):
"""
Pass data to the dummy XMPP client as if it came from an XMPP server.
If using a live connection, verify what the server has sent.
Arguments:
data -- String stanza XML to be received and processed by
the XMPP client or component.
stanza_class -- The stanza object class for verifying data received
by a live connection. Defaults to StanzaBase.
defaults -- A list of stanza interfaces with default values that
may interfere with comparisons.
use_values -- Indicates if stanza comparisons should test using
getStanzaValues() and setStanzaValues().
Defaults to True.
timeout -- Time to wait in seconds for data to be received by
a live connection.
"""
if self.xmpp.socket.is_live:
# we are working with a live connection, so we should
# verify what has been received instead of simulating
# receiving data.
recv_data = self.xmpp.socket.next_recv(timeout)
if recv_data is None:
return False
stanza = stanza_class(xml=self.parse_xml(recv_data))
return self.check(stanza_class, stanza, data,
defaults=defaults,
use_values=use_values)
else:
# place the data in the dummy socket receiving queue.
data = str(data)
self.xmpp.socket.recv_data(data)
def recv_header(self, sto='',
sfrom='',
sid='',
stream_ns="http://etherx.jabber.org/streams",
default_ns="jabber:client",
version="1.0",
xml_header=False,
timeout=1):
"""
Check that a given stream header was received.
Arguments:
sto -- The recipient of the stream header.
sfrom -- The agent sending the stream header.
sid -- The stream's id. Set to None to ignore.
stream_ns -- The namespace of the stream's root element.
default_ns -- The default stanza namespace.
version -- The stream version.
xml_header -- Indicates if the XML version header should be
appended before the stream header.
timeout -- Length of time to wait in seconds for a
response.
"""
header = self.make_header(sto, sfrom, sid,
stream_ns=stream_ns,
default_ns=default_ns,
version=version,
xml_header=xml_header)
recv_header = self.xmpp.socket.next_recv(timeout)
if recv_header is None:
raise ValueError("Socket did not return data.")
# Apply closing elements so that we can construct
# XML objects for comparison.
header2 = header + '</stream:stream>'
recv_header2 = recv_header + '</stream:stream>'
xml = self.parse_xml(header2)
recv_xml = self.parse_xml(recv_header2)
if sid is None:
# Ignore the id sent by the server since
# we can't know in advance what it will be.
if 'id' in recv_xml.attrib:
del recv_xml.attrib['id']
# Ignore the xml:lang attribute for now.
if 'xml:lang' in recv_xml.attrib:
del recv_xml.attrib['xml:lang']
xml_ns = 'http://www.w3.org/XML/1998/namespace'
if '{%s}lang' % xml_ns in recv_xml.attrib:
del recv_xml.attrib['{%s}lang' % xml_ns]
if recv_xml.getchildren:
# We received more than just the header
for xml in recv_xml.getchildren():
self.xmpp.socket.recv_data(tostring(xml))
attrib = recv_xml.attrib
recv_xml.clear()
recv_xml.attrib = attrib
self.failUnless(
self.compare(xml, recv_xml),
"Stream headers do not match:\nDesired:\n%s\nReceived:\n%s" % (
'%s %s' % (xml.tag, xml.attrib),
'%s %s' % (recv_xml.tag, recv_xml.attrib)))
def recv_feature(self, data, use_values=True, timeout=1):
"""
"""
if self.xmpp.socket.is_live:
# we are working with a live connection, so we should
# verify what has been received instead of simulating
# receiving data.
recv_data = self.xmpp.socket.next_recv(timeout)
if recv_data is None:
return False
xml = self.parse_xml(data)
recv_xml = self.parse_xml(recv_data)
self.failUnless(self.compare(xml, recv_xml),
"Features do not match.\nDesired:\n%s\nReceived:\n%s" % (
tostring(xml), tostring(recv_xml)))
else:
# place the data in the dummy socket receiving queue.
data = str(data)
self.xmpp.socket.recv_data(data)
def send_header(self, sto='',
sfrom='',
sid='',
stream_ns="http://etherx.jabber.org/streams",
default_ns="jabber:client",
version="1.0",
xml_header=False,
timeout=1):
"""
Check that a given stream header was sent.
Arguments:
sto -- The recipient of the stream header.
sfrom -- The agent sending the stream header.
sid -- The stream's id.
stream_ns -- The namespace of the stream's root element.
default_ns -- The default stanza namespace.
version -- The stream version.
xml_header -- Indicates if the XML version header should be
appended before the stream header.
timeout -- Length of time to wait in seconds for a
response.
"""
header = self.make_header(sto, sfrom, sid,
stream_ns=stream_ns,
default_ns=default_ns,
version=version,
xml_header=xml_header)
sent_header = self.xmpp.socket.next_sent(timeout)
if sent_header is None:
raise ValueError("Socket did not return data.")
# Apply closing elements so that we can construct
# XML objects for comparison.
header2 = header + '</stream:stream>'
sent_header2 = sent_header + b'</stream:stream>'
xml = self.parse_xml(header2)
sent_xml = self.parse_xml(sent_header2)
self.failUnless(
self.compare(xml, sent_xml),
"Stream headers do not match:\nDesired:\n%s\nSent:\n%s" % (
header, sent_header))
def send_feature(self, data, use_values=True, timeout=1):
"""
"""
sent_data = self.xmpp.socket.next_sent(timeout)
if sent_data is None:
return False
xml = self.parse_xml(data)
sent_xml = self.parse_xml(sent_data)
self.failUnless(self.compare(xml, sent_xml),
"Features do not match.\nDesired:\n%s\nSent:\n%s" % (
tostring(xml), tostring(sent_xml)))
def send(self, data, defaults=None,
use_values=True, timeout=.1):
"""
Check that the XMPP client sent the given stanza XML.
Extracts the next sent stanza and compares it with the given
XML using check.
Arguments:
stanza_class -- The class of the sent stanza object.
data -- The XML string of the expected Message stanza,
or an equivalent stanza object.
use_values -- Modifies the type of tests used by check_message.
defaults -- A list of stanza interfaces that have defaults
values which may interfere with comparisons.
timeout -- Time in seconds to wait for a stanza before
failing the check.
"""
if isinstance(data, str):
xml = self.parse_xml(data)
self.fix_namespaces(xml, 'jabber:client')
data = self.xmpp._build_stanza(xml, 'jabber:client')
sent = self.xmpp.socket.next_sent(timeout)
self.check(data, sent,
defaults=defaults,
use_values=use_values)
def stream_close(self):
"""
Disconnect the dummy XMPP client.
Can be safely called even if stream_start has not been called.
Must be placed in the tearDown method of a test class to ensure
that the XMPP client is disconnected after an error.
"""
if hasattr(self, 'xmpp') and self.xmpp is not None:
self.xmpp.socket.recv_data(self.xmpp.stream_footer)
self.xmpp.disconnect()
# ------------------------------------------------------------------
# XML Comparison and Cleanup
def fix_namespaces(self, xml, ns):
"""
Assign a namespace to an element and any children that
don't have a namespace.
Arguments:
xml -- The XML object to fix.
ns -- The namespace to add to the XML object.
"""
if xml.tag.startswith('{'):
return
xml.tag = '{%s}%s' % (ns, xml.tag)
for child in xml.getchildren():
self.fix_namespaces(child, ns)
def compare(self, xml, *other):
"""
Compare XML objects.
Arguments:
xml -- The XML object to compare against.
*other -- The list of XML objects to compare.
"""
if not other:
return False
# Compare multiple objects
if len(other) > 1:
for xml2 in other:
if not self.compare(xml, xml2):
return False
return True
other = other[0]
# Step 1: Check tags
if xml.tag != other.tag:
return False
# Step 2: Check attributes
if xml.attrib != other.attrib:
return False
# Step 3: Check text
if xml.text is None:
xml.text = ""
if other.text is None:
other.text = ""
xml.text = xml.text.strip()
other.text = other.text.strip()
if xml.text != other.text:
return False
# Step 4: Check children count
if len(xml.getchildren()) != len(other.getchildren()):
return False
# Step 5: Recursively check children
for child in xml:
child2s = other.findall("%s" % child.tag)
if child2s is None:
return False
for child2 in child2s:
if self.compare(child, child2):
break
else:
return False
# Step 6: Recursively check children the other way.
for child in other:
child2s = xml.findall("%s" % child.tag)
if child2s is None:
return False
for child2 in child2s:
if self.compare(child, child2):
break
else:
return False
# Everything matches
return True
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,012
|
Q: encrypt database username and password in java? Actually by type 4 db connection from java i am connecting to oracle database like this:
In dbconnection.java i have written below:
Class.forName("oracle.jdbc.driver.OracleDriver");
con=DriverManager.getConnection("jdbc:oracle:thin:@localhost:1521:database host name","database user name", "database user password");
return con;
Is there any method how can i write these 3 fields(database host, database username, database user password) in encrypted form in java class dbconnection.java like below:
Class.forName("oracle.jdbc.driver.OracleDriver");
con=DriverManager.getConnection("jdbc:oracle:thin:@localhost:1521:wrtwtr#$%$_rfwrw","regfwerfgwf", "%%5frfr^&%$%4");
return con;
and during database connection, these encrypted fields will be decrypted and will be connected to oracle database in actual names
Then in servlet i am calling like:
dbconnection db= new dbconnection();
Any help please
A: I'm not aware of any way to do that.
But I also don't see what doing this would achieve. Sure, the username and password are not shown in clear, but a bad guy who had access to those encrypted strings would be able to use them the same way as your program does.
A: Rather than trying to hide your connection details, you should provide a secure way to allow externally controlled applications (or instances of applications) to access your database.
An API is basically a middle layer between your database and an application you do not have control over. For example, facebook, rather than allowing direct access to their database, allows developers to access their data through an API. This means that applications can be authenticated (thus meaning they can be held responsible), and you can control explicitly what applications can and cannot see and edit.
Basically, through an API, you can protect your database while simultaneously keeping track of who is doing what (though the protection aspect is usually the main draw).
I must say though that sometimes, if you trust the people using the application, it's just easier to not worry about it. For example, if you work at a small company of competent, well meaning people, then it would likely be safe to allow the application to connect directly to the database.
If you are distributing your program to the general public though, or a large set of people whom you do not completely and totally trust, then you should not allow direct access, no matter what kind of precautions are taken.
Assume that you do figure out how to encrypt your credentials. At some point, you must still make the connection. What happens now if a user grabs the decryption/connection code, has the connection made, then inserts his own code after it? Suddenly he has access to your database. With an API, worst case, he could steal the API key and have limited, traceable, easily revokable access.
And besides, if you're allowing access to an API, you only allow users to do what you want them to do. So worst case, if he does figure out how to use the API directly, all he can do is what the program allows him to do anyway.
A: Passing encrypted data to the DriverManager is not an option.
You should pass the decrypted string into DriverManager. So somewhere you would have the user name and password encrypted and then decrypt them before passing them to the DriverManager
Then you would have other issues e.g. where to store private key etc, but as a first defence it would be better than using plaintext since my understanding is that you have some security requirement.
You did not mention what OS you are using. If you are using Windows I would suggest to use Windows based authentication (the connection to the database is authenticated against the current windows user).So no need to provide user name and password in the connection string. Check if Oracle supports this (have tried only with MS-SQL server.Hopefully the link I provided is useful).
This is IMHO the best option. For Linux there must be something equivalent.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,001
|
La manufacture J. Vieillard & Cie est une manufacture de faïencerie bordelaise, installée sur le quai de Bacalan. Elle succède à la manufacture David Jonhston en 1845. Elle est dirigée par Jules Vieillard jusqu'à sa mort en 1868 : ses fils Albert et Charles, qui partagent la gérance avec leur père depuis 1865, prennent la suite jusqu'à sa fermeture en 1895.
La faïence fine bordelaise au siècle
Dès le milieu du siècle, les fabriques de faïence européennes, concurrencées par les manufactures de porcelaine, se livrent à de nombreuses expérimentations pour imiter la finesse, la gamme colorée des décors et la qualité des pâtes de la porcelaine. Des fabricants anglais, plus particulièrement les potiers du Staffordshire, mettent au point une nouvelle céramique, une faïence fine dont la pâte est blanche, homogène, fine et opaque. Cette pâte, qui se cuit au grand feu, présente la plasticité nécessaire aux processus de fabrication en série et permet l'application des nouvelles techniques de décor que sont le transfert d'impressions et le moulage de décors appliqués.
Un céramiste, Pierre-Honoré Boudon de Saint-Amans (1774-1858) découvre ces innovations en Angleterre. C'est sous son égide que va débuter dès 1830 la fabrication de faïences fines à Bordeaux. Elle durera près de , jusqu'en 1895, en présentant une belle continuité de production à travers les trois manufactures successives : la manufacture de Lahens et Rateau (1830-1832), celle de David Johnston (1834-1845) et celle de Jules Vieillard et de ses successeurs (1845-1895).
Sous influence anglaise
Boudon de Saint-Amans entreprend de réinventer sur le sol français les procédés anglais, à la Manufacture de Sèvres, à celles de Creil, de Montereau, de Choisy, puis de nouveau à Sèvres, avant de croiser la route du négociant bordelais Jean-François Rateau. Celui-ci et son associé Pierre-Louis Lahens l'invitent à s'installer à Bordeaux où il arrive avec une équipe d'ouvriers parisiens.
Quand la manufacture est contrainte de fermer ses portes, un autre projet voit le jour à Bordeaux sous la conduite de David Johnston (1789-1853), un négociant d'origine irlandaise.
Il aménage sa manufacture de « faïences anglaises » dans d'anciens moulins à Bacalan, sous la direction technique de Boudon de Saint-Amans qui y restera jusqu'en 1837. On retrouve dans sa fabrication les modèles issus de moules créés pour Lahens et Rateau, en particulier des pièces moulées de feuilles de chêne ou de rosettes. Cette influence anglaise perdurera dans jusque dans les productions de Jules Vieillard (1817-1868).
Production
Décors imprimés régionalistes
Dans un contexte de concurrence accrue avec les nouvelles faïenceries françaises, David Johnston souhaite diversifier sa production. Pour ce faire il va exploiter l'une des propriétés de la faïence fine, qui est de restituer finement les décors imprimés, ce qui permet la mécanisation, donnée capitale de l'économie de l'entreprise. Encore faut-il s'entourer de bons artistes et de graveurs qui sauront mettre au point les techniques de transfert dans les meilleures conditions. David Johnston fait appel à des artistes bordelais, comme le dessinateur Pierre Lacour fils et les lithographes Jean-Baptiste Légé (mort en 1846) et Pierre Gorse (1816-1875), qui privilégient les représentations de la Gironde.
Influences de la porcelaine de Chine
Bien avant l'arrivée, dans les années 1870, d'Amédée de Caranza (1840-1912), directeur artistique qui mettra au service de la fabrique reprise par les fils de Jules Vieillard, outre une inventivité débridée, son goût pour l'art persan et sa profonde compréhension des arts graphiques japonais, la faïencerie bordelaise crée déjà des pièces marquées par le goût d'un orient de fantaisie, avec ses paysages exotiques, ses palmiers, saules pleureurs, jonques, oiseaux fabuleux et autres dragons. D'autres décors sont inspirés par la prestigieuse porcelaine de commande, importée de Chine au siècle par la Compagnie de Indes Orientales. Enfin, avec de nouveaux fours à houille construits en 1851, la faïencerie se diversifie dans la fabrication de porcelaine, cherchant alors à reproduire les éclats de l'or sur les harmonies de bleu et de corail des porcelaines « vieux japon » ou imari.
Décors peints régionalistes
La manufacture J. Vieillard & Cie trouve dans le régionalisme une source d'inspiration pour une production en marge de la fabrication industrielle.
Au musée des Arts décoratifs et du Design de Bordeaux, les pièces de Marie Gadou font preuve d'un sens de l'observation minutieux et d'une facture appliquée quand les vases de Louis Cabié d'un style plus enlevé se distinguent par leur naturalisme. Cabié utilise par ailleurs une technique proche de la barbotine, procédé de décoration élaboré à la manufacture de Sèvres pour la porcelaine qui permet d'obtenir un rendu proche de la peinture à l'huile, donnant l'impression qu'une toile tourne autour de la panse de ses vases.
Lieux d'exposition
Musée des Arts décoratifs et du Design de Bordeaux
Bibliographie
Brigitte Michaux, J. Vieillard & Cie. Éclectisme et japonisme, Catalogue de l'exposition, Bordeaux, musée des Arts décoratifs, 1986.
Jacqueline du Pasquier, J. Vieillard & Cie. Histoire de la faïence fine à Bordeaux. De l'anglomanie au rêve orientaliste, Bordeaux, Mollat, 2002.
De David Johnston à Jules Vieillard, l'ivresse Darrigade, Catalogue de l'exposition, Bordeaux, musée des Arts décoratifs et du Design, 2015.
Notes et références
Faïence
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#ifndef QVECTOR3D_H
#define QVECTOR3D_H
#include <QtCore/qpoint.h>
#include <QtCore/qmetatype.h>
QT_BEGIN_NAMESPACE
class QMatrix4x4;
class QVector2D;
class QVector4D;
#ifndef QT_NO_VECTOR3D
class Q_GUI_EXPORT QVector3D
{
public:
QVector3D();
QVector3D(float xpos, float ypos, float zpos);
explicit QVector3D(const QPoint& point);
explicit QVector3D(const QPointF& point);
#ifndef QT_NO_VECTOR2D
QVector3D(const QVector2D& vector);
QVector3D(const QVector2D& vector, float zpos);
#endif
#ifndef QT_NO_VECTOR4D
explicit QVector3D(const QVector4D& vector);
#endif
bool isNull() const;
float x() const;
float y() const;
float z() const;
void setX(float x);
void setY(float y);
void setZ(float z);
float length() const;
float lengthSquared() const;
QVector3D normalized() const;
void normalize();
QVector3D &operator+=(const QVector3D &vector);
QVector3D &operator-=(const QVector3D &vector);
QVector3D &operator*=(float factor);
QVector3D &operator*=(const QVector3D& vector);
QVector3D &operator/=(float divisor);
static float dotProduct(const QVector3D& v1, const QVector3D& v2);
static QVector3D crossProduct(const QVector3D& v1, const QVector3D& v2);
static QVector3D normal(const QVector3D& v1, const QVector3D& v2);
static QVector3D normal
(const QVector3D& v1, const QVector3D& v2, const QVector3D& v3);
float distanceToPoint(const QVector3D& point) const;
float distanceToPlane(const QVector3D& plane, const QVector3D& normal) const;
float distanceToPlane(const QVector3D& plane1, const QVector3D& plane2, const QVector3D& plane3) const;
float distanceToLine(const QVector3D& point, const QVector3D& direction) const;
friend inline bool operator==(const QVector3D &v1, const QVector3D &v2);
friend inline bool operator!=(const QVector3D &v1, const QVector3D &v2);
friend inline const QVector3D operator+(const QVector3D &v1, const QVector3D &v2);
friend inline const QVector3D operator-(const QVector3D &v1, const QVector3D &v2);
friend inline const QVector3D operator*(float factor, const QVector3D &vector);
friend inline const QVector3D operator*(const QVector3D &vector, float factor);
friend const QVector3D operator*(const QVector3D &v1, const QVector3D& v2);
friend inline const QVector3D operator-(const QVector3D &vector);
friend inline const QVector3D operator/(const QVector3D &vector, float divisor);
friend inline bool qFuzzyCompare(const QVector3D& v1, const QVector3D& v2);
#ifndef QT_NO_VECTOR2D
QVector2D toVector2D() const;
#endif
#ifndef QT_NO_VECTOR4D
QVector4D toVector4D() const;
#endif
QPoint toPoint() const;
QPointF toPointF() const;
operator QVariant() const;
private:
float xp, yp, zp;
friend class QVector2D;
friend class QVector4D;
#ifndef QT_NO_MATRIX4X4
friend QVector3D operator*(const QVector3D& vector, const QMatrix4x4& matrix);
friend QVector3D operator*(const QMatrix4x4& matrix, const QVector3D& vector);
#endif
};
Q_DECLARE_TYPEINFO(QVector3D, Q_MOVABLE_TYPE);
inline QVector3D::QVector3D() : xp(0.0f), yp(0.0f), zp(0.0f) {}
inline QVector3D::QVector3D(float xpos, float ypos, float zpos) : xp(xpos), yp(ypos), zp(zpos) {}
inline QVector3D::QVector3D(const QPoint& point) : xp(point.x()), yp(point.y()), zp(0.0f) {}
inline QVector3D::QVector3D(const QPointF& point) : xp(point.x()), yp(point.y()), zp(0.0f) {}
inline bool QVector3D::isNull() const
{
return qIsNull(xp) && qIsNull(yp) && qIsNull(zp);
}
inline float QVector3D::x() const { return xp; }
inline float QVector3D::y() const { return yp; }
inline float QVector3D::z() const { return zp; }
inline void QVector3D::setX(float aX) { xp = aX; }
inline void QVector3D::setY(float aY) { yp = aY; }
inline void QVector3D::setZ(float aZ) { zp = aZ; }
inline QVector3D &QVector3D::operator+=(const QVector3D &vector)
{
xp += vector.xp;
yp += vector.yp;
zp += vector.zp;
return *this;
}
inline QVector3D &QVector3D::operator-=(const QVector3D &vector)
{
xp -= vector.xp;
yp -= vector.yp;
zp -= vector.zp;
return *this;
}
inline QVector3D &QVector3D::operator*=(float factor)
{
xp *= factor;
yp *= factor;
zp *= factor;
return *this;
}
inline QVector3D &QVector3D::operator*=(const QVector3D& vector)
{
xp *= vector.xp;
yp *= vector.yp;
zp *= vector.zp;
return *this;
}
inline QVector3D &QVector3D::operator/=(float divisor)
{
xp /= divisor;
yp /= divisor;
zp /= divisor;
return *this;
}
inline bool operator==(const QVector3D &v1, const QVector3D &v2)
{
return v1.xp == v2.xp && v1.yp == v2.yp && v1.zp == v2.zp;
}
inline bool operator!=(const QVector3D &v1, const QVector3D &v2)
{
return v1.xp != v2.xp || v1.yp != v2.yp || v1.zp != v2.zp;
}
inline const QVector3D operator+(const QVector3D &v1, const QVector3D &v2)
{
return QVector3D(v1.xp + v2.xp, v1.yp + v2.yp, v1.zp + v2.zp);
}
inline const QVector3D operator-(const QVector3D &v1, const QVector3D &v2)
{
return QVector3D(v1.xp - v2.xp, v1.yp - v2.yp, v1.zp - v2.zp);
}
inline const QVector3D operator*(float factor, const QVector3D &vector)
{
return QVector3D(vector.xp * factor, vector.yp * factor, vector.zp * factor);
}
inline const QVector3D operator*(const QVector3D &vector, float factor)
{
return QVector3D(vector.xp * factor, vector.yp * factor, vector.zp * factor);
}
inline const QVector3D operator*(const QVector3D &v1, const QVector3D& v2)
{
return QVector3D(v1.xp * v2.xp, v1.yp * v2.yp, v1.zp * v2.zp);
}
inline const QVector3D operator-(const QVector3D &vector)
{
return QVector3D(-vector.xp, -vector.yp, -vector.zp);
}
inline const QVector3D operator/(const QVector3D &vector, float divisor)
{
return QVector3D(vector.xp / divisor, vector.yp / divisor, vector.zp / divisor);
}
inline bool qFuzzyCompare(const QVector3D& v1, const QVector3D& v2)
{
return qFuzzyCompare(v1.xp, v2.xp) &&
qFuzzyCompare(v1.yp, v2.yp) &&
qFuzzyCompare(v1.zp, v2.zp);
}
inline QPoint QVector3D::toPoint() const
{
return QPoint(qRound(xp), qRound(yp));
}
inline QPointF QVector3D::toPointF() const
{
return QPointF(qreal(xp), qreal(yp));
}
#ifndef QT_NO_DEBUG_STREAM
Q_GUI_EXPORT QDebug operator<<(QDebug dbg, const QVector3D &vector);
#endif
#ifndef QT_NO_DATASTREAM
Q_GUI_EXPORT QDataStream &operator<<(QDataStream &, const QVector3D &);
Q_GUI_EXPORT QDataStream &operator>>(QDataStream &, QVector3D &);
#endif
#endif
QT_END_NAMESPACE
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
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Tigers game caller Shepard hopes to be back at Comerica Park soon
by: Jack Doles, WOODTV.com staff
GRAND RAPIDS, Mich. (WOOD) — While Detroit Tigers players work to keep in shape for when games resume, fans — and Fox Sports Detroit play-by-play man Matt Shepard — wait impatiently.
"Man, I miss baseball like nobody else," Shepard, who will be in his second year behind the mic, told News 8 in a video call. "I miss being around the batting cage and watching hitters hit and talking to coaches. I miss working my way around the clubhouse and talking to players about everyday life, not just baseball. And I miss sitting in (manager) Ron Gardenhire's office and sharing a laugh or two."
Baseball is back in South Korea, where play-by-play guys like Shepard are calling games from home.
"That is really challenging," Shepard noted. "I have done that before for Big Ten Network. It's not easy. It's very challenging in baseball especially because you're really relying on the cameramen to find the ball for you. You don't get a great gauge on how well a ball is hit, either."
He's hoping that broadcasters for home teams will be working at ballparks when games start up again.
"The question I have is if they have an 82-game schedule, where will we be broadcasting the 41 road games from? Because of the possibility of expanded rosters, 30 and a taxi squad is what they've talked about. How will they want to handle us traveling with them?" he wondered. "How will broadcasting companies be able to pay for those big crews on the road? Those are all things that need to be decided."
==Above, Shepard talks about his first season calling games for the Tigers and what baseball may look like when it returns.==
Tags: Detroit Tigers, Home, Michigan, News, Sports
Previous EGR native seeks transfer after Bowling Green cuts baseball
Next As budget cuts loom, CMU discontinues men's track & field
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
Heterogeneous graphs consist of diverse types of nodes and relationships between nodes, and can comprehensively model many real-world complex systems, such as transportation systems, the World Wide Web and citation networks. Analytic techniques based on deep learning have been researched and applied to heterogeneous graphs in recent years~\cite{hinsurvey1,hinsurvey2}. In particular, heterogeneous graph neural networks (HGNNs) learn node representations by aggregating node attributes from the graph topologies and neighbors of the nodes. HGNNs have enjoyed great success on various graph analysis tasks, e.g., node classification~\cite{gtn,han}, node clustering, link prediction~\cite{magnn,hetgnn} and recommendation~\cite{recommendation}.
HGNNs have been so far developed for annotated graphs, and the existing HGNN methods are supervised or semi-supervised learning methods, i.e., they require node labels for learning node representations and training models~\cite{nshe,gtn,magnn,hgt}. Nevertheless, adequately annotating nodes typically requires domain-specific knowledge and is costly and time-consuming. The recent development of self-supervised learning has been adopted to address the lack of annotated data by mining intrinsic information hidden within the given data as supervised signals~\cite{dgi,gmi,dmgi}. In particular, contrastive learning, a representative self-supervised technique, has achieved competitive performance in computer vision~\cite{cv1,cv2}, natural language processing~\cite{nlp1,nlp2} and graph analysis~\cite{zs,grace}.
Contrastive learning on graphs aims at generating different contrastive views, maximizing the similarity between positive samples selected from the views, and minimizing the similarity between negative samples selected from the views to learn a rich representation of the nodes in a graph. The existing methods on homogeneous graphs often use data augmentations to generate contrastive views, including attribute augmentation (e.g., attribute masking~\cite{hgsl,mtisc}), structure augmentation (e.g., graph diffusion~\cite{Diffusion}) and hybrid augmentation (e.g., subgraph sampling~\cite{gcc}). However, because heterogeneous graphs contain multiple types of nodes and edges, it is infeasible to directly use data augmentations on homogeneous graphs to design contrastive views. To our best knowledge, there is only one attempt proposed to derive contrastive views for heterogeneous graphs~\cite{heco}, which designs contrastive views based solely on pre-defined structural properties (network schema and meta-paths) in the graph. But this approach is based on the assumption that graph topology is trustworthy, which is often violated in practice~\cite{hgsl}. Since the extraction of heterogeneous graphs usually needs to follow certain pre-defined rules, and real-world systems are large and complex, accompanied by various uncertain information, these inevitably introduce noises to graph descriptions. Importantly, both node attributes and graph topologies may be noisy, and the disparity between the attributes and the topology is typically inevitable. In supervised or semi-supervised learning, the presence of node labels may shield the influence of noises in the graph on the model performance. However, for unsupervised and self-supervised learning, the presence of noises has a great impact on the accuracy of the model and the robustness of the underlying methods. Therefore, it is imperative and urgent to develop an effective and robust contrastive learning approach for heterogeneous graphs.
Furthermore, for the noisy graph contrastive learning task, it would be ideal if we could trust node attributes when there is noise in the graph topology, and likewise, if we could trust the graph topology when there is noise in node attributes, i.e., it is desirable to learn from others' strengths to ameliorate one's weaknesses. However, in self-supervised learning, we cannot determine the source of noise, so it is difficult to mine effective information while reducing noise interference. In addition, heterogeneous graphs contain diverse and complex information, which makes it complex to explore key and effective information. Therefore, how to design contrastive views for heterogeneous graphs is very challenging, especially when the attributes and topology are both noisy to some extent as those in many real applications.
To address this problem, we propose here a novel and robust approach for \textbf{H}eterogeneous \textbf{G}raph reciprocal \textbf{C}ontrastive \textbf{L}earning, short-handed as \textbf{HGCL}, for heterogeneous graph learning. The existing methods design contrastive views by using data augmentations that destroy the original graph data or using structural properties alone. In contrast, HGCL comprehensively considers the original node attributes and graph topologies to construct contrastive views. It also considers the information from node attributes and graph topologies when selecting samples. In HGCL, attribute-guided view and topology-guided view are introduced separately to capture the effective information of node attributes and graph topologies to the greatest extent by adopting different fusion mechanisms. A flexible sample selection mechanism is further introduced to consider attributes similarity and topological structure correlation simultaneously and to form contrastive loss to promote the two views. By using two different fusion mechanisms, the effects of complex and diverse attributes are maximized in the attribute-guided view, and the structural characteristics of graphs are fully utilized in the topology-guided view. The use of two fusion mechanisms helps reduce the impact of noises and the disparity between the attributes and the topology on model performance. The contrastive samples jointly determined by attribute and topology are conducive to enhancing the quality of the samples to improve the discriminative power of the model and to mine the key common information in the attributes and topology to help supervise the model during contrastive training. Extensive experiments on node classification and node clustering tasks demonstrate the remarkable superiority of the proposed HGCL over state-of-the-art methods.
The rest of the paper is organized as follows. In Section~\ref{sec:relatedWork} we discuss related work. In Section~\ref{Preliminaries} we give formal definitions of the key terms used in this paper. We present the HGCL approach in Section~\ref{method} and conduct extensive experiments in Section~\ref{shiyan}. Finally, we conclude in Section~\ref{sec:conclusion}.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figure/kj2.eps}
\caption{Overview of the HGCL approach. The attribute-guided view uses different ways to regenerate homogeneous and heterogeneous edges and uses two encoding methods to aggregate homogeneous and heterogeneous neighbor information. The topology-guided view takes meta-path as prior knowledge, and aggregates messages across the same and different meta-paths to learn node representation. The reciprocal contrast integrates and enhances the two views through the high-quality sample selection mechanism.}\label{fig:datu}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figure/yigou.eps}
\caption{A toy example of a heterogeneous graph (DBLP) and an illustration of meta-paths.}
\label{toy}
\end{figure}
\section{Related Work}\label{sec:relatedWork}
\subsection{Heterogeneous Graph Neural Networks}
Most HGNN-based methods aim to learn node representations by aggregating the information from neighbor nodes of a heterogeneous graph while preserving the structure and semantic information. For example, HAN~\cite{han} introduces a hierarchical attention mechanism to aggregate the information from meta-path-based neighbors. MAGNN~\cite{magnn} has made further improvements, taking the intermediate nodes in the meta-path into consideration. GTN~\cite{gtn} no longer relies on artificially defined meta-paths, but designs methods to automatically learn the multi-hop relationship between nodes, and then aggregate messages based on this relationship. HGT~\cite{hgt} adopts relation-based mutual attention to learn node representations for web-scale heterogeneous graphs. The above methods are all based on semi-supervised, and recently some methods based on unsupervised have also been proposed. For example, HetGNN ~\cite{hetgnn} aggregates information by designing neighbor sampling. NSHE ~\cite{nshe} preserves the node pair similarity and network schema structure to learn node representations. Although the above methods have achieved good performance, they cannot mine supervisory information from intrinsic information hidden within the given data which is correctly a main focus in this work.
\subsection{Self-supervised Contrastive Learning}
Self-supervised learning has been popular in computer vision~\cite{cv1,cv2} and has also been extended to natural language processing~\cite{nlp1,nlp2} and graph representation learning~\cite{zs}. DGI~\cite{dgi} and GMI~\cite{gmi} are the earliest self-supervised methods proposed on graphs. As an extension of DGI on heterogeneous graphs, DMGI~\cite{dmgi} has excellent performance. Contrastive learning, a representative self-supervised technique has achieved competitive performance. Contrastive learning on graphs aims to construct different contrastive views and design different loss functions for training. The existing methods often use data augmentation (e.g., structural disturbance, attribute perturbation, and graph diffusion) to design a view and contrast it with the original graph. For example, GRACE ~\cite{grace} creates a view by removing edges and masking attributes and designs a loss function to contrast between node representations. GCA~\cite{gca} proposes an adaptive augmentation method for both edges and node attributes to extends the augmentation strategy of GRACE. GCC~\cite{gcc} samples multiple subgraphs of the same graph and contrasts these subgraphs. GraphCL~\cite{graphcl} uses different graph augmentations and uses a graph contrastive loss to make representations invariant to perturbation.
The methods mentioned above are all for homogeneous graph analysis, and the contrastive view is obtained through the disturbance of data, which destroys the original graph topologies or node attributes and does not consider the interference of noise. Different from homogeneous graphs, heterogeneous graphs contain complex topological structures and diverse node attributes. As a result, the design of views is more flexible and requires further consideration. At present, there has been only one attempt of using contrastive learning for heterogeneous graph analysis~\cite{heco}, which designs contrastive views based solely on structural properties and also ignores noise interference in both attributes and topology. Therefore, how to design a robust and efficient contrastive mechanism for heterogeneous graphs is important and necessary, while is a challenge due to the diversity and complexity of information contained in heterogeneous graphs.
\section{Preliminaries}\label{Preliminaries}
In this section, we start with some key terms used throughout this paper.
\textbf{Definition 1. Heterogeneous Graph.} A heterogeneous graph $\mathcal{G}=(\mathcal{V}, \mathcal{E}, \mathcal{X}, \mathcal{F}, \mathcal{R} )$ is composed of a set of nodes $\mathcal{V}$, an set of edges $\mathcal{E}$, a set of attributes $\mathcal{X}$ on nodes, a set of node types $\mathcal{F}$, and a set of edge types $\mathcal{R}$, where $\left| \mathcal{F} \right|+\left| \mathcal{R} \right|\ge 2$. Every node ${v} \in {\mathcal{V}}$ is associated with a node-type mapping function $\varphi: {\mathcal{V}} \rightarrow \mathcal{F}$, and every edge ${e} \in {\mathcal{E}}$ has an edge-type mapping function $\phi:{\mathcal{E}}\rightarrow\mathcal{R}$.
Take the DBLP citation network as an example (Fig.~\ref{toy}(a)). It includes four types of nodes (author, paper, venue and term), and three types of heterogeneous edges (author-paper, paper-venue and paper-term).
\textbf{Definition 2. Meta-path.} A meta-path $\mathcal{M}$ in a heterogeneous graph $\mathcal{G}$ is a path in the form of ${\mathcal{F}_{1}}\xrightarrow{{\mathcal{R}_{1}}}{\mathcal{F}_{2}}\xrightarrow{{\mathcal{R}_{2}}}...\xrightarrow{{\mathcal{R}_{l}}}{\mathcal{F}_{l+1}}$ (abbreviated
as ${\mathcal{F}_{1}}{\mathcal{F}_{2}}...{\mathcal{F}_{l+1}}$), where ${\mathcal{F}_{1}},{\mathcal{F}_{2}},...,{\mathcal{F}_{l+1}}\in \mathcal{F}$ and ${\mathcal{R}_{1}},{\mathcal{R}_{2}},...,{\mathcal{R}_{l}}\in \mathcal{R}$.
A meta-path describes a composite relation between two nodes in a heterogeneous graph. For example, the meta-path Author-Paper-Author (APA) represents that two author nodes have a co-author relationship (Fig.~\ref{toy}(b)).
\section{The Approach}\label{method}
We first briefly overview the HGCL approach and then discuss its major components, including the encoder for attribute-guided view, encoder for topology-guided view, and reciprocal contrastive optimization.
\subsection{Overview}
HGCL is a self-supervised contrastive learning approach for heterogeneous graphs. It adopts two contrastive views guided by node attributes and graph topologies and a contrastive sample selection mechanism to integrate and enhance the two views to learn rich node representation (Fig.~\ref{fig:datu}). It contains three components: an encoder for attribute-guided view, an encoder for topology-guided view, and a reciprocal contrastive optimization module. In the encoder for the attribute-guided view, we use the similarities between node attributes to regenerate the graph structure (homogeneous and heterogeneous edges) and use two encoding methods to aggregate messages across the nodes of the same and different types to learn node representation. In the encoder for the topology-guided view, we take the pre-defined structural property meta-path as prior knowledge, and use attention mechanism to aggregate messages across the same and different meta-paths to learn node representation. Finally, in reciprocal contrastive optimization, we simultaneously compute attribute similarity and meta-path-based topological correlation to define positive and negative samples, and optimize the proposed model by maximizing the agreement between the representations of the positive sample nodes.
\subsection{Encoder for Attribute-guided View}
Due to the interference of noise in the heterogeneous graph, propagating and aggregating attributes by the guidance of topology alone may be sub-optimal. To maximize the effects of diverse node attributes, an idea is to reconstruct the graph topologies using node attributes. To this end, two key issues need to be addressed, i.e., how to use node attributes to accurately build edges and how to make comprehensive use of different types of node attributes. In this section, we first use the similarity of node attributes to regenerate type-specific homogeneous edges for each type of nodes. We then employ a graph neural network (GNN) to obtain the initial representations of different types of nodes with corresponding homogeneous topology and attributes as input. Then, we use the obtained type-specific initial node representations to calculate the similarity between different types of nodes to regenerate heterogeneous edges. Finally, we adopt the attention mechanism~\cite{gat} to aggregate the information of different types of nodes to obtain the final node representations $Z_{attr}$ from the attribute-guided view .
\subsubsection{Generation of Type-specific Homogeneous Edges} To generate homogeneous edges of node type $f\in\mathcal{F}$, we first calculate the similarity matrix $S^f$ using node attribute matrix ${X^f}\in\mathcal{X}$. After condering several choices for the similarity measure (including Jaccard Similarity, Cosine Similarity and Gaussian Kernel), we adopt Cosine Similarity which computes the cosine function of the angle between two vectors to quantify their similarity since they have similar results. Given a pair of nodes $v_i^f$ and $v_j^f$ with their corresponding attribute vectors $x_i^f$ and $x_j^f$ respectively, the similarity ${s}_{ij}^f$ of the two nodes is defined as
\begin{equation}\label{sim}
s_{ij}^f = \frac{x_{i}^{f}}{{{\left\| x_{i}^{f} \right\|}_{2}}}\cdot \frac{x_{j}^{f}}{{{\left\| x_{j}^{f} \right\|}_{2}}},
\end{equation}
where the operation $\cdot$ is the dot product and ${\left\| \cdot \right\|}_{2}$ is the L2-norm. $s_{ij}^f$ is the $(i,j)$ element in the similarity matrix ${S}^f$. Considering that the underlying graph structure is sparse, we mask off (i.e., set to zero) those elements in ${S}^f$ which are smaller than a small non-negative threshold $\epsilon^{f}_{ho}$. With ${S}^f$, we choose the node pairs with non-zero similarity for each node to set edges. Then, type-specific homogeneous graph $G^f=(A^f,X^f)$ for node type $f$ can be obtained, where $A^f$ is the adjacency matrix.
\subsubsection{Message Aggregation of Type-specific Nodes} For each type of nodes, we have a corresponding type-specific homogeneous (sub)graph. We employ mean-based aggregator of GraphSAGE~\cite{sage} to aggregate messages on the graph to derive type-specific node representations. Given the graph $G^f=(A^f,X^f)$ for node type $f$, the representations of the $f$-th type of nodes can be expressed as
\begin{equation}\label{eq:att-view-gnn}
H^f = \operatorname{GraphSAGE}\left( A^f,X^f \right).
\end{equation}
After type-specific message aggregation, we can obtain $|\mathcal{F}|$ groups of node representations corresponding to $|\mathcal{F}|$ node types, denoted as $\{H^1,H^2,...,H^{|\mathcal{F}|}\}$.
\subsubsection{Generation of Different Types of Heterogeneous Edges} After fully acquiring the information of homogeneous neighbors, we further consider the information gain brought by heterogeneous neighbors. Therefore, we need to establish connections between different types of nodes. Here we use the node representation $H^{\mathcal{F}}$ obtained by the above process to calculate the similarity between different types of nodes to regenerate heterogeneous edges. We also use Cosine Similarity for similarity calculation in Eq.(\ref{sim}). Considering that different types of nodes have different feature spaces, we use the type-specific transformation matrix $W_f$ to project the features of different types of nodes into the same feature space, and then calculate the similarity
\begin{equation}\label{cos_he}
s_{ij}^{f_{i}f_{j}} = \operatorname{cos}(W_{f_{i}}h_{i}^{f_{i}}, W_{f_{j}}h_{j}^{f_{j}}).
\end{equation}
We also choose the heterogeneous node pairs whose similarities are greater than the threshold $\epsilon^{r}_{he}$ for each node to set edges, where $r$ is the type of heterogeneous edges. Also of note, in heterogeneous graph analysis there is typically only one type of nodes (target type of nodes) analyzed for downstream tasks, so we only regenerate heterogeneous edges between the target type of nodes and all other types of nodes.
\subsubsection{Message Aggregation between Different Types of Nodes} Through the above process, we obtain $|\mathcal{F}-1|$ kinds of heterogeneous neighbors for the target type of nodes. In this work, we apply the attention mechanism~\cite{gat} to calculate the importance relationships between the target type of nodes and other types of nodes, and then perform weighted message aggregation. Given the representation $h^t_i$ of a node $v_i$ in the target type $t$, and the representation $h^f_j$ of node $j$ in another type $f$ ($f\in{\mathcal{F},f\neq t}$), the importance coefficient between $h^t_i$ and $h^f_j$ can be formulated as
\begin{equation}
e_{i,j}^{t,f} = \sigma \left({\left(h^t_i\right)}^{\mathrm{T}}Wh^f_j\right),
\end{equation}
where $\sigma(\cdot)$ is an activation function, and $W$ the weight matrix.
After obtaining all the importance coefficients, we normalize them via softmax function to get the final coefficient
\begin{equation}\label{eq:att-view-attention}
\alpha_{i,j}^{t,f} = \operatorname{softmax}\left(e_{i,j}^{t,f}\right) = {\frac{\operatorname{exp}\left(e_{i,j}^{t,f}\right)} {\sum_{r \in N^f_i} \operatorname{exp}\left(e_{i,r}^{t,f}\right)}},
\end{equation}
where $N^f_i$ is the set of $f$-th type of neighbor nodes for $v_i$. Then, the $f$-th type information-based representation of target node $v_i$ can be formulated as
\begin{equation}\label{eq:att-view-agg}
z_{i}^{f}=\sigma \left( \sum\nolimits_{j\in N_{i}^{f}}{\alpha _{i,j}^{t,f}\cdot }h_{j}^{f} \right).
\end{equation}
For nodes of the $t$-th type, we can obtain $|\mathcal{F}| - 1$ groups of node representations by Eq.(\ref{eq:att-view-agg}), i.e., $\{Z^f\ |\ f\in\mathcal{F},\ f\neq t\}$, as well as $H^t$ by Eq.(\ref{eq:att-view-gnn}). We denote $H^t$ as $Z^t$ so that there are $|\mathcal{F}|$ groups of $t$-th type of node representations $\{Z^1, Z^2, ..., Z^{|\mathcal{F}|}\}$. Several methods can be used to generate final attribute-guided node representations, including the mean pooling and max pooling. In this work, we use a parameterized attention vector to calculate the normalized importance coefficient
\begin{equation}\label{eq:att-view-semantic-attention}
{{\beta }_{f}}=\frac{\exp \left({\frac{1}{\left| {{\mathcal{V}}_{f}} \right|}\sum\limits_{{{v}_{i}}\in {{\mathcal{V}}_{f}}}{{{q}^{T}_{f}}\cdot \sigma \left( {W}'\cdot z_{i}^{f}+{b}' \right)}}\right)}{\sum\nolimits_{f\in \mathcal{F}}{\exp \left({\frac{1}{\left| {{\mathcal{V}}_{f}} \right|}\sum\limits_{{{v}_{i}}\in {{\mathcal{V}}_{f}}}{{{q}^{T}_{f}}\cdot \sigma \left( {W}'\cdot z_{i}^{f}+{b}' \right)}}\right)}},
\end{equation}
where $W'$ and $b'$ are learnable parameter matrix and bias vector, $\sigma(\cdot)$ the activation function, $q_f$ the attention vector of $f$-th type of nodes, and $\mathcal{V}_f$ the set of $f$-th type of nodes. After getting the normalized importance coefficient ${\beta_\mathcal{F}}$, the final node representation of the attribute-guided view $Z^t_{attr}$ is given as
\begin{equation}
{{Z}_{attr}^t}=\sum\nolimits_{f\in \mathcal{F}}{{{\beta }_{f}}\cdot {{Z}^{f}}}.
\end{equation}
\subsection{Encoder for Topology-guided View}
As an important structural property of heterogeneous graphs, meta-paths play an indispensable role in modeling different semantic relationships in heterogeneous graphs. To learn a good node representation under the guidance of topology, the topology-guided encoder performs a hierarchical message aggregation process~\cite{han}, including message aggregation within the same meta-path and message aggregation between different meta-paths.
\subsubsection{Message Aggregation within the Same Meta-path} Given a meta-path $\mathcal{M}^p$, many node pairs may be connected by $\mathcal{M}^p$. For the target node $v_i$ and its meta-path $\mathcal{M}^p$ based neighbor nodes $N^{\mathcal{M}^p}_i$, we adopt a message aggregation process based on the attention mechanism and the weight coefficient between nodes $v_i$ and $v_j$ on $\mathcal{M}^p$ is formalized as
\begin{equation}\label{eq:nodeWeight}
\gamma _{ij}^{{{\mathcal{M}}^{p}}}=\frac{\exp \left( \sigma \left( \text{s}_{{{\mathcal{M}}^{p}}}^{T}\cdot [{{h}_{i}}||{{h}_{j}}] \right) \right)}{\sum\nolimits_{k\in N_{i}^{{{\mathcal{M}}^{q}}}}{\exp \left( \sigma \left( \text{s}_{{{\mathcal{M}}^{p}}}^{T}\cdot [{{h}_{i}}||{{h}_{k}}] \right) \right)}},
\end{equation}
where $\sigma$ is the activation function, $s_{{\mathcal{M}}^{p}}$ the learnable parameter vector specific to meta-path ${\mathcal{M}}^{p}$, and $||$ the concatenate operation. The representation of target node $v_i$ under meta-path $\mathcal{M}^q$ can be aggregated by the neighbor's representations with the corresponding coefficients as follows
\begin{equation}\label{eq:intraGroupAgg}
h_{i}^{{{\mathcal{M}}^{p}}}=\sigma \left( \sum\nolimits_{j\in N_{i}^{{{\mathcal{M}}^{p}}}}{\gamma _{ij}^{{{M}^{p}}}{{h}_{j}}} \right).
\end{equation}
\subsubsection{Message Aggregation between Different Meta-paths} In a heterogeneous graph, there normally exist multiple meta-paths $\{\mathcal{M}^1, \mathcal{M}^2, ..., \mathcal{M}^P\}$. Correspondingly, we can obtain $P$ groups of representations $\{H^{\mathcal{M}^1},H^{\mathcal{M}^2},...,H^{\mathcal{M}^p}\}$ by message aggregation within the same meta-path. We then also use the attention mechanism to combine semantic messages from different meta-paths to obtain the final representations $Z_{topo}$. The weighted aggregation between $\{\mathcal{M}^1, \mathcal{M}^2, ..., \mathcal{M}^P\}$ can then be formulated as
\begin{equation}
{{Z}_{topo}}=\sum\nolimits_{p=1}^{P}{{{\eta }^{{{\mathcal{M}}^{p}}}}\cdot {{H}^{{{\mathcal{M}}^{p}}}}}.
\end{equation}
where $\eta^{\mathcal{M}^p}$ is the weight coefficient of meta-path $\mathcal{M}^p$, which can be computed in the same way as Eq.(\ref{eq:att-view-semantic-attention}).
\subsection{Reciprocal Contrastive Optimization} After having the two view encoders on respective guidance of attributes and topology, we can compute node representations from these two views respectively. To learn the final rich node representation, we introduce a reciprocal contrastive mechanism to integrate and enhance the two views, which includes the definition of positive/negative samples and loss function for the contrastive optimization of the model.
\subsubsection{The Definition of Positive and Negative Sample Pairs} For computing contrastive loss, we define the positive and negative samples in the heterogeneous graph. We first use the representations of the same node under the above two views as a pair of positive samples like the existing methods~\cite{grace}. To improve the ability of self-supervised learning, we consider attribute similarity and topological correlation at the same time to expand positive samples, while ensuring the high quality of samples. We calculate the attribute similarity (Eq.(\ref{sim})) of the target type of nodes, and choose each node pair $<v_i , v_j>$ whose attribute similarity $s_{ij}$ is greater than a preset hyper-parameter threshold $\epsilon_{a}$ as the positive sample candidates. Then, we further determine the positive sample pairs by computing the correlation between the target node and neighbors based on the meta-path. Since different meta-paths in a heterogeneous graph often play different roles, the correlation is determined by the number of meta-paths between the target node and its neighbors on the meta-paths and further assign importance coefficients to meta-paths.
To be specific, given a target node $v_i$ and a meta-path-based node $v_j$, the correlation function ${\mathbb{T}}_i(\cdot)$ is defined as
\begin{equation}
{{\mathbb{T}}_{i}}(v_j)=\sum\nolimits_{p=1}^{P}{{{\delta }^{{{\mathcal{M}}^{p}}}}\cdot {\mathbbm{1}}\left( {{v}_{j}}\in N_{i}^{{{\mathcal{M}}^{p}}} \right)},
\end{equation}
where ${\mathbbm{1}}(\cdot)$ is the indicator function with the value of $0$ or $1$, ${\delta }_{\mathcal{M}^p}$ the importance coefficient of meta-path ${\mathcal{M}^p}$, and ${N^{\mathcal{M}^p}_i}$ the set of neighbors of node $v_i$ on meta-path ${\mathcal{M}^p}$. We choose a node pair $<v_i , v_j>$ whose topological correlation ${\mathbb{T}}_i(v_j)$ is greater than the preset hyper-parameter threshold $\epsilon_{t}$ as the positive sample candidates. Finally, the overall positive sample set of the target node $v_i$ need to satisfy both attribute similarity and topological correlation, defined as follows
\begin{equation}\label{eq:pp}
\mathbb{P}(i)=\{{{v}_{j}}\ |\ s_{ij}\ge \epsilon_{a} \cap {{\mathbb{T}}_{i}}({{v}_{j}})\ge \epsilon_{t},\ i\ne j\}.
\end{equation}
Similarly, the overall negative sample set of $v_i$ is
\begin{equation}
\mathbb{N}(i)=\{{{v}_{j}}\ |\ s_{ij}< \epsilon_{a} \cup {{\mathbb{T}}_{i}}({{v}_{j}})< \epsilon_{t},\ i\ne j\}.
\end{equation}
\subsubsection{Contrastive Loss} With the positive sample set $\mathbb{P}(i)$ and negative sample set $\mathbb{N}(i)$ obtained above, we optimize the model by maximizing the agreement between the representations of the positive sample nodes~\cite{grace}. The contrastive loss function for each view can be formulated as
\begin{equation}\label{eq:contrastiveInOneView}
\psi (Z,{Z}')=\frac{1}{|\mathcal{V}|}\sum\limits_{{{v}_{i}}\in \mathcal{V}}{-\log \frac{\sum\limits_{{{v}_{j}}\in \mathbb{P}(i)}{{{e}^{\left\langle {{z}_{i}},{{z}_{j}} \right\rangle }}}+\sum\limits_{{{v}_{j}}\in \mathbb{P}(i)\cup {{v}_{i}}}{{{e}^{\left\langle {{z}_{i}},z_j' \right\rangle }}}}{\sum\nolimits_{{{v}_{j}}\in \mathbb{P}(i)\cup \mathbb{N}(i)}{{{e}^{\left\langle {{z}_{i}},{{z}_{j}} \right\rangle }}+{{e}^{\left\langle {{z}_{i}},z_j' \right\rangle }}}}},
\end{equation}
where $\left\langle {{z}_{i}},{{z}_{i}'} \right\rangle =\cos ({{z}_{i}},{{z}_{i}'})/\tau$ and $\tau$ a temperature parameter. The overall loss function is then defined as the weighted average of the losses of two views, formally given by
\begin{equation}\label{eq:finalLoss}
{{\mathcal{L}}_{final}}=\lambda\cdot {\psi ({Z_{topo}},{Z_{attr}})}+(1-\lambda)\cdot{\psi ({Z_{attr}},{Z_{topo}})},
\end{equation}
where $\lambda$ is a weighted coefficient to balance these two parts. To this end, we use the concatenation of $Z_{attr}$ and $Z_{topo}$ to perform downstream tasks.
\section{Experiments}\label{shiyan}
We first discuss the experimental setup, including datasets, baseline methods, and detailed experimental settings. We then present the comparison results on node classification and node clustering. We discuss additional experiments to demonstrate the robustness and generality of the new HGCL approach and consider a parameter analysis.
\subsection{Experimental Setup}
\paragraph{Datasets}
To analyze the effectiveness of HGCL, we performed a comprehensive experimental analysis using three widely-used heterogeneous graph datasets (Table~\ref{tab:datasets}).
\begin{enumerate}
\item ACM\footnote{http://dl.acm.org/}~\cite{han}: We extracted a subset of ACM for 4019 papers, 7167 authors, and 60 subjects. We conducted experimental analysis on paper nodes in ACM dataset. The papers were labeled according to their fields.
\item Yelp\footnote{https://www.yelp.com/dataset}~\cite{yelp}: We extracted a subset from Yelp Open Dataset containing 2614 businesses, 1286 users, 4 services, and 9 rating levels. We conducted experimental analysis on business nodes in Yelp dataset. The business nodes were labeled by their categories.
\item DBLP\footnote{https://dblp.uni-trier.de}~\cite{dblp}: We extracted a subset of DBLP containing information of 4057 authors, 14328 papers, 8789 terms, and 20 venues. We conducted experimental analysis on author nodes in DBLP dataset. Authors were divided into four research areas.
\end{enumerate}
\paragraph{Baseline Methods}
We compared the new HGCL approach with nine state-of-the-art embedding methods. These methods include three semi-supervised methods (GAT~\cite{gat}, HAN~\cite{han}, and MAGNN~\cite{magnn}) and five unsupervised methods (Mp2vec~\cite{metapath2vec}, DGI~\cite{dgi}, GMI~\cite{gmi} DMGI~\cite{dmgi}, and HeCo~\cite{heco}) and can be divided into four methods for homogeneous graphs (GAT, DGI, GMI, and GAE) and five methods for heterogeneous graphs (HAN, MAGNN, Mp2vec, DMGI, and HeCo). Among these methods, HeCo is the only contrastive method for heterogeneous graphs as far as we know.
\begin{table}[t]
\centering
\caption{Statistics of datasets}\label{tab:datasets}
\footnotesize
\resizebox{0.91\linewidth}{!}{
\begin{tabular}{|c|l|l|l|l|}
\hline
Datasets & \multicolumn{1}{c|}{Nodes} & \multicolumn{1}{c|}{Edges} & \multicolumn{1}{c|}{Meta-paths} \\ \hline
ACM & \begin{tabular}[c]{@{}l@{}}Paper(P):4019\\ Author(A):7167\\ Subject(S):60\end{tabular} & \begin{tabular}[c]{@{}l@{}}P-P:9615\\ P-A:13407\\ P-S:4019\end{tabular} & \begin{tabular}[c]{@{}l@{}}PAP\\ PSP\end{tabular} \\ \hline
Yelp & \begin{tabular}[c]{@{}l@{}}Business(B):2614\\ User(U):1286\\ Service(S):4\\ Level(L):9\end{tabular} & \begin{tabular}[c]{@{}l@{}}B-U:30838\\ B-S:2614\\ B-L:2614\end{tabular} & \begin{tabular}[c]{@{}l@{}}BUB\\ BSB\\ BLB\end{tabular} \\ \hline
DBLP & \begin{tabular}[c]{@{}l@{}}Author(A):4057\\ Paper(P):14328\\ Term(T):8789\\ Venue(V):20\end{tabular} & \begin{tabular}[c]{@{}l@{}}A-P:19645\\ P-T:85810\\ P-V:14328\end{tabular} & \begin{tabular}[c]{@{}l@{}}APA\\ APTPA\\ APVPA\end{tabular} \\ \hline
\end{tabular}
}
\end{table}
\paragraph{Detailed Settings}
For a fair comparison of all methods, the embedding dimensions evaluated were set to 64. For semi-supervised methods (GAT, HAN and MAGNN), the labeled nodes were divided into training, validation, and testing sets in the ratio of 10\%, 10\%, and 80\% as done in existing works. For homogeneous methods (GAT, DGI, GMI and GAE), we tested all their meta-paths and report here the best performance of all methods compared.
For our HGCL approach, we use cross validation to set the parameters (details are provided in Table~\ref{tab:cs}) and adopt the Adam optimizer with a learning rate of 0.0005. In Table~\ref{tab:cs}, $\lambda$ is the loss weighted coefficient, $\epsilon_{ho}$ the attribute similarity threshold for regenerated homogeneous edges, $\epsilon_{he}$ the attribute similarity threshold for regenerated heterogeneous edges, $\epsilon_{a}$ the attribute similarity threshold for sample selection, $\delta $ the weight coefficient for meta-path, $\epsilon_{t}$ the topological correlation threshold for sample selection and $\tau$ the temperature parameter.
\subsection{Node Classification}
Node classification is a traditional task for the evaluation of the quality of learned node representations. After learning node representations, we adopted a linear support vector machine (SVM)~\cite{svm} classifier to classify nodes. Because the labels for the training and validation sets have been used in semi-supervised methods, to make a fair comparison, we only classified the nodes in test set in each dataset. We fed node representations to the SVM classifier with varying training ratios from 20\% to 80\%. We repeated experiments 10 times and report here the Macro-F1 and Micro-F1, which are commonly used evaluation metrics.
The results are shown in Table~\ref{tab:classification}, where the best results are in bold fonts and the second-best results are underlined. The new HGCL approach outperformed baseline methods compared under different training ratios, including three semi-supervised methods (GAT, HAN, and MAGNN) and the only contrastive learning method which contrasts the design view based on the structural properties alone (HeCo), in most cases considered. In particular, the proposeed HGCL outperformed HeCo by 15.40\% in Macro-f1 and 8.67\% in Micro-f1 on average. The superior performance of the new approach may be mainly due to the design of both attribute- and topology-guided views and the use of the high-quality sample selection mechanism to achieve mutual supervision and mutual enhancement of the two views, which thus makes it robust to noisy graphs. All GNN-based baseline methods employ a single fusion mechanism in which attributes are propagated and aggregated under the guidance of topology, causing noise in and disparity of attributes and topology to affect model performance.
\begin{table*}[t]
\caption{Performance evaluation of node classification on three datasets}\label{tab:classification}
\resizebox{\linewidth}{!}{
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Datasets} & \multirow{2}{*}{Metrics} & \multirow{2}{*}{Training} & \multicolumn{3}{c}{Semi-supervised} & \multicolumn{5}{c}{Unsupervised} \\ \cmidrule(r){4-6} \cmidrule(r){7-11}
& & & GAT & HAN & MAGNN & Mp2vec & DGI & DMGI & HeCo & HGCL \\ \hline
\multirow{8}{*}{ACM} & \multirow{4}{*}{Macro-F1} & 20\% & 0.8966 & 0.9062 & 0.8693 & 0.7011 & 0.9041 &\textbf{0.9222} & 0.8637 & \ul{ 0.9177} \\
& & 40\% & 0.8975 & 0.9102 & 0.8889 & 0.7043 & 0.9042 & \ul{0.9251} & 0.8784 & \textbf{0.9253} \\
& & 60\% & 0.8993 & 0.9128 & 0.8985 & 0.7073 & 0.9062 & \ul{0.9278} & 0.8863 & \textbf{0.9300} \\
& & 80\% & 0.8960 & 0.9150 & 0.9064 & 0.7113 & 0.9055 & \ul{0.9256} &0.8937 & \textbf{0.9328} \\ \cline{2-11}
& \multirow{4}{*}{Micro-F1} & 20\% & 0.8955 & 0.9056 & 0.8703 & 0.7444 & 0.9033 & \textbf{0.9207} &0.8696 & \ul{ 0.9163} \\
& & 40\% & 0.8968 & 0.9099 & 0.8895 & 0.7480 & 0.9034 & \ul{0.9236} &0.8807 & \textbf{0.9237} \\
& & 60\% & 0.8984 & 0.9123 & 0.8985 & 0.7522 & 0.9051 & \ul{0.9260} &0.8875 & \textbf{0.9290} \\
& & 80\% & 0.8950 & 0.9142 & 0.9061 & 0.7557 & 0.9044 & \ul{0.9238} &0.8949 & \textbf{0.9318} \\ \hline
\multirow{8}{*}{Yelp} & \multirow{4}{*}{Macro-F1} & 20\% & 0.5407 & 0.7724 & \ul{0.8676} & 0.5396 & 0.5407 & 0.7273 &0.5395 & \textbf{0.9133} \\
& & 40\% & 0.5407 & 0.7848 & \ul{ 0.8871} & 0.5400 & 0.5407 & 0.7381 &0.5399 & \textbf{0.9258} \\
& & 60\% & 0.5400 & 0.7858 & \ul{ 0.9018} & 0.5396 & 0.5400 & 0.7448 & 0.5396 & \textbf{ 0.9319} \\
& & 80\% & 0.5381 & 0.7893 & \ul{ 0.8991} & 0.5370 & 0.5381 & 0.7541 & 0.5372 & \textbf{0.9294} \\ \cline{2-11}
& \multirow{4}{*}{Micro-F1} & 20\% & 0.7306 & 0.7885 & \ul{ 0.8711} & 0.7289 & 0.7306 & 0.7833 & 0.7289 & \textbf{0.9053} \\
& & 40\% & 0.7314 & 0.7992 & \ul{ 0.8887} & 0.7295 & 0.7314 & 0.7893 & 0.7298 & \textbf{0.9189} \\
& & 60\% & 0.7296 & 0.7997 & \ul{ 0.9034} & 0.7297 & 0.7296 & 0.7934 & 0.7297 & \textbf{0.9258} \\
& & 80\% & 0.7281 & 0.8041 & \ul{ 0.9008} & 0.7278 & 0.7281 & 0.8000 & 0.7280 & \textbf{0.9246} \\ \hline
\multirow{8}{*}{DBLP} & \multirow{4}{*}{Macro-F1} & 20\% & 0.9040 & 0.9221 & \ul{ 0.9381} & 0.7666 & 0.8851 & 0.9290 & 0.9041 & \textbf{0.9401} \\
& & 40\% & 0.9061 & 0.9244 & \ul{ 0.9391} & 0.8214 & 0.8849 & 0.9296 & 0.9111 & \textbf{0.9420} \\
& & 60\% & 0.9073 & 0.9251 & \ul{ 0.9394} & 0.8425 & 0. 8861 & 0.9317 & 0.9158 & \textbf{0.9429} \\
& & 80\% & 0.9088 & 0.9271 & \ul{ 0.9417} & 0.8420 & 0.8856 & 0.9333 & 0.9151 & \textbf{0.9468} \\ \cline{2-11}
& \multirow{4}{*}{Micro-F1} & 20\% & 0.9105 & 0.9269 & \ul{ 0.9420} & 0.7761 & 0.8926 & 0.9339 & 0.9112 & \textbf{0.9446} \\
& & 40\% & 0.9126 & 0.9290 & \ul{ 0.9428} & 0.8289 & 0.8920 & 0.9344 & 0.9179 & \textbf{0.9460} \\
& & 60\% & 0.9135 & 0.9300 & \ul{ 0.9432} & 0.8502 & 0.8934 & 0.9364 & 0.9224 & \textbf{0.9470} \\
& & 80\% & 0.9148 & 0.9318 & \ul{ 0.9453} & 0.8495 & 0.8927 & 0.9378 & 0.9213 & \textbf{0.9503} \\ \hline
\end{tabular}
}
\end{table*}
\begin{center}
\begin{table*}[t]
\centering
\caption{Performance evaluation of node clustering on three datasets}\label{tab:clustering}
\footnotesize
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{cclccccccc}
\hline
Dataset & \multicolumn{2}{c}{Metrics} & Mp2vec & GAE & DGI & GMI & DMGI & HeCo & HGCL \\ \hline
\multirow{2}{*}{ACM} & \multicolumn{2}{c}{NMI} & 0.3765 & 0.4059 & 0.5183 & 0.3763 & 0.6065 & \ul{0.6316} & \textbf{0.6746} \\
& \multicolumn{2}{c}{ARI} & 0.3025 & 0.3319 & 0.4374 & 0.3022 & 0.5827 & \ul{0.6649} & \textbf{0.7125} \\ \hline
\multirow{2}{*}{Yelp} & \multicolumn{2}{c}{NMI} & 0.3890 & 0.3919 & \ul{ 0.3942} & \ul{0.3942} & 0.3690 &\ul{0.3942} & \textbf{ 0.4250} \\
& \multicolumn{2}{c}{ARI} & 0.4249 & 0.4257 & \ul{ 0.4262} & 0.4260 & 0.3346 & \ul{0.4262} & \textbf{0.4452} \\ \hline
\multirow{2}{*}{DBLP} & \multicolumn{2}{c}{NMI} & 0.5451 & 0.5257 & 0.6637 & 0.4101 & \ul{0.7388} & 0.6942 & \textbf{ 0.8031} \\
& \multicolumn{2}{c}{ARI} & 0.5815 & 0.4986 & 0.6838 & 0.4056 & \ul{ 0.7929} & 0.7483 & \textbf{0.8496} \\ \hline
\end{tabular}
}
\end{table*}
\end{center}
\vspace{-20pt}
\subsection{Node Clustering}
Our unsupervised HGCL approach is particularly suitable for this unsupervised task. For comparison, we chose six unsupervised methods (Mp2vec, GAE, DGI, GMI, DMGI, and HeCo) as baselines but excluded semi-supervised methods since they use the labels of some of the training data. We applied the $k$-Means algorithm 10 times to the learned node representations. We report the average normalized mutual information (NMI) and adjusted rand index (ARI) for comparison.
The experimental results are presented in Table~\ref{tab:clustering}. As shown, HGCL outperformed all baseline methods in terms of both NMI and ARI. For example, the proposeed HGCL outperformed the best baseline method by 4.60\% in NMI and 4.11\% in ARI on average. The superior performance of HGCL in node clustering over the state-of-the-art methods further demonstrates the effectiveness of our new method.
\begin{figure}[h]
\centering
\subfloat[Edge deletion]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/tuopu.eps}
\end{minipage}
}
\subfloat[Attribute masking]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/shuxing.eps}
\end{minipage}
}
\caption{The average result of node classification under different training ratios. We report Macro-F1 on ACM dataset as a case study.}\label{lubang}
\end{figure}
\subsection{Robustness Verification}
To evaluate the robustness of HGCL, we construct original graphs
with random edge deletion or random attribute masking. Specifically, for each pair of nodes in the original graph, we randomly remove (if an edge exists) an edge with a probability 25\%, 50\% or 75\%~\cite{dele}. For random attribute masking, we randomly mask a fraction of dimensions with zeros in node initial features, and set the masking ratio to 25\%, 50\% and 75\% for experiments respectively. As shown in Fig.~\ref{lubang}, compared with the baseline HeCo, which designs contrastive mechanism based solely on pre-defined structural properties in the graph, HGCL achieves better and more stable results in both scenarios. While HeCo has large fluctuations when the edge deletion probability increases, and fails completely with increasing attribute masking ratio, HGCL performs reasonably well. We think that this is because the proposed contrastive mechanism of mining key information in attributes and topology separately and integrating and enhancing them with each other reduces the interference of noise to the model and greatly improves the robustness.
\subsection{Ablation Study}
To gain deeper insights into the contributions of the two views used in our approach, we designed two variants of proposed HGCL, i.e., HGCL\_attr and HGCL\_topo. In HGCL\_attr, nodes are encoded only in attribute-guided view; for HGCL\_topo, nodes are only encoded in topology-guided view. We compared these two variants with the complete HGCL for the task of node classification as an example. As shown in Fig.~\ref{fig:lubang}, HGCL performed consistently better than its two variants on all three datasets. In addition, HGCL\_attr and HGCL\_topo exhibited different strengths on different datasets, suggesting the need to maximize and integrate their advantages to learn better node representations.
\begin{figure}[t]
\centering
\subfloat
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/xr_ma.eps}
\end{minipage}
}
\subfloat
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/xr_mi.eps}
\end{minipage}
}
\caption{The average result of node classification under different training ratios. We report Macro-F1 (on the left) and Micro-F1 (on the right ) on three dataset as the ablation study.}\label{fig:lubang}
\end{figure}
\begin{figure}[h]
\centering
\tiny
\subfloat[ACM]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{figure/L-ACM1.eps}
\end{minipage}
}
\subfloat[DBLP]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{figure/L-DBLP.eps}
\end{minipage}
}
\caption{Parameter sensitivity analysis of $\lambda$. We report the average result of the node classification task under different training ratios on the ACM and DBLP datasets.}\label{fig:lamuda}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.83\linewidth]{figure/mp1.eps}
\caption{Parameter sensitivity analysis of $\delta$ on ACM dataset. Shown are the average result of node classification under different training ratios. A warmer color denotes a higher accuracy. }\label{fig:metapath}
\end{figure}
\begin{figure}[h]
\centering
\subfloat[Yelp-$\epsilon^{B}_{ho}$]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/sim_ho1.eps}
\end{minipage}
}
\subfloat[Yelp-$\epsilon_{a}$]
{
\begin{minipage}[t]{0.24\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{figure/sim_ps1.eps}
\end{minipage}
}
\caption{Parameter sensitivity analysis of $\epsilon$. We report the average result of the node classification task under different training ratios on the Yelp dataset.}\label{fig:sim}
\end{figure}
\subsection{Parameter Analysis}
We investigated the sensitivity of three key hyperparameters: the weighted coefficient $\lambda$ of the loss function, the coefficient $\delta$ of meta-path and the attribute similarity threshold $\epsilon$. We report the average result of node classification with different training ratios on these three datasets.
The performance of the model first increases and then decreases with increase of $\lambda$, and different datasets correspond to different best $\lambda$, suggesting the need of mining information from attributes and topology respectively. We chose "PAP" and "PSP" used in the ACM dataset to analyze $\delta$. HGCL achieved a good performance when $0.1\le {{\delta }_{PAP}}\le 0.3$, and was also not so sensitive to ${{\delta }_{PSP}}$ (Fig.~\ref{fig:metapath}). This is reasonable since papers shared by collaborators tend to belong to the same field, and thus PAP is more important than PSP. Therefore, the well-designed meta-paths can help select high-quality positive samples to further improve the performance of HGCL. Note that we still needed to use a suitable value of ${{\delta }_{PSP}}$, because we set the threshold in Eq.(\ref{eq:pp}) as a constant, and ${{\delta }_{PAP}}$ and ${{\delta }_{PSP}}$ are positively correlated with $\epsilon_{t}$. To assess the impact of the attribute similarity threshold $\epsilon$ (taking $\epsilon^{B}_{ho}$ and $\epsilon_{a}$ in Yelp as an example) for regenerating edges and selecting contrastive samples, we study the model performance with various $\epsilon$ from 0.1 to 0.7 as shown in Fig.~\ref{fig:sim}. The accuracies also increase first and then decrease. It may probably be because that smaller $\epsilon$ may introduce more noisy edges.
\section{Conclusion}\label{sec:conclusion}
We proposed a novel and robust contrastive learning approach, named HGCL, for mining and analyzing heterogeneous graphs. HGCL adopts two contrastive views on the guidance of node attributes and graph topologies respectively and integrates and enhances the two views by reciprocally contrastive mechanism. The attribute- and topology-guided views employed different attribute and topology fusion mechanisms, which fully excavated the information and reduced noise interference in the graph. High-quality samples in reciprocal contrast further improved the discriminative power of the model. Extensive experimental results demonstrated the effectiveness of the new HGCL approach over the state-of-the-art methods.
\begin{center}
\begin{table*}[t]
\centering
\caption{The values of parameter used in HGCL.}\label{tab:cs}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{c|ccccccc}
\toprule
Datasets & $\lambda$ & $\epsilon_{ho}$ & $\epsilon_{he}$ & $\epsilon_{a}$ &$\delta $ & $\epsilon_{t}$ & $\tau$ \\ \midrule[0.6pt]
Yelp & 0.4 & B:0.3;U:0.5;S:0.5;L:0.5 &BU:0.5;BS:0.6;BL:0.5 & 0.6 &BUB:0.7;BSB:0.1;BLB:0.1 & 1.0 & 0.4 \\ \midrule[0.6pt]
ACM & 0.7 & P:0.8;A:0.5;S:0.5 &PA:0.6;PS:0.5 & 0.6 &PAP:0.2;PSP:0.4 & 1.0 & 0.4 \\ \midrule[0.6pt]
DBLP & 0.6 & A:0.6;P:1.0;T:0.7;V0.9 &AP;0.6;AT:0.6;AV:0.5 & 1.0 &APA:0.1;APTPA:0.1;APVPA:0.8 & 1.0 & 0.4 \\ \bottomrule
\end{tabular}
}
\end{table*}
\end{center}
\bibliographystyle{IEEEtran}
|
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Ipod Shuffle 31: Bob Dylan 04: Songs & Albums Quiz
My fourth Dylan quiz, this covers all facets of his recording career, including some rarities.
A multiple-choice quiz by berenlazarus. Estimated time: 7 mins.
Music D-G
berenlazarus
Last 3 plays: Guest 12 (4/10), Guest 92 (0/10), Guest 12 (5/10).
1. Slash plays guitar on what Bob Dylan recording? Hint
10,000 Men
Wiggle Wiggle
Handy Dandy
2. On what album does Dylan sing Latin?
Answer: (Album title)
3. During the Newport Folk Festival of 1965, Dylan plugged in and played a brief fifteen minute set. What song caused Pete Seeger to go back to the mixing board due to the sound, famously wishing he had an ax to cut the cable? Hint
It Takes A Lot To Laugh, It Takes A Train To Cry
Mr. Tambourine Man
4. Dylan has never been a strong singles artist. From the 1960s to the 1990s, What was Dylan's last Top Ten single on the Billboard Top 100?
Answer: (Song title, No Punctuation)
5. Who produced the 1986 recording "Band of the Hand"? Hint
Arthur Baker
Daniel Lanois
Don Was
6. Dylan cowrote two songs with Gerry Goffin, which were released on the 1996 album "Back Room Blood". The fist track was "Time To End This Masquerade". What was the second track? Hint
Sacred Heart of Stone
Tragedy of the Trade
Elysian Fields
Death to the Printed World
7. Dylan's tour with The Hawks (soon to be renamed The Band) from 1966 has long been the stuff of legend. However, for three decades the only official release of a live performance from this tour was a live recording of "Just Like Tom Thumb's Blues", a B-Side to a US single from "Blonde on Blonde". What was the A-Side? Hint
One Of Us Must Know (Sooner Or Later)
Just Like A Woman
Rainy Day Women #12 and 35
8. Dylan made a rare public appearance in 1968 with The Band, playing three Woody Guthrie songs. Which song did he NOT play at this performance? Hint
The Grand Coulee Dam
I Ain't Got No Home
Dear Mrs. Roosevelt
This Land is Your Land
9. On an officially released live album, Dylan, poking fun at his numerous bootleggers, introduces a then new song by saying "Here's an unrecorded song. See if you can guess which one it is". What song is it? Hint
Señor (Tales of Yankee Power)
Is Your Love in Vain
Precious Angel
Slow Train Coming
10. What song from a soundtrack released in 2000 did Dylan win an Oscar for?
Answer: (Song Title)
1. Slash plays guitar on what Bob Dylan recording?
Answer: Wiggle Wiggle
"Under the Red Sky" is rather the oddity in Dylan's catalog. Dylan normally does not have star-studded cameos on his albums, but "Under the Red Sky" is the exception. Slash, David Crosby, Elton John, Steve Ray Vaughan, Bruce Hornsby, George Harrison, and Al Kooper all play on the record.
Slash plays on "Wiggle Wiggle", which is one of Dylan's most notorious songs.
Answer: Christmas in the Heart
2009 was quite the surprising year for the Dylan faithful, getting "Together Through Life" in April 2009. Then, just a few months later, we get a (!) Christmas album called "Christmas in the Heart". The last time Dylan recorded albums so closely together was in 1970 with "Self Portrait" and "New Morning". The Christmas album is entirely comprised with old Christmas hymns, carols, and Holiday standards (though sadly no "Rudolph the Rednose Reindeer", though Dylan did record the Gene Autry track "Here Comes Santa Claus".)
If you ever want to hear Dylan sing (or more appropriately, mangle) Latin, he sings the first verse of ""O' Come All Ye Faithful" (or "Adeste Fideles") in that long dead language.
3. During the Newport Folk Festival of 1965, Dylan plugged in and played a brief fifteen minute set. What song caused Pete Seeger to go back to the mixing board due to the sound, famously wishing he had an ax to cut the cable?
Answer: Maggie's Farm
Dylan sang three opening songs. These were, in order:
1. Maggie's Farm
2. Like A Rolling Stone
3. It Takes A Lot To Laugh, It Takes A Train To Cry
Afterward, Dylan left the stage, and at Peter Yarrow's request, Dylan returned and played acoustic versions of "Mr. Tambourine Man" and "It's All Over Now, Baby Blue".
Pete Seeger said this in 2005: "There are reports of me being anti-him going electric at the '65 Newport Folk festival, but that's wrong. I was the MC that night. He was singing 'Maggie's Farm' and you couldn't understand a word because the mic was distorting his voice. I ran to the mixing desk and said, 'Fix the sound, it's terrible!' The guy said 'No, this is what the young people want.' And I did say that if I had an axe I'd cut the cable! But I wanted to hear the words. I didn't mind him going electric."
Al Kooper said this:
"The reason they booed is because he only played for 15 minutes and everybody else played for 45 minutes to an hour, and he was the headliner of the festival. [...] The fact that he was playing electric...I don't know. The Paul Butterfield Blues Band (who had played earlier) had played electric and the crowd didn't seem too incensed.
In 'Maggie's Farm,' the beat got turned around, so instead of playing and two and four, (drummer) Sam Lay was playing on one and three. That's an accident that can happen, and it did, so it was sort of a disaster."
Answer: Lay Lady Lay
Dylan has only had four top ten singles in the Billboard Top 100 (all in the 1960s), and has never had a Number One track. The singles are "Like a Rolling Stone" which peaked at Number Two, "Positively 4th Street" (Number Seven) "Rainy Day Women #12 & 35" (Number Two), and "Lay Lady Lay" (Number Seven).
His most successful single after the 1960s was the 1973 single "Knockin' on Heaven's Door", which peaked at Number Twelve.
5. Who produced the 1986 recording "Band of the Hand"?
Answer: Tom Petty
One of my personal favorite tracks from the 1980s, "Band of the Hand" is the theme for the 1986 film "Band of the Hand". Dylan was touring with Tom Petty and the Heart Breakers at the time, Dylan's backing band for the song. Petty produced the song. Stevie Nicks of Fleetwood Mac fame contributed background vocals. The single peaked at Number Twenty Eight on the Billboard Top 100.
6. Dylan cowrote two songs with Gerry Goffin, which were released on the 1996 album "Back Room Blood". The fist track was "Time To End This Masquerade". What was the second track?
Answer: Tragedy of the Trade
Dylan cowrote both tracks for Gerry Goffin and provides backup vocals and plays guitar on both tracks. The lyrics of "Time To End This Masquerade" feature the phrase "Back Room Blood", which would become the title of the album.
"Tragedy Of The Trade" was written by Bob Dylan, Gerry Goffin, and Barry Goldberg, while "Time To End This Masquerade" was written by Dylan and Goffin.
7. Dylan's tour with The Hawks (soon to be renamed The Band) from 1966 has long been the stuff of legend. However, for three decades the only official release of a live performance from this tour was a live recording of "Just Like Tom Thumb's Blues", a B-Side to a US single from "Blonde on Blonde". What was the A-Side?
Answer: I Want You
Though Dylan's 1966 tour become legendary, the first time Dylan and Columbia officially released a significant body of work from this tour was in 1998 with the release of the 1966 bootleg "The Bootleg Series 4: The 'Royal Albert Hall' Show". Since then, a smattering of other live recordings have appeared from this era, mostly on The Band's box set "A Musical History".
The 1966 show had long been circulating. There are numerous bootlegs of other shows circulating. The setlists between the shows were largely the same.
8. Dylan made a rare public appearance in 1968 with The Band, playing three Woody Guthrie songs. Which song did he NOT play at this performance?
Answer: This Land is Your Land
After Dylan's motorcycle crash in 1966, Dylan would not launch a major tour for eight years. However, he made a few sporadic appearances, the first of which was a performance with The Band in 1968 at a Woody Guthrie tribute show. Dylan backed with The Band performed these three songs in order:
1. "I Ain't Got No Home"
2. "Dear Mrs. Roosevelt"
3. "The Grand Coulee Dam"
9. On an officially released live album, Dylan, poking fun at his numerous bootleggers, introduces a then new song by saying "Here's an unrecorded song. See if you can guess which one it is". What song is it?
Answer: Is Your Love in Vain
In ten years, Dylan released four official live albums. In 1974 he released the double live set "Before the Flood". In 1976 he released "Hard Rain", which for over two decades was the only official live document of the Rolling Thunder Revue. In 1978 he released the double live set "Bob Dylan At Bodukhan", where he made aforementioned comment before going into "Is Your Love In Vain". If you ever wanted to hear Dylan do "Don't Think Twice" reggae style, here's your chance. Then in 1984 he released "Real Live", famous for including a heavily rewritten version of "Tangled Up In Blue".
Answer: Things Have Changed
While Dylan may have never had a Number One, in 2000 he won an Oscar for the song "Things Have Changed" from the Michael Douglas film "Wonder Boys". Since then, while on tour Dylan always has an Oscar statue sitting on one of the amps near him.
Source: Author berenlazarus
This quiz was reviewed by FunTrivia editor agony before going online.
1. Bob Dylan's Body Parts Easier
2. Bob Dylan Album Match Easier
3. Bob Dylan Quotes Average
4. Bob Dylan Average
5. And The Times They Were A-Changin' - Side One Average
6. Anatomy of a Song: "Sub'n Homesick Blues" Average
7. The Ragged Clown at the Gates of Eden Average
8. Bob Dylan - That's Funny! Average
9. And The Times They Were A-Changin' - Side Two Average
10. Mathematical Dylan Average
11. Characters in Bob Dylan Songs Average
12. Dylan's Characters Average
Music History People Albums Bob Dylan Elton John Rolling Stones 2000s Country Music Fleetwood Mac Guitar Blues Tom Petty 1990s Country Music Stevie Nicks 1980s Country Music George Harrison 1960s Country Music Woody Guthrie 1970s Country Music Live Albums
Explore Other Quizzes by berenlazarus
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Three sayings to characterize China's role in the global pandemic fight
Source:CGTN.COM Editor:Wang Xinjuan 2020-04-21 01:01:15
A team of Chinese medical experts arrive in Phnom Penh to help fight against the COVID-19 pandemic in Cambodia, March 23, 2020. /Xinhua
By Duan Fengyuan
No country can detach itself from such a severe pandemic.
Confirmed COVID-19 cases have exceeded 2.4 million globally, with the death toll surpassing 160,000, according to the latest data by Johns Hopkins University.
To win the human challenge as soon as possible, China and the international community have been working closely since the start of the outbreak. Their cooperation can be characterized by several idioms and proverbs.
Friendship in the blazing fire of a global outbreak
"A friend in need is a friend indeed." The old saying popular among the Chinese people has been demonstrated many times when China was at the most difficult time of its epidemic fight, as countries around the world extended helping hands.
Cambodian Prime Minister Samdech Techo Hun Sen paid a special visit to Beijing in February, which, as Chinese President Xi Jinping put it, demonstrated the unbreakable friendship and mutual trust between the two countries and showcased the essence of building a community of shared future.
A Japanese girl wearing a red Chinese cheongsam bows deeply to passers-by with a donation box in hands to raise money to help those in China affected by the virus on the Chinese Lantern Festival in Tokyo, Japan, February 8, 2020. /Xinhua
Heads of more than 160 countries and international organizations have shown their support through phone calls, messages or letters. For instance, the two heads of state of China and France have held three phone calls to express condolences and support to each other, and there have been many moving moments of cooperation against the virus between the two sides.
Pakistan, South Korea, Russia and many other countries have donated much-needed medical supplies – a total of 62 countries and seven international organizations had pledged epidemic prevention and control supplies to China as of March 2.
"Toss a peach, get back a plum," which means to return a favor. The Chinese government as well as Chinese enterprises and institutions have made every effort within their capability to help those in need overseas as the pandemic spreads globally.
There are four main channels through which China helps other affected countries, namely government-to-government assistance, cooperation on health technology, assistance at regional levels and non-governmental assistance.
A Chinese support team arrives in Belgrade, Serbia to help the country stem the spread of COVID-19, March 21, 2020. /CCTV
China's cooperation with the world in various forms:
- Providing medical supplies to over 140 countries and international organizations
- Donating 20 million U.S. dollars to the World Health Organization
- Sending 20 medical teams to 18 countries as of April 18
- Holding 83 video conferences with experts from 153 countries as of April 12
- Compiling and sharing latest plans and technical documents with over 180 countries and more than 10 international and regional organizations
- Exporting anti-epidemic supplies of gross export value reaching 10.2 billion yuan (about 1.44 billion U.S. dollars) from March 1 to April 4:
Masks: 3.86 billion
Protective gears: 37.52 million
Infrared thermometers: 2.41 million
Ventilators: 16,000
Testing kits: 2.84 million
Protective goggles: 8.41 million
"China's anti-virus external assistance is the most intensive and wide-ranging emergency humanitarian operation since the founding of the People's Republic of China in 1949," said Deng Boqing, deputy head of China International Development Cooperation Agency (CIDCA).
Moreover, China has participated actively in several global meetings to provide the international community with solutions and experience, such as the G20 virtual summit on COVID-19, the summit of the Association of Southeast Asian Nations (ASEAN), China, Japan and South Korea (ASEAN Plus Three or APT) on COVID-19, and a special ASEAN-China Foreign Ministers' meeting on the outbreak.
'Trust needed' in cooperation
Though saving lives is the most urgent and important thing right now for the world, some politicians and media outlets have tried to politicize China's sincere and substantial assistance, alleging "politics of generosity," "mask diplomacy" and "propaganda."
"Never gauge the heart of a gentleman with one's own standard." Another proverb needs to be stressed in the battle against the virus.
The fact is, as the CIDCA said, that China mainly takes the following factors into consideration when determining the anti-epidemic assistance program: the severity of the local outbreak, specific needs for assistance from relevant countries and the capabilities of the Chinese government itself.
"What has happened to the world and how should we respond?" That was the question posed to the world by President Xi at the UN Office at Geneva in 2017. Right now, the world is giving the answer in handling this serious public health crisis.
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{"url":"https:\/\/cvgmt.sns.it\/paper\/3573\/","text":"## A kinetic selection principle for curl-free vector fields of unit norm\n\ncreated by pegon on 06 Sep 2017\n\n[BibTeX]\n\nAccepted Paper\n\nInserted: 6 sep 2017\nLast Updated: 6 sep 2017\n\nJournal: Communications in PDE\nYear: 2017\n\nArXiv: 1701.02931 PDF\n\nAbstract:\n\nThis article is devoted to the generalization of results obtained in 2002 by Jabin, Otto and Perthame. In their article they proved that planar vector fields taking value into the unit sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here, we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of any norm. Under natural assumptions on the norm, namely smoothness and a qualitative convexity property, that is to be of power type $p$, we prove that planar vector fields taking value into the unit sphere of such a norm and satisfying a certain kinetic equation are locally $\\frac{1}{p-1}$-H\u00f6lder continuous. Furthermore we completely describe the behaviour of such a vector field around singular points as a vortex associated to the norm. As our kinetic equation implies for the vector field to be curl-free, this can be seen as a selection principle for curl-free vector fields valued in spheres of general norms which rules out line-like singularities.","date":"2022-12-09 21:35:50","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\": 0.7849581241607666, \"perplexity\": 576.8395012517582}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711552.8\/warc\/CC-MAIN-20221209213503-20221210003503-00563.warc.gz\"}"}
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\section{Introduction}
In recent years model-free valuation approaches for exotic derivatives attracted enormous attention. In such approaches the aim is to determine arbitrage-free price bounds for an exotic, and therefore not liquidly traded, option $\Phi$ while imposing no assumptions on the dynamics or probability distributions of a potential underlying stochastic model of the financial market. Put differently, to price $\Phi$ one allows for all arbitrage-free pricing models and associated pricing measures $\mathbb{Q}$, and computes the extreme prices for $\Phi$ as minimial and maximal expectations $\mathbb{E}_\mathbb{Q}[\Phi]$ among these models, resulting in a range of arbitrage-free prices.
In this way, the model-free pricing approach respects the not-quantifiable \emph{Knightian uncertainty} \cite{knight1921risk} of having chosen a wrong financial model for option valuation, which is particularly important in periods in which financial models calibrated to historical data do not depict the real behaviour of the market appropriately, for instance due to unforeseen financial crises.
However,
a major drawback of the model-free approach evidently is that the resultant range of possible arbitrage-free prices for $\Phi$ turns out to be too large and therefore the usefulness of the original model-free pricing approach in practice is limited, see also \cite{dolinsky2018super, neufeld2018buy}.
To decrease the range of possible arbitrage-free prices, one follows an inverse approach by inferring information from the market and then reducing the set of admissible models to those models that are consistent with the considered information. This market information usually is related to the prices of liquidly traded options (\cite{ brown2001robust, hobson1998robust, hobson2012model,neufeld2020model}), to market believes (\cite{bartl2019duality,cheridito2017duality,hou2018robust,neufeld2021model}), or to marginal distributions that are derived by using the Breeden--Litzenberger result (\cite{breeden1978prices, neufeld2022numerical, talponen2014note}), where the latter case refers to the so called martingale optimal transport problem (\cite{beiglbock2013model,beiglbock2016problem, beiglbock2017complete,cheridito2021martingale, dolinsky2014martingale, guo2019computational,guo2021path,guo2019local, henry2017model,liu2019compactness,tan2013optimal} to name but a few).
In this paper, we combine copula theory and martingale optimal transport to construct improved price bounds for multi-asset derivatives. To this end, we first utilize the well-known relationship between prices of certain derivatives depending on multiple underlying assets at a single time point and expectation operators defined in dependence of copulas and quasi-copulas (see \cite{Ansari-2021, Bernard-2019, Lux-2017,papapantoleon2020detection, Tankov-2011}). If the payoff function of the derivative fulfils certain monotonicity properties ($\Delta$-monotonicity, $\Delta$-antitonicity, or supermodularity), then the extreme expectations can be associated to so called Fr\'{e}chet-bounds (see for instance \cite{Nelsen-2006}) which then represent model-free price bounds for these derivatives. By following the approaches pursued in \cite{bartl2017marginal,Lux-2017,papapantoleon2020detection,Tankov-2011} one can further restrict the class of admissible (quasi-) copulas through the inclusion of additional market information which then leads to improved price bounds for the financial derivative of interest in a single period framework. We extend this single-period model-free pricing approach relying on copula theory to a multi-period setting by connecting it with model-free pricing approaches relying on martingale optimal transport theory.
Our paper contributes to the literature on model-independent pricing in various aspects. First, we show how price information on certain liquid derivatives, that depend on multiple assets, leads to restrictions on possible inter-asset dependencies expressed either in terms of correlations if prices of \emph{basket options} are observable, or through restrictions on the set of admissible copulas if prices of \emph{digital options} are observable. In the latter case the set of admissible copulas can then through Sklar's Theorem directly be associated with admissible pricing measures. These market implied dependence restrictions can be translated into linear equality and inequality constraints specifying the set of admissible pricing measures. To that end, we prove a general model-independent super-hedging duality result which allows to include these additional equality and inequality constraints.
This contribution can be seen in line with the various approaches that were recently established to improve model-free price bounds, see~\cite{eckstein2019martingale, lutkebohmert2019tightening, neufeld2021model, sester2019robust}. For improvements in the multi-asset case we refer to the recent contributions \cite{aquino2019bounds, eckstein2019robust,neufeld2020model, papapantoleon2020detection}. In \cite{neufeld2020model} algorithms were developed to exactly compute price bounds using market implied information in a single-period model. While most of the mentioned approaches yield tighter price bounds mainly through restrictions on admissible pricing measures based on the distributions of single underlying securities, our approach includes restrictions imposed on the inter-asset dependencies.
Second, we utilize the internal factor model approach, in which one assumes that the inter-asset dependencies of all assets with respect to a specific reference asset (this can for example be a stock index) are known or can be derived from price information. In this situation the maximal inter-asset dependencies can no longer be described by copulas and we therefore use the concept of quasi-copulas. Extending \cite[Theorem 1]{Ansari-Rueschendorf-2020} to quasi-copulas and introducing the supermodular ordering on the class of quasi-copulas by an application of a multivariate integration by parts formula (see \cite{Ansari-2021}), we show, in particular, that in this case the price bounds of a broad class of multi-asset derivatives can be computed as analytical expressions in dependence of the limiting quasi-copulas.
Finally, we provide several numerical examples based on simulated and real data to illustrate the significant improvement of price bounds when inter-asset dependencies are taken into account. More specifically, we show in many relevant cases how upper and lower price bounds can be substantially tightened when the set of admissible pricing measures is reduced due to market-implied dependencies.\\
The remainder of the paper is as follows. In Section~\ref{section_main_results} we present the underlying setting and derive an adjusted model-free pricing-hedging duality. Section~\ref{sec dependence modelling} introduces the concept of copulas and quasi-copulas and the most important associated results. In Section~\ref{sec improved price bounds} we explain how we can use price information of traded derivatives to derive restrictions on the set of resulting pricing measures and compatible copulas. In Section~\ref{section_numerics} we provide several examples illustrating how model-free price bounds can be computed within our approach and how existing conventional price bounds can be improved. The proofs of all mathematical statements are provided in Section~\ref{section_proofs}.
\section{Setting and Duality Result}\label{section_main_results}
The underlying problem of the present article is a model-independent approach to the pricing of financial derivatives depending on several assets.
At time $t_0\in \mathbb{R}$, we consider a financial market with $d\in \mathbb{N} \cap [2,\infty)$ securities with non-negative values $S_0^1,\dots,S_0^d\in \mathbb{R}_+$, and we denote by $S:=(S_{t_i}^k)^{k=1,\dots,d}_{i=1,\dots,n}$ their future values at times $t_1<t_2<\dots<t_n$ for $n\in \mathbb{N}$. We model $S$ by the canonical process on $(\mathbb{R}_+^{n d},\mathcal{B}(\mathbb{R}_+^{n d}))$, where $\mathcal{B}(\mathbb{R}_+^{n d})$ denotes the Borel-$\sigma$-algebra on $\mathbb{R}_+^{n d}$, i.e., the components of $S$ are defined via
\[
S_{t_i}^k:(x_{1}^1,\dots,x_{n}^d) \mapsto x_i^k.
\]
For simplicity, we normalize interest rates to zero and assume absence of dividends. This means, $S_{t_i}^k$ denotes the price of the $k$-th security at time $t_i$.
Further, we fix some payoff function $c:\mathbb{R}_+^{n d} \rightarrow \mathbb{R}$ of a financial derivative depending on $S$.
Our goal is to calculate an arbitrage-free price interval for $c$ in a model-independent way, i.e., by using only information that is implied by market prices without imposing any assumptions on the dynamics or joint distributions of $S$. Therefore, we proceed as follows to define our set of pricing measures.
\begin{enumerate}
\item[(i)] First, we observe for all $k=1,\dots,d$, $i=1,\dots,n$ prices of European call options written on the $k$-th security maturing at $t_i$ for a continuum of strikes. According to \cite{breeden1978prices} we can then infer the one-dimensional risk-neutral marginal distributions \(\mu_i^k\) of \(S_{t_i}^k\) from this data for all $i,k$. This means for all $i,k$ and for any pricing measure $\mathbb{Q}$ that we have $\mathbb{Q} \circ {S_{t_i}^k}^{-1}=\mu_i^k$, where $\mu_i^k$ has mean $S_0^k \in \mathbb{R}_+$.
Denote for each such $\mu=(\mu_i^k)_{1\leq i \leq n}^{1\leq k \leq d}$ by
\[
\Pi(\mu):=\left\{\mathbb{Q}\in \mathcal{P}(\mathbb{R}_+^{nd})~\middle|~\mathbb{Q} \circ {S_{t_i}^k}^{-1}=\mu_i^k \text{ for all } i,k \right\}
\]
the set of transport plans that consists of all Borel probability measures on \(\mathbb{R}_+^{nd}\), denoted by $\mathcal{P}(\mathbb{R}_+^{nd})$, with univariate marginals \(\mu_1^1,\ldots,\mu_n^d\) having finite first moments equal to $S_0^1,\dots,S_0^d$. Further, we denote by $F_i^k(\cdot) = \int_{-\infty}^ \cdot \,\D \mu_i^k$ the cumulative distribution function of $\mu_i^k$.
\item[(ii)] Moreover, to ensure absence of model-independent-arbitrage\footnote{in the sense of \cite[Definition 1.2.]{acciaio2016model}}, we assume that for every pricing measure $\mathbb{Q}$ the martingale property
\begin{equation}\label{eq_martingaleprop}
\mathbb{E}_\mathbb{Q}[S_{t_i}|S_{t_j},\dots,S_{t_1}]=S_{t_j} ~~~\mathbb{Q}\text{-a.s. and for all } t_j \leq t_i
\end{equation}
holds true, where \((S_{t_i})_{0\leq i\leq n}\) is the \(d\)-variate process with components \(S_{t_i}=(S_{t_i}^{k})^{k=1,\ldots,d}\,\) for $i=0,1,\dots,n$.
According to a straightforward extension of \cite[Lemma 2.3]{beiglbock2013model}, the equality in \eqref{eq_martingaleprop} may be rewritten as
\[
\int_{\mathbb{R}^{nd}_+} \Delta(x_1^1,\dots,x_j^d)(x_{j+1}^k-x_j^k) \,\D\mathbb{Q}(x_1^1,\dots,x_n^d)=0
\]
for all $k=1,\dots,d$, $j=0,\dots,n$ and $\Delta \in C_b(\mathbb{R}^{jd}_+)$, which is the class of continuous and bounded functions on \(\mathbb{R}_+^{jd}\,.\)
We denote by $\mathcal{M}(\mu)\subset \Pi(\mu) \subset \mathcal{P}(\mathbb{R}_+^{nd})$ the set of martingale measures on $\mathbb{R}^{nd}_+$ with fixed univariate marginal distributions $\mu = (\mu_i^k)_{1\leq i \leq n}^{1\leq k \leq d}$.
\item[(iii)] Besides the marginal distributions and the martingale property, we impose additional linear constraints that are implied by observations on the market. These constraints additionally restrict the dependence structure of the underlying assets $S$. More precisely, we consider linear equality constraints of the form
\begin{equation}\label{eq_eq_constraints}
\mathbb{E}_\mathbb{Q}[f_i^{\operatorname{eq}}(S)]=K_i^{\operatorname{eq}}
\end{equation}
for problem-tailored Borel-measurable functions $f_i^{\operatorname{eq}} :\mathbb{R}_+^{n d} \rightarrow \mathbb{R}$ and $K_i^{\operatorname{eq}} \in \mathbb{R}$ with $i$ in some index set $\mathcal{I^{\operatorname{eq}}}$. We will adjust the choices of $f_i^{\operatorname{eq}}$ to the specific problems. Additionally, we implement inequality constraints of the form
\begin{equation}\label{eq_ineq_constraints}
\mathbb{E}_\mathbb{Q}[f_i^{\operatorname{ineq}}(S)]\leq K_i^{\operatorname{ineq}}
\end{equation}
for Borel-measurable $f_i^{\operatorname{ineq}} :\mathbb{R}_+^{n d} \rightarrow \mathbb{R}$, $K_i^{\operatorname{ineq}} \in \mathbb{R}$, and $i \in \mathcal{I}^{\operatorname{ineq}}$. We impose only countably many equality and inequality constraints, i.e., $\mathcal{I^{\operatorname{eq}}}$ and $\mathcal{I^{\operatorname{ineq}}}$ are countable sets.
\end{enumerate}
The set of measures which fulfil these additional constraints is denoted by
\begin{equation}\label{eq_restriction}
\begin{aligned}
\mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},\mathcal{I^{\operatorname{eq}}},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\mathcal{I^{\operatorname{ineq}}}}(\mu):=\mathcal{M}(\mu) &\cap \left\{\mathbb{Q}\in \mathcal{P}(\mathbb{R}_+^{nd})~\middle|~\mathbb{E}_\mathbb{Q}[f_i^{\operatorname{eq}}(S)]=K_i^{\operatorname{eq}} \text{ for all } i\in \mathcal{I}^{\operatorname{eq}}\right\}\\
&\cap \left\{\mathbb{Q} \in \mathcal{P}(\mathbb{R}_+^{nd})~\middle|~\mathbb{E}_\mathbb{Q}[f_i^{\operatorname{ineq}}(S)]\leq K_i^{\operatorname{ineq}} \text{ for all } i\in \mathcal{I}^{\operatorname{ineq}}\right\}\,.
\end{aligned}
\end{equation}
For the sake of readability we abbreviate this set by $\mathcal{M}^{\mathrm{lin}}$ and consider it as our set of pricing measures.
An arbitrage-free and model-independent upper price bound for a payoff $c$ under the above mentioned equality and inequality constraints can then be obtained by pursuing two different approaches. First, in the primal approach, we consider the supremum of the expected values among all martingale models consistent with available price information on options and the imposed equality and inequality constraints given by
\begin{equation}\label{equ primal problem M_lin}
\overline{P}_{\mathcal{M}^{\mathrm{lin}}}:=\sup_{\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}} \mathbb{E}_\mathbb{Q} [c(S)].
\end{equation}
A corresponding lower price bound \(\underline{P}_{\mathcal{M}^{\mathrm{lin}}}\) can be obtained by considering the infimum over all measures $\mathbb{Q}\in\mathcal{M}^{\mathrm{lin}}$.
Second, in the dual approach, the fair price bounds of the derivative $c$ can be calculated by using trading strategies instead of pricing measures. For the upper bound we consider the problem of finding the cheapest super-replication price of $c$. More precisely, we consider strategies $\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}:\mathbb{R}_{+}^{nd} \rightarrow \mathbb{R}$ of the form
\begin{equation}\label{eq_dual_strats1}
\begin{aligned}
\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}(x_1^1,\dots,x_n^d):=\sum_{k=1}^d\sum_{i=1}^n u_i^k(x_i^k)&+\sum_{i=1}^{n-1}\sum_{k=1}^d\Delta_i^k(x_1^1,\dots,x_i^d) (x_{i+1}^k-x_i^k)\\
&+\sum_{i\in \mathcal{I}^{\operatorname{eq}}}\alpha_i\left(f_i^{\operatorname{eq}}(x_1^1,\dots,x_n^d)-K_i^{\operatorname{eq}}\right)\\
&+\sum_{i\in \mathcal{I}^{\operatorname{ineq}}}\beta_i\left(f_i^{\operatorname{ineq}}(x_1^1,\dots,x_n^d)-K_i^{\operatorname{ineq}}\right).
\end{aligned}
\end{equation}
with
\[
u_i^{k} \in \mathfrak{C}:=\left\{u\colon \mathbb{R}_+ \to \mathbb{R}~\middle|~ u(x)=a+bx+\sum_{i=1}^m c_i(x-d_i)_+\,, a,b,c_i,d_i\in \mathbb{R}\,, m\in \mathbb{N} \right\}
\]
and with each $\Delta_{i}^k \in C_b(\mathbb{R}_+^{id})$, $\alpha_i \in \mathbb{R}$, $\beta_i \in \mathbb{R}_+$ such that $\alpha_i=0$, $\beta_j=0$ for all but finitely many $i\in \mathcal{I}^{\operatorname{eq}}$, $j \in \mathcal{I}^{\operatorname{ineq}}$.
This means, we consider trading strategies allowing for static positions in the European options $u_i^k$, in derivatives with payoffs $f_i^{\operatorname{eq}}$ traded for price $K_i^{\operatorname{eq}}$ and long positions in $f_i^{\operatorname{ineq}}$ for a traded price not higher than $K_i^{\operatorname{ineq}}$. Moreover, we consider dynamic self-financing trading positions $\Delta_i^k$ in the underlying securities. In line with our model-free approach, we are interested in strategies which super-replicate the payoff of the derivative pointwise, i.e., for every possible path, independent of any associated probability. We use the notation $f\geq c$ to express pointwise inequalities, i.e., $f(x) \geq c(x)$ for all $x \in \mathbb{R}^{nd}_+$. The following result shows that - under mild assumptions - minimizing the prices of such super-replication strategies yields the same value as maximizing expectations w.r.t.\,measures from $\mathcal{M}^{\mathrm{lin}}$.
To this end, we set
\begin{align*}
\mathcal{S}:=\bigg\{\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)} ~\bigg|~ &\exists~u_i^{k} \in \mathfrak{C},\Delta_{i}^k \in C_b(\mathbb{R}_+^{id}),\alpha_i \in \mathbb{R},\beta_i \in \mathbb{R}_+ \\
&\text{with } \alpha_i=0, \beta_j=0\text{ for all but finitely many }i\in \mathcal{I}^{\operatorname{eq}},~ j \in \mathcal{I}^{\operatorname{ineq}}\\
&\hspace{6cm}\text{s.t. } \Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)} \geq c \bigg\},
\end{align*}
and denote by
\[
\underline{\mathcal{D}}_{\mathcal{S}}(c):=\inf_{\Psi\in \mathcal{S}}\left\{\sum_{k=1}^d \sum_{i=1}^n\mathbb{E}_{\mu_i^k}[u_i^k]\right\}
\]
the minimal price among all super-replicating strategies \(\Psi:=\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}\) for the payoff $c$. When there is no ambiguity about the payoff $c$, we abbreviate the notation as $\underline{\mathcal{D}}_{\mathcal{S}}$.
We define for $m \in \mathbb{N}$ by
\begin{align*}
C_{\operatorname{lin}}(\mathbb{R}_+^m)&:= \left\{f\colon \mathbb{R}_+^m \to \mathbb{R}~\middle|~ f\text{ continuous},~ \sup_{(x_1,\dots,x_m)\in \mathbb{R}_+^m} \frac{|f(x_1,\dots,x_m)|}{1+\sum_{i=1}^m x_i} < \infty \right\}\,,\\
L_{\operatorname{lin}}(\mathbb{R}_+^m)&:=\left\{f\colon \mathbb{R}_+^m \to \mathbb{R}~\middle|~ f\text{ lower semicontinuous},~\sup_{(x_1,\dots,x_m)\in \mathbb{R}_+^m} \frac{|f(x_1,\dots,x_m)|}{1+\sum_{i=1}^m x_i} < \infty \right\}\,,\\
U_{\operatorname{lin}}(\mathbb{R}_+^m)&:=\left\{f\colon \mathbb{R}_+^m \to \mathbb{R}~\middle|~ f\text{ upper semicontinuous},~\sup_{(x_1,\dots,x_m)\in \mathbb{R}_+^m} \frac{|f(x_1,\dots,x_m)|}{1+\sum_{i=1}^m x_i} < \infty \right\}
\end{align*}
the set of continuous, lower semicontinuous, and upper semicontinuous functions, respectively, with at most linear growth. We then adapt the model-independent super-hedging duality results from, e.g., \cite{acciaio2016model}, \cite{beiglbock2013model}, \cite{burzoni2017model}, \cite{cheridito2021martingale}, \cite{dolinsky2014martingale}, \cite{Rueschendorf-2019}, and \cite{zaev2015monge}, to our situation by formulating the following theorem.
\begin{thm}[Duality with additional constraints]\label{theaddcon}
\label{thm_duality_linear_constraints} ~\\ Assume that $c \in U_{\operatorname{lin}}(\mathbb{R}^{nd}_+)$, $f_i^{\operatorname{eq}}\in C_{\operatorname{lin}}(\mathbb{R}_+^{nd}),\) and \(f_j^{\operatorname{ineq}} \in L_{\operatorname{lin}}(\mathbb{R}_+^{nd})$ for all $i \in \mathcal{I}^{\operatorname{eq}}$, $j \in \mathcal{I}^{\operatorname{ineq}}$, and assume that $\mathcal{M}^{\mathrm{lin}}\neq \emptyset$. Then it holds
\begin{align}\label{gendures}
\overline{P}_{\mathcal{M}^{\mathrm{lin}}}= \underline{\mathcal{D}}_{\mathcal{S}}\,.
\end{align}
Moreover, there exists \(\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}\) such that
\begin{align}\label{eqmaxele}
\overline{P}_{\mathcal{M}^{\mathrm{lin}}} = \mathbb{E}_\mathbb{Q} [c(S)]\,.
\end{align}
\end{thm}
\begin{rem}
\label{rem_differences_martingale_property}\begin{itemize}
\item[(a)] The set $\mathcal{M}(\mu)$ of martingale measures with fixed univariate marginals is non-empty if and only if the $d$-dimensional marginals $(\mu_i^1,
\dots,\mu_i^d)_{i=1,\dots,n}$ increase in convex order\footnote{A finite set of probability measures $\{\mathbb{P}_1,\cdots,\mathbb{P}_n\}$ on $\mathbb{R}^d$ is said to increase in convex order if $\int_{\mathbb{R}^d} f(x) \D \mathbb{P}_i(x) \leq \int_{\mathbb{R}^d} f(x) \D \mathbb{P}_{i+1}(x)$ for all convex functions $f:\mathbb{R}^d \rightarrow \mathbb{R}$ and for all $i=1,\dots,n-1$ such that the integrals are finite.}, see \cite{strassen1965existence}. Thus, the latter condition is necessary for the non-emptiness of $\mathcal{M}^{\mathrm{lin}}(\mu)$.
To derive sufficient conditions for the non-emptiness of the set $\mathcal{M}^{\mathrm{lin}}$ we proceed as follows.
Assume $\mathcal{M}(\mu) \neq \emptyset$ and w.l.o.g. $\mathcal{I}^{\operatorname{eq}}=\mathbb{N}$, $\mathcal{I}^{\operatorname{ineq}}=\mathbb{N}$. We observe that if
\[
\inf_{\mathbb{Q} \in \mathcal{M}(\mu)}\mathbb{E}_\mathbb{Q}[f_1^{\operatorname{eq}}]\leq K_1^{\operatorname{eq}}\leq\sup_{\mathbb{Q} \in \mathcal{M}(\mu)} \mathbb{E}_\mathbb{Q}[f_1^{\operatorname{eq}}],
\]
then we get as a convex combination of minimal and maximal measure the existence of a measure $\mathbb{Q} \in \mathcal{M}(\mu)$ with $\mathbb{E}_\mathbb{Q}[f_1^{\operatorname{eq}}]=K_1^{\operatorname{eq}}$. We proceed inductively, and see that if for all $i=2,3,\dots,$ we have
\[
\inf_{\mathbb{Q} \in \mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},\{1,\dots,i-1\},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\emptyset}(\mu)} \mathbb{E}_\mathbb{Q}[f_i^{\operatorname{eq}}]
\leq K_i^{\operatorname{eq}} \leq \sup_{\mathbb{Q} \in \mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},\{1,\dots,i-1\},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\emptyset}(\mu)} \mathbb{E}_\mathbb{Q}[f_i^{\operatorname{eq}}],
\]
then it holds $\mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},
\mathcal{I}^{\operatorname{eq}},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\emptyset}(\mu) \neq \emptyset$.
In the same way we can check consistency of the inequality constraints. Thus, if we have
\[
\sup_{\mathbb{Q} \in \mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},\mathcal{I}^{\operatorname{eq}},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\emptyset}(\mu)} \mathbb{E}_\mathbb{Q}[f_1^{\operatorname{ineq}}]\leq K_1^{\operatorname{ineq}},
\]
and for all $i=2,3,\dots,$ that
\[
\sup_{\mathbb{Q} \in \mathcal{M}^{\mathrm{lin}}_{f_i^{\operatorname{eq}},K_i^{\operatorname{eq}},\mathcal{I}^{\operatorname{eq}},\atop f_i^{\operatorname{ineq}},K_i^{\operatorname{ineq}},\{1,\dots,i-1\}}(\mu)} \mathbb{E}_\mathbb{Q}[f_i^{\operatorname{ineq}}]\leq K_i^{\operatorname{ineq}},
\]
then it holds $\mathcal{M}^{\operatorname{lin}}\neq \emptyset$.
\item[(c)] \label{rem_differences_martingale_propertyB} In the one-period case, i.e., if \(n=1\,,\) the martingale property \eqref{eq_martingaleprop} only constrains the marginal distributions of \(S_{t_1}^k\) for \(k=1,\ldots,d\,\) but not the dependence structure of \((S_{t_1}^1,\ldots,S_{t_1}^d)\,,\) because \eqref{eq_martingaleprop} simplifies to the mean constraint $\mathbb{E}_\mathbb{Q}[S_{t_1}^k]=S_0^k$ for some deterministic values $S_0^k \in \mathbb{R}_+$ for $k =1,\dots,d$ representing today's spot values of the respective securities.
\end{itemize}
\end{rem}
A major contribution of this paper is the specification of situations, where the choices of $f_i^{\operatorname{eq}},~K_i^{\operatorname{eq}},~\mathcal{I}^{\operatorname{eq}}$ and $f_i^{\operatorname{ineq}},~K_i^{\operatorname{ineq}},~\mathcal{I}^{\operatorname{ineq}}$, respectively, can explicitly be inferred from market data. These specifications are given in detail in Section~\ref{sec improved price bounds} and require the concept of copulas and quasi-copulas which will be introduced in the following Section~\ref{sec dependence modelling}
\section{Dependence Modelling}\label{sec dependence modelling}
In this section, we introduce the basic notions for copulas, quasi-copulas, and dependence orderings which we use in Section \ref{sec improved price bounds} to restrict the inter-asset dependencies when prices of specific financial derivatives are given.
Further, we formulate in Theorem \ref{theqcub} a main result of this paper which extends the characterization of the supermodular ordering of upper products in internal factor models from \cite[Theorem 1]{Ansari-Rueschendorf-2020} to the case of a quasi-copula as upper bound. We build on this result in Section \ref{sec improved price bounds} to derive closed-form expressions for improved price bounds of supermodular payoff functions like basket options, when dependence information related to the setting of an internal factor model is available.
\subsection{Basic Notions}\label{section_copula_notions}\label{section_copula_main_results}
For the analysis of dependence structures, we consider some well-known function classes. We focus on functions with non-negative domain because we are only interested in those functions which can be interpreted as payoff functions depending on the non-negative underlying assets. Denote by \(\overline{\mathbb{R}}_+:=\mathbb{R}_+\cup\{\infty\}\) the extended real non-negative numbers.
\begin{defn}\label{def_DeltaMonotone_Antitone_Supermodular}
Let $m \in \mathbb{N}$. For a function \(f\colon {\overline{\mathbb{R}}}_{+}^m\supset \bigtimes_{i=1}^m[a_i,b_i] \to \mathbb{R}\,,\) define the difference operator \(\triangle_\varepsilon^i\,,\) \(\varepsilon>0\,,\) \(1\leq i \leq m\,,\) by
\begin{align*}
\triangle_\varepsilon^if(x):=f((x+\varepsilon e_i)\wedge b)-f(x),
\end{align*}
where $x\in \bigtimes_{i=1}^m[a_i,b_i]$, $b=(b_1,\dots,b_m)$, \(e_i\) denotes the \(i\)-th unit vector, and \(\wedge\) denotes the componentwise minimum.
\begin{itemize}
\item[(a)] The function \(f\) is \emph{\(\Delta\)-monotone} if for all $k \in \{1,\dots, m\}$, any subset \(J=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,m\}\,,\) and all \(\varepsilon_1,\ldots,\varepsilon_k>0\,,\) it holds that
\begin{align*}
\triangle^{i_1}_{\varepsilon_1}\cdots \triangle_{\varepsilon_k}^{i_k} f(x)\geq 0\,
\end{align*}
for all $x \in \bigtimes_{i=1}^m[a_i,b_i]$.
\item[(b)] The function \(f\) is \emph{$m$-increasing} if
\[
\triangle^{1}_{\varepsilon_1}\cdots \triangle_{\varepsilon_m}^{m} f(x)\geq 0\,
\]
for all $x \in \bigtimes_{i=1}^m[a_i,b_i]$ and all $\varepsilon_1,\ldots,\varepsilon_m>0$.
\item[(c)] The function \(f\) is \emph{\(\Delta\)-antitone} if
\begin{align*}
(-1)^k \triangle^{i_1}_{\varepsilon_1}\cdots \triangle_{\varepsilon_k}^{i_k} f(x)\geq 0
\end{align*}
for all $k \in \{1,\dots, m\}$, for all \(J=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,m\}\), and all \(\varepsilon_1,\ldots,\varepsilon_k>0\,.\)
\item[(d)] The function \(f\) is \emph{supermodular} if
\begin{align*}
\triangle^{i}_{\varepsilon_i} \triangle_{\varepsilon_j}^{j} f(x)\geq 0
\end{align*}
for all \(i\ne j\) and all \(\varepsilon_i,\varepsilon_j > 0\,.\)
\end{itemize}
We denote by \(\mathcal{F}_{\Delta}\,\) the set of \(\Delta\)-monotone functions, by \(\mathcal{F}_{\Delta}^-\) the set of \(\Delta\)-antitone functions, and by \(\mathcal{F}_{\operatorname{sm}}\) the set of supermodular functions.
\end{defn}
Note that, by definition, a \(\Delta\)-monotone function is componentwise increasing and a \(\Delta\)-antitone function is componentwise decreasing. Further, every \(\Delta\)-monotone or \(\Delta\)-antitone function is supermodular.
For the analysis of dependence structures, we introduce copulas and quasi-copulas as follows, compare also \cite{Nelsen-2011}.
\begin{defn}[Copula, quasi-copula]\label{def_quasicop}
Let \(m\in \mathbb{N}\,\).
\begin{itemize}
\item[(a)]
A function \(Q\colon [0,1]^m\to [0,1]\) is called \(m\)-variate \emph{quasi-copula} if it fulfils the following properties.
\begin{enumerate}[(i)]
\item \label{def_quasicop1} \(Q\) is \emph{grounded}, i.e., \(Q(u)=0\) if at least one coordinate of \(u\) is \(0\,,\)
\item \label{def_quasicop1b}\(Q\) has uniform marginals, i.e., \(Q(1,\ldots,1,u_i,1,\ldots,1)=u_i\) for all \(u_i\in[0,1]\) and \(1\leq i\leq m\,,\)
\item \label{def_quasicop2}\(Q\) is non-decreasing in each component,
\item \label{def_quasicop3}\(Q\) fulfils the Lipschitz condition \(|Q(v)-Q(u)|\leq \sum_{i=1}^m |v_i-u_i|\) for all \(u=(u_1,\ldots,u_m), \) \(v=(v_1,\ldots,v_m)\in [0,1]^m\,.\)
\end{enumerate}
\item[(b)] Further, a function $C:[0,1]^m \rightarrow [0,1]$ is called \(m\)-variate \emph{copula} if \(C\) fulfils \eqref{def_quasicop1}, \eqref{def_quasicop1b}, and
\begin{enumerate}[(i)]\setcounter{enumi}{4}
\item \label{def_quasicop4}\(C\) is \(m\)-increasing.
\end{enumerate}
\end{itemize}
We denote by \(\mathcal{C}_m\) the set of \(m\)-variate copulas and by \(\mathcal{Q}_m\) the set of \(m\)-variate quasi-copulas.
\end{defn}
Note that condition \eqref{def_quasicop4} implies \eqref{def_quasicop2} and together with \eqref{def_quasicop1} and \eqref{def_quasicop1b} also \eqref{def_quasicop3} in the above definition. Hence, it holds that \(\mathcal{C}_m\subset \mathcal{Q}_m\). We also allow the trivial case \(m=1\) which is considered in the definition of the quasi-expectation operator, see \eqref{defquexpop}.
The motivation to consider copulas stems from Sklar's Theorem, which decomposes an \(m\)-dimensional/ $m$-variate distribution function \(F\) into its univariate marginal distribution functions \(F_1,\ldots,F_m\) and some \(m\)-variate copula $C$ that describes the dependence structure by
\begin{equation}\label{eq_sklar}
F(x)=C(F_1(x_1),\ldots,F_m(x_m)),~~~x=(x_1,\ldots,x_m)\in \mathbb{R}^m\,.
\end{equation}
Conversely, for every copula \(C\in \mathcal{C}_m\) and for all univariate distribution functions \(F_1,\ldots,F_m\,,\) the function \(F\) in \eqref{eq_sklar} defines an \(m\)-variate distribution function,
see, e.g., \cite[Theorem 1.10.9]{Nelsen-2006}.
Therefore, if a random vector $X$ is distributed according to $F$, and if $C$ fulfils \eqref{eq_sklar}, then we say that \emph{$C$ is a copula of $X$}.
Denote by \(\mathcal{F}^m\) the class of \(m\)-variate distribution functions, and by \(\mathcal{F}^m_+\) the subclass of \(m\)-variate distribution functions \(F\) defined by \eqref{eq_sklar} such that \(F_i(0)=0\) for all \(1\leq i\leq m\,.\) Then, as an extension of \eqref{eq_sklar} to quasi-copulas, we define a quasi-distribution function as follows.
\begin{defn}[Quasi-distribution function]\label{defquadisfun}~\\
We call a function \(H\colon \mathbb{R}^m\to [0,1]\) \emph{quasi-distribution function} if there exist \(F_1,\ldots,F_m\in \mathcal{F}^1\) and \(Q\in \mathcal{Q}_m\) such that
\begin{align*}
H(x)=Q(F_1(x_1),\ldots,F_m(x_m))
\end{align*}
for all \(x=(x_1,\ldots,x_m)\in \mathbb{R}^m\,.\)
We also say $H$ is the quasi-distribution function of $Q$ w.r.t.\,$F_1,\dots,F_m$.
Denote by \(\mathcal{H}^m\) the class of \(m\)-variate quasi-distribution functions and by \(\mathcal{H}_+^m\) its subclass with \(F_i\in \mathcal{F}_+^1\) for all \(i\in \{1,\ldots,m\}\,.\)
\end{defn}
An important property of quasi-copulas is that for any subset \(\mathcal{N}\subset \mathcal{Q}_m\) of quasi-copulas the pointwise supremum $Q_{\mathcal{N}}$ defined by
\begin{align}\label{eqmxqc}
Q_{\mathcal{N}}(u):=\sup\{Q(u)\,|\, Q\in \mathcal{N}\},~\text{ for } u \in [0,1]^m,
\end{align}
is again a quasi-copula, see \cite[Theorem 2.2]{Nelsen-2004} (where the proof for \(m=2\) can be extended to arbitrary dimension \(m\geq 2\)). In contrast, the pointwise supremum of copulas is in general not a copula.
Denote by \(W^m\) and \(M^m\) the lower and upper Fr\'{e}chet bound, respectively, given by
\begin{align}\label{frebou}
W^m(u):=\max\left\{\sum_{i=1}^m u_i -m+1,0\right\} ~~~\text{and}~~~ M^m(u):=\min_{1\leq i\leq m} \{u_i\}
\end{align}
for \(u=(u_1,\ldots,u_m)\in [0,1]^m\,.\) Then, it holds that
\begin{align}\label{eqfrhobo}
W^m(u)\leq Q(u)\leq M^m(u),\quad u\in [0,1]^m
\end{align}
for all quasi-copulas \(Q\in \mathcal{Q}_m\). Further, \(M^m\) is a copula for all \(m\in \mathbb{N}\), wheras \(W^2\) is a copula and \(W^m\in \mathcal{Q}_m\setminus \mathcal{C}_m\) is a quasi-copula, but not a copula for \(m\geq 3\,,\) see, e.g., \cite{Nelsen-2006}.
In the following, to avoid technical difficulties, we will often consider functions \(f\colon \Theta^m\to \mathbb{R}\) that are continuous at the boundary of \(\Theta\,,\) where $\Theta\in \{\mathbb{R}_+,[0,1),[0,1]\}$, i.e., whenever \(a:=\inf(\Theta)\in \Theta\) and/or \(b:=\sup(\Theta)\in \Theta\,,\) respectively, then we will often require \(f\) to satisfy
\begin{align}\label{contboun1}
&\lim_{h\downarrow 0} f(x_1,\ldots,x_{i-1},a+h,x_{i+1},\ldots,x_m) = f(x_1,\ldots,x_{i-1},a,x_{i+1},\ldots,x_m)\\
&\nonumber \text{and/or} \\
\label{contboun2} &\lim_{h\downarrow 0} f(x_1,\ldots,x_{i-1},b-h,x_{i+1},\ldots,x_m) = f(x_1,\ldots,x_{i-1},b,x_{i+1},\ldots,x_m)\,,
\end{align}
respectively, for all \(i\in \{1,\ldots,m\}\) and for all \(x_j\in \Theta\,,\) \(j\ne i\,.\)
For the consideration of dependence orderings on the sets \(\mathcal{C}_m\) and \(\mathcal{Q}_m\,,\) we define for a bounded function $f\colon \Theta^m \to \mathbb{R}$ which is in each component either increasing or decreasing, which fulfils \eqref{contboun2}, and which is defined on \(\Theta\in \{[0,1),[0,1],\mathbb{R}_+,\mathbb{R}\}\,,\) its \emph{survival function} \(\widehat{f}\colon \Theta^m\to \mathbb{R}\) by\footnote{We stress that the sum in the definition of the survival function in \eqref{defsurvfun} also takes into account the summand \(J=\emptyset\).}
\begin{align}\label{defsurvfun}
\widehat{f}(x):=\sum_{J\subseteq \{1,\ldots,m\}} (-1)^{m-|J|} f(y)\,,
\end{align}
where \(y=(y_1,\ldots,y_m)\) satisfies for all \(i\in \{1,\ldots,m\}\) that \(y_i=\sup(\Theta)\) if \(i\in J\) and \(y_i=x_i\) if \(i\notin J\,\), and where we set
\begin{equation}\label{eq_def_limsup}
f(w_1,\ldots,w_{l-1}, \infty,w_{l+1},\ldots,w_m):=\lim_{z\to \infty} f(w_1,\ldots,w_{l-1},z,w_{l+1},\ldots,w_m).
\end{equation}
for all $(w_1,\dots,w_m) \in \Theta^m$, $l\in \{1,\dots,m\}$. For the cumulative distribution function \(F:\mathbb{R}^m\rightarrow [0,1]\) of a real-valued random vector \(X=(X_1,\ldots,X_m)\) on a probability space \((\Omega,\mathcal{A},\mathbb{P})\,,\) the associated survival function \(\widehat{F}\) is, in line with \eqref{defsurvfun}, due to the inclusion-exclusion principle given by\footnote{Note that, since copulas are also distribution functions, the identity in \eqref{survdisfun} also holds true for copulas.}
\begin{align}\label{survdisfun}
\widehat{F}(x)= \mathbb{P}(X_i> x_i ~\forall i\in \{1,\ldots,m\})= \int_{\mathbb{R}^m} \one_{\{z>x\}} \mathsf{\,d} F(z)\,\text{ for } x=(x_1,\dots,x_m)\in \mathbb{R}^m.
\end{align}
For a copula \(C \in \mathcal{C}_m\) and some measurable function $f:[0,1]^m \rightarrow \mathbb{R}$, we consider the expectation operator \(\psi_f(C)\) defined by
\begin{align}\label{defexpop_0}
\psi_f(C):=&\int_{[0,1]^m} f(u) \mathsf{\,d} C(u),
\end{align}
which is well-defined as $C\in \mathcal{C}_m$ induces a measure.
To derive price bounds under dependence information, we consider for a measurable (payoff) function \(c\colon \mathbb{R}_+^m\to \mathbb{R}\) and for some random vector \(X=(X_1,\ldots,X_m)\) on $\mathbb{R}_+^m$ with univariate marginal distribution functions $F_i, i =1,\dots,m$, the quantile transformed function defined by
\begin{align}\label{eqqutra}
c(F_1^{-1}(u_1),\ldots,F_{m}^{-1}(u_m))\,, ~~~(u_1,\ldots,u_m)\in [0,1]^m\,,
\end{align}
where, for a distribution function \(F\,,\) its generalized inverse \(F^{-1}\) is defined by
\[
F^{-1}(u):=\inf\{x\in \mathbb{R}_{+}\,, F(x)\geq u\}\,, \text{ for } u\in [0,1]\,.
\]
Then, for a copula \(C \in \mathcal{C}_m\) associated by \eqref{eq_sklar} with the distribution function $F(\cdot)=\mathbb{Q}(X\leq \cdot)$, for some measure \(\mathbb{Q}\in \mathcal{P}(\mathbb{R}^{m}_+)\,,\) we obtain
\begin{align}\label{defexpop}
\begin{split}
\psi_c^{(F_1,\ldots,F_m)}(C) := &\psi_{c\,\circ (F_1^{-1},\ldots,F_m^{-1})}(C) = \int_{[0,1]^m} c(F_1^{-1}(u_1),\ldots,F_{m}^{-1}(u_m)) \mathsf{\,d} C(u) \\
= &\int_{\mathbb{R}_+^m} c(x)\mathsf{\,d} C(F_1(x_1),\ldots,F_m(x_m))= \int_{\mathbb{R}_+^m} c(x)\mathsf{\,d} F(x) = \mathbb{E}_\mathbb{Q} [c(X)]\,,
\end{split}
\end{align}
where the third equality is a consequence of the transformation formula for Stieltjes integrals, see, e.g., \cite[Theorem (2)]{Winter-1997}. Hence, in financial contexts $\psi_c^{(F_1,\ldots,F_m)}(C) $ can be interpreted as the price of \(c\) under some pricing measure \(\mathbb{Q}\,\).
Next, we define the lower orthant, upper orthant, and concordance ordering on \(\mathcal{Q}_m\), compare also, e.g., \cite{Lux-2017}.
\begin{defn}[\(\leq_{\operatorname{lo}}\,,\) \(\leq_{\operatorname{uo}}\,,\) \(\leq_c\)]\label{def_orthant_orders}
Let $m \in \mathbb{N}$, and let $Q$, $Q'\in \mathcal{Q}_m$ be quasi-copulas.
\begin{itemize}
\item[(a)] The quasi-copula \(Q\) is smaller than \(Q'\) in the \emph{lower orthant order}, written \(Q\leq_{\operatorname{lo}} Q'\,,\) if \(Q(u)\leq Q'(u)\) for all \(u\in [0,1]^m\,.\)
\item[(b)] The quasi-copula \(Q\) is smaller than \(Q'\) in the \emph{upper orthant order}, written \(Q\leq_{\operatorname{uo}} Q'\,,\) if \(\widehat{Q}(u)\leq \widehat{Q}'(u)\) for all \(u\in [0,1]^m\,.\)
\item[(c)] The quasi-copula \(Q\) is smaller than \(Q'\) in the \emph{concordance order}, written \(Q\leq_{\operatorname{c}} Q'\,,\) if both \(Q\leq_{\operatorname{lo}} Q'\) and \(Q\leq_{\operatorname{uo}} Q'\) hold true.
\end{itemize}
\end{defn}
For the comparison of prices w.r.t.\,different market-implied dependence structures, we make use of the following characterization of the orthant orders on the class \(\mathcal{C}_m\,,\) see \cite{Rueschendorf-1980}, \cite{Lux-2017}.
\begin{prop}\label{prop_characterization_sm}
Let $m \in \mathbb{N}$. For copulas \(C_1,C_2\in \mathcal{C}_m\,,\) the lower orthant ordering \(C_1\leq_{\operatorname{lo}} C_2\) is equivalent to
\begin{align}\label{chaloord}
\psi_f(C_1)\leq \psi_f(C_2)~~~\text{for all }\Delta\text{-antitone functions }f\colon [0,1]^m\to \mathbb{R}
\end{align}
such that the expectations exist\footnote{We say the expectation $\psi_f(C)$ exists whenever the associated Stieltjes integral in \eqref{defexpop} is well-defined and finite.}.\\
The upper orthant ordering \(C_1\leq_{\operatorname{uo}} C_2\) is equivalent to
\begin{align}\label{chauoord}
\psi_f(C_1)\leq \psi_f(C_2)~~~\text{for all }\Delta\text{-monotone functions }f\colon [0,1]^m\to \mathbb{R}
\end{align}
such that the expectations exist.
\end{prop}
As a strengthening of the lower orthant and the upper orthant ordering, we also consider the \emph{supermodular ordering} \(C_1\leq_{\operatorname{sm}} C_2\) which is defined by
\begin{align}\label{supmodcop}
\psi_f(C_1)\leq \psi_f(C_2) ~~~\text{for all supermodular functions }f\colon [0,1]^m\to \mathbb{R}
\end{align}
such that the expectations exist.\\
Since $\mathcal{F}_{\Delta}^-\cup \mathcal{F}_{\Delta}\subset \mathcal{F}_{\operatorname{sm}}$, the supermodular ordering implies both orthant orderings.
An important property of all these orderings is the invariance under increasing transformations, i.e., \(\psi_f(C_1)\leq \psi_f(C_2)\) implies \(\psi_{h}(C_1)\leq \psi_h(C_2)\) for \(h=f\circ (g_1,\ldots,g_m)\,,\) with each \(g_i:[0,1] \rightarrow [0,1]\,,\) \(i=1,\ldots,m\,,\) non-decreasing and \(f\) supermodular (and, thus, also for \(f\) being \(\Delta\)-antitone or \(\Delta\)-monotone), see, e.g., \cite[Theorems 6.G.3(a) and 9.A.9(a)]{Shaked-Shanthikumar-2007}.
For an extension of \eqref{chaloord} -- \eqref{supmodcop} to quasi-copulas, a modification of the expectation operator in \eqref{defexpop} is necessary because, for \(Q\in \mathcal{Q}_m\,,\) the term \(\mathsf{\,d} Q\) does not generally define a signed measure to integrate against, see \cite[Theorem 4.1]{Fernandez-2011a}. However, in the case that \(f=c \circ (F_1^{-1},\dots,F_m^{-1})\) induces a signed measure, \eqref{defexpop} can be extended to quasi-copulas using integration by parts.
\begin{defn}[Measure-inducing functions]\label{def_measure_inducing}~\\
Let $m \in \mathbb{N}$, let \(g\colon \Theta^m \to \mathbb{R}\) be a function, where \(\Theta\in \{[0,1),[0,1],\mathbb{R}_+\}\). Denote by $\mathcal{B}(\Theta^m)$ the Borel \(\sigma\)-algebra on \(\Theta^m\,.\)
\begin{itemize}
\item[(a)] In the case that \(g\) is left-continuous\footnote{We call a multivariate function left-continuous/right-continuous if the function is componentwise left-continuous/componentwise right-continuous at every point.}, $g$ is said to be \emph{measure-inducing} if there exists a signed measure \(\eta_g\) on $\mathcal{B}(\Theta^m)$ such that
\begin{align}\label{eqindmeas}
\eta_g([x_1,x_1+\varepsilon_1)\times \cdots \times [x_m,x_m+\varepsilon_m)) = \triangle_{\varepsilon_1}^1 \cdots \triangle_{\varepsilon_m}^m g(x)
\end{align}
for all \(x=(x_1,\ldots,x_m)\in \Theta^m\) and \(\varepsilon_1,\ldots,\varepsilon_m\in \mathbb{R}_+\,.\)
\item[(b)] In the case that \(g\) is right-continuous \(g\) is said to be \emph{measure-inducing} if there exists a signed measure \(\eta_g\) on $\mathcal{B}(\Theta^m)$ such that
\begin{align}\label{eqindmeasu}
\eta_g((x_1,x_1+\varepsilon_1]\times \cdots \times (x_m,x_m+\varepsilon_m]) = \triangle_{\varepsilon_1}^1 \cdots \triangle_{\varepsilon_m}^m g(x)
\end{align}
for all \(x=(x_1,\ldots,x_m)\in \Theta^m\) and \(\varepsilon_1,\ldots,\varepsilon_m\in \mathbb{R}_+\,.\)
\item[(c)]
For \(I=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,m\}\,,\) the \emph{\(I\)-marginal} \(g_I\) of \(g\) is defined by
\begin{align}\label{deffmarg}
g_I\colon \Theta^{k} \ni (u_{i_1},\ldots,u_{i_k}) \to g(u_1,\ldots,u_m)\,, ~~~\text{where }u_j=0 ~\text{for all }j\notin I\,.
\end{align}
\end{itemize}
\end{defn}
In particular, by \eqref{eqindmeas} and \eqref{eqindmeasu}, for all \(I\subseteq\{1,\ldots,m\}\,,\) \(I\ne \emptyset\,,\) the \(I\)-marginal of every left-/ right-continuous \(\Delta\)-monotone function induces a non-negative measure, and the \(I\)-marginal of every left-/right-continuous \(\Delta\)-antitone function induces a non-negative measure on \(\mathcal{B}([0,1]^{|I|})\) if \(|I|\) is even and a non-positive measure if \(|I|\) is odd, see, e.g., \cite[Corollary 2.24]{Ansari-2021}, cf. \cite[Proposition 2.1]{papapantoleon2020detection}.
In general, a left-/right-continuous measure-inducing function defined on a compact domain like \([0,1]^m\) always induces a finite signed measure, and if all \(I\)-marginals are measure-inducing, this equivalently means that the function has bounded Hardy-Krause variation, see \cite[Theorem 2.12]{Ansari-2021}, see also \cite[Theorem 3 and, for a precise definition of the Hardy-Krause variation, Section 2.1]{Aistleitner-2015}\footnote{In this reference, the authors show the statement for a right-continuous function \(f\colon [0,1]^m\to \mathbb{R}\,.\) However, it also applies for a left-continuous function \(g\colon [0,1]^m \to \mathbb{R}\,\) by setting \(g(x)=f(1-x)\,.\)}.
We will often assume that for $\Theta\in \{\mathbb{R}_+,[0,1),[0,1]\}$ a measure-inducing function \(f\colon \Theta^m\to \mathbb{R}\) is continuous at the boundary of \(\Theta\). To this end, we denote by
\begin{align}\begin{split}
\mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\Theta):=&\{f\colon \Theta^m \to \mathbb{R}\mid f \text{ satisfies } \eqref{contboun1} \text{ and } \eqref{contboun2},\\
&~~~~~~~~~~~~~~~~~~~~~~~~ f_I \text{ is measure-inducing for all } I\subseteq \{1,\ldots,m\}\,, I\ne \emptyset\}
\end{split}
\end{align}
the class of measure-inducing functions that satisfy the continuity conditions at the boundary of the domain and for which all \(I\)-marginal functions are measure-inducing.
As an extension of \eqref{defexpop_0}, and to allow for integration w.r.t.\,induced measures according to Definition~\ref{def_measure_inducing}, we define for a left-/right-continuous and measure-inducing function \(g\colon \Theta^m \to \mathbb{R}\,,\) for \(\Theta\in \{[0,1),[0,1],\mathbb{R}_+,\mathbb{R}\}\,,\) with associated signed measure \(\eta_g\) on \(\mathcal{B}(\Theta^m)\) and for a \(\eta_g\)-integrable function \(q\colon \Theta^m\to \mathbb{R}\,,\) the extended expectation operator
\begin{align}\label{defexpop4}
\psi_q(g):=\int_{\Theta^m} q(x) \mathsf{\,d} \eta_g(x)\,.
\end{align}
To derive price bounds w.r.t. quasi-copulas, we will also use the integration by parts operator
\begin{align}\label{defqeoqc}
\pi_g(q) &:= \sum_{I\subseteq \{1,\ldots,m\} \atop I\ne \emptyset} \int_{\Theta^{|I|}} q_I \mathsf{\,d} \eta_{g_I}(u) + q(0,\ldots,0)\cdot g(0,\ldots,0),
\end{align}
whenever \(g\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\Theta^m)\,.\) By an application of a multivariate integration by parts formula for Lebesgue integrals, it holds for every left-continuous function \(\xi\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\Theta^m)\) and for every bounded, grounded, right-continuous, and measure-inducing function \(h\colon \Theta^m\to \mathbb{R}\) satisfying continuity conditions \eqref{contboun1} and \eqref{contboun2} that
\begin{align}\label{eqqop}
\psi_\xi(h)=\pi_\xi(\widehat{h})
\end{align}
whenever the integrals exist,
see \cite[Theorem 3.1]{Ansari-2021}.
In particular, for every copula \(C\in \mathcal{C}_m\) and thus by \eqref{eq_sklar} for every distribution function \(F\in \mathcal{F}_+^m\) it follows that
\begin{align}\label{qeopexopeq}
\psi_f(C) = \pi_f(\widehat{C}) && \text{and} && \int_{\mathbb{R}_+^m} c(x) \mathsf{\,d} F(x) = \psi_c(F)= \pi_c(\widehat{F}) \,,
\end{align}
whenever \(f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1]^m)\) and \(c\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\mathbb{R}_+^m)\) are left-continuous.
Similar to \eqref{defexpop}, the integration by parts operator satisfies the transformation
\begin{align}\label{margtrafgs}
\pi_c(h\circ (F_1,\ldots,F_m))=\pi_{c\,\circ(F_1^{-1},\ldots,F_m^{-1})}(h)
\end{align}
for all
functions \(g:=c\circ(F_1^{-1},\ldots,F_m^{-1})\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1)^m)\,,\) for all measurable functions \(h\colon [0,1]^m\to \mathbb{R}\,,\) and for all distribution function \(F_1,\ldots,F_m\in \mathcal{F}_+^1\,,\) whenever the integrals exist,
see \cite[Proposition 3.6]{Ansari-2021}.
By the following two convergence results from \cite[Theorem 3.7 and Corollary 3.12]{Ansari-2021}, the Lebesgue-integral of a measure-inducing function may be extended to the case where one integrates w.r.t. to a quasi-copula or a quasi-distribution function that does not induce a signed measure.
First, we assume that the underlying space is \([0,1]^m\) and thus compact. In this case, every measure-inducing function induces a finite signed measure.
\begin{prop}[Convergence on a compact domain (\cite{Ansari-2021}, Theorem 3.7) ]\label{propconcomdom}~\\
For all \(1\leq i \leq m\) and \(n\in \mathbb{N}\,,\) let \(F_{i,n}:[0,1] \rightarrow [0,1]\) be the distribution function of a distribution with finite support in \((0,1)\,.\) Let \(Q\in \mathcal{Q}_m\) be a quasi-copula and let \(g\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1]^m)\) be left-continuous.\\
If \(F_{i,n}(x)\to x\) for all \(x\in [0,1]\) and \(1\leq i \leq m\,,\) then
\(\pi_g(\widehat{Q})\) exists and
\begin{align*}
\lim_{n\to \infty}\psi_g({Q\circ(F_{1,n},\ldots,F_{m,n})}) = \pi_g(\widehat{Q})\,.
\end{align*}
\end{prop}
In the case that the underlying space is \([0,1)^m\) or \(\mathbb{R}_+^m\) and thus non-compact, a measure-inducing function does not necessarily induce a finite signed measure. Thus, the following convergence result requires an additional integrability condition.
\begin{prop}[Convergence on a non-compact domain (\cite{Ansari-2021}, Corollary 3.12) ]\label{propconncomdom}~\\
For \(1\leq i \leq m\) and \(n\in \mathbb{N}\,,\) let \(F_{i,n}: [0,1] \rightarrow [0,1]\) be the distribution function of a distribution with finite support in \((0,1)\,.\)
Let \(Q\in \mathcal{Q}_m\) be a quasi-copula and \(g\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1)^m)\) be left-continuous.
Assume that the Lebesgue-integral \(\int_0^1 g_I(u,\ldots,u) \mathsf{\,d} u\) exists for all \(I\subseteq\{1,\ldots,m\}\,,\) \(I\ne \emptyset\,.\)\\
If \(F_{i,n}(x)\to x\) for all \(x\in [0,1]\) and \(1\leq i \leq m\,,\) then
\(\pi_g(\widehat{Q})\) exists and
\begin{align*}
\lim_{n\to \infty}\psi_g({Q\circ(F_{1,n},\ldots,F_{m,n})}) = \pi_g(\widehat{Q})\,.
\end{align*}
\end{prop}
Due to Proposition \ref{propconncomdom} and the transformation formula \eqref{margtrafgs}, we obtain for every left-continuous, measure-inducing function \(c\colon \mathbb{R}_+^m \to \mathbb{R}\) and for all distribution functions {\(F_1,\ldots,F_m\in \mathcal{F}_+^1\)} such that \(g:=c\circ (F_1^{-1},\ldots,F_m^{-1})\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1)^m)\) the existence of
\begin{align}\label{reptraf}
\pi_c(\widehat{Q}\circ \left(F_1,\ldots,F_m)\right)= \pi_c\left(\reallywidehat{Q\circ(F_1,\ldots,F_m)}\right)=\pi_g(\widehat{Q})
\end{align}
whenever \(u\mapsto g_I(u,\ldots,u)\) is Lebesgue-integrable on \([0,1)\) for all \(I\subseteq\{1,\ldots,m\}\,,\) \(I\ne \emptyset\,.\) Then,
the approximation of \(\pi_g(\widehat{Q})\) in Proposition \ref{propconncomdom} by a sequence of Lebesgue integrals w.r.t.\,signed measures motivates to consider an extension of the Lebesgue integral to the case where a measure-inducing function is integrated w.r.t.\,a quasi-distribution function, see \cite[Remark 3.13]{Ansari-2021}, compare \cite{Lux-2017}.
\begin{defn}[Quasi-expectation]~\label{def_quasi_ex}\\
Let \(H=Q\circ (F_1,\ldots,F_m):\mathbb{R}_+^m \rightarrow [0,1]\) be a quasi-distribution function of $Q \in \mathcal{Q}_m$ w.r.t.\,$F_1,\dots,F_m \in \mathcal{F}_+^1$ and let \(c\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\mathbb{R}_+^m)\) be left-continuous. Then, the \emph{quasi-expectation} of \(c\) w.r.t.\,\(H\) is defined by
\begin{align}\label{defquexpop}
\int_{\mathbb{R}_+^m} c(x) \mathsf{\,d} H(x):=~ & \pi_c(\widehat{H}) :=\sum_{I\subseteq \{1,\ldots,m\} \atop I\ne \emptyset} \int_{\mathbb{R}_+^{|I|}} \widehat{H}_I(x)\mathsf{\,d} \eta_{c_I}(x) + c(0,\ldots,0)\\
\nonumber=& \sum_{k=1}^m\sum_{I\subseteq \{1,\ldots,m\} \atop I=\{i_1,\ldots,i_k\}} \int_{\mathbb{R}_+^{|I|}} \widehat{Q}_I(F_{i_1}(x_1),\ldots,F_{i_k}(x_{i_k}))\mathsf{\,d} \eta_{c_I}(x_1,\ldots,x_k) + c(0,\ldots,0)\,,
\end{align}
whenever the integrals exist.
We also write
\begin{align}\label{defquexpop1}
\pi_c^{\mu}(\widehat{Q}):= \pi_c^{(F_1,\ldots,F_m)}(\widehat{Q}):=\pi_c(\widehat{H}) = \pi_c(\widehat{Q}\circ (F_1,\ldots,F_m))
\end{align}
where, for \(\mu=(\mu_1,\ldots,\mu_m)\,,\) \(F_i\) is the distribution function of the marginal distribution \(\mu_i \in \mathcal{P}(\mathbb{R}_+)\,,\) \(1\leq i \leq m\,.\)
\end{defn}
Note that for \(1\leq i \leq m\,,\) every \(F_i\in \mathcal{F}_+^1\) is, by definition of \(\mathcal{F}_+^1\), increasing and fulfils \(F_i(0)=0\) which implies that \(\widehat{H}_I=\widehat{Q}_I\circ (F_1,\ldots,F_m)\) and thus proves the second equality in \eqref{reptraf}.
\begin{rem}
\begin{itemize}
\item[(a)] The integrand \(\widehat{H}_I\) in \eqref{defquexpop} is the \(I\)-marginal function of the survival function \(\widehat{H}\) of \(H\) and does, in general, not coincide with the survival function \(\widehat{H_I}\) of $H_I$, see \cite[Example 2.17]{Ansari-2021}.
\item[(b)] If \(c,F\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}(\mathbb{R}_+^m)\) are both right-continuous and have common jump discontinuities, then the formula \(\psi_c(F)=\pi_c(\widehat{F})\) in \eqref{qeopexopeq} is in general not valid, see \cite[Example 3.4]{Ansari-2021}.
\item[(c)] For a measure-inducing function \(c\,,\) the \(I\)-marginal measure \(\eta_c^I\) of the signed measure \(\eta_{c}\) does in general not coincide with the signed measure \(\eta_{c_I}\) induced by the \(I\)-marginal of \(c\,,\) see \cite[Example 2.7]{Ansari-2021}.
\end{itemize}
\end{rem}
In Theorem \ref{theloob}, we determine analytic expressions for improved price bound w.r.t.\,\(\Delta\)-antitone and \(\Delta\)-monotone payoff functions under dependence constraints applying the following extension of \eqref{chaloord} and \eqref{chauoord} to quasi-copulas.
\begin{prop}[\cite{Lux-2017}, Theorem 5.5]\label{charortord}
Let \(Q,Q'\in \mathcal{Q}_m\) be quasi-copulas. Then, it holds that
\begin{itemize}
\item[(a)] \label{charortord1} \(Q\leq_{\operatorname{lo}} Q'\) if and only if \(\pi_f(\widehat{Q})\leq \pi_f(\widehat{Q'})\) for all left-continuous \(\Delta\)-antitone functions \\$f:[0,1)^m\rightarrow\mathbb{R}$ such that the integrals exist.
\item[(b)] \label{charortord2}\(Q\leq_{\operatorname{uo}} Q'\) if and only if \(\pi_f(\widehat{Q})\leq \pi_f(\widehat{Q'})\) for all left-continuous \(\Delta\)-monotone functions \\$f:[0,1)^m\rightarrow\mathbb{R}$ such that the integrals exist.
\end{itemize}
\end{prop}
In Theorem \ref{corsmifm}, we also derive analytic expressions for improved price bounds w.r.t.\,supermodular payoff functions when dependence restrictions are imposed by quasi-copulas. This motivates to introduce the supermodular ordering on the class \(\mathcal{Q}_m\,.\) As usual, denote by \(C^k([0,1]^m)\equiv C^k\,,\) \(k\in \mathbb{N}\cup\{\infty\}\,,\) the class of functions \(f\colon [0,1]^m\to \mathbb{R}\) such that all (mixed) partial derivatives of order \(k\) exist and are continuous.
We make use of the following lemma.
\begin{lem}[\cite{Ansari-2021}, Corollary 2.19]\label{lemmind}
Let \(f\colon [0,1]^m\to \mathbb{R}\) be a \(C^m\)-function. Then \(f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1]^m)\,.\)
\end{lem}
Since the supermodular ordering on \(\mathcal{C}^m\) is generated by the class \(C^\infty\cap \mathcal{F}_{\operatorname{sm}}\) of smooth supermodular functions, see \cite[Theorem 3.2]{Denuit-2002}, and since all \(I\)-marginals of a smooth function induce a signed measure due to Lemma \ref{lemmind}, we can extend the supermodular ordering to \(\mathcal{Q}_m\) as follows.
\begin{defn}[Supermodular ordering for quasi-copulas]\label{def_sm_qc}~\\
Let \(Q,Q'\in \mathcal{Q}_m\,.\) Then \(Q\) is said to be smaller than \(Q'\) in the supermodular ordering, written \(Q\leq_{\operatorname{sm}} Q'\,,\) if \(\pi_f(\widehat{Q})\leq \pi_f(\widehat{Q'})\) for all supermodular and left-continuous functions $f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1]^m)$.
\end{defn}
Note that \(f\) in the above definition is defined on a compact domain and thus induces a finite measure. Hence, \(\pi_f(\widehat{Q})\) is finite for every quasi-copula \(Q\in \mathcal{Q}_m\,,\) compare Proposition \ref{propconcomdom}.
\begin{rem}
\begin{itemize}
\item[(a)]
As in the copula case, \(Q\leq_{\operatorname{sm}} Q'\) implies \(Q\leq_c Q'\) for \(Q,Q'\in \mathcal{Q}_m\,,\) and the converse holds true only for \(m\leq 2\,,\) cf.
\cite[Theorem 2.6]{Mueller-2000}.
\item[(b)] Similar to the extension of the supermodular ordering to quasi-copulas, several stochastic orderings on \(\mathcal{F}^m\) that have a generator\footnote{In this context the term \emph{generator} refers to a class of integrands that characterizes the integral-ordering.} consisting only of infinitely differentiable functions can be generalized to quasi-copulas and quasi-distribution functions. Some examples of such orderings are provided in \cite[Theorem 3.2]{Denuit-2002}.
\end{itemize}
\end{rem}
\subsection{Internal Factor Models}\label{sec IFM}
In this section, we consider internal factor models with bivariate dependence information. We first give a brief overview on the notion of the upper product of bivariate copulas which describes the worst case dependence structure (w.r.t.\,\(\leq_{\operatorname{sm}}\)) in partially specified factor models (PSFMs). As a main result, we then extend by Theorem \ref{theqcub} the upper product ordering result in \cite[Theorem 1]{Ansari-Rueschendorf-2020} to quasi-copulas. In Section \ref{sec improved price bounds} we build on this results to derive an
upper price bound for a supermodular payoff function which improves the comonotonic standard bound\footnote{For \(\mathbb{R}\)-valued random variables \(X_1,\ldots,X_m\) with distribution functions \(F_1,\ldots,F_m\,,\) the vector \(X^c:=(X_1^c,\ldots,X_m^c):=(F_1^{-1}(U),\ldots,F_m^{-1}(U))\,,\) with \(U\) uniformly distributed on \((0,1)\,,\) is comonotonic. It holds that \(X_i^c\stackrel{\mathrm{d}}= X_i\) for all \(i\) and the copula of \(X^c\) is the upper Fr{\'e}chet copula \(M^m\,.\) Hence, \(X^c\) has the maximal distribution (w.r.t.\,the supermodular ordering) in the class of all distributions with fixed marginals \(F_1,\ldots,F_m\,,\) and thus it is referred to as (comonotonic) standard upper bound.} based on knowledge of the marginal distributions.
In a PSFM, a random vector \(X=(X_i)_{1\leq i\leq m}:=(f_i(Z,\varepsilon_i))_{1\leq i\leq m}\,,\) is expressed through Borel-measurable functions $f_i \colon \mathbb{R}^{2}\to\mathbb{R}$
of an \(\mathbb{R}\)-valued random variable \(Z\) and \(\mathbb{R}\)-valued random variables \(\varepsilon_i\) for $i=1,\dots,m$, where the common (risk) factor \(Z\) is assumed to be independent of \((\varepsilon_i)_{1\leq i\leq m}\). Moreover, the bivariate distributions \((X_i,Z)_{1\leq i\leq m}\) are specified, i.e., the univariate distributions of \(X_i\) and \(Z\) as well as the common copula \(D^i\) are known for all \(1\leq i \leq m\,.\) However, in contrast to the usual independence assumption, the dependence structure among the vector of idiosyncratic risks \((\varepsilon_i)_{1\leq i \leq m}\) is not specified, see \cite{Bernard-2017}.
The maximal random vector (w.r.t. \(\leq_{\operatorname{sm}}\)) in the PSFM is given by the conditionally comonotonic vector \(X_Z^c:=\left(F_{X_i|Z}^{-1}(U)\right)_{1\leq i \leq m}\), where $F_{X_i|Z}^{-1}$ denotes the generalized inverse of the conditional distribution function of $X_i$ given $Z$ and where \(U\) is uniformly distributed on \((0,1)\) and independent of \(Z\,.\) If \(Z\) has a continuous distribution function, the copula of \(X_Z^c\), in the sense of \eqref{eq_sklar}, is given by the upper product \(\bigvee_{i=1}^m D^i \equiv D^1\vee \cdots \vee D^m\) of the bivariate copulas \(D^1,\ldots,D^m\) which is an \(m\)-copula defined by
\begin{align*}
\bigvee_{i=1}^m D^i(u) := \int_0^1 \min_{1\leq i\leq m}\{\partial_2 D^i(u_i,t)\} \mathsf{\,d} t~~~ \text{for } u=(u_1,\ldots,u_m)\in [0,1]^m\,,
\end{align*}
where \(\partial_2\) denotes the partial derivative w.r.t.\,the second component, see \cite{Ansari-Rueschendorf-2018}.
In a partially specified \emph{internal} factor model (IFM), it is assumed that the factor \(Z=X_1\) is a component of the vector \((X_1,\ldots,X_m)\) and, thus, the first bivariate dependence constraint \(D^1\) is imposed by the upper Fr\'{e}chet copula, i.e., \(D^1=M^2\,.\) For a bivariate copula \(E\in \mathcal{C}_2\,,\) the class of IFMs with dependence specifications \(D^k\leq_{\operatorname{lo}} E\,,\) \(k=2,\ldots,m\,,\) has a greatest element w.r.t. the supermodular ordering given by the \(m\)-variate upper product
\begin{align}\label{eqcococo}
M^2\vee E\vee \cdots \vee E \,(u)=E\left(\min_{2\leq i \leq m}\{u_i\},u_1\right)\,,
\end{align}
for \(u=(u_1,\ldots,u_m)\in [0,1]^m\,,\)
see \cite[Theorem 1]{Ansari-Rueschendorf-2020}.
In the following theorem, we generalize this result to dependence specifications \(D^k\leq_{\operatorname{lo}} Q_2\,,\) \(k=2,\ldots,m\,,\) for a fixed given bivariate quasi-copula \(Q_2\in \mathcal{Q}_2\) which serves as an upper bound for each $D^2\,,\ldots,D^m\,$. Thus, if $Q_2$ is implied by market price information, this allows us to incorporate bivariate dependence information inferred from market prices. This is of particular relevance as, according to Lemma~\ref{lemqcp}, such price information often only corresponds to a quasi-copula that serves as an upper bound for the dependence structure.
We make use of the property that, whenever a left-continuous function \(f\colon [0,1]^m\to \mathbb{R}\) is supermodular and componentwise increasing/componentwise decreasing and \(m\geq 2\,,\) the function \(\phi_f\colon [0,1]^2\to \mathbb{R}\) defined by
\begin{align}\label{defphif}
\phi_f(x_1,x_2):=f(x_2,x_1,\ldots,x_1)
\end{align
is \(\Delta\)-monotone/-antitone and, thus, for all \(I\subseteq\{1,\ldots,m\}\,,\) \(I\ne \emptyset\,,\) the \(I\)-marginal induces a positive measure by \eqref{eqindmeas}, see, e.g. \cite[Corollary 2.24]{Ansari-2021}. This enables us to establish an \(m\)-variate quasi-copula \(Q^*\) as an upper bound (w.r.t.\,the supermodular ordering) for the upper products \(M^2\vee D^2\vee \cdots \vee D^m\,,\) \(D^k\leq_{\operatorname{lo}} Q_2\,,\) which describe the worst case dependence structures in IFMs. The quasi-copula \(Q^* \in \mathcal{Q}_m\) given by
\begin{align}\label{deqcqs}
Q^*(u):=Q_2\left(\min_{2\leq i \leq m}\{u_i\},u_1\right)\,,~~~ u=(u_1,\ldots,u_m)\in [0,1]^m\,,
\end{align}
relates to the conditionally comonotonic structure in \eqref{eqcococo} and, thus, can be associated with the two-dimensional case. Note that the non-intuitive arrangement of the arguments in \eqref{defphif} and \eqref{deqcqs} is a consequence of the definition of the upper product where the copulas in the integrand are differentiated w.r.t.\,the second component.
\begin{thm}\label{theqcub}
Let \(D^2,\ldots,D^m\in \mathcal{C}_2\) be bivariate copulas and \(Q_2\in \mathcal{Q}_2\) a bivariate quasi-copula. Then, \(Q^*\) defined by \eqref{deqcqs}
is a quasi-copula, and the following statements are equivalent.
\begin{itemize}
\item[(a)] \label{theqcub1}\(D^i\leq_{\operatorname{lo}} Q_2\) for all \(2\leq i\leq m\,,\)
\item[(b)] \label{theqcub2}\(M^2\vee D^2\vee \ldots\vee D^m \leq_{\operatorname{lo}} Q^*\,,\)
\item[(c)] \label{theqcub2a}\(M^2\vee D^2\vee \ldots\vee D^m \leq_{\operatorname{uo}} Q^*\,,\)
\item[(d)] \label{theqcub2b}\(M^2\vee D^2\vee \ldots\vee D^m \leq_{\operatorname{c}} Q^*\,,\)
\item[(e)] \label{theqcub3}\(M^2\vee D^2\vee \ldots\vee D^m \leq_{\operatorname{sm}} Q^*\,.\)
\item[(f)] \label{theqcub6}\(\psi_f({M^2\vee D^2\vee \cdots \vee D^m})\leq \pi_{\phi_f}(\widehat{Q_2})\) \\for all
left-continuous, supermodular functions \(f\colon [0,1)^m\to \mathbb{R}\) which are componentwise increasing/componentwise decreasing such that \((\phi_f)_I\) is Lebesgue integrable on \([0,1)^{|I|}\) for \(I\subseteq\{1,2\}\,,\) \(I\ne \emptyset\,,\) where \(\phi_f\) is defined by \eqref{defphif}, and such that $f$ is lower bounded by some function which is integrable w.r.t.\,$M^2\vee D^2\vee \ldots\vee D^m$.
\end{itemize}
\end{thm}
\begin{rem}\label{remmaithean1}
\begin{itemize}
\item[(a)] \label{remmaithean1a} In the special case that \(Q_2=E\in \mathcal{C}_2\) is a copula, the upper bound \(Q^*\) in Theorem \ref{theqcub} simplifies by \eqref{deqcqs} and \eqref{eqcococo} to \(M^2\vee E\vee \cdots \vee E\,,\) which implies the result from
\cite[Theorem 1]{Ansari-Rueschendorf-2020}.
\item[(b)] \label{remmaithean1b} Let \(D^2,\ldots,D^m\in \mathcal{C}_2\) be copulas and \(Q^2,\ldots,Q^m\in \mathcal{Q}_2\) be quasi-copulas with \(D^i\leq_{\operatorname{lo}} Q^i\) for all \(i=2,\ldots, m\,.\) Then, \(Q_2\) defined by \(Q_2(u):=\max_{i=2,\dots,m}\{Q^i(u)\}\,,\) \(u\in [0,1]^2\,,\) is by \eqref{eqmxqc} a quasi-copula. Hence, Theorem \ref{theqcub}~(e) implies that
\begin{align*}
M^2\vee D^2\vee \cdots \vee D^m \leq_{\operatorname{sm}} Q^*
\end{align*}
with \(Q^*\) defined by \eqref{deqcqs}.
\item[(c)] Note that Theorem~\ref{theqcub}~(f) improves the standard bound \(\mathbb{E}[c(X_1^c,\ldots,X_m^c)]\) for the expectation \(\mathbb{E}[c(X_1,\ldots,X_m)]\) of a random vector \((X_1,\ldots,X_m)\) w.r.t.\,a continuous and supermodular payoff function \(c\colon [0,1]^m \to \mathbb{R}\)
if for all $k$ the copula of $(X_1,X_k)$ is upper bounded by $Q_2\in \mathcal{Q}_2$ in the lower orthant order, even if \(c\) is not measure-inducing. \\
As an example of such a payoff function, let \(K>0\). Then, the function \(c(u_1,\ldots,u_m):=\left(\sum_{i=1}^m u_i-K\right)_+\,,\) \((u_1,\ldots,u_m)\in [0,1]^m\,,\) is continuous and supermodular. But \(c\) is measure-inducing only if \(m\leq 2\), which can be seen from the fact that the lower Fr\'{e}chet bound \(W^m\) in \eqref{frebou} induces a signed measure if and only if \(m\leq 2\,,\) see \cite[Theorem 2.4]{Nelsen-2010}. However, since \(\phi_c\) given by \(\phi_c(u_1,u_2)=\left((m-1)u_1+u_2-K\right)_+\) is measure-inducing, we obtain by Theorem \ref{theqcub}~(f) that \(\pi_{\phi_c}(\widehat{Q_2})\) is an upper bound for \(\psi_c({M^2\vee D^2\vee \cdots \vee D^m})\) if \(D^i\leq_{\operatorname{lo}} Q_2\) for all \(i\in \{2,\ldots,m\}\,,\) see also Lemma~\ref{lembasopt}. In Example~\ref{exasmpf} we apply the characterization in Theorem~\ref{theqcub}~(f) and determine an improved upper price bound for a basket call option under dependence information related to an internal factor model.
\end{itemize}
\end{rem}
\section{Improved Price Bounds under Dependence Information}\label{sec improved price bounds}
In this section, we make use of the notions and results of Section~\ref{sec dependence modelling} to derive price bounds on financial derivatives under dependence restrictions.
First, we consider the case of \(\Delta\)-monotonic or \(\Delta\)-antitonic payoff functions and derive price bounds that take into account upper and lower quasi-copula bounds which can be inferred from market prices of multi-asset derivatives. These bounds are derived analogously to the approach from \cite{Lux-2017}. Through an application of the duality in Theorem \ref{theaddcon}, we show how these price bounds can significantly be improved when the martingale property is incorporated as a linear constraint, see also Example~\ref{exdelan} and Example~\ref{exadelmo}.
Further, we state an upper price bound for supermodular payoff functions, where we assume an internal factor model with additional dependence information derived from multi-asset options which restrict the bivariate \((1,k)\)-marginal copula of \(S_{t_i}=(S_{t_i}^1,\ldots,S_{t_i}^d)\) for all $k=2,\ldots, d$.
Finally, in Section~\ref{secadddepinreltocor}, we present various scenarios where dependence information related to risk-neutral correlation is inferred from option prices and incorporated by linear constraints restricting the class \(\mathcal{M}(\mu)\) of martingale measures with fixed marginal distributions.
\subsection{Dependence Information Through Copulas}\label{secadddep}
For any $i=1,\ldots,n,$ $k=1,\ldots, d$, and for any probability measure $\mathbb{Q}\in\mathcal{M}(\mu)$, we recall that we denote by $F^k_i$ the univariate marginal distribution function of the component $S_{t_i}^k$ under \(\mathbb{Q}\,.\) By Sklar's Theorem, the multivariate distribution function $F_\mathbb{Q}=\mathbb{Q}(S\leq \cdot)$ is decomposed by
\begin{align}\label{theSklar}
F_\mathbb{Q}(x)=C_{\mathbb{Q}}(F_1^1(x_{1}^1),\ldots,F_n^d(x_n^d)),\quad \mbox{for all}\quad x=(x_1^1,\ldots,x_n^d)\in\mathbb{R}^{nd}_+,
\end{align}
into the univariate marginal distribution functions $F_i^k$ and a copula $C_\mathbb{Q}\in \mathcal{C}_{nd}$ which describes the dependence structure among \(S\) under \(\mathbb{Q}\,.\)
As we will show in Section \ref{section_numerics}, when traded market prices of appropriate multi-asset options can be observed in the market, this allows to infer restrictions on the dependence structure of the underlying assets through (pointwise) upper and lower quasi-copula bounds \(\underline{Q}\) and \(\overline{Q}\) on the copula $C_\mathbb{Q}$.
In the sequel we explain how these bound can be included as inequality constraints in the Problem \eqref{equ primal problem M_lin} to derive improved price bounds for a given payoff $c$.
First, we discuss some relevant classes of models where the copulas of the models, with associated risk-neutral distributions in \(\mathcal{M}(\mu)\), are restricted w.r.t.\,the lower and upper orthant ordering, respectively.
\begin{enumerate}[1)]
\item \textbf{Copula bounds w.r.t.\,the orthant orders}\\
Let \(\underline{Q},\overline{Q}\in \mathcal{Q}_{nd}\) be \(nd\)-variate quasi-copulas such that \(\underline{Q}\leq_{\operatorname{lo}} \overline{Q}\,.\)
Then, we consider the class
\begin{align}\label{eqdefclloor}
\mathcal{M}^{\operatorname{lo}}_{\underline{Q},\overline{Q}} := \left\{\mathbb{Q}\in \mathcal{M}(\mu)\,\middle|\, \underline{Q} \leq_{\operatorname{lo}} C_{\mathbb{Q}}\leq_{\operatorname{lo}} \overline{Q}\right\}
\end{align}
of probability measures \(\mathbb{Q} \in \mathcal{M}(\mu)\) with univariate marginal distributions \(\mu=(\mu_i^k)_{1\leq i\leq n}^{1\leq k\leq d}\) such that the copula \(C_{\mathbb{Q}}\) is lower bounded by $\underline{Q}$ and upper bounded by $\overline{Q}$ w.r.t.\,the lower orthant ordering.
If the quasi-copula $\underline{Q}$ coincides with the lower Fr\'{e}chet bound \(W^{nd}\), or if $\overline{Q}$ coincides with the upper Fr\'{e}chet bound \(M^{nd}\), then the dependence structure in \eqref{eqdefclloor} is only restricted from one-side if both quasi copulas $\underline{Q}$ and $\overline{Q}$ coincide with the respective Fr\'{e}chet bounds, then no additional dependence restriction on the class \(\mathcal{M}(\mu)\) is imposed and we have that \(\mathcal{M}^{\operatorname{lo}}_{W^{nd},M^{nd}}=\mathcal{M}(\mu)\,.\) In the one-period case $n=1$, the martingale property reduces to a constraint on the expectation and, thus, the dependence structures in \(\mathcal{M}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}\) are only restricted by \(\underline{Q}\) and \(\overline{Q}\,.\)
The following observation turns out to be crucial for the implementation of copula constraints via trading strategies in Theorem~\ref{theloob}.
\begin{lem}\label{lem_indicator_constraints}
Let \(\underline{Q},\overline{Q}\in \mathcal{Q}_{nd}\) and define for $x \in \mathbb{R}^{nd}$ the functions $f_x:=\one_{\{\cdot \leq x\}},~g_x:=\one_{\{\cdot < x\}}$, \(\widetilde{f}_x:=\one_{\{\cdot\, > x\}},~\widetilde{g}_x:=\one_{\{\cdot\, \geq x\}}\) where the inequalities in the indicator functions are meant componentwise. Let $F_i^k(\cdot) = \int_{-\infty}^ \cdot \,\D \mu_i^k$, $1\leq k \leq d$, $1\leq i \leq n$, and let \(\mathds{Q_+}\) denote the set of non-negative rational numbers. Then the following holds.
\begin{itemize}
\item[(a)]
We have that \(\underline{Q} \leq_{\operatorname{lo}} C_{\mathbb{Q}}\leq_{\operatorname{lo}} \overline{Q}\) for $\mathbb{Q} \in \mathcal{M}(\mu)$ is equivalent to a countable number of inequality constraints of the form
\begin{align}\label{eqineqconstr}
\begin{array}{l}
\phantom{-}\mathbb{E}_{\mathbb{Q}} \left[g_x(S)\right]\leq \phantom{-}\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))\\[2mm]
\mathbb{E}_{\mathbb{Q}} \left[-f_x(S)\right]\leq -\underline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))
\end{array}
~\forall\, x=\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}\,.
\end{align}
\item[(b)]
We have that \(\underline{Q} \leq_{\operatorname{uo}} C_{\mathbb{Q}}\leq_{\operatorname{uo}} \overline{Q}\) for $\mathbb{Q} \in \mathcal{M}(\mu)$ is equivalent to a countable number of inequality constraints of the form
\begin{align}\label{eqineqconstr_uo}
\begin{array}{l}
\phantom{-}\mathbb{E}_{\mathbb{Q}} \left[\widetilde{f}_x(S)\right]\leq \phantom{-}\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))\\[2mm]
\mathbb{E}_{\mathbb{Q}} \left[-\widetilde{g}_x(S)\right]\leq -\underline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))
\end{array}
~\forall\, x=\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}\,.
\end{align}
\end{itemize}
\end{lem}
On account of Lemma~\ref{lem_indicator_constraints} the inequality constraints \eqref{eqineqconstr} specify $(f_i^{\operatorname{ineq}})_{i\in I^{\operatorname{ineq}}}$ and $(K_i^{\operatorname{ineq}})_{i\in I^{\operatorname{ineq}}}$ in \eqref{eq_restriction}. More precisely, e.g., in the case considered in Lemma~\ref{lem_indicator_constraints}~(a) we take into account
\begin{align*}
\left\{f_i^{\operatorname{ineq}}~\middle|~i\in I^{\operatorname{ineq}} \right\} &= \{g_x, x\in \mathds{Q}_+^{nd} \}\cup \{-f_x, x\in \mathds{Q}_+^{nd} \}\subset L_{\operatorname{lin}}(\mathbb{R}^{nd}),\\
\left\{ K_i^{\operatorname{ineq}}~\middle|~i\in I^{\operatorname{ineq}}\right\}&=\left\{\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))~\middle|~\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}\right\} \\
&\hspace{1cm}\cup \left\{-\underline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))~\middle|~\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}\right\}.
\end{align*}
In the following Theorem \ref{theloob}, which is partly a consequence of Theorem~\ref{thm_duality_linear_constraints}, we identify the upper price bound \(\overline{P}_{\mathcal{M}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}}\) of a payoff $c\in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})$ with the infimal price over all super-replicating strategies which involve trading in digital options with payoffs $g_x$, $-f_x$, \(x\in \mathds{Q}_+^{nd}\,.\) As a consequence of Definition~\ref{def_sm_qc}, in the case of a \(\Delta\)-antitone payoff function \(c\in \mathcal{F}_\Delta^-\), which is left-continuous and measure-inducing, an upper bound for \(\overline{P}_{\mathcal{M}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}}\) is given by a quasi-expectation w.r.t.\,\(\overline{Q}\,.\)
Similar to the model in \eqref{eqdefclloor}, by means of Lemma~\ref{lem_indicator_constraints}~(b) we also derive improved price bounds for the class
\begin{align*}
\mathcal{M}^{\operatorname{uo}}_{\underline{Q},\overline{Q}} := \{\mathbb{Q}\in \mathcal{M}(\mu)\,|\, \underline{Q} \leq_{\operatorname{uo}} C_{\mathbb{Q}}\leq_{\operatorname{uo}} \overline{Q}\}
\end{align*}
of risk-neutral distributions with dependence restrictions w.r.t. the upper orthant order. In this case we obtain, for every $\Delta$-monotone payoff function \(c\in \mathcal{F}_\Delta\) which is left-continuous and measure-inducing, an upper bound for \(\overline{P}_{\mathcal{M}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\) which is given by a quasi-expectation w.r.t.\,\(\overline{Q}\,.\) Examples of typical \(\Delta\)-antitone and \(\Delta\)-montone payoff functions are provided in Table \ref{table_payoff_functions_overview} (see also \cite[Table 1]{Lux-2017,Roncalli-2001,Tankov-2011}).
\begin{thm}[Upper price bounds with \(\leq_{\operatorname{lo}}\)- and \(\leq_{\operatorname{uo}}\)-contraints]\label{theloob}~\\
Let \(c\in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\,.\)
\begin{itemize}
\item[(a)] \label{theloob1} If \({\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}\ne \emptyset\,,\) then
\begin{align}\label{theloob1a}
\overline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}} &= \max_{\mathbb{Q}\in {\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}} \mathbb{E}_\mathbb{Q} \left[c(S)\right] = \underline{\mathcal{D}}_{\mathcal{S}}\,.
\end{align}
In particular, if \(c\in \mathcal{F}_\Delta^-\cap U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) is left-continuous, then
\begin{align}\label{eqlosb}
\overline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}} \leq \pi_c^{\mu} (\widehat{\overline{Q}})\,.
\end{align}
\item[(b)] \label{theloob2} If \({\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}\ne \emptyset\,,\) then
\begin{align*}
\overline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}} &= \max_{\mathbb{Q}\in {\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}} \mathbb{E}_\mathbb{Q} \left[c(S)\right] = \underline{\mathcal{D}}_{\mathcal{S}}\,.
\end{align*}
In particular, if \(c\in \mathcal{F}_\Delta\cap U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) is left-continuous, then
\begin{align}\label{equosb}
\overline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}} \leq \pi_c^{\mu} (\widehat{\overline{Q}})\,.
\end{align}
\end{itemize}
\end{thm}
\begin{rem}\label{remgenan}
\begin{itemize}
\item[(a)] Similar to Theorem \ref{theloob}, we also obtain dual lower bounds under consideration of the martingale property as well as lower bounds \(\pi_c^{\mu} (\widehat{\underline{Q}})\) and \(\pi_c^{\mu} (\widehat{\underline{Q}})\) for \(\underline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}}\) and \(\underline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\) in the case of a \(\Delta\)-antitone and a \(\Delta\)-monotone payoff function, respectively, which are left-continuous and measure-inducing.
\item[(b)] By incorporating the martingale condition as a linear constraint, Theorem \ref{theloob} improves the dual risk bounds considered in \cite[Theorem 3.2]{Rueschendorf-2019} as well as the quasi-copula bounds obtained in \cite{Lux-2017}.
\item[(c)] Since the martingale property also restricts the dependence structure of \(\mathcal{M}(\mu)\,,\) the class \({\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}\) might be empty for too restrictive choices of the quasi-copulas \(\underline{Q}\) and \(\overline{Q}\,.\) However, if the market is free of \emph{model-independent arbitrage}, compare \cite[Definition 1.2.]{acciaio2016model}, and if the dependence restrictions are inferred from option prices with continuous payoff functions, then there exists a martingale measure \(\mathbb{Q}\in \mathcal{M}(\mu)\) for which the bounds \(\underline{Q}\) and \(\overline{Q}\) for the copula of \(\mathbb{Q}\) are consistent. This follows by \cite[Theorem 1.3.]{acciaio2016model}, for which we additionally need to assume the existence of a convex superlinear payoff that can be bought.
\item[(d)] \label{remgenan1}
Note that by Definition~\ref{def_orthant_orders}~(b) we have
$$
{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}} =\left\{\mathbb{Q}\in \mathcal{M}(\mu)~\middle|~\widehat{\underline{Q}}(u)\leq \widehat{C_\mathbb{Q}}(u)\leq \widehat{\overline{Q}}(u) \text{ for all } u\in [0,1]^{nd} \right\}.
$$
This allows to extend \eqref{equosb} to bounded measurable functions $\widehat{\underline{Q}}, \widehat{\overline{Q}}$, defined on $[0,1]^{nd}$ which are not survival functions of quasi-copulas. Indeed, the identity in \eqref{equosb} is still valid due to the positivity of the measures \(\eta_{c_I}\) induced by the \(\Delta\)-monotone function \(c\) applied in \eqref{defquexpop} for the definition of $\pi_c^\mu$ and due to the pointwise upper bound \(\widehat{Q^I}\) for \(\widehat{C_\mathbb{Q}^I}\) for all \(I\subseteq \{1,\ldots,nd\}\,\), see also Example~\ref{exadelmo}
\end{itemize}
\end{rem}
\item \textbf{Upper bounds in internal factor models}\\
Liquidly traded options written on each pair of the underlying assets $S^1,\ldots, S^d$ which allow to derive inter-asset dependence information are often not available.
However, prices of derivatives written on a main reference asset \(S^1_{t_i}\) and another asset \(S^{k}_{t_i}\) at the same time \(t_i\) may be available for \(k=2,\ldots,d\,.\) For example, digital options on \(S^1\) and \(S^2\) as well as on \(S^1\) and \(S^3\) may be traded, but not on \(S^2\) and \(S^3\,.\) Therefore, we consider the case where prices of specific derivatives as a function of an asset \(S^1_{t_i}\) and assets \(S^{k}_{t_i}\) at time \(t_i\) are known for \(k=2,\ldots,d\,.\) This price information then implies by \cite[Theorem 3.1]{Lux-2017} an upper quasi-copula bound \(Q^k\in \mathcal{Q}_2\) (w.r.t.\,the lower orthant ordering) for the copula \(C_{\mathbb{Q}_i^{1,k}}\) of the risk-neutral distribution \(\mathbb{Q}_i^{1,k}\) of \((S_{t_i}^1,S_{t_i}^k)\,,\) \(k=2,\ldots,d\,,\) under some pricing measure \(\mathbb{Q} \in \mathcal{M}(\mu)\) because then the value of the copula \(C_{\mathbb{Q}_i^{1,k}}\) is known on a corresponding compact set, compare \eqref{eqdigoppri1} and \eqref{eqdigoppri3}. Here \(\mathbb{Q}_i^{1,k}\) denotes the bivariate \((1,k)\)-marginal distribution of \(\mathbb{Q}\) at time \(t_i\,.\)
Taking the pointwise maximum $Q_2:=\max_{k=2,\dots,d}Q^k$ over these quasi-copula bounds then yields a quasi-copula \(Q_2\in \mathcal{Q}_2\) as a pointwise upper bound for the associated copulas \(\{C_{\mathbb{Q}_i^{1,k}}\,, k=2,\ldots,d\},\) see Example~\ref{exasmpf}.
Given marginal distributions $\mu=(\mu_1^1,\dots,\mu_n^d)$ with $\mu_i^k\in \mathcal{P}(\mathbb{R}_+)$ for all $i=1,\dots,n$, $k=1,\dots,d$, a quasi copula $Q_2\in \mathcal{Q}_2$, and some time $t_i$, we consider the class
\begin{align}\label{eqdefclinfamo}
\mathcal{M}_{Q_2}^{\operatorname{IFM}}&:= \left\{\mathbb{Q}\in \mathcal{M}(\mu)\,\middle|\, C_{\mathbb{Q}_i^{1,k}}\leq_{\operatorname{lo}} Q_2 ~\text{for all } k=2,\ldots,d\right\}
\end{align}
of measures \(\mathbb{Q}\) in \(\mathcal{M}(\mu)\) such that for all $k=2,\dots,d$ the copula \(C_{\mathbb{Q}_i^{1,k}}\) associated with the bivariate component \((S_{t_i}^1,S_{t_i}^k)\) under \(\mathbb{Q}\) is upper bounded by \(Q_2\) w.r.t.\,the lower orthant ordering. In particular, this class is another example for a specification of the set $\mathcal{M}^{\operatorname{lin}}$. Elements of this class correspond to internal factor models (IFM) with bivariate dependence specification sets as described in Section \ref{sec IFM}. Here the market-implied dependence structure of \((S_{t_i}^1,\ldots,S_{t_i}^d)\) is restricted to the subclass of copulas having the property that the bivariate marginal copula \(C_{\mathbb{Q}_i^{1,k}}\) associated with \((S_{t_i}^1,S_{t_i}^k)\) belongs to the constrained specification set \(\{C\in \mathcal{C}_2\,, C\leq_{\operatorname{lo}} Q_2\}\) for all \(k=2,\ldots,d\,,\) see also \cite{Ansari-Rueschendorf-2020}.
If \(Q_2\) is the upper Fr\'{e}chet copula \(M^2\), no dependence restrictions are imposed, and it follows that \(\mathcal{M}_{Q_2}^{\operatorname{IFM}} = \mathcal{M}(\mu)\,.\)
As a consequence of Theorem \ref{theqcub}, the dependence structure of \(S_{t_i}=(S_{t_i}^1,\ldots,S_{t_i}^d)\) in \(\mathcal{M}_{Q_2}^{\operatorname{IFM}}\) has an upper bound w.r.t. \(\leq_{\operatorname{sm}}\) given by the quasi-copula in \eqref{deqcqs}.
This yields even for a supermodular payoff function \(c\in \mathcal{F}_{\operatorname{sm}}\) a representation of an upper bound for \(\overline{P}_{\mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}}\) in form of an analytic expression depending on the quasi copula \(Q_2\,.\)
To apply the duality result from Theorem~\ref{thm_duality_linear_constraints}, we denote by $\mathbb{R}_+^{nd}\ni x=(x_1^1,\dots,x_n^d)\mapsto \operatorname{proj}_i^k(x)=x_i^k \in \mathbb{R}_+$ the projection of $x$ onto its $(i,k)$-th component.
We take in this setting inequality constraints of the form
\begin{align*}
\mathbb{E}_\mathbb{Q} [g_{x,y}(S_{t_i}^1,S_{t_i}^k)]\leq Q_2(F_{i}^1(x),F_{i}^k(y))\,,~~~g_{x,y}(\cdot):=\one_{\{\cdot < (x,y)\}}\,, (x,y)\in \mathds{Q}_+^2\,, 2\leq k\leq d\,,
\end{align*}
into account, i.e., we have
\begin{align*}
\left\{f_j^{\operatorname{ineq}}~\middle|~ j\in I^{\operatorname{ineq}}\right\} &= \bigcup_{k=2}^d\left\{(g_{x,y}\circ\operatorname{proj}_i^1,g_{x,y}\circ\operatorname{proj}_i^k)~\middle|~ (x,y)\in \mathds{Q}_+^{2} \right\} \subset L_{\operatorname{lin}}(\mathbb{R}^{2}),\\
\left\{K_j^{\operatorname{ineq}}~\middle|~ j\in I^{\operatorname{ineq}}\right\}&=\bigcup_{k=2}^d\left\{Q_2(F_{i}^1(x),F_{i}^k(y))~\middle|~ (x,y)\in \mathds{Q}_+^{2} \right\}.
\end{align*}
Then we derive the following duality result.
\begin{thm}[Upper price bounds with contraints related to IFMs]\label{corsmifm}~\\
Assume that \(\mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}\ne \emptyset\,.\) Then, the following holds.
\begin{itemize}
\item[(a)]
Let \({c}\in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\), then
\begin{align}\label{eqcorsmifm1}
\overline{P}_{\mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}} &= \max_{\mathbb{Q}\in \mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}} \mathbb{E}_\mathbb{Q} [c(S)] = \underline{\mathcal{D}}_{\mathcal{S}}\,.
\end{align}
\item[(b)]
Let \(\widetilde{c}\in \mathcal{F}_{\operatorname{sm}}\cap C_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) be componentwise increasing/componentwise decreasing. Then, we have that the function $c:=\left(\widetilde{c} \circ \operatorname{proj}_i^1,\cdots, \widetilde{c} \circ \operatorname{proj}_i^d\right) \in \mathcal{F}_{\operatorname{sm}}\cap C_{\operatorname{lin}}(\mathbb{R}_+^{d})$ is also componentwise increasing/componentwise decreasing and
\begin{align}\label{eqcorsmifm2}
\overline{P}_{\mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}} \leq \pi_{\phi_{c\,\circ\left((F_i^1)^{-1},\ldots,(F_i^d)^{-1}\right)}} (\widehat{Q_2})\,\text{ for all } i=1,\dots,n,
\end{align}
with
\(\phi_{c\,\circ\left((F_i^1)^{-1},\ldots,(F_i^d)^{-1}\right)}\) defined by \eqref{defphif}.
\end{itemize}
\end{thm}
In particular, Theorem~\ref{corsmifm}~(b) can be applied to payoff functions that depend on multiple assets but only on one specific maturity.
\begin{rem}
\begin{enumerate}[(a)]
\item Theorem \ref{corsmifm} allows a closed-form representation of an upper price bound for a componentwise increasing/componentwise decreasing supermodular payoff function by an integral of the bivariate dependence constraint \(Q_2\,.\) Examples for common supermodular payoff functions which are componentwise increasing/componentwise decreasing are listed in Table~\ref{table_payoff_functions_overview}. In particular, if \(\varphi\colon \mathbb{R}\to \mathbb{R}\) is (increasing and) convex, then \(\varphi\left(\sum_{i=1}^d x_i\right)\) is (increasing and) supermodular.
\item A similar result as the equalitity in \eqref{eqcorsmifm1} holds true for lower bounds. However, an analogon of \eqref{eqcorsmifm2} for lower bounds cannot be obtained for general dimension since upper product ordering results are different from ordering results for lower products which correspond to lower bounds in partially specified factor models, see \cite{Ansari-Rueschendorf-2021}.
\end{enumerate}
\end{rem}
Table~\ref{table_payoff_functions_overview} shows several payoff functions of financial derivatives which fall into one of the previously considered function classes, compare also \cite[Table 1]{Lux-2017,Roncalli-2001,Tankov-2011}.
Note that all \(\Delta\)-monotone and \(\Delta\)-antitone functions are also supermodular.
\end{enumerate}
\begin{center}
\begin{table}[h!]
\begin{tabular}{lll} \toprule
Name &Payoff $c(x_1,\dots,x_m)$ & Type \\ \midrule
digital put on the maximum &\(\one_{\{\max_{1\leq i\leq d}\{x_i\}\leq K\}}\) &\(\Delta\)-antitone, supermodular \\
put on the maximum &\((K-\max_{1\leq i\leq m}\{x_i\})_+\)&\(\Delta\)-antitone, supermodular \\
short call on the maximum &\(-(\max_{1\leq i\leq m}\{x_i\}-K)_+\) &\(\Delta\)-antitone, supermodular \\ \midrule
digital call on the minimum &\(\one_{\{\min_{1\leq i\leq m}\{x_i\}\geq K\}}\) &\(\Delta\)-monotone, supermodular \\
call on the minimum &\((\min_{1\leq i\leq m}\{x_i\}-K)_+\)&\(\Delta\)-monotone, supermodular \\
short put on the minimum &\(-(K-\min_{1\leq i\leq m}\{x_i\})_+\) &\(\Delta\)-monotone, supermodular \\ \midrule
basket call & $(\sum_{i=1}^m x_i-K)_+$ &supermodular (for \(m=2\) \(\Delta\)-monotone) \\
basket put & $(K-\sum_{i=1}^m x_i)_+$ &supermodular (for \(m=2\) \(\Delta\)-antitone)\\ \midrule
\end{tabular}
\caption{Overview of \(\Delta\)-antitone, \(\Delta\)-monotone, and supermodular payoff functions.}\label{table_payoff_functions_overview}
\end{table}
\end{center}
\subsection{Dependence Information Related to Correlations}\label{secadddepinreltocor}
In addition to martingale and marginal constraints, we take into account additional market-implied dependence information related to the inter-asset correlation.
\begin{enumerate}[1.)]\setcounter{enumi}{2}
\item \textbf{Knowledge of risk-neutral correlations} \\
We incorporate additional information on the covariance of the assets.
This approach is motivated by the observation of basket options with payoff structure
\begin{equation}
(a_1S_{t_i}^k +a_2S_{t_j}^l-K)_+, \label{eq_basket_def}
\end{equation}
for some fixed weights $a_1,a_2 \in \mathbb{R}$ with $a_1 a_2 \neq 0$ and $k,l \in \{1,\dots,d\},~i,j\in \{1,\dots,n\}$.
If prices of such options are observable for all strikes $K\in \mathbb{R}$ and $\mathbb{Q} \in \mathcal{M}(\mu)$ is consistent\footnote{We call a measure $\mathbb{Q}$ consistent with a price $p$ for a derivative $c$ if it holds $\mathbb{E}_\mathbb{Q}[c]=p$.} with these prices, then we deduce by deriving\footnote{Here we implicitly assume that the prices are differentiable as a function of the strike. If this is not the case, then we consider instead the right derivative and still obtain a one-to-one relation between the risk-neutral distribution of the sum and the basket option prices. Compare for the case of call options e.g. \cite[Lemma 2.2]{hobson2011skorokhod} and the discussion thereafter.} w.r.t.\,$K$
\[
\frac{\partial}{\partial K}\mathbb{E}_\mathbb{Q}\left[(a_1S_{t_i}^k +a_2S_{t_j}^l-K)_+\right]=\mathbb{Q}\left(a_1S_{t_i}^k +a_2S_{t_j}^l \leq K\right)-1.
\]
We hence obtain the distribution of $a_1S_{t_i}^k +a_2S_{t_j}^l$. This allows in particular to compute its second moment. In addition, observe that
\begin{align} \label{eq_covariance_info_basket}
\mathbb{E}_\mathbb{Q}\left[S_{t_i}^kS_{t_j}^l\right]=\frac{\mathbb{E}_\mathbb{Q}\left[\left(a_1S_{t_i}^k +a_2S_{t_j}^l\right)^2\right]-a_1^2\mathbb{E}_\mathbb{Q}\left[\left(S_{t_i}^k\right)^2\right]-a_2^2\mathbb{E}_\mathbb{Q}\left[\left(S_{t_j}^l\right)^2\right]}{2a_1a_2},
\end{align}
assuming that the second moments of the marginal distributions exist.
Since $\mathbb{Q} \in \mathcal{M}(\mu)$, all values of the right-hand side from \eqref{eq_covariance_info_basket}
are known and hence so is the left-hand side.
Moreover, using the martingale property, we obtain that the correlation is given by
\begin{equation*}\label{eq_covariance}
\operatorname{Corr}_\mathbb{Q}\left(S_{t_i}^k,S_{t_j}^l\right)=\frac{\mathbb{E}_\mathbb{Q}\left[S_{t_i}^k S_{t_j}^l\right]-S_0^kS_0^l}{\sqrt{\mathbb{E}_{\mu_i^k}\Big[(S_{t_i}^k)^2\Big]-(S_0^k)^2}\sqrt{\mathbb{E}_{\mu_j^l}\Big[(S_{t_j}^l)^2\Big]-(S_0^l)^2}},
\end{equation*}
which by \eqref{eq_covariance_info_basket} is known, too, since $S_0^kS_0^l$ is some constant value. Therefore, price information on options of \eqref{eq_basket_def}-type for all strikes $K$ is sufficient to obtain information on the correlation between $S^k_{t_i}$ and $S^l_{t_j}$. To model the risk-neutral correlation $\rho_{ij}^{kl}:=\operatorname{Corr}_\mathbb{Q}\left(S_{t_i}^k,S_{t_j}^l\right) \in [-1,1]$ between $S_{t_i}^k$ and $S_{t_j}^l$ with respect to a measures $\mathbb{Q} \in \mathcal{M}(\mu)$, we specify the equality constraints in equation \eqref{eq_restriction} by $f_{(i,j,k,l)}^{\operatorname{eq}}\in C_{\operatorname{lin}}(\mathbb{R}_+^{nd})$ with
\begin{equation}\label{eq_correlation_constraints_1}
f_{(i,j,k,l)}^{\operatorname{eq}}(x_1^1,\ldots,x_n^d)=\frac{x_i^k x_j^l-S_0^kS_0^l }{\sqrt{\mathbb{E}_{\mu_i^k}\Big[(S_{t_i}^k)^2\Big]-(S_0^k)^2}\sqrt{\mathbb{E}_{\mu_j^l}\Big[(S_{t_j}^l)^2\Big]-(S_0^l)^2}},~~(x_1^1,\ldots,x_n^d) \in \mathbb{R}_+^{nd}.
\end{equation}
such that $\mathbb{E}_\mathbb{Q}\left[f_{(i,j,k,l)}^{\operatorname{eq}}\right]=K_{(i,j,k,l)}^{\operatorname{eq}}$,
where $K_{(i,j,k,l)}^{\operatorname{eq}}:=\rho_{ij}^{kl}$ for all measures $\mathbb{Q}$ consistent with the correlation structure.
In Example~\ref{exa_corr1}, we investigate two examples in the case $n=2,~d=2$ and include additional information on the correlation between the assets. This information leads to several constraints which restrict the set of possible pricing measures in different degrees and therefore effectively influence robust price bounds.
\item \textbf{Knowledge of the risk-neutral distribution of sums}\\
Next, we consider not just correlation information, but the entire information on the sum of the underlying assets. This corresponds to considering prices of basket options directly.
Then we specify $ f_{(i,j,k,l,m)}\in C_{\operatorname{lin}}(\mathbb{R}_+^{nd})$ in \eqref{eq_eq_constraints}
\[
f_{(i,j,k,l,m)}^{\operatorname{eq}}(x_1^1,\ldots,x_n^d)=(a_1^{(i,j,k,l)}x_i^k +a_2^{(i,j,k,l)} x_j^l-K_m)_+,~~(x_1^1,\ldots,x_n^d) \in \mathbb{R}_+^{nd},
\]
for all $i,j \in \{1,\dots,n\},\,\) \(k,l\in \{1,\ldots,d\},\,\) \(m\in \{1,\ldots,N_{(i,j,k,l)}\}\), and where $a_1^{(i,j,k,l)},~a_2^{(i,j,k,l)} \in \mathbb{R}$ denote the corresponding weights of the basket options under consideration, $K_m \in \mathbb{R}$ the strike of the option and $N_{(i,j,k,l)}\in \mathbb{N}$ corresponds to the amount of observable options for this asset-maturity combination.
Moreover, we denote by \(K_{(i,j,k,l,m)}^{\operatorname{eq}}\)
the price of the basket option with payoff function \(f_{(i,j,k,l,m)}^{\operatorname{eq}} \).
If the price information implied by basket options is consistent with risk-neutral correlations as considered in 3.), then respecting the prices instead of the correlations may lead to a further improvement of the price bounds, as not only the second moment of the underlying distribution is taken into account, see also Example~\ref{losing_information_baskets}.
\item \textbf{Risk-neutral correlation is constant over time}\\
In Section~\ref{sec_correlation_constant} we discuss situations for $d=2$ in which it is reasonable to assume for any $\mathbb{Q} \in \mathcal{M}(\mu)$ that
\begin{equation*}
\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)=\operatorname{Corr}_\mathbb{Q}(S_{t_j}^1,S_{t_j}^2)\text{ for all } i,j \in \{1,\dots,n\}.
\end{equation*}
This leads to equality constraints of the form
\begin{equation}\label{eq_correlation_constant_1}
\begin{aligned}
f^{\operatorname{eq}}_{(i,j)}(x_1^1,\dots,x_n^d)&=\Bigg(\tfrac{x_{i}^1x_i^2-S_{t_0}^1S_{t_0}^2}{\sqrt{\mathbb{E}_{\mu_i^1}\left[\left(S_{t_i}^1\right)^2\right]-\left(S_{t_0}^1\right)^2}\sqrt{\mathbb{E}_{\mu_i^2}\left[\left(S_{t_i}^2\right)^2\right]-\left(S_{t_0}^2\right)^2}}\\
&\hspace{1cm}-\tfrac{x_{j}^1x_{j}^2-S_{t_0}^1S_{t_0}^2}{\sqrt{\mathbb{E}_{\mu_j^1}\left[\left(S_{t_j}^1\right)^2\right]-\left(S_{t_0}^1\right)^2}\sqrt{\mathbb{E}_{\mu_j^2}\left[\left(S_{t_j}^2\right)^2\right]-\left(S_{t_0}^2\right)^2}}\Bigg),~~(x_1^1,\ldots,x_n^d) \in \mathbb{R}_+^{nd},
\end{aligned}
\end{equation}
and $K^{\operatorname{eq}}_{(i,j)}=0$ for all $i,j =1,\dots,n,~ i \leq j$.
\item \textbf{Risk-neutral correlation is bounded from below by the real world correlation}\\
In Section~\ref{sec_correlation_bounded}, we discuss situations in $d =2$ in which it makes sense to assume for every $\mathbb{Q} \in \mathcal{M}(\mu)$ that $\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)\geq \operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)$, where $\mathbb{P}\in \mathcal{P}(\mathbb{R}_+^{nd})$ denotes some underlying real-world measure.
Thus, we model the inequality constraints in \eqref{eq_ineq_constraints} by setting
\begin{equation}\label{eq_corr_constraint_below_1}
f^{\operatorname{ineq}}_{i}(x_1^1,\dots,x_n^d)= -\frac{x_i^1x_i^2-S_{t_0}^1S_{t_0}^2}{\sqrt{\mathbb{E}_{\mu_i^1}\left[\left(S_{t_i}^1\right)^2\right]-\left(S_{t_0}^1\right)^2}\sqrt{\mathbb{E}_{\mu_i^2}\left[\left(S_{t_i}^2\right)^2\right]-\left(S_{t_0}^2\right)^2}},~~(x_1^1,\ldots,x_n^d) \in \mathbb{R}_+^{nd},
\end{equation}
and $K^{\operatorname{ineq}}_{i}=-\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)$ for $i=1,\dots,n$, where $\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)$ can often be estimated empirically with statistical methods.
\end{enumerate}
\section{Examples and Numerics}\label{section_numerics}
In accordance with the scenarios described in Section~\ref{secadddep} and Section~\ref{secadddepinreltocor}, we provide several examples for the improvement of the upper multi-asset price bound \(\overline{P}_{\mathcal{M}(\mu)}\). This improvement is due to the consideration of appropriate market-implied dependence information. The following examples cover the case where a restriction on the dependence structure is imposed through (quasi-)~copulas as well as the case where additional information on the correlation is taken into account. The \emph{Python} codes for the numerical examples from this section are provided under \htmladdnormallink{https://github.com/juliansester/improved-dependence-pricing}{https://github.com/juliansester/improved-dependence-pricing}.
\subsection{Improved price bounds through copula bounds}\label{sec41a}
We consider for $K \in \mathbb{R}$ the payoff functions
\begin{align}\label{eqdefpof}
c_{1,K}(S_{t_1}^1,S_{t_1}^2,S_{t_2}^1,S_{t_2}^2)&:=\left(K-\max_{i,k\in \{1,2\}}\{S_{t_i}^k\}\right)_+\,,\\
\label{eqdefpof2} c_{2,K}(S_{t_1}^1,S_{t_1}^2,S_{t_1}^3,S_{t_2}^1,S_{t_2}^2,S_{t_2}^3) &:= \left( \min_{\substack{i=1,2 \\ k=1,2,3}}\{S_{t_i}^k\}- K \right)_+\,,\\
\label{eqdefpof3} c_{3,K}(S_{t_1}^1,S_{t_1}^2,S_{t_1}^3)&:=\left(\frac {S_{t_1}^1+S_{t_1}^2+S_{t_1}^3} 3-K\right)_+\,.
\end{align}
For every $K \in \mathbb{R}$, the payoff function \(c_{1,K}\) is \(\Delta\)-antitone, \(c_{2,K}\) is \(\Delta\)-monotone, and \(c_{3,K}\) is increasing and supermodular, but neither \(\Delta\)-antitone nor \(\Delta\)-monotone. For the sake of readability, we sometimes abbreviate \(c_i:=c_{i,K}\,,\) \(i=1,2,3\,.\) In the following, we apply Theorem \ref{theloob} and Theorem \ref{corsmifm} to determine price bounds for these options under consideration of the martingale property and of copula bounds for the risk-neutral distributions inferred from dependence information based on prices of some options.
More specifically, we determine price bounds by considering minimal and maximal expectations w.r.t.\,measures from $\mathcal{M}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}$, $\mathcal{M}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}$, and $\mathcal{M}_{Q_2}^{\operatorname{IFM}}$, respectively, where the quasi-copulas $\underline{Q}$, $\overline{Q}$, and \(Q_2\) are inferred from option prices \(p_i^{k\ell}(K')\) of digital options $d_i^{k\ell}(K')$ with a payoff function defined by
\begin{align}\label{obsop}
d_i^{k\ell}(K')\big(S_{t_i}^k,S_{t_i}^\ell\big):= \one_{\{\max\{S_{t_i}^k,S_{t_i}^\ell\}\leq K'\}} ,~~k,\ell \in \{1,2,3\}, i \in \{1,2\}.
\end{align}
We assume that prices $p_i^{k\ell}(K')$ are observed in the market for strikes $ K'\in \mathcal{K}:=\{K_1,\ldots,K_m\}$, where $m \in \mathbb{N}$ describes the number of observed digital options. Knowledge of such option prices restricts the set of consistent pricing measures and therefore, via Sklar's theorem, prescribes the values of the associated (survival) copula on a finite set. This set is implied by the choice of $\mathcal{K}$ and the marginal distributions \(\mu\,.\) In such a case, lower orthant and upper orthant copula bounds for the dependence structure of the underlying assets are given by the quasi-copulas obtained from the following two lemmas, see \cite[Theorem 3.1 and Proposition A.1]{Lux-2017}. Several examples are provided in this section. More generally, prices of options whose payoff function is increasing w.r.t.\,the lower or upper orthant ordering allow to infer bounds for the partially known copula of the underlying asset, see \cite[Theorem 3.3 and Proposition A.1]{Lux-2017} and Table \ref{table_payoff_functions_overview}.
\begin{lem}[\cite{Lux-2017},~Theorem 3.1]\label{lemqcp}
Let $m\in \mathbb{N}$, let \(\mathcal{T}\subseteq [0,1]^{m}\) be a compact set, and let \(Q\in \mathcal{Q}_m\,.\) Then, the set
\begin{align*}
\mathcal{Q}^{\mathcal{T},Q}:=\{Q'\in \mathcal{Q}_{m}\mid Q'(x)=Q(x)~\forall x\in \mathcal{T}\}
\end{align*}
satisfies for each $Q'\in \mathcal{Q}^{\mathcal{T},Q}$ that
\[
\underline{Q}^{\mathcal{T},Q}(u)\leq Q'(u)\leq \overline{Q}^{\mathcal{T},Q}(u)
\]
where
\begin{align*}
\underline{Q}^{\mathcal{T},Q}(u)&:=\max\left\{0\,,~\sum_{k=1}^m u_k -m + 1\,,~\max_{x\in \mathcal{T}}\left\{Q(x)-\sum_{k=1}^m (x_k-u_k)_+\right\}\right\}\,,\\
\overline{Q}^{\mathcal{T},Q}(u)&:=\min\left\{u_1,\ldots,u_m\,,~\min_{x\in \mathcal{T}}\left\{Q(x)+\sum_{k=1}^m (u_k-x_k)_+\right\}\right\}\,,
\end{align*}
and \(u=(u_1,\ldots,u_m)\in [0,1]^{m}\).
\end{lem}
We say \(\widehat{Q}\) is a \emph{quasi-survival function} if \(\widehat{Q}\) is the survival function of a quasi-copula \(Q\,\), in the sense of \eqref{defsurvfun}.
\begin{lem}[\cite{Lux-2017},~Proposition A.1]\label{lemqcpu}
Let $m \in \mathbb{N}$, let \(\mathcal{T}\subseteq [0,1]^{m}\) be a compact set, let \(\widehat{Q}\) be an \(m\)-variate quasi-survival function, and let \(\mathcal{C}^{\mathcal{T},\widehat{Q}}:=\{ C\in \mathcal{C}_{m} \mid \widehat{C}(x)=\widehat{Q}(x)~\forall x\in \mathcal{T}\}\,.\) Then the set
\begin{align*}
\widehat{\mathcal{C}}^{\mathcal{T},\widehat{Q}}:=\{ \widehat{C} \mid C\in \mathcal{C}^{\mathcal{T},\widehat{Q}}\}
\end{align*}
satisfies for each $C\in \mathcal{C}^{\mathcal{T},\widehat{Q}}$ that
\[
\underline{\widehat{Q}}^{\mathcal{T},\widehat{Q}}(u)\leq \widehat{C}(u)\leq \overline{\widehat{Q}}^{\mathcal{T},\widehat{Q}}(u),
\]
where
\begin{align*}
\underline{\widehat{Q}}^{\mathcal{T},\widehat{Q}}(u)&:=\max\left\{0\,,~1-\sum_{k=1}^m u_k \,,~\max_{x\in \mathcal{T}}\left\{\widehat{Q}(x)-\sum_{k=1}^m (u_k-x_k)_+\right\}\right\}\,,\\
\overline{\widehat{Q}}^{\mathcal{T},\widehat{Q}}(u)&:=\min\left\{1-u_1,\ldots,1-u_m\,,~\min_{x\in \mathcal{T}}\left\{\widehat{Q}(x)+\sum_{k=1}^m (x_k-u_k)_+\right\}\right\}\,,
\end{align*}
and \(u=(u_1,\ldots,u_m) \in [0,1]^{m}\).
\end{lem}
In a first example, we illustrate the behaviour of the improved price bounds for the option \(c_1\) in dependence on prices \(p_1^{12}(K')\) of digital options \(d_1^{12}(K')\,,\) \(K'\in \mathcal{K}\,,\) and under consideration of the martingale property.
\begin{exa}[\(\Delta\)-antitone payoff function]\label{exdelan}
We assume that
\begin{align*}
&S_{t_1}^1 \sim \mu_1^1=\mathcal{U}(\{8,10,12\}),&&S_{t_1}^2 \sim \mu_1^2=\mathcal{U}(\{8,10,12\}),\\
&S_{t_2}^1 \sim \mu_2^1=\mathcal{U}(\{7,9,11,13\}),&&S_{t_2}^2 \sim \mu_2^2=\mathcal{U}(\{4,7,10,13,16\}),
\end{align*}
where $\mathcal{U}$ is the discrete uniform distribution, and we choose $\mathcal{K}:=\{8,10\}$. Then, for each $K' \in \mathcal{K}$, knowledge of the price \(p_1^{12}(K')\) of the option \(d_1^{12}(K')\) means knowledge of the value \(C_1^{12}(F_1^1(K'),F_1^2(K'))\) of the copula \(C_1^{12}\) associated with \((S_{t_1}^1,S_{t_1}^2)\,,\) i.e.,
\begin{align}\label{eqdigoppri1}
C_1^{12}(F_1^1(K'),F_1^2(K'))=\mathbb{Q}(S_{t_1}^1\leq K', S_{t_1}^2\leq K') =\mathbb{E}_\mathbb{Q} [d_1^{12}(K')] =p_1^{12}(K')
\end{align}
for all \(\mathbb{Q}\in \mathcal{M}^{\operatorname{lin}}\) fulfilling the linear constraint \(\mathbb{E}_\mathbb{Q}[d_1^{12}(K')]=p_1^{12}(K')\,.\) This yields the compact set
\begin{align*}
\mathcal{T}:=\bigcup_{K'\in \mathcal{K}} \{(F_1^1(K'),F_1^2(K'),1,1)\} = \{(F_1^1(8),F_1^2(8),1,1)\} \cup \{(F_1^1(10),F_1^2(10),1,1)\},
\end{align*}
on which the value of the copula \(C\) of \((S_{t_1}^1,S_{t_1}^2,S_{t_2}^1,S_{t_2}^2)\) is prescribed. As a consequence of Lemma~\ref{lemqcp}, we obtain the lower and upper bounds \(\underline{Q}\) and \(\overline{Q}\) for \(C\) w.r.t.\,the lower orthant ordering given by the quasi-copulas
\begin{align*}
\underline{Q}(u):&=\max\left\{0\,, ~\sum_{l=1}^4 u_l - 3 ,\max_{K'\in \mathcal{K}}\Big\{p_1^{12}(K')-\sum_{l=1,2}(F_1^l(K')-u_l)_+\Big\}+u_3+u_4-2\right\}\,,\\
\overline{Q}(u):&=\min\left\{u_1,u_2,u_3,u_4,\min_{K'\in \mathcal{K}}\Big\{p_1^{12}(K')+\sum_{l=1,2}(u_l-F_1^l(K'))_+\Big\}\right\}\\
\end{align*}
for \(u=(u_1,u_2,u_3,u_4)\in [0,1]^4\,.\)
Now, we compute for $K \in \mathbb{R}$, the upper bound \(\pi_{{c_{1,K}}}^\mu(\widehat{\overline{Q}})\) in \eqref{eqlosb} for the price of the \(\Delta\)-antitone option $c_1=c_{1,K}$ specified in \eqref{eqdefpof} under knowledge of the digital option prices \(p_1^{12}(8)\) and \(p_1^{12}(10)\,.\) \\
For \(I=\{(i_1,k_1),\ldots,(i_m,k_m)\}\subset \{1,\ldots,n\}\times \{1,\ldots,d\}\,,\) abbreviate
\begin{align}\label{abbFI}
F^I(x_I):= \left(F_{i_1}^{k_1}(x_{i_1}^{k_1}),\ldots,F_{i_m}^{k_m}(x_{i_m}^{k_m})\right)\,,
\end{align}
where \(x_I=(x_{i_1}^{k_1},\ldots,x_{i_m}^{k_m})\in \mathbb{R}_+^{|I|}\,.\)
Then we obtain with Definition~\ref{def_quasi_ex}
\begin{align*}
\pi_{c_{1}}^{\mu}(\widehat{\overline{Q}}) &= \sum_{I\subseteq \{1,2\}\times\{1,2\}\atop I\ne \emptyset} \int_{\mathbb{R}_+^{|I|}} (\widehat{\overline{Q}})_I(F^I(x_I))\mathsf{\,d} \eta_{c_{1,I}}(x_I) + c_1(0,\ldots,0) \\
&= \sum_{I\subseteq \{1,2\}\times\{1,2\}\atop I\ne \emptyset} (-1)^{|I|} \int_0^K (\widehat{\overline{Q}})_I(F^I(x,\ldots,x))\mathsf{\,d} x + c_{1}(0,\ldots,0)\,,
\end{align*}
where \(c_{1,I}:=(c_1)_I\) denotes the \(I\)-marginal of \(c_1\,\). Here we use for the first equality that \(c_1\) is left-continuous, \(\Delta\)-antitone and thus measure-inducing.
For the second equality we apply formula (1) in Table \ref{table_quasiExp_overview}.
Figure~\ref{fig_c1_bounds} illustrates the improvement of the upper bound \(\overline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}}\) over the upper bound \(\pi_{c_{1}}^{\mu}(\overline{Q})\) obtained from Theorem \ref{theloob}~(b) for the payoff $c_{1,K}$ with strikes $K \in \{9,11\}$ in dependence on the digital option prices \(p_1^{12}(8)\) and \(p_1^{12}(10)\,.\) The bound \(\overline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}}\) additionally incorporates the martingale property as a linear constraint and is computed numerically with a linear programming approach. We observe that the upper price bound implied by digital option prices may improve the standard upper price significantly, in particular, if the observed prices of the digital options are low, and therefore a strong dependence information is entailed in the option prices. The martingale property improves the bound additionally to a small degree.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.8\textwidth]{inc/eps/discrete_copula_info_legend-eps-converted-to.pdf}
\includegraphics[width=1\textwidth]{inc/eps/discrete_copula_info-eps-converted-to.pdf}
\caption{Regarding Example \ref{exdelan}, this figure shows the impact of the knowledge of the prices of the digital options \(d_1^{12}(8)\) and \(d_1^{12}(10)\) on the upper price bound of the option \(c_{1,K}\) for strike \(K=9\) on the left and for strike \(K=11\) on the right, respectively.} \label{fig_c1_bounds}
\end{center}
\end{figure}
\end{exa}
In the following two examples, we determine upper price bounds for the options \(c_{2,K}\) and \(c_{3,K}\) for different strikes \(K\) under the assumption that prices of some digital options as specified in \eqref{obsop} are given. We generate prices $p_i^{k\ell}$, similar to \cite[Example 6.8]{Lux-2017}, by assuming an underlying multivariate Black-Scholes model \(S=(S_t^1,S^2_t,S^3_t)_{t\geq 0}\) with
\begin{align}\label{eqsimpro}
S^k_{t}=S_0^k\exp\left(-\tfrac {(\sigma^k)^2} 2 {t}+\sigma^k X_{t}^k\right),~k=1,2,3,~t\geq 0,
\end{align}
where \(X=(X_t^1,X_t^2,X_t^3)_{t\geq 0}\) is a Brownian motion with dependent components that are distributed \mbox{\((X_1^1,X_1^2,X_1^3)\sim N(0,\Sigma)\)} with covariance matrix
\begin{align*}
\Sigma=\begin{pmatrix} 1 & \rho_{12} & \rho_{13} \\
\rho_{12} & 1 & \rho_{23} \\
\rho_{13} & \rho_{23} & 1 \end{pmatrix}\,.
\end{align*}
We specify the parameters \(S_0^k\,,\) \(\sigma^k\,,\) and \(\rho_{k\ell}\,,\) as well as the set of strikes \(\mathcal{K}\) of the observed digital options in the respective examples.
In particular, we assume that the one-dimensional marginal distributions $\mu_i^k$ are consistent with model \eqref{eqsimpro}, i.e., that the securities $S_{t_i}^k$, $i=1,2$, $k=1,2,3$ are log-normally distributed.
\begin{exa}[\(\Delta\)-monotone payoff function]\label{exadelmo}~
We specify \(t_1=1\,,\) \(t_2=2\,,\) \(\sigma^k=0.5\) for all \(k=1,2,3\,,\) \(S_0^1=9\,,\) \(S_0^2=10\,,\) and \(S_0^3=11\) as well as the risk-neutral correlations \(\rho_{12}=\rho_{13}=\rho_{23}=0.8\,.\) Further, we assume that the price \(p_2^{k\ell}(K')\) of the digital option \(d_2^{k\ell}(K')\,,\) \(1\leq k < \ell\leq d =3\,,\) can be observed for strikes \(K'\in \mathcal{K}:=\{8,9,10,11,12\}\,.\)
Knowledge of such option prices \(p_2^{k\ell}(K')\) means knowledge of the value of the survival function \(\widehat{C}^{k\ell}\) associated with the copula \(C^{k\ell}\) of \((S_{t_2}^k,S_{t_2}^\ell)\) given by
\begin{align}
\nonumber
\widehat{C}^{k\ell}(F_2^k(K'),F_2^\ell(K')) &=C^{k\ell}(F_2^k(K'),F_2^\ell(K'))+1-F_2^k(K')-F_2^\ell(K')\\
\label{eqdigoppri2}&= \mathbb{E}_\mathbb{Q} [d_2^{k\ell}(K')] +1-F_2^k(K')-F_2^\ell(K') = p_2^{k\ell}+1-F_2^k(K')-F_2^\ell(K')
\end{align}
for all \(\mathbb{Q}\in \mathcal{M}^{\operatorname{lin}}\)
fulfilling the equality constraint \(\mathbb{E}_\mathbb{Q} [d_2^{k\ell}(K')]=p_2^{k\ell}(K')\) for \(K'\in \mathcal{K}\,.\)
Hence, we consider the set
\begin{align*}
\mathcal{T}:=\bigcup_{K'\in \mathcal{K}} &\left( \{(0,0,0,F_2^1(K'),F_2^2(K'),0)\} \cup \{(0,0,0,F_2^1(K'),0,F_2^3(K'))\}\phantom{\sum}\right.\\
& \hspace{4.4cm}\left.\phantom{\sum}\cup \{(0,0,0,0,F_2^2(K'),F_2^3(K'))\}\right),
\end{align*}
on which the values of the survival function \(\widehat{C}\) associated with the copula \(C\) of the random vector \((S_{t_1}^1,S_{t_1}^2,S_{t_1}^3,S_{t_2}^1,S_{t_2}^2,S_{t_2}^3)\) are prescribed.
We obtain from Lemma \ref{lemqcpu} pointwise a lower bound \(\underline{\widehat{Q}}\) and an upper bound \(\widehat{\overline{Q}}\) for \(\widehat{C}\,,\) where
\begin{align}
\nonumber \underline{\widehat{Q}}(u)&:=\max\Big\{0\,,~1-\sum_{i=1,2\,,\atop k=1,2,3} u_i^k \,,~\max_{\{j,k,\ell\}= \{1,2,3\}\,,\atop K'\in \mathcal{K}}\left\{p_2^{k\ell}(K')+1-F_2^k(K')-F_2^\ell(K')\right.\\
\nonumber &\left. \hspace{7cm} -\sum_{l=k,\ell} (u_2^l-F_2^l(K'))_+- u_2^j-\sum_{r=1}^3 u_1^r\Big\}\right\}\,,\\
\label{ex42eqovlQ}\widehat{\overline{Q}}(u)& := \min\Bigg\{1-u_1^1,\ldots,1-u_2^3\,,~\min_{1\leq k < \ell \leq 3\atop K'\in \mathcal{K}}\Big\{p_2^{k\ell}(K')+1-F_2^k(K')-F_2^\ell(K')\\
&\hspace{9cm}+\sum_{l=k,\ell} (F_2^l(K')-u_2^l)_+\Big\}\Bigg\}
\end{align}
for \(u=(u_1^1,u_1^2,u_1^3,u_2^1,u_2^2,u_2^3)\in [0,1]^6\,.\)
Now, for $K \in \mathbb{R}$, we compute the upper bound \(\pi_{c_{2}}^\mu(\widehat{\overline{Q}})\) in \eqref{equosb} for the price of the option \(c_2=c_{2,K}\) as specified in \eqref{eqdefpof2} under knowledge of the digital option prices \(p_2^{k\ell}(K')\,,\) \(1\leq k < \ell \leq 3\,,\) \(K'\in \mathcal{K}\,.\)
To compute \(\pi_{c_{2}}^\mu(\widehat{\overline{Q}})\) we apply Theorem~\ref{theloob}~(b)
and use that \(\widehat{\overline{Q}}\) is a pointwise upper bound for \(\widehat{C}\,\), see also Remark~\ref{remgenan}~(d).
Since $c_2$ is left-continuous and measure-inducing we obtain from formula (4) in Table \ref{table_quasiExp_overview} with the upper bound \(\widehat{\overline{Q}}\) in \eqref{ex42eqovlQ} that
\begin{align*}
\pi_{c_{2}}^\mu(\widehat{\overline{Q}}) &= \sum_{I\subseteq \{1,2\}\times \{1,2,3\}\atop I\ne \emptyset} \int_{\mathbb{R}_+^{|I|}} \widehat{\overline{Q}}_I(F^I(x_I))\mathsf{\,d} \eta_{{c_2}_I}(x_I) + c_{2,K}(0,\ldots,0)\\
&= \int_{K}^\infty \min\left\{1-F_1^1(x),\ldots,1-F_2^3(x),\min_{1\leq k < \ell \leq 3\atop K'\in \mathcal{K}}\Big\{p_2^{k\ell}(K')+1-F_2^k(K')-F_2^\ell(K')\phantom{\sum_{l=k,\ell} }\right.\\
& \hspace{9.2cm} \left. +\sum_{l=k,\ell} (F_2^l(K')-F_2^l(x))_+\Big\}\right\} \mathsf{\,d} x .
\end{align*}
Figure~\ref{exa_42_martingale_improvement} illustrates the price bounds \(\pi_{c_{2}}^{\mu}(\widehat{\overline{Q}})\) and \(\overline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\) for the option \(c_2=c_{2,K}\) obtained from Theorem \ref{theloob}~(b) in dependence of the strike $K$. Moreover, we illustrate the corresponding lower bounds. We observe that the bound which incorporates the martingale property improves the bound $\pi_{c_{2}}^\mu(\widehat{\overline{Q}})\,$ significantly. The price bound \(\overline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\) is computed through an adaption of the algorithm provided in \cite{aquino2019bounds}, which relies on a neural network approximation of the optimal dual hedging strategy. We observe that in this setting including information on prices of digital options improves the price bounds only slightly, whereas in combination with the martingale property the price bounds can be improved significantly.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\textwidth]{inc/eps/exa_42_martingale_improvement-eps-converted-to.pdf}
\caption{In the setting of Example~\ref{exadelmo}, the figure depicts different lower and upper price bounds of $c_{2,K}$ in dependence on the strike $K$. We show price bounds without knowledge of prices of digital options, price bounds which additionally respect the martingale property, price bounds which one obtains after the inclusion of price information of digital option prices as well as \(\underline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\), \(\overline{P}_{{\mathcal{M}}^{\operatorname{uo}}_{\underline{Q},\overline{Q}}}\) which take into account the martingale property and the prices of digital options.} \label{exa_42_martingale_improvement}
\end{center}
\end{figure}
\begin{center}
\begin{table}[h!]\resizebox{\linewidth}{!}{
\begin{tabular}{l|lll} \toprule
~& Payoff \(c(x_1,\ldots,x_m)\) & Monotonicity & \(\int g(x_{i_1},\ldots,x_{i_k})\mathsf{\,d} \eta_{c_I}\) \\ \midrule
\((1)\) & \((K-\max\{x_1,\ldots,x_m\})_+\) & \(c\) \(\Delta\)-antitonic & \(\begin{cases} \int_0^K g(x,\ldots,x)\mathsf{\,d} x\,, &\text{if } |I| \text{ even}\,, \\ -\int_0^K g(x,\ldots,x)\mathsf{\,d} x & \text{if } |I| \text{ odd}\end{cases}\) \\
\((2)\) & \((\max\{x_1,\ldots,x_m\}-K)_+\) &\(-c\) \(\Delta\)-antitonic & \(\begin{cases} -\int_K^\infty g(x,\ldots,x)\mathsf{\,d} x\,, &\text{if } |I| \text{ even}\,, \\ \int_K^\infty g(x,\ldots,x)\mathsf{\,d} x & \text{if } |I| \text{ odd}\end{cases}\) \\ \midrule
\((3)\) & \((K-\min\{x_1,\ldots,x_m\})_+\) & \(-c\) \(\Delta\)-monotonic & \(\begin{cases} \int_0^K g(x,\ldots,x)\mathsf{\,d} x\,, &\text{if } I=\{1,\ldots,m\} \\ 0 & \text{else}\end{cases}\) \\
\((4)\) & \((\min\{x_1,\ldots,x_m\}-K)_+\) &\(c\) \(\Delta\)-monotonic & \(\begin{cases} \int_K^\infty g(x,\ldots,x)\mathsf{\,d} x\,, &\text{if } I=\{1,\ldots,m\} \\ 0 & \text{else}\end{cases}\) \\ \midrule
\((5)\) & \(\one_{\{\min\{x_1,\ldots,x_m\}\geq K\}}\) & \(c\) \(\Delta\)-monotonic & \(\begin{cases} g(K,\ldots,K)\,, &\text{if } I=\{1,\ldots,m\} \\ 0 & \text{else}\end{cases}\) \\
\((6)\) & \(\one_{\{\max\{x_1,\ldots,x_m\}\leq K\}}\) & \(c\) \(\Delta\)-antitonic & \(\begin{cases} g(K,\ldots,K)\,, &\text{if } |I| \text{ even}\,, \\ -g(K,\ldots,K)\,, &\text{if } |I| \text{ odd}\end{cases}\) \\ \midrule
\((7)\) & \((\alpha x_1 + \beta x_2 - K)_+\,, \alpha,\beta>0\) & \(c\) \(\Delta\)-monotonic & \(\begin{cases} \int_0^K g(x/\alpha,(K-x)/\beta)\mathsf{\,d} x\,, &\text{if } I=\{1,2\}\,, \\ \int_{K}^\infty g(x/\alpha)\mathsf{\,d} x \,,& \text{if } I=\{1\}\\
\int_{K}^\infty g(x/\beta)\mathsf{\,d} x \,,& \text{if } I=\{2\}\,,\end{cases}\) \\
\((8)\) & \((K-\alpha x_1-\beta x_2)_+\,, \alpha,\beta>0\) &\(c\) \(\Delta\)-antitonic & \(\begin{cases} \int_0^K g(x/\alpha,(K-x)/\beta)\mathsf{\,d} x\,, &\text{if } I=\{1,2\}\,, \\ -\int^{K}_0 g(x/\alpha)\mathsf{\,d} x \,,& \text{if } I=\{1\}\\
-\int^{K}_0 g(x/\beta)\mathsf{\,d} x \,,& \text{if } I=\{2\}\,,\end{cases}\) \\ \midrule
\end{tabular}}
\caption{Examples of measure-inducing payoff functions \(c\colon \mathbb{R}_+^m\to \mathbb{R}\) and integrals w.r.t. the induced (marginal) measures, compare also \cite[Table 1]{Tankov-2011} for a similar table in the case $m=2$.}\label{table_quasiExp_overview}
\end{table}
\end{center}
\end{exa}
For the determination of an improved upper price bound for the basket call option \(c_3\) when dependence information is related to the setting of an internal factor model, we make use of the following lemma, whose proof is provided at the end of Section~\ref{section_proofs}.
\begin{lem}[European basket options]\label{lembasopt}~\\
Let \(\mathfrak{C}(x_1,\ldots,x_d)=\left(\sum_{i=1}^d \alpha_i x_i-K\right)_+\) be the payoff function of the European basket call option with strike $K \in \mathbb{R}$ and let \(\mathfrak{P}(x_1,\ldots,x_d) = \left(K-\sum_{i=1}^d \alpha_i x_i\right)_+\) be the payoff function of the European basket put option with weights \(\alpha_i>0\,,\) \(1\leq i\leq d\,,\) and strike $K \in \mathbb{R}$. Then the following statements hold true:
\begin{itemize}
\item[(a)] \label{lembasopt1} \(\mathfrak{P}\) and \(\mathfrak{C}\) are measure-inducing if and only if \(d\leq 2\,.\)
\item[(b)] \label{lembasopt2} Let \(F_1,\ldots,F_d\in \mathcal{F}_+^1\) be continuous with finite first moments. If \(D^2,\ldots,D^d\in \mathcal{C}_2\) and \(Q_2\in \mathcal{Q}_2\) with \(D^i\leq_{\operatorname{lo}} Q_2\) for \(2\leq i\leq d\,,\) then
\begin{align}
\label{upppbcaa}\psi_{\mathfrak{C}}^{(F_1,\ldots,F_d)} (M^2\vee D^2\vee \cdots \vee D^d)& \leq \pi_{\phi_{\mathfrak{C}}}^{(G,F_1)}(\widehat{Q_2}),~~~ \text{and}\\
\nonumber \psi_{\mathfrak{P}}^{(F_1,\ldots,F_d)} (M^2\vee D^2\vee \cdots \vee D^d) &\leq \pi_{\phi_{\mathfrak{P}}}^{(G,F_1)}(\widehat{Q_2}),
\end{align}
where \(G\) is the distribution function defined by its generalized inverse
\[
G^{-1}(u):=\frac{\sum_{i=2}^d \alpha_i F^{-1}_i(u)}{\sum_{i=2}^d \alpha_i },~u\in [0,1]\,,
\]
and where \(\phi_{\mathfrak{C}}\)
and \(\phi_{\mathfrak{P}}\) are
defined as in \eqref{defphif}.
\end{itemize}
\end{lem}
\begin{exa}[Supermodular payoff function]\label{exasmpf}
We determine the upper price bound \(\pi_{\phi_{c_3}}^{\mu_1}(\widehat{Q_2})\,,\) \(\mu_1=(\mu_1^k)^{k=1,2,3}\,,\) for the option \(c_{3}\) specified in \eqref{eqdefpof3} when the quasi-copula bound \(Q_2\) is inferred from prices \(p_1^{1\ell}(K')\) of the digital options \(d_1^{1\ell}(K')\,,\) \(\ell=2,3\,,\) in \eqref{obsop} for strikes \(K'\in \mathcal{K}=\{8.5,9,9.5,10,10.5\}\,.\) Note that \(c_{3}\) is a continuous supermodular payoff function which is componentwise increasing but neither measure-inducing nor \(\Delta\)-antitone nor \(\Delta\)-monotone, see Lemma \ref{lembasopt} and compare \cite[Example 3.9.4]{Mueller-Stoyan-2002}. However, the transformed function \(\phi_{c_{3}}\) given by \eqref{defphif} is measure-inducing because it is \(\Delta\)-monotone, compare Lemma \ref{lembasopt}~(a).
To generate option prices $p_1^{1\ell}(K')$ according to the underlying model from \eqref{eqsimpro}, we specify \(t_1=1\,,\) the volatility \(\sigma=1\,,\) the initial time asset values \(S_0^1=10\,,\) \(S_0^2=9\,,\) and \(S_0^3=11\). For the correlation, we consider the four different cases specified as follows
\[
(\rho_{12},\rho_{13})\in \{(-1,-1),(-0.5,-0.5),(0,0),(0.5,0.5)\}.
\]
First, we determine the quasi-copula \(Q_2\) in \eqref{eqdefclinfamo}. Knowledge of the option prices \(p_1^{1\ell}(K')\) means knowledge of the values of the copula \(C^{1\ell}\) associated with \((S_{1}^1,S_{1}^{\ell})\,,\) i.e.,
\begin{align}\label{eqdigoppri3}
C^{1\ell}(F_1^1(K'),F_1^\ell(K'))=\mathbb{Q}(S_{t_1}^1\leq K', S_{t_1}^\ell\leq K') =\mathbb{E}_\mathbb{Q} [d_1^{1\ell}(K')] =p_1^{1\ell}(K')
\end{align}
for all \(\mathbb{Q}\in \mathcal{M}^{\operatorname{lin}}\)
fulfilling the equality constraints $\mathbb{E}_\mathbb{Q} [d_1^{1\ell}(K')]=p_1^{1\ell}(K')$, $\ell=2,3$ and \(K'\in \mathcal{K}\,.\)
Therefore, we consider the compact sets
\begin{align*}
\mathcal{T}_{12}&:=\bigcup_{K'\in \mathcal{K}} \{(F_1^1(K'),F_1^2(K'))\}&& \text{and} && \mathcal{T}_{13}:=\bigcup_{K'\in \mathcal{K}} \{(F_1^1(K'),F_1^3(K'))\}
\end{align*}
on which the values of the copulas \(C^{12}\) of \((S_{t_1}^1,S_{t_1}^2)\) and \(C^{13}\) of \((S_{t_1}^1,S_{t_1}^3)\,,\) respectively, are prescribed.
Then, we obtain from Lemma~\ref{lemqcp} the (w.r.t. \(\leq_{\operatorname{lo}}\)) upper bound \(Q^{12}\) for \(C^{12}\) and \(Q^{13}\) for \(C^{13}\) given by the quasi-copulas
\begin{align*}
\overline{Q^{12}}(u)&:=\min\left\{u_1,u_2,\min_{K'\in \mathcal{K}}\left\{p_1^{12}(K')+\sum_{l=1,2}(u_l-F_1^l(K'))_+\right\}\right\}~~~\text{and}\\
\overline{Q^{13}}(v)&:=\min\left\{v_1,v_3,\min_{K'\in \mathcal{K}}\left\{p_1^{13}(K')+\sum_{l=1,3}(v_l-F_1^l(K'))_+\right\}\right\} \,, \\
\end{align*}
respectively, for \(u=(u_1,u_2),v=(v_1,v_3)\in [0,1]^2\,.\) Thus, we choose the bound \(Q_2\) in model \eqref{eqdefclinfamo} as
\begin{align}\label{defqzmax}
Q_2(u):=\max\{\overline{Q^{12}}(u),\overline{Q^{13}}(u)\}\,, ~~~ u\in [0,1]^2\,,
\end{align}
which is a quasi-copula, as discussed in \eqref{eqmxqc}.
Since \(c_3\) is the payoff function of a basket call option, the upper price bound in the setting of an internal factor model in \eqref{eqcorsmifm2} is given by \(\pi_{\phi_{c_3}}^{(G,F_1^1)}(\widehat{Q_2})\,,\) see \eqref{upppbcaa} in Lemma~\ref{lembasopt}, where \(G\) is the distribution function defined by its generalized inverse function $G^{-1}(u):= \frac{1}{2}(F_1^2)^{-1}(u)+\frac{1}{2}(F_1^3)^{-1}(u)$ for \(u\in [0,1]\,.\)
We obtain from \eqref{defquexpop} that
\begin{equation}\label{eq_exa45_computation}
\begin{aligned}
\pi_{\phi_{c_3}}^{(G,F_1^1)}(\widehat{Q_2})&= \int_{\mathbb{R}_+^2} \widehat{Q_2}\left(G(x_1),F_1^1(x_2)\right) \mathsf{\,d} \eta_{\phi_{c_3}}(x_1,x_2)
+ \int_{\mathbb{R}_+} (\widehat{Q_2})_{\{1\}}(G(x)) \mathsf{\,d} \eta_{({\phi_{c_3}})_{\{1\}}}(x)\\
&\hspace{6.6cm}+ \int_{\mathbb{R}_+} (\widehat{Q_2})_{\{2\}}(F_1^1(x)) \mathsf{\,d} \eta_{({\phi_{c_3}})_{\{2\}}}(x)+\phi_{c_3}(0,0)\\
&=\int_0^K \widehat{Q_2}\left(G\left(3/2 \cdot x\right),F_1^1(3 \cdot (K-x))\right) \mathsf{\,d} x +\int_{K}^{\infty} (1-G( 3/2 \cdot x)) \mathsf{\,d} x \\
&\hspace{6.8cm}+ \int_{K}^{\infty} (1-F_1^1(3\cdot x)) \mathsf{\,d} x
\end{aligned}
\end{equation}
where we apply formula (7) in Table \ref{table_quasiExp_overview} with $\alpha = \frac{2}{3}, \beta = \frac{1}{3}$ for the second equality.
In Figure~\ref{fig_exa43_improvement}, we illustrate the standard upper price bound \(\overline{P}_{\mathcal{M}}\) based on knowledge of the marginals and the improved upper price bounds \(\pi_{\phi_{c_{3}}}^{(G,F_1^1)}(\widehat{Q_2})\) for the payoff function \(c_3=c_{3,K}\) in dependence on the strike \(K\,.\) The improved bounds are inferred from prices of the digital option \(d_1^{1\ell}(K')\,,\) \(\ell=2,3\,,\) \(K'\in \mathcal{K}\,,\) which are computed according to the multivariate Black-Scholes model with dependent components explained by \eqref{eqsimpro}. For an illustration, we choose different specifications to model the dependencies between the components of the underlying Brownian motion expressed by the correlations \(\rho_{1\ell}\,,\) \(\ell=2,3\,.\) We observe that the higher the components are negatively correlated the more the price bounds get improved.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\textwidth]{inc/eps/exa_43_improvement_zoom-eps-converted-to.pdf}
\caption{Regarding Example~\ref{exasmpf}, we illustrate upper price bounds $\pi_{\phi_{c_{3}}}^{(G,F_1^1)}(\widehat{Q_2})$ for the basket put option $c_3=c_{3,K}$ in dependence on the strike $K$ for several correlations \(\rho_{1\ell}\) of the underlying Brownian motion in the Black-Scholes model from which the prices \(p_1^{1\ell}(K')\) of the digital options \(d_1^{1\ell}(K')\,,\) \(K'\in \mathcal{K}\,,\) \(\ell=2,3\,,\) are calculated. The quasi-copula \(Q_2\) is determined by \eqref{defqzmax} and the upper price bounds are computed by \eqref{eq_exa45_computation}.} \label{fig_exa43_improvement}
\end{center}
\end{figure}
In Figure~\ref{fig_exa43_3d}, we consider the behaviour of the price bound \(\pi_{\phi_{c_3,K}}^{(G,F_1^1)}(\widehat{Q_2})\) for strike \(K=6\) (left) and \(K=10\) (right) in dependence on the digital option prices $p_1^{12}(9)$ and $p_1^{13}(9)$.
We observe that smaller digital option prices imply a dependence relation which is far away from being comonotonic and thus lead to price bounds for $c_{3,K}$ which are significantly smaller than the upper price bound without price information which is implied by the upper Fr\'{e}chet copula corresponding to comonotonicity.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\textwidth]{inc/eps/continuous_copula_3d-eps-converted-to.pdf}
\caption{In the setting of Example~\ref{exasmpf}, the figure shows how the upper price bound $\pi_{\phi_{c_{3,K}}}^{(G,F_1^1)}(\widehat{Q_2})$ behaves for \(K=6\) (left) and \(K=10\) (right) under price information on $p_1^{12}(9)$ and $p_1^{13}(9)$. } \label{fig_exa43_3d}
\end{center}
\end{figure}
\end{exa}
\subsection{Improved price bounds through correlations}~
In this section we show within several examples how information on the risk-neutral correlation can improve model-independent price bounds of derivatives. Before discussing the improvement in explicit settings in Example~\ref{exa_corr1} and Example~\ref{losing_information_baskets}, we stress the influence of the chosen filtration for the martingale formulation on the set of admissible martingale measures and therefore on resultant price bounds, as discussed in Remark~\ref{rem_differences_martingale_property}~(a).
Suppose for all examples in this section that $n=d=2$ and that $S$ has the following marginal distributions
\begin{equation}\label{eq_marginals_correlation}
\begin{aligned}
&S_{t_1}^1 \sim \mu_1^1=\mathcal{U}(\{8,10,12\}),&&S_{t_1}^2 \sim \mu_1^2=\mathcal{U}(\{8,10,12\}),\\
&S_{t_2}^1 \sim \mu_2^1=\mathcal{U}(\{7,9,11,13\}),&&S_{t_2}^2 \sim \mu_2^2=\mathcal{U}(\{4,7,10,13,16\}).
\end{aligned}
\end{equation}
We consider, similar as in \cite[Example 5.12]{schmithals2018contributions} and \cite[Example 5.34]{schmithals2018contributions}, the following four payoff functions
\begin{align}\label{payofffc1c4}
\begin{split}
&c_4(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):=\left(1/4\cdot(S_{t_1}^1+S_{t_2}^1+S_{t_1}^2+S_{t_2}^2)-10\right)_+, \\
&c_5(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):=\left(10-\min\left\{S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2\right\}\right)_+, \\
&c_6(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):=\frac{1}{4}\left(S_{t_2}^2-S_{t_2}^1\right)_+\cdot\left(S_{t_1}^2-S_{t_1}^1\right)_+, \\
&c_7(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):=\left(\frac{S_{t_2}^1-S_{t_1}^1}{S_{t_1}^1}\right)^2\cdot \left(\frac{S_{t_2}^2-S_{t_1}^2}{S_{t_1}^2}\right)^2.
\end{split}
\end{align}
All numerical price bounds in this setting are computed using a linear programming approach, compare for further details e.g. \cite{guo2019computational} and \cite{henry2013automated}.
\begin{exa}\label{exa_corr1}
We use the marginals from \eqref{eq_marginals_correlation}, the payoffs $c_4,c_5,c_6$ and $c_7$ as specified in \eqref{payofffc1c4}, and we include additional information on the correlation between the two underlying securities $S^1$ and $S^2$. In a first investigation we study the influence of the risk-neutral correlation $\rho_{11}^{12}=\operatorname{Corr}_\mathbb{Q}(S_{t_1}^1,S_{t_1}^2)$ between the two underlying securities at time $t_1$ on the price bounds, i.e., we reduce the class of admissible pricing measure to such measures $\mathbb{Q} \in \mathcal{M}(\mu)$ such that $\rho_{11}^{12}=\operatorname{Corr}_\mathbb{Q}(S_{t_1}^1,S_{t_1}^2)$ holds true. Figure~\ref{fig_correlation_influence_1} shows the dependence of the price bounds for the payoffs $c_4,c_5,c_6$ and $c_7$ on the correlation coefficient at time $t_1$ (Panel a) and at time $t_2$ (Panel b).
In Figure~\ref{fig_3d_correlation} we further combine correlation information at times $t_1$ and $t_2$ and study the impact on the lower and upper price bound of $c_4, c_5,c_6$, and \(c_7\).
As a result, we obtain a significant improvement of the price bounds for each of the payoff functions.
\begin{figure}[h!]
\begin{center}
\subfigure[Price Bounds with correlation information at time $t_1$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_correlations_improvement_1-eps-converted-to.pdf}}
\subfigure[Price Bounds with correlation information at time $t_2$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_correlations_improvement_2-eps-converted-to.pdf}}
\end{center}
\caption{As explained in Example~\ref{exa_corr1}, we depict robust lower (blue) and upper (red) price bounds for the multi-asset derivatives $c_4,c_5,c_6$ and $c_7$ under additional information on the correlation between assets at time $t_1$ and $t_2$ respectively. The price bounds without additional correlation information are indicated by dashed lines.}\label{fig_correlation_influence_1}
\end{figure}
\begin{figure}[h!]
\begin{center}
\subfigure[Price Bounds of $c_4$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_3d_correlations_1-eps-converted-to.pdf}}
\subfigure[Price Bounds of $c_5$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_3d_correlations_2-eps-converted-to.pdf}}
\subfigure[Price Bounds of $c_6$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_3d_correlations_3-eps-converted-to.pdf}}
\subfigure[Price Bounds of $c_7$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_3d_correlations_4-eps-converted-to.pdf}}
\caption{This figure shows, in the setting of Example~\ref{exa_corr1}, the impact of combined information on the correlations between $S^1$ and $S^2$ at $t_1$ as well as at $t_2$ on the lower (blue) and upper (red) price bounds of derivatives \(c_4,c_5,c_6\), and \(c_7\,.\) The bounds without the consideration of additional information are indicated by colored wireframes.} \label{fig_3d_correlation}
\end{center}
\end{figure}
\end{exa}
\begin{exa}\label{losing_information_baskets}
We reconsider the marginal distributions from \eqref{eq_marginals_correlation} and the payoff functions from \eqref{payofffc1c4}. Measures $\widetilde{\mathbb{Q}} \in \mathcal{M}^{\operatorname{lin}}$ that are consistent with the given marginals and a specific correlation can be computed as solutions to linear equations implied from these marginal constraints, the martingale conditions, and the given correlation. We denote by $\operatorname{P}_{\widetilde{\mathbb{Q}},t_i}(K_j):=\mathbb{E}_{\widetilde{\mathbb{Q}}}\left[\left(1/2 \cdot S_{t_i}^1+1/2\cdot S_{t_i}^2-K_j\right)_+\right]$ the price of a basket option with strike $K_j$ and maturity $t_i$ in such a model. We compute prices $\operatorname{P}_{\widetilde{\mathbb{Q}},t_1}(K_j)$ for basket options for $100$ equidistant strikes $K_j$ between $5$ and $15$ expiring at time $t_1$, where the reference measure $\widetilde{\mathbb{Q}}$ is, for each specific level of given correlation, some element from $\mathcal{M}^{\operatorname{lin}}$. We incorporate this price information in addition to marginal and martingale condition (but no correlation constraint) by solving
\begin{equation}\label{eq_bound_price_basket}
\sup_{\mathbb{Q} \in \mathcal{M}(\mu) \cap {\left\{\mathbb{Q}: \mathbb{E}_\mathbb{Q}\left[\left(1/2 \cdot S_{t_i}^1+1/2\cdot S_{t_i}^2-K_j\right)_+\right]=\operatorname{P}_{\widetilde{\mathbb{Q}},t_i}(K_j) \text{ for }j=1,\dots,100\right\}}} \mathbb{E}_\mathbb{Q}[c]
\end{equation}
as well as the associated lower bound problem for $i=1,2$. To avoid a too strong dependence of the price bounds in \eqref{eq_bound_price_basket} on the single specific chosen measure $\widetilde{\mathbb{Q}}$, we compute the bounds in \eqref{eq_bound_price_basket} $10$ times and average them over $10$ different measures $\widetilde{\mathbb{Q}}$. In Figure~\ref{fig_correlation_loss} we also depict the model leading to the most extreme (i.e. either minimal or maximal) improvement by a separate line\footnote{To generate the $10$ different measures we consider for $i=1,\dots,10$ convex combinations $(i-1)/9 \cdot \mathbb{Q}_1+ (10-i)/9\cdot \mathbb{Q}_2$ of two measures $\mathbb{Q}_1,\mathbb{Q}_2$ which emerge as solutions of $\inf_{\mathbb{Q} \in \mathcal{M}^{\operatorname{lin}}}\mathbb{E}_\mathbb{Q}[|S_{t_1}^1+S_{t_2}^1-S_{t_1}^2-S_{t_2}^2|]$ and $\sup_{\mathbb{Q} \in \mathcal{M}^{\operatorname{lin}}}\mathbb{E}_\mathbb{Q}[|S_{t_1}^1+S_{t_2}^1-S_{t_1}^2-S_{t_2}^2|]$, respectively. This choice of the measures $\mathbb{Q}_1,\mathbb{Q}_2$ is completely arbitrary and it is of course also possible to use other measures, e.g., randomly chosen measures or an entirely different approach to average the bounds.}, and we compare the emerging price bounds with those when including correlation information instead of price information. \begin{figure}[h!]
\includegraphics[width=0.3\textwidth]{inc/eps/fig_correlations_improvement_loss_time1_legend-eps-converted-to.pdf}
\begin{center}
\subfigure[Price bounds with correlation information at time $t_1$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_correlations_improvement_loss_time1-eps-converted-to.pdf}}
\subfigure[Price bounds with correlation information at time $t_2$]{
\includegraphics[width=0.45\textwidth]{inc/eps/fig_correlations_improvement_loss_time2-eps-converted-to.pdf}}
\end{center}
\caption{In the setting of Example~\ref{losing_information_baskets}, the solid lines indicate the price bounds when just incorporating information on the corresponding correlations, the dotted lines indicate the improvement through including information on basket prices computed in models consistent with the indicated correlation. These prices are averaged over 10 different models that are in this discrete example associated to pricing measures that emerge as solutions to linear programs. The dashed lines indicate maximal lower bound and minimal upper price bounds obtained from the inclusion of price information on basket options.}\label{fig_correlation_loss}
\end{figure}
The results reveal that most information on the joint distribution of the sum is contained within the correlation. Some further improvement can be observed by directly incorporating the price information, especially at time $t_2$ as can be seen in Figure~\ref{fig_correlation_loss}~(b).
\end{exa}
\subsection{Additional market-implied assumptions}
In this section we study how to take into account several additional conditions that reflect observations made on financial markets. In contrast to the inclusion of conditions that are directly linked to the prices of basket options and/or other liquid options these conditions are rather implied by properties that can be observed on financial markets.
In addition to equality constraints for a pricing measure $\mathbb{Q}$, we will in the sequel also consider inequality constraints.
For sake of illustration, we formulate in the following all assumptions only for two underlying assets (i.e. the case $d=2$). It is then straightforward to generalize the implied conditions to a larger number of underlying securities.
\subsubsection{Correlation is constant over time}\label{sec_correlation_constant}
In this section, we assume the risk-neutral correlation between two securities to be constant over time, i.e.,
\begin{equation}\label{eq_correlation_constant}
\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)=\operatorname{Corr}_\mathbb{Q}(S_{t_j}^1,S_{t_j}^2)\text{ for all } i,j \in \{1,\dots,n\}.
\end{equation}
The study \cite{adams2017correlations} finds that real-world correlations can reasonably be considered to be constant over time. Based on an empirical analysis of pairs of $40$ stocks, bonds, commodities, and currencies, their findings imply that, for $26\%$ of the pairs, constant real-world correlations for the whole period \(2000\)--\(2014\) can be assumed. For $54\%$ of pairs there appeared exactly one break in the correlation relationship and for $10\%$ there were two breaks. Only for the remaining $10\%$ of the pairs there were three and more breaks in this $14$ year period. The breaks were mostly corresponding to respective crises.
Moreover, \cite{buss2012measuring} assume that $\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)-\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)=\alpha \left(1-\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)\right)$ for some constant $\alpha \in (0,1)$, where $\mathbb{P}$ denotes the underlying real-world probabilty measure\footnote{Note that real-world correlations can be estimated based on historical observations.}. In combination with \cite{adams2017correlations}, this motivates us to assume a constant risk-neutral correlation as in equation \eqref{eq_correlation_constant}.
Regarding all these empirical findings, equation \eqref{eq_correlation_constant} should not be assumed in all market situations, but can be a reasonable assumption when no break in the correlation relationship, e.g. due to a change of market behaviour, is expected or when the time period is short. We refer to \cite{ghosh2021forecasting,krauss2017deep} for a discussion of various time periods over the last $20$ years. We also stress that most options of interest have rather short maturities such that a breakdown in the correlation relationship until maturity is rather unlikely.
The condition \eqref{eq_correlation_constant} can be included as equality constraints in the dual formulation of the robust pricing problem, as shown in \eqref{eq_correlation_constant_1}.
\subsubsection{Correlation is bounded from below by the real world correlation}\label{sec_correlation_bounded}
Following the argumentation in \cite{buss2012measuring}, i.e., assuming the existence of some $\alpha \in (0,1)$ such that for all $i=1,\dots,n$, it holds $\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)-\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)=\alpha \left(1-\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)\right) \geq 0$, we obtain the condition
\begin{equation}\label{eq_correlation_constraint_below}
\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2) \geq \operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2) \text{ for all } i =1,\dots,n.
\end{equation}
Since in most situations, the right-hand side ${\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)}$ can be well estimated using historical data (see e.g. \cite{eun1984estimating}), we obtain a lower bound for the risk-neutral correlation $\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2)$.
In combination with the assumption of a time-independent correlation we assume the same lower bound for all risk-neutral correlations. In general, the higher the lower bound for the correlation, the more restrictive is the resulting linear constraint and consequently more significant improvement of robust price bounds can be expected.
\begin{rem}
The estimation of ${\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)}$ may be subject to uncertainty, such that ${\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)}$ lies within some confidence interval $[\underline{c},\overline{c}]$ with a pre-specified probability. In this case, we can substitute \eqref{eq_correlation_constraint_below} by
\begin{equation}\label{eq_corr_ineq_1}
\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2) \geq \underline{c} \text{ for all } i =1,\dots,n.
\end{equation}
\end{rem}
Equation \eqref{eq_corr_constraint_below_1} allows implementing dual strategies of the form described in \eqref{eq_dual_strats1} to incorporate \eqref{eq_corr_ineq_1}, where $\operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)$ is estimated.
We test in two numerical examples the effect of the additional constraints on the associated robust price bounds.
\begin{exa}\label{exa_improvement_additional_info_table}
We suppose again that $S=(S^1,S^2)$ has the marginal distributions as specified in \eqref{eq_marginals_correlation}, and we consider the four payoff functions in \eqref{payofffc1c4}.
In Table \ref{tbl_improv} we summarize the improvement obtained through incorporating different additional conditions. Price bounds that are improved under additional assumptions are written bold. The results are computed using a linear programming approach.
\begin{table}[h!]
\begin{center}{
\begin{tabular}{lcc} \toprule
& $\inf_{\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}} \mathbb{E}_\mathbb{Q} [c_i(S)]$ & $\sup_{\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}} \mathbb{E}_\mathbb{Q} [c_i(S)]$\\
\midrule
\hspace{1cm}No additional assumptions \\
$c_4$ &$0.25$ & $1.0111$ \\
$c_5$ &$1.9611$ & $3.2167$\\
$c_6$&$0$ & $1.9778$ \\
$c_7$ & $0.0012$ & $0.0207$ \\
\midrule
\hspace{1cm}Constant correlation \\
$c_4$ &$\textbf{0.2781}$ & $\textbf{0.9781}$\\
$c_5$ &$1.9611 $ & $\textbf{3.198}$ \\
$c_6$&$\textbf{0.0795}$ & $1.9778 $ \\
$c_7$ & $0.0012$ & $0.0207$ \\
\midrule
\hspace{1cm} Correlation lower bounded by $-0.5$ \\
$c_4$ &$\textbf{0.3179}$ & $1.0111 $ \\
$c_5$ &$1.9611 $ & $\textbf{3.1615 }$ \\
$c_6$&$0 $ & $1.9778 $ \\
$c_7$ & $0.0012$ & $0.0207$ \\
\midrule
\hspace{1cm} Correlation lower bounded by $0.5$ \\
$c_4$ &$\textbf{0.5375}$ & $1.0111$\\
$c_5$ &$1.9611 $ & $\textbf{2.9714}$ \\
$c_6$ &$0 $ & $\textbf{0.8083}$ \\
$c_7$ & $0.0012$ & $0.0207$ \\
\midrule
\hspace{1cm}Constant correlation \\ \hspace{1cm} lower bounded by $-0.5$ \\
$c_4$ &$\textbf{0.329}$ & $\textbf{0.9781}$ \\
$c_5$ &$1.9611$ & $\textbf{3.1615 }$ \\
$c_6$&$\textbf{0.0795}$ & $1.9778 $ \\
$c_7$ & $0.0012 $ & $0.0207$ \\
\midrule
\hspace{1cm}Constant correlation \\\hspace{1cm} lower bounded by $0.5$ \\
$c_4$ &$\textbf{0.639}$ & $\textbf{0.9781}$ \\
$c_5$ &$1.9611 $ & $\textbf{2.893}$ \\
$c_6$ &$\textbf{0.0795}$ & $\textbf{0.6784}$ \\
$c_7$ & $\textbf{0.0014}$ & $0.0207$ \\
\bottomrule
\end{tabular}
}
\caption{Improvement of the price bounds described in Example~\ref{exa_improvement_additional_info_table} under different additional assumptions}\label{tbl_improv}
\end{center}
\end{table}
\\
Although the marginal distributions possess a quite simple discrete structure, the results described in Table~\ref{tbl_improv} allow several important insights concerning the effect of additional constraints on the resultant price bounds. First, we observe that the improvements are highly payoff-dependent. Indeed, while the price bounds for $c_7$ are barely affected through the inclusion of additional constraints, the price bounds for $c_4$ can be improved strongly by any kind of constraint we investigated. Second, the improvements can either concern only the lower bound (e.g. correlation constrained from below by -$0.5$ for $c_4$), only the upper bound (correlation constrained from below by $0.5$ for $c_6$) or affect both bounds (constant correlation for $c_4$). Third, a combination of different constraints can improve the price bounds even more than the sum of the improvements of both constraints when considered separately (upper bound of $c_5$ in the case that the correlation is constant and constrained from below by $0.5$).
\end{exa}
\subsection{Real-world examples}
In this section we study price bounds of multi-asset derivatives with underlying marginal distributions that are implied from real market data. In particular, we study how the price bounds behave under additional constraints on the joint distributions.
\subsubsection*{Deriving the marginals}
On $t_0=$ 17th August 2020 we observe prices of put and call options written on $S^1:=$ the stock of \emph{Apple Inc.} and on $S^2:=$ the stock of \emph{Microsoft Corp.} We take into account options with maturities lying $11$ days and $32$ days ahead respectively. This means we set
$$
t_1 - t_0 = 11/365\quad\mbox{and}\quad t_2 - t_0 = 32/365.
$$
We consider mid prices of call and put options, i.e., we take the average of bid and ask prices. These prices are then cleaned in two ways: the mid prices shall not allow for static arbitrage (call prices should decrease w.r.t.\,increasing strikes, put prices should increase w.r.t.\,increasing strikes). Further, we exclude butterfly arbitrage involving these prices, basically meaning prices as a function of the strikes should possess a convex shape.
After having cleaned the prices we apply the Breeden-Litzenberger result\footnote{We refer also to \cite{talponen2014note} for a multidimensional version of \cite{breeden1978prices}, as well as \cite{neufeld2022numerical} for a non-asymptotic version of \cite{breeden1978prices}, \cite{talponen2014note}.} in \cite{breeden1978prices} to obtain marginal distributions associated to the underlying securities at maturities $t_1,t_2$. The density of the marginals can be computed as the second derivative of the prices w.r.t. the strikes. For this step, to approximate the second derivative, we use the finite differences method, i.e., given strikes $(K_j)_{j=1,\dots,N_{\operatorname{strikes}}}$ with $N_{\operatorname{strikes}} \in \mathbb{N}$ and mid (call or put) prices $\left(\operatorname{P}(K_{j},t_i)\right)_{j=1,\dots,N_{\operatorname{strikes}}}$, the time-$t_i$ density $\operatorname{p}_i(K_j)$ evaluated at $K_j$ for $j =2,\dots,N_{\operatorname{strikes}}$ is approximated by
$$
\frac{\partial^2 \operatorname{P}(K,t_i)}{\partial K^2}\big|_{K=K_j} \approx \operatorname{p}_i(K_j):=\frac{\operatorname{P}(K_{j+1},t_i)-2\operatorname{P}(K_{j},t_i)+\operatorname{P}(K_{j-1},t_i)}{(K_{j+1}-K_{j-1})^2}
$$
and we further set $\operatorname{p}_i(K_1)=\operatorname{p}_i(K_{N_{\operatorname{strikes}}})=0$.
We then approximate the one-dimensional marginal distribution of the asset through
\begin{equation}\label{eq_marginals_data}
S_{t_i}^k \sim \frac{1}{\sum_{j=1}^{N_{\operatorname{strikes}}} \operatorname{p}_i(K_j)} {\sum_{j=1}^{N_{\operatorname{strikes}}}} \delta_{K_j}\operatorname{p}_i(K_j) ~~~ \text{ for }i,k=1,2,
\end{equation}
where $\delta_{K_j}$ denotes the Dirac measure at point $K_j$.
This leads to the densities displayed in Figure~\ref{fig_marginal_densities} by linear interpolation.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.85\textwidth]{inc/eps/fig_marginal_densities-eps-converted-to.pdf}
\end{center}
\caption{Approximated $t_1$- and $t_2$-densities of $S^1$ (\emph{Apple Inc.}) and of $S^2$ (\emph{Microsoft Corp.}) approximated by vanilla option prices observed on $t_0=$ $17$th August 2020.}\label{fig_marginal_densities}
\end{figure}
To ensure an increasing convex order of the marginals of each stock we equalize the means of $S_{t_1}^j,S_{t_2}^j$ for $j = 1,2$, compare also \cite{alfonsi2019sampling}. Finally we apply $\mathcal{U}$-quantization introduced in \cite[Section 2.4.]{baker2012martingales} in a similar way as in \cite[Section 3]{neufeld2021deep} such that each marginal is supported on $20$ values which can then be implemented into a linear program to compute robust price bounds. Additionally we remark that we neglect interest rates and dividend yields for these rather short maturities.
\subsubsection*{Computation of price bounds under correlation information}
We study the payoff functions of derivatives \(c_4,c_5,c_6,\) and \(c_7\) given by \eqref{payofffc1c4}, where we modify $c_4$ and $c_5$ by considering a strike of $250$, i.e., we have
\begin{align*}
c_4(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):&=\left(1/4\cdot(S_{t_1}^1+S_{t_2}^1+S_{t_1}^2+S_{t_2}^2)-250\right)_+,\\
c_5(S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2):&=\left(250-\min\left\{S_{t_1}^1,S_{t_2}^1,S_{t_1}^2,S_{t_2}^2\right\}\right)_+.
\end{align*}
In Figure~\ref{fig_real_data_correlations_improvement}~(a) and Figure~\ref{fig_real_data_correlations_improvement}~(b) we display the influence of information on the time-$t_1$ and time-$t_2$ correlation, respectively, on the price bounds of these derivatives.
As elaborated, such information can be extracted from prices of basket options, if observable. Since we have no access to price quotes of basket options, we instead show the improvement obtained if certain levels of correlation are given as an input.
As already observed in the examples with artificial marginals, in general, the improvement of the price bounds becomes stronger for information concerning the time $t_2$ correlation.
\begin{figure}[h!]
\begin{center}
\subfigure[Price bounds with correlation information at time $t_1$]{\includegraphics[width=0.48\textwidth]{inc/eps/fig_real_data_correlations_improvement_1-eps-converted-to.pdf}}
\subfigure[Price bounds with correlation information at time $t_2$]{\includegraphics[width=0.48\textwidth]{inc/eps/fig_real_data_correlations_improvement_2-eps-converted-to.pdf}}
\end{center}
\caption{The price bounds of the options $c_4,c_5,c_6,c_7$ in dependence of correlation information (either regarding time $t_1$ or time $t_2$) while the marginal distributions are implied by vanilla option prices written on Apple and Microsoft.}\label{fig_real_data_correlations_improvement}
\end{figure}
\subsubsection{Computation of price bounds under additional assumptions}
Eventually, we investigate the influence of additional assumptions on the price bounds.
We first observe that, under the real-world measure $\mathbb{P}$, which is here set to be the empirical measure based on historical data from $2$ January $2018$ until
$17$th August $2020$, the stocks of \emph{Apple Inc.}\,and \emph{Microsoft Corp.}\,seem to be highly correlated.
The idea is to make use of this apparently strong relation between the two assets to obtain tighter price bounds for derivatives $c_i$ written on both assets by using only such pricing measures that are consistent with an assumption on the strictly positive correlation.
To obtain an indication for the level of the correlation between the two assets in an $11$ and $32$ day period, we consider the empirical bivariate return distribution of the two assets in an observation period ranging from $2$nd January $2018$ until $22$ July $2020$. From this empirical distribution we simulate in a bootstrapping approach $100,000$ paths of length $11$ and $32$ respectively, and then compute the respective correlation coefficients. We obtain an estimation of $0.7948$ for $\operatorname{Corr}_\mathbb{P}(S_{t_1}^1,S_{t_1}^2)$ and an estimation of $0.7952$ for $\operatorname{Corr}_\mathbb{P}(S_{t_2}^1,S_{t_2}^2)$.
Thus, evidence is provided to include the weaker assumption $\operatorname{Corr}_\mathbb{Q}(S_{t_i}^1,S_{t_i}^2) \geq \operatorname{Corr}_\mathbb{P}(S_{t_i}^1,S_{t_i}^2)\geq 0.75$ for $i=1,2$ and pricing measures $\mathbb{Q}$. Here, $0.75$ can obviously be substituted by any other number associated to another degree of physical correlation that is believed to be more accurate. The higher this number, the more improvement of the price bounds can be expected. However, to compute price bounds among consistent martingale measures $\mathbb{Q}$, this number must lie within the interval of correlations that are consistent with the marginals.
According to Remark~\ref{rem_differences_martingale_property}, the bounds of this interval can be computed through minimizing and maximizing
\[
\mathbb{Q} \mapsto \mathbb{E}_\mathbb{Q}\left[\frac{S_{t_i}^1S_{t_i}^2-S_{t_0}^1S_{t_0}^2}{\sqrt{\mathbb{E}_{\mu_i^1}[\left(S_{t_i}^1\right)^2]-\left(S_{t_0}^1\right)^2}\sqrt{\mathbb{E}_{\mu_i^2}[\left(S_{t_i}^2\right)^2]-\left(S_{t_0}^2\right)^2}}\right]
\]
w.r.t.\,measures $\mathbb{Q} \in \mathcal{M}(\mu)$. We compute the bounds as solutions of linear programming problems using the empirical distributions derived in \eqref{eq_marginals_data} and obtain at $t_1$ the interval $[-0.588,0.799]$ and at $t_2$ the interval of possible prices given by $[-0.487,0.8490]$.
In Table~\ref{tbl_improv_real_data} we display the results revealing that indeed the assumption on a lower bound of the correlation has a strong impact on the quality of the price bounds. The assumption on constant correlations has only in combination with the assumption of the lower bound of the correlation an influence on the lower bound of $c_4$, while for the other price bounds we cannot report any influence of the assumption of constant correlations. For sake of readability, price bounds showing improvement in comparison with the original bounds are displayed in bold characters.
\begin{table}[h!]
\begin{center}
{
\begin{tabular}{lcc} \toprule
& $\inf_{\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}} \mathbb{E}_\mathbb{Q} [c_i(S)]$ & $\sup_{\mathbb{Q}\in \mathcal{M}^{\mathrm{lin}}} \mathbb{E}_\mathbb{Q} [c_i(S)]$\\
\midrule
\hspace{2cm}No additional assumptions \\
$c_4$&$31.9339$ & $37.7576 $ \\
$c_5$ &$55.5357$ & $ 66.5152 $ \\
$c_6$&$0.0 $ & $31.6319 $ \\
$c_7$ & $0.0$ & $0.0116$ \\
\midrule
\hspace{2cm}Constant correlation \\
$c_4$&$\textbf{31.9344}$ & $37.7576 $ \\
$c_5$ &$55.5357$ & $ \textbf{66.3507} $ \\
$c_6$&$0.0 $ & $31.6319 $ \\
$c_7$ & $0.0$ & $0.0116$ \\
\midrule
\hspace{2cm}Correlation lower bounded by $0.75$\\
$c_4$ &$\textbf{34.4798}$ & $37.7576 $ \\
$c_5$ &$\textbf{ 55.5359}$ & $\textbf{64.0369 } $ \\
$c_6$ &$0.0$ & $\textbf{2.1662} $ \\
$c_7$ & $0.0$ & $0.0116 $ \\
\bottomrule
\hspace{2cm}Constant correlation \\\hspace{2cm}$\&$ lower bounded by $0.75$\\
$c_4$ &$\textbf{35.3573}$ & $37.7576 $ \\
$c_5$ &$\textbf{55.5398}$ & $\textbf{60.6242 } $ \\
$c_6$ &$0.0$ & $\textbf{2.1662} $ \\
$c_7$ & $\textbf{0.0023}$ & $0.0116 $ \\
\bottomrule
\end{tabular}
}
\caption{Improvement of the price bounds under different additional assumptions}\label{tbl_improv_real_data}
\end{center}
\end{table}
\section{Proofs}\label{section_proofs}
In this section, we provide all proofs that were omitted in the main part of the paper.
The proof of the duality result in Theorem \ref{theaddcon} is based on the following version of the Monge--Kantorovich duality, see \cite[Proposition 2.1]{beiglbock2013model} and \cite[Chapter 5]{villani2008optimal}.
\begin{lem}[Monge--Kantorovich duality]\label{lemkanduathe}~\\
Let $c \in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})$. Then, the following holds.
\begin{align*}
\sup_{\pi\in\Pi(\mu)}\left\{\int_{\mathbb{R}_+^d} c(x)\mathsf{\,d} \pi(x) \right\} = \inf\left\{\sum_{i=1}^n\sum_{k=1}^d \int_{\mathbb{R}_+} u_i^k(x_i^k) \mathsf{\,d} \mu_i^k (x_i^k) ~\middle|~ u_1^1 \oplus \cdots \oplus u_n^d\geq c\,, ~ u_i^k\in \mathfrak{C}\right\}\,,
\end{align*}
where \(u_1^1\oplus \cdots \oplus u_n^d (x):=\sum_{i=1}^n\sum_{k=1}^d u_i^k(x_i^k)\) for \(x=(x_1^1,\ldots,x_n^d)\in \mathbb{R}_+^{nd}\,.\)
\end{lem}
We will also apply the following classical minimax theorem by Ky--Fan, see, e.g., \cite[Lemma 3.1]{Rueschendorf-2019}\footnote{We adapted the original formulation to this equivalent formulation which considers upper semicontinuous concave functions of the first argument instead of lower semicontinuous and convex functions.}.
\begin{lem}[Minimax theorem]\label{lem_kyfan}
Let \(B_1\) be a compact convex subset of a topological vector space \(V_1\), and let \(B_2\) be a convex subset of a vector space \(V_2\,.\) If \(f\colon B_1\times B_2\to \mathbb{R}\) has the properties that
\begin{itemize}
\item[(a)] \(f(\cdot,b_2)\) is upper semicontinuous and concave on \(B_1\) for all \(b_2\in B_2\,,\)
\item[(b)] \(f(b_1,\cdot)\) is convex on \(B_2\) for all \(b_1\in B_1\,,\)
\end{itemize}
then
\begin{align*}
\sup_{b_1\in B_1}\inf_{b_2\in B_2} f(b_1,b_2) = \inf_{b_2\in B_2} \sup_{b_1\in B_1} f(b_1,b_2)\,.
\end{align*}
\end{lem}
To verify the lower semicontinuity of $\mathcal{M}^{\operatorname{lin}} \ni \mathbb{Q} \mapsto \mathbb{E}_\mathbb{Q}[c]$ and to prove the compactness of the set \(\mathcal{M}^{\mathrm{lin}}\) we make use of the following continuity result, see \cite[Lemma 2.2]{beiglbock2013model} and \cite[Lemma 4.3]{villani2008optimal}.
\begin{lem}\label{lemcont}
Let \(f\colon \mathbb{R}^{nd} \to \mathbb{R}\) be (lower/upper semi-)continuous and linearly bounded. Then, the mapping
\begin{align*}
\pi \mapsto \int_{\mathbb{R}^{nd}} f(x) \mathsf{\,d} \pi(x)
\end{align*}
is (lower/upper semi-)continuous on \(\Pi(\mu)\) in the weak topology.
\end{lem}
\begin{proof}[Proof of Theorem~\ref{theaddcon}]\label{protheaddcon}
To prove the duality in \eqref{gendures}, we abbreviate
\begin{align*}
\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}(x)&:=\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}(x)-\sum_{i=1}^n\sum_{k=1}^d u_i^k(x_i^k)\\
\phantom{:}= \sum_{i=1}^{n-1}\sum_{k=1}^d \Delta_i^k&(x_1,\ldots,x_i)(x_{i+1}^k-x_i^k) + \sum_{i \in \mathcal{I}^{\operatorname{eq}}}\alpha_i (f_i^{\operatorname{eq }}(x)-K_i^{\operatorname{eq }}) + \sum_{i \in \mathcal{I}^{\operatorname{ineq}}} \beta_i (f_i^{\operatorname{ineq }}(x)-K_i^{\operatorname{ineq }})
\end{align*}
for $x=(x_1,\ldots,x_n)=(x_1^1,\ldots,x_n^d)\in \mathbb{R}_+^{nd}$, $\Delta_i^k \in C_b(\mathbb{R}_+^{id})$, $\alpha_i \in \mathbb{R}$, $\beta_i \in \mathbb{R}_+$, \(i=1,\dots,n,k=1,\dots,d\,\) such that $\alpha_i=0, \beta_j=0$ for all but finitely many $i\in \mathcal{I}^{\operatorname{eq}},~ j \in \mathcal{I}^{\operatorname{ineq}}$. Note that $\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)} \in L_{\operatorname{lin}}(\mathbb{R}_+^{nd})$, because all \(\Delta_i^k\) and \(f_i^{\operatorname{eq }}\) are linearly bounded and continuous for all $i,k$, and all \(\beta_if_i^{\operatorname{ineq }}\) are linearly bounded and lower semicontinuous. Therefore, we obtain that
\begin{align}
\nonumber \underline{\mathcal{D}}_{\mathcal{S}} &= \inf_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}~\inf_{\Psi_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)}\geq ~c} \left\{\sum_{i=1}^n \sum_{k=1}^d \int_{\mathbb{R}_+} u_i^k(x_i^k) \mathsf{\,d} \mu_i^k(x_i^k)\right\}\\
\nonumber &= \inf_{(u_i^k),(\Delta_i^k),(\alpha_i),(\beta_i)} ~\inf_{u_1^1\oplus \cdots \oplus u_n^d \geq ~c -\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)} }\left\{\sum_{i=1}^n \sum_{k=1}^d \int_{\mathbb{R}_+} u_i^k(x_i^k) \mathsf{\,d} \mu_i^k(x_i^k)\right\}\\
\label{eqproduth1}&= \inf_{(\Delta_i^k),(\alpha_i),(\beta_i)} ~ \sup_{\pi\in \Pi(\mu)}\left\{ \int_{\mathbb{R}_+^{nd}} \left( c(x)-\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)} (x) \right)\mathsf{\,d} \pi(x) \right\}\\
\label{eqproduth2}&= \sup_{\pi \in \Pi(\mu)} ~ \inf_{(\Delta_i^k),(\alpha_i),(\beta_i)} \left\{ \int_{\mathbb{R}_+^{nd}} c(x) \mathsf{\,d} \pi(x) - \int_{\mathbb{R}_+^{nd}} \sum_{i=1}^{n-1}\sum_{k=1}^d \Delta_i^k(x_1,\ldots,x_i)(x_{i+1}^k-x_i^k) \mathsf{\,d} \pi(x) \right. \\
\label{eqproduth3}&\left.~~~~~~~~~~~~~~~~~~~- \int_{\mathbb{R}_+^{nd}} \sum_{i \in \mathcal{I}^{\operatorname{eq}}} \alpha_i (f_i^{\operatorname{eq }}(x)-K_i^{\operatorname{eq }}) \mathsf{\,d} \pi(x) - \int_{\mathbb{R}_+^{nd}} \sum_{i \in \mathcal{I}^{\operatorname{ineq}}} \beta_i (f_i^{\operatorname{ineq }}(x)-K_i^{\operatorname{ineq }}) \mathsf{\,d} \pi(x)\right\}\\
\label{eqproduth4}&= \sup_{\mathbb{Q}\in \mathcal{M}^{\operatorname{lin}}} \int_{\mathbb{R}_+^{nd}} c(x) \mathsf{\,d} \mathbb{Q}(x) = \overline{P}_{\mathcal{M}^{\operatorname{lin}}}.
\end{align}
Indeed, Equation \eqref{eqproduth1} is a consequence of the Monge--Kantorovich duality (see Lemma \ref{lemkanduathe}) using that \(c-\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}\in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) because \(c\) and \(\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}\) are both linearly bounded and $c$ is upper semicontinuous, whereas $\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}$ is lower semicontinuous.
For Equation \eqref{eqproduth2}, we apply Lemma~\ref{lem_kyfan} to the compact convex set \(B_1=\Pi(\mu)\,,\) the convex set
\begin{align*}B_2=\bigg\{ (\Delta_i^k),(\alpha_i),(\beta_i) \in \left(C_b(\mathbb{R}_+^{d})\times \cdots \times C_b(\mathbb{R}_+^{(n-1)d})\right) \times \mathbb{R}^{|\mathcal{I}^{\operatorname{eq }}|}\times \mathbb{R}^{|\mathcal{I}^{\operatorname{ineq }}|} ~&\text{ s.t.}~\\ &\hspace{-8cm}\alpha_i=0,~\beta_j=0\text{ for all but finitely many } i\in \mathcal{I}^{\operatorname{eq}},~j \in \mathcal{I}^{\operatorname{ineq}} \bigg\},
\end{align*}
and the function \(f\) given by
\begin{align*}
f(\pi,((\Delta_i^k),(\alpha_i),(\beta_i))):=\int_{\mathbb{R}_+^d} \left(c(x)- \Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}(x)\right) \mathsf{\,d} \pi(x)\,.
\end{align*}
The compactness of \(\Pi(\mu)\) is meant w.r.t.\,the weak topology and can be obtained from \cite[Lemma 4.4]{villani2008optimal} and Prokhorov's Theorem.
Due to Lemma \ref{lemcont}, \(f\) is upper semicontinuous in \(\pi\) because \(c-\Phi_{(\Delta_i^k),(\alpha_i),(\beta_i)}\) is upper semicontinuous and linearly bounded.
Further, \(f\) is linear, and thus concave in $\pi$, as well as linear w.r.t.\,\(((\Delta_i^k),(\alpha_i),(\beta_i))$ and hence convex.
For the equality in \eqref{eqproduth4}, we note that \(\int c \mathsf{\,d} \pi\) is uniformly bounded in \(\pi\in \Pi(\mu)\) using that all \(\mu_i^k\) have finite first moments and that $c$ is linearly bounded. We observe that the second integral in \eqref{eqproduth2} vanishes whenever \(\pi\in \Pi(\mu)\) is a martingale. If \(\pi\in \Pi(\mu)\) fulfils the equality constraints \(\mathbb{E}_\pi [f_i^{\operatorname{eq }}]=K_i^{\operatorname{eq }}\) for all \(i\,,\) the first integral in \eqref{eqproduth3} is \(0\,,\) and if \(\pi\in \Pi(\mu)\) fulfils the inequality constraints \(\mathbb{E}_\pi [f_i^{\operatorname{ineq }}]\leq K_i^{\operatorname{ineq }}\) for all \(i,\) the second integral in \eqref{eqproduth3} is non-positive using that \(\beta_i\geq 0\,.\) Hence, for \(\pi\in \mathcal{M}^{\operatorname{lin}}\,,\) the infimum of the expression in the curly brackets is given by $\int c(x)\mathsf{\,d} \pi(x)>-\infty$. If \(\pi\in \Pi(\mu)\) is not a martingale or does not fulfil one of the equality or inequality constraints, then there exist \(\Delta_i^k\,,\) \(\alpha_i\,,\) and \(\beta_i\,,\) respectively, such that at least one of the corresponding integrals is positive. By scaling, we conclude that in this case the infimum over \((\Delta_i^k),(\alpha_i),(\beta_i)\) is \(-\infty\,.\)
Next, we prove that the supremum is attained. By \cite[Proposition 2.4]{beiglbock2013model}, the set \(\mathcal{M}(\mu)\) is compact in the weak topology. We show that \(\mathcal{M}^{\operatorname{lin}}\) is a closed subset of \(\mathcal{M}(\mu)\,.\) Let \((\pi_m)_{m\in \mathbb{N}}\subset \mathcal{M}^{\operatorname{lin}}\) be a sequence that converges weakly to some $\pi \in \mathcal{M}^{\operatorname{lin}}$.
Then, Lemma \ref{lemcont} implies for all $i \in \mathcal{I}^{\operatorname{eq}}$ that \(K_i^{\operatorname{eq }}=\mathbb{E}_{\pi_n} [f_i^{\operatorname{eq }}]\to \mathbb{E}_\pi [f_i^{\operatorname{eq }}]\) as \(n\to \infty\,,\) and, thus, \(\mathbb{E}_\pi [f_i^{\operatorname{eq }}]=K_i^{\operatorname{eq }}\,,\) where we use that \(f_i^{\operatorname{eq }}\) is linearly bounded and continuous. For the inequality constraints, we obtain from Lemma \ref{lemcont} that for all $i \in \mathcal{I}^{\operatorname{ineq}}$
\begin{align*}
\mathbb{E}_\pi [f_i^{\operatorname{ineq }}] \leq \liminf_{n\to \infty} \mathbb{E}_{\pi_n} [f_i^{\operatorname{ineq }}] \leq K_i^{\operatorname{ineq }}
\end{align*}
using that \(f_i^{\operatorname{ineq }}\) is linearly bounded and lower semicontinuous.
Hence, also \(\mathcal{M}^{\operatorname{lin}}\) is compact. Now, let \((\mathbb{Q}_m)_m\subset \mathcal{M}^{\operatorname{lin}}\) be a sequence such that \(\overline{P}_{\mathcal{M}^{\operatorname{lin}}} \leq \mathbb{E}_{\mathbb{Q}_m}[c] + \frac 1 m\) for all \(m\in \mathbb{N}\,.\) Since \(\mathcal{M}^{\operatorname{lin}}\) is compact, there exists a measure \(\mathbb{Q}^*\in \mathcal{M}^{\operatorname{lin}}\) and a subsequence \((\mathbb{Q}_{m_k})_{k \in \mathbb{N}}\) which converges weakly to \( \mathbb{Q}^*\,.\) Since \(c\) is upper semicontinuous and linearly bounded, we obtain from Lemma \ref{lemcont} that \(\limsup_{k\to \infty} \mathbb{E}_{\mathbb{Q}_{m_k}} [c] \leq \mathbb{E}_{\mathbb{Q}^*}[c]\,.\) This implies that \(\overline{P}_{\mathcal{M}^{\operatorname{lin}}}=\mathbb{E}_{\mathbb{Q}^*}[c]\,.\)\\
\end{proof}
For the proof of Theorem \ref{theqcub}, we apply the following lemmas.
\begin{lem}\label{lemsufu}
Let $m \in \mathbb{N}$, let $Q_2 \in \mathcal{Q}_2$. Then, it holds that \(Q^*\in \mathcal{Q}_m\) defined in \eqref{deqcqs} is a quasi-copula with survival function \(\widehat{Q^*}\) given by
\begin{align*}
\widehat{Q^*}(u_1,\ldots,u_m)=\widehat{Q_2}\left(\max_{2\leq i\leq m} \{u_i\},u_1\right)\,, ~~~(u_1,\ldots,u_m)\in [0,1]^m\,.
\end{align*}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lemsufu}]
The function \(Q^*\) fulfils the defining properties of a quasi-copula because \(Q_2\) is a quasi-copula.\\Since survival functions do not depend on the order of the arguments, w.l.o.g.\, we may for \((u_1,\ldots,u_m)\in [0,1]^m\) consider the case that \(u_2\geq \ldots\geq u_m\,.\)
Then, it holds true that
\begin{equation}
\label{eq_long_equation_1}
\begin{aligned}
\widehat{Q^*}(u_1,\ldots,u_m)&= \sum_{I\subseteq \{1,\ldots,m\}\atop v_i:=1 ~\forall i\in I\,,~v_i:=u_i~\forall i\notin I} (-1)^{m-|I|} Q^*(v_1,\ldots,v_m)\\
&=1-\sum_{J\subseteq \{1,\ldots,m\}, J\ne \emptyset\,,\atop v_i:=u_i ~\forall i\in J\,, ~v_i:=1 ~\forall i\notin J} (-1)^{|J|+1} Q^*(v_1,\ldots,v_m)\\
&= 1 - u_1
- \sum_{ J\subseteq \{1,\ldots,m\}, \atop 1\in J, |J|\geq 2} (-1)^{|J|+1} Q_2\left(\min_{j\in J\setminus \{1\}} \{u_j\} , u_1\right)\\
& ~~~~~~ - \sum_{ J \subseteq \{2,\ldots,m\}, J \neq \emptyset} (-1)^{|J|+1} \min_{j\in J}\{ u_j\} \\
&= 1 - u_1
- \sum_{k=2}^m \sum_{j=0}^{k-2} (-1)^{j+1} \binom{k-2}{j} Q_2(u_k,u_1)\\
& ~~~~~~- \sum_{k=2}^m \sum_{j=0}^{k-2} (-1)^{j} \binom{k-2}{j} u_k\\
&= 1-u_1+ Q_2(u_2,u_1)-u_2\\
&= 1- u_1-\max_{2\leq i\leq m}\{u_i\} + Q_2(\max_{2\leq i\leq m}\{u_i\},u_1)\\
&= \widehat{Q_2}(\max_{2\leq i\leq m} u_i,u_1).
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_long_equation_1} holds, note that the first and last equality follow from the definition of the survival function in \eqref{defsurvfun}. For the second equality, we sum over \(J=\{1,\ldots,m\}\setminus I\) and use that $Q^*(1,\dots,1)=1$. The third equality follows from the definition of \(Q^*\) and the uniform marginal property of Definition \ref{def_quasicop}~(b) for quasi-copulas. The fourth equality holds true for the following reason: in the first sum, we consider for every $k=2,\ldots,m$, the subsets $J\subseteq \{1,\ldots,k\}$ with $1,k\in J$. Then $k$ is the maximal element of $J$ and hence $\min_{j\in J\setminus \{1\}}\{u_j\}=u_k$. There are \(\binom{k-2}{j}\) subsets of \(\{2,\ldots,k-1\}\) with \(j\) elements and we have $|J|=j+2$. In the second sum, we consider subsets $J\subseteq \{2,\ldots,k\}$ with $k\in J$ for every $k=2,\ldots,m$. Here again there are \(\binom{k-2}{j}\) subsets of \(\{2,\ldots,k-1\}\) with \(j\) elements but now $|J|=j+1.$ The fifth equality follows from the symmetry of the binomial coefficients given by \(\sum_{i=0}^N \binom{N}{i} (-1)^i = \one_{\{N=0\}}\) for $N \in \mathbb{N}$.
\end{proof}
Denote by \(\mathbb{K}_n^m:=\{\tfrac 1 {n+1},\tfrac 2{n+1},\ldots,\tfrac {n}{n+1}\}^m \subset [0,1]^m\) the canonical \(m\)-dimensional \(n\)-grid with edge length \(\tfrac 1 {n+1}\) contained in \([0,1]^m\,.\) Denote by \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m):=\{(\tfrac 1 {n+1},\ldots,\tfrac 1 {n+1}),\ldots,(\frac n {n+1},\ldots,\frac n {n+1})\}\) the diagonal of \(\mathbb{K}_n^m\,.\) For a finite signed measure \(\sigma\) on \([0,1]^m\,,\) we define by
\begin{equation}\label{eq_def_g_sigma}
G_\sigma(u_1,\ldots,u_m):=\sigma([0,u_1],\ldots,[0,u_m])\,, \quad (u_1,\dots,u_m) \in [0,1]^m\,,
\end{equation}
its \emph{measure generating function}.
Conversely, by embedding $\mathbb{K}_n^m \subseteq [0,1]^m$ and identifying $f: \mathbb{K}_n^m \rightarrow \mathbb{R}$ with $\overline{f}:[0,1]^m \rightarrow \mathbb{R}$ defined by $ \overline{f}(x_1,\dots,x_m):= f \left(\tfrac{\lfloor x_1\cdot (n +1)\rfloor}{n+1} \wedge \tfrac{n}{n+1},\dots,\tfrac{\lfloor x_n\cdot (n +1)\rfloor}{n+1} \wedge \tfrac{n}{n+1}\right)$ where $\lfloor x \rfloor := \max \{ n \in \mathbb{N}_0: n \leq x \}$, $x \in \mathbb{R}_+$, we see that
every function $f:\mathbb{K}_n^m \to \mathbb{R}$ has bounded Hardy--Krause variation as $\mathbb{K}_n^m \subseteq [0,1]^m$ is discrete, and hence induces a finite signed measure on $\mathbb{K}_n^m$. We refer to the discussion after Definition~\ref{def_measure_inducing}, see also, e.g. \cite[Theorem 3]{Aistleitner-2015} or \cite[Theorem 3.29]{folland1999real}.
\begin{lem}\label{lemmadia}
Let $m,n \in \mathbb{N}$, and let \(\sigma\) be a finite signed measure on \(\mathbb{K}_n^m\,.\) Then the following statements hold true:
\begin{itemize}
\item[(a)] \label{lemmadia1} There exists a finite signed measure \(\mu\) on \(\mathbb{K}_n^1\) such that
\begin{align}\label{eqmadia}
G_\sigma(u_1,\ldots,u_m)=G_\mu(\min_{i=1,\ldots,m}\{u_i\})~~~ \text{for all }(u_1,\ldots,u_m)\in \mathbb{K}_n^m\,,
\end{align}
if and only if the mass of \(\sigma\) is concentrated on the diagonal of \(\mathbb{K}_n^m\,\), denoted by \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\), i.e., \(\sigma(\{x\})=0\) for all \(x\in \mathbb{K}_n^m\setminus \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\)
In this case $\mu$ is defined by
\begin{equation}\label{eq_def_mu_sigma}
\mu([0,u]):=\sigma\left([0,u]\times \cdots \times [0,u]\right),~u \in [0,1].
\end{equation}
\item[(b)] \label{lemmadia2} Assume that \(\sigma(\mathbb{K}_n^m)=1\,.\) In the case that \(\sigma\) fulfils \eqref{eqmadia} for some signed measure \(\mu\) on \(\mathbb{K}_n^1\,,\) it follows that
\begin{align}\label{eqmadia2}
\int_{[0,1]^m} f(u_1,\ldots,u_m) \mathsf{\,d} G_\sigma (u_1,\ldots,u_m) = \int_{[0,1]} f(v,\ldots,v) \mathsf{\,d} G_\mu(v)
\end{align}
for all \(\sigma\)-integrable functions \(f\colon [0,1]^m \to \mathbb{R}\,.\)
\end{itemize}
\end{lem}
\begin{proof}
To show (a), first assume that \eqref{eqmadia} holds and let \(x=(x_1,\ldots,x_m)\in \mathbb{K}_n^m\setminus \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\) Then,
\begin{align*}
\sigma(\{x\})&= \triangle_{1/n}^1 \cdots \triangle_{1/n}^m G_\sigma(x_1,\ldots,x_m)\\
&= \triangle_{1/n}^1\cdots \triangle_{1/n}^m G_\mu\left(\min_{i\in \{1,\ldots,m\}}\{x_i\}\right) = 0\,,
\end{align*}
because there exists \(j\in \{1,\ldots,m\}\) such that \(x_j>\min_{i\in \{1,\ldots,m\}}\{x_i\}\) and thus
\begin{align*}
\triangle_{1/n}^j G_\mu\left(\min_{i\in \{1,\ldots,m\}}\{x_i\}\right)= G_\mu\left(\min_{i\ne j}\{x_i\}\right)-G_\mu\left(\min_{i\ne j}\{x_i\}\right) = 0\,.
\end{align*}
For the reverse direction, assume that the mass associated with \(\sigma\) is concentrated on \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\). Then \(\mu\) defined by \(\mu([0,u]):=\sigma([0,u]\times \cdots \times [0,u])\,,\) \(u\in \mathbb{K}_n^1\,,\) is a signed measure with the property that
\begin{align*}
G_\sigma(u_1,\ldots,u_m)&=\sigma([0,u_1]\times \cdots \times [0,u_m]) \\&= \sigma([0,\min_{i=1,\ldots,m} \{u_i\}]^m) = \mu([0,\min_{i=1,\ldots,m} \{u_i\}]) = G_\mu(\min_{i=1,\ldots,m} \{u_i\})\,,
\end{align*}
where the second equality holds true because \(\sigma(\{x\})=0\) for all \(x\in \mathbb{K}_n^m\setminus \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\)
To show (b), let us first consider the case where \(\sigma\) is a probability measure, i.e., all mass (which is by (a) distributed on \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\)) is non-negative.
Let $U_1,\dots,U_m$ be random variables on a probability space $(\Omega,\mathcal{A},\sigma)$ such that $(U_1,\dots,U_m) \sim \sigma$. Since $\sigma$ is concentrated on the diagonal, we have that $U_i \stackrel{\mathrm{d}}= U_j$ and that $U_i,U_j$ are comonotone for all $i,j\in \{ 1,\dots,m\}$. Hence, with $U: \stackrel{\mathrm{d}}= U_1$, we obtain $(U_1,\ldots,U_m)\stackrel{\mathrm{d}}= (U,\ldots,U)$.
This implies
\begin{align}\label{eqmadia3}
\begin{split}
\int_{[0,1]^m} f(u_1,\ldots,u_m)\mathsf{\,d} G_\sigma (u_1,\ldots,u_m) &= \int_{\Omega} f(U_1,\ldots,U_m)\mathsf{\,d} P\\
&= \int_{\Omega}f(U,\ldots,U)\mathsf{\,d} P = \int_{[0,1]} f(v,\ldots,v)\mathsf{\,d} G_\mu(v)\,,
\end{split}
\end{align}
which proves \eqref{eqmadia2} in the case where \(\sigma\) is a probability measure.
Now, consider the general case where \(\sigma\) is a finite signed measure satisfying $\sigma(\mathbb{K}_n^m)=1$. By \eqref{lemmadia1} all mass of \(\sigma\) is concentrated on \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\) which is a finite set. So, there exists \(M\in \mathbb{N}\) such that \(\sigma(x)\geq -M\) for all \(x\in \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\) Denote by \(\sigma^u\) the uniform distribution on \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,,\) i.e., \(\sigma^u(\{x\})=\tfrac 1 n\) for all \(x\in \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\) and \(\sigma^u(\{x\})=0\) for all \(x\in \mathbb{K}_n^m\setminus \mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\) Then, since $\sigma(\mathbb{K}_n^m)=1$,
\begin{equation}\label{eq_definition_sigma^M}
\sigma^M:=\frac{nM \sigma^u+\sigma}{nM+1}
\end{equation}
defines a probability measure on \(\mathbb{K}_n^m\) with non-negative mass and which is concentrated on \(\mathop{\mathrm{diag}}\nolimits(\mathbb{K}_n^m)\,.\) Then, by \eqref{eq_def_mu_sigma}, the measure \(\mu^M\) defined by
\begin{align*}
\mu^M([0,v]):= \sigma^M([0,v]\times \cdots\times [0,v])\,,
\end{align*}
is related to \(\sigma^M\) by \(G_{\sigma^M}(u_1,\ldots,u_m)=G_{\mu^M}(\min_{i=1,\ldots,m}\{u_i\})\,,\) \((u_1,\ldots,u_m)\in [0,1]^m\,.\) Hence, we obtain by \eqref{eqmadia3} that
\begin{align*}
\int_{[0,1]^m} f(u_1,\ldots,u_m)\mathsf{\,d} \sigma^M(u_1,\ldots,u_m) = \int_{[0,1]} f(u,\ldots,u) \mathsf{\,d} \mu^M(u)\,.
\end{align*}
For \(\mu^u\) defined by \(\mu^u([0,v]):=\sigma^u([0,v]\times \cdots \times [0,v])\,,\) \(v\in [0,1]\,,\) we obtain, by using \eqref{eq_definition_sigma^M}, the identity \(\mu=(nM+1) \mu^M-nM \mu^u\,\). This yields
\begin{align*}
\int_{[0,1]^m} f(u) \mathsf{\,d} \sigma(u) &= (nM+1) \int_{[0,1]^m} f(u)\mathsf{\,d} \sigma^M(u) - nM \int_{[0,1]^m} f(u)\mathsf{\,d} \sigma^u(u) \\
&= (nM+1) \int_{[0,1]} f(v,\cdots,v) \mathsf{\,d} \mu^M(v)-nM \int_{[0,1]} f(v,\cdots,v) \mathsf{\,d} \mu^u(v) \\
&= \int_{[0,1]} f(v,\cdots,v) \mathsf{\,d} \mu(v)\,,
\end{align*}
which proves \eqref{eqmadia2}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theqcub}.]\label{protheamr}
\underline{(f) \(\Longrightarrow\) (c)}: For any fixed \(u=(u_1,\ldots,u_m)\in (0,1)^m\,,\) let $f(x):= \one_{\{u<x\}}$, $\widetilde{f}(x):= \one_{\{u
\leq x\}}$, $x=(x_1,\ldots,x_m)\in [0,1)^m\,,$ and let \((\Phi_n)_{n \in \mathbb{N}}\) be a sequence of \(N(u,I_m/n)\)-distribution functions, i.e., \(\Phi_n\) is the distribution function of the \(m\)-variate normal distribution with mean vector \(u\) and covariance matrix \(I_m/n\,,\) where \(I_m\) denotes the \((m\times m)\)-unit matrix.
Then, \(\Phi_n\) is \(\Delta\)-monotone and, thus, supermodular and measure-inducing for all \(n \in \mathbb{N}\,.\)
Note that \(\eta_{\Phi_n}\to \eta_{\widetilde{f}}=\delta_{\{u\}}\) weakly as \(n\to \infty\,,\) where \(\delta_{\{u\}}\) denotes the one-point probability measure in \(u\).
Moreover, note that $\eta_{\widetilde{f}}=\eta_f$.
Further, \(\phi_f\) defined via \eqref{defphif} by
\begin{align}\label{eqonpomeas}
\phi_f(x_1,x_2)=f(x_2,x_1,\ldots,x_1)=\one_{\{u_1<x_2,\max_{2\leq i\leq m}\{u_i\} < x_1\}}
\end{align}
is componentwise left-continuous and induces the one-point probability measure \(\eta_{\phi_f} = \delta_{\{\max_{2\leq i\leq m}\{u_i\},u_1\}} \,.\)
Thus, for the survival function of the upper product, it follows by \eqref{survdisfun}, and since $M^2\vee D^2\vee \cdots \vee D^m$ is a copula, that
\begin{equation}\label{eq_m_dddd_hat_proof_eq_1}
\begin{aligned}
\reallywidehat{M^2\vee D^2\vee \cdots \vee D^m}\,(u) &= \int_{[0,1]^m} \one_{\{u < v\}} \mathsf{\,d} (M^2\vee D^2\vee \cdots \vee D^m)(v) \\
&= \lim_{n\to \infty} \int_{[0,1]^m} \Phi_n(v) \mathsf{\,d} (M^2\vee D^2\vee \cdots \vee D^m)(v) \\
& = \lim_{n\to \infty} \psi_{\Phi_n}(M^2\vee D^2\vee \cdots \vee D^m)\\
&\leq \lim_{n\to \infty} \pi_{\phi_{\Phi_n}} (\widehat{Q_2})\\
&= \lim_{n\to \infty} \sum_{I\subseteq \{1,2\}\atop I\ne \emptyset} \int_{[0,1]^{|I|}} (\widehat{Q_{2}})_I(v) \mathsf{\,d} \eta_{(\phi_{\Phi_n})_I}(v) + \phi_{\Phi_n}(0,0)\\
&= \int_{[0,1]^2} \widehat{Q_2}(u) \mathsf{\,d} \eta_{\phi_f}(u)\\
&= \widehat{Q_2}(\max_{2\leq i \leq m}\{u_i\},u_1)\\
&= \widehat{Q^*}(u_1,\ldots,u_m)\,,
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_m_dddd_hat_proof_eq_1} holds, note that the second equality follows from the dominated convergence theorem and by using that $M^2\vee D^2\vee \cdots \vee D^m$ is continuous. The third equality is due to \eqref{defexpop_0} using that the upper product is a copula and thus continuous and measure-inducing.
The inequality holds by assumption using that \(u\to \Phi_n(u,\ldots,u)\) is Lebesgue-integrable and that \(\phi_{\Phi_n}\) is \(\Delta\)-monotone and thus measure-inducing.
The fourth equality is given by \eqref{defquexpop}. For the fifth equality, we apply that \((\eta_{\phi_{\Phi_n}})_n\) converges weakly to \(\eta_{\phi_f}\,,\) and that the measures \(\eta_{(\phi_{\Phi_n})_{\{1\}}}\) and \(\eta_{(\phi_{\Phi_n})_{\{2\}}}\) induced by the marginals of \(\phi_{\Phi_n}\) converge weakly to the null-measure because, as $u \in (0,1)^m$, \((\phi_{\Phi_n})_{\{1\}}(x)=\phi_{\Phi_n}(x,0)\to 0=\phi_f(x,0)= (\phi_f)_{\{1\}}(x)\) for all \(x\in [0,1]^m\,,\) and similarly for \((\eta_{(\phi_{\Phi_n})_{\{2\}}})_n\,.\) Further, we use that \(\phi_{\Phi_n}(0,0)\to 0=\phi_f(0,0)\,.\) The sixth equality follows from \eqref{eqonpomeas}, and the last equality holds due to Lemma \ref{lemsufu}.
Since $u \mapsto \reallywidehat{M^2\vee D^2\vee \cdots \vee D^m}\,(u)$ and $u \mapsto \widehat{Q^*}(u)$ are both continuous on $[0,1]^m$, we obtain that \eqref{eq_m_dddd_hat_proof_eq_1} holds also for $u\in [0,1]^m$.\\
\underline{(c) \(\Longrightarrow\)(a)}:
For \(i\in\{2,\ldots,m\}\) let \(u=(u_1,\ldots,u_m)\in [0,1]^m\) with \(u_j=0\) for all \mbox{\(j\in \{2,\ldots,m\}\setminus \{i\}\,.\)} Then, the survival function of \(Q_2\) satisfies that
\begin{equation}
\label{eq_iii_i_proof_equalities_1}
\begin{aligned}
\widehat{Q_2}(u_i,u_1) &=\widehat{Q_2} \left(\max_{2\leq j\leq m}\{u_j\},u_1 \right)\\
&= \widehat{Q^*}(u_1,\ldots,u_m)\\
&\geq \reallywidehat{M^2 \vee D^2 \vee \cdots \vee D^m}\,(u) \\
&= 1-\int_0^1 \max\{\one_{\{u_1>t\}},\partial_2 D^2(u_2,t),\ldots,\partial_2 D^m(u_m,t)\} \mathsf{\,d} t\\
&= 1- u_1 - \int_{u_1}^1 \max_{2\leq j\leq m}\{\partial_2 D^j(u_j,t)\} \mathsf{\,d} t \\
&= 1- u_1 - u_i + D^i(u_i,u_1)\,.
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_iii_i_proof_equalities_1} holds, note that the second equality follows with Lemma \ref{lemsufu}. The inequality holds by assumption (c). The third equality follows with \cite[Proposition 2.4~(viii)]{Ansari-Rueschendorf-2018} by using that \(\partial_2 M^2(u_1,t)=\one_{\{u_1>t\}}\) for all $t\in [0,1]$ with \(t\ne u_1\,.\) The fourth equality is a consequence of \(0\leq \partial_2 D^j(u_j,t)\leq 1\) for Lebesgue-almost all \(t\in [0,1]\) and for \(j=2,\ldots m\,,\) see \cite[Theorem 2.2.7]{Nelsen-2006}. The last equality holds true by \cite[Theorem 2.2.7]{Nelsen-2006} because \(\partial_2 D^i(u_i,t)\geq \partial_2 D^i(0,t)=\partial_2D^j(u_j,t)\) for Lebesgue-almost all \(t\in [0,1]\) and for all \(j\ne i\,,\) using that \(D^2,\ldots,D^m\) are copulas. Hence, it follows that
\begin{align*}
D^i(u_i,u_1)\leq \widehat{Q_2}(u_i,u_1)-1+u_1+u_i = Q_2(u_i,u_1)\,.
\end{align*}
\underline{(a) \(\Longrightarrow\) (b):}
For \(u=(u_1,\ldots,u_m)\in [0,1]^m\,,\) we have
\begin{equation}\label{eq_i_ii}
\begin{aligned}
M^2\vee D^2\vee \ldots\vee D^m ~(u)&=\int_0^{u_1}\min_{2\leq i \leq m}\{\partial_2 D^i(u_i,t)\} \mathsf{\,d} t\\
&\leq \min_{2\leq i \leq m}\{D^i(u_i,u_1)\}\\
&\leq \min_{2\leq i \leq m}\{Q_2(u_i,u_1)\}\\
&= Q_2(\min_{2\leq i \leq m}\{u_i\},u_1) = Q^*(u).
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_i_ii} holds, note that the first equality follows from the definition of the upper product, from \(\partial_2 M^2(u_1,t)=\one_{\{u_1>t\}}\) and by \(0 \leq \partial_2 D^i(u_i,t) \leq 1\) for all \(i=1,\dots,m\) and for Lebesgue-almost all \(t\in [0,1]\,.\) The first inequality is a consequence of Jensen's inequality, the fundamental theorem of calculus, and property \eqref{def_quasicop1} in Definition \ref{def_quasicop} of copulas. The second inequality holds by assumption (a). The second equality follows because \(Q_2\) is a quasi-copula and, thus, non-decreasing in each argument.\\
\underline{(b) \(\Longrightarrow\) (a):}
Let $i\in \{2,\ldots,m\}\,.\) For \(u=(u_1,\ldots,u_m)\in [0,1]^m\) such that \(u_j=1\) for all \(j\in \{2,\ldots,m\}\setminus \{i\}\) it follows that
\begin{align*}
D^i(u_i,u_1) = M^2\vee D^2\vee \cdots \vee D^m \,(u) \leq Q^*(u) = Q_2(u_i,u_1)\,,
\end{align*}
where the first equality is given by \cite[Proposition 2.4~(iv),(vi)]{Ansari-Rueschendorf-2018} and the inequality holds by assumption~(b).\\
\underline{((b) and (c)) \(\Longleftrightarrow\) (d):} This holds by the definition of the concordance ordering.\\
\underline{(a)\(\Longrightarrow\)(e)}:
We extend the proof of the main result in \cite[Chapter 3]{Ansari-Rueschendorf-2020} to a quasi-copula \(Q_2\in \mathcal{Q}_2\) instead of a copula \(E\in \mathcal{C}_2\,,\) cf.\,Remark \ref{remmaithean1}~(a).
Analogously, we first prove the statement in a discretized version using that all discretized copulas and quasi-copulas induce (signed) measures with finite support.
Then, we show the statement by an approximation of the discretized version, which differs from the proof of \cite[Theorem 1]{Ansari-Rueschendorf-2020} because we need to apply the quasi-expectation operator w.r.t.\,a quasi-copula instead of the expectation w.r.t.\,a probability measure.
For the first step, we make use of the same ideas and concepts as in the first part of the proof of \cite[Theorem 1]{Ansari-Rueschendorf-2020}, namely applying mass transfer theory from \cite{Mueller-2013} which requires a discretization of the distributions to a finite grid as follows. For \(n\in \mathbb{N}\) and \(m\geq 1\) denote by
\begin{align*}
\mathbb{G}_{n}^m:&= \left\{(\tfrac {i_1} n,\ldots,\tfrac {i_m} n) ~\middle|~ i_k\in \{1,\ldots,n\} \text{ for all } k\in \{1,\ldots,m\}\right\}\,,\\
\mathbb{G}_{n,0}^m:&= \left\{(\tfrac {i_1} n,\ldots,\tfrac {i_m} n) ~\middle|~ i_k\in \{0,\ldots,n\} \text{ for all } k\in \{1,\ldots,m\}\right\}
\end{align*}
the (extended) uniform unit \emph{\(n\)-grid} of dimension \(m\) with edge length \(\tfrac 1 n\,.\)
For the discretization of copulas and quasi-copulas, we use the concept of a \emph{(signed) $n$-grid $m$-copula} \(D\colon [0,1]^m \to \mathbb{R}\) which is the measure-generating function (see \eqref{eq_def_g_sigma}) of a (signed) measure \(\mu\) on \(\mathbb{G}_{n}^m\) that satisfies for all \(u=(u_1,\dots,u_m)\in [0,1]^m\)
\begin{enumerate}[(i)]
\item \(D(u)=D\left(\frac{\lfloor n u_1 \rfloor}{n},\dots,\frac{\lfloor n u_m \rfloor}{n}\right)=\mu\left([0,\frac{\lfloor n u_1 \rfloor}{n}]\times \cdots \times [0,\frac{\lfloor n u_m \rfloor}{n}]\right)\),
\item for all \(i=1,\ldots,m\,,\) it holds \(D(u)=\tfrac k n\) for all \(k=0,\ldots,n\,,\) if \(u_i=\tfrac k n\) and \(u_j=1\) for all \(j\ne i\,,\)
\end{enumerate}
where \(\lfloor\cdot\rfloor\) is the componentwise floor function. Denote by \(\mathcal{C}_{m,n}\) (respectively \(\mathcal{C}_{m,n}^s\)) the set of all (signed) \(n\)-grid \(m\)-copulas. Note that, as discussed after \eqref{eq_def_g_sigma}, a (signed) \(n\)-grid \(m\)-copula induces a (signed) measure with support on the finite grid \(\mathbb{G}_n^m\,.\)
Hence, for every $m$-variate quasi-copula \(Q\in \mathcal{Q}_m\,,\) the \emph{canonical \(n\)-grid quasi-copula} \(\mathbb{G}_n(Q')\) defined by
\begin{equation}\label{eq_G_n_eq_q_prime}
\mathbb{G}_n(Q')(u):=Q'(\tfrac{\lfloor nu \rfloor}{n})\,,~~~u\in [0,1]^m\,,
\end{equation}
induces a signed measure.
For a function \(g\colon [0,1]^m\to \mathbb{R}\,,\) denote the difference operator of length \(\tfrac 1 n\) w.r.t.\,the \(i\)-th variable by
\begin{equation}\label{eq_definition_differencfes_proof}
\Delta_n^i g(u):= g(u)-g(\max\{u-\tfrac 1 n e_i,0\})\,,
\end{equation}
where \(e_i\) is the \(i\)-th unit vector. Then, we define the upper product \(\bigvee \colon (\mathcal{C}_{2,n})^m \to \mathcal{C}_{m,n}\) for grid copulas \(D_n^1,\ldots,D_n^m\in \mathcal{C}_{2,n}\) by
\begin{align}\label{defdisuppprod}
\bigvee_{i=1}^m D_n^i (u_1,\ldots,u_m):&=\sum_{k=1}^n \min_{1\leq i \leq m}\left\{\Delta_n^2 D_n^i(u_i,\tfrac k n)\right\}\\
\nonumber&=\frac 1 n\sum_{k=1}^n \min_{1\leq i \leq m}\left\{n\Delta_n^2 D_n^i(u_i,\tfrac k n)\right\}
\end{align}
for \((u_1,\ldots,u_m)\in [0,1]^m\,.\) To see that indeed $\bigvee_{i=1}^m D_n^i \in \mathcal{C}_{m,n}$ note for the uniform marginal property~(i) that for all \(k=0,\ldots,n\,,\) if \(u_i=\tfrac k n\) and \(u_j=1\) for all $j\in \{1,\dots,m\}\setminus\{i\}$, then, by definition
\begin{align*}
\bigvee_{\ell=1}^m D_n^\ell (u_1,\ldots,u_m) &= \sum_{j=1}^n \min \left\{\min_{1\leq \ell \leq m, \atop \ell \neq i}\left\{D_n^\ell(1,\tfrac{j}{n})-D_n^\ell(1,\tfrac{j-1}{n})\right\},~ D_n^i(\tfrac{k}{n},\tfrac{j}{n})-D_n^i(\tfrac{k}{n},\tfrac{j-1}{n}) \right\} \\
&= \sum_{j=1}^n \min \left\{\frac{1}{n},~D_n^i(\tfrac{k}{n},\tfrac{j}{n})-D_n^i(\tfrac{k}{n},\tfrac{j-1}{n}) \right \} \\
&=\sum_{j=1}^n \left\{D_n^i\left(\frac{k}{n},\frac{j}{n}\right)-D_n^i\left(\frac{k}{n},\frac{j-1}{n}\right)\right\} = D_n^i\left(\frac{k}{n},1\right)-D_n^i\left(\frac{k}{n},0\right) = \frac{k}{n}\,,
\end{align*}
where the last equality follows from the fact that the measure induced by \(D_n^i\) is a measure on \(\mathbb{G}_n^2\), and hence \(D_n^i(\tfrac k n,0)=0\). For the third equality, we use that
\begin{align*}
D_n^i(\tfrac k n, \tfrac j n)-D_n^i(\tfrac k n, \tfrac {j-1} n) &= \mu_n^i([0,\tfrac k n]\times (\tfrac {j-1} n , \tfrac j n]) \\
&\leq \mu_n^i(([0,1]\times (\tfrac {j-1} n , \tfrac j n]) = D_n^i(1,\tfrac j n)-D_n^i(1,\tfrac {j-1}n) = \tfrac 1 n,
\end{align*}
where here $\mu_n^i$ is the measure induced by the grid copula \(D_n^i\).
Moreover to see~(ii), note that the upper product $\bigvee_{i=1}^m D_n^i$ is the measure generating function of a measure $\mu$ on $\mathbb{G}_{n,0}^m$ defined by
\[
G_{\mu}(u_1,\dots,u_m):=\bigvee_{i=1}^m D_n^i (u_1,\ldots,u_m),~~(u_1,\dots,u_m) \in [0,1]^m,
\]
since every $D_n^i$, $i=1,\dots,m$ is the measure generating function of a measure $\mu_n^i$ on $\mathbb{G}_{n,0}^2$,
compare also \eqref{eq_def_g_sigma} and below.
The upper product for signed grid copulas \(D^1,\ldots,D^m\) is defined analogously where \(\bigvee_{i=1}^m D_n^i\in \mathcal{C}_{m,n}^s\) whenever \(\Delta_n^2 D_n^i(\cdot,t)\leq \tfrac 1 n\) for all \(i\in \{1,\ldots,m\}\) and \(t\in [0,1]\,.\)
Let \(D_n^i:=\mathbb{G}_n(D^i)\), $M_n^2:=\mathbb{G}_n(M^2)$, \(Q_{2,n}:=\mathbb{G}_n(Q_2)\), and \(Q_n^*:=\mathbb{G}_n(Q^*)\) be the canonical \(n\)-grid (quasi-)copulas of \(D^i\,,\) \(i=2,\ldots,m\,,\) $M^2$, \(Q_2\), and \(Q^*\,.\) Then it holds that \(Q_n^*(u)=Q_{2,n}(\min_{2\leq i \leq m}\{u_i\},u_1)\) and \(D_n^i(u_1,u_i)\leq Q_{2,n}(u_1,u_i)\) for all \(u=(u_1,\ldots,u_m)\in [0,1]^m\) and \(i\in \{2,\ldots,m\}\,.\)
Now, define for \(n\in \mathbb{N}\) the finite sequence \((Q_{n,k}^*)_{0\leq k \leq n}\) of signed \(n\)-grid quasi-copulas iteratively by
\begin{align*}
Q_{n,0}^*&:=M_n^2\vee D_n^2\vee\cdots \vee D_n^m\,,
\end{align*}
and, for $ 1\leq k \leq n$, by $Q_{n,k}^*$ via
\begin{equation}\label{eq_definition_grid_quasi_copula_differences}
\begin{aligned}
Q_{n,k}^*(u)&:=\sum_{l\leq \lfloor nu_1\rfloor\atop l\in \mathbb{N}_0} \Delta_n^1
Q_{n,k}^*(\tfrac l n,u_2,\ldots,u_m) ~\text{for} ~u=(u_1,\ldots,u_m)\in \mathbb{G}_{n}^m\,, ~\text{where}\\
\Delta_n^1 Q_{n,k}^*(u)&:=
\begin{cases}
\Delta_n^1 Q_{n}^* (u_1,\ldots,u_m) & \text{if }u_1\leq \frac k n\,,\\
\Delta_n^1 M^2_n\vee D^2_n\vee \ldots\vee D^m_n (u)
-\big[\Delta_n^1 Q_{n}^*(u_1-\tfrac 1 n,u_2,\ldots,u_m)&\\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\Delta_n^1 Q_{n,k-1}^*(u_1-\tfrac 1 n,u_2,\ldots,u_m)\big]& \text{if } u_1 = \frac {k+1} n\,,\\
\Delta_n^1 M^2_n\vee D_n^2\vee\cdots \vee D_n^m (u) &\text{if } u_1 > \frac {k+1} n\,.
\end{cases}
\end{aligned}
\end{equation}
Then, for \(1\leq k \leq n\,,\) we have by construction for all \(u=(u_1,\ldots,u_m)\in \mathbb{G}_n^m\) with \(u_1\leq \tfrac k n\)
that
\begin{equation}\label{eq_q*nk}
\begin{aligned}
Q_{n,k}^*(u)&=\sum_{l\leq \lfloor nu_1\rfloor\atop l\in \mathbb{N}_0} \Delta_n^1 Q_{n}^*(\tfrac l n,u_2,\ldots,u_m)\\
&=\sum_{l\leq \lfloor nu_1\rfloor\atop l\in \mathbb{N}_0} Q_{n}^*(\tfrac{l} {n},u_2,\ldots,u_m)-Q_{n}^*\left(\max\left\{\tfrac{l-1}{n},0\right\},u_2,\ldots,u_m\right)\\
&=Q_{n}^*(\tfrac{\lfloor nu_1\rfloor}{n},u_2,\ldots,u_m)-Q_{n}^*\left(0,u_2,\ldots,u_m\right)\\
&=Q_{n}^*(u)-Q_n^*\left(0,u_2,\ldots,u_m\right)\\
&=Q_{n}^*(u).
\end{aligned}
\end{equation}
Indeed to see that \eqref{eq_q*nk} holds, note that the first equality holds by the definition in \eqref{eq_definition_grid_quasi_copula_differences} and since $\tfrac{\lfloor nu_1\rfloor}{n} \leq \tfrac{k}{n}$ by assumption. The second equality is a consequence of \eqref{eq_definition_differencfes_proof}. The third equality follows since the sum is a telescoping sum. The fourth equality follows by definition of the $n$-grid quasi copula $Q_n^*$ which coincides on the grid $\mathbb{G}_n^m$ with the original copula $Q^*$. The last equality follows since quasi-copulas are grounded according to Definition~\ref{def_quasicop}~(a)~(i). Then by definition of \(Q_{n,0}^*\) and by Equation \eqref{eq_q*nk}, it follows that
\begin{equation}\label{eq_appendix_proof1}
\begin{aligned}
Q_{n,0}^*&=M_n^2\vee D_n^2\vee\cdots \vee D_n^m\,,\\
Q_{n,n}^*&= Q_n^*\,.
\end{aligned}
\end{equation}
Noting that the upper product in \eqref{eqcococo} and the quasi-copula \(Q^*\) in \eqref{deqcqs} have a similar form, we obtain exactly in the same way as in the proof of \cite[Theorem 1]{Ansari-Rueschendorf-2020} by means of mass transfer theory by \cite{Mueller-2013} that
\begin{align}\label{eqppp}
Q_{n,k-1}^*\leq_{\operatorname{sm}} Q_{n,k}^*~~~\text{for all }1\leq k \leq n ~\text{and for all } n\in \mathbb{N}\,,
\end{align}
Note that the supermodular ordering can also be defined w.r.t.\,finite signed measures \(\nu_1\) and \(\nu_2\) with finite support because the inequality \(\int f \mathsf{\,d} \nu_1\leq \int f \mathsf{\,d} \nu_2\) depends only the difference \(\nu_2-\nu_1\) by \(\int f \mathsf{\,d} (\nu_2-\nu_1)\geq 0\) for \(f\in \mathcal{F}_{\operatorname{sm}}\,.\)
So, the comparison in \eqref{eqppp} is well-defined because the expressions on both sides are signed grid copulas which correspond to signed measures with finite support \(\mathbb{G}_n^m\,.\) Hence, each \(Q_{n,k}^*\) induces a signed measure which we can integrate against.
Due to the transitivity of the supermodular ordering, \eqref{eq_appendix_proof1} and \eqref{eqppp} imply that \(M_n^2\vee D_n^2\vee\cdots \vee D_n^m\leq_{\operatorname{sm}} Q_n^*\) for all \(n\in \mathbb{N}\,,\) i.e.,
\begin{align}\label{mrdisve}
\int_{[0,1]^m} f(u)\mathsf{\,d} (M_n^2\vee D_n^2\vee\cdots \vee D_n^m)(u) \leq \int_{[0,1]^m} f(u) \mathsf{\,d} Q_n^*(u)~~~\,,
\end{align}
$\text{for all }f\in \mathcal{F}_{\operatorname{sm}}$ such that the integrals exist. This proves the statement in the discretized version for grid copulas.
For the second step, let \(f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1]^m)\) be a left-continuous and supermodular function.
Note that \(f\) is bounded because it is measure-inducing and defined on a compact domain.
In the first step, we chose for notational conveniences the grid \(\mathbb{G}_n^m=\{\tfrac 1 n,\ldots,\tfrac {n-1}n,1\}^m\) as support of the discretized copulas and quasi-copulas
\begin{align}\label{disqqr}
\begin{split}
C_n&:= M_n^2\vee D_n^2 \vee\cdots \vee D_n^m = M^2\vee D^2\vee\cdots\vee D^m\circ(F_n,\ldots,F_n) ~~~\text{and}\\
Q_n^*&\phantom{:}= Q^*\circ (F_n,\ldots,F_n)\,,
\end{split}
\end{align}
respectively, where \(F_n:[0,1]\rightarrow [0,1]$ is now defined as
\begin{align}\label{dffn}
F_n(x)&= \begin{cases}
0 & \text{if } x<\tfrac 1 {n+1}\,,\\
\tfrac k n &\text{if } x \in \big[\tfrac k {n+1}, \tfrac {k+1}{n+1}\big)\,, k=1,\ldots,n\,,\\
1& \text{if } x\geq 1\,.
\end{cases}
\end{align}
Note that the range of \(F_n\) is also \(\{0,\tfrac 1 n,\ldots,\tfrac {n-1}n,1\}\,.\)
Then, \(C_n\) and \(Q_n^*\) in \eqref{disqqr} are distributions with finite support on the grid \(\mathbb{K}_n^m:=\{\tfrac 1 {n+1},\ldots,\tfrac n {n+1}\}^m\,.\) Applying mass transfer theory analogously to the first step, we also obtain \eqref{mrdisve}, now for the discretization w.r.t.\, $F_n$ defined by \eqref{disqqr}, i.e., we have $ \int_{[0,1]^m} f(u) \mathsf{\,d} C_n(u) \leq \int_{[0,1]^m} f(u) \mathsf{\,d} Q_n^*(u)$. Then, it follows that
\begin{equation}\label{eq_pi_f}
\begin{aligned}
\pi_f(M^2\vee D^2\vee \cdots \vee D^m) &=
\sum_{I\subseteq \{1,\ldots,m\} \atop I\ne \emptyset} \int_{[0,1]^m} \lim_{n\to \infty} (\widehat{C_n})_I(u) \mathsf{\,d} \eta_{f_I}(u) + f(0,\ldots,0) \\
&= \lim_{n\to \infty} \sum_{I\subseteq \{1,\ldots,m\} \atop I\ne \emptyset} \int_{[0,1]^m} (\widehat{C_n})_I(u) \mathsf{\,d} \eta_{f_I}(u) + f(0,\ldots,0)\\
&=\lim_{n\to \infty} \int_{[0,1]^m} f(u) \mathsf{\,d} C_n(u) \\
& \leq \lim_{n\to \infty} \int_{[0,1]^m} f(u) \mathsf{\,d} Q_n^*(u) \\
&= \lim_{n\to \infty} \psi_{Q^*\circ(F_n,\ldots,F_n)}(f)\\
& = \pi_f(\widehat{Q^*})\,.
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_pi_f} holds, note that for the first equality, we apply \eqref{qeopexopeq} using that \(f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}\) and thus \(f_I\) induce a finite signed measures, \(I\subseteq\{1,\ldots,m\}\,.\) Further, we apply that the grid approximation \(C_n\) converges weakly and, thus, pointwise to \eqref{defquexpop} see \cite[Proposition 2.12]{Ansari-Rueschendorf-2018}, using that \(M^2\vee D^2\vee \cdots \vee D^m\) is a copula and thus continuous. Moreover, we use that $\reallywidehat{M^2\vee D^2\vee \cdots \vee D^m}(0,\cdots,0)=1$.
The second equality holds due to the dominated convergence theorem applying again that \(f\) induces a finite signed measure.
The third and fourth equality follow from \eqref{eqqop} using that the discretized copula \(C_n\) and the discretized quasi-copula \(Q_n^* = Q^*\circ(F_n\ldots,F_n)\) are right-continuous, grounded, bounded, measure-inducing, and fulfil the continuity conditions \eqref{contboun1} and \eqref{contboun2}.
The last equality follows from Proposition \ref{propconcomdom}.
The inequality is a consequence of the discretized supermodular ordering result \eqref{mrdisve} in the modified version discretizing w.r.t. the grid \(\mathbb{K}_n^m\,.\)
\underline{(e) \(\Longrightarrow\) (f)}:
Let \(f\colon[0,1)^m\to \mathbb{R}\) be lower bounded by some $M^2\vee D^2\vee \cdots \vee D^m$-integrable function, left-continuous, supermodular, and componentwise increasing/componentwise decreasing
such that \((\phi_f)_I\) is Lebesgue integrable on \([0,1)^{|I|}\) for \(I\subseteq\{1,2\}\,,\) \(I\ne \emptyset\,,\)
For \(n\in \mathbb{N}\,,\) let \(F_n\) be the distribution function given by \eqref{dffn}. We first show that
\begin{align}\label{azw}
\pi_{f\circ(F_n^{-1},\ldots,F_n^{-1})} (\widehat{Q^*}) = \pi_{\phi_f \circ (F_n^{-1},F_n^{-1})}(\widehat{Q_2})\,.
\end{align}
Define \(Q_{(n)}(u_1,\ldots,u_m):=Q^*(F_n(u_1),\ldots,F_n(u_m))\) and \(Q_{2,(n)}(u_1,u_2):=Q_2(F_n(u_2),F_n(u_1))\) for \(u_1,\ldots,u_m\in [0,1]\,.\) Then \(Q_{(n)}\) and \(Q_{2,(n)}\) induce by \eqref{eqindmeasu} finite signed measures on \(\mathcal{B}([0,1]^m)\) and \(\mathcal{B}([0,1]^2)\) with mass concentrated on the \(n\)-grid
\(\mathbb{K}_n^m=\{\tfrac 1 {n+1},\ldots,\tfrac {n} {n+1}\}^m\) and \(\mathbb{K}_n^2=\{\tfrac 1 {n+1},\ldots,\tfrac {n} {n+1}\}^2\,,\) respectively.
For \(u_1\in \mathbb{K}_n^1=\{\tfrac 1 {n+1},\ldots,\tfrac {n} {n+1}\}\,,\) consider the conditional measure generating functions \(Q_{(n)}^{u_1}\) and \(Q_{2,(n)}^{u_1}\) given by
\begin{align*}
Q_{(n)}^{u_1}(u_2,\ldots,u_m) &:= n \cdot \left[Q_{(n)}(u_1+\tfrac 1 {n+1},u_2,\ldots,u_m)- Q_{(n)} (u_1,u_2,\ldots,u_m)\right]\,,\\
Q_{2,(n)}^{u_1}(u_2) &:= n \cdot \left[Q_{2,(n)}(u_1+\tfrac 1 {n+1},u_2)- Q_{2,(n)}(u_1,u_2)\right]\,.
\end{align*}
Then we obtain that
\begin{equation}\label{eq_pi_f_Finverse}
\begin{aligned}
\pi_{f\circ (F_n^{-1},\ldots,F_n^{-1})}(\widehat{Q^*}) &=\pi_f(\widehat{Q^*}\circ (F_n,\ldots,F_n))\\
&= \int_{[0,1]^m} f(u) \mathsf{\,d} Q_{(n)}(u)\\
&= \int_{[0,1]}\int_{[0,1]^{m-1}} f(u_1,u_2,\ldots,u_m) \mathsf{\,d} Q_{(n)}^{u_1}(u_2,\ldots,u_m) \mathsf{\,d} F_n(u_1)\\
&= \int_{[0,1]}\int_{[0,1]^{m-1}} f(u_1,u_2,\ldots,u_m) \mathsf{\,d} Q_{2,(n)}^{u_1}(\min_{2\leq i \leq m}\{u_i\}) \mathsf{\,d} F_n(u_1)\\
&= \int_{[0,1]}\int_{[0,1]^{m-1}} f(u_1,v,\ldots,v) \mathsf{\,d} Q_{2,(n)}^{u_1}(v) \mathsf{\,d} F_n(u_1)\\
&= \int_{[0,1]^m} \phi_f(v,u_1)\mathsf{\,d} Q_{2,(n)}(v,u_1)\\
&= \pi_{\phi_f}(\widehat{Q_2}\circ (F_n,F_n))\\
&=\pi_{\phi_f\circ (F_n^{-1},F_n^{-1})}(\widehat{Q_2})\,.
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_pi_f_Finverse} holds, note that the first and last equality follow from the marginal transformation formula \eqref{margtrafgs}. Since \(Q_{(n)}\) and \(Q_{2,(n)}\) are defined on \([0,1)^m\) and \([0,1)^2\,,\) respectively, they are grounded and satisfy \(\widehat{Q^*}\circ (F_n,\ldots,F_n)=\widehat{Q_{(n)}}\) and \(\widehat{Q_2}\circ (F_n,F_n)=\widehat{Q_{2,(n)}}\). Hence, the second and seventh equality follows with \eqref{eqqop} using that \(Q_{(n)}\) and \(Q_{2,(n)}\) are right-continuous and measure-inducing. The third and sixth equality hold true by the disintegration theorem applied on the positive part and the negative part of the Hahn-Jordan decomposition of the signed measures induced by \(Q_{(n)}\) and \(Q_{2,(n)}\,,\) respectively. The fourth equality follows from \(Q^*(u_1,u_2,\ldots,u_m)=Q_2(\min_{2\leq i \leq m}\{u_i\},u_1)\,.\) The fifth equality holds by Lemma \ref{lemmadia} using that \(Q_{(n)}^{u_1}\) and \(Q_{2,(n)}^{u_1}\) are measure generating functions of signed measures with \(Q_{(n)}^{u_1}(u_2,\ldots,u_m)=Q_{2,(n)}^{u_1}(\min_{i=\{2,\ldots,m\}}\{u_i\})\) for all \(u_1\in \mathbb{K}_n^1\) and for all \((u_2,\ldots,u_m)\in \mathbb{K}_n^{m-1}\,.\)
As a consequence of \eqref{azw}, we now obtain that
\begin{equation}
\label{appslbc}
\begin{aligned}
\begin{split}
\psi_f(M^2\vee D^2\vee \cdots \vee D^m) &\leq \liminf_{n\to \infty} \psi_{f\circ (F_n^{-1},\ldots,F_n^{-1})} (M^2\vee D^2\vee \cdots \vee D^m) \\
&= \liminf_{n\to \infty} \pi_{f\circ (F_n^{-1},\ldots,F_n^{-1})} (\reallywidehat{M^2\vee D^2\vee \cdots \vee D^m}) \\
& \leq \liminf_{n\to \infty} \pi_{f\circ (F_n^{-1},\ldots,F_n^{-1})}(\widehat{Q^*})\\
&= \liminf_{n\to \infty} \pi_{\phi_f\circ (F_n^{-1},F_n^{-1})}(\widehat{Q_2}) \\
&= \liminf_{n\to \infty} \pi_{\phi_f}(\widehat{Q_2}\circ(F_n,F_n))\\
& = \liminf_{n\to \infty} \psi_{\phi_f}({Q_2\circ (F_n,F_n)})= \pi_{\phi_f}(\widehat{Q_2})\,,
\end{split}
\end{aligned}
\end{equation}
Indeed, to see that \eqref{appslbc} holds observe that the first inequality follows by an application of Fatou's Lemma using that \(f\) is lower bounded by some integrable function.
The first equality is a consequence of \eqref{qeopexopeq}. The second inequality holds true by assumption using that \(f\circ (F_n^{-1},\ldots,F_n^{-1})\) is left-continuous, supermodular, and measure-inducing. The third equality follows from the marginal transformation formula \eqref{margtrafgs}. The fourth equality follows from \eqref{eqqop} noting that \(Q_2\circ (F_n,F_n)\) is grounded. The last equality is a consequence of Proposition \ref{propconncomdom} using that \(\phi_f\in \mathcal{F}_{\operatorname{mi}}^{\operatorname{c},\operatorname{l}}([0,1)^2)\) is left-continuous and \((\phi_f)_I\) is Lebesgue integrable for \(I\subseteq \{1,2\}\,,\) \(I\ne \emptyset\,,\) as well as \(F_n(x)\to x\) for all \(x\in [0,1]\,.\)
This proves (f).
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem_indicator_constraints}]
We only prove the assertion of Lemma~\ref{lem_indicator_constraints}~(a). The assertion of Lemma~\ref{lem_indicator_constraints}~(b) follows analogously.
We observe that for each sequence $(x^{(N)})_{N \in \mathbb{N}} \subset \mathds{Q}_+^{nd}$ with $x^{(N)} \downarrow x\in \mathds{Q}_+^{nd}$ for $N \rightarrow \infty$ we have for all $\mathbb{Q} \in \mathcal{M}(\mu)$ that
\begin{equation}\label{eq_ineq_gxfx_1}
\lim_{N \rightarrow \infty} \mathbb{E}_\mathbb{Q}\left[g_{x^{(N)}}(S)\right]=\lim_{N \rightarrow \infty}\mathbb{Q}\left(S<x^{(N)}\right)=\mathbb{Q} \left(S\leq x\right)=\mathbb{E}_\mathbb{Q}[f_x(S)]
\end{equation}
and
\begin{equation}\label{eq_ineq_gxfx_2}
\lim_{N \rightarrow \infty} \overline{Q}\left(F_1^1\left({x_1^1}^{(N)}\right),\ldots,F_n^d\left({x_n^d}^{(N)}\right)\right) =\overline{Q}\left(F_1^1({x_1^1}),\ldots,F_n^d({x_n^d})\right).
\end{equation}
Thus, if we have
\begin{equation}\label{eq_ineq_1}
\mathbb{E}_{\mathbb{Q}} \left[g_x(S)\right]\leq \phantom{-}\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d)) \text{ for all } x=\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd},
\end{equation}
then we may choose for each $x\in \mathds{Q}_+^{nd}$ a sequence $(x^{(N)})_{N \in \mathbb{N}} \subset \mathds{Q}_+^{nd}$ with $x^{(N)} \downarrow x\in \mathds{Q}_+^{nd}$ for $N \rightarrow \infty$, and it follows with \eqref{eq_ineq_gxfx_1} and \eqref{eq_ineq_gxfx_2} that
\begin{equation}\label{eq_ineq_2}
\mathbb{E}_{\mathbb{Q}} \left[f_x(S)\right]\leq \phantom{-}\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))\text{ for all } x=\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}.
\end{equation}
Moreover, \eqref{eq_ineq_2} implies \eqref{eq_ineq_1} by definition of the respective indicator functions, thus \eqref{eq_ineq_1} and \eqref{eq_ineq_2} are equivalent.
The assertion follows, since \(\underline{Q} \leq_{\operatorname{lo}} C_{\mathbb{Q}}\leq_{\operatorname{lo}} \overline{Q}\) is, by definition of the lower orthant order, equivalent to
\begin{align}
\begin{array}{l}
\phantom{-}\mathbb{E}_{\mathbb{Q}} \left[f_x(S)\right]\leq \phantom{-}\overline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))\\[2mm]
\mathbb{E}_{\mathbb{Q}} \left[-f_x(S)\right]\leq -\underline{Q}(F_1^1(x_1^1),\ldots,F_n^d(x_n^d))
\end{array}
~\forall\, x=\left(x_1^1,\ldots,x_n^d\right)\in \mathds{Q}_+^{nd}\,.
\end{align}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theloob}]
We only prove part (a), part~(b) follows analogously.\
Equation \eqref{theloob1a} is a consequence of Theorem \ref{theaddcon} and Lemma~\ref{lem_indicator_constraints}.
Inequality \eqref{eqlosb} follows from
\begin{equation}\label{eq_ineq_thm42_proof}
\overline{P}_{{\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}} = \sup_{\mathbb{Q}\in {\mathcal{M}}^{\operatorname{lo}}_{\underline{Q},\overline{Q}}} \mathbb{E}_\mathbb{Q}[c(S)] \leq \sup_{Q\in \mathcal{Q}_{nd}\atop \underline{Q}\leq_{\operatorname{lo}} Q\leq_{\operatorname{lo}} \overline{Q}} \pi_c^{\mu}(\widehat{Q})=\pi_c^{\mu}(\widehat{\overline{Q}}),
\end{equation}
where we neglect the martingale property and the requirement that $\mathbb{Q}$ needs to be a probability measure for the inequality in \eqref{eq_ineq_thm42_proof}. The last equality is a consequence of Proposition \ref{charortord} noting that, by Proposition \ref{propconncomdom}, \(\pi_c^{\mu}(\widehat{\overline{Q}})\) exists because
\begin{align*}
\int_0^1 |c_I((F_{i}^{j_1})^{-1}(u),\ldots,(F_{i}^{j_k})^{-1}(u)) |\mathsf{\,d} u &\leq \alpha \int_0^1 \left(1+ \sum_{j\in I} | (F_i^j)^{-1}(u)|\right) \mathsf{\,d} u \\
&=\alpha \left(1+\sum_{j \in I} \mathbb{E}_{\mu_i^j}[|S_{t_i}^j|]\right) < \infty
\end{align*}
for all \(I=\{j_1,\ldots,j_k\}\subseteq \{1,\ldots,nd\}\,,\) \(I\ne \emptyset\,,\) and for some \(\alpha>0\) using that \(c\in U_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) and using that the first moments of \(\mu\) exist.
\end{proof}
For the proof of Theorem \ref{corsmifm}, we formulate an auxiliary lemma based, for some fixed time \(t_i\) and a quasi-copula \(Q\in \mathcal{Q}_d\,,\) on the class
\begin{align*}
\overline{\mathcal{M}}_{Q,t_i}^{\operatorname{lo}}(\mu):=\{\mathbb{Q}\in \mathcal{M}(\mu)\, |\, C_{\mathbb{Q}_i}\leq_{\operatorname{lo}} Q\}
\end{align*}
of probability measures \(\mathbb{Q} \in \mathcal{M}(\mu)\) such that the copula \(C_{\mathbb{Q}_i}\in \mathcal{C}_d\) at time \(t_i\) defined by
\[
C_{\mathbb{Q}_i}(F_i^1(x_1),\ldots,F_i^d(x_d))=\mathbb{Q}(S_{t_i}^1\leq x_1, \dots,S_{t_i}^d\leq x_d),~~~x=(x_1,\ldots,x_d)\in \mathbb{R}^d\,,
\]
is upper bounded by \(Q\in \mathcal{Q}_d\) w.r.t.\,the lower orthant ordering, and where $\mathbb{Q}_i = \mathbb{Q} \circ S_{t_i}^{-1}$.
Note that we have \(\overline{\mathcal{M}}^{\operatorname{lo}}_{M^d,t_i}(\mu)=\mathcal{M}(\mu)\) in the case that no additional dependence restriction is included.
\begin{lem}\label{lemifm}
For \(Q_2\in \mathcal{Q}_2\,,\) let \(Q^*\in \mathcal{Q}_d\) be the \(d\)-variate quasi-copula given by \eqref{deqcqs}. Then it holds for all $i =1,\dots,n$ that
\begin{align*}
\mathcal{M}_{Q_2,t_i}^{\operatorname{IFM}}(\mu) = \overline{\mathcal{M}}_{Q^*,t_i}^{\operatorname{lo}}(\mu)\,.
\end{align*}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lemifm}]
For $i \in \{1,\dots,n\}$ let \(\mathbb{Q}\in \mathcal{M}_{Q_2,t_i}^{\operatorname{IFM}}(\mu)\) and \(D^k=C_{\mathbb{Q}_i^{1k}}\in \mathcal{C}_2\) be the copula associated with the bivariate \((1,k)\)-marginal
\(\mathbb{Q}_i^{1k}\in \mathcal{P}(\mathbb{R}_+^2)\) of \(\mathbb{Q}_i=\mathbb{Q}\circ S_{t_i}^{-1}\,,\) \(2\leq k\leq d\,.\)
Then, \cite[Proposition 2.4 (i)]{Ansari-Rueschendorf-2018} and Theorem \ref{theqcub} imply \(C_{\mathbb{Q}_i} \leq_{\operatorname{lo}} M^2\vee D^2\vee \cdots \vee D^d\leq_{\operatorname{lo}} Q^*\,,\)
which means that \(\mathbb{Q}\in \overline{\mathcal{M}}_{Q^*,t_i}^{\operatorname{lo}}(\mu)\,.\)\\
For the reverse inclusion, let \(\widetilde{\mathbb{Q}}\in \overline{\mathcal{M}}_{Q^*,t_i}^{\operatorname{lo}}(\mu)\,.\) From the closure of the lower orthant ordering under marginalization\footnote{The lower orthant ordering is closed under marginalization in the sense that \(Q\leq_{\operatorname{lo}}Q'\,,\) \(Q,Q'\in \mathcal{Q}_d\,,\) implies \(Q^I(u_1,\ldots,u_k)\leq_{\operatorname{lo}} Q'^I(u_1,\ldots,u_k)\) for all \(k=1,\ldots,d\,,\) \(I=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,d\}\,,\) and \(u_1,\ldots,u_k\in [0,1]^k\,,\) where the \(Q^I\,,\) and analogously \(Q'^I\,,\) is defined by \(Q^I(u_{i_1},\ldots,u_{i_k}):=Q(u_1,\ldots,u_d)\) for all \(u_1,\ldots,u_d)\in [0,1]^d\) with \(u_j=1\) whenever \(j\ne I\,.\)} we obtain for the copula of the bivariate \((1,k)\)-marginal distribution of \(\mathbb{Q}_i\) that \(C_{\mathbb{Q}_{i}^{1k}}\leq_{\operatorname{lo}} Q_2\,,\) \(2\leq k\leq d\,,\) which means that \(\widetilde{\mathbb{Q}}\in \mathcal{M}_{Q_2}^{\operatorname{IFM}}(\mu)\,.\)
\end{proof}
\begin{proof}[Proof of Theorem~\ref{corsmifm}]
The statement \eqref{eqcorsmifm1} in Theorem~\ref{corsmifm}~(a) follows from Theorem \ref{theaddcon} and Lemma~\ref{lem_indicator_constraints} with the inequality constraints
\begin{align*}
\mathbb{E}_\mathbb{Q} [g_{x,y}(S_{t_i}^1,S_{t_i}^k)]\leq Q_2(F_{i}^1(x),F_{i}^k(y))\,,~~~g_{x,y}:=\one_{\{\cdot < (x,y)\}}\,, (x,y)\in \mathds{Q}_+^2\,, 2\leq k\leq d\,.
\end{align*}
To prove \eqref{eqcorsmifm2}, first note that the definition of super-modularity \(\widetilde{c}\in \mathcal{F}_{\operatorname{sm}} \cap C_{\operatorname{lin}}(\mathbb{R}_+^{nd})\) and that $\widetilde{c}$ is componentwise increasing/componentwise decreasing implies that $c=\left(\widetilde{c} \circ \operatorname{proj}_i^1,\cdots, \widetilde{c} \circ \operatorname{proj}_i^d\right) \in \mathcal{F}_{\operatorname{sm}}\cap C_{\operatorname{lin}}(\mathbb{R}_+^{d})$ and that $c$ is componentwise increasing/componentwise decreasing. Moreover, we obtain by Lemma \ref{lemifm} that
\begin{equation}
\label{eqprcorsmifm}
\begin{aligned}%
\overline{P}_{\mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}} &= \sup_{\mathbb{Q}\in \mathcal{M}_{Q_2}^{\operatorname{\operatorname{IFM}}}(\mu)} \mathbb{E}_\mathbb{Q} \left[c(S_{t_i}^1,\ldots,S_{t_i}^d)\right]
= \sup_{\mathbb{Q}\in \overline{\mathcal{M}}_{Q^*,t_i}^{\operatorname{lo}}(\mu)} \mathbb{E}_\mathbb{Q} \left[c(S_{t_i}^1,\ldots,S_{t_i}^d)\right]\\
&\leq \sup_{C \leq_{\operatorname{lo}} Q^*} \psi_c^{(F_i^1,\ldots,F_i^d)}(C)
\leq \sup_{C=M^2\vee D^2\vee \cdots \vee D^d\,,\atop D^k\leq_{\operatorname{lo}} Q_2\,, k=2,\ldots,d} \psi_c^{(F_i^1,\ldots,F_i^d)}(C)\leq \pi_{\phi_{c\,\circ\left((F_i^1)^{-1},\ldots,(F_i^d)^{-1}\right)}} (\widehat{Q_2})\,.
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eqprcorsmifm} holds, observe that we neglect the martingale property for the first inequality.
For the second inequality, let \(D^k\in \mathcal{C}_2\,,\) \(2\leq k\leq d\,,\) such that the transposed copula \((D^k)'\in \mathcal{C}_2\) (defined by \((D^k)'(u,v):=D^k(v,u)\) for $u,v \in [0,1]$) is the bivariate \((1,k)\)-marginal copula of \(C\,.\) Then, \(C\leq_{\operatorname{lo}} Q^*\) implies \(D^k\leq_{\operatorname{lo}} Q_2\,.\) Since the upper product \(M^2\vee D^2\vee \cdots \vee D^d\) is the greatest element w.r.t.\,\(\leq_{\operatorname{sm}}\) in the class of copulas with bivariate \((1,k)\)-marginal specifications \(D^k\,,\) \(2\leq k\leq d\,,\) see \cite[Proposition 2.4]{Ansari-Rueschendorf-2018}, it follows that \(C\leq_{\operatorname{sm}} M^2\vee D^2\vee \cdots \vee D^d\,.\) This implies the second inequality using that \(c\) is supermodular. The last inequality is a consequence of Theorem \ref{theqcub}~(f) using that \(u\mapsto c_I\,\circ\left((F_i^{j_1})^{-1},\ldots,(F_i^{j_k})^{-1}\right) (u,\ldots,u)\) is Lebesgue-integrable for all \(I=\{j_1,\ldots,j_k\}\subseteq\{1,\ldots,d\}\,,\) \(I\ne \emptyset\,,\) since \(c\in C_{\operatorname{lin}}(\mathbb{R}_+^{d})\) and the first moments of \(F_i^1,\ldots,F_i^d\) exist, see \eqref{eq_ineq_thm42_proof}. Moreover, we use that $c\in C_{\operatorname{lin}}(\mathbb{R}_+^{d})$ which implies that $c\circ ((F_i^1)^{-1},\dots,(F_i^d)^{-1})$ is lower bounded by a function of the form $(x_1,\dots,x_d)\mapsto K\bigg(1+\sum_{j=1}^d\big|(F_i^j)^{-1}(x_j)\big|\bigg)$ for $K \in \mathbb{R}$, which is $M^2\vee D^2\vee \cdots \vee D^d$-integrable due to the existing first moments of the marginals.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lembasopt}:]
(a):
If \(d\leq 2\,,\) then \(\mathfrak{C}\) and \(\mathfrak{P}\) induce signed measures, see Table \ref{table_quasiExp_overview}. If \(d\geq 3\,,\) then \(\mathfrak{P}\) and \(\mathfrak{C}\) do not induce signed measures because they can be linearly transformed into the lower Fr\'{e}chet bound \(W^d\) in \eqref{frebou} which is a quasi-copula that does not induce a signed measure, see \cite[Theorem 2.4]{Nelsen-2010}.
(b): We show the statement for the payoff function \(\mathfrak{C}\,.\) The proof for \(\mathfrak{P}\) follows analogously.
By definition of \(\phi\) and \(G^{-1}\,,\) we have for all $x_1,x_2 \in \mathbb{R}$ that
\begin{align*}
\phi_{\mathfrak{C}\circ(F_1^{-1},\ldots,F_d^{-1})}(x_1,x_2)&= \left(\alpha_1F_1^{-1}(x_2)+\sum_{i=2}^d \alpha_i F_{i}^{-1}(x_1)-K\right)_+\\
&=\left(\alpha_1F_1^{-1}(x_2)+\sum_{i=2}^d \alpha_i G^{-1}(x_1)-K\right)_+=\phi_{\mathfrak{C}}\circ(G^{-1},F_1^{-1})(x_1,x_2)\,.
\end{align*}
First, consider the special case where for all $i=1,\dots,d$ the generalized inverse distribution functions \(F_i^{-1}\) are continuous on the range of $F_i$, which is equivalent to \(F_i\) being strictly increasing. Then, we obtain that
\begin{equation}
\label{eq_psi_pi_proof}
\begin{aligned}
\psi_{\mathfrak{C}}^{(F_1,\ldots,F_d)} (M^2\vee D^2\vee \cdots \vee D^d) &= \psi_{\mathfrak{C}\circ(F_1^{-1},\ldots,F_d^{-1})}(M^2\vee D^2\vee \cdots \vee D^d)\\
&\leq
\pi_{\phi_{\mathfrak{C}}\circ(G^{-1},F_1^{-1})}(\widehat{Q_2})\\
&= \pi_{\phi_{\mathfrak{C}}}^{(G,F_1)}(\widehat{Q_2})\,,
\end{aligned}
\end{equation}
where the first equality is given by \eqref{defexpop}. The inequality follows from Theorem \ref{theqcub}~(f) using that \(\mathfrak{C}\circ (F_1^{-1},\ldots,F_d^{-1})\) is continuous and increasing supermodular applying that it is an increasing transformations of the increasing supermodular function \(\mathfrak{C}\,.\) Note that \(\int_0^1 \mathfrak{C}\circ (F_1^{-1},\ldots,F_d^{-1})(u,\ldots,u) \mathsf{\,d} u\) exists because \(\mathfrak{C}\in C_{\operatorname{lin}}(\mathbb{R}_+^m)\) and the first moments of \(F_1,\ldots,F_d\) exist. Further, we use that \(\phi_{\mathfrak{C}}\) and thus \(\phi_{\mathfrak{C}}\circ (G^{-1},F_1^{-1})\) are continuous and hence, by (a), measure-inducing. The last equality follows from \eqref{defquexpop}.
In the case that for $i \in \mathbb{N}$ \(F_i\) is continuous (but not \(F_i^{-1}\)), approximate \(F_i\) pointwise by a sequence \((F_{i,n})_{n\in \mathbb{N}}\) of strictly increasing distribution functions supported on $\mathbb{R}_+$ with finite first moments such that $F_{i,n} \to F_i$, \(F_{i,n}\geq F_i\) pointwise and \(\int_{\mathbb{R}_+} x \mathsf{\,d} F_{i,n}(x)\to \int_{\mathbb{R}_+} x \mathsf{\,d} F_i(x)\) as \(n\to \infty\,.\) Note that for all \(i\in \{1,\ldots,d\}\,,\) the first moment of \(F_i\) exists by assumption.
We approximate $G$ pointwise by a sequence $(G_n)_{n \in \mathbb{N}}$ with \(G_n(x)\geq G(x)\) for all \(n\) and such that $G_n \rightarrow G$ pointwise for $n \rightarrow \infty$. Consider for \((U_1,\ldots,U_d)\sim M^2 \vee D^2 \vee \cdots \vee D^d\,,\) the random variables \(X_{n,i}:=F_{n,i}^{-1}(U_i)\) and \(X_i:=F_i^{-1}(U_i)\,.\) Then, by Scheffé's lemma, we have that $X_{n,i}$ converge in $L^1$ to $X_i$ for all $i$. This implies also \(\sum_{i=1}^d X_{n,i} \to \sum_{i=1}^d X_i\) in \(L^1\) (see, e.g., \cite[Theorem 6.25]{Klenke-2020}) and thus for \(X_n:=(X_{n,1},\ldots,X_{n,d})\) and \(X:=(X_1,\ldots,X_d)\,,\) we obtain \(\mathfrak{C}(X_n)\to \mathfrak{C}(X)\) in \(L^1\,.\)
Since by Sklar's Theorem \(X_n\sim M^2\vee D^2\vee \cdots \vee D^d(F_{1,n},\ldots,F_{d,n})\) and \(X\sim M^2\vee D^2\vee \cdots \vee D^d(F_{1},\ldots,F_{d})\,,\) we then obtain that\begin{equation}
\label{eq_C_P_proof_1}
\begin{aligned}
\psi_{\mathfrak{C}}^{(F_1,\ldots,F_d)} &(M^2\vee D^2\vee \cdots \vee D^d)\\
&=\int \mathfrak{C}(x) \mathsf{\,d} (M^2\vee D^2\vee \cdots \vee D^d)(F_1(x_1),\ldots,F_d(x_d)) \\
&= \lim_{n\to \infty} \int \mathfrak{C}(x) \mathsf{\,d} (M^2\vee D^2\vee \cdots \vee D^d)(F_{1,n}(x_1),\ldots,F_{n,d}(x_d))\\
&= \lim_{n\to \infty} \psi_{\mathfrak{C}}^{F_{1,n},\ldots,F_{d,n}} (M^2\vee D^2\vee \cdots \vee D^d)\\
&\leq \lim_{n\to \infty} \pi_{\phi_{\mathfrak{C}}}^{(G_n,F_{1,n})} (\widehat{Q_2})\\
&= \lim_{n\to \infty} \pi_{\phi_{\mathfrak{C}}} (\widehat{Q_2}\circ (G_n,F_{1,n}))\\
&= \pi_{\phi_{\mathfrak{C}}} (\widehat{Q_2}\circ (G,F_{1}))\\
&= \pi_{\phi_{\mathfrak{C}}}^{(G,F_{1})} (\widehat{Q_2})
\end{aligned}
\end{equation}
Indeed, to see that \eqref{eq_C_P_proof_1} holds, note that the third equality is given by \eqref{defexpop}. The inequality follows from the special case \eqref{eq_psi_pi_proof} where \(F_{i,n}^{-1}\) is continuous and integrable. The fourth and the last equality hold due to the notation in \eqref{defquexpop}, and the fifth equality is a consequence of the dominated convergence theorem using that \(\phi_{\mathfrak{C}}\) induces a positive measure and that \(F_{i,n}(x)\geq F_i(x)\) and \(G_n(x)\geq G(x)\) for all \(n\) and \(x\in \mathbb{R}\) as well as \(F_{i,n}(x)\to F_i(x),~G_n(x) \to G(x)\) for all \(x\in \mathbb{R}\) using the continuity of \(F_i\,,\) \(i\in \{1,\ldots,d\}\,\) and of $G$.
\end{proof}
\vspace{0.7cm}
\section*{Acknowledgments}
\noindent
Ariel Neufeld gratefully acknowledges the financial support by the Nanyang Assistant Professorship Grant (NAP Grant) \emph{Machine Learning based Algorithms in Finance and Insurance}.
|
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<h1 class="post-title">BREAKING: Disney to offer "Third Shift Magic Hours" after-after-hours event</h1>
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<time class="post-date" datetime="2016-04-01">01 April 2016</time> on <a href="http://www.mouseguests.com/tag/Disney-World">Disney World</a>, <a href="http://www.mouseguests.com/tag/Magic-Kingdom"> Magic Kingdom</a>, <a href="http://www.mouseguests.com/tag/Newz"> Newz</a>
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<div class="paragraph">
<p>As you may know, Disney has offered Extra Magic Hours for years as a perk to guests staying in a Disney resort. And recently Disney has announced "Disney After Hours", a hard-ticket event that gives visitors access to the Magic Kingdom after the park closes for meeting characters, shorter lines at select attractions, and drinks and ice cream. In addition, we’ve just learned about "Third Shift Magic Hours", a new hard-ticket after-after-hours event coming to the Magic Kingdom.</p>
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<p>"Third Shift Magic Hours" will offer guests a unique opportunity to see the park in ways that virtually nobody ever sees it. Once all of the crowds are gone and the park is completely closed, you’ll be able to participate in unique interactive experiences, including (but not limited to): Mickey’s Custodial Adventure, Stitch’s Street Sweep Sensation, and Pooh’s Potty Purification Parade.</p>
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<p>Disney spokesperson, Josh Inyu, tells us that this is a prime example of giving guests what they want: "As we survey Disney World visitors, two themes stand out. Guests want longer hours and want the parks to be clean and well-maintained. This new premium experience addresses both of those concerns."</p>
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<p>It’s our understanding that while this experience will be initially exclusive to the Magic Kingdom, if successful it may be offered at all Disney parks and resorts.</p>
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<p>The price tag for this one of a kind experience is $69, which includes unlimited bottles of water, a grey commemorative jumpsuit, and one-size-fits-all latex gloves. Dates for "Third Shift Magic Hours" will be announced soon. Due to Florida labor laws, "Third Shift Magic Hours" will be limited to visitors 16 years and older.</p>
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\section{Introduction}
Computer vision has witnessed many impressive breakthroughs over the past decades in image classification~\cite{krizhevsky2012imagenet,he2016deep}, image segmentation~\cite{long2015fully}, and object detection~\cite{girshick2014rich} by applying convolutional neural networks to large-scale, labeled datasets, often exceeding human performance. These systems give outputs such as class labels, segmentation masks, or bounding boxes, but it would be more natural for humans to interact with these systems through natural language. To this end, the research community has introduced various multi-modal tasks, such as image captioning~\cite{xu2015show}, referring expressions~\cite{kazemzadeh2014referitgame}, visual question-answering~\cite{antol2015vqa,malinowski2015ask}, visual reasoning~\cite{johnson2017clevr}, and visual dialogue~\cite{de2017guesswhat,das2017visual}.
These tasks require models to effectively integrate information from both vision and language.
One common approach is to process both modalities independently with large unimodal networks before combining them through concatenation~\cite{malinowski2015ask}, element-wise product~\cite{kim2016multimodal,lu2016hierarchical}, or bilinear pooling~\cite{fukui2016multimodal}. Inspired by the success of attention in machine translation~\cite{bahdanau2014neural}, several works have proposed to incorporate various forms of spatial attention to bias models towards focusing on question-specific image regions~\cite{xu2015show,xu2016ask}. However, spatial attention sometimes only gives modest improvements over simple baselines for visual question answering~\cite{jabri16vqa} and can struggle on questions involving multi-step reasoning~\cite{johnson2017clevr}.
More recently,~\cite{devries2017modulating,perez2018film} introduced Feature-wise Linear Modulation (FiLM) layers as a promising approach for vision-and-language tasks. These layers apply a per-channel scaling and shifting to a convolutional network's visual features, conditioned on an external input such as language, e.g.\@, captions, questions, or full dialogues. Such feature-wise affine transformations allow models to dynamically highlight the key visual features for the task at hand. The parameters of FiLM layers which scale and shift features or feature maps are determined by a separate network, the so-called \emph{FiLM generator}, which predicts these parameters using the external conditioning input. Within various architectures, FiLM has outperformed prior state-of-art for visual question-answering~\cite{devries2017modulating,perez2018film}, multi-modal translation~\cite{delbrouck2017modulating}, and language-guided image segmentation~\cite{rupprecht2018guide}.
\begin{figure}[t]
\begin{minipage}{0.30\textwidth}
\centering
\includegraphics[width=0.85\textwidth]{img/frisbee}
\end{minipage}
\begin{minipage}{0.68\textwidth}
\begin{tabular}{llll}
\toprule
\textit{ReferIt\xspace} & \quad\quad & \textit{GuessWhat?!\xspace} & \\
\midrule
- The girl with a sweater & & Is it a person? &\quad Yes\\
- The fourth person & & Is it a girl? &\quad Yes\\
- The girl holding a white & & Does she have a blue&\quad No\\
frisbee & & frisbee? &\\
\end{tabular}
\end{minipage}
\caption{The ReferIt\xspace task identifies a selected object (in the bounding box) using a single expression, while in GuessWhat?!\xspace, a speaker localizes the object with a series of yes or no questions.}
\label{fig:example}
\end{figure}
However, the best way to design the FiLM generator is still an open question. For visual question-answering and visual reasoning, prior work uses single-hop FiLM generators that predict all FiLM parameters at once~\cite{perez2018film,devries2017modulating}. That is, a Recurrent Neural Network (RNN) sequentially processes input language tokens and then outputs all FiLM parameters via a Multi-Layer Perceptron (MLP). In this paper, we argue that using a \textit{Multi-hop FiLM Generator} is better suited for tasks involving longer input sequences and multi-step reasoning such as dialogue. Even for shorter input sequence tasks, single-hop FiLM generators can require a large RNN to achieve strong performance; on the CLEVR visual reasoning task~\cite{johnson2017clevr} which only involves a small vocabulary and templated questions, the FiLM generator in~\cite{perez2018film} uses an RNN with 4096 hidden units that comprises almost 90\% of the model's parameters. Models with Multi-hop FiLM Generators may thus be easier to scale to more difficult tasks involving human-generated language involving larger vocabularies and more ambiguity.
As an intuitive example, consider the dialogue in Fig.~\ref{fig:example} through which one speaker localizes the second girl in the image, the one who does not ``have a blue frisbee.'' For this task, a single-hop model must determine upfront what steps of reasoning to carry out over the image and in what order; thus, it might decide in a single shot to highlight feature maps throughout the visual network detecting either non-blue colors or girls.
In contrast, a multi-hop model may first determine the most immediate step of reasoning necessary (i.e.\@, locate the girls), highlight the relevant visual features, and then determine the next immediate step of reasoning necessary (i.e.\@, locate the blue frisbee), and so on.
While it may be appropriate to reason in either way, the latter approach may scale better to longer language inputs and/or or to ambiguous images where the full sequence of reasoning steps is hard to determine upfront, which can even be further enhanced by having intermediate feedback while processing the image.
In this paper, we therefore explore several approaches to generating FiLM parameters in multiple hops. These approaches introduce an intermediate context embedding that controls the language and visual processing, and they alternate between updating the context embedding via an attention mechanism over the language sequence (and optionally by incorporating image activations) and predicting the FiLM parameters. We evaluate Multi-hop FiLM\xspace generation on ReferIt\xspace~\cite{kazemzadeh2014referitgame} and GuessWhat?!\xspace~\cite{de2017guesswhat}, two vision-and-language tasks illustrated in Fig.~\ref{fig:example}. We show that Multi-hop FiLM\xspace models significantly outperform their single-hop counterparts and prior state-of-the-art for the longer input sequence, dialogue-based GuessWhat?!\xspace task while matching the state-of-the-art performance of other models on ReferIt\xspace. Our best GuessWhat?!\xspace model only updates the context embedding using the language input, while for ReferIt\xspace, incorporating visual feedback to update the context embedding improves performance.
\noindent In summary, this paper makes the following contributions:
\begin{itemize}
\item We introduce the Multi-hop FiLM\xspace architecture and demonstrate that our approach matches or significantly improves state-of-the-art on the GuessWhat?!\xspace Oracle task, GuessWhat?!\xspace Guesser task, and ReferIt\xspace Guesser task.
\item We show Multi-hop FiLM\xspace models outperforms their single-hop counterparts on vision-and-language tasks involving complex visual reasoning.
\item We find that updating the context embedding of Multi-hop FiLM Generator\xspace based on visual feedback may be helpful in some cases, such as for tasks which do not include object category labels like ReferIt\xspace.
\end{itemize}
\section{Background}
In this section, we explain the prerequisites to understanding our model: RNNs, attention mechanisms, and FiLM. We subsequently use these building blocks to propose a Multi-hop FiLM\xspace model.
\subsection{Recurrent Neural Networks}
One common approach in natural language processing is to use a Recurrent Neural Network (RNN) to encode some linguistic input sequence $l$ into a fixed-size embedding. The input (such as a question or dialogue) consists of a sequence of words $\omega_{1:T}$ of length $T$, where each word $\omega_t$ is contained within a predefined vocabulary $\mathcal{V}$. We embed each input token via a learned look-up table $e$ and obtain a dense word-embedding $\bm{e}_{\omega_t} = e(\omega_t)$.
The sequence of embeddings $\{\bm{e}_{\omega_t}\}_{t=1}^T$ is then fed to a RNN, which produces a sequence of hidden states $\{\bm{s}_{t}\}_{t=1}^T$ by repeatedly applying a transition function $f$: $
\bm{s}_{t+1} = f(\bm{s}_{t}, \bm{e}_{\omega_t})
$
To better handle long-term dependencies in the input sequence, we use a Gated Recurrent Unit (GRU)~\cite{chung2014empirical} with layer normalization~\cite{ba2016layer} as transition function. In this work, we use a bidirectional GRU, which consists of one forward GRU, producing hidden states $\overrightarrow{\bm{s}_{t}}$ by running from $\omega_1$ to $\omega_T$, and a second backward GRU, producing states $\overleftarrow{\bm{s}_{t}}$ by running from $\omega_T$ to $\omega_1$. We concatenate both unidirectional GRU states $\bm{s}_{t} = [ \overrightarrow{\bm{s}_{t}}; \overleftarrow{\bm{s}_{t}} ]$ at each step $t$ to get a final GRU state, which we then use as the compressed embedding $\bm{e}_l$ of the linguistic sequence $l$.
\subsection{Attention}
The form of attention we consider was first introduced in the context of machine translation~\cite{bahdanau2014neural,luong2015effective}. This mechanism takes a weighted average of the hidden states of an encoding RNN based on their relevance to a decoding RNN at various decoding time steps. Subsequent \textit{spatial} attention mechanisms have extended the original mechanism to image captioning~\cite{xu2015show} and other vision-and-language tasks~\cite{xu2016ask,kim2016hadamard}.
More formally, given an arbitrary linguistic embedding $\bm{e}_l$ and image activations $\bm{F}_{w,h,c}$ where $w$, $h$, $c$ are the width, height, and channel indices, respectively, of the image features $\bm{F}$ at one layer, we obtain a final visual embedding $\bm{e}_v$ as follows:
\begin{align}
\small
\xi_{w, h} = MLP(g(\bm{F_{w,h,\cdot}},\bm{e_l})) \;;\quad
\alpha_{w, h} = \frac{\exp(\xi_{w, h})}{\sum_{w',h'} \exp(\xi_{w',h'})} \quad;\;
\bm{e}_{v} = \sum_{w, h} \alpha_{w, h} \bm{F_{w,h,\cdot}},
\end{align}
where $MLP$ is a multi-layer perceptron and $g(.,.)$ is an arbitrary fusion mechanism (concatenation, element-wise product, etc.). We will use Multi-modal Low-rank Bilinear (MLB) attention~\cite{kim2016hadamard} which defines $g(.,.)$ as:
\begin{equation}
g(\bm{F}_{w,h,\cdot}, \bm{e_l}) = \tanh(\bm{U}^{T}\bm{F}_{w, h, \cdot}) \circ \tanh(\bm{V}^{T}\bm{e_l}) ,
\end{equation}
where $\circ$ denotes an element-wise product and where $\bm{U}$ and $\bm{V}$ are trainable weight matrices. We choose MLB attention because it is parameter efficient and has shown strong empirical performance~\cite{kim2016hadamard,kafle2017visual}.
\subsection{Feature-wise Linear Modulation}
Feature-wise Linear Modulation was introduced in the context of image stylization~\cite{dumoulin2017learned} and extended and shown to be highly effective for multi-modal tasks such as visual question-answering~\cite{devries2017modulating,perez2018film,delbrouck2017modulating}.
A Feature-wise Linear Modulation (FiLM) layer applies a per-channel scaling and shifting to the convolutional feature maps. Such layers are parameter efficient (only two scalars per feature map) while still retaining high capacity, as they are able to scale up or down, zero-out, or negate whole feature maps. In vision-and-language tasks, another network, the so-called FiLM generator $h$, predicts these modulating parameters from the linguistic input $\bm{e}_l$. More formally, a FiLM layer computes a modulated feature map $\bm{\hat{F}}_{w,h,c}$ as follows:
\begin{align}
[\;\bm{\gamma} \;;\; \bm{\beta}\;] = h(\bm{e}_l) \qquad ; \qquad
\bm{\hat{F}}_{.,.,c} = \gamma_{c}\bm{F}_{.,.,c} + \beta_{c},
\end{align}
where $\bm{\gamma}$ and $\bm{\beta}$ are the scaling and shifting parameters which modulate the activations of the original feature map $\bm{F}_{.,.,c}$. We will use the superscript $k \in [1;K]$ to refer to the $k^{th}$ FiLM layer in the network.
FiLM layers may be inserted throughout the hierarchy of a convolutional network, either pre-trained and fixed~\cite{de2017guesswhat} or trained from scratch~\cite{perez2018film}.
Prior FiLM-based models~\cite{devries2017modulating,perez2018film,delbrouck2017modulating} have used a single-hop FiLM generator to predict the FiLM parameters in all layers, e.g.\@, an MLP which takes the language embedding $\bm{e}_{l}$ as input~\cite{devries2017modulating,perez2018film,delbrouck2017modulating}.
\section{Multi-hop FiLM\xspace}
\label{sec:mem_cell}
In this section, we introduce the Multi-hop FiLM\xspace architecture (shown in Fig.~\ref{fig:film}) to predict the parameters of FiLM layers in an iterative fashion, to better scale to longer input sequences such as in dialogue. Another motivation was to better disantangle the linguistic reasoning from the visual one by iteratively attending to both pipelines.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\linewidth]{img/FiLM.pdf}
\caption{\label{fig:film} The Multi-hop FiLM\xspace architecture, illustrating inputs (green), layers (blue), and activations (purple). In contrast, Single-hop FiLM\xspace models predict FiLM parameters directly from $\bm{e}_{l,T}$.}
\end{figure}
We introduce a context vector $\bm{c}^k$ that acts as a controller for the linguistic and visual pipelines. We initialize the context vector with the final state of a bidirectional RNN $\bm{s}_T$ and repeat the following procedure for each of the FiLM layers in sequence (from lowest to highest convolutional layer): first, the context vector is updated by performing attention over RNN states (extracting relevant language information), and second, the context is used to predict a layer's FiLM parameters (dynamically modulating the visual information).
Thus, the context vector enables the model to perform multi-hop reasoning over the linguistic pipeline while iteratively modulating the image features.
More formally, the context vector is computed as follows:
\begin{align}
\begin{cases}
\bm{c}^0 = \bm{s}_T \\
\bm{c}^{k} = \sum_{t} \kappa^k_{t}(\bm{c}^{k-1},\bm{s}_{t}) \bm{s}_{t},
\end{cases}
\end{align}
where:
\begin{equation}
\kappa^k_{t}(\bm{c}^{k-1},\bm{s}_{t}) = \frac{\exp(\chi^k_{t})}{\sum_{t} \exp(\chi^k_{t})} \qquad ; \qquad \chi^k_{t}(\bm{c}^{k-1},\bm{s}_{t}) = MLP_{Attn}(g'(\bm{c}^k,\bm{s}_{t})),
\end{equation}
where the dependence of $\chi^k_{t}$ on $(\bm{c}^{k-1},\bm{s}_{t})$ may be omitted to simplify notation. $MLP_{Attn}$ is a network (shared across layers) which aids in producing attention weights. $g'$ can be any fusion mechanism that facilitates selecting the relevant context to attend to; here we use a simple dot-product following~\cite{luong2015effective}, so $g'(\bm{c}^k,\bm{s}_{t}) = \bm{c}^k\circ\bm{s}_{t}$ . Finally, FiLM is carried out using a layer-dependent neural network $MLP_{FiLM}^k$:
\begin{align}
[\;\bm{\gamma}^k \;;\; \bm{\beta}^k\;] = MLP_{FiLM}^k(\bm{c}^k) \qquad ; \qquad
\bm{\hat{F}}^k_{w,h,c} = \gamma^k_{c}\bm{F}^k_{.,.,c} + \beta^k_{c}.
\end{align}
As a regularization, we append a normalization-layer~\cite{ba2016layer} on top of the context vector after each attention step.
\emph{External information.}
Some tasks provide additional information which may be used to further improve the visual modulation. For instance, GuessWhat?!\xspace provides spatial features of the ground truth object to models which must answer questions about that object. Our model incorporates such features by concatenating them to the context vector before generating FiLM parameters.
\emph{Visual feedback.}
Inspired by the co-attention mechanism~\cite{lu2016hierarchical,zhuang2017parallel}, we also explore incorporating visual feedback into the Multi-hop FiLM\xspace architecture. To do so, we first extract the image or crop features $\bm{F}^k$ (immediately before modulation) and apply a global mean-pooling over spatial dimensions. We then concatenate this visual state into the context vector $\bm{c}^k$ before generating the next set of FiLM parameters.
\section{Experiments}
In this section, we first introduce the ReferIt\xspace and GuessWhat?!\xspace datasets and respective tasks and then describe our overall Multi-hop FiLM\xspace architecture\footnote{The code and hyperparameters are available at \url{https://github.com/GuessWhatGame}}.
\subsection{Dataset}
\emph{ReferIt\xspace}~\cite{kazemzadeh2014referitgame,yu2016modeling} is a cooperative two-player game. The first player (the Oracle) selects an object in a rich visual scene, for which they must generate an expression that refers to it (e.g.\@, ``the person eating ice cream''). Based on this expression, the second player (the Guesser) must then select an object within the image. There are four ReferIt\xspace datasets exist: RefClef, RefCOCO, RefCOCO+ and RefCOCOg. The first dataset contains 130K references over 20K images from the ImageClef dataset~\cite{Mller:2012:IEE:2462702}, while the three other datasets respectively contain 142K, 142K and 86K references over 20K, 20k and 27K images from the MSCOCO dataset~\cite{lin2014microsoft}. Each dataset has small differences. RefCOCO and RefClef were constructed using different image sets. RefCOCO+ forbids certain words to prevent object references from being too simplistic, and RefCOCOg only relies on images containing 2-4 objects from the same category. RefCOCOg also contains longer and more complex sentences than RefCOCO (8.4 vs. 3.5 average words). Here, we will show results on both the Guesser and Oracle tasks.
\emph{GuessWhat?!\xspace}~\cite{de2017guesswhat} is a cooperative three-agent game in which players see the picture of a rich visual scene with several objects. One player (the Oracle) is randomly assigned an object in the scene. The second player (Questioner) aims to ask a series of yes-no questions to the Oracle to collect enough evidence to allow the third player (Guesser) to correctly locate the object in the image. The GuessWhat?!\xspace dataset is composed of 131K successful natural language dialogues containing 650k question-answer pairs on over 63K images from MSCOCO~\cite{lin2014microsoft}. Dialogues contain 5.2 question-answer pairs and 34.4 words on average. Here, we will focus on the Guesser and Oracle tasks.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{img/Oracle_model_by_modality2}
\caption{Overall model, consisting of a visual pipeline (red and yellow) and linguistic pipeline (blue) and incorporating additional contextual information (green).}
\label{fig:oracle model}
\end{figure}
\subsection{Task Descriptions}
\emph{Game Features.} Both games consist of triplets $(\mathcal{I}, l, o)$, where $\mathcal{I} \in \mathbb{R}^{3\times M\times N}$ is an RGB image and $l$ is some language input (i.e.\@, a series of words) describing an object $o$ in $\mathcal{I}$.
The object $o$ is defined by an object category, a pixel-wise segmentation, an RGB crop of $\mathcal{I}$ based on bounding box information, and hand-crafted spatial information $\bm{x}_{spatial}$, where
\begin{equation}
\bm{x}_{spatial} = [ x_{min}, y_{min}, x_{max}, y_{max}, x_{center}, y_{center}, w_{box}, h_{box}]
\end{equation}
We replace words with two or fewer occurrences with an ${<}unk{>}$ token.
\emph{The Oracle task.} Given an image $\mathcal{I}$, an object $o$, a question $q$, and a sequence $\delta$ of previous question-answer pairs $(\bm{q},a)_{{1:\delta}}$ where $a \in \{\mbox{Yes},\mbox{No},\mbox{N/A}\}$, the oracle's task is to produce an answer $a$ that correctly answers the question $q$.
\emph{The Guesser task.} Given an image $\mathcal{I}$, a list of objects $O = o_{1:\Phi}$, a target object $o^* \in O$ and the dialogue $\mathcal{D}$, the guesser needs to output a probability $\sigma_\phi$ that each object $o_\phi$ is the target object $o^*$. Following~\cite{hu2016natural}, the Guesser is evaluated by selecting the object with the highest probability of being correct. Note that even if the individual probabilities $\sigma_\phi$ are between 0 and 1, their sum can be greater than 1. More formally, the Guesser loss and error are computed as follows:
\begin{equation}
L_{Guesser} = \frac{-1}{N_{games}}\sum_n^{N_{games}} \frac{1}{\Phi^n} \sum_\phi^\Phi \text{log}(p(o^*|\mathcal{I}^n, o^n_\phi, \mathcal{D}^n))
\end{equation}
\begin{equation}
E_{Guesser} = \frac{-1}{N_{games}}\sum_n^{N_{games}} \mathbb{1}( o^* \not = o_{ \argmax_\phi \sigma^n_\phi})
\end{equation}
where $\mathbb{1}$ is the indicator function and $\Phi^n$ the number of objects in the $n^{th}$ game.
\subsection{Model}
We use similar models for both ReferIt\xspace and GuessWhat?!\xspace and provide its architectural details in this subsection.
\emph{Object embedding} The object category is fed into a dense look-up table $\bm{e}_{cat}$, and the spatial information is scaled to [-1;1] before being up-sampled via non-linear projection to $\bm{e}_{spat}$. We do not use the object category in ReferIt\xspace models.
\emph{Visual Pipeline} We first resized the image and object crop to $448\times 448$ before extracting $14\times 14\times 1024$ dimensional features from a ResNet-152~\cite{he2016deep} (block3) pre-trained on ImageNet~\cite{russakovsky2015imagenet}. Following~\cite{perez2018film}, we feed these features to a $3\times3$ convolution layer with Batch Normalization~\cite{ioffe2015batch} and Rectified Linear Unit~\cite{nair2010rectified} (ReLU). We then stack four modulated residual blocks (shown in Fig~\ref{fig:film}), each producing a set of feature maps $\bm{F}^k$ via (in order) a $1\times 1$ convolutional layer (128 units), ReLU activations, a $3\times 3$ convolutional layer (128 units), and an untrainable Batch Normalization layer. The residual block then modulates $\bm{F}^k$ with a FiLM layer to get $\bm{\hat{F}}^k$, before again applying ReLU activations. Lastly, a residual connection sums the activations of both ReLU outputs. After the last residual block, we use a $1\times1$ convolution layer (512 units) with Batch Normalization and ReLU followed by MLB attention~\cite{kim2016hadamard} (256 units and 1 glimpse) to obtain the final embedding $\bm{e}_v$. Note our model uses two independent visual pipeline modules: one to extract modulated image features $\bm{e}^{img}_v$, one to extract modulated crop features $\bm{e}^{crop}_v$.
To incorporate spatial information, we concatenate two coordinate feature maps indicating relative x and y spatial position (scaled to $[-1, 1]$) with the image features before each convolution layer (except for convolutional layers followed by FiLM layers). In addition, the pixel-wise segmentations $S \in \{0,1\}^{M\times N}$ are rescaled to $14\times 14$ floating point masks before being concatenated to the feature maps.
\emph{Linguistic Pipeline} We compute the language embedding by using a word-embedding look-up (200 dimensions) with dropout followed by a Bi-GRU ($512\times2$units) with Layer Normalization~\cite{ba2016layer}. As described in Section~\ref{sec:mem_cell}, we initialize the context vector with the last RNN state $\bm{c}^0=\bm{s}_{T}$. We then attend to the other Bi-GRU states via an attention mechanism with a linear projection and ReLU activations and regularize the new context vector with Layer Normalization.
\emph{FiLM parameter generation} We concatenate spatial information $\bm{e}_{spat}$ and object category information $\bm{e}_{cat}$ to the context vector. In some experiments, we also concatenate a fourth embedding consisting of intermediate visual features $\bm{F}^k$ after mean-pooling. Finally, we use a linear projection to map the embedding to FiLM parameters.
\emph{Final Layers} We first generate our final embedding by concatenating the output of the visual pipelines $\bm{e}_{final} = [\bm{e}^{img}_v$ ; $\bm{e}^{crop}_v]$ before applying a linear projection (512 units) with ReLU and a softmax layer.
\emph{Training Process} We train our model end-to-end with Adam~\cite{kingma2014adam} (learning rate $3e^{-4}$), dropout ($0.5$), weight decay ($5e^{-6}$) for convolutional network layers, and a batch size of 64. We report results after early stopping on the validation set with a maximum of 15 epochs.
\subsection{Baselines}
In our experiments, we re-implement several baseline models to benchmark the performance of our models.
The standard \emph{Baseline NN} simply concatenates the image and object crop features after mean pooling, the linguistic embedding, and the spatial embedding and the category embedding (GuessWhat?!\xspace only), passing those features to the same final layers described in our proposed model. We refer to a model which uses the MLB attention mechanism to pool the visual features as \emph{Baseline NN+MLB}.
We also implement a \emph{Single-hop FiLM\xspace} mechanism which is equivalent to setting all context vectors equal to the last state of the Bi-GRU $\bm{e}_{l,T}$. Finally, we experiment with injecting intermediate visual features into the FiLM Generator input, and we refer to the model as \emph{Multi-hop FiLM\xspace~(+img)}.
\begin{table}[t]
\caption{ReferIt\xspace Guesser Error.}
\label{tab:referit_res}
\centering
\scriptsize
\begin{tabular}{l|x{1.2cm}x{1.1cm}x{1.1cm}|x{1.1cm}x{1.1cm}x{1.1cm}|x{1.5cm}}
\toprule
\textbf{Referit} & \multicolumn{3}{c|}{RefCOCO} & \multicolumn{3}{c|}{RefCOCO+} &\multicolumn{1}{c}{RefCOCOg}\\
Split by & \multicolumn{3}{c|}{(unc)} & \multicolumn{3}{c|}{(unc)} &\multicolumn{1}{c}{(google)} \\
Report on& Valid& TestA & TestB & Valid & TestA & TestB & Val \\
\midrule
MMI~\cite{nagaraja2016modeling} & - &71.7\% & 71.1\% & - & 58.4\% & 51.2\% & 59.3\% \\
visDif + MMI~\cite{yu2016modeling} & - &74.6\% & 76.6\% & - & 59.2\% & 55.6\% & 64.0\% \\
NEG Bag~\cite{nagaraja2016modeling} & - &75.6\% & 78.0\% & - & - & - & 68.4\% \\
Joint-SLR~\cite{yu2017joint} & 78.9\% &78.0\% & 80.7\% & 61.9\% & 64.0\% & 59.2\% & - \\
PLAN~\cite{zhuang2017parallel} & 81.7\% &80.8\% & 81.3\% & 64.2\% & 66.3\% & 61.5\% & 69.5\% \\
MAttN~\cite{yu2018mattnet} & $\bm{85.7}$\% &85.3\% & $\bm{84.6}$\% & 71.0\% & 75.1\% & $\bm{66.2}$\% & - \\
\midrule
Baseline NN+MLB & 77.6\% &79.6\%&77.2\% &60.8\% & 59.7\% &66.2\% & 63.1\% \\
Single-hop FiLM\xspace & 83.4\% &85.8\% &80.9\% &72.1\% & 77.3\% &63.9\% & 67.8\% \\
Multi-hop FiLM\xspace & 83.5\% &86.5\% &81.3\% &73.4\% & 77.7\% &64.5\% & 69.8\% \\
Multi-hop FiLM\xspace (+img) & 84.9\% &$\bm{87.4}$\% &83.1\% &$\bm{73.8\%}$&$\bm{78.7}$\% &65.8\% & $\bm{71.5}$\% \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Results}
\emph{ReferIt\xspace Guesser} We report the best test error of the outlined methods on the ReferIt\xspace Guesser task in Tab.~\ref{tab:referit_res}. Note that RefCOCO and RefCOCO+ split test sets into TestA and TestB, only including expression referring towards people and objects, respectively. We do not report~\cite{yu2018mattnet} and~\cite{yu2017joint} scores on RefCOCOg as the authors use a different split (umd). Our initial baseline achieves 77.6\%, 60.8\%, 63.1\%, 73.4\% on the RefCOCO, RefCOCO+, RefCOCOg, RefClef datasets, respectively, performing comparably to state-of-the-art models. We observe a significant improvements using a FiLM-based architecture, jumping to 84.9\%, 87.4\%, 73.8\%, 71.5\%, respectively, and outperforming most prior methods and achieving comparably performance with the concurrent MAttN~\cite{yu2018mattnet} model. Interestingly, MAttN and Multi-hop FiLM\xspace are built in two different manners; while the former has three specialized reasoning blocks, our model uses a generic feature modulation approach. These architectural differences surface when examining test splits: MAttN achieves excellent results on referring expression towards objects while Multi-hop FiLM\xspace performs better on referring expressions towards people.
\begin{table*}[t]
\caption{\label{tab:oracle_results} GuessWhat?!\xspace Oracle Error by Model and Input Type.}
\centering
\scriptsize
\begin{tabular}{l|x{1.1cm}x{1.1cm}x{1.1cm}x{1.1cm}x{1.1cm} | x{ 1.5cm} }
\toprule
\textbf{Oracle Models} & Quest. & Dial. & Object & Image & Crop & Test Error \\
\midrule
Dominant class (``no'') & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & 50.9\% \\
Question only~\cite{de2017guesswhat} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & 41.2\% \\
Image only~\cite{de2017guesswhat} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & 46.7\% \\
Crop only~\cite{de2017guesswhat} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & 43.0\% \\
\midrule
No-Vision (Quest.)~\cite{de2017guesswhat} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} &21.5\% \\
No-Vision (Dial.) & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} &20.6\% \\
\midrule
Baseline NN (Quest.) & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 23.3\%\\
Baseline NN (Dial.) & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 22.4\%\\
Baseline NN + MLB (Quest.) & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 21.8\%\\
Baseline NN + MLB (Dial.) & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 21.1\%\\
\midrule
MODERN~\cite{devries2017modulating} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} &19.5\% \\
\midrule
Single-hop FiLM\xspace (Quest.) & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 17.8\%\\
Single-hop FiLM\xspace (Dial.) & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 17.6\%\\
Multi-hop FiLM\xspace & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \bf{16.9\%} \\
Multi-hop FiLM\xspace (+img) & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & 17.1\% \\
\bottomrule
\end{tabular}
\end{table*}
\emph{GuessWhat?!\xspace Oracle} We report the best test error of several variants of GuessWhat?!\xspace Oracle models in Tab.~\ref{tab:oracle_results}. First, we baseline any visual or language biases by predicting the Oracle's target answer using only the image (46.7\% error) or the question (41.1\% error). As first reported in~\cite{de2017guesswhat}, we observe that the baseline methods perform worse when integrating the image and crop inputs (21.1\%) rather than solely using the object category and spatial location (20.6\%). On the other hand, concatenating previous question-answer pairs to answer the current question is beneficial in our experiments. Finally, using Single-hop FiLM\xspace reduces the error to 17.6\% and Multi-hop FiLM\xspace further to 16.9\%, outperforming the previous best model by 2.4\%.
\emph{GuessWhat?!\xspace Guesser} We provide the best test error of the outlined methods on the GuessWhat?!\xspace Guesser task in Tab.~\ref{tab:guesser_results}. As a baseline, we find that random object selection achieves an error rate of 82.9\%. Our initial model baseline performs significantly worse (38.3\%) than concurrent models (36.6\%), highlighting that successfully jointly integrating crop and image features is far from trivial. However, Single-hop FiLM\xspace manages to lower the error to 35.6\%. Finally, Multi-hop FiLM\xspace architecture outperforms other models with a final error of 30.5\%.
\section{Discussion}
\emph{Single-hop FiLM\xspace \textit{vs.} Multi-hop FiLM\xspace} In the GuessWhat?!\xspace task, Multi-hop FiLM\xspace outperforms Single-hop FiLM\xspace by 6.1\% on the Guesser task but only 0.7\% on the Oracle task. We think that the small performance gain for the Oracle task is due to the nature of the task; to answer the current question, it is often not necessary to look at previous question-answer pairs, and in most cases this task does not require a long chain of reasoning. On the other hand, the Guesser task needs to gather information across the whole dialogue in order to correctly retrieve the object, and it is therefore more likely to benefit from multi-hop reasoning. The same trend can be observed for ReferIt\xspace. Single-hop FiLM\xspace and Multi-hop FiLM\xspace perform similarly on RefClef and RefCOCO, while we observe 1.3\% and 2\% gains on RefCOCO+ and RefCOCOg, respectively. This pattern of performance is intuitive, as the former datasets consist of shorter referring expressions (3.5 average words) than the latter (8.4 average words in RefCOCOg), and the latter datasets also consist of richer, more complex referring expressions due e.g.\@ to taboo words (RefCOCO+).
In short, our experiments demonstrate that Multi-hop FiLM\xspace is better able reason over complex linguistic sequences.
\begin{table*}[t]
\caption{GuessWhat?!\xspace Guesser Error.}
\label{tab:guesser_results}
\begin{minipage}{0.36\linewidth}
\centering
\scriptsize
\begin{tabular}{l|x{1.5cm}}
\toprule
\textbf{Guesser Error} & {Test Error} \\
\midrule
Random & 82.9\% \\
\midrule
LSTM~\cite{de2017guesswhat} &38.7\% \\
LSTM + Img~\cite{de2017guesswhat} &39.5\% \\
PLAN~\cite{zhuang2017parallel} &36.6\% \\
\midrule
Base NN + MLB (crop)& 38.3\%\\
Single-hop FiLM\xspace & 35.6\%\\
Multi-hop FiLM\xspace &\bf{30.5\%} \\
\bottomrule
\end{tabular}
\end{minipage}%
\quad
\begin{minipage}{0.62\linewidth}
\centering
\scriptsize
\begin{tabular}{l|x{1.2cm}x{1.2cm}x{1.45cm}}
\toprule
\textbf{Guesser Error} & {Crop} & {Image} & {Crop+Img} \\
\midrule
Baseline NN & 38.3\% & 40.0\% & 45.1\% \\
Single-hop FiLM\xspace & 35.3\% & 35.7\% & 35.6\% \\
Multi-hop FiLM\xspace & 32.3\% & 35.0\% & $\bf{30.5\%}$ \\
Multi-hop FiLM\xspace (no categ.) & 33.1\% & 40\% & 33.4\%\\
\bottomrule
\end{tabular}
\end{minipage}%
\end{table*}
\emph{Reasoning mechanism}
We conduct several experiments to better understand our method. First, we assess whether Multi-hop FiLM\xspace performs better because of increased network capacity. We remove the attention mechanism over the linguistic sequence and update the context vector via a shared MLP. We observe that this change significantly hurts performance across all tasks, e.g.\@, increasing the Multi-hop FiLM\xspace error of the Guesser from 30.5 to 37.3\%.
Second, we investigate how the model attends to GuessWhat?!\xspace dialogues for the Oracle and Guesser tasks, providing more insight into how to the model reasons over the language input. We first look at the top activation in the (crop) attention layers to observe where the most prominent information is. Note that similar trends are observed for the image pipeline. As one would expect, the Oracle is focused on a specific word in the last question 99.5\% of the time, one which is crucial to answer the question at hand. However, this ratio drops to 65\% in the Guesser task, suggesting the model is reasoning in a different way. If we then extract the top 3 activations per layer, the attention points to ${<}yes{>}$ or ${<}no{>}$ tokens (respectively) at least once, 50\% of the time for the Oracle and Guesser, showing that the attention is able to correctly split the dialogue into question-answer pairs. Finally, we plot the attention masks for each FiLM layer to have a better intuition of this reasoning process in Fig.~\ref{fig:attention}.
\emph{Crop \textit{vs.} Image.} We also evaluate the impact of using the image and/or crop on the final error for the Guesser task~\ref{tab:guesser_results}. Using the image alone (while still including object category and spatial information) performs worse than using the crop. However, using image and crop together inarguably gives the lowest errors, though prior work has not always used the crop due to architecture-specific GPU limitations~\cite{devries2017modulating}.
\emph{Visual feedback} We explore whether adding visual feedback to the context embedding improves performance. While it has little effect on the GuessWhat?!\xspace Oracle and Guesser tasks, it improves the accuracy on ReferIt\xspace by 1-2\%. Note that ReferIt\xspace does not include class labels of the selected object, so the visual feedback might act as a surrogate for this information. To further investigate this hypothesis, we remove the object category from the GuessWhat?!\xspace task and report results in Tab.~\ref{tab:oracle_results_nocat} in the supplementary material. In this setup, we indeed observe a relative improvement 0.4\% on the Oracle task, further confirming this hypothesis.
\emph{Pointing Task} In GuessWhat?!\xspace, the Guesser must select an object from among a list of items. A more natural task would be to have the Guesser directly point out the object as a human might. Thus, in the supplementary material, we introduce this task and provide initial baselines (Tab.~\ref{sec:pointing}) which include FiLM models. This task shows ample room for improvement with a best test error of 84.0\%.
\begin{figure}[t]
\centering
\subfloat{\includegraphics[width=0.44\linewidth]{img/tie}}
\quad
\subfloat{\includegraphics[width=0.45\linewidth]{img/dog}}
\caption{\label{fig:attention} Guesser (left) and Oracle (right) attention visualizations for the visual pipeline which processes the object crop.}
\end{figure}
\section{Related Work}
The ReferIt\xspace game~\cite{kazemzadeh2014referitgame} has been a testbed for various vision-and-language tasks over the past years, including object retrieval~\cite{nagaraja2016modeling,yu2016modeling,yu2017joint,zhuang2017parallel,luo2017comprehension,yu2018mattnet}, semantic image segmentation~\cite{hu2016segmentation,rohrbach2016grounding}, and generating referring descriptions~\cite{yu2016modeling,luo2017comprehension,yu2017joint}.
To tackle object retrieval,~\cite{nagaraja2016modeling,yu2016modeling,yu2018mattnet} extract additional visual features such as relative object locations and~\cite{yu2017joint,luo2017comprehension} use reinforcement learning to iteratively train the object retrieval and description generation models.
Closer to our work,~\cite{hu2016natural,zhuang2017parallel} use the full image and the object crop to locate the correct object. While some previous work relies on task-specific modules~\cite{yu2016modeling,yu2018mattnet}, our approach is general and can be easily extended to other vision-and-language tasks.
The GuessWhat?!\xspace game~\cite{de2017guesswhat} can be seen as a dialogue version of the ReferIt\xspace game, one which additionally draws on visual question answering ability.
\cite{strub2017end,lee2018answerer,zhu2017interactive} make headway on the dialogue generation task via reinforcement learning. However, these approaches are
bottlenecked by the accuracy of Oracle and Guesser models, despite existing modeling advances~\cite{zhuang2017parallel,devries2017modulating}; accurate Oracle and Guesser models are crucial for providing a meaningful learning signal for dialogue generation models, so we believe the Multi-hop FiLM\xspace architecture will facilitate high quality dialogue generation as well.
A special case of Feature-wise Linear Modulation was first successfully applied to image style transfer~\cite{dumoulin2017learned}, whose approach modulates image features according to some image style (i.e.\@, cubism or impressionism).
\cite{devries2017modulating} extended this approach to vision-and-language tasks, injecting FiLM-like layers along the entire visual pipeline of a pre-trained ResNet.
\cite{perez2018film} demonstrates that a convolutional network with FiLM layers achieves strong performance on CLEVR~\cite{johnson2017clevr}, a task that focuses on answering reasoning-oriented, multi-step questions about synthetic images.
Subsequent work has demonstrated that FiLM and variants thereof are effective for video object segmentation where the conditioning input is the first image's segmentation (instead of language)~\cite{yang2018osmn} and language-guided image segmentation~\cite{rupprecht2018guide}. Even more broadly,~\cite{dumoulin2018feature-wise} overviews the strength of FiLM-related methods across machine learning domains, ranging from reinforcement learning to generative modeling to domain adaptation.
There are other notable models that decompose reasoning into different modules. For instance, Neural Turing Machines~\cite{graves2014neural,graves2016hybrid} divide a model into a controller with read and write units. Memory networks use an attention mechanism to answer a query by reasoning over a linguistic knowledge base~\cite{weston2014memory,sukhbaatar2015end} or image features~\cite{xiong2016dynamic}. A memory network updates a query vector by performing several attention hops over the memory before outputting a final answer from this query vector. Although Multi-hop FiLM\xspace computes a similar context vector, this intermediate embedding is used to predict FiLM parameters rather than the final answer. Thus, Multi-hop FiLM\xspace includes a second reasoning step over the image.
Closer to our work, \cite{arad2018compositional} designed networks composed of Memory, Attention, and Control (MAC) cells to perform visual reasoning. Similar to Neural Turing Machines, each MAC cell is composed of a control unit that attends over the language input, a read unit that attends over the image and a write unit that fuses both pipelines. Though conceptually similar to Multi-hop FiLM\xspace models, Compositional Attention Networks differ structurally, for instance using a dynamic neural architecture and relying on spatial attention rather than FiLM.
\section{Conclusion}
In this paper, we introduce a new way to exploit Feature-wise Linear Modulation (FiLM) layers for vision-and-language tasks. Our approach generates the parameters of FiLM layers going up the visual pipeline by attending to the language input in multiple hops rather than all at once.
We show Multi-hop FiLM Generator\xspace architectures are better able to handle longer sequences than their single-hop counterparts. We outperform state-of-the-art vision-and-language models significantly on the long input sequence GuessWhat?!\xspace tasks, while maintaining state-of-the-art performance for the shorter input sequence ReferIt\xspace task. Finally, this Multi-hop FiLM Generator\xspace approach uses few problem-specific priors, and thus we believe it can extended to a variety of vision-and-language tasks, particularly those requiring complex visual reasoning.
\paragraph*{Acknowledgements}
The authors would like to acknowledge the stimulating research environment of the SequeL Team. We also thank Vincent Dumoulin for helpful discussions.
We acknowledge the following agencies for research funding and computing support: Project BabyRobot (H2020-ICT-24-2015, grant agreement no.687831), CHISTERA IGLU and CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies 2015-2020, NSERC, Calcul Qu\'{e}bec, Compute Canada, the Canada Research Chairs, and CIFAR.
\bibliographystyle{splncs04}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Function-level code representation learning aims to learn continuous distributed vectors that represent the semantics of code snippets \citep{alon2019code2vec}, which has led to dramatic empirical improvements on a variety of code-related tasks such as code search, clone detection, code summarization, etc. To learn function-level code representation on unlabeled code corpus with self-supervised objectives, recent works \citep{jain2021contrastive,bui2021self,ding2022towards,wang2022code}
propose contrastive pre-training methods for programming language.
In their contrastive pre-training, they usually pull positive code pairs together in representation space and regard different codes as negative pairs via pushing their representation apart. However, they ignore the potential relevance between codes since different programs in a large code corpus may have some similarities. For example, an ascending sort program and a descending sort program are somewhat similar since they both sort their input in a certain order. More seriously, there are a lot of duplications in code corpus~\citep{code_duplication_1,code_duplication_2}, which can cause the "false negative" problem and deteriorate the model~\citep{cv_false_negative_1,cv_false_negative_2}. The other problem of current code contrastive pre-training methods is their positive sample construction.
ContraCode \citep{jain2021contrastive} and Corder \citep{bui2021self} design code transformation algorithms like variable renaming and dead code insertion to generate semantically equivalent programs as positive samples, while Code-MVP \citep{wang2022code} leverages code structures like abstract syntax tree (AST) and control flow graphs (CFG) to transform a program to different variants.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{images/code_transformation_v1.pdf}
\caption{An example of applying variable renaming.}
\label{fig:code_trans}
\vskip -0.2in
\end{figure}
However, as shown in Figure \ref{fig:code_trans}, these methods usually result in generated positive samples with highly similar structures (e.g. double loop statements with a conditional statement) to the original program. To pull such positive pairs closer in representation space, the model will tend to learn function-level code representation from superficial code structure rather than substantial code semantics.
To address these limitations, we present \textbf{SCodeR}, a \textbf{S}oft-labeled contrastive pre-training framework with two positive sample construction methods to learn function-level \textbf{Code} \textbf{R}epresentation.
The soft-labeled contrastive pre-training framework can obtain relevance scores between samples and the original program as soft-labels in an iterative adversarial manner to improve code representation.
Specifically, we first leverage hard-negative samples from contrastive pre-training to fool discriminators that can explore finer-grained token-level interactions, while discriminators learn to distinguish them and predict relevance scores among samples as soft-labels for contrastive pre-training.
Through this adversarial iteration, discriminators can provide progressive feedback to improve code contrastive pre-training through soft-labels.
As for positive sample construction, we propose to utilize code comment and abstract syntax sub-tree of the source code to construct positive samples for SCodeR pre-training. Generally, user-written code comments highly describe the function of a source code like "sort the input array" in Figure \ref{fig:code_trans}, which provides crucial semantic information for the model to capture code semantics.
Besides the comment, the code itself also contains rich information. To further explore the intra-code correlation and contextual knowledge for code contrastive pre-training, we randomly select a piece of code via AST like the conditional statement of Figure \ref{fig:code_trans} and its context as a positive pair. These positive pairs require the model to understand code semantics and learn to infer the selected code based on its context and can help the model learn representation from code semantics.
We evaluate SCodeR on four code-related downstream tasks over seven datasets, including code search, clone detection, zero-shot code-to-code search, and markdown ordering in python notebooks.
Results show that SCodeR achieves state-of-the-art performance and ablation studies demonstrate the effectiveness of positive sample construction and soft-labeled contrastive pre-training. We release the codes and resources at \url{https://github.com/microsoft/AR2/tree/main/SCodeR}.
\section{Related Works}
\paragraph{Pre-trained Models for Programming Language.}
With the great success of pre-trained models in natural language processing field \citep{devlin2018bert,lewis2019bart,raffel2019exploring,brown2020language}, recent works attempt to apply pre-training techniques on programming languages to facilitate the development of code intelligence. \citet{kanade2019pre} pre-train CuBERT on a large-scale Python corpus using masked language modeling (MLM) and next sentence prediction objectives. \citet{feng2020codebert} pre-train CodeBERT on code-text pairs in six programming languages via MLM and replaced token detection objectives to support text-code related tasks such as code search. GraphCodeBERT \cite{guo2020graphcodebert} leverages data flow as additional semantic information to enhance code representation.
To support code completion, \citet{svyatkovskiy2020intellicode} and \citet{lu2021codexglue} respectively propose GPT-C and CodeGPT. Both of them are decoder-only models and pre-trained by unidirectional language modeling.
Some recent works \citep{ahmad2021unified,wang2021codet5,guo2022unixcoder} explore unified pre-trained models to support both understanding and generation tasks. PLBART \citep{ahmad2021unified} and CodeT5 \citep{wang2021codet5} are based on the encoder-decoder framework. PLBART uses denoising objective to pre-train the model and CodeT5 considers the crucial token type information from identifiers.
However, these pre-trained models usually result in poor function-level code representation \citep{guo2022unixcoder} due to the anisotropy representation issue \citep{li2020sentence}.
In this work, we mainly investigate how to learn function-level code semantic representations.
\paragraph{Contrastive Pre-training for Code Representation.}
To learn function-level code semantic representation, several attempts have been made to leverage contrastive pre-training on programming languages. ContraCode \citep{jain2021contrastive} and Corder \citep{bui2021self} design transformation algorithms like variable renaming and dead code insertion to generate semantically equivalent programs as positive instances, while \citet{ding2022towards} design structure-guided code transformation algorithms that inject real-world security bugs to build hard negative pairs for contrastive pre-training. Instead of using semantic-preserving program transformations, SynCoBERT \cite{wang2022syncobert} and Code-MVP \citep{wang2022code} construct the positive pairs
through the compilation process of programs like AST and CFG.
However, these works usually generate positive samples with highly similar structures as the original program.
To distinguish these positive samples from candidates,
the model might learn code representation from code surface forms according to hand-written patterns, instead of code semantics.
In this paper, we propose to utilize the comment and abstract syntax sub-tree of the code to construct positive samples and present a method to obtain relevance scores among samples as soft-labels for contrastive pre-training.
\section{Positive Sample Construction}
\label{sec:positive}
In this section, we describe how to construct positive pairs for SCodeR.
Different from previous works that design transformation algorithms to generate semantically equivalent but highly similar programs, we propose to leverage comment and abstract syntax sub-tree of the code for positive sample construction to encourage the model to capture semantic information.
\subsection{Code Comment}
User-written code comments usually summarize the functionality of the codes and provide crucial semantic information about the source code. Taking the code in Figure~\ref{fig:code_asst} as an example, the comment "sort the input array" highly describes the goal of the code and can help the model to understand code semantics from the natural language. Therefore, we take source code $c$ with the corresponding comment $t$ as positive pair $(t,c)$.
Such positive pairs not only enable the model to understand the code semantics but also align the representation of different programming languages with a unified natural language description as a pivot.
\subsection{Abstract Syntax Sub-Tree}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{images/ASST.pdf}
\caption{An ASST example of bubble sort.}
\label{fig:code_asst}
\end{figure}
Besides the comment, the code itself also contains rich information. To further explore the intra-code correlation and contextual knowledge for contrastive pre-training, we propose a method, called \textbf{A}bstract \textbf{S}yntax \textbf{S}ub-\textbf{T}ree Extraction (\textbf{ASST}), that leverages the abstract syntax sub-tree of the source code to construct positive code pairs.
We give an example of a Python code with its AST in Figure~\ref{fig:code_asst}. We first randomly select the sub-tree of the AST like "if statement", and then take the corresponding code of the sub-tree and the remaining code as positive code pairs. The procedure of extraction is illustrated in Algorithm~\ref{alg:asst}.
Specifically, we first pre-define a set $N$ of node types whose sub-trees can be used to construct positive pairs. The set mainly consists of statement-level types like "for\_statement" that usually contain a complete and functional code snippet. We then start from a randomly selected leaf node (line 1-2) and find an eligible node in the pre-defined set $N$ along the direction of the root node (line 3-10). Finally, we take the corresponding code $s$ (i.e. leaf children) of the eligible node and the remaining code context $\tilde{s}$ as a positive code pairs $(s,\tilde{s})$ for contrastive pre-training. To avoid extracting those code spans that are too short or meaningless, we set a minimum length $l_{min}$ for the extracted code spans $s$.
While transformation-based methods generate programs with similar structures, the structures of positive code pairs generated by ASST are different since they belong to different parts of a program. Meanwhile, they are logically relevant because they compose a program of full functionality. To estimate which code is complementary to a given code context in contrastive pre-training, the model needs to understand code semantics and learn to reason based on its context, which encourages the model to understand code semantics.
\begin{algorithm}[t]
\small
\caption{Abstract Syntax Sub-Tree Extraction}
\label{alg:asst}
\begin{algorithmic}[1]
\REQUIRE The AST $T$ of a code $c$ and pre-defined selectable node types $N$.
\STATE Collect leaf children $C$ of nodes whose types are in $N$
\STATE Randomly sample a node $a$ from $C$
\WHILE{True}
\STATE $s \gets$ the corresponding code of $a$
\IF{$length(s) \geq l_{min}$ and $a\in N$}
\RETURN $s$
\ELSE
\STATE $a \gets$ the parent node of $a$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
There are similar mechanisms to learn text representation such as Inversed Cloze Task (ICT) \citep{lee2019latent} that takes a random span of natural language tokens and their context as a positive pair. However, ICT cannot be directly applied to code because code has an explicit structure. If we randomly select code spans on token-level, the selected code spans might be ungrammatical such as \ \textit{``for i''}, which will mislead the model to focus on structural matching rather than semantic matching.
\section{Soft-Labeled Contrastive Pre-training}
Previous code contrastive pre-training methods usually take different programs in a code corpus as negative pairs and push them apart in the representation space.
However, different programs in an unlabeled code corpus may have some similarities.
Taking a program that sorts the input in ascending order as an example, even though another "descendingly sort" program is not semantically equal with it, they both sort their input in a certain order and thus are somewhat similar. Another problem is the "false negative" issue~\citep{cv_false_negative_1,cv_false_negative_2} due to the duplication in the code corpus~\citep{code_duplication_1,code_duplication_2}.
To alleviate these problems, we propose soft-labeled contrastive pre-training framework that uses relevance scores between different samples as soft-labels to learn function-level code representation.
\subsection{Overview}
The soft-labeled contrastive pre-training framework involves three components: (1) A dual-encoder ${G}_{\theta}$ that aims to learn function-level code representation (2) Two discriminators $D_{\phi}$ and $D_{\psi}$ that calculate relevance scores between two inputs for text-code and code-code pairs, respectively.
These components compute the similarity between two samples $(x,y)$ as follows:
\begin{align}
G_{\theta}(x,y) &= E_{\theta}(x)^T {E}_{\theta}(y)
\\
D_{\phi}(x,y) &= \mathbf{w_{\phi}}^T E_{\phi}\left(\left[x;y\right]\right)
\\
D_{\psi}(x,y) &= \mathbf{w_{\psi}}^T E_{\psi}\left(\left[x;y\right]\right)
\end{align}
where $E_{\theta}$, $E_{\phi}$ and $E_{\psi}$ are multi-layer Transformer \citep{vaswani2017attention} encoders with mean-pooling. $w_{\phi}$ ($w_{\psi}$) is a linear layer to obtain similarity score and $[\cdot; \cdot]$ indicates the concatenation operator. If the input $(x,y)$ is a text-code pair, we use $D_{\phi}$ to calculate the similarity, otherwise we use $D_{\psi}$.
While the dual-encoder encodes samples separately, discriminators take the concatenation of two samples as the input and fully explore finer-grained token-level interactions through the self-attention mechanism, which can predict more accurate relevance scores between two samples. Therefore, we propose to utilize relevance scores from discriminators as soft-labels to help the encoder $E_{\theta}$ learn better code representation.
\begin{algorithm}[t]
\small
\caption{Soft-Labeled contrastive pre-training}
\label{alg:scl}
\begin{algorithmic}[1]
\REQUIRE A dual-encoder $G_\theta$, two discriminators $D_{\phi(\psi)}$, and a set $X$ of positive pairs with a unlabeled code corpus $C$.
\STATE Initialize the dual-encoder and discriminators.
\STATE Train the warm-up dual-encoder.
\STATE Get top-$K$ negative codes $C^{x_k}_{hard}$ from $C$ for each positive pair $(x_k,x_{k}^+) \in X$ using the dual-encoder.
\FOR{$i$ in $1\cdots I$}
\FOR{Discriminators training step}
\STATE Sample hard negative codes from $C^{x_k}_{hard}$.
\STATE Update parameters of discriminators $D_\phi$ and $D_\psi$.
\ENDFOR
\FOR{Dual-encoder training step}
\STATE Sample hard negative codes from $C^{x_k}_{hard}$ and obtain relevance scores from discriminators.
\STATE Update parameters of the dual-encoder $G_\theta$.
\ENDFOR
\STATE Refresh Top-$K$ negative codes $C^{x_k}_{hard}$ using new $G_\theta$.
\ENDFOR
\end{algorithmic}
\end{algorithm}
We show the detailed illustration of our proposed soft-labeled contrastive pre-training in Algorithm~\ref{alg:scl}. Specifically, we first initialize all encoders with a pre-trained model like UniXcoder \citep{guo2022unixcoder} and follow \citet{code_retriever} to train a warm-up dual-encoder using a simple strategy where negative samples come from other positive pairs in the same batch $\mathbb{X}_b$ (line 1-2 of Algorithm \ref{alg:scl}). The loss is calculated as follows,
\begin{equation}
p_\theta(x^+|x,\mathbb{X}_b)=\frac{e^{G_{\theta}(x,x^+)}}{\sum_{x' \in \mathbb{X}_b}e^{G_{\theta}(x,x')}}
\end{equation}
\begin{equation}
L_{warm}^{\theta}=-\log p_\theta(x^+|x,\mathbb{X}_b),
\end{equation}
where $(x,x^+) \in X$ is a positive pair as described by Section \ref{sec:positive}.
We then iteratively alternate two training procedures: (1) The dual-encoder is used to obtain hard-negative codes to train the discriminators (line 5-8). (2) The optimized discriminators predict relevance scores among samples as soft-labels to improve the dual-encoder (line 9-12). Through this iterative training, the dual-encoder gradually produces harder negative samples to train better discriminators, whereas the discriminators provide better progressive feedback to improve the dual-encoder. The details about training procedures for the discriminators and dual-encoder will be described next.
\subsection{Discriminators Training}
\label{sec:train_cross_encoder}
Given a text $x$ from positive text-code pairs $(x,x^+)$, the discriminator $D_{\phi}$ is optimized by maximizing the log likelihood of selecting positive code $x^+$ from candidates $\mathbb{X}$ as follows,
\begin{equation}
p_\phi(x^+|x,\mathbb{X})=\frac{e^{D_{\phi}(x,x^+)}}{\sum_{x' \in \mathbb{X}}e^{D_{\phi}(x,x')}}
\end{equation}
\begin{equation}
L^\phi=-log p_\phi(x^+|x,\mathbb{X}),
\end{equation}
where $\mathbb{X}$ is the set of negative codes $\mathbb{X}^-$ with a positive code $x^+$. If $x$ is a code from positive code-code pairs, the calculation of $p_\psi$ and $L^\psi$ are analogous to $p_\phi$ and $L^\phi$, respectively.
To better train discriminators, we take those hard-negative examples that are not positive samples but closed to the original example $x$ in the vector space as the negative candidates $\mathbb{X}^-$. In practice, we first get the top-K code samples that are closest to $x$ using $G_\theta$ as the distance function and randomly sample examples from them to obtain a subset $\mathbb{X}^-$.
\subsection{Dual-Encoder Training}
After training discriminators, we utilize relevance scores predicted by discriminators as soft-labels and follow \citet{ar2} to use adversarial and distillation losses to optimize the dual-encoder.
\begin{table*}[t]
\small
\centering
\begin{tabular}{@{}l|ccccccc|c|c@{}}
\toprule
\textbf{Dataset} & \multicolumn{7}{c|}{\textbf{CSN}} & \textbf{AdvTest} & \textbf{CosQA} \\ \midrule
\textbf{Lang} & \textbf{Ruby} & \textbf{Javascript} & \textbf{Go} & \textbf{Python} & \textbf{Java} & \textbf{PHP} & \textbf{Average} & \textbf{Python} & \textbf{Python} \\ \midrule
CodeBERT
& 67.9 & 62.0 & 88.2 & 67.2 & 67.6 & 62.8 &69.3 & 27.2&64.7 \\
GraphCodeBERT
& 70.3 & 64.4 & 89.7 & 69.2 & 69.1 & 64.9 &71.3 & 35.2 & 67.5\\
SyncoBERT
& 72.2 & 67.7 & 91.3 & 72.4 & 72.3 & 67.8 &74.0 &38.1 & - \\
CodeRetriever
& 75.3 & 69.5 & 91.6 & 73.3 & 74.0 & 68.2 & 75.3 & 43.0 & 69.6\\
Code-MVP
&- &- &- &- &- &- &- &40.4 &72.1 \\
UniXcoder
&74.0 & 68.4 &91.5 &72.0 & 72.6 & 67.6 & 74.4 & 41.3 & 70.1 \\
SCodeR & \textbf{77.5} & \textbf{72.0} & \textbf{92.7} & \textbf{74.2} & \textbf{74.8} & \textbf{69.2} & \textbf{76.7} & \textbf{45.5} & \textbf{74.5}\\
\bottomrule
\end{tabular}
\vspace{-5pt}
\caption{The comparison on code search task. The results of compared models are from their original papers.}
\label{result_code_search}
\vspace{-10pt}
\end{table*}
\paragraph{Adversarial loss:}
\begin{equation}
\label{eq_adv}
{L}^{\theta}_{adv}=-\sum_{x^-\in \mathbb{X}^-}w(x^-)* \log p_{\theta}(x^- | x, \mathbb{X}^- )
\end{equation}
where $w(x^-)$ is
$-\log p_\phi( x^+|x,\{x^+,x^-\})$ if $x$ is a text otherwise $w$ is $-\log p_\psi( x^+|x,\{x^+,x^-\})$. We apply the same approach to obtain hard-negative candidates $\mathbb{X}^-$ as described in Section~\ref{sec:train_cross_encoder}.
When optimizing $G_\theta$, $w$ in Equation~\ref{eq_adv} is a constant and adjusts weight for each negative example.
When $-\log p_{\phi(\psi)}( x^+|x,\{x^+,x^-\})$ is small, i.e. discriminators predict that $x$ and $x^-$ are semantically relevant, $w$ will be a high weight and force $G_\theta$ to draw the representation of $x$ and $x^-$ closer among $\mathbb{X}^-$.
Since we optimize the dual-encoder on negative codes under different weight $w$, the representation of negative codes with high relevance score will be closer to $x$, and those with low relevance score will be pushed away.
\paragraph{Distillation loss:}
\begin{equation}
\begin{aligned}
L^{\theta}_{distill} & = H(p_{\phi(\psi)}(\cdot |x,\mathbb{X}), p_{\theta}(\cdot | x,\mathbb{X}) )
\end{aligned}
\label{eq_distill}
\end{equation}
We also use a distillation loss function~\citep{knowledge_distillation_hinton} to encourage the dual-encoder to fit the probability distribution of discriminators over $\{x^+\}\cup\mathbb{X}^-$ using KL divergence loss $H$.
Through $L^{\theta}_{distill}$, we can inject discriminators' knowledge into the dual-encoder by soft-labels $p_{\phi(\psi)}$.
\paragraph{Training Objective of Dual-Encoder}
The overall loss function of the dual-encoder is the integration of adversarial loss and distillation loss as follows, where $\lambda$ is a pre-defined hyper-parameter.
\begin{equation}
L^\theta = \lambda * L^{\theta}_{adv} + (1-\lambda) * L^{\theta}_{distill}
\end{equation}
Through $L_\theta$, we can provide discriminators' progressive feedback to the dual-encoder through soft-labels.
After this adversarial iteration, we will use $E_\theta$ to serve for downstream tasks.
\section{Experiment}
\subsection{Model Comparison}
We compare SCodeR with various state-of-the-art pre-trained models. \textbf{RoBERTa}~\citep{liu2019roberta} is pre-trained on text corpus by masked language model (MLM). \textbf{CodeBERT}~\citep{feng2020codebert} is pre-trained on large scale code corpus with MLM and replaced token detection. \textbf{GraphCodeBERT}~\citep{guo2020graphcodebert} is based on CodeBERT and integrates the data flow information to enhance code representation. \textbf{PLBART}~\citep{ahmad2021unified} is adapted from the BART~\citep{lewis2019bart} architecture and pre-trained using denoising objective on Java, Python and stackoverflow corpus. \textbf{CodeT5}~\citep{wang2021codet5} is based on the T5~\citep{nl_t5} architecture, considering the identifier token information and applying multi-task learning. \textbf{UniXcoder}~\citep{guo2022unixcoder} is adapted from the UniLM~\citep{unilm} architecture, pretrained by different tasks (understanding and generation) on unified cross-modal data (code, AST and text). We also compare SCodeR with those code pre-trained models that utilize contrastive pre-training. \textbf{SynCoBERT}~\citep{wang2022syncobert} and \textbf{Code-MVP}~\citep{wang2022code} construct positive pairs through multiple views of code like AST and CFG. \textbf{Corder}~\citep{bui2021self} and \textbf{DISCO}~\citep{ding2022towards} construct positive code pairs from semantic-preserving transformations, and the latter additionally uses bug-injected codes as hard negatives. \textbf{CodeRetriever}~\citep{code_retriever} builds code-code pairs by corresponding documents and function name automatically.
For fair comparison, we use the same model architecture, pre-training corpus, and downstream hyper-parameters as previous works~\citep{code_retriever,guo2022unixcoder}. To accelerate the training process, we initialize dual-encoder and discriminators with the released parameters of UniXcoder~\citep{guo2022unixcoder}.
More details about pre-training and fine-tuning can be found in the Appendix~\ref{sec:appendix_pretrain_hyper_parameters} and \ref{sec:appendix_finetuning_hyper_parameters}.
\subsection{Natural Language Code Search}
Given a natural language query as the input, code search aims to retrieve the most semantically relevant code from a collection of code candidates. We conduct experiments on CSN~\citep{guo2020graphcodebert}, AdvTest~\citep{lu2021codexglue} and CosQA~\citep{huang2021cosqa} to evaluate SCodeR. CSN contains six programming languages, including Ruby, Javascript, Python, Java, PHP and Go. The dataset is constructed from CodeSearchNet Dataset~\citep{husain2019codesearchnet} and noisy queries with low quality are filtered. AdvTest normalizes the function name and variable name of python code and thus is more challenging. The queries of CosQA are from Microsoft Bing search engine, which makes it closer to real-world code search scenario. Following previous works~\citep{feng2020codebert,guo2020graphcodebert,guo2022unixcoder}, we adopt Mean Reciprocal Rank (MRR)~\citep{mrr1} as the evaluation metric.
The results are shown in Table~\ref{result_code_search}. We can see that SCodeR outperforms previous code pre-trained models and achieves the new state-of-the-art performance on all datasets.
Specifically, SCodeR outperforms UniXcoder by 2.3 points on the CSN dataset, and improves over state-of-the-art models about 2.5 points on AdvTest and CosQA datasets, which demonstrates the effectiveness of SCodeR.
\subsection{Code Clone Detection}
\begin{table}[h]
\centering
\small
\setlength{\tabcolsep}{0.5mm}
\begin{tabular}{@{}lcccc@{}}
\toprule
& \multicolumn{1}{l}{\textbf{POJ-104}} & \multicolumn{3}{c}{\textbf{BigCloneBench}} \\ \midrule
& \multicolumn{1}{l}{MAP@R} & \multicolumn{1}{l}{Recall} & \multicolumn{1}{l}{Precision} & \multicolumn{1}{l}{F1-score} \\
RoBERTa & 76.67 & 95.1 & 87.8 & 91.3 \\
CodeBERT & 82.67 & 94.7 & 93.4 & 94.1 \\
GraphCodeBERT & 85.16 & 94.8 & 95.2 & 95.0 \\
SyncoBERT & 88.24 & - &- &- \\
CodeRetriever & 88.85 & -&- &- \\
Corder &84.10 &- &- &- \\
DISCO &82.77 &94.6 &94.2 &94.4 \\
PLBART & 86.27 & 94.8 & 92.5 & 93.6 \\
CodeT5-base & 88.65 & 94.8 & 94.7 & 95.0 \\
UniXcoder & 90.52 & 92.9 & \textbf{97.6} & 95.2 \\
SCodeR & \textbf{92.45} & \textbf{96.2} & 94.5 & \textbf{95.3} \\ \bottomrule
\end{tabular}
\caption{Performance on code clone detection. The results of compared models are from their original papers.}
\label{result_code_clone}
\end{table}
Code clone detection aims to identify the semantic similarity between two codes. We consider POJ-104~\citep{poj_104} and BigCloneBench~\citep{bcb} to evaluate SCodeR. POJ-104 dataset (C/C++) consists of codes from online judge (OJ) system. It aims to find the semantically similar codes given a code as query and evaluates by Mean Average Precision (MAP). BigCloneBench dataset (Java) is to judge whether two codes are similar and evaluates by Precision, Recall, and F1-score.
We show the results in Table~\ref{result_code_clone}.
Compared with previous pre-trained models, SCodeR achieves the overall best performance on both datasets. On POJ-104 dataset, SCodeR surpasses all other methods. Specifically, SCodeR outperforms UniXcoder by 1.93 points. Although the pre-training corpus does not cover C/C++ programming languages, the superior performance reflects that SCodeR learns better general code knowledge.
On the BigCloneBench dataset, SCodeR also achieves comparable performance.
These results show that SCodeR learns better function-level code representation for code clone detection.
\subsection{Zero-Shot Code-to-Code Search}
\begin{table*}[t]
\small
\centering
\begin{tabular}{@{}l|ccc|ccc|ccc|c@{}}
\toprule
\textbf{Query PL} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{\textbf{Ruby}} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{\textbf{Python}} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{\textbf{Java}} & \multicolumn{1}{l|}{} & \multirow{2}{*}{\textbf{Overall}} \\ \cmidrule(r){1-10}
\textbf{Target PL} & \multicolumn{1}{l}{\textbf{Ruby}} & \multicolumn{1}{l}{\textbf{Python}} & \multicolumn{1}{l|}{\textbf{Java}} & \multicolumn{1}{l}{\textbf{Ruby}} & \multicolumn{1}{l}{\textbf{Python}} & \multicolumn{1}{l|}{\textbf{Java}} & \multicolumn{1}{l}{\textbf{Ruby}} & \multicolumn{1}{l}{\textbf{Python}} & \multicolumn{1}{l|}{\textbf{Java}} & \\ \midrule
CodeBERT & 13.55 & 3.18 & 0.71 & 3.12 & 14.39 & 0.96 & 0.55& 0.42& 7.62& 4.94 \\
GraphCodeBERT & 17.01 & 9.29 & 6.38 & 5.01 & 19.34 & 6.92 & 1.77& 3.50& 13.31& 9.17\\
PLBART & 18.60 & 10.76 & 1.90 & 8.27 & 19.55 & 1.98 & 1.47& 1.27& 10.41& 8.25 \\
CodeT5-base & 18.22 & 10.02 & 1.81 & 8.74 & 17.83 & 1.58 & 1.13& 0.81& 10.18& 7.81 \\
UniXcoder& 29.05 & 26.36 & 15.16 & 23.96 & 30.15 & 15.07 & 13.61& 14.53& 16.12& 20.45 \\
SCodeR& \bf{33.87} & \bf{30.25} & \bf{17.10} & \bf{26.48} & \bf{33.02} & \bf{16.95} & \bf{16.5}& \bf{19.06}& \bf{18.87}& \bf{23.57} \\\bottomrule
\end{tabular}
\caption{The comparison on zero-shot code-to-code search. Baselines' results are reported by \citet{guo2022unixcoder}.}
\label{result_zero_shot_code_to_code_search}
\vspace{-10pt}
\end{table*}
We also evaluate SCodeR in zero-shot code-to-code search. Given a code snippet as query, the task aims to find semantically similar codes from a collection of code candidates in zero-shot setting. Since the annotation of code-to-code search is labor-intensive and costly~\citep{bigclonebench,code_retriever}, the zero-shot performance can indicate the model's utility in real-world scenario, where a lot of programming languages do not have an annotated dataset for code-to-code search.
We follow \citet{guo2022unixcoder} to conduct the experiment on CodeNet~\citep{codenet} and evaluate models using MAP score.
The results are listed in Table~\ref{result_zero_shot_code_to_code_search}. The first and the second row correspond to query and target programming languages.
We can see that SCodeR outperforms all other compared models and improves over the state-of-the-art model, i.e. UniXcoder, by 3.12 average absolute points. Meanwhile, SCodeR has a consistent improvement on the cross-PL setting, which can help users to translate programs from one PL to another via retrieving semantically relevant codes.
\begin{table}[t]
\centering
\small
\begin{tabular}{@{}lc@{}}
\toprule
Model & Kendall's Tau \\ \midrule
CodeBERT~\citep{feng2020codebert} & 81.9 \\
GraphCodeBERT~\citep{guo2020graphcodebert}\hspace{-15pt} & 84.7 \\
PLBART~\citep{ahmad2021unified} & 84.7 \\
CodeT5~\citep{wang2021codet5} & 84.7 \\
UniXcoder~\citep{guo2022unixcoder} & 85.9 \\
SCodeR & \textbf{86.6} \\ \bottomrule
\end{tabular}
\caption{Experiment results on markdown ordering in python notebooks.}
\label{table:result_notebook_reorder}
\end{table}
\begin{table*}[h]
\begin{center}
\begin{small}
\begin{tabular}{lccccccc}
\toprule
Methods & Ruby & Javascript & Go & Python & Java & Php & Overall\\
\midrule
Baseline &74.0 & 68.4 &91.5 &72.0 & 72.6 & 67.6 & 74.4 \\
\hdashline
Baseline + Code Transformation & 74.5 & 68.7 & 91.9 & 72.2 & 72.6 & 67.7 & 74.6 \\
Baseline + ASST & 76.1 & 70.1 & 92.1 & 73.0 &73.3 &68.1&75.4 \\
\hdashline
Baseline + ASST + Code Comment & 76.2 &71.2 &92.2&73.4&73.7&68.5&75.9\\
Baseline + ASST + Code Comment + Soft-Labled & \textbf{77.5}&\textbf{72.0}&\textbf{92.7}&\textbf{74.2}&\textbf{74.9}&\textbf{69.2}&\textbf{76.8}\\
\bottomrule
\end{tabular}
\caption{Ablation study on natural language code search.}
\label{ablation_stdudy_code_search}
\vspace{-10pt}
\end{small}
\end{center}
\end{table*}
\subsection{Markdown Ordering in Python Notebooks}
This task is to reconstruct the order of markdown cells in a given notebook according to the ordered code cells. We conduct experiments on the dataset provided by Kaggle\footnote{https://www.kaggle.com/competitions/AI4Code/overview} and use the official evaluation metric, \textit{Kendall's tau ($\tau$)}. It is computed as $1-2*N/\tbinom{n}{r}$ where $N$ is the number of pairs
in the predicted sequence with incorrect relative order and $n$ is the sequence length.
We take the normalized markdown cell's position in a given notebook as labels for each markdown cell (0$\sim$1), and solve this task as a regression task. To test performance of function-level code representation, we use pre-trained models to encode each cell to function-level representation as features. We use a randomly initialized Transformer that takes extracted features of cells in the python notebook to predict position of each cell. Note that parameters of pre-trained models are fixed in the fine-tuning procedure, and thus the performance of this task depends on function-level feature extracted from pre-trained models.
We show the results in Table~\ref{table:result_notebook_reorder}. SCodeR outperforms other pre-trained models and achieves 0.5 points higher than UniXcoder. This indicates that SCodeR learns better representation for both code and natural language comments, and can help better understand the fine-grained relationship of codes and comments in the python notebook.
\subsection{Analysis}
\paragraph{Ablation Study}
To evaluate the effect of our positive sample construction methods and soft-labeled contrastive pre-training framework, we conduct ablation study on the CSN dataset and take the pre-trained model with no enhancement as the baseline (i.e. UniXcoder). At first, we individually compare the proposed ASST with the transformation-based positive sample construction method~\citep{contracode,bui2021self}.
Notice that previous works do not apply their transformation-based methods on all six programming languages covered by our pre-training corpus. For fair comparison and keeping the pre-training corpus consistent, we follow~\citet{reacc} to implement the widely used transformations including variable renaming and dead code insertion on six programming languages by ourselves.
Then, we add the remaining modules of SCodeR to evaluate their performance. The results are shown in Table~\ref{ablation_stdudy_code_search}.
Compared with transformation-based methods, we can see that our positive sample construction (ASST) achieves better performance.
Meanwhile, positive pairs from ASST can bring significant improvement over the baseline, which reflects its effectiveness. After using the text-code pairs, the performance improves over 0.5 points, which shows that code comments provide rich semantic information to help model learn better code representation. When adding soft-labeled contrastive pre-training, the model performance increases by 0.9 points, which demonstrates that applying relevance among samples as soft-labels for contrastive learning can further improve code representation.
\paragraph{Effect of AST-based Splitting}
\begin{table}[h]
\centering
\small
\begin{tabular}{@{}lcccc@{}}
\toprule
& \textbf{Ruby} & \textbf{Python} & \textbf{Java} & \textbf{Overall} \\ \midrule
SCodeR & \textbf{33.87} & \textbf{33.02} &\textbf{18.87} &\textbf{28.6}\\
SCodeR$_T$ & 32.97 & 30.78 & 18.01 & 27.2 \\
SCodeR$_L$ & 33.22 & 31.31 & 18.23 & 27.6 \\ \bottomrule
\end{tabular}
\caption{Experiment results of different strategies of code splitting on zero-shot code-to-code search. SCodeR$_T$ and SCodeR$_L$ use token-level and line-level ICT to replace ASST.}
\label{table:different_code_splitting}
\end{table}
We conduct experiments on zero-shot code-to-code search to analyze the effect of AST-based splitting strategy of ASST by comparing ASST with two variants of splitting strategy. The first strategy is token-level ICT that takes a random span of code tokens and their context as positive pairs. The second strategy is line-level ICT that considers random consecutive code lines and the remaining lines as positive pairs. Compared with our AST-based splitting method, these two splitting strategies will cause ungrammatical codes and mislead the model to focus on structural matching rather than semantic matching. The results are shown in Table~\ref{table:different_code_splitting} and we can see that the two variants of splitting strategy lead to worse performance, which shows the effectiveness of our AST-based splitting method.
\paragraph{Case Study}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{images/case_study_8.pdf}
\caption{Case study on the discriminator. The soft-label is the relevance scores between the comment and codes from the discriminator, $p_{\phi}(\cdot |x,\mathbb{X})$.}
\label{fig:case_study}
\vskip -0.2in
\end{figure}
We give a case study in Figure~\ref{fig:case_study} to show the importance of soft-labels for contrastive pre-training. The figure includes one paired code and two other codes with soft-labels provided by the discriminators. We can see that the soft-label of negative $\text{Code}^-_1$ is close to 0 since the code is unrelated with the comment and the discriminators can predict correct relevance score between them for contrastive pre-training.
$\text{Code}^-_2$ is a false negative that has the same functionality as $\text{Code}^+$ and should be assigned similar weights when we apply contrastive pre-training. As we can see in the figure, the discriminator can understand code semantics and provide similar soft-labels (i.e 0.5284 vs 0.4713) about $\text{Code}^+$ and $\text{Code}^-_2$ for contrastive pre-training, which can alleviate the influence of false negative issue and learn better code representation through soft-labels.
\section{Conclusion}
In this paper, we present SCodeR to learn function-level code representation with soft-labeled contrastive pre-training. To alleviate the "false negative" issue in code corpus, we propose a soft-labeled contrastive pre-training framework that takes relevance scores among samples as soft-labels for contrastive pre-training in an iterative adversarial manner. Besides, we propose to utilize code comment and abstract syntax sub-tree of the source code to build positive samples that can facilitate the model to capture semantic information from the source code. Experimental results show that SCodeR achieves state-of-the-art performance on four code-related tasks over seven datasets. Further ablation studies show the effectiveness of our soft-labeled contrastive pre-training framework and positive sample construction methods.
\section*{Limitations}
There are two limitations of this work:
1) In the adversarial iteration, we introduce discriminators to provide soft-labels for the training of dual-encoder, which increases GPU memory occupation. To solve it, we can obtain these soft-labels offline, which may complicate the pipeline of data processing.
2) We only use UniXcoder as the backbone model in the experiments due to the computation resources limitation.
We leave pre-training based on other code pre-trained models like CodeBERT~\citep{feng2020codebert}, GraphCodeBERT~\citep{guo2020graphcodebert} and Codex~\citep{codex} as future work.
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{"url":"https:\/\/physics.stackexchange.com\/questions\/162409\/wavelength-of-x-rays-in-a-tube","text":"# Wavelength of X-rays in a tube\n\nIf I increase the accelerating potential in an X-ray tube the wavelength of the characteristic X-rays do not change.\n\nIs this because when an electron beam strikes the target in an X-ray tube, part of its kinetic energy is converted into X-ray energy?\n\n\u2022 Actually, the wavelength of xrays produced is a function of the energy electrons have when hitting the anode. Perhaps you are seeing secondary re-emissions? \u2013\u00a0Olin Lathrop Jan 30 '15 at 14:48\n\nSo changing the energy of the incident photons may well change the intensity of the fluorescence peak, but it will not change the wavelength. Typically a commercial X-ray tube, e.g. for use in a diffractometer, will have the electron energy tuned to maximise the intensity of the required fluorescence peak (usually the $K^\\alpha$.","date":"2020-02-22 23:59:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.7555258274078369, \"perplexity\": 756.8270514551971}, \"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-2020-10\/segments\/1581875145729.69\/warc\/CC-MAIN-20200222211056-20200223001056-00486.warc.gz\"}"}
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{"url":"https:\/\/chemistry.stackexchange.com\/questions\/133731\/type-of-structural-isomerism","text":"# Type of structural isomerism\n\nWhat is the type of structural isomerism in which isotopes move around?\n\nExamples\n\n1. $$\\ce{CH2D-NH2}$$ and $$\\ce{CH3-NHD}$$\n2. $$\\ce{CH2D-NH-CH3}$$ and $$\\ce{CH3-ND-CH3}$$\n3. $$\\ce{CH2D-NH-CH2CH3}$$ and $$\\ce{CH3-NH-CHD-CH3}$$ and $$\\ce{CH3-NH-CH2-CH2D}$$\n\nhere $$\\ce{D}$$ refers to deutrium.\n\nMy progress: So I know that there are various types of isomerism, like\n\n1. Tautomerism - Differ in the position of protons and electrons. Usually I see that this occurs with changes in bond orders. So I don't think it will be this.\n2. Ring-chain isomerism - When there is a difference in the number of carbons in a ring\n3. Functional isomerism - When the functional group changes. I don't know if $$\\ce{-NH2}$$ is a different functional group from $$\\ce{-NHD}$$.\n4. Metamerism - This is when there is a change in carbon chain on both sides of a bridging group. I don't know if a change in isotopes would count as a change in carbon chain.\n5. Chain isomerism - When the carbon skeleton changes. Here the skeleton carbon is the same.\n6. Positional isomerism - When a functional group attaches itself on different carbon of the same chain. Not applicable here.\n\nThis is not a homework question. This question was to fill the gap in my knowledge of isomerism with isotopes.\n\nIsotopomers or isotopic isomers are isomers with isotopic atoms, having the same number of each isotope of each element but differing in their positions. The result is that the molecules are either constitutional isomers or stereoisomers solely based on isotopic location. For example, $$\\ce{CH3CHDCH3}$$ and $$\\ce{CH3CH2CH2D}$$ are a pair of constitutional isotopomers of propane","date":"2022-01-17 23:41:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 12, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6041662096977234, \"perplexity\": 856.9107671743967}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320300624.10\/warc\/CC-MAIN-20220117212242-20220118002242-00319.warc.gz\"}"}
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Q: Purpose of IP addresses on tun interface? --ifconfig option, which configures local and peer IP address for point-to-point tunnel interface, is a mandatory configuration option for OpenVPN. However, what is the purpose of local and peer IP addresses? Only purpose I can think of is that one can use those IP addresses to route traffic to tunnel. For example ip route add 10.10.3.0/24 via 192.168.1.2 dev tun0 where 192.168.1.2 is the peer IP address. However, for point-to-point links, one should be able to route traffic directly onto interface, e.g. ip route add 10.10.3.0/24 dev tun0.
A: Historically in IPv4, unnumbered interfaces were not possible. The only possible way to configured a point-to-point interface was with a local address and a remote address. The only way to route some other IP address through the point-to-point interface in question was to install a route using the interface's remote address as the gateway (there was no option in the route command or in the kernel to specify the destination by interface name). In those days, you'd see oodles and oodles of precious IP addresses consumed as interface addresses on SLIP connections.
Even now, an IPv4 interface requires at least some kind of address configured on it if only to serve as the source address for ICMP errors sourced from that interface (and that's how traceroute detects which interface on a router was traversed).
But what you can do now (and couldn't do historically) is make the interface unnumbered. That means giving it the same IP address as some other interface on the system. Usually a stable always-up interface (such as a loopback interface or the router's "main" interface) serves as the donor for the IP address. As for the remote IP address, you do not have to supply one at all (but see next paragraph). Without a remote IP address specified, you can use ip route ... dev <interface> instead of ip route ... via <address> to direct traffic into the interface, as you know. All in all, this can lead to a tremendous savings in IP addresses. (As an aside, it's interesting to note that while in UNIX an unnumbered interface is implicit in the fact that the local IP address is not unique, on other operating systems such as Cisco's IOS, it's quite explicit: the command is ip unnumbered <donor-interface-name>.
Notwithstanding this, it appears that in OpenVPN, specifying the remote IP address is not optional. However, what you can do is specify a somewhat arbitrary remote address. For example, if the other end of the tunnel is another router, you can use the remote router's loopback or "main" IP address (basically reflecting an unnumbered configuration at the remote end), thus consuming no IP addresses for the tunnel. If you are willing to make the routing table look a little bit ugly, you can actually use a completely bogus IP address for the ifconfig remote address; everything will work fine and the bogus address will never even appear anywhere (even in traceroutes).
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A digital version of Deepak Chopra is coming to your phone as a chatbot
Digital Deepak Chopra is an AI bot that will give you advice.
The AI Foundation
A digital version of Dr. Deepak Chopra will soon be able to give you advice whenever you need it through an app on your phone.
Chopra, the famous writer and teacher, has partnered with San Francisco-based software company The AI Foundation to create an advanced, personalized artificial intelligence adviser called "Digital Deepak" that will launch on phones in early 2020.
"My hope is to reach a critical mass —at least 1 billion people," Dr. Chopra said. "The AI will live longer than me and be able to speak to my grandkids', grandkids', grandkids."
A preview of the Digital Deepak app was presented last month in a first-ever public demonstration on stage at Dr. Chopra's annual Sages & Scientists Symposium, and CNBC had a chance to try it out.
"At the AI Foundation, we are driven by the idea that AI has incredible potential to benefit humanity," said Dr. Lars Buttler, CEO of The AI Foundation. "Digital Deepak exemplifies our mission to promote positive global uses of AI. [Digital Deepak] is a digital representation of an actual human being in your pocket."
Here's what it's like.
Digital Deepak
The Digital Deepak app's interface operates similar to a video call, with the user on camera in the top right corner of the device. After it greets the new user, it informs them that it will serve as their "infinite well-being guide."
It tracks facial expressions and tone of voice to answer questions, share insights and lead users through one-on-one meditative exercises geared toward improving various aspects of the user's life, from financial wellness to fitness and health.
There are still some bugs, however. Response times are still slightly delayed, which caused Digital Deepak to sometimes talk over me.
How 'Digital Deepak' learns
Digital Deepak is trained with information exclusively from books, letters, speeches and other content Dr. Chopra provides the AI Foundation. If it is asked something he should know, but does not yet understand it will respond, "Thank you for making me smarter, let me get back to you."
Digital Deepak also learns about the user through regular use. The AI develops a more personal relationship with you over time and will eventually greet you by name or talk about previous discussions.
However, the AI only learns new capabilities directly from the real Dr. Chopra. If the AI is asked to do something it is not meant for, like discussing politics or religion, it will simply respond, "Let's stay on course."
"Ultimately, the AI is trained to stick to its guns," said Dr. Buttler. "It does not learn directly from the audience."
This article was originally published by Cnbc.com. Read the original article here.
A tiny wage increase could have prevented 13,800 deaths in 6 years
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Home » National University Selects EBSCO Discovery...
National University Selects EBSCO Discovery Service™
posted by kmcevoy on Tue, 02/08/2011 - 12:13
~Improved Access to Full-Text Resources, a Simple Search Interface with Advanced Options and Mobile Access—National University Selects EBSCO Discovery Service~
IPSWICH, Mass. — February 8, 2011 — National University, the second-largest private, nonprofit institution of higher education in California and the flagship university of the National University System, has selected EBSCO Discovery Service™ (EDS) from EBSCO Publishing to provide the University with a way to deliver its large collection to the end user and to make its library resources easier to find.
National University researched a variety of options from next generation catalogs to resource portals as well as discovery services—more than 11 products in all—as it sought to improve access to the library's full-text resources. National University was looking to find an academic library resource that had a simple search interface but still had advanced options to limit search results.
National University's librarians wanted a service that would provide users with more access to the library collection to better support research projects and assignments. Students were asking for a simple search experience that was closer to a traditional Web search. In deciding on EBSCO Discovery Service, the university was able to leverage its large eBook collection (more than 135,000 titles) and search the full-text of its collection alongside research databases and other library resources.
Senior Analyst for Research and Electronic Resources at the National University Library, Betty Kellogg, says EBSCO Discovery Service proved itself in side-by-side trials with other services. "In our trials with test databases we received more full-text relevant results with EBSCO Discovery Service than other products."
While relevancy was an important differentiator, the strong metadata available within EDS, its ease of access and the customizability were also important factors according to Kellogg. "We liked that resources such as JSTOR and LexisNexis Academic could be searched through EBSCO Discovery Service, reducing our silos of separate information and that EDS was available in a mobile version. The option to create links on the search page to preselected databases from any vendor or subject collections to help students focus their results was also appealing."
EDS will be part of a new library website design and a key component of a library initiative to serve students accessing the library via mobile devises. Kellogg says EDS will allow students to search the library without needing to know the names and descriptions of all the databases in the collection. The library is anticipating an increase in database usage since students will be searching across all resources and returning more relevant results from underused or previously hard to find resources.
EBSCO Discovery Service creates a unified, customized index of an institution's information resources, and an easy, yet powerful means of accessing all of that content from a single search box—searching made even more powerful because of the quality of metadata and depth and breadth of coverage.
The Base Index for EBSCO Discovery Service forms the foundation upon which each EDS subscribing library builds out its custom collection. Beginning with the Base Index, each institution extends the reach of EDS by adding appropriate resources including its catalog, institutional repositories, EBSCOhost and other databases, and additional content sources to which it subscribes. It is this combination that allows a single, comprehensive, custom solution for discovering the value of any library's collection.
The EDS Base Index is comprised of metadata from the world's foremost information providers. At present, the EDS Base Index represents content from approximately 20,000 providers in addition to metadata from another 70,000 book publishers. Although constantly growing, today the EDS Base Index provides metadata for nearly 50,000 magazines & journals, approximately 825,000 CDs & DVDs, nearly six million books, more than 100 million newspaper articles, more than 400,000 conference proceedings and hundreds of thousands of additional information sources from various source-types.
About EBSCO Publishing
EBSCO Publishing is the world's premier database aggregator, offering a suite of more than 300 full-text and secondary research databases. Through a library of tens of thousands of full-text journals, magazines, books, monographs, reports and various other publication types from renowned publishers, EBSCO serves the content needs of all researchers (Academic, Medical, K-12, Public Library, Corporate, Government, etc.). The company's product lines include proprietary databases such as Academic Search™, Business Source®, CINAHL®, DynaMed™, Literary Reference Center™, MasterFILE™, NoveList®, SocINDEX™ and SPORTDiscus™ as well as dozens of leading licensed databases such as ATLA Religion Database™, EconLit, Inspec®, MEDLINE®, MLA International Bibliography, The Philosopher's Index™, PsycARTICLES® and PsycINFO®. Databases are powered by EBSCOhost®, the most-used for-fee electronic resource in libraries around the world. EBSCO is the provider of EBSCO Discovery Service™ a core collection of locally-indexed metadata creating a unified index of an institution's resources within a single, customizable search point providing everything the researcher needs in one place—fast, simple access to the library's full text content, deeper indexing and more full-text searching of more journals and magazines than any other discovery service (www.ebscohost.com/discovery). For more information, visit the EBSCO Publishing Web site at: www.ebscohost.com, or contact: [email protected].
EBSCO Publishing is a division of EBSCO Industries Inc., one of the largest privately held companies in the United States.
Kathleen McEvoy
(800) 653-2726 ext. 2594
EBSCO Publishing
discovery solutions
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Q: Developing functional thinking. I am taking this course on Scala, at first it seemed easy with trivial exercises but gradually its becoming more of a mathematical course which seems to require an experienced mathematical mind. Giving an example of one of the exercises:
Quoting from the exercise:
Representation
We will work with sets of integers.
As an example to motivate our representation, how would you represent
the set of all negative integers? You cannot list them all… one way
would be so say: if you give me an integer, I can tell you whether
it's in the set or not: for 3, I say 'no'; for -1, I say yes.
Mathematically, we call the function which takes an integer as
argument and which returns a boolean indicating whether the given
integer belongs to a set, the characteristic function of the set. For
example, we can characterize the set of negative integers by the
characteristic function (x: Int) => x < 0.
Therefore, we choose to represent a set by its characteristic function
and define a type alias for this representation:
type Set = Int => Boolean
Using this representation, we define a function that tests for the
presence of a value in a set:
def contains(s: Set, elem: Int): Boolean = s(elem)
Code:
type Set = Int => Boolean
def singletonSet(elem: Int): Set =
(elem: Int) => (elem == elem)
def contains(s: Set, elem: Int): Boolean =
s(elem)
def union(s1:Set, s2:Set): Set =
(elem) => contains(s1, elem) || contains(s2, elem)
def intersect(s1:Set, s2:Set): Set =
(elem) => contains(s1, elem) && contains(s2, elem)
def diff(s1:Set, s2:Set): Set =
(elem) => contains(s1, elem) && !contains(s2, elem)
def filter(s1:Set, p: Int => Boolean): Set =
(elem) => contains(s1, elem) && p(elem)
def forall(s: Set, p: Int => Boolean): Boolean = {
def inforall(lo: Int): Boolean = {
if (lo > 1000) true
else if (contains(s, lo))
p(lo) && inforall(lo + 1)
else
inforall(lo + 1)
}
inforall(-1000) // [-1000, 1000]
}
def negate(p: Int => Boolean): Int => Boolean =
(x) => !p(x)
// A predicate is true for some elements of the set but
// its not false for all of them.
def exists(s: Set, p: Int => Boolean): Boolean =
// negative predicate should not be true forall elems
forall(s, negate(p)) != true
def map(s: Set, f: Int => Int): Set =
(y: Int) => exists(s, x => f(x) == y)
Having said this much, I need to commit that I couldn't solve the exists and map and I took help from Google, but I know the issues I faced while solving the above exercise:
*
*Functions like union, intersect felt like magic I was able to write them without thinking much but when I tried to understand them my head started spinning, it felt like a paradox when you can solve the problem without thinking less about it.
*Functions like exists and map felt like a non-trivial task to me even after seeing their solution I doubt that I could have thought like that.
My question is how to develop thinking habit to solve problems like these what are the sources that could help me? Is there any mental model that could help me visualize the solution or thinking process, or at least what not to do while solving problem in this domain?
A: I'm not very good at coding, and I have been trying to get better. While watching videos and such is helpful, it is not always helpful when I am trying to solve a problem. I have found that Project Euler has helped me learn how to solve a problem (and when I have a problem with my code, I google python syntax or ask a question and keep going). I also try to solve problems on here that I think I could take a crack at with code. The more problems I solve, the more commands I have enough of a command of to think of using on the next problem, and so I'm able to try harder problems.
The other thing that has been important for me is the way of thinking about the problem. I try to start by making sure I understand the problem, and then I try to define the constants (variables) and equations I'll need. Then I try to work on the method I want to use to get the result, and then, if necessary, I'll make it look pretty. I guess that's my "mental model".
Hope this helps!
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Kontospora är ett släkte av svampar. Kontospora ingår i divisionen sporsäcksvampar och riket svampar.
Källor
Sporsäcksvampar
Kontospora
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Arla Foods UK Delivers 2.3% First Half Revenue Growth
Despite globally challenging market conditions including the impact of a continued weak pound and changes in fat and protein prices, Arla Foods UK delivered a steady start to the first half of the year. A 6.7 per cent volume driven branded growth contributed to an overall net revenue growth of 2.3 per cent, comparative to the first half of 2017, increasing revenue to £961m1 (€1.093bn).
The 6.7 per cent strategic branded sales volume for Arla Foods UK was supported by the core Arla® brand at 9.32 per cent growth, as well as Lurpak®at 2.22 per cent growth and Castello® at 8.82 per cent growth in the UK. Within the Arla portfolio, Arla's foodservice range Arla Pro grew by 52.82 per cent whilst sales of B.O.B (Best of Both) grew at 38.12 per cent.
Ash Amirahmadi.
Ash Amirahmadi, managing director of Arla Foods UK, comments, "Year on year, Arla's portfolio continues to lead the way in driving growth across the UK dairy sector. Our decision to invest to build capability in the growing channels of foodservice, online and convenience is paying off and will continue to do so as we further develop the UK business."
At a Global level across the farmer owned co-operative, total Arla Group revenue grew by 2.2 per cent to €5.1 billion, supported by higher strategic branded sales volumes, which were up 3.0 per cent due to strong performance by the Arla® brand, Lurpak®, Puck® and Castello®.
Each year, Arla targets an annual net profit share in the range of 2.8 to 3.2 per cent of revenue. This allows the company to balance its retained capital for future investments and provide a supplementary payment to farmer owners, while continuing to pay out the highest possible share of our ongoing profit to our farmer owners via the prepaid milk price throughout the year.
In the first half year of 2018, Arla delivered a net profit share of 2.2 per cent, up from last half year's profit share of 2.1 per cent. The company expects to reach its full year net profit share to be within the 2.8 to 3.2 per cent target range.
Early in 2018, the three year transformation and efficiency programme Calcium was announced by Arla's management to reduce cost by over €400 million across the group. For the full-year of 2018, Arla expects Calcium to deliver a positive contribution of at least €50 million, up from the initially forecasted €30 million.
"It was a tough start to 2018 as we took urgent action to repair our bottom line. This urgency delivered a positive result as we were able to improve our profitability and the performance of our milk price over the period. However, there is more work to be done as we continue to relentlessly execute our transformation programme, Calcium, which will further improve our performance," says Peder Tuborgh, chief executive of Arla Foods.
"I'm pleased to say that Calcium is starting to deliver – every week we see steady progress across the programme. Our top management, leaders and employees are working adamantly on each initiative that will enable us to pay a more competitive milk price to our farmers, compete more effectively in the markets and categories we operate in worldwide, and boost the strategic investments that will sustain our long term profitable growth," says CFO in Arla Foods, Natalie Knight, chief financial officer of Arla Foods.
Of the €400 million Calcium is expected to deliver by 2021, Arla aims to direct €300 million to its farmer owners via the prepaid milk price while reinvesting €100 million into areas that fuel growth.
Full Year Expectations
Jan Toft Nørgaard, chairman of Arla Foods.
For the full year, Arla will remain focused on balancing branded growth in a higher retail price environment. Continued execution of the transformation programme Calcium will be the company's other key priority. Arla's revenue outlook for the full year 2018 is expected to be €10.0-10.5 billion and Arla's branded growth is expected to be at the high-end of the previously communicated target range of 1.0-3.5 per cent.
Net profit share for 2018 is expected in the targeted 2.8-3.2 range.
A proposal to pay out the entire 2018 net profit of Arla Group has been prepared by Arla's board due to the positive development of Arla Foods' balance sheet. The board also recognizes that many Arla farmers are facing a tough financial situation due to this summer's drought in Europe, and that it is in Arla's best interest for this year's net profit to be paid out to the farmers.
"As a farmer-owned dairy company, we care deeply about the livelihood of our farmers and we recognize that this summer's drought in Europe has been extraordinary. We are proposing that extraordinary measures be taken in this situation, and the board is satisfied with the positive development of the company's balance sheet, which makes this proposal possible." says Jan Toft Nørgaard, chairman of Arla Foods.
Lidl and Liffey Meats Will See Bord Bia Quality Approved Irish Beef Exported to 17 Countries Around Europe and to the US
AG Barr Continues to Build For Profitable Long-term Growth
Bright Food to Acquire 60% of Weetabix
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Q: How to create cookies for login name and password in mvc3 I want to store login name and password in cookies when click on a "remember me" checkbox. My login.cshtml looks like input type="text" , id="loginName" and input type="password" id = "password". How can I do this?
A: Try this
// create a cookie
HttpCookie loginName = new HttpCookie("loginName", model.LoginName);
loginName.Expires = DateTime.Today.AddMonths(12);
Response.Cookies.Add(loginName);
HttpCookie password= new HttpCookie("password", model.Password);
password.Expires = DateTime.Today.AddMonths(12);
Response.Cookies.Add(password);
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{
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Superhuman Flies Sky-High, Powered By Brand Positioning
Superhuman is red-hot.
The buzz surrounding Superman has much to do with its approach to branding and marketing.
First, let's take a take back.
Superhuman is an email app built on top of Gmail that includes some interesting features.
So why the buzz?
Two words: brand positioning.
First, it touts itself as "The Fastest Email Experience Ever"
But there's more: Superhuman proclaims that has "rebuilt the inbox from the ground up to make you brilliant at what you do. We specifically designed it for those of you who want the best."
Within the boring and utilitarian email world, Superhuman stands out. It's cool, seemingly different…..and not for everyone.
To use Superman, you need to request access and then pass the audition. It's a brilliant tactic: exclusivity makes Superhuman that much more intriguing.
It explains why Superhuman charges $30/month when most of its competitors (e.g. Gmail, Outlook, Yahoo) are free.
There are clearly people who are happy to pay for a well-designed email app that is worth $360/year.
That's the power of brand positioning.
Superhuman has differentiated itself within a competitive landscape. That's impressive.
Of course, Superhuman's ability to select its users and charge $30/month is supported by raising $33-million in venture capital.
Having lots of capital makes it easy to experiment with a new concept. If I was a betting man, however, it would not be surprising to see Superhuman launch a "lite" service to attract new customers.
For more, check out this review on The Verge.
Marketing Spark is hosted by Mark Evans. the principal with ME Consulting, which works with fast-growing B2B companies to do marketing better or differently.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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F96hpr0Iz+NFpArqtd68M32YURqC9glH/RWibYzsxS+JZTsKGvQ411Upgcq7yAWGR2eupOk1fqvLITn3Wn86FM4Xq4Iaefa8gnBY80D7kYscH5uxG4nNTUZp/G6tct+R0qwZtZYVSfAfV8wC/U8Tlx0ZIepUEPTVMhy7/j4Tceg=
Exelon Executive Wins Lifetime Achievement Award for Contributions to the Energy Industry
Elizabeth "Betsy" Moler is first woman to receive this honor
NEW YORK - Elizabeth "Betsy" Moler, recently retired executive vice president of government affairs and public policy at Exelon, received the Platts Lifetime Achievement Award at the 12th Annual Platts Global Energy Awards yesterday in New York. The first woman to win the award, Moler was recognized for her contributions as a preeminent voice on energy policy throughout her 40-year career in Washington, D.C.
"I truly appreciate the decade of leadership Betsy provided to Exelon," said Exelon chairman and CEO John W. Rowe. "Her contributions, both to Exelon and the energy industry, have been invaluable. She is truly deserving of this honor."
In her 10 years at Exelon, Moler served as a vital resource to public officials concerned about energy and environmental policy, testifying before Congress and FERC numerous times and working closely with the utility and NGO communities in support of energy and climate legislation.
"Ms. Moler is recognized for her outstanding contributions to the development and future of the energy industry over the course of her long and varied career," said Larry Neal, president of Platts, a leading global energy and metals information provider and host of the awards program. "Based on her tireless work to shape effective, sensible energy policy both on and off Capitol Hill, our panel of judges agreed that Ms. Moler exemplifies the best our industry has to offer in strategic vision, leadership and innovation."
Before joining Exelon, Moler served under former President Clinton as deputy secretary of the U.S. Department of Energy and was the principal architect of the Clinton Administration's Comprehensive Electricity Competition Act, presented to Congress in June 1998. Prior to that, Moler served as a commissioner for the Federal Energy Regulatory Commission, for which former President Reagan nominated her in 1988. Former Presidents George H.W. Bush and Clinton reappointed Moler, making her the only commission member appointed by three different presidents. Clinton designated her to serve as the commission's chair in 1993.
Moler is credited with the strategic vision and ultimate implementation of competitive electricity markets as the best framework for meeting important economic and environmental policy objectives. As the chair of FERC, Moler led the effort that resulted in the successful restructuring of both the interstate natural gas industry and the wholesale electricity industry. Her work ushered in a new era of organized competitive markets, which has aided the transition to a lower-carbon economy, promoted energy efficiency and demand response, spurred growth in energy supply and generated significant customer savings.
In addition, Moler has been a vocal advocate for encouraging women to take leadership roles in the energy industry. She is an active supporter of the Women's Council on Energy and the Environment, which honored her with its Woman of the Year Award in 1996 and 1998.
Moler started her career as a staff member for 20 years on Capitol Hill, first as a staff assistant in the office of Senator Mike Gravel of Alaska, and later on the staff of the Senate Energy and Natural Resources Committee as counsel for both Chairman Henry M. "Scoop" Jackson of Washington state and Chairman J. Bennett Johnston of Louisiana. During that time, she was the principal staff member responsible for all natural gas issues and helped craft the Natural Gas Policy Act of 1978.
Nominees for the Lifetime Achievement Award are evaluated in the areas of integrity, achievements, strategic vision, leadership, innovation and peer recognition. Winners are selected by an international judging panel of past industry executives, academics, former energy regulators and efficiency experts.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
The structure of complex networks highly affect the critical behavior
of different cooperative models \cite{Dorogovtsev} and the nonlinear
dynamical process that take place on the network \cite{Motter_rev}.
In particular both the directionality of the links which suggest a non
symmetric interaction \cite{Grad,Synchr,Galla} and the local loop
structure \cite{Klemm} of the network which correlates neighboring nodes has important dynamical consequences. In fact directionality of links
becomes particularly important when a transport process of mass or information
takes place in the network \cite{Grad} and the loop structure in these directed networks are
crucial for assessing the networks' robustness characteristics and determining
the load distribution.
Directed networks are ubiquitous in both man-made and natural systems. Some
examples of directed networks are the Texas power-grid, the World-Wide-Web, the
foodwebs and in biological networks, such as the metabolic network, the
transcription network and the neural network.
The local structure of directed network is radically different from
the structure of their undirected version \cite{last}.While many
undirected networks are characterized but large clustering
coefficient \cite{swn} and large number of short loops \cite{Loops,loop_lungo} this is not a general trend for directed networks.
For example the $C. elegans$ neural network has a
over-representation of short loops compared to a randomized network if
the direction of the links is not considered while it has an
under-representation of the number of loops when the
direction of the links is taken into account \cite{last}.
Nevertheless, while counting small loops is a given network is a
relatively easy computation, counting large loops in a real world
network is a very hard task. In fact the
number of large loops can, and usually does grow exponentially with the number
of nodes $N$ in the network. The known efficient exhaustive
algorithms \cite{Johnson,Tarjan} for counting loops still have a time bound of
$O(N*M*(L+1))$ where $N,M,L$ are respectively the number of nodes, links and
loops in the network. This task becomes computationally inapplicable for
counting large loops in many real networks. Two different approaches for the
study of long loops have been proposed: devising MonteCarlo algorithms, or
using Belief-Propagation (BP) algorithms. The two approaches have both been
pursued in the case of undirected networks \cite{Loopy,Algorithm,Circuits}. The BP
algorithm~\cite{Algorithm} is a heuristic algorithm which does not have sampling
bias as the MonteCarlo algorithm~\cite{Loopy} does and is observed to give
good results as the size of the network increases.
In this paper we generalize the BP algorithm proposed by
\cite{Algorithm,Circuits} to directed networks. We analytically derive the
outcome of the algorithm in an ensemble of random uncorrelated networks with
given degree sequence of in/out degrees in agreement with the prediction for
the average number of nodes in this ensemble \cite{last}. We finally study the
particular limitations of the algorithm for small network sizes and small
number of loops in the graph. The paper is divided into four further sections.
In Section 2, we derive the BP algorithm for directed networks following the
similar steps as described in \cite{Circuits}. In Section 3, we derive the
distribution of the small loops in uncorrelated random ensembles. In Sections 4
and 5, we describe the steps in the algorithm and its application to a few
characteristic directed networks.
\section{Derivation of the BP algorithm}
Given a network G of $N$ nodes and $M$ links,
we define a partition function $Z(u)$ as the generating functions of
the number ${\cal N}_L$ of loops of length $L$ in the network,
\begin{equation}
Z(u)=\sum_Lu^L {\cal N}_L(G).
\end{equation}
Starting with this partition function, we can define a free energy $f(u)$ and an
entropy $\sigma(\ell)$ of the loops of length $L=N\ell$$\sigma(\ell)$ as
the following:
\begin{eqnarray}
f(u)=\frac{1}{N}\ln Z(u)\nonumber \\
\sigma(\ell)=\frac{1}{N}\ln{\cal N}_{L=\ell N}.
\end{eqnarray}
For each directed link in the network, $l=\avg{ij}$ from node $i$ to node $j$,
if we define a variable $S_l = 0, 1$ which indicates if a given loop passes
through the link $l$, the partition function $Z(u)$ can then be written as
\begin{equation}
Z(u)=\sum_{\{S_l\}}w(\{S_l\})u^{\sum_{l=1}^M S_l},
\label{Z1}
\end{equation}
where $w(\{S_l\})$ is an indicator function of the loops, i.e. it is 1 if
the variables $S_l=1$ have a support which form a closed loop, and it is zero
otherwise. As in References~\cite{Algorithm,Circuits} we take for simplicity a relaxed local
form of the indicator function $w(\{S_l\})$ which is 1 also if the assignment
of the link variables $S_l$ is compatible with a few disconnected loops.
In particular we take $w(\{S_l\})$ as
\begin{equation}
w(\{S_l\})=\prod_{i=1}^N w_i(\{S\}_i)
\end{equation}
where $\{S\}_i=\{S_{\avg{ij}}\}_{j\in \partial i}$, and $\partial i$ indicates
the set of nodes either pointing to $i$ or pointed by $i$ and where $w_i(\{S\}_i) $ is defined as
\begin{eqnarray}
w_i(\{S\}_i)=\left\{\begin{array}{lcr}1&\mbox{if}\ \sum_{j\in \partial_{+}i}S_{\avg{ij}}=1
\ \mbox{and}\ \sum_{j\in \partial_{-}i}S_{\avg{ij}}=1\nonumber\\
1&\mbox{if}\ \sum_{j\in \partial_{+}i}S_{\avg{ij}}=0\ \mbox{and}\
\sum_{j\in \partial_{-}i}S_{\avg{ij}}=0\nonumber \\
0&\mbox{otherwise}\end{array} \right.
\end{eqnarray}
with $\partial_+i$ and $\partial_{-}i$ indicating the set of nodes $j$ which
points to $i$ or which are pointed by $i$, respectively.
Finding the free energy $f(u)$ associated with the partition function
$(\ref{Z1})$ can be cast into finding normalized distributions
$p_v(\{S_l\})$ which minimize the Kullback distance
\begin{equation}
F_{Gibbs}[p_v]=\sum_{\{S_l\}}p_v(\{S_l\})\ln\left(\frac{p_v(\{S_l\})}{w(\{S_l\})u^{\sum_lS_l}}\right).
\end{equation}
In fact it is straightforward to show that $F_{Gibbs}$ assumes its minimal value
when $p_v(\{S_l\})=w(\{S_l\})u^{\sum_lS_l}/Z$. If the given network is a tree, the
trial distribution $p_v(\underline{S})$ takes the form
\begin{equation}
p(\{S_l\})=\left(\prod_{l} p_l(S_l)\right)^{-1}\left(\prod_i
p_i(\underline {S}_i)\right).
\label{tree}
\end{equation}
with $p_{l}(S_l)$ and $p_i(\{S\}_i)$ being the marginal distributions
\begin{eqnarray}
p_{\ell}(S_{\ell})&=&\sum_{\{S_l\} \backslash S_l}p(\{S_l\})
\nonumber \\
p_i(\{S\}_i)&=&\sum_{\{S\} \backslash \{S\}_i}p(\{S\}).
\end{eqnarray}
In a real case, when the network is not a tree, we can always take a variational approach and try a
given trial distribution of the form (\ref{tree}).
After taking this variational approach, we then have to minimize the Bethe free
energy $F_{Bethe}$ as
\begin{equation}
F_{Bethe}[\{p_i\},\{p_l\}]=\sum_i \sum_{\{S\}_i\backslash
S_l}p_i(\{S\}_i)\ln\left(\frac{p_i(\{S\}_i)}{w_i(\{S\}_i)}\right)-\sum_l\sum_{S_l}p_l(S_l)\ln(p_l(S_l)u^{S_l}).
\end{equation}
For each link $l\avg{ij}$ starting from $i$ and ending in $j$, there are the constraints
\begin{eqnarray}
p_l(S_l)&=&\sum_{\{S\}_i}p_i(\{S\}_i)\nonumber \\
p_l(S_l)&=& \sum_{\{S\}_j}p_j(\{S\}_j).
\label{constraint}
\end{eqnarray}
Introducing the Lagrangian multipliers enforcing the conditions
$(\ref{constraint})$ and the normalization of the probabilities it is
easy to show that a possible parametrization of the marginals
is the following,
\begin{eqnarray}
p_l(S_l)&=&\frac{1}{C_l}(u y_{i\rightarrow j} \hat{y}_{j\rightarrow i})^{{S}_l}\nonumber
\\
p_i(\{S\}_i)&=&\frac{1}{C_i}w_i(\{S\}_i)\prod_{j\in
\partial_{+}i}(u y_{i\rightarrow j})^{S_{<ij>}}\prod_{j\in
\partial_{-}i}(u \hat{y}_{i\rightarrow j})^{S_{<ij>}}.
\end{eqnarray}
For every directed link $\avg{ij}$ from node $i$ to node $j$
the values of the messages $y_{i\rightarrow j}$ and $\hat{y}_{j \rightarrow i}$ are
fixed by the constraints in Eq.~$(\ref{constraint})$ to satisfy the following
BP equations:
\begin{eqnarray}
y_{i\rightarrow j}=\frac{u\sum_{k\in \partial_{-} i} y_{k\rightarrow
i}}{1+u^2\sum_{k'\in\partial_{+}(i)\backslash j}\hat{y}_{k'\rightarrow i}\sum_{k\in
\partial_{-} i} y_{k\rightarrow i}}\nonumber \\
\hat{y}_{j\rightarrow i}=\frac{u\sum_{k\in \partial_{+} j} \hat{y}_{k\rightarrow
j}}{1+u^2\sum_{k'\in\partial_{-}j\backslash i}{y}_{k'\rightarrow j}\sum_{k\in
\partial_{+} i} \hat{y}_{k\rightarrow j}}.
\label{BP}
\end{eqnarray}
The normalization constants for the marginals is consequently given by
\begin{eqnarray}
C_l&=&1+u y_{i\rightarrow j}\hat{y}_{j\rightarrow i}.\nonumber \\
C_i&=&1+u^2\sum_{k'\in\partial_{-}i}{y}_{k'\rightarrow i}\sum_{k\in
\partial_{+} i} \hat{y}_{k\rightarrow i}.
\end{eqnarray}
The Bethe free energy density $f_{Bethe}=\frac{1}{N}F_{Bethe}$ becomes
\begin{equation}
Nf_{Bethe}(u)=-\sum_{l=1}^M\ln C_l+\sum_{i=1}^N\ln C_i.
\label{f}
\end{equation}
For any given value of $u$ the loops length is given by
\begin{equation}
\ell(u)=\frac{1}{N}\sum_{l=1}^Mp_l(1)=\frac{1}{N}\sum_l\frac{u y_{i\rightarrow
j}\hat{y}_{j\rightarrow i}}{1+u y_{i\rightarrow j}\hat{y}_{j\rightarrow i}}.
\label{ell}
\end{equation}
The function $\ell(u)$ can be inverted giving the function $u(\ell)$
and finally proving an expression for the entropy of the loops in the
graph under a Bethe variational approach,
\begin{equation}
\sigma_{Bethe}(\ell)=f(u(\ell))-\ell\ln u(\ell).
\label{sigma}
\end{equation}
\section{Derivation of the typical number of short loops in random directed network with given degree sequence}
We consider an ensemble of random directed networks with given
degree sequence $\{k^{in}_i,k^{out}_i\}~\forall i=1,\dots, N$. If the
maximal in/out connectivities $K^{in}$/$K^{out}$ of the network satisfy the inequality
$K^{in}K^{out}<(\avg{k_{in}})N$, the network is uncorrelated.
By $q_{k_{in},k_{out}}$ we indicated the degree distribution of the
ensemble.
In Ref. \cite{last} an expression for the average number ${\cal N}_L$
of small loops was given,
\begin{equation}
\avg{{\cal N}_L}\simeq \frac{1}{L}\left(\frac{\avg{k_{in}k_{out}}}{\avg{k_{in}}}\right)^L
\end{equation}
valid as long as
\begin{equation}
L\ll N\frac{\avg{k_{in}k_{out}}^2}{\avg{(k_{in}k_{out})^2}}.
\label{bound}
\end{equation}
Is an interesting exercise to see what is the distribution of the
number of small
loops in the ensemble of directed networks by solving the BP equation
for a random directed ensemble in parallel with the distribution found
in the undirected case \cite{Circuits}.
In a directed network ensemble the BP messages $y$
and $\hat{y}$ along each link are equally distributed depending only on the
value of $u$. Given the BP equations $(\ref{BP})$, the distribution $P(y;u)$ of the
field $y$ has to satisfy the self-consistent equation
\begin{eqnarray}
P(y;u)&=&
\sum_{k_{out}=1}^{\infty}\frac{k_{out}}{\avg{k_{out}}}q_{0,k_{out}}\delta(y)+\sum_{k_{in}=1}^{\infty}
\sum_{k_{out}=1}^{\infty}\frac{k_{out}}{\avg{k_{out}}}q_{k_{in},k_{out}}\nonumber \\
& &\int_{0}^{\infty} dy_1 P(y_1;u) \dots, dy_{k_{in}}P(y_{k_{in}};u)\nonumber \\
& & \int_0^{\infty}d\hat{y}_1 P(\hat{y}_1;u) \dots
d\hat{y}_{k_{out}}P(\hat{y}_{k_{out}};u)\delta(y-g_k(\{y\},\{\hat{y}\}))
\label{uno}
\end{eqnarray}
with
\begin{eqnarray}
g_1=u y_1\nonumber \\
g_k=\frac{u\sum_{k\in \partial_{-} i} y_{k\rightarrow
i}}{1+u^2\sum_{k'\in\partial_{+}(i)\backslash j}\hat{y}_{k'\rightarrow i}\sum_{k\in
\partial_{-} i} y_{k\rightarrow i}} \ \mbox{for}\ k\geq2.
\end{eqnarray}
In fact, given a random edge the probability that its starting node
$i$ has connectivity $(k_{out},k_{in})$ is given by
$\frac{k_{out}}{\avg{k_{out}}}q_{k_{in},k_{out}}$.
The fields $\hat{y}$ have to satisfy a similar recursive equation, i.e.
\begin{eqnarray}
P(\hat{y};u)&=& \sum_{k_{in}=1}^{\infty}
\frac{k_{in}}{\avg{k_{in}}}q_{k_{in},0}\delta(y)+\sum_{k_{in}=1}^{\infty}
\sum_{k_{out}=1}^{\infty}\frac{k_{in}}{\avg{k_{in}}}q_{k_{in},k_{out}}\nonumber \\
& &\int_{0}^{\infty} dy_1 P(y_1;u) \dots
du_{k_{in}}y_{k_{in}}P(y_{k_{in}};u)\nonumber \\
& &\int_0^{\infty}d\hat{y}_1 P(\hat{y}_1;u) \dots
du_{k_{out}}\hat{y}_{k_{out}}P(\hat{y}_{k_{out}+};u)\delta(y-\hat{g}_k(\{y\},\{\hat{y}\}))
\label{due}
\end{eqnarray}
with
\begin{eqnarray}
\hat{g}_1=u \hat{y}_1\nonumber \\
\hat{g}_k=\frac{u\sum_{k\in \partial_{+} i} \hat{y}_{k\rightarrow
i}}{1+u^2\sum_{k'\in\partial_{+}(i)\backslash j}\hat{y}_{k'\rightarrow i}\sum_{k\in
\partial_{-} i} y_{k\rightarrow i}} \ \mbox{for}\ k\geq2.
\end{eqnarray}
For a given small value of $ u=u_m+\epsilon$, the two coupled equations in
Eq.~$(\ref{uno})$ and Eq.~$(\ref{due})$ become independent.
By proceeding as in \cite{Circuits}, we find that the number of small
loops in the ensemble is given by
\begin{equation}
\avg{N_L}\simeq\frac{1}{L}\left(\frac{\avg{k_{in}k_{out}}}{\avg{k_{in}}}\right)^L
\label{short}
\end{equation}
with Poisson fluctuations for loops of size $L\ll\log(N)$. For larger
loop sizes up to the boundary limit given by $(\ref{bound})$, the average number of loops in the ensemble is still given
by $(\ref{short})$ but with significant fluctuations in the number of loops.
\section{The BP algorithm}
The study of the partition function Eq. $(\ref{Z1})$ carried on in
Section 2 is such that a new algorithm for counting large loops in a
directed network can be formulated. In particular,
given a network with $N$ nodes and $M$ links, the algorithm is:
\begin{itemize}
\item
Initialize the messages $y_{i\rightarrow j}$, $\hat{y}_{j\rightarrow i}$ for every
directed link between $i$ and $j$ to random values.
\item
For every value of $u$, iterate the BP equations in Eq.~$(\ref{BP})$
\begin{eqnarray}
y_{i\rightarrow j}=\frac{u\sum_{k\in \partial_{-} i} y_{k\rightarrow
i}}{1+u^2\sum_{k'\in\partial_{+}(i)\backslash j}\hat{y}_{k'\rightarrow i}\sum_{k\in
\partial_{-} i} y_{k\rightarrow i}}\nonumber \\
\hat{y}_{j\rightarrow i}=\frac{u\sum_{k\in \partial_{+} j} \hat{y}_{k\rightarrow
j}}{1+u^2\sum_{k'\in\partial_{-}j\backslash i}{y}_{k'\rightarrow j}\sum_{k\in
\partial_{+} i} \hat{y}_{k\rightarrow j}}.
\end{eqnarray}
until convergence.
\item
Calculate $\ell(u)$ and $f(u)$ from Eqn's $(\ref{ell})$ and $(\ref{f})$
which we recall here for convenience
\begin{equation}
\ell(u)=\frac{1}{N}\sum_{l=1}^Mp_l(1)=\frac{1}{N}\sum_l\frac{u y_{i\rightarrow
j}\hat{y}_{j\rightarrow i}}{1+u y_{i\rightarrow j}\hat{y}_{j\rightarrow i}}.
\end{equation}
\begin{eqnarray}
Nf_{Bethe}(u)&=&-\sum_{l=1}^M\ln \left(1+u y_{i\rightarrow j}\hat{y}_{j\rightarrow
i}\right)\nonumber \\
&& +\sum_{i=1}^N\ln \left(1+u^2\sum_{k'\in\partial_{-}i}{y}_{k'\rightarrow i}\sum_{k\in
\partial_{+} i} \hat{y}_{k\rightarrow i}\right).
\end{eqnarray}
\item
Evaluate $\sigma(\ell)$ by Eq. $(\ref{sigma})$ which again we repeat
here for convenience
\begin{equation}
\sigma_{Bethe}(\ell(u))=f(u)-\ell(u)\ln u.
\label{sigma}
\end{equation}
\end{itemize}
\begin{figure}
\includegraphics[width=6.5cm, height=5.5cm]{chesapeake}
\caption{Entropy $\sigma(L/N)$ of the loops of length $L$ for the
real {\it Chesapeake} food-web (solid line) and the entropy of the
loops counted by exact enumeration (diamods). }
\label{chesapeake}
\end{figure}
\section{Application of the algorithm to real directed networks}
We applied the formulated algorithm to a large set of directed networks
\cite{nnets}. For some of these networks we calculated the number of
loops ${\cal N}_L$ of lenght $L$ directly by exact enumeration
\cite{Tarjan}.
We then compare the entropy of the loops
$\sigma(\ell)$ find by the BP algorithm with
the entropy of the loops $\sigma_0(\ell)$ find by directed enumeration
of the number of loops
\begin{equation}
\sigma_0(\ell)=\frac{1}{N}\ln({\cal N}_{L=\ell N}^{\mbox{exact}})
\end{equation}
We note that for the foodweb with
small number of nodes the algorithm does not provide a good approximation for
the number of loops present in the graph. A dramatic example is the
Chesapeake foodweb. In this case we were able to count all the loops in the
network exhaustively since the network contains very few loops.
In this case the BP algorithm since the loops are few the BP
algorithm highly overestimates the largest loop in the
network (See Figure \ref{chesapeake}). In fact it predict a largest loop of lend $L_{max}=12$ where
the largest loop is of length $L_{max}=7$. This effect is observed to be present also in the undirected BP algorithm
\cite{Algorithm}.
\begin{figure}
\includegraphics[width=7.5cm, height=5.5cm]{celegans}
\caption{Entropy $\sigma(L/N)$ of the loops of length $L$ for the
real {\it C.elegans} neural network (solid line) and the entropy of the loops counted by
exact enumeration for small loops (small diamods). \\}
\label{celegans}
\end{figure}
The discrepancy is predicted to be strong only in cases where the
size of the network is small and the number of loops in the network is
small just as in the Chesapeake case. When the network has a larger number of
loops and the entropy of the loops is larger, much better results are expected.
In the case of the {\it C. elegans} neural network ($N=306$) the entropy for
small number of loops is overlapping with the results of exact enumeration as
it can clearly be seen in Figure \ref{celegans}.
We further compare the results of the algorithm on a given network and
on randomized network ensemble.
A typical example is the metabolic network of {\it E. coli}
\cite{nnets} in which we could compare the entropy provided by the BP
algorithm with the entropy of a series of $100$ random network with
the same degree distribution.
\section{Conclusions}
In conclusion we provide a new algorithm for counting large loops in
directed network.
The algorithm is predicted to give good results only for large
networks size $N$. In the paper we demonstrate cases in which it fails
to predict the right entropy and loop structure due to the small size of the
network. We propose to study the significance of loops structure in
large networks by comparing the results of the algorithm on real
networks and randomized networks when networks are large an the number
of loops in the network are also large.
\section{Acknowledgment}
We acknowledge G. Semerjian and A. E. Motter for interesting discussions.
\\
\begin{figure}
\includegraphics[width=7.5cm, height=5.5cm]{met}
\caption{Entropy $\sigma(L/N)$ of the loops of length $L$ for the
real metabolic network and average entropy of the loops in the
randomized network ensemble with same degree sequence. }
\label{metabolic}
\end{figure}
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{"url":"http:\/\/tex.stackexchange.com\/questions\/34939\/axis-with-trigonometric-labels-in-pgfplots\/34958","text":"# Axis with trigonometric labels in PGFPlots\n\nHow do I make the axis labels use multiples of \\pi in PGFPlots?\n\n-\nSee the example in pgfplots manual pages 262-263 (manual for version 1.5, of July 29 2011) for a solution. (Assuming you're after the labels on the x-axis.) \u2013\u00a0 Torbj\u00f8rn T. Nov 15 '11 at 16:31\nI think you mean \"display one axis\". If so, which one? Can you clarify the question a bit? Also, it would help to show the code that produced your figure, not just the figure. \u2013\u00a0 Alan Munn Nov 15 '11 at 16:31\nDoes this post help? \u2013\u00a0 cmhughes Nov 15 '11 at 16:32\nIt is not clear what you are looking for. Please compose a MWE that illustrates the problem including the \\documentclass and the appropriate packages so that those trying to help don't have to recreate it. \u2013\u00a0 Peter Grill Nov 15 '11 at 16:34\nDid you look at Pgfplots with custom axis markers? I think they solve your question. May be not an automatic solution, but a solution which looks like the one at page 264-265 (section 4.21) of pgfplots manual. \u2013\u00a0 Ignasi Nov 15 '11 at 16:36\n\nAs mentioned in the comments, this is basically the same as Pgfplots with custom axis markers. All you need to do is to specify where you want the xtick={...} and how you want each of them labelled via xticklabels={...}.\n\n## Method 1: Explicit Labels:\n\nExplicit labels can be specified using xticklabels. The one complication that comes about is that sometimes the label overlaps with the plot as is the case for -\\pi and 2pi in the blue graph. I have not found a elegant way to fix that so I just manually add some spacing to those labels to tweak them as I did the red graph:\n\n## Method 2: Scaled Axis Labels:\n\nAn alternate is to scale the x-axis labels in terms of multiples of pi, and show that the x axis labels are multiples of pi. This solution is based on Spikes solution, so you should up vote that if you prefer this version. I prefer to label this as part of the axis (brown graph), but others might prefer to display it as in the cyan graph:\n\n## xticklabels:\n\nIf you want a tick mark, but not a corresponding label you can simply place an empty label as in or just better just use a double comma ,, to skip it being labelled. For instance, if the labels at +\\pi and -\\pi are not desired, simply replace those labels with spaces (extra spaces here are just to point out where the gap is):\n\nxticklabels={$-2\\pi$, $-\\frac{3\\pi}{2}$, , $-\\frac{\\pi}{2}$,\n$\\frac{\\pi}{2}$, , $\\frac{3\\pi}{2}$, $2\\pi$}\n\n\n## xtick:\n\nNote that two methods of specifying where the tick marks go are used in the code. One is to explicitly list them as\n\nxtick={-6.28318, -4.7123889, -3.14159, -1.5708, 1.5708, 3.14159, 4.7123889, 6.28318}\n\n\nThis is used in the first two examples so that the correspondence between the xtick and xticklabels is easier to see. The second two use the more compact method:\n\nxtick={-6.28318, -4.7123889, ..., 6.28318}\n\n\n## Code:\n\n\\documentclass[border=2pt,tightpage]{standalone}\n\\usepackage{pgfplots}\n\n% Grouping the common style settings here to make the code below easier to read\n\\pgfkeys{\/pgfplots\/Axis Style\/.style={\nwidth=13.5cm, height=5cm,\naxis x line=center,\naxis y line=middle,\nsamples=100,\nymin=-1.5, ymax=1.5,\nxmin=-7.0, xmax=7.0,\ndomain=-2*pi:2*pi\n}}\n\n\\begin{document}\n\\begin{tikzpicture}\n\\begin{axis}[\nAxis Style,\nxtick={\n-6.28318, -4.7123889, -3.14159, -1.5708,\n1.5708, 3.14159, 4.7123889, 6.28318\n},\nxticklabels={\n$-2\\pi$, $-\\frac{3\\pi}{2}$, $-\\pi$, $\\frac{\\pi}{2}$,\n$\\frac{\\pi}{2}$, $\\pi$, $\\frac{3\\pi}{2}$, $2\\pi$\n}\n]\n\\addplot [mark=none, ultra thick, blue] {sin(deg(x))};\n\\end{axis}\n\\end{tikzpicture}\n\n\\begin{tikzpicture}\n\\begin{axis}[\nAxis Style,\nxtick={\n-6.28318, -4.7123889, -3.14159, -1.5708,\n1.5708, 3.14159, 4.7123889, 6.28318\n},\nxticklabels={\n$-2\\pi$, $-\\frac{3\\pi}{2}$, $-\\pi\\hspace{0.30cm}$, $\\frac{\\pi}{2}$,\n$\\frac{\\pi}{2}$, $\\pi\\hspace{0.10cm}$, $\\frac{3\\pi}{2}$, $\\hspace{0.25cm} 2\\pi$\n}\n]\n\\addplot [mark=none, ultra thick, red] {sin(deg(x))};\n\\end{axis}\n\\end{tikzpicture}\n\n\\begin{tikzpicture}\n\\begin{axis}[\nAxis Style,\nxtick={-6.28318, -4.7123889, ..., 6.28318},\nscaled x ticks={real:3.1415},\nxtick scale label code\/.code={},\nxlabel={$x \\thinspace [\\times \\pi]$}\n]\n\\addplot [mark=none, ultra thick, brown] {sin(deg(x))};\n\\end{axis}\n\\end{tikzpicture}\n\n\\begin{tikzpicture}\n\\begin{axis}[\nAxis Style,\nxtick={-6.28318, -4.7123889, ..., 6.28318},\nscaled x ticks={real:3.1415},\nxtick scale label code\/.code={$[\\times \\pi]$},\nxlabel={$x$}\n]\n\\addplot [mark=none, ultra thick, cyan] {sin(deg(x))};\n\\end{axis}\n\\end{tikzpicture}\n\\end{document}\n\n-\n+1 Is there a way to specify xtick using a \\foreach loop? This would be useful if one wanted to have different fractions of \\pi. \u2013\u00a0 cmhughes Nov 15 '11 at 20:14\n@cmhughes: I had attempted something similar and manually add the tick labels but have not come up with an elegant solution. See my related posts: Using macro defined lists in tikz pgfplot, and PGFplots foreach equivalent to TikZ's with multiple variables separated by a slash. \u2013\u00a0 Peter Grill Nov 15 '11 at 20:23\n@cmhughes: Also related: Macro to access a specific member of a list \u2013\u00a0 Peter Grill Nov 15 '11 at 20:25\n@cmhughes: See alternate solution which does not require you to specify each point explicitly. \u2013\u00a0 Peter Grill Nov 16 '11 at 1:47","date":"2013-12-18 07:53:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9279280304908752, \"perplexity\": 2183.4427159962192}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-48\/segments\/1387345758214\/warc\/CC-MAIN-20131218054918-00013-ip-10-33-133-15.ec2.internal.warc.gz\"}"}
| null | null |
The reptilian Damascus steel ring features a snake scale pattern created due to the unique manufacturing process of this American-Forged Damascus steel ring. These rings feature a true-raised Damascus steel profile as opposed to the profile of faux Damascus steel where it is laser etched into the ring. Featuring an inverted rounded profile this ring will be a truly unique addition to anyone's collection.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,884
|
\section{Background}
\label{sec:background}
To address the fault diagnosis problem, a data-driven framework
is proposed to model data from different fault classes and
quantify fault diagnosis performance. A summary of the relevant results and
definitions from previous work \cite{eriksson2013method} is presented here.
\subsection{Modeling Fault Classes} \label{sec:fault_modeling}
Let $\bar{r} = (r_{1}, r_{2}, \ldots, r_{n})$ denote a set of $n$
signals or features, for example residuals. A sample of $\bar{r}$ at time index $t$,
denoted $\bar{r}_t$, belongs to one of $m$ known fault classes $\{f_1, f_2, \ldots, f_m\}$.
The fault-free class is denoted $NF$ (No Fault).
To capture the impact of model uncertainties and measurement noise, each dataset
$\{\bar{r}_1, \bar{r}_2, \bar{r}_3, \ldots\}$ is partitioned into batches of size $N$, for example $R = \{\bar{r}_{t-N+1}, \bar{r}_{t-N+2}, \ldots, \bar{r}_{t}\}$,
where the distribution of data in each batch is modeled as a probability density function (pdf) $p = p(R)$. An illustration
of the one-dimensional case is shown in Figure~\ref{fig:batch_pdfs}. The figure shows partitioned
time-series data and the estimated Gaussian distribution for each batch. For comparison, the distribution
for each batch is also estimated using a kernel density estimator. It is assumed that all samples in one batch belong to
the same fault class.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{batch_pdfs-crop}
\caption{An illustration of the one-dimensional residual which is partitioned into consecutive batches and the estimated data distribution of each batch. The distribution is estimated using a Gaussian distribution and a kernel density estimation. The upper plot shows the estimated Gaussian distribution using all samples in each batch and the lower plot the estimated distribution after removing 10\% of the outliers in each batch.}
\label{fig:batch_pdfs}
\end{figure}
The pdf $p(R)$ of each batch $R$ varies depending on different
system operating conditions, such as, operating point and fault realization. Let $\Omega_i
= \Omega_i(\bar{r})$ denote the set of pdfs, based on the feature set $\bar{r}$, that can be explained by fault $f_i$ where
$p(R) \in \Omega_i(\bar{r})$ is used to denote one pdf $p(R)$ in the set.
For example, if $p \sim \mathcal{N}(\mu, \Sigma)$ is a multivariate normal distribution, the
mean estimate $\hat{\mu}$ and covariance estimate $\hat{\Sigma}$ can be obtained from a batch of $N$ samples, $R = \{\bar{r}_1, \bar{r}_2, \ldots, \bar{r}_N\}$, as
\begin{equation}
\hat{\mu} = \frac{1}{N}\sum_{t=1}^N \bar{r}_t \quad \quad \hat{\Sigma} = \frac{1}{N-1}\sum_{t=1}^N (\bar{r}_t - \hat{\mu})(\bar{r}_t - \hat{\mu})^{\text{T}}
\end{equation}
The following definition is used to model each fault class.
\begin{defn}[Fault mode]
Let $\{\bar{r}_1, \bar{r}_2, \bar{r}_3, \ldots\}$ be a set of time series data, when fault $f_i$ is present, that is
partitioned into a set of consecutive batches where each batch is represented by a pdf $p(R)$.
A fault mode is defined by the set $\Omega_i(\bar{r})$
with corresponding pdfs $p(R) \in \Omega_i(\bar{r})$ for each batch that can be explained by fault class $f_i$.
\end{defn}
To simplify notation, $p$ and $\Omega_i$ are used where the dependence on
$R$ and $\bar{r}$, respectively, are omitted. Different fault classes $f_i$ are represented by different modes
$\Omega_i$ where $\Omega_{NF}$ is used to denote the fault-free mode.
Note that each fault mode $\Omega_i$ is modeled independently and
there can be pdfs that belong to multiple fault modes, i.e., different fault realizations can result
in the same distribution of data in one batch.
Detectability and isolability of different fault classes depend on if there are
observations (pdfs) that can be explained by one fault class but not another, i.e., a fault
class $f_i$ can be isolated from another fault $f_j$ if there is a $p \in \Omega_i$
such that $p \not\in \Omega_j$.
\begin{defn}[Fault isolability]
A fault class $f_i$ is isolable from another fault class $f_j$ if
$\Omega_i\setminus\Omega_j \neq \emptyset$. If a fault $f_i$ is isolable from the
fault-free class, then the fault is said to be detectable.
\label{defn:isolability}
\end{defn}
Using Definition~\ref{defn:isolability}, it is possible to analyze fault detection and isolation
performance by comparing the different sets $\Omega_i$ where $i = 1,2, \ldots, m$ \cite{jung2015analysis}.
Note that, even though fault modes are isolable from each other it does not mean that
all pdfs $p \in \Omega_i$ can distinguish $f_i$ from $f_j$. These ambiguities are caused by, for example,
model uncertainties and sensor noise or lack of fault excitation. One illustration of fault excitation is leakage
detection where the flow through the orifice, and thus detection performance, depends on the pressure
difference since a small difference will result in a small leakage flow which can be difficult to detect.
\subsection{Fault Classification by Rejection of Fault Hypotheses}
Assuming that faults can be small or have varying impact on the feature set $\bar{r}$,
each fault class $f_j$ is modeled such that the nominal mode
$\Omega_{NF} \subseteq \Omega_j$. An implication is that it is possible to distinguish
faults from nominal behavior but not vice versa. This is consistent with the principles of consistency-based
fault isolation algorithms, such as \cite{de1987diagnosing}, where $p \in \Omega_{NF}$
can be explained by a fault free system but also that
the system is faulty (in case the fault has not yet been detected).
Instead of selecting the
most likely target class, the set of plausible fault hypotheses is computed by rejecting fault classes that
cannot explain data. A fault is detected when the fault-free class is rejected, i.e., when
$p \not\in \Omega_{NF}$. Similarly, fault classification is not performed by selecting the
most likely fault class but by rejecting fault hypotheses, e.g., a fault class $f_j$ is rejected if
$p \not\in \Omega_{\text{j}}$. This principle will be used here when performing data-driven fault isolation.
\subsection{Quantitative Fault Diagnosis Analysis}
The similarities between the different fault modes $\{\Omega_1, \Omega_2, \ldots, \Omega_m\}$
can be used to analyze fault diagnosis performance for a given system. However,
only analyzing qualitative performance, such as fault isolability in Definition~\ref{defn:isolability},
does not give sufficient information regarding how easy it is to detect and isolate different faults.
If feature distributions are used to model each fault class, one way to quantify fault diagnosis
performance is to use the KL divergence to measure the similarity between two pdfs \cite{eriksson2013method}.
The KL divergence can be used as a similarity measure between pdfs
and is defined as \cite{kullback1951information}
\begin{equation}
K(p\|q) = \int p \log \left( \frac{p}{q}\right) dp = \mathbb{E}_p\left[ \log \frac{p}{q}\right]
\label{eq:kl}
\end{equation}
where $\mathbb{E}_p\left[\cdot\right]$ denotes the expected value given the pdf $p$.
From a fault diagnosis perspective, \eqref{eq:kl} can be interpreted as the
expected value of a log-likelihood ratio test determining if $\bar{r}$ is drawn from a
distribution with a pdf $p$ or $q$ when $p$ is the true density function. If $p$ and $q$ are two pdfs
representing two different fault realizations, the larger the value of $K(p\|q)$
the easier it is to distinguish $p$ from $q$ when $p$ is true. However, since each fault can
have different realizations, quantitative isolation performance of fault $f_i$
with realization $p$ from another fault $f_j$ is defined by the smallest value of $K(p\|q)$ for all
$q \in \Omega_j$. This measure is proposed in \cite{eriksson2013method},
called distinguishability, and is defined as
\begin{equation}
\mathcal{D}_{i,j}^*(p) = \min_{q \in \Omega_j} K(p \| q)
\label{eq:dij_true}
\end{equation}
If the distribution of batch $\bar{r}$ is described by
pdf $p$ when fault $f_i$ is present, i.e., $p \in \Omega_i$, the distinguishability
measure $\mathcal{D}_{i,j}^*(p)$ quantifies how easy it is to isolate
$f_i$ from $f_j$. A large value of \eqref{eq:dij_true} corresponds to an easier isolation
problem \cite{jung2020sensor}. Fault detection performance
is denoted $\mathcal{D}_{i,NF}^*(p)$. The distinguishability measure
$\mathcal{D}_{i,j}^*(p)$ is non-negative and equal to zero if and only if
$p \in \Omega_j$, i.e., when it is not possible to isolate from fault class $f_j$.
Another property of the distinguishability measure is that if
$\Omega_{NF} \subseteq \Omega_{j}$, then \cite{eriksson2013method}
\begin{equation}
\mathcal{D}_{i,NF}^*(p) \geq \mathcal{D}_{i,j}^*(p)
\label{eq:dij_det_isol}
\end{equation}
This result can be interpreted as that it is easier to detect a fault $f_i$ than to
isolate it from another fault $f_j$.
If $p \sim \mathcal{N}(\mu_p, \Sigma_p)$ and
$q \sim \mathcal{N}(\mu_q, \Sigma_q)$ are two $n$-dimensional
multivariate normal distributions with known mean vectors, $\mu_p, \mu_q \in \mathbb{R}^n$, and
covariance matrices, $\Sigma_p, \Sigma_q \in \mathbb{R}^{n \times n}$,
$K(p \| q)$ can be computed analytically as \cite{bishop2006pattern}
\begin{equation} \label{eq:analytical_KLD}
K(p \| q)=
\frac{1}{2} \left[ \Tr \left (\Sigma_q^{-1}\Sigma_{p}\right ) + (\mu_q - \mu_p)^\intercal \Sigma_q^{-1}(\mu_q - \mu_p) - n + \log \left(\frac{\det \Sigma_{q}}{\det \Sigma_{p}} \right) \right]
\end{equation}
Even though $K(p\|q)$ can be computed analytically if $p$ and $q$ are multivariate normally
distributed, other probability distributions, such as Gaussian mixture models \cite{bishop2006pattern},
can be used to model the distribution of $\bar{r}$. Gaussian mixture models
are more flexible than multivariate normal distributions but have no analytical expression for the KL divergence.
Even though there are methods to numerically approximate \eqref{eq:kl}, see for example \cite{hershey2007approximating},
it is a tradeoff between model fit and evaluation cost of $\mathcal{D}_{i,j}^*(p)$.
For illustration, Figure~\ref{fig:batch_pdfs} shows an example of residual data from the case study.
The upper plot shows that the normal distribution can approximate the mean and variation in
batch data, except when there are outliers, and the variance is overestimated. One solution is to
remove the extreme values, e.g., 10\% of the samples in each batch, by considering
them as outliers before estimating the normal distribution. The lower plot in Figure~\ref{fig:batch_pdfs}
shows the resulting normal distributions and it is visible that the estimated variance better captures
the variation in data represented by a kernel density estimator. In this work, it is assumed that data
in each batch can be represented by a multivariate normal distribution.
\section{Case study}
\label{sec:casestudy}
The diagnostic framework is evaluated by using experimental data collected from
an engine test bench, see Figure~\ref{fig:engine_test_bench}. The engine is a commercial, turbo charged, four-cylinder,
internal combustion engine from Volvo Cars. The sensor and actuator setup are the standard
commercial configuration for the engine \cite{jung2018combining}.
Figure~\ref{fig:engine_schematic} shows a schematic view of the engine along
with the monitored signals where $y$ denotes sensor measurements and $u$
denotes actuator signals. The system represents the air path through the engine
and is an interesting case study for fault
diagnosis because of its non-linear dynamic behavior and wide operating range.
In addition, the coupling from the exhaust flow to the air intake by the turbocharger
complicates fault isolation since the effect of a fault anywhere in the system will
affect the behavior of many other components in the whole system.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\linewidth]{Figures/engine_test_bench.jpg}
\caption{The engine test bench which was used for data collection. The engine is a commercial four-cylinder combustion engine with standard sensor and actuator configuration \citep{jung2020residual}.}
\label{fig:engine_test_bench}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.9\linewidth}{!}{\includegraphics{Figures/engine_model}}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\fill[white!0!white] (0.4,0) rectangle (0.6,0.105);
\fill[white!0!white] (0.7917,0.5) rectangle (1.0,0.64);
\draw[->] (0.22, 0.22) -- (0.17, 0.21) node[left]{$y_{pic}$};
\draw[->] (0.22, 0.26) -- (0.17, 0.27) node[left]{$y_{Tic}$};
\draw[->] (0.68, 0.21) -- (0.73, 0.20) node[right]{$y_{pim}$};
\draw[->] (0.19, 0.80) -- (0.12, 0.79) node[below]{$y_{Waf}$};
\draw[->] (0.53, 0.36) -- (0.48, 0.35) node[left]{$y_{\omega}$};
\draw[->] (0.46, 0.10) -- (0.47, 0.06) node[right]{$y_{xpos}$};
\draw[] (0.00, 0.58) node[right]{$y_{pamb}$};
\draw[] (0.00, 0.52) node[right]{$y_{Tamb}$};
\draw[<-] (0.7917,0.62) -- (0.8417,0.61) node[right]{$u_{wg}$};
\draw[<-] (0.53, 0.39) -- (0.48, 0.41) node[left]{$u_{mf}$};
\end{scope}
\end{tikzpicture}
\caption{A schematic of the model of the air flow through the model. Available output signals are sensors $y$ and actuators $u$. The figure is used with permission from \cite{Eriksson2002}.}
\label{fig:engine_schematic}
\end{figure}
\subsection{Data Collection}
Engine sensor data are collected from various operating scenarios including different types of faults and fault magnitudes.
The fault classes include four multiplicative sensor faults, a leakage in the intake manifold after the throttle, as well as nominal
system operation, see Table~\ref{tab:fault_classes}.
The sensor faults are introduced by altering the sensor output gain in the engine control system.
Since the errors are injected in this way, the faulty signal output is used in the engine control
scheme which gives a more realistic fault realization compared to if the error is simulated in the data
using post processing. Each sensor fault is injected by multiplying the measured variable $x_i$ in
each sensor $y$ by a factor $\theta$ such that the resulting output is given as
$y = (1 + \theta) x$ where $\theta$ is the fault size and $\theta = 0$ corresponds to the
nominal case. The leakage in the intake manifold is introduced by opening valves with different diameters
during operation.
\begin{table}[h!]
\caption{Fault classes considered in the case study. All sensor faults are induced as multiplicative faults.}
\label{tab:fault_classes}
\centering
\begin{tabular}{cl}
\hline
Fault Class & Description\\
\hline
$ NF $ & Fault-free class \\
$ f_{ypim} $ & Fault in intake manifold pressure sensor \\
$ f_{ypic} $ & Fault in intercooler pressure sensor \\
$ f_{ywaf} $ & Fault in air-mass flow sensor \\
$ f_{iml} $ & Leakage in the intake manifold \\
\hline
\end{tabular}
\end{table}
Each dataset was collected during transient operation following the class~3 Worldwide harmonized Light-duty
vehicles Test Cycle (\textsc{WLTC}), which is part of the World harmonized
Light-duty vehicles Test Procedure (\textsc{WLTP}) \cite{Tutuianu2013}.
The cycle is shown in Figure~\ref{fig:WLTC} and is used since it covers a variety of operating
conditions. One dataset has been collected for each fault class and fault size in Table~\ref{tab:fault_modes}
resulting in 26 datasets (24 fault scenarios and two fault-free datasets). Each fault is introduced in
the dataset after approximately two minutes of the driving cycle.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Figures/wltp-crop}
\caption{Speed profile for the WLTC class 3 test cycle. The dashed line represents the time when a fault is introduced in each fault scenario.}
\label{fig:WLTC}
\end{figure}
\begin{table}[h!]
\caption{Fault classes and known magnitudes (i.e., $\theta$ for multiplicative sensor faults and leakage diameters) represented in training data. Data from the leakage $f_{iml}$ have been collected from two known diameters of the orifice.}
\label{tab:fault_modes}
\centering
\begin{tabular}{ccccccccc}
\hline
Fault Class & \multicolumn{8}{c}{Fault magnitudes} \\
\hline
$ NF $ & \\
$ f_{ypim} $ & -20\%& -15\% & -10\% & -5\% & 5\% & 10\% & 15\% & \\
$ f_{ypic} $ & -20\% & -15\% & -10\% & -5\% & 5\% & 10\% & 15\% &\\
$ f_{ywaf} $ & -20\% & -15\% & -10\% & -5\% & 5\% & 10\% & 15\% & 20\%\\
$ f_{iml} $ & 4mm & 6mm\\
\hline
\end{tabular}
\end{table}
\subsection{Residual Generation}
The proposed method can be applied to any set of features to be used for
fault diagnosis. In dynamic systems operated in various transient operating
conditions, such as the engine, using raw sensor data as features requires that a data-driven
classifier captures these dynamics since these signals can vary significantly over time.
Here, a set of four residual generators $\bar{r} = (r_1, r_2, r_3, r_4)$ is generated
by comparing predictions from a set of Recurrent Neural Networks (RNN) with the corresponding sensor
outputs, see Figure~\ref{fig:residual}, that will be used as features for fault diagnosis in the case study. A
summary of the set of residual generators used in the case study is
presented here. For the interested reader, a more detailed description is given in \cite{jung2020residual}.
\tikzstyle{block} = [rectangle, draw, fill=blue!10,
text width=4em, text centered, minimum height=3.0em]
\tikzstyle{wide_block} = [rectangle, draw, fill=blue!10,
text width=6.0em, text centered, minimum height=2.5em]
\tikzstyle{line} = [draw, -latex']
\tikzstyle{sum} = [draw, fill=blue!10, circle, node distance=1cm]
\begin{figure}[h!]
\centering
\begin{tikzpicture}[node distance=0.2cm and 0.2cm]
\node [wide_block] (system) {\includegraphics[width=55pt]{Figures/engine_test_bench}};
\node [block, below= of system] (model) {\scriptsize Model \\ \vspace{0.1cm} $\begin{aligned} \dot{\hat{x}} &= g(\hat{x}, u) \\ \hat{y} &= h(\hat{x},u) \end{aligned}$};
\draw ($(system.east) + (1.5,0.0)$) node[sum] (sum) {+};
\draw[line, ultra thick] ($(system.west) + (-1.0,0.40)$) node[above]{\scriptsize $f$} -- ($(system.west) + (-0.4,0.40)$) -- ($(system.west) + (0.0,0.40)$);
\draw[line, ultra thick] ($(system.west) + (-1.0,-0.40)$) node[above]{\scriptsize $u$} -- ($(system.west) + (-0.75,-0.40)$) -- ($(system.west) + (0.0,-0.40)$);
\draw[line, ultra thick] ($(system.west) + (-0.5,-0.40)$) |- ($(model.west)$);
\draw[line, ultra thick] (system) -- node[above]{\scriptsize $y$} (sum);
\draw[line, ultra thick] (model) -- ($(model.east) + (0.75,0.0)$) -- ($(model.east) + (1.25,0.0)$) node[above]{\scriptsize $\hat{y}$} -| (sum);
\draw[line, ultra thick] (sum) -- node[above]{\scriptsize $r$} ($(sum.east) + (1.00,0.0)$);
\draw[] ($(sum.east) + (-0.60,-0.60)$) node{$-$};
\end{tikzpicture}
\caption{An example of a residual $r$ comparing measurements from the system $y(t)$ with model predictions $\hat{y}$.}
\label{fig:residual}
\end{figure}
The prediction performance of two of the four residual generators is shown
in Figure~\ref{fig:r3_nf} and Figure~\ref{fig:r4_nf}, respectively. The figures show that the residual
generators filter out most of the system dynamics and have a small relative
prediction error. To show the impact of different faults on the residual output,
three of the four residuals are plotted against each other for different fault
classes in Figure~\ref{fig:residual_faults}. The different faults are
projected into different directions in the residual space which indicates that it is
possible to distinguish between these faults. However, some fault
classes are partially overlapping, e.g., a fault in the sensor measuring pressure
after the intake manifold, $f_{ypim}$, and a leakage in the intake manifold,
$f_{iml}$. It is expected that it is more difficult to distinguish between these
two faults since they are related to the pressure after the throttle.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Figures/r3_nf-crop}
\caption{The upper plot compares data from sensor $y_{pic}$ and model predictions from an RNN regression model. The lower plot shows the resulting residual $r_3$.}
\label{fig:r3_nf}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Figures/r4_nf-crop}
\caption{The upper plot compares data from sensor $y_{pim}$ and model predictions from an RNN regression model. The lower plot shows the resulting residual $r_4$.}
\label{fig:r4_nf}
\end{figure}
\section{Concluding Remarks and Future Works}
\label{sec:conclusions}
A data-driven framework for fault diagnosis of technical systems and time-series data is
proposed that can handle imbalanced training data and unknown faults. The KL divergence
is used as a similarity measure when evaluating if new data can be explained by that class
or not. An advantage of the proposed $\mathcal{D}_j$ classifier, with respect to other black-box models,
is interpretability, where the quantitative performance analysis and modeling of different
fault modes can give valuable insights about the nature of different faults, and the ability
to integrate fault size estimation within the proposed framework. The open set fault
classification algorithm consists of a set of one-class classifiers modeling each fault
class which makes it possible to identify all plausible fault hypotheses including scenarios
with likely unknown faults. Instead of sample-by-sample classification, the KL divergence is used to
classify if a batch of data can be explained by a given fault class or not. Experiments using
real datasets from an internal combustion engine test bench show that the proposed framework
can predict which faults that are easy to classify. They also show that the $\mathcal{D}_j$ classifiers
can classify faults that were difficult to distinguish using the 1-SVM classifiers and that
it is possible to give an accurate estimation of the fault size without the need of a
parametric model of the fault.
Estimating pdfs from training data becomes complicated when the number of
features grows. For future works, the objective is to adapt the proposed
methods for large-scale problems by using distributed fault diagnosis techniques
but also reduce computation complexity of, e.g. \eqref{eq:dij} when training data
grows, by using different heuristics or systematic search algorithms.
Another interesting continuation of this work is to extend the proposed methods
for applications in condition monitoring and prognostics to be used for predicting
system degradation rate by using batch data to track changes in fault size.
\section{Data-driven Fault Size Estimation}
\label{sec:estimation}
The method presented in Section~\ref{sec:classification} provides
a means to classify new data, but it does not give any information
about the severity of these faults. If each pdf $q_k \in \hat{\Omega}_j$
has a known fault size $\theta_k$, this information can be utilized to
estimate the size of new realizations of fault $f_j$ by comparing how
similar the distribution of new data is to training data. One approach that has
been suggested in
\cite{Grezmak2019} is to model faults into qualitative classes, such as
\{normal, slight, large\}. Another way, which is a method that is largely
unexplored, is to find a quantitative severity estimation $\hat{\theta}$.
Here, it is assumed that two pdfs $p$ and $q$ are from the same fault class
$f_i$ and have similar fault sizes $\theta_p \simeq \theta_q$ should also be
similar in a KL divergence sense, i.e., $K(p\|q)$ should be small. Fault size
estimation is formulated as a convex optimization problem by using the
KL divergence as a dissimilarity measure.
The fault size of a pdf $p$ is estimated by finding a representation
of $ p $ using a set of training distributions. Assume that for a given $p$,
$f_j \in D$ is a fault hypothesis. Then, the fault size $\theta$ of the corresponding fault
class $ f_j $ is estimated from a linear combination of pdfs
$\{q_1,q_2,\dots,q_M\} = \hat{\Omega}_j$, where $M = |\hat{\Omega}_j|$,
by solving the following optimization problem:
\begin{equation}
\begin{aligned}
\lambda^*_{1},...,\lambda^*_{M} = \arg\min_{\lambda_{1},...,\lambda_{M}} &&& K(p\|\lambda_{1}q_{1} + ... + \lambda_{M}q_{M})\\
\text{s.t.} &&& \sum_{k = 1}^{M} \lambda_{k} = 1 \\
&&& \lambda_{k}\geq 0, \quad \forall k = 1,2, \ldots, M
\end{aligned}\label{eq:lambda_est}
\end{equation}
The fault size estimate $ \hat{\theta} $ is then computed as a weighted sum of the
fault sizes $ \theta_1, \theta_2, \ldots, \theta_M $, corresponding to the pdfs
$ q_1, q_2, \ldots q_M $, as $\hat{\theta} = \sum_{k=1}^M \lambda^*_k \theta_k$.
The estimate $ \hat{p} $ in \eqref{eq:lambda_est} is obtained by using all pdfs
$q_k \in \hat{\Omega}_j $ in the optimization. This is an ineffective strategy
since it increases the computational cost by adding numerous distributions $q_k$
likely to correspond to $\lambda_k = 0$. If multiple datasets have been collected from a variety of
severities and conditions, it is unlikely that batches of new data would have a distribution that
is similar to all pdfs in the training set. Using this line of reasoning, only a
small subset of all pdfs is reasonably of interest when formulating the optimization problem \eqref{eq:lambda_est}.
Let $\{q_{(1)}, q_{(2)}, \ldots, q_{(M)}\} = \hat{\Omega}_j$ denote an ordered set of all elements
such that $K(p\|q_{(1)}) \leq K(p\|q_{(2)}) \leq \ldots \leq K(p\|q_{(M)})$. Then $\hat{\Omega}_j^l \subseteq \hat{\Omega}_j$
is defined as the first $l$ elements in the ordered set to reduce the number of parameters in \eqref{eq:lambda_est}.
The parameter $ l \leq M $ is the cardinality of $ \hat{\Omega}_j^l $ and is calibrated
to include the subset of elements in $\hat{\Omega}_j^l$ that are "similar" to $p$, i.e., $K(p\|q)$ is
relatively small and similar in value to $K(p\|q_{(1)})$.
If $q_{(1)}, q_{(2)}, \ldots, q_{(l)}$ are multivariate normal distributions then
$\sum_{k=1}^{l} \lambda_k q_{(k)}$ is a Gaussian mixture model \cite{hastie2009elements}.
Here, Monte Carlo sampling is used to estimate $K(p\| q )$, where $q = \sum_{k=1}^{l} \lambda_k q_{(k)}$, as
\begin{equation} \label{eq:kld_mc}
K_{MC}(p || q) = \frac{1}{v} \sum_{\gamma = 1}^{v} \log\left( \frac{p(x_\gamma)}{q(x_\gamma)} \right)
\end{equation}
by generating $v$ samples $\{ x_\gamma \}_{\gamma = 1}^{v}$
from $p$ to approximate the integration in~\eqref{eq:kl}.
By the law of large numbers $\lim_{v \to \infty} K_{MC}(p||q) = K(p||q)$.
A closer examination of the actual upper and lower bounds of this approximation is found in
\cite{durrieu2012lower}.
Applying \eqref{eq:kld_mc} in \eqref{eq:lambda_est} gives the updated algorithm:
\begin{equation}
\begin{aligned}
\lambda^*_{1},...,\lambda^*_{l} = &&& \\
\arg\min_{\lambda_{1},...,\lambda_{l}} &&& \frac{1}{v} \sum_{\gamma = 1}^{v} \log \left( \frac{p(x_\gamma)}{\lambda_{1}q_{(1)}(x_\gamma) + ... + \lambda_{l}q_{(l)}(x_\gamma)} \right)\\
\text{s.t.} &&& \sum_{k = 1}^{l} \lambda_{k} = 1 \quad \lambda_{k}\geq 0, \quad \forall k = 1,2, \ldots, l
\end{aligned}
\label{eq:KLD_minimization}
\end{equation}
where $\{q_{(1)}, q_{(2)}, \ldots, q_{(l)}\} = \Omega_j^l$ and the fault size is estimated as $\hat{\theta} = \sum_{k=1}^l \lambda^*_k \theta_k$. Note that if the diagnosis output $D$ contains multiple fault hypotheses $f_j$, a fault size is estimated for each fault hypothesis by solving \eqref{eq:KLD_minimization} for each $f_j \in D$.
\section{Evaluation}
\label{sec:evaluation}
The proposed methods for quantitative fault diagnosis analysis and
open-set fault classification are evaluated using data from the engine
case study. Residual data from all fault scenarios in Table~\ref{tab:fault_modes}
are partitioned into batches where each batch is used to estimate a multivariate
normal distribution. Different batch sizes are tested to evaluate the effect on
classification performance.
First, the distinguishability measure \eqref{eq:dij} is used to evaluate fault
detection and isolation performance. Then, the proposed $\mathcal{D}_j$
classifier is evaluated, including classification of unknown faults and fault size
estimation.
\subsection{Data Processing}
To analyze how the batch size will impact fault diagnosis performance, outputs from the
four residual generators are partitioned into batches of various lengths in the interval
of 50 - 300 samples. For each interval, the mean and covariance
matrix of a four-dimensional multivariate normal distribution are estimated.
To evaluate the modeling assumption that batch data are multivariate normal distributed,
the distribution of residual data is compared to the estimated normal distribution.
In Figure~\ref{fig:eval_normal_assumption}, samples from the four residuals are plotted pairwise
against each other together with an ellipse representing the estimated covariance with 95\%
confidence interval. Each column represents one batch of data and each row represent one combination of
residual outputs. The blue ellipses represent the covariance estimated from all samples.
To avoid an overestimation of the covariance due to outliers, the orange ellipses show the
estimated covariances after removing 10\% of the outliers in each batch. These two approaches of estimating the pdfs
of each batch will be further discussed in later sections. It is visible that the assumption that data is
multivariate normal distribution is an approximation, especially when there are model uncertainties
and outliers in residual data. Still, it gives some information about data distribution and correlation
between features which can be used for fault classification.
\begin{figure}
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.93\columnwidth}{!}{\includegraphics{Figures/eval_normal_assumption-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\end{scope}
\end{tikzpicture}
\caption{Residual data from four different batches compared to estimated multivariate normal distributions. The blue ellipses
correspond to the covariance estimated using the whole batch data. The orange ellipses are estimated using a subset of
batch data after removing 10\% of the samples that are outliers.}
\label{fig:eval_normal_assumption}
\end{figure}
Before designing the set of $\mathcal{D}_j$ classifiers, the set of estimated pdfs
for each fault class is then randomly split into a training and validation
set where $67\%$ are used for training. The training set from each fault class
is used to model each fault mode $\hat{\Omega}_i$ for the different fault
classes represented in training data, see Table~\ref{tab:fault_modes}. The training
set covers different realizations and magnitudes of each fault. Note that pdfs
estimated from the fault-free case are included in all fault modes, i.e.,
$\hat{\Omega}_{NF} \subseteq \hat{\Omega}_{j}$ for all $j = 1,2, \ldots, m$.
\subsection{Evaluating Fault Diagnosis Performance}
The first step of the analysis is to evaluate the set of modeled fault modes to quantify
how easy it is to distinguish between the different fault classes. In the analysis of fault diagnosis
performance using the distinguishability measure, all available datasets from the different fault
classes are used. The distinguishability measure
is evaluated for all pdfs $p \in \hat{\Omega}_i$ with respect to all other fault modes and
the distributions of $\mathcal{D}_{i,j}(p)$ values for different fault magnitudes when the batch size $N = 100$ are plotted in
Figure~\ref{fig:dij_faults}. The subplot at position $(i,j)$ shows the distribution of $\mathcal{D}_{i,j}(p)$
for all $p \in \hat{\Omega}_i$. The marks on each vertical line represent the
$10\%$, $25\%$, $50\%$, $75\%$, and $90\%$ quantiles.
The results show that detection and isolation performance, in general, improves with
increasing fault magnitude and that all faults are distinguishable from
each other. In addition, fault $f_{ypic}$ should be the easiest of
the three sensor faults to distinguish while, e.g., $f_{ywaf}$ is more difficult since the
distinguishability measure is significantly smaller. Another observation is that the
distinguishability measure is not symmetric, i.e., it might not be as easy
to distinguish mode $f_i$ from $f_j$ as the other way around \citep{eriksson2013method}. For example, it is easier
to distinguish $f_{ypic}$ from $f_{ypim}$ than vice versa which is shown by that the distinguishability
measure is larger, see Figure~\ref{fig:dij_faults}. Also, it is easier to distinguish each fault mode
from the fault-free mode, see the leftmost column in Figure~\ref{fig:dij_faults}, than to distinguish from the other
fault modes, which is consistent with \eqref{eq:dij_det_isol}.
\begin{figure}
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.93\columnwidth}{!}{\includegraphics{Figures/dij_faults-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\draw (-0.03,0.90) node{\rotatebox{90}{\small $f_{ypic}$}};
\draw (-0.03,0.64) node{\rotatebox{90}{\small $f_{ypim}$}};
\draw (-0.03,0.40) node{\rotatebox{90}{\small $f_{ywaf}$}};
\draw (-0.03,0.13) node{\rotatebox{90}{\small $f_{iml}$}};
\draw (0.11,-0.04) node{\small $NF$};
\draw (0.31,-0.04) node{\small $f_{ypic}$};
\draw (0.52,-0.04) node{\small $f_{ypim}$};
\draw (0.71,-0.04) node{\small $f_{ywaf}$};
\draw (0.91,-0.04) node{\small $f_{iml}$};
\end{scope}
\end{tikzpicture}
\caption{Evaluating the distinguishability measure \eqref{eq:dij} between fault modes (modeled using all data) as function of fault size. Each vertical line shows the quantiles $\{10\%, 25\%, 50\%, 75\%, 90\%\}$ of the distribution of distinguishability measures. Each plot $(i,j)$ shows that fault $f_i$ is easier to distinguish from fault mode $f_j$ with increasing fault size which is expected. Note that the distinguishability measure is non-symmetric, e.g., it is easier to distinguish $f_{ypic}$ from $f_{ypim}$ (or $f_{ywaf}$) than vice versa.}
\label{fig:dij_faults}
\end{figure}
Some of the results are summarized in Table~\ref{tab:Dij_varying_batch} corresponding to
detection performance of the different fault classes. The tables show the mean value of
$\mathcal{D}_{i,j}(p)$ when $p$ are estimated from batches of different sizes
(between 50 and 300 samples). It is visible that $\mathcal{D}_{i,j}(p)$ increases
for fault sizes with higher distinguishability values when increasing the batch size, while it is stable
or slightly decreasing when the distinguishability measure is small. In general, longer batch sizes
make it easier to distinguish fault $f_{ypic}$ while it becomes more difficult for the other three fault
classes. One explanation is that shorter batches can make it easier to distinguish the impact of the
fault when fault excitation and residual noise level varies over time.
{
\setlength{\tabcolsep}{3pt}
\begin{table}[h!]
\caption{Comparison of mean values of distinguishability measure of fault detection as a function of fault size and used batch size when estimating pdfs.}
\label{tab:Dij_varying_batch}
\centering
\footnotesize
\begin{tabular}{ c | c c c c }
\multicolumn{5}{c}{$\mathcal{D}_{fypic,NF}$} \\
\hline
$\theta$ \textbackslash $N$ & 50 & 100 & 200 & 300 \\
\hline
-20\% & 37.7 & 45.9 & 69.5 & 77.1 \\
-15\% & 23.7 & 27.8 & 39.7 & 43.0 \\
-10\% & 13.4 & 14.8 & 18.6 & 19.1 \\
-5\% & 7.7 & 7.2 & 7.0 & 6.6 \\
5\% & 9.1 & 7.1 & 7.0 & 7.3 \\
10\%& 17.0 & 14.9 & 18.3 & 20.9 \\
15\%& 31.0 & 28.8 & 40.0 & 45.6 \\
\hline
\end{tabular}\quad\quad\quad%
\begin{tabular}{ c | c c c c }
\multicolumn{5}{c}{$\mathcal{D}_{fypim,NF}$} \\
\hline
$\theta$ \textbackslash $N$ & 50 & 100 & 200 & 300 \\
\hline
-20\% & 9.5 & 9.4 &10.1 & 10.7 \\
-15\% & 6.7 & 6.5 & 6.8 & 7.1 \\
-10\% & 5.1 & 4.6 & 4.2 & 4.2 \\
-5\% & 2.6 & 2.2 & 1.8 & 1.6\\
5\% & 3.2 & 2.8 & 1.7 & 2.3 \\
10\%& 4.7 & 4.2 & 3.9 & 3.8 \\
15\%& 6.3 & 6.0 & 6.0 & 6.1 \\
\hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{ c | c c c c }
\multicolumn{5}{c}{$\mathcal{D}_{fywaf,NF}$} \\
\hline
$\theta$ \textbackslash $N$ & 50 & 100 & 200 & 300 \\
\hline
-20\% & 3.9 & 3.3 & 3.0 & 2.9 \\
-15\% & 3.7 & 3.1 & 2.6 & 2.4 \\
-10\% & 3.1 & 2.7 & 2.1 & 1.9 \\
-5\% & 2.4 & 2.0 & 1.8 & 1.6 \\
5\% & 1.8 & 1.4 & 1.1 & 0.8 \\
10\%& 2.2 & 1.9 & 1.4 & 1.1 \\
15\%& 2.6 & 2.1 & 1.6 & 1.4 \\
20\%& 3.0 & 2.5 & 2.1 & 2.0 \\
\hline
\end{tabular}\quad\quad\quad%
\begin{tabular}{ c | c c c c }
\multicolumn{5}{c}{$\mathcal{D}_{fiml,NF}$} \\
\hline
$\theta$ \textbackslash $N$ & 50 & 100 & 200 & 300 \\
\hline
4mm& 4.0 & 3.6 & 3.0 & 2.6 \\
6mm& 6.9 & 6.7 & 6.4 & 5.8 \\
\hline
\end{tabular}
\end{table}
}
\subsection{Fault Classification}
The open set fault classification algorithm described in Section~\ref{sec:classification}
is implemented where a $\mathcal{D}_j$ classifier is trained for each
known fault class $f_j$, as described in Section~\ref{sec:classification}.
A threshold is selected based
on the distribution of the within-class distinguishability measure \eqref{eq:within_class_dij}
using kernel density estimation to have a $5\%$ outlier rate. The calibrated thresholds for each fault class
are presented in Table~\ref{tab:J}. For comparison, the kernel density estimations of both training
and validation data for one fault class are shown in Figure~\ref{fig:ksdensity_J}.
\begin{table}[h!]
\caption{Example of calibrated thresholds $J_j$ for each $\mathcal{D}_j$ classifier.}
\label{tab:J}
\centering
\begin{tabular}{ c c c c c }
\hline
$J_{NF}$ & $J_{fypic}$ & $J_{fypim}$ & $J_{fywaf}$ & $J_{fiml}$ \\
\hline
1.98 & 3.03 & 2.43 & 2.36 & 2.31 \\
\hline
\end{tabular}
\end{table}
\subsubsection{Classification of known fault classes}
First, the set of $\mathcal{D}_j$ classifiers are evaluated using pdfs
from the known fault classes in the validation set. Since each batch of residual data is modeled as a multivariate normal distribution,
estimation of the covariance matrix can be sensitive to outliers in the residual outputs. Therefore, an additional set of
$\mathcal{D}_j$ classifiers are trained using trimmed estimates of the covariance matrix by first removing 10\% of
the outliers, denoted $\mathcal{D}_j^{trim}$. For comparison,
two sets of one-class support vector machines (1SVM) \cite{scholkopf2000support} are trained.
The 1SVM classifiers are implemented using the function \texttt{fitcsvm} in \texttt{Matlab} and
their kernel parameters are fit to training data using a subsampling heuristic \cite{matlab2018b}.
When analyzing the results of the $\mathcal{D}_j$ classifier, the false alarm
rate ($<2\%$) was lower compared to the outlier rate of $5\%$ when selecting the threshold $J_j$.
Therefore, an outlier rate of 2\% was selected when training the 1SVM classifiers to give more
comparable results. The first 1SVM classifier uses the mean of the pdfs as input, referred to as
1SVM-$\mu$ and the second set uses the raw residual data as input, referred to as 1SVM-r.
Results are presented when the pdfs are estimated
using two different batch sizes: 50 samples and 300 samples. Ideally, the probability of
rejecting a fault class should be small for the true class and large for all other fault classes.
The probabilities of rejecting each fault class given data from different fault
realizations using the three different open set fault classifiers are shown in Figure~\ref{fig:classification_50}
using batch size 50 and Figure~\ref{fig:classification_300} using batch size 300, respectively.
The curves show the mean values of 10 Monte Carlo evaluations.
\begin{figure}
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.93\columnwidth}{!}{\includegraphics{Figures/classification_50-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\draw (-0.03,0.90) node{\rotatebox{90}{\small $f_{ypic}$}};
\draw (-0.03,0.64) node{\rotatebox{90}{\small $f_{ypim}$}};
\draw (-0.03,0.40) node{\rotatebox{90}{\small $f_{ywaf}$}};
\draw (-0.03,0.13) node{\rotatebox{90}{\small $f_{iml}$}};
\draw (0.11,-0.04) node{\small $NF$};
\draw (0.31,-0.04) node{\small $f_{ypic}$};
\draw (0.52,-0.04) node{\small $f_{ypim}$};
\draw (0.71,-0.04) node{\small $f_{ywaf}$};
\draw (0.91,-0.04) node{\small $f_{iml}$};
\end{scope}
\end{tikzpicture}
\caption{Monte Carlo evaluation of detection and isolation performance using a batch size of 50 samples. Each curve shows the average of 10 runs. The decision boundary is selected using an approximate $ 2 \% $ training outlier rate. Each subplot at position $(i,j)$ shows the probability of rejecting fault class $f_j$ when a fault $f_i$ occurs as function of fault size. The $\mathcal{D}_j$ classifier using trimmed covariance estimates is labelled $\mathcal{D}_j^{trim}$.}
\label{fig:classification_50}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.93\columnwidth}{!}{\includegraphics{Figures/classification_300-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\draw (-0.03,0.90) node{\rotatebox{90}{\small $f_{ypic}$}};
\draw (-0.03,0.64) node{\rotatebox{90}{\small $f_{ypim}$}};
\draw (-0.03,0.40) node{\rotatebox{90}{\small $f_{ywaf}$}};
\draw (-0.03,0.13) node{\rotatebox{90}{\small $f_{iml}$}};
\draw (0.11,-0.04) node{\small $NF$};
\draw (0.31,-0.04) node{\small $f_{ypic}$};
\draw (0.52,-0.04) node{\small $f_{ypim}$};
\draw (0.71,-0.04) node{\small $f_{ywaf}$};
\draw (0.91,-0.04) node{\small $f_{iml}$};
\end{scope}
\end{tikzpicture}
\caption{Monte Carlo evaluation of detection and isolation performance using a batch size of 300 samples. Each curve shows the average of 10 runs. The decision boundary is selected using an approximate $ 2 \% $ training outlier rate. Each subplot at position $(i,j)$ shows the probability of rejecting fault class $f_j$ when a fault $f_i$ occurs as function of fault size.}
\label{fig:classification_300}
\end{figure}
Ideally, probability of rejection should be 100\%, for all non-zero fault sizes except when
classifying data from the same fault class that the classifier has been trained on.
In that case, probability of rejection should be as small as possible because this would otherwise
mean that the true fault class is rejected.
Classification performance is consistent with the previous results in Figure~\ref{fig:dij_faults}
showing $f_{ypic}$ is easiest to diagnose, since the probabilities to reject the wrong fault hypotheses
are higher compared to the other fault scenarios. The most difficult fault to diagnose is $f_{ywaf}$.
The results in Table~\ref{tab:Dij_varying_batch} are also consistent with the analysis in
Figure~\ref{fig:dij_faults} since classification performance of the $\mathcal{D}_j$ classifier
is better for smaller batch sizes.
When comparing the results between the
four algorithms in Figure~\ref{fig:classification_50} and Figure~\ref{fig:classification_300}, the 1SVM-r
classifier has the overall worst performance while 1SVM-$\mu$ improves with increasing batch size.
However, using trimmed estimates of the covariance matrix in the $\mathcal{D}_j^{trim}$ classifiers improve
classification performance for both shorter and longer batch sizes. Table~\ref{tab:classifiers_varying_batch}
shows the detection performance values to compare $\mathcal{D}_j$, $\mathcal{D}_j^{trim}$, and 1SVM-$\mu$,
for the two different batch sizes. These results indicate that outliers in residual data could explain the reduced
performance of the original $\mathcal{D}_j$ classifiers for longer batch sizes. Thus, one solution to improve
classification performance is to apply robust estimation of the normal distribution parameters.
{
\setlength{\tabcolsep}{1.5pt}
\begin{table}[h!]
\caption{Results from comparing fault detection accuracy in Figure~\ref{fig:classification_50} and Figure~\ref{fig:classification_300} of $\mathcal{D}_j$ classifier, $\mathcal{D}_j^{trim}$, and 1SVM-$\mu$, respectively, when a fault is present, and false alarm rate when fault size is $0\%$.}
\label{tab:classifiers_varying_batch}
\centering
\tiny
\begin{tabular}{ c | c c c | c c c}
$f_{ypic}$ & \multicolumn{3}{c}{ batch size 50} & \multicolumn{3}{c}{ batch size 300} \\
\hline
$\theta$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$\\
\hline
-20\% & 99.0\% & 99.0\%& 99.0\% & 98.6\% & 98.6\% & 98.6\% \\
-15\% & 100\% & 100\%& 100\% & 100\% & 100\% &100\%\\
-10\% & 100\% & 100\%& 100\% & 100\% & 100\% &100\% \\
-5\% & 96.1\% & 95.5\%& 98.2\% & 97.6\% & 98.1\% & 98.6\%\\
0\% & 1.5\% & 1.4\%& 2.0\% & 2.1\% & 1.1\% & 1.4\% \\
5\% & 87.4\% & 86.9\%& 91.0\% & 96.7\% & 96.4\% & 100\% \\
10\%& 100\% & 100\%& 100\% & 100\%& 100\% & 100\%\\
15\%& 100\% & 100\% & 100\% & 100\%& 100\% & 100\%\\
\hline
\end{tabular} \hfil
\begin{tabular}{ c | c c c | c c c}
$f_{ypim}$ & \multicolumn{3}{c}{ batch size 50} & \multicolumn{3}{c}{ batch size 300} \\
\hline
$\theta$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$\\
\hline
-20\% & 98.9\% & 100\%& 100\% & 100\% & 100\% & 100\%\\
-15\% & 93.4\% & 96.1\%& 96.3\% & 97.4\% & 97.6\% & 97.4\%\\
-10\% & 84.6\% & 91.8\%& 60.8\% & 85.2\% & 98.0\% & 98.2\%\\
-5\% & 50.8\% & 51.7\%& 40.6\% & 38.4\% & 45.3\% & 67.1\%\\
0\% & 1.3\% & 1.2\%& 2.2\% & 1.5\% & 1.3\% & 1.1\%\\
5\% & 53.5\% & 60.9\% & 34.6\% & 42.1\% & 53.4\% & 59.0\%\\
10\%& 71.6\% & 77.6\% & 52.1\% & 65.0\% & 77.6\% & 99.2\%\\
15\%& 85.5\% & 89.3\% & 93.5\% & 90.8\% & 96.0\% & 100\%\\
\hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{ c | c c c | c c c}
$f_{ywaf}$ & \multicolumn{3}{c}{ batch size 50} & \multicolumn{3}{c}{ batch size 300} \\
\hline
$\theta$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$ & $\mathcal{D}_j$ & $\mathcal{D}_j^{trim} $ & 1SVM-$\mu$\\
\hline
-20\% & 74.8\% & 83.7\%& 49.6\% & 63.7\%& 87.9\% & 89.4\%\\
-15\% & 71.6\% & 81.5\%& 49.6\% & 59.5\%& 84.3\% & 79.3\%\\
-10\% & 58.1\% & 69.6\%& 45.8\% & 44.9\%& 56.9\% & 68.1\%\\
-5\% & 42.2\% & 45.3\%& 33.6\% & 33.7\%& 38.6\% & 55.9\%\\
0\% & 1.1\% & 1.3\%& 2.8\% & 2.0\% & 1.4\%& 1.7\%\\
5\% & 31.6\% & 28.3\%& 15.0\% & 10.7\%& 12.2\%&30.1\%\\
10\%& 38.3\% & 43.1\%& 18.2\% & 21.6\%& 30.0\%& 33.2\%\\
15\%& 46.6\% & 50.7\%& 31.6\% & 25.5\%& 32.1\%&47.2\%\\
20\%& 53.0\% & 60.7\%& 45.5\% & 36.7\%& 49.0\%& 67.9\%\\
\hline
\end{tabular}\hfill%
\begin{tabular}{ c | c c c | c c c}
$f_{iml}$ & \multicolumn{3}{c}{ batch size 50} & \multicolumn{3}{c}{ batch size 300} \\
\hline
$\theta$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$ & $\mathcal{D}_j$ & $\mathcal{D}_{j}^{trim}$ & 1SVM-$\mu$\\
\hline
0mm & 1.5\% & 1.5\%& 2.1\% & 0.9\% & 1.4\% & 1.4\% \\
4mm & 62.3\% & 65.9\% & 78.4\% & 53.6\% & 64.6\% & 93.6\%\\
6mm & 80.8\% & 85.2\%& 95.3\% & 93.1\% & 97.8\% & 100\% \\
\hline
\end{tabular}
\end{table}
}
Another interesting observation is that $\mathcal{D}_j$ and $\mathcal{D}_j^{trim}$
are significantly better at distinguishing fault $f_{iml}$ from $f_{pim}$ compared to the 1SVM classifiers.
These faults are expected to be difficult to distinguish from each other since data from the two classes are overlapping,
as illustrated in Figure~\ref{fig:residual_faults}. However, classifying distributions instead of only sample means, makes it
possible to distinguish between the two classes, which is expected from the quantitative analysis in Figure~~\ref{fig:dij_faults}.
The results from the experiments show that approximating residual data using a multivariate normal distribution is sufficient
in this case study to distinguish between the different fault classes. However, it is likely that classification accuracy can be
improved by using a more flexible model to estimate the distribution, at the expense of increased computational cost. Another
approach to improve classification accuracy, especially detection of small faults, is to compare the diagnosis output over consecutive
batches. Since more and more data will be collected over time, the classification accuracy in Figure~\ref{fig:classification_50}
and Figure~\ref{fig:classification_300}, respectively, can most likely be improved by allowing a longer time to detect.
An advantage of using the $\mathcal{D}_j$ classifiers is that less memory is needed to store
information, since it is sufficient to store distribution parameters and not the original batch data.
If different remote diagnosis solutions are used which have access to more computational
capabilities for data analysis compared to what is available in an on-board diagnosis system,
it is also relevant to minimize the amount of transmitted data \cite{song2005remote,langarica2019industrial}.
A limitation of the $\mathcal{D}_j$ classifier is that evaluating \eqref{eq:dij} corresponds to a nearest
neighbor search problem, which can be computationally heavy when the cardinality of the set $\hat{\Omega}_j$
is large. Parallelization and the use of different heuristics could help
to prune the search space and thus, significantly reduce the computation time, see for example \cite{cayton2008fast}.
\subsubsection{Classification of unknown fault class}
Fault classification is here performed by rejecting fault hypotheses, as described in
Section~\ref{sec:computing_fault_hypotheses} where each $\mathcal{D}_j$~classifier
is used to determine if fault class $f_j$ can explain the distribution of batch data or not.
Note that each pdf in the validation set is evaluated against each known fault class,
independently of the other fault classes. Thus, the performance of classifying
unknown fault classes is based on the probabilities that all known fault classes are rejected.
This can be evaluated by using the previous results in Figures~\ref{fig:classification_50} and
\ref{fig:classification_300}. A scenario where one of the fault classes is assumed
to be unknown can be evaluated by ignoring the results in the column that corresponds to the
$\mathcal{D}_j$~classifier which models the unknown fault class.
In the first example, an unknown fault scenario is evaluated using validation data from fault $f_{ypic}$, i.e.,
it is assumed that there are no training data from that fault.
To evaluate the probability that the known fault classes, $NF$, $f_{ypim}$, and $f_{ywaf}$,
are correctly rejected in this scenario, the first row is analyzed in
Figure~\ref{fig:classification_50} (and Figure~\ref{fig:classification_300}) while ignoring the second
column which corresponds to the $\mathcal{D}_j$ classifier modeling fault $f_{ypic}$. In this
case all classifiers have more than $80\%$ probability of rejecting the $NF$ class for all realizations
of $f_{ypic}$ in validation data. Similarly, the other known fault classes have a high probability
of correctly being rejected, except for the 1SVM-$r$ classifier modeling fault $f_{ypim}$ which has a low
probability of rejecting $f_{ypim}$ for scenarios when $f_{ypic}$ is of size $5\%$. This shows
that $f_{ypic}$ is likely to be correctly classified as an unknown fault since all known fault classes
will be rejected with high probability. Note that the probabilities of rejecting each fault class are
evaluated by classifying one pdf. The probability of correctly classifying smaller faults can be
significantly improved by evaluating multiple pdfs corresponding to consecutive batches and
compare the rate that each fault class is rejected with respect to the probability that it is falsely
rejected when it is the true fault class.
As a second example, fault $f_{iml}$ is simulated as an unknown fault while the other fault classes
in the training set are known. Then, the last row is
analyzed in Figure~\ref{fig:classification_50} (and Figure~\ref{fig:classification_300}) while ignoring the
last column. When comparing the probability of correctly rejecting each known fault class in the last row
the 1SVM-$\mu$ performs better than the $\mathcal{D}_j(p)$ and $\mathcal{D}_j(p)^{trim}$
classifiers to detect the fault (rejecting the $NF$ class). The 1SVM-$r$ classifier modeling the $NF$
class has significantly worse performance, especially for the 4mm leakage.
However, when analyzing the probability of rejecting the other known fault classes, $f_{ypic}$, $f_{ypim}$,
and $f_{ywaf}$, the $\mathcal{D}_j(p)$ and $\mathcal{D}_j(p)^{trim}$ classifiers have an overall
better performance compared to 1SVM-$\mu$ and 1SVM-$r$, especially for the 4mm leakage.
The 1SVM-$r$ is not able to reject $f_{ypic}$ and $f_{ypim}$, for any of the leakage sizes
since the probabilities of correctly rejecting those fault classes are not significantly higher than the probability
that each classifier is falsely rejecting the true fault class. For the larger batch size, the 1SVM-$\mu$
performs significantly better for the 6mm leakage compared to the 4mm leakage but is not able to
reject $f_{ypim}$.
The two examples show the ability of the proposed open set fault classification algorithm, using
$\mathcal{D}_j(p)$ or $\mathcal{D}_j(p)^{trim}$ classifiers, to identify unknown fault classes. The
examples also illustrate the computational benefits since there is no need of recalibrating
the whole set of $\mathcal{D}_j$ classifiers for the existing set of known fault classes when
updating one classifier $\mathcal{D}_j$ with new training data or when including a new classifier
for a new fault class.
\subsection{Fault Size Estimation}
The fault estimation algorithm \eqref{eq:lambda_est} described in Section~\ref{sec:estimation}
is applied to the validation set when the corresponding fault size of each pdf in the training set is known. The numerical KL divergence
\eqref{eq:kld_mc} is evaluated using 1000 Monte Carlo samples where the cost function is minimized using the
10 pdfs in training data with the smallest $\mathcal{D}_j(p)$ values. The fault size estimation for each pdf $p$ using
\eqref{eq:lambda_est} is denoted $\hat{\theta}_{KL}$.
The prediction results
for each fault class are shown in Table \ref{tab:theta_estimation} by presenting the 10\% and 90\% quantiles
of the fault size predictions. To evaluate the proposed data-driven fault size estimation algorithm \eqref{eq:lambda_est},
it is compared to estimating the fault size $\hat{\theta}_{mean}$ by computing the mean fault size of the 10 pdfs with
the smallest $\mathcal{D}_j(p)$ values. For the proposed algorithm, the true fault sizes are within the intervals
while for the estimate $\hat{\theta}_{mean}$, the true fault sizes are outside the intervals for the largest realizations of $f_{ywaf}$.
The algorithm \eqref{eq:lambda_est}, in general, has a narrower interval compared to using the mean estimate.
Note that a limitation of the evaluated methods is that they are not capable of estimating fault sizes beyond
what is available in training data.
The prediction error correlates with the analysis of distinguishability between
different fault modes in Figure~\ref{fig:dij_faults}. The intervals are smaller for $f_{ypic}$
while $f_{ywaf}$ has the largest intervals. One solution to improve the estimation
accuracy over time is to look at the distribution of the estimated fault size over consecutive
batches.
\setlength{\tabcolsep}{1.5pt}
\begin{table}[h!]
\caption{Results from using the fault size estimation algorithm \eqref{eq:lambda_est} on engine residual data. For each fault size $\theta$, the intervals represent the 10\% and 90\% quantiles of the estimates $\hat{\theta}_{KL}$ that are computed using \eqref{eq:KLD_minimization} and $\hat{\theta}_{mean}$ which is the mean fault size computed from the 10 pdfs with smallest $\mathcal{D}_j(p)$ values.}
\label{tab:theta_estimation}
\centering
{
\footnotesize
\begin{tabular}{ c | c | c}
\multicolumn{3}{c}{ $f_{ypic}$} \\
\hline
$\theta$ & $\hat{\theta}_{KL}$ & $\hat{\theta}_{mean}$ \\
\hline
-20\% & $[-20.0\%, -18.4\%]$ & $[-20.0\%, -14.5\%]$ \\
-15\% & $[-15.0\%, -14.8\%]$ & $[-15.5\%, -11.4\%]$ \\
-10\% & $[-10.0\%, -10.0\%]$ & $[-10.0\%, -8.0\%]$ \\
-5\% & $[-5.0\%, -5.0\%]$ & $[-5.5\%, -5.0\%]$ \\
0\% & $[-1.4\%, 0.0\%]$ & $[-2.0\%, 0.5\%]$ \\
5\% & $[5.0\%, 5.0\%]$ & $[4.5\%, 5.0\%]$ \\
10\%& $[10.0\%, 10.0\%]$ & $[8.5\%, 10.0\%]$ \\
15\%& $[15.0\%, 15.0\%]$ & $[13.0\%, 15.0\%]$ \\
\hline
\end{tabular} \hfil
\begin{tabular}{ c | c | c}
\multicolumn{3}{c}{ $f_{ypim}$} \\
\hline
$\theta$ & $\hat{\theta}_{KL}$ & $\hat{\theta}_{mean}$ \\
\hline
-20\% & $[-20.0\%, -17.2\%]$ & $[-20.0\%, -17.0\%]$ \\
-15\% & $[-16.2\%, -12.3\%]$ & $[-16.5\%, -12.5\%]$ \\
-10\% & $[-12.8\%, -8.4\%]$ & $[-12.5\%, -7.0\%]$ \\
-5\% & $[-6.1\%, -1.4\%]$ & $[-7.2\%, -1.5\%]$ \\
0\% & $[-3.2\%, 1.8\%]$ & $[-3.0\%, 2.0\%]$ \\
5\% & $[3.2\%, 9.2\%]$ & $[2.1\%, 9.5\%]$ \\
10\%& $[8.6\%, 12.6\%]$ & $[8.1\%, 12.0\%]$ \\
15\%& $[10.8\%, 15.0\%]$ & $[10.0\%, 15.0\%]$ \\
\hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{ c | c | c}
\multicolumn{3}{c}{ $f_{ywaf}$} \\
\hline
$\theta$ & $\hat{\theta}_{KL}$ & $\hat{\theta}_{mean}$ \\
\hline
-20\% & $[-20.0\%, -12.0\%]$ & $[-19.5\%, -11.6\%]$ \\
-15\% & $[-17.4\%, -11.3\%]$ & $[-17.5\%, -7.9\%]$ \\
-10\% & $[-13.5\%, -6.3\%]$ & $[-14.0\%, -5.7\%]$ \\
-5\% & $[-10.1\%, -0.1\%]$ & $[11.0\%, 1.7\%]$ \\
0\% & $[-2.6\%, 7.9\%]$ & $[-2.0\%, 7.5\%]$ \\
5\% & $[1.3\%, 10.3\%]$ & $[-1.5\%, 9.0\%]$ \\
10\%& $[4.4\%, 13.5\%]$ & $[5.0\%, 13.0\%]$ \\
15\%& $[7.9\%, 17.6\%]$ & $[7.0\%, 16.5\%]$ \\
20\%& $[10.2\%, 20.0\%]$ & $[8.2\%, 18.3\%]$ \\
\hline
\end{tabular}
}
\end{table}
\section{Introduction}
Fault diagnosis of technical systems deals with the problem of detecting and isolating faults
by comparing model predictions of nominal system behavior and data from sensors
mounted on the monitored system \cite{jung2018combining}. Early detection
of faults and identifying their root cause are important to improve system
reliability and to be able to select suitable counter measures.
Two common approaches of fault diagnosis are model-based and data-driven \cite{gao2015survey}.
Model-based fault diagnosis relies on a mathematical model describing the nominal
system behavior, where the model is derived based on physical insights about the system.
Residuals are computed by comparing model predictions and sensor data to detect
inconsistencies caused by faults \cite{isermann2005model,jiang2020optimized}.
Data-driven fault diagnosis uses training data from different operating conditions and
faulty scenarios to capture the relationship between a set of input and
output signals \cite{dai2013model}. The output signal could be a feature or sensor value to be
predicted, which is referred to as regression, or the class label that input
data belong to, referred to as classification \cite{hastie2009elements}.
Fault detection and isolation are complicated by prediction inaccuracies
and measurement noise \cite{eriksson2013method}. For nonlinear dynamic systems,
the sensor data distribution varies due to different operating conditions which requires
complex data-driven classifiers to capture the distribution of different classes to
distinguish between faults. One solution, see for example \cite{jung2020residual}, is to use
residual generators to compute residual outputs as features to filter out the system dynamics
before classifying faults.
Even though system dynamics can be filtered out in feature data, the distribution of faulty
data is not only dependent on fault class but also the realization of the fault, e.g., fault magnitude
and excitation. Thus, collecting representative training data from various fault scenarios that
can occur in the system is a complicated task, especially when developing a diagnosis system
during early system life when failures are still rare \cite{theissler2017detecting}.
General-purpose models for supervised multi-class classification, such as
Random Forests and Neural Networks, have a risk of misclassifications
when training data are limited since these methods assume that training data
are representative of all data classes. In many technical applications, it is
necessary that a diagnosis system can handle limited, and imbalanced, training data
and is able to detect likely unknown fault scenarios that would require special attention by an operator
or technician \cite{theissler2017detecting}. One application is, for example, computer aided
troubleshooting where a priority list of plausible fault hypotheses can guide a technician when
identifying the faulty component \cite{pernestaal2012modeling}. It is also important to update the models
over time as more training data become available to improve classification performance
\cite{jung2018combining,sankavaram2015incremental,dong2017method}.
Another complicating factor of data-driven fault diagnosis is class overlapping \cite{lee2018overlap}
when different faults have similar impact on the system behavior. One example is when trying to classify
small faults at an early stage, which can result in classification ambiguities \cite{campagner2020three}.
This is illustrated in Figure~\ref{fig:residual_faults} where real data from a set of
residual based fault detectors for an internal combustion engine test bench \cite{jung2020residual},
are plotted against each other. The figure shows that fault-free data (No Fault - $NF$) and data from the different fault
classes $f_i$ are overlapping, especially data from small faults close to the origin.
It can also be different fault classes that have similar impact on system behavior, e.g., the faults $f_{ypim}$ and $f_{yiml}$
that are both related to the engine intake manifold. Thus, it is not desirable to compute only the most likely fault
hypothesis (class label), since this could fail to identify the true fault, but instead
find all plausible fault hypotheses that can explain the observed data.
Note that this is different from multi-label classification since each sample belongs to one
fault class but can be explained by multiple classes \cite{campagner2020three}.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Figures/residual_faults-crop}
\caption{The figure shows residual data from the engine case study where data from
three of four residual generators are plotted against each other. The different colors
correspond to data from different fault classes where $NF$ represents the fault-free class.}
\label{fig:residual_faults}
\end{figure}
Different classifiers that can handle overlapping classes have been proposed in, for example,
\cite{lee2018overlap}. Because of overlapping fault classes, analysis methods that can quantify how
easy it is to distinguish between different fault classes are necessary during the diagnosis system
design process \cite{eriksson2013method}. Applying quantitative analysis early during the
system development phase can be used to predict if, for example, fault diagnosis performance
requirements can be met. Some of the most common performance analysis methods of multi-class
classifiers are cross-validation and confusion matrices, see e.g. \cite{hastie2009elements}. Note that these methods are used
to evaluate the performance of a given classifier rather than analyzing the distribution of data from different
fault classes. There are also methods, such as \cite{van2008visualizing}, that can help to visually
analyze multi-dimensional data. However, it is not obvious how to use this information to quantitatively
measure separation between different classes.
Besides fault detection and isolation, an important task of diagnosis systems is to track system
degradation, for example by continuously estimating the size, or severity, of a fault that is present
in the system. Accurate information about fault size is important for predictive maintenance
\cite{larsson2014gas}, prognostics \cite{daigle2012comparison}, and fault reconstruction
algorithms \cite{yan2007nonlinear}.
Several model-based techniques for fault size estimation have been proposed, see e.g.
\cite{daigle2012comparison}. Still, it is a non-trivial problem to estimate the fault size when no fault models are available.
The main objective of this work is to develop a framework for data-driven analysis and
classification for fault diagnosis of dynamic systems. The first contribution is a quantitative analysis method
for data-driven fault classification using the Kullback-Leibler (KL) divergence to
model data from different fault classes and to analyze fault detection and isolation performance
for a given set of features. The second contribution is a data-driven open set classification algorithm
designed for fault diagnosis applications that can handle: limited training data, unknown fault
classes, and overlapping regions of different classes. A third contribution is a data-driven fault size estimation
algorithm, using the same modeling framework, when training data are available with known fault
sizes. As a case study, real data are collected from an internal combustion engine test bench which is
operated during both nominal and faulty operation \cite{jung2018combining}.
The outline of this paper is as follows. First, the problem statement is presented
is Section~\ref{sec:problemformulation}. Related research is summarized in
Section~\ref{sec:relatedresearch} and some background to fault diagnosis and
quantitative fault detection and isolation analysis is given in
Section~\ref{sec:background}. Then, the proposed framework for data-driven
quantitative analysis is presented in Section~\ref{sec:modeling}, the proposed
open set fault classification algorithm in Section~\ref{sec:classification}, and the
fault size estimation algorithm in Section~\ref{sec:estimation}. The internal combustion
engine case study
is described in Section~\ref{sec:casestudy} and the results of the experiments are presented in
Section~\ref{sec:evaluation}. Finally, some conclusions are summarized in Section~\ref{sec:conclusions}.
\section{}
\bibliographystyle{elsarticle-num}
\section{Open Set Classification for Fault Diagnosis}
\label{sec:classification}
Since there could be multiple fault modes that could explain an observation
$\bar{r}$ with pdf $p$, a conventional multi-class classifier is not suitable.
To handle both that training data is not representative of all fault classes and
that the objective is to identify not only the most likely diagnosis candidate but all
diagnosis candidates, an open set fault classification algorithm is developed.
A one-class classifier is designed for each fault class $f_i$ based on the modeling
of fault modes and the approximated distinguishability measure \eqref{eq:dij}, such that
a pdf $p$ can be explained by fault class $f_i$ if
\begin{equation}
\mathcal{D}_{i}(p) = \min_{q \in \hat{\Omega}_i} K(p \| q) \leq J_i
\label{eq:fi_classifier}
\end{equation}
where $J_i$ is a threshold. The set $\hat{\Omega}_i$ is estimated from training
data. Here, residual data is partitioned into fixed intervals and $p$ is estimated as
a multivariate normal distributed fit to each interval. Note that more flexible
distributions could be used at the cost of increased computational cost and
numerical approximation of computing \eqref{eq:fi_classifier}.
\subsection{Fault Mode Data Clustering}
Evaluating \eqref{eq:fi_classifier} requires a search over all elements in $\hat{\Omega}^c_i$.
Hierarchical clustering can be used to divide the search space such that
\eqref{eq:fi_classifier} can be solved without having to evaluate all pdfs in $\hat{\Omega}^c_i$.
For example, the set $\hat{\Omega}_i$ can be clustered using a subset
$\hat{\Omega}^c_i \subseteq \hat{\Omega}_i$ such that
$\min_{q \in \hat{\Omega}^c_i} K(p \| q) \leq J_i$ for all
$p \in \hat{\Omega}_i$. This subset can be approximated using, for example,
the generalized $ k $-median clustering suggested in \cite{Ackerman2010clustering}.
Then, \eqref{eq:fi_classifier} can be evaluated by first evaluating all $\hat{\Omega}^c_i$
and then only continue evaluating the elements in the corresponding clusters
that are close to $p$.
\subsection{Summary}
\label{sec:classification:summary}
For each new $p$:
\begin{itemize}
\item If $\mathcal{D}_{NF}(p) \leq J_i$ then $NF$ is the diagnosis output
\item else, for each fault mode $f_i$:
\begin{itemize}
\item $f_i$ is a diagnosis candidate if $\mathcal{D}_i(p) \leq J_i$
\end{itemize}
\item else if $\mathcal{D}_i(p) > J_i$ for all fault modes $f_i$, then the unknown fault case is the diagnosis candidate
\end{itemize}
The open set fault classifier can be improved incrementally as new data from different fault scenarios are collected and labelled, by updating the corresponding fault classes accordingly.
\section{Approximated Distinguishability Measure Using Training Data}
\label{sec:modeling}
In many applications, the sets $\Omega_j$ are either partially, or completely, unknown because
training data only consist of a limited number of fault realizations. This means that
the distinguishability measure \eqref{eq:dij_true} cannot be used. Instead, an
approximated distinguishability measure is proposed where each fault mode
is estimated from training data.
\subsection{Distinguishability Measure for Data-Driven Analysis}
If training data are correctly labelled, partitioning the data into batches can be used to estimate
pdfs belonging to different fault classes. The estimated pdfs belonging to
fault class $f_j$ can be used to make a lower approximation
of the true fault mode $\Omega_j$ denoted
$\hat{\Omega}_j \subseteq \Omega_j$. Then, an approximation of
\eqref{eq:dij_true} can be computed as
\begin{equation}
\mathcal{D}_{i,j}(p) = \min_{q \in \hat{\Omega}_j} K(p \| q)
\label{eq:dij}
\end{equation}
i.e., distinguishability is computed based on the set of already observed
realizations of each fault $f_j$ available in training data.
The approximate distinguishability measure \eqref{eq:dij} is illustrated in
Figure~\ref{fig:minkl} where three pdfs, $p_1$, $p_2$,
and $p_3$, are compared to a set of pdfs that is used to represent a fault mode $\hat{\Omega}_j$. The dashed lines show each
pdf $q$ that minimizes $\eqref{eq:dij}$ for each $p_k$. The lower plot shows
the computed KL divergence from one pdf, $p_2$, to all $q \in \hat{\Omega}_j$
which, in general, increases for pdfs $q$ that are located further away from $p_2$.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.95\columnwidth}{!}{\includegraphics{Figures/minkl-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\draw (0.19,0.82) node{ $p_1$};
\draw (0.42,0.91) node{ $p_2$};
\draw (0.82,0.84) node{ $p_3$};
\draw (0.42,0.365) node{ $p_2$};
\draw (-0.02,0.79) node{\rotatebox{90}{$r_2$}};
\draw (0.515,0.52) node{$r_1$};
\draw (-0.02,0.24) node{\rotatebox{90}{$r_2$}};
\draw (0.515,-0.03) node{$r_1$};
\end{scope}
\end{tikzpicture}
\caption{The upper plot shows an illustration of the approximate distinguishability measure
\eqref{eq:dij} from a set of pdfs, $p_1$, $p_2$, and $p_3$, to a fault mode $\hat{\Omega}_j$.
The dashed lines show each pdf $q$ that minimizes $\eqref{eq:dij}$ for each $p_i$.
For illustration, the computed KL divergence for each $q \in \hat{\Omega}_j$ is written
in the lower plot given $p_2$.}
\label{fig:minkl}
\end{figure}
Since $\hat{\Omega}_j \subseteq \Omega_j$, the relation between
\eqref{eq:dij_true} and \eqref{eq:dij} is given by the inequality
$0 \leq \mathcal{D}_{i,j}^*(p) \leq \mathcal{D}_{i,j}(p)$
which is derived from
\begin{equation}
\mathcal{D}_{i,j}^*(p) = \min_{q \in \Omega_j} K(p \| q) \leq \min_{q \in \hat{\Omega}_j} K(p \| q) = \mathcal{D}_{i,j}(p)
\end{equation}
The approximation \eqref{eq:dij} gives an upper bound of how easy it is to
reject $f_j$ for $f_i$ given $p$. Note that \eqref{eq:dij_det_isol} also holds for \eqref{eq:dij}, i.e.,
$\mathcal{D}_{i,NF}(p) \geq \mathcal{D}_{i,j}(p)$.
\section{Open-Set Fault Classification Using Distinguishability}
\label{sec:classification}
Since different faults can have similar impact on system operation, it is
relevant to not only select the most likely fault class but to identify all fault
classes that can explain a set of observations. Here, a set of $m$ one-class
classifiers is used to model data from each of the $m$ fault classes to see if
each class can explain the observation or not. If a pdf $p$ cannot be explained
by any of the known fault classes, i.e., $p \not\in \hat{\Omega}_j$ for all $f_j$,
it belongs to an unknown fault class. Note that there can be two types of unknown
faults \cite{scheirer2014probability}:
\begin{itemize}
\item Data belong to a new unknown fault class, or
\item Data is a new realization of a known fault class $f_j$, i.e., $p \in \Omega_j \setminus \hat{\Omega}_j$.
\end{itemize}
These unknown fault scenarios require extra attention, for example, by a technician, to identify
the root cause and correctly label data to be used for updating the corresponding fault mode.
\subsection{One-class Classification}
Here, a one-class classifier is proposed using the approximate distinguishability measure \eqref{eq:dij}
to tests if a new pdf $p$ can be explained by a fault mode $f_j$ or not. The proposed classifier is
denoted $\mathcal{D}_j(p) = \min_{q \in \hat{\Omega}_j} K(p \| q)$ with respect to \eqref{eq:dij} to
emphasize that the class label of $p$ is unknown. Hereafter, the classifier modeling a fault mode $f_j$
will be referred to in the text as $\mathcal{D}_j$.
An open set classification algorithm for fault diagnosis is then formulated using one $\mathcal{D}_j$
classifier for each known fault class $f_j$. Fault classification is then performed such that the fault
hypothesis $f_j$ is rejected if
\begin{equation}
\mathcal{D}_j(p) > J_j
\label{eq:Dj_J}
\end{equation}
where $J_j$ is a threshold. A small threshold $J_j$ increases the risk of falsely rejecting the true fault
class while a large threshold $J_j$ means that fault $f_j$ is more likely to be a wrong hypothesis
increasing fault classification ambiguity.
\subsection{Tuning of The One-Class Classifier Thresholds Using Within-Class Distinguishability}
A tuning strategy of the threshold $J_j$ in \eqref{eq:Dj_J} is proposed such that most of the
pdf's $p \in \hat{\Omega}_j$ should be explained by fault class $f_j$ if $p$ is removed
from $\hat{\Omega}_j$. Let
\begin{equation}
\mathcal{D}_{j,j}(p) = \min_{q \in (\hat{\Omega}_j \setminus \{p\})} K(p \| q)
\label{eq:within_class_dij}
\end{equation}
which is here referred to as \emph{within-class distinguishability}. Analyzing the
distribution of $\mathcal{D}_{j,j}(p)$ for all $p \in \hat{\Omega}_j$ can be used
to select a threshold. The distribution will have non-negative support and is here
approximated using a kernel density estimation method \cite{hastie2009elements} as illustrated
in Figure~~\ref{fig:ksdensity_J}. Note that \eqref{eq:within_class_dij} does not state anything
about the relation between the sets $\hat{\Omega}_j$ and $\Omega_j$ but
rather gives information about how scattered training data are from that fault class.
Let $\Phi(x)$ denote the cumulative density
function (cdf) of the estimated distribution and let $\alpha$ denote a desired
false alarm rate. Then, the threshold $J_j$ is selected such that
$\Phi(J_j) = 1-\alpha$. The lower plot in Figure~~\ref{fig:ksdensity_J} illustrates
selecting a threshold corresponding to $\alpha = 5\%$.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\resizebox{0.95\columnwidth}{!}{\includegraphics{Figures/ksdensity_J-crop}}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\draw (0.52,-0.06) node{$\mathcal{D}_{j,j}(p)$};
\draw (-0.03, 0.77) node{\rotatebox{90}{pdf}};
\draw (-0.03, 0.25) node{\rotatebox{90}{cdf}};
\end{scope}
\end{tikzpicture}
\caption{Kernel density estimation of within-class distinguishability for the fault-free class (pdf in upper plot and cdf in lower plot). Dashed line represents threshold $J_{NF}$ tuned to have a $5\%$ outlier rate.}
\label{fig:ksdensity_J}
\end{figure}
\subsection{Computing Fault Hypotheses}
\label{sec:computing_fault_hypotheses}
If $\mathcal{D}_j(p)$ is large, it means that $p$ is not likely to be explained by fault $f_j$.
If $\mathcal{D}_j(p)$ is large for all fault classes $f_j$, this means that no known fault class can explain $p$,
thus indicating the occurrence of an unknown fault class.
These results give a systematic approach to compute fault hypotheses, including
the unknown fault case, by evaluating and comparing $\mathcal{D}_j(p)$ for each known fault class
$f_j$.
The classifier (diagnosis) output $D$ given an observation $p$ is determined using the following decision rule:
\begin{enumerate}
\item If $\mathcal{D}_{NF}(p) \leq J_{NF}$ then $D = \{NF\}$, i.e., the output of the classifier is that the system is fault-free.
\item If $\mathcal{D}_{NF}(p) > J_{NF}$ then $D = \{f_j : \mathcal{D}_{j}(p) < J_{j}\}$, i.e., the output is the set of all known
fault classes that can explain $p$.
\item If $\mathcal{D}_{j}(p) > J_{j}$ for all known fault classes $f_j$, i.e., no known fault class can explain $p$, then
$D = \{f_x\}$ where $f_x$ denotes an unknown fault class.
\end{enumerate}
An advantage of testing the fault hypothesis for each fault class individually, is that there is no bias in
the diagnosis output if the fault models are trained using imbalanced datasets. A fault $f_j$ is a diagnosis candidate if $\mathcal{D}_j(p)$
does not exceed the threshold $J_j$. Note that when the output $D$ contains at least one known fault class,
it is still plausible that an unknown fault type has occurred, thus $f_x$ is always a plausible fault hypothesis.
However, $f_x$ is only explicitly stated as a diagnosis output if $\mathcal{D}_j(p)$ is large for all $f_j$.
If the $NF$ class is not rejected, i.e., if $\mathcal{D}_{NF}(p) \leq J_{NF}$, there could still
be an undetected fault present in the system since $\hat{\Omega}_{NF} \subseteq \hat{\Omega}_j$
for all $f_j$. However, if the $NF$ class is not rejected, it is more likely that the system is fault-free
than faulty. Therefore, the only diagnosis output in this case is that the system is fault-free.
\section{Problem Statement}
\label{sec:problemformulation}
The main objective is to develop a data-driven framework for supervised machine learning and
data-driven analysis to systematically address the complicating factors of data-driven fault diagnosis
of non-linear dynamic systems. The characterizing properties of the data-driven fault diagnosis problem are
summarized in the following bullets:
\begin{itemize}
\item Training data are not representative of all relevant fault realizations
\item There are both known and unknown fault classes
\item Feature data from different fault classes are overlapping
\item The distribution of feature data is not only depending on fault class, but also fault magnitude and excitation
\end{itemize}
Even though there has been significant work done in data-driven fault diagnosis and machine learning,
it is still an open question how to address all these complicating factors in a common framework.
In the problem formulation, it is assumed that a set of $n$ features has been developed to monitor
a non-linear dynamic system. The set of computed features is time-series data, for example residuals
computed from sensor outputs, that can be used to detect faults in the system. It is assumed that
a set of training data has been collected from different nominal and faulty scenarios where the
samples are correctly labelled. Labelling mainly considers which fault class each sample belongs to,
but it will also be investigated what can be done when information about fault sizes is available as well.
It is assumed that the available training data fulfill the characterizing properties listed above.
The first objective of this work is to develop a data-driven framework for modeling of different fault
classes. To analyze the properties of training data, a quantitative method will be developed that is
able to analyze the ability to classify the different faults. The proposed method should be able to
quantify how easy it is to distinguish between different fault classes which can give valuable
insights about the nature of different faults.
The second objective is to develop a fault classification algorithm that is designed to detect
and classify known fault classes that can explain the observations but also identify scenarios with
unknown faults. Because feature data is assumed to overlap between different classes, the classifier
should be designed such that the computed fault hypotheses are based on the characterizing
properties of training data to avoid misclassifications.
The third objective considers the case when training data are collected from different fault realizations with
known fault magnitudes, e.g., a given sensor bias or leakage diameter. The purpose is to use the proposed
framework to estimate the magnitude of a detected known fault to track the system health when no
mathematical model is available to estimate the fault.
\section{Related Research}
\label{sec:relatedresearch}
There has been an increasing focus on data-driven classification algorithms that can handle
unknown classes, which is referred to as open set classification, see e.g. \cite{scheirer2013toward},
and overlapping classes, see e.g. \cite{lee2018overlap}. However, it is still an open problem
how to address all these mentioned characteristics of data-driven fault diagnosis problems.
Open set classification has been used in, e.g., computer vision, to deal with unknown
classes not covered by training data \cite{scheirer2013toward}.
Different algorithms have been developed to solve the open set classification problem,
for example Weibull-calibrated support vector machines \cite{scheirer2014probability}
and extreme value machines \cite{rudd2018extreme}.
Different data-driven approaches have been
proposed for open set fault classification to handle both known and unknown fault scenarios,
for example, one-class support vector machines \cite{jung2018combining,jung2020data}, conditional
Gaussian network \cite{atoui2019single}, ensemble methods \cite{theissler2017detecting}, and
Hidden Markov Models \cite{yan2018fault}. Several papers have proposed open set classification
algorithms for machinery condition monitoring, see for example \cite{tian2018subspace,wang2021novel,yu2021deep}.
The authors of \cite{tian2018subspace} propose a hybrid approach combining different data-driven
methods and subspace learning.
In \cite{wang2021novel}, a neural network with residual learning blocks is proposed and \cite{yu2021deep}
proposes a deep learning neural networks-based approach to handle the situations where training
data and test data are collected from different operating conditions.
In \cite{michau2019domain}, a domain adaptation open set classification approach is proposed
combining auto-encoders, a proposed homothety loss, and an origin discriminator, to monitor a
system using training data from other similar systems.
With respect to previous works, the proposed data-driven framework can also be used to evaluate
fault diagnosis performance given a set of features. The same framework is used to design an open set fault
classification algorithm that handles overlapping classes, including scenarios with unknown faults,
and estimates the fault sizes of known fault classes by evaluating the probability distribution of
feature data.
Quantitative fault detection and isolation analysis has been considered in, for example,
\cite{eriksson2013method,li2020gap}. In \cite{eriksson2013method}, the KL divergence
is used to analyze time-discrete linear descriptor models. Similar approaches have also been used
in e.g. \cite{fu2020data} and \cite{gienger2020robust}. In \cite{li2020gap}, diagnosis performance is
measured based on the distance between different kernel subspaces. One application of
quantitative analysis is sensor selection, see for example \cite{jung2020sensor,jiang2019multi}.
The KL divergence has also been used for fault detection, see for example \cite{yan2018fault,chen2018improved}.
With respect to these mentioned works, a data-driven framework is proposed here for quantitative
fault diagnosis performance analysis, open set fault classification, and fault size estimation.
Developing a mathematical model of complex systems with sufficient accuracy
for fault diagnosis purposes is a time-consuming process \cite{naderi2017data,shen2020hybrid}.
This has motivated the use of machine learning and data-driven fault diagnosis
methods to instead learn the system behavior from collected data. Still,
model-based techniques can be used to derive useful features for system
monitoring, see e.g. \cite{larsson2014gas}. One example is model-based
residuals that can filter out system dynamics and improve signal fault-to-noise ratio
which reduces the need for complex data-driven classification models to distinguish
between different fault classes \cite{jung2018combining}.
Several recent papers consider hybrid diagnosis system designs for dynamic systems
combining, e.g., model based residual generators and machine learning. In
\cite{luo2009integrated}, model-based residuals and observers are used for fault detection in
an automotive braking system where a hybrid fault isolation approach is proposed combining
model-based fault isolation and support vector machines. The same braking system case study
is also considered in \cite{slimani2018fusion} where an ensemble classifier is proposed combining
both model-based and data-driven fault detectors. In \cite{khorasgani2018methodology}, both
sensor data and residual data are used as input to a tree augmented naive Bayes fault classifier.
In \cite{zhang2019knowledge}, feature selection using neural networks is applied before training
the fault classifiers. In \cite{tidriri2018generic}, model-based residuals and sensor data are used
as inputs to a Bayesian network to perform fault classification and in \cite{matei2018classification}
model data features are extracted and fed into a neural network classifier. With respect
to previous work, the proposed diagnosis system can identify unknown fault scenarios
and estimate the fault size of known faults.
Fault size estimation is important for tracking system degradation \cite{larsson2014gas}, prognostics
\cite{daigle2012comparison} and fault reconstruction \cite{yan2007nonlinear}.
In \cite{wan2016data}, a robust linear receding horizon model-based approach is proposed for
fault estimation for linear discrete-time state space models with additive faults. In \cite{naderi2018data},
a data-driven fault estimation algorithm is evaluated on an aircraft gas turbine case study
where a linear state-space model, is estimated from data, and then used to estimate the fault size
by assuming additive fault models. In these mentioned works, the faults are included as parameters
in the system model and estimated using model-based techniques. With respect to these works, a
data-driven fault size estimation algorithm is proposed when no model of the fault is available.
Some work on data-driven fault size estimation has been done, see for
example \cite{Sawalhi2007,Guo2016}. In \cite{Sawalhi2007}, faults are assumed to
appear as pulses in the time domain data which is inherently tied to the bearing case.
In \cite{Guo2016}, Paris' formula \cite{Paris1963}, estimating crack growth in bearings, is
used to interpolate between distributions from known fault sizes. A hybrid method is
proposed in \cite{ezzat2020model} for structural health monitoring combining finite
element models and machine learning to estimate fault location and severity. In \cite{zhang2020incipient},
a fault detection and estimation scheme are proposed for incipient faults using the Jensen-Shannon
divergence measure. With respect to previous work, fault size estimation is formulated as an
optimization problem using the KL divergence and training data from different types of fault realizations.
|
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|
{"url":"http:\/\/questiun.com\/10577\/how-many-cubic-metres-sand-cement-mix-will-need-cover-area-405?show=10623","text":"Area.\n\nYou actually wants to calculate the volume in cubic meters.\n\nVolume for this solid, a cobblestone rectangle, is calculated simply multiplying area of the base by its height.\n\nReplacing the given values in the formula, we get\n\nV = ABh = 90 x 405 x 5 cm3\n\nV = 182250 cm3\n\nNow we convert to cubic meters by dividing by 100x100x100 = 100,000.\n\nV = 182250\/100,000 = 1.8225 m3 (cubic meters) (answer)\n\nso, you will need approximately 2 m3 of sand cement mix.\n\nImage from: somatematica.com.br","date":"2019-02-19 05:05:42","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8420391082763672, \"perplexity\": 3888.7086613335973}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247489343.24\/warc\/CC-MAIN-20190219041222-20190219063222-00414.warc.gz\"}"}
| null | null |
Q: Sprites with html and css I've got some code that uses sprites, this is designed to look like a horizontal navigation bar. My question is: for my code shown below, does this method actually cause the end user to download the png image 8 times or just the once.
And if it is actually once, is there a better method to do this?
CSS Code
#facebook, #twitter, #linkedin, #googleplus {
width: 64px;
height: 64px;
display: inline-block;
}
#facebook {
background: url("images/sprite.png") 0 0;
}
#twitter {
background: url("images/sprite.png") -64px 0;
}
#linkedin {
background: url("images/sprite.png") -128px 0;
}
#googleplus {
background: url("images/sprite.png") -192px 0;
}
#facebook:hover {
background: url("images/sprite.png") 0 -64px;
}
#twitter:hover {
background: url("images/sprite.png") -64px -64px;
}
#linkedin:hover {
background: url("images/sprite.png") -128px -64px;
}
#googleplus:hover {
background: url("images/sprite.png") -192px -64px;
}
HTML Code:
<nav>
<a id="facebook" href="https://www.facebook.com"></a>
<a id="twitter" href="https://www.twitter.com"></a>
<a id="linkedin" href="https://www.linkedin.com"></a>
<a id="googleplus" href="https://plus.google.com"></a>
</nav>
A: The request will happen only once and the image will be cached. The image will be redrawn again, but not by requesting the server. It requests the server only once. But it is always a good practise to have the url() in the master class or a single instance:
#facebook, #twitter, #linkedin, #googleplus {
background: url("images/sprite.png");
}
Moreover it is better to use background-position alone, instead of background, also you don't need to reuse the background: url() again in :hover.
A: It'll be downloaded only once, and this is the main reason to use CSS sprite sheets, it will be downloaded once only and with only one HTTP request instead of 8 HTTP requests if you provide an image for each icon with normal and hover states.
And no, this is how it is implemented and you're doing it correctly.
Resources:
*
*https://css-tricks.com/css-sprites/
*http://www.w3schools.com/css/css_image_sprites.asp
*https://developer.mozilla.org/en-US/docs/Web/CSS/CSS_Images/Implementing_image_sprites_in_CSS
Edit:
As Praveen Kumar mentioned you only write the url() once for all corresponding rules and with background-position shift the image either horizontally or vertically depending on your image
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 4,304
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L'Union sportive Morlaàs Rugby est un club de rugby français situé à Morlaàs (Pyrénées-Atlantiques). Il évolue actuellement en Fédérale 2.
Histoire
L'Union Morlanaise sportive et patriotique fut créée en 1911. Elle avait pour but d'organiser les activités sportives, mais aussi la préparation militaire.
C'est en 1932 que la section rugby voit le jour et que naît l'Union Sportive Morlanaise Rugby. Les premières tribunes seront construites en 1962. D'autres, plus modernes et plus spacieuses verront le jour en 1975. Cette même année, l'équipe première accède à la deuxième division.
L'équipe fanion se hissera en première division (groupe B) en 1997.
À l'issue de la saison 2006-2007, l'US Morlaàs termine en tête de sa poule de Fédérale 2 et obtient le droit de jouer en Fédérale 1 (soit la troisième division nationale) pour la saison 2007-2008. Le club avait déjà accédé à ce niveau en 1994-1995 et 2003-2004 pour redescendre après une seule saison.
Avec pour objectif le maintien en Fédérale 1 lors de la saison 2007-2008, le club a recruté intelligemment en signant trois anciens professionnels de Top 14 et de Pro D2 : Sébastien Bria, Guillaume Chasseriau et Jean-Marc Souverbie. En s'appuyant également sur des jeunes pousses du club et quelques bons joueurs des clubs alentour, Morlaàs souhaite pérenniser son statut d'équipe de Fédérale 1, pour la première fois de son histoire.
Palmarès
2004 Vice-champion de France Nationale B
Joueurs et personnalités du club
Entraîneurs
Michel Crauste
Joueurs célèbres
Sébastien Bria (pilier) : ancien joueur de la Section paloise, du Stade montois et du Tarbes Pyrénées rugby ;
Guillaume Chasseriau () : ancien joueur de La Rochelle, Pau et Mont-de-Marsan ;
Jean-Marc Souverbie (arrière / ailier) : international A' et ancien joueur de Pau, Bègles et Perpignan.
Stephen Sparks (pilier gauche) : ancien joueur de Saracens, La Rochelle, Castres olympique, Section paloise, seul pilier de Fédérale 1 à accomplir vaillamment et avec panache ses à tous les matches. Auteur notamment d'un "seul contre tous" à la fin du match contre Cahors pendant la saison Fédérale 1 2009-2010;
Arthur Iturria International A, formé à l'Ecole de Rugby de Morlaàs.
Julien Jacquot : Pilier, l'ancien capitaine de la Section Paloise a connu la Pro D2 ainsi que le TOP 14, avant de rejoindre l'USM
Thomas Seradin : Talonneur, combattant reconnu, il est passé par Nay, Pau, Lourdes.
Sébastien Laulhé : Flanker, formé à Morlaàs qui est maintenant joueur de Pro D2 du côté d'Angoulème (SA XV).
Romain Chabat : International 7's , après avoir goûté à la PROD2 au SA XV, il joue actuellement en F1 à Anglet.
Sylvain Granget: 15 ou 9, ancien joueur d'Idron et D'Espéchede XV. Formé à l'école du jeu morlanaise, a connu la F1 et la F2. Buteur très fiable, il est surtout le grand spécialiste de la chistéra. Surnoms : Granyet ou Petit G.
Francois Sarthou : Numéro 8 - Section Paloise, Lestelle XV, Aviron Bayonnais, Fenix rugby Zaragoza. International Béarnais A
Pierrick Sébie (centre): ancien joueur de la Section paloise
Frédéric Izaguirre (ailier, arrière) : ancien joueur de Hendaye, meilleur scoreur 2007-2008 de fédérale 3 à Hendaye
Baptiste Chaudière (demi d'ouverture, centre, ailier, arrière) : ancien joueur de Hendaye, élu meilleur joueur de la saison 2007-2008 de fédérale 3 à Hendaye, élu meilleur joueur et révélation de l'année 2008-2009 à Morlaàs.
Thibault Debaes : International -20 ans, il évolue au sein de la Section Paloise en Espoirs.
Liens externes
Site officiel
itsrugby.fr
Notes et références
Sport à Pau
Club de rugby à XV en Béarn
|
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| 9,604
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Q: if an ajax request brings in a chunk of javascript code, should I expect it to be executable? if an ajax request brings in a chunk of javascrpt code, should I expect it to execute?
for example, if an ajax request brings in a another ajax request and deposits it in the page, how can i make it functional?
$("select#select_host").change(function(){
$.ajax({
url: '<?php echo $sn; ?>/admini/list/is_active/'+$(this).val(),
type: 'get',
asynch: 'false',
dataType: 'text' ,
success: function(response) {
$("#list_hosts").html(response);
}
});
});
A: The technic you're talking about is called JSONP and is used all the time to bypass the same-origin-policy.
What you'll be doing, basically would be to insert <script> tags with actual Javascript per AJAX call.
Hope it points you in the right direction.
A: To paraphrase Ian Malcolm, it's not so much whether you can, it's whether you should.
There are two ways you can execute code that has come from a server and it generally depends on where that code came from.
Javascript has a function called eval(), which takes a string and executes it as Javascript. The main concern with this is when you can't be sure what that string contains. In the world of Internet security, most people would argue that you can never be sure of incoming data and so you should never use eval on incoming data.
The other method is JSONP. This method allows you to pull data from remote sources. It does this by generating a <SCRIPT> tag which pulls in remote Javascript. Your call to the JSONP source will usually include a callback function that is called upon receipt of the JSONP data, giving your local code access to it.
Without knowing precisely what you wish to achieve, I don't want to make any assumptions, but let me make some suggestions.
Calling code on demand from the server can by a little heavyweight. I can't think of many (read - any) scenario whereby calling bespoke functions from a server would be the best approach. However, I can imagine a scenario whereby you have a large set of potential functions that you might want to execute and you don't want to load them all into the browser at one time. I'd suggest two approaches:
*
*Perform the functions on the server. Just send you data to the server in some format and have it return a result set. Leave the processing to the server
*If you need a function that you don't currently have available on the browser, load in the script file containing that function, then call the function in your Javascript. Think of those files as Dynamically Loadable Libraries that you call in as and when you need them. You're limited to your own server using this method which improves security (if you trust that your own server isn't going to send malicious code.
In short, I'd argue that a design that requires you to immediately take text from a server and execute it as code may indicate a problem in that design. But you didn't come here for a design review, so hopefully those suggestions will help you find a good approach.
A: Generally, you should be able to call "eval()" with the resultant data as the parameter, and it will run the code. HOWEVER, using eval is strongly discouraged for security reasons. For example, remember to NEVER EVER evaluate code that you get from the user.
You should look into other ways to do what you want to do.
A: You have multiple ways to get to the goal. I´d advise you to take a closer look at
jQuery.live. With the .live handler you needn´t to evaluate the returning code again. It is done by jQuery.
My second advise is: Use JSON.
From PHP you can return an array with json_encode() and use the array by jQuery.each. By this, you can write the list, without eval() ;-)
("select#select_host").change(function(){
var data = {'is_active':$(this).val()};
// Use jquery.post, it´s simple and plain
jQuery.post('/admini/list', data, function (JSON) {
jQuery.each(JSON.data, function(key, list)
{
$("#list_hosts").append('<li>'+list+'</li>');
});
}, "json");
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 245
|
layout: default
---
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,075
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Q: Restrict user access from "main area" Dynamics 365 on prem We've created a new "App" which is the first app so far in our solution. The idea is that a group of users will have access to this app only and they will no longer have access to the "main area" of the application. Some of the forms for certain entities will look differently than the others etc.
(With "main area" I mean the "normal" area that exists by default more or less).
So, it was easy to give this group of users access to the new app because they have a particular security role and we've assigned this role to the app.
But now the question:
How do we restrict this group of users to only have access to this app? That is, when they browse to dynamics they should immediately "land" in the new app and should not be able to reach the "main area" anymore.
I feel that since the "default Dynamics"-app doesn't have this "Manage Roles" option I'm not sure what is the best approach to remove this particular security role from default app.
A: When you say "main area" it means "Dynamics 365 - custom". This is always accessible with url like https://xyz.crm.dynamics.com/main.aspx for everyone.
Under: Settings - Application - My Apps
It can be either hidden in Left navigation for all users excluding System Admin role or visible for all (unlike other apps).
As of today, the only way you can stop users accessing it is by train them to bookmark the app url like https://xyz.crm4.dynamcis.com/Apps/yourcustomApp to land directly & not the https://xyz.crm.dynamics.com to avoid confusion till MS enhance it.
Reference
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| 9,785
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Q: Is there a better way to say, "explain to themselves"? In the Common Core Standard for Mathematics, it says.."
"1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning
of a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. ..."
In my attempt to re-writing this, MS word reports that the grammar is incorrect because of "themselves"; MS word would prefer I use "them".
Here is what I've come up with. Can someone offer an alternative and/or explain why it is wrong?
"Confidently, dive right into interpreting a problem, and explain it to themselves to look for entry points to a solution."
A: The sentence is not incorrect. Thus, you do not need to re-write it. Word processor grammar checkers don't actually speak English; instead, they use rules and heuristics that have exceptions that are correct but get flagged as incorrect.
Your version's MS Word's grammar checker is probably using some kind of heuristic to flag the word "themselves" because it could be a misused word, especially following the word "to." There are few situations where that could be correct, but many situations where "to them" would be correct--but MS Word isn't paying attention to the verb that comes before "to themselves" in order to know that this instance of "to themselves" is correct (although odd).
This Google NGram shows how much more frequently the phrase "to them" occurs compared to "to themselves."
So the upshot is that you can't trust grammar checkers to be correct in all situations. They're just computer programs, and they are doing what they are told to do.
(Incidentally, the version of MS Word that I am using right now does not flag "themselves" as incorrect in the example sentence in your question.)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,934
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Q: iPhone 6 simulator doesn't work properly My code for ball going to end of bottom of screen and then regenerate back at top of screen...
if(ball.center.y >666){
ball.center=CGPointMake(ONE.center.x, -30);
In iPhone 6 simulator, the farthest the ball goes down is ball.center.y >585 and only then it regenerates. But how come it doesn't go all the way down to ball.center.y >666?
I have retina 4.7HD and retina 5.5HD launch images placed.
Whether I use auto layout and size classes or no auto layout and no size classes the ball doesn't go lower than ball.center.y >585.
Please help, i don't know how to solve this problem.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,180
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Harishankar Road railway station is a railway station on the East Coast Railway network in the state of Odisha, India. It serves Lathore village. Its code is HSK. It has two platforms. Passenger, Express and Superfast trains halt at Harishankar Road railway station.
Major trains
Korba–Visakhapatnam Express
Puri–Durg Express
Samata Express
Bilaspur–Tirupati Express
See also
Balangir district
References
Railway stations in Balangir district
Sambalpur railway division
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,833
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Amazon's Alexa Still Keeps Your Voice Files After You Delete Them
Sun, July 7, 2019 at 4:42 AM
Despite being a truly ingenious idea which continues to be a great help to folks, Amazon's Alexa has only managed to raise suspicion amongst many of us. We previously reported on the digital voice assistant device recording a couple's private conversation. Moreover, the same couple reported that the voice-controlled device even laughed at them while they were having a confidential discussion. Evidently, there remain a few things about Alexa that raises the hairs on the skin. And now, reports by HYPEBEAST have only increased the level of suspicion surrounding the Amazon product by indicating that the voice assistant still retains your recorded audio files after you delete them. According to the aforementioned news outlet, the news came after the billion dollar company sent a letter to Senator Chris Coons.
Stephen Brashear/Getty Images
Herein, the latter delved into the company's privacy practices when it comes to Alexa. They addressed the question posed by Coons on how long the company held recordings and transcripts from the Echo. "We retain customers' voice recordings and transcripts until the customer chooses to delete them." Yet in another instance, they confirmed that they "do not store the audio of Alexa's response. However, we may still retain other records of customers' Alexa interactions, including records of actions Alexa took in response to the customer's request." Croons addressed the urging matter and pressed the company to let users know that "user voice interactions with Alexa are not deleted from all of Amazon's servers, even after a user has deleted a recording of his or her voice." Senator Croons believed it was a service owed to the American people. You can read the full letter here.
via: https://www.hotnewhiphop.com/amazons-alexa-still-keeps-your-voice-files-after-you-delete-them-news.85047.html
article, Amazon's Alexa Still Keeps Your Voice Files After You Delete Them
"Stunna 4 Vegas & DaBaby Deliver "Fresh Prince Of Bel-Air" Visual For "Ashley" Single"
"Apryl Jones Praises Fizz & Shares Strained Relationship With Omarion"
Rahiem Shabazz
Sniper Squad DJs
Bianca Bee
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,651
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{"url":"http:\/\/math.stackexchange.com\/questions\/42961\/if-g-is-a-locally-compact-hausdorff-group-when-does-g-z-have-a-probability","text":"# If $G$ is a locally compact Hausdorff group, when does $G\/Z$ have a probability Haar measure?\n\nI am reading an introductory material about topological groups and the question in the tittle comes up. Due this Proposition\n\nProposition. A locally compact Hausdorff topological group $G$ is compact, if and only if, $\\mu(G)<+\\infty\\qquad$ ($\\mu$ is the Haar measure of $G$).\n\nit is enough to know what are the groups $G$ for which $G\/Z$, where $Z$ is the center of $G$, is a compact group. Are those groups well-known ?\n\n-\nYou are asking about the classification up to isomorphism of (topological) central extensions $0\\to A\\to G\\to K\\to1$ with $A$ locally compact abelian and $K$ compact. These extensions with $K$ and $A$ fixed are classified by the second continuous cohomology group $H_{c}^2(K,A)$. Phrasing it this way it is rather clear that full answer is rather impossible. For specific (nice) groups $K$ (e.g. semi-simple Lie) and $A$ not too bad (e.g. $\\mathbb{R,Z,R\/Z}$ and finite products thereof) the answer should be known but I don't have the relevant texts (Guichardet, Borel-Wallach) handy at the moment. \u2013\u00a0 t.b. Jun 3 '11 at 6:28\nIn my first sentence I was a bit ambiguous. I meant: you are asking in particular about the classification up to isomorphism of all $G$ arising as such topological central extensions. \u2013\u00a0 t.b. Jun 3 '11 at 6:37\nHi Theo, thanks for all the informations. It is simple to give an example where $G\/Z$ is not compact ? \u2013\u00a0 Leandro Jun 3 '11 at 7:05\nHi Leandro This should be a comment rather than an answer, but I can't currently login from this computer. A very simple example of a non-compact quotient group is $PSL_2 (\\mathbb{R}) = SL_2 (\\mathbb{R})\/\\{\\pm 1\\}$ where $\\pm 1$ is the center of $SL_2 (\\mathbb{R})$ consisting of $\\pm$ the identity matrix. More generally, if $G$ is a non-compact semi-simple linear Lie group then $G\/Z(G)$ will be non-compact. I hope this answers your follow-up question. Theo \u2013\u00a0 t.b. Jun 3 '11 at 8:57\nHi Theo, thanks a lot. I first thought about some matriz groups but I wrong convinced myself that they were compacts. The collection of example you provided let more clear that is in pratical very hard to classify all the groups with that property. \u2013\u00a0 Leandro Jun 3 '11 at 16:16","date":"2015-04-21 05:52:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8171496987342834, \"perplexity\": 359.55123504633417}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246640124.22\/warc\/CC-MAIN-20150417045720-00189-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
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\section{Introduction}
Image-to-image translation aims to learn a mapping between different groups of visually distinguishable images.
Based on the success of generative adversarial networks \cite{goodfellow2014generative}, recent studies have shown a remarkable ability to learn and render intricate appearance changes across image classes \cite{zhu2017unpaired, kim2017learning, yi2017dualgan, liu2017unsupervised, choi2018stargan, liu2019few, choi2020stargan}.
These approaches rely on domain labels to specify visually different groups, which often significantly limits the scope of applications.
It's because inconsistent or coarse labels easily degrade translation results while collecting high-quality labels is usually expensive.
To overcome such limitations, more recent studies explore an unsupervised approach \cite{bahng2020exploring, baek2020rethinking}.
These methods leverage image clustering techniques to identify distinct groups of images without ground-truth labels.
Specifically, they produce pseudo-labels of unlabeled images by clustering algorithms and learn a mapping between these estimated groups using a multi-domain discriminator.
While these methods have shown promising results, it is usually difficult to produce accurate clusters when the data distribution is unbalanced or the predefined configurations are misleading.
For example, the clustering methods require the user to specify the number of clusters, which is generally difficult to choose without prior knowledge, despite having a significant impact on the results.
Instead, we propose an alternative, rather straightforward way to circumvent these limitations.
Our key idea is to supervise the image translation process with visual similarity captured by contrastive learning \cite{hadsell2006dimensionality, he2020momentum, chen2020simple} instead of learning separate domain distribution.
Contrastive learning aims to learn a useful representation by contrasting between positive pairs and negative pairs.
Through the contrasting process, the representations of similar samples are mapped closely together while dissimilar samples are mapped further away.
We observe that the contrastive learning captures meaningful visual features that form the unique appearance of individual image, which we call a \textit{style}.
Conversely, we call the characteristics shared among all images (e.g., pose) a \textit{structure}.
We enable effective, unsupervised image-to-image translation using contrastive learning to transfer styles between images.
Specifically, we introduce a contrastive discriminator that has two branches, one for contrastive representation and another for typical GAN logit.
During training, the discriminator is provided with not only real and fake images but also an augmented version of real images to learn the contrastive representation.
The representation is utilized to induce an output image to have a style similar to a given reference image.
Furthermore, we introduce a patch aggregation method that effectively mitigates the structure loss problem of output images.
Empirical results demonstrate the advantage of our method over existing unsupervised baselines in terms of visual quality and translation accuracy.
Moreover, we analyze the latent style space to verify whether the space learns semantically meaningful styles.
Through extensive ablation studies, we investigate the effect of individual components including data augmentation policy and contrastive style matching strategies for further insights.
\section{Related Work}
\subsection{Multi-domain image-to-image translation}
Most of the existing image-to-image translation methods are based on the conditional GAN \cite{mirza2014conditional}.
They learn domain-specific discriminators to identify whether the generated image belongs to the target domain while imposing additional constraints like cycle consistency \cite{zhu2017unpaired} to preserve domain-invariant characteristics.
While earlier methods consider a bijective mapping between two domains \cite{zhu2017unpaired, liu2017unsupervised, kim2017learning, yi2017dualgan}, more recent works present multi-domain methods that enable translation across multiple domains with a unified model \cite{choi2018stargan, liu2019few, choi2020stargan}.
These methods employ an auxiliary domain classifier \cite{choi2018stargan} or a multi-task discriminator \cite{liu2019few, choi2020stargan} for scalable multi-domain translation.
While successful, these methods rely on a vast quantity of domain labels, which often becomes a serious bottleneck.
To reduce such appetite for labels, more recent studies propose fully unsupervised methods that leverage pseudo-labels acquired by the image clustering methods \cite{bahng2020exploring, baek2020rethinking}.
However these methods easily yield unintended translation results if the clustering algorithms fail to produce consistent clusters.
Our work differs in that we present a model that implicitly learns and transfers visual features without explicit domain separation.
\subsection{Style transfer}
Our work can be seen as a photorealistic style transfer task as we aim to transfer the style of an image to another while preserving its structure.
Photorealistic style transfer methods attempt to transform an image into its stylized version reflecting arbitrary reference images \cite{luan2017deep, li2018closed, yoo2019photorealistic}.
However, these works presume style as color distributions and mainly focus on local color change rather than capturing semantically meaningful features.
Recently, SwappingAutoencoder \cite{park2020swapping} presented a code swap-based autoencoder that transfers texture information between images by learning co-occurrence patch statistics between images.
While it succeeds in capturing local semantic features, it often fails to capture higher-level semantics such as the complex appearance of animals, since the model highly relies on the local patch statistics.
In contrast, our model learns distinctive visual features that form higher-level semantics by contrastive learning.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=1.0\linewidth]{figures/overview.png}
\caption{An overview of our proposed method, CLUIT. The method consists of three modules: a generator $G$, a style encoder $E$, and a contrastive discriminator $D$. Using a latent style code from the encoder, the generator synthesizes an output image. Input and reference images are randomly sampled within a batch when training. The discriminator learns the real image distribution from adversarial loss. At the same time, it exploits data augmentation techniques to learn image representations through contrastive loss. The generator is guided to synthesize an image with a visual style similar to the reference image through contrastive match loss.}
\label{fig:overview}
\end{figure*}
\subsection{Unsupervised representation learning}
Our work is also closely related to unsupervised representation learning techniques in that we aim to capture and transfer such meaningful visual features without supervision.
Various pretext tasks are designed for this purpose, such as image reconstruction \cite{hinton2006reducing}, denoising \cite{vincent2008extracting}, inpainting \cite{pathak2016context}, colorization \cite{larsson2017colorization}, context prediction \cite{noroozi2016unsupervised, santa2017deeppermnet}, and geometric prediction \cite{noroozi2017representation, liu2019exploiting}.
More recent approaches focus on learning representations that maximize the mutual information between examples that are assumed to be closely related.
These methods employ a contrastive loss to attract the associated examples closer than other negatives \cite{oord2018representation, bachman2019learning, he2020momentum, chen2020simple}.
Through linear evaluation and various down-stream tasks, contrastive learning has shown the ability to capture complex visual features.
In this work, we present a method that effectively utilizes those capabilities in an image-to-image translation setting, which has not been explored before.
\section{Method}
Given an unlabeled set of images $X \subset \mathbb{R}^{H \times W \times C}$, we aim to translate an input image $x_o \in X$ to appear like a reference image $x_r \in X$ while preserving the class-invariant characteristics (i.e., pose).
To this end, we propose a contrastive learning based unsupervised image-to-image translation framework, namely \textit{CLUIT}, consisting of three modules: (1) a contrastive discriminator $D$ that learns visual styles of images while guiding a generator to synthesize realistic images, (2) a style encoder $E$ that encodes an image into latent style code and (3) a generator $G$ that translates the input image $x_o$ into a corresponding output reflecting the encoded style of the reference image $x_r$.
Figure \ref{fig:overview} illustrates an overview of our framework.
\subsection{Contrastive discriminator}
Our discriminator $D$ is designed to have two output branches above shared layers, each of which produces a contrastive representation $v = D_{\text{ct}}(x)$ and a typical GAN logit $D_{\text{adv}}(x)$ for a given image $x$.
The contrastive representation learns the visual styles of images by maximizing the mutual information between an image $x$ and its augmented version $x^+$ in contrast to other negative images $x^-$ within the dataset.
Specifically, the images $x$, $x^+$, and $N$ negatives are mapped into $K$-dimensional vectors $v, v^+ \in \mathbb{R}^K$ and $v_i^- \in \mathbb{R}^{K}$, respectively.
The vectors are normalized to prevent collapsing and then used for ($N + 1$)-way classification as follows:
\begin{equation} \label{eq:loss-ct-d}
\mathcal{L}_{\text{ct}}^D = - \log \frac{
\textstyle \exp(v \cdot v^+ / \tau)
}{
\textstyle \exp(v \cdot v^+ / \tau)
+
\textstyle \sum_{i=1}^N \exp(v \cdot v^-_i / \tau)
},
\end{equation}
where $\tau$ is a temperature scale factor.
We maintain a large dictionary of negative examples using a memory bank architecture following MOCO \cite{he2020momentum}.
The contrastive representation provides proper guidance for the generator to transfer styles between images, which we describe in Section \ref{section:ct}.
We also impose the adversarial loss to encourage the generator to synthesize realistic images:
\begin{equation} \label{eq:loss-adv}
\mathcal{L}_{\text{adv}} =
\mathbb{E} \left[ \log D_{\text{adv}}(x_o) \right]
+
\mathbb{E} \left[\log(1 - D_{\text{adv}}(x_g)) \right],
\end{equation}
where $x_g$ is a synthesized image which we describe in the following subsection.
\subsection{Synthesizing a hybrid image with locality-preserving generator}
During training, we randomly sample two images from dataset and use one image as an input image and another as a reference image that provides a style reference.
We notate the input and the reference image as $x_o$ and $x_r$, respectively.
In practice, we use the first half of batch images as the input images and the rest as the reference images.
A style encoder $E$ is used to extract the style information from the reference image $x_r$ by encoding the image into corresponding latent style code $t_r = E(x_r)$.
Our generator $G$ learns to translate the input image $x_o$ into the output image $x_g = G(x_o, t_r)$ using the encoded style code $t_r$.
We adopt the locality-preserving architecture as our generator, which uses adaptive instance normalization (AdaIN \cite{huang2017arbitrary}) to inject the latent style code $t_r$ into $G$.
We also generate a reconstructed version of the input image $x_o$ to impose the cycle consistency constraint \cite{zhu2017unpaired} to ensure that the generated image $x_g$ preserves the class-invariant characteristics (i.e., pose) of the input image:
\begin{equation} \label{eq:loss-cyc}
\mathcal{L}_{\text{cyc}} = \mathbb{E} \left[ \left\| x_o - G(x_g, t_o) \right\|_1 \right].
\end{equation}
\subsection{Style transfer via contrastive style match}
\label{section:ct}
Most of the existing methods transfer the class-specific style by learning class-specific discriminators to identify whether the output image is from the given target class.
We instead impose a style matching constraint defined on the contrastive representation space of the discriminator.
We use the same form of contrastive loss as used for learning the discriminator but compute the loss using the contrastive representations of the output image $x_g$ and the reference image $x_r$ to guide $x_g$ to have a style similar to $x_r$:
\begin{equation} \label{eq:loss-ct-g}
\mathcal{L}_{\text{ct}}^G = \mathbb{E} \left[
- \log \frac{
\textstyle \exp(v_g \cdot v_r / \tau)
}{
\textstyle \exp(v_g \cdot v_r / \tau)
+
\textstyle \sum_{i=1}^N \exp(v_g \cdot v^-_i / \tau)
}
\right],
\end{equation}
where $v_g = D_{\text{ct}}(x_g)$ and $v_r = D_{\text{ct}}(x_r)$ denote the respective contrastive representation of $x_g$ and $x_r$.
The negatives are sampled from the same dictionary used to learn the discriminator.
Through the contrastive style match, the model can transfer the style of the reference image to the resulting image.
However, we observe that the model often suffers from the \textit{structure loss} problem where the output images emulate the reference images rather than preserving the structure of the input images (see Figure \ref{fig:matching}).
To alleviate this problem, we propose using a \textit{patch aggregation} based style match.
The key idea is to use multiple local patch representations instead of a single full-image representation to prevent the output images from emulating the reference.
Specifically, $M$ patches of size 1/8 to 1 of the full image dimension are randomly selected from both the output and the reference image, and compute the style match loss using the mean representation of patches in each image:
\begin{equation}
v = \frac{1}{M} \sum_{i=1}^M D_{\text{ct}}(x^{(i)}),
\end{equation}
where $x^{(i)}$ denote $i$-th patch from image $x$.
\subsection{Full objective}
Finally, our full objective functions can be written as:
\begin{equation} \label{eq:full}
\begin{split}
\mathcal{L}_D &= -\mathcal{L}_{\text{adv}} + \lambda_{\text{ct}}^D \mathcal{L}_{\text{ct}}^D,
\\
\mathcal{L}_{G, E} &= \mathcal{L}_{\text{adv}} + \lambda_{\text{cyc}} \mathcal{L}_{\text{cyc}} + \lambda_{\text{ct}}^G \mathcal{L}_{\text{ct}}^G,
\end{split}
\end{equation}
where $\lambda_{\text{ct}}^D$, $\lambda_{\text{ct}}^G$ and $\lambda_{\text{cyc}}$ are hyper-parameters for each term.
\section{Experiments}
In this section, we describe the evaluation setups and experimental results.
We compare the proposed method to the state-of-the-art methods and present an additional analysis of our method for further insights.
\vspace{0.2cm}
\noindent \textbf{Datasets.}
We evaluate our method on the following datasets and all the images are resized to $128 \times 128$ resolution for a fair comparison.
AFHQ \cite{choi2020stargan} contains 15,000 high-quality animal face images which are roughly categorized into three classes (i.e., cat, dog, and wildlife).
CelebA \cite{liu2015deep} contains 202,599 face images of celebrities annotated with 40 binary facial attributes. We initially center-crop the images to $178 \times 178$ size and then resize them to $128 \times 128$.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=1.0\linewidth]{figures/comparison-afhq.png}
\caption{Qualitative comparison of the proposed method and other baselines in AFHQ dataset. Each method translates the input images (left-most column of each grid) into corresponding output, reflecting the styles of the reference images (top-most row). The images are generated by each method specified in the left.}
\label{fig:comparison-afhq}
\end{figure*}
\vspace{0.2cm}
\noindent \textbf{Evaluation protocol.}
All the methods translate an input image into its corresponding output given a single reference image.
We use the test set images from each dataset as the input images and each input image is translated using 10 reference images randomly sampled from the test set resulting in 10 output images.
We use the same set of random reference images for all the competing approaches.
We evaluate both the visual quality and the translation accuracy of generated images.
To measure the visual quality of generated images, we adopt Frechet inception distance (FID) \cite{heusel2017gans} which empirically estimates the distribution of real and generated images using pre-trained Inception-v3 \cite{simonyan2014very}.
In AFHQ dataset, we report the mean of class-wise FID using the ground-truth labels of reference images to penalize the case when a model produces realistic reconstructions of input images.
Furthermore, we report the translation accuracy measured by the pre-trained ResNet-50 \cite{he2016deep} classifier to determine whether the translation output belongs to the reference image's domain.
For CelebA dataset, we select 12 binary attributes (see Figure \ref{fig:attribute} for more details) that are considered invariant to the structure of images so that we can verify the transferability of the pose-invariant style.
\vspace{0.2cm}
\noindent \textbf{Baselines.}
We use TUNIT \cite{baek2020rethinking} and SwapAE \cite{park2020swapping} as our unsupervised baselines.
In the case of TUNIT, the number of clusters must be specified in advance, but it is difficult to know the optimal number for each dataset.
Therefore, we train the model with varying numbers ($k = 3, 10, 30$) of clusters.
We also report the results of StarGAN-v2 \cite{choi2020stargan} in AFHQ dataset to confirm the results of the leading supervised method.
\vspace{0.2cm}
\noindent \textbf{Implementation.}
We set $\lambda_{\text{cyc}} = 1$ and $\lambda_{\text{ct}}^D = \lambda_{\text{ct}}^G = 0.1$ in Equation \ref{eq:full}.
We use $M = 4$ patches in all experiments except for the contrastive style match ablation.
The batch size is set to 32.
We adopt the non-saturating adversarial loss \cite{goodfellow2014generative} with lazy $R_1$ regularization \cite{mescheder2018training} using $\gamma = 1$ for CelebA and $\gamma = 10$ for AFHQ.
We use Adam \cite{kingma2014adam} optimizer with $\beta_1 = 0$ and $\beta_2 = 0.99$.
The learning rate is set to $5 \times 10^{-5}$.
We train the model for 100,000 iterations in total.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=1.0\linewidth]{figures/comparison-celeba.png}
\caption{Qualitative comparison of the proposed method and other baselines in CelebA dataset. Each method translates the input images (left-most column of each grid) into corresponding output, reflecting the styles of the reference images (top-most row). The images are generated by each method specified in the left.}
\label{fig:comparison-celeba}
\end{figure*}
\subsection{Comparison to baselines}
The representative results of CLUIT are presented in Figure \ref{fig:teaser}.
The results show that CLUIT can successfully transform the input images into corresponding outputs, reflecting distinctive visual styles of reference images.
Figure \ref{fig:comparison-afhq} illustrates the comparison results with the baselines in AFHQ dataset.
All the methods generate clear images, but they differ significantly in terms of the style being transferred.
As expected, StarGAN-v2 successfully transfers the style of reference images since it utilizes the ground truth labels to learn class-specific distribution.
However, we observe that StarGAN-v2 often fails to capture intra-class styles if the images are coarsely labeled.
For example, the "wildlife" class contains a much wider variety of animals than the other classes (i.e., "cat" and "dog").
In this case, StarGAN-v2 translates an image into a rather random species in the "wildlife" category, not reflecting a given reference image.
In Figure \ref{fig:comparison-afhq}, StarGAN-v2 translates a cat into a fox even if a leopard is given as a reference.
The results of TUNIT have high variances depending on the number of pre-defined clusters.
When the number of clusters is small ($k = 3$), TUNIT only allows the local texture or color change rather than overall appearance.
Even with the best performing configuration ($k = 10$), the model often fails to differentiate between visually similar species such as cats and foxes.
In the case of SwapAE, only the color and texture distributions are matched with the reference images.
On the other hand, our method successfully transfers the distinctive features, including overall shapes and local details (e.g., shape of a face part, fur pattern, skin color, etc.).
\begin{table}
\centering
\begin{tabular}{|l|r|r|r|r|}
\hline
\multirow[b]{2}{*}{Method} & \multicolumn{2}{|c|}{AFHQ} & \multicolumn{2}{c|}{CelebA} \\
\cline{2-5}
& mFID$\downarrow$ & Acc$\uparrow$ & FID$\downarrow$ & Acc$\uparrow$ \\
\hline
\hline
StarGan-v2* & 14.6 & 97.0 & - & - \\
\hline
TUNIT-3 & 82.1 & 40.5 & 8.3 & 77.3 \\
TUNIT-10 & 39.5 & 77.6 & 5.3 & 78.6 \\
TUNIT-30 & 48.9 & 66.7 & 6.1 & 81.7 \\
SwapAE & 65.3 & 52.5 & 6.2 & 77.5 \\
CLUIT (ours) & \textbf{17.8} & \textbf{96.1} & \textbf{5.0} & \textbf{90.1} \\
\hline
\end{tabular}
\caption{Quantitative comparison of the proposed method and other baselines. StarGAN-v2* is a supervised model trained with ground-truth labels. $K$ in TUNIT-$K$ means the predefined number of clusters. $\uparrow$ means larger numbers are better, $\downarrow$ means smaller numbers are better.}
\label{tab:quantitative}
\end{table}
As shown in Table \ref{tab:quantitative}, our method outperforms all the unsupervised baselines by a large margin even comparable to the supervised baseline, in terms of class-specific visual quality (mFID) and translation accuracy.
\begin{figure}[!tbp]
\centering
\includegraphics[width=1.0\linewidth]{figures/attribute.png}
\caption{Translation accuracy of individual attributes in CelebA dataset. 12 attributes considered to be invariant of different poses are selected. Note that we show the average accuracy of hair color related attributes for brevity. They are BlackHair, BlondHair, BrownHair, and GrayHair.}
\label{fig:attribute}
\end{figure}
For CelebA (Figure \ref{fig:comparison-celeba}), our method renders distinctive visual features of reference images in various poses.
For other unsupervised baselines, the visual features are conveyed, but they are rather partial or incomplete, often resulting in unrealistic images.
For example, all the methods modify the hair color following the reference images but only CLUIT succeeds in drawing other hair details such as hair texture and hair length (e.g., long blond wavy hair or short black wavy hair).
Other details, including eyeglasses, beards, wrinkles, and make-up styles are also successfully transferred to render different views with consistent styles rather than random blends.
Improvements in terms of both FID and translation accuracy support our observation (Table \ref{tab:quantitative}).
Figure \ref{fig:attribute} illustrates the translation accuracy of individual attributes.
We observe that the abstract style-related attributes (e.g., gender or age) show larger margins than relatively low-level attributes (e.g., hair color) or regional attributes (e.g., goatee).
This observation demonstrates that CLUIT captures and transfers higher-level visual styles via contrastive learning.
\subsection{Analysis and ablation study}
In this section, we first analyze the latent style space to verify whether the style encoder learns a smooth and semantically meaningful feature space without any direct constraints.
We also conduct an ablation study to understand the effect of different data augmentation schemes and patch aggregation methods in contrastive style match.
\vspace{0.2cm}
\noindent \textbf{Latent style space analysis.}
There are two desired properties for latent style codes.
First, they should learn meaningful visual features.
Also, the learned visual features should be invariant to different structures.
To figure out if these properties are satisfied, we conduct two additional analysis tasks; similarity search and linear interpolation.
\begin{figure}[!tp]
\centering
\includegraphics[width=1.0\linewidth]{figures/retrieval.png}
\caption{Similarity search results on AFHQ dataset. The first column shows query images, and the next five columns show the search results sorted by the cosine similarity of latent style codes. We can observe that style codes successfully represent high-level class information such as species.}
\label{fig:retrieval}
\end{figure}
For similarity search, all test images are encoded into corresponding latent style codes and normalized to calculate the cosine similarity between pairs of images.
Figure \ref{fig:retrieval} illustrates the images sorted by the similarity with given query images.
We observe that even if the pose or background is different, images of the same species and similar style as the query image are retrieved.
The results illustrate that the style codes capture such visual features invariant to structural changes.
Figure \ref{fig:interpolation} demonstrates the linear interpolation results by keeping the input image fixed while interpolating the style code between the input and given reference image.
The interpolated samples smoothly change towards the reference species while preserving the overall structure, such as the position of face parts and the head direction.
\begin{figure}[!tp]
\centering
\includegraphics[width=1.0\linewidth]{figures/interpolation.png}
\caption{Linear interpolation result on AFHQ dataset. The first and last columns show the input and reference images, respectively. The middle four columns show images generated by interpolating the style codes of the input and the reference while keeping the input image fixed. The results indicate that as the portion of the style code from the reference increases, high-level visual features such as ear and nose shapes gradually change.}
\label{fig:interpolation}
\end{figure}
\begin{table}
\centering
\begin{tabular}{|l|c|r|r|}
\hline
Augmentation & Crop Size & mFID$\downarrow$ & Accuracy$\uparrow$ \\
\hline
\hline
Color & 0.125 - 1.0 & 21.25 & 94.9 \\
Color + Affine & 0.125 - 1.0 & \textbf{17.84} & 96.1 \\
Color + Affine & 0.5 - 1.0 & 18.12 & \textbf{96.6} \\
Color + Affine & 0.9 - 1.0 & 20.71 & 95.8 \\
\hline
\end{tabular}
\caption{Performance of CLUIT with different data augmentation schemes. The results indicate that CLUIT benefits from affine transformations and access to small crop size.}
\label{tab:augmentation}
\end{table}
\vspace{0.2cm}
\noindent \textbf{Data augmentation.}
Table \ref{tab:augmentation} shows the effect of different data augmentation schemes.
We start with a set of transformations that includes random cropping, horizontal flipping, and color distortion, following the state-of-the-art unsupervised representation learning \cite{chen2020simple}.
While affine transformations are known to be relatively less useful with contrastive learning, we observe that our model benefits from them in the context of image-to-image translation.
This is because the translation task aims to transfer geometry-invariant features to images with an arbitrary pose.
Thus, geometric transformations help the model to generalize better.
The affine transformations used for experiments are rotation, shear, and shift transformation.
Random crop size also affects the results.
Access to small size patches (i.e., 1/8 of image size) helps the model learn local patterns such as eye shape and fur pattern.
Using only large patches decreases the visual quality and translation accuracy.
We use random cropping with a crop size of 0.125-1.0, horizontal flipping, affine transformation, and color transformation as our default set of transformations for all experiments.
\begin{table}
\centering
\begin{tabular}{|l|r|r|}
\hline
Matching Strategy & mFID$\downarrow$ & Accuracy$\uparrow$ \\
\hline
\hline
Full Image & 19.08 & 95.8 \\
PatchMean-1 & 20.43 & 93.0 \\
PatchMean-4 & \textbf{17.84} & 96.1 \\
PatchMean-8 & 17.98 & \textbf{96.4} \\
\hline
\end{tabular}
\caption{Performance of CLUIT with different contrastive style match schemes. $K$ in PatchMean-$K$ denotes the number of patches sampled for patch aggregation. The results show that the patch aggregation are effective not only for mitigating structure loss problem, but also for translation capabilities.}
\label{tab:matching}
\end{table}
\vspace{0.2cm}
\noindent \textbf{Contrastive style match.}
Figure \ref{fig:matching} illustrates that the patch aggregation effectively mitigates the structure loss in the output images.
When calculating the style match loss using the representation of the full images, the resulting image is more related to the structure of the reference images (e.g., head size, body position) rather than maintaining the structure of the input images.
Conversely, when patch aggregation is applied, the output images consistently retain the structure of the input images.
In addition, loss of structure tends to introduce more artifacts, resulting in slight performance degradation (see Table \ref{tab:matching}).
We also confirm that using more than a certain amount of patches will benefit performance.
\section{Conclusion}
In this paper, we introduce an unsupervised image-to-image translation method that learns to transfer distinctive styles between images.
To this end, we designed a specialized multi-task discriminator that learns meaningful visual representation via contrastive learning.
The image translation process is supervised by the learned visual representation of the discriminator therefore does not require domain labels or their estimates.
The proposed framework sets the model free from errors propagated from inconsistent or coarse labels or inaccurate estimation of class labels, leading to better visual quality and translation accuracy.
Experimental results show that our CLUIT outperforms the state-of-the-art unsupervised image-to-image translation methods and is even comparable to the leading supervised method.
\begin{figure}[!tbp]
\centering
\includegraphics[width=1.0\linewidth]{figures/matching.png}
\caption{Comparison results of different style-matching strategies. The second line is the result of applying the patch aggregation, and the third line used the representation of the full image match. Without the patch aggregation, the output often tends to be a modified version of the reference, ignoring the structure of the input.}
\label{fig:matching}
\end{figure}
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The Debates are the report—transcribed, edited, and corrected—of what is said in the House. The Journals are the official record of the decisions and other transactions of the House. The Order Paper and Notice Paper contains the listing of all items that may be brought forward on a particular sitting day, and notices for upcoming items.
To access Debates and Journals from 1867 to 1993, please visit the Canadian Parliamentary Historical Resources portal.
41st Parliament, 1st Session (June 2, 2011 - September 13, 2013)
Order Paper and Notice Paper
User Guide XML PDF
EDITED HANSARD • NUMBER 179
Helping Families in Need Act
The Acting Speaker (Mr. Barry Devolin)
Hon. Jim Flaherty (10:05)
Hon. Jim Flaherty
Mr. Brad Butt (Mississauga—Streetsville, CPC) (10:10) (10:15) (10:20)
Ms. Chris Charlton (Hamilton Mountain, NDP) (10:25)
Mr. Brad Butt
Mr. Kevin Lamoureux (Winnipeg North, Lib.) (10:30)
Mr. John Carmichael (Don Valley West, CPC)
Ms. Chris Charlton (Hamilton Mountain, NDP) (10:35) (10:40) (10:45) (10:50)
Mr. Harold Albrecht (Kitchener—Conestoga, CPC)
Ms. Chris Charlton (10:55)
Mr. Rodger Cuzner (Cape Breton—Canso, Lib.)
Ms. Chris Charlton
STATEMENTS BY MEMBERS
Canada-Ukraine Parliamentary Program
Mr. Peter Goldring (Edmonton East, Ind. Cons.)
Mr. Ed Holder (London West, CPC)
Hamilton Steel Workers' Cenotaph
Mr. Wayne Marston (Hamilton East—Stoney Creek, NDP)
Queen's Diamond Jubilee Medal
Mr. Ed Komarnicki (Souris—Moose Mountain, CPC)
Family Doctor Week (11:05)
Hon. Carolyn Bennett (St. Paul's, Lib.)
Hockey Night in Leeds and Grenville
Mr. Gordon Brown (Leeds—Grenville, CPC)
Mrs. Anne-Marie Day (Charlesbourg—Haute-Saint-Charles, NDP)
Portage—Lisgar
Ms. Candice Bergen (Portage—Lisgar, CPC)
Canadian Young Scientist Journal Awards
Mr. Chungsen Leung (Willowdale, CPC)
Birthday Congratulations (11:10)
Mr. Dan Harris (Scarborough Southwest, NDP)
Mr. Dan Albas (Okanagan—Coquihalla, CPC)
Media Literacy Week
Mr. Charlie Angus (Timmins—James Bay, NDP)
Mr. Jacques Gourde (Lotbinière—Chutes-de-la-Chaudière, CPC)
Hon. Irwin Cotler (Mount Royal, Lib.)
New Democratic Party of Canada
Mr. Wladyslaw Lizon (Mississauga East—Cooksville, CPC)
Ms. Jinny Jogindera Sims (Newton—North Delta, NDP)
Mr. Costas Menegakis (Richmond Hill, CPC)
Ms. Nycole Turmel (Hull—Aylmer, NDP)
Ms. Kellie Leitch (Parliamentary Secretary to the Minister of Human Resources and Skills Development and to the Minister of Labour, CPC) (11:20)
Ms. Kellie Leitch (Parliamentary Secretary to the Minister of Human Resources and Skills Development and to the Minister of Labour, CPC)
Hon. James Moore (Minister of Canadian Heritage and Official Languages, CPC)
Ms. Megan Leslie (Halifax, NDP)
Parks Canada
Hon. Peter Kent (Minister of the Environment, CPC)
Correctional Service Canada (11:25)
Hon. Bob Rae (Toronto Centre, Lib.)
Hon. Bob Rae
Mr. Randall Garrison (Esquimalt—Juan de Fuca, NDP)
Hon. Vic Toews (Minister of Public Safety, CPC) (11:30)
Hon. Vic Toews (Minister of Public Safety, CPC)
Ms. Françoise Boivin (Gatineau, NDP)
Hon. Rob Nicholson (Minister of Justice and Attorney General of Canada, CPC)
Mr. Robert Chisholm (Dartmouth—Cole Harbour, NDP)
Hon. Rob Nicholson (Minister of Justice and Attorney General of Canada, CPC) (11:35)
Hon. Peter Penashue (Minister of Intergovernmental Affairs and President of the Queen's Privy Council for Canada, CPC)
41st General Election
Hon. Peter Van Loan (Leader of the Government in the House of Commons, CPC)
Mr. Alexandre Boulerice (Rosemont—La Petite-Patrie, NDP)
Mr. Pierre Poilievre (Parliamentary Secretary to the Minister of Transport, Infrastructure and Communities and for the Federal Economic Development Agency for Southern Ontario, CPC)
Mr. Craig Scott (Toronto—Danforth, NDP)
Hon. Peter Van Loan (Leader of the Government in the House of Commons, CPC) (11:40)
Correctional Service Canada
Mr. Scott Andrews (Avalon, Lib.)
Hon. Peter Penashue
Ms. Hélène LeBlanc (LaSalle—Émard, NDP)
Hon. Christian Paradis (Minister of Industry and Minister of State (Agriculture), CPC) (11:45)
Mr. Gerald Keddy (Parliamentary Secretary to the Minister of International Trade, for the Atlantic Canada Opportunities Agency and for the Atlantic Gateway, CPC)
Ms. Linda Duncan (Edmonton—Strathcona, NDP)
Hon. Tony Clement (President of the Treasury Board and Minister for the Federal Economic Development Initiative for Northern Ontario, CPC)
Mr. Denis Blanchette (Louis-Hébert, NDP)
Mr. Rob Clarke (Desnethé—Missinippi—Churchill River, CPC)
Hon. Steven Blaney (Minister of Veterans Affairs, CPC) (11:50)
Mr. François Pilon (Laval—Les Îles, NDP)
Hon. Denis Lebel (Minister of Transport, Infrastructure and Communities and Minister of the Economic Development Agency of Canada for the Regions of Quebec, CPC)
Mr. Nathan Cullen (Skeena—Bulkley Valley, NDP)
Hon. Denis Lebel (Minister of Transport, Infrastructure and Communities and Minister of the Economic Development Agency of Canada for the Regions of Quebec, CPC) (11:55)
Mr. Sean Casey (Charlottetown, Lib.)
Hon. Steven Blaney (Minister of Veterans Affairs, CPC)
Hon. Dominic LeBlanc (Beauséjour, Lib.)
Ms. Lysane Blanchette-Lamothe (Pierrefonds—Dollard, NDP)
Hon. Christian Paradis (Minister of Industry and Minister of State (Agriculture), CPC)
Mr. Pierre Dionne Labelle (Rivière-du-Nord, NDP)
The Acting Speaker (Mr. Barry Devolin) (12:00)
Mr. Blake Richards (Wild Rose, CPC)
Ms. Kirsty Duncan (Etobicoke North, Lib.)
Hon. Leona Aglukkaq (Minister of Health and Minister of the Canadian Northern Economic Development Agency, CPC)
Mr. Jean-François Larose (Repentigny, NDP)
Hon. Jim Flaherty (Minister of Finance, CPC)
Sealing Industry
Mrs. Tilly O'Neill Gordon (Miramichi, CPC)
Hon. Gail Shea (Minister of National Revenue, CPC) (12:05)
Mrs. Sadia Groguhé (Saint-Lambert, NDP)
Mr. Rick Dykstra (Parliamentary Secretary to the Minister of Citizenship and Immigration, CPC)
Mr. André Bellavance (Richmond—Arthabaska, BQ)
M. Nathan Cullen (Skeena—Bulkley Valley, NPD)
ROUTINE PROCEEDINGS
Supplementary Estimates (B), 2012-13
Performance Reports, 2011-12
Government Response to Petitions (12:15)
Mr. Tom Lukiwski (Parliamentary Secretary to the Leader of the Government in the House of Commons, CPC)
Interparliamentary Delegations
Mr. Mike Wallace (Burlington, CPC)
Hon. Michael Chong (Wellington—Halton Hills, CPC)
Mr. Yvon Godin (Acadie—Bathurst, NDP)
National Capital Act
(Motions deemed adopted, bill read the first time and printed)
Mr. Yvon Godin (12:20)
Justice and Human Rights
Experimental Lakes Area
Mr. Mathieu Ravignat (Pontiac, NDP)
Ms. Mylène Freeman (Argenteuil—Papineau—Mirabel, NDP)
Ms. Rathika Sitsabaiesan (Scarborough—Rouge River, NDP)
Mr. Kevin Lamoureux
Ms. Rathika Sitsabaiesan
Ms. Marie-Claude Morin (Saint-Hyacinthe—Bagot, NDP)
Ms. Irene Mathyssen (London—Fanshawe, NDP)
Mr. David McGuinty (Ottawa South, Lib.)
Experimental Lakes Area (12:30)
Mr. Kevin Lamoureux (Winnipeg North, Lib.)
Ms. Elizabeth May (Saanich—Gulf Islands, GP)
Sex Selection
Mr. Mark Warawa (Langley, CPC)
Mr. Tom Lukiwski
Financial Literacy Leader Act
Mr. Matthew Dubé (Chambly—Borduas, NDP) (12:35) (12:40) (12:45)
Mr. Matthew Dubé
Mr. François Choquette (Drummond, NDP) (12:55)
Mr. Pierre-Luc Dusseault (Sherbrooke, NDP)
Mr. Kevin Lamoureux (Winnipeg North, Lib.) (13:00) (13:05) (13:10) (13:15) (13:20)
Mr. Pierre-Luc Dusseault (Sherbrooke, NDP) (13:25)
Mr. Sean Casey
PRIVATE MEMBERS' BUSINESS
House of Commons Debates
41st PARLIAMENT
OFFICIAL REPORT (HANSARD)
Speaker: The Honourable Andrew Scheer
The House met at 10 a.m.
[Government Orders]
The House proceeded to the consideration of Bill C-44, An Act to amend the Canada Labour Code and the Employment Insurance Act and to make consequential amendments to the Income Tax Act and the Income Tax Regulations, as reported (without amendment) from the committee.
[Table of Contents]
The Acting Speaker (Mr. Barry Devolin):
There being no motions at report stage, the House will now proceed without debate to the putting of the question on the motion to concur in the bill at report stage.
Hon. Jim Flaherty (for the Minister of Human Resources, Skills and Social Development)
moved that the bill be concurred in.
When shall the bill be read the third time? By leave, now?
Some hon. members: Agreed.
moved that the bill be read the third time and passed.
Mr. Brad Butt (Mississauga—Streetsville, CPC):
Mr. Speaker, I am grateful for the opportunity to speak in the House today to this absolutely fantastic legislation. The helping families in need act is a bill that would provide help to parents who are going through some of the most difficult times in their lives.
As a new member of Parliament, I very much remember one of the first constituent meetings I had after being elected. I heard from the father of a very young girl who was very stricken with cancer. Our party had made a promise in the election campaign that we would bring in changes to the employment insurance system that would allow parents of critically ill children to claim benefits while they were spending much needed time with their child. I remember him sharing his story, saying how much of a benefit it would have been in their case if his wife had been able to take the time off work and claim EI benefits, enabling her to stay with their very sick child. I am delighted to report that the child has had a great recovery and is doing well.
That is just one example of one family out of thousands that the changes proposed in Bill C-44 would help. As one individual member of Parliament, to hear that story and to be able to rise in the House today and talk about it, knowing that the bill would make a significant difference to families like that, makes me immensely proud.
The proposed legislation supports the implementation of three initiatives: the new federal income support for parents of murdered or missing children, the new employment insurance benefit for parents of critically ill children and enhanced access to EI sickness benefits for parents who fall ill while they are receiving EI parental benefits.
The title of the bill says it all. It is about helping Canadian families. It is about supporting them through situations that are both financially and emotionally difficult.
As announced by the Prime Minister earlier this spring, we will be creating a new benefit for the parents of murdered or missing children. We heard at committee from a number of parents whose lives had been affected by the fact that their child was missing, had been missing and, in some cases, had been murdered. I cannot even imagine as a parent going through that. We listened to the testimony and heard from these parents indicating how much this benefit would have helped them in their situation. Even though they continued to grieve, and they will each and every day for that murdered child, the ability to take time off to spend with the remaining members of the family, to have that level of support, and to know that their job will be secure through the amendments that are proposed in the bill to the Canada Labour Code, will make a significant difference to the lives of those people who showed the courage to come before the human resources skills and social development committee and share their stories with all of us. I know every member of the committee, both opposition and government members, were very moved by the testimony of those individuals.
As a father of two wonderful daughters, I understand how important family is and a parent's desire to protect our children. The loss or disappearance of a child as a result of a criminal act can only be described as the most difficult experience a parent could ever go through.
The new federal income support for parents of murdered or missing children, which would provide parents with financial support of up to $350 per week for up to 35 weeks, is a major step forward. I am delighted, at report stage, that all members of the committee recommended that the bill come back here today for further and final debate, and hopefully passage. We all recognize the tremendous benefit that this support would provide to these families.
The helping families in need act would also amend the Canada Labour Code to protect the jobs of parents who temporarily leave a federally regulated job to cope with the death or disappearance of a child as a result of a suspected Criminal Code offence. We know that the new income support and the knowledge that their job is also protected would help ease the pressure on parents in this unimaginable situation.
Since Canadians first elected a Conservative government, we have been devoted in our support for victims of crime, despite the fact that the opposition parties continually vote against our measures to strengthen victims' rights. However, the bill does transcend party politics. It would provide support for Canadians going through something that is so personal, so devastating and so profound that only those who have been touched by this kind of tragedy themselves will ever truly understand it.
We have already heard a number of very touching speeches in the House by many members of Parliament, including the member of Parliament for Brant. I want to thank him for what I think was a very moving speech and for sharing his family's personal story yesterday in the House on the bill. It just goes to show that there are 308 people who get elected to this place who all come from different parts of the country, different families and so on. Many of us wind up being touched on a personal level by some of these tragedies in life because we know of a relative with a critically ill child or we know of a family where a child has gone missing. Unfortunately, there are some of us who also do know families who have been touched by the fact that a child has been murdered.
I am glad to hear that my hon. colleagues from the Liberal and NDP parties will be supporting the bill, even though the NDP did initially vote against the ways and means motion to get it moving. Let me review some of the measures we have taken to help families, even though the opposition parties either opposed or delayed many of these great initiatives.
Our government has expanded the eligibility for compassionate care benefits to include people considered as family by the person who is ill. We have allowed self-employed workers to opt into the employment insurance program to be able to receive maternity, parental, sickness and compassionate care benefits.
In speaking to a number of self-employed people, I know how much in my riding of Mississauga—Streetsville they very much appreciate the fact that they can opt into the EI system. This is especially true of women entrepreneurs because they will then be eligible for maternity and parental leave benefits while they are caring for their newborn, a time when they may not be able to continue to run their own business directly. We have also improved access to EI parental benefits for military families and for foster families who make a demonstrable commitment to adopt the child in their care.
As a former board member of the Peel Children's Aid Society, I know how important it is for foster families to be able to spend the time they need with a new child they have adopted into their family and to be eligible to collect EI benefits just as if it were their own, biological, newborn child. What a great initiative we have brought forward to encourage more families to foster and to adopt children who need wonderful homes.
Our government is committed to making targeted, common sense changes to the EI program to support hard-working Canadian families. We are doing this through the helping families in need act. This support will help people at a time when they most desperately need it.
While we are fortunate to live in one of the safest countries in the world, we are not immune to violence. It is unthinkable but every year in Canada some 1,100 children are reported abducted and about 100 children are murdered. I cannot even begin to imagine what it might be like to lose someone I love more than anything in the world, especially in something as senseless as a child abduction or a murder. It is unthinkable, and families who have to deal with that reality deserve our support.
Our government is taking steps to help families who are dealing with these traumas. We have heard from Canadians that this help is needed and it is long overdue. That is why our government made the campaign commitment to provide support. I am delighted to say we are following through on it right now. This income support will help these parents take time off work to address legal issues and to begin their emotional recovery.
Fortunately, many employers grant unpaid leave to parents in these situations. I think there would be very few employers who would not empathize with their employee and his or her family going through this, and those companies are often very generous in their support. However, it is important that we, as legislators, also make sure that we are doing what we need to do within the laws of Canada and within our employment insurance system to also provide our support.
Fortunately, while it helps parents focus on their families, most parents cannot afford to go without an income for extended periods of time. They also need to know that their job will still be there when they are ready to come back.
We are the one party that is always putting the rights of victims ahead of criminals. Our record has been very clear on this issue. In 2007, we provided $52 million to strengthen the federal victims strategy. In budget 2011, there was an additional $26 million for this initiative. Since Canadians elected our government, we have brought in much needed legislation to protect the victims of crime as well as to ensure that those who do commit crimes pay the price. We will continue to deliver on our commitment to protect all Canadians.
As the parent of two daughters aged 13 and 8, the thought of losing one of them is unimaginable. We talk to families who have been through that situation and all members of the House empathize with what they go through. We as the government have an obligation to support and help families cope during these tragedies, however horrible the circumstances. By providing this much-needed financial support and job protection for parents, we can at least give them some time to begin to heal.
I want to quote a number of different representatives and organizations that support the bill and its speedy passage. Dan Demers from the Canadian Cancer Society stated:
This new EI support will allow parents to focus on caring for their child rather than worrying about how to pay the bills and can they keep their jobs. These programs will strengthen Canadian families and provide them the flexibility and the security they need to help keep their lives as normal as possible through a very difficult time.
Terri Odeneal, executive director of the Comox Valley Hospice Society, stated, "By extending these benefits, we can ensure that parents can focus on caring for their child during this difficult time rather than worrying about financial issues".
Sharon Ruth, who is the mother of a critically-ill child, in a press conference this past September stated:
Every time there was an election, all the efforts died on the order paper and we had to start again. The truth is that I had help from each party, but it wasn't until our country finally got a majority government that I'm standing here today with all of you on the brink of what I hope will be revolutionary change to help those families that are in need and most vulnerable.
I will quote Sharon Ruth again, because she speaks very passionately about where we are and where we need to go. She said:
I want to thank [the minister] who has a genuine concern for family and their suffering, for receiving myself and Colleen and Edwina Eddy last November and to listening to what we had to say. She believed that changes needed to be made and worked toward making this day happen.
In conclusion, I want to go back to one of the first face-to-face constituency meetings I had with a resident of the great riding of Mississauga—Streetsville, a father who shared his story with me. He said, "Brad, you guys need to keep your election commitment. You need to get this measure enacted".
Nothing gives me greater honour and privilege today than to stand in the House to speak to Bill C-44 and tell that father that we have delivered.
Ms. Chris Charlton (Hamilton Mountain, NDP):
Mr. Speaker, I thank my colleague for his work on committee. We sat on HUMA together and reviewed this bill in some detail.
I noticed the member could not resist taking some partisan jabs. I will try not to reply in kind, but I want to point out that he ended by suggesting that his party had kept its commitment with respect to the electoral platform where indeed the Conservatives promised in 2011 to " provide enhanced EI benefits to parents of murdered or missing children".
We on this side of the House support that commitment and will vote in favour of Bill C-44. However, the member neglected to say that the Conservatives' commitment also said that "Funding for this measure will come from general revenue, not EI premiums". This is part of the commitment they did not keep.
We know from other debates in the House that the EI system is not serving Canadians well. Only four out of ten Canadians can access EI. We know that successive Liberal and Conservative governments have stolen $57 billion out of the EI fund to pay down the debt and deficit rather than provide the much needed benefits for people who, frankly, paid into the system, which is only workers and employers.
Now we have an about face. We are adding something else to the draw on the EI fund, which is we now have to pay for this bill out of the EI fund as well.
Could the member talk for a minute about why the Conservatives flip-flopped on their commitment that this program would not be funded by EI?
Mr. Brad Butt:
Mr. Speaker, all party members at the human resources committee have worked very hard in reviewing the bill. I want to express my thanks on behalf of our government to the opposition parties for their support of the bill.
Opposition members have raised some issues where they believe some changes would be appropriate. We have listened and we believe that this is the right way to go. Based on the best information that we get from the very professional and hard-working bureaucrats within the ministry, we believe the right way to go is to have part of the benefits in the bill paid through the EI fund and part of the benefits for missing and murdered children paid out of the general revenue fund. That is consistent with the compassionate care and sick leave benefits that we presently have in the EI system.
One of the things I did not get a chance to mention in my speech, which did come up at committee, was the reason for the 37 weeks of the entitlement under critically ill and whether it should be longer. In fact, it can be longer. A family can claim the 37 weeks, plus the 15 weeks for a sick benefit, plus 6 weeks for compassionate care leave. That is a tremendous amount of time for families to claim benefits, especially when they are workers who have already paid into the EI fund.
Mr. Kevin Lamoureux (Winnipeg North, Lib.):
Mr. Speaker, the Liberal Party will be support the bill as well. We have recognized this as a very important issue. We care about our children and we want to be there in a very real and tangible way, which is one of the reasons we support the bill.
In the past, through electoral elections, we have talked about expanding this to look at, for example, gravely ill parents or siblings and how employment insurance might be able to assist them. At the end of the day, the government has an obligation to apply that compassion and caring attitude to those individuals as well. We hope the government will act on that.
However, with this legislation, there would be an obligation to carry on discussions and dialogue with provincial jurisdictions. Many would argue that there has been a great deal of concern regarding the Minister of Intergovernmental Affairs. Could the member provide some comment on how that minister needs to play a strong role in working with the provinces so we can realize the maximum benefits of the bill?
Mr. Speaker, I thank my colleague and the members of the Liberal Party as well for indicating that they will be supporting the bill. I thank the hon. member for Cape Breton—Canso who is the Liberal Party member on the committee. He, too, just like the member for Hamilton Mountain, has worked very hard on the bill, reviewing it at committee and ensuring that it is a strong bill to go forward and ensuring that it will help families.
After this bill is passed, it would be the role of all members of Parliament to have conversations with our respective provincial colleagues to get them to adopt similar labour code changes in their provincial labour legislation to mirror the changes that we bring forward to the Canada Labour Code, to ensure that the 90% of the people in our country who are not covered under the Canada Labour Code but are covered by employment standards and labour legislation in the provinces will also be able to get the benefit that federally-regulated business and federal government employees will get through the changes in Bill C-44 with respect to the Canada Labour Code.
Mr. John Carmichael (Don Valley West, CPC):
Mr. Speaker, the families in need act is clearly a compassionate act that is long overdue. I am delighted to be in the House today to hear members of all parties agree that they are in support of this compassionate and necessary legislation.
As a parent three times and a grandparent of four little ones, I cannot imagine the stress on a family when confronted with a missing or murdered child. We have all been touched in the House. I doubt there is anyone who has not been touched by someone who has lost a child. We have seen the grief and the difficulties the families face.
This bill addresses important needs to take certain issues off the table so families can focus on the stress at hand of dealing with a missing or murdered child and the grief that goes with it.
I would like to commend my colleague for his speech this morning. I felt he addressed the issues very well and obviously with great compassion. We have talked about job security, but would the member perhaps go a little deeper on the issue surrounding federal income support? It is something that is very important in helping families to deal with these issues at such a time as they are being confronted with them.
Mr. Speaker, I thank my colleague from Don Valley West very much for his support of the legislation. I am not at the grandparent level yet, but I see the joy in my parents and inlaws' eyes when they interact with my daughters. Hopefully my day will come when I can enjoy grandchildren as well.
The member asked a specific question about the federal income support for parents of murdered or missing children. Beginning on January 1, 2013, this new grant will provide $350 per week for up to 35 weeks to parents of murdered or missing children, if they are less than 18 years of age, whose death or disappearance is the result of a suspected Criminal Code offence.
To receive this taxable grant, the affected parents will need to have earned a minimum level of income in the previous calendar year of $6,500 and take leave from their employment.
Again, this is a support mechanism for families is not directly related to the EI fund, but is related to another benefit to recognize the fact that in the case of missing or murdered children, it is different and in a lot of cases it is finite. It is done because the child has been murdered and the family needs the support mechanisms and the longer time to deal with that issue.
This is a major step forward for these families that need and require this level of support.
Mr. Speaker, I am pleased to rise in the House again to continue the debate on Bill C-44, An Act to amend the Canada Labour Code and the Employment Insurance Act and to make consequential amendments to the Income Tax Act and the Income Tax Regulations. Although the title itself does not tell us much, the bill would make a series of improvements, most of them through the employment insurance program, to Canadian families that desperately need the support of their government. For that reason, as I indicated at second reading, my NDP colleagues and I are pleased to support the bill.
However, we take our role as the official opposition seriously. We hold the government to account and even when we agree with the intent of a particular piece of government legislation, we will work hard to ensure it is the very best bill that it can be.
To that end, we went into the committee hearings on Bill C-44 hoping to make the process work. Committee is where we have the opportunity to go through a bill clause-by-clause to question the minister, or in this case ministers, responsible for the bill and to hear testimony from both experts in the field and from individuals who would be impacted by the proposed legislative changes. We then go through the bill with a fine tooth comb to address concerns because, with the current government in particular, the devil is often in the details.
Because we support Bill C-44, we went into committee hoping that a spirit of co-operation would prevail and that we would collaborate to make the necessary improvements to give parents of critically ill, murdered or missing children the support they so desperately needed. I cannot tell members how profoundly disappointed I was when the government members on committee reverted to the old caricature of themselves and refused to entertain a single amendment proposed by opposition MPs. Honestly, it was a disgrace.
I will give one example of an area where we could have made an important improvement to the bill.
In order for parents of murdered or missing children to receive the government grant, they would have to have earned $6,500. Presumably, that threshold was set to show some kind of attachment to the labour force. Leaving aside the broader question of whether it is necessary to prove such attachment in the first place, I asked the Minister of Human Resources and Skills Development in committee why she chose to adopt a threshold based on earnings as opposed to hours worked. Obviously, under her rules, somebody who worked for minimum wage would have to work many more hours to qualify for the benefit than somebody who made $150 an hour. Why would we create such an unequal threshold when people at the lower end of the wage scale would likely need the financial support even more than those at the higher end?
I encouraged the minister to explore other ways of proving attachment to the labour force. The minister responded by saying that they could not use hours worked to prove attachment to the labour force because:
—that would not be compatible with the eligibility of self-employed workers who have opted into the EI system. Their eligibility for EI special benefits is based on financial figures, on dollars earned, because we cannot measure their hours. There's no way to validate that.
Really? We do not trust the self-employed to report accurately, so we will punish those workers who earn minimum wage or work part-time because the government cannot create a nuanced enough system to ensure that the bill is fair to everyone. Really? Is that what the government is saying to parents of murdered and missing children?
We in the opposition could not move the necessary amendments in committee because they would have been ruled out of order. However, the minister had, and still has, the opportunity to right this wrong. Changing the eligibility criteria is the right thing to do and it would not throw the government into financial crisis.
Let us be clear about the numbers here. According to the Canadian Police Information Centre, there were 25 abductions by strangers in 2012. Helping 25 families will not break the bank, but even if it did, it is absolutely the right thing to do and the minister should not be creating artificial barriers by means testing eligibility for support. The Conservatives' failure to reconsider these provisions is an absolute disgrace and belies the spin that they are sincere about wanting to help families in crisis.
I would say the same thing about the other amendments my NDP colleagues and I were pushing for in committee. I know I will not have time to repeat them all in the House today, but let me continue to highlight some of the most obvious areas where we could and should have found common cause.
I will begin with the most egregious example where the Conservatives' strict adherence to talking points trumped common sense. Clause 5 of Bill C-44 states that leave for critical illness would end on the last day of the week that the child died. New Democrats tried to move a modest amendment that would have extended that leave for another two weeks after the child's death to give the parents time to grieve and to bury their child.
Our proposed amendment was supported by the Canadian Association of Social Workers, Ronald McDonald House Charities, the Canadian Association for Community Living and the Canadian Labour Congress. Fred Phelps, the executive director of the Canadian Association of Social Workers was almost incredulous when he asked the committee, "would compassion not dictate that families require time after death to mourn and bury their child?" For most Canadians, the answer would have been a resounding yes, but sadly, compassion does not appear to be the government's forte and the bill is proceeding unamended.
Let me give another example. The bill as it currently stands defines children as those under 18 years of age. Why is that? In many cases children are defined not by age but by their dependency on their parents. For example, many dental and health insurance plans cover so-called children until they reach the age of 18 or they complete school; 18 is not a hard and fast cutoff. I would argue that this should be the case in Bill C-44 as well. Particularly, it is essential that the definition of child be expanded beyond the age of 18 for disabled children.
As the minister herself acknowledged, the criteria she used were emotional dependency and emotional maturity. Clearly, those criteria would apply to some disabled Canadians who may well be over the age of 18 but for whom the emotional attachment to their parents is every bit as real as for those children who fit the current definition in the act.
As Tyler Hnatuk, representing the Canadian Association for Community Living made clear at committee:
...caregiving responsibilities for parents of children with disabilities often continue much longer in life than for other families, and so certainly I want to recognize the need and the duties that carry on throughout a lifetime.
The parenting of a child with a severe disability is a lifetime commitment.
It would have been easy for us to allow for the expansion of the definition of the word "child" in this bill. That is why the New Democrats on the committee moved an amendment that would add child to the list of terms that could be defined through regulations, which would allow the government to expand the definition to include dependent children over the age of 18. Again, we are talking about a very small number of families who would be impacted, but for those families the concern is very real.
It is not good enough for the Conservatives to vote no just because the amendment came from the NDP. The Conservatives should have put partisanship aside and acted in the best interests of Canadians. That is what they were elected to do, but if they cannot even do it on a bill that has all-party support, how are we ever going to make the committee process work on the more contentious matters that are referred to our committee? Is it really that foreign a concept to the government that detailed scrutiny of its bills may actually lead to better legislation? Committee work used to be an integral part of the legislative process under governments of all stripes, but under the Conservative Prime Minister, that work is wholly devalued.
Here is another example. Clause 6 of the bill provides leave for the parents of murdered or missing children. We all support that provision, of course, but would it not make sense to allow parents to take that leave on a flexible basis rather than mandating that it be taken in consecutive weeks? We were not suggesting that the total number of weeks be increased. We simply wanted to allow parents to apportion their leave to suit their personal circumstances. Oftentimes their dealings with the judicial system occurred months down the road. Why would we not allow them to use some of their leave time during that critical time? Again, that flexibility found broad support among the witnesses who gave testimony before our committee.
Let me give a sampling from the very people whom this bill is intended to help. When asked whether it would be helpful to create flexibility with respect to the leave provisions of the bill, here is what they said.
Mr. Bruno Serre, whose daughter Brigitte was murdered in January 2006 at the age of 17 during her shift at a gas station in Montreal, said:
I think that would be a very good thing.
For example, if this happened to someone and, after 10 weeks, they felt ready, they could return to work. In my case, I went back after five weeks, but I wasn't really capable.
So it would help to have hours or weeks banked. Five probably would have been used and then there would be 30 left, which could be used over the years. But there should be no expiry date. For example, it could be decided that the recipient would have one year to use these 35 weeks, as is sometimes the case in the government. Instead, this should be spread out over two or three years. Some trials can take place three years later.
If someone has used all the weeks and the trial comes up, that person will relive the tragedy. When the trial comes up, you relive the day when you learned about the death of your loved one. So there are other steps to take. If the person doesn't have any weeks left, he or she will have to go through the same situation again that happened at the very start. That person will be lost and unable to work.
Being able to bank the weeks for later would be a very good solution.
Christiane Sirois concurred. Ms. Sirois' son was kidnapped on November 1, 1984, when he was 8 years old. When asked about the desirability of creating greater flexibility she said:
My answer is yes, without hesitation. I support what Mr. Serre said: there should be banked hours, should a person need them.
This doesn't apply for me. I haven't found my son, but I can put myself in the shoes of people who have found their child. I do not dream about finding him alive after 28 years, you can be sure. But I understand. I am suspended. What will happen when I find his little eight-year-old body or what's left of it? This will happen one day, for sure. I will relive 28 years stored up in my memory. It is important to be prepared for this, that is certain.
That is why it is crucial that these victims have a minimum amount of financial assistance to help them survive. Because listen carefully: you don't really live with this, but you survive.
Lastly, let me add the words of Ms. Céline Hotte, whose life changed forever when her daughter was murdered. Here is what Ms. Hotte told us in committee:
For 10 to 17 years after the events, I had to deal with the perpetrator's parole requests and the issue of halfway houses. To contest these requests, you need to put together a file. This takes signatures from people in the village where he lived. This isn't easy to do. You also have to read about everything he did in prison. This isn't easy. You cannot talk to him—that's not what I want to do anyway. You have to read the reports. He never followed the recommendations. Each time, it put me right back into the situation I had gone through.
Clearly there is widespread agreement that every circumstance is different and that there must be enough flexibility to allow for accommodation.
That is certainly the conclusion drawn by Canada's Federal Ombudsman for Victims of Crime, who in her testimony also encouraged the government to allow for flexibility. She focused particularly on the administration of justice and the court process:
We know that if there is a murder, the court case may be several years down the road, so to provide an option and some flexibility—for example, a parent may choose to take a certain amount of time at the time of the crime, and then, if the criminal court process is two years down the road, they may need to have time then as well. Also, in some cases the person responsible may not be apprehended for a while. I'm just saying adding that flexibility would provide parents of murdered and missing children an opportunity to take the time when it's appropriate for them, when they need that time.
Clearly, there was broad-based consensus about what needed to be done to make Bill C-44 as effective as possible for the people it was intended to help. The only people offside were the members of the Conservative caucus, presumably at the direction of the minister responsible and the Prime Minister.
I make these points more in sadness than in anger. We had the perfect opportunity to improve a bill that we all agree is worth supporting. This did not need to be an exercise in rigid partisanship, where the Conservative members of the committee automatically oppose anything proposed by the NDP. Frankly, the victims of crime deserve better. The parents of critically ill children deserve better. The Canadian public deserves better. They deserve a concerted effort from all their elected officials to make Parliament work. In this instance, on Bill C-44, the Conservative members on the human resources committee let Canadians down.
Now at this point members may well wonder whether we will continue to support the bill. Let me reassure them. My NDP colleagues and I will of course vote in favour of Bill C-44. My point is simply that we could have achieved more, that we could have improved the bill in meaningful ways, but that we failed to seize that important opportunity.
That does not mean that baby steps in the right direction are not worth taking.
In fact, as I said at the time of second reading, there are parts of Bill C-44 that were lifted directly from my own private member's Bill C-362. Let me just review which those are.
First, one of the proposals included in the government's bill would amend the Employment Insurance Act to allow mothers and fathers currently on parental leave to access EI sickness benefits if they fall ill during their parental leave. This is a welcome and long-overdue amendment. There are few Canadians who would disagree that new parents who are very often already stretched both physically and financially should not be penalized if they become ill while on parental leave.
I am a little puzzled, though, as to why the minister would have stopped short of extending this benefit further. If she appreciates the injustice of denying sickness benefits to those whose circumstances change while on parental leave, then why did she fail to apply the same consideration and logic to workers who are laid off while on parental leave? Why would we solve one injustice and at the same time wilfully ignore the other?
My bill does take that extra step. It would fix that wrong. It recognizes that those on parental leave, the very same physically and emotionally drained new parents who sometimes become ill while on parental leave, can find that they have been downsized or laid off, through no fault of their own, while on parental leave.
As it currently stands, parents in that situation are denied benefits. Inexplicably, the government is content to leave them twisting in the wind, unsupported by even the meagre support provided by EI.
On the upside, my private member's bill also includes provisions to cover the self-employed in this benefit arrangement. I am pleased to see that the government has at least adopted them.
I do want to reflect for a moment on whether the EI program is the best vehicle for delivering the larger package of supports contemplated by Bill C-44. As members can tell from my phrasing, I obviously do not think it is. It bears pointing out that at one time the government agreed with me.
As recently as 2011, the Conservative Party platform stated, "Funding for this measure will come from general revenue, not EI premiums". The Conservatives were right to adopt that approach. Whether one is a waged worker, senior manager, professional or stay-at-home parent, the devastation of a critically ill child is the same. All Canadians who find themselves caring for their seriously ill child are incurring a myriad of expenses that go beyond lost wages, and they all deserve our support.
What happened to make the government change its mind? The grant for parents of murdered and missing children will be paid from general revenues and not through EI. However, with respect to critically ill children, the Conservatives have ignored their election promise and are paying for their commitment through EI. I don't need to remind anyone in the House that the EI fund is not the government's money. It is a fund to which only workers and employers contribute. Therefore, for the government to draw on that pool of money to create a photo op on a policy announcement, no matter how positive, is surely beyond the pale.
I know my time is almost up, but I ask that the House indulge me for one more minute so that I can make a final point.
New Democrats support the bill. It is not a question of ideology or partisan politics; it is about assisting families in their time of need. However, let us be clear. The bill does not go far enough to help the families of missing and murdered children, nor the parents of kids who are critically ill.
Also, the bill does not go far enough in making reforms to EI. These measures completely fail to address the greatest challenge with EI, which is the lack of access for unemployed Canadians.
The bill will clearly pass, and by all means let us do it quickly, because we have to get on with tackling the larger question of comprehensive EI reform. We must make EI accessible and effective for all Canadians. Nothing less will do.
Mr. Harold Albrecht (Kitchener—Conestoga, CPC):
Mr. Speaker, I thank my colleague for her speech and for the indication that her party will indeed be supporting the bill.
One of the most rewarding opportunities I have had since being elected to Parliament almost seven years ago was to co-chair, along with her colleague from Windsor—Tecumseh, a parliamentary committee on palliative and compassionate care. We had the opportunity to go across Canada and listen to dozens of witnesses who gave input on the issue of palliative care. One of the issues was dealing with gravely ill children. I would like to read an excerpt from Sharon Ruth with respect to her daughter Colleen Ruth.
Governments must support and invest in families during these tragically difficult times. The long term socio-economic benefits and returns of supporting families are far greater than the supposed cost savings that result from a politics of inertia. Doing nothing simply raises the toll of broken individuals and families. Colleen is living proof that there are gaps in our social and support systems that need to be updated. I am asking you to extend compassionate leave benefits to at least 26 weeks in a 52 week period. I am also asking that you change the qualifying criteria to "gravely ill" as opposed to "significant risk of death".
I ask my colleague this. Rather than focusing on some of the things that, yes, we could improve, could she just acknowledge that the 35-week benefit is much better than the 26 weeks that many were requesting? Indeed, not only is it 35 weeks, but my colleague from Mississauga—Streetsville pointed out many other positive initiatives. I wonder if she could acknowledge that.
Ms. Chris Charlton:
Mr. Speaker, I want to acknowledge my colleague's work with the palliative care caucus.
Of course, I agree that baby steps are better than no steps. That to me is a given. However, I do want to take our responsibility here as parliamentarians seriously. There is a reason that we refer bills to committee after second reading. Second reading simply suggests that we have agreed to a bill in principle. We then send it to committee so that we can do the hard work and listen to expert testimony and learn from people's individual experiences about how we can make legislation the very best it can possibly be so that we are actually serving the people whom it is intended to help.
We did that at committee. We heard very moving testimony. It took incredible strength for some of the witnesses to share their personal stories with us. All I was suggesting is that we did not do them justice. The changes that they were asking for were not of a huge magnitude. They were not very costly. Let us be clear: there are not that many children in Canada who in any given year are murdered or go missing. However, for the families who are impacted, their lives change forever. They were looking to their government and to our committee to help them find the support they so desperately need. We had that opportunity in committee and we let those families down. Therefore, I do not think we should be proud of the work our committee accomplished on the bill.
Mr. Rodger Cuzner (Cape Breton—Canso, Lib.):
Mr. Speaker, my colleague's speech did an excellent job of summarizing the ascent of the bill, where we are today and maybe the opportunity that was missed. The Liberal Party wholeheartedly supports the spirit and intent of the bill, but there were some opportunities missed along the way.
Stephen Moreau was the lawyer who represented Natalya Rougas in the case that was presented to the Supreme Court and was ruled on. One thing that was brought out during the debate was the stacking provisions. The court ruling in 2011 said that those provisions on receiving maternity leave and then sick leave were already in legislation, as intended by parliamentarians who passed the law in 2002.
I would like the member's comments on Mr. Moreau's testimony during the hearings.
Mr. Speaker, that is absolutely what Mr. Moreau said in committee. I do want to point out that what Mr. Moreau was talking about, rightly, was the stacking of special benefits.
At no point did our committee address the stacking of those special benefits with regular EI. Of course, that is one of the most fundamental problems with our EI system. For example, if someone is on maternity leave and intending to go back to work, but their company shuts down and they are laid off or downsized while on maternity leave, it is not possible now to stack the maternity benefits with regular EI.
That is something that is absolutely critical for us as legislators to look at doing. That is exactly what my private member's bill proposes. I look forward to all members of the House supporting that bill when it comes forward.
The time for government orders has expired. The hon. member for Hamilton Mountain will have four minutes remaining for questions and comments when this matter returns before the House.
[Statements by Members]
Mr. Peter Goldring (Edmonton East, Ind. Cons.):
Mr. Speaker, I wish to recognize 33 youthful delegates who have visited with us for the past seven weeks. They are here in members' offices to gain valuable perspectives on Canada's most important democratic institution, the Parliament of Canada.
These young people, representing the Canada-Ukraine Parliamentary program, embody the highest ideals of achievement and community service. They are the future leaders of Ukraine, young people like Yaroslav Barkov from my office.
Canada and Ukraine are inextricably linked forever by prior migration. Fully one in 30 Canadians is of Ukrainian descent, as are my wife, daughters and granddaughters. Ukraine holds a special place in the hearts of Canadians. Canada was the first country in the western world to accord diplomatic recognition in 1991 to an independent Ukraine.
As the young emissaries depart, we wish them well and say to them, Menohaya Leeta.
Mr. Ed Holder (London West, CPC):
Mr. Speaker, this past weekend I participated in MADD London's 25th anniversary red ribbon campaign.
Held at London's Airport Kia dealership, Mothers Against Drunk Driving launched its annual challenge to Londoners not to get behind the wheel if they drink. London Police Chief Brad Duncan reminded us that the RIDE program would be out in full force to help put a stop to totally preventable, heart-wrenching accidents and deaths.
We heard a tragic story from Mary Rodrigues. Mary was driving through an intersection when her car was T-boned by a drunk driver. Her son Alex was in the car. Alex was killed. Alex was four months old. Mary challenged us to honour her son by following the simplest of rules: if someone drinks, they should not drive.
Project red ribbon runs until the first Monday after New Year and will see volunteers distribute millions of red ribbons to the public to attach to their vehicles and keychains. It pays tribute to the more than 1,000 Canadians who do not need to die every year because of selfish, stupid and impaired drivers. We have to make drinking and driving as socially unacceptable as lighting up a cigarette in a non-smoking area. If we cannot do it for ourselves, we should do it for Alex.
Mr. Wayne Marston (Hamilton East—Stoney Creek, NDP):
Mr. Speaker, as Canadians across the country take part in Remembrance Day services this week, I would like to bring to the attention of this House a special ceremony in Hamilton.
In Hamilton in 1959, a cenotaph was unveiled on the grounds of the former Stelco plant. That day more than 800 steel workers, who were members of the Stelco War Veterans Association, marched in that initial dedication ceremony. The cenotaph as been the site of Remembrance Day services ever since.
This year's ceremony will have new significance because U.S. Steel Canada recently completed the revitalization of this very important cenotaph dedicated to the 3,000 steel workers who became war veterans and their families. The cenotaph's renovations include the complete refurbishment of the plaques with the names of the 172 steel workers who gave their lives in the service of Canada.
I call on the members of the House to join me in remembering those brave steel workers who sacrificed their lives in the service of Canada.
Mr. Ed Komarnicki (Souris—Moose Mountain, CPC):
Mr. Speaker, I rise today to acknowledge the life and work of Robert (Bob) Burns of Estevan, Saskatchewan, in the constituency of Souris—Moose Mountain.
Robert was born in 1930 in the hills of southwestern Saskatchewan in the Wood Mountain area. Sports like baseball, softball, skating, curling and hockey were always a big part of his life. During his years, he worked as a grain buyer, car salesman and for 25 years as a retail salesman with Sears Canada.
He always believed in being a team player and devoted much of his life to helping youngsters become involved in the sport of hockey and ball. He picked them up and at times fed them kept them at his home, and he coached and taught them not only about sports but also important life lessons. He believed that to be successful in sports and in life, one had to give 100%, hold one's head high and be a good sport.
It is these great attributes and qualities in Robert and his life's contribution to the many communities in Souris—Moose Mountain that are recognized in the awarding of a Queen Elizabeth II Diamond Jubilee Medal to him. I congratulate Bob. Way to go.
Family Doctor Week
Hon. Carolyn Bennett (St. Paul's, Lib.):
Mr. Speaker, November 12 to November 17 is Family Doctor Week in Canada.
Family Doctor Week proudly acknowledges the outstanding contributions of Canadian family doctors and dedication to their patients and delivery of high quality care.
We have heard from patients just how much they value their family doctors and how important their help is when it comes to providing the necessary care to patients and their families.
In particular, I would like to pay tribute to Dr. Calvin Gutkin, who is retiring as executive director and CEO of the College of Family Physicians of Canada.
For the past 16 years, he has worked tirelessly on behalf of family doctors and all Canadians who want to see improvements in family medicine across the country.
He supported the successful evolution of the Family Medicine Forum, and the enhancement of medical student and resident training as the future of family medicine, and he spearheaded the development of the National Physician Survey. Most recently, he helped author, and is currently advocating for, the Patient's Medical Home as the next significant step in enhancing family practice care in Canada.
Cal is a true believer in family medicine. We thank him for his endeavours and wish him well in the future.
Mr. Gordon Brown (Leeds—Grenville, CPC):
Mr. Speaker, on November 1, I hosted my sixth annual Hockey Night in Leeds and Grenville. This charity hockey match features former NHL stars, local dignitaries and members of Parliament on the ice raising money for the United Way of Leeds and Grenville. The game ended in a tie, but I am pleased to report that the real winner was the United Way, as $112,000 was raised for its campaign.
I want to thank the donors who made this game and its results possible, and the fans who came out to support this great cause. In particular, I would also like to thank Crystal Sled and executive director Judi Baril and her team at the United Way for their tremendous organizing efforts, and special thanks go to former NHL star Doug Smith, who was the honorary chair of this event.
The event was held in Athens this year, with the support of that community's council, headed up by Mayor Herb Scott. Finally, George and Kevin Tackaberry of Tackaberry and Sons deserve special mention. They made a very large donation and then topped it off to bring the total to $112,000.
Mrs. Anne-Marie Day (Charlesbourg—Haute-Saint-Charles, NDP):
Mr. Speaker, first of all, I would like to point out that the Conservatives hold the shameful record of muzzling the members and cutting off debate 29 times since the beginning of this Parliament.
I would like to take this opportunity to urge the government to take action on employment insurance. For the past few weeks, it has been using surreptitious means to pressure and harass job seekers who are already feeling the stress of not being able to make ends meet.
On the one hand, the government is going after unemployed workers like a desperate vulture and forcing them to send from three to five CVs a week, under the pretext that there is a labour shortage. On the other hand, we have the facts: Statistics Canada has said that the Canadian economy is not producing enough jobs, and the Parliamentary Budget Officer tells us that cuts in 2012 will eliminate another 125,000 jobs.
Canadians need jobs. The government needs to stop wasting its time hounding unemployed workers and do what it was elected to do.
The government wants unemployed Canadians to work. The solution is simple: create jobs.
Ms. Candice Bergen (Portage—Lisgar, CPC):
Mr. Speaker, as we all prepare for possible boundary changes in our ridings, I would like to bring the attention of this House to a location of immense importance in western Canada.
The prime meridian sits on a small plot of land in the northeast corner of my riding of Portage—Lisgar. This is where, beginning 1872, the dominion surveyor measured, mile by mile, on all points of the compass, over 200 million acres of land from Manitoba to British Columbia.
Immigrants and settlers moved onto the land and words section, half-section and quarter-section came into our vocabulary. Our roads, our farms, our towns and villages all designated from this one simple beginning.
The measure of any great nation is the strength of its history and the character of its people. The prime meridian is an important starting point for the western Canada we know and love. Today, a cairn marks this historic spot in my riding and preserves its history for centuries to come.
Mr. Chungsen Leung (Willowdale, CPC):
Mr. Speaker, it is a pleasure to rise today to share with the House the success of some of our young scientists for their candidacy for the Canadian Young Scientist Journal best paper awards.
As members may know, the Canadian Young Scientist Journal located in my riding of Willowdale provides a unique opportunity to high school students to express their views on a broad range of scientific topics and introduce themselves to the national scientific community.
The competition helps young people to discover the innovative potential of tomorrow through a platform that allows them to discuss their ideas with their peers and academic communities. The success of these young scientists will surely lead them on the path to university or further research.
Today I congratulate Abhishek Chakraborty, Brittney Allen, Carlos Xu, Alexandra Ficht and Rachel McLay. They have worked hard to come up with some of the best papers in this cross-Canada competition.
These aspiring young scientists are truly leading citizens of our nation and the world.
Mr. Speaker, I am sure you and all hon. members join me today in recognizing the achievement of these young scientists.
Birthday Congratulations
Mr. Dan Harris (Scarborough Southwest, NDP):
Mr. Speaker, it is my great privilege to rise today to pay tribute to Stephen Lewis on the occasion of his 75th birthday on November 11. I am very proud to follow in the footsteps of New Democrats. like Stephen Lewis, who have represented my community. Stephen was the MPP for Scarborough West for 15 years, 8 of them as leader of the Ontario NDP.
Among his many accomplishments, Stephen served as Canada's ambassador to the United Nations and the UN's Special Envoy for HIV/AIDS in Africa, work that he continues today through the Stephen Lewis Foundation. Anyone who has heard him speak knows of his oratory skills and mastery of the English language. They also know of his passion: his passion for Africa; his passion for women's rights and equality; and his passion for a just society.
On behalf of the official opposition and, indeed, all New Democrats, I wish Stephen Lewis a happy 75th birthday and I extend our best wishes for many more to come.
Mr. Dan Albas (Okanagan—Coquihalla, CPC):
Mr. Speaker, the 70th anniversary of the Dieppe raid in August was a powerful reminder of the bravery and dedication that Canadians demonstrated during the Second World War. Of the nearly 5,000 Canadians who participated, only 2,210 returned to England, many of them wounded. Tragically, 916 Canadians died as a result.
Those who participated in the Dieppe Raid were among the more than one million Canadians who served during the Second World War. Their efforts helped ensure victory was achieved. During the Second World War, approximately 55,000 Canadians were wounded and more than 45,000 gave their lives in the fight for freedom. This was a remarkable sacrifice for a young nation with a population of only 11 million people.
It is our duty to preserve the legacy these courageous men and women granted us and ensure their selfless dedication is never forgotten. We must pass the torch of remembrance on to future generations so they, too, can understand and appreciate the importance of commemoration.
Mr. Charlie Angus (Timmins—James Bay, NDP):
Mr. Speaker, the digital world offers an incredible opportunity for free expression, social engagement and learning and yet, along with this unprecedented access come serious issues, particularly for young people who are navigating the world of chat pages and social media. This is why privacy online is a right. Privacy matters.
This is the message of the National Media Literacy Week. The New Democrats are very proud to support the seventh annual Media Literacy Week that was launched by MediaSmarts which, along with the Canadian Teachers' Federation and 70 other organizations, are working with young people, teachers and community leaders to raise awareness and understanding of digital literacy online.
We call on the government to get proactive, to ensure that the rights of young people online are protected and to ensure that the Privacy Commissioner has the tools she needs to protect the rights of Canadians in the digital realm.
Mr. Jacques Gourde (Lotbinière—Chutes-de-la-Chaudière, CPC):
Mr. Speaker, I would like to thank everyone who so generously shared their touching poems, thoughts, stories and feelings on the Records & Collections page of the Veterans Affairs website. I would like to quote an excerpt from a poem by Madeleine Heckbert entitled Remember:
Remember them:
The war they waged,
The lives they saved,
The freedom they gave.
Remember them.
Remember that for justice,
For love of country,
Many perished.
These children we cherished,
They fought for justice.
Remember their bravery,
Their strength and invincibility.
We depended on them
And they on us.
Some gave their lives
While others still live.
Our love for them thrives.
Our thanks we still give.
I would like to join with Ms. Heckbert in saying: lest we forget.
Hon. Irwin Cotler (Mount Royal, Lib.):
Mr. Speaker, I rise on the eve of the 74th anniversary of Kristallnacht, the Night of Broken Glass, when, in 1938, Nazi troops stormed Jewish neighbourhoods in Germany, vandalizing Jewish homes and businesses, desecrating synagogues and cemeteries, arresting some 30,000 Jews, murdering others and burning books, as a precursor to burning people and the atrocities to come.
We commemorate Kristallnacht on the eve also of Remembrance Day when we honour the brave men and women who fought for Canada and for freedom, including those instrumental in defeating the Third Reich.
This week is also Holocaust Education Week, wherein the lessons of the Holocaust are transmitted to generations of Canadians will resonate globally with Canada as incoming chair of the International Task Force for Holocaust Remembrance and Education.
The lessons of the Holocaust are universal. We remember the dangers associated with cultures of hate and the importance of speaking out against injustice.
Tonight, I will attend a commemoration of Kristallnacht and, on Sunday, Remembrance Day ceremonies in my riding. As we remember the horrors of the past, let us ensure that the world will never experience such horrors again.
Mr. Wladyslaw Lizon (Mississauga East—Cooksville, CPC):
Mr. Speaker, next week, members of Parliament will travel back to their ridings to speak with their constituents on the things that matter most to them. I can assure members that we will not hear a call for higher taxes.
Canadians are pleased with our government's low tax plan for jobs and growth and for lowering the price on almost everything with our 2% cut to GST.
What they do not want to see is the NDP's proposed carbon tax that would raise the price on almost everything. Its sneaky tax scheme would have Canadians spend more of their hard-earned tax dollars on gas, groceries and electricity.
Why does the leader of the NDP want hard-working Canadians to pay for his $21 billion carbon tax?
Ms. Jinny Jogindera Sims (Newton—North Delta, NDP):
Mr. Speaker, this week will be busy back in my riding of Newton—North Delta. Not only will I have the pleasure of participating in multiple events but I will also have to explain the $8 billion Conservative money grab. That is right, for the last six years Conservatives have been sneaking increases to user fees for Canadians. They are now twice as high as they were a decade ago. What are the Conservatives gouging Canadians for? Youth exchange programs, maps, family reunification, even divorce, and I could go on. There are many more.
What is worse is that almost all of their budget cuts are to the very services Canadians rely on. Increased user fees and blind and reckless cuts, it is a double whammy for Canadians and their wallets. That is why New Democrats vote against the Conservatives' irresponsible agenda.
Mr. Costas Menegakis (Richmond Hill, CPC):
Mr. Speaker, in May of 2011 the NDP misled Canadians when it titled its election platform, "Give families a break". In fact, this could not be further from the truth. What it should have called its platform is, "Give families a $21 billion carbon tax". At least then it would be accurate.
It is located right on page 4 in the same platform in black and white, an NDP carbon tax scheme that would raise the price of all things Canadian families need, such as groceries, gas, electricity and heat. In effect, this NDP tax gets tough on Canadian families, not giving them a break as the opposition would have Canadians believe.
Thankfully, Canadian voters gave our Conservative government a mandate to keep taxes low, grow jobs and really give Canadian families the break they so rightfully deserve.
[Oral Questions]
Ms. Nycole Turmel (Hull—Aylmer, NDP):
Mr. Speaker, today, we learned that the Minister of Human Resources and Skills Development will announce a plan to privatize social services. One of the models that she wants to use to justify the government's massive cuts to public services is that of Goldman Sachs, the commercial bank accused of fraud in the American real estate scandal no less.
Can the Conservatives tell us exactly what services they are going to privatize and why those services are no longer part of the government's plans?
Ms. Kellie Leitch (Parliamentary Secretary to the Minister of Human Resources and Skills Development and to the Minister of Labour, CPC):
Mr. Speaker, our government has been focused on making sure that communities and families are well supported. That is exactly what this initiative is doing, very unlike the NDP's approach.
Whether we put forward initiatives such as the enabling accessibility fund or whether it is the registered disabilities savings plan, these are initiatives to help Canadians and help communities, unlike the NDP's approach where it wants to tax individuals to make sure that we cannot provide those programs for them.
Mr. Speaker, the government's disengagement is not going to put more money in people's pockets.
The other model that the minister is using is that of the Conservative Party in the United Kingdom. The people of Great Britain opposed it, and their government had to rethink its plan four times.
What the minister is proposing here is merely a public relations exercise to justify new cuts to services for Canadians. Why imitate a plan that failed so miserably in the United Kingdom?
Mr. Speaker, we actually have a plan in this country. We have the economic action plan, and we are putting it in place to make sure that Canadians have jobs. We want to make sure that Canadians are able to provide for themselves and build their communities. That is exactly what we are doing with this initiative.
I look forward to the NDP supporting the economic action plan, but apparently it is not willing to do that. It is not willing to help create Canadian jobs, as we have done with 820,000 net new jobs.
Mr. Speaker, the Parliamentary Budget Officer says that 85% of Conservative cuts are to front-line staff and services Canadians rely on. Service Canada is already in bad shape. One in four EI applications are not being processed on time. The majority of calls are not being answered.
Why is the only Conservative response to this more privatization?
Hon. James Moore (Minister of Canadian Heritage and Official Languages, CPC):
Mr. Speaker, that is clearly not the case. We have indeed put forward more resources to ensure that Canadians have the services when and where they need them, all across this country. We have done so in a way that is economically efficient and responsible, and reacts to the needs of communities that are diverse all across the country.
Just because things are being done differently does not mean that the NDP members have to put on their tinfoil hats and come up with conspiracy theories. The reality is that Canadians need these services, and we are providing them. We have increased services to those communities that are most in need. We will continue to do so because they are indeed the services that Canadians have come to expect.
Ms. Megan Leslie (Halifax, NDP):
Mr. Speaker, the proposed Conservative model for fiscal restraint is to focus their cuts on the services that Canadians rely on and then use Goldman Sachs as a model for prudent management. What is motivating the Conservatives here is not actually any concern about social services, it is a PR stunt to make up for the fact that they are making these reckless cuts.
Can the Conservatives tell us what in the world they hope to learn about social service delivery from investment banks?
Mr. Speaker, as I mentioned before, our government recognizes that we have to take steps to ensure we enable communities to tackle local problems. That is exactly what these initiatives do. They allow communities to deal with their local issues so that they can be effective in supporting families.
It is very unlike the NDP approach, which wants to tax and spend, making sure that Canadians do not have the funds in their pockets to help with their local communities. We are about creating jobs. We are about making sure we build communities. I encourage the NDP to support the economic action plan, which helps create those jobs and build those communities.
Mr. Speaker, today we also learned that Conservatives will stop maintaining cross-country ski trails in many of our national parks. They include Prince Albert National Park in Saskatchewan, Riding Mountain National Park in Manitoba, and Elk Island National Park in Alberta. This is another example of cuts to direct services.
Can the Conservatives tell us if they plan to use a private company to deliver these services as well, perhaps Goldman Sachs?
Hon. Peter Kent (Minister of the Environment, CPC):
Mr. Speaker, no government has done more when it comes to protecting our great natural spaces. National parks and historic sites will remain open this winter. Parks Canada is working with volunteer groups to provide assistance and equipment for ski tracking and trail maintenance.
Hon. Bob Rae (Toronto Centre, Lib.):
Mr. Speaker, to the Minister of Public Safety, yesterday he told the House, in the course of an answer, that he in fact had been instrumental in instructing officials to settle the civil suit between Ashley Smith's family and the Government of Canada.
What facts was the minister aware of that led him to reach that conclusion? Second, does the government have other videos in its possession with respect to the treatment of Ashley Smith? Does it have other videos in its possession with respect to the treatment of other people in our prison system?
Mr. Speaker, I think the Minister of Public Safety was very clear on this matter yesterday. Of course, the treatment of Ashley Smith is indeed a tragedy. That is why the coroner's inquest is ongoing. The Minister of Public Safety has instructed all those associated with the inquiry to provide all the information necessary so that we can not only get to the bottom of the Ashley Smith incident specifically, but also more broadly ensure that those who are in custody going forward are treated with respect and appropriate techniques that are greatly improved, based on the tragedy that we saw in the treatment of Ashley Smith.
Mr. Speaker, it is odd that the minister who is aware of the situation did not answer the question, but I am willing to accept the other minister's response on behalf of the government as long as he provides the specific answers we need. I asked the minister very specific questions, and I think that we are entitled to receive answers that are just as specific.
What facts led the government to settle with Ashley Smith's family? Is the government in possession of any other videos with respect to the treatment of Ashley Smith? Is the government in possession of any other videos with respect to the treatment of other mentally ill people in the Canadian prison system?
I would remind all hon. members not to refer to who is or is not in the chamber.
The hon. Minister of Canadian Heritage.
Mr. Speaker, the member's first question was clear, and my answer was also clear.
The minister said it yesterday, and I am repeating it today: we gave instructions that all the necessary information be given to investigators so that, in the future, the judicial process produces more responsible outcomes than what we saw in the videos of Ashley Smith.
Mr. Speaker, with great respect, the Minister of Public Safety is in the House. That is why I asked him the question. He is here.
Some hon. members: Oh, oh!
Order, please. What I said was that members should not refer to who is or is not in the chamber.
The hon. member for Toronto Centre may proceed.
Hon. Bob Rae:
Mr. Speaker, I will not refer to the fact that the minister is not answering the question that I asked him. However, I will ask the same minister, why is he not answering this question?
I am asking him a very specific question. What facts led him to settle with Ashley Smith's family? What videos did the government have in its possession? Those are very clear questions. What other videos does he have in his possession? What other photographs does he have in his possession? What other things does he have in his possession about the mistreatment of mental patients who are in our prisons?
Mr. Speaker, we take this particular case and the broad subject very seriously. That is why we are working with the inquest to ensure that all the information is being given forward.
More broadly, we want to ensure that Canadians understand that we have invested into our regime for supporting mental health across the country, not only in the corrections system but beyond that. We want to ensure that the tragedy and the cruelty, frankly, that Ashley Smith faced is not repeated in the future. That is why the minister has taken the responsible action to ensure that all the information is coming forward.
Mr. Randall Garrison (Esquimalt—Juan de Fuca, NDP):
Mr. Speaker, if the government and the member for Toronto Centre paid any attention to the correctional investigator, they would know he already reviewed more than 60 hours of tape with regard to Ashley Smith and made recommendations more than two years ago. When the Minister of Public Safety was asked to implement those recommendations to ensure that no one else faces the same fate as Ashley Smith, the minister responded with an absurd partisan attack.
Ashley was a victim of the system, a system unprepared to deal with mental illness. This time could the minister spare us the talking points and platitudes. Will he apologize to Ashley Smith's family and will he commit today to implementing the correctional investigator's recommendations?
Hon. Vic Toews (Minister of Public Safety, CPC):
Mr. Speaker, I note the member is reading from his talking points. What I am prepared to say in response to his talking points is that this tragedy continues to show that individuals with mental health issues do not belong in prisons but rather in professional health facilities.
At the same time, our government continues to take concrete steps on the issue of mental health in prisons. Since 2006, we have invested nearly $90 million in mental health for prisoners and we continue to take action to improve access to mental health treatment and training for staff.
Mr. Speaker, once again, the minister just admitted that individuals with mental illnesses do not belong in prison. However, on the Conservatives' watch the intake of inmates with mental illnesses has nearly doubled.
The system is not equipped to deal with this problem and the Conservatives refuse to fix it. Instead, they tied the hands of judges with mandatory minimums, cut the Correctional Service Canada budget and ignored the consequences inside and outside of prisons.
Again, I ask the same simple question. Will the minister listen to the correctional investigator and implement his recommendations?
Mr. Speaker, our police and judges do the work that they are assigned to do by Parliament under the Criminal Code. In that context, people with mental health issues have landed up not only in our federal penitentiaries but in our provincial jails.
We sat down last week with the provincial ministers to find a better way of dealing with those who have mental health problems. I know the member is concerned about this. I can tell him that all of the ministers right across Canada are very concerned about coming up with a solution to this.
Ms. Françoise Boivin (Gatineau, NDP):
Mr. Speaker, it is not just Ashley Smith who has not had justice. Let us look at what is happening in Alberta where judicial resources are stretched to the breaking point.
The case of an accused child molester was thrown out of court for unreasonable delays after more than three years in the court system. The Conservatives' failure to provide adequate judicial resources has real consequences.
Will the minister step up and fulfill Alberta's request for additional judges?
Hon. Rob Nicholson (Minister of Justice and Attorney General of Canada, CPC):
Mr. Speaker, we consider all requests from our provincial counterparts. Quite frankly, I am pleased to see the opposition has discovered the justice file. It is an important one and one that we have made a priority over the years.
I hope this new interest will extend to supporting the legislation that we have brought forward to crack down on criminals and stand up for victims in our country. I hope we get that from the opposition. It would be great.
Mr. Speaker, they say they are against bullying, but when it comes to justice issues they respond with bully tactics.
Last week, at the meeting of ministers of justice, the provinces and territories voiced their concerns about cuts to youth justice programs, underfunding of legal aid and the shortage of police officers, especially in first nations communities.
They also discussed the lack of consultation about federal crime bills. The federal government does not consult them at all.
Will the Minister of Justice address the problems identified by the provinces and territories?
Mr. Speaker, we have been fixing these problems over the years with no help whatsoever from the NDP and their colleagues.
Under the leadership of our Minister of Finance, transfers to the provinces have been up 43% since the year 2006. That is very impressive. We support the provinces and non-governmental organizations. We have programs at the federal level, all better designed to tackle crime in our country and assist victims across Canada. This is what we should do and we should have the support of everyone in the House on that.
Mr. Robert Chisholm (Dartmouth—Cole Harbour, NDP):
Mr. Speaker, on matters of justice in the provincial government, maybe the Minister of Intergovernmental Affairs could work with the provinces to try and help solve some of these problems, because the Prime Minister has been alienating the provinces, even refusing to attend an upcoming first ministers meeting in Halifax. In fact, he will not even sit down with the premiers to discuss our shaky economy.
Will the Minister of Intergovernmental Affairs be attending the first ministers meeting in Halifax? Will he give us some idea about his plan to correct the problems in the relationship between the provinces and the federal government?
Mr. Speaker, I certainly hope the hon. member did not miss this, but just last week we met with our provincial counterparts. The Minister of Public Safety and I sat down with our provincial and territorial counterparts. We had a great discussion with respect to areas of mutual concern such as cracking down on crime, updating the Criminal Code, standing up for victims and law-abiding Canadians. It was a very fruitful, very productive discussion. I was very pleased to have it. I hope that has the support of everyone here.
Mr. Speaker, we are really trying to help the minister from Labrador, but he is not really doing much. He gets a limo, a driver and lots of staff. He gets a very generous pay from the taxpayer and for that he is supposed to be able to stand and explain himself. However, we will go with something simpler.
Why was 79% of the travel expenses as minister spent in his own province? That is on top of the $18,000 in free flights from his campaign. We have a simple question. Is he using his ministerial dollars and taxpayer dollars to perpetually campaign around his riding?
Hon. Peter Penashue (Minister of Intergovernmental Affairs and President of the Queen's Privy Council for Canada, CPC):
Mr. Speaker, our government is committed to strong relations with the provinces and territories based on true respect for their jurisdictions. I work with my counterparts to ensure that a strong relationship continues to grow. I also make it a priority to ensure that our government does its part in fostering jobs and growth in Newfoundland and Labrador and across Canada. That is what we do and that is what we will continue to do.
Mr. Speaker, yes, but what he has not done is explain why he is blowing all this money in his riding like he has not explained about his campaign spending. The member is like a poster boy for why we need stronger laws at Elections Canada. Let us get on to that.
This past week, Elections Canada praised the New Democratic Party for pushing for stronger powers for the Chief Electoral Officer to deal with voter fraud. Meanwhile, we also learned from new affidavits that it was the Conservative Party headquarters that had the call list that went out for voter suppression.
What are Conservatives doing? Are they going to sit and wait for the next RCMP raid against Conservative Party headquarters, or are they going to come clean with the taxpayers?
Hon. Peter Van Loan (Leader of the Government in the House of Commons, CPC):
Mr. Speaker, the Minister of Intergovernmental Affairs has made it clear that we need to improve our electoral laws. That is why he is going to bring forward legislation to do that.
However, the problem is not always that we need to improve our electoral laws. The problem is that some parties like the NDP do not even follow our election finance laws, having accepted hundreds of thousands of dollars in illegal donations that they then were required to pay back. They decided behind closed doors what was going on with Elections Canada. They did not disclose it to the House or to Canadians. They are the ones who have to be accountable to Canadians for this and they are not willing to do that.
Mr. Alexandre Boulerice (Rosemont—La Petite-Patrie, NDP):
Mr. Speaker, I would think that, having been found guilty of using the in and out scheme and having the RCMP raid party headquarters, the Conservatives would be a little more humble.
After an investigation lasting a year and a half, voters are eager to find out who is behind the biggest voter fraud in the history of Canada. But the Conservatives are dragging their feet and refuse to give Elections Canada more powers.
It is odd. It is as though they had something to hide and were trying to buy time. It is more than a little suspicious in light of the fact that the Conservative Party headquarters provided the call list for fraudulent calls.
When will they stop covering for the fraudsters? When will they take their heads out of the sand? When will they give the powers—
The hon. Parliamentary Secretary to the Minister of Transport.
Mr. Pierre Poilievre (Parliamentary Secretary to the Minister of Transport, Infrastructure and Communities and for the Federal Economic Development Agency for Southern Ontario, CPC):
Mr. Speaker, those members start out by attacking the minister for spending time in his constituency and reaching out to the people of Labrador. Perhaps if the member for Timmins—James Bay spent as much time reaching out to his constituents, he would know that they did not want him to break his word and vote in favour of the much despised Liberal long gun registry. However, at the same time, he has a seat mate who will not even answer the simple question of whether he believes his province should still be in the country. I will give him a chance to stand and declare it—
Order, please. The hon. member for Toronto—Danforth.
Mr. Craig Scott (Toronto—Danforth, NDP):
Mr. Speaker, in a federal court affidavit, the COO of RMG is clear. Scripts were constructed to inform voters of polling changes in the last week of the campaign. The calls were based wholly on data provided by the Conservative Party. Such unreported data is crucial for Elections Canada to complete its investigation, but there is nothing in the Elections Act to require such reporting.
Will Conservatives support my bill that would enforce good reporting of voter telecommunications during campaigns, yes or no?
Mr. Speaker, the Minister of State for Democratic Reform has made it quite clear that we need to improve Canada's elections laws. That is why he will bring forward legislation to do exactly that. We continue to gather input from other sources.
Just the other day I saw media reports of a further report from the chair of Elections Canada suggesting that we should be considering other changes. He will look comprehensively at what needs to be changed and bring forward proposals that will address those problems so we will not have the kind of issues we have had with the Liberals with loans that are not being paid back from their leadership contestants that are effectively illegal donations and the kind of illegal donations—
Order, please. The hon. member for Toronto Centre.
Mr. Speaker, my question is again for the Minister of Public Safety. Could the minister tell us what facts and documentation in possession of the government led it to settle out of court with Ashley Smith's family? Does the government still have in its possession other videos and documentation with respect to Ashley Smith going back further than just one year? Will the government guarantee that it will not destroy the other videos and documentation in its possession with respect to the treatment of people who have mental challenges and mental problems who have ended up in Canada's prisons?
Mr. Speaker, any of the records the leader of the third party is asking for would be in the custody of Correctional Service Canada. As members know, the government has clearly directed Correctional Service Canada to work with the coroner's inquest. This matter is being dealt with in a public forum and in a fully transparent way.
Mr. Scott Andrews (Avalon, Lib.):
Mr. Speaker, we all know that the member for Labrador is drowning in his election irregularities. The management of the relationship between the federal government and the provinces is a major job and the minister is simply not doing his job. His own website shows only three meetings with provincial counterparts, and that was all in 2011. God only knows what he has been doing in 2012. He has been ripping off taxpayers by not doing his job.
Will the minister admit to not doing his job and resign immediately?
Mr. Speaker, sometimes I just cannot believe how rude and how bullish those people can be.
Order, please. The Minister of Intergovernmental Affairs.
Hon. Peter Penashue:
Mr. Speaker, our government has a strong relationship with provincial and territorial governments. I meet with counterparts regularly and focus on the strength of the province and growing Canada's economy. New exploration and investment are occurring across Canada, especially in Labrador.
In my role as Minister of Intergovernmental Affairs, I get to share these success stories with people from coast to coast to coast. I am working hard to ensure that all Canadians benefit.
Mr. Speaker, while the government is slashing front-line services to Canadians, it is spending a record half a billion dollars on lawyers, an increase of almost 40% in the past three years alone. Instead of doing the right thing, it is telling people it will see them in court. The Conservatives are using the courts to quash first nations child rights and to fight with veterans over the clawback of their military pensions.
When will the government work with Canadians on finding real solutions rather than spending millions of dollars on lawyers?
Mr. Speaker, at any given time the federal government is involved with nearly 50,000 litigation files. The costs are based on a number of cases, including fighting on behalf of residential school survivors as well as fighting against big tobacco.
We make no apologies for defending the rights of Canadians in court.
Ms. Hélène LeBlanc (LaSalle—Émard, NDP):
Mr. Speaker, the chief energy researcher at CNOOC recently gave a speech in Beijing. He warned oil industry executives that if Canada did not approve the construction of pipelines to China, the oil sands would become outdated.
The Nexen deal has not even been approved and CNOOC is already announcing its intentions to get greater access to the oil sands.
Why are the Conservatives letting themselves be intimidated by this Chinese state-owned company that only wants to help itself to our energy resources?
Hon. Christian Paradis (Minister of Industry and Minister of State (Agriculture), CPC):
Mr. Speaker, with all due respect to the member, there was a lot of fabrication in her question.
Obviously, every decision regarding foreign investment is always made in the best interests of Canadians. As for the potential CNOOC-Nexen deal, it is a matter of determining whether there is a net benefit to Canada. I repeat that the deal is now being carefully examined.
Mr. Speaker, the Conservatives seem to have a hard time understanding their new investment agreement with China.
Do the Conservatives realize that under this agreement, giving national treatment to Chinese state-owned companies means that they have the same rights as those given to Canadian companies with respect to purchasing new oil leases and expanding their operations in Canada?
Mr. Gerald Keddy (Parliamentary Secretary to the Minister of International Trade, for the Atlantic Canada Opportunities Agency and for the Atlantic Gateway, CPC):
Mr. Speaker, as I explained yesterday, the FIPA between Canada and China is about recognizing Chinese investment rights in Canada and Canadian investment rights in China. It is as simple as that. There is nothing nefarious about this.
I would ask the hon. member, though, why the NDP consistently opposes trade. Why would it allow members of the NDP, like the member for British Columbia Southern Interior, to say that trade threatens the very existence of our nation? Where does that mindset come from and what does that say about the future of the NDP?
Ms. Linda Duncan (Edmonton—Strathcona, NDP):
Mr. Speaker, the Standing Committee on Government Operations and Estimates worked diligently to identify key reforms to improve the capacity of MPs to deliver their constitutional duty to scrutinize government spending, reforms based on the advice of renowned parliamentary experts. Despite two similar reports over the last decade, the government voted to throw these reforms in the dustbin.
Why is the government refusing to make the very reforms needed to usher in the once-promised open, transparent, participatory democracy?
Hon. Tony Clement (President of the Treasury Board and Minister for the Federal Economic Development Initiative for Northern Ontario, CPC):
Mr. Speaker, the Government of Canada overwhelmingly agreed with the recommendations from that particular committee. Specifically, I can say to the House that there is a direct relationship now, a new structure, for approving spending that will provide taxpayers and parliamentarians with a clear and traceable line between spending approvals and specific government programs. No other government has done this. We are proud of the initiatives that we have taken to improve transparency and improve the ability of parliamentarians to review the budget.
Mr. Denis Blanchette (Louis-Hébert, NDP):
Mr. Speaker, in any case, they did not listen to the key recommendation with regard to the Parliamentary Budget Officer.
At the Standing Committee on Government Operations and Estimates, the Conservatives contributed to drafting and adopting the committee report, which suggested many ways to make government finances more transparent. The committee spent months and months working on this.
But yesterday, instead of standing up, doing their job and defending the report, the Conservatives supported an irresponsible amendment to shelve it and undo their own work.
Can any of the Conservative members of the committee explain this about-face?
The President of the Treasury Board.
Mr. Speaker, as I said, we agreed with most of this committee's and the report's recommendations. It is important to have more transparency and accountability. We accept this report. It is important in order to have members' support for the budget. We accept most of the recommendations.
Mr. Rob Clarke (Desnethé—Missinippi—Churchill River, CPC):
Mr. Speaker, Remembrance Day is a time for all Canadians to remember the sacrifices of Canada's fallen. As parliamentarians, we are privileged to stand in the House regardless of political differences and represent the very democracy of Canada's veterans who fought and gave their lives for us.
Would the Minister of Veterans Affairs comment on the importance of recognizing the services and sacrifices of Canada's veterans?
Hon. Steven Blaney (Minister of Veterans Affairs, CPC):
Mr. Speaker, the member for Desnethé—Missinippi—Churchill River, a former RCMP officer, is absolutely right. We as Canadian parliamentarians owe a tribute to our veterans because it is due to their sacrifice that we can stand for democracy in this very House.
On the weekend, I invite all parliamentarians to take the time to thank the veterans, reach out to them, lay a wreath and say thank you. Canada thanks you for the sacrifices you made for our country.
Mr. François Pilon (Laval—Les Îles, NDP):
Mr. Speaker, because of bad Conservative policies, only four lakes in Quebec are now protected.
I would reply to my hon. colleague from Argenteuil—Papineau—Mirabel that the Minister of Transport, Infrastructure and Communities has even said that he would not go everywhere and into every café to consult people. He believes that consultation with representatives is enough.
Can the minister tell us who in Sorel finds it acceptable that Lac Saint-Pierre is no longer protected, even though it is home to the largest archipelago in the St. Lawrence and the largest heronry in North America?
Hon. Denis Lebel (Minister of Transport, Infrastructure and Communities and Minister of the Economic Development Agency of Canada for the Regions of Quebec, CPC):
Mr. Speaker, my colleague is talking about herons, while we on the other hand are talking about boats. That makes all the difference. We are talking about two completely different things. What we want to talk about is navigation. We want to manage any area where navigation is used. Environment officials will manage the environment and Fisheries and Oceans people will manage their department. The opposition members are mixing everything up.
As for navigation, we will continue to ensure that what needs to be done is done for Canada's economy. The municipalities and provinces have all asked us to eliminate delays in awarding contracts and issuing mandates.
Mr. Speaker, the Muskoka minister made sure that his lakes and rivers were protected in the budget, just not other important waterways that support recreation and fishing.
The Bowmanville Creek in the riding of Durham is a great example. The creek has recently seen salmon stocks improve due to local efforts. Sadly, the Conservatives' new budget bill would remove protection from this waterway. Bridges and impediments would go through without federal review.
Why did the Conservatives abandon the protection of this waterway?
Mr. Speaker, we have not abandoned the protection of waterways. We have many rules and acts to protect that. In our department, we have to protect and control navigation. That is what we have done.
I want to be clear when I say that 98% of the applications received have never posed a threat to navigation. That means that they want to invest money for absolutely nothing. This is not the case. This would not be a responsible use of taxpayers dollars and shows just how much change is needed. That is why we are reforming the act. Now our resources can focus on waterways navigation.
Mr. Nathan Cullen (Skeena—Bulkley Valley, NDP):
Mr. Speaker, the aftershocks of the massive 7.7 magnitude earthquake that struck Haida Gwaii over a week ago was still shaking the B.C. coast last night. It was a close call, far too close.
Instead of cutting programs that help community emergency response, as the Conservatives are doing, we need to ensure that every protection is available to our towns and villages in the event of a major natural disaster.
Will the minister commit today to work with the people of Haida Gwaii and make all the necessary investments to keep our communities safe?
Mr. Speaker, the majority of emergencies in Canada are local in nature and managed by local or provincial governments. We, in fact, have worked very closely with local governments.
We are now focusing on some of the things that the federal government can do better, for example, the mitigation program that was announced by the Prime Minister for Saskatchewan, Manitoba and Quebec, the $99 million to assist with the flooding. We are also in overall discussions with the provinces about a national mitigation program, and I hope I can get the support of that member for this.
Mr. Speaker, the hard truth about the Conservatives' recent monster budget bill is an affront to democracy and a direct attack on our environment. By warping a law meant to protect our rivers into a pipeline promotions act, by ignoring the good people of Victoria, Haida Gwaii and the coastal communities of British Columbia, this Conservative government is threatening the very resources we rely upon.
When will B.C. Conservative MPs actually stand up for our beautiful province in this place? When will they stand up for British Columbia and stop selling it out?
Mr. Speaker, the bill and its amendments will improve navigation everywhere in Canada, including in British Columbia, and do not in any way eliminate the other departments' responsibilities.
All projects must continue to comply with federal legislation, such as the Canadian Environmental Protection Act, the Canadian Environmental Assessment Act, the Fisheries Act, the Migratory Birds Convention Act, 1994, and many others.
Many measures will continue to protect nature. And we will look after navigation.
Mr. Sean Casey (Charlottetown, Lib.):
Mr. Speaker, let us look at government assistance for veterans' funerals. Serving members of the Canadian Forces get $12,700. Some traditional vets get $3,600. Modern-day veterans get zero. Veterans who earn more than $1,000 a month get zero.
It is time that we embrace one veteran, one standard. When will the government treat all of our veterans fairly and evenly?
Mr. Speaker, I must say that it is a big job looking after our veterans after years of Liberal neglect.
For six years, our government has invested in our veterans. I can reassure them that we have no intention of making cuts to programs, as the Liberals did to the program that we are talking about.
Every week, every month, we continue to improve the quality of life of our veterans by providing them with better services, because they deserve that.
Hon. Dominic LeBlanc (Beauséjour, Lib.):
Mr. Speaker, I received petitions signed by thousands of people who are against the Conservatives' proposed changes to employment insurance. These people understand very well that there are no jobs in the middle of the winter where we come from and that the Prime Minister is punishing them because he thinks that these people might just be lazy.
Why does the Prime Minister refuse to meet with these workers and why does the government refuse to change its plan that will punish both the people working in seasonal industries and their employers?
Mr. Speaker, our government has been focused on ensuring that Canadians have an opportunity to have jobs in their local areas that meet their local qualifications and their qualifications overall.
We have created 820,000 net new jobs in this country since the downturn in the economy, the recession. That is why we are focused on ensuring that Canadians have opportunities for jobs.
Just like in my riding of Simcoe—Grey, where seasonal workers have great opportunities, we are building those opportunities for Canadians in the future.
Ms. Lysane Blanchette-Lamothe (Pierrefonds—Dollard, NDP):
Mr. Speaker, Resolute Forest Products workers and retirees are worried that their pension fund is in jeopardy. Earlier this week, the company announced the elimination of 111 jobs at the plant in Grand-Mère, and in Clermont, the workers still do not know whether they will return to work.
Canada is one of the only OECD countries that provides no protection for workers' pension funds when an employer goes bankrupt or becomes insolvent.
What are the Conservatives waiting for? When will we see a bill to protect the pension funds of Canadian workers?
Mr. Speaker, that is completely false. First of all, I offer my sympathies to those who are facing these challenges.
Fortunately, these people can count on a government that has taken action in the past: we improved protections for workers, we changed the legislation to ensure that regular pension payments are made before other creditors are paid, and we also expanded the wage earner protection program, to pay severance pay up to $3,400.
I know that the member was not here during the previous Parliament, but her party voted against these changes, and that is shameful.
Mr. Pierre Dionne Labelle (Rivière-du-Nord, NDP):
Mr. Speaker, this week, Luxorama in Saint-Jérôme filed for bankruptcy without notice under the Bankruptcy and Insolvency Act, brutally dismissing 200 employees. With a just a few weeks left before the holidays, the employees are devastated. Not only are they losing their jobs without warning, but they are also finding out that the average wait time in Quebec to get the first employment insurance cheque is 39 days, when the wait time should not exceed 28 days.
For weeks I have been listening to the Minister of Human Resources tell us that she is working on improving service delivery. I am sorry, but this minister has to stop taking Canadians for fools. She is not working—
The Parliamentary Secretary to the Minister of Human Resources.
Mr. Speaker, our government's top priority is to improve job creation and economic growth. Service Canada will use more automation to continue to work on responding better to Canadians and serving them in a quick and efficient manner.
Service Canada continues to provide improvements and update its operations so that Canadians have opportunities to access our programming.
We are using taxpayers dollars the best. We want to ensure that Canadians have opportunities to access all programming available to them.
Mr. Blake Richards (Wild Rose, CPC):
Mr. Speaker, yesterday our nation's exporters were disappointed when the NDP voted against Canada's free trade agreement with Panama. After having delayed the agreement for more than two years, this was obviously no surprise. In fact, the leader of the opposition has said that, if given the opportunity, he would repeal agreements that promote trade, increase Canadian exports and protect Canadians doing business abroad.
Could the parliamentary secretary please share with the House how our government is standing up for the interests of Canada's exporters?
Mr. Speaker, the hon. member for Wild Rose is correct. The NDP's vote against our agreement with Panama was predictable. This is the same party that sent an anti-trade mission to Washington to lobby against Canadian jobs. It is the same party that has consistently opposed our government's initiatives to increase trade and promote Canadian exports.
While the NDP talks down Canada to the world, our Conservative government will continue to open new opportunities to grow Canadian exports and create Canadian jobs.
Ms. Kirsty Duncan (Etobicoke North, Lib.):
Mr. Speaker, on Tuesday, 52 Conservative senators voted en masse to block MS sufferers from testifying before a Senate committee on Bill S-204. Can anyone imagine a committee hearing on cancer not calling cancer sufferers? Given this slap in the face to MS patients, will the government commit, if S-204 comes to the House, to allow MS sufferers to testify? Will it give MS patients back their voice?
Hon. Leona Aglukkaq (Minister of Health and Minister of the Canadian Northern Economic Development Agency, CPC):
Mr. Speaker, our government recognizes the difficulties and heartbreak faced by the thousands of MS patients and their families across Canada. We were clear all along that we are committed to funding a clinical trial for CCSVI once all the necessary medical and ethical standards were met. That is why last month at the health ministers meetings in Halifax I announced that, after a rigorous review process, clinical trials for CCSVI have been accepted and recruitment for patients to participate will begin this month.
Mr. Jean-François Larose (Repentigny, NDP):
Mr. Speaker, the Standing Committee on Government Operations and Estimates just completed a study on public-private partnerships. P3s are a form of insidious privatization and, as we are seeing right now in Quebec, the risk of collusion and corruption is huge if we allow private enterprise too much involvement in the management of public affairs.
One of the recommendations that we would like to see included in the final report is ensuring greater transparency and oversight when it comes to P3 contracts.
Will this government follow that recommendation?
Hon. Jim Flaherty (Minister of Finance, CPC):
Mr. Speaker, PPP Canada Inc. has been remarkably successful and is announcing more and more partnerships, not only with certain municipalities but with aboriginal groups and first nations in Canada, with projects from coast to coast to coast. This is a crown corporation that operates in an objective arm's length manner. It reviews applications on the basis of merit. We do not comment, of course, on applications under review.
I appreciate the interest of the member opposite in PPP Canada Inc. It does provide greater value to taxpayers and helps meet Canada's infrastructure needs.
Mrs. Tilly O'Neill Gordon (Miramichi, CPC):
Mr. Speaker, our government has long supported Canadian sealers and their families while Hollywood activists have spread false information about the humaneness and practice of the hunt.
Could the Minister of Fisheries and Oceans please inform the House of a recent study regarding the seal hunt and its humane practices, as well as our government's support for the industry?
Hon. Gail Shea (Minister of National Revenue, CPC):
Mr. Speaker, I thank the member for Miramichi for raising this issue. Two veterinary researchers who have observed the hunt first-hand over several seasons have published a report indicating "...there is no reliable evidence that the Canadian harp seal hunt differs from other forms of exploitation of wildlife resources from the perspective of animal welfare".
With a record high number of seals and highly regulated humane practices used in the hunt, the government will continue to defend Canadian sealers and their families against those who do not understand the practice of the hunt or how important it is to the traditional way of life in many communities.
Mrs. Sadia Groguhé (Saint-Lambert, NDP):
Mr. Speaker, Conservative cuts to services and centralized processing of immigration applications are having a terrible impact on wait times for international adoptions.
Our constituency offices are overwhelmed by complaints from Canadians who are fed up with having to wait longer and longer for their international adoptions to go through.
Every additional month spent away from their new Canadian parents is harmful to the physical and psychological development of these children.
Can the Minister of Citizenship, Immigration and Multiculturalism tell us how he plans to fix this problem, which was created by the Conservatives in the first place?
Mr. Rick Dykstra (Parliamentary Secretary to the Minister of Citizenship and Immigration, CPC):
Mr. Speaker, the ministry and this government recognize the importance that Canadian families place on being united with adopted children from abroad as quickly and efficiently as possible. As the member may know, in all cases they are done on a priority basis.
There is no reason for anyone in the House to stand up and accuse this government of not doing what is absolutely right when it comes to adoption. It is about priority, placement and doing the job properly.
Mr. André Bellavance (Richmond—Arthabaska, BQ):
Mr. Speaker, yesterday, the Minister of Justice accused us of overlooking all his fine accomplishments, but he is the one who overlooked our question.
As the minister knows, he can already add terrorist groups to a list to freeze or seize their assets, and he can require banks to prevent these groups from accessing their accounts.
Why not do the same thing with the Mafia and criminal gangs who, as we have seen in the media, use the Port of Montreal as a gateway for their criminal activity because of the weakness of the inspection process there?
Is the minister willing to consider an anti-Mafia law and a list of banned criminal organizations such as the one we already have for terrorist groups? What is good for the goose must also be good for the gander.
Mr. Speaker, since forming government we have passed legislation to address auto theft, ID theft, fraud and white-collar crime. All of these are directed at criminal organizations that turn a profit at the expense of Canadians.
We have had lots of legislation dealing with organized crime. If the hon. member wants to put the mob out of business, his party should start supporting our efforts to do so.
Order. This concludes question period for today.
Although we are following a Friday schedule, I believe the hon. opposition House leader has his regular Thursday question today.
[ Business of the House]
M. Nathan Cullen (Skeena—Bulkley Valley, NPD):
Mr. Speaker, I am honoured to rise on behalf of the official opposition to ask the government what it has planned for the House for the remainder of this week and next week.
As MPs head back to their ridings to mark the solemn occasion of Remembrance Day, I want to take a moment to acknowledge the ultimate price that has been paid all too many times by men and women, affecting all too many families, in the name of Canadians and our most cherished rights and freedoms at home and abroad. We will never forget their sacrifice particularly over the week of Remembrance Day ceremonies.
I have two specific questions for the House leader today. First, what does he have planned for the House in the days following the Remembrance Day constituency week? Second, given the absolute disaster and sham of a process that the government has set up on the committees study of its monster budget bill in which a number of ministers, parliamentary secretaries and government MPs have contradicted one another as to what the process actually is, does the government leader simply expect more from his most senior Minister of Finance on something so important as the budget of Canada, or is this sham of a process going to suffice from now until the budget's reintroduction in the House?
Mr. Speaker, this afternoon, before we depart to our constituencies and events for Remembrance Day where most of us will be participating in remembrance services in our ridings, we will resume third reading debate on Bill C-28, the financial literacy leader act.
The week of November 19 will continue to see a lot of important action at the House committee level, where we are looking at the budget implementation act, Bill C-45, the jobs and growth act, as it advances through the legislative process. The finance committee is supported by 10 other committees looking at it and all together they will conclude the review of this very important bill and the very important job creation and economic measures that are laid out, measures that were first put before Parliament back in our March budget.
Meanwhile, on Monday the House will continue the third reading debate of Bill C-44, the helping families in need act, which we started this morning. Given support for the bill from all corners of the House, I hope it will pass that day so the Senate can pass it before the end of the year.
After Bill C-44, it is our intention to take up the report stage and third reading of Bill S-11, the safe food for Canadians act, which was reported back from the agriculture committee yesterday. I hope we will see strong interest in passing that bill quickly, just as we did for second reading.
Once that bill passes on Monday, the House will return to third reading of Bill C-28, the Financial Literacy Leader Act, if we do not finish the debate today.
That will be followed by second reading of Bill S-8, the Safe Drinking Water for First Nations Act. On Tuesday, Wednesday and Friday, the chamber will consider report stage and third reading of Bill C-27, the First Nations Financial Transparency Act, which was also reported back from committee yesterday.
I should also advise the House that on Tuesday when we return from the Remembrance Day week, immediately after question period I will call ways and means Motion No. 14 respecting some technical amendments to tax laws. Let me assure the House that there should be no doubt about this, but the opposition will no doubt be disappointed. This motion will definitely not implement the New Democrats' $21.5 billion job-killing carbon tax.
Finally, on Thursday before question period, the House will resume second reading debate of Bill S-8 and after question period we will take up Bill S-2, the family homes on reserves and matrimonial interests or rights act, also at second reading.
[Routine Proceedings]
A message from His Excellency the Governor General transmitting supplementary estimates (B) for the financial year ending March 31, 2013, was presented by the President of the Treasury Board and read by the Speaker to the House.
Mr. Speaker, I have the honour to table in both official languages, on behalf of 95 departments and agencies, the performance reports for 2011-12.
I invite members to access the performance reports at www.tbs-sct.gc.ca/dpr-rmr.
Mr. Tom Lukiwski (Parliamentary Secretary to the Leader of the Government in the House of Commons, CPC):
Mr. Speaker, pursuant to Standing Order 36(8), I have the honour to table, in both official languages, the government's response to 43 petitions.
Mr. Mike Wallace (Burlington, CPC):
Mr. Speaker, pursuant to Standing Order 34(1) I have the honour to present to the House, in both official languages, the report of the Canadian Parliamentary Delegation of the Canada-Japan Inter-Parliamentary Group respecting its participation at the 18th bilateral meeting with the Japan-Canada Diet Friendship League, which was held in Tokyo and Tohoku region, Japan, from May 20 to 25.
Hon. Michael Chong (Wellington—Halton Hills, CPC):
Mr. Speaker, I have the honour to table, in both official languages, the second report of the Standing Committee on Official Languages concerning the assessment of the roadmap and the improvement of programs and service delivery.
Pursuant to Standing Order 109 of the House of Commons the committee requests that the government table a comprehensive response to this report.
Mr. Yvon Godin (Acadie—Bathurst, NDP):
Mr. Speaker, this was the longest study on official languages undertaken by Parliament. I can certainly say that it was longer than the study of the budget. It took a year and several months to conduct this review.
We approve this report, but we have some reservations. We are going to table a supplementary opinion to address certain omissions, inaccuracies and weaknesses in the scope of the recommendations, particularly with regard to funding, seniors, the media, health, the language industry, literacy, second language, early childhood and other areas.
We recommend that the government renew this roadmap in 2013 and take it seriously. When we met with community representatives, they told us that they really wanted the roadmap to be renewed for another term in 2013.
moved for leave to introduce Bill C-465, An Act to amend the National Capital Act (Gatineau Park).
She said: Mr. Speaker, I am pleased to introduce my bill to protect Gatineau Park, and I thank the member for Pontiac for supporting its introduction.
Gatineau park's size, beauty and importance make it a real national treasure. It is one of the most visited parks in Canada, yet it does not currently benefit from protections that would allow us to ensure it will be preserved for future generations.
My bill would fix that by making some amendments to the National Capital Act. It would enshrine the park's boundaries in the act, give these boundaries parliamentary protection and prohibit the sale of public lands located within the park.
If we want to leave a healthy park to future generations that looks like the park we know today, we must take action now. I urge all members of all of the parties represented in the House to support my bill at second reading.
Mr. Yvon Godin:
Mr. Speaker, there have been consultations among all parties and I believe that if you seek it you would find unanimous consent for the following motion:
That the Standing Committee on Justice and Human Rights be the committee for the purposes of section 533.1 of the Criminal Code.
Does the hon. member have the unanimous consent of the House to move the motion?
Some hon. members: No.
Mr. Speaker, I move, pursuant to Standing Order 56.1, and seconded by the President of the Treasury Board:
The House has heard the terms of the motion. Will those members who object to the motion please rise in their places.
And fewer than 25 members having risen:
The Acting Speaker (Mr. Barry Devolin): Fewer than 25 members having risen, the motion is adopted.
Mr. Mathieu Ravignat (Pontiac, NDP):
Mr. Speaker, I have two petitions to present today.
The first petition asks the government to recognize the importance of the ELA to the Government of Canada's mandate to study, preserve and protect aquatic ecosystems and reverse the decision to close the ELA research station and continue to staff and provide financial resources to the ELA.
Mr. Speaker, the second petition calls upon the House to pass Bill C-398, without significant amendment, to facilitate the immediate and sustainable flow of lifesaving generic medicines to developing countries.
Mr. Speaker, I have a petition to present today about the Experimental Lakes Area.
The petitioners who signed this petition ask the government to recognize the importance of the Experimental Lakes Area. They ask the government to reserve the decision to close the Experimental Lakes Area and also to continue to provide adequate resources to the Experimental Lakes Area.
Ms. Mylène Freeman (Argenteuil—Papineau—Mirabel, NDP):
Mr. Speaker, I am pleased to present this petition today. People in my riding and across Canada have signed the petition, which calls on Parliament to support my motion M-400 to protect water and public health in our rural communities.
This motion urges the government to support Canadians who need to upgrade their septic systems in order to protect the environment, public health and the vitality of our rural communities and in the interest of addressing the rural-urban divide.
I am proud to say that my motion has received a great deal of support from municipalities, citizens and environmental groups all over Canada, as well as the FCM.
Ms. Rathika Sitsabaiesan (Scarborough—Rouge River, NDP):
Mr. Speaker, I rise today to present two petitions on behalf of the residents of Scarborough—Rouge River as well as people in other—
Mr. Kevin Lamoureux:
Mr. Speaker, I rise on a point of order. Traditionally, on petitions, the Speaker will often look around and acknowledge members from different parties as opposed to one political party consistently. The lights on the ceiling are out over at this end and maybe that is preventing our members from being seen.
The hon. member is quite correct that the Speaker sometimes does that but it is not a requirement. It is the prerogative of the Chair to recognize members. As such, I will continue to do that. The hon. member for Scarborough—Rouge River.
Ms. Rathika Sitsabaiesan:
Mr. Speaker, the first petition I present calls upon the Government of Canada to urge the United Nations to immediately establish an independent, international and impartial mechanism to ensure truth, accountability and justice in Sri Lanka following the United Nations report of the Secretary General's panel of experts on accountability in Sri Lanka, who found credible allegations, which have proved and indicate that, during the final stages of Sri Lanka's civil war, a wide range of serious violations of international humanitarian and human rights laws were committed.
Mr. Speaker, in my second petition, the petitioners call upon the Government of Canada to repeal Bill C-31, which they call the punishing refugees act, and return to the framework of the Balanced Refugee Reform Act, passed with the support of all parties in the previous Parliament, as the current bill, Bill C-31, concentrates way too much power in the hands of one minister.
Ms. Marie-Claude Morin (Saint-Hyacinthe—Bagot, NDP):
Mr. Speaker, I have the honour today to present a petition signed by people from across Canada, from all social classes and of all ages, who are urging the government to take action and adopt Bill C-400, which would establish a national housing strategy.
Ms. Irene Mathyssen (London—Fanshawe, NDP):
Mr. Speaker, I have two petitions today.
The first petition is from petitioners who are concerned about the Canada-EU comprehensive trade agreement. Their concern is that sub-national governments and public agencies will not be able to procure locally and that would have a negative impact on our economy.
They ask the government to exclude sub-national governments and public agencies, and also to have negotiations regarding the Canada-EU comprehensive free trade agreement suspended until there can be a national consultation.
Mr. Speaker, the second petition is in regard to the changes to old age security benefits and eligibility for those benefits .
The petitioners note that OAS and GIS are very important to the financial welfare of Canadians and to force Canadians to work two years longer deprives them of their ability to have a secure retirement.
The petitioners ask the Government of Canada to maintain the retirement age for old age security at 65 and to make the required investments in guaranteed income supplement that would ensure all Canadians are able to retire in dignity and to keep them out of poverty.
Mr. Speaker, I am pleased to present a petition on CCSVI, particularly as, on Tuesday, 52 Conservative senators voted en masse to block MS sufferers from testifying before a Senate committee on Bill S-204.
I am wondering if anyone can imagine cancer sufferers not being allowed to present at a committee studying cancer. Why did the government shut down the voices of Canadian MS patients?
The petitioners call for the Minister of Health to undertake phase III clinical trials on an urgent basis in multiple centres across Canada and to require follow-up care for Canadian MS patients.
Mr. David McGuinty (Ottawa South, Lib.):
Mr. Speaker, I rise today to table petitions on behalf of my constituent, Janet Wilkinson, and the Grandmothers Advocacy Network.
The petitions have been signed by hundreds of local residents who are urging the government to pass Bill C-398 without significant amendment to facilitate the immediate and sustainable flow of life-saving generic medicines to developing countries.
I am pleased to table these petitions this afternoon and look forward to the government's response.
Mr. Speaker, it is with pleasure that I table a petition today with regard to the freshwater research station. The petitioners ask that the government recognize the importance of the ELA to the Government of Canada's mandate to study, preserve and protect the aquatic ecosystem and, ultimately, for it to therefore reverse the cutbacks.
Ms. Elizabeth May (Saanich—Gulf Islands, GP):
Mr. Speaker, I rise today to present two petitions. The first is from residents of British Columbia and Ontario. It is particularly timely given the reports that are coming out of the United States of the extensive use of robocalls to tell voters in various states that they could vote after the U.S. election had already ended on Wednesday. We need to get to the bottom of this.
The petitioners call upon the Canadian Parliament to launch an immediate public inquiry into the use of these calls in Canada. I think that all parties in the House would like to make sure that any methods of voter suppression are eliminated from Canadian elections.
Mr. Speaker, the second petition is from residents of Alberta, British Columbia, Saskatchewan and Quebec. The petitioners call upon the House to ensure that the investment treaty already signed between Canada and China not be ratified. There is no duty on the Prime Minister's part to ratify the treaty. The public has been clear. Many Canadians are concerned that we not ratify a treaty that binds Canada for up to 31 years.
Mr. Mark Warawa (Langley, CPC):
Mr. Speaker, I am honoured to present a petition on discrimination against females. The petition highlights the fact that CBC aired an investigation on gender selection, revealing that ultrasound clinics were revealing the sex of an unborn child, and that often, if it were a girl, the pregnancy were terminated. Gender selection has been strongly condemned by all parties within the House. Ninety-two per cent of Canadians believe that sex selection pregnancy termination should be illegal. Along with the Society of Obstetricians and Gynecologists, they vehemently oppose this practice. The petitioners call upon the House to condemn, not endorse, this type of activity.
Mr. Speaker, the following questions will be answered today: Nos. 902, 905, 909, 917, 931 and 935.
The Acting Speaker (Mr. Barry Devolin): Is that agreed?
Question No. 902--Mr. Mathieu Ravignat:
With regard to the early retirement packages in place for employees affected by reductions in the federal public service: (a) what proportion of public service employees are between 50 and 54 years old, as a figure and as a percentage; and (b) what proportion of public service employees between 50 and 54 years old have accumulated (i) over 25 years of service, (ii) over 30 years of service?
Mr. Speaker, Treasury Board Secretariat is tasked with responding to this question on behalf of the government. This information is taken from the regional pay system as of March 31, 2012.
The federal public service includes the core public administration (departments and organizations listed in schedules I and IV of the Financial Administration Act) and separate agencies (federal agencies listed in schedule V of the Financial Administration Act).
These figures include only employees appointed to indeterminate positions. Employees appointed to term positions, casual employees and students are not included.
Note that similar information regarding the age of public servants is posted on the Treasury Board Secretariat website. Those figures include employees appointed to term positions, casual employees and students. These employees were not included in the response to Q-902 since the question requests statistics regarding tenure.
In response to (a), with respect to employees appointed to indeterminate positions within the federal public service, as of March 31, 2012, there were 44,509 employees between the ages of 50 and 54 years old in the public service. This figure represented 18.4% of the total number of employees in the public service at that time.
In response to (b)(i), with respect to employees appointed to indeterminate positions within the federal public service, as of March 31, 2012, 40.4% of public service employees between the ages of 50 and 54 years old had accumulated over 25 years of service.
In response to (b)(ii), with respect to employees appointed to indeterminate positions within the federal public service, as of March 31, 2012, 20.3% of public service employees between the ages of 50 and 54 years old had accumulated over 30 years of service.
Question No. 905--Ms. Rosane Doré Lefebvre:
With regard to the Correctional Service of Canada and the closure of the Kingston and Leclerc penitentiaries and of the Kingston Regional Treatment Centre: (a) what does the $120 million in savings announced by the Minister of Public Safety include and on what is that figure based; and (b) how many cells are lost by the closure of these institutions?
Mr. Speaker, in response to (a), CSC's budget will be $120 million less at the end of implementation. This reflects the savings from salaries, operating and maintenance, as well as savings realized from the addition of new cells.
In response to (b), CSC has a comprehensive plan to safely move offenders impacted by these closures to other institutions. Many institutions in the Ontario region are undergoing infrastructure expansions to better manage the complex and diverse offender population. Budget 2012 was clear: the government has not built a single new prison since 2006 and has no intention of building any new prisons.
Maximum security inmates will remain maximum security inmates and be placed in appropriate facilities at this level. The same will apply for medium security offenders.
As of September 25, 2012, the rated capacity of Kingston Penitentiary is 421 cells; the rated capacity of Leclerc Institution is 481 cells; and the rated capacity of Kingston Regional Treatment Centre is 143 cells.
Question No. 909--Hon. Geoff Regan:
With respect to Sable Island National Park Reserve: (a) what are the particulars of all costs associated with its establishment since January 1, 2009; and (b) how many visitors have visited the Reserve in each operating season since its establishment?
Mr. Speaker, in response to (a), Parks Canada has spent $391,679 on the establishment of Sable Island National Park Reserve since January 1, 2009, including consultations with the Mik'maw, as well as the negotiation of a 2010 memorandum of understanding and the 2011 national park establishment agreement with the Province of Nova Scotia.
In response to (b), there have been a total of 125 visitors since Parks Canada assumed operations of Sable Island National Park Reserve in April 2012.
Question No. 917--Hon. Mark Eyking:
With respect to the Canadian International Development Agency: (a) what was the total cost to produce and print the Agency's fourth annual report, entitled "Development for Results 2010-11"; (b) what are the details of those costs; and (c) how many copies of the report were produced?
Hon. Julian Fantino (Minister of International Cooperation, CPC):
Mr. Speaker, in response to (a), through the international assistance envelope, CIDA is focusing its international assistance on three thematic sectors: sustainable economic growth, food security, children and youth, and humanitarian assistance. CIDA is ensuring that Canadian tax dollars are delivering value for money and making a real difference in the lives of the people it intends to help.
For example taxpayer's investments have caused a 53% increase in treatments for pregnant women with HIV in South and Western Africa since 2005, a six-fold increase in the amount of businesses accessing microfinance in Indonesia, and 205,000 women and newborns benefiting from improved health services in Burkina Faso.
The total cost to produce and print the Canadian International Development Agency's fourth annual report "Development for Results 2010–2011" was $17,834.11.
In response to (b) and (c), the translation was done by the Translation Bureau and cost $8,562.60; the in-house design team led the design; and the printing of the 1,250 copies of the 176-page report was done by Dollco Integrated Print Solutions and cost $9,271.51.
Question No. 931--Ms. Jean Crowder:
With regard to the proposed First Nations Education Act: (a) what is the government's timeline for consultation with stakeholders, (b) what is the government's plan for meaningful consultations with stakeholders, (c) with which First Nations organizations and communities does the government intend to consult, (d) with which other stakeholders does the government intend to consult, and (e) what is the government's plan to meaningfully incorporate and address input from stakeholders in the legislative drafting process?
Hon. John Duncan (Minister of Aboriginal Affairs and Northern Development, CPC):
Mr. Speaker, with regard to (a), economic action plan 2012 committed the government to having a first nation education act in place by September 2014.
In the coming months, the Government of Canada will be working with first nation and provincial partners to determine the path forward on first nation education through intensive but time-limited consultation.
In response to (b), as committed to in economic action plan 2012, the government will work with willing partners to introduce a new first nation education act and have it in place by September 2014.
The government acknowledges that improving the educational outcomes of first nation students is a shared responsibility in which governments, first nations, educators, families and students all play an important role. That is why, in the coming months, the Government of Canada will be working with first nation and provincial partners to determine the path forward on first nation education.
The Government will develop legislation through intensive, but time-limited consultation with first nations.
In response to (c), improving the educational outcomes of first nation students is a shared responsibility in which governments, first nations, educators, families and students all play an important role. The Government will develop legislation through intensive, but time-limited consultation with first nations.
In response to (d), improving the educational outcomes of first nation students is a shared responsibility: first nations, educators, families and students all play an important role. We look forward to the assistance of experts from provinces and first nations, including educators, families, and students, in the development of legislation.
In response to (e), the government has committed to developing this legislation through intensive consultation with first nations and other stakeholders across the country. We look forward to the input and perspectives of provinces and first nations in the development of legislation.
Question No. 935--Mr. Dany Morin:
With regard to the cuts to the National Council of Welfare (NCW): (a) are there any studies on the spinoff benefits for the community of this council, and, if so, what are they; (b) are there any studies on the impact of these cuts on the community and on the government, and, if so, what are they; (c) how much do these cuts represent for fiscal years 2012-2013 and 2013-2014 in (i) dollars; (ii) percentage of the government's operating budget; (d) how many jobs will be lost; (e) how many employees will be transferred; (f) following the closure of the NCW, what will the government's sources be on the issues of (i) poverty, (ii) social exclusion, (iii) reducing inequalities; (g) will organizations and groups that used NCW research be compensated for this loss?
Hon. Diane Finley (Minister of Human Resources and Skills Development, CPC):
Mr. Speaker, with regard to (a), the department has not carried out any studies on these benefits.
With regard to the NCW, and (b) in particular, the government is putting its policy resources to best use and in the most efficient manner. There are many non-governmental organizations that provide comparable independent advice and research on poverty and other related issues.
It should also be noted that the Caledon Institute of Social Policy, a private, non-profit, social policy think tank, has announced plans to continue the data and analysis of the NCW's two key reports, "Welfare Incomes" and "Poverty Profile". The Caledon Institute has indicated that it will seek continued cooperation and input from the provinces and territories.
Information on provincial social assistance rates remains publicly available from each province.
With regard to (c)(i), the economic action plan 2012 provides for savings to begin in 2013-2014. The NCW wind-down will realize savings of $1.1 million in 2013-2014 and in future years. As for (c)(ii), the government's operating budget for 2013-2014 was not yet available at the time of this written question.
With regard to (d) and the NCW wind-down, 16 part-time Governor in Council appointments, which include one chairperson and up to 15 members, have ceased to hold office. In addition, nine full-time equivalent public service positions have been eliminated.
With regard to (e), the department has applied all the provisions of workforce adjustment agreements to help manage impacts on employees and to facilitate employment continuity, when possible.
With regard to (f), HRSDC conducts its own research and analysis into poverty, social exclusion and on reducing inequalities. Departmental officials analyze existing and emerging trends and provide the minister with evidence-based advice. The department uses a wide variety of resources internal and external to government. Other federal institutions, such as Statistics Canada, carry out research on related topics.
With regard to (g), the Government of Canada has maintained key NCW research papers and reports on the HRSDC website. All NCW research papers and reports have also been archived by Library and Archives Canada.
Mr. Speaker, if Questions Nos. 903, 912, 941 and 951 could be made orders for returns, these returns would be tabled immediately.
Question No. 903--Mrs. Anne-Marie Day:
With regard to Employment Insurance (EI) for each calendar year since 2000: (a) how many applications for regular EI benefits have been submitted; (b) how many applications for regular EI benefits have been approved; (c) how many applications for regular EI benefits have been rejected, broken down by reason for rejection; (d) what was the average time for processing claims for regular EI benefits; (e) how many applications for special EI benefits have been submitted, broken down by benefit type; (f) how many applications for special EI benefits have been approved, broken down by benefit type; (g) how many applications for special EI benefits were rejected, broken down by reason for rejection; and (h) what was the average time for processing claims for special EI benefits, broken down by benefit type?
(Return tabled)
Question No. 912--Mr. Scott Simms:
With regard to Library and Archives Canada, what are the particulars of each use of the exhibition space on the main floor since January 1, 2006, including (i) the purpose, (ii) date, (iv) duration, (v) organization using the space?
Question No. 941--Hon. Scott Brison:
With regard to fisheries: since September 1, 2011, how many briefs, letters, or presentations were submitted concerning the fleet separation policy, or the owner-operator principle, to: (i) the Minister of Fisheries and Oceans, (ii) the Minister of Intergovernmental Affairs and President of the Queen's Privy Council for Canada, (iii) the Associate Minister of National Defence and Minister of State (Atlantic Canada Opportunities Agency) (La Francophonie), (iv) the Minister of National Revenue, (v) the Minister of National Defence, broken down by the numbers submitted by provincial or territorial governments, municipal or regional governments, businesses, industry associations, trade unions, Aboriginal organizations, other organizations, and individuals?
With regard to the Canadian Coast Guard, what are the details of the $1.4 billion in spending which has been committed in the past six years, as mentioned in the August 24, 2012, press release by the Department of Fisheries and Oceans, under the headline "New Canadian Coast Guard Ship Vladykov in St. John's", and, in particular, what is the nature, anticipated timeline and location of each project, investment or purchase which makes up this amount?
Mr. Tom Lukiwski:
Mr. Speaker, I ask that the remaining questions be allowed to stand.
The House resumed from October 31 consideration of the motion that Bill C-28, An Act to amend the Financial Consumer Agency of Canada Act, be read a third time and passed.
Mr. Matthew Dubé (Chambly—Borduas, NDP):
Mr. Speaker, despite a certain level of enthusiasm, financial literacy does not necessarily seem the most exciting of subjects. That being said, it is still an extremely important subject. It is important for young people who are beginning to save their money and to understand what is involved in investing and having money, and it is important for retired seniors who must manage their pensions and a significantly lower income now that they are no longer working and are taking their well-deserved retirement.
Following consultations and a study, the government introduced Bill C-28, a bill that would create the position of a leader who would be responsible for the development of financial literacy for Canadians.
Initially, we opposed this bill for the reasons I will set out. However, we will support it at third reading for the reasons that I am also going to take this opportunity to explain.
First of all, we kept up our opposition for a number of reasons. The first point, and it is not the least of them, is that the position would not necessarily be bilingual. As a matter of fact, my colleague from Louis-Saint-Laurent recently put forward a first-rate bill. She has done an outstanding job on this issue. However, even though the incumbent of the position would not be an officer of Parliament, it is necessary that he or she be bilingual. This person will have to interact with people from all linguistic communities, including those in Quebec, where there is a high concentration of people whose mother tongue is French, in New Brunswick, in Eastern Ontario and elsewhere. We must be able to allow people to learn financial literacy skills in the official language of their choice. This is a very important point.
We put forward an amendment to this effect in committee and, unfortunately, it was rejected by the government. Nevertheless, we want to support the bill now because it is a very good start, but we regret that this position, which should have a mandatory bilingual designation, does not. This is a very simple, but a very important, requirement.
The second point, which led to a certain degree of consternation among our members and caused us great concern, is the fact that the people who studied this issue recommended that a board be set up to engage in consultations with the financial literacy leader. The board would be made up of people representing a variety of sectors, such as the financial education sector, unions, financial institutions and so on. The board would engage in consultations with the financial literacy leader and create a system to consult people from all walks of life. This is very important when speaking of financial literacy.
Despite the important role played by financial institutions, when teaching financial literacy to Canadians, it is important not to do so in a way that will benefit only the financial institutions. Financial literacy must consider the diverse realities of individuals and various sectors, such as the unions, and the importance of pensions, for instance. When we talk about financial institutions, there is the size of the investment. When we talk about the people who are in a position to educate Canadians and so on, I think it is very important to have a board involved.
Now we are told, following the committee study, that it is not necessary that the board be established as part of the bill. Just the same, we are concerned about the political will that currently exists on this issue.
I think that the board should be established in the bill. We would require this. Why should we wait? Why should we just hope that the board will be established? I think that if we could include the establishment of the board in the bill, we would already be ahead of the game. We are very concerned about this, but not enough to oppose the passage of the bill. I think we will be taking a big step in the right direction in order to establish a position for beginning Canadians' financial literacy education.
These are the reasons why we were opposed to the bill, but we also believe that we are heading in the right direction. We are comfortable with giving it our support at the moment, but we will continue to push for improvements in the measures in the bill. Now that I have an opportunity to talk about financial literacy and Bill C-28, it would be unfortunate if I missed my chance to explain how this affects the people in my riding.
First of all, I am going to refer to a resident in my riding who works for a financial institution and with whom I often have an opportunity to chat. He has become a very good friend. His name is Jacques Rémy and he is the general manager of the Caisse populaire Desjardins in Beloeil–Mont-Saint-Hilaire. When I sat down with him in my riding one Saturday evening for supper, I had a chance to discuss various problems faced by the residents with regard to their pensions, their retirement and their investments.
One of the major problems at the moment is debt. Many residents are going deeper and deeper into debt. It may not necessarily be the fault of the individual, but there is not enough financial literacy. I do not want to lay the blame on Canadians, but these days, there are so many opportunities to invest, to contribute to savings plans, and to use several credit cards with all the rewards they offer, and all this may be quite difficult to manage. Considering that the cost of living is going up, just like everything else, it can be very easy to wind up in debt, and this has become a very serious problem. As I said, I had a chance to talk about this with Mr. Rémy, who is the general manager of a financial institution, and he is seeing this more and more often.
For instance, in a riding like mine, there is a great deal of hidden poverty. We are a suburb, there are beautiful houses, people sometimes have two or three cars in their driveway, but this does not necessarily mean that they do not have problems with debt that could lead to poverty. As the representative for these people, I think it is so important to promote financial literacy in every way we can, and this is an idea or a value that I share with my colleagues. This is why we are going to support this positive first step, while bearing in mind the extremely important fact that we have to continue improving the measures and the systems already in place.
There is another anecdote involving financial literacy that I would like to tell. Every year, a seniors fair is held in Chambly. This is a very interesting and important event in our community. The agencies serving seniors get together with members of both the National Assembly of Quebec and Parliament in Ottawa. We and the agencies can set up booths where we distribute brochures and give out other types of information, and there are presentations given by community stakeholders.
Last year, when I attended the fair for the first time—I went this year, too—there was an extremely interesting presentation on financial literacy. It focused on, as I said at the beginning of my speech, the best way to manage our pensions, our RRSPs and all the financial resources we have, the money we save and for which we work very hard over the years.
In the case of our seniors, this is well-deserved financial support. Promoting financial literacy and helping our seniors by ensuring that they are able to use the educational tools we can offer them as a government and as representatives is a worthwhile undertaking. Often when we speak of financial literacy, it comes down to explaining how important it is.
When we speak about financial literacy, one of the traps we often fall into is thinking that people are to blame, that they are not educated and not able to manage their money. I cannot emphasize enough that this is absolutely not the issue we are discussing here today. The problem is that people feel overwhelmed by the various investment options available to them.
Seniors often face this problem. They frequently receive telephone calls at home from people offering different types of services. As an aside, these calls are often not legitimate. This is a big problem. It is one of the reasons why we are supporting the government's bill to impose harsher penalties on people who commit crimes against seniors.
That being said, there are many new measures that allow people to invest and to retire relatively easily. It is really important for seniors to be able to rely on someone. That someone could be the person in the position we are creating today, or when the bill is voted on, of course.
This is a troubling issue that also applies to young people. I have spoken about seniors, I have talked about their experience, but there are also young people who are beginning their working lives, who are beginning to learn what it means to have an income and money to spend.
When you are young—I have enough experience to talk about it and to say that we have all been there—you eventually reach a point in your life where you have some independence and you have money to spend for the first time. Once you reach that point, you want to know how to get the most for your money, how to spoil yourself a little, if you can, and also how to make sensible and responsible decisions.
Here again, I am repeating myself, but it is so important to underscore it: I am not trying to say that young people are not able to make sensible decisions or be responsible, but the point is that more and more frequently credit cards are being offered to very young people. When you are very young, you try to learn how to invest in an RRSP, how to invest in that famous retirement pyramid, which consists of RRSPs as well as contributions to a retirement plan. When you reach this point in your life, it is very important to have meaningful support from the government and from various agencies, which will be possible with the creation of a new position with the mandate of promoting financial literacy.
I have been talking about this issue for several minutes now. It is interesting because this shows that financial literacy can mean different things to different people. From my comments, it is clear that this can mean different things to different people of all ages.
We in the NDP firmly believe that a concrete definition of the term "financial literacy" is needed. When a position is created whose mandate is to promote financial literacy and educate citizens, it is crucial that we have a clear definition, as was the case with the infamous "net benefit". Thus, once greater clarification and precision are brought to the definition, then we can have a leader, someone who is responsible for and able to properly manage the file.
In the past, the lack of a clear definition was one of the reasons we had decided to oppose the bill. However, the pluses outweigh the minuses at this point. Although we would like to see more clarification at this stage, we hope to achieve that in the coming months and years, as this matter evolves. We do still have this concern, and it is very important that it be raised here today.
While I am on the topic of our change in position on this bill, I would also like to explain the other reason we reached this conclusion; it was because of the work done in committee, particularly by our party's consumer protection critic. Financial literacy is extremely important for the protection of consumers, and the hon. member for Sudbury has done an excellent job on this. He meticulously explained to us that when the Standing Committee on Finance began examining this bill and this issue, many witnesses called for the same things that the NDP has been calling for. They had the same concerns and raised the same points that I just mentioned in my speech.
Looking at the work done in committee, it is clear that the witnesses were able to explain to the members the importance of moving forward by taking this first step.
After hearing this testimony, we think it would be a big mistake not to support this first step in the right direction. This testimony also allowed us to confirm the problems with this bill.
This is a good opportunity to emphasize the committee work and the importance of inviting expert witnesses from different backgrounds. Various testimony was given by a diverse group of witnesses, including economists, people representing financial institutions and people representing unions. In my opinion, bringing all these people together to have a serious discussion about something that affects us all, without allowing the discussion to focus too much on one topic and not enough on another, allows us to have a clear view of the overall picture.
Another thing that is very important in all this is ensuring that everyone's interest is served. Last fall and this past spring, we debated a bill whose number escapes me, unfortunately, that would implement another retirement savings account much like an RRSP. When we were discussing this bill, many concerns were raised about the various existing retirement accounts and plans.
We do not want to fall into the same trap. Far too often, people have watched the companies they worked for declare bankruptcy and have ended up losing their pension. We do not want the retirement plans and pensions of people who have worked so hard for so long to be tied to the fate of a company that mismanages its investments and ends up going under. In 2008, at the height of the last recession, this type of situation happened at an alarming rate in the U.S.
That is why we should make financial literacy more of a priority. Doing so would help us start a conversation on these retirement plans, on RRSPs and on all the measures available to us as individuals and workers, regardless of where we work. This allows us to recognize the risks of these measures, so that we may proceed safely. There is nothing worse than working for years, investing and contributing to various retirement plans only to lose that money because of bad decisions made by people at the head of various companies.
That is why we want to be very careful. It is very important to educate Canadians so they have the tools they need to make good decisions about their investments and for their retirement.
I would like to talk about my own experience. As I mentioned earlier, retirement may seem very far away for young people of a certain age who are just starting to work, to have some money and to have these kinds of opportunities. People in their twenties do not think about their pension, but I believe that it is very important to start thinking about it. I always say that pensions are very important to our seniors and those who retire, but they are also important to our young people. We have to realize this and set aside our preconceived ideas about this affecting one group of people more than another or only affecting people of a certain age.
This is an extremely important matter, and I cannot stress that enough. For that reason, we support this bill and we will continue to improve these measures and work on this, so that Canadians can make good investments and have adequate financial security.
Mr. Speaker, it is great to see that the federal government has taken an interest in financial literacy to the degree that it has introduced a bill that would, in good part at least, move us forward a little. There is a lot more it could be doing. Part of that responsibility is to work with the different provinces. Provincial governments play a huge role in this whole issue of financial literacy. We could ultimately even pass that down to places such as our school divisions.
Does the member see a role for other ministers, specifically the minister responsible for intergovernmental relations, in taking this important file and advancing it with all provinces in Canada?
Mr. Matthew Dubé:
Mr. Speaker, I thank my colleague for the question. He is absolutely right.
In this kind of file, we must definitely work closely with the provinces because they have different programs, which are part of the great retirement investment pyramid. I am from Quebec, which has its own pension plan. That is exactly why we must work with the provinces.
My colleague asked me whether the Minister of Intergovernmental Affairs should do something about this. I am sorry to say this but, unfortunately, the minister is not really up to the job these days. However, that is another debate.
My colleague also mentioned schools. He is quite right about that. I spoke about the importance of educating young people who are just heading into the work world. It is very important to have a board made up of people from various backgrounds. We must ensure that educators do not represent only the financial institutions, although they do have a role to play, but that they come from all backgrounds, in order to have a balanced approach that better represents reality.
Mr. François Choquette (Drummond, NDP):
Mr. Speaker, I want to congratulate and thank the hon. member for Chambly—Borduas for his excellent speech. Financial literacy is not easy to explain. His concrete examples and the excellent work he did in his riding allowed him to explain to us in detail the importance of this bill.
I have a background in education. It is indeed very important to educate and inform people and to ensure that information is distributed, present and available to them so that they can make better decisions, whether we are talking about seniors or young people, as my colleague rightly said. As far as young people are concerned, this is important in terms of their retirement and the fact that they have a higher debt load because of their maxed-out credit cards that charge exorbitant fees, for example.
The hon. member also mentioned the excellent work done by the committee. This work is not always easy. For example, in the case of Bill C-45, we should have had independent studies in various committees of the profound changes being made to various laws, instead of depending on the power of the Standing Committee on Finance.
I want to congratulate my colleague and ask him the following question. Does he think the bill will help improve Canadians' financial knowledge?
Mr. Speaker, I thank the member for his question. I was very happy to see that we were able to change and adapt our position after we heard from the experts in committee. That is the purpose of committees and it shows just how important they are.
As for his question regarding our support for the bill and whether it is truly a good thing, as I have said many times, it is a step in the right direction, but there is still a lot to be done. At the same time, like with Bill C-44, which we debated this morning, our support should not be misinterpreted. If we support a bill, it does not necessarily mean that we are happy with it and do not think there is still a lot of work to do. There is a difference there and it is important to point that out.
I also thank my colleague for sharing the education perspective. That is extremely important. As I have said many times, when we educate people, we must be very careful, because we must respect their intelligence, their own responsibilities and their ability to invest. We must not fall into the trap of the one-way street, meaning that we must not talk only about savings. We must make people understand the realities of our financial systems and show them how they can use these systems to secure their retirement and bring in positive returns on their investments.
Mr. Pierre-Luc Dusseault (Sherbrooke, NDP):
Mr. Speaker, I thank the member for his speech.
As a party, the NDP has always worked towards eliminating poverty in Canada. I would like to hear my colleague's comments on that. If people were better informed of their financial rights and if they had more answers and clearer information on finance in general, would that reduce poverty?
Mr. Speaker, I would like to thank my colleague for his question because this is exactly why we are saying that there is still a long way to go.
This is the kind of education, the kind of work that a government and MPs can do, in co-operation with all the other authorities, to help people escape poverty and make the most of the means at their disposal. Bearing in mind the objective of eliminating poverty, something that we as members of Parliament are trying to do, we must continue pushing ahead with this issue, with developing the concept and clarifying the definition of financial literacy and the mandate of the financial literacy leader, the position this bill would create. It is completely relevant.
As I said in my speech when I commented on debt, very often, people who are in debt do not appear to be poor. This problem is specific to poverty, particularly in my riding, Chambly—Borduas. It is a very dangerous path. It is also what is known as a slippery slope: people can get into debt, and that can exacerbate the problem of poverty.
We can resolve the problem of debt, or at least we can do our part. We will certainly not resolve it overnight. However, once people can get proper financial literacy education, we will be able to do our part to try to resolve this issue, which is increasingly present in our society and which, I believe, is a form of hidden poverty that is of great concern.
Mr. Speaker, it is with pleasure that I rise to speak to Bill C-28. Off-hand, there are a couple of good reasons to speak in favour of the bill.
First, it is a step forward, as we acknowledge within the Liberal Party. We did have concerns during second reading, but many of those concerns were addressed at committee. We are pleased with the responses that we heard at committee and in the other discussions that have taken place since second reading, which ultimately bring us to where we are today. We recognize that the bill does take us forward.
The other thing that I like about this particular bill is that it points out the difference in good part between the government and the Liberal Party. On the one hand, the Conservatives have recognized that there is a need to deal with the issue, but on the other hand, they are not coming up with any sort of comprehensive plan. They did not really show evidence of any meetings with stakeholders prior to the bill coming to the House at first reading. It is almost as if they had an idea, or someone within the party had an idea, and they kept it to themselves and then tabled the bill.
Within the Liberal Party, we believe that the bill could have been so much better. Had the government actually held some consultations with different stakeholders, I believe that we would be looking at a much more detailed plan today as to where we should be going.
Financial literacy and education is critically important. We have to look at education in terms of it going beyond the House of Commons simply passing Bill C-28. That is why I referred to the stakeholders.
I might have referred in the past to the fact that I was the education critic in the Province of Manitoba. There are many battles that take place within education about what should and should not be part of the curriculum. People want to emphasize, for example, the importance of language arts and mathematics, and justifiably so. When our students graduate from high school, we want them to have a basic understanding of the language arts, mathematics and a number of other areas.
At the end of the day, we are suggesting that financial literacy is critically important. However, when we look at how we would disseminate that information, it is a huge mistake for us not to take into consideration the important role that other stakeholders have to play. That is why I bring up the whole area of education.
When we talk about leadership, we want leadership from the government in dealing with this very important issue. However, we are not just hoping to see legislation before the House, but also a government that is actively promoting and encouraging dialogue with the different stakeholders.
That is why I posed my question in regards to the Minister of Intergovernmental Affairs, because this is somewhat time sensitive. This is an interesting issue and we brought it up today in question period. We must ask ourselves: To what degree does that particular minister now have responsibility in regard to this legislation?
It would appear that the bill has the support of all members of the House and will pass. I do not know for sure, but we will find out when the bill is actually voted on. However, what do we do after we have passed the legislation?
What is the next step? I understand and appreciate that the primary purpose of this legislation is to create a financial literacy leader. That is great, but along with that we need to recognize the importance of education and reaching out to the different stakeholders.
It is not good enough to just say, "Here is the idea, let us put it to the House and have the House vote on it" and then leave it at that. There is a responsibility for officials, like the Minister of Intergovernmental Affairs, to go out and meet with different representatives of government, different levels of government, to say, "Here is what is happening in the House. Here is the type of thing we think we could move forward on. What do you think as a provincial entity?"
We need to recognize that different departments are involved. A provincial jurisdiction will have a department of consumer and corporate affairs. There will also be departments of education and other departments that one might want to consider. There may even be different departments within the national government itself that would have a vested interest in this whole subject matter.
I have referred to school divisions. Even our municipalities, whether large municipalities or cities, or small, rural municipalities, all have a vested interest, even in distributing information on tax rolls, and so forth, so that people can understand what a tax roll is and the obligations to pay a property tax and how that is done through automatic banking if one chooses.
There are all sorts of reasons why all these different government agencies have a vested interest. I would suggest that we need go even beyond that. We have to look at the private sector and the important role it has to play. It is not just about the banking industry or our top banks, because some of the most progressive policy today regarding finances and consumer awareness is actually coming from our credit unions, our co-ops. Those are private companies or corporations that have seen the merit of consumer awareness or financial literacy. The private sector obviously has a critical role to play in this, and we need to encourage that ongoing support.
I did not make reference to the non-profit sector, but obviously that sector also has a role to play. I remember meeting with representatives of Winnipeg Harvest, the largest food distribution centre for people who just do not have the ability to pay for all of their food and have to go to Winnipeg Harvest as a result. I have talked to representatives of Winnipeg Harvest about the issue of finances. Non-profit organizations have a great talent pool, including social planning councils.
Whether it is government, non-profit agencies or the private sector, all have a role to play in financial literacy. Nevertheless, I would suggest that there is only one real authority with the most significant leadership role to play, and that of course is here within the House of Commons
We need to see a government that is committed to doing more than just bringing in Bill C-28 to creates a financial literacy leader. We need to ensure there is a lot more than just that. Nonetheless, it is good that the government has brought this bill before us today and we will be supporting it at third reading, because see it as a move forward.
We believe that it will add to the importance of education on a very important issue. However, we do not want the government members to sit back and say they have done enough because there is a lot more that needs to be done.
Things have changed dramatically. It was not that long ago that teenagers would open up a bank account by providing a couple of pieces of ID. They could deposit or take money out of the bank and it was pretty simple. Loans and credit cards were more challenging at that early age, but the point is that they went to the bank. All of a sudden we have phone banking systems where you can register your bills and make payments and do transactions over the telephone. Prior to that we could make automatic deposits or withdrawals to pay our monthly bills, and now it is through the website. A vast majority now do their banking online.
I should be careful when I say a vast majority because I do not know that for a fact. I suspect as we continue to move forward, we will get a good, solid majority of people banking online. However, I know there are many people today who refuse to use the technology, sometimes for very good reason. They choose to have a face-to-face connection with the teller or to walk to the bank.
Things are changing and they are changing rapidly. We need to recognize that change. With that change all sorts of other issues arise such as credit cards and the amount of money paid in interest and service charges on credit cards. If a 19-year-old or even a 40-year-old is given a credit card, especially at this time of the year with consumer spending expected to increase significantly, those credit cards are very attractive little pieces of plastic. It does not take too much to accrue a significant amount of money on that piece of plastic.
Now stores have gift cards. People purchase them as a Christmas gift or as a holiday gift. Many gift cards have a short period of time before they expire and many consumers are not aware of that. These are the types of things that have an impact. This is why it is so very important that we recognize education is of critical importance. We need to ensure that whether people are 16 years old in high school, or 30 years old working on a factory floor or working in an office, there is an advocate talking about what is happening and what consumers can do to protect themselves. We must ensure that there is someone who is on the consumers' side, ensuring their rights and that they are not being taken advantage of or exploited.
In 2001 we saw the creation of the Financial Consumer Agency of Canada, which I thought was a great initiative of the former prime minister, Jean Chrétien, and his government. It was in direct response to what was happening in the real world. Gift cards were coming out big time back then and there was very little consumer knowledge about them. At least the government back in 2001 recognized the importance of consumer education and created that agency. There is a great deal to be learned if we go to its website. I have encouraged constituents to visit it.
We recognize that the government has seen the agency's merit. It has proven itself because it has now stood the test of time. Even though the Liberals brought it in, the Conservatives have now been in for a number of years and they have recognized the value of the agency because the legislation that we have before us today, Bill C-28, would create a financial literacy leader who would be reporting to the Commissioner of the Financial Consumer Agency of Canada.
Today, under the Conservative banner, we see the Government of Canada recognizing that what Jean Chrétien and the Liberals created back in 2001 was a good idea. We find that governments at different levels are on board with respect to that particular agency. I believe that tying the new literacy leader to this agency will be a good thing and it will give more credibility to the financial literacy leader and the office that no doubt will follow.
I know that Canadians are very concerned about debt and the overall debt that Canadians have today, and we should be concerned. Mark Carney, Governor of the Bank of Canada, raised the issue of just how much debt there really is. The number that I have heard is $1.63 for every dollar of annual income. That is significant. From what I understand, the Governor of the Bank of Canada has highlighted the point that it is a record high for consumer debt.
The government does have to take some responsibility for that record, and I do not say that lightly. It was the current Minister of Finance who introduced the 40-year mortgage, which did not even require a minimum down payment. Even though the Conservatives have learned their lesson and are bringing it back to 25 years, that 40-year mortgage contributed to the overall debt ratio that Canadians have today.
The bottom line is that government does play a role. Financial literacy is important and we in the Liberal Party have recognized that. We are supporting the bill because it does move us forward, although not very much, but we are prepared to support it at third reading.
We encourage the government to do more. If it wants some good ideas, it can always turn to the opposition members, particularly members of the Liberal Party who would be more than happy to share our ideas. We recognize how important it is to talk about financial literacy.
Mr. Speaker, I want to pick up on the very last point that my colleague made with regard to the level of household debt in our country. There is no question that the level of household debt has ballooned and this is one of the reasons that undoubtedly prompted the government to say there was a need for greater financial literacy. What is ironic is that the level of household debt was caused, in part, by the decision of the Conservative government to introduce 40-year mortgages with no down payments.
My hon. colleague said that the appointment of a financial literacy officer was a good first step. Given that the situation we are in is not one that is entirely separate from the policies of the government, what measures would he propose because there is more that we could do? Could he go through some of the measures that we could undertake over and above the appointment of a financial literacy officer?
Mr. Speaker, the most important thing the government could do would be to have a ministerial meeting on the issue of financial literacy. The government has an obligation to not only recognize an issue, but to consult with people, different organizations, whether it is the government or private sector, to see how it can take a more collective approach to ensure we maximize the amount of financial literacy programming. Not only would people in schools be educated, but there would also be the potential of educating people within the working environment. Whether one is 40 years old or 16 years old, that individual is being taught some sort of financial literacy. Hopefully the financial literacy officer would take a look at those types of initiatives to encourage that.
I made reference to the $1.63 of debt for every $1 annually that a person has. That is very close to what it was for the Americans at the time in which they had the huge housing crisis. I am not trying to raise a red flag unnecessarily, but I would suggest that people should take note that the Governor of the Bank of Canada, Mark Carney, raised the issue. We should be listening and taking appropriate action, especially on government policy.
Mr. Speaker, I am pleased to ask another question on this issue.
I have to say that the committee did a fine job. I was not there, but I heard about it. The NDP proposed six amendments in all, while the Liberals and the Conservatives did not suggest one single amendment.
One of the amendments put forward by the NDP was that the legislation should express explicitly that the incumbent of this position should be bilingual. As we have seen in the case of the Auditor General of Canada, a unilingual anglophone holds this position. However, right now, it seems he is making an effort at least to learn the second official language.
As I did not attend the committee, I would like to know whether the Liberals supported this amendment to ensure that the legislation states explicitly that the position should be filled by someone who is bilingual.
Mr. Speaker, the Liberal Party has made a very strong statement on the Auditor General, second to no other statement, saying that in positions of this nature it is a mistake not to have someone who is bilingual. I hope the Conservatives have learned a lesson on that particular issue. We will have to wait and see what they do.
In regard to the amendments, after bills pass out of second reading the Liberal Party approaches committees very seriously. We listen to what people have to say. In some bills, amendments are moved and in others they are not. With the first omnibus bill, the Liberals introduced over 400 amendments inside the House on third reading and the NDP did zero. It fluctuates. Sometimes we move amendments. It depends on the critics. A lot depends on what other amendments are being moved. Sometimes an amendment that is moved by one opposition party has the support of the other opposition party.
If there is a good idea and we feel it would make the bill better, we would support the amendment. It is unfortunate that the Conservatives did not make some of the changes some of their colleagues suggested. I suspect we might have even voted for some of those amendments. As I acknowledged at the very beginning, the bill would be just a small step in an area in which we need to make some significant steps.
Mr. Sean Casey:
Mr. Speaker, the hon. member indicated that a key part to increasing financial literacy would be co-operation among the provinces.
When we look at the record of the Conservative government in terms of dealing with the provinces, we have a Prime Minister who will not meet with the provincial ministers as a group. We have a Minister of Intergovernmental Affairs who does all his work either in Ottawa or in Labrador. We have a Minister of Finance who says to the provinces, "Here is what you are getting for health care". That is the Conservatives' idea of federal-provincial co-operation. We have changes to old age security that would increase the welfare rolls in the provinces; changes in EI that are going to increase the welfare rolls in the provinces; changes in crime legislation that are going to increase costs in the provinces.
Given the state of federal-provincial relations in this country, could the member explain his level of optimism that this would achieve its objective, given the necessity of working together between the provincial and federal levels of government to get it done?
Mr. Speaker, I started my speech by saying that one of the things I like about the bill is that it shows a difference between the Conservatives and the Liberal Party of Canada.
The members of the Liberal Party of Canada believe that in order to adequately deal with financial literacy in our country, in order to get the job done and get it done right, we need to work with the provinces, the different stakeholders including school divisions and so forth, and the private sector. That means having ministers who actually go out and meet with their provincial counterparts, the Prime Minister having a first ministers' meeting and so forth.
I regret having to interrupt the hon. member for Winnipeg North at this time. It being 1:30 p.m., the House will now proceed to the consideration of private members' business as listed on today's order paper.
[Private Members' Business]
The House proceeded to the consideration of Bill C-370, An Act to amend the Canada National Parks Act (St. Lawrence Islands National Park of Canada) as reported (without amendment) from the committee.
There being no motions at report stage, the House will now proceed, without debate, to the putting of the question on the motion to concur in the bill at report stage.
moved that, the bill be concurred in.
The Acting Speaker (Mr. Barry Devolin): All those in favour of the motion will please say yea.
Some hon. members: Yea.
The Acting Speaker (Mr. Barry Devolin): All those opposed will please say nay.
Some hon. members: Nay.
The Acting Speaker (Mr. Barry Devolin): In my opinion the yeas have it.
The Acting Speaker (Mr. Barry Devolin): Pursuant to Standing Order 98, the recorded division stands deferred until Wednesday, November 21, immediately before the time provided for private members' business.
It being 1:31 p.m., pursuant to an order made Tuesday, September 25, the House stands adjourned until Monday, November 19 at 11 a.m., pursuant to Standing Orders 28(2) and 24(1).
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\section{Clumping Factors and the Photon Budget for Reionization}
\label{sec:ClumpingFactors}
\subsection{Clumping Factor Analysis of Madau}
\label{Madau}
In this section we begin our examination of Equation \eqref{eq:ndot} from \cite{MadauEtAl1999} as an accurate predictor of when reionization completes, focusing on the clumping factor. While it is true that the Madau-type analysis was not designed to predict the precise redshift for reionization completion, only the ionization rate density needed to maintain the IGM in an ionized state after reionization has completed, it is effectively being used in this way when it is applied to galaxy populations at increasingly higher redshifts $z=6-7$ (cf. \cite{FanEtAl2006, RobertsonEtAl2013}).
Our methodology is the following. The simulation supplies $\dot{N}_{sim}(z)$ ionizing photons, which increases with decreasing redshift because the SFRD increases with decreasing redshift. Equation \eqref{eq:ndot} poses a minimum requirement on the ionizing emissivity to maintain the IGM in an ionized state at given redshift z. This requirement decreases with decreasing redshift due to the strong z dependence. We look to see if the box becomes fully ionized when these two curves cross; i.e., when $\dot{N}_{sim} \geq \dot{\mathcal{N}}_{ion} $. In subsequent sections we do this for more recent definitions of the clumping factor that have been introduced by various authors, in roughly chronological order.
The way the clumping factor is introduced and used, is to estimate the amount of recombination that radiation has to overcome, in order to keep the universe ionized \citep{GnedinOstriker1997,ValageasSilk1999,MadauEtAl1999,FanEtAl2006}. In a homogeneous universe, the hydrogen recombination rate is also homogeneous, and is a simple function of the mean density, ionization fraction, and temperature. The clumping factor is a correction factor to account for density inhomogeneities induced by structure formation, although in principle inhomogeneties in ionization fraction and temperature are also important.
The most common definition for the clumping factor is:
\begin{equation}
C=\frac{\langle n_\mathrm{H\,II}^2 \rangle}{\langle n_\mathrm{H\,II} \rangle^2}
\label{eq:clumpingfactor}
\end{equation}
Where the $\langle\rangle$ brackets denotes an average over the simulation volume. To see where this comes from lets look at the change of $n_\mathrm{H\,II}$ with respect to time due to recombinations:
\begin{align}
\label{eq:recombtime}
\frac{\partial n_\mathrm{H\,II}}{\partial t}
&= -n_\mathrm{e} n_\mathrm{H\,II}\alpha_B(T)\notag\\
\frac{\partial n_\mathrm{H\,II}}{n_\mathrm{H\,II}}
&= -\partial t n_\mathrm{e}\alpha_B(T)\notag\\
\int^{n_f}_{n_i}\frac{\partial n_\mathrm{H\,II}}{n_\mathrm{H\,II}}
&= -\int^{t_f}_{t_i}\partial t n_\mathrm{e}\alpha_B(T)\notag\\
ln\left(\frac{n_f}{n_i}\right)
&= -(t_f-t_i)n_\mathrm{e}\alpha_B(T), \notag\\
\frac{n_f}{n_i}
&= exp(-t_{rec}n_\mathrm{e}\alpha_B)
\end{align}
In the last step, we have set $(t_f-t_i)$ to be $t_{rec}$. This leads to
\begin{equation}
t_{rec} = [n_\mathrm{e}\alpha_B(T)]^{-1}
\label{recombtime}
\end{equation}
being the characteristic time when the fraction $n_f/n_i = 1/e$. Using this expression for the recombination time, one can rewrite the right hand side of the equation as
\begin{align}
\label{eq:trec}
\frac{\partial n_\mathrm{H\,II}}{\partial t}
&= - n_\mathrm{H\,II}n_\mathrm{e}\alpha_B(T) = - n_\mathrm{H\,II} / t_{rec}\notag\\
&= - n_\mathrm{H\,II} (1+2\chi) n_\mathrm{H\,II} \alpha_B(T) \notag\\
&= - n_\mathrm{H\,II}^2 (1+2\chi) \alpha_B(T) \notag\\
\end{align}
where in the last two steps, following \cite{MadauEtAl1999}, we replace $n_\mathrm{e}$ with $(1+2\chi)n_\mathrm{H\,II}$ assuming helium is fully ionized. Here $\chi$ is the cosmic fraction of helium. Taking the volume average we have:
\begin{align}
\label{eq:trecpart2}
\langle \frac{\partial n_\mathrm{H\,II}}{\partial t} \rangle
&= - \langle n_\mathrm{H\,II}^2 (1+2\chi) \alpha_B(T) \rangle \notag\\
&= - \langle n_\mathrm{H\,II}^2 \rangle (1+2\chi) \alpha_B \notag\\
&= - \langle n_\mathrm{H\,II} \rangle^2 (1+2\chi) \alpha_B C\notag\\
&= - \langle n_\mathrm{H\,II} \rangle /\bar{t}_{rec}
\end{align}
In the above we have made the oft-used assumption of a uniform IGM temperature of $10^4$K, making the Case B recombination coefficient, $\alpha_B$ a constant. Note this is not physically justified, but since the temperature of the IGM is not well determined observationally, it is a useful approximation, and one that is embedded in Equation \eqref{eq:ndot}. With this simplifying assumption, when taking the volume average on both sides of the equation, we may rewrite the result in the same form as the first line in Equation \eqref{eq:trec}. Therefore, the effective recombination time can be written as
\begin{equation}
\bar{t}_{rec} = t_\mathrm{Madau} \equiv [(1+2\chi)\langle n_\mathrm{H\,II} \rangle \alpha_B C]^{-1}
\label{eq:tmadau}
\end{equation}
This expression is the same as Equation (20) of \cite{MadauEtAl1999} if we substitute $\langle n_\mathrm{H\,II} \rangle$ for $\bar{n}_\mathrm{H}$. In the case of a fully ionized universe these two quantities are equivalent. We note that $t_\mathrm{Madau}$ is not at all the volume average of $t_{rec}$ but is $\langle t_{rec}^{-1} \rangle ^{-1}C^{-1}$, which weights regions
with the {\em shortest} recombination times; i.e. regions at the mean density and above. If we now make the
{\em ansatz} $\dot{\mathcal{N}}_{ion} \times \bar{t}_{rec} = \bar{n}_\mathrm{H}(0)$, we may derive Equation (26) in \cite{MadauEtAl1999}, updated by \cite{FanEtAl2006}, repeated here for convenience:
\begin{equation}
\label{eq:updatedNdot}
\dot{\mathcal{N}}(z)=10^{51.2}s^{-1}Mpc^{-3}\left(\frac{C}{30}\right)\left(\frac{\Omega_\mathrm{b} h^2}{0.02}\right)^{2}\left(\frac{1+z}{6}\right)^{3}.
\end{equation}
This equation gives an estimate of the ionizing photon production rate density (in units of s$^{-1}$Mpc$^{-3}$comoving) that is needed to balance the recombination rate density (the right-hand-side of Equation \eqref{eq:updatedNdot}) in a completely ionized universe. Values for $C$ ranging $\sim$10-30 are often quoted from earlier hydrodynamical simulations such as \cite{GnedinOstriker1997}, and $\sim 3$ for more recent work following \cite{PawlikEtAl2009, RaicevicTheuns2011, ShullEtAl2012, FinlatorEtAl2012} and the methods there.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig8-eps-converted-to.pdf}
\caption{Ionizing photon production rate density and various estimates of the recombination rate density versus redshift. The blue curve labeled ``$\dot{N}_{sim}$'' is the measured photon production rate density averaged over the entire simulation volume. The green curve labeled ``$\dot{R}_\mathrm{H\,II}$'' is the recombination rate density estimate from using the clumping factor calculated with Equation \eqref{eq:clumpingfactor} substituted in Equation \eqref{eq:updatedNdot}. The red curve labeled ``$\dot{R}_b$'' is Equation \eqref{eq:updatedNdot} evaluated using a clumping factor calculated from the baryon density. The black curve labeled ``$\dot{R}_\mathrm{dm}$'' is using a clumping factor calculated with dark matter density.}
\label{unthresholded}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig9-eps-converted-to.pdf}
\caption{Unthresholded clumping factors used in Fig. \ref{unthresholded}. $C_{HII}, C_b, C_{dm}$ are calculated from the unthresholded H \footnotesize{II}, baryon, and dark matter densities, respectively.}
\label{unthreshclumping}
\end{figure}
We follow these earlier studies using our own simulation data. In Figure \ref{unthresholded} we plot the ionizing photon production rate density and recombination rate density from our fiducial simulation. The curve in blue labeled $\dot{N}_{sim}$ is the photon production rate density from the simulation, calculated using a time average of the volume integrated ionizing emissivity $\eta$ (Equation \eqref{eq:emissivity}) divided by the average energy per photon which we obtain directly from the SED. The other three curves plot Equation \eqref{eq:updatedNdot} for three methods for calculating $C$: green uses the H {\footnotesize II} density directly (Equation \eqref{eq:clumpingfactor}); red uses the baryon density $C=\langle \rho^2_b \rangle / \langle \rho_b \rangle^2$; and black uses the dark matter density $C=\langle \rho^2_{dm} \rangle / \langle \rho_{dm} \rangle^2$. In all cases no thresholding is being applied (the effect of threholding is examined in the next section); the averages are done over every cell in the simulation including those inside the virial radii of galaxies. The H {\footnotesize II} curve drops sharply with decreasing redshift because $C$ is large when the H {\footnotesize II} distribution is patchy. The baryon and dark matter curves track one another for $z > 6$ because the clumping factors are nearly the same, but begin to separate after overlap as the baryon clumping factor drops due to Jeans smoothing.
Where the ionization and recombination rate density lines cross is roughly when we expect the universe to become highly ionized.
If we define the end of the EoR as when 99.9\% of the volume has reached the Well Ionized level, then our simulation reaches that point around $z\sim5.8$ according to Figure \ref{linearIonized}. The $\dot{N}_{sim}$ curve crosses the $\dot{R}_\mathrm{H\,II}$ curve at $z\sim6.2$. This is somewhat reassuring since we are counting every ionizing photon emitted and every recombination, at lease insofar as Equation \eqref{eq:updatedNdot} provides a good estimate of that. The recombination rate density curves using clumping factors computed from the baryon and dark matter densities curves cross the $\dot{N}_{sim}$ curve at a somewhat higher redshift of $z \approx 6.6$. By following the original methodology of using the clumping factor to estimate recombinations, we find that the clumping factor calculated with the H {\footnotesize II} density field to be the closest predictor for the end of EoR in our simulation.
The photon budget that enabled us to reach different levels of ionization is plotted in Figure \ref{unthreshphotonbudget}.
Here we plot the evolution of the ionized volume fraction versus $\gamma_{ion}/H=\int dt \dot{N}_{sim} / \bar{n}_\mathrm{H}(0)$. So, for the same definition for the end of EoR, we see that we need $\sim$4 photons per hydrogen atom to achieve. This cannot be considered a converged result because this estimate includes the dense gas inside galaxies, which is not well resolved in our simulation. Even though a small fraction of the baryons reside inside galaxies, due to the short recombination time many ionizing photons are required to keep the gas ionized. Since we have not resolved the internal structure of galaxies, and higher resolution would likely result in higher density gas, we must consider $\gamma_{ion}/H=4$ a lower bound. We eliminate this issue in the next subsection by excluding the dense gas in halos from the calculation.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig10-eps-converted-to.pdf}
\caption{Ionized volume fraction as a function of the number of ionizing photons emitted per H atom averaged over the entire simulation volume (including inside halos) for three different ionization levels: $f_i \geq 0.1$ (blue line); $f_i \geq 0.999$ (green line); $f_i \geq 0.99999$ (red line). Compare with Fig. \ref{threshphotonbudget} which excludes gas inside halos.}
\label{unthreshphotonbudget}
\end{figure}
\subsection{Quantitative Analysis of Recombinations}
As the clumping factor method grew in popularity, various authors have applied thresholds of one form or another to improve upon its accuracy in predicting the recombination rate density needed to maintain an ionized universe. When thresholds are applied, parts of the volume are excluded from the photon counting analysis. \cite{PawlikEtAl2009, RaicevicTheuns2011} and others, limit the calculation of the clumping factor to the low density IGM by using $\Delta_b$ thresholds, usually set at 100. They threshold out gas in virialized halos and the self-shielded collapsed objects, because radiation does not penetrate these objects, or they recombine too fast, which leaves them neutral and not contributing to recombinations in the IGM. More recently \cite{ShullEtAl2012} has also thresholded out void regions ($\Delta_b < 1$), arguing that they do not contribute appreciably to the total recombinations due to their long recombination times.
To investigate the contribution of gas of different density to the total recombination rate density, we plot in Figure \ref{recomb}, three quantities dealing with recombinations in our simulation. In the left column we have a 2D distribution plot of recombination rate density $\dot{R} = n_\mathrm{H\,II}n_e\alpha_B(T)$ divided by ionization rate density $\Gamma_\mathrm{H\,I}^{ph}n_\mathrm{H\,I}$ versus baryon overdensity $\Delta_b$, where
\begin{align}
\label{eq:photoionization}
\Gamma_\mathrm{H\,I}^{ph} &= \frac{c E}{h} \left[\int_{\nu_\mathrm{H\,I}}^{\infty}
\frac{\sigma_\mathrm{H\,I}(\nu) \chi_E(\nu)}{\nu}\,\mathrm d\nu \right] \bigg /
\left[\int_{\nu_\mathrm{H\,I}}^{\infty} \chi_E(\nu)\,\mathrm d\nu\right].
\end{align}
Here, $\sigma_\mathrm{H\,I}(\nu)$ and $\nu_\mathrm{H\,I}$ are the ionization cross section and ionization threshold for H {\footnotesize I}, respectively, and
$h$ is Planck's constant (Paper I). In the middle column we plot the relative bin contribution to the total recombination rate density versus $\Delta_b$. We draw vertical lines at $\Delta_b$=1 and 100, and in the legend box calculate the cumulative contribution to total reionizations to those thresholds. In the right column, we plot the cell recombination time divided by the Hubble time versus $\Delta_b$. All three columns evolve with descreasing redshift from top to bottom.
\begin{figure*}[!tp]
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11a-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11b-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11c-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11d-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11e-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11f-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11g-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11h-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11i-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11j-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11k-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11l-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11m-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11n-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 5mm 8mm 0mm 0mm, clip, width=1.0\textwidth]{fig11o-eps-converted-to.pdf}
\end{minipage}
\\
\caption{Quantifying recombination information. Left column is a 2D distribution of recombination rate density divided by ionization rate density versus overdensity. Middle column is plot relative bin contribution to the total recombination rate density versus overdensity bins. The lines show the cumulative of all previous bins. Blue line is at $\Delta_b$=100, red line is at $\Delta_b$=1. Right column is plot of recombination time divide by Hubble time versus overdensity. All three columns evolve with decreasing redshift from top to bottom.}
\label{recomb}
\end{figure*}
At $z\sim9$, in the left column of Figure \ref{recomb}, we see that even though there are regions of the volume that are in approximate ionization equilibrium (indicated by the horizontal distribution near 10$^0$), there is a wide distribution of cells far out of equilibrium, some even off by $\sim120$ orders of magnitude. The middle column shows that about 37\% of all recombinations happen below a $\Delta_b$ of 100, and about 3.2\% happen below $\Delta_b$ of 1. The phase diagram in the right column shows that there is a bimodal distribution of cells in terms of their recombination time normalized by Hubble time. The top concentration of cells are more neutral, having long recombination times, and the lower concentration of cells are photoionized, having smaller recombination times. The recombination time is lower for the ionized cells simply because there are more free electrons available to recombine with protons. The blue cloud at low $\Delta_b$ and high $t_{rec}/t_\mathrm{Hubble}$ are the small number of cells that are shock heated to $T >$10$^6$K by supernova feedback. Due to this high temperature, even though there are more free electrons their recombination times remain long.
At $z\sim7$, more of the volume has reached the Well Ionized level, and we see the size of the out of equilibrium distribution shrink in the left column. Now the maximum is only $\sim37$ orders of magnitude higher compared to equilibrium. The middle column shows about 40\% of total recombinations are happening below $\Delta_b$ of 100, and about 4.2\% happens below $\Delta_b$ of 1. In the right column, we see roughly equal numbers of cells in the upper (more neutral) distribution as compared to the lower (more ionized) distribution, whereas the top was much greater in numbers before. As more cells become ionized to a high degree, their recombination time will decrease and their cell counts will shift to the lower distribution.
At $z\sim6$, looking at the left column, most of the cells are now in equilibrium. This is indicated by the peak of the distribution in red, being near zero on the y-axis. The maximum of the distribution is now less than 19 orders of magnitude apart from equilibrium. The middle column showing 30\% to 3.8\% recombinations below $\Delta_b$ of 100 and 1, respectively. The right column shows that the majority of the cells are now in the more ionized distribution and have a low recombination time. This can be verified by looking at the same redshift in Figure \ref{NeutralPhase}, where most of the cells are at the Well Ionized level compared to fewer before.
At $z\sim5.5$, after the entire volume has become Well Ionized, and the vertical spread of the distribution has collapsed to about an order of magnitude away from equilibrium with the vast majority of the cells in equilibrium. The fraction of recombinations are 25\% and 4\% below $\Delta_b$ of 100 and 1, respectively. Looking at the recombination time to Hubble time, we no longer see the bimodal distribution of neutral cells and highly ionized cells, we only see the bottom distribution of highly ionized cells now. The small distribution of shock heated gas is still present, but now seem more prominent with the absence of the neutral distribution.
At $z\sim5$, on the left column, the few cells that are in the low density void, which were recombining slower than ionizing are now all near equilibrium. Cells that are higher in $\Delta_b$ are more likely to be above equilibrium. In the middle column, we see the fraction of recombinations are 16\% and 2.9\% for region below $\Delta_b$ of 100 and 1, respectively. Not much has changed in the recombination time column except there are fewer cells above the $\Delta_b$ of 10$^4$, possibly due to effect of Jeans smoothing.
We see that there is no real one-to-one correspondence between overdensity and the quantities we show on the y-axis. That is because in a given panel, we are only seeing two dimensions of a multidimensional physical process that depends on locality to sources of radiation, the behavior of said sources at a given moment, the local density of neutral and ionized gas, temperature, among others. It is helpful to speak about the average behavior in any given overdensity as we have done, but we should always keep in mind that the average may not be as representative of the wider distribution as we may think.
\subsection{Investigating Thresholded Clumping Factor Analyses}
\subsubsection{Excluding Halos}
\label{ExcludingHalos}
We saw in \S\ref{Madau} that using the unthresholded H {\footnotesize II} density field to calculate $C$ via Equation \eqref{eq:clumpingfactor} yields a reasonably good estimate of when reionization completes (Figure \ref{unthresholded}). This is perhaps not surprising since we count every ionizing photon emitted and every recombination to the accuracy of Equation \eqref{eq:updatedNdot}. Possible sources of disagreement between theory and simulation are: (1) inaccuracies in estimating the recombination rate density using Equation \eqref{eq:updatedNdot}; (2) breakdown of the ``instantaneous approximation'' used to derive Equation \eqref{eq:updatedNdot} due to history-dependent effects; (3) finite propagation time for I-fronts to cross voids; and (4) numerical inaccuracies. Regarding possibility (4) we note that our mathematical formalism is photon conserving, and that our I-front tests in Paper I show that I-fronts propagate at the correct speed, which is an indication that numerical photon conservation is good.
To investigate whether improved estimates of the recombination rate density will improve the agreement, we follow the practice of some recent investigators \citep{PawlikEtAl2009, RaicevicTheuns2011} and threshold out dense gas bound to halos, leaving only the diffuse IGM to consider. The motivation for this is that since we are only interested in the photon budget required to maintain the diffuse IGM in an ionized state, by excluding the complicated astrophysics within halos we have a simpler problem to model and resolve numerically.
To proceed we must calculate the ionization and recombination rate densities outside of collapsed objects. We estimate the number of ionizing photons escaping halos by multiplying $\dot{N}_{sim}(z)$ by a global escape fraction $\bar{f}_{esc}(z)$ derived in \S\ref{escape} and plotted in Figure \ref{RadEscFraction}:
\begin{equation}
\dot{N}_{IGM}(z)=\bar{f}_{esc}(z)\dot{N}_{sim}(z)
\label{eq:ndot_igm}
\end{equation}
\noindent
The recombination rate density outside of halos is calculated using Equation \eqref{eq:updatedNdot} where now the clumping factor is thresholded such that only cells for which $\Delta_b < 100$ contribute to the sum. As in Figure \ref{unthresholded} we plot three curves for the recombination rate density calculated using Equation \eqref{eq:updatedNdot} using H {\footnotesize II}, baryons, and dark matter density fields. These are plotted in Figure \ref{thresholded} as green, red, and black curves, respectively. We see that the recombination rate density based on the singly thresholded H {\footnotesize II} (labeled $\dot{R}_\mathrm{tH\,II}$) and on the thresholded dark matter (labeled $\dot{R}_\mathrm{tdm}$) curve cross the ionizing emissivity curve labeled ``$\dot{N}_\mathrm{IGM}$'' at $z \approx 6.7$ in Figure \ref{thresholded}, whereas the thresholded baryon density curve (labeled $\dot{R}_{tb}$) crosses ``$\dot{N}_\mathrm{IGM}$'' at $z \sim 7.2$. Taking the doubly-thresholded H {\footnotesize II} curve as the best estimate for the recombination rate density, we find that restricting the analysis to only IGM gas yields poorer agreement than the simpler, global model of Madau, which at first blush is a perplexing result. By thresholding out the gas in galaxies we have isolated the thing we care about: the ionization balance of the IGM. Why then should the implied redshift of reionization completion become worse compared to the analysis in \S\ref{Madau}? We defer addressing this question until later sections.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig12-eps-converted-to.pdf}
\caption{Same quantities as Figure \ref{unthresholded}, except now the ``$\dot{N}_\mathrm{IGM}$'' curve is the number of ionizing photons which escape into the IGM (see \S\ref{escape}). The recombination rate densities with a subscript that begins with ``t" are calculated as described in the caption for Figure \ref{unthresholded}, except that the clumping factors are computed excluding regions satisfying $\Delta_b > 100$. The curve labelled $\dot{R}_{ttHII}$ is calculated from Equation (22) using the doubly-thresholded clumping factor $C_{ttHII}$ defined in Figure \ref{threshclumping}.}
\label{thresholded}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig13-eps-converted-to.pdf}
\caption{Thresholded clumping factors used in Fig. \ref{thresholded}. $C_{tHII}, C_{tb}, C_{tdm}$ are calculated using thresholded H II, baryon, and dark matter density fields, respectively, where only cells satisfying $\Delta_b < 100$ contribute. $C_{ttHII}$ is calculated from the H II density where only cells satisfying $\Delta_b < 100$ and $f_i > 0.1$ contribute.}
\label{threshclumping}
\end{figure}
Finally, we ask how many ionizing photons per H atom are required to convert the neutral gas residing outside halos to a well ionized state. We repeat the analysis of Figure \ref{unthreshphotonbudget} and show the result in Figure \ref{threshphotonbudget}. We see that the effect of counting only escaped photons on the photon budget is significant. Previously, we summed $\dot{N}_{sim}(z)$ and divided by the total number of hydrogen atoms in the simulation volume, and used that as our progress variable.
In Figure \ref{threshphotonbudget} we sum $\dot{N}_{IGM}(z)$ and divide by the number of hydrogen atoms in the thresholded volume, and use that as our progress variable.
Instead of needing $\sim$4 to ionize the IGM, now we only need $\sim$2 photons per hydrogen atom for 99.9\% of the universe to reach Well Ionized level. This result supports the ``photon starved'' reionization scenario discussed by \cite{BoltonHaehnelt2007}.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig14-eps-converted-to.pdf}
\caption{Ionized volume fraction as a function of the number of ionizing photons emitted per H atom averaged over the entire simulation volume (excluding gas inside halos) for three different ionization levels: $f_i \geq 0.1$ (blue line); $f_i \geq 0.999$ (green line); $f_i \geq 0.99999$ (red line). Compare with Fig. \ref{unthreshphotonbudget} which includes gas inside halos.}
\label{threshphotonbudget}
\end{figure}
\subsubsection{Including Temperature Corrections}
\label{IncludingTemperatureCorrections}
During the preparation of this paper, a new way of estimating the recombinations in the IGM appeared in the literature. The authors \citep{ShullEtAl2012,FinlatorEtAl2012} reformulated the expression for the clumping factor taking the temperature dependence of the recombination rate into account. We briefly investigate their methods here. In order for the calculation of the clumping factor to take only IGM gas that is ionized but recombining, several additional thresholds were applied. Equation (15) in \cite{ShullEtAl2012} is a new expression for the clumping factor, similar in form to \cite{Gnedin2000},
\begin{equation}
C_\mathrm{RR}=\frac{\langle n_e n_\mathrm{H\,II}\alpha_B(T) \rangle}{\langle n_e \rangle \langle n_\mathrm{H\,II} \rangle \langle \alpha_B(T) \rangle}
\label{eq:CRR}
\end{equation}
with the following thresholds applied: 1$<\Delta_b<$100, 300K$<$$T$$<10^5$K, Z$<10^{-6}$Z$_\odot$, $x_e$$>$0.05. Here, Z is metalicity and $x_e$ is the ionized fraction. The reason that a lower limit threshold is applied to the baryon overdensity, the authors argued, is because very little recombinations happen there, due to the low density. \cite{ShullEtAl2012} also provide a new formulation for ionizing photon rate density that uses this definition of the clumping factor, in their Equation (10),
\begin{align}
\frac{dN}{dt}=4.6\times 10^{50}\mathrm{s}^{-1}\mathrm{Mpc}^{-3}\notag\\
\times \(\frac{(1+z)}{8}\)^3 T_4^{-0.845}\(\frac{C}{3}\)
\label{eq:ShullNdot}
\end{align}
Here, T$_4$ is mean IGM temperature measured in units of 10$^4$K.
Equation \eqref{eq:ShullNdot} is proposed as an improvement over Equation \eqref{eq:ndot}. To see if this is the case we used our data to evaluate the clumping factor C$_\mathrm{RR}$ and then used Equation \eqref{eq:ShullNdot} to calculate ionizing photon rate density versus redshift needed to maintain an ionized IGM. The result is shown in Figure \ref{Shull}. The curve labeled $\dot{R}_\mathrm{RR,T4}$ in green uses the average temperature, in units of 10$^4$K, of the region that satisfies the C$_\mathrm{RR}$ thresholds for T$_4$ in Equation \eqref{eq:ShullNdot}. The curve $\dot{R}_\mathrm{RR}$ uses 1 in place of T$_4$ in Equation \eqref{eq:ShullNdot}, essentially fixing the IGM temperature to a constant 10$^4$K. The green curve is lower than the red curve because the average temperature in the simulation is higher than $10^4$K. The blue curve labeled $\dot{N}_{IGM}$ is as defined previously. We see that Equation \eqref{eq:ShullNdot} predicts that reionization completes at significantly higher redshifts than exhibited by the simulation, calling into question the validity of the analysis.
We find it curious that as the clumping factor analysis is refined through physically well-motivated modifications, it yields predictions for the redshift of reionization completion that become worse and worse, moving to higher redshift rather than lower redshift. This suggests that there is something fundamentally wrong with the whole approach, and that the seemingly good agreement found in \S\ref{Madau} was fortuitous. One worrisome aspect about the utility of Equation \eqref{eq:ShullNdot} is that the fraction of simulation volume included in the C$_\mathrm{RR}$ thresholds is actually quite small. This is illustrated in Figure \ref{volumefracCRR}. The included volume grows from 3\% at $z=9$ to only 23\% of the simulation volume by overlap. One wonders about the validity of making global statements about reionization based on such a restricted sample of the IGM. It is also unclear how we should interpret the redshift at which lines across in Figure \ref{Shull}. Should we interpret it as the redshift below which an ionization rate given by Equation \eqref{eq:ShullNdot} can keep the whole volume ionized, or only the fraction of the volume satisfying the thresholds? If it is the former, how do we account for the time it takes for I-fronts to cross neutral voids?
At this point the reader may rightfully claim that the Madau-type analysis was never meant to predict the precise redshift for reionization completion, only the ionization rate density needed to maintain the IGM in an ionized state after reionization has completed. We would agree with that. However it is effectively being used in this way when it is applied to galaxy populations at increasingly higher redshifts $z=6-7$ (cf. \cite{FanEtAl2006, RobertsonEtAl2013}). Our investigations indicate that formulae such as Equation \eqref{eq:ndot} and \eqref{eq:ShullNdot} are not reliable estimates of when reionization completes. In \S\ref{Discussion} we examine whether they can be usefully applied at lower redshifts, as originally intended.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig15-eps-converted-to.pdf}
\caption{Ionizing photon injection rate density in the IGM from the simulation $\dot{N}_{IGM}$ versus the predictions of Equation \eqref{eq:ShullNdot}, evaluated with two choices for the clumping factor which take temperature corrections into account. The curve labeled ``$\dot{R}_\mathrm{RR,T4}$'' is from Equation \eqref{eq:ShullNdot}, with T$_4$ being the average temperature in C$_\mathrm{RR}$ region in units of 10$^4$K. The curve ``$\dot{R}_\mathrm{RR}$'' is calculated the same way as $\dot{R}_\mathrm{RR,T4}$ except now T$_4$ is set to 1 in Equation \eqref{eq:ShullNdot}, for an effective IGM temperature of 10$^4$K.}
\label{Shull}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig16-eps-converted-to.pdf}
\caption{Evolution of the volume filling fraction with redshift of regions satisfying the C$_\mathrm{RR}$ thresholding criteria.}
\label{volumefracCRR}
\end{figure}
\subsection{Comparing Clumping Factors}
\label{ClumpingFactorEvolution}
For ease of comparison we collect into one plot all the H {\footnotesize II} clumping factors used in the previous sections. The unthresholded H {\footnotesize II} calculated using Equation \eqref{eq:clumpingfactor} is denoted C$_\mathrm{H\,II}$. The singly thresholded clumping factor is denoted C$_\mathrm{tH\,II}$, in which the threshold $\Delta_b<100$ is being applied. The curve labeled C$_\mathrm{RR}$ plots the evolution of Equation \eqref{eq:CRR} with the following thresholds: 1$<\Delta_b<$100, 300K$<$$T$$<10^5$K, $x_e$$>$0.05. For comparison we also plot a doubly thresholded H {\footnotesize II} clumping factor denoted C$_\mathrm{ttH\,II}$ with thresholds $\Delta_b<100$ and $x_e>0.05$, which can be thought of as the clumping factor inside H {\footnotesize II} regions excluding the dense gas in halos.
We see a clear trend that as more thresholds are applied the lower the value of the clumping factor goes. This is because as more regions of the volume are excluded from the averaging process the remaining regions are more homogeneous exhibiting less variations. If no thresholds are applied, the H {\footnotesize II} clumping factor starts around 200 at $z\sim9$ (Figure \ref{unthresholded}). Such high values arise because when the first couple of ionizing sources created high H {\footnotesize II}, they are localized and spread far apart, making the H {\footnotesize II} density very clumpy. As more of the universe is ionized, the H {\footnotesize II} density becomes more homogeneous. We see the single and double thresholded H {\footnotesize II} clumping factors become the same after overlap with a value of $\sim 4.5$ because the second threshold $x_e>0.05$ is satisfied everywhere.
The clumping factor that is not based on the H {\footnotesize II} density alone is C$_\mathrm{RR}$. We see from Equation \eqref{eq:CRR}, C$_\mathrm{RR}$ depends on electron number density, H {\footnotesize II} number density, and the case B hydrogen recombinationation coefficient $\alpha_B(T)$, which is itself dependent on the gas temperature T (fit to Table 2.7 in \cite{OsterbrockFerland2006} implemented in Enzo). $\alpha_B(T)$ depends on T to a negative power and this causes Equation \eqref{eq:CRR} to sometimes have a very low numerator compared to the denominator. This as well as the exclusion of gas in the voids leads to the low clumping factor value of $\sim 2$ we see in the graph. It is very possible to have a value that is smaller than unity, which can lead to even more confusion with the original definition of the clumping factor in Equation \eqref{eq:clumpingfactor}. There, the clumping factor can only have a value of greater than 1, and 1 occurs only in the case of homogeneous distribution of the gas number density.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig17-eps-converted-to.pdf}
\caption{Various clumping factors versus redshift. C$_\mathrm{H\,II}$ is Equation \eqref{eq:clumpingfactor} used in $\dot{R}_\mathrm{H\,II}$ curve in Figure \ref{unthresholded}, C$_\mathrm{tH\,II}$ is used in $\dot{R}_\mathrm{tH\,II}$ curve in Figure \ref{thresholded}, C$_\mathrm{ttH\,II}$ is clumping factor with two thresholds applied, $\Delta_b < 100$ and $f_i>0.1$, shown here solely for comparison. C$_\mathrm{RR}$ is the value of recombination rate clumping factor from Equation \eqref{eq:CRR} with the 5 thresholds applied.}
\label{ClumpingFactors}
\end{figure}
\section{Summary and Conclusions}
\label{Conclusions}
We now summarize our main results.
\begin{enumerate}
\item
We use a fully self-consistent simulation including self-gravity, dark
matter dynamics, cosmological hydrodynamics, chemical ionization and
flux limited diffusion radiation transport, to look at the epoch of hydrogen
reionization in detail. By tuning our star formation recipe to approximately match the observed high redshift star formation rate density and galaxy luminosity function, we have created a fully coupled radiation hydrodynamical realization of hydrogen reionization which begins to ionize at $z \approx 10$ and completes at $z \approx 5.8$ without further tuning. While our goal is not the detailed prediction of the redshift of ionization completion, the simulation is realistic enough to analyze in detail the role of recombinations in the clumpy IGM on the progress of reionization.
\item
We find that roughly 2 ionizing photons per H atom are required to convert the neutral IGM to a well ionized state ($f_i>0.999$), which supports the ``photon starved'' reionization scenario discussed by \cite{BoltonHaehnelt2007}.
\item
Reionization proceeds initially ``inside-out", meaning that regions of higher mean density ionize first, consistent with previous studies. However the late stages of reionization are better characterized as ``outside-in" as isolated neutral islands are swept over by externally driven I-fronts. Intermediate stages of reionization exhibit both characteristics as I-fronts propagate from dense regions to voids to filaments of moderate overdensity. In general, the appropriateness of a given descriptor depends on the level of ionization of the gas, and the reionization process is rather more complicated that these simple descriptions imply.
\item
The evolution of the ionized volume fraction with time $Q_{H {\footnotesize II}~}(z)$ depends on the level of ionization chosen to define a parcel of gas as ionized. The curves for ionization fractions $f_i = 0.1$ and $f_i =0.999$ are very similar, but the curve for $f_i =0.99999$ is significantly lower at a given redshift, amounting to a delay of $\Delta z \approx 1$ relative to the other curves for $Q_{H {\footnotesize II}~} \ll 1$, smoothly decreasing to 0 as the redshift of overlap is approached.
\item
Before overlap, 30-40\% of the total recombinations occur outside halos in our simulation, where this refers to gas with $\Delta_b < 100$. After overlap, this fraction decreases to 20\% and continues to decrease to lower redshifts.
\item
Before and after overlap, 3-4\% of the total recombinations occur in voids (defined as $\Delta_b < 1$.) While this is a small fraction of all recombinations, it is about 10\% of the recombinations in the IGM before overlap, increasing to about 20\% by $z=5$. The contribution of voids to the ionization balance of the IGM is therefore not negligible.
\item
The formula for the ionizing photon production rate needed to maintain the IGM in an ionized state derived by \cite{MadauEtAl1999} (Eq. \ref{eq:ndot}) should not be used to predict the epoch of reionization completion because it ignores history-dependent terms in the global ionization balance which are not ignorable. While not originally intended for this purpose, it is being used by observers to assess whether increasingly higher redshift populations of star forming galaxies can account for the ionized state of the IGM. A direct application of the formula to our simulation predicts an overlap redshift of $z=7.4$ compared to the actual value of $z=5.8$.
\item
Estimating the recombination rate density in the IGM before overlap through the use of clumping factors based on density alone is unreliable because it ignores large variations in local ionization state and temperature which increase the effective recombination time compared to density-based estimates. For a currently popular value of the clumping factor $C=3$ \citep{ShullEtAl2012}, the formula for $\bar{t}_{rec}$ from \cite{MadauEtAl1999}(Eq. \ref{eq:tmadau}) understimates by $2\times$ at all redshifts the effective recombination time measured directly from the simulation. If we adjust $C$ downward so that Eq. \ref{eq:tmadau} matches $t_{rec,eff}$ from the simulation, then it is too low by 60\% at $z=6$ due to the aforementioned effects.
\item
The assumption that $\bar{t}_{rec}/t \ll 1$ which underlies the derivation of Eq. \ref{eq:ndot} is never valid over the range of reionization redshifts explored by our simulation (Fig. \ref{treceffhubble}). Depending on how $\bar{t}_{rec}$ is evaluated, $\bar{t}_{rec}/t$ increases from $0.3-0.4$ at $z=9.7$ to $\geq 1$ at overlap. This means that an instantaneous analysis of the ionization balance in the IGM post overlap is invalid because recombination times are so long.
\item
Retaining time-dependent effects is important for the creation of analytic models of global reionization. The analytic model for the evolution of $Q_{H {\footnotesize II}~}$ introduced by \cite{MadauEtAl1999}(Eq. \ref{eq:dQdt}) retains important time-dependent effects, and predicts well the shape of our simulated curve, but overpredicts $Q_{H {\footnotesize II}~}$ at all redshifts because it does not take into account that reionization begins in overdense regions consistent with the inside-out paradigm. It also assumes every emitted ionizing photon results in a prompt photoionization, which is not true in our simulation at late times $Q_{H {\footnotesize II}~}>0.5$. The Madau model, which ignores these effects, predicts a universe which reionizes too soon by $\Delta z \approx 1$. When we introduce correction factors for these effects into Eq. \ref{eq:dQdtdbg} the simulation and model curves agree to approximately 1\% accuracy. We recommend researchers use Eq. \ref{eq:dQdtdbg} for future analytic studies of reionization.
\item
Finally, we present in Figs. \ref{deltabvsQfit5}, \ref{treceffvszfit}, and \ref{RatiovsQfit} fitting functions for the overdensity correction $\delta_b(Q)$, the effective recombination time derived from our simulation, and the ionization efficiency parameter $\gamma(Q)$ which may be useful for other researchers in the field.
\end{enumerate}
This research was partially supported by National Science Foundation grants AST-0808184 and AST-1109243
and Department of Energy INCITE award AST025 to MLN and DRR. Simulations were performed on the {\em Kraken}
supercomputer operated for the Extreme Science and Engineering Discovery Environment (XSEDE)
by the National Institute for Computational Science (NICS), ORNL with support from XRAC allocation MCA-TG98N020 to MLN.
MLN, DRR and GS would like to especially acknowledge the tireless devotion to this project by our co-author Robert Harkness
who passed away shortly before this manuscript was completed.
\section{Discussion}
\label{Discussion}
\subsection{Significance of our Main Results}
We have carried out a fully-coupled radiation hydrodynamic cosmological simulation of hydrogen reionization by stellar sources using an efficient flux-limited diffusion radiation transport solver coupled to the Enzo code (Paper I). This method has the virtue of a high degree of scalability with respect to the number of sources, which allows us to simulate reionization in large cosmological volumes including hydrodynamic and radiative feedback effects self-consistently. In this paper we have presented first results from a simulation in a cosmological volume of modest size--20 Mpc comoving--to investigate the detailed radiative transfer, nonequilibrium photoionization, photoheating and recombination processes that operate during reionization and dictate its progress. In a future paper we apply our method to larger volumes to examine the large scale structure of reionization, evolution of the bubble size distribution, etc.
The simulation presented here is carried out on a uniform mesh of $800^3$ cells and with an equivalent number of dark matter particles. As such, the mass resolution is sufficiently high to evolve a dark matter halo population which is complete down to ($M_{halo} \approx 10^8 M_{\odot}$) which cools via H and He atomic lines. However, a spatial resolution of 25 kpc comoving poorly resolves internal processes within early galaxies, but does an excellent job of resolving the Jeans length in the photoionized IGM \citep{BryanEtAl1999}. Our simulation is most appropriately thought of as a high redshift IGM simulation which evolves an inhomogeneous ionizing radiation field sourced by star-forming early galaxies. Star formation is modeled using a modified version of the Cen \& Ostriker (1992) recipe that can be tuned to reproduce the observed star formation rate density (SFRD) \citep{SmithEtAl2011}. We have tuned our simulation to roughly match the observed SFRD \citep{BouwensEtAl2011,RobertsonEtAl2013} for $z\geq 7$, but due to the small boxsize, it somewhat underpredicts the SFRD for $z < 7$. Our simulation also matches the observed $z=6$ galaxy luminosity function well, which gives us some confidence that our ionizing souce population is representative of the real universe. However a substantial fraction of our ionizing flux comes from sources that are too faint to be observed; we defer a discussion of this topic to Paper III in this series (So et al., {\em in prep.})
Our goal was not to predict the precise redshift of ionization completion, as this would depend on details such as escape fraction of ionizing radiation from galaxies and their stellar populations that we do not model directly. Rather our goal was to examine the mechanics of reionization in its early, intermediate, and late phases within a model which is calibrated to the observed source population. Nonetheless, we present a model in which reionization completes at $z\approx 6$, consistent with observations.
At early and intermediate times we find that reionization proceeds ``inside-out", confirming the results of many previous investigations \citep{Gnedin2000,RazoumovEtAl2002,SokasianEtAl2003,FurlanettoEtAl2004,IlievEtAl2006,TracCen2007,TracEtAl2008}. However, at late times isolated islands of neutral gas are ionized from the outside-in as they have no internal sources of ionization. Even this characterization is somewhat oversimplified when {\em degree of ionization} is considered, as we discussed in Sec. \ref{IOOI}. It accurately depicts how reionization proceeds for a low degree of ionization (> 0). However for high degrees of ionization, ``inside-out-middle" is more appropriate, as filaments lag behind low and high density regions, as discussed by \cite{FinlatorEtAl2009}.
Our most interesting findings concerns the widely used analytic model of reionization introduced by \cite{MadauEtAl1999}. Both the instantaneous (Equation \ref{eq:ndot}) and time-dependent (Equation \ref{eq:dQdt}) versions of this model underpredict the time (overpredict the redshift) when reionization completes, when applied to our simulation. There are two reasons for this having to do with the detailed mechanics of reionization at early and late times respectively. At early times, I-fronts are propagating in regions of higher density than the cosmic mean since the first sources are highly biased. Higher densities translate into slower bubble expansion rates, retarding $Q_{H {\footnotesize II}~}(z)$ relative to a solution which assumes the cosmic mean density (Figure \ref{Qeffv2}). At late times, which we loosely define as $Q_{H {\footnotesize II}~} > 0.5$, conversion of ionizing photons into new ionized hydrogen atoms becomes inefficient. This can be seen by forming this ratio directly from the simulation data (Figure \ref{Ndot_Ratio}), or by defining a global H {\footnotesize I}~ ionization parameter (Equation \eqref{eq:IP} and Figure \ref{IP}). The consequence of this dropping ionization efficiency, which is as low as 0.05 at overlap in our simulation, is to further retard $Q_{H {\footnotesize II}~}(z)$ relative to a solution which assumes an ionization efficiency of unity (Figure \ref{Qeffv3}).
We have introduced a modified version of \cite{MadauEtAl1999}'s time-dependent analytic reionization model in Equation \eqref{eq:dQdtdbg}. Modifications which correct for the above-mentioned effects apply to the source term only, {\em not to the recombination term}. These corrections are therefore totally independent of issues like clumping factors and the temperature of the IGM, which enter into the characteristic recombination time of the IGM. The modifications are introduced as correction factors to the mean density of baryons in the vicinity of ionizing sources at early times ($\delta_b$), and the conversion efficiency of ionizing photons emitted to H {\footnotesize I}~ photoionization rate at late times ($\gamma$). Fits of these two correction factors versus $Q_{H {\footnotesize II}~}$ are presented in Figures \ref{deltabvsQfit5} and \ref{RatiovsQfit} for consumption by other researchers. At this point we do not know how general these results are. However we have indications based on another simulation we have analyzed with a softer source SED that the functional forms are representative of this class of reionization model.
The significance of these results to high redshift galaxy observers is the following. Setting $Q_{H {\footnotesize II}~} = 1$ and $\delta_b = 1$ in Equation \eqref{eq:dQdtdbg}, we derive
\begin{equation}
\dot{n}_{ion} = \frac{1}{\gamma} \frac{\bar{n}_H}{\bar{t}_{rec}}.
\label{eq:ndotgamma}
\end{equation}
This differs from the usual expression used to assess whether a given ionizing photon injection rate can maintain an ionized IGM by the factor $1/\gamma$, which is a factor of $\sim 20$ at overlap in our simulation. If this result is correct, then it means that the required UV luminosity density to maintain an ionized IGM has been underestimated by a factor of approximately 20. However, a more precise statement would be that the UV luminosity density required to maintain the IGM in a {\em highly ionized state; $f_n =10^{-5}$} is 20 times higher than what has been previously estimated. Lower levels of UV luminosity density than that specified in Equation \eqref{eq:ndotgamma} could still maintain the IGM in an ionized state, but one with a higher neutral fraction.
As we showed in Figure \ref{treceffhubble}, the effective recombination time at and after overlap in our model is comparable to the Hubble time, whether we use the Madau formula to evaluate it for reasonable values for the clumping factor, or we evaluate it directly from our simulation data. This fact casts in doubt the entire instantaneous photon counting argument which is the basis of Equation \ref{eq:ndot}, and the equation becomes less useful for the purposes to which it has been applied (e.g., Robertson et al. 2013). It means that the ionization state of the IGM has a memory on the timescale of $\bar{t}_{rec}$ which is always a significant fraction of $t_{Hubble}$ before overlap, and of order the Hubble time after overlap. We therefore recommend observers use the time-dependent version Equation \eqref{eq:dQdtdbg} in future assessments of high redshift galaxy populations and their role in reionization.
\subsection{Limitations of the Simulation}
We conclude this section with a brief discussion of the known limitations of our simulation and a comparison of our results with others in the published literature. First the limitations. The principal limitation is the use of a uniform grid, which prevents us from resolving processes occuring inside galaxy halos. The main defect this introduces is an inability to calculate the ionizing escape fraction directly, as is done in some high resolution simulations; e.g., \cite{WiseCen2009,FernandezShull2011}. In our simulation, we calibrate our star formation recipe to match the observed SFRD, and then use that that to calculate UV feedback cell-by-cell via Equation \eqref{eq:emissivity}. We use a value for $\epsilon_{UV}$ taken from \cite{RicottiEtAl2002} for an unattenuated low metallicity stellar population. We underestimate the amount of internal attenuation of ionizing flux due to our limited resolution within halos, and we do not incorporate an explicit escape fraction parameter in Equation \eqref{eq:emissivity}. Effectively, we assume $f_{esc}(ISM)=1$. Using a lower value for $f_{esc}$ would result in a lower overlap redshift \citep{PetkovaSpringel2011a}. Clearly, it would be desirable to vary this parameter in future studies.
A second limitation of our simulation is that we have presented only one realization in a relatively small box. Previous studies have shown that H {\footnotesize II}~ bubbles reach a characteristic size of $\sim 10$ Mpc comoving in the lates stages of reionization \citep{FurlanettoEtAl2004,ZahnEtAl2007,ShinEtAl2008}. At 20 Mpc on a side, our box is scarcely larger than this. Therefore one can ask how robust our results are to boxsize. We have addressed this by carrying out a simulation of identical physics, spatial, and mass resolution in a volume 64 times as large as the one described in this paper. The simulation is carried out in a box 80 Mpc on a side on a uniform mesh of $3200^3$ cells, and with an equivalent number of dark matter particles. Results of this simulation will be presented in a forthcoming paper (So et al., in preparation). For the present we merely state that the $Q_{H {\footnotesize II}~}(z)$ curve for the $800^3$ simulation falls within the $\pm 1 \sigma$ band for the larger simulation, where this band is obtained by subdividing the large simulation into 64 cubes of size 20 Mpc on a side, and calculating the mean and standard deviation. While the larger box begins to ionize at a slightly earlier redshift, due to the presence of higher sigma peaks forming galaxies, both simulations complete reionization at the same redshift, $z_{reion} = 5.8$. The $Q_{H {\footnotesize II}~}(z)$ curve for the $800^3$ simulation is near the lower edge of the band, which means that at intermediate redshifts ($7 \leq z \leq 8$), where the difference is largest, the small box simulation underestimates the fraction of the volume that is ionized by about 20\%, with differences smoothly decreasing to lower and higher redshift.
A third limitation is that our SFRD systematically deviates from observations below $z \sim 7$, flattening and then decreasing slightly, rather than continuing to rise (Figure \ref{SFR}). The large box simulation does not show this effect, but rather tracks the observed SFRD over the entire range of redshifts. The difference in the mean SFRD between the large and small box simulations increases smoothly from 0.1 dex at $z=9$ to 0.3 dex at $z=6$. The higher levels of star formation in the large box simulation accounts for the higher ionized volume fraction at intermediate redshifts. Nonetheless, the two simulations complete reionization at virtually the same redshift, which is a curious result which we address in a subsequent paper.
Another limitation of our method is the use of flux-limited diffusion (FLD) to transport radiation. It is well known that FLD does not cast shadows behind opaque blobs. This could potentially overestimate how rapidly the IGM ionizes, and hence overestimate $z_{reion}$. In Paper I we showed through a direct comparison between FLD and an adaptive ray tracing method incorporated in the {\em Enzo} code on a standard test problem that the differences in the volume- and mass-weighted ionized volume fraction are small. This was for a rather small volume with a small number of ionizing sources. The differences will likely be even smaller as larger volumes containing larger numbers of sources are considered. At the present time, no fully-coupled radiation hydrodynamic simulations of reionization using ray tracing in large volumes are available to compare our method against, to confirm or deny this conjecture.
\subsection{Comparison with Other Self-Consistent Simulations}
Finally, we compare our results to the results of several recent fully-coupled simulations of reionization including hydrodynamics, star formation, and radiative transfer. \cite{PetkovaSpringel2011a} simulated a (10 Mpc/h)$^3$ volume with the {\tt Gadget-2} code coupled to a variable tensor Eddington factor moment method for the ionizing radiation field sourced by star forming galaxies. They carried out a suite of simulations with $2 \times 128^3$ gas and dark matter particles, varying the ionizing escape fraction and the mean energy per photon from hot, young stars. The also performed one simulation at $2 \times 256^3$ resolution to check for convergence. Our simulation has 80/10 times superior mass resolution as their $128^3/256^3$ simulations. Because {\tt Gadget} is a Lagrangian code, our Eulerian simulation has 8/16 times lower resolution in the highest density regions, but 4.46/2.23 times higher resolution at mean density, and even higher resolution compared to the {\tt Gadget} simulations in low density voids. Our method also has a more accurate adaptive subcycling timestepping scheme for the coupled radiation-ionization-energy equations, obviating the need to model nonequilibrium effects by means of a gas heating parameter $\epsilon$.
Morphologically, our results are qualitatively similar, as are the neutral hydrogen fraction versus overdensity phase diagrams. As might be expected from the two methods, the phase diagrams show some differences at the highest and lowest overdensities which is likely a resolution effect. The SFRD in the \cite{PetkovaSpringel2011a} simulation is about an order of magnitude higher than observed, making a direct comparison on $Q_{H {\footnotesize II}~}(z)$ somewhat problematic. However, since they vary the ionizing escape fraction, we can roughly compare their $f_{esc}=0.1$ case with our results. Their model completes reionization at $z \approx 5$ compared to our own which completes at $z \approx 5.8$. They plot the quantity $log[1-Q_{H {\footnotesize II}~}(z)]$, which makes the end of reionization look abrupt. We plot $Q_{H {\footnotesize II}~}(z)$, which makes the end of reionization look slow. When we plot $log[1-Q_{H {\footnotesize II}~}(z)]$ using our data, it looks very similar to their curves, and shows a rapid plunge in the average neutral fraction at late times. \cite{PetkovaSpringel2011a} do not compare with the predictions of the \cite{MadauEtAl1999} model, nor do they investigate the evolution of clumping factors, recombination times, or the number of photons per H atom to achieve overlap as we do. We do not investigate the properties of the $z=3$ IGM via Lyman $\alpha$ forest statistics, as they do. Therefore further comparisons are not possible at this time.
\cite{FinlatorEtAl2012} examined some of the same issues we have, hence a comparison with their results is informative. They carried out a suite of {\tt Gadget-2} simulations in small volumes (3, 6)Mpc/h coupled to a variable tensor Eddington factor moment method. Unlike \cite{PetkovaSpringel2011a}, the radiation transport is solved on a uniform Cartesian grid, rather than evaluated using the SPH formalism. The results presented in \cite{FinlatorEtAl2012} use $2 \times 256^3$ dark matter and gas particles, which given their small volumes, yields a similar mass resolution to our simulation, superior spatial resolution in high density regions, and slightly coarser spatial resolution at mean density and below. However, their radiation transport is done on coarse $16^3$ mesh, which in their fiducial run is $536$ comoving kpc $\approx 20 \times$ as coarse as ours. Their simulation thus coarse-grains the radiation field relative to the density field, which necessitates the introduction of a sub (radiation) grid model for unresolved self-shielded gas (i.e., Lyman limit systems). The effect of their subgrid model is to remove some gas in the overdensity regime $1 \leq \Delta_b \leq 50$ in the calculation of the H {\footnotesize II}~ clumping factor, thereby lowering it. Since our radiation field is evolved on the same grid as the density field, we have not included an explicit subgrid model for unresolved self-shielded gas. Lyman limit systems, with neutral column densities of $\sim 10^{17}$ cm$^{-2}$, have a characteristic size of 10 physical kpc \citep{Schaye2001,McQuinnEtAl2011}. At $z=6$ this is 70 comoving kpc, which is resolved by 3 grid cells in our simulation. While this is lower than one would ideally like (5-10 cells), we believe we can make an apples-to-apples comparison between our resolution-matched simulation results and Finlator et al.'s results.
Our results are in broad agreement with those of \cite{FinlatorEtAl2012}, with some minor quantitative differences. We both find that the unthresholded baryon clumping factor $C_b$ significantly overestimates the clumping in ionized gas at redshifts approaching overlap, and therefore that it should not be used to estimate the mean recombination rate in the IGM. We confirm their findings that properly accounting for the ionization state and temperature of gas of moderate overdensities lowers the clumping factor to less than $\approx 6$ (in our case less than 5; see Figure \ref{ClumpingFactors}). Finlator et al. quote a value for $C_{H {\footnotesize II}~}$ of 4.9 at $z=6$ taking self-shielding into account, which is in good agreement with our value of $C_{ttH {\footnotesize II}~} \approx 4.8$. However, they favor a lower value for $C$ of 2.7-3.3 taking temperature corrections into account. This can be compared with our value for $C_{RR} \approx 2.3$, which includes temperature corrections but also excludes gas with $\Delta_b<1$. Including this low density gas, as Finlator et al. do, would raise this value somewhat since a larger range of densities enter into the average. We conclude therefore that clumping factors derived from our simulation are in good agreement with those reported by \cite{FinlatorEtAl2012}.
We find that approximately 2 photons per hydrogen atom ($\gamma/H\approx 2$) are required to reionize gas satisfying $\Delta_b<100$--our proxy for the fluctuating IGM. \cite{FinlatorEtAl2012} quote a model-dependent value for $\gamma/H$ which depends on the redshift at which the IGM becomes photoheated and thereby Jeans smoothed (their Fig. 7). For $z=6$, $\gamma/H \approx 5$, significantly higher than our number evaluated directly from the simulation. However, for $z=8$, when our box is already significantly ionized, $\gamma/H \approx 3$. Because there are many model-dependent assumptions that go into the Finlator et al. estimate, we consider this reasonably good agreement. However we point out that our estimate is the first to be derived from a self-consistent simulation of reionization with no subgrid models aside from the star formation/radiative feedback recipe.
Finally, \cite{FinlatorEtAl2012} compare $Q_{H {\footnotesize I}~}(z)=1-Q_{H {\footnotesize II}~}(z)$ for their fiducial model with the time-dependent model of \cite{MadauEtAl1999}. They point out the sensitivity of the redshift of overlap on the choice of clumping factor, which enters into the recombination time, and showed that $C_{H {\footnotesize II}~}$ provides better agreement with theory at early times than $C_b$, consistent with our findings. Since small discrepancies in $Q_{H {\footnotesize II}~}(z)$ at early times are masked by plotting $Q_{H {\footnotesize I}~}(z)$, Finlator et al. did not discover the need for our overdensity correction $\delta_b$. Similar to us, they found that even with the best clumping factor estimate the analytic model predicts that reionization completes earlier than the simulation by $\Delta z \approx 1$. They ascribe this delay to finite speed-of-light effects (which can only account for $\Delta z =0.1$), while we ascribe it to nonequilibrium ionization effects. \cite{FinlatorEtAl2012} did not propose modifications to the \cite{MadauEtAl1999} model to improve agreement with simulation, as we do in Equation \eqref{eq:dQdtdbg}.
\section{A Global Estimate for Circumgalactic Absorption of Ionizing Radiation}
\label{escape}
The ionizing escape fraction from galaxies is an important parameter in models of reionization. Typically, one thinks about the escape fraction as a property of individual galaxies, determined by the absorption of ionizing radiation on small scales in the ISM. However it is interesting to ask whether there is significant absorption in the denser Circumgalactic Medium (CGM) surrounding galaxies. If we write the total escape fraction as the product of escape fractions, then $f_{esc}=f_{esc}(ISM)f_{esc}(CGM)$. Here we use our simulation to derive an estimate of the globally averaged escape fraction as a function of redshift due to the circumgalactic medium $\bar{f}_{esc}(CGM)$.
Recall from Sec. \S\ref{Method} that the halo escape fraction is not a model input parameter, but is rather an ouput since the equation of radiative transfer is solved throughout the computational domain. Our halos are not well resolved internally, and so we are underestimating the amount absorption of ionizing radiation on galaxy ISM scales. However if significant absorption occurs on scales of the virial radius or larger, then that would be simulated reasonably accurately. In the following we assume this is the case, and present results that can be taken to be an upper limit on the total escape fraction (ISM+CGM).
Rather than measure the escape fraction halo by halo and take the average over all halos, we use a simpler method. Since we know every ionization requires an ionizing photon, and we have the ionization rate density as a field defined at every grid cell, then we can estimate $\bar{f}_{esc}(CGM)$ as follows (hereafter we drop the CGM modifier with the reader's understanding that this is what we are estimating):
\begin{equation}
\bar{f}_{esc}(I_t) = \int_{V_t} n_\mathrm{H\,I}\Gamma_\mathrm{H\,I}^{ph} d^3x \bigg / \int_{V} n_\mathrm{H\,I}\Gamma_\mathrm{H\,I}^{ph} d^3x ,
\label{eq:fesc}
\end{equation}
\\where $\Gamma_\mathrm{H\,I}^{ph}$ is evaluated cell by cell via Equation \eqref{eq:photoionization}, $V$ is the simulation volume and $V_t$ denotes the integration includes only cells which satisfy $\Delta_b < 100.$ In other words, $\bar{f}_{esc}$ is the ratio of the number of ionizations in the IGM, as defined by the overdensity threshold, to the total number of ionizations in the volume. The modifier $I_t$ refers to this method of estimating $\bar{f}_{esc}$ (a superior method is presented below).
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig18-eps-converted-to.pdf}
\caption{Estimate of the globally averaged ionizing radiation escape fraction due to circumgalactic absorption $\bar{f}_{esc}(I_t)$ computed as the ratio of the volume integrated ionization rate in the IGM ($\Delta_b < 100$) divided by the total ionization rate (Eq. \eqref{eq:fesc}). }
\label{EscFraction}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig19-eps-converted-to.pdf}
\caption{Evolution of the volume averaged rate densities for: (1) ionizing photons injected into the IGM ($\dot{N}_\mathrm{IGM}$), (2) gas photoionization ($\dot{N}_t$), and (3) gas recombination ($\dot{R}_t$) integrated over the singly thresholded volume $V_t$ defined as $\Delta_b<100$. The ionization rate density curve tracks the photon injection rate density curve in the photon starved regime at high redshifts, but begins to fall below it as the globally averaged ionization parameter approaches unity (Fig. \ref{IP}). After overlap, in the photon abundant regime, the ionization rate density is $\sim 20\times$ the photon injection rate density, but comes into balance with the recombination rate density.}
\label{sanitycheckrate}
\end{figure}
The result is plotted in Fig. \ref{EscFraction}. At high redshifts the escape fraction is high and relatively constant at $\bar{f}_{esc} \sim 0.65-0.7$. As overlap is approached $\bar{f}_{esc}$ drops considerably, reaching values of $\sim 0.2$ by $z=5.$ There is no obvious reason why the escape fraction should drop so dramatically at the epoch of overlap. To investigate this properly would require a statisical analysis of individual halo escape fractions, which we defer to a subsequent paper. Perhaps this is an artifact of how we are estimating $\bar{f}_{esc}$. While it is true that every ionization requires and ionizing photon in the photon starved regime (i.e., before overlap), after overlap the volume becomes optically thin to ionizing radiation, and it is not true that every ionizing photon causes an ionization in the box. This is illustrated in Fig. \ref{sanitycheckrate}.
The curve labeled $\dot{N}_t$ is the actual ionization rate density measured in the simulation averaged over the entire 20 Mpc cubic volume satisfying the overdensity threshold $\Delta_b < 100$; i.e. precisely the numerator of Eq. \eqref{eq:fesc} divided by 20$^3$. The curve labeled $\dot{R}_t$ is the recombination rate density averaged over the same volume; i.e.
\begin{equation}
\dot{R}_t = \int_{V_t} n_e n_\mathrm{H\,II}\alpha_B(T) d^3x .
\label{eq:totrecombst}
\end{equation}
We see that ionization rate density $\dot{N}_t$ grows with redshift and reaches a maximum at $z \approx 6.5$, and then drops by roughly 0.8 dex by overlap completion at $z=5.8$. It continues to decrease thereafter. The reason for this sudden drop is that after overlap there are very few neutral atoms left to ionize ($n_\mathrm{H\,I}/n_\mathrm{H} \sim 10^{-5}$).
This can be illustrated by considering the {\em global ionization parameter}, which is the number of ionizing photons per neutral H atom $\Gamma_{IP} = \langle n_{ph}\rangle/\langle n_{H I} \rangle$ averaged over the entire volume. Specifically, we integrate the grey radiation energy density divided by the mean photon energy $\bar{\epsilon}$ over the singly thresholded volume, and divide by the number of H {\footnotesize I} atoms in the same volume:
\begin{equation}
\Gamma_{IP} = \int_{V_t} (E/\bar{\epsilon}) d^3x \bigg / \int_{V_t} n_{HI} d^3x.
\label{eq:IP}
\end{equation}
We see from Fig. \ref{IP} that $\Gamma_{IP}$ grows from $\sim 10^{-3}$ at $z=10$ to unity at $z\approx 6.5$ just before overlap. Thereafter $\Gamma_{IP}$ grows very rapidly, reaching a value around $10^5$ at the overlap redshift, and leveling off at around $10^6$ below that. From the standpoint of the global ionization parameter, reionization begins photon starved but completes photon abundant.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig20-eps-converted-to.pdf}
\caption{Redshift evolution of the global H {\footnotesize I}~ ionization parameter as defined in Eq. \eqref{eq:IP}. }
\label{IP}
\end{figure}
Returning to Fig. \ref{sanitycheckrate} we see that the recombination rate density $\dot{R}_t$ curve tracks the ionization rate density curve to $z\sim 7$, but is about 0.7 dex lower in magnitude, as it must be if the ionized volume filling fraction is to grow. As overlap is approached ionizations and recombinations come into balance, but the recombination rate density has dropped considerably since it reached its maximum value at $z \approx 6.5$. This is also the redshift at which the ionization rate achieves a maximum, and when the global ionization parameter reaches unity. We also observe that the $f_{esc}$ curve in Fig. \ref{EscFraction} begins its precipitous drop at this redshift. We believe all of these events signal the rapid rise in the global ionization parameter below $z=6.5$, and not some change in the escape fraction of young galaxies.
Counting the fraction of all ionizations occuring outside halos is not a reliable estimate of the escape fraction for $\Gamma_{IP} \gg 1$ because it does not count the photons in the radiation field that have nothing to ionize. Therefore we need to modify Eq. \eqref{eq:fesc} to include photons which build up of the radiation field:
\begin{equation}
\bar{f}_{esc} = \int_{V_t} (n_\mathrm{H\,I}\Gamma_\mathrm{H\,I}^{ph} + \frac{1}{\bar{\epsilon}}\frac{dE}{dt}) d^3x \bigg / \int_{V} (\eta/\bar{\epsilon}) d^3x .
\label{eq:fescimproved}
\end{equation}
\\Here the numerator is the rate at which ionizing photons are causing ionizations in the IGM and building up the UV background, and the denominator is volume integrated ionizing photon production rate.
Fig. \ref{RadEscFraction} plots $\bar{f}_{esc}$ calculated according to Eq. \eqref{eq:fescimproved}. Each contribution to $\bar{f}_{esc}$ is plotted separately, as well as the sum. We see that $\bar{f}_{esc}$ is roughly constant with redshift with a value of around 0.6. We see that as the contribution due to ionizations declines below $z\sim 7$, the contribution due to the change in radiation background intensity increases in a compensating fashion. This confirms our earlier suspicions and gives us a better estimate of the mean circumgalactic attenuation of ionizing radiation from young galaxies.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig21-eps-converted-to.pdf}
\caption{Redshift evolution of the globally averaged escape fraction contribution from circumgalactic absorption as estimated by the number of ionizations occuring in the IGM and the buildup of the ionizing radiation background. The curves labeled $\bar{f}_{esc}(I_t), \bar{f}_{esc}(\dot{E})$ plot the contributions of the first and second terms in Eq. \eqref{eq:fescimproved}, while the curve labeled $\bar{f}_{esc}$ plots their sum.}
\label{RadEscFraction}
\end{figure}
To complete the picture we plot in Fig. \ref{sanitycheckrate} the number density of ionizing photons escaping into the IGM, calculated as $\dot{N}_{IGM}=\bar{f}_{esc}\dot{N}_{sim}$, where $\dot{N}_{sim}$ is the ionizing photon production rate in the simulation, and $\bar{f}_{esc}$ is the improved estimate for the escape fraction calculated using Equation \eqref{eq:fescimproved}. We see that at high redshifts the $\dot{N}_{IGM}$ and $\dot{N}_t$ track each other closely. This tells us two things. First, that reionization at high redshifts when $Q_{\mathrm{H\,II}} \ll 1$ is photon starved, in the sense that every ionizing photon emitted results in an ionization. And second that our estimate of $\bar{f}_{esc}$ is reasonably accurate at these redshifts. However, as redshift decreases, the two curves systematically begin to deviate from one another in the sense that $\dot{N}_t < \dot{N}_{IGM}$. Beginning at $z = 6.5$ the ionization rate density begins to decrease while the ionizing photon production rate into the IGM continues to rise.
After overlap the large disparity between the $\dot{N}_{IGM}$ and $\dot{N}_t$ curves can then be understood as saying that the IGM becomes photon abundant.
The ratio of ionization rate density and the photon injection rate into the IGM is plotted in Fig. \ref{Ndot_Ratio}. The ratio is unity initially, and slowly decreases until $z\approx 7$, and then drops rapidly as overlap is approached. After overlap the ratio is about 0.05. In other words, after overlap, the photon production rate is about 20$\times$ the ionization rate in a volume averaged sense. Since the ionization and recombination rates are in balance after overlap, we conclude that the volume averaged photon injection rate is about 20$\times$ the recombination rate.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig22-eps-converted-to.pdf}
\caption{Ratio of the volume integrated photoionization rate in the IGM $\dot{N}_t$ to the integrated photon injection rate into the IGM $\dot{N}_{IGM}$, where the IGM is defined as cells with $\Delta_b < 100$. The ratio is near unity initilly, remains high until $z\approx 7$ ($Q_{HII} \approx 0.5$), and then drops rapidly as overlap is approached and the IGM becomes highly ionized.}
\label{Ndot_Ratio}
\end{figure}
\section{Introduction}
\label{sec:introduction}
The Epoch of Reionization (EoR) is an active area of research observationally,
theoretically, and computationally. Observations constrain the tail end of hydrogen reionization
to the redshift range $z=6-8$ \citep{RobertsonEtAl2010}. These observations include the presence of Gunn-Peterson
troughs in the Ly $\alpha$ absorption spectra of high redshift quasars \citep{FanEtAl2006}, and
the strong evolution of Lyman $\alpha$ emitter luminosity function (Robertson et al. 2010 and references
therein.)
Observations from the WMAP and Planck satellites tell us that the universe was substantially
ionized by $z \approx 10$ but can say little about the
reionization history or topology \citep{JarosikEtAl2011,Planck2013}.
High redshift 21cm observations hold forth great promise of elucidating the details of this transition \citep{BarkanaLoeb2007, PritchardLoeb2012}, but these results are still in the future.
It is believed that early star forming galaxies provided the bulk of the UV photons responsible for
reionization \citep{RobertsonEtAl2010,RobertsonEtAl2013}, but early QSOs may have also contributed \citep{MadauEtAl1999, BoltonHaehnelt2007, HaardtMadau2012}. The ``galaxy reionizer" hypothesis has been greatly strengthened by the recent advances in the study of high redshift galaxies afforded by the IR-sensitive Wide Field Camera 3 (WFC3) aboard the Hubble Space Telescope \citep[e.g.][]{RobertsonEtAl2010, RobertsonEtAl2013, BouwensEtAl2011, BouwensEtAl2011b, OeschEtAl2013}. Within uncertainties, the luminosity function of $z=6$ Lyman break galaxies (LBGs) appears to be sufficient to account for reionization at that redshift from a photon counting argument \citep{BoltonHaehnelt2007, RobertsonEtAl2010, BouwensEtAl2012}. Among the observational uncertainties are the faint-end slope of the galaxy luminosity function \citep{WiseCen2009,LabbeEtAl2010,BouwensEtAl2012}, the spectral energy distribution of the stellar population \citep{CowieEtAl2009,WillotEtAl2010,HaardtMadau2012}, and the escape fraction of ionizing photons \citep{WyitheEtAl2010, YajimaEtAl2011, MitraEtAl2013}. Among the theoretical uncertainties are the number of ionizing photons per H atom required to bring the neutral IGM to its highly ionized state by $z=6$, the clumping factor correction to the mean IGM recombination time \citep{PawlikEtAl2009, RaicevicTheuns2011, FinlatorEtAl2012, ShullEtAl2012, RobertsonEtAl2013},
and the contribution of Pop III stars and accreting black holes to the early and late stages of reionization \citep{BoltonHaehnelt2007,TracGnedin2011,AhnEtAl2012}.
When assessing whether an observed population of high-z galaxies is capable of
reionizing the universe (e.g., Robertson et al. 2013), observers often use the criterion derived
by \cite{MadauEtAl1999} for the ionzing photon volume density $\dot{\mathcal{N}}_{ion}$ necessary
to maintain the clumpy IGM in an ionized state:
\begin{align}
\label{eq:ndot}
\dot{\mathcal{N}}_{ion}(z) &= \frac{\bar{n}_\mathrm{H}(0)}{\bar{t}_{rec}(z)} = (10^{51.2}s^{-1}Mpc^{-3})\left(\frac{C}{30}\right) \notag\\
&\times \left(\frac{1+z}{6}\right)^3 \left(\frac{\Omega_b h_{50}^2}{0.08}\right)^2,
\end{align}
where $\bar{n}_\mathrm{H}(0)$ is the mean comoving number density of H atoms, $C \equiv \langle n^2_\mathrm{H\,II} \rangle /\langle n_\mathrm{H\,II} \rangle ^2$ is the H {\footnotesize II} clumping factor (angle brackets denote volume average over a suitably large volume that the average is globally meaningful),
and the rest of the symbols have their usual meaning.
The origin of this formula is a simple photon counting argument, which says
that in order to maintain ionization at a given redshift $z$,
the number of ionizing photons emitted in
a large volume of the universe multiplied by a characteristic recombination
time, denoted $\bar{t}_{rec}$, must equal the number of hydrogen atoms: $\dot{\mathcal{N}}_{ion} \times \bar{t}_{rec} = \bar{n}_\mathrm{H}(0)$.
The clumping factor enters as a correction factor to account for the density inhomogeneties in the IGM
induced by structure formation. We note that $\bar{t}_{rec}$ is not the volume average of the local recombination time of the
ionized plasma, as this would heavily weight regions with the {\em longest} recombination
times; i.e. voids. A proper derivation of Equation \eqref{eq:ndot} shows that $\bar{t}_{rec} \propto \langle t_{rec}^{-1} \rangle ^{-1}$, which weights regions
with the {\em shortest} recombination times; i.e. regions at the mean density and above.
Equation \eqref{eq:ndot} is based on a number of simplifying assumptions discussed by
\cite{MadauEtAl1999}, including the assumption $\bar{t}_{rec} \ll t$. It is this assumption that allows history-dependent effects to be ignored, and a quasi-instantaneous analysis of the
photon budget for reionization to be done. The validity of this assumption is naturally
redshift dependent, but it is also dependent upon the adopted definition of $\bar{t}_{rec}$. A second comment about Equation \eqref{eq:ndot} is that it does not ask how many ionizing photons per H atom are required to convert a neutral IGM to a fully ionized one, only how many are required to {\em maintain} the IGM in an ionized state. Because the recombination time is short at high redshifts, it is expected that this number is greater than one.
In this paper we examine these and related topics within the context of a direct numerical simulation of cosmic reionization based on a new flux-limited diffusion radiation transport solver installed in the {\em Enzo} code \citep{NormanEtAl2013} (hereafter Paper I).
Our approach self-consistently couples all the relevant physical processes
(gas dynamics, dark matter dynamics, self-gravity, star formation/feedback,
radiative transfer, nonequilibrium ionization/recombination, heating and cooling) and evolves the
system of coupled equations on the same high resolution mesh. We refer to this
approach as {\em direct numerical simulation} or {\em resolution matched}, in contrast to previous approaches
which decouple and coarse-grain the radiative transfer and ionization balance
calculations relative to the underlying dynamical calculation.
Our method is
scalable with respect to the number of radiation sources, size of the mesh, and the
number of computer processors employed.
This scalability permits us to simulate cosmological reionization in large cosmological
volumes (L $\sim100$ Mpc) while directly modeling the sources and sinks of ionizing
radiation,
including radiative feedback effects such as photoevaporation of gas from halos,
Jeans smoothing of the IGM, and enhanced recombination due to small scale clumping.
In this the first of several application papers, we investigate in a volume of modest size (L=$20$ Mpc) the mechanics of reionization from stellar sources forming in high-$z$ galaxies, the role of gas clumping, recombinations, and the photon budget required to complete reionization.
By analyzing this simulation we are able to critically examine the validity of Equation \eqref{eq:ndot} as a predictor of when the end of EoR will occur, and we can calculate the integrated number of ionizing photons
per H atom needed to ionize the simulated volume $\gamma_{ion}/H=\int dt \dot{\mathcal{N}}_{ion} / \bar{n}_\mathrm{H}(0)$. Ignoring recombinations within the virial radii of collapsed halos, we find $\gamma_{ion}/H \approx 2$. This result
supports the ``photon starved'' reionization scenario discussed by \cite{BoltonHaehnelt2007}.
We also examine whether modern revisions to Equation \eqref{eq:ndot} using alternatively defined
clumping factors \citep{PawlikEtAl2009, RaicevicTheuns2011, FinlatorEtAl2012, ShullEtAl2012} are
improvements over the original. We find they systematically overestimate the redshift of reionization completion $z_{reion}$ because the condition $\bar{t}_{rec}/t \ll 1$ is never obeyed. We study the accuracy and validity of the time-dependent analytic model of \cite{MadauEtAl1999}, and find that while it is in better agreement with the simulation, it also overestimates $z_{reion}$ because it ignores important corrections to the ionization term at early and late times.
This paper is organized as follows: in \S\ref{Method} we discuss the design criteria
for the simulation and briefly outline the basic equations and implementation of the FLD radiation transport model, referring the
reader to Paper I for a more complete description of the numerical algorithms and tests.
In \S\ref{GeneralResults}, we present some general features of the simulation and demonstrate its broad consistency with observed star formation rate density and high redshift
galaxy luminosity function.
In \S\ref{sec:ClumpingFactors} we examine the accuracy of different clumping factor approaches to estimating the redshift of complete reionization.
In \S\ref{escape} we derive a global estimate for the circumgalactic absorption of ionizing radiation from our simulation.
In \S\ref{Qdot} we test a simple analytic model for the evolution of the ionized volume fraction $Q_\mathrm{H\,II}$ and present an improvement to the model which better agrees with our simulation.
In \S\ref{Discussion} we
discuss implications of our results on the current understanding of
reionization. And finally, in \S\ref{Conclusions} we end with a
summary of our main results and conclusions.
\section{Method}
\label{Method}
\subsection{Simulation Goals and Parameters}
We use the Enzo code \citep{TheEnzoCollaboration}, augmented with a flux-limited diffusion radiative transfer solver and a parameterized model of star formation and feedback \citep{NormanEtAl2013} to simulate inhomogeneous hydrogen reionization in a 20 Mpc comoving box in a WMAP7 $\Lambda$CDM cosmological model. Details of the numerical methods and tests are provided in Paper I. Here we briefly describe the simulation's scientific goals and design considerations to put it into perspective with other reionization simulations. For completeness, the physical equations we solve and the treatment of the ionizing sources and radiation field are included below.
Our principle
goal is to simulate the physical processes occuring in the IGM outside the virial radii of high redshift galaxies in a {\em representative} realization of inhomogenous reionization. We wish to simulate the early, intermediate, and late phases of reionization in a radiation hydrodynamic cosmological framework so that we may study the nonequilibrium ionization/recombination processes in the IGM at reasonably high resolution self-consistently coupled to the dynamics. In this way we can study such effects as optically thick heating behind the I-fronts \citep{AbelHaehnelt1999}, Jeans smoothing \citep{ShapiroEtAl1994,Gnedin2000b}, photoevaporation of dense gas in halos \citep{ShapiroEtAl2004}, and nonequilibrium effects in the low density voids. Because we carry out our simulation on a fixed Eulerian grid, we do not resolve the internal processes of protogalaxies very well. In this sense, our simulation is not converged on all scales. Nonetheless Equations \eqref{eq:gravity} to \eqref{eq:cons_radiation} are solved everywhere on the mesh self-consistently, including ionization/recombination and radiative transfer inside protogalaxies. The escape of ionizing radiation from galaxies to the IGM is thus simulated directly, and not introduced as a parameter. We use a star formation recipe that can be tuned to closely reproduce the observed high-$z$ galaxy luminosity function (LF), star formation rate density (SFRD), and redshift of reionization completion. This gives us confidence that we are simulating IGM processes in a realistic scenario of reionization.
We simulate a WMAP7 \citep{JarosikEtAl2011} $\Lambda$CDM cosmological model with the following parameters:
$\Omega_{\Lambda} = 0.73$, $\Omega_m = 0.27$, $\Omega_b = 0.047$,
$h = 0.7$, $\sigma_8 = 0.82$, $n_s = 0.95$, where the symbols have their usual meanings.
A Gaussian random
field is initialized at $z=99$ using the {\em Enzo} initial conditions generator {\em inits}
using the \cite{EisensteinHu1999} fits to the transfer functions.The simulation is performed in a comoving volume of (20 Mpc)$^3$ with a grid
resolution of $800^3$ and the same number of dark matter particles. This yields a comoving spatial resolution of 25 kpc and
dark matter particle mass of $4.8 \times 10^5 M_{\odot}$. This resolution yields a dark matter halo mass function that
is complete down to $M_h = 10^8 M_{\odot}$, which is by design, since this is the mass scale below
which gas cooling becomes inefficient. However, due to our limited boxsize, our halo mass function is incomplete above
$M_h \approx 10^{11} M_{\odot}$ (see Figure \ref{HMF}). In a forthcoming paper we will report on a simulation of identical design and resolution as this one, but in a volume 64 times as large, which contains the rarer, more massive halos. With regard to resolving the diffuse IGM, our $25$ kpc resolution equals the value recommended by \cite{BryanEtAl1999} to converge on the properties of the Ly $\alpha$ forest at lower redshifts, is $3\times$ better than the optically thin high resolution IGM simulation described in \cite{ShullEtAl2012}, and nearly $4\times$ better than the inhomogeneus reionization simulation described in \cite{TracEtAl2008}.
As described below in \S\ref{starformationandfeedback}, we use a parameterized model of star formation calibrated to observations of high redshift galaxies. The star formation efficiency parameter $f_*$ is adjusted to match the observed star formation rate density in the interval $6 \leq z \leq 10$ from \cite{BouwensEtAl2011}.
The simulation consumed 255,000 core-hrs running on 512 cores of the Cray XT5 system {\em Kraken} operated by the
National Institute for Computational Science at ORNL.
\subsection{Governing Equations}
\label{GoverningEquations}
The equations of cosmological radiation hydrodynamics implemented in the Enzo code used for this research are given by the following system of partial differential equations (Paper I):
\begin{align}
\nabla^2 \phi &= \frac{4\pi g}{a}(\rho_b + \rho_\mathrm{dm} - \langle \rho \rangle),
\label{eq:gravity}\\
\partial_\mathrm{t} \rho_b + \frac1a {\bf v}_b \cdot \nabla
\rho_b &= -\frac1a \rho_b \nabla\cdot{\bf v}_b -\dot{\rho}_{SF},
\label{eq:cons_mass}\\
\partial_\mathrm{t} {\bf v}_b + \frac1a\({\bf v}_b\cdot\nabla\){\bf v}_b &=
-\frac{\dot{a}}{a}{\bf v}_b - \frac{1}{a\rho_b}\nabla p - \frac1a
\nabla\phi,
\label{eq:cons_momentum}\\
\partial_\mathrm{t} e + \frac1a{\bf v}_b\cdot\nabla e &=
- \frac{2\dot{a}}{a}e
- \frac{1}{a\rho_b}\nabla\cdot\left(p{\bf v}_b\right) \nonumber\\
&- \frac1a{\bf v}_b\cdot\nabla\phi + G - \Lambda + \dot{e}_{SF}
\label{eq:cons_energy}\\
\partial_\mathrm{t} {\tt n}_\mathrm{i} + \frac{1}{a}\nabla\cdot\({\tt n}_\mathrm{i}{\bf v}_b\) &=
\alpha_\mathrm{i,j} {\tt n}_\mathrm{e} {\tt n}_\mathrm{j} - {\tt n}_\mathrm{i} \Gamma_\mathrm{i}^{ph}, \qquad \nonumber\\
&i=1,\ldots,N_\mathrm{s}
\label{eq:chemical_ionization}\\
\partial_\mathrm{t} E + \frac1a \nabla\cdot\(E {\bf v}_b\) &=
\nabla\cdot\(D\nabla E\) - \frac{\dot{a}}{a}E \nonumber\\
&- c \kappa E + \eta
\label{eq:cons_radiation}
\end{align}
Equation \eqref{eq:gravity} describes the modified gravitational
potential $\phi$ due to baryon density $\rho_\mathrm{b}$ and dark matter
density $\rho_\mathrm{dm}$, with $a$ being the cosmological scale factor, $g$
being the gravitational constant, and $\langle \rho \rangle$ being the
cosmic mean density. The collisionless dark matter density
$\rho_\mathrm{dm}$ is evolved using the Particle Mesh method (equation not
shown above), as described in
\citealt{HockneyEastwood1988, TheEnzoCollaboration}.
Equations \eqref{eq:cons_mass}, \eqref{eq:cons_momentum} and
\eqref{eq:cons_energy} are conservation of mass, momentum and energy,
respectively, in a comoving coordinate system \citep{BryanEtAl1995,TheEnzoCollaboration}.
In the above equations, ${\bf v}_b\equiv a(t)\dot{{\bf x}}$ is the proper
peculiar baryonic velocity, $p$ is the proper pressure, $e$ is the
total energy per unit mass, and $G$ and $\Lambda$ are the heating and
cooling coefficients. Equation \eqref{eq:chemical_ionization}
describes the chemical balance between the different ionization
species (in this paper we used H {\footnotesize I},
H {\footnotesize II}, He {\footnotesize I}, He {\footnotesize II},
He {\footnotesize III} densities) and electron density. Here, ${\tt n}_\mathrm{i}$ is the
comoving number density of the $i^{th}$ chemical species, ${\tt n}_\mathrm{e}$ is
the electron number density, ${\tt n}_\mathrm{j}$ is the ion that reacts with
species $i$, and $\alpha_\mathrm{i,j}$ are the reaction rate coefficient
between species $i$ and $j$ \citep{AbelEtAl1997, HuiGnedin1997}, and
finally $\Gamma^{ph}_\mathrm{i}$ is the photoionization rate for species $i$.
\subsection{Radiation Transport}
\label{RadiationTransport}
Equation \eqref{eq:cons_radiation} describes radiation transport in the Flux Limited
Diffusion (FLD) approximation in an expanding
cosmological volume \citep{ReynoldsEtAl2009,NormanEtAl2013}. $E$ is the
comoving grey radiation energy density. The {\em flux limiter} $D$ is
a function of $E$, $\nabla E$, and the opacity $\kappa$
\citep{Morel2000}, and has the form:
\begin{align}
D &= \mbox{diag}\left(D_1, D_2, D_3\right), \quad\mbox{where} \\
D_\mathrm{i} &= c \(9\kappa^2 + R_\mathrm{i}^2\)^{-1/2},\quad\mbox{and} \\
R_\mathrm{i} &= \max\left\{\frac{|\partial_\mathrm{x_i} E|}{E},10^{-20}\right\}
\end{align}
In the calculation of the grey energy density $E$, we assume
$E_\nu(\mathbf{x},t,\nu)=\tilde{E}(\mathbf{x},t)\,\chi_E(\nu)$, therefore:
\begin{align}
\label{eq:grey_definition}
E(\mathbf{x},t) &= \int_{\nu_1}^{\infty} E_\nu(\mathbf{x},t,\nu)\,\mathrm d\nu \nonumber \\
&=\tilde{E}(\mathbf{x},t) \int_{\nu_1}^{\infty} \chi_E(\nu)\,\mathrm d\nu,
\end{align}
Which separates the dependence of $E$ on coordinate $\mathbf{x}$ and
time $t$ from frequency $\nu$. Here $\chi_E$ is the spectral energy
distribution (SED) taken to be that of a Pop II stellar population
similiar to one from \citep{RicottiEtAl2002}.
\subsection{Star Formation and Feedback}
\label{starformationandfeedback}
Because star formation occurs on scales not resolved by our uniform mesh simulation,
we rely on a subgrid model which we calibrate to observations of star formation in high
redshift galaxies. The subgrid model is a variant of the \cite{CenOstriker1992}
prescription with two important modifications as described in \cite{SmithEtAl2011}. In the original \cite{CenOstriker1992} recipe, a computational cell forms a collisionless ``star particle" if a number of criteria are met: the baryon density exceeds a certain numerical threshold; the gas velocity divergence is negative, indicating collapse; the local cooling time is less than the dynamical time; and the cell mass exceeds the Jeans mass. In our implementation, the last criterion is removed because it is always met in large scale, fixed-grid simulations, and the overdensity threshold is taken to be $\rho_b/(\rho_{c,0}(1+z)^3) > 100$, where $\rho_{c,0}$ is the critical density at $z=0$. If the three remaining criteria are met, then a star particle representing a large collection of stars is formed in that timestep and grid cell with a total mass
\begin{equation}
m_* = f_* m_{cell} \frac{\Delta t}{t_{dyn}},
\end{equation}
where $f_*$ is an efficiency parameter we adjust to match observations of the cosmic star formation rate density (SFRD) \citep{BouwensEtAl2011}, $m_{cell}$ is the cell baryon mass, $t_{dyn}$ is the dynamical time of the combined baryon and dark matter fluid, and $\Delta t$ is the hydrodynamical timestep. An equivalent amount of mass is removed from the grid cell to maintain mass conservation.
Although the star particle is formed instantaneously (i.e., within one timestep), the conversion of removed gas into stars is assumed to proceed over a longer timescale, namely $t_{dyn}$, which more accurately reflects the gradual process of star formation. In time $\Delta t$, the amount
of mass from a star particle converted into newly formed stars is given by
\begin{equation}
\Delta m_{SF} = m_* \frac{\Delta t}{t_{dyn}} \frac{t-t_*}{t_{dyn}} e^{-(t-t_*)/t_{dyn}},
\end{equation}
where $t$ is the current time and $t_*$ is the formation time of the star particle. To make the
connection with Equation \eqref{eq:cons_momentum}, we have $\dot{\rho}_{SF} =\Delta m_{SF}/(V_{cell}\Delta t)$,
where $V_{cell}$ is the volume of the grid cell.
Stellar feedback consists of the injection of thermal energy, gas, and radiation
to the grid, all in proportion to $\Delta m_{SF}$. The thermal energy $\Delta e_{SF}$ and gas
mass $\Delta m_g$ returned to the grid are given by
\begin{equation}
\Delta e_{SF} = \Delta m_{SF} c^2 \epsilon_{SN}, \qquad
\Delta m_g = \Delta m_{SF} f_{m*},
\end{equation}
where $c$ is the speed of light, $\epsilon_{SN}$ is the supernova energy efficiency parameter, and $f_{m*}=0.25$ is the fraction of the stellar mass returned to the grid as gas. Rather than add
the energy and gas to the cell containing the star particle, as was done in
the original \cite{CenOstriker1992} paper, we distribute it evenly among the cell and its
26 nearest neighbors to prevent overcooling. As shown by \cite{SmithEtAl2011}, this
results in a star formation recipe which can be tuned to reproduce the observed SFRD. This is critical for us, as we use the observed high redshift SFRD to calibrate our reionization simulations.
To calculate the radiation feedback, we define an emissivity field $\eta(x)$ on the grid which accumulates
the instantaneous emissivities $\eta_i(t)$ of all the star particles within each cell. To calculate the contribution of each star particle $i$ at time $t$ we assume an equation of the same form for supernova energy feedback, but with a different energy conversion efficiency factor $\epsilon_{UV}$. Therefore
\begin{equation}
\label{eq:emissivity}
\eta= \sum_\mathrm{i}\epsilon_\mathrm{uv}\frac{\Delta m_\mathrm{SF} c^2}{V_\mathrm{cell}\Delta t}
\end{equation}
Emissivity $\eta$ is in units of erg s$^{-1}$cm$^{-3}$. The UV efficiency factor $\epsilon_\mathrm{uv}$ is taken from \cite{RicottiEtAl2002} as 4$\pi\times 1.1 \times 10^{-5}$, where the factor $4\pi$ comes from the conversion from mean intensity to radiation energy density.
\subsection{Data Analysis}
\label{DataAnalysis}
Due to the enormous amount of data produced by the simulation (one output file is about 100 GB),
we needed a scalable tool suited to the task of organizing and manipulating
the data into human readable form. We use the analysis software tool \texttt{yt} \citep{TurkEtAl2011} specifically created for doing this type of vital task. It is a python based software tool that does ``Detailed data analysis and visualizations,
written by working astrophysicists and designed for pragmatic analysis needs."
\texttt{yt} is open source and publicly available at http://yt-project.org.
\section{An Improved Model for the Evolution of $Q_\mathrm{H\,II}$}
\label{Qdot}
In this section we compare the evolution of the ionized volume fraction $Q_\mathrm{H\,II}$ from our simulation with the analytic model introduced by \cite{MadauEtAl1999}. We are motivated to do this because as we have seen from \S\ref{sec:ClumpingFactors}, Equation \eqref{eq:ndot} is not a useful predictor of when $Q_\mathrm{H\,II}$ reaches unity. We therefore want to investigate the accuracy of the time dependent model from which Equation \eqref{eq:ndot} is derived as a limiting case.
\cite{MadauEtAl1999} derived the following ODE for the evolution of $Q_\mathrm{H\,II}$ (their Equation 20):
\begin{equation}
\label{eq:dQdt}
\frac{dQ_\mathrm{H\,II}}{dt} = \frac{\dot{n}_{ion}}{\bar{n}_\mathrm{H}}-\frac{Q_\mathrm{H\,II}}{\bar{t}_{rec}}
\end{equation}
where $\dot{n}_{ion}$ is ionizing photon injection rate, $\bar{n}_\mathrm{H}$ is the mean density of H atoms in the universe, and $\bar{t}_{rec}$ is some characteristic recombination time taking the clumpiness of the IGM into account. For a constant clumping factor and comoving emissivity \cite{MadauEtAl1999} show that
\begin{equation}
Q_\mathrm{H\,II}(t) \approx \frac{\dot{n}_{ion}}{\bar{n}_\mathrm{H}} \bar{t}_{rec}
\end{equation}
Setting $Q=1$ one arrives at $\dot{n}_{ion}\bar{t}_{rec}=\bar{n}_\mathrm{H}$, the basis for deriving Equation \eqref{eq:ndot}. \cite{MadauEtAl1999} state that this relation should still be valid provided the clumping factor and comoving emissivity are slowly varying on a timescale of $\bar{t}_{rec}$. We utilize the differential form for our comparison because our emissivity is not a constant value, nor is it slowly varying on a recombination time as $Q \rightarrow 1$, as we show below.
A practical issue when testing Equation \eqref{eq:dQdt} is how $\bar{t}_{rec}$ should be evaluated when $Q<1$, and in particular when $Q\ll 1$. In the limit $Q\ll 1$ one is dealing with isolated H {\footnotesize II} regions evolving under the influence of local conditions. Yet the definition for $\bar{t}_{rec}$ in Equation \eqref{eq:tmadau} invokes {\em global} values for $C$ and $\langle n_\mathrm{H\,II} \rangle$. Should these quantitles be evaluated locally only within ionized regions? Or are global estimates good enough? In particular, since \cite{MadauEtAl1999}'s Equation (20) uses $\bar{n}_\mathrm{H}$ as a proxy for $\langle n_\mathrm{H\,II} \rangle$, what is the appropriate value for $C$ to use?
A second practical issue is what to take for $\dot{n}_{ion}$. This is commonly understood to be the rate at which ionizing photons are injected into the IGM (e.g., Haardt \& Madau 2012, \S9.3), which in our parlance is $\dot{N}_{IGM}$. Or should we take the actual ionization rate density measured in the simulation $\dot{N}_t$? As we saw in the previous section, these two rates diverge as overlap is approached, and differ by more than an order of magnitude after overlap (Fig. \ref{Ndot_Ratio}).
To examine these issues we plot in Figure \ref{Qeffv1} $Q(z)$ from our simulation, as well as theoretical curves obtained by integrating Equation \eqref{eq:dQdt} under various assumptions. The curve labelled $Q(sim)$ is the ionized volume fraction from our simulation that is at least 99.9\% ionized (Well Ionized). The other four curves are obtained by integrating Equation \eqref{eq:dQdt} setting $\dot{n}_{ion}=\dot{N}_t$ for various choices for $\bar{t}_{rec}$ (we investigate the $\dot{n}_{ion}=\dot{N}_{IGM}$ case at the end of this section.) The integral is approximated by summing a piecewise linear interpolation of the two terms on the RHS of Equation \eqref{eq:dQdt} using the trapezoidal rule:
\begin{align}
Q(t) &= \int_{t*}^{t} \frac{dQ}{dt}dt \approx \sum \frac{dQ}{dt} \Delta t \notag\\
&= \sum_{i}(\mathrm{Term_1} - \mathrm{Term_2})_i \Delta t_i
\label{eq:integration}
\end{align}
where $t*$ is the time when the first star forms in the simulation.
The curve labeled $Q(\langle t_{rec}\rangle)$ uses the volume averaged recombination time (volume average of Equation \ref{recombtime}). The two curves labeled $Q(t_\mathrm{Madau})$ use Equation \eqref{eq:tmadau} to evaluate $\bar{t}_{rec}$ for $C=2$ and $3$, substituting $\bar{n}_\mathrm{H}$ for $\langle n_\mathrm{H\,II} \rangle$ and assuming a constant T=10$^4$K for the IGM.
The curve labeled $Q(t_{rec,eff})$ uses the effective recombination time definition
\begin{equation}
\label{eq:treceff}
\bar{t}_{rec}=t_{rec,eff}\equiv \frac{\langle n_\mathrm{H\,II}\rangle}{\langle n_\mathrm{H\,II} n_e \alpha_B(T)\rangle}
\end{equation}
This particular definition makes the last line of Equation \eqref{eq:trecpart2} true trivially, with no assumption about the IGM temperature or ionization state of the hydrogen. It involves no {\em ad hoc} clumping factors, and represents the actual appropriately averaged recombination time in the simulation. All the above volume averaged quantities have the threshold of $\Delta_b<100$ applied, and thus exclude dense gas bound to halos.
Several of the curves derived from integrating $\frac{dQ}{dt}$ reach values above unity at the end of the overlapping phase. While it is physically impossible to have $Q>1$ it is not mathematically forbidden, and so we show the complete curves because they give us some insight about the relative contribution of the recombination term (Term$_2$) as compared to the ionization term (Term$_1$).
The $Q(\langle t_{rec}\rangle)$ curve ionizes the quickest, reaching $Q=1$ at $z\sim 6.5$, which is substantially before the simulation which achieves it at $z\approx 5.8$. The reason for this, as we will analyze shortly, is that recombinations play essentially no role in this model. The $Q(t_{rec,eff})$ curve has the same shape as the $Q(sim)$, but is everywhere higher, and crosses $Q=1$ at $z\sim 6.1$. Given that this integration uses the actual ionization rate density and effective recombination time in the simulation, this discrepancy demands an explanation. We address this below. Finally the $Q(t_\mathrm{Madau})$ curves do not match the shape of the $Q(sim)$ curve, ionizing more quickly at early times, and exhibiting a maximum value for $Q$ at $z \sim 6$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig23a-eps-converted-to.pdf}
\includegraphics[width=0.5\textwidth]{fig23b-eps-converted-to.pdf}
\caption{{\em Top}: Comparison of the evolution of the ionized volume fraction Q from our simulation with the analytic model introduced by \cite{MadauEtAl1999}. Q(sim) is calculated directly from counting the cells satisfying the Well Ionized threshold of $f_i>0.999$. The other curves are calculated from integrating Equation \eqref{eq:integration} with the different expressions for $\bar{t}_{rec}$ in Term$_2$, as described in the text. {\em Bottom}: Plot of Term$_1$ and Term$_2$ individually using the different expressions for $\bar{t}_{rec}$.}
\label{Qeffv1}
\end{figure}
To understand this behavior more fully we plot in Figure \ref{Qeffv1} {\em bottom} the values for Term$_1$ and Term$_2$ in Equation \eqref{eq:dQdt}.
The blue curve is Term$_1$ of Equation \eqref{eq:dQdt}. The other four curves plot Term$_2$ with their respective values for $\bar{t}_{rec}$. The ionization curve dominates all the recombination curves at high redshifts, and reaches a maximum at $z\sim 6.5$.
This is a partial reflection of the plateauing and subsequent decline of the SFRD shown in Figure \ref{SFR}. More fundamentally, it is a reflection of the rapid drop in the neutral fraction of the IGM as overlap is approached.
The curve using the volume averaged recombination time $\langle t_{rec}\rangle$ yields such low values compared to the others that we multiply it by 100 to make it more visible. Although this is not the relevant recombination time to use, since it weights low density regions, it is effectively the limiting case $\bar{t}_{rec}\rightarrow \infty$. We can therefore interpret the blue curve in Figure \ref{Qeffv1}a as an integration of the ionization term only. It is significantly higher than the $Q(sim)$ curve, suggesting that recombinations are important in the simulation at some level. The ionization term dominates the recombination term by factors of $6-10$ in the $t_{rec,eff}$ curve until just before overlap, and the two terms come into balance after overlap. The two $t_\mathrm{Madau}$ recombination curves are subdominant to the ionization term until $z \sim 6$, and at lower redshifts they become dominant. This explains the turnaround in the corresponding $Q$ curves in Figure \ref{Qeffv1}a.
The differences in the magnitude of the recombination curves in Figure \ref{Qeffv1}b, especially at higher redshifts, is directly attributable to the magnitude of $\bar{t}_{rec}$. For completeness we plot $\bar{t}_{rec}$ versus redshift in Figure \ref{treceffhubble}, both unnormalized and normalized by $t_\mathrm{Hubble}$. In addition to the three curves for $t_{rec,eff}$ and $t_\mathrm{Madau}$ for $C=2, 3$, we also plot $t_\mathrm{Madau}$ for $C=C_\mathrm{ttH\,II}$ and $C=C_\mathrm{tdm}$. We see that all the curves with the exception of the Madau formula curve using the thresholded dark matter clumping factor exhibit an increasing recombination time with decreasing redshift, in line with our expections. The latter curve shows the opposite trend, which is due to the fact that the dark matter clumping factor increases with decreasing redshift, even if it is thresholded to exclude halos (see Figure \ref{thresholded} bottom). Among the remaining curves the $t_{rec,eff}$ has the highest values, and increases more sharply than the $t_\mathrm{Madau}$ curves due to the temperature of the IGM. To demonstrate that, we plot one additional curve (dashed curve) for $t_{rec,eff}$ evaluated assuming a constant $T=10^4$K in the recombination rate coefficient.
We now comment on the often-made assumption in reionization models that $\bar{t}_{rec} \ll t$. \cite{MadauEtAl1999} make this assumption in order to derive Equation \eqref{eq:ndot}. It is this assumption that allows for an instantaneous analysis of the photon budget to maintain the universe in an ionized state while ignoring history dependent effects.
Referring to Figure \ref{treceffhubble}b we see this is never true for $t_{rec,eff}$ and it is not true for $t_\mathrm{Madau}$ at redshifts approaching overlap for any sensible value of $C$. We therefore conclude that history-dependent effects cannot be ignored, and that this is the reason Equations \eqref{eq:ndot}, \eqref{eq:updatedNdot} and \eqref{eq:ShullNdot} mis-predict the epoch of reionization completion. For the same reason applying these formulae at lower redshifts is highly suspect.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig24a-eps-converted-to.pdf}
\includegraphics[width=0.5\textwidth]{fig24b-eps-converted-to.pdf}
\caption{{\em Top}: Recombination time versus redshift, for various expressions for $\bar{t}_{rec}$ as described in the text. Curve labeled $t_{rec,eff}$ is the characteristic recombination time measured directly in the simulation. Curves labeled $t_{Madau}$ evaluate Eq. \eqref{eq:tmadau} for various choices for the clumping factor C. {\em Bottom}: Recombination time versus redshift normalized by the Hubble time, for various expressions for $\bar{t}_{rec}$.}
\label{treceffhubble}
\end{figure}
Returning to the discrepancy between the $Q(sim)$ and $Q(t_{rec,eff})$ curves in Figure \ref{Qeffv1}a, since the most sensible choice for $t_{rec}$ did not give us satisfactory agreement, we wondered what the origin of the discrepancy could be. Since we have shown that recombinations are relatively unimportant at high redshifts, but that the discrepancy is already present at high redshifts, the only possibility is that there is something wrong with the first term of Equation \eqref{eq:integration}. When looking at the derivation for Equation \eqref{eq:dQdt} in \cite{MadauEtAl1999}, it is stated that it ``approximately holds for every isolated source of ionizing photon in the IGM.'' That got us to think that our calculation of $\bar{n}_\mathrm{H}$ may be off from what is originally intended if it is a global average over the entire simulation box. Since the original $\frac{dQ}{dt}$ is derived from the analytical Str\"{o}mgren sphere model, it assumed a single ionizing source at the center of the volume, and the the average density of the box is just the uniform density everywhere, we thought that might be the discrepancy. In an Inside-out model, I-fronts are not initially propagating in a gas with an average density given by $\bar{n}_H$, but somewhat higher density. Would agreement improve if instead of using $\bar{n}_H$ in the first term of Equation \eqref{eq:dQdt}, we used the local average density?
We therefore modify Equation \eqref{eq:dQdt} as follows:
\begin{equation}
\frac{dQ}{dt} = \frac{\dot{n}_{ion}}{\delta_b\bar{n}_\mathrm{H}}-\frac{Q}{\bar{t}_{rec}}
\label{eq:dQdtdb}
\end{equation}
where we have introduced in the denominator of the first term a factor $\delta_b \geq 1$ which corrects for the higher mean density within ionized bubbles. We measure $\delta_b$ from each redshift output as follows: $\delta_b = \langle \rho_b\rangle_{tt}/\langle \rho_b\rangle_{t}$. The volume average $\langle\rangle$ with subscript $t$ is the usual $\Delta_b<100$ threshold, the double subscript $tt$ indicates the additional threshold of $x_e>0.1$. Thus $\delta_b$ is the average baryon overdensity within Ionized regions excluding gas inside halos. Figure \ref{deltabvsQfit5} shows a plot of $\delta_b$ versus $Q$ together with a simple fitting formula which fits the data extremely well over the domain $0.01 \leq Q \leq 1$.
To see if this formulation improves agreement with our simulated data, in Figure \ref{Qeffv2} we integrate Equation \eqref{eq:dQdtdb} again setting $\dot{n}_{ion}=\dot{N}_t$ and using $t_{rec,eff}$ to evaluate the second term. For comparison we show the curve obtained setting $\delta_b=1$, which repeats a curve already presented in Figure \ref{Qeffv1}. Although the simulated and integrated analytic model curves do not agree exactly, the $Q(\delta_b,t_{rec,eff})$ curve shows much better agreement with the simulation, with error on the order of 1\% instead of 10\%.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig25-eps-converted-to.pdf}
\caption{Improved agreement between theory and simulation. Green and blue curves are as in Fig. \ref{Qeffv1}. Red curve is obtained by integrating modified evolution equation for Q taking into account the overdensity effect of Inside-out reionization (Equation \eqref{eq:dQdtdb}).}
\label{Qeffv2}
\end{figure}
By not assuming a constant emissivity and using the modified differential form in determining the volume filling fraction of Equation \eqref{eq:dQdtdb}, we are able to more accurately model the evolution of the simulated volume filling fraction of H {\footnotesize II} to the Well Ionized level. For completeness we plot in Figure \ref{treceffvszfit} the evolution of $t_{rec,eff}$ used in the above integration, including a reasonably good fit to the data.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig26-eps-converted-to.pdf}
\caption{Mean baryon overdensity of ionized gas as a function of the ionized volume filling fraction Q. Blue points are measured in the simulation by averaging over the doubly thresholded cells obeying $\Delta_b<100$ and $x_e > 0.1$. Red curve is a fit to the data.}
\label{deltabvsQfit5}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig27-eps-converted-to.pdf}
\caption{Analytic fit to $t_{rec,eff}$ (red line) , evaluated using simulation data (blue points) via Equation \eqref{eq:treceff}.}
\label{treceffvszfit}
\end{figure}
Finally, we return to the question of what is the appropriate choice for $\dot{n}_{ion}$ in Equation \eqref{eq:dQdtdb}. This is commonly taken to be the rate at which ionizing photons are injected into the IGM (e.g., Haardt \& Madau 2012, \S9.3), because this can be connected to the observed UV luminosity density $\rho_{UV}$ by the formula $\dot{n}_{ion}=f_{esc}\xi_{ion}\rho_{UV}$, where $f_{esc}$ is the escape fraction for ionizing radiation, and $\xi_{ion}$ is the rate of ionizing photons per unit UV (1500 \AA{}) luminosity for the stellar population \citep{RobertsonEtAl2013}. However we have obtained excellent agreement between simulation and Equation \eqref{eq:dQdtdb} using the mean ionization rate density in the IGM $\dot{N}_t$, which differs from the ionizing photon injection rate density $\dot{N}_{IGM}$ as $Q \rightarrow 1$. In Figure \ref{Qeffv3} we show the result of integrating Equations \eqref{eq:dQdt} and \eqref{eq:dQdtdb} with the choice $\dot{n}_{ion}=\dot{N}_{IGM}$, as originally proposed by \citep{MadauEtAl1999}. Also plotted in Figure \ref{Qeffv3} is $Q(sim)$ (blue line) and our best agreeing model (green line). The red line ignores the $\delta_b$ correction, and deviates to the high side of $Q(sim)$ almost immediately, for reasons we discussed earlier. It crosses $Q=1$ at $z\approx 6.6$, which is too early by $\Delta z =0.8$. The teal line includes the $\delta_b$ correction, and tracks the $Q(sim)$ closely to $z \approx 7$, and thereafter deviates on the high side. It crosses $Q=1$ at $z\approx 6.4$, which is too early by $\Delta z =0.6$. Both curves show an accelerated change in $Q$ as z decreases, which is characteristic of standard analytic ionization models (e.g., Haardt \& Madau 2012, Fig.14). By contrast, the simulation and our best fit model using $\dot{n}_{ion}=\dot{N}_t$ show a decelerated change in $Q(z)$ as $Q \rightarrow 1$. This is clearly due to the fact that the ration of ionizations to emitted photons decreases as $Q \rightarrow 1$, as illustrated in Figure 22. The consequence of this flattening in the $Q(z)$ curve is a delay in redshift of overlap of $\Delta z=0.6-0.8$, relative to the predictions of Equations \eqref{eq:dQdtdb} and \eqref{eq:dQdt}, respectively, using the photon injection rate as the source term.
We have seen above that the ionization rate density is the appropriate quantity to use to source the $dQ/dt$ equation, independent of $\delta_b$ corrections. Because the ionization rate density is not directly observable, but since $\dot{n}_{ion}$ can be derived from observables, we introduce a correction factor to convert from one to the other. Defining
\begin{equation}
\gamma \equiv \frac{\langle n_{HI}\Gamma_{HI}^{ph}\rangle}{\dot{n}_{ion}} = \frac{\dot{N}_t}{\dot{n}_{ion}}
\label{gamma}
\end{equation}
\\where the angle brackets denote an average over the singly thresholded volume (IGM), then we can recast Equation \eqref{eq:dQdtdb} into a form useful for observers:
\begin{equation}
\frac{dQ}{dt} = \frac{\gamma\dot{n}_{ion}}{\delta_b\bar{n}_\mathrm{H}}-\frac{Q}{\bar{t}_{rec}},
\label{eq:dQdtdbg}
\end{equation}
\\where $\gamma$ and $\delta_b$ are functions of $Q$.
In Fig. \ref{RatiovsQfit} we plot data values for $\gamma(Q)$ taken from our simulation, as well as a simple powerlaw fit. The fit is not meant to be definitive, but merely illustrative. More simulations need to be performed under various circumstances, and better fits made, to see whether our $\gamma(Q)$ is approximately universal, or merely anecdotal.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig28-eps-converted-to.pdf}
\caption{Dependence of analytic models on the choice for $\dot{n}_{ion}$. Red and teal curves assume $\dot{n}_{ion}=\dot{N}_{IGM}$; i.e., the photon injection rate into the IGM. Green curve assumes $\dot{n}_{ion}=\dot{N}_{t}$; i.e., the measured photoionization rate in the IGM. Blue curve is $Q(sim)$--the measured ionized volume filling fraction in the simulation. The green and teal curves take into account the overdensity effect of inside-out reionization (Equation \eqref{eq:dQdtdbg}), while the red curve assumes $\delta_b=1$. All models assume $\bar{t}_{rec}=t_{rec,eff}$ as measured in the simulation (Fig. \ref{treceffvszfit}.}
\label{Qeffv3}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig29-eps-converted-to.pdf}
\caption{Ratio of the volume averaged H {\footnotesize I}~ photoionization rate to photon injection rate in the IGM as a function of $Q$. Data points are measured from the simulation; line is a simple powerlaw fit. }
\label{RatiovsQfit}
\end{figure}
\section{General Results}
\label{GeneralResults}
\begin{figure*}[!tp]
\begin{minipage}[h]{0.5\linewidth}
\centering
\includegraphics[trim = 15mm 5mm 0mm 15mm, clip, width=1.0\textwidth]{fig1a-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.5\linewidth}
\centering
\includegraphics[trim = 15mm 5mm 0mm 15mm, clip, width=1.0\textwidth]{fig1b-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}[h]{0.5\linewidth}
\centering
\includegraphics[trim = 15mm 5mm 0mm 15mm, clip, width=1.0\textwidth]{fig1c-eps-converted-to.pdf}
\end{minipage}
\hspace*{-4.00mm}
\begin{minipage}[h]{0.5\linewidth}
\centering
\includegraphics[trim = 15mm 5mm 0mm 15mm, clip, width=1.0\textwidth]{fig1d-eps-converted-to.pdf}
\end{minipage}
\caption{H {\footnotesize I} density on slices through the 20 Mpc volume showing the growth,
percolation, and final overlap of H II regions. Panels show $z=9.18, 8.0,
7.0, 6.1$. The box becomes fully ionized at $z=5.8$ as the last neutral islands
are overrun by the I-fronts. Regions of extremely low H {\footnotesize I} density are shock-heated
bubbles due to supernova feedback.}
\label{HI_slices}
\end{figure*}
Here we first present the basic properties of the simulation before delving into specific topics in subsequent sections. The star formation and feedback parameters for this simulation are $f_* =0.1, f_{m*}=0.25, \epsilon_{SN}=10^{-5}, \epsilon_{UV}=1.38 \times 10^{-4}$. Figure \ref{HI_slices} shows the reionization process as it proceeds through growth, percolation, and final overlap of ionized hydrogen (H {\footnotesize II}) regions driven by ionizing radiation from star forming galaxies. We plot the neutral hydrogen (H {\footnotesize I}) density on a slice through the densest cell in the volume at redshifts $z=9.18, 8.0, 7.0, 6.1$. At $z=9.18$ several isolated quasi-spherical I-fronts are intersected by the slice plane. These grow and have begun to merge by $z=8.0$. By $z=7.0$ the toplogy is beginning to invert, in that there are now isolated peninsula of H {\footnotesize I} gas embedded in an otherwise ionized IGM. By $z=6.1$ the remaining neutral island has almost disappeared as it is being irradiated from all sides. We can also see in the figure small patches of extremely low H {\footnotesize I} density; these correspond to bubbles of shock heated gas near galaxies heated to above $10^6$K in temperature by supernova feedback.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig2-eps-converted-to.pdf}
\caption{Evolution of the ionized volume fraction versus redshift for hydrogen ionized to less than 1 neutral in 10$^3$ atoms. As redshift decreases, the volume filling fraction grows rapidly until around redshift of 6, at which time the rate of growth slows significantly as the last neutral island is ionized . The sensitivity of this curve to ionization level is discussed in \S\ref{QuantitativeLanguage}.}
\label{Ion1E3}
\end{figure}
Figure \ref{Ion1E3} plots the evolution of the ionized volume fraction $Q_\mathrm{H\,II}$ versus redshift. Here a cell is called ionized if $\rho_\mathrm{H\,II}/\rho_\mathrm{H} \geq 0.999$ (In \S\ref{QuantitativeLanguage} we discuss the sensitivity of this curve to level of ionization.) The first ionizing sources turn on at $z \sim 10$ in this simulation. The ionized volume fraction rises rapidly, reaching 0.5 at $z \approx 6.8$, 0.95 at $z \approx 6.0$, and near unity at $z \approx 5.8$. We compare this evolution with the predictions of the simple analytic model introduced by \cite{MadauEtAl1999} in \S\ref{Qdot}. For now we only draw attention to the flattening of the curve in the redshift interval $5.8 \leq z \leq 6$. This is the signature of neutral islands being ionized by I-fronts converging in 3D, as opposed to being ionized by internal sources.
Our simulation was not designed to complete reionization by a certain fiducial redshift. Rather we adjusted our star formation efficiency parameter $f_*$ so that we can approximately match the star formation rate density (SFRD) in \citep{BouwensEtAl2011}. Our SFRD is shown in Figure \ref{SFR}, along with the Bouwens data, plotted without error bars. For reference we also include the fitting function described in \citep{HaardtMadau2012}. This shows that our simulated universe is one that produces approximately the same amount of stars in a given comoving volume, albeit a bit low relative to the data. We also note that the SFRD begins to flatten out at $z \approx 6.5$, and even turns over after overlap at $z \approx 5.8$, rather than continue to rise as indicated by the data points. This is an artifact of the small box size as a simulation completed in a 80 Mpc comoving on a side box with identical physics, mass, and spatial resolution and star formation/feedback parameters does not show this slowing down of the SFRD. This will be reported on in a future paper.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig3-eps-converted-to.pdf}
\caption{A comparison of simulated and observed star formation rate densities (SFRD) in units of M$_\odot$yr$^{-1}$Mpc$^{-3}$ comoving. Blue curve labeled ``This Work'' is from our 20 Mpc / $800^3$ simulation, and ``Bouwens et al 2011'' are observationally derived data points from \cite{BouwensEtAl2011b} plotted without error bars. The leveling off of the simulated SFRD is an artifact of the small volume as a simulation carried out with identical physics, mass, and spatial resolution but in 64 times the volume does not show this effect.}
\label{SFR}
\end{figure}
To check and make sure that our simulation is giving us a fair representation of the universe, we plot several more quantities and look for any anomalies. In Figure \ref{HMF}, we see that our halo mass function at redshift of $z \sim 6$ matches well with the Warren fit implemented in \texttt{yt} \citep{WarrenEtAl2006,TurkEtAl2011}. The mass function captures haloes down to $\sim$10$^8$M$_\odot$, which as previously stated was a simulation design criterion. The haloes are found by first running the parallelHOP halo finder installed in \texttt{yt} \citep{SkoryEtAl2010}, then taking the linked list of dark matter particles for each halo and wrapping the region around them in an ellipsoidal 3D container introduced in \texttt{yt} 2.4. The 3D container enables the query of the fluid quantities of the haloes, such as baryonic, emissivity, radiation contents in addition to the particle information. Since the dark matter particles used are $\sim 5 \times$ 10$^5$M$_\odot$, the 10$^8$M$_\odot$ dark matter haloes are considered to be resolved \citep{TrentiEtAl2010}.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig4-eps-converted-to.pdf}
\caption{The dark matter halo mass function from our simulation (blue line). Green line is the fit from \citep{WarrenEtAl2006}. Our low-mass HMF is reasonably complete down to $M_{halo} \approx 10^8 M_{\odot}$; i.e. halos believed to form stars efficiently due to atomic line cooling. Incompleteness at the high mass end is due to the limited volume sampled.}
\label{HMF}
\end{figure}
As a final check that our ionizing source population is not wildly unrepresentative of the observed universe, in Figure \ref{scaledLF} we plot the luminosity function of our simulated galaxies at $z=6.1$ along side the observational data points from Table 5 of \citep{BouwensEtAl2007}. The points in red are the bolometric luminosities for our galaxy population calculated directly from the $z=6.1$ halo catalogue. To calculate the luminosity of a given halo we sum the emissivity field within the 3D ellipsoidal containers defined by the halos' dark matter particles. Our error bars are taken using one standard deviation of luminosity in the mass bins. Although this is not proof that our simulation is matching observations exactly, it does lend support that our realization of reionization is being driven by sources not too dissimilar to those observed and is sufficient for the purposes of this study.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig5-eps-converted-to.pdf}
\caption{Bolometric luminosity function derived from our simulation data (red), compared with observational data points (blue) from \citep{BouwensEtAl2007}.}
\label{scaledLF}
\end{figure}
\subsection{Quantitative Language}
\label{QuantitativeLanguage}
Earlier works on reionization such as \cite{ValageasSilk1999,Gnedin2000,MiraldaEscudeEtAl2000,IlievEtAl2006} speak of a two phase medium composed of completely neutral and completely ionized hydrogen gas, while more recent works \citep{CiardiEtAl2003,ZahnEtAl2007,ShinEtAl2008,PetkovaSpringel2011a,FinlatorEtAl2012} begin to consider the {\em degree of ionization} within ionized gas. The simplification of considering a two phase medium helps reduce the simulation complexity and the language needed to describe the results. However, as simulations become more sophisticated, the two phase paradigm becomes ill-suited to convey the wealth of information contained in the larger and more detailed simulations.
As people begin to describe the new simulations, the old paradigm lingers and causes ambiguities. As a case in point, consider the ionized volume filling fraction versus redshift, one of the simplest quantitative metrics of any reionization simulation. Within the framework of a two-phase medium, this is uniquely defined at any redshift. For a simulation such as ours which tracks the ionization state in every cell, the volume filling fraction depends on the degree of ionization, as illustrated in Figure \ref{linearIonized}.
This figure shows the evolution of the volume filling fraction of ionized gas which exceeds a minimum local ionization fraction $f_i \equiv \rho_\mathrm{H\,II}/\rho_\mathrm{H}$. The three thresholds are $f_i=$ 0.1, 0.999, and 0.99999 and are labelled 10\%, 1E3, 1E5, respectively in Figure \ref{linearIonized}.
We choose three specific levels not because we think they are more important than others, but because it suits our later narrative and gives a range values. With the ionization state tracked by the simulation, we see that it is now ambiguous to ask at what redshift 50\% of the volume is ionized. In our simulation this occurs at $z \approx$ 7, 6.8 and 6.5 for $f_i$=0.1, 0.999, and 0.99999, respectively.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig6a-eps-converted-to.pdf}
\includegraphics[width=0.5\textwidth]{fig6b-eps-converted-to.pdf}
\caption{Volume filling fraction of ionized gas versus redshift for three ionized fraction thresholds. {\em Top} linscale; {\em Bottom} logscale. The three ionization levels are ``10\%'' in blue: fractional volume that have more than 1 ionized hydrogen atom per 10 hydrogen atoms. ``1E3'' in green: fractional volume that have less than 1 neutral hydrogen atom per 10$^3$ hydrogen atoms. ``1E5'' in red: fractional volume that have less than 1neutral hydrogen atom per 10$^5$ hydrogen atoms.}
\label{linearIonized}
\end{figure}
In the rest of this paper we will often report results as a function of these three ionization fraction thresholds.
To make the text easier to read we will use the terms ``Ionized'' to designate $f_i$=0.1, ``Well Ionized'' to designate $f_i$=0.999, and ``Fully Ionized'' to designate $f_i$=0.99999 ionization levels.
\subsection{Inside-out or Outside-in}
\label{IOOI}
Besides specifying the amount of ionized volume and levels of ionization, another area where quantitative language is useful is in the description of the reionization history. Since the Outside-in model was proposed by \cite{MiraldaEscudeEtAl2000}, there is gathering support for the opposing view of the Inside-out model by \cite{SokasianEtAl2003,FurlanettoEtAl2004,IlievEtAl2006} to name a few. In \cite{FinlatorEtAl2009}, the authors go even further and add to the lexicon ``Inside-outside-middle'', trying to describe the rich detail in a reionization scenario. The basic Inside-out picture is that galaxies form in the peaks of the dark matter density field and drive expanding H {\footnotesize II} regions into their surroundings ({\em expansion phase}).
These H {\footnotesize II} regions are initially isolated, but begin to merge into larger, Mpc-scale H {\footnotesize II} regions due to the clustering of the galaxy distribution ({\em percolation phase}). Driven by a steadily increasing global star formation rate and recombination time (due to cosmic expansion) this process goes on until H {\footnotesize II} regions completely fill the volume ({\em overlap phase}). In this picture, rare peaks in the density field ionize first while regions of lower density ionize later from local sources that themselves formed later.
To investigate how reionization progresses in regions of different density, we plot in Figure \ref{NeutralPhase} the hydrogen neutral fraction ($\rho_\mathrm{H\,I}/\rho_\mathrm{H}$) versus overdensity $\Delta_b\equiv\rho_b/\langle\rho_b\rangle$ in the left column, and in the right column a slice of the gas temperature, with redshift decreasing from top to bottom. One would expect if inside-out ionization is the case, that the neutral fraction of higher density region should drop down more quickly than lower density regions. Below, we will describe each row of the figure in more detail.
Looking at the redshift $z=10$ row, we see in the gas temperature slice that two isolated regions of ionization appear due to UV feedback from new stars, indicated by the $\sim$10$^4$K gas . These regions correspond to places on the neutral fraction vs. overdensity phase plot where a small amount of volume emerges around $\Delta_b$ of $10^{-1}-10^1$, reaching Well Ionized to Fully Ionized levels. The T $\sim$10$^7$K region corresponds to the extended tail of very low neutral fraction gas in the left column, and indicates gas shock heated by supernova feedback. Although the cell count of shock heated gas will grow, it remains orders of magnitude smaller compared to the photoionized regions that we will emphasize. Even at this early stage, there are high density regions above $\Delta_b$ of 10$^2$-10$^3$ that are Well Ionized; this is due to their close proximity to the ionizing sources, supporting the Inside-out paradigm.
Looking at the next row of figures at a redshift of $z=7$, we see that the volume of Well Ionized regions has increased greatly, and so has the shock heated region in the phase plot. We also see that most, but not all the $\Delta_b > 10^2$ cells have reached the Well Ionized level. Although a large portion of the volume is in the Well Ionized regime, the majority of the volume (the red pixels) is still neutral, as we can see in the corresponding temperature slice plot. Most of the volume is still well under 10$^4$K, where we expect the temperature to hover around once the ionization front has passed through the region and the gas has had time to come into photoionization thermal eqilibrium.
By a redshift of $z=6.1$, we see from the left column that the region that is ionized beyond the Fully Ionized level (an irony in terms, which means there is definitely room for improvement in the naming convention), dominates the simulation volume. There are still some regions not yet consumed by the ionization front, that is seen on the top of the neutral fraction plot and on the right according to the temperature slice.
The next row at redshift of $z=5.5$ is after the entire volume has been swept over by ionization fronts. Most of the volume is beyond the Well Ionized level, except for a few cells around $\Delta_b \sim 10^2$. There are also some cells that are still neutral around $\Delta_b \sim 10^4$. They remain neutral because their densities are so high, leading to high recombination rates. Over time these cells will shift up and down the neutral fraction plot with waves of star formation and supernova explosions since they are likely close to the source of the radiation and kinetic energy.
The last row of Figure \ref{NeutralPhase} is at redshift $z\sim5$, where we can see that the previous few cells that have yet to reach Well Ionized levels around $\Delta_b \sim 10^2-10^3$ have now disappeared. The cells that have not reached Well Ionized level before are cells where either the radiation is not strong enough due to shielding effects or the density is so high the gas recombines quickly even after being ionized. After the ionization front has passed though and highly ionized the IGM, there is little material left to shield against the radiation background and we see all but the densest few cells become Well Ionized. The high density region reaching the same ionization level after the under dense void, would fit well with the description for the Outside-in model. Note, that the remaining cells that finally reached Well Ionized levels, are orders of magnitude smaller in total volume compare to the rest of the cells at the same density. So if we call cells of $\Delta_b \sim 10^2$ filaments, not all dense filaments get Well Ionized until late in the EoR. Before the volume is filled with radiation, these dense filaments are able to remain relatively neutral.
Unfortunately, the evolution of these redshift panels is not enough to capture the propagation of radiation fronts from the initial sources, but they do convey the overall ionization history of the universe. The panels suggest that the region surrounding the ionization sources, whether they are dense cores, filaments, or voids, are all affected by the radiation on roughly the same time scale. However, the degree to which they are ionized is different. It is this difference, that is the key to answering the original question, whether the universe ionize inside-out or outside-in.
When focusing on the ionization of the IGM, lets for a moment neglect the $\Delta_b \sim10^4$ cells that shift ionization level with waves of star formation which comprise a tiny fraction of the volume. If we use the ``Ionized'' level to characterize something as completely ionized and draw the line for neutral fraction at 10\%, then the universe reaches end of EoR before $z \sim 5.5$. Since radiation propagates from sources outward, that would correspond to the Inside-out picture. If we were to instead draw the completely ionized line at ``Well Ionized'' level, then we can see that even at $z \sim 5.5$, there is a small peak in the dense region of the phase diagram ($\Delta_b \sim 2.4\times10^2$) that has yet to reach below the line to be considered completely ionized. This would correspond to the Outside-in picture which reaches end of EoR sometime before $z \sim 5$ (or Inside-outside-middle if one uses the \cite{FinlatorEtAl2009} terminology and considers the neutral peak to be a part of the filaments). And finally, if we were to draw the line at the ``Fully Ionized'' level, the universe has yet to ionize even for regions that are only 10$\times$ over dense. Thus the ionization history is a story with many perspectives, and it really depends on how the story teller draws the line as to whether Inside-out, Outside-in, or Inside-outside-middle is a better qualitative description.
\begin{figure*}[!tp]
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 7mm 9mm 1mm 7mm, clip, width=1.0\textwidth]{fig7a-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7b-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7c-eps-converted-to.pdf}
\end{minipage}fi
\vspace*{-2.00mm}\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 7mm 9mm 1mm 7mm, clip, width=1.0\textwidth]{fig7d-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7e-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7f-eps-converted-to.pdf}
\end{minipage}
\vspace*{-2.00mm}\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 7mm 9mm 1mm 7mm, clip, width=1.0\textwidth]{fig7g-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7h-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7i-eps-converted-to.pdf}
\end{minipage}
\vspace*{-2.00mm}\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 7mm 9mm 1mm 7mm, clip, width=1.0\textwidth]{fig7j-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7k-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7l-eps-converted-to.pdf}
\end{minipage}
\vspace*{-2.00mm}\\
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 7mm 9mm 1mm 7mm, clip, width=1.0\textwidth]{fig7m-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7n-eps-converted-to.pdf}
\end{minipage}
\hspace*{-2.00mm}
\begin{minipage}[h]{0.33\linewidth}
\centering
\includegraphics[trim = 10mm 0mm 7mm 7mm, clip, width=1.0\textwidth]{fig7o-eps-converted-to.pdf}
\end{minipage}
\caption{{\em Left}: Phase diagram of neutral hydrogen fraction versus baryon overdensity with decreasing redshift from top to bottom. {\em Middle}: Slices of Log Temperature [K] through a region that remained mostly neutral until just before overlap at redshift of $\sim$5.8. {\em Right}: Slices of neutral hydrogen fraction through the same region as before. Please refer to \S\ref{IOOI} for detailed description.}
\label{NeutralPhase}
\end{figure*}
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\newcommand{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}
\newcommand{Current address: Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'{e} de Toulouse, CNRS, UPS, Toulouse, France}{Current address: Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'{e} de Toulouse, CNRS, UPS, Toulouse, France}
\begin{document}
\title{Excited states, symmetry breaking, and unphysical solutions on the CASSCF energy landscape}
\title{Excited states, symmetry breaking, and unphysical solutions in state-specific CASSCF theory}
\date{\today}
\author{Antoine \surname{Marie}}
\affiliation{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}
\affiliation{Current address: Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'{e} de Toulouse, CNRS, UPS, Toulouse, France}
\author{Hugh G.~A.~\surname{Burton}}
\email{hgaburton@gmail.com}
\affiliation{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}
\begin{abstract}
\begin{wrapfigure}[10]{r}{0.38\textwidth}
\flushleft
\vspace{-0.4cm}
\hspace{-1.7cm}
\fbox{\includegraphics[width=0.38\textwidth]{toc.png}}
\end{wrapfigure}
State-specific electronic structure theory provides a route towards balanced excited-state
wave functions by exploiting higher-energy stationary points of the electronic energy.
Multiconfigurational wave function approximations can describe both
closed- and open-shell excited states and avoid the issues associated with state-averaged approaches.
We investigate the existence of higher-energy solutions in complete active space
self-consistent field (CASSCF) theory and characterise their topological properties.
We demonstrate that state-specific approximations can provide accurate higher-energy excited states
in \ce{H2} (6-31G) with more compact active spaces than would be required in a state-averaged formalism.
We then elucidate the unphysical stationary points, demonstrating that they arise from
redundant orbitals when the active space is too large, or symmetry breaking when the active space is
too small.
Furthermore, we investigate the conical intersection in \ce{CH2} (6-31G) and the avoided crossing
in \ce{LiF} (6-31G), revealing the severity of root flipping and demonstrating that state-specific
solutions can behave quasi-diabatically or adiabatically.
These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages
and challenges of practical state-specific calculations.
\end{abstract}
\maketitle
\linepenalty1000
\raggedbottom
\section{Introduction}
\label{sec:intro}
Electronic excited states are fundamentally higher-energy solutions to the time-independent
Schr\"{o}dinger equation.
``State-specific'' representations can be identified using
higher-energy stationary points of the electronic energy landscape.\cite{Burton2022a}
The exact excited states in full configuration interaction (FCI) correspond to energy
saddle points and the number of downhill Hessian eigenvalues increases with each energy
level.\cite{Olsen1982,Golab1983,Olsen1983,Golab1985,Burton2022a,Bacalis2016,Bacalis2020}
Higher-energy stationary points also exist in non-linear wave function approximations,
but the development of practical state-specific methods has been hindered by the
challenges of non-ground-state optimisation, the non-linearity of the electronic energy
landscape, and the presence of unphysical solutions.
Instead, the workhorse of modern excited-state electronic struture theory is linear-response time-dependent
density functional theory (LR-TDDFT), which predicts excitation energies from the response
of the ground-state electron density to a weak external perturbation.\cite{Runge1984,Dreuw2005,Burke2005}
Despite its computational efficiency, LR-TDDFT inherits the failures of approximate Kohn-Sham (KS)
exchange-correlation functionals, creating large errors for bond dissociation or open-shell electronic states.\cite{Hait2019}
Furthermore, the ubiquitous adiabatic approximation excludes double excitations and their associated
avoided crossings.\cite{Burke2005,Maitra2004}
Alternative single-reference methods, such as algebraic diagrammatic construction\cite{Schirmer1982,Dreuw2015}
(ADC) and equation-of-motion coupled cluster\cite{Stanton1993,Krylov2008} (EOM-CC)
can provide more accurate excitation energies at a greater
computational cost, but depend strongly on the quality of the reference determinant.
The strong influence of the ground-state orbitals can also create an unbalanced description of charge transfer
and Rydberg excitations,\cite{Tozer2003,Dreuw2004}
where significant electronic relaxation can occur.\cite{McLachlan1964,Runge1984,Dreuw2005,Burke2005,Bartlett2012,HelmichParis2019}
These challenges have encouraged
researchers to revisit excited state-specific approximations.
For higher-energy SCF calculations ($\Delta$SCF),
this progress has been catalysed by the development of new optimisation algorithms
that avoid variational collapse to the ground state, including the maximum overlap method,\cite{Gilbert2008,Barca2014,Barca2018}
square-gradient optimisation,\cite{Hait2021,Hait2020} state-targeted energy projection,\cite{CarterFenk2020}
quasi-Newton direct orbital optimisation,\cite{Levi2020,Levi2020a,Ivanov2021}
and generalised variational principles.\cite{Shea2017}
Recent calculations have shown that higher-energy Hartree--Fock (HF) and KS-DFT solutions
can accurately describe charge transfer and double excitations at a low computational cost.\cite{Gilbert2008,Hait2020}
Beyond SCF approximations, higher-energy variational or projective coupled-cluster ($\Delta$CC) solutions
can provide more accurate double and double-core excitations by incorporating dynamic electron
correlation.\cite{Jankowski1994,Jankowski1994a,PiecuchBook,Mayhall2010,Lee2019,Kossoski2021,Marie2021a}
While $\Delta$SCF and $\Delta$CC are successful for double and charge transfer excitations,
these single-reference methods cannot describe open-shell excited states and statically correlated ground states.
The onset of this failure usually becomes apparent through spin contamination,\cite{Burton2021a,Burton2018}
spontaneous symmetry breaking,\cite{Coulson1949,Fukutome1973,Fukutome1974,Fukutome1974a,Burton2021a,Barca2014,Hait2019,Ye2017}
and additional unphysical
solutions.\cite{Burton2021a,Jankowski1994,Jankowski1994a,PiecuchBook,Mayhall2010,Kossoski2021,Marie2021a}
Furthermore, the solutions of interest can disappear as the molecular structure changes,
creating discontinuous excited-state energy surfaces or
gradients.\cite{Thom2008,Marie2021a,Burton2021a,Burton2020,Jensen2018,Vaucher2017,Dong2020}
Multiconfigurational SCF (MCSCF) methods,\cite{SzaboBook} particularly the
complete-active space self-consistent field (CASSCF) formulation,\cite{Das1966,Roos1980a,Roos1980b}
are the state-of-the-art for describing statically correlated electronic systems.\cite{RoosBook}
The CASSCF wave function is a linear expansion of all the configurations
that can be constructed from a set of partially occupied ``active orbitals'',
and the energy is optimised with respect to the configuration interaction (CI) and orbital coefficients simultaneously.\cite{Roos1980a}
It has long been known that higher-energy MCSCF solutions can represent electronic
excited states,\cite{Das1973,Krauss1976,Bauschlicher1978,Bauschlicher1980a,Bauschlicher1980b,Bauschlicher1980c}
and that multiple symmetry-broken CASSCF solutions can occur for an inadequate active space.\cite{Guihery1997,Angeli2003}
More recently, MCSCF expansions truncated to single excitations have shown
promise for singly excited charge transfer states,\cite{Shea2018,Shea2020,Zhao2020a,Zhao2020b,Hardikar2020}
while state-specific configuration interaction with higher degrees of truncation can
handle challenging multireference problems, singly-, and doubly-excited states.\cite{Kossoski2022a}
However, the strong coupling between the orbital and CI degrees of freedom makes the optimisation
challenging, and second-order optimisation algorithms are generally required to reach
convergence in practice.%
\cite{Dalgaard1978,Dalgaard1979,Yeager1979,Lengsfield1980,Seigbahn1981,Werner1981,Werner1985,Yeager1980a,Yeager1980b,Jorgensen1981,Yeager1982,Sun2017,Kreplin2019,Kreplin2020}
Extensive research in the 1980s focused on characterising higher-energy MCSCF solutions.
It was originally suggested that an $n$th excited state approximation should be the
$n$th state in the configuration expansion.\cite{Lengsfield1980}
However, this requirement is often not achieved, resulting in ``root flipping''.\cite{Das1973,Werner1981,Olsen1982}
Furthermore, several stationary points satisfying this condition can often be identified.\cite{Golab1983,Golab1985,Rizzo1990}
The enormous complexity of the multiconfigurational solution space led Golab \textit{et al.}~{}\ to conclude that
\textit{``selecting an MCSCF stationary point is a very severe problem.''}\cite{Golab1983}
Instead, the state-averaged (SA) approach is generally used, where a weighted average energy of the
$n$ lowest CI states constructed from one set of orbitals is optimised.\cite{Werner1981}
While this approach has become the method of choice for excited-state CASSCF, it has several disadvantages:
discontinuities can occur on the SA-CASSCF potential energy surface if two states
require orbitals with significantly different character;\cite{Zaitsevskii1994}
the number of states is limited by the size of the active space; large active spaces
are required to target high-lying states; and the Helmann-Feynamn theorem
cannot be applied to compute nuclear gradients because individual
SA-CASSCF solutions are not stationary points of the energy.
Recently, the limitations of SA-CASSCF and the development of non-ground-state SCF optimisation
algorithms has inspired several new investigations into state-specific CASSCF excited states.
In particular, Neuscamman and co-workers have developed generalized
variational principles\cite{Tran2020,Hanscam2022} and the $W\Gamma$ approach inspired by MOM-SCF,\cite{Tran2019}
demonstrating that the issues of root flipping and variational collapse to the ground state
can be successfully avoided.
Despite these advances, we still do not have a complete understanding of the multiple stationary points on the
SS-CASSCF energy landscape and several practical questions remain.
For example, how many stationary points are there and how does this change with the
active space or basis set size?
Where do unphysical solutions arise, what are their characteristics, and
when does symmetry breaking occur?
And finally, do state-specific excitations behave diabatically or adiabatically as the molecular structure
evolves?
Our aim in this work is to answer these questions and establish
a theoretical foundation for practical excited state-specific calculations.
Using second-order optimisation techniques, we investigate the existence and properties
of multiple CASSCF solutions in typical molecular systems.
Our numerical optimisation exploits analytic gradients and second derivatives of the CASSCF energy,
and the relevant differential geometry is summarised below.
Using these techniques, we comprehensively enumerate the multiple CASSCF solutions in
\ce{H2} (6-31G) and characterise the resulting unphysical solutions.
We find that state-specific calculations can accurately describe high-lying excitations
with fewer active orbitals than state-averaged formalisms,
and reveal that multiple solutions can arise from active spaces that are too large or too small.
We then investigate the conical intersection in \ce{CH2} (6-31G) and the
avoided crossing of \ce{LiF} (6-31G), demonstrating the importance and difficulty
of selecting the correct physical solution.
\section{Exploring the multiconfigurational energy landscape}
\label{sec:theoretical}
\subsection{Defining the CASSCF wave function}
\label{subsec:CASSCF}
A multiconfigurational wave function is defined as the linear combination of $M$ Slater determinants
\begin{equation}
\ket{\Psi_k} = \sum_{I=1}^{M} C_{I k} \ket{\Phi_I},
\label{eq:CIwfn}
\end{equation}
where $\ket{\Phi_I}$ represents different configurations built from a common set of
molecular orbitals (MO) $\phi_p(\bm{x})$ and the $C_{Ik}$ are the variable
CI coefficients for state $k$.\cite{HelgakerBook}
Here, $\bm{x} = (\bm{r},\sigma)$ is the combined spatial and spin electronic coordinate.
The MOs are constructed as linear combinations of $n$ (nonorthogonal) atomic orbitals (AO) $\chi_{\mu} (\bm{x})$ as
\begin{equation}
\label{eq:mo}
\phi_p(\bm{x}) = \sum^{n}_\mu \chi_{\mu}(\bm{x})\, c_{\cdot p}^{\mu \cdot},
\end{equation}
where we use the nonorthogonal tensor notation of Ref.~\onlinecite{HeadGordon1998} and the
$c_{\cdot p}^{\mu \cdot}$ denote the variable MO coefficients.
Normalisation of the wave function, and orthogonalisation of the MOs, is guaranteed by the constraints
\begin{equation}
\sum_{I=1}^{M} \abs{C_I}^2 = 1
\quad\text{and}\quad
\sum_{\mu=1}^{n} (c^*)_{p \cdot}^{\cdot \mu}\, \braket*{\chi_\mu}{\chi_\nu}\, c_{\cdot q}^{\nu \cdot} = \delta_{pq},
\label{eq:constraint}
\end{equation}
where $\braket*{\chi_\mu}{\chi_\nu}$ denotes the AO overlap matrix elements.
We will only consider wave functions where $C_{Ik}$ and $c_{\cdot p}^{\mu \cdot}$ are real.
When every electronic configuration for a finite basis set is included in an FCI expansion,
the global minimum on the parametrised
electronic energy landscape corresponds to the exact ground state.\cite{Burton2022a}
Excited states form saddle points of the energy and the number of downhill directions
increases with each excitation.\cite{Burton2022a,Bacalis2016,Bacalis2020,Golab1983}
The FCI wave function is invariant to unitary transformations of the MOs, but
the number of configurations scales exponentially with the system size.
The complete active space (CAS) framework builds a truncated expansion using every configuration
within a set of ``active orbitals'' that describe the dominant static electron correlation.\cite{Roos1980a}
The orbitals are partitioned into inactive and virtual orbitals that are doubly occupied or empty in every
configuration, respectively, and active orbitals with varying occupations.
Simultaneously optimising the energy with respect to the orbital and CI coefficients leads to the state-specific CASSCF
approach and gives true stationary points of the electronic energy.\cite{Roos1980a,Roos1980b,Seigbahn1981}
If the CASSCF wave function targeting the $k$th excited state is represented by the $k$th eigenstate of the
corresponding CAS-CI expansion, then the Hylleraas-Undheim-MacDonald theorem\cite{Hylleraas1930b,MacDonald1933}
also provides a upper bound to the excited-state energy.\cite{Golab1983}
\subsection{Differential geometry of the CASSCF energy}
\label{subsec:differentialGeometry}
We exploit an exponential form of the CASSCF wave function that conserves the
orthogonality constraints [Eq.~\eqref{eq:constraint}].\cite{Dalgaard1978,Yeager1979}
Starting from an initial CASSCF wave function $\ket{\Psi_0}$, an arbitrary step
can be defined using unitary transformations as
\begin{equation}
\label{eq:mcscf_wf}
\ket{\Psi}=\mathrm{e}^{\Hat{R}} \mathrm{e}^{\hS}\ket{\Psi_0},
\end{equation}
where $\mathrm{e}^{\Hat{R}}$ and $\mathrm{e}^{\hS}$ account for orbital relaxation and transformations of
the CI component, respectively.
The $\Hat{R}$ operator is anti-Hermitian and is defined using the second-quantised
creation and annihilation operators for the current MOs as\cite{Dalgaard1978,Douady1980}
\begin{equation}
\Hat{R} = \sum_{p>q} R_{pq}\, \hE^{-}_{pq},
\label{eq:Rop}
\end{equation}
where the spin-adapted one-body anti-Hermitian replacement operators are\cite{HelgakerBook}
\begin{equation}
\hE^{-}_{pq} = \sum_{\sigma \in \{\uparrow, \downarrow \} } \cre{q\sigma}\ani{p\sigma} - \cre{p\sigma}\ani{q\sigma}.
\end{equation}
The invariance of the energy with respect to inactive-inactive, active-active,
and virtual-virtual orbital transformations
means that $R_{pq}$ can be further restricted to only excitations between different
sub-blocks.
Similarly, $\mathrm{e}^{\hS}$ performs a unitary transformation between the CI
component of $\ket{\Psi_0}$ and the remaining orthogonal states $\ket{\Psi_{K}}$
in the current CASCI space, with $\hS$ defined as\cite{Yeager1979}
\begin{equation}
\hS = \sum_{K \neq 0} S_{K} \Big( \ket{\Psi_K}\bra{\Psi_0}-\ket{\Psi_0}\bra{\Psi_K} \Big).
\label{eq:Sop}
\end{equation}
Using the exponential parametrisation, the CASSCF energy can be expressed as
\begin{equation}
\label{eq:casscf_energy}
E(\bm{R},\bm{S}) = \mel{\Psi_0}{\mathrm{e}^{-\hS} \mathrm{e}^{-\Hat{R}} \hH \mathrm{e}^{\Hat{R}} \mathrm{e}^{\hS}}{\Psi_0},
\end{equation}
where $\bm{R}$ and $\bm{S}$ are vectors that gather the $R_{pq}$ and $S_K$ coefficients
in the orbital and CI transformations, respectively, and $\hH$ is the electronic Hamiltonian.
Stationary points of $E$, corresponding to optimal CASSCF solutions, then occur when
the gradients with respect to orbital and CI transformations are simultaneously zero.
Performing a Baker--Campbell--Hausdorff expansion of the energy to
second order gives\cite{Yeager1979}
\begin{equation}
\begin{split}
E \approx &\mel*{\Psi_0}{\hH}{\Psi_0}
+ \mel*{\Psi_0}{[\hH, (\Hat{R} + \hS)]}{\Psi_0}
\\
&+ \frac{1}{2} \mel*{\Psi}{[[\hH,(\Hat{R} + \hS)],(\Hat{R} + \hS)]}{\Psi_0} + \dots
\end{split}
\end{equation}
Expressions for the first- and second-derivates of the energy
can then be identified as
\begin{subequations}
\begin{align}
\left.\frac{\partial E}{\partial R_{pq}}\right|_{\bm{R},\bm{S}=\bm{0}} &= \mel*{\Psi_0}{[\hH, \hE^{-}_{pq}]}{\Psi_0},
\label{eq:gradOrb}
\\
\left.\frac{\partial E}{\partial S_{K}}\right|_{\bm{R},\bm{S}=\bm{0}} &= 2 \mel*{\Psi_0}{\hH}{\Psi_K},
\label{eq:gradCI}
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\left.\frac{\partial^2 E}{\partial R_{pq}\partial R_{rs}}\right|_{\bm{R},\bm{S}=\bm{0}}
&= \frac{1}{2}(1+P_{pq,rs}) \mel*{\Psi_0}{[[\hH, \hE^{-}_{pq}],\hE^{-}_{rs}]}{\Psi_0},
\\
\left.\frac{\partial^2 E}{\partial R_{pq}\partial S_{K}}\right|_{\bm{R},\bm{S}=\bm{0}}
&=
\mel*{\Psi_0}{[\hH, \hE^{-}_{pq}]}{\Psi_K},
\\
\left.\frac{\partial E}{\partial S_{L} \partial S_{K}}\right|_{\bm{R},\bm{S}=\bm{0}}
&= 2 \mel*{\Psi_K}{\hH - E_0}{\Psi_L},
\end{align}
\end{subequations}
where $E_0$ is the energy at $\bm{R}, \bm{S} = \bm{0}$, $P_{pq,rs}$ permutes the $(pq)$ and $(rs)$ indices,
and the Hermiticity of $\hH$ and $[\hH, \hE_{pq}^{-}]$ have been exploited.
Explicit formulae for these expressions have been summarised elsewhere [see Ref.~\onlinecite{Olsen1983}]
but are given in the Supporting Information (Section~S1) for completeness.
Note that
the first and second derivatives can only be computed
when $\bm{R} = \bm{0}$ and $\bm{S} = \bm{0}$.\cite{Douady1980}
Therefore, after taking a step in the parameter space, the energy gradient and Hessian
must be computed using the new MOs and CI vectors corresponding to the updated wave function.
A similar shift in the reference state after each step is also required for second-order
HF optimisation algorithms.\cite{Voorhis2002,Burton2021a}
\subsection{Characterising distinct solutions}
\label{subsec:distinguishSolutions}
The invariance to unitary transformations within each orbital partition
means that the same CASSCF wave function can be identified with different CI or MO coefficients.
We use the overlap between two stationary solutions $\ket{^x\Psi}$ and $\ket{^w\Psi}$
to define a positive semidefinite distance metric
\begin{equation}
d(x,w) = 1 - \abs{\braket*{^x\Psi}{^w\Psi}\,}.
\end{equation}
The overlap for two arbitrary CI wave functions with $M_x$ and $M_w$
configurations, respectively, is given by
\begin{equation}
\braket*{^x\Psi}{^w\Psi}
=
\sum_{I=1}^{M_x} \sum_{J=1}^{M_w} \,^{x}C^{*}_{I} \braket*{^x\Phi_I}{^w\Phi_J} \,^{w}C_{J}.
\end{equation}
Since $\ket{^x\Psi}$ and $\ket{^w\Psi}$ have different sets of MOs, evaluating the
overlap matrix elements $\braket*{^x\Phi_I}{^w\Phi_J}$ requires a nonorthogonal framework.
We compute these matrix elements using the extended nonorthogonal Wick's theory,\cite{Burton2021c,Burton2022c}
which avoids the computationally expensive generalized Slater--Condon rules.\cite{MayerBook}
To understand the MOs in a CASSCF solution, we canonicalise the inactive and virtual
orbitals and construct natural orbitals within the active space.
The canonical inactive and virtual orbitals, and their associated orbital energies,
are identified by diagonalising the relevant sub-blocks of the
Fock matrix, defined as\cite{HelgakerBook}
\begin{equation}
F_{pq} = h_{pq} + \sum_{rs} \gamma_{rs}\qty(\ceri{pq}{sr} - \frac{1}{2}\ceri{pr}{sq}).
\label{eq:general_fock}
\end{equation}
Here, $\gamma_{pq}$ denotes the one-body reduced density matrix elements
in the MO basis, $h_{rq}$ are the one-electron Hamiltonian matrix elements, and $\ceri{pq}{rs}$ are
the two-electron repulsion integrals.
The natural orbitals within the active space are the eigenvectors of the one-body reduced
density matrix and their eigenvalues are the occupation numbers $n_p$.\cite{Lowdin1955a}
\subsection{Optimization techniques}
\label{subsec:optimization}
Since we are concerned with understanding the CASSCF solution space, we require an algorithm
capable of converging arbitrary stationary points on the energy landscape, including minima and
higher-index saddle points.
Higher-energy CASSCF stationary points are notoriously difficult to converge due to the strong coupling
between the orbital and CI degrees of freedom,\cite{Das1973,Yeager1979,Bauschlicher1980b,Bauschlicher1980c,Yeager1980a}
and the possibility of root flipping in the configuration space.\cite{Docken1972,Werner1981}
Therefore, we employ second-order techniques that introduce the orbital-CI coupling through
the analytic Hessian matrix of second derivatives.
These algorithms are too computationally expensive to be practical for larger systems,
but they are sufficient for understanding the CASSCF solutions in small molecules.
We search for multiple solutions using several initial guesses generated using random
orbital and CI rotations from the ground state HF solution.
The eigenvector-following technique with analytic gradient and Hessian information was used to
target stationary points with a particular Hessian index.\cite{Cerjan1981,Wales1990}
While this method has been described in detail elsewhere [see Ref.~\onlinecite{WalesBook}],
we include a summary in the Supporting Information (Section~S2).
Related mode-following methods have previously been applied to locate higher-energy
electronic stationary points in multiconfigurational\cite{Olsen1982,Golab1983,Olsen1983,Hoffmann2002}
and single-determinant\cite{Burton2021a} SCF calculations.
The convergence behaviour was further improved with a modified trust region approach based
on the dogleg method.\cite{NocedalBook}
Trust region methods are a well-established approach for controlling the convergence
of second-order methods in CASSCF calculations.\cite{Yeager1980a,Yeager1980b,Jorgensen1981,Yeager1982}
Once a set of stationary points have been identified, their evolution with changes
in the molecular structure can be determined
by using the optimised orbital and CI coefficients at one geometry
to define an initial guess at the next geometry.
Since the Hessian index may not be conserved along a reaction coordinate,\cite{Olsen1982}
these subsequent calculations are performed using a trust region Newton--Raphson
algorithm, as described in the Supporting Information (Section~S3).
We have implemented this numerical optimisation in an extension to the
PySCF software package.\cite{PySCFb}
The convergence threshold for the root-mean-squared value of the gradient amplitudes was
universally set to $10^{-8}\,\mathrm{E_h}$.
The canonical and natural orbitals for stationary points were
subsequently computed using PySCF and visualised using VMD.\cite{VMD}
All other graphical figures were created using Mathematica~12.0.\cite{Mathematica}
\section{Results and Discussion}
\subsection{Molecular \ce{H2} dissocation}
\label{sec:results}
We start by considering the \ce{H2} binding curve using the 6-31G basis set.\cite{Ditchfield1971}
To identify all the CASSCF\,(2,2) solutions, a comprehensive search was performed using up to 1000
random starting points for target Hessian indices from 0 to 16.
Solutions were identified near the equilibrium geometry $R=\SI{1.0}{\bohr}$ and the dissociation
limit $R=\SI{6.0}{\bohr}$, and were then traced over all bond lengths, as shown in Fig.~\ref{fig:h2_binding_curve}.
We believe that we have found every stationary point on the landscape, although the nature
of non-convex optimisation means that this can never be guaranteed.
To the best of our knowledge, this study is the first comprehensive enumeration of the CASSCF solutions
for a molecular system.
\subsubsection{Excitations near equilibrium}
\begin{figure*}
\includegraphics[width=0.88\linewidth]{figure1}
\caption{State-specific CASSCF\,(2,2) stationary points can be identified for every
excited FCI state in \ce{H2}.
Additional solutions can also be found that dissociate to an unphysical electronic state.}
\label{fig:h2_binding_curve}
\end{figure*}
Near the equilibrium geometry, the ground state of \ce{H2} can be accurately described using a single
reference approximation.
We have identified 25 stationary points on the CASSCF\,(2,2) energy landscape, corresponding to
19 singlet solutions and 6 triplet solutions (Table~\ref{tab:tab_1}).
\begin{table}[htb]
\caption{Energies of \ce{H2} at $R=1~a_0$ using the 6-31G basis set for various formalism:
FCI, SA-CASSCF\,(2,2), SA-CASSCF\,(3,2), and SS-CASSCF\,(2,2).}
\label{tab:tab_1}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
State & FCI & SA(2,2) & SA(3,2) & SS(2,2) & $\expval*{S^2}$ & Index\\
\hline
0 & -1.09897 & -1.07170 & -1.08924 & -1.09225 & 0 & 0 \\
& & & & -1.08569 & 0 & 1 \\
& & & & -1.07871 & 0 & 2 \\
1 & -0.57616 & -0.57166 & -0.57406 & -0.57417 & 2 & 1 \\
2 & -0.46395 & -0.43494 & -0.44196 & -0.46368 & 0 & 2 \\
3 & -0.28180 & & -0.27990 & -0.27990 & 2 & 2 \\
4 & -0.07450 & & -0.06164 & -0.05946 & 0 & 3 \\
5 & \hd0.32015 & \hd0.33066 & \hd0.32624 & \hd0.31914 & 0 & 3 \\
& & & & \hd0.31821 & 0 & 2 \\
& & & & \hd0.31844 & 0 & 2 \\
& & & & \hd0.32440 & 0 & 3 \\
6 & \hd0.51519 & & \hd0.51654 & \hd0.51638 & 2 & 3 \\
7 & \hd0.57224 & & \hd0.61682 & \hd0.61429 & 0 & 4 \\
8 & \hd0.62520 & & & \hd0.62401 & 2 & 3 \\
9 & \hd0.86353 & & \hd0.86876 & \hd0.86392 & 0 & 5 \\
& & & & \hd0.85673 & 0 & 4 \\
& & & & \hd0.86266 & 0 & 4 \\
10 & \hd0.96373 & & & \hd0.91147 & 0 & 4 \\
11 & \hd1.30761 & & & \hd1.30572 & 2 & 4 \\
12 & \hd1.46479 & & & \hd1.45704 & 0 & 5 \\
13 & \hd1.61884 & & & \hd1.61685 & 2 & 5 \\
14 & \hd1.81277 & & & \hd1.80747 & 0 & 6 \\
15 & \hd2.71948 & & & \hd2.71766 & 0 & 7 \\
& & & & \hd2.70046 & 0 & 6 \\
& & & & \hd2.69883 & 0 & 5 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Each of the exact FCI states has a corresponding SS-CASSCF\,(2,2) counterpart, and the
energetic agreement between these solutions is consistent for all excitations.
We have also found several additional solutions that appear to be less accurate
approximations to the exact states, which will be characterised in Sections~\ref{subsubsec:closedshell}
and \ref{subsubsec:openshell}.
In comparison, the SA-CASSCF\,(2,2) approach can only describe the lowest triplet and the three
lowest singlet states, while increasing the number of active orbitals to a (3,2) active space
provides an approximation to the lowest nine excitations.
These results demonstrate two important features of state-specific calculations.
Firstly, they can describe more excited states than state-averaged calculations by defining the
active space using only orbitals that are relevant for a particular excitation.
This property allows higher energy excitations to be predicted while avoiding
large active spaces and the associated increase in the configuration space.
An upper bound to the exact excited state energy is only provided by
stationary points that correspond to the correct excitation within the CASCI configuration
space,\cite{Golab1983}
although more accurate energies are generally preferred even if they are not variational.
Secondly, bespoke orbital optimisation for each state-specific solutions can give more
accurate total energies for the excited states compared to the state-averaged approach.
For example, the mean absolute deviations (MAD) for the lowest four states are $2.5\,\mathrm{mE_h}$
and $17.8\,\mathrm{mE_h}$ for the state-specific and state-averaged CASSCF\,(2,2) approaches, respectively.
Using analytic second derivatives of the energy also allows the nature of
SS-CASSCF\,(2,2) stationary points to be characterised according to their number of
downhill directions.
The corresponding Hessian index for each solution is listed in Table~\ref{tab:tab_1}.
It is known that the exact $n$-th excited state should have $n$ downhill directions.\cite{Olsen1983,Olsen1982,Burton2022a}
We find that the SS-CASSCF\,(2,2) excited states are all saddle points on the electronic energy
landscape and the Hessian index generally increases with the energy,
in common with the observations for other theoretical approximations.\cite{Gilbert2008,Hait2021,Burton2021a,Kossoski2021}
However, except for the lowest three exact states, the Hessian index does not provide a reliable
indicator of the corresponding exact excitation index.
This mismatch must always occur for higher-lying excited states as the
approximate CASSCF\,(2,2) wave function has fewer degrees of freedom than the exact formulation.
Consequently, if we only consider stationary points of the correct Hessian index, then
we must forgo the advantages of capturing state-specific excitations outside
the state-averaged active space.
\subsubsection{Multiple ground state solutions}
\label{subsubsec:closedshell}
While Table~\ref{tab:tab_1} shows that a SS-CASSCF\,(2,2) approximation can be identified
for each exact eigenstate, we also find additional state-specific solutions.
In particular, there are three close-lying stationary points that can be considered as
approximations to the ground state, with Hessian indices of 0, 1, and 2 in order of
ascending energy.
This pattern of multiple solutions is repeated for the $(2{\upsigma_\textrm{g}})^2$ and $(2{\upsigma_\textrm{u}})^2$
singlet configurations, while the other closed-shell $(1{\upsigma_\textrm{u}})^2$
configuration exhibits four close-lying solutions.
Choosing the most physical solution for each eigenstate presents a challenge
for state-specific CASSCF approaches.
Therefore, it is important that we understand their mathematical origins and physical differences.
\begin{figure*}
\includegraphics[width=0.85\linewidth]{figure2.pdf}
\caption{There are three SS-CASSCF(2,2) solutions that represent the exact ground state in \ce{H2}.
(\textsf{\textbf{A}}) Comparison of the natural orbitals for each ground-state solution at $R=\SI{1.0}{\bohr}$.
(\textsf{\textbf{B}}) Only the lowest-energy solution dissociates correctly, while the higher-energy
solutions mirror the restricted Hartree--Fock binding curve.}
\label{fig:multigs_h2}
\end{figure*}
The natural orbitals in the active space provides a clear explanation for the multiple \ce{H2} ground-state solutions.
Figure~\ref{fig:multigs_h2}\textcolor{blue}{A} compares the natural orbitals and occupation numbers for the three
lowest-energy singlet stationary points.
Since the ground state at the equilibrium geometry can be relatively well approximated by
a single closed-shell Slater determinant, the active space for
each of these solutions includes a $(1{\upsigma_\textrm{g}})$-like natural orbital that is almost completely
doubly occupied.
This natural orbital dominates the electronic wave function and the corresponding energies
are all relatively close approximations to the exact ground state.
However, the second active orbital, which is almost completely unoccupied, is different for each
solution, corresponding to a $(1{\upsigma_\textrm{u}})$, $(2{\upsigma_\textrm{g}})$, or $(2{\upsigma_\textrm{u}})$ orbital as the energy increases,
respectively.
These higher-energy stationary points have downhill orbital rotations that interconvert the
multiple ground state solutions and correspond to the negative eigenvalues of the Hessian.
Different choices for the nearly unoccupied active orbital have only a small effect on
the total \ce{H2} energy near equilibrium.
However, the incorrect choice of the active space becomes very significant as
the bond is stretched towards dissociation.
Only the $\{1{\upsigma_\textrm{g}},1{\upsigma_\textrm{u}}\}$ active space can correctly dissociate into the $\mathrm{H(1s)\cdots H(1s)}$
ground state of the dissociated fragments (Fig.~\ref{fig:multigs_h2}\textcolor{blue}{B}).
In contrast, the binding curves for the $\{ 1{\upsigma_\textrm{g}},2{\upsigma_\textrm{g}} \}$ and $\{ 1{\upsigma_\textrm{g}},2{\upsigma_\textrm{u}} \}$ solutions
mirrors the RHF energy as the corresponding wave functions are close to a single Slater
determinant at all geometries, with $(1{\upsigma_\textrm{g}})$ occupation numbers at dissociation of 1.997 and 1.999, respectively.
Notably, the stationary points preserve the character of the active orbitals along the potential energy
surface, suggesting that SS-CASSCF solutions exhibit some degree of diabatic character.
The same pattern of solutions is observed for the other closed-shell solutions.
However, the $(1{\upsigma_\textrm{u}})^2$ configuration exhibits an additional multiple solution
where the nearly unoccupied active orbital corresponds to a symmetry-broken
$2\mathrm{s}$-like orbital localised on either
the left or right \ce{H} atom.
This symmetry breaking results in a two-fold degenerate pair of stationary points.
These results indicate that additional solutions can arise from the free choice of virtual orbitals
when the active space is larger than required for the degree of static correlation.
Malrieu and co-works elegantly summarised this phenomenon by stating that
\textit{``the so-called valence CASSCF wave function does not necessarily keep a valence character when the wave function concentrates
on a closed-shell valence bond structure''.}\cite{Meras1990}
Therefore, we expect that the number of ground state solutions will increase combinatorially
with the number of active orbitals or the basis set size,
and the number of unphysical solutions can grow for larger active spaces even though
the correct ground state solution will become more accurate.
Table~\ref{tab:H2_6_311g} demonstrates this increase for \ce{H2} using the 6-311G basis set
with three basis functions for each hydrogen atom.\cite{Krishnan1980}
Taking the (3,2) active space as an example, there are two redundant active orbitals beyond the
$1{\upsigma_\textrm{g}}$ that must be chosen from the five remaining orbitals, giving a total of ten solutions through
the binomial coefficient $\binom{5}{2}=10$.
The relative energy ordering of these additional solutions will depend on the amount of dynamic
correlation captured by the redundant active orbitals, which may not correspond with the same
orbital required to capture the static correlation in the dissociation limit.
This phenomenon has previously been described for \ce{MgO}, where oxygen-centred orbitals are preferred over
the magnesium d orbitals,\cite{Hanscam2022} and transition metal compounds where
non-valence $\mathrm{d}$ orbitals may be preferred over certain valence d orbitals.\cite{Andersson1992b}
\begin{table}[htb]
\caption{Close-lying ground-state $(n,2)$ SS-CASSCF energies ($\mathrm{E_h}$) of \ce{H2} at $R=1~a_0$
using the 6-311G basis set for various active space size $n$.}
\label{tab:H2_6_311g}
\begin{ruledtabular}
\begin{tabular}{llllll}
SS(1,2): HF & -1.08025 & & & &\\
\hline
SS(2,2) & -1.09429 & -1.08866 & -1.08074 & -1.08033 & -1.08026 \\
\hline
SS(3,2) & -1.10195 & -1.09500 & -1.09436 & -1.09429 & -1.08904 \\
& -1.08886 & -1.08867 & -1.08082 & -1.08075 & -1.08034 \\
\hline
SS(4,2) & -1.10251 & -1.10212 & -1.10196 & -1.09507 & -1.09500 \\
& -1.09437 & -1.08923 & -1.08905 & -1.08886 & -1.08083 \\
\hline
SS(5,2) & -1.10267 & -1.10251 & -1.10213 & -1.09507 & -1.08924 \\
\hline
SS(6,2): FCI & -1.10267 & & & &\\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsubsection{Open-shell singlet and triplet excitations}
\label{subsubsec:openshell}
The low-lying open-shell triplet and singlet $(1{\upsigma_\textrm{g}})^{1} (1{\upsigma_\textrm{u}})^{1}$ configurations
are represented by only one SS-CASSCF\,(2,2) solution across the full
binding curve (Fig.~\ref{fig:h2_binding_curve}).
These single solutions arise because all the active orbitals are required to describe the two-configurational
static correlation and there is no flexibility for multiple solutions to exist.
In addition, SS-CASSCF\,(2,2) gives an accurate representation of the open-shell
$(1\upsigma_\textrm{g/u})^{1} (2 \upsigma_\textrm{g/u})^{1}$ configurations.
However, the accuracy of these solutions deteriorates in the dissociation limit, where additional
symmetry broken solutions can be identified (Fig.~\ref{fig:h2_symmetry_breaking}).
These additional solutions break spatial symmetry and spontaneously
appear at instability thresholds that are multi-configurational analogues to the Coulson--Fischer
points\cite{Coulson1949} in HF theory.\cite{Fukutome1973,Fukutome1975,Mestechkin1978,Mestechkin1979,Mestechkin1988}
Each stationary point is a pure singlet or triplet state and has a two-fold degeneracy,
reflecting the left-right symmetry of the molecule.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure3.pdf}
\caption{Spontaneous symmetry breaking occurs when the active space is not large enough to capture
all the important configurations in the physical wave function, as illustrated for the $1\mathrm{s}\,2\mathrm{s}$
states in the dissociation of \ce{H2} (6-31G).}
\label{fig:h2_symmetry_breaking}
\end{figure}
The origin of this symmetry breaking is explained by considering the correlation processes
involved in the excited dissociation limit.
These excited states dissociate to hydrogenic $(1\mathrm{s})^1 (2\mathrm{s})^1$ configurations, where the occupied
$1\mathrm{s}$ and $2\mathrm{s}$ orbitals can either be on the same or different atomic centres.
Taking the latter case as an example, the corresponding open-shell singlet wave function
at large nuclear separations has the form
\begin{equation}
\begin{split}
\ket{\Psi} &= \frac{1}{2}
\Big(\ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{R}} + \ket{1\mathrm{s}_\text{R}2\mathrm{s}_\text{L}}\Big)
\Big(\ket{\alpha\beta} - \ket{\beta \alpha}\Big).
\end{split}
\label{eq:opensinglet}
\end{equation}
Correctly describing this wave function requires an active space with four spatial orbitals
$\{1\mathrm{s}_\text{L}, 1\mathrm{s}_\text{R},2\mathrm{s}_\text{L}, 2\mathrm{s}_\text{R} \}$, or equivalently $\{1{\upsigma_\textrm{g}}, 1{\upsigma_\textrm{u}}, 2{\upsigma_\textrm{g}}, 2{\upsigma_\textrm{u}}\}$,
and thus the SS-CASSCF\,(2,2) approximation is insufficient for these correlation mechanisms.
Instead, the symmetry breaking reduces the SS-CASSCF\,(2,2) wave function to a subset of the dominant configurations,
e.g.\
\begin{equation}
\ket*{\tilde{\Psi}} = \frac{1}{\sqrt{2}} \ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{R}}
\Big(\ket{\alpha\beta} - \ket{\beta \alpha} \Big).
\end{equation}
The CASSCF configurations corresponding to each symmetry-broken solution are assigned in Table~\ref{tab:h2_sym_breaking}.
\begin{table}[h]
\begin{ruledtabular}
\begin{tabular}{cccc}
State & Energy / $\mathrm{E_h}$ & $\expval*{S^2}$ & Configuration \\ \hline
A & \hphantom{-}0.543355 & 0.00
& $\begin{cases}%
\ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{L}} \qty(\ket{\alpha\beta} - \ket{\beta \alpha} ) \\
\ket{1\mathrm{s}_\text{R}2\mathrm{s}_\text{R}} \qty(\ket{\alpha\beta} - \ket{\beta \alpha} )
\end{cases}$
\\ \hline
B & \hphantom{-}0.293363 & 2.00
& $\begin{cases}%
\ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{L}} \qty(\ket{\alpha\beta} + \ket{\beta \alpha} ) \\
\ket{1\mathrm{s}_\text{R}2\mathrm{s}_\text{R}} \qty(\ket{\alpha\beta} + \ket{\beta \alpha} )
\end{cases}$
\\ \hline
C & -0.037221 & 0.00
& $\begin{cases}%
\ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{R}} \qty(\ket{\alpha\beta} - \ket{\beta \alpha} ) \\
\ket{1\mathrm{s}_\text{R}2\mathrm{s}_\text{L}} \qty(\ket{\alpha\beta} - \ket{\beta \alpha} )
\end{cases}$
\\ \hline
D & -0.037499 & 2.00
& $\begin{cases}%
\ket{1\mathrm{s}_\text{L}2\mathrm{s}_\text{R}} \qty(\ket{\alpha\beta} + \ket{\beta \alpha} ) \\
\ket{1\mathrm{s}_\text{R}2\mathrm{s}_\text{L}} \qty(\ket{\alpha\beta} + \ket{\beta \alpha} )
\end{cases}$
\end{tabular}
\end{ruledtabular}
\caption{The symmetry-broken CASSCF\,(2,2) solutions in the dissociation of \ce{H2}
are two-fold degenerate and represent dominant configurations in the
exact excitations.}
\label{tab:h2_sym_breaking}
\end{table}
This ``pinning'' of the wave function onto a particular electronic configuration is directly analogous to the
symmetry breaking phenomena observed in HF theory\cite{Trail2003,Burton2021b}
and demonstrates that the active space is too small to fully account for the static correlation.
From the energy landscape perspective, the onset of symmetry-broken CASSCF\,(2,2) states
is associated with a change in the Hessian index for the associated symmetry-pure solutions.
For example, the symmetry-broken state D (Table~\ref{tab:h2_sym_breaking}) emerges from the symmetry-pure
$(1\upsigma_\mathrm{g})^1 (2\upsigma_\mathrm{g})^1$
triplet state at an instability threshold close to $R = \SI{2.28}{\bohr}$.
The Hessian index of the symmetry-pure state changes from 2 to 3 at this point, while the
symmetry-broken solutions form index-2 saddle points, leading to a higher-index analogue of a
cusp catastrophe.\cite{GilmoreBook,Burton2018,Burton2021a}
Practically, the emergence of a zero Hessian eigenvalue at these instability thresholds may hinder
the numerical optimisation of second-order techniques onto these higher-energy stationary points.
It is also interesting to note that,
while the symmetry-broken solutions describe two degenerate FCI states at dissociation,
they only connect to one of the corresponding symmetry-pure solutions in the equilibrium region.
Consequently, one cannot rely on these additional solutions to obtain an accurate and
continuous representation of every excited state across all geometries.
\subsection{Conical Intersection in methylene}
We next consider the bending mode of methylene, which has a diradical ground state with $^3\mathrm{B}_1$
symmetry and a low-lying $1\,^1\mathrm{A}_1$ excited state.
The bond length was fixed to the value $R(\ce{C-H}) = \SI{2.11}{\bohr}$ identified by
Bauschlicher and Taylor\cite{Bauschlicher1986a,Bauschlicher1986b}
and the 6-31G basis set was used.\cite{Ditchfield1971}
Methylene has a long history as a benchmark for electronic structure theory.\cite{Schaefer1986}
One of the primary questions is the description of the conical intersection between
the low-lying $^3\mathrm{B}_1$ and $1\,^{1}\mathrm{A}_1$ states.
\begin{figure*}[htb]
\includegraphics[width=\linewidth]{figure4.pdf}
\caption{Low-lying SS-CASSCF\,(2,2) in the bending mode of methylene represent the
$1\,^3\mathrm{B}_1$ and $1\,^1\mathrm{A}_1$ configurations.
(\textbf{\textsf{A}}) Both states remain local minima (solid purple) for a short region beyond the conical intersection before
becoming an index-1 saddle point (solid cyan).
An additional spin-contaminated index-1 saddle point (dashed cyan) connects the two instability thresholds (black dots).
Two degenerate local minima exist everywhere along the bending curve (dashed purple) with an active space containing
C-H bonding $\upsigma$ and antibonding $\upsigma^{*}$ orbitals.
(\textbf{\textsf{B}}--\textbf{\textsf{E}}) The natural orbitals at a bond angle of $103.7^{\circ}$ are illustrated
for each solution.}
\label{fig:methylene_bend}
\end{figure*}
\subsubsection{Local minima for the minimal (2,2) active space}
A minimal two-configuration wave function is required to qualitatively describe
both the lowest-energy singlet $\mathrm{S}_0$ ($^1\mathrm{A}_1$)
and diradical triplet $\mathrm{T}_0$ ($^3\mathrm{B}_1$) states.\cite{Bauschlicher1986a}
Therefore, we begin by analysing the SS-CASSCF\,(2,2) energy landscape.
The $\mathrm{S}_0$ and $\mathrm{T}_0$ are the ground state for small and large bond angles, respectively,
and provide an example of a conical intersection separating the two regimes.
At bond angles of $76^\circ$, $102^\circ$ and $130^\circ$, a large number of stationary points can be identified
with a variety of Hessian indices.
Therefore, we simplify our analysis by focussing on a subset of low-energy solutions that
resemble the desired physical states (Fig.~\ref{fig:methylene_bend}).
The energetic minimum of the $\mathrm{S}_0$ state occurs at a bond angle of $103.7^\circ$.
While the $\mathrm{S}_0$ state is the first excited state at this geometry, we find that
the corresponding SS-CASSCF\,(2,2) stationary point is a local minimum rather than an index-1 saddle point.
This incorrect Hessian index arises from a root flip in the configuration space,
where the singlet state is the ground state for the corresponding active orbitals.
When the bond angle increases, this singlet state eventually becomes an index-1 saddle point.
Similarly, when the bond angle decreases from $103.7^\circ$,
the $\mathrm{T}_0$ state remains a local minimum beyond the conical intersection where it becomes the first excited state.
This process behaves like an unphysical hysteresis, where the ground state remains a local minimum for a
small region after a conical intersection before becoming an index-1 saddle point at an instability threshold.
An additional index-1 saddle point can be identified that connects these two solutions and coalesces with
each local minimum at the two instability thresholds.
This unphysical index-1 stationary point is two-fold degenerate,
has symmetry-pure spatial orbitals, but is spin contaminated with
an $\expval*{\hat{S}^2}$ value that changes continuously from 0 to 2 as it connects the singlet and
triplet states.
Similar patterns of coalescing solutions have been observed in single determinant
SCF approximations,\cite{Burton2018,Zarotiadis2020,Fukutome1975,Huynh2019}
particularly in the generalised HF representation of conical intersections between
states with different $\expval*{\hat{S}_z}$ values.\cite{Jimenez-Hoyos2011}
In contrast to symmetry-broken SCF solutions, the spin contamination observed here arises from the
mixture of singlet and triplet states in the configurational part of the CASSCF wave function.
Since the $\mathrm{S}_0$ solution has only one significantly occupied active orbital,
we predict the existence of closely-related solutions that have alternative redundant orbitals
with $n_\text{occ} \approx 0$.
Indeed, there are a pair of degenerate local minima that lie slightly lower in energy
than the $\mathrm{S}_0$ solution.
In contrast to the \ce{H2} ground state, including the inactive space means that methylene has multiple doubly occupied orbitals,
and thus the active orbital with $n_\text{occ} \approx 2$ may also change between different solutions.
The active orbitals for these symmetry-broken solutions are localised bonding $\upsigma$ and
anti-bonding $\upsigma^*$ orbitals for one of the two \ce{C-H} bonds, and the degeneracy accounts for the
two possible ways to localise onto one bond.
Notably, the symmetry breaking here is associated with an active space that is too large, in contrast
to \ce{H2} where symmetry breaking arises from an insufficient active space for the static correlation.
These solutions are local minima across all the bond angles considered.
While they provide an accurate energy for the $\mathrm{S}_0$ state near the singlet equilibrium structure, this
deteriorates for large angles as the active space cannot describe the diradical
open-shell $^1\Sigma_\text{g}^{+}$ state at the linear geometry.
Their existence indicates that the \ce{C-H} $\upsigma/\upsigma^{*}$ configurations provide
an important contribution to static correlation and should ideally be included in the active space,
as suggested by Bauschlicher and Taylor.\cite{Bauschlicher1986a,Bauschlicher1986b}
\subsubsection{Full valence active space}
\begin{figure}[htb]
\includegraphics[width=\linewidth]{figure5.pdf}
\caption{Low-lying SS-CASSCF\,(6,6) for the bending mode of methylene
represent the $1\,^3\mathrm{B}_1$ and $1\,^1\mathrm{A}_1$ configurations.
The full valence (6,6) active space introduces more unphysical solutions, but does not
remove the spin-contaminated solution that arises at the conical intersection.}
\label{fig:methylene_6_6}
\end{figure}
\begin{figure*}[htb]
\includegraphics[width=\linewidth]{figure6.pdf}
\caption{Comparison of the active orbitals for the two lowest energy triplet CASSCF solutions for \ce{CH2} (6-31G)
using the full valence (6,6) active space at a bond angle of $102^\circ$.
(\textbf{A}) The active orbitals for the local minimum represent the chemically intuitive valence space.
(\textbf{B}) For the unphysical index-1 saddle point, one of the antibonding \ce{C-H} $\upsigma^*$
orbitals with $n_\text{occ} \approx 0$ is replaced by a carbon $3 \mathrm{p}$ orbital with $n_\text{occ} = 0.0029$.
The remaining $\upsigma$ and $\upsigma^{*}$ orbitals localise onto the \ce{C-H} bonds.}
\label{fig:ch2_6-6_orbitals}
\end{figure*}
Using the full-valence (6,6) active space, we find that the symmetry-pure singlet state is now
correctly represented by an index-1 saddle point at a bond angle of $\SI{102}{\degree}$
(Fig.~\ref{fig:methylene_6_6}).
The unique downhill direction corresponds to a rotation in the configuration space only, as expected for
the first excited state.
Despite the larger active space, a root flip still occurs as the states approach the singlet--triplet
conical intersection at $\SI{82.2}{\degree}$, with the singlet state becoming a local minimum
at $\SI{89.6}{\degree}$ and the triplet state becoming an index-1 saddle point at $\SI{77.0}{\degree}$.
Like the (2,2) active space, a degenerate pair of unphysical, spin-contaminated index-1 saddle
points connect the solutions that cross at the conical intersection.
This phenomenon occurs because the orbital optimisation can lower the energy of the target excited state
below the corresponding ground state configuration when the energy gap becomes small.
Therefore, while larger active spaces will reduce the range of molecular
geometries affected, these
unphysical local minima will be common for state-specific conical intersections.
While the larger active space alleviates root flipping, it also causes
more unphysical solutions associated with redundant active orbitals.
For example, the triplet ground state (dominated by two configurations) is represented
by one SS-CASSCF\,(2,2) solution, but there are several higher-energy solutions in
the (6,6) active space.
Analogously to the \ce{H2} ground state, these additional solutions have a higher Hessian index,
with two index-1 and one index-2 saddle points represented in Fig.~\ref{fig:methylene_6_6}.
Again, the main difference from the true ground state are the
active orbitals with occupation numbers close to zero, as illustrated
for the global minimum and lowest-energy index-1 saddle point in Fig.~\ref{fig:ch2_6-6_orbitals}.
Furthermore, we find an additional local minimum and index-1 saddle point that represent the
$^1\mathrm{A_1}$ state.
While all the triplet solutions give approximately the same equilibrium bond angle,
the unphysical stationary points shift the conical intersection to coincide with the
singlet equilibrium geometry.
This qualitative change in the energy surface would create a near-barrierless decay from the
singlet excited state to the triplet ground state, demonstrating the importance of verifying the physicality of
state-specific solutions.
\subsection{Avoided crossing in Lithium Fluoride}
\label{sec:avoided}
\subsubsection{Physicality of multiple solutions}
\begin{figure*}[htb]
\includegraphics[width=\linewidth]{figure7.pdf}
\caption{(\textbf{A}) The SS-CASSCF\,(2,2) appoach gives many solutions for the \ce{LiF} binding curve (6-31G)
when the ground or excited state is dominated by a single configuration.
Ground- and excited-state solutions with a suitable active space (\textbf{B} and \textbf{D}) behave adiabatically
at the avoided crossing (cyan lines).
Additional solutions with unsuitable active orbitals can represent either the
ionic equilibrium configuration (\textbf{C}) or the covalent dissociation configuration (\textbf{E}),
and behave quasi-diabatically at the avoided crossing (purple lines).
The active orbitals are plotted at $R(\ce{Li-F})=\SI{4}{\bohr}$.
Exact FCI and SA(2)-CASSCF\,(2,2) data are taken from Ref.~\onlinecite{Burton2020}.}
\label{fig:LiF_6-31g}
\end{figure*}
The \ce{LiF} binding curve provides a typical example of an avoided crossing.
The ground state has ionic character at equilibrium, but becomes a covalent state with almost no dipole moment
in the dissociation limit.
Multiple HF solutions are known to behave ``quasi-diabatically'' and cross each other at the
physical avoided crossing.\cite{Thom2009,Burton2020}
On the other hand, Bauschlicher and Langhoff demonstrated that this avoided crossing can lead to
discontinuities in the CASSCF ground- and excited-state energy surfaces.\cite{Bauschlicher1988}
Here, we start by considering the state-specific singlet CASSCF solutions in the 6-31G basis set.
Using the minimal (2,2) active, we search for stationary points with
Hessian indices of 0 to 10 at $R(\ce{Li-F}) = \SI{2.75}{\bohr}$ (near the equilibrium geometry), using
1000 random starting points for each index.
The active space for the SS-CASSCF global minimum contains the valence bonding $\upsigma$ and anti-bonding $\upsigma^{*}$
orbitals with occupation numbers close to 2 and 0, respectively (Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{B}).
Because the exact wave function is dominated by a single closed-shell configuration,
there are many additional solutions that are close to the ground-state energy at the equilibrium geometry.
For example, the second lowest energy solution has an active space containing the out-of-plane
fluorine $2\mathrm{p_{x/y}}$ and $3\mathrm{p_{x/y}}$ orbitals with occupation numbers close to 2 and 0,
respectively (Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{C}).
This active space accounts for the radial correlation on the fluorine atom, providing a more balanced
description of \ce{F} and \ce{F-}.\cite{Bauschlicher1988}
In contrast, the exact excited state is more multiconfigurational at short bond lengths and is accurately represented
by only one solution (Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{D}), alongside a spurious symmetry-broken
solution with diradical character (Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{E}).
These characteristics are reversed for bond lengths longer than the avoided crossing,
where the excited state has closed-shell character with a large number of solutions
and the ground state is represented by only two solutions.
State-specific CASSCF solutions can behave both quasi-diabatically and adiabatically in the vicinity
of the avoided crossing.
As the bond length changes, the unphysical solutions do not have the correct active orbitals
to capture the strong correlation at the avoided crossing.
Therefore, the two lowest-energy unphysical solutions intersect quasi-diabatically
(dark purple in Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{A}, corresponding to the solutions
in Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{C} and \ref{fig:LiF_6-31g}\textcolor{blue}{E}).
On the other hand, the physically meaningful solutions behave adiabatically and correctly avoid each other
(cyan in Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{A}).
In principle, a linear expansion of both the quasi-diabatic and adiabatic states may provide a
more accurate representation of the avoided crossing by introducing some of the dynamic correlation
captured by the unphysical solutions.
This expansion would require a multiconfigurational variant of nonorthogonal CI,\cite{Thom2009} where the
Hamiltonian and overlap matrix elements can be efficiently computed using the nonorthogonal framework
developed in Refs.~\onlinecite{Burton2021c,Burton2022c}.
While a complete description of the avoided crossing requires dynamic correlation,\cite{Malrieu1995}
the advantage of state-specific orbital relaxation is still clear in the dissociation limit.
The physical SS-CASSCF excited state tends towards the exact FCI energy for the separated \ce{Li^+ \cdots F^{-}}
configuration, while state-averaged calculations (with an equal weighting for the two states)
overestimates the energy of this excited state (Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{A}).
In this SS-CASSCF solution, the $\upsigma$ and $\upsigma^{*}$ orbitals
(Fig.~\ref{fig:LiF_6-31g}\textcolor{blue}{A}) both localise to give $\mathrm{2p_z}$ orbitals that
accurately represent the \ce{F^{-}} anion.
Consequently, as expected, the state-specific formalism provides a more accurate representation
of this charge transfer excitation than a state-averaged approach.
\subsubsection{Elucidating the Bauschlicher--Langhoff discontinuity}
\label{sec:BLdiscont}
The seminal CASSCF investigation of \ce{LiF}, by Bauschlicher and Langhoff,
highlighted the presence of a discontinuity in
the ground-state dipole moment in the vicinty of the avoided crossing.\cite{Bauschlicher1988}
This discontinuity is a signature of a discontinuity in the wave function, which manifests
as a cusp in the corresponding energy surface.
This phenomenon, which we name the ``Bauschlicher--Langhoff discontinuity'', has long been
used as key evidence for the potential issues of state-specific calculations in the vicinity
of an avoided crossing.
Malrieu and co-workers attributed its origin to a near degeneracy between the closed-shell ionic and
the open-shell covalent configurations, and described a lower-energy covalent state that
emerges from a potential symmetry-breaking point as the bond length increases.\cite{Meras1990}
The framework developed here, and the advance in computing over the past 30 years,
now allows this topological characterisation to be rigorously tested.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure8.pdf}
\caption{Topology of the low-energy SS-CASSCF\,$(2,2)$ solutions near the Bausclicher--Langhoff
discontinuity in \ce{LiF},\cite{Bauschlicher1988,Meras1990}
using the basis set defined in Ref.~\onlinecite{Bauschlicher1988}.
A cusp in the ground-state energy occurs when two local minima cross, while the covalent structure
coalesces with an index-1 saddle point at a pair annihilation point (black dot).}
\label{fig:LiF_custom}
\end{figure}
To identify the relevant solutions, we searched for minima and index-1 saddle points
at a bond length of $\SI{8.50}{\bohr}$ using 1000 random starting points, a $(2,2)$ active space,
and the original basis set described in Ref.~\onlinecite{Bauschlicher1988}.
At $R(\ce{Li-F})=\SI{8.5}{\bohr}$, the global minimum corresponds to the covalent structure
identified in Ref.~\onlinecite{Meras1990}.
In addition, two local minima and two index-1 saddle points exist at higher energies,
representing the ionic configurations (Fig.~\ref{fig:LiF_custom}).
As the bond length is shortened, there is a crossing between the lowest energy ionic and covalent
minima near $R(\ce{Li-F})=\SI{7.4}{\bohr}$, which we believe corresponds to the previously
described discontinuity.\cite{Bauschlicher1988,Meras1990}
Topologically, two non-degenerate minima cannot
coalesce without the presence of an index-1 saddle point, and thus the disappearance described
by Malrieu and co-workers cannot be the full picture.\cite{Meras1990}
Instead, we find that the covalent structure crosses the two lowest-energy local minima and
eventually coalesces with an index-1 saddle point representing the ionic configurations.
Following the downhill directions from this index-1 saddle points reveals that it connects the
covalent local minimum with the lowest-energy ionic local minimum.
Furthermore, the downhill Hessian eigenvector has significant orbital and CI components, which highlights
the strong coupling between the different degrees of freedom in the vicinity of the avoided crossing.
Both solutions disappear at this point (black dot in Fig.~\ref{fig:LiF_custom}),
and thus there is no quasi-diabatic covalent solution at shorter bond lengths.
In the mathematical framework of catastrophe theory,\cite{ThomBook} this type of coalescence
can be classified as a fold catastrophe, or a pair annihilation point.
Singularities in this class have previously been identified and characterised for multiple HF solutions,\cite{Fukutome1975}
where they most commonly occur in asymmetric molecules, for example \ce{LiF},\cite{Thom2009,Burton2020}
\ce{H-Z}\cite{Burton2018} (for a partial nuclear charge $Z$), and ethylene analogues.\cite{Burton2018}
The discontinuous jump in the energy at the pair annihilation point in \ce{LiF} will create issues for
calculations that attempt to follow the covalent solution across multiple bond lengths, making
these solutions unsuitable for techniques such as \textit{ab initio} molecular dynamics.
Furthermore, since the lowest energy covalent and ionic local minima cross rather than coalesce,
the gradient of the global minimum energy at the crossing point is discontinuous and
there is an unphysical cusp in the resulting energy surface.
The absence of this pair annihilation point using 6-31G compared to Bauschlicher and Langhoff's basis set
demonstrates how the topology of multiple CASSCF solutions can be affected by the AO basis.
We suspect that these differences arise from the subtle changes in the underlying energy landscape
that affect the relative stability of different solutions.
However, these results demonstrate the danger of generalising conclusions from one basis set
to another, even for the same molecule.
\section{Concluding Remarks}
\label{sec:conclusion}
State-specific approximations promise to provide a more balanced representation of electronic
excitations by independently optimising both the ground- and excited-state wave functions.
In this work, we have investigated the energy landscape for excited state-specific stationary points
in the multi-configurational CASSCF approach.
We have shown how state-specific approximations can accurately describe high-energy and charge transfer
excitations, beyond the reach of state-averaged calculations with small active spaces.
However, the CASSCF energy landscape can have a large number of stationary points, which complicates
the selection and interpretation of physically relevant solutions.
Multiple stationary points in state-specific CASSCF calculations arise through two primary mechanisms.
Firstly, many solutions occur when the active space is too large for the static correlation that
must be described.
In this case, the redundant active orbitals with $n_\text{occ} \approx 0$ can be interchanged with
virtual orbitals without significantly changing the energy, creating a series of stationary points with
an increasing number of downhill Hessian eigendirections.
Active orbitals with $n_\text{occ} \approx 2$ can be interchanged with doubly occupied inactive orbitals
in a similar fashion.
On the other hand, symmetry broken solutions occur when the active space is too small to describe
the static correlation mechanisms, causing the CASSCF wave function to become ``pinned''
onto a subset of the configurations in the exact wave function.
These results demonstrate the importance of finding a ``Goldilocks region'', where the active space
is neither too large or too small, but just right.
Unphysical solutions can have important consequences for the resulting potential energy surfaces.
For example, while choosing the wrong active space only introduces a small energy error when the wave function
is dominated by a single closed-shell configuration,
it can prevent the CASSCF wave function from correctly capturing
static correlation when the molecular structure changes.
The active space for stationary points does not change significantly along a reaction coordinate,
meaning that the incorrect active orbitals remain for all geometries.
For ground-state calculations, one can rely on following downhill directions away from saddle points
to obtain a more suitable local minimum, hopefully with the best active space.
However, it is hard to predict which Hessian index will give the most physical stationary point for an excited state,
and thus choosing the most accurate excited-state stationary point is challenging
without prior chemical intuition.
It has long been known that the right choice of active orbitals is key to the success of CASSCF,
but the current results demonstrate the severity of this challenge for state-specific excitations.
In addition, we have investigated the topology of SS-CASSCF\,$(2,2)$ solutions near the singlet-triplet
conical intersection in \ce{CH2} and the covalent-ionic avoided crossing in \ce{LiF}.
We observe unphysical root flipping where the \ce{CH2} excited state solution is a local minimum near
the conical intersection, before becoming an index-1 saddle point further along the reaction trajectory.
This phenomenon occurs because the state-specific orbital optimisation artificially stabilises
the local minima, and is still present in the full valence $(6,6)$ active space.
Furthermore, the change in Hessian index is associated with an additional spin-contaminated index-1 saddle point
that connects the singlet and triplet stationary points.
The presence of zero Hessian eigenvalues at these instability thresholds may cause numerical
issues for second-order optimisation algorithms.
On the other hand, for the \ce{LiF} avoided crossing, we have observed the coalescence of
the local covalent minimum with an index-1 saddle point representing the ionic state, which both
disappear entirely at shorter bond lengths.
While this pairwise coalescence depends on the basis set, it would catastrophically affect
the applicability of SS-CASSCF for generating smooth and continuous potential energy surfaces.
Moving forwards, SS-CASSCF calculations must overcome the troublesome issues of multiple solutions.
Practical solutions may rely on the identification of suitable initial guesses from more black-box
techniques, or by focussing on
optimisation algorithms that target desirable excited-state physical properties
(e.g.\ dipole moments), such as the generalized variational principles developed by Hanscam and
Neuscamman.\cite{Hanscam2022}
Alternatively, more bespoke excited-state wave function \textit{ans\"{a}tze}, such as
minimal configuration state functions\cite{Kossoski2022a}
or excited-state mean-field theory,\cite{Shea2018,Hardikar2020,Shea2020}
may remove unphysical solutions associated with redundant active orbitals and avoid the disappearance
of solutions at pairwise coalescence points.
Surmounting these issues will allow the benefits of state-specific calculations for computing
excited states, with bespoke orbitals and small active spaces, to be fully realised.
\section*{Supporting Information}
Derivation of the gradient and second-derivatives for the CASSCF energy,
description of eigenvector-following and Newton--Raphson optimisation algorithms used (PDF).
\section*{Acknowledgements}
H.G.A.B was supported by New College, Oxford, through the Astor Junior Research Fellowship.
The authors thank David Tew for support and computing resources.
\section*{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,108
|
package org.olat.presentation.portfolio.artefacts.collect;
import java.util.Date;
import java.util.List;
import org.olat.data.portfolio.artefact.AbstractArtefact;
import org.olat.lms.commons.context.BusinessControlFactory;
import org.olat.lms.commons.context.ContextEntry;
import org.olat.presentation.framework.core.UserRequest;
import org.olat.presentation.framework.core.components.form.flexible.FormItemContainer;
import org.olat.presentation.framework.core.components.form.flexible.elements.RichTextElement;
import org.olat.presentation.framework.core.components.form.flexible.elements.StaticTextElement;
import org.olat.presentation.framework.core.components.form.flexible.elements.TextElement;
import org.olat.presentation.framework.core.components.form.flexible.impl.Form;
import org.olat.presentation.framework.core.components.form.flexible.impl.FormBasicController;
import org.olat.presentation.framework.core.control.Controller;
import org.olat.presentation.framework.core.control.WindowControl;
import org.olat.presentation.framework.core.control.generic.wizard.StepFormBasicController;
import org.olat.presentation.framework.core.control.generic.wizard.StepsEvent;
import org.olat.presentation.framework.core.control.generic.wizard.StepsRunContext;
import org.olat.presentation.framework.core.translator.PackageTranslator;
import org.olat.presentation.portfolio.artefacts.run.EPArtefactViewController;
import org.olat.system.commons.Formatter;
import org.olat.system.commons.StringHelper;
import org.olat.system.event.Event;
/**
* Description:<br>
* first collection step, collecting title and description of an artefact
* <P>
* Initial Date: 01.11.2010 <br>
*
* @author Roman Haag, roman.haag@frentix.com, http://www.frentix.com
*/
public class EPCollectStepForm00 extends StepFormBasicController {
private TextElement title;
private RichTextElement descript;
private final AbstractArtefact artefact;
private boolean simpleMode = false;
public EPCollectStepForm00(final UserRequest ureq, final WindowControl wControl, final Form rootForm, final StepsRunContext runContext, final int layout,
final String customLayoutPageName, final AbstractArtefact artefact) {
super(ureq, wControl, rootForm, runContext, layout, customLayoutPageName);
// set fallback translator to re-use given strings
final PackageTranslator pt = new PackageTranslator(EPArtefactViewController.class.getPackage().getName(), ureq.getLocale(), getTranslator());
this.flc.setTranslator(pt);
this.artefact = artefact;
initForm(this.flc, this, ureq);
}
// this constructor is used when editing an artefact, therefore the form doesn't show all fields!
public EPCollectStepForm00(final UserRequest ureq, final WindowControl wControl, final AbstractArtefact artefact) {
super(ureq, wControl, FormBasicController.LAYOUT_VERTICAL);
// set fallback translator to re-use given strings
final PackageTranslator pt = new PackageTranslator(EPArtefactViewController.class.getPackage().getName(), ureq.getLocale(), getTranslator());
this.flc.setTranslator(pt);
this.artefact = artefact;
this.simpleMode = true;
initForm(ureq);
}
@Override
protected void initForm(final FormItemContainer formLayout, @SuppressWarnings("unused") final Controller listener, final UserRequest ureq) {
title = uifactory.addTextElement("title", "artefact.title", 500, artefact.getTitle(), formLayout);
title.setMandatory(true);
title.setNotEmptyCheck("artefact.title.not.empty");
title.setNotLongerThanCheck(512, "artefact.title.too.long");
title.setVisible(!simpleMode);
descript = uifactory.addRichTextElementForStringDataMinimalistic("description", "artefact.description", artefact.getDescription(), 7, -1, false, formLayout,
ureq.getUserSession(), getWindowControl());
descript.setExtDelay(true);
descript.setMaxLength(4000);
descript.setNotLongerThanCheck(4000, "artefact.description.too.long");
final String artSource = artefact.getSource();
if (StringHelper.containsNonWhitespace(artSource) && !simpleMode) {
uifactory.addStaticTextElement("artefact.source", artSource, formLayout);
}
Date artDate = artefact.getCreationDate();
if (artDate == null) {
artDate = new Date();
}
final StaticTextElement date = uifactory.addStaticTextElement("artefact.date", Formatter.getInstance(getLocale()).formatDateAndTime(artDate), formLayout);
date.setVisible(!simpleMode);
final String busPath = artefact.getBusinessPath();
if (StringHelper.containsNonWhitespace(busPath) && !simpleMode) {
final BusinessControlFactory bCF = BusinessControlFactory.getInstance();
final List<ContextEntry> ceList = bCF.createCEListFromString(busPath);
final String busLink = bCF.getAsURIString(ceList, true);
if (StringHelper.containsNonWhitespace(busLink)) {
final String finalPath = "<a href=\"" + busLink + "\">" + busLink + "</a>";
uifactory.addStaticTextElementWithTrustedText("artefact.link", finalPath, formLayout);
}
}
if (!isUsedInStepWizzard()) {
// add form buttons
uifactory.addFormSubmitButton("stepform.submit", formLayout);
}
}
@Override
protected void doDispose() {
// nothing
}
@Override
protected void formOK(final UserRequest ureq) {
artefact.setTitle(title.getValue());
artefact.setDescription(descript.getValue());
// either save values to runContext or do persist them
// directly, if form is used outside step-context
if (isUsedInStepWizzard()) {
addToRunContext("artefact", artefact);
if (artefact.getFileSourceContainer() != null) {
addToRunContext("tempArtFolder", artefact.getFileSourceContainer());
}
fireEvent(ureq, StepsEvent.ACTIVATE_NEXT);
} else {
fireEvent(ureq, Event.DONE_EVENT);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 14
|
Q: Instantaneous Coulomb interaction in QED It seems I am stuck with a (at a first sight) trivial problem.
It's from the "Quarks and Leptons" (Halzen, Martin) book page $141$, where one considers the following integral:
$$\tag{1} T_{fi} = -i\int \!d^4x \, J_0^A(t_A,\vec{x}_A)\,J_0^B(t_A,\vec{x}_A)\frac{1}{|\vec{q}|^2}. $$
In equation $(1)$, $J_0^A$ and $J_0^B$ are the zeroth component of two electron currents:
$$J_\mu(x) = j_\mu\mathrm{exp}[(p_f-p_i)\cdot x].$$
Now, according to the authors, one can rewrite $(1)$ by making use of the Fourier transform
$$\tag{2} \frac{1}{|q|^2} = \int\! d^3x\, e^{i\vec{q}\cdot\vec{x}}\frac{1}{4\pi|\vec{x}|}, $$
to the following
$$ \tag{3} T_{fi}^{Coul} = -i\int \!dt_A\int d^3x_A\int d^3x_B \, \frac{J_0^A(t,\vec{x}_B)\,J_0^B(t,\vec{x}_B)}{4\pi|\vec{x}_B-\vec{x}_A|}. $$
Equation $(3)$ is then interpreted as the instantaneous$^1$ Coulomb interaction between the charges of the particles, $J_0^A$ and $J_0^B$.
The derivation of this is given in the answer below.
$^1$I.e. interaction without retardation at time $t_A$.
A: It seems to me that there is a typo in the book. Your starting equation should be the following:
\begin{equation}
T_{fi} = -i\int \frac{d\omega \,d^{3}\textbf{q}}{(2\pi)^{4}} \tilde{A}(\textbf{q},\omega)\tilde{B}(-\textbf{q},-\omega) \frac{1}{|\textbf{q}|^{2}},
\end{equation}
where $\tilde{A}$ denotes the Fourier transform of $A$.
Then, using
\begin{equation}
\tilde{f}(\textbf{q},\omega) = \int dt\, d^{3}\textbf{x} \,f(\textbf{x},t)\, e^{-i(\textbf{q}\cdot \textbf{x}-\omega t)}
\end{equation}
will lead to the desired result.
A: I suspected that one needed to go back to the definition of the currents and indeed, in doing so one can derive the result. Here's a short version.
The electron current is defined as [see equation (6.6) in 1]
$$\tag{1}J_\mu(x) = -e\bar{u}_f\gamma_\mu u_i \times\mathrm{exp}[(p_f-p_i)\cdot x]$$
which we write as
$$\tag{2}J_\mu(x) = j_\mu\mathrm{exp}[(p_f-p_i)\cdot x]. $$
We will also need to use
$$\tag{3}q = p_i^A-p_f^A = p_f^B-p_i^B$$
Then the integral $(1)$ in the original post can be written
$$ \tag{4} T_{fi} = -i\int \!dt_Ad^3x_A d^3x \,\, j^Aj^B e^{i(p_f^{A0}-p_i^{A0})t_A}e^{i(p_f^{B0}-p_i^{B0})t_A}\frac{1}{|\vec{x}|}e^{i\vec{q}\cdot\vec{x}}. $$
Now shifting $\vec{x}=\vec{x}_B-\vec{x}_A$ with $d^3x=d^3x_B$ and using $(3)$ and
$$(\vec{p}_f^A-\vec{p}_i^A)\cdot(\vec{x}_B-\vec{x}_A) = -(\vec{p}_f^B-\vec{p}_i^B)\cdot\vec{x}_B-(\vec{p}_f^A-\vec{p}_i^A)\cdot\vec{x}_A, $$
equation $(4)$ becomes
$$\tag{3} T_{fi} = -i\int \!dt_A\int d^3x_A\int d^3x_B \, \frac{J_0^A(t_A,\vec{x}_A)\,J_0^B(t_A,\vec{x}_B)}{4\pi|\vec{x}_B-\vec{x}_A|}, $$
where $J_0^A(t_A,\vec{x}_A)$ corresponds to the OP's $A(t_A,\vec{x}_A)$ and so on.
References: See appendix of J. H. Field, Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 518
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I'm looking to upgrade to a mobo with discrete nvidia graphics. (41W1201) As far as I can tell, it will fit. But the biggest difference I can see is the that the one I am upgrading from is a 14W board, while the one I want to upgrade to is a 15W board.
So my question, in short, is can I upgrade from the 41W1199 to the 41W1201 without killing anything?
New motherboards for laptops usually cost more than the laptop itself is worth, therefore its not cost-effective.
Everything on a laptop is geared for the original configuration and uses unique connectors that aren't always transferable to a board from the same company. Plus there's no guarantee its going to work anyway as the new board may require a different power connector or voltage.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 624
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Networking…whatever your preconceived notions are about the subject, we are here to share some tips and skills that you can implement to effectively network with others!
You may be an extravert who loves and thrives off of talking to people, or you may be that introvert that hates crowded rooms. While some may view networking as inauthentic and forced, we are here to change your perception of networking.
First of all, you need to change your mind set about networking! It is NOT about YOU! A common problem people face when networking with others is that you are constantly thinking about what YOU are going to say next. Think of the acronym FROG. Ask questions about the person's family, recreation, occupation, and goals. It is more beneficial to ask questions in that order; if you skip the first two and start with asking someone what is his occupation, you have missed the opportunity to connect with him on other topics!
By asking these questions, you can lend your services/help this person to overcome his challenge. It also allows him to do his own introspection/self-reflection.
Why should you network? It allows you to build relationships, develop a network of individuals across different industries, create generosity, and hone in on your communication skills. A great book that we recommend that helps you with your communication skills is The Seven Levels of Communication by Michael J. Maher. The most important takeaways from this book are that you should not wait for a life changing event to change your life, use your meeting times as a way to connect others together, and focus mainly on people.
With all of these great tips in mind, think about networking in a positive and uplifting way! Networking is not about what you can get from someone, but rather what you can offer to someone.
Growing your network will help you grow your business. Once you make the investment in yourself to begin meaningful networking, you can reinvest in your business by using these connections to expand it! The greatest asset you have is yourself, and your greatest investment is your business.
What Wealth Without Wall Street can help you with is what strategies you can use to constantly reinvest in yourself and your business.
Are you great at networking, or do you have any other suggestions for successful networking? We would love to view your comments in the comments section below!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,942
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Q: jQuery toggle radio/parent style will also toggle neighboring radios of different names To first give you an idea of what my HTML is like, I have a page with 3 groups of radio buttons, each group of radios is named by their container (ie. Page1, Page2, Page3). So the first group of radios all share the name "page1" and the second share the name "page2", etc.
So I'm using jQuery to toggle radio buttons and the class of the containing table for that radio (ie. addClass, removeClass on toggle). The toggle works fine, and so does the adding and removing of the class on the toggle. My problem is if I have a radio from page2 or page3 selected and then select/reselect a page1 radio, the jQuery will remove the class of the page2 and page3 checked radios styles.
Here is a link to the JSfiddle Demo.
And heres a look at the jQuery I'm using.
$(function() {
$('table').click(function(event) {
$(this).closest('.container').addClass('selected').parent().siblings().each(function() {
$(this).find('.container').removeClass('selected');
});
if(event.target.type != "radio") {
var that = $(this).find('input:radio');
that.attr('checked', !that.is(':checked'));
if (that.is(':checked')) {
that.closest('table').addClass('selected');
}
else {
that.closest('table').removeClass('selected');
}
}
});
});
A: the .parent().siblings() is picking up the other pages because you don't have a div around page 1 if you add that it solves the problem http://jsfiddle.net/5YnBq/1/
A: Well I am not 100% on this but try event.preventDefault();
I think it is bubbling up so when it targets input:radio it uses the outer most table so it would basically target all of the radio buttons. I have not tested this because the fiddle was not working for me after I clicked continue but give it a try and see what happens
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,299
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package org.syftkog.web.test.framework;
import java.util.ArrayList;
import org.syftkog.web.test.framework.retry.RetryTestContext;
import org.apache.commons.lang3.builder.ToStringBuilder;
import org.openqa.selenium.remote.DesiredCapabilities;
import org.testng.ITestResult;
import org.testng.SkipException;
/**
*
* @author BenjaminLimb
*/
public class TestCaseContext implements WrapsTestCaseContext, HasDriver, TagContext, HasStepLogger, DriverContext<TestCaseContext>, RetryTestContext<TestCaseContext> {
private TestCaseParameters testCaseParameters;
private transient DriverFactory driverFactory;
private transient Driver driver;
private transient StepLogger stepLogger = new StepLogger();
private transient int currentAttempt = 1;
private String name;
private transient ITestResult testResult;
private ArrayList<String> tags;
/**
*
* @param params
*/
public TestCaseContext(TestCaseParameters params) {
this.testCaseParameters = params;
}
/**
*
* @param bvp
*/
public TestCaseContext(BrowserVersionPlatform bvp) {
testCaseParameters = new TestCaseParameters(bvp);
}
/**
*
*/
public TestCaseContext() {
testCaseParameters = new TestCaseParameters();
}
/**
*
* @return
*/
@Override
public DriverFactory getDriverFactory() {
// if no Driver factory is specified, use the singleton instance.
if (driverFactory == null) {
driverFactory = DriverFactory.getInstance();
}
return driverFactory;
}
/**
*
* @param driverFactory
* @return
*/
@Override
public TestCaseContext setDriverFactory(DriverFactory driverFactory) {
this.driverFactory = driverFactory;
return this;
}
/**
*
* @return
*/
@Override
public StepLogger getStepLogger() {
return stepLogger;
}
/**
*
* @param stepLogger
*/
public void setStepLogger(StepLogger stepLogger) {
this.stepLogger = stepLogger;
}
/**
*
* @return
*/
@Override
public Integer getMaxRetryCount() {
if (parameters().getMaxAttempts() != null) {
return this.parameters().getMaxAttempts() +1;
} else {
return null;
}
}
/**
*
* @return
*/
@Override
public Boolean isDriverInitialized() {
return driver != null;
}
/**
*
* @return
*/
@Override
public Driver getDriver() {
if (driver == null) {
driver = getDriverFactory().getDriver(this);
}
return driver;
}
/**
*
* @param driver
* @return
*/
@Override
public TestCaseContext setDriver(Driver driver) {
this.driver = driver;
return this;
}
/**
*
* @return
*/
@Override
public TestCaseContext getWrappedTestCastContext() {
return this;
}
/**
*
* @return
*/
public TestCaseParameters parameters() {
return testCaseParameters;
}
/**
*
* @param testCaseParameters
*/
public void setTestCaseParameters(TestCaseParameters testCaseParameters) {
this.testCaseParameters = testCaseParameters;
}
@Override
public String toString() {
return ToStringBuilder.reflectionToString(this);
}
/**
*
* @return
*/
public int getCurrentAttempt() {
return currentAttempt;
}
/**
*
* @return
*/
@Override
public TestCaseContext retry() {
this.getStepLogger().log("RETRYING");
driver = null; // If we retry, then we must get a new driver becasue the driver was quit or dismissed.
currentAttempt++;
return this;
}
/**
*
* @return
*/
public String getName() {
return name;
}
/**
*
* @param name
*/
public void setName(String name) {
this.name = name;
}
/**
*
* @return
*/
public ITestResult getTestResult() {
return testResult;
}
/**
*
* @param testResult
*/
public void setTestResult(ITestResult testResult) {
this.testResult = testResult;
}
/**
*
* @return
*/
public String getTestName() {
return testResult.getTestClass().getName() + "." + testResult.getMethod().getMethodName();
}
/**
*
* @return
*/
@Override
public ArrayList<String> getTags() {
ArrayList<String> tags = new ArrayList<>();
// if (testResult != null) {
// tags.add(testResult.getTestClass().getName() + "." + testResult.getMethod().getMethodName());
// }
tags.addAll(parameters().getTags());
return tags;
}
/**
*
* @param tag
*/
@Override
public void addTag(String tag) {
tags.add(tag);
}
/**
*
* @return
*/
public DesiredCapabilities toDesiredCapabilities() {
DesiredCapabilities caps = testCaseParameters.toDesiredCapabilities();
caps.setCapability("name", name);
return caps;
}
/**
*
* @param env
*/
public void assertEnvironment(EnvironmentType env) {
if (parameters().getEnvironment().getEnvironmentType() != env) {
throw new SkipException("This test is restricted to run in " + env);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,946
|
\section{Introduction}
It is being increasingly realized by those engaged in
search for supersymmetry (SUSY) that the principle of R-parity
conservation\cite{s1,s13}, assumed to be sacrosanct in
the prevalent search strategies, is not in practice
inviolable\cite{s2}. The R-parity of a particle is defined by
$R=(-1)^{2S+3B+L}$ and can be violated either by baryon-number
(B) breaking or by lepton-number (L) breaking\cite{s2}.
Proton decay experiments have set stringent
restrictions on the violations of the first and
the second generations of the baryon-numbers, but
the existent experiment data do not impose
so stringent restriction neither on lepton-number violations
nor on the third generations of baryon-numbers. In particular,
for some cosmology models to explain the baryongenesis,
it requests lepton-numbers not to be conserved, that
the intensive studies of the supersymmetry models without R-parity,
through lepton-number violations and/or the violation
for the third generation of baryon-number, have attracted quite a lot
of attentions recently[3-22]. As far as the
literature is concerned, besides those the so-called
basis-independent studies of the R-parity violations\cite{s7},
the models with lepton-numbers being broken are characterized
by certain `special' Lagrangian which has
`bilinear'\cite{s21,s11,chf,s5,s55} and/or
trilinear\cite{s5,s55,s3,s4,s6,s10} R-parity
violations explicitly in
superpotential and/or the SUSY soft-breaking terms,
and by the violations spontaneously generated
by nonzero vacuum expectation values (VEVs) of sneutrinos[3-9].
The supersymmetry models without R-parity can also be arranged
well that there will be no contradiction with all the existent
experimental data[9-22]. Respect to the very general case,
the MSSM without R-parity through various possible
lepton-number violations simultaneously has not
been investigated thoroughly.
In general, the minimal supersymmetric standard model (MSSM)
(R-parity is conserved) has the following
general form for the superpotential in terms of superfields:
\begin{eqnarray}
{\cal W}_{MSSM} &=& \mu\varepsilon_{ij} \hat{H}_{i}^{1} \hat{H}_{j}^{2}
+ l_{I}\varepsilon_{ij} \hat{H}_{i}^{1} \hat{L}_{j}^{I} \hat{R}^{I} -
u_{I}(\hat{H}_{1}^{2} C^{JI*}\hat{Q}_{2}^{J} - \hat{H}_{2}^{2}
\hat{Q}_{1}^{I}
)\hat{U}^{I} \nonumber \\
& & - d_{I}(\hat{H}_{1}^{1}\hat{Q}_{2}^{I} - \hat{H}_{2}^{1} C^{IJ}
\hat{Q}_{1}^{J})\hat{D}^{I}.
\label{eq-1}
\end{eqnarray}
Here $\hat{H}^{1}$, $\hat{H}^{2}$ are Higgs superfields;
$\hat{Q}^{I}$ and $\hat{L}^{I}$ being quark and lepton superfields
(I=1, 2, 3 is the index of generation), all are
in doublet of the weak SU(2) respectively. The rest superfields:
$\hat{U}^{I}$ and $\hat{D}^{I}$ being quark superfields
and $\hat{R}^{I}$ charged lepton ones, but in singlet
of the weak SU(2). Here the indices i, j are contracted
in a general way for the SU(2) group, and $C^{IJ}$ $(I, J=1, 2, 3)$
are the elements of the CKM matrix.
When R-breaking interactions are incorporated,
the superpotential will be modified as the
follows:
\begin{equation}
{\cal W} = {\cal W}_{MSSM} + {\cal W}_{L} + {\cal W}_{B}
\label{eq-2}
\end{equation}
with
\begin{eqnarray}
{\cal W}_{L} &=& \varepsilon_{ij} [ \lambda_{IJK} \hat{L}_{i}^{I}
\hat{L}_{j}^{J}
\hat{R}^{K} + \lambda_{IJK}^{\prime}\hat{L}_{i}^{I} \hat{Q}_{j}^{J} \hat{D}
^{K} + \epsilon_{I} \hat{H}_{i}^{2} \hat{L}_{j}^{I} ] \nonumber \\
{\cal W}_{B} &=& \lambda_{IJK}^{\prime\prime}
\hat{U}^{I} \hat{D}^{J} \hat{D}^{K}.
\label{eq-3}
\end{eqnarray}
Considering the stringent constraint by proton decay experiments
on the violations for the first and the second generation
baryon-numbers, many authors would like to focus on the third
generation baryon-number\cite{s5,s55,s24,s22}, but the other
authors would like to examine the effects of the broken
lepton-numbers. Here we will suppress ${\cal W}_{B}$ for all
generations totally. The first two terms in ${\cal W}_{L}$ in
Eq.\ (\ref{eq-3}) have received a lot of
consideration, and many restrictions on them have been
derived from existing experimental
data\cite{s7,s3,s4,s6,s25,se1}.
However, the term $\epsilon_{I}\varepsilon_{ij}
\hat{H}_{i}^{2}\hat{L}_{j}^{J}$ is also a viable agent for
R-parity breaking. It is particularly interesting because it
with proper SUSY soft-breaking terms
can result in observable factors that cannot
be effected by the trilinear terms alone. One of these
distinctive effects which we would like to mention
here is that, at tree level the
lightest neutralino can decay invisibly into three neutrinos,
which is not possible if only the trilinear terms in
${\cal W}_{L}$ are presented. In addition, such as
non-zero vacuum expectation values (VEVs)
of some sneutrinos or/and bilinear violation terms
will cause `fresh' mixing and different phenomenology etc,
the interesting results are obtained\cite{s21,s11,chf,s5,s55}.
Whereas what happens to the most
general case where all possible lepton-number violations
(not only from superpotential but also from the terms
causing SUSY soft breaking) are simultaneously involved,
is still an interesting problem to be investigated, thus
we would like to turn to the problem in the paper.
Indeed in this kind model there is a freedom in principle for
defining the superfields, thus remarkable problems, how big
of the freedom and how to recognize two different
`parameterization' manners of the same model among
this kind of models, emerge. Therefore we will start
with the general SUSY version
and keep all the possible R-parity violation terms in the
superpotential ${\cal W}_{L}$ and in the SUSY soft-breaking
Lagrangian properly, then to work out the Lagrangian in
`component version'\footnote{In fact, we may consider
the resultant effective theory as a SUSY superfield version,
renormalized at the energy-scale of SUSY breaking.
Thus the full renormalization at weak-interaction energy-scale
on the parameters in the effective Lagrangian should be made
only `further' in component version.}.
All possible mixing and Feynman rules for
further precise studies of the phenomenology of
the R-parity violation effective theory will be given
precisely. The straightforward deductions for these
pursers are omitted and lengthy formulas are put into
Appendices. Moreover we put the problem aside that the effective
theory may have a more fundamental origin at a comparatively
high energy scale, although it is interesting and
the effective theory may be helpful to trace out some
clue on the problem\footnote{Some parameters maybe vanish
due to higher symmetries if one relates the theory to
a specific GUT model\cite{s9}, whereas the investigation here
is still applicable as long as to set the
corresponding parameters to vanish.}. As for the freedom
for defining the superfields, we would also take one section
to demonstrate how big it is precisely and make some suggestions
on it for later conveniences in later applications.
The paper is organized as follows: in Sect.II, we describe the
basic ingredient of the SUSY without R-parity through various
lepton-number breaking. The mass matrices of the CP-even Higgs,
CP-odd Higgs and charged Higgs are derived precisely. As an
important result, relations for CP-even and CP-odd Higgs masses
as those in the bilinear case\cite{chf}, are recovered.
For completeness, we also give the mixing matrix of charginos and
charged leptons, and that of neutralinos and neutrinos.
In Sect.III, the Feynman rules for the interactions
relevant to R-parity violation, i.e., those of
the Higgs bosons (sleptons) with the gauge bosons, and the
charginos, neutralinos with gauge bosons or Higgs bosons
(sleptons) are presented. The self interactions of the Higgs
and the interactions of chargino (neutralino)-squark-quark
are also given. In Sect.IV, we examine the
freedom\cite{s7,s25,se1} for re-defining the Higgs
superfield and the $n-$lepton superfields which are
relevant to the R-parity violation. We precisely show the
equivalence for two superficially different parameterizations
generated by two sets of the $n+1$ superfields, if the
two sets of the $n+1$ superfields may be connected by a $U(n+1)$
transformation exactly, hence the $U(n+1)$ transformation can be
understood as a freedom for redefining the Higgs and lepton
superfields at very beginning. In Sect.V, we try to consider
the comparatively interesting `particle spectrum'
numerically under a few further reasonable assumptions on
the parameter space of the model partly for simplifying the
practical calculations.
\section{The physical masses in the MSSM without R-parity}
Generally the lepton-number violations in a MSSM not only cause
R-parity broken but also make quite a lot of fresh
and interesting mixings between particles and sparticles.
Let us examine the subject for the model with various
lepton-number violations in this section.
Since those parts, such as gauge, matter and the
gauge-matter interactions etc, in the model are the same
as the MSSM, thus we will omit them in the paper everywhere
except special needs.
As stated above, we are to consider the superpotential,
(to combine Eq.(1) and Eq.(2)):
\begin{eqnarray}
{\cal W} &=& \mu \varepsilon_{ij} \hat{H}_{i}^{1} \hat{H}_{j}^{2}
+ l_{I}\varepsilon_{ij}\hat{H}_{i}^{1}\hat{L}_{j}^{I}\hat{R}^{I} -
u^{I}(\hat{H}_{1}^{2} C^{JI*}\hat{Q}_{2}^{J} -
\hat{H}_{2}^{2}\hat{Q}_{1}^{I})\hat{U}^{I} \nonumber \\
& & - d^{I}(\hat{H}_{1}^{1}\hat{Q}_{2}^{I} - \hat{H}_{2}^{1} C^{IJ}
\hat{Q}_{1}^{J})\hat{D}^{I} +
\varepsilon_{ij} [ \lambda_{IJK} \hat{L}_{i}^{I} \hat{L}_{j}^{J}
\hat{R}^{K} + \epsilon_{I} \hat{H}_{i}^{2} \hat{L}_{j}^{I}] \nonumber \\
&& + \lambda_{IJK}^{\prime}(\hat{L}_{1}^{I}\hat{Q}_{2}^{J}\delta_{JK} -
\hat{L}_{2}^{I} C^{JK}
\hat{Q}_{1}^{K})\hat{D}^{J}
\label{eq-4}
\end{eqnarray}
with $\mu$, $\epsilon_{I}$ are the parameters with units of mass,
$u^{I}$, $d^{I}$ and $l^{I}$ are the Yukawa couplings as in the
MSSM with R-parity, and the parameters $\lambda_{IJK}$,
$\lambda_{IJK}^{\prime}$ describe the trilinear R-parity violation.
Since now we consider the case with three families thus the
subscripts $I,J,K=1,2,3$.
To break the SUSY so as to have a correct phenomenology,
the general soft SUSY-breaking terms are introduced
accordingly:
\begin{eqnarray}
{\cal L}_{soft} & = & -m_{H^{1}}^{2}H_{i}^{1*}H_{i}^{1} -
m_{H^{2}}^{2} H_{i}^{2*}
H_{i}^{2}-m_{L^{I}}^{2} \tilde{L}_{i}^{I*} \tilde{L}_{i}^{I} -
m_{R^{I}}^{2}\tilde{R}^{I*}
\tilde{R}^{I} \nonumber \\
&& -\sum\limits_{I}m_{HL^{I}}^{2}(H_{i}^{1*}\tilde{L}_{i}^{I}
+ H_{i}^{1}\tilde{L}_{i}^{I*}) - \sum\limits_{I\neq J}
m_{L^{IJ}}^{2}\tilde{L}_{i}^{I*}\tilde{L}_{i}^{J} - \sum\limits_{I\neq J}
m_{R^{IJ}}^{2}\tilde{R}^{I*}\tilde{R}^{J} \nonumber \\
& & -m_{Q^{I}}^{2} \tilde{Q}_{i}^{I*} \tilde{Q}_{i}^{I} - m_{D^{I}}^{2}
\tilde{D}
^{I*} \tilde{D}^{I} - m_{U^{I}}^{2}\tilde{U}^{I*} \tilde{U}^{I} + (m_{1}
\lambda_{B}
\lambda_{B} \nonumber \\
& & + m_{2}\lambda_{A}^{i}\lambda_{A}^{i} +
m_{3} \lambda_{G}^{a}\lambda_{G}^{a} + h.c.) +
\{ B \varepsilon_{ij}H_{i}^{1}H_{j}^{2} +
B_{I}\varepsilon_{ij}H_{i}^{2}\tilde{L}_{j}^{I} \nonumber \\
& & + \varepsilon_{ij} l_{sI} H_{i}^{1}\tilde{L}_{j}^{I}\tilde{R}^{I} +
d_{sI} (-H_{1}^{1}\tilde{Q}_{2}^{I} + C^{IK}H_{2}^{1}\tilde{Q}_{1}^{K})
\tilde{D}^{I} \nonumber \\
& &+ u_{sI} (-C^{KI*}H_{1}^{2}\tilde{Q}_{2}^{I} +
H_{2}^{2}\tilde{Q}_{1}^{I})\tilde{U}^{I} +
\varepsilon_{ij}\lambda_{IJK}^{S}\tilde{L}_{i}^{I}\tilde{L}_{j}^{J}
\tilde{R}^{K} \nonumber \\
& & + \lambda_{IJK}^{S^{\prime}}(\tilde{L}_{1}^{I}\tilde{Q}_{2}^{J}
\delta^{JK} - \tilde{L}_{2}^{I}C^{JK}\tilde{Q}_{1}^{J})\tilde{D}^{K} + h.c.\}
\label{eq-5}
\end{eqnarray}
where $m_{H^{1}}^{2}$, $m_{H^{2}}^{2}$, $m_{L^{I}}^{2}$, $m_{R^{I}}^{2}$,
$m_{Q^{I}}^{2}$,
$m_{D^{I}}^{2}$, $m_{U^{I}}^{2}$, $B$ and $B_{I}$ are the 'bare" mass
parameters while $m_{3}$, $m_{2}$, $m_{1}$ denote the masses of
$\lambda_{G}^{a}$, $\lambda_{A}^{i}$ and $\lambda_{B}$,
the $SU(3)\times SU(2) \times U(1)$ gauginos.
$d_{sI}$, $u_{sI}$, $l_{sI}$ $(I=1,2,3)$ and $\lambda_{IJK}^{S}$,
$\lambda_{IJK}^{S^{\prime}}$
are the soft breaking
parameters that make necessary mass splitting between the quarks,
leptons and their supersymmetric partners. To correspond to the
superpotential in Eq.\ (\ref{eq-4}), all the possible lepton number
violation terms for breaking SUSY softly are involved in
Eq.\ (\ref{eq-5}).
In general, the scalar potential of the model can be written as
\begin{eqnarray}
V &=& \sum_{i} |\frac{\partial{\cal W}}{\partial A_{i}}|^{2} + V_{D} +
V_{soft} \nonumber \\
&=& V_{F} + V_{D} + V_{soft}
\label{eq-6}
\end{eqnarray}
where $A_{i} (i=\cdots)$ denote scalar components, $V_{D}$
the usual D-terms, $V_{soft}$ just the SUSY soft breaking terms given in
Eq.\ (\ref{eq-5}). Using the superpotential Eq.\ (\ref{eq-4}) and the soft
breaking terms Eq.\ (\ref{eq-5}), we can write down the scalar
potential precisely with the following forms:
\begin{eqnarray}
V_{F} &=& |\frac{\partial{\cal W}}{\partial H^{1}}|^{2} + |\frac{\partial
{\cal W}}{\partial H^{2}}|^{2} + |\frac{\partial{\cal W}}{\partial
\tilde{L}^{I}
}|^{2} + |\frac{\partial{\cal W}}{\partial \tilde{R}^{I}}|^{2} +
|\frac{\partial{\cal W}}{\partial \tilde{Q}^{I}}|^{2} +
|\frac{\partial{\cal W}}{\partial \tilde{U}^{I}}|^{2} +
|\frac{\partial{\cal W}}{\partial \tilde{D}^{I}}|^{2} \; . \nonumber \\
\end{eqnarray}
As MSSM but more general, the electroweak symmetry in the model
is broken spontaneously if the two Higgs
doublets $H^{1}$, $H^{2}$, and the sleptons as well
acquire nonzero vacuum expectation values (VEVs):
\begin{equation}
H^{1}=
\left(
\begin{array}{c}
\frac{1}{\sqrt{2}}(\chi_{1}^{0} + \upsilon_{1} + i\phi_{1}^{0}) \\
H_{2}^{1} \end{array} \right)
\label{H1-vacuum}
\end{equation}
\begin{equation}
H^{2}=
\left(
\begin{array}{c}
H_{1}^{2} \\
\frac{1}{\sqrt{2}}(\chi_{2}^{0} + \upsilon_{2} + i\phi_{2}^{0})
\end{array} \right)
\label{H2-vacuum}
\end{equation}
and
\begin{equation}
\tilde{L}^{I} =
\left(
\begin{array}{c}
\frac{1}{\sqrt{2}}(\chi_{\tilde{\nu}_{I}}^{0} +
\upsilon_{\tilde{\nu}_{I}} + i\phi_{\tilde{\nu}_{I}}^{0}) \\
\tilde{L}_{2}^{I} \end{array} \right)
\label{L3-vacuum}
\end{equation}
where $\tilde{L}^{I}$ denote the slepton SU(2)
doublets, and $I=e$, $\mu$, $\tau$, i.e. the indices of the
three families for leptons. From
Eqs.\ (\ref{eq-6},\ref{H1-vacuum},\ref{H2-vacuum},\ref{L3-vacuum})
it is easy to find the scalar potential includes the linear terms as
follows:
\begin{equation}
V_{tadpole} = t_{1}^{0}\chi_{1}^{0} + t_{2}^{0} \chi_{2}^{0}
+ t_{\tilde{\nu}_{e}}^{0} \chi_{\tilde{\nu}_{e}}^{0} +
t_{\tilde{\nu}_{\mu}}^{0} \chi_{\tilde{\nu}_{\mu}}^{0} +
t_{\tilde{\nu}_{\tau}}^{0} \chi_{\tilde{\nu}_{\tau}}^{0}
\label{tadpole0}
\end{equation}
where
\begin{eqnarray}
t_{1}^{0} &=& \frac{1}{8}(g^{2} +
g^{\prime^{2}})\upsilon_{1}(\upsilon_{1}^{2} - \upsilon_{2}^{2} +
\sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2})
+ |\mu|^{2}\upsilon_{1} + m_{H^{1}}^{2}\upsilon_{1} \nonumber \\
&& +\sum\limits_{I}m_{HL^{I}}^{2}\upsilon_{\tilde{\nu}_{\tau}}-
B \upsilon_{2} -
\sum_{I} \mu \epsilon_{I}\upsilon_{\tilde{\nu}_{I}}, \nonumber \\
t_{2}^{0} &=& -\frac{1}{8}(g^{2} +
g^{\prime^{2}})\upsilon_{2}(\upsilon_{1}^{2} -
\upsilon_{2}^{2} + \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2})
+ |\mu|^{2}\upsilon_{2} + m_{H^{2}}^{2}\upsilon_{2} -
B \upsilon_{1} + \sum_{I}\epsilon_{I}^{2}\upsilon_{2} \nonumber \\
& & +\sum_{I} B_{I}\upsilon_{\tilde{\nu}_{I}},
\nonumber \\
t_{\tilde{\nu}_{I}}^{0} &=& \frac{1}{8}(g^{2} +
g^{\prime^{2}})\upsilon_{\tilde{\nu}_{I}}(\upsilon_{1}^{2} -
\upsilon_{2}^{2} +
\sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2})
+m_{L^{I}}^{2}\upsilon_{\tilde{\nu}_{I}} +
\epsilon_{I}\sum_{J}\epsilon_{J}\upsilon_{\tilde{\nu}_{J}} \nonumber \\
& & - \mu\epsilon_{I}
\upsilon_{1} + B_{I}\upsilon_{2} + m_{HL^{I}}^{2}\upsilon_{1}
+\sum\limits_{J\neq I}m_{L^{IJ}}^{2}\upsilon_{\tilde{\nu}_{J}}.
\label{tadpole1}
\end{eqnarray}
Here $t_{i}^{0}$ ($i=1$, $2$, $\tilde{\nu}_{e}$, $\tilde{\nu}_{\mu}$,
$\tilde{\nu}_{\tau}$)
are tadpoles at `tree level', thus the true VEVs
of the neutral scalar fields should satisfy the condition $t_{i}^{0}=0$,
($i=1$, $2$, $\tilde{\nu}_{e}$, $\tilde{\nu}_{\mu}$,
$\tilde{\nu}_{\tau}$), therefore we obtain:
\begin{eqnarray}
m_{H^{1}}^{2} &=& -\Bigg(|\mu|^{2} - \sum_{I}\bigg(\epsilon_{I}
\mu - m_{HL^{I}}^{2}\bigg)\frac{\upsilon_{\tilde{\nu}_{I}}}{\upsilon_{1}} -
B\frac{\upsilon_{2}}{\upsilon_{1}}
+ \frac{1}{8}\bigg(g^{2} + g^{\prime^{2}}\bigg)
\bigg(\upsilon_{1}^{2} - \upsilon_{2}^{2}
+ \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2}\bigg)\Bigg), \nonumber \\
m_{H^{2}}^{2} &=& -\Bigg(|\mu|^{2} + \sum_{I}\epsilon_{I}^{2} +
\sum_{I}B_{I}\frac{\upsilon_{\tilde{\nu}_{I}}}{\upsilon_{2}} -
B\frac{\upsilon_{1}}{\upsilon_{2}} -
\frac{1}{8}\bigg(g^{2} + g^{\prime^{2}}\bigg)\bigg(\upsilon_{1}^{2}
- \upsilon_{2}^{2} +
\sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2}\bigg)\Bigg), \nonumber \\
m_{L^{I}}^{2} &=& -\Bigg(\frac{1}{8}\bigg(g^{2} + g^{\prime^{2}}\bigg)
\bigg(\upsilon_{1}^{2}
- \upsilon_{2}^{2} + \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{2}\bigg)
+ \epsilon_{I}\sum_{J}\epsilon_{J}\frac{\upsilon_{\tilde{\nu}_{J}}}
{\upsilon_{\tilde{\nu}_{I}}} -
\epsilon_{I}\mu\frac{\upsilon_{1}}{\upsilon_{\tilde{\nu}_{I}}} \nonumber \\
& & +B_{I}\frac{\upsilon_{2}}{
\upsilon_{\tilde{\nu}_{I}}} + m_{HL^{I}}^{2}\frac{\upsilon_{1}}
{\upsilon_{\tilde{\nu}_{I}}} + \sum\limits_{J\neq I}m_{L^{IJ}}^{2}
\frac{\upsilon_{\tilde{\nu}_{J}}}{\upsilon_{\tilde{\nu}_{I}}}
\Bigg), \hspace{5mm} (I=e, \mu, \tau). \nonumber \\
\label{masspara}
\end{eqnarray}
For convenience, later on we will call all of these scalar bosons ($H^{1}$,
$H^{2}$ and $\tilde{L}^{I}$) as `Higgs'.
As for the scalar sector, the Higgs-boson mass-matrices
squared may be obtained by:
\begin{equation}
{\cal M}_{ij}^{2} = \frac{\partial^{2} V}{\partial \phi_{i}
\phi_{j}}|_{minimum},
\label{getsmatrix}
\end{equation}
here `minimum' means to evaluate the values at $<H_{1}^{1}> =
\frac{\upsilon_{1}}{\sqrt{2}}$, $<H_{2}^{2}> =
\frac{\upsilon_{2}}{\sqrt{2}}$,
$<\tilde{L}_{1}^{I}> = \frac{\upsilon_{\tilde{\nu}_{I}}}{\sqrt{2}}$ and
$<A_{i}>=0$ ($A_{i}$ represent all the other scalar fields).
Note that the matrices of the CP-even and the CP-odd scalar
bosons both are $5\times 5$ as we have the sneutrinos
which correspond to three left-handed neutrinos;,
whereas the matrix of the charged Higgs is $8\times 8$
as we have the charged sleptons which correspond to
three left-handed and three right-handed charged leptons.
Now let us summarize the results and try to
make the matrices diagonal in the following subsections.
\subsection{The mass matrices for Higgs}
The mass terms of the
CP-even Higgs from the scalar potential Eq.\ (\ref{eq-6}):
\begin{equation}
{\cal L}_{m}^{even} = - \Phi_{even}^{\dag} {\cal M}_{even}^{2} \Phi_{even}
\label{l-add1}
\end{equation}
are obtained with the interaction CP-even Higgs fields
$\Phi_{even}^{T} = (\chi_{1}^{0}$, $\chi_{2}^{0}$,
$\chi_{\tilde{\nu}_{e}}^{0}$, $\chi_{\tilde{\nu}_{\mu}}^{0}$,
$\chi_{\tilde{\nu}_{\tau}}^{0})$
as basis, and the corresponding mass matrix is
\begin{equation}
{\cal M}_{even}^{2} = \left(
\begin{array}{ccccc}
r_{11} & -e_{12} - B & e_{13} - \mu\epsilon_{1} &
e_{14} - \mu\epsilon_{2} & e_{15} - \mu\epsilon_{3} \\
-e_{12} - B & r_{22} & -e_{23} + B_{1} & -e_{24} +
B_{2} & -e_{25} + B_{3} \\
e_{13} - \mu\epsilon_{1} & -e_{23} + B_{1} & r_{33} &
e_{34} + \epsilon_{1}\epsilon_{2} & e_{35} + \epsilon_{1}\epsilon_{3} \\
e_{14} - \mu\epsilon_{2} & -e_{24} + B_{2} & e_{34} +
\epsilon_{1}\epsilon_{2} & r_{44} & e_{45} + \epsilon_{2}\epsilon_{3} \\
e_{15} - \mu\epsilon_{3} & -e_{25} + B_{3} & e_{35} +
\epsilon_{1}\epsilon_{3} & e_{45} + \epsilon_{2}\epsilon_{3} & r_{55}
\end{array}
\right)\; .
\label{matrix-even}
\end{equation}
The parameters appearing in the matrix elements are defined in Appendix A.
Note that when obtaining the above mass matrix, the Eq.\ (\ref{masspara})
is used.
The physical CP-even `Higgs' $H_i^0$ (eigenvalues
and corresponding eigenstates) are obtained
by means of a standard method to make the matrix
Eq.\ (\ref{matrix-even}) diagonal.
Namely we may find a unitary matrix $Z_{even}$:
\begin{equation}
H_{i}^{0} = \sum_{j=1}^{5}Z_{even}^{ij}\chi_{j}^{0}\;,
\label{masseven}
\end{equation}
where $Z^{ij}_{even}$ ($i$, $j=1$, $2$, $3$, $4$, $5$) are the
elements of the matrix that converts the mass matrix
Eq.\ (\ref{matrix-even}) into a diagonal one:
$${\cal L}_{m}^{even} = - \Phi_{even}^{\dag}
\cdot {\cal M}_{even}^{2} \cdot \Phi_{even} =
- {\cal H}^{0 \dag} \cdot {\cal M'}_{even}^{2} \cdot {\cal H}^0\;,$$
where
$$ {\cal M'}_{even}^2 = Z_{even} \cdot {\cal M}_{even}^2 \cdot
Z_{even}^\dag = diag (m_{H^0_1}^2, m_{H^0_2}^2, m_{H^0_3}^2,
m_{H^0_5}^2, m_{H^0_1}^2)\; .$$
Thus $\Phi_{even}$ are the `interaction fields' and ${\cal
H}^0$ are the physical fields (the eigenstates of the mass matrix).
As for the CP-odd sector of the Higgs, with the `interaction'
basis $\Phi_{odd}^{T} =
(\phi_{1}^{0}$, $\phi_{2}^{0}$, $\phi_{\tilde{\nu}_{e}}^{0}$,
$\phi_{\tilde{\nu}_{\mu}}^{0}$, $\phi_{\tilde{\nu}_{\tau}}^{0})$,
the mass matrix for the CP-odd `Higgs' can be written as:
\begin{equation}
{\cal M}_{odd}^{2} =
\left(
\begin{array}{ccccc}
s_{11} & B & -\mu\epsilon_{1}+ m_{HL^{1}}^{2} &
-\mu\epsilon_{2}+ m_{HL^{2}}^{2} & -\mu\epsilon_{3}+ m_{HL^{3}}^{2} \\
B & s_{22} & -B_{1} &
-B_{2} & -B_{3} \\
-\mu\epsilon_{1}+ m_{HL^{1}}^{2} & -B_{1} &
s_{33} & \epsilon_{1}\epsilon_{2}+ m_{L^{12}}^{2} &
\epsilon_{1}\epsilon_{3}+ m_{L^{13}}^{2} \\
-\mu\epsilon_{2}+ m_{HL^{2}}^{2} & -B_{2} &
\epsilon_{1}\epsilon_{2}+ m_{L^{12}}^{2} & s_{44} &
\epsilon_{2}\epsilon_{3}+ m_{L^{23}}^{2} \\
-\mu\epsilon_{3}+ m_{HL^{3}}^{2} & -B_{3} &
\epsilon_{1}\epsilon_{3}+ m_{L^{13}}^{2} &
\epsilon_{2}\epsilon_{3}+ m_{L^{23}}^{2} & s_{55}
\end{array}
\right)\; . \label{massodd}
\end{equation}
The parameters appearing in the matrix elements
are defined precisely in Appendix A.
To be different from the CP-even sector,
it is easy, as Ref. \cite{s11}
from Eq.\ (\ref{massodd}), to find a neutral
Goldstone boson (with zero eigenvalue):
\begin{eqnarray}
H_{6}^{0} &=& \sum_{i=1}^{5} Z_{odd}^{1i} \phi_{i}^{0} \nonumber \\
&=& \frac{1}{\upsilon}(\upsilon_{1}\phi_{1}^{0} -
\upsilon_{2}\phi_{2}^{0} +
\upsilon_{\tilde{\nu}_{e}}\phi_{\tilde{\nu}_{e}}^{0}
+ \upsilon_{\tilde{\nu}_{\mu}}\phi_{\tilde{\nu}_{\mu}}^{0} +
\upsilon_{\tilde{\nu}_{\tau}}\phi_{\tilde{\nu}_{\tau}}^{0}),
\label{ngold}
\end{eqnarray}
which is indispensable for spontaneously breaking the EW gauge symmetry.
Here the $\upsilon=\sqrt{\upsilon_{1}^{2} + \upsilon_{2}^{2} +
\sum\limits_{I}\upsilon_{\tilde{\nu}_{I}}^{2}}$ and similar
to the R-parity conserved MSSM, the mass of $Z$-boson
$m_{Z} = \frac{\sqrt{g^{2} + g^{\prime^{2}}}}{2}\upsilon$ is kept.
The other four massive neutral bosons can be written as:
\begin{equation}
H_{5+i}(i=2, 3, 4, 5)=\sum_{j=1}^{5}Z_{odd}^{ij}\phi_{j}^{0}
\label{oddhiggs}
\end{equation}
where again $Z^{ij}_{odd}$ ($i$, $j=1$, $2$, $3$, $4$, $5)$ is
the matrix elements and the matrix converts the interaction fields
into the physical ones.
From the eigenvalue equations for CP-even and CP-odd
`Higgs' and the identities of the model, similar to
the case of Ref.\cite{chf},
it is not very difficult to find two independent
relations for the eigenvalues as follows:
\begin{eqnarray}
\sum_{i=1}^{5}m_{H_{i}}^{2} &=& \sum_{i=2}^{5}m_{H_{5+i}}^{2}
+ m_{Z}^{2}, \nonumber \\
\prod_{i=1}^{5}m_{H_{i}}^{2} &=& \Bigg[\frac{\upsilon_{1}^{2} -
\upsilon_{2}^{2} +
\sum\limits_{I=1}^{3}\upsilon_{\tilde{\nu}_{I}}^{2}}
{\upsilon^{2}}\Bigg]^{2} m_{Z}^{2} \prod_{i=2}^{5}m_{H_{5+i}}^{2}.
\label{massrelation}
\end{eqnarray}
The first relation of Eq.\ (\ref{massrelation})
is obtained by relating the traces of the two
neutral Higgs mass matrices (CP-even and CP-odd)
and the second is relating the determinants of the
mass matrices. Note: there is a Goldstone in CP-odd
sector, thus to obtain the second relation of
Eq.\ (\ref{massrelation}), the Goldstone mode must
have been taken away already, and it is reason why
the multi-product in the r.h.s. of the equation is
start from $2$ (According to the convention here the
number $1$ corresponds to the Goldstone).
If we introduce the following notations:
\begin{eqnarray}
& & \upsilon_{1} = \upsilon\cos\beta\cos\theta_{\upsilon} \; ,
\nonumber \\
& & \upsilon_{2} = \upsilon\sin\beta \; , \nonumber \\
& & \sqrt{\sum\limits_{I=1}^{3}\upsilon_{\nu_{I}}^{2}} =
\upsilon\cos\beta\sin\theta_{\upsilon} \; ,
\label{defineang}
\end{eqnarray}
the second relation of Eq.\ (\ref{massrelation}) becomes:
\begin{equation}
\prod_{i=1}^{5}m_{H_{i}}^{2} = \cos^{2}2\beta m_{Z}^{2}
\prod_{i=2}^{5}m_{H_{5+i}}^{2}\;.
\label{massrelation1}
\end{equation}
The first relation of Eq.\ (\ref{massrelation}) was obtained in
Ref.\cite{s12} firstly. The second relation of Eq.\ (\ref{massrelation})
was obtained in Ref.\cite{chf} firstly in special case of the bilinear
R-parity violating.
The two equations are independent, and restrict the masses of
the neutral `Higgs' bosons substantially. As discussed in Ref.\cite{chf},
for instance, with Eq.\ (\ref{massrelation}),
Eq.\ (\ref{massrelation1}) and simple algebra reduction,
the upper limit on the mass of the lightest Higgs at tree level
\begin{equation}
m_{H_{1}}^{2} \leq
m_{H_{n}}^{2}\Bigg(\frac{m_{Z}^{2}\cos^{2}2\beta}{m_{H_{n}}^{2}}
\Bigg)^{\frac{1}{n-1}}
\frac{1-\frac{1}{n-1}\frac{m_{Z}^{2}}{m_{H_{n}}^{2}}}{1-\frac{1}{n-1}
\bigg(\frac{m_{Z}^{2}\cos^{2}2\beta}{m_{H_{n}}^{2}}\bigg)^{\frac{1}{n-1}}},
\label{bound-masshiggs}
\end{equation}
is obtained straightforwardly.
Here $n \geq 2$ is the number of the CP-even `Higgs' (the `original'
Higgs and the sneutrinos), $m_{H_{1}}$ is the mass of the lightest
one among them, whereas $m_{H_{n}}$ is the heaviest one.
On Eq.\ (\ref{bound-masshiggs}) two points
should be noted:
\begin{itemize}
\item When $n=2$ or $m_{H_{1}}^{2} = \cdots = m_{H_{n}}^{2} =
m_{H_{n+2}}^{2}=\cdots =
m_{H_{n+n}}^{2}=m_{Z}^{2}$, and $\cos^{2}2\beta=1$, the symbol "="
is established.
\item In the case of MSSM with R-parity i.e. n=2,
$m_{H_{1}}=m_{Z}^{2}\cos^{2}2\beta\frac{1-\frac{m_{Z}^{2}}{m_{H_{2}}
^{2}}}{1-\frac{m_{Z}^{2}}{m_{H_{2}}^{2}}\cos^{2}2\beta}
\leq m_{Z}^{2}\cos^{2}2\beta $ is recovered.
\end{itemize}
In present case when $n > 2$, such a strong constraint on the
lightest Higgs mass $m_{H_{1}}$ at tree level as that for the
R-parity conserved MSSM\cite{s13,s19}
$$m_{H_{1}}^{2} \leq m_{Z}^{2}\cos^{2} 2\beta \leq m_{Z}^{2}$$
cannot be obtained.
In the MSSM with R-parity, the radiative corrections make
the mass of the lightest Higgs larger than that of tree level
when completing one-loop corrections and leading two-loop
corrections of ${\cal O}(\alpha\alpha_{s})$
are included\cite{sg1}. For instance the
Ref.\cite{sg2} by precise loop calculations sets the
limit on the lightest Higgs
mass: $m_{H_{1}^{0}}\leq 132$GeV.
In the MSSM without R-parity, as indicated here there is no such a
stringent restriction on the lightest Higgs at tree level
as R-parity conserved one, hence one can quite be sure that the
`theoretical' bound on the lightest Higgs mass must be loosened
a lot (i.e. it can be heavier than the restriction from MSSM with
R-parity conservation), especially, when loop corrections are
involved . In Section V we will
show the indications of Eqs.(\ref{massrelation},
\ref{massrelation1}) on the lightest Higgs mass more precisely
numerically.
\subsection{The mass matrix for charged Higgs\label{app13}}
With the interaction basis $\Phi_{c}=(H_{2}^{1*}$, $H_{1}^{2}$,
$\tilde{L}_{2}^{1*}$, $\tilde{L}_{2}^{2*}$, $\tilde{L}_{2}^{3*}$,
$\tilde{R}^{1}$, $\tilde{R}^{2}$, $\tilde{R}^{3})$
\footnote{The model which we are considering here is that there
is no right-handed neutrinos at all thus there are neutrinos'
SUSY partners corresponding to the lift-handed ones, but as for
charged leptons, there are not only left-handed ones but
also right-handed ones, thus the numbers of `charged Higgs'
are 8 instead of 5 for the `neutral Higgs' (more than 3 in
three generations of leptons.}
and Eq.\ (\ref{eq-6}),
it is easy to obtain the following mass terms for charged
`Higgs':
\begin{equation}
{\cal L}_{m}^{C} = -\Phi_{c}^{\dag}{\cal M}_{c}^{2}\Phi_{c},
\label{eq-20}
\end{equation}
the symmetric matrix ${\cal M}_{c}^{2}$ is given as Appendix A.
Making the mass matrix diagonal, a zero mass Goldstone boson state:
\begin{eqnarray}
H_{1}^{+} &=& \sum_{i=1}^{8}Z_{c}^{1i}\Phi_{c}^{i} \nonumber \\
&=& \frac{1}{\upsilon}(\upsilon_{1}H_{2}^{1*} - \upsilon_{2}H_{1}^{2} +
\upsilon_{\tilde{\nu}_{e}}\tilde{L}_{2}^{1*} +
\upsilon_{\tilde{\nu}_{\mu}}\tilde{L}_{2}^{2*} +
\upsilon_{\tilde{\nu}_{\tau}}\tilde{L}_{2}^{3*})
\label{cgold}
\end{eqnarray}
is obtained. Together with its charge conjugate state $H_{1}^{-}$
are needed to break electroweak symmetry and give $W^{\pm}$ bosons
masses. With the transformation matrix $Z_{c}^{ij}$ (to convert
the interaction fields into
the physical eigenstates), the other seven physical eigenstates
$H_{i}^{+}$ $(i=2$, $3$, $4$, $5$, $6$, $7$, $8)$ can be expressed as:
\begin{equation}
H_{i}^{+} = \sum_{j=1}^{8}Z_{c}^{ij}\Phi^{c}_{j} \hspace{0.2in}(i, j=1,
\cdots, 8).
\label{charhiggs}
\end{equation}
\subsection{The mixing of neutralinos and neutrinos:}
Due to the lepton number violations in the model, fresh and
interesting mixing of neutralinos-neutrinos and charginos-charged leptons
may happen. We devote two subsections to outline the mixing and
solve them numerically late in Sect.V. The piece of Lagrangian
responsible for the mixing of neutralinos and neutrinos is:
\begin{eqnarray}
{\cal L}_{\chi_{i}^{0}}^{mass} &=& \{
ig\sqrt{2}T_{ij}^{a}\lambda^{a}\psi_{j}A_{i}^{*} -
\frac{1}{2}\frac{\partial^{2}
{\cal W}}{\partial A_{i} \partial A_{j}}\psi_{i}\psi_{j} + h.c. \} +
m_{1}(\lambda_{B}\lambda_{B} + h.c.) + \nonumber \\
& & m_{2}(\lambda_{A}^{i}\lambda_{A}^{i} + h.c.)
\label{massneutra}
\end{eqnarray}
where ${\cal W}$ is given by Eq.\ (\ref{eq-4}). $T^{a}$ are the
generators of the SU(2)$\times$U(1) gauge group and $\psi$,
$A_{i}$ stand for generic two-component fermion and scalar fields.
Writing down the Eq.\ (\ref{massneutra}) explicitly, we obtain:
\begin{equation}
{\cal L}_{\chi_{i}^{0}}^{mass} = -\frac{1}{2}(\Phi^{0})^{T}
{\cal M}_{N}\Phi^{0} + h.c.
\label{massneutra1}
\end{equation}
with the interaction basis $(\Phi^{0})^{T} = (-i\lambda_{B}$,
$-i\lambda_{A}^{3}$,
$\psi_{H^{1}}^{1}$, $\psi_{H^{2}}^{2}$, $\nu_{e_{L}}$, $\nu_{\mu_{L}}$,
$\nu_{\tau_{L}})$ and
\begin{equation}
{\cal M}_{N} = \left(
\begin{array}{ccccccc}
2m_{1} & 0 & -\frac{1}{2}g^{\prime}\upsilon_{1} &
\frac{1}{2}g^{\prime}\upsilon_{2} &
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{e}} &
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\mu}} &
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\tau}} \\
0 & 2m_{2} & \frac{1}{2}g\upsilon_{1} & -\frac{1}{2}g\upsilon_{2} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{e}}
& \frac{1}{2}g\upsilon_{\tilde{\nu}_{\mu}} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{\tau}} \\
-\frac{1}{2}g^{\prime}\upsilon_{1} & \frac{1}{2}g\upsilon_{1} & 0 &
-\frac{1}{2}\mu & 0 & 0 & 0\\
\frac{1}{2}g^{\prime}\upsilon_{2} & -\frac{1}{2}g\upsilon_{2} &
-\frac{1}{2}\mu
& 0 & \frac{1}{2}\epsilon_{1}
& \frac{1}{2}\epsilon_{2} & \frac{1}{2}\epsilon_{3} \\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{e}} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{e}} & 0 & \frac{1}{2}\epsilon_{1}
& 0 & 0 & 0\\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\mu}} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{\mu}} & 0 & \frac{1}{2}\epsilon_{2}
& 0 & 0 & 0\\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\tau}} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{\tau}} & 0 & \frac{1}{2}\epsilon_{3}
& 0 & 0 & 0
\end{array} \right)\; .
\label{neutralino-matrix}
\end{equation}
The mixing has the formulation:
\begin{eqnarray}
& & -i\lambda_{B} = Z_{N}^{1i}\tilde{\chi}_{i}^{0},
\hspace{5mm}
-i\lambda_{A}^{3} = Z_{N}^{2i}\tilde{\chi}_{i}^{0},
\hspace{5mm}
\psi_{H^{1}}^{1} = Z_{N}^{3i}\tilde{\chi}_{i}^{0}, \nonumber \\
& &\psi_{H^{2}}^{2} = Z_{N}^{4i}\tilde{\chi}_{i}^{0},
\hspace{9mm}
\nu_{e_{L}} = Z_{N}^{5i}\tilde{\chi}_{i}^{0},
\hspace{9mm}
\nu_{\mu_{L}} = Z_{N}^{6i}\tilde{\chi}_{i}^{0}, \nonumber \\
& &\nu_{\tau_{L}} = Z_{N}^{7i}\tilde{\chi}_{i}^{0}
\label{mixing-neutralino}
\end{eqnarray}
and the transformation matrix $Z_{N}$ has the property
\begin{eqnarray}
Z_{N}^{T}{\cal M}_{N}Z_{N} &=& diag(m_{\tilde{\kappa}_{1}^{0}},
m_{\tilde{\kappa}_{2}^{0}},
m_{\tilde{\kappa}_{3}^{0}}, m_{\tilde{\kappa}_{4}^{0}},
m_{\nu_{e}},m_{\nu_{\mu}},m_{\nu_{\tau}}).
\label{define-ZN}
\end{eqnarray}
For convenience as in Ref.\cite{s13}, we formulate all the neutral
fermions into four component Majorana spinors as follows:
\begin{equation}
\nu_{e}=
\left( \begin{array}{c}
\tilde{\chi}_{5}^{0} \\
\bar{\tilde{\chi}}_{5}^{0} \end{array}
\right), \label{define-eneutrino}
\end{equation}
\begin{equation}
\nu_{\mu}=
\left( \begin{array}{c}
\tilde{\chi}_{6}^{0} \\
\bar{\tilde{\chi}}_{6}^{0} \end{array}
\right), \label{define-muneutrino}
\end{equation}
\begin{equation}
\nu_{\tau}=
\left( \begin{array}{c}
\tilde{\chi}_{7}^{0} \\
\bar{\tilde{\chi}}_{7}^{0} \end{array}
\right), \label{define-tauneutrino}
\end{equation}
\begin{equation}
\kappa_{i}^{0}(i=1, 2, 3, 4)=
\left( \begin{array}{c}
\tilde{\chi}_{i}^{0} \\
\bar{\tilde{\chi}}_{i}^{0} \end{array}
\right). \label{define-neutralino}
\end{equation}
It is easy from Eq.\ (\ref{neutralino-matrix})
to find that only one type
of neutrinos obtains mass from the mixing
at tree level, as pointed out by Ref. \cite{se6} firstly,
and we will assume it is $\tau$-neutrino naively.
One of the stringent restrictions comes from the bound
that the mass of $\tau$-neutrino should be less than
20 MeV\cite{s14}. Late on for convenience, we will call the
mixtures of neutralinos and
neutrinos as `neutralinos' shortly as long as there is no confusion.
\subsection{The mixing of charginos and charged leptons}
Similar to the mixing of neutralinos and neutrino, charginos
mix with the charged leptons and form a set of physical
charged fermions: $e^{-}$, $\mu^{-}$, $\tau^{-}$, $\kappa_{1}^{\pm}$,
$\kappa_{2}^{\pm}$. In the interaction basis, $\Psi^{+T}
=(-i\lambda^{+}$, $\tilde{H}_{2}^{1}$, $e_{R}^{+}$, $\mu_{R}^{+}$,
$\tau_{R}^{+})$ and $\Psi^{-T}=(-i\lambda^{-}$, $\tilde{H}_{1}^{2}$,
$e_{L}^{-}$, $\mu_{L}^{-}$, $\tau_{L}^{-})$, the charged fermion mass terms
of the Lagrangian have a general formulation\cite{s15}:
\begin{equation}
{\cal L}_{\chi_{i}^{\pm}}^{mass} = -\Psi^{-T}{\cal M}_{C} \Psi^{+} + h.c.
\label{masscharg1}
\end{equation}
and the mass matrix:
\begin{equation}
{\cal M}_{C} = \left(
\begin{array}{ccccc}
2m_{2} & \frac{e\upsilon_{2}}{\sqrt{2}S_{W}} & 0 & 0 & 0 \\
\frac{e\upsilon_{1}}{\sqrt{2}S_{W}} & \mu &
\frac{l_{1}\upsilon_{\tilde{\nu}_{e}}}{\sqrt{2}} &
\frac{l_{2}\upsilon_{\tilde{\nu}_{\mu}}}{\sqrt{2}} & \frac{l_{3}
\upsilon_{\tilde{\nu}_{\tau}}}{\sqrt{2}} \\
\frac{e\upsilon_{\tilde{\nu}_{e}}}{\sqrt{2}S_{W}} &
-\epsilon_{1} & \frac{l_{1}\upsilon_{1}}{\sqrt{2}}+
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I11}\upsilon_{\tilde{\nu}_{I}} &
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I12}\upsilon_{\tilde{\nu}_{I}} &
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I13}\upsilon_{\tilde{\nu}_{I}} \\
\frac{e\upsilon_{\tilde{\nu}_{\mu}}}{\sqrt{2}S_{W}} & -\epsilon_{2} &
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I21}\upsilon_{\tilde{\nu}_{I}} &
\frac{l_{2}\upsilon_{1}}{\sqrt{2}} +\frac{1}{\sqrt{2}}\sum\limits_{I}
\lambda_{I22}
\upsilon_{\tilde{\nu}_{I}} & \frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I23}
\upsilon_{\tilde{\nu}_{I}} \\
\frac{e\upsilon_{\tilde{\nu}_{\tau}}}{\sqrt{2}S_{W}} & -\epsilon_{3} &
\frac{1}{\sqrt{2}}
\sum\limits_{I}\lambda_{I31}\upsilon_{\tilde{\nu}_{I}} &
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I32}\upsilon_{\tilde{\nu}_{I}} &
\frac{l_{3}\upsilon_{1}}{\sqrt{2}} +
\frac{1}{\sqrt{2}}\sum\limits_{I}\lambda_{I33}
\upsilon_{\tilde{\nu}_{I}}
\end{array} \right)\; .
\label{chargino-matrix}
\end{equation}
Here $S_{W}=\sin\theta_{W}$ and $\lambda^{\pm} = \frac{\lambda_{A}^{1}
\mp i\lambda_{A}^{2}}{\sqrt{2}}$. Generally
two mixing matrices $Z_{+}$ and $Z_{-}$ can be obtained
by making the mass matrix ${\cal M}_{c}$ diagonal in a similar
way as in the SM to make the mass matrix of quark diagonal i.e.
the product $(Z_{+})^{T}{\cal M}_{C}Z_{-}$ turns to a diagonal matrix:
\begin{equation}
(Z_{+})^{T}{\cal M}_{C}Z_{-}= \left(
\begin{array}{ccccc}
m_{\kappa_{1}^{-}} & 0 & 0 & 0 &0 \\
0 & m_{\kappa_{2}^{-}} & 0 & 0 & 0 \\
0 & 0 & m_{e} & 0 & 0 \\
0 & 0 & 0 & m_{\mu} & 0 \\
0 & 0 & 0 & 0 & m_{\tau}
\end{array} \right)\; .
\label{dig-chargino}
\end{equation}
We denote the mass eigenstates with $\tilde{\chi}$ as follows:
\begin{eqnarray}
&&-i\lambda_{A}^{\pm} = Z_{\pm}^{1i}\tilde{\chi}_{i}^{\pm}, \hspace{3mm}
\psi_{H^{2}}^{1} = Z_{+}^{2i}\tilde{\chi}_{i}^{+}, \nonumber \\
&&\psi_{H^{1}}^{2} = Z_{-}^{2i}\tilde{\chi}_{i}^{-}, \hspace{6mm}
e_{L} = Z_{-}^{3i}\tilde{\chi}_{i}^{-}, \nonumber \\
&&e_{R} = Z_{+}^{3i}\tilde{\chi}_{i}^{+}, \hspace{8mm}
\mu_{L} = Z_{-}^{4i}\tilde{\chi}_{i}^{-}, \nonumber \\
&&\mu_{R} = Z_{+}^{4i}\tilde{\chi}_{i}^{+}, \hspace{8mm}
\tau_{L} = Z_{-}^{5i}\tilde{\chi}_{i}^{-}, \nonumber \\
&&\tau_{R} = Z_{+}^{5i}\tilde{\chi}_{i}^{+}.
\label{mixing-chargino}
\end{eqnarray}
The four-component fermions are defined as:
\begin{equation}
\kappa_{i}^{+}(i=1, 2, 3, 4, 5)=
\left( \begin{array}{c}
\tilde{\chi}_{i}^{+} \\
\bar{\tilde{\chi}}_{i}^{-} \end{array}
\right), \label{define-chargino}
\end{equation}
where $\kappa_{1}^{\pm}$, $\kappa_{2}^{\pm}$ are the usual charginos
and $\kappa_{i}^{\pm}$ ($i=3$, $4$, $5$) correspond to $e$, $\mu$ and
$\tau$ lepton respectively. For convenience,
late on we will call the mixtures of charginos and charged leptons as
`charginos' shortly sometimes.
Due to the trilinear terms with coefficients $\lambda'_{IJK}$ (a lepton
superfield couples to two quark superfields) in Eq.\ (\ref{eq-4}),
the squark mixing is also affected by the lepton number breaking
interactions. Since it is not the main subject of this paper, besides
being discussed elsewhere\cite{cchf}, we outline the effects in
the simplest case in Appendix B briefly.
According to the above analysis, we have achieved the
formulations of the mass spectrum of
the neutralinos-neutrinos, charginos-charged leptons,
neutral Higgs-sneutrinos and charged Higgs-charged sleptons.
Since the vertices of the interactions are also important,
thus in the next section we will give
the Feynman rules of the model, which are new to those of
the MSSM with R-parity conserved.
\section{The Feynman rules for the R-parity violating interactions}
We have discussed the various masses of the MSSM with R-parity violation.
Now, we are discussing the Feynman rules for the model that are
new to those in MSSM with R-parity conserved.
For convenience in loop calculations, we work out here the rules
only in t$^{\prime}$Hooft-Feynman gauge\cite{s16},
which has the gauge fixed terms:
\begin{eqnarray}
{\cal L}_{GF} &=& -\frac{1}{2\xi}\Big(\partial^{\mu}A_{\mu}^{3} + \xi
M_{Z}C_{W}H_{6}^{0}\Big)^{2} - \frac{1}{2\xi}\Big(\partial^{
\mu}B_{\mu} - \xi M_{Z}S_{W}H_{6}^{0}\Big)^{2} -
\frac{1}{2\xi}\bigg(\partial^{\mu}A_{\mu}^{1} \nonumber \\
& &+\frac{i}{\sqrt{2}}\xi M_{W}\Big(
H_{1}^{+} - H_{1}^{-}\Big)\bigg)^{2} -
\frac{1}{2\xi}\bigg(\partial^{\mu}A_{\mu}^{2} +
\frac{1}{\sqrt{2}}\xi M_{W}\Big(H_{1}^{+} + H_{1}^{-}\Big)\bigg)^{2}
\nonumber \\
&=& \Bigg\{-\frac{1}{2\xi}(\partial^{\mu}Z_{\mu})^{2} -
\frac{1}{2\xi}(\partial^{\mu}F_{\mu})^{2} - \frac{1}{\xi}
(\partial^{\mu}W_{\mu}^{+})(\partial^{\mu}W_{\mu}^{-}) \Bigg\} -
\Bigg\{M_{Z}H_{6}^{0}\partial^{\mu}Z_{\mu} \nonumber \\
& & +iM_{W}\Big(H_{1}^{+}
\partial^{\mu}W_{\mu}^{-} - H_{1}^{-}\partial^{\mu}W_{\mu}^{+}\Big)\Bigg\}
- \Bigg\{ \frac{1}{2}\xi M_{Z}^{2}H_{6}^{0^{2}} -
\xi M_{W}^{2}H_{1}^{+}H_{1}^{-}
\Bigg\},
\label{gauge-fixed}
\end{eqnarray}
where $C_{W}=\cos\theta_{W}$ and $H_{6}^{0}$, $H_{1}^{\pm}$
are defined as the above. By inserting the expressions
Eq.\ (\ref{gauge-fixed}) into Lagrangian, the desired vertices for the
Higgs bosons are obtained. We assume the relevant parameters are real,
i.e. at this moment only CP being conserved is considered, one may find
that $H_{1}$, $H_{2}$, $H_{3}^{0}$ $H_{4}^{0}$, $H_{5}^{0}$ are scalars
but $H_{6}^{0}$, $H_{7}^{0}$, $H_{8}^{0}$, $H_{9}^{0}$, $H_{10}^{0}$
pseudoscalar.
\subsection{Feynman rules for Higgs (slepton)- gauge boson interactions}
Let us compute the vertices of Higgs (slepton)- gauge bosons in the
model precisely. The interaction terms of Higgs bosons and
gauge bosons are:
\begin{eqnarray}
{\cal L}_{int}^{1} &=& -\sum_{I}({\cal D}_{\mu}\tilde{L}^{I\dag}{\cal
D}^{\mu}\tilde{L}^{I} - {\cal D}_{\mu}\tilde{R}^{I*}{\cal D}^{\mu}
\tilde{R}^{I}) - {\cal D}_{\mu}H^{1\dag}{\cal D}^{\mu}H^{1} - {\cal
D}_{\mu}H^{2\dag}{\cal D}^{\mu}H^{2} \nonumber \\
&=&\sum_{I}\Bigg\{ \bigg[i\tilde{L}^{I\dag}\Big(g\frac{\tau^{i}}{2}
A_{\mu}^{i} -
\frac{1}{2}g^{\prime}B_{\mu}\Big)\partial^{\mu}\tilde{L}^{I} +
h.c. \bigg] - \tilde{L}^{I\dag}\Big(g\frac{\tau^{i}}{2}A_{\mu}^{i}
\nonumber \\
&& - \frac{1}{2}g^{\prime}B_{\mu}\Big) \Big(g\frac{\tau^{j}}{2}A^{j\mu} -
\frac{1}{2}g^{\prime}B^{\mu}\Big)\tilde{L}^{I} + \Big( ig^{\prime}
B_{\mu}\tilde{R}^{I*}\partial^{\mu}\tilde{R}^{I} \nonumber \\
&& +h.c. \Big) - g^{\prime^{2}}\tilde{R}^{I*}\tilde{R}^{I}B_{\mu}B^{\mu}
\Bigg\} + \Bigg\{ H^{1\dag}\Big(g\frac{\tau^{i}}{2}A_{\mu}^{i} -
\frac{1}{2}g^{\prime}B_{\mu}\Big)\partial^{\mu}H^{1} \nonumber \\
& &+ h.c. \Bigg\}
- H^{1\dag}\Big(g\frac{\tau^{i}}{2}A_{\mu}^{i} -
\frac{1}{2}g^{\prime}B_{\mu}\Big) \Big(g\frac{\tau^{j}}{2}A^{j\mu} -
\frac{1}{2}g^{\prime}B^{\mu}\Big)H^{1} \nonumber \\
& &+\Bigg\{ H^{2\dag}\Big(g\frac{\tau^{i}}{2}A_{\mu}^{i} -
\frac{1}{2}g^{\prime}B_{\mu}\Big)\partial^{\mu}H^{2} + h.c. \Bigg\}
- H^{2\dag}\Big(g\frac{\tau^{i}}{2}A_{\mu}^{i} \nonumber \\
& &- \frac{1}{2}g^{\prime}B_{\mu}\Big) \Big(g\frac{\tau^{j}}{2}A^{j\mu} -
\frac{1}{2}g^{\prime}B^{\mu}\Big)H^{2} \nonumber \\
&=& {\cal L}_{SSV} + {\cal L}_{SVV} + {\cal L}_{SSVV},
\label{interaction1}
\end{eqnarray}
here ${\cal L}_{SSV}$, ${\cal L}_{SVV}$ and ${\cal L}_{SSVV}$,
are the relevant interaction terms. For the Feynman rules
and convenience in practical applications they are precisely rewritten
in the physical bases which were obtained in the previous section.
Since the precise formulas of ${\cal L}_{SSV}$, ${\cal L}_{SVV}$ and
${\cal L}_{SSVV}$ are lengthy, so we put them into Appendix C.
Instead, let us summarize the relevant Feynman rules
in Fig.\ \ref{fig1} $\sim$ \ref{fig4} and
emphasize a few features about them.
First, the presence of the vertices
$Z_{\mu}H_{i}H_{5+j}^{0}$ $(i$, $j=1$, $2$,
$3$, $4$, $5)$ and the forbiddance of the vertices
$Z_{\mu}H_{i}H_{j}^{0}$ and $Z_{\mu}H_{5+i}H_{5+j}^{0}$
($i$, $j=1$, $2$, $3$, $4$, $5$) are due to their CP nature.
Second, besides
the $W_{\mu}^{+}Z^{\mu}H_{1}^{-}$ ($H_{1}^{-}$ is just
the charged Goldstone
boson) interaction, there are not vertices $W_{\mu}^{+}Z^{\mu}H_{i}^{-}$
$(i=2$, $3$, $4$, $5$, $6$, $7$, $8)$ at tree level,
that is the same as the MSSM with R-parity being conserved and
the general two-Higgs doublet models.
\subsection{Self-couplings of the Higgs bosons (sleptons)}
It is a straightforward calculation
by inserting Eqs.\ (\ref{masseven},
\ref{ngold},
\ref{oddhiggs}, \ref{cgold}, \ref{charhiggs})
into Eqs.\ (\ref{eq-6}), to obtain the desired interaction
terms. Similar to the interactions of gauge-Higgs (slepton)
bosons, we split the Lagrangian into pieces:
\begin{equation}
{\cal L}_{int}^{S} = {\cal L}_{SSS} + {\cal L}_{SSSS}
\end{equation}
where ${\cal L}_{SSS}$ represents trilinear coupling terms,
and ${\cal L}_{SSSS}$ four scalar boson coupling terms. The trilinear pieces
are most interesting. If the masses of the scalars are appropriate,
the decays of one Higgs boson into other two Higgs bosons may occur.
After tedious computation, we obtain:
\begin{eqnarray}
{\cal L}_{SSS} &=& -\frac{g^{2} +
g^{\prime^{2}}}{8}A_{even}^{ij}B_{even}^{k}H_{i}H_{j}^{0}H_{k}^{0} -
\frac{g^{2} + g^{\prime^{2}}}{8}A_{odd}^{ij}B_{even}^{k}H_{5+i}H_{5+j}^{0}
H_{k}^{0} \nonumber \\
& & -A_{ec}^{kij}H_{k}^{0}
H_{i}^{-}H_{j}^{+} + iA_{oc}^{kij}H_{5+k}^{0}H_{i}^{-}H_{j}^{+}
\label{vertex-sss}
\end{eqnarray}
and
\begin{eqnarray}
{\cal L}_{SSSS} &=& -\frac{g^{2} +
g^{\prime^{2}}}{32}A_{even}^{ij}A_{even}^{kl}H_{i}H_{j}^{0}H_{k}^{0}H_{l}^{0}
- \frac{g^{2} +
g^{\prime^{2}}}{32}A_{odd}^{ij}A_{odd}^{kl}H_{5+i}
H_{5+j}^{0}H_{5+k}^{0}H_{5+l}^{0} \nonumber \\
& & -\frac{g^{2} +
g^{\prime^{2}}}{16}A_{even}^{ij}A_{odd}^{kl}
H_{i}H_{j}^{0}H_{5+k}^{0}H_{5+l}^{0} -
{\cal A}_{ec}^{klij}H_{k}^{0}H_{l}^{0}H_{i}^{-}H_{j}^{+} \nonumber \\
& &-{\cal A}_{oc}^{klij}H_{5+k}^{0}H_{5+l}^{0}H_{i}^{-}H_{j}^{+} -
i {\cal A}_{eoc}^{klij}H_{k}^{0}H_{5+l}^{0}H_{i}^{-}H_{j}^{+} -
{\cal A}_{cc}^{klij}H_{k}^{-}H_{l}^{+}H_{i}^{-}H_{j}^{+}
\label{vertex-ssss}
\end{eqnarray}
with
\begin{eqnarray}
A_{even}^{ij} &=& \sum_{\alpha=1}^{5}Z_{even}^{i \alpha}Z_{even}^{j
\alpha} - 2Z_{even}^{i 2}Z_{even}^{j 2}, \nonumber \\
A_{odd}^{ij} &=& \sum_{\alpha=1}^{5}Z_{odd}^{i \alpha}Z_{odd}^{j \alpha}
- 2Z_{odd}^{i 2}Z_{odd}^{j 2}, \nonumber \\
B_{even}^{i} &=& \upsilon_{1}Z_{even}^{i 1} - \upsilon_{2}Z_{even}^{i 2}
+ \sum_{I}\upsilon_{\tilde{\nu}_{I}}Z_{even}^{i I+2} .
\end{eqnarray}
The definitions of $A_{ec}^{kij}$, $A_{oc}^{kij}$, ${\cal A}_{ec}^{klij}$,
${\cal A}_{oc}^{klij}$, ${\cal A}_{eoc}^{klij}$ and ${\cal A}_{cc}^{ijkl}$
are lengthy, so we put them in Appendix C.
The Feynman rules are summarized in Fig.\ref{fig5} and Fig.\ \ref{fig6}.
Note that the lepton number violations have
led to very complicated form for the ${\cal L}_{SSS}$
and ${\cal L}_{SSSS}$.
\subsection{The R-parity violation couplings of Higgs}
In this subsection we compute the R-parity violation
couplings of Higgs i.e. the Higgs couplings to charginos
(charged lepton) and neutralinos
(neutrinos). After spontaneous breaking of the
gauge symmetry SU(2)$\times$U(1), the gauginos, higgsinos and
leptons with the
same electric charge may mix as described in Section II. Let us
proceed to compute the interaction
$S\tilde{\kappa}_{i}^{0}\tilde{\kappa}_{j}^{0}$
(Higgs-neutralinos-neutralinos interactions).
The interactions (in two-component spinors)\cite{s13} are:
\begin{eqnarray}
{\cal L}_{S\kappa\kappa} &=&
i\sqrt{2}g\Big(H^{1\dag}\frac{\tau^{i}}{2}\lambda_{A}^{i}\psi_{H^{1}} -
\bar{\psi}_{H^{1}}\frac{\tau^{i}}{2}
\bar{\lambda}_{A}^{i}H^{1}\Big) -
i\sqrt{2}g^{\prime}\Big(\frac{1}{2}H^{1\dag}\psi_{H^{1}}\lambda_{B} -
\frac{1}{2}\bar{\lambda}_{B}\bar{\psi}_{H^{1}}
H^{1}\Big) \nonumber \\
& & + i\sqrt{2}g\Big(H^{2\dag}\frac{\tau^{i}}{2}\lambda_{A}^{i}\psi_{H^{2}}
- \bar{\psi}_{H^{2}}\frac{\tau^{i}}{2}
\bar{\lambda}_{A}^{i}H^{2}\Big) +
i\sqrt{2}g^{\prime}\Big(\frac{1}{2}H^{2\dag}\psi_{H^{2}}\lambda_{B} -
\frac{1}{2}\bar{\lambda}_{B}\bar{\psi}_{H^{2}}
H^{2}\Big) \nonumber \\
& & + i\sqrt{2}\tilde{L}^{I\dag}\bigg(g\frac{\tau^{i}}{2}
\lambda_{A}^{i}\psi_{L^{I}} -
\frac{1}{2}g^{\prime}\lambda_{B}\psi_{L^{I}}\bigg) -
i\sqrt{2}\tilde{L}^{I}\bigg(g\frac{\tau^{i}}{2}
\bar{\lambda}_{A}^{i}\bar{\psi}_{L^{I}} -
\frac{1}{2}g^{\prime}\bar{\lambda}_{B}
\bar{\psi}_{L^{I}}\bigg) \nonumber \\
& & + i\sqrt{2}g^{\prime}\tilde{R}^{I\dag}\lambda_{B}\psi_{R^{I}} -
i\sqrt{2}g^{\prime}\tilde{R}^{I}\bar{\lambda}_{B}\bar{\psi}_{R^{I}} -
\frac{1}{2}l_{I}\varepsilon_{ij}\bigg(\psi_{H^{1}}^{i}
\psi_{L^{I}}^{j}\tilde{R}^{I} +
\psi_{H^{1}}^{i}\psi_{R^{I}}\tilde{L}_{j}^{I}
\nonumber \\
& & + \psi_{R^{I}}\psi_{L^{I}}^{j}H_{i}^{1} + h.c.\bigg)
-\frac{1}{2}\lambda_{IJK}\varepsilon_{ij}\bigg(\psi_{L^{I}}^{i}
\psi_{L^{J}}^{j}
\tilde{R}^{K} + \psi_{L^{I}}^{i}\psi_{R^{K}}\tilde{L}_{j}^{J} \nonumber \\
&& + \psi_{L^{J}}^{j}\psi_{R^{K}}\tilde{L}_{i}^{I} + h.c.\bigg)\; .
\label{snn-tcom}
\end{eqnarray}
We sketch the derivation for the vertices,
$S\tilde{\kappa}_{i}^{0}\tilde{\kappa}_{j}^{0}$ etc. Starting with the
Eq.\ (\ref{snn-tcom}), we convert the pieces from the two-component spinor
notation into four-component spinor notation first,
then using the definitions in
Eq.\ (\ref{define-eneutrino} $\sim$ \ref{define-neutralino})
and Eq.\ (\ref{define-chargino}), we obtain:
\begin{eqnarray}
{\cal L}_{S\kappa\kappa} &=& \frac{\sqrt{g^{2} + g^{\prime^{2}}}}{2}\bigg[
C_{snn}^{ijm}H_{i}\bar{\kappa}_{j}^{0}P_{L}\kappa_{m}^{0}
+C_{snn}^{ijm*}H_{i}\bar{\kappa}_{j}^{0}P_{R}\kappa_{m}^{0}\bigg]
\nonumber \\
& & +
\frac{g}{\sqrt{2}}\bigg[C_{skk}^{ijm}H_{i}\bar{\kappa}_{m}^{+}P_{L}
\kappa_{j}^{+}
+C_{skk}^{ijm*}H_{i}\bar{\kappa}_{j}^{+}P_{R}\kappa_{m}^{+}
\bigg] \nonumber \\
& & + i\frac{\sqrt{g^{2} + g^{\prime^{2}}}}{2}\bigg[
C_{onn}^{ijm}H_{5+i}\bar{\kappa}_{j}^{0}P_{R}\kappa_{m}^{0}
-C_{onn}^{ijm*}H_{5+i}\bar{\kappa}_{m}^{0}P_{L}\kappa_{j}^{0}\bigg]
\nonumber \\
& & +i\frac{g}{\sqrt{2}}\bigg[C_{okk}^{ijm}H_{5+i}
\bar{\kappa}_{m}^{+}P_{L}\kappa_{j}
^{+}
-C_{okk}^{ijm*}H_{5+i}\bar{\kappa}_{m}^{+}P_{R}\kappa_{j}^{+} \bigg]
\nonumber
\\
&&+ \sqrt{g^{2} + g^{\prime^{2}}}\bigg[C_{Lnk}^{ijm}
\bar{\kappa}_{j}^{+}P_{L}\kappa_{m}^{0}H_{i}^{+}
-C_{Rnk}^{ijm}\bar{\kappa}_{j}^{+}P_{R}\kappa_{m}^{0}H_{i}^{+}\bigg]
\label{vertex-snn}
\end{eqnarray}
and the coefficients $C_{snn}^{ijm}$, $C_{Lnk}^{ijm}$, $C_{Rnk}^{ijm}$ and
$C_{skk}^{ijm}$, being lengthy, are put in Appendix D
Here $P_{L,R} = \frac{1 \pm \gamma_{5}}{2}$ are project operators and
the transformation matrices $Z_{\pm}$, $Z_{N}$ are defined in Sect.II.
The corresponding Feynman rules are summarized in Fig.\ \ref{fig7}.
Note here: as for $\kappa_{i}^{0}$ being a Majorana fermion,
the useful identity
\begin{equation}
\bar{\kappa}_{j}^{0}(1 \pm \gamma_{5})\kappa_{k}^{0} =
\bar{\kappa}_{k}^{0}(1 \pm \gamma_{5})\kappa_{j}^{0} ,
\label{majorana-protet}
\end{equation}
holds for the anticommuting four-component Majorana spinors
always, that the $H_{i}\bar{\kappa}_{j}^{0}\kappa_{k}^{0}$
interaction can be rearranged symmetrically
under the interchange of the indices $j$ and $k$.
Since $\nu_{e}$ $(e)$, $\nu_{\mu}$ $(\mu)$ and
$\nu_{\tau}$ $(\tau)$ should be
identified with the lightest three `neutralinos' (`charginos') in the
model, there must be some fresh and interesting
phenomena relevant to them, e.g.
$\kappa_{i}^{0}$ $(i=1,2,3,4) \rightarrow \tau H_{j}^{+}$ $(j=2$,$3$,
$\cdots$, $8)$, $\kappa_{i}^{0}$ $(i=1,2,3,4) \rightarrow \nu_{e,\mu,\tau}
H_{j}^{0}$ $(j=1$, $2$, $\cdots$, $5)$ etc may occur, if the
masses are suitable (the phase space is allowed).
Namely, these interactions without R-parity conservation may
induce new rare processes\cite{s5,s55,s4,s6,se6,s20}.
\subsection{The R-parity violation couplings of gauge bosons}
In this subsection we focus on the R-parity
violation couplings of the gauge bosons
($W$, $Z$, $\gamma$) i.e. the couplings of the gauge bosons
($W$, $Z$, $\gamma$) to the charginos
(charged leptons) and neutralinos (neutrinos).
Since we identify the three types of charged leptons (neutrinos)
with the three lightest charginos (neutralinos), the restrictions
relating to them from the present experiments must
be considered carefully. The relevant interactions
come from the following pieces of Lagrangian:
\begin{eqnarray}
{\cal L}_{int}^{gcn} &=& -i\bar{\lambda}_{A}^{i}\bar{\sigma}^{\mu}{\cal
D}_{\mu}\lambda_{A}^{i} - i\bar{\lambda}_{B}\bar{\sigma}^{\mu}
{\cal D}_{\mu}\lambda_{B} - i\bar{\psi}_{H^{1}}\bar{\sigma}^{\mu}{\cal
D}_{\mu}\psi_{H^{1}} - i\bar{\psi}_{H^{2}}\bar{\sigma}^{\mu}
{\cal D}_{\mu}\psi_{H^{2}} - i\bar{\psi}_{L^{I}}\bar{\sigma}^{\mu}{\cal
D}_{\mu}\psi_{L^{I}} \nonumber \\
& & -i\bar{\psi}_{R^{I}}\bar{\sigma}^{\mu}{\cal D}_{\mu}\psi_{R^{I}}
\label{lang-gcn}
\end{eqnarray}
with
\begin{eqnarray}
{\cal D}_{\mu}\lambda_{A}^{1} &=& \partial_{\mu}\lambda_{A}^{1} -
gA_{\mu}^{2}\lambda_{A}^{3} + gA_{\mu}^{3}\lambda_{A}^{2} ,\nonumber \\
{\cal D}_{\mu}\lambda_{A}^{2} &=& \partial_{\mu}\lambda_{A}^{2} -
gA_{\mu}^{3}\lambda_{A}^{1} + gA_{\mu}^{1}\lambda_{A}^{3} ,\nonumber \\
{\cal D}_{\mu}\lambda_{A}^{3} &=& \partial_{\mu}\lambda_{A}^{3} -
gA_{\mu}^{1}\lambda_{A}^{2} + gA_{\mu}^{2}\lambda_{A}^{1} ,\nonumber \\
{\cal D}_{\mu}\lambda_{B} &=& \partial_{\mu}\lambda_{B} ,\nonumber \\
{\cal D}_{\mu}\psi_{H^{1}} &=& (\partial_{\mu} + igA_{\mu}^{i}
\frac{\tau^{i}}{2}
- \frac{i}{2}g^{\prime} B_{\mu})\psi_{H^{1}} ,\nonumber \\
{\cal D}_{\mu}\psi_{H^{2}} &=& (\partial_{\mu} + igA_{\mu}^{i}
\frac{\tau^{i}}{2}
+ \frac{i}{2}g^{\prime} B_{\mu})\psi_{H^{2}} ,\nonumber \\
{\cal D}_{\mu}\psi_{L^{I}} &=& (\partial_{\mu} + igA_{\mu}^{i}
\frac{\tau^{i}}{2}
- \frac{i}{2}g^{\prime} B_{\mu})\psi_{L^{I}} ,\nonumber \\
{\cal D}_{\mu}\psi_{R^{I}} &=& (\partial_{\mu} + ig^{\prime}
B_{\mu})\psi_{R^{I}}\; .
\label{con-partial}
\end{eqnarray}
Similar to the couplings in ${\cal L}_{S\kappa\kappa}$, we convert
all spinors in
Eq.\ (\ref{lang-gcn}) into four component ones and with
Eq.\ (\ref{define-neutralino})
and Eq.\ (\ref{define-chargino}), then we obtain:
\begin{eqnarray}
{\cal L}_{int}^{gcn}
&=&\bigg\{\sqrt{g^{2}+g^{\prime^{2}}}\sin\theta_{W}\cos\theta_{W}A_{\mu}
\bar{\kappa}_{i}^{+}\gamma^{\mu}\kappa_{i}^{+} -
\sqrt{g^{2}+g^{\prime^{2}}}Z_{\mu}\bar{\kappa}_{i}^{+}
\bigg[\cos^{2}\theta_{W}\delta_{ij}\gamma^{\mu} \nonumber \\
& &+\frac{1}{2}\Big(Z_{-}^{*i 2}Z_{-}^{j 2} +
\sum_{I=1}^{3}Z_{-}^{*i 2+I}Z_{-}^{j 2+I}\Big)\gamma^{\mu}P_{R}
\nonumber \\
&& + \Big(\frac{1}{2}Z_{+}^{*i 2}Z_{+}^{j 2} -
\sum_{I=1}^{3}Z_{+}^{*i 2+I}Z_{+}^{j 2+I}\Big)
\gamma^{\mu}P_{L}\bigg]\kappa_{j}^{+} \bigg\} \nonumber \\
& &+\bigg\{ g\bar{\kappa}_{j}^{+}\bigg[ \Big(-Z_{+}^{*i 1}Z_{N}^{j 2} +
\frac{1}{\sqrt{2}}Z_{+}^{*i 2}
Z_{N}^{j 4}\Big)\gamma^{\mu}P_{L} + \bigg(Z_{N}^{*i 2}Z_{-}^{j 1}
\nonumber \\
& &+\frac{1}{\sqrt{2}}\Big(Z_{N}^{*i 3}Z_{-}^{j 2} +
\sum_{I=1}^{3}Z_{N}^{*i 4+I}Z_{-}^{j 2+I}\Big)\bigg)
\gamma^{\mu}P_{R}\bigg]\kappa_{i}^{0}W_{\mu}^{+} + h.c. \bigg\} \nonumber \\
& &+\frac{\sqrt{g^{2}+g^{\prime^{2}}}}{2}
\bar{\kappa}_{i}^{0}\gamma^{\mu}\Bigg\{\bigg[Z_{N}^{*i 4}Z_{N}^{j 4} -
\Big(Z_{N}^{*i 3}Z_{N}^{j 3} +
\sum_{\alpha=5}^{7}Z_{N}^{*i \alpha}Z_{N}^{j \alpha}\Big)\bigg]P_{L}
\nonumber \\
&& - \bigg[Z_{N}^{*i 4}Z_{N}^{j 4} - \Big(Z_{N}^{*i 3}Z_{N}^{j 3} +
\sum_{\alpha=5}^{7}Z_{N}^{*i \alpha}Z_{N}^{j \alpha}\Big)\bigg]P_{R}
\Bigg\}\kappa_{j}^{0}Z_{\mu} \;.
\label{vertex-gcn}
\end{eqnarray}
The corresponding Feynman rules are summarized in Fig.\ \ref{fig8}.
Since we identify the three lightest neutralinos (charginos) with three
types of neutrinos (charged leptons), some features about
Eq.\ (\ref{vertex-gcn}) should be emphasized:
\begin{itemize}
\item At tree level for the $\gamma$-$\kappa$-$\kappa$ vertices,
there is no lepton flavor-changing current interaction, that
is the same as
that in the SM and MSSM with R-parity.
\item At tree level for the $Z$-$\kappa$-$\kappa$ vertices, there
are lepton flavor-changing current interactions, that is different
from the MSSM with R-parity.
\item Similar to the $Z$-$\kappa$-$\kappa$ vertices, there are
the vertices such as $W\tau\nu_{e}$, which are forbidden in the MSSM
with R-parity.
\end{itemize}
\subsection{The R-parity violation couplings of quarks and/or squarks}
In this subsection we focus the R-parity violation couplings of
quarks and/or squarks i.e. pursue the Feynman rules for the interactions
of quarks and scalar-quarks with charginos (charged leptons) and
neutralinos (neutrinos) e.g. the $\tilde{Q}q\kappa_{i}^{\pm}$
vertices. Because the mixing of neutrinos (charged leptons) and
original neutralinos (charginos), so the vertices will lead to certain
interesting phenomenology, thus it is interesting to write them
down precisely.
Of the vertices, they can be divided into two categories:
the supersymmetric analogies of the $q\bar{q}W^{\pm}$ and $q\bar{q}Z$
interactions and the supersymmetric analogy of the $q\bar{q}H$
interaction which is proportional to quark mass and
depends on the properties of the Higgs bosons in the model.
Let us consider the $\bar{q}q\kappa_{i}^{\pm}$ interactions first.
In two-component spinors they are as the follows:
\begin{eqnarray}
{\cal L}_{\tilde{Q}q\kappa^{\pm}} &=& ig\bigg(C^{IJ}\tilde{Q}_{2}^{I*}
\lambda_{A}^{-}\psi_{Q_{1}}^{J} + C^{IJ*}\tilde{Q}_{1}^{J*}\lambda_{A}^{+}
\psi_{Q_{2}}^{I}\bigg) -
ig\bigg(C^{IJ*}\tilde{Q}_{2}^{I}\bar{\lambda}_{A}^{-}\bar{\psi}_{Q_{1}}^{J} +
C^{IJ}\tilde{Q}_{1}^{J}\bar{\lambda}_{A}^{+}\bar{\psi}_{Q_{2}}^{I}\bigg)
\nonumber \\
& & - \frac{d^{I}}{2}\bigg(C^{IJ}
\psi_{H^{1}}^{2}\psi_{Q^{1}}^{J}\tilde{D}^{I}
+ C^{IJ}\psi_{H^{1}}^{2}\tilde{Q}_{1}^{J}\psi_{D}^{I} +
h.c.\bigg) +
\frac{u^{I}}{2}\bigg(C^{JI*}\psi_{H^{2}}^{1}\psi_{Q^{2}}^{J}\tilde{U}^{I}
\nonumber \\
& & + C^{JI*}\psi_{H^{2}}^{1}\psi_{U}^{I}\tilde{Q}_{2}^{J} + h.c.\bigg)
-\frac{1}{2}\lambda_{IJK}
\bigg(C^{JK}\psi_{L^{I}}^{2}\psi_{Q^{1}}^{K}\tilde{D}^{J} \nonumber \\
&& + C^{JK}\psi_{L^{I}}^{2}\tilde{Q}_{1}^{K}\psi_{D}^{J} + h.c.\bigg)\; .
\label{Qqc-tcom}
\end{eqnarray}
As discussed above, we convert the two-component spinors into
four-component spinors:
\begin{eqnarray}
{\cal L}_{\tilde{Q}q\kappa^{\pm}} &=&
C^{IJ}\bar{\kappa}_{j}^{+}\Bigg\{\bigg(-gZ_{D_{I}}^{i 1}Z_{-}^{j 1} +
\frac{d^{I}}{2}Z_{D_{I}}^{i 2}Z_{-}^{j 2} +
\frac{1}{2}\lambda_{KIJ}^{\prime}Z_{D_{I}}^{i 2}Z_{-}^{j 2+K}\bigg)P_{L}
\nonumber \\
&& +
\frac{u^{J}}{2}Z_{+}^{j 2*}Z_{D_{I}}^{i 1}P_{R}
\Bigg\}\psi_{u^{I}}\tilde{D}_{I i}^{+} +
C^{IJ*}\bar{\kappa}_{j}^{-}\Bigg\{\bigg(-gZ_{U_{J}}^{i 1}Z_{+}^{j 1} +
\frac{u^{J}}{2}Z_{U_{J}}^{i 2}Z_{+}^{j 2}\bigg)P_{L} \nonumber \\
& &-\bigg(\frac{d^{I}}{2}Z_{U_{J}}^{j 1*}Z_{-}^{j 2*} +
\frac{1}{2}\lambda_{KIJ}^{\prime}Z_{U_{J}}^{j 1*}
Z_{-}^{j 2+K*}\bigg)P_{R}\Bigg\}\psi_{d^{J}}\tilde{U}_{Ji}^{-}
+ h.c.\; .
\label{Qqc-fcom}
\end{eqnarray}
Here $\psi_{u^{I}}$, $\psi_{d^{I}}$ are four-component quark spinors of
the I-th generation. The $\kappa_{j}^{-}$ is a charged-conjugate state
of $\kappa_{j}^{+}$, and $\kappa_{j}^{+}$ is defined by
Eq.\ (\ref{define-chargino}).
For the $\tilde{Q}q\kappa_{i}^{0}$ interactions, in two-component notation
they are:
\begin{eqnarray}
{\cal L}_{\tilde{Q}q\kappa_{i}^{0}} &=& i\sqrt{2}\tilde{Q}^{I*}\left(
g\frac{\tau^{3}}{2}\lambda_{A}^{3} +
\frac{1}{6}g^{\prime}\lambda_{B}\right)
\psi_{Q}^{I} -
i\sqrt{2}\tilde{Q}^{I}\left(g\frac{\tau^{3}}{2}\bar{\lambda}_{A}^{3} +
\frac{1}{6}g^{\prime}\bar{\lambda}_{B}\right)
\bar{\psi}_{Q}^{I} \nonumber \\
& & -i\frac{2\sqrt{2}}{3}g^{\prime}\tilde{U}^{I*}\lambda_{B}\psi_{U}^{I} +
i\frac{2\sqrt{2}}{3}g^{\prime}\tilde{U}^{I}\bar{\lambda}_{B}
\bar{\psi}_{U}^{I} + i\frac{\sqrt{2}}{3}g^{\prime}
\tilde{D}^{I*}\lambda_{B}\psi_{D}^{I} -
i\frac{\sqrt{2}}{3}g^{\prime}\tilde{D}^{I}
\bar{\lambda}_{B}\bar{\psi}_{D}^{I} \nonumber \\
& & +\frac{d^{I}}{2}\left\{\psi_{H^{1}}^{1}\psi_{Q^{2}}^{I}\tilde{D}^{I} +
\psi_{H^{1}}^{1}\psi_{D}^{I}\tilde{Q}_{2}^{I} + h.c.\right\} -
\frac{u^{I}}{2}\left\{\psi_{H^{2}}^{2}\psi_{Q^{1}}^{I}\tilde{U}^{I}
+ \psi_{H^{2}}^{2}\psi_{U}^{I}\tilde{Q}_{1}^{I} + h.c.\right\} \nonumber \\
& & +
\frac{1}{2}\lambda_{IJJ}^{\prime}
\left\{\psi_{L^{I}}^{1}\psi_{Q^{2}}^{J}\tilde{D}^{J} +
\psi_{L^{I}}^{1}\psi_{D}^{J}\tilde{Q}_{2}^{J} + h.c.\right\} \; .
\label{Qqn-tcom}
\end{eqnarray}
After converting into four-component notation
straightforward and using the definition for neutralino mass eigenstates,
we have:
\begin{eqnarray}
{\cal L}_{\tilde{Q}q\kappa_{i}^{0}} &=& \kappa_{j}^{0}\left\{
\left[\frac{e}{\sqrt{2}\sin\theta_{W}
\cos\theta_{W}}Z_{U^{I}}^{i 1*}\left(\cos\theta_{W}
Z_{N}^{i 2} + \frac{1}{3}\sin\theta_{W}Z_{N}^{j 1}\right) -
\frac{u^{I}}{2}Z_{U^{I}}^{i 1*}Z_{N}^{j 4*}
\right]P_{L} \right. \nonumber \\
& &+\left. \left[\frac{2\sqrt{2}}{3}g^{\prime}
Z_{U^{I}}^{i 2*}Z_{N}^{j 1} -
\frac{u^{I}}{2}Z_{U^{I}}^{i 1*}Z_{N}^{j 4*}\right]P_{R}\right\}
\psi_{u^{I}}\tilde{U}_{I,i}^{-} +
\bar{\kappa}_{j}^{0}\left\{\left[
\frac{e}{\sqrt{2}\sin\theta_{W}\cos\theta_{W}}
Z_{D^{I}}^{i 1} \right. \right. \nonumber \\
& &\left. \left.\left(-\cos\theta_{W}Z_{N}^{i 2} + \frac{1}{3}
\sin\theta_{W}Z_{N}^{j 1}\right) +
\frac{d^{I}}{2}Z_{D^{I}}^{i 2}Z_{N}^{j 3} +
\frac{1}{2}\lambda_{KIJ}Z_{D^{I}}^{i 2}Z_{N}^{j 4+K} \right]P_{L} \right.
\nonumber \\
& & + \left. \left[-\frac{\sqrt{2}}{3}g^{\prime} Z_{D^{I}}^{i 2}
Z_{N}^{j 1} + \frac{d^{I}}{2}Z_{D^{I}}^{i 1*}Z_{N}^{j 3*} +
\frac{1}{2}\lambda_{KIJ}Z_{D^{I}}^{i 1*}Z_{N}^{j 4+K*}\right]P_{R}
\right\}\psi_{d^{I}}\tilde{D}_{I i}^{+} + h.c. \;.
\label{Qqn-fcom}
\end{eqnarray}
Thus the Feynman rules for the concerned interactions may
be depicted exactly as the last two diagrams in Fig.\ \ref{fig9}.
\section{Various R-parity breaking models and the freedom for
redefining the superfields}
As stated at beginning, we are working in a
very general model, where the R-parity is broken
via various lepton-number violations with possible parameters
explicitly and spontaneously, then an interesting problem
is raised. Namely, we should realize possible freedom in
representing the model and should fix it properly. In fact, there
is some confusion on the freedom in literature.
In this section we focus the problem carefully.
If the superpotential and the soft SUSY breaking were
switched off, the MSSM models would turn to have a $U(n+1)$
global symmetry, i.e. in the case the `down' type of
Higgs boson $H^{1}$ and the leptons chiral superfields $L^{I}
(I=e, \mu, \tau; n=3)$ can be composed as a `vector', and under
a transformation as $(H^{1}, L^{I}) \rightarrow U\cdot
(H^{1}, L^{I})$, $U\in U(n+1=4)$, the theory would be invariant.
Due to the invariance, the quantum numbers of leptons
would become meaningless. Whereas the $U(4)$ symmetry is
completely broken down when switching
on the superpotential and the SUSY breaking terms as well,
then two possibilities happen: a) If the superpotential and the
soft SUSY breaking terms in the Lagrangian of the model
conserve each lepton number respectively as a global symmetry,
the lepton quantum numbers are fixed so the lepton numbers make
senses. In fact this case is just the MSSM with R-parity. b)
If they are switched on, but break the lepton numbers,
although the $U(4)$ symmetry is lost, instead a freedom to
re-define the three lepton superfields and the down type
Higgs is raised in representing the model. Namely if all
the terms in the superpotential and in the soft SUSY breaking
terms undergo a $U(4)$ transformation accordingly, i.e. the
$U(4)$ acts as redefining the lepton and Higgs superfields in the
model, then superficially the VEVs of the sneutrinos, the mass
matrices and the relevant couplings are changed accordingly,
whereas the physics is not changed.
The MSSMs without R-parity may be constructed at beginning with
very different parameters even very different assumptions naively,
but they may be equivalent exactly i.e. they are just the same one
of the models. Indeed in the case b) with such broken
lepton-number superfields and complicated soft SUSY-breaking terms,
it is not so straightforward to see the freedom, so
it is an important and non-trivial task.
Let us examine the problem now.
To bare the task to realize the freedom for the MSSM
without R-parity in mind, in the section we precisely show
the equivalence of the models which are related to each other
by $U(n+1)$ ($n$ is the family number with broken lepton
number) transformations. Namely the $U(n+1)$ transformations
can be understood as a freedom for re-defining the
relevant superfields. Finally in this section
we propose two suggestions which may be considered as
`conventions' for possible choices to fix the freedom.
Of the two, one is `to rotate away' the nonzero vacuum
expectation values (VEVs) of all the three generations
of sneutrinos, this is emphasized by Refs.\cite{se6,s25,se1},
the other one is to `rotate away' the bilinear terms in the
superpotential\cite{s21,s5,s55,s6,s9}.
Note that in the second case, in general,
the bilinear R-parity violation terms in soft breaking SUSY
terms so nonzero VEVs of the sneutrinos still may exist.
We would emphasize here that based on the whole effective
Lagrangian we may compute the spectrum (mixing) of the content
particles of the model precisely, and may have a global view
of the model too. Furthermore, having the precise Lagrangian
one may easy connect the effective one to a more fundamental
theory, thus we would not do the problem such as in the
references\cite{s7} where from the very beginning only the
`basis-indepedent' parameters are focused on so as to investigate
the phenomenology of the model. Whereas in the present way, we
should examine the freedom in defining the fields carefully, and
make all the parameters being substantial. In fact, when all
possible terms (not only in superpotential but also
in SUSY soft-breaking and D-terms) are involved
the problem is not so transparent to realize the freedom.
Here we take the general case, that the three lepton-numbers
are broken, as an example to examine the freedom. In fact,
the freedom is $U(n+1=4)$ ($n$: the number of
violated lepton-numbers) in defining the superfields
within the extented MSSM as shown in the follows.
The $U(4)$
\begin{equation}
X^{\dagger}X = X X^{\dagger} = I,
\label{eq29}
\end{equation}
and $X\in U(4)$ is $4\times 4$ matrix, $I$ is the unit matrix.
The $U(4)$ matrix `acting' on the `old' Higgs and lepton
superfields $\hat{H}^{1}, \hat{L}^{1}, \hat{L}^{2}, \hat{L}^{3}$
is defined as:
\begin{equation}
\left(
\begin{array}{c}
\hat{H}^{a^{1}} \\
\hat{L}^{a^{1}} \\
\hat{L}^{a^{2}} \\
\hat{L}^{a^{3}} \\ \end{array} \right) = X
\left(
\begin{array}{c}
\hat{H}^{1} \\
\hat{L}^{1} \\
\hat{L}^{2} \\
\hat{L}^{3} \\ \end{array} \right),
\label{eq30}
\end{equation}
where $\hat{H}^{a^{1}}, \hat{L}^{a^{1}}, \hat{L}^{a^{2}},
\hat{L}^{a^{3}}$ are `new' Higgs-lepton superfields.
As the D-terms in the model are invariant under the $U(4)$
transformation, so we need not consider them at all for
present purpose. As for the superpotential,
having the $U(4)$ transformation performed, it
turns into the following form accordingly
by means of the `new' superfields (with a upper-suffix
as in Eq.\ (\ref{eq30})):
\begin{eqnarray}
&W =& \varepsilon_{ij}\mu^{a}\hat{H}_{i}^{a^{1}}\hat{H}_{j}^{2}
+ \varepsilon_{ij}
\epsilon_{I}^{a}\hat{H}_{i}^{2}\hat{L}_{j}^{a^{I}} +
\varepsilon_{ij}h_{IJ}^{l}
\hat{H}_{i}^{a^{1}}\hat{L}_{j}^{a^{I}}\hat{R}^{J} \nonumber \\
&&- u_{I}(\hat{H}_{1}^{2}C^{JI*}
\hat{Q}_{2}^{J} - \hat{H}_{2}^{2}\hat{Q}_{1}^{J}\delta^{IJ})\hat{U}^{I}
\nonumber \\
& & -h_{IJ}^{d}(\hat{H}_{1}^{a^{1}}\hat{Q}_{2}^{J}\delta^{IJ} -
\hat{H}_{2}^{a^{1}}
\hat{Q}_{1}^{J}C^{IJ})\hat{D}^{I} + \varepsilon_{ij}
\lambda_{IJK}^{a}\hat{L}_{i}^{a^{I}}
\hat{L}_{j}^{a^{J}}\hat{R}^{K} \nonumber \\
& & + \lambda_{IJK}^{a^{\prime}}(\hat{L}_{1}^{a^{K}}
\hat{Q}_{2}^{J}\delta{IJ} -
\hat{L}_{2}^{a^{K}}\hat{Q}_{1}^{J}C^{IJ})\hat{D}^{I}
+ \lambda_{IJK}^{\prime\prime}\hat{U}^{I}\hat{D}^{J}\hat{D}^{K},
\label{eq31}
\end{eqnarray}
and the `new' coefficients are related to the `old'
ones through the $U(4)$ matrix elements precisely:
\begin{eqnarray}
&h_{IJ}^{l} =& l_{J}(X_{11}^{*}X_{(I+1)(J+1)}^{*} -
X_{(I+1)1}^{*}X_{1(J+1)}^{*}) \nonumber \\
& &+ \sum\limits_{KM}
\lambda_{KMJ}(X_{1(K+1)}^{*}X_{(I+1)(M+1)}^{*} \nonumber \\
&& - X_{(I+1)(K+1)}^{*}X_{1(M+1)}^{*}) , \label{eq32} \\
&h_{IJ}^{d} =& d_{J}X_{11}^{*} -
\sum\limits_{K}\lambda_{KIJ}^{\prime}X_{1(K+1)}^{*} ,
\label{eq33} \\
&\lambda_{IJK}^{a} =& l_{K}X_{(I+1)1}^{*}X_{(J+1)(K+1)}^{*} \nonumber \\
&& + \sum\limits_{MN}\lambda_{MNK}
X_{(I+1)(M+1)}^{*}X_{(J+1)(N+1)}^{*} , \label{eq34} \\
&\lambda_{IJK}^{a^{\prime}} =& -d_{I}X_{(I+1)1}^{*} + \sum\limits_{M}
\lambda_{MKJ}^{\prime}
X_{(I+1)(M+1)}^{*} ,\label{eq35} \\
&h_{I}^{a} =& l_{I}X_{11}^{*}X_{1(I+1)}^{*} + \sum\limits_{JK}
\lambda_{KJI}X_{1(K+1)}^{*}X_{1(J+1)}^{*}
, \label{eq36} \\
&\mu^{a} =& X_{11}^{*}\mu - \sum\limits_{I}\epsilon_{I}X_{1(I+1)}^{*} ,
\label{eq37} \\
&\epsilon_{I}^{a} =& -X_{(I+1)1}^{*}\mu +
\sum\limits_{J}X_{(I+1)(J+1)}^{*}\epsilon_{J}. \label{eq38}
\end{eqnarray}
Here $X_{\alpha\beta}^* (\alpha, \beta=1, 2, 3, 4)$ are the
complex conjugations of $X_{\alpha\beta}$, and $(I, J, K,
M, N=1, 2, 3)$ always.
As the same way, the soft SUSY-breaking terms
in Lagrangian now correspondingly become:
\begin{eqnarray}
{\cal L}_{soft} &=& - M_{11}^{s^{2}}H_{i}^{a^{1*}}H_{i}^{a^{1}} -
\sum\limits_{I}M_{1(I+1)}^{s^{2}}
(H_{i}^{a^{1*}}\tilde{L}_{i}^{a^{I}} +
H_{i}^{a^{1}}\tilde{L}_{i}^{a^{I*}}) \nonumber \\
& & - \sum\limits_{IJ}
M_{(I+1)(J+1)}^{s^{2}}\tilde{L}_{i}^{a^{I*}}\tilde{L}_{i}^{a^{J}}
-m_{R^{I}}^{2}\tilde{R}^{I*}\tilde{R}^{I} -m_{Q^{I}}^{2} \tilde{Q}_{i}^{I*}
\tilde{Q}_{i}^{I} \nonumber \\
&& - m_{D^{I}}^{2}
\tilde{D}
^{I*} \tilde{D}^{I} - m_{U^{I}}^{2}\tilde{U}^{I*} \tilde{U}^{I} + (m_{1}
\lambda_{B}\lambda_{B}+ m_{2}\lambda_{A}^{i}\lambda_{A}^{i} \nonumber \\
& & + m_{3} \lambda_{G}^{a}\lambda_{
G}^{a} + h.c.) + \{\varepsilon_{ij}B^{a} H_{i}^{a^{1}}H_{j}^{2} +
\varepsilon_{ij}B_{I}^{a} H_{i}^{2}\tilde{L}_{j}^{a^{I}}
\nonumber \\
& &+\varepsilon_{ij}l_{s_{IJ}}^{a}H_{i}^{a^{1}}
\tilde{L}_{j}^{a^{I}}\tilde{R}^{J}
- d_{s_{IJ}}^{a}(H_{1}^{a^{1}}\tilde{Q}_{2}^{J}\delta^{IJ}
- H_{2}^{a^{1}}\tilde{
Q}_{1}C^{IJ})\tilde{D}^{I} \nonumber \\
& & + u_{s_{I}}(-C^{KI*}H_{1}^{2}\tilde{Q}_{2}^{I} +
H_{2}^{2}\tilde{Q}_{1}^{I})\tilde{U}^{I} +
\varepsilon_{ij}\lambda_{IJK}^{s_{a}}
\tilde{L}_{i}^{a^{I}}\tilde{L}_{i}^{a^{J}}\tilde{R}^{K} \nonumber \\
& & +\lambda_{KIJ}^{s_{a}^{\prime}}(\tilde{L}_{1}\tilde{Q}_{2}^{J}\delta^{IJ}
-\tilde{L}_{2}^{a^{K}}\tilde{Q}_{1}^{J}C^{IJ})\tilde{D}^{I} \nonumber \\
&&+\lambda_{IJK}^{s^{\prime\prime}}\tilde{U}^{I}\tilde{D}^{J}\tilde{D}^{K}
+ h.c. \}. \label{39}
\end{eqnarray}
Here the 'new' soft breaking parameters are defined as
\begin{eqnarray}
&M_{s_{\alpha\beta}}^{2}=& m_{H^{1}}^{2}X_{\alpha 1}^{*}X_{\beta 1}^{*} +
\sum\limits_{I}m_{L^{I}}^{2}X_{\alpha (I+1)}^{*}X_{\beta (I+1)}^{*} +
\sum\limits_{I\neq J}m_{L^{IJ}}^{2}X_{\alpha
(I+1)}^{*}X_{\beta (J+1)}^{*}
\nonumber \\
&& + \sum\limits_{I}m_{HL^{I}}^{2}X_{\alpha 1}^{*}X_{\beta (I+1)}^{*}
, \label{eq40} \\
&B^{a} = & B X_{11}^{*} -
\sum\limits_{I}B_{I}X_{1(I+1)}^{*},\label{eq41} \\
&B_{I}^{a} =& -B X_{(I+1)1}^{*} + \sum\limits_{J}B_{J}
X_{(I+1)(J+1)}^{*}, \label{eq42} \\
&l_{s_{MK}}^{a} =&\sum\limits_{s_{I}}(-X_{(M+1)1}^{*}X_{1(I+1)}^{*}+
X_{11}^{*}X_{(M+1)(I+1)}^{*})\delta_{IK} \nonumber \\
& & + \sum\limits_{IJ}\lambda_{IJK}^{s}(X_{(M+1)(I+1)}^{*}X_{1(J+1)}^{*}
+ X_{1(I+1)}^{*}X_{(M+1)(J+1)}^{*} , \label{eq43} \\
& d_{s_{IJ}}^{a} =&d_{s_{I}}X_{11}^{*} - \sum\limits_{K}
\lambda_{KIJ}^{s^{\prime}}X_{1(K+1)}^{*} , \label{eq44} \\
&\lambda_{IJK}^{s_{a}} = & l_{s_{K}}X_{(I+1)1}^{*}X_{(J+1)(K+1)}^{*}
\nonumber \\
&& + \sum\limits_{MN}\lambda_{MNK}^{s}X_{(I+1)(M+
1)}^{*}X_{(J+1)(N+1)}^{*} , \label{eq45} \\
&\lambda_{KIJ}^{s_{a}^{\prime}} =& -d_{s_{I}}X_{(K+1)1}^{*} +
\sum\limits_{M}\lambda_{MJI}^{s^{\prime}}X_{(K+1)(M+1)}^{*}, \label{eq46} \\
&h_{I}^{s_{a}} =& l_{s_{I}}X_{11}^{*}X_{1(I+1)}^{*} +
\sum\limits_{JK}\lambda_{KJI}X_{1(K+1)}^{*}X_{1(J+1)}^{*}. \label{eq47}
\end{eqnarray}
Now, the $H^{a^{1}}$ and slepton acquire
vacuum expectation values (VEVs):
\begin{equation}
H^{a^{1}}=
\left(
\begin{array}{c}
\frac{1}{\sqrt{2}}(\chi_{1}^{a} + \upsilon_{1}^{a} + i\phi_{1}^{a}) \\
H_{2}^{a^{1}} \end{array} \right)
\label{eq48}
\end{equation}
\begin{equation}
H^{2}=
\left(
\begin{array}{c}
H_{1}^{2} \\
\frac{1}{\sqrt{2}}(\chi_{2}^{0} + \upsilon_{2} + i\phi_{2}^{0})
\end{array} \right)
\label{eq49}\\[2mm]
\end{equation}
and
\begin{equation}
\tilde{L}^{a^{I}} =
\left(
\begin{array}{c}
\frac{1}{\sqrt{2}}(\chi_{\tilde{\nu}_{I}}^{a} +
\upsilon_{\tilde{\nu}_{I}}^{a} +
i\phi_{\tilde{\nu}_{I}}^{a}) \\
\tilde{L}_{2}^{a^{I}} \end{array} \right)
\label{eq50}\\[2mm]
\end{equation}
where
\begin{eqnarray}
\upsilon_{1}^{a} &=& X_{11}\upsilon_{1} + \sum\limits_{J}X_{1(J+1)}
\upsilon_{\tilde{\nu}_{J}} , \nonumber \\
\upsilon_{\tilde{\nu}_{I}}^{a} &=& X_{(I+1)1}\upsilon_{1} +
\sum\limits_{J}X_{(I+1)(J+1)}\upsilon_{\tilde{\nu}_{J}}. \label{eq51}
\end{eqnarray}
As before by expanding the scalar potential, the tadpole terms become:
\begin{equation}
V_{tadpole} = t_{1}^{0}\chi_{1}^{0} + t_{2}^{0} \chi_{2}^{0} +
t_{\tilde{\nu}_{1}}^{0} \chi_{\tilde{\nu}_{1}}^{0} +
t_{\tilde{\nu}_{2}}^{0} \chi_{\tilde{\nu}_{2}}^{0} +
t_{\tilde{\nu}_{3}}^{0} \chi_{\tilde{\nu}_{3}}^{0}
\label{eq52}
\end{equation}
where
\begin{eqnarray}
t_{1}^{a} &=& \frac{1}{8}(g^{2} + g^{'2})
\upsilon_{1}^{a}(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} + \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{a^{2}})
+ \mu^{2}\upsilon_{1}^{a} - B^{a} \upsilon_{2} \nonumber \\
& & -\sum_{I} \mu^{a} \epsilon_{I}^{a}\upsilon_{\tilde{\nu}_{I}}^{a}+
M_{s_{11}}^{2}\upsilon_{1}^{a}
+ \sum\limits_{I}M_{s_{1(I+1)}}^{2}\upsilon_{\tilde{\nu}_{I}}^{a}
\nonumber \\
t_{2}^{a} &=& -\frac{1}{8}(g^{2} + g^{'2})\upsilon_{2}
(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} + \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{a^{2}})
+ \mu^{a^{2}}\upsilon_{2} - B^{a} \upsilon_{1}^{a} \nonumber \\
& & + \sum_{I}\epsilon_{I}^{a^{2}}\upsilon_{2}
+\sum_{I} B_{I}^{a}\upsilon_{\tilde{\nu}_{I}}^{a} +
m_{H^{2}}^{2}\upsilon_{2}, \nonumber \\
t_{\tilde{\nu}_{I}}^{a} &=& \frac{1}{8}(g^{2} +
g^{\prime^{2}})\upsilon_{\tilde{\nu}_{I}}^{a}(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} +
\sum_{I}\upsilon_{\tilde{\nu}_{I}}^{a^{2}}) +
\epsilon_{I}^{a}\sum_{J}\epsilon_{J}^{a}
\upsilon_{\tilde{\nu}_{J}}^{a} \nonumber \\
& & - \mu^{a}\epsilon_{I}^{a}\upsilon_{1}^{a} +
B_{I}^{a}\upsilon_{2} + M_{s_{(I+1)(I+1)}}^{2}
\upsilon_{\tilde{\nu}_{I}}^{a} +
M_{s_{1(I+1)}}^{2}\upsilon_{1}^{a} + \nonumber \\
& &\sum\limits_{J \neq I}M_{s_{(I+1)(J+1)}}^{2}
\upsilon_{\tilde{\nu}_{J}}^{a}.
\label{eq53}
\end{eqnarray}
Namely in the new basis $(\phi_{1}^{a}$, $\phi_{2}^{0}$,
$\phi_{\tilde{\nu}_{1}}^{a}$,
$\phi_{\tilde{\nu}_{2}}^{a}$, $\phi_{\tilde{\nu}_{3}}^{a})$, the
mass matrix of the CP-odd Higgs becomes
\begin{equation}
M_{odd}^{a^{2}} = \left(
\begin{array}{ccccc}
s_{11}^{a} & B^{a} & M_{s_{12}}^{2} - \mu^{a}\epsilon_{1}^{a} &
M_{s_{13}}^{2} - \mu^{a}\epsilon_{2}^{a} & M_{s_{14}}^{2} -
\mu^{a}\epsilon_{3}^{a} \\
B^{a} & s_{22}^{a} & -B_{1}^{a} & -B_{2}^{a} &
-B_{3}^{a} \\
M_{s_{12}}^{2} - \mu^{a}\epsilon_{1}^{a} &
-B_{1}^{a} & s_{33}^{a} &
\epsilon_{1}^{a}\epsilon_{2}^{a} + M_{s_{23}}^{2} &
\epsilon_{1}^{a}\epsilon_{3}^{a}
+ M_{s_{24}}^{2} \\
M_{s_{13}}^{2} - \mu^{a}\epsilon_{2}^{a} & -B_{2}^{a} &
\epsilon_{1}^{a}\epsilon_{2}^{a} + M_{s_{23}}^{2} & s_{44}^{a} &
\epsilon_{2}^{a}\epsilon_{3}^{a}
+ M_{s_{34}}^{2} \\
M_{s_{14}}^{2} - \mu^{a}\epsilon_{3}^{a} & -B_{3}^{a} &
\epsilon_{1}^{a}
\epsilon_{3}^{a} + M_{s_{24}}^{2} & \epsilon_{2}^{a}\epsilon_{3}^{a} +
M_{s_{34}}^{2} & s_{55}^{a}
\end{array} \right).
\label{eq54}
\end{equation}
The definitions for the parameters:
\begin{eqnarray}
&&s_{11}^{a} = \frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} +
\sum_{I}\upsilon_{\tilde{\nu}_{I}}^{a^{2}}) + \mu^{a^{2}}
+ M_{s_{11}}^{2} \nonumber \\
&&= \sum_{I}\mu^{a}\epsilon_{I}^{a}\frac{\upsilon_{\tilde{\nu}_{I}}^{a}}
{\upsilon_{1}^{a}} +
B^{a}\frac{\upsilon_{2}}{\upsilon_{1}^{a}} -\sum\limits_{I}
M_{s_{\small{1(I+1)}}}^{2}, \nonumber \\
&&s_{22}^{a} = -\frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} +
\upsilon_{\tilde{\nu}_{I}}^{a^{2}}) + \mu^{a^{2}} +
\sum_{I}\epsilon_{I}^{a^{2}} + m_{H^{2}}^{2} \nonumber \\
&&= B^{a}\frac{\upsilon_{1}^{a}}{\upsilon_{2}} -
\sum_{I}B_{I}^{a}\frac{\upsilon_{\tilde{\nu}_{I}}^{a}}
{\upsilon_{2}}, \nonumber \\
&&s_{\small{(I+2)(I+2)}}^{a} = \frac{g^{2} +
g^{\prime^{2}}}{8}(\upsilon_{1}^{a^{2}} -
\upsilon_{2}^{2} + \sum_{I}\upsilon_{\tilde{\nu}_{I}}^{a^{2}}) +
\epsilon_{I}^{a^{2}} + M_{s_{\small{(I+1)(I+1)}}}^{2} \nonumber \\
& &= \mu^{a}\epsilon_{I}^{a}\frac{\upsilon_{1}^{a}}
{\upsilon_{\tilde{\nu}_{e}}^{a}}
- B_{I}^{a}\frac{\upsilon_{2}}
{\upsilon_{\tilde{\nu}_{I}}^{a}}
- \sum\limits_{J \neq I}(\epsilon_{I}^{a}\epsilon_{J}^{a}
+ M_{s_{\small{(I+1)(J+1)}}}^{2})
\frac{\upsilon_{\tilde{\nu}_{J}}^{a}}{\upsilon_{\tilde{\nu}_{I}}^{a}}
\;\; . \nonumber \\
\label{eq55}
\end{eqnarray}
Using Eqs.\ (\ref{eq32}) $\sim$ \ (\ref{eq38}) and Eqs.\ (\ref{eq40})
$\sim$ \ (\ref{eq47})
properly, we find
\begin{equation}
M_{odd}^{a^{2}} = T_{1}^{-1} M_{odd}^{2} T_{1}
\label{eq561}
\end{equation}
and the unitary matrix $T_1$ is defined:
\begin{equation}
T_{1} = \left(
\begin{array}{ccccc}
X_{11}^{*} & 0 & X_{21}^{*} & X_{31}^{*} & X_{41}^{*} \\
0 & 1 & 0 & 0 & 0 \\
X_{12}^{*} & 0 & X_{22}^{*} & X_{32}^{*} & X_{42}^{*} \\
X_{13}^{*} & 0 & X_{23}^{*} & X_{33}^{*} & X_{43}^{*} \\
X_{14}^{*} & 0 & X_{24}^{*} & X_{34}^{*} & X_{44}^{*}
\end{array} \right)\;\;.
\label{eq56}
\end{equation}
It is known from Eq.\ (\ref{eq561}) that
$M_{odd}^{a^{2}}$ and
$M_{odd}^{2}$ have the same eigenvalues and the eigenstates of them
are related by ${\cal X}^\alpha=T_1 {\cal Y}^\alpha$, if ${\cal X}^\alpha$
presents an eigenstate of $M_{odd}^{2}$ and ${\cal Y}^\alpha$ presents
that of $M_{odd}^{a^{2}}$ for the same eigenvalue $m_\alpha^2$.
For the $CP$-even `Higgs', in the same way one can find
$$M_{even}^{a^{2}} = T_{1}^{-1}M_{even}^{2}T_{1}$$
here $T_{1}$ is the same as that in Eq.\ (\ref{eq56}). So the same
conclusion on the `new' and `old' relation of the eigenvalues
and the eigenstates as that in the $CP$-odd case is obtained.
As for the mass matrix of charged Higgs, the case is more complicated
because the right handed sleptons should be included in the
mixing . In the basis
$\Phi_{c}=(H_{2}^{a^{1*}}$, $H_{1}^{2}$, $\tilde{L}_{2}^{a^{1*}}$,
$\tilde{L}_{2}^{a^{2*}}$, $\tilde{L}_{2}^{a^{3*}}$, $\tilde{R}^{1}$,
$\tilde{R}^{2}$, $\tilde{R}^{3})$, one find:
\begin{equation}
M_{c}^{a^{2}} = T_{2}^{-1}M_{c}^{2} T_{2},
\label{eq61}
\end{equation}
and the unitary transformation matrix $T_{2}$ now is
\begin{equation}
T_{2}= \left(
\begin{array}{cccccccc}
X_{11}^{*} & 0 & X_{21}^{*} & X_{31}^{*} & X_{41}^{*} & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\
X_{12}^{*} & 0 & X_{22}^{*} & X_{32}^{*} & X_{42}^{*} & 0 & 0 & 0 \\
X_{13}^{*} & 0 & X_{23}^{*} & X_{33}^{*} & X_{43}^{*} & 0 & 0 & 0 \\
X_{14}^{*} & 0 & X_{24}^{*} & X_{34}^{*} & X_{44}^{*} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array} \right)\; .
\label{eq62}
\end{equation}
The same conclusion is obtained as the above neutral Higgs cases.
Now, let us consider the neutral fermions. In the basis
$(\Phi^{0})^{T} = (-\lambda_{B}$,
$-i\lambda_{A}^{3}$, $\psi_{H^{a^{1}}}^{1}$,
$\psi_{H^{2}}^{2}$, $\nu_{e_{L}^{a}}$,
$\nu_{\mu_{L}^{a}}$, $\nu_{\tau_{L}^{a}})$,
the mass matrix of neutralino-neutrino is written as
\begin{equation}
{\cal M}_{N}^{a} = \left(
\begin{array}{ccccccc}
2m_{1} & 0 & -\frac{1}{2}g^{\prime}\upsilon_{1}^{a} & \frac{1}{2}g^{\prime}
\upsilon_{2} & -\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{e}}^{a} &
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\mu}}^{a} & -\frac{1}{2}g^{\prime}
\upsilon_{\tilde{\nu}_{\tau}}^{a} \\
0 & 2m_{2} & \frac{1}{2}g\upsilon_{1}^{a} & -\frac{1}{2}g\upsilon_{2} &
\frac{1}{2}g\upsilon_{\tilde{\nu}_{e}}^{a}
& \frac{1}{2}g\upsilon_{\tilde{\nu}_{\mu}}^{a} & \frac{1}{2}g
\upsilon_{\tilde{\nu}_{\tau}}^{a} \\
-\frac{1}{2}g^{\prime}\upsilon_{1}^{a} & \frac{1}{2}g\upsilon_{1}^{a} & 0 &
-\frac{1}{2}\mu^{a} & 0 & 0 & 0\\
\frac{1}{2}g^{\prime}\upsilon_{2} & -\frac{1}{2}g\upsilon_{2}
& -\frac{1}{2}\mu^{a} & 0 & \frac{1}{2}\epsilon_{1}^{a}
& \frac{1}{2}\epsilon_{2}^{a} & \frac{1}{2}\epsilon_{3}^{a} \\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{e}}^{a} & \frac{1}{2}g
\upsilon_{\tilde{\nu}_{e}}^{a} & 0 & \frac{1}{2}\epsilon_{1}^{a} & 0
& 0 & 0\\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\mu}}^{a} & \frac{1}{2}g
\upsilon_{\tilde{\nu}_{\mu}}^{a} & 0 & \frac{1}{2}\epsilon_{2}^{a} & 0 &
0 & 0\\
-\frac{1}{2}g^{\prime}\upsilon_{\tilde{\nu}_{\tau}}^{a} & \frac{1}{2}g
\upsilon_{\tilde{\nu}_{\tau}}^{a} & 0 & \frac{1}{2}\epsilon_{3}^{a} & 0
& 0 & 0
\end{array} \right),
\label{eq63}
\end{equation}
and we find
\begin{equation}
M_{N}^{a} = T_{3}^{-1}M_{N}T_{3}.
\label{eq64}
\end{equation}
The $T_{3}$ unitary matrix is defined:
\begin{equation}
T_{3} = \left(
\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & X_{11}^{*} & 0 & X_{21}^{*} & X_{31}^{*} & X_{41}^{*} \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & X_{12}^{*} & 0 & X_{22}^{*} & X_{32}^{*} & X_{42}^{*} \\
0 & 0 & X_{13}^{*} & 0 & X_{23}^{*} & X_{33}^{*} & X_{43}^{*} \\
0 & 0 & X_{14}^{*} & 0 & X_{24}^{*} & X_{34}^{*} & X_{44}^{*}
\end{array} \right).
\label{eq65}
\end{equation}
Once more the same conclusion on the relations of `new' and `old' eigenvalues
and eigenstates is reached.
As for the mixing of chargino-charged lepton, the mass matrix from the
Lagrangian now can be related as follows:
\begin{equation}
{\cal M}_{\small{C}}^{a} = T_{4}^{-1}{\cal M}_{\small{C_{T}}} I
\label{eq66}
\end{equation}
with
\begin{equation}
T_{4} = \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & X_{11}^{*} & X_{21}^{*} & X_{31}^{*} & X_{41}^{*} \\
0 & X_{12}^{*} & X_{22}^{*} & X_{32}^{*} & X_{42}^{*} \\
0 & X_{13}^{*} & X_{23}^{*} & X_{33}^{*} & X_{43}^{*} \\
0 & X_{14}^{*} & X_{24}^{*} & X_{34}^{*} & X_{44}^{*}
\end{array} \right)\; .
\label{eq67}
\end{equation}
Since charged fermions are considered here,
the situation is a little complicated.
We need to make the mass matrix diagonal as the case of SM
for quarks i.e. first to diagonalize the mass squared matrix
(the combination of the matrix and its conjugate), whereas, owing to
the relation Eq.\ (\ref{eq66}), the same conclusion can be obtained
too as the above.
Furthermore, it is easy to check the interaction terms are the same
no matter to start with what an `old' Lagrangian or a `new' Lagrangian:
as long as the vertices for the model all turn to represent by means of
their eigenvalues (physical value) and corresponding eigenstates
(physical states) coordinately, the equivalence for the interactions
can be seen clearly. Therefore, the $U(4)$ transformation
Eq.\ (\ref{eq30}) indeed is shown a freedom for
defining the superfields, and the problem
how to fix a model of R-parity violation MSSM emerges.
To solve this problem, we would like to suggest two
`conventions' for choices: a) with the freedom to
rotate away the VEVs for all sneutrinos; b) with the
freedom to rotate away all the bilinear terms of R-parity
violation in superpotential. Note that one can apply the freedom
only once that is to mean one can make either a) or b)
but cannot do both successfully in a general case.
In fact, besides the two choices we suggest here, there are many
choices to fix the freedom. For instance, one may rotate
part of the trilinear lepton-superfield terms or the
linear lepton-superfield terms which couple to the
quark superfields properly in Eq.\ (\ref{eq-4}),
for specific convenience.
Now let us show the convention a) first: in fact,
Refs.\cite{se6,s25,se1} may be considered as the case.
For convenience, let us define
the angles $\theta, \phi, \xi$ if all the VEVs are real\footnote{If
the VEVs of sneutrinos are not real but complex values
i.e. there are spontaneous CP violation in the model through
the VEVs, the discussion here is still valid. Only the change
of the discussion is from a rotation into a
unitary $U(4)$ transformation.}:
\begin{eqnarray}
&&\sin\theta=\frac{v_{\tilde\nu_e}}{\sqrt{v_1^2+
v_{\tilde\nu_e}^2}}\;, \nonumber \\
&&\sin\phi=\frac{v_{\tilde\nu_\mu}}{\sqrt{v_1^2+
v_{\tilde\nu_e}^2+v_{\tilde\nu_\mu}^2}}\; , \nonumber \\
&&\sin\xi=\frac{v_{\tilde\nu_\tau}}
{\sqrt{v_1^2+v_{\tilde\nu_e}^2+v_{\tilde\nu_\mu}^2+
v_{\tilde\nu_\tau}^2}} \; .
\label{bas1}
\end{eqnarray}
Indeed, anyone of the R-parity
violation MSSMs with nonzero VEVs of sneutrinos may be rotated
to the one where only the Higgs superfield
$\hat {H}^1$ has nonzero VEV ($v_1\neq 0$; $v_{\tilde\nu_I}=0$,
with $I=e, \mu, \tau$). The
`rotation matrix' (here the $U(4)$ transformation `degenerates'
just to a rotation) can be decomposed into three rotations as below:
\begin{equation}
X=R_1 \cdot R_2 \cdot R_3 \;.
\label{rot1}
\end{equation}
Here
$$ R_1= \left(
\begin{array}{cccc}
\cos\xi & 0 & 0 & \sin\xi\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\sin\xi & 0 & 0 & \cos\xi
\end{array} \right) \; ,$$
$$ R_2= \left(
\begin{array}{cccc}
\cos\phi & 0 & \sin\phi & 0\\
0 & 1 & 0 & 0 \\
-\sin\phi & 0 & \cos\phi & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)\; , $$
and
$$ R_3= \left(
\begin{array}{cccc}
\cos\theta & \sin\theta& 0 & 0 \\
-\sin\theta & \cos\theta & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)\; . $$
It is easy further to check that
\begin{eqnarray}
H_{6}^{0} &=&\frac{1}{\upsilon}\Big( \upsilon_{d}\phi_{1}^{a^{0}}
- \upsilon_{2}
\phi_{2}^{0} \Big) \nonumber \\
&=&\frac{1}{\upsilon}\Big(\upsilon_{1}\phi_{1}^{0} -
\upsilon_{2}\phi_{2}^{0} +
\upsilon_{\tilde{\nu}_{e}}\phi_{\tilde{\nu}_{e}}^{0}
+ \upsilon_{\tilde{\nu}_{\mu}}\phi_{\tilde{\nu}_{\mu}}^{0} +
\upsilon_{\tilde{\nu}_{\tau}}\phi_{\tilde{\nu}_{\tau}}^{0}\Big)
\hspace{1mm},
\end{eqnarray}
and
\begin{eqnarray}
H_{1}^{+}& = &\frac{1}{\upsilon}\Big( \upsilon_{d}H_{2}^{1^{a}}
- \upsilon_{2}
H_{1}^{2*} \Big) \nonumber \\
& = &\frac{1}{\upsilon}(\upsilon_{1}H_{2}^{1*} -
\upsilon_{2}H_{1}^{2} +
\upsilon_{\tilde{\nu}_{e}}\tilde{L}_{2}^{1*} +
\upsilon_{\tilde{\nu}_{\mu}}\tilde{L}_{2}^{2*} +
\upsilon_{\tilde{\nu}_{\tau}}\tilde{L}_{2}^{3*}).
\end{eqnarray}
are just the Goldstones for spontaneously breaking the EW gauge symmetry.
The rest parts of the models can be checked without difficulty, but to
shorten the paper we will not show them here precisely.
The second `convention' b), which is suggested above, can be
`realized' from anyone of the R-parity violation MSSMs by a proper
rotation which is similar to the above,
if the coefficients $\epsilon_I$ of the bilinear
terms in the superpotential are real, otherwise an
according $U(4)$ transformation instead of the rotation
to complete the purpose. For convenience in various application
let us present the rotation precisely as below:
\begin{equation}
X'=R'_1 \cdot R'_2 \cdot R'_3 \;.
\label{rot2}
\end{equation}
Here
$$ R'_1= \left(
\begin{array}{cccc}
\cos\xi' & 0 & 0 & \sin\xi'\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\sin\xi' & 0 & 0 & \cos\xi'
\end{array} \right) \; ,$$
$$ R'_2= \left(
\begin{array}{cccc}
\cos\phi' & 0 & \sin\phi' & 0\\
0 & 1 & 0 & 0 \\
-\sin\phi' & 0 & \cos\phi' & 0 \\
0 & 0 & 0 & 1
\end{array} \right)\; , $$
and
$$ R'_3= \left(
\begin{array}{cccc}
\cos\theta' & \sin\theta' & 0 & 0 \\
-\sin\theta' & \cos\theta' & 0 & ) \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)\; , $$
with
\begin{eqnarray}
&&\sin\theta'=\frac{\mu }{\sqrt{\epsilon_1^2 +
\mu^2}}\;\;,
\sin\phi'=\frac{\mu }{\sqrt{\epsilon_1^2 +
\epsilon_2^2 + \mu^2}}\;, \nonumber \\
&&\sin\xi'=\frac{\mu}{\sqrt{\epsilon_1^2 +
\epsilon_2^2 + \epsilon_3^2 + \mu^2}}\;.
\label{bas2}
\end{eqnarray}
After the rotation, in superpotential only the term
$\mu'\varepsilon_{ij}{\hat H}_i^1{\hat H}_j^2$
with $\mu'=
\sqrt{\mu^2+\epsilon_1^2+\epsilon_2^2
+\epsilon_3^2}$ is survived and the other
bilinear terms disappear totally.
Before closing this section, we would like to emphasize again:
if one would like to compare different R-parity violation MSSMs
and to draw any definite conclusion, he must
fix the freedom in defining the four superfields (three leptons and the
relevant Higgs which has the same quantum numbers as those
of leptons) first, and then carry on the comparisons. Otherwise,
the obtained surface `differences' can be attributed to a
different definition on the superfields totally or partly.
For convenience in applications and not only to fix the freedom
for redefining the superfields, we will further `simplify'
the parameterization in various ways elsewhere\cite{cchf}.
\section{Numerical results}
In this section, to be reference results for further studies,
we analyze the masses of neutral Higgs
and charginos numerically and show their values
in proper ways. We have obtained the mass matrices
by setting the three type sneutrinos
with non-zero vacuum expectation values and $\epsilon_{i} \neq 0$ $(i=1$,
$2$, $3)$. However, the matrices are quite big that may obscure the
typical features. To simplify the `problem' and to deduct
the parameters, we assume only those terms
which related the third generation (only $\tau$-lepton
number) of lepton number is broken, but those
to the first two generations are not relevant
i.e. the terms relating to
the `first two generation lepton-numbers' disappear correspondly.
Furthermore through fixing the freedom for redefining the
superfields as discussed in the previous section,
for the `survived' trilinear terms relevant to the third
generation leptons in superpotential and SUSY soft breaking
terms, we will `rotate away' them as possible as one can, that
only the trilinear terms $\varepsilon_{ij}\lambda_{333}
\hat {L}^3_i \hat{L}^3_j\hat{R}^3$ in Eq.\ (\ref{eq-4})
and $\varepsilon_{ij}\lambda_{333} \tilde {L}^3_i
\tilde {L}^3_j\tilde {R}^3$ in Eq.\ (\ref{eq-5}) are kept.
Namely in the Section for the numerical calculation, we restrict
ourselves to compute the case that the VEV of $\tau$-sneutrino
is nonzero, the bilinear terms relevant to $\tau$-family
lepton as well as the two trilinear terms
$\varepsilon_{ij}\lambda_{333} \hat {L}^3_i \hat{L}^3_j\hat{R}^3$
and $\varepsilon_{ij}\lambda_{333} \tilde {L}^3_i
\tilde {L}^3_j\tilde {R}^3$ are present.
Two reasons to make such an assumption that only
$\tau$-lepton number is violated:
\begin{itemize}
\item Under the assumption, we think the main feature will
not be lost too much but the mass matrices will turn much simple.
\item According to experimental indications,
the $\tau$-neutrino may be the heaviest among the three type
neutrinos, and so far the constraints for the $\tau$ lepton
rare decays are comparatively loose etc, i.e. the third
generation of leptons probably are special.
\end{itemize}
In the numerical calculations below, input parameters are chosen as:
$\alpha = \frac{e^{2}}{4\pi}=\frac{1}{128}$, $M_{Z}=91.19$GeV,
$M_{W}=80.23$GeV, $m_{\tau}=1.77$GeV, but for the parameters $m_{1}$, $m_{2}$,
we assume $m_{1} = m_{2} =250$GeV and the upper limit
on $\tau$-neutrino mass $m_{\nu_{\tau}} \leq 10$MeV is taken into account
seriously.
Now let us consider the masses of the charginos first,
when $\epsilon_{1} = \epsilon_{2} =0$, and
$\upsilon_{\tilde{\nu}_{e}} =
\upsilon_{\tilde{\nu}_{\mu}}=0$, the Eq.\ (\ref{chargino-matrix}) becomes:
\begin{equation}
{\cal M}_{C} = \left(
\begin{array}{ccc}
2m_{2} & \frac{e\upsilon_{2}}{\sqrt{2}S_{W}} & 0 \\
\frac{e\upsilon_{1}}{\sqrt{2}S_{W}} & \mu &
\frac{l_{3}\upsilon_{\tilde{\nu}_{\tau}}}{\sqrt{2}} \\
\frac{e\upsilon_{\tilde{\nu}_{\tau}}}{\sqrt{2}S_{W}} & \epsilon_{3} &
\frac{l_{3}\upsilon_{1}}{\sqrt{2}}+\frac{\lambda_{333}
\upsilon_{\tilde{\nu}_{\tau}}}{\sqrt{2}}
\end{array} \right).
\label{rchargino-matrix}
\end{equation}
Because $m_{\tau}^{2}$ should be identified
as the lightest eigenvalue of the
matrix ${\cal M}_{C}^{\dag}{\cal M}_{C}$, we should take it
as an eigenvalue away first so as not to conflict
the measurement of $\tau$ lepton mass.
After taking the eigenvalue $m_{\tau}^{2}$
away, the survived eigenvalue equation becomes:
\begin{equation}
\lambda^{2} - {\cal A}_{C}\lambda + {\cal B}_{C} = 0,
\end{equation}
and
\begin{eqnarray}
{\cal A}_{C} &=& X^{2} + Y^{2} + 4m_{2}^{2} + \frac{l_{3}^{2}\bigg(
\upsilon_{1}^{2} + \upsilon_{\tilde{\nu}_{\tau}}^{2}\bigg)
+ \lambda_{333}^{2}\upsilon_{\tilde{\nu}_{\tau}}^{2}}{2}
+ \frac{e^{2}\upsilon^{2}}{2S_{W}^{2}}, \nonumber \\
{\cal B}_{C} &=& \frac{2l_{3}^{2}}{m_{\tau}^{2}}
\Bigg\{m_{2}\upsilon\cos\beta Y
\bigg(1+\frac{\lambda_{333}}{l_{3}}\sin\theta_{\upsilon}\bigg) \nonumber \\
&& +\frac{e^{2}}{4S_{W}^{2}}
\upsilon^{3}\cos^{2}\beta\sin\beta \bigg(\sin^{2}\theta_{\upsilon} -
\cos^{2}\theta_{\upsilon}\bigg) \Bigg\}^{2},
\label{coeffi3}
\end{eqnarray}
with the parameters $X$, $Y$ are defined by
\begin{eqnarray}
X &=& \epsilon_{3}\cos\theta_{\upsilon} + \mu\sin\theta_{\upsilon},
\nonumber \\
Y &=& -\epsilon_{3}\sin\theta_{\upsilon} + \mu\cos\theta_{\upsilon} ,
\label{define-XY}
\end{eqnarray}
Therefore the masses of the other two charginos are expressed as:
\begin{equation}
m_{\kappa_{1,2}^{\pm}}^{2} = \frac{1}{2}\bigg\{ {\cal A}_{C} \mp
\sqrt{{\cal A}_{C}^{2} - 4{\cal B}_{C}}\bigg\}.
\label{charginomass}
\end{equation}
The auxiliary parameter $l_{3}$ can be fixed by the condition that
$Det|m_{\tau}^{2} - {\cal M}_{c}^{\dag}{\cal M}_{c}|=0$.
When the values of $m_{1}$, $m_{2}$,
$\tan\beta$, $\tan\theta_{\upsilon}$ and Y are fixed, the value of X
will be fixed by the mass of $\tau$-neutrino. Trying
to take $m_{\nu_{\tau}} = 0.1$ MeV, we plot the mass of the
lightest charginos versus with Y in Fig.\ \ref{fig10}. The two
lines in the figure correspond to
$\lambda_{333}=0$ and $\lambda_{333}=0.5$ respectively.
In the figure, we
find that the trilinear effect on the chargino masses is small when
$\tan\beta >> 1$ and $\tan\theta_{\upsilon} < 1$. In Fig.\ \ref{fig10}(c),
the line correspond to $\lambda_{333}=0.5$ is coincide with the
line for $\lambda_{333}=0.5$. When the
$\tan\beta \sim 1$ and $\tan\beta > 1$, the difference between
$\lambda_{333}= 0.5$ and $\lambda_{333}=0$ is large. In the case,
$\sqrt{\upsilon_{1}^{2} + \upsilon_{\tilde{\nu_{\tau}}}^{2}} \sim
\upsilon_{2}$ and $\upsilon_{1} < \upsilon_{\tilde{\nu_{\tau}}}$ and the
effect of $\lambda_{333} \upsilon_{\tilde{\nu_{\tau}}}$ on the lightest
chargino mass cannot be neglected. For comparison and considering the
results obtained at Super-K for neutrino oscillations, with a smaller
neutrino mass $m_{\nu_{\tau}} = 10 $eV but the same parameters
being taken, we do the numerical calculation once more. The obtained
curves are different from those in Fig.\ \ref{fig10} by certain amount
but not qualitatively and we plot them in Fig.\ \ref{fig11}.
Now, as for the mass-matrices of the neutral Higgs, under the same
assumption, the one for CP-even Higgs is truncated to:
\begin{equation}
{\cal M}_{even}^{2} = \left(
\begin{array}{ccc}
r_{11} & -e_{12} - B &
e_{15} - \mu\epsilon_{3} \\
-e_{12} - B & r_{22} &
-e_{25} + B_{3} \\
e_{15} -
\mu\epsilon_{3} & -e_{25} + B_{3} & r_{33}
\end{array}
\right)
\label{rmatrix-even}
\end{equation}
with
\begin{eqnarray}
r_{11} &=& \frac{g^{2} + g^{\prime^{2}}}{8}(3\upsilon_{1}^{2} -
\upsilon_{2}^{2}
+ \upsilon_{\tilde{\nu}_{\tau}}^{2}) +
|\mu|^{2} + m_{H^{1}}^{2} \nonumber \\
&=& \frac{g^{2}+g^{\prime^{2}}}{4}\upsilon_{1}^{2} +
\bigg(\mu\epsilon_{3}-m_{HL^{3}}^{2}\bigg)
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{1}}
+ B\frac{\upsilon_{2}}{\upsilon_{1}} , \nonumber \\
r_{22} &=& \frac{g^{2} + g^{\prime^{2}}}{8}(-\upsilon_{1}^{2} +
3\upsilon_{2}^{2} - \upsilon_{\tilde{\nu}_{\tau}}^{2}) +
|\mu|^{2} + |\epsilon_{3}|^{2} + m_{H^{2}}^{2} \nonumber \\
&=& \frac{g^{2}+g^{\prime^{2}}}{4}\upsilon_{2}^{2} +
B\frac{\upsilon_{1}}{\upsilon_{2}}
- B_{3}\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{2}},
\nonumber \\
r_{33} &=& \frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{2} - \upsilon_{2}^{2}
+ 3\upsilon_{\tilde{\nu}_{\tau}}^{2}) +
|\epsilon_{3}|^{2} + m_{L^{3}}^{2} \nonumber \\
&=& \frac{g^{2}+g^{\prime^{2}}}{4}\upsilon_{\tilde{\nu}_{\tau}}^{2} +
\bigg(\mu\epsilon_{3}-m_{HL^{3}}^{2}\bigg)\frac{\upsilon_{1}}{\upsilon_{
\tilde{\nu}_{\tau}}}
- B_{3}\frac{\upsilon_{2}}{\upsilon_{\tilde{\nu}_{\tau}}}.
\label{r123}
\end{eqnarray}
and $e_{12}$, $e_{15}$, $e_{25}$ are defined in Eq.\ (\ref{eij}).
Whereas the mass matrix of CP-odd Higgs is truncated to:
\begin{equation}
{\cal M}_{odd}^{2} =
\left(
\begin{array}{ccc}
s_{11} & B & -\mu\epsilon_{3}+m_{HL^{3}}^{2} \\
B & s_{22} & -B_{3} \\
-\mu\epsilon_{3}+m_{HL^{3}}^{2} & -B_{3} & s_{33}
\end{array}
\right) \label{rmassodd}
\end{equation}
with
\begin{eqnarray}
s_{11} &=& \frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{2} -
\upsilon_{2}^{2}
+ \upsilon_{\tilde{\nu}_{\tau}}^{2}) + |\mu|^{2}
+ m_{H^{1}}^{2} \nonumber \\
&=& \bigg(\mu\epsilon_{3}-m_{HL^{3}}^{2}\bigg)
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{1}} +
B\frac{\upsilon_{2}}{\upsilon_{1}}, \nonumber \\
s_{22} &=& -\frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{2} - \upsilon_{2}^{2}
+ \upsilon_{\tilde{\nu}_{\tau}}^{2}) + |\mu|^{2}
+|\epsilon_{3}|^{2} + m_{H^{2}}^{2} \nonumber \\
&=& B\frac{\upsilon_{1}}{\upsilon_{2}} -
B_{3}\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{2}}, \nonumber
\\
s_{33} &=& \frac{g^{2} + g^{\prime^{2}}}{8}(\upsilon_{1}^{2} - \upsilon_{2}^{2}
+ \upsilon_{\tilde{\nu}_{\tau}}^{2}) +
|\epsilon_{3}|^{2} + m_{L^{3}}^{2} \nonumber \\
&=& \bigg(\mu\epsilon_{3}-m_{HL^{3}}^{2}\bigg)
\frac{\upsilon_{1}}{\upsilon_{\tilde{\nu}_{\tau}}} -
B_{3}\frac{\upsilon_{2}}{\upsilon_{
\tilde{\nu}_{\tau}}}.
\label{s123}
\end{eqnarray}
Introducing the following auxiliary variables:
\begin{eqnarray}
X_{s} &=& B ,\nonumber \\
Y_{s} &=& \mu\epsilon_{3}-m_{HL^{3}}^{2},\nonumber \\
Z_{s} &=& B_{3},
\label{xyzs}
\end{eqnarray}
the masses of the neutral Higgs can be expressed by the parameters
$X_{s}$, $Y_{s}$, $Z_{s}$
and $\tan\beta$, $\tan\theta_{\upsilon}$.
For the masses of CP-odd Higgs,
the masses of the two CP-odd Higgs are given by
\begin{equation}
m_{\small{H_{3+2,3}^{0}}}^{2} = \frac{1}{2}({\cal A} \mp \sqrt{{\cal A}^{2} - 4
{\cal B}} )
\label{massodd1}
\end{equation}
if we define:
\begin{eqnarray}
{\cal A} &=& X_{s} (\frac{\upsilon_{1}}{\upsilon_{2}} +
\frac{\upsilon_{2}}{\upsilon_{1}}) + Y_{s} (\frac{
\upsilon_{1}}{\upsilon_{\tilde{\nu}_{\tau}}} +
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{1}}) - Z_{s}(\frac{\upsilon_{2}}{
\upsilon_{\tilde{\nu}_{\tau}}} +
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{2}}) ,\nonumber \\
{\cal B} &=& - Y_{s}Z_{s} (\frac{\upsilon_{1}}{\upsilon_{2}} +
\frac{\upsilon_{2}}{\upsilon_{1}}) -
X_{s}Z_{s}(\frac{\upsilon_{1}}{\upsilon_{\tilde{\nu}_{\tau}}} +
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{1}}) + X_{s}Y_{s}
(\frac{\upsilon_{2}}{\upsilon_{\tilde{\nu}_{\tau}}} +
\frac{\upsilon_{\tilde{\nu}_{\tau}}}{\upsilon_{2}}) + \nonumber \\
& &
X_{s}Y_{s}\frac{\upsilon_{1}^{2}}{\upsilon_{2}\upsilon_{\tilde{\nu}_{\tau}}} -
X_{s}Z_{s}\frac{\upsilon_{
2}^{2}}{\upsilon_{1}\upsilon_{\tilde{\nu}_{\tau}}} -
Y_{s}Z_{s} \frac{\upsilon_{\tilde{\nu}_{\tau}}^{2}}{\upsilon_{1}
\upsilon_{2}}.
\label{calab}
\end{eqnarray}
In the numerical calculation, we have taken the parameter
$\sqrt{|X_{s}|} = 500$GeV. In Fig.\ \ref{fig12}, we plot the
mass of the lightest CP-even Higgs versus the parameter
$\sqrt{|Y_{s}|}$. The three lines correspond to
$\sqrt{|Z_{s}|}=60$GeV, $150$GeV and $250$GeV respectively.
From the Fig.\ \ref{fig12}, we find the mass of the lightest
CP-even Higgs turns small when the parameter $\sqrt{|Y_{s}|}$
turns large. In Fig.\ \ref{fig13}, we plot the
mass of the lightest CP-even Higgs versus the parameter
$\sqrt{|Z_{s}|}$. The three lines correspond to
$\sqrt{|Y_{s}|}=60$GeV, $300$GeV and $400$GeV respectively.
From the Fig.\ \ref{fig13}, we find the mass of the lightest
Higgs turns large, as the parameter $\sqrt{|Z_{s}|}$ changes
large. From the numerical calculations, we can find
certain parameter space that at tree level the lightest Higgs
mass can be $m_{H_{1}^{0}} \geq 132$GeV, thus for
the supersymmetry model without R-parity one cannot obtain
such a stringent limit on the lightest Higgs mass as that in the
MSSM with R-parity.
As shown above, the results obtained by our numerical and formulation
analysis, both confirm the difference from the MSSM with R-parity:
the upper bound of the lightest CP-even Higgs mass of the MSSM without
R-parity is loosened a lot.
In summary, besides the formal analysis and clarifying
the confusion on the freedom for the redefining the fields,
with the assumption that only $\tau$-lepton number is broken, we have
calculated the mass spectra in the MSSM without R-parity numerically.
From the restriction on the neutrino mass: even
$m_{\nu_{\tau}} \leq 10$ eV, we cannot rule out the possibilities
with large $\epsilon_{3}$. The Feynman rules have been derived in the
$\prime$t-Hooft Feynman gauge which are convenient when studying the
phenomenology beyond tree level of the model. Here, we would like
to point out some references have analyzed the $0\nu \beta\beta$-decay
in the model\cite{s22} and may obtain certain new constraints about
the upper limits on the first generation R-parity violating parameters,
such as $\epsilon_{1}$ and $\upsilon_{\tilde{\nu}_{e}}$;
whereas for the other two generations, there are no such
serious restrictions on the R-parity violating parameters.
\vspace{20mm}
{\Large\bf Acknowledgment} We would like to thank Prof. T. Han and
Prof. Z.-Y. Zhao for valuable discussions on the topics of the paper.
One (C.-H. C) of the authors would like to thank Prof. Edmond L. Berger
for warm hospitality at ANL (Argonne) and some suggestions
during his visit. This work was
supported in part by the National Natural Science
Foundation of China and the Grant No. LWLZ-1298
of the Chinese Academy of Sciences.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,745
|
Q: how to publish a feed of posts with a certain custom field value? In addition to the normal feed that wordpress produces, i need to publish an additional feed of posts that have a certain custom field value.
I'm a bit lost on where to start. Any tip?
A: Something like this will add a /feed/special to your site.
add_feed('special','do_special_feed');
function do_special_feed() {
query_posts(array(
'meta_key' => 'key',
'meta_value' => 'value',
));
do_feed_rss2(false);
}
After putting it in a plugin or your theme's functions.php or what have you, you'll need to visit the Settings->Permalinks page at least once and save the settings, to get it to rebuild the permalink rules.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,898
|
HENRY STUBBE
AND THE BEGINNINGS OF ISLAM
HENRY STUBBE
_AND THE_
BEGINNINGS
OF ISLAM
_The Originall & Progress of Mahometanism_
EDITED AND INTRODUCED BY
NABIL MATAR
COLUMBIA UNIVERSITY PRESS
_NEW YORK_
Columbia University Press
_Publishers Since 1893_
New York Chichester, West Sussex
cup.columbia.edu
Copyright © 2014 Columbia University Press
All rights reserved
E-ISBN 978-0-231-52736-1
Library of Congress Cataloging-in-Publication Data
Stubbe, Henry, 1632–1676.
[Account of the rise and progress of Mahometanism]
Henry Stubbe and the beginnings of Islam : the Originall & progress of Mahometanism / edited and introduced by Nabil Matar.
pages cm
Includes bibliographical references and index.
ISBN 978-0-231-15664-6 (cloth : alk. paper)—ISBN 978-0-231-52736-1 (e-book)
1. Islam—Early works to 1800. 2. Stubbe, Henry, 1632–1676. I. Matar, N. I. (Nabil I.),
1949– II. Title.
BP160.S7 2014
297.09—dc23
2013018042
A Columbia University Press E-book.
CUP would be pleased to hear about your reading experience with this e-book at cup-ebook@columbia.edu.
Cover art: From _Historia Orientalis_ (1660).
_Andover-Harvard Theological Library, Harvard University_.
Cover design: Milenda Nan Ok Lee
References to Web sites (URLs) were accurate at the time of writing. Neither the author nor Columbia University Press is responsible for URLs that may have expired or changed since the manuscript was prepared.
For Mohammad Asfour
Light upon light,
God guides to His light.
—QUR'ĀN 24:35 (KHALIDI'S TRANSLATION)
CONTENTS
ACKNOWLEDGMENTS
Introduction
_The "Copernican Revolution" of Henry Stubbe_
The Life of Henry Stubbe
_The Originall & Progress of Mahometanism_
Arabic Studies in England
'Ῑsa: The Qur'ānic Jesus
The Prophet Muḥammad
"Let us then fancy the gallant Aly"
Islam and Empire
The Printed and Manuscript Sources
_Editorial Policy_
_THE ORIGINALL & PROGRESS OF MAHOMETANISM_
NOTES
BIBLIOGRAPHY
INDEX
ACKNOWLEDGMENTS
This project started as a result of incisive questions from two talented graduate students, both of whom have now completed their dissertations: Josh Mabie and Eric Carlson. In a course on "Britain and the Islamic Mediterranean," we read the 1911 edition of Henry Stubbe's treatise on Islam, but on a number of occasions I found myself unable to address adequately some of the issues they raised. When I visited England later that year, I decided to consult the manuscript versions of the treatise. I then realized the need for a new edition.
I was awarded a Grant-in-Aid from the University of Minnesota that allowed me to spend time at the British Library, the Senate Library of the University of London, and the Bodleian. To the staff at these libraries, I am deeply grateful. Closer to home, the staff of the James Ford Bell Library at the University of Minnesota have been most supportive: Dr. Marguerite Ragnow and Ms. Margaret Borg. So too were the ILL staff at Wilson Library. As always, the staff at the Houghton Library, Harvard University, some of whom I have known since my first visit in 1988, were most gracious.
I wish to thank Heather Krebs, a former graduate student, who typed the manuscripts. She faced a herculean task, which she completed with masterful accuracy. I also want to thank my tireless student assistant, Katie Sisneros, who helped in formatting and proofing as well as for preparing the index. I know it was not easy. I consulted many colleagues and friends to whom I am thankful: Professor Wadad Kadi (Arabic) who spent many hours with me; Professor Dominic Baker-Smith (Latin), Professor Philip Sellew (Greek), Professor Spencer Cole (Latin), Professor Marco Perale (Greek), and Mr. Gabriel Fuchs who focused on the longer Latin passages. To all: thank you. Any shortcomings that might remain are, of course, mine.
I am thankful to the Center for Early Modern History, director Sarah Chambers, at the University of Minnesota and the Center for Near Eastern Studies at the University of Chicago for giving me the opportunity to present a lecture on my initial findings on Stubbe. I was honored by the UK arm of the Association of Muslim Social Scientists, director Anas Hajj Ali, and the Alwalid bin Talal Centre for Islamic Studies at the University of Cambridge, director Yasir Suleiman, with the Building Bridges Award, which afforded me another occasion to develop my thoughts in a lecture on Henry Stubbe.
On a personal note, I would like to thank the friends without whom I could not have completed this work: A. A. Baramki; Professor Muhammad Shaheen, University of Jordan, a deeply cherished friend, who met with me, in Amman and in Oxford, to offer suggestions and insights; Professor Jeanne Kilde, director of the Program of Religious Studies at the University of Minnesota, whose initiative and drive are ever an inspiration to me; and Professor Wadad Kadi, whose enduring friendship and vast scholarship have been my sustenance in the icy Minnesota "Polite." And from the British side of the pond, I would like to thank the many friends I have in London: Dina Matar and John Taysom for their wonderful hospitality; David Brooks, my Cambridge friend of yore; Samira Kawar, my student during my first teaching job at the University of Jordan in Amman, and her husband Yacoub Douani; Patrick Spottiswoode of the Globe Theatre, a dear friend and always an inspiration; Basim Ziadeh, friend from childhood in Beirut, and Riad Nourallah of AUB days.
As always, I remember Selim Kemal and Rudy Stoeckel, friends of Beirut, Cambridge, and Melbourne, Florida—towns of memory. And so too, the doe-eyed girl of the green.
And forever in the pictures around me in my study: Abraham and Hady, may the Lord ever shine His face on you... and forgive you the sarcasm you inflicted on me for wasting taxpayers' money on editing a seventeenth-century document; my sister Inaam, resisting retirement in Amman; Suheil Farouqui, a fellow traveler on the road to Jerusalem; GH always, always cherished and thanked; and, of course, Galina, _al ḥabība_.
*** * ***
This book is dedicated to Mohammad Asfour.
It was his from the moment I started working on it.
With Mohammad, I shared the first office in my first job at the University of Jordan in 1976. Poet, translator, and teacher, Mohammad is unforgettable. I have met many of his former students in various parts of the world—at the University of Wisconsin, on a flight to Rabat, in the Reading Room of the British Library: there was always wonder in their eyes as we talked about him. I too had fallen under his spell: a man for whom language is sacred, and literature, both Arabic and English, an Ariadne's thread in the labyrinth of exile.
From him I learned about beauty and holiness, about Islamic devotion and Arabic prosody, about Byron and Jabra Ibrahim Jabra. And he continues to be a mentor: every so often, I pester him with questions and queries, and he responds, with grace and learning and illumination. As he approaches retirement, and with Um Firās by his side, I hope he will record his last journey from 'Ayn Ghazāl and the diaspora of Palestine.
INTRODUCTION
_The "Copernican Revolution" of Henry Stubbe_
UROPEAN MEDIEVAL representations of the Prophet Muḥammad and of the beginnings of Islam were uniformly negative, as Norman Daniel showed in his magisterial _Islam and the West: The Making of an Image_ (1960). Although Nicholas of Cusa described the Prophet as merely a man in error ( _Cribratio Alkorani_ , ca. 1458–64), European writers always viewed Muḥammad and the Qur'ān from an oppositional perspective. John Tolan has shown that from Theodor Bibliander's _Machvmetis Sarracanorvm principis vita_ (Basel, 1543) to Humphrey Prideaux's _The true nature of imposture fully display'd in the life of Mahomet_ (1697) there was not a single European text that attempted to present a historically accurate biography of the Prophet and of the beginnings of Islam. Rather, and as the conflict among Christian denominations surged in Western Europe—Catholics, Lutherans/Protestants, Socinians, Deists, and others—writers dragged Muḥammad into the fray treating him as a forerunner of the Protestant heresy, an ally of the pope, a proto-Socinian, or an "atheist." Which is why Henry Stubbe (1632–76) is important: he was the "exception" to all early modern writers on Islam.
This little-known physician who spent the last years of his life in between Stratford-upon-Avon and Bath undertook a Copernican Revolution (in Kant's use of the phrase) in the study of Islam. For Kant, the phrase served as a metaphor for the shift in the position of the observer that made possible a new astronomy and for him a new epistemology. In the study of Islam Stubbe moved away from Euro-Christian sources to the canon of Arabic histories and chronicles in Latin translation, instituting thereby a sharp methodological and historiographical break with the past. In just under sixty thousand words, Stubbe presented the first heavily annotated biography of the man who had given birth to "Islamism" as well as the first English description of 'Ali, the Prophet's cousin. But, Stubbe knew that it was the message in "Alkoran" that was of paramount importance: his encomium on Islam in the last pages of his treatise is unparalleled in early modern European writing.
In the seventeenth century, three Arabic histories about early Islam were translated into Latin: accounts by Jirjis ibn al-'Amīd al-Makīn, Sa'īd ibn al-Baṭrīq/Eutychius, and Gregorios Abū al-Faraj/Ibn al-'Ibrī, all of which had relied on important Muslim historians such as Muḥammad ibn Jarīr al-Ṭabarī, 'Abdallah ibn'Umar al-Bayḍāwī, Muḥammad ibn 'abd al-Karīm al-Shahrastānī, and others. A master of Greek and Latin, Stubbe consulted these translations closely and realized how sources indigenous to Islamic civilization could lead to a new understanding of contested history and a reassessment of the most misrepresented man in early modern European religious thought: the Prophet Muḥammad. These sources were widely available, having been published in the academic powerhouses of Oxford and Leiden, Paris and Basel. But among all the English writers about Islam and the "Mahometans" in the seventeenth century, from preachers to travelers, from theologians to comparative historians, only Stubbe consulted these sources to produce a detailed history of the beginnings of Islam, _The Originall & Progress of Mahometanism_.
Many were the readers who perused and copied Stubbe's treatise, as evidenced by the numerous manuscript versions of the treatise that have survived and the others that are known to have been lost. But, to date, all scholars who have written about Stubbe's treatise have relied on the Lahore 1911 edition of the work, _The Rise and Progress of Mahometanism_ , published by Hafiz Mahmud Khan Shairani—with the exception of J. A. I. Champion's 2010 study. This edition was the composite of three "authors": the original text by Stubbe, "improvements" on the text by Charles Hornby in 1705, and the editorial excisions by Shairani of passages which were deemed "not polite." This mix of hands renders the edition unreliable. Only by focusing on the earliest complete manuscript of _The Rise and Progress of Mahometanism_ , which I am renaming _The Originall & Progress of Mahometanism_, ca. 1701, will it be possible to examine the actual words, or close to the actual words, that Stubbe wrote. Because this manuscript is not an autograph, there can be no absolute certainty, but the sections that Charles Blount copied in 1678 and included in letters to Thomas Hobbes and to the earl of Rochester (which appeared in print in 1693 and 1695) and the three manuscript fragments that survive from the latter part of the century at the British Library all correspond exactly to this manuscript, with inevitable scribal variances.
A century before Edward Gibbon, Stubbe recognized how integrated "Muslim history [was] with that of the Roman and Byzantine empires." Islam was not an appendage to Greco-Roman civilization, but a fresh start, "a revolucion" in world history (fol. 48). It was a religion that returned to the purity of monotheism that had been lost amidst the theological controversies of the "Jews Judaizing Arabians Judaizing Christians... Jacobites Nestorians Arrians Trinitarians Manichees Montanists Sabeans & Idolaters" (fol. 119). In this respect the importance of Stubbe should not be underestimated: he was the first writer in English to use Arabic and non-Chalcedonian sources to develop a largely accurate interpretation of the beginnings of Islam and of the life of its founder. Although he fell into some of the errors and misrepresentations about Muḥammad that were endemic among European writers, he carefully referenced his sources to show where he had found his information. Toward the end of the treatise, and perhaps after further reading, Stubbe corrected some of his views and presented a heroic portrait of a "great prophet," thereby refuting the "foolish relations our authors give of their [Muslim] prophet and religion" (fol. 126). To a very large extent, Stubbe realized this goal in his treatise not by discovering new manuscripts or by learning new languages but by reading what was already available in print, in Latin, and in the libraries and bookstores. Thus his "Copernican Revolution."
THE LIFE OF HENRY STUBBE
Henry Stubbe was born on 28 February 1632 in Partney, Lincolnshire, to "anabaptistically inclin'd" parents, as his first biographer, Anthony Wood, recalled. At the age of ten, and after fleeing with his mother from Tredagh, Ireland, in the wake of the Uprising (1641), he was admitted to Westminster School where he studied under Richard Busby. Through the patronage of Sir Henry Vane the younger, one of the leaders of the Independents, he matriculated at Christ Church, Oxford, in 1649, and, along with Humphrey Prideaux, future dean of Norwich, "presumably attended [Edward] Pococke's Arabic lectures." After graduating BA in 1653, he joined up with the Cromwellian army in Scotland until 1655. On his return to England, he settled in Oxford, and in a number of letters to Thomas Hobbes he praised _Leviathan_ , "so great a work," which he had "read all over." On 13 January 1657 he reported to Hobbes that he was dedicating four hours "each day to ye translation of ye Leviathan" into Latin. With the help of John Owen, the Independent Dean of Christ Church (1651–58), Stubbe was appointed Second Keeper at the Bodleian Library, serving under Thomas Barlow. He remained in office "as a hired hand of Dr Owen" until 1659, when he was ejected, after which he moved to Stratford-upon-Avon "to practise the faculty of physic."
By then Stubbe had become active in writing. His first letters and publications show him as a confrontational and rather pedantic man, with some interest in English literature, citing in the course of his treatises Abraham Cowley's poetry and the plays of Ben Jonson. He was, as Wood wrote in _Athenai Oxonienses_ (1691–92), "the most noted Latinist and Grecian of his age." In this early stage of his writing career, Stubbe did not show an interest in Islam, but, in 1659, upon disagreeing with William Prynne, the Presbyterian polemist, he ridiculed him for supporting monarchy and associated his ideas with practices of the Ottoman government: "I hope the Assembly of Lincolns-Inne will keep a Fast for the good success and prosperity of the Turke, that so they may have the best of Governments, a Monarchy." Stubbe published other treatises in that year, revealing a knack for extensive citations from a wide array of sources. He had, as Wood explained, "a most prodigious memory." Stubbe proved faithful to his patron, Sir Henry Vane, who was maligned in the dangerously transitional year of 1659 (he would be executed in June 1662) and wrote to defend him against the accusation of "Socinianism."
After the Restoration Stubbe conformed to the Anglican establishment and became an ardent supporter of the Stuart king and of the Church of England. In 1661 he went to practice medicine in Jamaica, having secured the posting through the assistance of Sir Alexander Fraizer, the king's first physician. After his return to England in 1665, he pursued his "practice in the Countrey" of Warwick, while keeping himself informed about the intellectual changes in the metropolis. Stubbe rejected the new scientific method promoted by the Royal Society because of its emphasis on utilitarian experimentation and its total disregard of historical learning, and he set out to refute the ideology that Thomas Sprat promulgated, in _History of the Royal Society_ (1667), and which was taken up by Robert Boyle and Joseph Glanvill in the latter's _Plus Ultra_ (1670). The result was a series of acrimonious exchanges that appeared in print in the late 1660s and early 1670s. "I was sensible of the injuries he [Glanvill] doth unto the dead," Stubbe wrote, "the affronts he puts upon the living, the contempt wherewith he decries the University Learning and those Studies by which Christianity hath been supported against the Arrians, the Jews, the Mahometans, and of late the Papists and Socinians." Stubbe emphasized that the universities, with their historical traditions of knowledge, were crucial to science—contrary to the views of the Royal Society that saw the universities as antiquated. The "Utility of the Ancient and Established Method of Medicaments used in Physick," Stubbe asserted, revealed how much the "innovations" of the "Institution of the Royal Society" were really not innovations at all, but reworkings of past demonstrations. The desire to show that he was au courant with, although opposed to, the new "virtuosi" and his intense zeal to challenge them drove Stubbe to conduct experiments even on himself. Thus, "in January last 1669, I had another occasion to bleed... I took also some of the pure citrine Serum of my blood, which tasted not very salt." He was not against the experimental method, he proclaimed, just against the skepticism it generated regarding the past.
In June 1670, Stubbe wrote to his friend N. N. how "during the late times, because I would joyn with no party in a Church, they imagined that I could be of no Religion." Perhaps in his attempt to show that he _did_ have religion, Stubbe combined his attacks on the Royal Society with a strong defense of the Anglican establishment and its theological appeal to the first "Three Creeds, and four general Councils, or thirty nine Articles." Stubbe feared that the importance given by the Latitudinarian members of the society to reason in theological matters would expose England to Catholicism and "furnish the Spaniard with better and more advantageous Opportunities." And so, he vehemently defended the "Monarchy and Religion of this land, the welfare of the Church or State" while continuing his attacks on Sprat, Glanvill and others for affronting him in their writings. These attacks may help to explain why he was the butt of satire in Samuel Butler's "The Elephant in the Moon" (ca. 1671).
In the course of his attacks Stubbe reminded his adversaries of Muslim contributions to learning. Instead of disposing with the knowledge and languages of the ancients, as the new scientists urged, "the King," wrote Stubbe, "should erect certain Schools in all the principal Cities, wherein the Arabick tongue should be taught: that so by this means there may be such among his Subjects, as shall be able to Dispute with the Turks, Moors, and Persians." Knowledge about the languages and heritage of other civilizations was needed in the defense and consolidation of England. That was why, continued Stubbe, medieval monks had fervently studied the "learning of the Sarracens"—in order to refute them—a position that recalled the words of Alexander Ross in justifying his 1649 translation of the Qur'ān to English.
When the Third Anglo-Dutch War (1672–74) broke out, Stubbe turned away from the Royal Society and its danger to attack the Dutch. Conscripted by Secretary of State Arlington to write in defense of the king's policy, Stubbe published a treatise showing how the Protestant Dutch had been more harmful to England than the Muslim North Africans. In _A Justification of the Present War against the United Netherlands_ (1672) he stated: "If we look upon the number and quality of the injuries which we have received from the Dutch, the Turks of Algiers and Tunis are less offensive, and less perfidious. If we consider the courses by which the Dutch attacque us, the Algerines are the more supportable to an English spirit, since they act by force, and open piracy; what the Hollanders do by finess and deceipt." To defend the king, Stubbe marshaled his usual flare for citations from learned tomes and presented arguments explaining and justifying the king's actions—not only in waging war on Holland but also in issuing the Declaration of Indulgence (both took place in March 1672). The declaration generated so much opposition that the king had to withdraw it a year later, at which time Stubbe wrote a second treatise, _A further iustification of the present war against the United Netherlands illustrated with several sculptures_ (early 1673). Stubbe denounced the "Sectaries" who were opposed to the declaration and turned to the history of the early church in search of quotations, allusions, and references that would justify the actions of his king. He selected passages from the declaration and demonstrated how, in each passage, Charles II was following the "Declaration of Constantine the great, concerning a general Indulgence." Stubbe highlighted the doctrinal confusions of early Christians and how important Constantine and Theodosius were in enforcing religious authority during times of fissure and heresy. The second part of the treatise was a vitriolic attack on Holland, in which Stubbe surveyed the history of anti-English Dutch activities dating back to the reign of Queen Elizabeth. In praising the courage and enterprise of English seamen, he described them as "our Legionaries, our Janizaries, and Mammelucks."
While writing _Further iustification_ and earlier preparing _An Epistolary Discourse Concerning Phlebotomy_ for publication (1671), Stubbe started _The Originall & Progress of Mahometanism_. The reference in the latter treatise to Lancelot Addison's _West Barbary, or A Short Narrative of the Revolutions of the Kingdoms of Fez and Morocco_ (Oxford, 1671) shows that Stubbe started writing after that date; and the reference to Wilhelm Schickard's _Jus regium hebraeorum. E tenerbris Rabbinicis erutum, & luci donatum_, which reappeared in 1674 (with heavy annotations, unlike the 1625 edition), suggests that he was still working on the treatise then. Islamic history was becoming part of his study, and in his attack on Andrew Marvell's _The Rehearsal Transpros'd_ (1672) he referred in his opening to "Saracenical Histories" and rejected the allegation that "Mahomet" had had "two Companions, which clubb'ed with Him, in making the Alchoran." Clearly, Stubbe had become steeped in Latin texts about Islam, for not only did he defend the Prophet, but in the opening of _Originall_ he praised the (unnamed) man who had given rise to "Mahometanism":
Nothing was more mild than his Speech nothing more courteous & obliging {than his carriage} he could dexterously accommodate himself to all Ages humours & degrees He knew how to pay his Submissions to the great without Servility and to bee complacent to the meaner Sort without abasing himself. He had a ready wit {a penetrating and discerning Judgemt} & such an Elocution as no Arabian before or since hath ever equaled when he pleased he could be facetious without prejudice to his Grandeur; he pfectly understood the Art of placing his favours aright he could distinguish betwixt the deserts the inclinations & the interests of men he could penetrate into their Genius's & intenciõns without employing vulgar Espialls or Seeming himself to mind any such thing.
(Fol. 2)
After this opening, Stubbe moved to the two parts of his study of Islam. Part 1 focused on the beginnings of Christianity, from its messianic origins in Judaism and its subsequent doctrinal fissures to the century that saw the birth of Muḥammad (fols. 1-49). In writing this part, Stubbe relied on the "higher criticism" of the Bible by the foremost exegetes of his century, both continental and English, and presented an alternative history of the "original and progress" of Christianity—one that differed markedly from the account in _Further iustification_. The parallels in style and references between _Further iustification_ and _Originall_ are clear, especially the long discussion of the Novatians, the Donatists, and the Arians, as well as some of the turns of phrases. Stubbe was working on the two treatises simultaneously, but with different goals in mind—perhaps confirming what Wood said about him: "So dexterous was his pen, whether pro or con, that few or none could equal, answer or come near him." In _Originall_ Stubbe turned the material about Constantine around and presented a devastating critique of Christian historiography in the centuries before the rise of Islam.
Similar critiques had been written by Anglican clerics about early Christian historiography, but always with the Church of Rome as target. The polarization between Anglicans and Catholics intensified during the Restoration over the place of reason and of authority in religion. Taking their lead from continental biblical scholars such as Isaac Casaubon, and form English polemists such as William Chillingworth, especially his influential _The Religion of Protestants_ (1638), Anglicans emphasized the place of reason over the infallible authority of the pope. In such a context the study of church records and councils became central for Anglicans as they tried to prove the errors in Catholic claims. Thomas Traherne, a contemporary of Stubbe's (and also from Oxford, Brasenose College) published (anonymously) his _Roman Forgeries_ in 1673 in which he criticized the claims of the Catholic Church to ecclesiastical preeminence, arguing that the first 420 years of the Christian Church had been reliably documented, until the subversions and forgeries of the papacy. In his work Traherne built on the foundations of formidable Anglican apologetics such as Bishop John Jewel's _Defence of the Apologie of the Churche of Englande_ (1567) and Thomas James's _A Treatise of the Corruption of Scripture_ (1611).
Stubbe was a rationalist, but not a Socinian, and, unlike John Milton in _Of True Religion, Haeresie, Schism, Toleration_ (1673), he was not interested in attacking "popery" much as he feared Catholic influence on the king. Instead, and armed with a rationalist/Hobbesian approach, he turned to examine the origins of Christian theology through the writings of the earliest church historians. His was not a theological but a historiographical battle, which is why he consulted the writings of Eusebius, Socrates, Zosimus, and other historians because they showed, in his view, how the Christianity of the first centuries so "degenerated into such a kind of paganism" (fol. 41) that it lost its original message. "All that is written contrary hereunto are palpable untruths" (fol. 33), he asserted, supported by the scholarship of the Dutch classicist, Gerard Vossius, who had discovered many a "forgery" in the records of early Christianity (fol. 44). While in _Further iustification_ Stubbe praised Constantine, in _Originall_ , and in direct opposition to Anglican sentiments in favor of the emperor and the veracity of early church councils (as expressed by Traherne and later by Andrew Marvell in the 1676 _A Short Historical Essay_ ), Stubbe described Constantine as a "Bastard" whose "Sword was his title" (fol. 32). The records of the councils, he stated, were completely unreliable, as Eusebius demonstrated (fol. 43). Roman imperial power, in the persons of Constantine and Theodosius, had consolidated Christianity by force, thereby diverting the Gospel message to the variety of sects and schisms that plagued the six centuries before the birth of Muḥammad:
It may phaps seem strange that the generall descripcõn of the primitive Christians wch is here represented, should differ so much from the usuall Accts thereof wch are given by the Divines & Vulgar Historians, but in Answer hereunto I desire the Reader to consider first the grounds & proofs wch I go upon, and if the Authors be good the Citations true & indisputable, if the progress of Christianity be such as is conformable to the constant Course of human Affairs & great Revolucõns that then he would not oppose me, by discourses of Miraculous Accidents unimaginable effusions of the Holy Ghost & such like Harangues.
(Fols. 42–43)
Having completed the first part of _Originall_ (fols. 1–49), and using the same revisionist methodology, Stubbe turned to the second part—to study Islamic history in the manner he had studied early Christian history. This second part focused on Muḥammad and the revelation of Islam and it is in two sections:
a. The first section (fols. 49–107) includes chapters 3–7 in the University of London manuscript and corresponds to a fragment that has survived from the late seventeenth century: BL Sloane 1709. This fragment is a complete and separate pamphlet, with its own pagination, suggesting that it was written as a unit on its own.
b. The second section of the University of London manuscript, fols. 107–142, also survives in two fragments in BL Sloane 1786. Like BL Sloane 1709, these fragments are written in the same hand, on the same quality of paper, and stand as units on their own, each with its own title: fols. 107–113: "Concerning the Justice of the Mahometan Warrs & that Mahomet did not propagate his Doctrine by the Sword/with a vindication of Mahomett's Carriage towards the Christians"; fols. 114–128: "Concerning the Christian Additions"; and fols. 129–142: and "As to their opinions concerning God, purgatory, Judgmt & paradise they are these." In this section Stubbe relied heavily on the work of his friend and mentor Edward Pococke, who had been translating and commenting on Arabic histories for the previous twenty years. Heavily annotated with references to al-Makīn and Abū al-Faraj, , this second section focuses on the errors of "European Xtians" regarding the history of Islam (fol. 121). Importantly, and while the first section saw Stubbe including a few negative references to the Prophet, in this section Stubbe presents the Prophet and the revelation of Islam in wholly admiring terms: although his emphasis remained on the Prophet as a political and military leader, Stubbe showed how Muḥammad's actions had been determined by his historical context and by his goal of inspiring his followers towards empire. In so doing, Stubbe rebutted Euro-Christian errors about the Prophet and the miracles that were falsely ascribed to him: "Behold the simplicity of the Christians then who were deluded, and thought to delude by such fopperies as these" (fol. 122). In writing this part Stubbe often replaced _Mahometanism_ with the term _Islam/Islamism_ and _Mahometan/Mahometans_ with _Moslemin._
While working on _Originall_ , and in July 1672, Stubbe wrote but did not publish "An enquiry into the Supremacy spiritual of the Kings of England, occasioned by a proviso in the late Act of Parliament against conventicles." Stubbe was serving as the king's publicist (and receiving handsome payments for his services), and he was clearly in total support of the king's policies. But his opposition to Catholic influence on King Charles II was so intense that, after the passing of the Test Act and the removal of the duke of York as lord high admiral (June 1673), Stubbe published, anonymously in the _Paris Gazette_ , a two-page attack on the marriage of the duke to the Catholic Mary of Modena (between 20 and 27 October 1673). Like many of his Anglican compatriots, Stubbe feared Catholic power and, while remaining supportive of the royal and ecclesiastical establishments of the monarchy and the Anglican Church, he feared that the king's brother would beget Catholic children to inherit the throne from his heirless sibling. As a result of this publication, on 30 October 1673, a warrant was issued to John Dawson "to take into custody Dr. Henry Stubbe for seditious discourses and printing and publishing unlicensed papers." Anthony Wood explained that Stubbe was "hurried in the dark from one private prison to another, threatened with hanging, and was put to a great deal of charge."
After his release, Stubbe quickly tried to ingratiate himself with the secretary of state and on 30 November 1673 he published a translation of Jaques Godefroy's _The History of the United Provinces of Achaia_ , a third attack on the Dutch and a further defense of the king. The brush with the law might have alienated Stubbe from political involvement, for in the years that followed he dedicated himself to his medical practice and to the local scene. In 1674, he described in letters to the earl of Kent some of the ribaldries in Bath: The Duchess of Portsmouth "is frolicksom in the Bath shews her feet & leggs above water. He also wrote about the wines of the spa city. Meanwhile, he kept up his scholarship, visiting the Bodleian to consult books he did not own and meeting with Pococke, whom he often mentioned by name in _Originall_. Wading into Christian and Islamic history to work on his treatise must have taken much of his time, but he continued living and working in Stratford-upon-Avon, in summer maintaining a practice in Bath. On 12 July 1676, while traveling from Bath to Bristol to care for a patient, having had a bit too much to drink, he fell off his horse and drowned. He was buried in St. Peter's and St. Paul's Church in Bath. Joseph Glanvill, his intellectual archenemy, who a few years earlier had described him as "this crackt Fop of W... rwick," gave, according to Wood, an "indifferent" funeral sermon. Not surprising, neither gravestone nor memorial has survived of Henry Stubbe in the abbey. But there is one of Glanvill.
_THE ORIGINALL & PROGRESS OF MAHOMETANISM_
Stubbe has received well-deserved attention in studies of Islam in modern scholarship and has been recognized for the boldness of his thought. P. M. Holt saw Stubbe's interest in Islam as a product of the mid-seventeenth-century civil wars, while James R. Jacob argued that _Originall_ reflected the change in Restoration England that gave rise to a "secular conception of history" inspired by Hobbes. Stubbe, added Jacob, wrote his treatise after he began to identify with the radical movement in English religious thought that included figures like John Milton, Andrew Marvell, Algernon Sidney, and Lord Shaftesbury. For Jacob, _Originall_ was intended as a message to Charles II about proper governance at the same time that it could be viewed as the link between "radical Protestantism" and Deism. Citing Jacob, Christopher Hill agreed, as did Justin A. I. Champion: the _Originall_ is part of the "radical" religious developments that led to the "early English deists," and belongs in the trajectory that led to the Socinian tracts of the 1690s and to John Toland's _Nazarenus_ (1718). Humberto Garcia argued that the beginnings of the Enlightenment in England can be traced, in some measure, to Stubbe and his views on Islam.
These views are cogent, but they do not explain Stubbe's fascination with the Prophet Muḥammad. Nor do they explain his attack on the sectaries whom Marvell had defended and his implicit alignment with Samuel Parker, the formidable Anglican conservative, whom Marvell had ridiculed in his _Rehearsal Transpros'd_ (1672 and 1673). Neither is it clear how Stubbe's position regarding the Prophet could reflect the "religious and theological exuberance of the Interregnum" (Holt), when the printing of the first English translation of the Qur'ān in 1649 caused an angry reaction from Interregnum authorities—along with the vicious attack on Muḥammad in the "Caveat" by Alexander Ross. Nor why Stubbe's supposed "radicalism" extended to praising Islam: after all, Stubbe wrote in praise of Muḥammad in a manner that no other Restoration writer did—not even Milton. On the contrary, in _Paradise Lost_ , Milton demonized Islam by comparing Satan to the Ottoman Sultan. No other English writer who upheld politically "radical" views about English politics wrote about Islam and Muḥammad with the same admiration and scholarly erudition that Stubbe did, nor did any other writer, either in England or on the continent, establish a place for Islam in the historical sequence of monotheism as interpreted in the Qur'ān: from Abraham through "Ismael" to Muḥammad. By deliberately ignoring Isaac, Stubbe upheld the Muslim narrative of prophetic history.
C. E. Bosworth argued that Stubbe's rejection of Trinitarianism stemmed from admiration of the Great Tew Circle, especially Lord Falkland and William Chillingworth, both of whom are mentioned in _Originall_. Although they may not, as H. John McLachlan observed, "have been antitrinitarian in theology, their Latitudinarianism may be regarded as a step in the direction of Arianism and Socinianism." Jacob and Champion concurred with Bosworth, confirming Stubbe's place within the rising trends of Socinianism, Deism, and Whiggism in the Restoration period. As Champion noted, Socinian tracts were widely available in the 1660s and '70s, and writers in the 1690s such as Arthur Bury, William Freke, and Stephen Nye identified "Unitarianism with monotheistic Islam." But there is no evidence in any work that Stubbe admired Socinianism: actually, he attacked it from the time he defended Sir Henry Vane to shortly before he began writing _Originall_ : "The Socinians multiply upon us," he complained in 1670. That Stubbe would laboriously research a vast corpus of writings about Islam, that he would openly and unambiguously praise "Mahomet" and the Qur'ān, that he would elaborate on the role of 'Ali as the missionary of Islam, and that he would do all that to present Islam as a "standard against which to measure current Christian practice and the current conduct of Christian princes—and no doubt Charles II and the English church," as Jacob urges—is unlikely. After all, Stubbe bluntly praised the Prophet as an able legislator, admired Muslim toleration of minorities, and defended the laws of the Qur'ān against usury and wine. Nor did Stubbe write an imaginary satire in the manner of later texts that used an Islamic mouthpiece to criticize contemporary Europe, such as Giovanni Paolo Marana's _Letters writ by a Turkish Spy_ (1684–86) or Montesquieu's _Lettres persanes_ (1721). _Originall_ is a work of meticulous historical revisionism, presenting "a thorough defence of their [Muslims] sentiments," as Thomas Magney wrote about the treatise just under half a century after Stubbe's death.
Further, Stubbe's attitude toward Islam went beyond anything that even the Socinians proposed to the Moroccan ambassador in 1682. As the surviving account shows, the Socinian delegation did not praise the Qur'ān but rather wanted to show the Muslim visitor the common errors in the Bible and in the Qur'ān. But Stubbe did not criticize Islam or the Qur'ān in his treatise. At the same time, had he written his treatise to advise Charles II, Stubbe would not have hesitated to state that openly. He was "a very bold man," as Wood confirmed, and "utter'd any thing that came into his mind, not only among his companions, but in public coffee-houses (of which he was a great frequenter) and would often speak his mind of particular persons, then accidentally present, without examining the company he was in." He did not mince his words, nor did he appeal to the "rhetoric of subterfuge" or the "art of theological lying" or treated Islam as "a beating stick with which to attack Christian revelation and the clerical establishment." He wrote about Islam because he came to see Islam and the Prophet in a new light.
In regard to Deism: Stubbe's discussion of Islam led him to statements about religion that could have been taken from the pages of Lord Herbert of Cherbury's _De Veritate_ (1624): "He taught his followers to abolish Idolatry every where And that all the world was obliged to the profession of these truths that there was one God, that he had no Associates, that there was a providence & a retribution hereafter proportionate to the good or evil Actions of Men" (fol. 108). But, no writer _before_ Stubbe had argued for a "Unitarian-Islamic syncretism." The first Deist and Socinian writers in England argued for a religious system based on reason: but they did not appeal to Islam as a model. And when Restoration Anglican theologians like Edward Stillingfleet, John Tillotson, Isaac Barrow, and Joseph Glanvill wrote to show that the Bible and "science" were compatible, not a single one of them praised Islam or Muḥammad. On the contrary, Barrow wrote a savage attack on the Prophet. Perhaps most revealing of Stubbe's intellectual and theological position is the absence of reference in _Originall_ to any of the aforementioned English theologians, nor, for that matter, to any Socinian writers. Rather, his constant references were not to "radicals" but to his mentor at Christ Church, Edward Pococke, a nonradical of deep Anglican piety; to al-Makīn, Ibn al-Baṭrīq, and Abū al-Faraj; and to "Judicious Protestants" (fol. 45) like Isaac Casaubon, Joseph Scaliger, John Selden, Hugo Grotius, Johann Hottinger, Claudius Salmasius (of anti-Milton fame), and G. J. Vossius. Had he been a generation younger, Stubbe might well have been part of the "republic of letters" that included many of those authors whose works he admired and cited.
In 1678 Charles Blount, an avowed Deist, lifted pages from _Originall_ and included them in two letters, one to Hobbes and the other to the earl of Rochester. Significantly, in neither of these letters did Blount use any material from the Islamic sections of the treatise, nor did he allude to a "Unitarian-Islamic" association. At the end of the century, Humphrey Prideaux associated Islam with Deism and attacked them both; if he had Stubbe's treatise in mind, it is strange he did not mention it, since he was meticulous in referencing his sources. Critics have associated Stubbe with the anti-Trinitarianism of Deism because of the treatise that appears after _Originall_ in the 1701 University of London manuscript, _An Epistle from Achmet Benabdalla a Learned Moor concerning the Christian Religion_. This refutation of Trinitarian Christianity—which appeals for evidence to the Qur'ān and to the various doctrinal conflicts of European Christendom—was written in 1612 by "Ahmet Ben-Abdala" in Latin and addressed to the Prince of Orange who had hosted the Moroccan ambassador. But the treatise was added in the manuscript after Stubbe's death; earlier Thomas Erpenius had thought of publishing it and John Selden had owned a copy of a Latin portion of the manuscript without them becoming accused of Deism.
At the outset of the _Originall_ , Stubbe presented the goal of his treatise: to study Islam in its historical context. To do so, he explained his methodology. There was always "a series of preceding causes which principally" contributed to change. History, he wrote, worked according to the laws of causality: "This is certain that when the previous dispositions intervene, a slight occasion oftentimes a meer casualty, opportunity taken hold of & wisely prosecuted, will produce those Revolutions which otherwise no human Sagacity or Courage could accomplish" (fol. 4). And so to understand Islam there was need to examine the Jewish and Christian background that gave rise to it. While other writers about Islam had been aware of that history, Johann Hottinger in particular, only Stubbe viewed that history as a heterodox cause to a monotheistic end. As he explained, late Judaism developed expectations of a temporal Messiah to rule over Jew and Gentile, to which early Petrine Christianity adhered. But from Paul on and with Constantine and Theodosius, Christianity changed in regard to the belief in the Messiah, the theology of the sacraments, the role of the clergy, and the godhood of Christ. Christianity, wrote Stubbe, was corrupted by imperial power and diverted from the message expressed in the Acts of the Apostles, the only extant document that captured the ideals of the Christ movement. Such causes gave rise to Islam—not as anti-Christian, as European theologians viewed it, but as post-Christian.
Stubbe knew that such views were dangerous, but he was clearly growing angry with some of his Anglican compatriots who were waxing lyrical about the purity of the church: Thomas Traherne, Paul Rycaut, and others. Stubbe emphasized that the Anglican Church, which he fervently defended, recognized the problems in the foundational documents of Christianity. These writings were "so depraved that the Church of England and generally the Protestants reject the authority of them and admit no general councils after that of Chalcedon under the Emperor Martianus." And, to make matters worse, there had never been a study of the earliest sources of Christianity: "I must add that the Church History of ye primitive times seems mainly deduced from the Latin & Greek writers who give no Acct. either of the Syriack or Judaizing Churches so that we hear no news of the latter till St. Jerom & Epiphanius came to represent them as Hereticks for adhering to the same Doctrine & Discipline wch St. Peter & St. James & all the Apostles (except Paul) had instructed them in" (fol. 44). And to those who would disagree with him, Stubbe answered: "what soever is alledged agt me must be out of suspected or spurious Writers partiall in their own case & ignorant either for the want of learning or want of books & Opportunities to be informed aright or a prejudicate opinion blinding their Judgmts: I conceive the Credit of what I write ought to seem most valid, because 'tis consonant to ye Acts of the Apostles & the reall existence of things" (fol. 43).
It is in the light of the extensive exposé of Christian historiography that Stubbe's discussion of Islam should be read. Islam was not a religious aberration or an erratic heresy, but part of the logic of history that had started with Judaism and Christianity. And it was a logic that Stubbe discovered after reading of Christian Arabic chronicles (in Latin translation).
ARABIC STUDIES IN ENGLAND
The seventeenth century witnessed an increase in the Arabic, and, to a lesser extent, Turkish and Persian, manuscripts that became available to English and continental writers. The first catalogue of the Bodleian Library in 1605 included only a few works by Averroes and Avicenna, and in 1615 the Arabist William Bedwell complained that the only copy of the Qur'ān he could consult "was so badly imprinted, that in verie many places, I was constrained to diuine and guesse." By mid-1630s, however, and as Samuel Hartlib mentioned in his _Ephemerides_ , the "world" of Arabic manuscripts had been "brought thither," that is, to Oxford. The change occurred during the tenure of William Laud as archbishop of Canterbury and vice-chancellor of Oxford. Laud encouraged the purchase of Eastern manuscripts for the Bodleian and established a chair of Arabic in 1636. From 1655 to 1670 the first professor of Arabic, Edward Pococke Sr., was able to train "almost single-handedly a new generation of Arabists" at the university. Pococke, who died in 1691, was at the center of a group of English thinkers who wrote about (and against) Islam. He was the student of William Bedwell (d. 1632) and Matthias Pasor (d. 1658); the friend of John Gregory (d. 1646), John Selden (d. 1654, whose collection of oriental manuscript augmented the Oxford collection after his death), and John Greaves (d. 1652); and the teacher of Henry Stubbe (d. 1676), John Locke (d. 1704), Humphrey Prideaux (d. 1724), and Samuel Clarke (d. 1729). His _Specimen_ also influenced the French orientalist Richard Simon (d. 1712).
The Laudian oriental collection and the disciplines that were developed to study it showed Britons, including Stubbe, what the swarthy "Moors" and the "Mahometan Turks" and sword-wielding "Saracens" had produced: a vast civilization that had adopted and adapted the same Greco-Latin legacy that Britons claimed as their own classical patrimony. Archbishop Laud, although not a scholar himself, recognized that there was a "great deale of Learning and that very fit and necessary to be knowne," a view upheld by Pococke: "the Progress" that the Arabs had made "in ingenious Studies," he wrote, "was so great, that they hardly came behind the Greeks themselves." In the polyglot Bible prepared by Brian Walton in the mid-1650s, there was praise for the intellectual legacy of the Arabs in "Marocum, Gessum, Septae, Hobbedae, Constantinae, Tuneteum, Tripoli, Alexandria, Alcairo, Basor, & Cusa in quibus millia studiocorum omnium facultarum"/in which there were thousands eager to learn every discipline. Not only did the libraries in those cities contain Arabic books, wrote Walton, but Greek and Latin, too, that had been translated into Arabic. The importance of Arabic could not be ignored—which explains the attraction to the language and its output not only in Britain but also on the continent, ranging from Thomas Erpenius's use of Arabic to write to Bedwell—and signing his name, Touma ibn Erpin—to the three-thousand-page _Lexicon Arabico-Latinum_ by his Dutch compatriot Jacob Golius in 1653.
Earlier, in 1645, the Maronite priest, Abraham Ecchellensis/Ibrahīm al-Ḥaqilānī, published his _Concilii Nicaeni Praefatio_ in which he showed how Arabic texts had preserved much that had been lost in Latin and Greek and that the "Oriental tradition is much stronger" than the Western one regarding accuracy in church records and history. Al-Ḥaqilānī corrected the errors of scholars ranging from Nicholas of Cusa to Cardinal Bellarmin who, ignorant of Arabic, had mistakenly understood a number of Qur'ānic verses as urging Muslims to worship the Prophet. Like his fellow Maronites, Jibrā'īl al-Ṣuhyūnī/Gabriel Sionita and Yūḥanna al-Ḥaṣrūnī/John Hesronita, al-Ḥaqilānī introduced Arabic writings about Islam to Christian European readers. Such emphasis on the value of the Arabic legacy was confirmed by an Egyptian Copt by the name of "Yusuf al-Misri al-Gibti" who coedited, and published in 1672 through the Sheldonian press in Oxford, a huge tome of early church council resolutions. Stubbe referred to al-Gibti in _Originall_ (fol. 116).
The Christian Arabic writers were a great discovery to Stubbe. "It is certain," Stubbe wrote, "that the Christians which lived under the Mahometans, as Elmacin & others, do mention Mahomet wth great respect as Mahomet of glorious Memory, and Mahumetes sup quo pax & benedictio"—unlike the European "others" who see in him the "Antechrist" (fol. 118). Stubbe determined that the study of the beginnings of Islam should be conducted through sources written in the language of Islam and by writers belonging to the world of Islam. But, for a non-Arabic reader, there were no Latin translations of Islamic history and theology by Muslim authors. Although there were numerous geographical, alchemical, and medical treatises by Muslim writers that were available in Arabic and in Latin translation, there were no oriental sources on the Prophet Muḥammad available in Latin other than the chronicles by the Christian Arabic authors. These writers described the prophetic calling of Muḥammad without the kind of vituperation that dominated medieval and early modern Euro-Christian writings. While an uninformed writer like Samuel Purchas could well believe that a "Christian Arabian" would attack the "words and Phrases" of the Qur'ān, Stubbe realized that the Christian Arabic writers had no reason to calumniate Muḥammad who was a "great Honorer of Isa so he [Muḥammad] alwaies express'd a great Reverence for them, And 'tis concerning them that he Sayes that Isa their prophet shall save them in the last day" (fol. 114) Since Muḥammad respected Christ, the Christians living among the Muslims respected Muḥammad, who in turn respected them. "That the Arabian Xtians were men of just Strict deportmt appears from hence, that Mahomet Saith of them that one might safely intrust them wth any sume of money they would restore it again" (fol. 114–15).
And so Stubbe turned to these Christian Arabic authors, all of whom drew on a non-Latin tradition of Christian history and, in two cases, non-Chalcedonian. These authors were the following—in the order of their seventeenth-century appearance in print:
_Jibr ā'īl al-Ṣuhyūnī/Gabriel Sionita_
Sionita (1577–1648) was a Maronite priest from Lebanon who had studied in Rome at the Maronite College (est. 1584), and later moved to France. Together with a fellow Maronite, Yūḥanna al-Ḥaṣrūnī (d. 1626), he translated the abridged geography of al-Idrīsī (d. 1166) which had been published in Arabic in Rome (1592). Their translation appeared in Paris in 1619, confusedly entitled _Geographia nubiensis_. They then added a treatise, "De Nonnvllis Orientalivm Vrbibvs, Nec non indigenarvm Religione ac moribus tractatus brevis," of sixteen short chapters about oriental religions and mores as well as about cities such as Baghdad, Damascus, "De Bochara Avicenna patria," Mecca and Medina, along with their native Lebanon and its Maronite patriarchate. They relied, as they stated, on a number of Muslim scholars, notably Muḥammad ibn Qāsim (whom Stubbe often mentions). They included descriptions of the theology and culture of Islam: thus the chapters on the "origo, fraus, dominium, sepulchrum, vxores, liberi, socij"/origin, fraud, rule, tomb, wives, children, allies of Muḥammad, on the Jacobite and Nestorian heresies in the east (and the Greek "schisma"), and on the funerary rites of Muslims and Christians.
Sionita and Hesronita's description of Muḥammad was hostile: perhaps it could not have been otherwise, given that their book was published in Paris under royal auspices, "Summa Priuilegij." But the two authors expressed great pride in the Arabic language and the intellectual legacy that had been developed throughout Islamic history, in "philosophiae, Medicinae, Astrologiae, legum, [and] Rhetorices." They recorded the names of many great historical figures who had flourished and written in Arabic. What is significant is that they mixed Muslim with Christian Arabic writers without distinguishing them. Thus there were Muslims such as "philosophi nobilissimi, Auerroes, Algazeles, Abū-Becr, Alfarabius" as well as Christians, such as "Abu-Zaid Ben-Honain, qui Euclidem in Arabicam linguam translutit, & Thabet Ben-Corra, qui eundem Euclidem à, multis mendis correxit, atque commentariis anno ab Hegyra 282 illustrauit"/who [corrected?] many mistakes from this same Euclid and showed in his commentaries it was the 282nd year after the Hijra. The Lebanese authors presented the two groups of writers as part of the intellectual legacy to which they belonged, and which they were eager to present to their European coreligionists. Two decades later, in 1641, and in line with the Maronites' eagerness to present Arabic learning to European readers, Ibrahim al-Ḥaqilāni translated and published in Paris _Synopsis propositorum sapientiae Arabum philosophorum_. He dedicated the book to Cardinal Richelieu.
In 1630 Sionita translated from Arabic into Latin a short text that had been brought to Europe by the Capuchin Pacifique Scaliger: _al-'Ahd wa-l-shur ūṭ al-latī sharra'ahā Muḥammad rasūl Allah li-ahl al-milla al-nuṣrāniyya/Testamentum et pactiones initae inter Mohamedem et Christianae fidei cultores_ (Paris, 1630, reprinted in 1638 and 1655). It consisted of treaties between the early Muslim armies and the Christians of the Byzantine East. Interestingly, in his translation Sionita used the phrase "Mahomedem Apostolum Dei"/Muḥammad the Apostle of God (which was removed by the printer from the title page), and he opened with "Mahomedes à Deo missus ad omnes homines erudientos"/Muḥammad sent by God to instruct all mankind. Nevertheless, the treaties were not about the Prophet, but about the history of Islamic protection of Christians during the early conquests, a protection that had allowed for the continuity of Christian society in the Islamic polity. Stubbe, who mentioned Sionita in the _Originall_ , specifically referred to those treaties because they supported the view that Islam did not spread by the sword and that Muḥammad had made a place for the Jews and the Christians within the bounds of his "dominion": "There is extant a Compact or League betwixt Mahomet & the Xtians, published in France by Gabriel Sconita & reprinted by Johannes Fabricius a Dantzicker, wch. is by him afirmed to be mostAuthentick, & mention'd by Selden tho Grotius takes it to be but a figment of the Xtians, that they might gain favour wth the Moselmen" (fol. 110). Significantly, Stubbe did not accept the authenticity of the treaties, judging them "Suppositious" (which scholars also question today). But he saw that they confirmed the spirit expressed in the Qur'ān which contained "Sundry passages... wherein he [Muḥammad] permits the unbelievers to hold their own religion, & declares that every of them [ _sic_ ] Jew Xtian or other might bee Saved if he hold that there was one God Creator, a day of Judgemt.. & lived justly & uprightly" (fol. 110). Notwithstanding the dubious history of those treaties, Stubbe wanted his English readers to consult them, which is why he directed them to the English translation in Paul Rycaut's "Relacion of the Turkish Governmt l. 2, c.2" (fol. 110). Like his mentor Pococke, Stubbe admired Sionita (whom Pococke had met in Paris), and he relied heavily on al-Idrīsī's geography for information about Arabia. But Sionita and Hesronita remained less useful to his project than other Christian Arabic writers who had belonged to non-Catholic backgrounds: al-Makīn, who was a Copt, Ibn al-Baṭrīq, who was a Melkite, and Abū al-Faraj, who was a Jacobite. Importantly for Stubbe, these writers had been part of Islamic society in the medieval East, and it was in the Eastern traditions of Christianity, which had developed their own and differing theologies, that Stubbe met with a view of Muḥammad that was found nowhere else in Western Christian thought. Specifically, from al-Makīn and Abū al-Faraj Stubbe was able to compile an account of the last ten years in the life of the Prophet—the first ever in English to present the battles, strategies, and negotiations of Muḥammad, with considerable admiration.
_Jirjis ibn al-'Am īd al-Makīn_
Al-Makīn (ca. 1205–73) was the author of _T ārīkh al-Muslimīn/History_ of the Muslims, which had been translated into Latin by Thomas Erpenius, the Dutch orientalist (who had been introduced to Arabic studies by William Bedwell), and published in 1625 in Leiden as _Historia Saracenica_ , the "first historical work in Arabic to be published in Europe." Al-Makīn wrote a history of the world, starting with the creation, and ending with the accession to power of the Mamluk sultan of Egypt, Baybars, in 1260. In the Latin translation Erpenius did not include the parts preceding the life of the Prophet Muḥammad and began his text with his birth, giving the subtitle (which is not al-Makīn's) as _Res Gestae Mvslimorvm, a inde Mvhammede primo Imperij & Religionis Muslimicae auctore_. Al-Makīn stated at the outset that he followed the account of the Prophet by al-Ṭabarī, the ninth-century historian whose _T ārīkh al-rusul wa-l-mulūk_/History of Messengers and Kings was, and remains, one of the most important histories of early Islam. As Pococke wrote, al-Ṭabarī was _thiqat f ī naqlihi wa tārīkhihi/an_ authority in his historical writings. What is striking is that al-Makīn's section on the Prophet is not very long, but it was intensively quoted by Stubbe, as also by Pococke, John Gregory, and Johann Hottinger. Al-Makīn's work was a standard reference among scholars from Oxford to Basel.
Erpenius opened his translation of al-Makīn's account by keeping the Islamic invocation, "In nomine Dei misericordia miseratoris" an opening that Christian Arabic writers sometimes used. Adopting the words from Islam, al-Makīn found no problem in proclaiming with his fellow Arabic-speaking Muslims the name of God, the merciful, the compassionate. Al-Makīn then turned in the first chapter to the "Primus Muslimorum Imperator"/the first emperor of the Muslims, with Erpenius oddly translating the plural _umar ā'_ as the singular "Imperator," a word that has no Arabic root, about "Muhammed Abūlcasimus, gloriose memorie"/of glorious memory, a close but not literal translation of _ṣ alla Allah 'alayhi wa sallam_ (Muḥammad was known as Abū Qāsim). Al-Makīn continued with information about the life of the Prophet, emphasizing famous events mentioned in al-Ṭabarī about "nativitatem ejus, genealogiam, & res gestas, donee fugit Medinam: deinde autem bella ejus, & victorias, omnemque fortunam, donee é vita excessit"/his nativity, genealogy, and other events leading to his flight from Mecca: then his wars and victories, and all the fortunes of his life, until his death. It was Muḥammad, according to al-Ṭabarī, and al-Makīn repeated unreservedly, who first revealed the religion of Islam/azhara _d īn al-Islam_: "Religionem Islamisimi... primum manifestavit & observavit." Notwithstanding his Christian faith, al-Makīn carefully described the prophetic calling of Muḥammad, confirming how Khadīja, the Prophet's wife, had been first to accept his prophethood, "Prima in prophetiam ejus credidit Chadiga"/Khadīja was the first to credit him with prophecy.
Al-Makīn used the Hijri calendar, and only sometimes the Coptic calendar (starting in AD 284), and he took the side of the "Arabs" as he described their wars with the Byzantines. Such commitment on the part of al-Makīn to Islamic history did not surprise Stubbe: al-Makīn had been "Secretary of State to one of their [Muslim] Princes" (fol. 110), and early in his life he had commanded Muslim armies in Egypt and Syria. Al-Makīn confirmed for Stubbe that the Prophet "was, God's prayer and peace on him well mannered, gentle in speech. He visited his companions as they visited him, and he kissed their faces as they kissed his. He consoled the weak and praised the strong and was compassionate to the poor, and whoever asked him for anything, received what the Prophet could give him, or received a helpful word." Of the Arabic Christian writers, al-Makīn furnished Stubbe with the longest, and most "Islamic," perspective on the Prophet. Towards the end of _Originall_ where Stubbe turned to a year-by-year life of the Prophet after the Hijra, he followed al-Makīn's chronology and description (as did Pococke and Hottinger, too).
Very important for Stubbe was the history of the Christian community during the early years of Islam. Al-Makīn emphasized how favorable the Prophet was to the Christians of Arabia who came to see him: "A dignitary from among the Christians came to him [the Prophet], so he stood up to welcome and host him. They asked him about that [his action] to which he replied 'Treat well the Copts of Egypt, for you have relatives from among them' [since one of his wives was a Copt]. And he said, 'He who mistreats a _dhimm ī_ will be punished on the day of judgment.' And he said, 'He who hurts a _dhimm ī_ hurts me.'" The Arabic text missed a few prepositions in the printing, which may explain why the translation into Latin omitted some words. Also the Latin added some marginalia that did not appear in the Arabic, although they served to emphasize Muḥammad's protectiveness of Christians: "Affectus Muhamadis erga Christianos." In his _Originall_ , Stubbe echoed those views about the treatment of Christians, because he found the information about the Prophet's openness to Christians (and to Jews) very important in the context of his presentation of Islam:
Thus Mahomet Ben Achmed expounds him, Elmomin who collected his History of the Saracens out of the best Mahometan Writers and was himself Secretary of State to one of their Princes, avowed that Mahomet did give protecciõn & Security to the Pagans Magicians & Jews & Xtians also. wch swore fealty to him & paid him yearly tribute, Moreover that he Sent Omar to the Xtians to assure them that they should live Securely under his dominion. And that he would esteem their lives as the lives of his Moslemin & of their Goods as the goods of those others.
(Fol. 110)
A page earlier, Stubbe quoted the whole Pact of 'Umar which confirmed for him a point he repeated often in his treatise: that Islam accepted other religious communities and did not try to forcibly convert or expel those communities. It was a point that the next Christian Arabic writer whom he consulted repeated: Ibn al-Baṭrīq.
_Aft īshyus/Sa'īd ibn al-Baṭrīq/Eutychius_
Ibn al-Baṭrīq (877–940) was the author of annals of the Church of Alexandria, _Kit āb al-Tārīkh al-Majmū'_. A short selection about the history of the Alexandrian patriarchs had been translated by John Selden and published in a bilingual Arabic and Latin text, with copious notes, in London in 1642 (the first substantial Arabic printing in England). Selden, "the chief of learned men reputed in this land," as John Milton described him in _Areopagitica_ (1644), urged his friend Pococke to translate the whole text because it was "considered by learned Men abroad [Erpenius and Casaubon] as a very useful Work." Pococke did not agree since he viewed the history as unreliable; still, he went ahead and in 1654 produced a translation of _Eutychii Patriarchae Alexandrini Annalium_ , followed by another edition with the title _Nazm al-Jawhar/Contextio gemmarum, sive, Eutychii patriarchae Alexandrini annales_ (Oxford, 1656). Another Oxford edition appeared in 1658/59.
Sa'īd ibn al-Baṭrīq was an Egyptian Melkite and patriarch of Alexandria. Disagreeing with Pococke, Stubbe viewed him as a "Historian of good Credit" (fol. 31), perhaps echoing Selden, who had described Ibn al-Baṭrīq as the "Egyptian Bede." Ibn al-Baṭrīq's history covered the time from Adam to AD 938, with specific focus on the Christian Eastern (in his case, Egyptian) communities and their encounters with the expanding empires of the Persians, the Byzantines, and the Muslims. Although he had little to say about the Prophet Muḥammad—but always added words of praise whenever he mentioned him—he started the history of the rise of Islam with reference to the migration to Yathrib/Medina, correctly recognizing the historical decisiveness of that event. He summarized the lives of the first four caliphs and later dynasties, focusing in great detail on the actions of the second caliph 'Umar, since it was during his reign that pivotal contacts with Christians occurred.
Ibn al-Baṭrīq described the many theological controversies that had bedeviled Christian history. After all, as the brief biography stated, he had lived at a time of much "dissidia"/conflicts. He then described the confrontations between the Monophysites and the Melkites and the plots and collusions of bishops and patriarchs as they vied for ecclesiastical office and doctrinal supremacy. Stubbe quoted him: "Ismael Ibn Ali a Mahometan Historian, that at the Nicene Council in the 20th year of the Raign of Constantine there were assembled 2048. Bishops out of wch he Chose 318.... It is granted by ye Orientall Historians of the Church that Such a Number of Bishops was sumoned as the Mahometans specify, So Saith Eutychius & Josephus an Egyptian Presbyter in his preface to the Arabick version of the Councils" (fol. 116). In his attempt to learn about the beginning of Islam, Stubbe was turning to historians other than those recognized in the Western Latin legacy. The fact that those historians were Christians of the Arabic tradition gave him ammunition for his argument: that the forgeries on which some church doctrines had been based can be exposed by the Christian historians who had lived among the Muslims. Ibn al-Baṭrīq showed how the divisions, disputations, and conflicts among the various Christian communities were resolved by Constantine's imperial authority, resulting in the alienation of the non-Melkite Christians from the centralizing power of the Constantinople-based patriarchate. The history of Ibn al-Baṭrīq showed Stubbe the Christian doctrinal confusions that preceded, and explained, the rise of Islam—a chief argument in _Originall_.
Ibn al-Baṭrīq presented Stubbe with a historiography that emphasized the continuing distinctiveness of the Christians in the context of the rise and expansion of Islam. The Egyptian writer saw Christianity within the continuum of the Islamic empires (Umayyad and Abbasid)—so much so that in his writing he adopted numerous phrases and images from the Arabic of the Qur'ān. Very importantly, Stubbe found in Ibn al-Baṭrīq historical descriptions about how Christians interacted with the early Muslim conquerors: Ibn al-Baṭrīq emphasized how the Byzantines fought against the Arabs, while other Christians, including the Copts, did not. Muslim-Christian relations had been more accommodating than the battles that Stubbe witnessed during the mid-seventeenth-century civil wars in England, Scotland, and Ireland: which is why he must have paused at Ibn al-Baṭrīq's account about the peace that "Chaledo Ebn Walid"/Khālid ibn al-Walīd granted to Damascus and how much the Muslim leader tried to put an end to bloodshed. Most striking for Stubbe would have been the meeting between the Caliph 'Umar and the patriarch of Jerusalem, Sofronius, and the subsequent treaty of peace: "In the Name of God Mercifull & gracious, from Omar ye Son of Alchittabi to the Inhabitants of the City Elia Security & protection is granted as to their psons Children Wives Estates. & all their Churches that they bee neither destroyed alienated nor prohibited the Xtians to resort to" (fol. 109).
Ibn al-Baṭrīq presented Stubbe with information about Islamic protection of Christians and their religious holy places, institutions, and traditions. The history of Islamic conquest had set the tone for accepting the Christians and granting them "securitate ipsis & pace pacta"/security to these very people and an agreed-upon peace, and although violations occurred, as Ibn al-Baṭrīq bitterly noted, Christian life continued—as the thirteenth-century Christmas sermon published in Arabic and in Latin in 1656 showed. The beautiful Arabic and the rhyming prose of the "Homile in Natalem Chrsiti" by "Petre Sancto, Catholico, Patriarcha. D. Elia Tertio, vulgo dicta Ibn Hadit" was lost on Stubbe, but the translation conveyed a deep emotional intensity on the part of the celebrant and his congregation—all of whom were living in the midst of the Islamic polity.
_Gregorios Ab ū al-Faraj_
Abū al-Faraj (1226–86), author of _T ārīkh mukhtaṣar al-duwal/A Short History of Dynasties_, was so respected in his Muslim society that although "Christianus erat," wrote Edward Pococke, "à quo tamen didicerunt multi è Muslemorum eximiè doctis"/although he was a Christian, many of the _fu ḍalā' al-Muslimīn/_the dignitaries of the Muslims revered his work. He was of "the Sect of the Jacobites," wrote Humphrey Prideaux in 1697, "an Author of eminent note in the East, as well among Mahometans as Christians." From Gregorios/Abū al-Faraj/Ibn al'Ibri/Bar Hebraeus, Edward Pococke had published a translation of thirty pages in 1650: _Specimen historiae Arabvm: sive, Gregorii Abul Farajii Malatiensis, de Origine & Moribus Arabum_ (Oxford), to which he added over two hundred pages of dense notes, producing a scholarly compendium about early Islam. Over a decade later, Pococke translated the whole book, _Historia compendiosa dynastiarvm authore Gregorio Abul-Pharajio_ (Oxford, 1663), which was reproduced in 1672 under the title of _Historia Orientalis: Authore Gregorio Abul-Pharajio_ (Oxford).
Abū al-Faraj included a historically balanced description of the life of the Prophet Muḥammad, confirming for Stubbe the traditional Islamic narrative about birth, parentage, marriage, flight, and consolidation of the Islamic polity. Abū al-Faraj lived in Aleppo for some time, supported in his bid for the patriarchate by the Damascus-based al-Mālik al-Nāṣir. In 1258 Hulago attacked Aleppo, and so Abū al-Faraj was sent to implore him to spare the city. Hulago did not heed him. After witnessing the horrors of conquerors, he, like the other Christian Arabic writers, realized how benevolent Islamic rule had been and therefore, in his history, conveyed a respectful view of Muḥammad, "Author Legis Islamitica Mohammed Ebn Abdallah." And so, like al-Makīn, he opened his volume with the Islamic invocation, "In nomine Dei Miseratoris Misericordes" and then, as he recounted the life of the Prophet, he mentioned the famous episode from the Sīra/biography about the monk Bahīrā, who declared that the boy Muḥammad would be famous all around the world: "In the future, he said, this boy will enjoy greatness, and his fame will spread in east and west, for when he arrived here, a cloud covered him with its shade." Quite striking a few lines later were the words that Abū al-Faraj used regarding the prophetic revelation: " _azhara_ al-da'wa,"/"prophetae munus sibi arrogavit"—the same verb that had been used by al-Makīn, " _azhara_." Two pages later Abū al-Faraj cited the words of Abū Sufyān to 'Abbās after Muḥammad's entry into conquered Mecca in 630. Abū Sufyān, the erstwhile enemy, converted to Islam in order to save his life, and when he saw the armies of Muḥammad he turned to the Prophet's uncle, 'Abbās, who had also initially opposed him and told him: "'Your nephew has become a great king.' To which the uncle replied, 'Be quiet [ _way ḥak_], it is Prophecy'/'Imo vero, Prophetia est.' And he replied, yes. Respondit ille, 'Esto igitur.'"
Like the other writers, Abū al-Faraj did not question the prophetic role of Muḥammad. After all, Muḥammad was nearly taken by his followers to be buried in Jerusalem, "locum scil.in quo sepulti essent Prophetae"/where prophets are buried. This Arabic admission, coupled with the history of inter-Christian rivalries which Abū al-Faraj described, produced in Stubbe the conviction that the Christianity which Muḥammad encountered had been corrupted and corrupt: that is why Stubbe compared the Qurayshites to the Sadducees (fol. 69), and the early Muslim victims of Qurayshite persecution to the early Christian martyrs. As Jesus sought to "reform" Judaism (a verb that Stubbe used consistently), so did Muḥammad seek to correct the errors of the Jews and the Christians. And to protect them: "the Christians who had been so persecuted by Cosros & finding their Condition very uncertain among the Arabians according to the humours or interests of the Government were very Glad of his [Muḥammad] rise & magnified his undertaking" (fol. 66).
No other seventeenth-century writer about Islam or about the Prophet Muḥammad used the Arabic Christian writings in the informative manner that Stubbe did. Neither great poets like Milton or Dryden, nor great theologians like Barrow, Baxter, Tillotson, nor travel writers like Rycaut and Maundrell, nor historians of comparative religion like Pagitt and Ross were willing to give credit to the Arabic historians—even though their works had been edited, translated, and published. On the contrary, even those who are known to have read the Arabic/Latin sources persisted in their hostility to Islam. Thomas Smith, who read Ibn al-Baṭrīq and Abū al-Faraj, attacked Islam violently in his _Remarks upon the manners, religion and government of the Turks_ (London, 1678; Latin versions in 1672 and 1674); Lancelot Addison, an Anglican clergyman, who wrote about Morocco after serving as chaplain in Tangier, and who mentioned al-Makīin and Abū al-Faraj, produced a vicious biography of the Prophet in 1679, as did Humphrey Prideaux, bishop of Durham, in 1697, whose _The True Nature of Imposture fully display'd in the Life of Mahomet_ (London, 1697) was a vitriolic attack on the Prophet, notwithstanding Prideaux's reading and quoting of the Arabic writers.
The Arabic/Latin texts demonstrate that there was a scholarly body of writings in the seventeenth century in England and on the continent that diverged markedly from the negative representation of Muḥammad and Islam. These writings could have played a significant role in disaggregating hostile representations and in furnishing new information based on fresh scholarship. But only Henry Stubbe turned to those texts that drew "on the narration of the Mahometans or Arabick Xtians" and not on "the European Xtians" (fol. 121) and rewrote the history of Islam. By giving credence to the Christian Arab writers, Stubbe brought them to the center of Islamic-Christian history, making them indispensable interlocutors who challenged Western historiography and the Western canon.
'ῙSA: THE QUR'ĀNIC JESUS
At the end of _Originall_ there is a crossed out paragraph in which Stubbe showed how God could not have expected believers to subscribe to such an "impossible" doctrine of "Father the Son & the Holy Spirit" (fol. 142). Stubbe was thinking in terms of the Qur'ānic Jesus whom he met in numerous texts against Islam. But as was his wont, Stubbe selected from these attacks the information that supported his point of view: that Jesus was historically closer to the Qur'ānic portrait than to the theological doctrine that had developed around him in the Christian church. In this context of an anti-Trinitarian Jesus, the originality of Stubbe becomes striking: while his contemporary John Milton was writing his _De Doctrina Christiana_ and invoking Christian history in his defense of a nonincarnational Jesus, and while Socinians appealed to "reason," Stubbe turned to Islamic sources for support.
One writer whom Stubbe read carefully in regard to the figure of Jesus in the Qur'ān was Levinus Warner and his _Compendium Historicum Eorum quae Mahummednai de Christo_ (1648). Warner wrote to challenge the Muslim view of Christ (the title continues, _Et praecipuis aliquot religionis Christianae capitibus tradiderunt/They_ relate in particular chapters some things [about] Christian religious practice), but still, he gave a detailed description of the "Muslim" Jesus: the meaning of his name, the virginity of his mother, his disciples/ _al- ḥuwariyyūn_, his prophetic lineage from Abraham, and the conundrum of his execution—all from passages in the Qur'ān. Using an Arabic text of the Qur'ān (alongside a Latin translation), and focusing on the verses that dealt with Jesus and the naṣārā/Christians, Warner "amass'd abundance of Testimonys of some Mahometan Doctors, who make honourable mention of our Lord Jesus Christ," as the Dutch orientalist, Adrian Reeland, explained over half a century later.
In his address to the reader, Warner stated that although the Qur'ān rejected the Christology of Jesus, it did not reject his prophetic/messianic role in history. Warner did not accept the refutations of the divinity of Jesus that appear in the Qur'ān and in other Muslim sources, but in presenting those refutations he emphasized how the Qur'ān still recognized that Jesus was "natus non sit, nisi solo Dei verbo"/was not born except by the word of God alone. Warner also linked the annunciation to Mary that appears in the Sura of 'Imrān ("ô Maria, Deus tibi annunciate VERBUM suum: nomen eius Christus"/O Mary, God announces to you His word, whose name is Christ) to the first verse of the Gospel of John, "In principio erat verbum/In the beginning was the word." Throughout his short treatise Warner added various stories from the vast compendium of the Hadith about the ascetical piety of Jesus. The Muslims had been "blind" to the doctrine of the incarnation, but they had many "vestiges" of truth regarding the prophetic role of Christ. It is from Warner, whom he mentions specifically, that Stubbe borrowed some of the sayings and proverbs of Jesus in the _Originall_.
Stubbe noted the high regard accorded to Jesus in the Qur'ān, and perhaps because he wanted to distance himself from the Nicene Jesus, he used the Qur'ānic name, 'Īsa, throughout his treatise: "(So they call Jesus Christ & So I shall name him in the subsequent story)" (fol. 59). At no point did Stubbe use the name Jesus. For a scholar who was quite familiar with the apocryphal literature about Jesus in the Christian tradition, it may well be that the Hadith stories confirmed his suspicions: that the Jesus of the Arabian tradition, who found his way into the prophetic revelation in Mecca and Medina and subsequently into Islamic piety, was the nondivine figure venerated by the Christian Arabs at the time of the Prophet Muḥammad. That is why, Stubbe believed, Muḥammad continued the teachings of Jesus and why the Qur'ān confirmed that "Isa was his predecessor [Muḥammad's] & taught ye same doctrine" (fol. 120). At the same time, Jesus spoke in language that invokes the Qur'ān: at the end of an invocation by 'Īsa ibn Maryam that Selden reproduced in his edition of Eutychius, Jesus appeals to God in words that are Islamic in their resonance: "Neque praesice mihi eum qui non miserebitur mei, Pro misericordia tua, O Miserantissime miserescentium"/Do not send against me those who are not merciful to me, you who are most merciful and compassionate. Since Muḥammad did not bring a new revelation, and since the Qur'ān confirmed the original teachings of Jesus, then, concluded Stubbe, it was "but Justice to stile him a Xtian" (fol. 120).
In the vast corpus of Hadith about 'Īsa, from the eighth century on, the unwavering emphasis of the tradition is on the poverty of Jesus, his humility and contempt of the world, his celibacy and love of God—the same Jesus whom Stubbe advocated. Quite striking in this context of the Islamic 'Īsa is Stubbe's summary of a story that recalls Chaucer's "Pardoner's Tale" about the three men who found a treasure and in their greed killed each other for it. Jesus commented at the end: "Behold how these three suffered by it these are deceased & left behind what they thought to bee owners of woe unto him that seeks riches in this world" (fol. 117). Stubbe found this story in another text by Warner: _Proverbiorum et sententiarum Persicarum Centuria collecta_. Even more striking is that it is the same story about Jesus that al-Ḥasan al-Yūsi, a contemporary jurist in Morocco (1631–1691), used. The same Jesus was in the Maghrib and in Stratford-upon-Avon—and in much of Islamic piety. It is very unlikely that Stubbe knew much about the image of Jesus in contemporary Islam other than what he read in Latin translation, but it is significant that all the stories about Jesus he included in his treatise came from the Islamic tradition, with very few references to the Gospels.
THE PROPHET MUḤAMMAD
So how does the Prophet Muḥammad appear in the _Originall?_ Stubbe presented the first historical biography of the Prophet in English, based on the Qur'ān and on "the formal words in wch ye Mehometans express themselves" (fol. 127), very likely meaning the Hadith and the biography (all of which had been excerpted and discussed in the works of Selden, Pococke, Hottinger, and others). It was the first account of Muḥammad in England that was chronologically presented and not theologically argued. Stubbe ignored the hostile literary and dramatic representations of Muḥammad in European writings and concluded his treatise with a year-by-year description of events until the death and burial of the Prophet. Influenced by Hobbes and the secular view of history (as Jacob noted), Stubbe showed that Muḥammad and the beginnings of Islam should be contextualized in Arabia's sixth/seventh-century religions and societies: thus his emphasis on continuity between many Judaic and Islamic teachings and on the fact that the military victories of Muḥammad and the consolidation of the new faith were a result of intelligent and inspired leadership, as had been the case with the Hebrew prophets. Stubbe showed that Islam made perfect sense as a religion: it was not the "scourge of God" to sinful Christians, but a continuation of revelation, a monotheism that prevailed over polytheism as a result of the Prophet's tenacity, the "heavenly wisdom" of the revelation (fol. 70), the support of the four successors, especially his cousin 'Ali, and the perseverance of the "Moslemin."
From the outset, Stubbe was interested in examining how the Prophet, "a fiercely opposed" man of low social standing ("mean estate," fol. 1), was able to transform the Arabs from tribes to empire through the power of revelation. Stubbe wanted his English readers to view the Prophet in the manner in which he had been seen by Arab Christians, who had deeply respected him. That is why the Christian Arabic writers became for Stubbe the measure of truth: whatever they did not mention, he did not accept. None of them, for instance, ever mentioned the frequently repeated "Fable" that Muḥammad had been inspired by a pigeon that he claimed as the "Holy Ghost" (fol. 121). Following Pococke, who in his notes to _Specimen_ had rejected this allegation, or the other allegation about Muḥammad's tomb being suspended in midair, Stubbe denounced such credulity in European writers. He also added his own insights: no pigeon, he explained, could have been trained to perch near Muḥammad's ear without rousing the suspicion of his followers and detractors. What clinched Stubbe's "empirical" refutation of this allegation, however, was that no "Christian of the Arabians mention it" (the pigeon) (fol. 121).
Stubbe was highly selective in his information. Although he carefully read the universal history of Marcus Boxhornius (published 1652), for instance, he picked what he thought fitting, ignoring the derogatory material that was always included in European texts about Muḥammad. Some of these denigrations included references to Muḥammad as a camel driver, poor and illiterate: "homo pauper & mercaturam exercens cum camelis," wrote Boxhornius. In regard to the last, Stubbe reminded his readers of the humble backgrounds of many of the Hebrew prophets and of Jesus and his carpentry. He emphasized also how the Prophet had been theologically and intellectually versatile by mentioning the journeys of Muḥammad, all the way to Egypt and Spain, regions that had not been associated with the Prophet during his lifetime. Given such travel and exposure, Muḥammad was not as outlandish a man as he appeared in the "great untruths" of Euro-Christian sources. Rather, added Stubbe, he was like any of the "Nobles of Venice or Genoa" (fol. 57), a merchant and a traveler with acute observation. For Stubbe, these Renaissance cities were not much different from the cities of Arabia: Sionita had described Mecca as "vrbs Arabiae Matrix nobilissima," while Erpenius had observed in his 1613 oration on the dignity of the Arabic language that Mecca was comparable to Amsterdam "Meccam, Amsterdami nostri magnitudine emporium." Stubbe repeated the analogy by Erpenius.
In following this line of argument, Stubbe was using the method that his contemporary Edward Stillingfleet had used in historicizing the Prophet Moses. In _Origines Sacrae or, A rational account of the grounds of Christian faith, as to the truth and divine authority of the Scriptures, and the matters therein contained_ (1662), the Anglican divine wrote against the skeptics who were raising questions whether a man as uncouth and unexposed as Moses could have written the Pentateuch. In chapter 2 of the second book, Stillingfleet examined the early education of Moses, emphasizing the latter's mastery of "Mathematical, Natural, Divine, and Moral learning of Egypt, their Political wisdom most considerable." Although there is no scriptural information to support this view, Stillingfleet emphasized that Moses was a historian and a law-giver, thereby proving the "certainty" of his writings, and the divine "truth of Scripture." Stillingfleet confronted the detractors who rejected the Mosaic authorship of the Scriptures in the same way that Stubbe argued against those who attacked Muḥammad's role in the Qur'ān, and, as Stillingfleet did not doubt that the Pentateuch was the divine Scripture written by Moses, neither did Stubbe doubt that the Qur'ān consisted of "Surats" which "the angel Gabriel presently" brought him, as he had earlier brought them to "Edris Noah Abraham Ismaell Moses Isa" (fols. 100, 84).
It is in this context of approximating Muḥammad with Moses that Stubbe's challenge to the illiteracy of the Prophet should be read. From the start, Muslim exegetes had appealed to the illiteracy of the Prophet in confirmation of the divine revelation of the Qur'ān. Stubbe mentioned this view, emphasizing that the Arabians believed him to be "Nabian Ommian, that is the illiterate prophet" (fol. 123). But Stubbe knew that European authors always used this reference to illiteracy as proof of the falsity of "Alcoran": since Muḥammad was illiterate, writers argued, he could not but have sought help in formulating his revelation from local Jews and Christians. Since "the Author himself being no Linguist or Scholar, nay, not able to read or write," wrote Alexander Ross in 1649, Muḥammad could not but have sought help from the Nestorian monk Sergius in formulating his "lyes and sensless follies." Because Muḥammad was "illiterate," wrote Sir William Temple contemporaneously with Stubbe, he produced "a wild Fanatick Rhapsody of his [Muḥammad's] Visions or Dreams." It did not help that very few men of those who attacked the Qur'ān could read it in Arabic, but rather relied on the poor Latin translation by Robert of Ketton (in the twelfth century), the poorer translation into French by André du Ryer (in 1647), and the poorest of all translations, to English by Alexander Ross in 1649, which Stubbe denounced (fol. 139).
Nor could Stubbe read Arabic. But he was intent on refuting such accusations. And so, to turn the tables on the detractors, Stubbe argued for a literate and cultured Prophet. He did so by showing his readers that in Muslim exegesis the adjective _umiyy_ could refer not only to illiteracy but also to Mecca, mother of the villages, _"ommal koras"_ (fol. 123). He learned this information from the careful discussion by Pococke (which he summarized). But, whereas Pococke was the philologist searching for alternative meanings to _umiyy_ , Stubbe was the polemist. Stubbe knew that illiteracy was a sine qua non in Muslim historiography, and thus, he continued, notwithstanding the Prophet's literacy, the Qur'ān was still a revelation from God. He recalled the words in the translation of the Sūra of Yūsuf/Joseph by Erpenius earlier in the century: "per verbum Dei intelligent suam quae Coranum ipsis dicitur, & quàm Muhamed ijs persuasit coelitus ad se demissam"/"by the word of God they understand their own [biblia/book], which they call the Coran and which Muḥammad persuaded them had been sent down to him from heaven." Also, in Pococke's _Porta Mosis_ , Stubbe came across a reference to al-Ghazali's discussion of Muḥammad's _"luminis prophetici"/n ūr al-nubuwwa_/light of prophecy. Stubbe concluded that, whatever the level of the Prophet's literacy, it remained separate from authorship because the Qur'ān was "inimitable," a word that he repeated twice in the treatise (fols. 100, 138). It was a word that recalled the description of the Qur'ān by Abū al-Faraj, "eloquentissimi," and a doctrine that was upheld by all Muslims. All the early companions of the Prophet, confirmed Stubbe, believed the "Coran" to be "derived from God" and full of "heavenly wisdom" (fols. 61, 70). Toward the end of the manuscript, Stubbe affirmed that for Muslims only what was mentioned in "the Alcoran" was foundational to Islam (fol. 140) because the Qur'ān was a "Standing Miracle" (fol. 141).
The three BL Sloane fragments are especially important in this context because they show a transformation in Stubbe's view of the Prophet. It is quite possible that after Stubbe finished the first fragment (BL Sloane 1709, fols. 94r–115r) he circulated it among readers, who objected to his largely favorable view of the Prophet. Although he had mentioned some of the medieval falsities about Muḥammad, Stubbe had still presented an admiring portrait. So, and not untypical of Subbe, he wrote the second two fragments (BL Sloane 1786, fols. 181r–185r and 186r–190r) in response to those readers where he confronted them with a totally positive view. These latter fragments show Stubbe in an argumentative mode, sometimes reminding readers that he had already mentioned some of the discussion points earlier. Mostly, he was eager to show his readers a more accurate view of the Prophet from what he had earlier written.
_The Prophet in BL Sloane 1709_
Without having access to Arabic primary sources that would furnish a more accurate view, Stubbe made some mistakes in this fragment, such as claiming a Qur'ānic origin for a dialogue between Muḥammad and a Jew (fols. 81–82), which was mentioned in many of the texts he read, or treating Abū Bakr as the Prophet's uncle. Eager to present a "secular" view of the Prophet, in the manner he viewed other prophets, Stubbe emphasized the personal initiative of Muḥammad and his astuteness of "designe" (fols. 63, 64, 66). But he mentioned that the Prophet had concocted prophecy (fol. 60), "intitling God and the angel Gabriel to his dictates" (fol. 65); on another occasion, he wrote of "venery" (fol. 63), although a scribe could have made a mistake in this particular case. Such views were not uncommon among European orientalists: even the learned Hottinger, whose study of the Prophet's history was both detailed and extensive, called Muḥammad, on every page of the printed text, the "pseudopr.." Stubbe was not immune to such views, and in this fragment there is a sense that he was still searching for a full interpretation of the Prophet. Chiefly, however, Stubbe was trying to approach Muḥammad as a historical figure: his interest was more in the religious and military achievement than in the prophetic background. What he wanted to show his readers was that Muḥammad's actions could be perfectly understood from a historical perspective: the Prophet had been a great strategist, warrior, and negotiator, whose success lay in harnessing the power of revelation to create the Islamic polity.
_The Prophet in BL Sloane 1786_
Stubbe continued searching and, in the two fragments which conclude _Originall_ , he arrived at his final view of Islam. He started by demolishing the widely held belief that Islam spread its message by the sword by referring to the Christian Arabic writers who had left important documents about Islamic history. He also brought in stories that demonstrated Islamic reverence to Jesus and he defended Muḥammad against accusations of sensuality, emphasizing the "stoical" element in Islam, as in Christianity (fol. 134). In these fragments Stubbe gives the impression that he had a list of all the criticisms that had been leveled at him—and he was refuting them, one after another. He praised Muḥammad for opposing usury, wine, and divination, and he discussed the thorny issue of the Prophet's miracles, which were commonly ridiculed in contemporary European writings. While some believed those miracles, so much so that Abū al-Faraj referred to them, Stubbe emphasized that the Qur'ān did not confirm them and that many Muslims rejected them because the only miracle that was to be credited to the Prophet was the Qur'ān (fols. 140–141). Earlier, Stubbe made the case for Muḥammad as a legislator superior to Moses (fol. 136) and praised him in the highest terms that a Christian writer in the seventeenth century could: in his piety before God, Muḥammad was like the historical (not theological) Jesus, for he, Muḥammad, told Christians "that such as beleived in Isa ought to live according to his pcepts with great humility piety & unconcernedness for ye pomp and Vanities of this world" (fol. 115). Stubbe then introduced a number of sayings by Jesus from the Hadith which had been preached to Christians by the Prophet (fol. 117). Perhaps thinking of this analogy between Jesus and Muḥammad, where the latter confirmed the teaching of the former, Stubbe described Muḥammad toward the end of the treatise as "a great prophet" (fol. 137)—the same phrase that described Jesus in the Gospel of Luke 7:16. No English writer had ever made that analogy—twice (fol. 61).
Nor had any writer praised the Qur'ān as did Stubbe. A contemporary such as Richard Baxter was willing to concede that "God hath made use of Mahumet as a great Scourge of Idolaters of the World," but he had produced "an Alcoran" that was "a Rhapsody of Nonsence and Confusion." Stubbe moved beyond such invective to open praise, and, having learned from Pococke that the Qur'ān was revealed in "Arabic verse," he used the only word available to him in English, "poem," as a translation of "surat." But, the "poem" was not "Nonsence"; rather, affirmed Stubbe, "God by Mahomet took a better Course by leaving to Mankind one lasting Miracle, the truth whereof should in all ages bee Satisfactory & Convincing" (fol. 138). Cognizant of the criticisms that European Christian writers leveled against the Qur'ān, he asserted: "I have often reflected upon the Excepcõns made by the Xtians agt t the Alcoran & find them to bee no other than what may be argued wth ye same strength against our bible. And what the Christians say for themselves will fully Justify the Alcoran" (fol. 139).
Interestingly, Stubbe paraphrased various verses from the Qur'ān and sometimes integrated them into his text:
The Alcoran, a transcendent miracle, & wch is more one that is permanent, from generation to generation. Nor is there any lasting Miracle of ye prophet, excepting that whereunto he appealed, challenging all the Wits of Arabia (&Arabia did then abound wth thousands whose chief study was eloquence & poetry) to make one Chapter or more that might compare therewith & thereby demonstrated to the most incredulous, the truth of his prophesy. And God Said concerning it, that if all Men & Angels should combine to write any thing like it, they should fail in their enterprize.
(Fols. 138–139)
Although Stubbe was meticulous in his recording his sources, he did not mention that the last sentence was a Qur'ānic verse: 17:88. His readers, thus, would have taken the Qur'ānic assertion as his own conviction And it was a conviction on which Stubbe elaborated. Having read Pococke's translation of passages from al-Ghazali's _Tarjamat 'q īdat ahl al-sunna_, he found no qualms in presenting that creed in his treaties: it is in the Qur'ān that God is revealed, His oneness, omnipotence, and omniscience; the power of God over all the creation; the prophetic continuity in His messengers to humankind; reward and punishment; and the salvation of the damned (apocatastasis):
That God is one God that there is none other that he hath no equall no Son nor Associate. That his Eternity hath neither beginning nor end that 'tis impossible to explain properly his Attributes and yt no intellect can comprehend the Extent of his Dominion. That contemplative Men may conjecture at his being by the daily occurents on earth, but never understand his Essence. That ye Heavens are his Throne, the Earth his footstool, but that the Governmt. of both is no trouble to him: That he is Omnipotent Omniscient Omnipresent, who sits upon the Universall Throne by his Essence, & by his Understanding penetrates into all things: That his providence disposeth of all Affairs below, neither doth any thing fall out not the Corn grow not the Grass wither but according to the Decrees of his eternall pdestination That whatsoever Man doth ascribe to him or imagine to bee in him it is eternall, & those attributes do not argue any Composition or istinction [of] {in} his being: That all things in this World good or evill befall us according to his Will: that the beginnings progress & Conclusion of all Emergencies depend absolutely upon him: and that he determined from all Eternity whatsoever should come to pass, That his knowledge extends to the deepest Secrets, That nothing happens agt or not according to his pleasure, that in all Matters to think to Will to do depends upon him. That the Souls of Men are imortall, That those who are pserved by Faith & the intercession of the Apostles of God, Moses Isa Mahomet from Sin) do upon death live in happyness untill ye resurrection & day of Judgemt that those who are more or less wicked, must in the Grave in a kind of Purgatory undrgo some torments until the last day and there wth more or less difficulty they shall be Saved, but that nothing of Evil how little soever shall escape unpunished nor any thing of good how small soever pass unrewarded.
This is the Sum of the Mahometan Religion.
(Fol. 129)
There is nothing comparable to Stubbe's breadth regarding Islam in any contemporary English or European text. While he made mistakes and sometimes slipped hostile remarks about the Prophet, he was careful about his presentation of Islamic revelation as an expression of absolute monotheism.
"LET US THEN FANCY THE GALLANT ALY"
Stubbe's reading of the Arabic histories showed the active role that the first four caliphs played in the consolidation of Islam. Although he mentioned Abū Bakr, the first caliph, on a number of occasions, especially his role in ensuring the compilation of the Qur'ān, it was 'Ali who caught Stubbe's attention as the best preacher of Islam. Muḥammad was the Prophet and 'Ali represented the piety and the determination that the Qur'ān inspired.
Stubbe's description of 'Ali is both fascinating and unusual: fascinating because Stubbe dramatized the activities of 'Ali during the initial stages of Islam's development, a dramatization that recalls the 'Ali of the Morisco warrior saga of the sixteenth century. Stubbe was also unusual because he described the Prophet's cousin without associating him with Persia and its tradition of Shi'ism—the theological and political locus for 'Ali. Pococke included some notes about Shi'ism and the concept of the imamate in his notes to _Specimen_ , and in the _Present State of the Ottoman Empire_ (1670) Paul Rycaut wrote about _"Mahomet_ and _Hali_ , that is, the Turk and the Persian." But Stubbe ignored such references which dominated English plays, chronicles, and travelogues as he ignored the supposed rivalry in prophetic calling that some writers claimed between the two cousins. Actually, Stubbe's 'Ali united Muslims—and ignited in them the spirit of religious unity to fight their enemies: the Byzantines and the Sassanids. 'Ali was not the divider between Turk and Persian, but the pan-Islamist calling on all Muslims to confront those who had subjugated them: "lett us not live devided under more petty princes then we have tribes lett us all unite into one monarchy as we are all of one language and one parentage we are all ageien's all Ismaellites the Same Hegira will suite with all the Same Crescents is our comon Standard" (fol. 86).
An important influence on Stubbe's view of 'Ali was Pococke, who had found in 'Ali "a man of such account with that impostor [Mahomet], not only for his valour, but knowledge too, that he was wont to declare, that if all the learning of the Arabians were destroyed, it might be found again in 'Ali, as in a living library." Earlier, Sionita had also praised 'Ali, "simul cum Mohamede Moslemannica lege fuit imbutus, quam ob causam saepe dicere solebat, ego sum primus Moslemannus"/'Ali was instructed in Muslim law with Muḥammad, for which reason he was accustomed to say often, 'I am the first Muslim.' And also al-Makīn:
Averfabatur mundana, ac Deum valde timebat: multus erat in dandis eleemosynis: justus, atq; humilis, defensor veræ religionis; acutus valde, & multū pollens eruditione, quippe speculartivis scientiis & practicis instructus: audax & audcaciâ celebris, liberalis, optimæ indolis & naturæ./He distanced himself from the world, feared God, just and humble, defender of the true faith, learned and perceptive, well instructed in practical and scientific knowledge, brave and famous for his bravery, generous, good natured.
Abū al-Faraj mentioned the story about 'Ali carrying the gates of Khaybar, while Hottinger listed the miracles associated with 'Ali: "ei multa miracula tribuunt." In 1651 the work of Ibn al-Rāhib was published in Latin in Paris by Hesronita. Ibn al-Rāhib, another Jacobite Arabic writer of the thirteenth century (a contemporary of al-Makīn), wrote a history that was translated as _Chronicon Orientale_. In describing the rise of Islam, after having written about the Rūm and others, Ibn al-Rāhib presented a chronology of events, starting with the migration of the Prophet, but with only a few words about Muḥammad. Although Ibn al-Rāhib was noncommittal about Muḥammad, he praised Abū Bakr as a pious ascetic, renouncing worldly pleasures and taking from the treasury only three dirhams per day. 'Ali, however, received the highest praise; he was a man contemptuous of worldly things, dedicated to the free bestowal of alms, and a fierce advocate of his religion.
This image of 'Ali holding the world in contempt was repeated by Stubbe: "Aly... had a contempt for the world it's glory & pomp, he feared God much gave many Alms" (fol. 60). But the 'Ali of the _Originall_ is much more than the model of a God-fearing man. Stubbe turned for information about 'Ali to Adam Olearius, who had traveled between 1633–1639 to Moscow/Russia and Isfahan/Persia as part of a commercial mission. During his travels Olearius learned about the meanings and histories of Shi'ite rituals, especially "Auschur, or solemn Feast, in memory of Haly, their great Saint and Patron," which he described with a high degree of accuracy. Because Olearius had read al-Makīn, he drew a line of demarcation between the historical 'Ali and the 'Ali of Persian veneration—a line that Stubbe also observed. "Aly," wrote Olearius, "did not change anything in the Alcoran, and though he gave several Interpretations to the words of Mahomet, and explicated the sense of his Law, yet did he submit to his Authority, where it was clear, and where the Text admitted no explication, in so much that this occasion'd no change in the Religion." In similar vein, Stubbe confirmed 'Ali as a fellow fighter with Muḥammad against the idolaters. Even the Turks, noted Olearius, acknowledged 'Ali "a near Kinsman of Mahomet's, that he is truly an Imam, or Saint, and that he led a very exemplary life; and particularly that he was valiant."
'Ali also came to Stubbe's attention through his proverbs in the collection of his works by al-Sharīf al-Raḍī (AD 970–1015), later known as _Nahj al-Bal āgha_. The proverbs are in the hundreds, and in 1629 Jacob Golius published a selection of them in Arabic without a Latin translation: _Shadhra min kal ām 'al-'Arab ayy ba'ḍ amthāl 'Ali al-khalīfa/Proverbia quaedam Alis, imperatoris Muslimici_. Golius treated the proverbs as moralistic aphorisms, without theological content, and so he presented them to readers as part of the wisdom of the "imperator" of Araby. His publication may have been instrumental in alerting Pococke to the importance of 'Ali. As the 1740 biography of Pococke by Leonard Twells shows, the Oxford Arabist became deeply interested in the proverbs: "The book, which he first undertook to read on, was the Proverbs of Ali, the fourth Emperor of the Saracens, and the cousin german and son-in-law of Mahomet: a man of such account with that impostor, not only for his valour, but knowledge too, that he was wont to declare, that if all the learning of the Arabians were destroyed, it might be found again in Ali, as in a living library." Pococke translated many proverbs, and, soon after his appointment to the chair of Arabic at Oxford in 1636, he lectured on them—although he did not publish the translation, nor did another later translator, Thomas Smith. But a large number of these proverbs were included by Hottinger in _Historia Orientalis_ under headings such as piety, humility, patience, justice, and others. Stubbe knew Hottinger's work well and may have been thinking of him when he referred to the proverbs in his manuscript: "Mahometan sayings having some of them ascribed unto Aly" (BL Harleian 1876, fol. 133).
Stubbe realized that 'Ali, unlike Muḥammad, whose prophetic revelation could prove objectionable to his readers, would not provoke a kneejerk reaction from worried Christians. Stubbe thus turned to 'Ali as a "neutral" medium by which to present Islam. In the rather long speeches that Stubbe put in the mouth of 'Ali in _Originall_ , the echoes of the proverbs ring clear: If you prefer the next world to this one, you win; friendliness in a face is a second beauty; reliance on God is enough; the ornament of men is their civility; belligerence in a man will destroy him; fear of God clears the heart; the best wealth is that which is spent for God; and many others. 'Ali became Stubbe's mouthpiece of Islam: where Muḥammad was the political and prophetic leader, 'Ali was the commentator and teacher. He was the pious observer who accepted the revelation to his cousin and eloquently propagated it.
'Ali continued to attract orientalists long after Stubbe's death. In 1717, a selection of 'Ali's proverbs was translated and published by the professor of Arabic at Cambridge, Simon Ockley: _Sentences of Ali Son-in-Law of Mahomet_ (London). Ockley had great admiration for the Arabic legacy and repeatedly denounced his countrymen for their ignorance of that legacy. "What we here present the Reader with," he wrote in his preface to the translation, "is a little Collection of Wise Sentences, calculated for the Direction of a Man's Conduct in Affairs of the greatest Consideration, and are of the same Nature as the Proverbs, and Ecclesisasticus. / / They are called the Sentences of Ali the Son of Abu Taleb. The whole Book is, as near as I can guess, not much less than our New Testament." As an Arabic scholar, however, Ockley was aware that 'Ali had not been the author of all the proverbs. Still, he explained, the proverbs are important because even if 'Ali had just collected them, he was a contemporary of the Prophet Muḥammad. He concluded: "The Sentences are full, and to the Purpose: They breathe a Spirit of pure Devotion, Strictness of Life, and express the greatest Gravity, and a most profound Experience in all the Affairs of Human Life.... There is enough [in the Sentences], even in this little Handful, to vindicate, in the Judgment of any Man of Sense, the poor injured Arabians, from the Imputation of that gross Ignorance fastned upon them by Modern Novices."
It was an imputation that Stubbe had rejected in his _Originall_ , and, in presenting a devout, heroic, committed, and decisive 'Ali, Stubbe showed his admiration for the man most closely associated with the Prophet Muḥammad and the rise of Islam. 'Ali's was a piety that continued to be admired by English orientalists: George Sale, in his commentary on Sura 76, mentioned the abstinence and poverty of 'Ali, who did not have enough provisions at home to fulfill a vow to God after his children, Hasan and Hussayn, were cured of sickness. Edward Gibbon, impressed with Ockley's translation, wrote that 'Ali "united the qualifications of a poet, a soldier, and a saint: his wisdom still breathes in a collection of moral and religious sayings; and every antagonist, in the combats of the tongue or of the sword, was subdued by his eloquence and valour." Their views were in line with the writings of Henry Stubbe, the first English writer to "fancy Aly."
ISLAM AND EMPIRE
Many Euro-Christians who visited, or did not visit, the Ottoman Empire wrote about the plight of the Christian minorities and how they were living in fear and ignorance; others expressed hostility, describing the Eastern Christians as intellectually and theologically superstitious, which is why they needed to be converted to Protestantism or Catholicism. Stubbe quoted from Grotius's _De Veritate Religionis Christianae_ a long passage about the deplorable conditions of the Eastern Christians under "Mahomet [who had] propagated his Doctrine by the Sword" (fols. 115–116), but he did not use the London 1632 English translation perhaps because the frontispiece showed a "Turke" with a sword in his hand. The caption read: "The Turke stands with his sword in his hand, by which he defends his Religion, that sprang from Mahomet, a false Prophet, foretold in general by Christ, Mat. 24:5-24 also a halfe Moone." The author of the preface to the 1649 translation of the Qur'ān had also invoked this treatise for exactly the same reason: that Islam was a religion of the sword. But, unlike the author of the preface, Stubbe did not turn for his information about Eastern Christians to theologians, but to travelers such as Adam Olearius, Paul Rycaut, and Henry Blount, all of whom included descriptions of the Christian populations of the Persian and Ottoman empires in their works. Stubbe thus repudiated Grotius (who had not traveled east) by introducing into the discourse about Eastern Christians an unusual contrast: between the condition of the American Indians under the Iberians, and of other slaves whom "wee keep," and the condition of Christians under Muslim rule.
Stubbe did not need to elaborate on the plight of the Indians or the slaves under the Spaniards; such knowledge had become widely familiar in England. It was usual to compare the tyranny of the Spaniards "towards the poore Indians who neuer offend them" with the tyranny of the Turks towards all "those fall into their hands." Stubbe differed, and instead he contrasted the plight of the Indians with the "Musarabick Christians"/Arabized Christians under Muslim rule in Spain, "who alwaies lived quietly & Safely under them & others in their other Kingdoms & Dominions, An inviolate Justice being preserved towards them, and tho' the rich & potent Nobility & Rulers were destroyed or reduced to nothing wch was don to prevent future Rebellions." Not only did Muslims protect Christians at the time of the early conquests by the caliphs, but also in current times, as confirmed by one of the greatest scholars of Europe, Joseph Scaliger: "Yet 'tis observed by Scaliger," continued Stubbe, "& 'tis an assured truth that the vulgar Greeks live in a better Condiciõn undr the Turk at present then they did under their own Emperors when there were perpetuall Murders practiced on their Princes & tyranny on their people, But they are now Secure from Injury if they pay their Taxes" (fols. 109–10). Such protection, as well as Muslim concern for Christian well-being, was not, Stubbe realized, a matter of Muslim whim but of binding Qur'ānic law. John Selden, whose _De Jure Naturale et Gentium_ he often cited, praised Qur'ānic theology, which promised mercy and reward for the Christians: "mercedem autem ibi memoratam nuncupat ille... compensationes operum suorum." Stubbe concluded with a view that proved prophetic: "& 'tis more the interest of ye princes & Nobles then of the people at present wch keeps all Europe from submitting to the Turks" (fol. 110). Contemporary Christian communities in Europe were being attracted to Islam, and, just a few years after Stubbe's death, some Protestant groups supported the Turks in their attack on Vienna (1683).
The Arab and Ottoman Empires had been spread by war and the sword, but, unlike the European empires, which had also spread by war and the sword, they had not forcibly converted the native populations. Rather they had protected them. Stubbe recognized that his was the age of empires and that England might just be starting to think in such terms, especially after the celebration of England's future glory in John Dryden's _Annus Mirabilis_ (1667). Christian and Muslim, Spanish and Ottoman and Persian—all had built, and were continuing to expand, empires. But where the Christian empires enslaved and deculturalized the conquered populations, the Ottoman Muslims did not. Although slavery was practiced by all empires and all religions, at least Muslims did not enslave fellow Muslims, and "the successors" of the early European empires, presumably the English, should not be casting the first stone since they deliberately prevented Indians from finding Christ in order to keep them as slaves. Though "the principles of the Xtians seem to condemn Slavery, yet in Portugall & other places, it is frequently practiced and perhaps the Xtian Laws & Customs agt. such usage had no higher rise then Ecclesiasticall & civil policy, which the successors have indiscretely (& not out of Conscience duly inform'd) retain'd still" (fol. 113).
Stubbe learned that the acceptance on the part of Muslims of the religious Other was an ongoing practice. In Olearius he read about Armenian Christians in the Persian Empire of the 1630s—about patriarchs and congregations, all enjoying freedom to trade, pray, and increase, walking in processions with bishops parading their crosses and banners and wearing "Pontifical Robes; with Wax-Candles in their hands." Acceptance was also a matter of commonality of belief: Muslim rulers shared with their Christian subjects the expectation of the return of 'Īsa—a messianic finale that must have resonated with the eschatological and millenarian excitement of Britons during the civil wars. Pococke, ever Stubbe's mentor, described the signs of the end in Islamic eschatology and noted that the final moment would witness "Descensus Jesu in terram... apud turrim albam ad parté Damasci orientalem"/the descent of Jesus to earth near the white minaret east of Damascus. It was perhaps that Islamic openness to Christ/Christians that, Stubbe observed, made European princes willing to join forces with Muslim armies and to invade Christian regions. Religion, observed Stubbe sardonically in _Further iustification_ had never been a divisive factor when it came to the advancement of empire. "How often," he asked rhetorically, "did the Emperours of Constantinople, the Kings of Spain and France, contract for the assistance of the Saracens against Christians?" It was hypocritical to trade and cooperate with Muslims and then denounce their religion, or to praise their "government" but decry "Mahometanism," as Robert South preached in his 1660 London sermon, "Ecclesiastical Policy the Best Policy: or Religion the Best Reason of State."
It is tempting to treat Stubbe's _Originall_ as a continuation of the letter exchange with John Locke. After reading Stubbe's _Essay in Defence of the Good Old Cause_ (1659), Locke wrote Stubbe a letter in which he expressed "admiration" for the "strength and vigour" of the style, but complained about the extent to which Stubbe was willing to go in advocating toleration (in this case for the Quakers). At that point in time, Locke still believed that religious differences in the state would result in anarchy and violence. It may well be that in _Originall_ Stubbe was addressing Locke, and other proponents of toleration, as Christopher Hill argues, by writing the history of Muḥammad, a wise legislator, who had established an empire with absolute toleration. Whether Locke read Stubbe's treatise or not is not known, but it is significant that, thirty years after disagreeing with Stubbe over toleration, he wrote his _Letter on Toleration_ , which became one of the cornerstones of the Enlightenment. In the _Letter_ , and in the sequels, Locke argued for the toleration and endenization of Muslims in the Stuart monarchy—for granting Muslims (and Jews and pagans) the same status that Stubbe had so admired about Muḥammad's toleration of Christians and Jews.
*** * ***
Stubbe read what others read about Islam, but he was selective in what he adopted from the works of scholars, all of whom, including Selden, Hottinger, and Pococke, were negative in their views about Muḥammad. Although he himself had fallen prey to some misinformation about Muḥammad in the early stages of writing his treatise, and although he inherited a dramatic, theological, and poetic legacy of relentless bigotry toward Islam, Stubbe was able to change his views and present, for the first time in English, a well-documented history.
Stubbe showed how long-held bigotries could be overcome by consulting indigenous sources, thereby decentering the historiographical perspective. It may well be that Stubbe shifted away from the Eurocentric sources of knowledge about Islam because he was not a clergyman. In the seventeenth century, experts on the Christian and Islamic East, with its various languages and religious communities, were ordained clergymen, educated at Oxford or Cambridge, and restricted in their writings by their (Anglican) church allegiance. Stubbe was a physician, and in his last years he seemed to have been a successful one. And so, as he treated the maladies of patients, he turned to treat the malady of ignorance which he diagnosed, as he would have a disease, through careful examination of symptoms. On many occasions in the treatise, Stubbe used his medical knowledge to support his arguments: for him, the historian, like the physician, was to rely on research: and as the treatment of diseases did not discriminate on the basis of culture or geography, so the treatment of history. As he pored over the tomes of English Edward Pococke, Swiss Johann Hottinger and Isaac Casaubon, French Claudius Salmasius, English John Selden, Dutch Thomas Erpenius, and the chronicles of al-Makīn, ibn al-Batrīq, and Abū al-Faraj, Stubbe became the physician trying to find the cure for the disease of ignorance. It is no coincidence that the _Panarion_ of Epiphanius, that fifth-century compendium of pre-Islamic Jewish and Christian heresies, which Stubbe constantly cited, was subtitled _Contra octoginta haereses opus, Panarium, sive Arcula, aut Capsula medica appelatum/the_ Medicine Chest [panarion] against Heresies.
Henry Stubbe belongs to that century in English and continental history when Arab-Islamic manuscripts made an impact on European thought. They were collected, edited, translated, and integrated into early modern intellectual activity, sometimes accepted, sometimes rejected, and sometimes adapted into discussions of biblical history, philosophy, philology, law, geography, and mathematics. But the Arabic/Latin histories of al-Makīn, Ibn al-Baṭrīq, and Abū al-Faraj were the first published sources about Islam to become available to non-Arabic readers in Western Europe. All previous information about Islam had derived from Greek and Latin sources, which is why these indigenous Arabic texts presented a view of Islam that was different from everything before them. Still, no English theologian or playwright, poet or Sunday preacher, translator or chronicler of world religions turned to them for an alternative view of Muḥammad and Islam in the manner Stubbe did. Perhaps it was difficult to do so: Britons of Stubbe's generation had experienced civil wars of fierce religious polarization, massacre, and desecration, and during the Restoration period the "Great Persecution" was inflicted on the nonconformists, with draconian laws curtailing mobility, education, and livelihood. A few years after Stubbe's death, England was gripped by the hysteria of the Popish Plot and the intimidation, false accusation, torture, and execution of Catholics. In 1682 three witches were burned.
Early modern English society was still caught in religious and denominational exclusivity and violence. It would have been very unlikely that "Mahometanism" or "Mahometans" could have been treated differently, which helps to explain why _The Originall & Progress of Mahometanism_ was not published—although it was read and copied and, among a few, Charles Hornby for one, it was admired. But it had been a very controversial project: as the text shows, Stubbe repeatedly defended himself by introducing his first-person voice against readers among whom he circulated the manuscript. He was not dogmatic in his views, and recognized the limitations of his sources and of his knowledge: he made a number of historical and factual mistakes, only because his sources made those mistakes. In order to deal with the vast diversity of information in his sources, he often presented multiple views in a series of conjectures and then, to make a determination, he appealed to his medical knowledge (as in the case of the Prophet's alleged epilepsy); invoked logic and common sense, especially against "Christian fables"; and showed how many of the features of Islam denounced by his contemporaries – such as polygamy, militarism, and the literalism of the joys of the afterlife – were present in the history of Judaism and Christianity (fols. 111–114). Although Stubbe viewed Muslims as erring in "the manner than in the object of their devotion" (fol. 127), he recognized Islam as a religion and a polity that had been inspired by a prophetic legislator, who, with his cousin, began an empire the sun of which was "no sooner... elevated above ye horizon but it was in its meridian" (fol. 107).
But in Restoration England there was not yet a place for a positive and historically accurate presentation of the beginnings of Islam. Henry Stubbe was alone among his contemporaries who found in the Arabic/Latin sources a compelling historical alternative to misrepresentation—thus his "Copernican Revolution" about the "great prophet" of Islam (fol. 137).
THE PRINTED AND MANUSCRIPT SOURCES
_Editorial Policy_
HE _ORIGINALL_ SURVIVES in fragments as well as in toto. Although the text was not printed until 1911, three excerpts had appeared in print in 1693 and 1695, in letters by Charles Blount to the earl of Rochester (published twice), and to Thomas Hobbes.
MATERIAL COPIED AND PRINTED BY CHARLES BLOUNT
December 1678 and 1693: letter from Blount to Thomas Hobbes ("Arrians") appeared in Charles Blount's _The Oracles of Reason_ (London, 1693), 97–105. The letter corresponds to fols. 38–41 in University of London MS 537. The letter is reproduced in Thomas Hobbes, _The Correspondence_ , ed. Noel Malcolm (Oxford, 1994), 2:759–763.
December 1678 and 1693: letter from Blount to the earl of Rochester ("republic") appeared in _The Oracles of Reason_ , 157–166, and in Charles Blount's _Miscellaneous Works_ (1695), 158–168. The letter corresponds to fols. 3–8 in University of London MS 537. The letter is reproduced in Jeremy Treglown, _The Letters of John Wilmot, Earl of Rochester_ (Oxford, 1980), 206–213.
Blount sometimes made alterations in punctuation, word order, and omission of biblical references. Where differences suggest change of meaning, they are listed in the notes that follow.
MANUSCRIPTS
_Late Seventeenth Century. British Library Sloane 1709 and Sloane 1786_
The first fragment in BL Sloane 1709, fols. 94r–115r, corresponds to fols. 48–107 in MS 537 of the Senate Library, University of London. This fragment contains the unit on "The hist. of the Saracens & of Mahomet." The writer/scribe used folded papers and wrote on four-page clusters. He then numbered them 1–12. This pagination suggests an autonomous unit. The second fragment in BL Sloane 1786, fols. 186r–190r, corresponds to fols. 107–118 in MS 537 and contains the unit "Concerning the Justice of ye Mahometan Warrs." Like the previous fragment, this unit appears as autonomous. The third fragment in Sloane 1786, fols. 181r–185v, corresponds to fols. 130–142 in MS 537. This fragment begins in the middle of a sentence because the preceding folios have been lost; the first surviving folio has "fol. 3" at the top left-hand side of the page. It includes the chief part of "concerning God, purgatory, Judgement & paradise." This fragment appears at the end of MS 537 and was written at a later stage—since Stubbe refers in it to his previous discussion of "Aly."
All three Sloane fragments are by the same hand, written densely on the same kind of folios, and with frequent marginalia. In comparing this hand with the hand in the letters to Hobbes (BL Letters from Stubbe to Hobbes, MS 32553, fol. 5 and fol. 25v), most probably by Stubbe himself, it is clear that the two are not the same.
While it is difficult to speculate about the time or place in which Stubbe wrote the material in the fragments, he must have worked on it when he had access to books that he did not own in his library—or at least were not listed in BL Sloane 35, the posthumous inventory of his books. The units relied heavily on Erpenius's translation of al-Makīn (1625) and Pococke's _Specimen_ (1650): neither of these appears in the inventory. But others do appear: Hottinger's _Historia Orientalis_ (1651), _Geographia Nubiensis_ (1619), and works by Selden and Salmasius.
1701, _University of London, Senate House Library, MS 537_
Complete manuscript, University of London, 537. Since this manuscript is edited in this volume, I requested from the Senate House Library information about its provenance. Ms. Tansy Barton kindly sent me the following paragraph:
This MS. bears the bookplate of the Rev. John Disney, D.D., of The Hyde, Ingatestone Essex. Items 1559, 1562, and 1564 disposed of with other property of Disney's at Sotheby's in April 1817 were all versions of this same work, but none corresponds with the present MS. Two manuscripts at least in the British Museum, Harl. 1876 and 6189, contain longer versions of the main text of MS. 537, but neither includes the letter. "An account of the life of Mahomet... from an MS. copied by Charles Hornby of the Pipe Office in 1705", was edited and published by Hafiz Mahmud Khan Shairani in 1911. It would seem therefore that MS. 537 is the earliest dated MS. copy of this text to survive. The gift of New College, Hampstead, 1960.
There was no further information on its provenance.
The inner leaf of the manuscript includes reference to "Charles Hornby" and "January 3rd 1701." See Champion's 2010 study for information about Hornby.
Two hands appear in this manuscript of the treatise. There is another treatise that follows upon Stubbe's with new pagination and in a different and very elegant hand.
Hand A: this hand is of the scribe who copied the whole manuscript. The scribe abbreviated words ("Xtians" for "Christians"), was inconsistent in spelling (spelling "Islamism," "Islanisme," and "Islamisme"), and was not familiar with the terminology he was copying—thus wrote "Reblah" instead of what must have been "Keblah"/ _qibla_. He copied mechanically and did not pay attention to meaning: thus "fund a Mentall" (fol. 74). The scribe paid little attention to new sentences, capitalizations, or other stylistic or syntactical markers. There are duplications of words, some blank spaces, and many mistakes in subject-verb agreement. There are numerous mistakes in the Latin passages, and the Greek words are sometimes illegible. Still, the scribe made some corrections, either inserting them above the lines, or adding them in the margins. Some of the corrections were significant: for instance, in fol. 28, the scribe corrected "Jews" to "Gentiles"—by deleting the first word and writing the second above it; fol. 30, "rights" is corrected to "rites."
Hand B: this hand added chapter divisions and titles, as if preparing the manuscript for publication. This hand also indicated the sections that Charles Blount had copied and made page references to another (lost) manuscript. The publication of the Blount letters in 1693 and 1695 shows that although Blount had copied from another version of the manuscript with its own stylistic variants, the text was taken from Stubbe's _Originall_.
Toward the end of the manuscript there were references to folio numbers, written in Hand A, next to specific paragraphs or sections. It is not clear what the purpose was—unless they indicated passages that were to be copied in, or were copied from, another manuscript.
This manuscript does not include the heavy marginal notes of the Sloane fragments. But the scribe either used the Sloane manuscripts or the Sloane scribe and this scribe were using the same (lost) manuscript/s. The difference between this manuscript and the Sloane manuscripts is that this manuscript contracts sentences and sometimes omits whole sentences or phrases. Occasionally, the scribe of MS 537 corrects egregious mistakes: Sloane 1709 fol. 115r writes that it was the will of Muḥammad "yt Moslemin should be deceived"; this manuscript corrects the sentence that it was the will of Muḥammad that "the Moslemin should be undeceived" (fol. 106). What suggests a link between this manuscript and the two Sloane fragments is the following:
1. Many of the stylistic features of the Sloane manuscripts appear unchanged in this manuscript, such as the opening and closing of parentheses for exactly the same (but not all) sentences.
2. On some occasions the Sloane scribe writes a word and then deletes it to replace it with another. MS 537 does not include the deleted word, but copies the replacement word that appears in Sloane.
3. On numerous occasions, the scribe of the Sloane manuscripts and this manuscript start paragraphs at the same point.
4. The Sloane fragments include extensive comments in the margin—some of which appear as part of the text in this manuscript.
5. Transliterations of Arabic words/names are the same in Sloane and in this manuscript.
6. The scribe of this manuscript did not know Greek and either copied what he saw or left blank spaces where Greek appears in later manuscripts. His Latin was as erratic in spelling and punctuation as the English.
This manuscript ends on fol. 142. It is followed by a new paragraph that is not completed and is crossed out.
The organization of the material in this manuscript is as follows (there is no title):
Fols. 1–48: History of Christianity
Fols. 48–56: "Chap 3. A Brief Account of Arabia & the Saracens/The History of the Saracens & of Mahomet."
Fols. 56–65: "Chap 4. The Transactions from the birth of Mahomet to his Flight from Mecca."
Fols. 65–91: "Chap 5. Mahomet's Conduct at Medina the Embassy of Aly to the Agarens & Saracens."
Fols. 91–101: "Chap VI. The Return of Aly & the Wars of Mahomet."
Fols. 101–107: "Chap VII Mahomet's last Pilgrimage his Death & Burial."
Fols. 107–113: "Concerning the Justice of the Mahometan Warrs & that Mahomet did not propagate his Doctrine by the Sword/with a vindication of Mahometts Carriage towards the Christians."
Fols. 114–128: "Concerning the Christian Additions."
Fols. 129–142: "As to their opinions concerning God, purgatory, Judgmt & paradise they are these."
_Early Eighteenth Century (?), British Library Harleian 1876_
Complete manuscript, BL Harleian 1876 ("The Life of Mahomet" on spine). The Sloane scribe of BL 1709 and 1786 and the scribe of this manuscript were copying from the same manuscript, or the scribe of this manuscript was using the Sloane fragments along with another manuscript to produce a complete copy of _Originall_. The marginalia of references and notes are the same in this manuscript as in the BL Sloane fragments. But this manuscript "improves" on both the BL fragments and the University of London manuscript. One example will suffice.
University of London MS 537, fol. 47:
It appears by Pauls carriage Acts 23.6. that he did Act by Somwhat like to Judging In his proceedings, how else could he cry Christianity assuring each of them singly that he was in the truth, and that afterwards when Paul was dead, each of 'em ptended his Religion to bee the true Religion derived from Paul whence arose great Feuds amongst them.
BL Harleian 1876, fols. 64–65:
It appears by Paul's carry age, Acts 23.6. that he did act by somewhat like unto Jugling in his proceedings. How else could he cry there, He was a Pharisee, & called in question of the hope of the Resurrection of the Dead? And in his Preaching unto the Jews, he became as a Jew, that he might gain the Jews; to them that were under the Law, as under the Law, that he might gain them that are under the Law; to them that are without Law, as without Law, (being not without Law to God, but under the Law to Christ, that he might gain them that are without Law; To the weak, he became as weak, & became all things to all men, that he might by all means save some. 1 Cor. 9.20, 21. This behavior of Paul, though it multiplied the number of Christians, yet it did lay the foundation of perpetual Schisms & Heresys; for they would not relinquish, as erroneous or evil, those Tenets or Usages, which he without reprehension indulged them in, & complyed actually with himself. When he had layd in them his Foundation, That Jesus was the Messiah, he permitted any Superstition, in Wood, Hay or Stubble, any variety of Doctrines, not ending in direct Idolatry; assuring his Confidents, that notwithstanding this they might be saved, 1 Cor 3. 11, 12, 13, 14, 15. And who knows how sincere, or how complacential he was in his Writing, whose Deportment otherwise **< 65>** was thus related? I remember a Mahometan Story of Achmed Ben Edris, that Paul instructed Three Princes in Religion, & taught each of them a different Christianity, assuring each of them singly, that he was in the Truth;* & that afterwards when Paul was dead, each of them pretended his Religion derived from Paul, whence arose great Feuds amongst them.
This manuscript is similar to University of London MS 537 in that words that were changed in the last dated manuscript (BL Harleian 6189, 7 July 1718) are not changed in this manuscript or in MS 537. For instance, MS 537 has "criminal" (fol. 19), which appears in BL Harleian 1876, but in BL Harleian 6189 the word is changed to "Ceremoniall" and a different hand adds in the margin: "Criminal... d over it Ceremonial" (fol. 33) suggesting that the scribe of BL Harleian 6189 was using a manuscript that had changed "criminal" to "ceremonial." At the same time, MS 537 includes words and sentences that are in BL Harleian 1876 but not in BL Harleian 6189. Sometimes the Sloane scribe starts a marginal comment, but then deletes or discontinues it (fol. 186r "This practice of his," which appears in full in BL Harleian 1876).
The scribe of this manuscript started by adding the references to the sources at the time of copying the manuscript. But then, on fol. 22, he stopped and later returned to add the references, along with longer comments, in a smaller script. He then went over the manuscript again and made a few additions: fol. 42: "Dr Pococke, histo. Arab. P. 212, 313" after "Pococke" the word "specim." is added above the line (see also fols. 46, 48, 49). Whether the comments were Stubbe's cannot be determined, but there is no reason not to think they were his. It would not have been easy for someone else—a scribe, for instance—to locate the page and chapter references. At the same time, and in all his publications, Stubbe added precise references to his sources.
Both the BL Sloane fragments and BL Harleian 1876 include marginalia referring to the sources, although a small number is missing from Sloane. But when both scribes record a reference, it is exactly the same title, chapter, and page. Neither manuscript introduces a different page or chapter, but on a few occasions BL Harleian 1876 enters a reference the BL Sloane fragments do not—perhaps suggesting a hasty or careless Sloane scribe; after all, the folios in BL Sloane 1709 and 1786 are tightly packed, unlike BL Harleian 1876, which is written in a clear and beautiful hand, with wide line spaces, in a manuscript dedicated solely to _Originall_ (unlike the BL Sloane fragments, which are included among other manuscripts; Sloane 1709 consists of "Miscellaneous Pieces").
In some cases what appears as marginalia in BL Harleian 1876 appears in brackets within the text in BL Sloane (cf. BL Sloane fol. 187r and BL Harleian fol. 225). The titles in BL Sloane and BL Harleian are the same: "Concerning the Christian Additions" (Sloane 1786 fol. 187v and Harleian fol. 230). But it is interesting to note the titles of the unit from Grotius:
BL Sloane 1709, fol. 188r: "Grotius de veritate relig. Christ. 1.6. cum notis"
University of London MS 537, fol. 115: "Grot. De Veritate Rel. Christianae lib.6"
BL Harleian1876, fol. 207 "Grotius de veritat. Relig.Christian. lib.6. cum notis
Was "cum notis" forgotten or did the scribe use another manuscript?
The title and the organization of the material in this manuscript are as follows:
"An Account of the Life of Mahomet" [title]
Fols. 1–55 (the unit on Christian history)
Fol. 55 "The History of ye Saracens and of Mahomet"
Blank folio
Fols. 57–100: "A generall Preface to the account of the originall & progress of Mahometanisme"
Blank folios
Fols. 103–193: "The History of the Saracens and of Mahomet"
Blank folio
Fols. 195–203: "Concerning the justice of the Mahometan wars & that Mahomet did not propagate his doctrine by the sword"
Blank folio
Fols. 205–210: "Concerning the Christians Additions"
1705. _Bodleian MS Eng. Misc. c. 309_
Complete manuscript, Bodleian MS Eng. Misc. c. 309. This manuscript was used by Shairani, although he re-arranged its chapters in his printed edition. It is the longest of all existing manuscripts, as it was rewritten and heavily "improved" by Hornby, who relied on a lost manuscript.
It is likely that Hornby wanted to publish this manuscript by Stubbe, whose name he mentions on the title page as the "supposed" author. Evidently, the manuscript was circulating, but there was no definite knowledge about its author. Hornby wrote elegantly, gave clear and separate titles of chapters, but he did not include the annotations that appeared in the BL Sloane and the Harleian 1876 manuscripts. Another hand entered few of the references in bolder ink, more in the first than in the second part of the treatise; toward the end there is barely any reference.
Hornby "improved" the text by consulting further sources and adding material—but he did not contradict Stubbe's views, nor the general tenor about Muḥammad or Islam. One comparison of entries from University of London MS 537, BL Harleian 6189, and Oxford MS Eng.. Misc. c. 309 will suffice to show the range of intrusion on the part of Hornby. He introduced a reference to John Gregory M.A. 1631, also of Christ Church, Oxford, which does not appear in the other manuscripts:
MS 537 UNIVERSITY OF LONDON, SENATE HOUSE LIBRARY
I do not find any understanding Author who doth controvert the Elegancy of the Alcoran, & it hath this advantage over the Xtian Bible, that being a poem, there is a greater Liberty allowed to fiction figurative expressions & Allegories then is allow'd of in prose also defects in Chronology & Errors in History are here tolerable, tho' for my part I beleive that many of the incoherencies & Chronologicall & Historical defaults are voluntary, partly because the vulgar being prepossessed wth them (in many cases this is evident to have dissented thereform would have been prejudicial to his Aims the universall Credit of the Errors being likely to overbear ye reall truth of things, partly because it was a received tradition among the Jews & Judaizing Xtians (& 'tis now made use of as an Apology for our Scripture that the Spirit of God in the prophets is not confined to the Gramatical Rules ordinary Methods.
MSS BL HARLEIAN 6189 AND BL HARLEIAN 1876
I do not find any understanding author who doth controvert [fol. 301 in BL Harleian 6189] the Elegancy of the Alcoran; And it hath this advantage over the Christian Bible, that being a Poem, there is greater liberty allowed for fiction, figurative expression and Allegories, than is allowed of in prose. Also defects in Chronology, and Errors in History are here tolerable: though for my part, I believe that many of the incoherences and Chronological or Historical defaults _were voluntary_ : partly because the vulgar being prepossesed with them, (in many cases this is evident) to have [fol. BL 96 Harleian 1876] dissented therefrom, would have been prejudicial to his Aim, the universal credit of the Errors being likely to overbear the real truth of things; partly because it was a received Tradition amongst the Jews, and Judaizing Christians, and 'tis now made use of as an Apology for our Scripture, that the Spirit of God in the Prophets, is not confined to Grammatical Rules or ordinary methods:
MS OXFORD ENG. MISC. C. 309 AND BL HARLEIAN 6189; ADDITIONS IN THE OXFORD MANUSCRIPT ARE IN BOLD; THE ITALICIZIED PASSAGES ARE IN BL HARLEIAN 6189
**¶ The truth is** I do not find any understanding author who doth controvert [fol. 301 in Harleian 6189] the Elegancy of the Alcoran, **it being generally esteemed as the standard of the Arabian Language and Eloquence, but they raise great exceptions against it for incoherency & confusion Errors in History &** [fol. 138 in Oxford Eng. Misc. c. 309] **chronology and charge it with numberless trifles fables and absurdities.**
**The late learned Mr. John Gregory in the preface to his works has this passage. "I was (says he) asked once by an able and understanding man whether the Alcoran as it is of it's self had so much in it as to work any thing upon a rational belief: I said yes. Thus much only I required that the believer should be brought up first under the engagement of that book. That which is everywhere called Religion hath more of Interest and the strong impressions of Education, then perhaps we consider otherwise for the book it's self it is taken for the greater part out of our Scripture, and would not appear altogether so ill if it were look'd upon in it's own Text, or through a good Translation."**
**We see this learned Man had not so ill an opinion of the Alcoran, and we shall likewise find upon examination that those who have most diligently perused that and the other books of the Mahometans, have abated much of the general prejudices of the Christians against that Religion and it's Author and entertained more favourable thoughts of both, then others whose aversion is kept up by their ignorance. If we look upon the Alcoran with the same indifferency as upon any other book we shall find that** _And_ it hath this advantage over the Christian Bible, that being a Poem, there is greater liberty allowed for fiction[, **Parables** ], figurative expression and Allegories, than is allowed of in prose. Also defects in Chronology, and Errors in History are here tolerable: though for my part, I believe that many of the incoherences and Chronological or Historical defaults [ **are here tolerable, tho' I believe most of those we call so Mahomet grounded upon the ancient Accounts in the Books of the Arabians or the general Traditions among them, and upon the Apocryphal books of the Jews and heretical Christians** ] _were voluntary: partly because the vulgar being prepossesed with them_ , (in many cases this is evident) [ **which being respectively received as Authentic by those of the several Religions and the vulgar prepossessed with them,** ] to have dissented therefrom, would have been prejudicial to his Aim, the universal credit of the Errors being likely to overbear the real truth of things; [ **Many of the mistakes and incoherencies therein might be voluntary** ] _partly_ because it was a received Tradition amongst the Jews, and [fol. 139 in Oxford Eng. Misc. C. 309t] Judaizing Christians, and 'tis now made use of as an Apology for our Scripture, that the Spirit of God in the Prophets, is not confined to Grammatical Rules or ordinary methods.
Gregory lived a short life, and his work appeared posthumously—and then went through numerous editions, attesting to the popularity of his polylinguistic research. That Hornby brought him into his improvement shows to what degree Stubbe was following in the scholarship about Muḥammad and Islam that emanated from Christ Church, the college founded by Cardinal Wolsey where the first chair of Hebrew had been established.
This manuscript formed the basis of the Shairani edition. But Shairani edited out passages of which he did not approve..
The title and the organization of the material in the manuscript are as follows:
"An Account of the Rise and Progress of Mahometanism with the life of Mahomet and a vindication of him and his Religion from the Calumnies of the Christians Supposed to be Written by Dr. Stubb Copied by C.H. Anno Dni 1705 With some variations and additions" [title, on a separate sheet]
The Contents [on two separate pages]
Chapter 1. An Introduction to the History of Mahomet, conteyning an Account of the State of Judaism, and Christianity, from the time of Jesus Christ to the birth of Mahomet pa. 1
Chapter 2. The Authors Apology for the foregoing Account of the primitive Christians. 47
Chapter 3. A brief Account of Arabia and the Saracens. 54
Chapter 4. Conteyning the Transactions from the birth of Mahomet to his flight from Meccha. 65
Chapter 5. Mahomet's Conduct at Medina, the Embassy of Aly to the Agarens & Saracens. 78
Chapter 6. The Return of Aly, & the Wars of Mahomet. 104
Chapter 7. Mahomet's last Pilgrimage, his Death and Burial 115
Chapter 8. The Character of Mahomet, and fabulous Inventions of the Christians concerning him & his Religion. 123
Chapter 9. Of the Alcoran, and Miracles of Mahomet, the prophesies concerning him, and a brief Account of his Religion and Policy. 136
Chapter 10. Concerning the Justice of the Mahometan Wars with a Vindication of Mahomet's Carriage towards the Christians, and that he did not propagate his Doctrine by the Sword. 258
_1718. British Library Harleian 6189_
Complete manuscript: British Library Harleian 6189 ("History of Mahometanism" on spine). The date is given at the end of the manuscript in the same hand that transcribed the whole text: 7 July 1718. As noted previously, the scribe was using a manuscript that included corrections. On several occasions, the scribe referred to those corrections: "These words were underlined by the Corrector" (fol. 34); "Interlined by the corrector of the Or.[iginal]" (fol. 35). These notes by the scribe point to a different set of manuscripts that were not available to the copyists in cluster introduced heretofore.
Hornby and the copyist of this manuscript must have used similar versions of the _Originall_. The word "Metaphysicks" appears in BL Harleian 1876 (fol. 42), but in University of London MS 537 and in this manuscript the word is "Mathematicks" (fols. 35 and 60 respectively), which is the same as in Hornby (fol. 60). On other occasions the scribe used the same word/s that appear in MS 537 and BL Harleian 1876: thus "revolutions" appears in BL Harleian 1876 (fol. 47), but in this manuscript it appears as "Resolutions" (fol. 66). There are passages in University of London MS 537 and in BL Harleian 1876 that do not appear in this manuscript. Was the scribe of this manuscript "editing" out passages, as for instance in the description of Muhammad, which is omitted from the beginning of this manuscript?
He had a ready Wit, & such an Elocution as no Arabian before or since hath ever equalled: whensoever He pleased He could be facetious without prejudice to his Grandeur; He perfectly understood the Art of placing his favours aright. He could distinguish betwixt the deserts, the inclinations, & the interests of Men, He could penetrate into their Genius's & Intentions, without employing vulgar [3] espialls, or seeming Himself to mind any such thing: In fine, such was his whole deportment, so was his naturall freedome tempered with a befitting reservedness, that He instructed others not to importune him with unbecoming proposalls, but never suffered any to understand what it was, to be denyed.
(BL Harleian 1876, Fols. 2–3)
The title and the organization of the material in this manuscript are as follows:
"The Rise and Progress of Mahometanism" [title]
Fols. 1–78: "The Rise and Progress of Mahometanism"
Fols. 78–216: "Of Mahomet and the Saracens"
Fols. 216–230: "Concerning the Justice of Mahometan Wars and that Mahomet did not propagate his Doctrine by the Sword"
Fols. 230–240: "Additions Concerning the Christians"
Fols. 240–308: "A generall preface to the account of the Original of Mahomentanism Progress"
_Lost Manuscripts_
In University of London MS 537 and in BL Harleian 6189 the scribes make reference to manuscripts they were using. These manuscripts are presumed lost. Hand B, in the former manuscript, mentions a "blew book" (fol. 113). There may well have been other manuscripts of Stubbe's treatise.
_Final Remarks_
Based on these manuscripts, it is possible to venture the following clusters:
BL Sloane, University of London MS 537 and BL Harleian 1876 Oxford MS Eng. Misc. c 309 and BL Harleian 6189
There are no two manuscripts that are alike. Each scribe deleted, added, and rewrote passages in his own style, at the same time that he left other passages unchanged.
The two manuscripts that are closest to each other are BL Sloane and University of London MS 537.
The other manuscripts exhibit significant differences: as shown, BL Harleian 6189 has a different sequence of chapters from Bodleian MS Eng. Misc. C 309 at the same time that the latter deletes/excludes passages that appear in University of London MS 537 and in BL Harleian 6189.
THE PRESENT EDITION
The present edition is a modernized version of the earliest complete version of _Originall_ : University of London MS 537. My goal is to make Stubbe's text as accessible as possible to today's reader while preserving its seventeenth-century syntax and style.
I have regulated all spelling in accordance with current usage (American). Also, I changed "then" to "than" (when appropriate); "their" to "there," expanded ampersands and contractions ("enthron'd" to "enthroned," "'tis" to "it is"), added or removed punctuation, introduced or removed capitalization, closed parentheses, removed double letters ("originall" to "original" and "ff" to "f"), and introduced new paragraph divisions. I have indented speeches and long quotations and added inverted commas to statements. I also regularized the spelling of place names that the scribe confused: "Pallestine," "palestine," "Palestine" and others. I also modernized them to correspond to today's geographical usage: thus "Affrick" becomes "Africa," and others.
I have not deleted or added words, even when the sense is unclear.
In the case of Latin and Greek phrases and passages: it is unlikely that in the original version of the manuscript by Stubbe, who was a master of those languages, there would have been mistakes in spelling. The scribe, however, made a vast amount of mistakes. I have corrected those mistakes by quoting the original Greek and Latin sources Stubbe consulted.
In the case of Arabic names and phrases: Stubbe knew no Arabic and therefore could not determine the accurate spelling of words. Relying on the works of numerous Arabists and scholars, he was confused by their different spellings of the same words. In frustration, he sometimes listed the variants: "the Caab Alccab Caaba Kabe Cabea" (MS 537, fol. 53). The scribe must have been equally confused, and so in the manuscript the same word is sometimes spelled differently, even on the same page. There are, however, words that are always spelled the same: "Medina," "Zamzam," "Omar," and "Edris." And there are names that are spelled consistently the same, but then are changed for a few pages, after which the old spelling is resumed ("Ismael" and "Ismaell").
On the first occurrence of words or names with variants, I have listed the variants in the note. In order to make the text as accessible as possible, I have chosen the spelling that is closest to today's pronunciation and used it throughout the edition. In the case of names that appear infrequently, I have retained the originals and explained them in the notes.
I have left unchanged but italicized all the Arabic words, names, and phrases that Stubbe transliterated. These transliterations are significant for the following reasons:
1. In some cases the transliteration shows the exact source from which Stubbe borrowed it. Most of his transliterations of names and phrases came from Pococke, Hottinger, and Selden.
2. Words such as _Musulman_ and _Moslemin; Coran_ and _Alkoran; Islamism, Islanism_ , and _Islamisme_ and other variants: by bringing names and titles from the Latin texts into his treatise, Stubbe may have introduced them into English for the first time.
3. In specific cases, Stubbe deliberately kept the Arabic nomenclature to emphasize the Islamic context. While he used "Abraham" and "David" and other biblical/Hebrew names in their English form, he used "Ismael," rather than Ishmael, and "Isa" rather than Jesus (even though his sources used "Jesu"), and "Edris" rather than Enoch (or alongside). Further, he sometimes used the Arabic word where an English word would have been quite satisfactory: _sallah_ and a few others.
I have included reference to the marginalia made by the two hands on the manuscript, as well as all significant variants and notes from the Sloane fragments and BL Harleian 1876 and 6189. In my notes I have corrected and elaborated on the sources mentioned in the marginalia
The transcript of MS 537 is available through Columbia University Press.
_The Originall & Progress of Mahometanism_
HENRY STUBBE
TABLE OF CONTENTS
Folios refer to the pagination in the University of London Senate House manuscript. In the text itself they appear in bold between pointed brackets.
Fols. 1–48: History of early Christianity
Fols. 48–56: A brief account of Arabia and the Saracens/The history of the Saracens and of Mahomet
Fols. 56–65: The transactions from the birth of Mahomet to his flight from Mecca
Fols. 65–91: Mahomet's conduct at Medina, the embassy of Aly to the Hagarenes and the Saracens
Fols. 91–101: The return of Aly and the wars of Mahomet
Fols. 101–107: Mahomet's last pilgrimage, his death and burial
Fols. 107–113: Concerning the justice of the Mahometan wars, and that Mahomet did not propagate his doctrine by the sword/with a vindication of Mahomet's carriage towards the Christians
Fols. 114–128: Concerning the Christian additions
Fols. 129–142: As to their opinions concerning God, purgatory, judgment, and paradise
INTEND TO WRITE of one of the greatest transactions the world hath ever yet been acquainted with: "The Original and Progress of Mahometanism," wherein a new religion was introduced into the world to the desolation, in a manner, of paganism, Judaism and Christianity, which hath now maintained itself above a thousand years and has increased its extent and proselytes over more than a fifth part of the known earth. Whereas Judaism, including all its colonies, was never equal thereunto, nor perhaps Christianity itself, if we consider the condition of it either before Constantine, or even to the days of Theodosius (during all which time as the senate of Rome so the greatest part of the empire were pagans), or afterwards, when uniformity was settled. But the inundation of the Arian Goths and the general irreligion, impiety, and division into sects, some whereof were idolaters, do not permit me to think that true and fervent Christianity was so far diffused as Mahometanism is at present.
The same narration includes in it the rise of an empire greater than any of the four so famed monarchies, erected in a barren poor country in the midst of two potent princes, one reigning over the Eastern Christians, the other over the Persians: and all this to be brought about in the compass of a few years by a man of a mean estate, fiercely opposed, and slenderly befriended.
By this time your curiosity prompts you to search after the physiognomy of this extraordinary person. This great soul was lodged in a body of a middle size: he was no giant nor did his stature equal that of an Almain Cimber whose bulk amazed the old Romans. He had a large head, a brown **< 2>** complexion but fresh color, his beard long and black but not gray, a grave aspect wherein the awfulness of majesty seemed to be tempered with admirable sweetness which at once imprinted in the beholders respect, reverence, and love. His eyes were quick and sparkling. He had very handsome legs, an incomparable mien, easy motion and every action of his had a grace so peculiar that it was impossible to see him with indifference. The Arabians compare him to the purest streams of some river gently gliding along, which arrest and delight the eyes of every approaching passenger.
Nothing was more mild than his speech, nothing more courteous and obliging than his carriage. He could dexterously accommodate himself to all ages, humors and degrees. He knew how to pay his submissions to the great without servility and to be complacent to the meaner sort without abasing himself. He had a ready wit, a penetrating and discerning judgment and such an elocution as no Arabian before or since hath ever equaled. When he pleased he could be facetious without prejudice to his grandeur: he perfectly understood the art of placing his favors aright. He could distinguish betwixt the deserts, the inclinations, and the interests of men; he could penetrate into their geniuses and intentions without employing vulgar espials or seeming himself to mind any such thing.
In fine, such was his whole deportment.
So was his natural freedom tempered with a befitting reservedness as instructed others not to importune him with unbecoming proposals, but never suffered any to understand what it was to be denied. Besides all those embellishments and qualifications, he had a great strength and agility of body, an indefatigable industry, an undaunted courage such as never forsook him in the greatest dangers. He was much addicted to ride the best and most warlike horses, **< 3>** and since every action of great men is remarkable and often carries a presage of future accidents, I shall relate one. He being once mounted on a brave but unruly courser, his friends desired him to forsake his back; but whether it were that he duly apprehended his own skill and abilities, or his great spirit thought it more fitting to contemn than acknowledge a danger into which he had rashly engaged himself, he denied the request, adding that it became the timorous and effeminate to have their horses exactly managed for them: that a generous and true Arab could not be surprised with an untamed steed, that the intractableness of his horse added to his pleasure, as a storm delights an intelligent pilot since it gives him an occasion to discover that skill which could not be manifested otherwise, and rewards the danger and trouble by an accession of glory.
Behold the character of that man who hath gained so much upon the esteem of one part of the world and filled the rest with astonishment.
But to discover the means by which he achieved those great things is a matter of more difficulty, and in order thereto you must consider what it was that disposed the people to such a change and what gave beginning thereunto. Prudent persons distinguish cautiously betwixt those two circumstances and know that the bravest actions do frequently miscarry under very happy pretenses or beginnings, in case the antecedent causes be not proportionate to the design. Never any republic did dwindle into a monarchy or any kingdom alter into an aristocracy or commonweal without a series of preceding causes which principally contributed thereto. Never had Caesar established himself, nor Brutus erected a senate: and if you inquire why the first Brutus could expel Tarquin and the second could not overthrow Augustus and Antony; why Lycurgus, **< 4>** Solon, and others could establish those governments which others have in vain attempted to settle in Genoa, Florence and other places, you will find it to arise from hence: that some considering those antecedent causes which secretly and securely incline to a change took the advantage thereof, whilst the others did only regard the speciousness or justice of their pretensions without a mature examination of what was principally to be observed. This is certain: that when the previous dispositions intervene, a slight occasion, oftentimes a mere casualty, opportunity taken hold of and wisely prosecuted, will produce those revolutions which otherwise no human sagacity or courage could accomplish.
I cannot find any authentic ground to believe that the sects among the Jews were more ancient than the days of the Maccabees, but arose after that Antiochus had subdued Jerusalem and reduced the generality of the Jews to paganism; and (the better to confirm his conquests) erected there an academy, placing the Pythagorean, Platonic, and Epicurean philosophers there. This I conceive to have been the original of the Pharisees, Sadducees, and Essenes. Though afterwards, when the Maccabees had made an edict against and anathematized all that taught their children the Greek philosophy, one party did honest their tenets by entitling them to Sadoc and Baithos, and the others from a Cabbala derived from Eleazar and Moses successively.
The introduction of those sects and of that Cabbala occasioned that exposition of the prophecy of Jacob (Genesis 49:20): "The scepter shall not depart from Judah nor a lawgiver from between his feet until Shiloh come, and unto him shall the gathering of the people be." From whence they did according to that fantastical Cabbala imagine that when so ever the scepter should depart from Judah and the dominion thereof cease, that then there should arise a Messiah of the line of David (this was no general opinion: for how then could any have imagined Herod the Great to have been the Messiah; and how could Josephus fix that character upon Vespasian) who should **< 5>** restore the empire and glory of Israel, and all nations should bow and submit to his scepter?
I do not read that the Jews harbored any such exposition during the captivity under Nebuchadnezzar; albeit that the scepter so departed at that time from the tribe of Judah and house of David that it never was resettled therein. After their return to Jerusalem, no such thing is spoken of when Antiochus Epiphanes subdued them, profaned their temple, destroyed their laws and rites, and left them nothing of a scepter or lawgiver—during all which time, although they had the same prophecies and scripture, there is no news of any expected Messiah. But after that, the curiosity of the rabbis involved them in the pursuance of mystical numbers, and Pythagorically or Cabbalistically to explain them, according to the Gematria. Then was discovered that Shiloh and Messiah consisted of letters which make up the same numerals, and therefore a mysterious promise of a redeemer was insinuated thereby. And the prophecy of Balaam (Numbers 24:17) concerning a star out of Jacob and a scepter rising out of Israel with a multitude of other predictions (which the condition of their nation made them otherwise to despair of) must be fulfilled under this Messiah.
I name no other prophesies because either they are general and indefinitely expressed as to the time of their accomplishment, or else inexplicable for obscurity and uncertain as to authority, as the weeks of Daniel, which book the Jews reckon among their _hagiographa_ or sacred (but not canonical) books. And also this prophesy had a contradictory one (Jeremiah 22:30), where it is said of Coniah, no man of his seed shall prosper, sitting upon the throne of David and ruling any more in Judah. And Ezekiel 22:26–27: "Thus saith the Lord God: Remove the diadem, and take off the crown: this shall not be the same: exalt him that is low, and abase him that is high. I will overturn, overturn, overturn it, and it shall be no more, until he come whose right it is; and I will give it him."
The aforesaid obscure prophecy, which did not take effect at first until the reign of David, and which suffered such a variety **< 6>** of interruptions, seems to have fallen under this interpretation in the days of Herod the Great, whom the Jews so hated for his usurpation upon the Maccabee Levitical family and his general cruelties; he was particularly detested by the Cabbalistical Pharisees. That to keep up their rancor against him and his lineage, and to alienate the people from him, I could easily imagine this to have been a contrivance, wither perhaps was Herod displeased with the interpretation of the prophesy after that the Herodians had accommodated it to him and made him the Messiah, who after their conquest and ignominy under Pompey, had restored the Jews to a great reputation and strength, rebuilt the temple, and found some who could deduce his pedigree from the thigh of Jacob, as directly as David and Solomon. This sense of the prophecy being inculcated into the people, and all those Jews or strangers or proselytes, which resorted to Jerusalem at the great festivals from Alexandria, Antioch, Babylon, and all those parts where the Jews had any colonies, there was an universal expectation of the Messiah to come (I except the Herodians) which continued amongst them ever after and possesses the Jews (our Jews are but the remains of the Pharisees) to this day.
Their impatience for his appearance seems to have been less under Herod the Great (there being no mention of false Messiahs then), perhaps because the prophecy was not so clear and convincing whilst Herod was king since the scepter and legislative power seemed to be still in Judea. Though swayed by an Idumean proselyte, the priesthood continued, the temple flourished, and there was a prince of the Sanhedrin, Rabbi Hillel of the lineage of David. But ten years after the birth of Christ, when Archelaus was banished to Vienna, and Judea reduced into the form of a province, the scepter seemed then to be entirely departed from Judah. The kingdom was now become a part of the government of Syria and ruled by a procurator, who taxed them severely. Then the sense of their miseries made the people more credulous, and whether they more easily believed what they earnestly desired might happen, or that the malcontents did the more frequently and diligently insinuate into the multitude that opinion, there arose then sundry false Messiahs, and the world was big with expectation raised by the Jews in every country who had used the intelligence **< 7>** from their common metropolis (Jerusalem) that the great prince was coming who should reestablish the Jewish monarchy and bring peace and happiness to all the earth.
Those circumstances made way for the reception of Christ, and the miracles he did (miracles were the only demonstration to the Jews, Mark 8:11). Convincing the people that he was the Messiah, they never stayed till he should declare himself to be so (I think he never directly told any so but the woman of Samaria, John 4:26) or evince his genealogy from David. For though some mean persons called him the Son of David, and the populace by that title did cry "Hosanna" unto him, yet did he acquiesce in terming himself the Son of Man, but esteemed him a prophet Elias, Jeremiah, and even the Messiah. And when he made his cavalcade upon an asinego, they cried him up as the descendant of King David. But his untimely apprehension and death, together with his neglect to improve that inclination of the people to make him king, did allay the affections of the Jews towards him, disappoint all their hopes, and so exasperated them that they, who had been a part of his retinue in that intrado of his, called for his execution and adjudged him by common suffrage to be crucified. His disciples fled; the apostles distrusted and sufficiently testified their unbelief by not crediting his resurrection. After that he was risen again, and they, assured thereof, they assume their former hopes of a temporal Messiah, and the last question they propose to him is: "Lord, wilt thou at this time restore the Kingdom to Israel" (Acts 1:6).
After his assumption into heaven, they attend in Jerusalem the coming of the Holy Ghost which seized in them and gave them the gift of tongues for a season: whereby they preached to the Jews, Elamites, Parthians, Alexandrians et cetera (those Salmasius shows not to be absolute strangers, natives of those countries, but Jews planted there) as also the proselytes. Those surprised with the miracles of the cloven tongues and gift of languages, being possessed with the desire and hopes of a Messiah, and being there ascertained by Peter that Jesus whom Pilate had crucified was the Lord and Christ (Acts 2:36), were to the number of three thousand immediately baptized in his name. And such as were to depart, when they came to their colonies, did divulge the **< 8>** tidings and engage other Jews and proselytes to the same belief, the apostles themselves going about and also ordaining others to preach the glad tidings of a Messiah come who, though dead, was risen again (according to the obscure prediction of David) for the salvation of Israel, and whose second appearance would perfect the happiness of all nations, as well Jews as Gentiles.
That we may the better understand the way whereby those glad tidings were spread, it is requisite to be informed of the condition of the Jews in those days. It will not injure any relation if I translate a piece of a letter written by Agrippa, a king in Jewry (though not of Jerusalem), to Caius Caligula on behalf of the holy city, and published by Philo in his embassy to the emperor:
Jerusalem is the metropolis not only of the country of Judea, but of many other places, by reason of the colonies translated thence at several times, either in adjacent territories, as Egypt, Phoenicia, Syria, Caelo-Syria, or those more remote, as Pamphylia, Cilicia, most parts of Asia as far as Bithynia, and the inmost parts of Pontus. Nor is Europe exempt from this jurisdiction, the Jews being planted in Thessalia, Boetia, Macedonia, Etolia, Attica, Argos, Corinth, throughout Peloponnesus, especially in the principal parts of that isthmus. Nor is the continent only replenished with this nation: they have settled themselves in the chiefest isles as Euboea, Cyprus, and Crete. I mention not those beyond Euphrates, every place where the soil is rich, except a small part of Babylon and some parcels in other principalities inhabited by Jews.
To illustrate this further, let us consider the number and interest of the Jews at Alexandria. What the glory and power of the Alexandrian and Egyptian Jews was, it is easy to understand out of Josephus and Philo. They were exceeding numerous there, the chiefest dignitaries, as well military as civil, were vested in them. They had a peculiar temple built for them at Heliopolis and Onias. For their High Priest (though deserted before the time of Christ), they had always their distinct ruler under the Egyptian kings, chosen by themselves out of their senate to rule them for life, being **< 9>** styled Ἀλαβάρχης, Γενάρχης, Ἐθνάρχης And you may easily guess at their splendor and number by this relation of the rabbi: he that has not seen the cathedral church (or chief synagogue) at Alexandria never saw the real glory of Israel. It was like a royal palace: there were two porticos by which to enter into it; there were in it seventy chairs adorned with gold and jewels according to the number of the elders; and a wooden pulpit in the midst thereof, wherein stood the bishop of the synagogue. And when the law was read after the pronouncing of every benediction, a sign was given by the shaking of a handkerchief for the people to say "Amen." They did not there sit promiscuously, but men of several ranks and professions did sit in distinct places. There were once so many Jews there that the multitude was double to what went first out of Egypt.
In the next place, let us consider the multitude at Babylon and the neighboring territories, of those which went up with Ezra, Nehemiah, and Zorobabel. The number was but small, and those consisting of two tribes principally Judah and Benjamin with four orders only of the priests and Levites. Josephus informs us that the rest of the multitude of Israelites chose rather to remain in the Babylonian territories, wherefore only two tribes are to be found in Asia and Europe subjected to the Roman Empire. The other ten tribes continue beyond Euphrates until this day, being an infinite people not to be numbered.
This account, how ten tribes remained, ought not to seem strange to those who consider how St. James writes his catholic epistle to the twelve tribes which were scattered abroad; nor to them which believe that the seventy-two interpreters of the Bible were chosen out of the twelve tribes, or who give any credit to the itinerary of Benjamin Tudelensis, a Jew who met with many of those Israelites that were captivated by Salmonasar in Media and other neighboring countries. Many of the Israelites fled into Judea also removed thence with the Levites upon the schism, and the rest continued in the dominions of Ashur till Nebuchadnezzar **< 10>** brought the Jews into the like captivity and scattered them all over his dominions, which comprised one hundred and twenty-seven provinces (Esther 8:9). And it is not to be doubted but the interest of the Jews was very great there about the time of Christ, since at the time of Benjamin's travels about five hundred years ago, they had so great a power and jurisdiction there, since they could be contradistinguished from the Palestine Jews, have their own Targum and Talmud, and their republics and universities in _Soria_ , Pumbeditha and Nehardea continuing till the times of Theodosius, Arcadius, and Honorius, besides an infinity of synagogues and a reverence universally to the rulers, equivalent to what was shown unto the Christian bishops and clergy.
All those came up frequently to the great festivals at Jerusalem, as Salmasius shows out of Acts 2:9, where the old Jews from the place of their residence are called Parthians, Medes, Elamites. And besides this inestimable number of Jews, there was a multitude of proselytes whom they continually converted to entire Judaism. For not only Abraham but all the Jews did perpetually endeavor to draw others to their religion. Idumeans were made proselytes in the days of Hyrcanus; in the Babylonian Empire, many of the nations turned Jews (Esther 8:17). The Talmudists reckon upon Nero, Cæsar, and Antoninus Pius as proselytes. But to evince their number better, let us learn from Salmasius that the originary Jews did never use the Septuagint in their synagogues, but that was made use of by the proselyte Jews and their posterity at Alexandria and elsewhere. The Hellenists, mentioned in the Acts, were no other; and the deacons elected there to provide for the widows were of that number, and being a proselyte of Antioch. That they were Jewish proselytes appears from hence: that the gospel had not then been preached to the Gentiles.
How diligent the Pharisees were to engage new converts the gospel tells us, and to descend so after ages, which is of some importance to the subsequent discourse. Dio Cassius tells us that in the time of Adrian when Bar Kokhba acted the Messiah, many nations joined with that imposter and the Jews; so that the whole world was in a commotion, which cannot be understood of any but entire proselytes. For the Jews would not have mingled with others. And after that, under the Christian emperors, our codes and ecclesiastical constitutions inform us that they retained the custom of inveigling proselytes. This being the condition of the Jews, and all the nation, however dispersed, **< 11>** being prepared aforehand to entertain any tidings of a Messiah who should advance the throne of David to an universal monarchy, it is not to be wondered that Christianity was so soon spread over the whole earth.
But wherein consisted this primitive Christianity which was there diffused? Certainly, the principal tenet which gained upon the spirits of all men was the doctrine of the coming of the Messiah. And it is evident that this was the fundamental article, from whence the Christians had at first their name in Antioch, and which they propagated everywhere as the sum of their religion: that Jesus who was crucified was the true Messiah, that he was risen again, and would return in glory to restore Israel and establish truth and peace throughout the earth.
The first part is apparent from these texts. The second seems demonstrated hence: that not only the Jews but the Christians were millenaries and did believe and expect the temporal reign of the Messiah and the union of the Jews and Gentiles under one most happy monarchy. Not one of the two first ages did dissent from this opinion; they which opposed it never quoted any for themselves before Dionysius Alexandrinus who lived at least 250 years after Christ. Of this opinion was Justin Martyr and (as he says) all Christians that were exactly Orthodox; Irenæus sets it down directly for a tradition and relates the very words which Christ used when he taught the doctrine.
And if this tenet were not an universal tradition in the most primitive times, I profess I know not what article of our faith will be found to be such upon the most diligent research.
This doctrine was taught by the consent of the most eminent Fathers of the first ages without any opposition from their contemporaries, and was delivered by them not as doctors but as witnesses, not as their own opinion but as apostolic tradition. This tenet has been so fully handled by Dr. Mead (not to mention some others) that I might have declined the allegation of those impartial and able witnesses—the Lord Falkland and Mr. Chillingworth. These were the principal tenets of those that were the first Christians, and from whence they were denominated. As to the subordinate doctrinals, they were no other than these: that the Messiah being already come, and since his ascension, being upon his return, in order to the recollection of Israel, the reestablishing of that kingdom and uniting of all nations under one scepter, a scepter of righteousness and truth; that all persons ought to prepare themselves for this holy kingdom of the Messiah and of heaven, and to relinquish all idolatry and wickedness, to repent **< 12>** of their sins unfeignedly, and to submit to those laws under the obedience whereof God had concluded mankind though in sundry manners, there being one obligation upon the Jews and entire proselytes, and another upon the rest of men, who were not under that dispensation but subjected to the seven commandments of Noah, and, by due observing thereof, might render themselves capable of a portion in the future life and be sufficiently qualified for the kingdom of Christ on earth: it being the custom of the Jews always to make proselytes where so ever they lived. And if they prevailed not so far, at least to reduce them from idolatry and Gentilism to the observation of that law of nature which they esteemed all the progeny of Noah and such as were not of the Jewish profession to be obliged to.
It is no wonder if upon the persuasion that the Messiah were born, and returning again in glory, that some became apostles and others evangelists and teachers, both to Jews and Gentiles. The partition wall and distinction was to be taken away when all the world should become subjects to the same prince, who should extend his favors first to the Jews and then to the Gentiles, the greatest prerogatives and privileges appertaining to the Jews as the peculiar people and children of the promise.
We find in the first churches a distinction between Christians that were Jews and entire proselytes and those that were Gentiles or uncircumcised. The first had their apostles, all except Paul being of that number, and of Peter it is particularly said that he had the apostleship of the Jews committed to him, as Paul was charged with that of the Gentiles (Galatians 2:9). And of those which were scattered upon the persecution as far as Phoenicia, Cyprus, and Antioch, they preached to the Jews only (Acts 11:19), though amongst the Jews there we find this distinction: that some of them being entire proselytes, but not understanding Hebrew nor reverencing that holy language so much as the originary Jews, they spoke Greek and used the Septuagint in their synagogues. Those are the Grecians spoken of (Acts 11:20) as Salmasius well observes. And such were the churches in Jerusalem to whom the apostles appointed deacons, all proselytes. Of them that preached to the Gentiles, some taught them the necessity of circumcision and becoming entire proselytes: thus did Peter (Galatians 2:14) and others (Acts 15:2), and even Paul circumcised Timothy though the son of a gentile father (Acts 16:3). Sixteen bishops of Jerusalem were successively circumcised, saith Sulpicius Severus; and even those who derived their pedigree so as to show they were of the kindred of Christ, called _deposyini_ , were always of the circumcision **< 13>** (Eusebeus, _Ecclesiastica Historia_ liber 1, caput 7).
From hence we may frame to ourselves a prospect of the primitive judaizing church, since it is certain that they were zealous as to the Mosaical Law (Acts 21) and lived in a perfect conformity to the legal rites. It is not to be doubted but their religion and doctrinals varied much from ours: such a sacrament as we make baptism to be, they had none, the Jewish baptism extending only to proselytes when newly made and their present family, not successive posterity—except we take it in a general sense for washing, as Luke 11:38. And so they might baptize either arbitrarily upon some great occasion, as at the preaching of John, or out of respect to legal or superstitious Pharisaical uncleanness. To this alludes the apostle when he tells the Hebrews of the doctrine of baptism (Hebrews 6:2). They that were circumcised resorted at usual times to the public temple service (Acts 3.1): they paid vows, offered sacrifices, and walked orderly, keeping the law, and yet were believers (Acts 21:20) et cetera. And there is not any sign that they were separated from the other Jews or were accounted heretics upon any other account than that they held Jesus to be the Messiah and taught in his name (Acts 14:17–18).
As to that other sacrament of the Lord's Supper, neither did they use that otherwise than Christ had done, as a Judaical rite used either at the Passover or constantly at meals, the cup of blessing being then distributed by the master of the household, and the bread broken and distributed. The use of red wine, the breaking of the bread formally, and the distributing of it, the very names and rites are the same which were usual among the Jews. Nor was this ever done in the Jewish synagogues but at home. And so it is recorded they continued daily in the temple and, breaking bread at their own houses, did eat their meat with gladness and singleness of heart (Acts 2:46). It is very probable that they added to the usual benediction of the bread and wine some commemorations in honor of the Messiah, which was no innovation or schismatical act since every rabbi might enlarge the synagogue worship or private devotion of his disciples in that manner. And it was usual for them so to do, wherefore this could give no distaste. And if we may believe that they imitated Christ in the celebration of his Last Supper, as that we may (for what was read of the Lord was delivered unto them (1 Corinthians 11:23), we must believe that all the ceremonies of the Jews were entirely retained by them at such time, seeing that in the evangelists we find nothing done in the Lord's Supper but what the Jewish rituals prescribe. **< 14>**
Scaliger avows it:" Ea omnia quae Evangelio traduntur, in ritibus Judaeorum sine ulla discrepantia eodem modo praecepta esse," with whom doth Buxtorf and those that are most versed in the rabbinical learning agree. But that they did never believe Christ to be the natural son of God by eternal generation, or any tenet depending thereon, or prayed unto him, or believed the Holy Ghost, or the trinity of persons in one deity, is as evident as it is that the Jews and they did expect no such Messiah, and the introducing such doctrines would have been capital among them as tending to blasphemy and polytheism. It was blasphemy adjudged in Christ to say that he should sit at the right hand of power—that power being esteemed an incommunicable attribute of God. And so suffered Stephen (Acts 7:56–57) though Jesus did not upon the adjuration of Caiaphas say that he was Christ the Son of God (Matthew 26:63), and albeit it is manifest that the term "Son of God" was not unusual amongst the Jews so that they bestowed it on men, yet did they not import thereby any real divinity in the person (nor did Caiaphas in this adjuration mean so), but an extraordinary perfection lodged in humanity and hyperbolically expressed.
Neither is to be believed that they conceived that Christ's death had put an end to the ceremonial law as consisting of types and fading shadows since they obstinately retained them so long after. And, which is most considerable, during this time they were instructed and governed by the apostles and their immediate successors. Such was the condition of the Judaizing Christians amongst which it is further remarkable that as the Jews originally did use the Hebrew Bible in their synagogues (howsoever that they expounded it in Syriac and Chaldean as they do now in Spanish, Italian, and according to the language which the auditory best understands), and the proselytes did follow the Septuagint and had a Greek liturgy, so it happened in Christianity: the Jewish converts did use the Hebrew Bible, and others adhered to the Septuagint, and as they hated each other before by reason of the use of the Septuagint, so they seem to have retained the same passion and animosity under the gospel. And the murmuring of the Greeks (Acts 6:1) perhaps derived its beginning from hence—the Hebrews not relieving the widows of those others, whereupon the **< 15>** Hellenist proselytes had seven deacons chosen out of their number to attend to that care.
The Hellenists did relate miracles concerning their version and feigned a tale of seventy cells in which each translator finished his version, and upon comparing, they were found to be the same word for word. But the Jews say that darkness was upon the face of the earth in the time of Ptolemy when that translation was made and in the month of Thebeth kept a fast to testify their sorrow and resentments for it. And though the Hellenists did reside in Judea or resort thither from Alexandria and Antioch, yet they held synagogues distinct from those of the Hebrews. Thus we have the synagogue of the Libertines, of the Cyrenians, Alexandrians, et cetera (Acts 6:9). Such were the Jewish synagogues of which Justin Martyr and Tertullian speak, in which the Septuagint was read; such was that in Cæsarea the Metropolis of Judea, whereof we read in the Hiero-solymitan Talmud that Rabbi Levi went to Cæsarea, and hearing them reading the lesson in Greek, he would have hindered them, but Rabbi Jesse was angry and said: "Must not he read at all who cannot read Hebrew? Let him read in any language that he understands and he discharges his duty."
This distinction of synagogues upon the same account was introduced amongst the Christians. For though the Hellenists and Gentiles did use the Septuagint and Greek service, the Hebrews did not so, nor used they the same gospels with the other. Such were the Nazarenes who lived at Cæsarea Borea, and elsewhere, who used the Hebrew Bible and either a gospel peculiar called Evangeluim Nazareorum or Evangeluim Secundum Hebraeos or at least the Gospel of Matthew written in Hebrew, but with that discrepancy from the others, that the church hath rejected it as apocryphal, sophisticated, with sundry fables and otherwise corrupted. But the Nazarenes and Ebionites, the remains of the Judaizing churches, did repute that as the only authentic **< 16>** gospel, as Epiphanius relates.
And from hence I conceive arose that division whereby some declared themselves to be of the synagogues of Paul, others of Apollo, and others of Cephas. The apostle of the circumcision did retain the Jewish rites, the Hebrew Scriptures and Hebrew Gospel (according to what descended to the Nazarenes at Pella Borea, et cetera) with a regimen exactly Judaical. But Apollo, being an Alexandrian Jew (or rather Jewish proselyte), as one would guess by his name, used (no doubt) the Septuagint and such books as composed the canon at Alexandria, and in all probability did introduce in his synagogues a conformity with the Alexandrians' rites and government, adding thereunto that Jesus was Christ which is all that I find he preached (Acts 18:28). But Paul who dealt with the Gentiles and did not reduce them under the Judaical law and circumcision, nor enforced them to any uniformity, but became all things unto all (by way of condescension) that he might gain them to Christ, whence his synagogues or churches must needs have varied exceedingly from those erected by Peter and Apollo, which gave occasion to the distinction at Corinth, some being of Paul, some of Apollo, and some of Cephas (1 Corinthians 1:12).
From hence, if one will frame unto himself a prospect of the first Christianity, he must imagine to himself distinct synagogues of the original Jews and Hellenist proselytes none of those subordinate to the same governors, but as independent as were the Jewish synagogues everywhere, each synagogue having its peculiar bishop or angel of the church and ruling presbyters which are termed in the civil law and by the Jews _archisynagogi_ and _presbyteri_ , though perhaps the _nasi_ or patriarch at Jerusalem might have one universal superintendency over them as he had a power to exact money from all the Jewish synagogues in the **< 17>** east and west. The officers which were sent to gather his _aurum coronarium_ or tax were called "Apostles." And in imitation thereof did Christ institute his apostles.
The whole constitution of the primitive clergy relates to the Jewish synagogue not to the hierarchy. The presbyters were not priests but laymen set apart to their office by imposition of hands: no temples, no altars, no sacrifices were known in those days, the name of priest unheard of then. Before the destruction of the temple at Jerusalem, the said apostles did go forth and collect the said money as shekels paid to the temple which was afterward converted into _aurum coronarium_. In the subsequent ages, there is mention of the _patriarchi minores_ to whom the Theodosian Code gives the title of _spectabiles_ (as the other grand patriarchs are styled _illustres_ and _clarissimi_ ), as also _primates_. There is also mention of the _hieres_ whose office is not known now except it relate to the _cohen_ now in use among the Jews.
How ancient are the lesser patriarchs (who seemed to have ruled over the _archisynagogi_ and presbyters of particular synagogues) I cannot tell. But since all learned men do now agree that the Christian Church was governed according to the pattern of the Jewish synagogue, there can be little doubt but that every officer of the Christian synagogues resembled those of the others as well in office as name; and that as they retained the rites and customs of the Jewish synagogue in all other things (as to structure), not building their synagogues east and west but west and east so that the coming in was at the east and they prayed to the west (so is Saint Peter's Church, built now at Rome), in **< 18>** observing the Sabbath, paschal, circumcision et cetera—so they did in their government. And it is observable that, even at this day, though the Jewish synagogues agree in the substance of their service, yet for the particulars thereof, there is a great discrepancy amongst them in several countries, and so there was in the primitive times amongst the Christians.
It is very possible also that the Judaizing Christians were at first subordinate to the Jewish patriarch and primates, having none of their own, before the Jews were animated against them and anathematized them as they were at first called by one common denomination of Jews. And some of them frequented the Jewish synagogues, the tenet of Christianity rendering them only _mimes_ or heretics not separatists (as St. Jerome says), and they continued to do so in his time, through all the synagogues of the eastern Jews.
We may further collect out of the different sects of the Jews (some of every one whereof embraced Christianity) the differences to have been as great in the Christian synagogues as in those of the Jews where the Sadducees, Essenes, Pharisees and Samaritans made sects. Each retained their opinions mixed with the doctrine of the Messiah. And here came those Judaizing sects of which such a number is recounted by Epiphanius. We may also imagine a great diversity betwixt the Jews living in the land of promise and those which lived out of it: for the Jews did not take themselves (nor do now) to be obliged to the Mosaical **< 19>** rites, much less any temple worship, out of Palestine. The criminal laws have no coercive power out of these bounds, the Paschal Lamb is not slain in any other country, all ground besides that holy Canaan is too impure and profane for such services and rites, though they did by authority of their rabbis frame to themselves many succedaneous rites and retain circumcision, the Judaical doctrines, and an opinion of their particular holiness above the Gentiles. Of these Judaizing Christians or believing Jews (being reckoned not as heretics but good Christians) doth Origen speak as continuing to his time as the Gentiles that were converted to the belief of the Messiah, though Paul were the apostle of the uncircumcision and did not reduce them under the Mosaical Law and rites. Yet being originally an Hebrew, it is easy to observe that in the settling of the church government, and in the penalty of excommunication, he did introduce in their church several Judaical constitutions, and also accommodated the pagan ceremonies frequently thereunto.
The intromission of the Gentiles by baptism was no Jewish rite in proselytes of the uncircumcision and can only be looked on as a particular washing from uncleanness such as was that of John Baptist or in imitation of that pagan rite so frequently used in case of enormous sins to wash them away by bathing in a river to which the poet Ovid alludes: "Ah nimium faciles, qui tristia crimina caedis/fluminea tolli posse putatis aqua! Sed tamen, antiqui ne nescius ordinis erres," Tristia, liber 2. And Virgil: "Tu genitor, cape sacra manu patriosque penatis; me bello e tanto digressum et caede recenti attrectare nefas, donec me flumine uiuo abluero." **< 20>**
It is most certain that baptism heretofore was not administered by aspersion as now, but by a total immersion. And as to the baptizing of children, if it were used by any in the first ages, it is condemned by Tertullian and others, and can vouch no precedents or precepts out of scripture. The discontinued usage of baptizing for the dead hath more to show for it since it is mentioned in scripture and not condemned in the Greek Church. Even Nazianzen was not baptized till thirty-three years old, albeit a bishop's son, nor was Valentinian, the Emperor Theodosius, although descended of Christian parents, nor Saint Ambrose, nor Constantine and his son Constantius: so that we may not wonder to read that sundry heretical Christians did reject baptism totally, perhaps it having never been used in their churches. Such were the Seleuciani, Hermiani, Pauliciani, and that ancient and numerous sect of the Manicheans. Even the Jacobites did not use baptism, but with a hot iron imprinted the sign of the cross in the forehead of their partisans.
If I may be allowed to guess at the original of baptism, I would derive it from the pagan custom aforesaid of washing away expiatorily in rivers the most enormous sins; in the doing whereof as the pagans were very tender, so the Christians were more frank, as Zosimus relates of Constantine the Great. And the baptism of children from hence: that because the Romans used the eighth and ninth day to devote their infants unto the _dea mundina_ and give them names then, and the Greeks had theirs **< 21>** to the same purpose on the tenth day after their birth, therefore the Christians out of compliance with that superstition of the vulgar did hereby incline them to initiate them unto Christ. Such condescension in other cases as in the observation of Christmas, New Year's Day, Mayday, Shrovetide and the pre-vigils and wakes of saints, and the form of churches, the praying to the east, processions about parish bounds, the denomination of the clergy by the titles of _antistites, pontifices, sacerdotes_ , and the churches by the name of _templa_ and _aedes_ , the shaving of the clergy, the surplice, the antiphons, and a thousand other things observable in the ancient Gentile Christianity had no other original. Even the sacrament of the Lord's Supper and all its rites seem to be deduced hence: the festivals of the pagan gods were usually suppers. This was so at the first: there was great feasting at them; so in this case at first, they were performed in the temples; so was this at the first, and so continues to be still. All the names of the pagan mysteries are fixed on this sacrament and its rites, et cetera: the procedure from the _catechumeni_ to _competentes_ and then to _fideles_ , the preparation before it, and all austerities so resembling that, they easily show whence they were derived.
But withal I must add that whereas those mysteries were not everywhere the same (for in the mysteries of Mithra they gave to the initiated a cup of water and some bread with some accessional forms of words), so neither were the rites of this Christian sacrament everywhere the same. Where the reverence of the mysteries was greatest and most solemn and accompanied with greater mortifications, there the Christians were more strict. Where it seemed rather substituted to the pagan **< 22>** festival suppers, there they were more jocund and the κυριακόν δεῖπνον was no other amongst the Christians than these suppers paganical δεῖπνα, or _pontifi-cum canæ_ of the Gentiles. In some places, their ἑταιρεῖαι or assemblies of men at festivals—these were allowed everywhere in Greece and Alexandria and usual amongst the Gentiles. They are termed συσσίτια, θιάσοι, ἔρανοι, and those which assembled there εἰλαπίναι, et cetera. They were either held upon a religious accommodation as the συμβολή—or merely for pleasure and conversation as the rest. Each person contributed his part or share to the defraying the expenses, and that contribution was called συμβολή— _symbola_ or _symbolum_. The associates had a kind of admission amongst themselves and the manner of assembling gained it the name of ἑταιρία and κοινωνία, and at those meetings there was usually laid up either the overplus of the money collected or perhaps some further collection made for any distressed member of the society or against such contingencies. These were publicly allowed by authority and held monthly. They were oftentimes held in the temples and also in other appointed places.
He that will compare this with that account which we have of the Christians' Love Feast or _agape_ and considers that though the magistrates did not usually allow of private combinations or meetings, and yet approved these if they extended not to the danger of the public, will think that the κυριακὸν δεῖπνον and the κοινωνία were but appellations of some such _sodalitium_ or fraternity. And this seems apparent from the communion of the Corinthians where every man brought in his contribution of food and wine and eat and drank thereof. **< 23>** The fault which the apostle doth blame in them is that the communists did not import what they brought into the rest of the fraternity, but each fed upon his own _symbola_ so that the poorer did rise hungry and the richer did riot it. He tells the Corinthians that if they will eat apart, they may do it at home, not in the church. This procedure was contrary to the rules of such... , and such as were poorer were put to shame and slighted by reason of the meanness of their contributions. He speaks there of the sacrament, and if we compare that and with the _agape_ we must conclude the _agape_ and communion all one. It is no wonder then if we read of several ways of celebrating this supper amongst the ancients. Even in the apostles' time some communicated with the pagans at their festivals and to the honor of Christ did drink the cup of devils, and did partake of the table of devils (1 Corinthians 10:20–21): from whence we may observe that the rites were the same though they differed in the objects of their devotion.
This being the duality of their rites and ceremonies, let us take a view of the doctrines which the Gentiles were converted unto. These seem to be principally contained in the grand tenet that Jesus was the Messiah of the Jews who was to unite Jew and Gentile under one temporal monarchy. To qualify the Gentiles as befitting subjects for that celestial prince, they were to repent of their sins, renounce idolatry, and entirely to obey the seven commandments under which the Jews did believe all the Gentiles were to be concluded if they would have any person in the life to come or kingdom of heaven. This seems most evident from the apostolical decree made by the synod at **< 24>** Jerusalem wherein there was a contest about what obligation the believing were to be brought under, whether they ought to be circumcised and instructed in the whole law of Moses to keep it, or whether they should be subjected to the seven commandments of Noah only. The synod concludes upon the latter. It is manifest from that synod that of these which entertained the doctrine of the Messiah and so became Christians, some were Pharisees and retained the opinions and traditions of their sect, together with the doctrine of Christ. They were not heretics if Ebion and Cerinthus were the persons that occasioned.
It is most certain they which held the opinion were then in good esteem with the church and are said to be the same of the sect of the Pharisees which believed (Acts 15:5), and they were a part in that apostolical synod (though overruled). How else could there have been such συζήτησις—so great a dispute there, as the text avers (Acts 15:7). So there were of the Sadducees that professed Christianity and believed Jesus to be the Messiah yet denied the resurrection. Of such does Justin Martyr speak in his dialogue with Tryphon the Jew, when he reckons up as a third part of Christendom those which were called Christians yet denied the resurrection and tenet of the Chiliasts. It is true he esteems them as wicked and heretical persons, but reckons them Christians. Seeing then that though the converts did retain their former opinions generally, and that Christianity itself was but a reformation of Judaism (as Mr. Selden more than once inculcates): to understand the decision we **< 25>** must consult the doctrine of the Jews as to this point.
Now it is a most received tenet amongst them that a Gentile continuing a Gentile was obliged to nothing but the law of nature contained in those seven precepts, and not to the Mosaical Law at all, and that it is needless if we consider the subsequent passage only as an instance of things commonly known to be necessary. Had the synod meant any other way, what a confusion would there have risen upon so great an innovation in the Jewish tenets. How can we imagine that they which were so zealous for the same would have acquiesced therein? How should they suffer this additional clause out of the Mosaical Law to be extended to the uncircumcised: that is to abstain from things strangled, which was permitted to the Gentiles (Deuteronomy 14:21). But that clause is held to be spurious in the text by St. Ambrose and others, and really it is incredible that it should ever have been in that decree. Nor can it be reputed a necessary point in the decree to abstain from things offered unto idols except we understand thereby to abstain from idolatry. For Paul decides that point otherwise to the Corinthians: that it was simply lawful for them to eat things offered to idols and that it is an indifferent matter, except in case of scandal, as Heideggerus doth demonstrate out of the place.
I do therefore take it for granted that this was the fundamental doctrine of the Gentile Christians. But I must further add that as the Jews retained their tenets and usages under Christianity, so did the Gentiles many of theirs. Thus Pantaenus and Clement Alexandrinus mixed Stoicism with Christianity; Origen and others, Platonism and Peripateticism, and I have read of Cynical and Epicurean Christians. It is also to be noted **< 26>** that as the Judaizing Christians were offended with Peter for going to Cornelius a Gentile (Acts 11:2), so neither did they come to the same assemblies or communicate with them except it were upon extraordinary occasion and by a paramount apostolical procedure. Thus Peter having at first associated with the Gentile Christians at Antioch and eaten with them: when some came unto him from James, he withdrew and separated himself, fearing them which were of the circumcision. And the other Jews dissembled likewise with him, insomuch that Barnabas was carried away with their dissimulation (Galatians 2:12–13), which distance continued afterwards in the Judaizing churches. For albeit that the Jews held that the pious amongst the Gentiles might be saved, yet did they esteem them as unclean and such as they might not freely converse or eat with (Acts 10:28).
It may be questioned whether the Gentile Christians did not believe the deity of Christ. They were so accustomed to the deifying and conferring divine honors and worship upon men that it is not to be doubted but many did believe him to be a god in the pagan sense, as other heroes were reputed. And thus Pliny in his inquisition after Christianity found that they did sing certain hymns to Christ _quasi deo_ , as if he were a god. And Tertullian relating the same thing says: they did sing hymns _Christo et Deo_. Nor can there be any doubt thereof or that there were many pieces of poetry composed by the brethren which ascribed a divinity to Christ. But if there be any ground for that assertion of Artemon, Apolonides, Hermophilus, and Theodotus, the most learned, subtle, and philosophical disputants (though styled heretics) of the ancient Christians, that all the first Christians, and even the apostles themselves, were taught and did teach that Christ was a mere man (which was their tenet) and that the truth of this doctrine **< 27>** was continued in the church until the days of Pope Victor who was the thirteenth bishop of Rome after Peter, and that Zephyrinus his successor did alter and corrupt that truth—if it be true which the Arians said that none but idiots and simple persons believed any such thing, and that till the decision at Nicaea the more knowing Christians did not hold him to be really God.
If we may conceive that they were firmly taught that there was but one God, and that they were too dull to comprehend or invent those subtle distinctions of essence and person, consubstantiation, eternal generation, and if it be certain that the Fathers after the Nicene Council were not agreed concerning the meaning of those uncouth words, and that the world was long after dissatisfied with the use of them, and that such as Gregory Nazianzen and Basil were shy how they taught the deity of the Holy Ghost or of Christ or touched upon the Trinity, _homousianism_ ; and if we reflect upon the Creed intituled to the apostles and certainly very ancient that there is no intimation hereof in it; if we take notice how differently the Fathers explicate themselves upon that point, and how much the other works of Athanasius do differ from the Creed which goes under his name, we may very well doubt concerning their judgment if not conclude the contrary.
As to their rites and church government, the apostle of the uncircumcision who became all things to all men that he might gain them to Christ did comply with their weakness and prejudice. Many pagan usages and superstitions were retained; their church government was much modeled according to the pagan usages in their temples and sodalities. It is not impossible that they might have had in some places as well priests as temples dedicated and altars; the distinction of their churches into the _ordo_ and _plebs_ savors of the Gentile customs in the civil government of their cities. It is not to be believed that the Jewish apostles did appoint or ordain officers for them: **< 28>** the Jews did not use to ordain Jews for judges to the stranger proselytes. Neither would they certainly appoint them rulers in their churches. Besides, it is notorious that the first bishops were elected by the suffrage of the people and might be deposed by them, that they might upon a dissension break into a subdivision of episcopacies and erect two or three or more in one city. And of bishops there was no subordination but a parity.
It is easy to imagine that during the numerousness, grandeur, and power of the Jews in Palestine, Egypt, and Babylon, and throughout all Greece, and whilst all the Gentiles expected a Messiah of the Jews to whom they were to be subjected, they did pay a great respect unto the Jews and were much swayed by their dictates. But about fifty-two years after the destruction and government of the Jews under Titus, there arose another Messiah amongst the Jews who accommodated to himself the prophecy of Balaam and styled himself _Bencochab_ , or the son of the star. The famed Rabbi Akibba joined with him and saluted him as the King Messiah. Great was his power and a bloody war did he wage with the Romans and the Emperor Adrian. After three years, he and four hundred thousand Jews were miserably slain by the emperor and those that escaped were so angry with their present Messiah that they termed him _Barcozabh_ or _Cuzzibha_ , the Son of a Lie. Adrian marched with his victorious army to Alexandria (where the Jews in favor of _Bencochab_ had destroyed the Romans) and put to death an infinite of the Jews there. The multitude which the Jews say was slain in that war is scarcely to be believed. He put down their synagogues everywhere, so that it is not to be credited that the Jews did anywhere after this appear embodied in the Roman Empire.
And now we may conceive naturally that the Christians must **< 29>** totally disclaim the Jews and pretend only to a spiritual Messiah. They could not have preserved themselves but by such an action, and undoubtedly, not long after that, we find mention of priests, temples, and the rites of the church do evidently comply with paganism. What befell the Judaizing churches I know not, but they became in little esteem and sank at last under the name of Ebionites and other heretics. This revolution had a mighty influence upon Christianity, and Adrian in his letter to Servianus, wherein he gives him an account of Egypt, doth avow that all the Christians, besides their devotion to Christ, did worship Serapis. "Illi, qui Serapim colunt, christiani sunt et devoti sunt Serapi, qui se Christi episcopos dicunt. Nemo illic archi-synagogus Judaeorum, nemo Samarites, nemo christianorum presbyter; non mathematicus, non aruspex, non aliptes. Ipse ille patriarcha, cum Egyptum venerit, ab aliis Serapidem adorare, ab aliis cogitur Christum."
Some would have this letter to be false or full of untruths since it seems incredible that the Christians should do so. Besides, since the great patriarch of the Jews did not come into Egypt at all, and it is certain the Christians had then no patriarch at Alexandria or elsewhere, how can this passage be verified? I am of a contrary judgment and do believe that the bishop of Alexandria by reason of the greatness of his power and splendor was called analogically or by way of flattery a patriarch. This appears from Eutychius in his _Origines Alexandrini_ and from other oriental writers who speak of St. George as son to the patriarch of Alexandria, though really in his time (under Diocletian) there were none. As to the mixing of pagan worship with Christianity, not to say what the _thurificatores_ (one whereof was a pope) did **< 30>** upon compulsion, if ever there were such a legion as that formed, fulminating in the army of the Emperor Adrian or the _Legio Thebea_ (both famed in ecclesiastical story) or any legionary soldiers that professed Christianity, I am confident they never had any dispensation from worshipping the Roman eagles: "Sequerentur Romanas acquilas propria legionum numina." And I am the more confirmed in this sentiment since under the Christian emperors the imperial banner called _labarum_ was worshiped in like manner.
I am sure that the Egyptian Christians were not so scrupulous afterwards but that they procured to themselves and executed the office of _archierosyna_ whose power was to superintend over and manage the pagan temples, festivals, rites, and whole religion of Egypt. And this they continued to do until Theodosius the Great did prohibit them to do so, AD 389. It is no less strange a thing that the Christian emperors should for a long time be and wear the habit of the _pontifices maximi_ ; that the senators, which were Christians at Rome, should as they went to the senate be present at the sacrifices of the Altar of Victory, about the removal whereof Symmachus and the majority of the senate, being pagans, did proffer a complaint. I shall illustrate this point with a strange relation out of Eutychius who was one of the patriarchs of Alexandria. It is thus: When Alexander, the predecessor of Athanasius, was first made patriarch of that place, he found there a temple and a great brazen idol dedicated to one Michael and much frequented by the pagans, and the inhabitants of Alexandria and Egypt did keep a great festival in honor of this idol on the twelfth day of the month **< 31>** Haturi and offered up many sacrifices thereto. The patriarch had a great mind to abolish this idolatry, but met with much opposition about it. At last he prevailed by subtlety: he told them that his idol was an insignificant statue, but if they would perform the same devotion and offer up the like sacrifices to Michael the Archangel, he would intercede with God for them and procure them greater benefits than that idol could. Whereupon he broke the idol in pieces and shaped it into a cross, and called the temple St. Michael's Church—which church was afterwards called Cæsarea, and was burnt when the western army took Alexandria. And the festivals and sacrifices were continued in honor of St. Michael, and, even still, the _Cophite_ Christians in _Misrar_ , or Grand Cairo, and in Alexandria do celebrate the festival of St. Michael on that day and offer many sacrifices unto him.
The relator is a historian of good credit, but I do not remember to have read the like sacrificing to have been performed by Christians elsewhere, though any man conversant in antiquity knows that a multitude of pagan usages crept in among the Christians. And though they did not sacrifice, yet they brought to their priests at the altar the first fruits, as was formerly practiced to the rural gods. And rather the objects of the devotion were changed than the things abolished: the same festivals were retained in a manner to the honor of Christ, the Virgin Mary, or the saints, which were performed before to Mercury, Venus _genetrix_ , Bacchus, and the rural deities. As the aforesaid calamities of the Jews did make a great alteration in Christianity, so did the frequent persecutions by the Roman emperors against them who looked upon them as no good subjects since they expected a temporal Messiah and oftentimes disclaimed all subjection to the pagan magistrates, sometimes rebelling, as in the reign of Diocletian did the Christians of Alexandria and the adjacent countries under St. George and in France under Amandus and Ӕlianus. They looked upon them generally as enemies to the received and established religion or idolatry of the empire and feared the consequences of a change therein. Besides that, the **< 32>** people hated them. Upon that account, everywhere, many bickerings and tumults happened thereupon. They did also look upon the mortification and monasticness of the Christians as inconsistent with the government, enfeebling men's minds and alienating them from military employments, the sinews of empire, and spoiling trade, by decrying luxury and all excess as well as in diminishing the sale of cattle and other commodities used in the pagan solemnities.
There, amongst many others (which Papinianus as I remember is said to have digested into seven books concerning the justice of punishing the Christians), were the motives upon which these persecutors went. And though Christianity were not extirpated, yet it changed much its complexion. The opinion of a temporal Messiah was laid aside, subjection to the pagan magistrate preached, many dissolute and enormous assemblies disowned and declared heretical. The Christians fought for the gentile emperors and watched at the temples to defend them, declared them to be no martyrs who disturbed or demolished them. Much of their rigor and strictness was abolished or preserved only in a few monasteries. As the Christians suffered this alteration, they were infected by the conversation and superstitions of the pagans. So those on the other side became much altered by mixing with the Christians: they were inclined to a contempt of their gods and an indifferency in their religion; they were exasperated at the threats of their priests and the expensiveness of their rites and devotions. The discipline of the Roman legions being extinct and the armies composed most of foreign men of mercenary spirits and no friends to the established religion, the soldiery beheld opulent priests and vestals together with their colleges with an envious eye and cared not if a new religion were introduced, so that they might share the spoils of the old.
In this juncture I find Constantine to have made himself emperor. Right he had none, being a bastard and not elected nor admitted by the senate. His sword was his title; success warranted it. His soldiers were not more assured of his courage and conduct than animated by the hopes of honor and riches, which the conquest of Italy and change of religion and government would instate them in. He subverted the power of the senate, removed the seat of the empire, altered much of the religion, and gained most of the sacerdotal lands and revenues by the change. He was no Christian in profession **< 33>** till a few days before he died. He never was at prayer among the _catecheumeni_ till then, nor so much as baptized, and without that initiating sacrament. It is not to be imagined that he could be instructed in, or admitted to, those doctrines and acts of nearer communion. All that is written contrary hereunto are palpable untruths or deeds of flattery.
It is true his mother seemed zealous for Christianity and built many churches, and he, out of his spoils, allotted some to pious uses thereby to amend the condition of the Christian clergy and oblige them to him. He endeavored the reducing of Christianity into one uniform doctrine: he assembled the Council at Nicaea and there framed a confession of faith and by new honor gave great luster to the church and ensured a secular power everywhere by advancing the ecclesiastical and that of the Christian bishops. These were spies and checks to his governors, and since Rome and Alexandria were the two places that had most influence upon his empire, he and his successors advanced those prelates to a kind of princely dignity that they might gain the greater veneration among the people and equal the splendor of the pagan priests. Then began temples to be dedicated with as much solemnity by the Christians as ever anywhere by the pagans and entitled to the apostles, martyrs, and angels for magnificence and largeness. They were equal to those of the heathens and, as in the fabrics and dedication of the churches, the resemblance of paganism was introduced. So the ecclesiastical government was made parallel to it. You read now not only of temples and altars, but the bishops, _sacerdotes, antistites, sacræ legis_ , et cetera. As the heathenish religion was supported by a priesthood under the _Pontifix Maximus_ and his college, consisting of the _provinciarum sacerdotes_ ( _Asiarchæ, Syriarchæ, et cetera_ ) which were also called _sacerdotes_ , the _flamens_ (whose power exceeded not their city or town), the _ministri, prefecti_ , and _hierophantæ agrorum_ which attended in country villages—just such was the reglement of the Christian Church and the jurisdiction in a manner equal.
Thus was the empire to be balanced, but, withal, the Christian emperors strengthened themselves by favoring the Jews, whose aversion from idolatry was as great or greater than that of the Christians. The interest they had in Persia did preserve them still in great splendor, notwithstanding the desolation which Hadrian had brought upon them. And they are a sort of people that so adhere and support each other that they are not to be destroyed. They began to spread again, and Christian emperors gave them not only freedom of religion but permitted their patriarch to gather his _aurum coronarium_ in the East and **< 34>** Western empires and to live with very great pomp and command. The rulers of the synagogues, lesser patriarchs, presbyters, and others concerned in the government of that nation, were exempted from civil and personal duties and employments by the decrees of Constantine, Constantius, Valentinian, Valens, Theodosius, and Arcadius. They were not only freed from affronts and contumelies, their synagogues protected, but the same respect paid to their rulers was shown to the Christian bishops and hierarchy (thus Arcadius, _Codex Theodosianus_ , 16. 8. 13):
Judaei fuerint obstricti ceremoniis suis: nos interea in conservandis eorum privilegiis veteres imitemur, quorum sanctionibus defi-nitum est, ut privilegia his, qui inlustrium patriarcharum ditioni subjecti sunt (archisynagogis patriarchisque ac presbyteris ceterisque, qui in eius religionis sacramento versantur, nutu nostri numinis preseverent ea, quæ venerandae Christianæ legis primis clericis Sanctimonia deferuntur. Id enim & divi principes Constantinus, Constantius, Valentinianus & Valens, divino arbitrio decreverunt—(AD 397).
Afterwards Theodosius the Younger confirmed their privileges, AD 412. But by reason of several enormities and misdemeanors which fell out by reason of their grandeur, and for reasons of state, the patriarchship was abrogated, AD 414. Yet the particular patriarchs and rulers of the synagogues gather for their own use the said pension of _aurum coronarium_ till it was annexed to the imperial exchequer by Theodosius the Younger, AD 429. In this flourishing condition did the Jews spread themselves in great numbers under a regular government over the Eastern and Western empire. Both Palestines and Egypt, Arabia, Babylon, and Persia were replenished with them, their rabbis flourished, their Talmuds and Targums were compiled about this time, and Judaism reduced to that system in which we at present find it in the Empire.
Now we see those grand religions all in a flourishing and powerful condition. The pagans whose interest was continually undermined by the emperors and imperial governors in several provinces and by the Jews and Christians: whatsoever laws were made in favor of them and for their liberty were either directly **< 35>** violated or pretenses daily sought to give color to that injustice. And whether they, being driven to despair, might entertain foreign correspondency is to me unknown but not improbable. It is most certain that they fomented discord among the Christians and on all occasions (as Athanasius particularly relates of the contests at Alexandria) they did abet the Arians and adhere unto them openly; their philosophers did dispute for the Arians in the Council of Nicaea. And as Alexandria had been the seat of the ethnic learning—philosophy natural, moral, and political, physic and mathematics being there most eminently professed and taught—so from hence they were propagated over the Eastern Empire. And the philosophers there did insensibly engage their Christian scholars into several heterodoxies according as their genius inclined each of them, to Platonism or Peripateticism.
It is most certain that the Arians, and all the subdivisions of that numerous sect were profoundly learned in those sciences, and that Origen derived his knowledge from an education under them and the benefit of their libraries there. The Christians had great encouragements and immunities et cetera to support them, and great privileges were enacted for such as turned to that religion and penalties frequently decreased and oftentimes rigorously inflicted on the pagans. But what contributed much to the prejudice of Christianity was their divisions among themselves. Besides the petty sects occasioned by pure ignorance, folly, or madness (of which kinds the catalogues of heretics do present us with many) which were easily extinguished by the imperial power or fell of themselves. There were three potent sects which did give a great check to the more facile and complacent Christianity of the Empire: the Donatists possessed in a manner all Africa and had some hold in Italy; the Arians possessed in great part the Eastern Empire; and the Novatians were with great repute for purity and piety diffused everywhere.
These were all settled under their episcopal reglements with distinct churches. The Donatists began in Africa upon this occasion: after that persecution ceased against the Christians, those whom either zeal or passion had made obstinate sufferers for Christianity detested and refused communion with such as either had delivered up their Bibles and holy writers up to the pagan emperors in obedience to their decrees (these were called _traditores_ ), or had exempted themselves from danger by paying a sum of money (which were called _libellatici_ ), or had offered incense and complied for the season with prevailing and persecuting paganism (which were **< 36>** called _thurificatores_ ). And a bishop who either had or was said to have delivered up his Bible, being surreptitiously chosen and ordained at Carthage for that city, the other bishops partly in vindication of their rights which the ecclesiastical constitutions had vested them in, partly out of zeal against such a betrayer of Christianity, declined his communion, excommunicated all that adhered unto him, and ordained another bishop of the place. The people generally adhered to the Donatists. Great was the schism, and two hundred and thirty bishops owned that party, whilst the bishop of Rome and the Italian clergy (who had been universally involved in the like compliance) did adhere to the other, as also did the imperialists at finding that party more pliable and suitable to their political ends and inclined to such a lax discipline that the Gentiles might upon easy terms be admitted to their churches.
The Donatists, exasperated hereby and finding no favor (or rather as they said no justice) from Constantine the Great, sought and found protection in Julian the Apostate by reason of their submissive and invidious address. After Julian the succeeding emperor did oppose and oppress them rigorously, fining them, confiscating their goods, estreating their churches, banishing many, and in a manner outlawing them. But as the resolution with which they suffered did ingratiate them with the populace (who are prone to think they are unjustly oppressed who bear their punishments bravely and with pretense to martyrdom for religion), so the arguments with which they defended their cause being very conformable to the strict Christianity professed in those parts and, rebaptizing those of that side which were converted to them, did imprint in the people every way an opinion or suspicion that the imperialists were scarcely to be reckoned among the number of Christians. The Donatists continued till after the days of Honorius, and, being reduced to great distress by their persecution, it is not to be wondered if many outrages were committed by those who saw themselves, their families and relations, utterly undone by such as, contrary to the general tenets of the Christians, employed force against them. And as it is usual with men in despair, and as Africans disposed to revenge to join with any third party, it is no wonder the conquest of Africa proved easy to the Goths, who were afterwards ejected after a long and fierce war by Justinian, and the Arians no less oppressed than the Donatists before.
And so I leave Africa in a fair way to comply with the designs of Mahomet, the constant wars having made them valiant and bold, their humor malicious, their sufferings angry, and their religion indifferent to any novelty. For during those contraventions, for want of instruction by their own clergy or the **< 37>** imperialists who minded little but ease and ambition (and were extremely ignorant), it is easy to be imagined that they retained only a sense of some generals in Christianity and had nothing of ecclesiastical government or order amongst them: no sacraments, no not of baptism, except they conformed, but this notion so whispered and insinuated into them that the conformists were not truly Christians that it seems to have been the great article of their creed.
The Novatians had their origin from the like occasion. All the Italian clergy, except Novatus and two more (as I remember) had consented to idolatry in the time of Diocletian and Maximinus and, upon the change of condition afterwards, returned with their followers to Christianity. This act of the Italian clergy (the bishop of Rome being involved in it) gave a beginning to the complacent and indulgent Christians of those ages, whereby the discipline of the church was ruined and men admitted and readmitted upon easy terms. Novatus and his adherents who had not apostatized would not communicate with such nor readmit them upon any penance: not that they thought it impossible for such to be saved, but that it was not in the power of the church upon any terms of repentance to associate with them, according to that of the Apostle (Hebrews 6:4–6). These were the Puritans, as I may call them, of those ages: they were men of a strict life and withal of a peaceable disposition. They were orthodox in their judgment about the Trinity which made those that adhered to the Nicene Council to show them great respect, to own their bishops and protect them, who seem of all sects alone not to have intermeddled with the public affairs and revolutions and acquiesced. So in a toleration as never that I know to have endeavored aright to the prejudice of other sects, or sinisterly to advance their own party, they continued till the ruin of the Eastern Empire. And their successions are recorded in our church history, and their oppression under the bishop of Rome is condemned by Socrates. Although this sect did raise no faction in the Empire, yet it is easy to conjecture that their continuance in open schism, the grounds whereof were known and asserted publicly in the Eastern and Western Empire everywhere, and their exemplary lives (in which whatever there was of innocence and true zeal in primitive Christianity seemed then to be lodged and preserved) must needs have added to the contempt of the imperial religion, as if it were corrupted in its discipline and purity, and so far have strengthened the dissenting parties.
The Arians were so powerful a sect in the Empire that the followers of **< 38>** the Nicene Council were not equal to them either in number, splendor, interest, or riches, if you will believe the learned Petavius and others. They did offer to be tried by the Fathers that preceded the Nicene Council. At the Nicene Council they were rather condemned by a party there by the general consent of the Christian Church. For Constantine, out of above 2,000 bishops assembled, excluded all but 318, nor were there perhaps (for accounts vary) all bishops that made the council. They were all of a judgment at first and so parties rather than judges. The Arians had not the freedom to dispute their case, and the emperor Constantine was so little satisfied with their prescription that he recalled Arius soon and, a little before his death, was baptized by an Arian bishop. Constantine and Valens were professed Arians, not to mention the Goths. Valentinian, Theodosius, and other emperors protected and honored them with military and civil commands. Their doctrine was not only confirmed by 8 councils, which were at divers times assembled at Tyre, at Sardis, at Syrinium, where 660 bishops were of their opinion—but three of name which held the contrary, but that they also punished others, their adversaries, who were of a contrary opinion to them with confiscations, banishments, and other grievous punishments. Whether the power of their chieftains, the riches of their churches, the magnificence of their worship (they first brought music into the church), the fame of their learning and pretensions to reason, which is always an invidious plea, did raise jealousy in the emperors and hatred against them in the Trinitarians, or what most contributed to their first depression and persecution I know not. Since to persecute for religion was by the Trinitarians—Athanasius, Hilary, and others—then accounted an Arian and unchristian tenet, it is not to be doubted, but after the days of Theodosius the reason of state did prevail most toward their subversion, lest they should join with the Goths who possessed themselves of Italy, Spain, Africa, and other provinces. And terrible to the Byzantine Empire, whatsoever it was, it is easy to comprehend that the depression of them did facilitate the conquest of the Goths. And if you will credit Salvian, the Goths were very pious in their way, mild to the conquered, just in their dealings, so that the wickedness of the Christian rulers of provinces, their depredations upon the people, and the insolence of the foreign soldiers by which they ruled, made even the Trinitarians willingly submit to the dominion of the Goths and prefer it before that of the Eastern Empire.
I come now to the Trinitarians **< 39>** who I cannot but represent as enemies to all human learning. They had canons forbidding them to read any ethnic book, and a pique which disposed them to destroy all they met with of that kind. Thus we may suppose them universally ignorant, except some few who appear to be somewhat knowing; and as were the pastors so were the people. Their religion consisted rather in an outside service than inward piety or knowledge; their faith was in a manner implicit, the mysteries of religion (such I call the doctrine of the Trinity and its dependencies) were scarce ever mentioned to them in sermons, much less explicated. Hence the vulgar became prone to embrace superstition, to credit miracles how ridiculous or fabulous so ever. Visions, allegories, allusions to texts were convincing arguments and no demonstration like a feigned story and legend or what might be interpreted a judgment upon a heretic. As to the imperial courts, I know not well what religion to install them into. They did long wear the habit of Roman _pontifices maximi_ and, after Gratian and his successor had laid that aside, they exercised their power. You will find a hundred times in the Theodosian and Justinian codes that they assumed the titles of _nostrum numen, aeternitas, perennitas_ and that they made their predecessors deceasing to be reputed _divi_. They continued the Circensian games, the obscenities of the theater and scenical women, with a multitude of other idolatrous and even brutal practices for which one would be astounded to read laws made in the Theodosian Code.
As to matters of religion, the emperors enacted what they pleased about it and imposed it on the generality. For synodical decrees did not bind; others then were willing or present and consenting in those days. And you may meet with accounts of the Christian faith in imperial edicts enjoined to be believed by Theodosius as well as Justinian so that I may reckon amongst the Trinitarians a sort of people who were of the court religion and believed as their prince ordained, living unconfined by the dictates of the then declining church. The Trinitarians, though they had resolved upon and subscribed unto the Nicene Council and embraced those forms of speech which are now in use, yet they could not tell what was meant by them: the Latin Church allowed not of three persons but of three hypostases, the Greek Church approved of three hypostases not of three persons, and difficult it was for them to explicate _ousia_ or essence. These hard words produced a subdivision amongst them, consisting of Nestorians and Eutychians: the Nestorians believing the divinity **< 40>** of Christ held that he was made up of two different persons and so perfect God and perfect man; the Eutychians averred that Christ had but one nature and that upon the hypostatic union the deity and humanity were so blended together by confusion of properties and substances that one person endowed with one will did emerge thence.
Those two sects were of great power in the Eastern Church, and though they were both condemned in the third and fourth general councils, yet did they spread far and near throughout Palestine, Egypt, the Kingdom of Abyssinia, and all Persia. Each of them had their patriarchs, bishops, churches, contradistinct from the Melkites. And now we see the face of Christianity thus represented: those that adhered to the Council of Chalcedon and subscribed to it, as did all the imperial clergy, were called Melkites (that is to say men of the king's religion) by the Nestorians and the Eutychians. The authors of these two sects were learned men and potent bishops: Eutychius was patriarch of Constantinople, and with him joined Dioscorus, patriarch of Alexandria, and Severus, patriarch of Antioch, and Jacobus Baradeus from whom the Jacobites are denominated at this day. Nestorius was also patriarch of Constantinople, and his sect very much diffused.
The truth is: such was the ignorance of the people, and such the debauchery of the ages then, that if a man did but live a pious, strict life, with great mortifications and outward devotion, and were but an eloquent preacher, he might in any place of the Eastern Empire make a potent sect instantly. And to show how ignorant the clergy were in the general Council of Chalcedon in the time of Marcianus the emperor, take notice that at that time the Greek tongue was so well understood at Rome, and the Latin in Greece, that the bishops of both countries, which were 630, were glad to speak by interpreters. Yea, in the very same Council of Chalcedon, the emperor made one speech in Greek for the one part, and another in Latin for the other, the matter of both being the same. The Council of Jerusalem made certain creeds both in Greek and Latin. The pope's legates at the Council of Ephesus had their interpreter to expound the words, and Celestine's letters were there read. The Acts tell us how the bishops desired they should be translated into Greek and read over again, insomuch that the Romish legates had almost made a controversy of it, fearing lest they should prejudice the papal dignity by such an act and alleging therefore how it was the ancient custom to propose the bulls **< 41>** of the See Apostolic in Latin only and that might now suffice—whereupon those poor Greek bishops were in danger not to have understood the pope's Latin.
But the legates were at length content with reason. It was evidenced to them that the major part could not understand a word of Latin. But the prettiest of all is Pope Celestine's excuse to Nestorius for his so long delaying answering his letters, the ground being this: that he could not by any means get his Greek construed any sooner. Pope Gregory the First ingeniously confesses to the bishop of Thessaly that he understood not a jot of his Greek. It is very probable that the proverb of honest Accursius was even then in use: "Græcum est; non legitur."
This was the condition of Christianity in which Justinian the emperor found it about the year 540. He, by conquest in Africa, subdued the Arian Goths and Vandals there as also in Italy, established and enforced the Trinitarian religion by severe laws, suppressed all the different sects and religions in the empire, abusing the Jews, suppressing the Arians and all other heretics (except it be true that he favored the Eutychians). He reigned thirty-nine years. After him succeeded Valentinus, Justinius II, Tiberius, Mauritius, Phocas, and Heraclius, in whose days arose Mahomet. It is observable that all these times were so corrupt and Christianity so depraved that the Church of England and generally the Protestants reject the authority of them and admit no general councils after that of Chalcedon under the Emperor Martianus. Their reigns suggest nothing considerable to the subsequent discourse.
But that Christianity was then degenerated into such a kind of paganism as wanted nothing but the ancient sacrifices and professed polytheism, and, even as to the latter, there wanted not some who did make three gods of the Trinity. Others made a goddess of the Virgin Mary. The reverence to the saints differed little from that of the pagans to their heroes and lesser gods, and images were brought into churches then, though not by public authority. And it is no less remarkable that obscure persons had several times been promoted by fraud or indifferent means to the Empire. Also the Emperor Mauritius, having reigned long and gained much upon the esteem of his own subjects and on the Persians, to whose King Chosroes he had married his daughter, and he had thereupon turned Christian, was barbarously murdered by a conspiracy of Phocas and the bishop of Rome, the last being incensed **< 42>** against Mauritius because he had permitted John, bishop of Constantinople, to assume the title of oeucumenical bishop, which title the pope declared to be antichristian. But Phocas, being made emperor, returned this acknowledgment to the bishop of Rome (who had solemnly owned him and given him repute) that he should be head of the church and universal bishop, AD 612. Chosroes, exasperated at the death of his father-in-law, who together with his wife and children were cruelly butchered, and abominating the Christians whose great prelate should countenance such an act, he renounced Christianity, destroyed all the Christians in his country who would not renounce the Melkites and turn Nestorians, which gave an occasion to Nestorianism to spread itself far and near in Persia and those oriental countries. Their patriarch resided in Mesopotamia at _Musall_ , or _Mawsell_ , which is supposed to be built nigh the old city of Nineveh. He invaded Syria and Palestine, sacked Antioch and Jerusalem, carried away multitudes captive into Persia, and excited the Jews to a rebellion in Palestine. Phocas, having reigned eight years, was slain by Heraclius who made himself to be chosen emperor by the soldiers. Chosroes, having disobliged his subjects in turning Christians and being afterwards unfortunate in his wars, was deposed by his son Syroes and murdered by him. He lived but a year and was succeeded by Hormisdas.
Whilst the Grecian Empire was thus unsettled by the frequent change of emperors, and the detestable means by which Phocas had gained the throne, having much alienated many from the love of Christianity, and the Eastern Church was divided into factions by the means of the bishop of Rome promoting his new authority there, and the Nestorians and Jacobites or Eutychians multiplying under their several patriarchs to the great disturbance of the church, anathematizing and being anathematized; and whilst that Persia was broken by intestine divisions and wars, and the people indifferent as to their princes, who should rule them, and divided by the mixture of Jews and Christians spread among them in great numbers everywhere, Mahomet arose and began the empire of the Saracens and a new religion.
It may perhaps seem strange that the general description of the primitive Christians which is here represented should differ so much from the usual accounts thereof which are given by the divines and vulgar historians. But in answer hereunto, I desire the reader to consider first the grounds and **< 43>** proofs which I go upon, and if the authors be good, the citations true and indisputable, if the progress of Christianity be such as is conformable to the constant course of human affairs and great revolutions, that then he would not oppose me, by discourses of miraculous accidents, unimaginable effusions of the Holy Ghost, and such like harangues as no reason can comprehend nor example parallel. Secondly: let him consider that since we are destitute of any solid chronicles or Christian annals before the days of Constantine the Great (for so Eusebius saith), and that he could find nothing substantial to establish an ecclesiastical history upon except the Acts of the Apostles, and that whatsoever is alleged against me must be out of suspected or spurious writers, partial in their own case and ignorant either for the want of learning or want of books and opportunities to be informed aright, or a prejudiced opinion blinding their judgments, I conceive the credit of what I write ought to seem most valid, because it is consonant to the Acts of the Apostles and the real existence of things.
It ought to seem a complete refutation of them all: that if what I say be true as undoubtedly it is, the contrary must be false. If it be further urged that the apologies of the ancient Christians are inconsistent with the relations I make, and that so great a deformity in the tenets and practices of the first Christians can never be reconciled to what they say, I answer that the apologies ought to be looked upon no otherwise than as rhetorical pleas and the defenses of advocates for their clients, wherein all things are managed as much to the advantage of the defendant as it is possible, and that it is most evident that this course was taken by those Christian Fathers. Neither indeed do I want testimonies of their imprudence whereby to satisfy any man that they would not scruple at palpable untruths if they might derive any benefit from thence. Justin Martyr in his apology to Antoninus Pius the emperor (not to mention that Irenæus, Eusebius, and Tertullian do relate the same) doth aver that Simon Magus did miracles at Rome and that he had a statue erected to him in the reign of Claudius Caesar with this inscription, "Simoni Deo Sancto," of which relation there is not a word true. And was not this most impudently done to obtrude a narration upon the Roman emperor which all Rome must know to be false? And even Justin who lived at Rome must have understood it **< 44>** to be so.
So Apollinaris, bishop of Hierapolis, and Tertullian, in their apologies, say that the emperor Marcus Antoninus, being reduced to great straits for want of water in Germany, and in danger to be taken thereupon by his enemies, the Christian legion called Legio Fulminea, did obtain such a return from God to their prayers, that at the same time a plentiful shower supplied the Roman army, whilst thunder and lightning destroyed the hostile Germans; whereupon the said emperor should write to the senate to decree that the Christians should not be molested, that any man might turn Christian, and that no governor of any province should divert any man or turn him from being a Christian. All which story is a mere forgery, as Vossius hath demonstrated at large and most imprudently urged to those who were certain of the contrary. I shall instance but in one more case and that is of the Sybils, whom Justin Martyr allegeth in favor of Christianity, as also doth Constantine in this oration in behalf of it. Yet are those Sibylline prophesies supposititious and feigned by the Christians as Casaubon, Blondel, Valesius, and others do acknowledge. A thousand such pious frauds might be instanced in, but that many learned men within this last century have saved me the trouble of prosecuting any such discourse.
I must add that the church history of the primitive times seems mainly deduced from the Latin and Greek writers who give no account either of the Syriac or Judaizing churches so that we hear no news of the latter till St. Jerome and Epiphanius came to represent them as heretics for adhering to the same doctrine and discipline which St. Peter and St. James and all the apostles (except Paul) had instructed them in, and wherein they had not been controlled during the lives of those who first founded the church. What... authority had power to do it, or how doth it appear that they had either corrupted their tenets or depraved their gospel, the Jews being so tenacious of tradition, and those being men that pursued no designs or worldly interests besides, were they ever heard by an indifferent judge or general council of all Christendom? No affairs were in so unlikely a posture as to that matter amongst the Greek and Latin Christians that, if they had been convened legally and fairly, not one of their adversaries could have understood what they said or judged of their allegations otherwise than according to their own prejudiced opinions.
But to speak more closely: it is evident that the first Christian Fathers when they would magnify the number of their converts or adherents would not only bring into their catalogue such pagans as opposed idolatry, though they no way pretended unto **< 45>** Christ and Christianity, but even all such as did profess it in any way and under how great variety so ever of rites or tenets—just as the Jews did reckon the Sadducees, Pharisees, and Essenes and other lesser sects in the number of Jews and as the modern Christians do compute in the Eastern Church, the Grecians, Melkites or Syrians, Georgians or Muscovites, Nestorians, Indians or Christians of St. Thomas, Jacobites, Cophites, Armenians, Abyssinians, and Maronites, and in the west Papists, Calvinists, Lutherans, Anabaptists, Socinians, and such like even to the inhabitants of Liefland where they would make an estimate of Christianity and its extent. Nor doth there want reason for this procedure, for as we denominate men and estimate those to be of human race who have the general resemblance, propagation, speech, and laughter, how different so ever their morality and rationality be: so in Christianity the external profession in general doth entitle them unto the appellation. And if all persons grossly ignorant—demi-Jews, demi-pagans, demi-philosophers, and such like as posterity has concluded under heresy—may not come under this notion, I do not know where to find the primitive church or Fathers. Since most if not all the Fathers of the three first centuries come under this number and the martyrs were no others, it is a caution suggested to us by the learned Casaubon and subscribed unto by all men of understanding that the Fathers do most frequently mistake in point of history as well those of the Greek as Latin Church (either through ignorance and inadvertency or because they inconsiderately need use of any story that seemed to make for their advantage in their sermons or writings). And that they deserve as little credit in reference to matters of faith is apparent by writings of Mr. Dale, Hottinger, et cetera, the sole consideration whereof made the judicious Protestants reject their authority and pay them but a precarious veneration. As to the subsequent condition of Christianity, that which we seem descended from is no other than a mixture of the religion of the Essenes, the Jewish heretics, and of the Egyptian _therapeutæ_ which likewise were Jews, but not Essenes, together with the superadded tenet of Jesus being the Messiah and of other doctrines derived from the Gentile philosophy, and of certain paganical rites and ceremonies.
It is apparent that Eusebius and Epiphanius reckon upon these Essenes and _therapeutæ_ as Christians, disciples of St. Mark, and the great ornament of the profession. Monkery deduces its original from them, and their lives were the patterns and precedents of those Christians who are renowned for the austere practice of piety, which the first apologies represent. It is apparent that the first Christians **< 46>** from whom we are derived were Alexandrian proselytes which, retaining the name of Jews, did adhere unto the Septuagint. And hereupon these Christians whether of the Latin or Greek Church had no other Bible than that, or a version of it, and from hence they read those books which after ages called apocryphal. Neither had our gospel any other original than from Alexandrians or other Hellenists. It is confessed that Luke was of Alexandria; that Mark also was such is most probable. And, as to the others, it is evident that either they are but versions or not to be entitled to any other beginning than that some Hellenists published them in the name of Matthew and John. For Matthew may certainly be said to have written in the common Syriac which was then the Hebrew tongue. The Gospel of Mark is thought to have been dictated by Peter and only translated by him into Greek as most of the ancients inform us. And the apostles did not understand Greek at all: the gift of tongues lasted but for that time, and all the sacred books except what bears the name of Paul (who did understand Greek) are but translations or counterfeits performed by unknown persons whose fidelity or integrity those questioned who rejected them or who embraced them.
But it is manifest that they are written by most illiterate persons. As to the Greek tongue,
Pro vero sane tenendum est, omnes fere discipulos Christi & Apostolos, ut erant idiotæ & plebaei, piscatores nimirum, nautæ & portitores, non aliam novisse linguam praeter vernaculam, hoc est Galilaeum, & Syriacum idioma quod in illa regione obtinebat. Etsi enim multi in Syria & Iudæa Græcè sciebant, hoc ad infimæ plebis homines nihil attinebat qui vernaculam tantum noverant, Græcè prorsus ignari.... Scribebant igitur Apostoli idiomate suo, & lingua sibi familiari & vernacula, quæ protinus à Syris ἑλληνίζουσι vel Graecis ipsis ad fidem conversis, quos cecum habebant Euangelii praerdicandi adjutores & administros, in Græcum transferebantur. De quibusdam hoc certò' compertum est, de aliis ignoratum, quia non proditum: de omnibus tamen verisimile est, quia de quibusdam verum est. Non enim disparatio eorum, qui gente ac genere & vocatione ac munere pares.... Quod ad Novi Testamenti libros attinet, ea causa quoque asseri potest cur multum diverso ab elegantiore & puriori Hellenisimi loquendi genere conscripti sunt. Ab idiotis quippe partim compositos dicere licet, partim à metaphrastis, & ipsis non ad modum Græci sermonis peritis.
I have transcribed this passage out of Salmasius (whose entire discourse upon the Sixth Question there deserves to be read). Not that I believe that either the **< 47>** Gospel of Mark or John were penned originally in Syriac (for then the Judaizing Christians would have had them as well as those of Matthew), or that the Epistles of Peter, James, and John (or that the Hebrews which is but a translation said to be made by Clement or some other) were ever penned in that language, which the supposed authors only knew, but to evidence how little certainty we have of their original and authenticness. And if they were not derived from those Hellenistical Jews, I know as little whence to fetch them as when they were written. It is true that Paul did understand Greek, but his epistles were as little regarded as his person amongst the Judaizing Christians. And they had as bad an opinion of him as the Jews themselves. It appears by Paul's carriage (Acts 23:6) that he did act by somewhat like to juggling in his proceedings. How else could he cry Christianity assuring each of them singly that he was in the truth and that afterwards, when Paul was dead, each of them pretended his religion to be the true religion derived from Paul, whence arose great feuds amongst them.
To pass from this discourse to that other concerning the ignorance of the first Christians and their enmity to all ethnic learning: It appears that in the days of Christ none but the vulgar sort (the Galileans were the worst among the Jews) did believe in him. The wise, the rulers, were such as the truth of the gospel was hidden from. And in the days of Paul, not many wise men after the flesh, not many mighty, not many noble were called, but the foolish things of the world were chosen, 1 Corinthians 1:26. And truly afterwards, till near the time of Constantine, very few of any better rank or intellectuals did embrace Christianity, which made the heathens upbraid the Christians as men that gained only on children, women, and the poorer and more ignorant sort of persons. Nor did they pretend to learning as appears out of Lactantius, Annobius, Minucius Felix. Nay, they were enemies to all human learning, as appears by the old constitution of Clement whereby all the books of the Gentiles are prohibited (Clement Romanus l: 1 Constit. caput 6). And the Council of Carthage did prohibit the clergy to read any such books, et cetera.
What I have said is notoriously true as to this point and needs no further proof except that I illustrate the condition of Christianity by that passage of **< 48>** Avicenna who, being to relate the nature of medicinal simples according to the Greek Alphabet, says that he follows _Alphabeta Barbarorum_.
I proceed now to the particular narration of the birth and actions of Mahomet whose rise, that we may the better understand, it is necessary that we consider the situation of Arabia and search into the original of the Saracens, a nation not mentioned by the ancient Greeks or Romans, and of whom there is no account given by the Christians contemporary to Mahomet.
_CHAPTER 3_
A BREIF ACCOUNT OF ARABIA AND THE SARACENS
_The History of the Saracens and of Mahomet_
HE BETTER TO UNDERSTAND this particular history of the Saracens and Hagarenes, it is necessary we make a relation of the Arabians in general. For the general situation, constitution, and religion thereof had an influence upon the Mahometan revolution, and the whole religion of Arabia was interposed in the production of it. The Arabians received their denomination not from Arabus, a son of Apollo, as the Latins imagine, but from Araba, one of the provinces of what is vulgarly called Arabia, situated near Medina where it is thought Ismael did first seat himself. But this Araba or Arabum or Arabia usually includes all that peninsula which was inhabited by the Arabians. These divide their country into several kingdoms whose names being unknown to the European geographers I shall forbear to use as much as I can and shall acquiesce in the vulgar distinction of it into Arabia Fælix, Petra, and Deserta.
It is of great compass and extent, having on the west the Red Sea for its bounds, on the south the ocean, on the east the Persian Gulf, and on the north Syria and the River Euphrates. It is as large or larger than Spain, France, Germany, and Italy. Pliny reckons a great part of Mesopotamia within the precincts of Arabia and the Arabians to be styled Syrians. And, as to this last, there is this ground for it that Arabian _Salihenses_ , called _Al Dajannuini_ , did conquer Syria. And a second time that province was reduced under the obedience and possession of the Ghassanian Arabs which came out of Yemen, or Arabia Fælix, and were of the tribe of Azdenses. And Aretas (whose lieutenant at Damascus sought to apprehend Paul, 2 Corinthians 11:32) must probably have been the first or one of their kings, if Christian and Saracen chronologers agree.
Arabia Petra (so called from Petra the chief city) is bounded on the west **< 49>** by the inmost nook of the Red Sea and Egypt, on the north by Palestine and Cælo-Syria, on the east by Arabia Deserta, on the south by a track of mountains which disjoins it from Arabia Fælix. This country Pliny, Strabo, and Ptolemy call Nabatea the Desert. Arabia Deserta is bounded on the west by Petra Cælo-Syria, on the north by Euphrates, on the east by sundry mountains which divide it from the country of Babylon, on the south also it is severed from Arabia Fælix by a ridge of mountains. As for Arabia Fælix, it runs out like a peninsula between the Arabian Gulf (or Red Sea) and the Gulf of Persia. It is the largest of the three divisions and said to contain in compass 3,504 miles. It is called by Solinus and others Ayman, by the Arabians, Yemen.
What share Arabia had in the Chaldean monarchy I know not, nor whether it were ever all reduced under the obedience of one sovereign. Most certain it is that the Arabians were divided into tribes and were as exact in preserving their genealogies and marrying in their own tribes as ever were the Jews. And as to their ancient religion and learning: though the accounts thereof be but slender, they not having had the use of writing and letters till or little before the birth of Mahomet, yet it may be conjectured that the astronomy, astrology, and other knowledge of the Persian Magi and Chaldeans was derived originally from them. There want not some who essay to prove this, but to insist thereon would not be to our purpose. Their language at first seems to have been little different from the Hebrew or at least Syriac, until one Yaarab introduced the Arabic.
Of the Arabians there are said to be two sects; the pure Arabians who are said to be descended from one Joktan or Kahtan (the son of Saleh, the son of Shem, the son of Noah) and the _Mosta-Arabs_ or denizened Arabians. For Ismael being ejected by Abraham came into Arabia, seated at Yathrib (since called Medina), and married into the tribe of Jorham, of the pure Arabians who lived in Yemen or Arabia Fælix, and from him descended the Coreischites and other _Mostaarabick_ tribes, who, notwithstanding, are not able to deduce their pedigrees with any certainty from Ismael, but from Adnanus who was of his race. Ismael marrying into the tribe of Jorham conformed himself to the language, manners, and way of living practiced by his new relations, and so he and his posterity became incorporated into the Arabians. And the tribes of the _Mostaarabes_ and Jorham did diffuse themselves into several provinces without retaining any national distinction but what arose from their genealogies or governments and language. For, as to this last, it is said that the progeny of Joktan or Jorhamide, did differ in dialect from the **< 50>** Ismaelites, the former using the Arabic dialect of Hamyar, which Ismael corrected and reformed into another called pure Arabism or the dialect of the Coreischites.
The progeny of Joktan had at first the preeminence and a kind of rule which they might well challenge not only upon the account that they received Ismael as a stranger, and his progeny were but _Mostaarabs_ , as because of their strength, number, and riches which arose from the fertility and opulence of their country, viz. Yemen, whereas the Ismaelites were possessed of Yathrib and the desert Arabia. The inhabitants of Yemen were called Sabei or Sabii from Saba the son of Yashab, the son of Yaarab, the son of Joktan, concerning whose religion it is necessary that I speak somewhat.
They avowed that their religion was exceeding ancient, that it was descended from Enoch (whom they call Edris) and Seth, and they pretended to have the books of Seth preserved from all antiquity down to them. This undoubtedly was the religion which Abraham professed when he was in Harran in Mesopotamia before God is said to have reclaimed him. This was the religion of the Nabathei, or inhabitants of Arabia Petra and of the Chaldeans, and the inhabitants of Harran or Carr in Mesopotamia. Nay, it was diffused over the face of the whole earth as Maimonides said, and as Abulfeda. It was the most ancient of all religions. They did believe there was but one great God, whom they called the Lord of Lords, and in their disputations they alleged most strong arguments for the unity of the Godhead. The chief God they called _Olla_ or _Alla taall_ , the highest or greatest God. Besides this chief God, they had other lesser gods to whom they attributed no intrinsic, essential, underived power, but only an efficacy communicated by the supreme deity, whereby men were intermediately influenced and ruled, whereupon they did adore them with a secondary, divine worship as mediators and intercessors for them. And this was their way of address to the great God: "I give myself to thy service; thou hast no companion but such as are in thy subjection, and thine is all that is devoted and offered to them."
It was their opinion that there could be no communication between the divine essence and man but by some intermediate beings, and that the pure invisible spiritual substances were employed to this end, and that in a subordination to their influences it was necessary that there should be other intermediate visible bodies. For those, some of the Sabii did feign _sacella_ or mansions: such are the gross bodies of the celestial planets (for those they principally addressed themselves to, though many had equal reverence for the fixed stars), which they imagined to be animated by those intelligences, as our bodies are by their **< 51>** souls. Upon this account, they observed diligently their houses and stations, rising, setting, conjunctions, oppositions, benevolent or malevolent aspects; they assigned to them days, nights, and hours; they ascribed unto these figures and shapes, subjected regions and particular persons to them. And agreeably hereto, they made prayers, incantations, and seals. As for example if, upon Saturn's day (which was their last day of the week denominating each day from the planets as we do), a man came to pray to him at the first hour having his seal on, made according to art and suitable to that planet, and he clothed in a fitting garb and made use of a convenient form of prayer: whatever he shall ask that is in the disposal of Saturn, it shall be granted to him; so for the rest of the planets whom they called lords and gods. And by this ascent from the creatures to the creator, from visible to invisible things, they did intermediately proceed to the intelligences and supreme deity.
And from hence arose the fabric of talismans of which there was such use among the Arabians and Egyptians, and whereto they attribute such power, and by which Apollonius Tyaneus is said (by the Christians) to have affected his miracles. Upon the same sentiments did others of the Sabii proceed, who yet went higher to erect statues and images to those lesser deities to intercede for them, they supposing it necessary for man (who is liable to so many contingent necessities) to have his mediator always ready that he might have recourse to him. And that these planetary bodies (the chapels of the glorious intelligences) were itinerant and moveable, sometimes rising, sometimes setting, sometimes continuing under/over hemisphere; whereupon they proposed to detain their influence and preserve their benevolent power by lodging it in some statue or image made of a metal and figure convenient to this or that planet; the days, hours, degrees, minutes, and all other circumstances being astrologically observed in order thereto, and such sigillations, prayers incantations, suffumigations, attire, et cetera, being used as are appropriate. Hereby they did apprehend they should always have their mediators ready to assist them; and these images they called vicegerents in reference to the celestial mediators by whose interposition a man was to propitiate their superior planets.
This is the sum of their religion and the foundation of all that idolatry which diffused itself thence into Chaldea, Egypt, and all parts of the world: the Chaldean discipline being the same with this, and the Persian magi having no other origin. Not to multiply my discourse into two many particulars, I shall in what is to come **< 52>** observe only what may conduce to my principal design:
The Sabii rejected the Jewish canon or sacred books and relied on those which they deduced from Edris, Seth, and other their patriarchs and prophets.
They say that Noah preached against their images and worship of mediators, and therefore they generally speak against him.
They say Abraham opposed their image worship and talismans, and therefore he was banished by them.
That many people in the Levitical Law were purposely enacted by Moses in opposition to the Sabii that the Israelites might not be ensnared in their ways.
They shave their heads close, abstain from the blood of animals, esteeming it the food of demons (though some want food thereon that they might contract an affinity and correspondence with them).
Though the principal place for their devotions was near the city of Harran or Carr, which Abulfeda calls the city of the Sabii, yet did they preserve a great esteem and reverence for the Caaba at Mecca. They kept sundry fasts, whereof one consisted of thirty days.
They continued in repute not only to the days of Gregory Nazianzen the inhabitants of whose diocese were generally Sabii, but even to the rise of Mahomet who often mentions them.
As to the inhabitants of Arabia Petra, I have already showed that they were of the religion of the Sabii. I shall only add that they were esteemed the meanest and most despicable tribes, insomuch that _Alnabat_ or _Nabateus_ in vulgar signified a mean and despicable person.
I come now to speak of the inhabitants of Arabia the desert which, as it had been the seat of Ismael, so the inhabitants pretended to a nobility even above those of Yemen. Of all the tribes of the Ismaelites, the Coreischites were the most illustrious, their military condition made them proud, and their poverty and inaccessibleness had hindered them from being conquered, so that their speech was more refined, and their glory less stained than that of the other provinces. But there were other grounds for their preeminence: for a Coreischite signified proverbially a gentleman of quality, as did _Alnabat_ an inferior person. And it is the general tenet of the Arabians that the Coreischites are the most noble of all the Arabian tribes, as Erpenius and Hottinger assure me, and notwithstanding the distinction of the pure Arabian and _Mostaarabs_ (which Abulfeda saith was the most received tenet of the Saracens of his time). Yet there want not good authors who believe the ancient tribes of the pure Arabians to have been extinct, and that all the Arabian tribes **< 53>** tribes recorded as in being, since the Saracen records are strangers to the country, some being called _Mota Arabs_ , the progeny of Joktan, others _Mostaarabs_ as coming from a remote place. Nay, they say that Joktan was of the race of Ismael and certain it is (they own) that none of them can deduce their genealogy beyond Adnanus, a descendant of Ismael.
If so, the Coreischites must have been the noblest tribe of all. However, it is certain that at the time of Mahomet they were the most illustrious tribe, for they were possessed of Mecca the metropolis of Arabia Deserta, and which the Arabians call the Mother of their Cities, and inhabited in the center of Arabia, and, which is more, had the keeping and were a kind of priests ( _aedili_ ) of the _AlCaaba, Caaba, Kabe_ or _Cabea_ , a temple reverenced universally by the Arabians and the chief place of their devotions. It is said that the tribe of Chozaah had the keeping of it once (they were of the number of the tribes of Yemen and forced by an inundation to alter their dwelling) and were esteemed thereupon the chief of the tribes, but that one Abu Gabshan sold it to one Kosa, a Coreischite, for a rundlet of wine. He repented heartily of his foolish bargain, which gave occasion to the Arabian proverb: "More vexed than Abu Gabshan."
The Caaba was so called either from its eminence and height or because it was a square building. It is also called _Albait Alharam_ , or the prohibited house of refuge. They report that Adam, being cast out of paradise, desired of God that he might build such a house as he had seen in heaven towards which he might pray, and which he might compass about in his devotions, as the angels and blessed spirits do about that celestial mansion that, thereupon, God sent down a glorious light embodied and shaped in the form and model which Adam desired, towards which he might pray and compass it about. After the decease of Adam, his son Seth did erect a fabric of clay and stone in that place, according to that model which, being destroyed by the deluge, Abraham and Ismael built a temple by the command of God in the same place, an angel showing them the ground and model. This temple was kept up by the succeeding Arabians, but while it was in the possession of the tribe of Coza, idols were first brought into it (also Hobalus, Asaphus, and Nayela) by King Amrus, who being of Yemen and a Sabian. That they were talismanic is no doubt. Afterwards Abu Corb Assad, a king of Yemen, did beautify it with stately curtains about seven hundred years before the birth of Mahomet. So great, so general was their veneration for this temple that when the tribe of Gatsan built another temple to themselves in imitation of the Caaba, the other tribes made war upon them for that and destroyed that schismatical edifice. **< 54>**
In the temple there were a multitude of idols undoubtedly accommodated to the superstition of every tribe. Within was Abraham's statue, attended with a multitude of angels and prophets, and on the outside were ranked 360 idols, whereof Hobal (which perhaps imports _Ha-ball_ , the great Baal or Bel), Asaf, and Nayela. Hobal was the chief, being a red statue shaped like a man holding seven unfeathered arrows in his hand; Asaf was also like a man; and Nayela like a woman. These two are said to have been turned into stones for having committed fornication together in the Caaba. Hither did all the tribes make pilgrimages and, in order to it, did solemnly devote themselves. Sometimes they went round the temple in a kind of procession and came betwixt Safa and Meriah (which are the same with Asaf and Nayela). This Safa was a blue stone at the foot of the mountain of Abikobais, and Meriah a great stone near the mountain of Koaikaban. They were distant from each other (about 780 cubits). They professed their reverence to the great god and some also to his associates as each particular person fancied—saying: "Thou hast no companion or associate besides—who is at thy disposal together with all that is devoted to him."
Thus they went through each station and offered their gifts and cast stones in certain places. And all the Arabians did agree to set certain months apart for this religious performance in which it was unlawful for any tribe to make war on the other or for any man to molest another. This they all did—except the tribes Tai and Cathaam and some of the race of Alhareth Eb. Caab. For those people made no pilgrimages nor reverenced the Caaba nor observed religiously any months or place as sacred.
This was the condition of Arabia, if we abstract from particular conquests and the mixtures of religion arising from the Jews who dwell among them, ever since the dispersion of Babylon, and the Christians whom either Peter converted to Christian Judaism or which fled thither upon several persecutions by the Roman emperors in Egypt and Syria. Of the Christians, it is notorious that the Nestorians and Jacobites or Eutychians were dispersed through those provinces, and the Arians were propagated that way since the generality of those provinces were infected from the Academy at Alexandria and the neighboring bishops. Nothing was more tenacious of their old rites than the Arabians and, withal, none more prone to admit of novel opinion under the specious color of religion. It is natural for men by cohabitation to infect each other with a mixture of devotion, as diseases are propagated **< 55>** by contagion, and we may believe that the Arabians, being possessed with an opinion of their being descended from Abraham and Ismael, did pay a great respect to the Jews who were spread through their country—yet not so as to relinquish their worship and pilgrimages to the Caaba. It was always a part of their religion to circumcise (as did the Cholchi and Egyptians) but not till thirteen years as Ismael was circumcised. About seven hundred years before Mahomet, those of Yemen did turn Jews, and about seventy years before Mahomet their King Du Nowas did compel all to Judaism and burnt the dissenters, many whereof were such mongrel Christians as that age usually produced.
Whereupon, to vindicate the Christians, the king of the Abyssinians invaded Yemen, conquered the country, and reigned there seventy-two years. The second of the Abyssinian princes or governments there, Abrahah Alashram, viceroy to the Negush, built a Christian church there to divert the people from their pilgrimages to the Caaba. But the generality of the inhabitants retained obstinately their ancient reverence for Mecca, and one of them, easing himself in this new church and otherwise defiling it, Abrahah swore he would destroy the Caaba. He marched with his army for that purpose directly for Mecca, but, when he approached Mecca, they say the elephant he rode on, called Mahud, kneeled down and refused to go forward, though he went cheerfully any other way: and withal a flock of birds carrying stones as big as peas in one their beak and two in their claws so weighty that they killed therewith his followers striking through helmets, men, and elephants so that the survivors relinquished their journey. And the Arabians, who usually computed their years from some remarkable accident, began a new account, from the year of the elephant.
After Mahomet was born, and not long before he begun to declare himself, the inhabitants of Yemen addressed themselves to the Greek emperor Heraclius for aid against the Abyssinians, but he refused it as being unsettled in his empire. Anusherwan, King of Persia, did relieve them and gained so much power in Yemen that he appointed them their kings out of the natives. It is to be noted that the Christian writers call him Cosroes whom the Arabians and Persians call Anusherwan, for Cosra or Cosroes was a common name to all the kings of Arabia Petra and Deserta.
I find they embraced Christianity in the time of Alnooman, the son of Almondar, and according to our ecclesiastical history under a queen called Mawia in the reign of Valentinian and Valens. And although they were Trinitarians, and in behalf of them made war upon the Arians, afterwards **< 56>** they turned Jacobites, and during the reign of Justinian they made war upon the Christians in their behalf. Upon the murder of the Emperor Mauritius, Cosroes, king of Persia, compelled all the Christians in his dominions (and Arabia was in a manner subject to him) to turn Nestorians upon pain of death, and erected them a patriarch at Mausell, so that we may believe the Arabian Christians or such as were fled thither at several times to have turned to that opinion. But, notwithstanding these revolutions, the reverence for the Caaba did in great part continue, it being incident to human nature oftentimes to turn their religion in obedience to their princes. But inveterate superstitions are not so soon exterminated as is the outward profession of religion, it being evident that most of the Coreischites retained their idolatry. Nor was the Caaba destroyed with its idols all this while.
_CHAPTER 4_
THE TRANSACTIONS FROM THE BIRTH OF MAHOMET TO HIS FLIGHT FROM MECCA
URING THOSE TRANSACTIONS, Mahomet was born about the year of Christ 580. Some place it in 570, others in 600, others in 620; but I follow the most probable account since it is generally agreed that he was forty years old in 620, at which time he began his prophesy. He was of the most noble tribe of the Coreischites. His father's name was Abdalla, his mother Amena or Emena, both of that tribe. He was born at Mecca; his father was curious to have his nativity calculated, and it was predicted that he should be exceedingly advanced by the propagation of a new law and monarchy. Not long after his birth, his father died, as did his mother when he was about six years old, whereupon his grandfather Abdolmutleb took care of him and he dying about two years after, his uncle Abutaleb undertook his education.
The Mahometans report several miracles which happened at his birth, but would be tedious to relate here. During his infancy he gave pregnant signs of a singular nature, great wit, and good behavior. His uncle gave him all the instruction the country could yield, which being then divided into sundry religions of the Christians, Trinitarians, Jacobites, Nestorians, Arians, Idolaters, and Jews, and those each as it were refracting other (as contrary elements do upon mixture), he was not ignorant of what each of those religions held. Abutaleb his uncle, having occasion to go to Jerusalem and not only to Damascus, carried him with him where he was more perfectly instructed in the principles and various sects of Christianity. As he went to **< 57>** Damascus, about Bosra, as soon as he saw him was astonished and seizing upon his hand said it was a prodigious youth, that, as he came along, he saw a cloud overshadow him and that the renown of that stripling should fill both eastern and western world.
Being returned from thence, as well educated as was possible for his uncle to breed him, and being educated to all the martial exercises of the Arabians, at twenty years old he gave great proofs of his valor and conduct in a fight between the Coreischites and the tribe Kais Ailan, which happened in the month Moharra, at what time it was unlawful for the Arabians to fight, it being the time of pilgrimage to the Caaba, whence the Arabians called it _dies sceleris_ , or the wicked day.
After this, his military genius not permitting him to live idle at home while many of his countrymen served in the Christian armies, he went into the field under his uncle Abubacr who commanded a brigade under the Christians. For albeit the Arabians were not tributaries or subjects to the Christians or Persians, yet their country lying between both empires, some of their princes confederated with one and some with the other, according as their situation made them obnoxious, and served either prince upon certain conditions. Here he accomplished himself in civil and military prudence and withal discovered the divisions and weaknesses of the Christians. During his being abroad in Syria, a noble lady being a widow (and who, for riches and birth, was courted by sundry Arabians, princes) falling in love with him (or directed by a vision as was pretended), invites him to relinquish the war and live with her, promising him a noble maintenance and to accommodate him in order to further travels with a large cargo that he might improve his intellectuals and estate together. He who thought (according to the opinion of his country) that merchandise might very well consist with nobility accepted of the overture. Nor is this more to his disparagement than it is to the nobles of Venice or Genoa; such were Vespasian, Pertinax, Tarquinius, Priscus among the Romans and even of late Spinola.
Being thus accommodated, he made several advantageous expeditions to Alexandria and other parts of Egypt. And curiosity or ambition prompting his great spirit, he undertook a voyage into Africa from whence he passed into Spain (AD 605), where he found the kingdom unsettled in religion. For the Goths, having been from the beginning of their Christianity zealous Arians, were forced to turn Trinitarians (AD 589) by their King Ricaredus. And it not being so easy to extirpate inveterate opinions as to alter the profession of them, the **< 58>** populace (and many others) retained their former sentiments. And after the death of Ricaredus, endeavors had been made (AD 603) to resettle Arianism. He finding it in this condition is said to have endeavored to instill into that nation some of his principles and had embroiled them. But the return of St. Isidore from Rome (whose esteem and power in that country was very great) enforced him to return the same way he came.
This voyage gave him an opportunity of seeing the weakness, the secret animosities, factions of the Christians, not only in Spain but Africa (where were the remains of the Donatists and Arian Vandals) and in Egypt. In his voyage I suppose he discovered the plantation and use of rice wherewith he acquainted his countrymen, recommending it to them as an excellent nourishing durable food: for which he was honored that it is still a tradition among them that Mahomet, being in paradise (before his assuming to be a prophet) and compassing the throne of God, fell into a sweat and one drop falling from him to the earth produced rice. Another became a rose.
After his return, Chadija married him. And it being the custom of the Arabians that the husband should endow the wife upon marriage, since his fortunes were not proportionable to the quality and riches of Chadija, I find that Abutaleb did make her a dowry of twenty camels and twelve ounces of gold, adding this speech:
Glory be to God who hath caused us to descend from Abraham and to be of the race of Ismael, and hath given us the holy land to possess, and the Caaba to keep where to pilgrims from all places resort; who hath also made us judges and rulers in our country. Mahomet, the son of Abdalla, my nephew, with whom none of the Coreischites can compare for virtue, bravery, glory, understanding, and wit, although his riches do not equal his birth and accomplishments (in truth riches are transitory as a shadow, and are lent to us by heaven so as to be recalled when Allah pleaseth) is passionately in love with Chadija, the daughter of Chowailed, and she likewise with him: whatsoever is demanded by way of dowry I will see it settled.
I think I follow the most probable story by placing Chadija in Syria, though she were a Coreischite. But it makes nothing to the prejudice of my narration, if Chadija be supposed to live in Mecca and there (upon a dream) fall in love with him and invite him, upon a large salary, to live with her and oversee her estate and conduct her merchandise into Syria and Egypt. The Arabians acknowledge the poverty of their prophet, and, for his being retained in her service, they plead that it hath often been the fortune of such as God hath designed **< 59>** for His prophets and the greatest dignities that they should arise from servitude to empire and by a long practiced obedience learn to command; that Joseph was a servant in Egypt, and Moses in Midian; that nobility is not extinguished by poverty; that Noah was a carpenter; that Isa (so they call Jesus Christ and so I shall name him in the subsequent story) followed the same trade. That since the nobility of his extraction is not questionable, it is malice and envy to upbraid him with this employment, as if it had been servile and mean; that this objection did not become the followers of Isa; that Mahomet had rendered eminent testimonies of his valor in the wars and might have been rich if he would have been less generous or more rapacious and extorting.
That whereas they said Mahomet was but a camel driver: there is nothing of contumely in that if we consider that wealth of the Arabians consisted much in camels; that the most illustrious and rich nobles of that country did usually attend their own business and drive or feed them themselves; that it is not imaginable he was the most inferior servant of Chadija, his warlike spirit would not have submitted to it, especially his uncle being of such quality and riches, or that Chadija would put him to any such inferior duties whom from the beginning she designed for her husband, dissembling only her passion for a time.
Mahomet, having added to the splendor of his family the riches and power of Chadija, entertained no mean thoughts of himself. She had an uncle called Warekeh or Varkah who understood Hebrew or Syriac well and could write the character. Him she did employ to teach Mahomet to write (for amongst the Coreischites scarce any could write at all). The invention of Arabic letters was then but new, one Moramer had found them out, one Ebn Mooklah three hundred years after Mahomet, and after him Ali Ebn Bozea perfected them as they are now. Of the Arabian tribe thence they were propagated to the Cendian tribe, and by one Bashar introduced at Mecca a little before the original of Mahomet's usurpation. Mahomet immediately acquainted himself with this new character, and it added much to his repute that he seemed ignorant of nothing. At Jerusalem he had been perfectly instructed in Christianity and had in sundry places conversed with the Nestorians, Jacobites, and Arians: by them he had been informed of the vanity of idols and talismans, **< 60>** of the unity of the Godhead, providence, virtue, and vice, the tenets of the Jews and Jewish Christians.
Thus he became very well versed in the scripture, the doctrine of the Old and New Testament, the traditions of the several sorts of Christians, especially of those Judaizing Churches which Peter had planted all along from Jerusalem to Babylon and all Mesopotamia. Nor is this all which I find written of him: for Rodericus Toletanus assures me that he was well acquainted with natural philosophy. In fine, the Arabians had such a repute for his universal knowledge that they believe he understood all things. Though the Arabians did much affect the glory of being eloquent and excellent poets, yet in those qualities did Mahomet surpass them all, not only in sublimity of thoughts, quaintness of speech, wittiness of his parables or apologies, but in choice of words and phrases. To all which as his great learning and education had much contributed, so it was an artifice of his to conceal those arts by which he had attained those excellencies. Being asked how he attained to so refined a language, rather than discover the means he had used, he told them that he had learned it from the Angel Gabriel who had taught him the dialect of Ismael himself.
Thus Mahomet gained upon the admiration and esteem of all men, and, by way of gratitude to his uncle Abutaleb, he contracted a particular friendship with young Ali his son and instructed him in writing and all manner of knowledge.
Ali was of a brown complexion, a little man with a belly somewhat large. He had a contempt for the world, its glory and pomp. He feared God much, gave many alms, was just in all his actions, humble and affable, of an exceeding quick wit, of an ingenuity that was uncommon. He was exceeding learned, not only in those sciences which terminate in speculation, but those which tend to practice and are the useful arts of life and supports of civil society. He had a great dexterity in managing the great house and a courage so extraordinary that it seemed to approach to rashness. In his company did Mahomet pass much of his time, and Ali was so surprised at the extraordinary abilities of his cousin that he did believe him to be no less than a prophet. It was no unusual thing for prophets to arise and to be owned in Arabia: the common traditions of the nation and the Sabian **< 61>** principles did incline them to believe it possible that, under certain configurations of the stars, a prophet might be born and that he might do great miracles.
It is most certain that those people were much addicted to judiciary astrology, and this is one tradition of it. The nativity of Mahomet had been calculated, and it had been predicted by a Jewish Genethlia that he should be a great prophet and prince. Abu Maasor said of him in defense of those (as Elmain) who write that he was born in the latter end of the night: "Necessariò Muhammedem fuisse natum in fine noctis, quando libra medium cœli teneret, media nocte verò meridiem transiisset signum tauri aliâs enim Prophetiam & principatum ei competere non potuisse." Nor was it held unusual among them for a prophet to bring his Coran, or sacred writ, derived from God. So they held that Edris, and Seth did so, Zaradast or Zoroaster, Moses, also his laws, and Isa and his apostles their gospels, so did Mani who gave a beginning to the Manichees, a sect diffused through Persia and Arabia even at that time, which they avowed to be the incorrupt word of God. Nor were the Manichees singular herein: the Gnostics, Nicolatians, Valentinians, Montanists had their sacred writs, which they reverenced, and contemned as spurious the testament or canon of the Trinitarians; and that the Nazarenes and churches planted by Peter had their peculiar gospel I have already showed.
Whilst the esteem of Mahomet thus began to grow, and that the people beheld him as an extraordinary person, the death of Mauritius with its odious circumstances, having taken off from the value of Christianity, and Chosroes having destroyed all the Christians in Persia or made them revolt to Nestorianism, and having wanted and made desolate Palestine, Jerusalem, Syria, and Egypt, and those Arabians who usually adhered to the Christians and served for pay, being discontented that Mauritius first, and then Phocas, denied them their salaries, and that they were involved in the troubles and calamities which Chosroes brought upon the Christians, Mahomet saw a fit opportunity to erect a new empire among the Coreischites. It did not a little contribute thereto that the other Arabians of Yemen had joined with Chosroes and shared in the rich booties of Egypt, which seems to have raised envy in those of Arabia the Desert. And afterwards, Heraclius having murdered Phocas and gained the empire, he marched against Chosroes, won three general battles, carried on the war into Persia, having in his army Abubacr and a brigade of the Scenites or desert Arabians. These misfortunes did create such troubles in Persia that Chosroes resigned his kingdom to Medorses, one of his sons, but Sirces, his other son, being disgusted at that, puts his father to death, assumes the crown, makes an ignominious peace.
Now were the Arabians of Yemen miserable, divided and **< 62>** broken into factions, and disquieted with the troublesome consequence of so disastrous a war. Those Scenites who had served valiantly under Heraclius, finding no acknowledgments proportionate to their merits, were sufficiently discontented. Mahomet gains to his friendship Abubacr surnamed the just, and Omar, and Othman, all persons of great power and esteem among the Coreischites, and men of mortified lives, so unconcerned in ambitious aims and private ends that they were able to give any party a luster and to any cause the face of justice and piety. Now began all Arabia the Desert to ring with the fame of Mahomet and Othman and Ali being his secretaries; his divine poems were divulged, nor was anything ever read with greater applause. The Arabians were great admirers of poetry (songs were their pandects, their laws and chronicles were contained in them), and when any poet had gained renown above others, all his tribe kept a public festival: the drums were beaten, processions made to him by men, women, and children to congratulate him. This, as it added to the glory and interest of Mahomet, so it was much to his advantage that in token of his humility and to show how great an enemy he was to luxury, he always wore woolen garments (a garb said to be used by the ancient prophets, and then much used by the ascetics or such Christians as pretended to austerity of life), and that he lodged commonly upon mattresses, whence he had that double appellation of the man clothed with woolen and the man that lodgeth on the mattress.
The Christians say he lodged thus because he was troubled with epileptic fits. But what is that to his garb or lodging? These fits seize the patient in any place, and most seldom when they are laid down. In the Arabian writers, though Christian, I find no such account of him nor can I believe it, because he was much addicted to venery and so able therein to gratify forty women in one night, whereas nothing is more inconsistent with, or pernicious in, that disease than immoderate venery. Besides, it is a disease not to be dissembled and in which no dissimulation can be used. I grant he might either naturally or by some other unknown means frequently fall into ecstasies, and lie entranced, but as this differs much from the falling sickness, so it was no incredible accident among the Arabians who might have learned from the Jews and Christians concerning the ecstasies of the old prophets and of Paul. But it was a common tenet with the Arabians that some men might fall into such raptures, and might converse with angels: I find of late that Cardan and his father could fall into them when they pleased. And I am apt to believe that the illustrious Cardinal Ximenes did the like sometimes.
Let us then imagine that in all those cases, nothing befell, nothing was done by Mahomet but what served to imprint in the people an opinion **< 63>** that he was a prophet, which he the more fomented in them by framing his poems to the great God and magnifying him and frequently, in public, crying _"Allah Allah Howa Cabah Allah, God, God, the great God."_ Having given these testimonies of his piety, and in his discourses enlarging in defense of the unity of the deity, he began to inveigh against the mediator gods or idols erected in and about the Caaba and write a _surat_ or chapter of the Alcoran called "Anaan," in which he introduceth God, complaining that the Arabians did pay really more honor to the associate gods than to Him, and that they robbed Him of that veneration which was due to Him. He told them that the Caaba was the temple of the great God, that Abraham and Ismael had dedicated it to His worship alone, that the introduction of idols was a novel practice, that the prophets and patriarchs, especially Abraham, Isaac, and their father Ismael, did worship God without associating any with Him; that all associating of others with the great God, either in worship or in essence or both, was idolatry and therefore the Coreischites and other Arabians that did worship these idols were idolaters. So were those Christians who either held a trinity of persons or trinity of gods or did hold the deity of the Virgin Mary. So also were the Jews who did associate Ozair or Ezra to the great God, saying that he was the son of God.
Among those, for whom the Arabians had a traditional reverence, who though he were not a prophet, yet was he for his witty apothegms and fables of as great credit as any of the prophets in a manner, there was one named Lockman or Lusman (most of the learned imagine him to be Ӕsop, the author of the vulgar fables). But Mahomet either feigned or met with other stories of him and fixed him in the time of King David. Him he bringeth in giving this character to his son. "Oh son, do not thou join with God any companion."
Those latter discourses occasioned Mahomet a great deal of trouble, for the chief of the Coreischites, the rulers of Mecca, and others that were devoted to idolatry and Sabiism began to resent these proceedings, and some out of religion opposed him and his followers, others out of interest, fearing that this doctrine might destroy the glory of the Caaba and prevent the usual resort to it and so extinguish the repute of the Coreischites, whose honor and profit seemed now at stake. Mahomet strengthened his interest by a marriage with the daughter of Abubacr and used all manner of insinuation and address to increase his party. But the number of those that firmly adhered to him was but small, and he placed so little consideration in the respects of the vulgar not cemented to him by a religious tie that he would not adventure his design upon their affection, it being too mutable to build any dangerous lasting design upon. He made some small sallies into the countries and endeavored **< 64>** to draw them into his party. But though he could not so effect his designs as to engage them into his religion or defense, yet by his demeanor and eloquent and sage apothegms he contrived them all in the opinion that he was a prophet.
He daily spread abroad relations of his discourses with God and his conferences with the angel Gabriel and used such a sagacity in discovering all plots and counsels held against him that his followers believed God almighty did reveal all to him. And Abutaleb did vigorously protect him, forbidding any to approach the presence of the prophet having a sword or any offensive weapon about him. The number of his adherents were now increased to forty when Abutaleb died at the age of eighty years. The reputation of that prudent and ancient person, and who died in the religion of his country, was a great support to Mahomet. Yet did not his decease abate the courage of Mahomet. He considered well the juncture of affairs that whilst the Christians and Persians were so embroiled, and the Arabians so divided into several factions and more religions, it became him to protract his designs. For great attempts like great trees must have a deep root or every contrast overthrows them.
Not long after, there came to him seventy-three resolute men from Awas and Chezra and two women. They came with great devotion and took oath unto the prophet that they would live and die in the profession of the faith of Ismael their common parent and patriarch and first propagator (under Abraham) of the Mahometan faith, who now denominates his followers Moslemin, that is, such as believe in God alone. He knew well enough the importance of giving his party a specious appellation: his adversaries he calls associates, very invidiously if we consider what influence those appellations would have upon such as could look no further than the names and appearances of things. Out of these last assistants, he chose twelve as principals or doctors, who, departing, did inveigle the inhabitants of Medina that they resolutely declared for him to assist him.
Medina, then called Yathrib, was the second city of Arabia Deserta, distant from Mecca 10 days' journey or 270 miles and the ways difficult to pass. It was seated conveniently so that he might upon any occasion retire into Yemen or Arabia Fælix (which it borders upon) or otherwise draw any converts or aid thence or even out of Persia. And it is observable that the more remote the people are from the court, church, or chief cities, the less devotion they retain for them. And the frontiers, by reason of the mixture of passengers and conflux of several strangers, are more civil and gentle to all comers than the more inland countries. Mahomet, understanding now the multitudes that would be faithful to him **< 65>** in Medina, demeans himself with more haughtiness than before at Mecca. He declares that he was the apostle of God sent to revive the glory of Ismael and the religion anciently professed amongst them. He commands all people to relinquish idolatry, to destroy their idols and worship only one God. He declares his faith to be the true faith, yet so that all which believed Moses or Isa might be saved, and should be protected, paying a moderate tribute. He enjoins all to believe the truth of the apostles and prophets, and to receive the sacred books, not declaring which were they particularly in favor of the Christians. He approved their laws and declared Christ to be the spirit of God and the word of God.
The Christians, finding such a declaration, entertained a favorable opinion of him, resorted unto him and recommended themselves to his most benign protection, and took a cartel of security from him. He commended them for opposing idolatry, as he did also the Jews, never mentioning any of the patriarchs, or Moses, Isa, or the apostles but with this honorable addition: "God's peace be upon them."
These tidings engaged the Coreischites against him, seeing all Arabia to divide. Hereupon Mahomet commands his followers to depart for Medina and promiseth to follow them. He retained with him Abubacr and Ali, sending the rest before. All of them arrived safe at Medina, Mahomet lodged at the house of Chalid Abiol till he had built a temple and a house for himself.
_CHAPTER 5_
MAHOMET'S CONDUCT AT MEDINA, THE EMBASSY OF ALI TO THE HAGARENES AND SARACENS
ND NOW WE SEE HIM seated at Medina. There he erects a prophetical monarchy, and entitling God and the angel Gabriel to his dictates, he imprinted a greater awe thereof in his followers and was more absolutely obeyed than force or terror could otherwise make him. He declares that after Moses, the Jewish state being corrupt and apostatized from the law given by Moses and grown wicked in their lives, that Isa was sent to reform them and all the world, by a spirit of meekness; that the Jews persecuted and would have crucified him, but that the divine providence substituted a phantasm in his stead, and so he was only put to death in effigy, being really translated into paradise that he should save all at the last day who believe in him and mortify themselves to the world, should observe his precepts. That God, finding that the mildness and gentleness of Isa had not proved effectual, had now sent him who was the comforter promised by him to protect his followers from further persecution, to propagate the doctrine of the prophets and of Isa: all which taught that there was only one God ruling the world by His providence, a rewarder of the just and punisher of evil doers, that God having patiently expected the issue of Isa's preaching and of his apostles, and finding it fruitless, even the Christians being much **< 66>** apostatized, and some lapsed into open image worship and idolatry, he was now sent to enforce all men to the truth (Islamism) and to make war upon them till they confessed one great God.
The Christians, who had been so persecuted by Chosroes, and finding their condition very uncertain among the Arabians according to the humors or interests of the governors, were very glad of his rise and magnified his undertaking. No less pleased were the Jews who were reduced to a mean condition by reason that the wars of Persia, and the animosities and jealousies of the Greek emperors, had abolished all grandeur and extinguished universities, patriarchs, and governments. They hoped, by fomenting Mahomet so, to embroil Arabia that they might draw advantage from thence to aggrandize themselves and either at last destroy Mahomet or see him reduced to such straits as that he should turn to them and become Jew. But Mahomet was too politic to be deceived by the Jews. However he cajoled them, he trusted them least of all.
He considered with himself that to pursue his designs an army was necessary and consequently money to raise and maintain them: to preserve an army in command and to render it serviceable discipline was requisite. And, that he might multiply his soldiers, he penned a _surat_ or chapter exciting and obliging them all to promote Islamism with fire and sword and made it meritorious to die in such quarrels.
The security which he gave to the Jews and Christians that they might live under him without molestation brought a great deal of riches into the public treasury, and these securities were observed with so inviolate a faith that it was a great invitation to the next neighbors to come under his government and for those afar off to wish him prosperity and increase of empire. It was the excellency of the Arabians always to be the best and most active horsemen in the world. Though their horses were lean, they were bold and well managed; they were fed with small sustenance and could endure the want of water in extraordinary manner. Their exercises were on horseback to shoot an arrow and spurring their horses to catch it before it came to the ground; to see an arrow shot at them and to avoid it by stooping or hanging on either side of the horse as occasion required and immediately put themselves again in a posture of defense. They would ride a full career and yet gather up javelins or arrows, which lay on the ground; they would hit the least visible mark with sling, bow, or javelin.
Water was their constant drink, their food coarse bread milk (new or sour), cheese, the flesh of goats or camels, pulse, and especially rice which he recommended to them as the most nourishing and venereal food in the world, saying that he himself did in one night **< 67>** gratify forty women after a supper of rice. Their diet was without luxury, and the same at home and abroad, in field and in garrison, by which his forces were preserved more healthful and were sustained less burthensome and expensive wherever they marched or quartered. Whereas the forces of an enemy not so disciplined would exasperate the country to a revolt by oppressing them to support their riot and mutiny, or disband upon any straits such as his men would not be sensible of, he frequently preached to the people in the temple of Medina leaning upon a column made of the body of a palm tree. But afterwards he had a desk made of tamarisk wood with two steps to ascend into it.
In Medina he wrote at sundry times the greatest part of his Alcoran. Upon every occasion that his dictates might be more authentic, he published a _surat_ or chapter of it, some whereof were at first very brief, not exceeding one period. He constituted for his emirs or generals four friends: Abubacr, Omar, Othman, and Ali. And here, he gave Ali his daughter Phatemia in marriage. Many resorted to him and became Muslemin, for I do not read that he armed and disciplined any others, not thinking it safe to put arms into the hands of new and unsettled friends. Besides that, the populace thought a happiness to be freed from so laborious a militia and other molestations upon paying a moderate tribute, having liberty to attend their employments and enjoy their own religion with security, provided it were not idolatrous. Many of the Christians even of the monks, perceiving with what reverence he spoke of Isa, now acknowledged him to be the word and spirit of God, though otherwise a mere man born miraculously of a virgin, and how much he preached up acts of mercy, justice, and did embrace Islamism. And so did sundry of the Jewish priests.
It had been always from the time of Ismael (if not before) the custom of the Arabians to circumcise their children, as Ismael was, at thirteen years old. Mahomet not only continued this custom among the Ismaelites, but extends it to all that would turn Muslemin. This was not done out of complacence to the Jews who circumcise on the eighth day, but to continue an inveterate use of the Arabians, the neglect whereof would have begot a great distaste in that people, and the imposing whereof upon foreigners, becoming Musulmen, was justified by the Mosaic precedent of circumcising proselytes. Mahomet's thoughts being wholly bent upon the introduction of a new religion and empire, he had Mecca frequently in his memory. He considered that place as the center of Arabia and metropolis of Arabia the Desert. He knew of what importance it was for a prince to be master of the chief city in his dominions, and that the sovereignty of Arabia **< 68>** were half gained, if could possess himself of that place. He had gained the esteem of the populace, who reverenced him as a prophet and were satisfied of the truth of the miracles related of him. They admired his poetry, perpetually sang them and thought it a great honor to their tribe and city to have so eminent a person reside among them. They were witnesses of this valor and piety and saw his deportment and the doctrine he spread to be such as they needed, not fear oppression from his cruelty, extortion from his avarice, nor tyranny from his government. Tyranny consists not in the unlimitedness of power, but in the extravagant use of it.
The military men or Scenites and nomads saw in his design all that might oblige them to him since his religion would involve them in a perpetual war which would furnish them with opportunities to gain honor and riches. And the other inhabitants, which were artisans and tradesmen, saw that under him they should find all encouragement and protection since they should neither be compelled to war nor exhausted with burdens. They saw the resort to the Caaba lessened and, if these divisions continued, the advantages they derived thence and the glory they retained thereupon would all vanish, and they saw the troubles and hazard of a nation broken into factions and each party wherein was weak and, which is worse, dissolute and insolent. They saw that though the rulers and prevailing party in Mecca were against Mahomet, yet the most upright, just, and popular were for him, that the prophet retired from Mecca out of tenderness to the people lest they should be embroiled in civil dissentions and the holy Caaba defiled with blood, that Abubacr, Omar, Othman, and Ali had relinquished all to adhere to him. And, at Mecca, they saw continual objects of their commiseration: their friends imprisoned and tortured for befriending and retaining a veneration for the prophet. Balal, Zohaib, Cabbab, Ammar, Abes, Abu Handen, and Sohail with many others were cast into prison and used outrageously.
The resolution with which they underwent those torments wrought effectually upon the commonality who pitied those that suffered so gallantly and could not hear with astonishment these words echoed out by the martyrs: "God God, the great God, and Mahomet his Apostle."
They saw that the Coreischites, which opposed Mahomet, were in profession absolute idolaters and, to support their grandeur and render their religion more mysterious and awful, find that God being all sufficient needed not any outward testimonies of their devotion, besides that they might be secure of His benignity by freely propitiating these associates. But they saw that in **< 69>** reality those Coreischites were men of no religion, perfect Sadducees (and it is in vain to attempt to excite others to a cordial defense of a religion which they themselves appear not to believe heartily). Moreover, there had fallen out an accident since the birth of Mahomet which was fresh in their memories and I suppose contributed to his reception. One Gawius Abu Abdoluzza, a priest or sacristan to one of their principal idols, having one day by negligence not shut up the door, two foxes came in and pissed upon the statue. He coming in, and perceiving what had happened, thought that idol was unable to help others, who could not avenge himself of the paltry foxes, and broke out into this expression: "Is he the lord upon whose head the fox pissed? Surely, he is despicable upon whom foxes do piss. Oh ye tribe of Salem (those were they that worshiped that idol), assuredly this statue doth you neither good nor hurt. He neither procures nor hinders your happiness." And, having said so, he broke the idol to pieces. Mahomet honored this man with a memorial in his Alcoran and changed his name, which signifies an erroneous person or son of a worshipper of Uzza (an Arabian deity) into Rashed Ebn Aba Rabehi—that is an orthodox person, the son of one that worshipped his Lord.
Whilst affairs were in this posture at Mecca, and the Coreischites exceedingly distracted, Mahomet seemed to have a happy prospect of his designs, his only difficulty was to adjust the interest of Mecca and of the Caaba with his new religion. In order hereto, he declares that his journey to Medina was not a flight from, or desertion of, Mecca, but a religious pilgrimage to that place where Ismael first settled himself and whence the Coreischites were issued originally, and calls it the Hegira or journey taken out of devotion. And his four companions (together with the others that accompanied them) were entitled _Almo Hajerin_ or the devout pilgrims. In this appellation he cunningly made use of a paranomasia in which he alludes to the nation of the Hagarenes, the Arab _Elhagiar_ , a warlike and potent people living thereabouts, called by Strabo and Ptolemy Ἀγραῖοι, _Agrei_ , and their chief city Agra. Of these there is mention made in the Chronicles that the Hagarenes were overcome by the Raabenites, 1 Chronicles 5:10. Such an artifice and anagrammatical allusion he made use of in styling his sect _Aleslam_ which by transposing the letters becomes _Ismael_.
These Hagarenes are mentioned in the Roman history for their valor. Trajan, having carried his conquests over all Persia as far as the Indies, was repulsed here with great damage, and neither he nor Severus could subdue them. Of this nation of the Arabians did Mahomet make particular account **< 70>** and invite them to Muslemisme, for it was his mode only to send to princes to invite them thereto, and they commonly submitted. But I am apt to think arguments and motives were made use of by his emissaries, and in an affair of this importance which required so much address as was requisite in the nonage of the empire. That it might not be dissolved before it was settled, I imagine Ali to have been employed in the negotiation, who was most dexterous in such affairs, who best understood the sense of the prophet and knew when to explain and when to intrigue his speeches. Besides, his youthful courage and address in horsemanship made him most fit to excite this warlike nation to so great an enterprise as the advancing the glory of Ismael and his descendants under the conduct of Mahomet.
Let us then fancy the gallant Ali mounted upon as good a horse as that he used afterwards called Duldall (which carried the miller from Medina to Katchan in Persia in one night which is above two hundred leagues), with a small but brave troop of Moslemin, martial in their aspect, grave in their speech and carriage, exact in their discipline and obedience, armed not for show but service. And though they valued their tulipants as diadems, yet they more esteemed the goodness of their swords, by which they seemed to design for each man a royalty. Such was their reverence to their commander that one would have thought they had been all slaves and could not retain a bold spirit under so imperious a general. But the dexterity of the prophet showed that tyranny that the prudent may be absolute without tyranny and without regret or enfeebling the spirits of the most valiant; that the arts of government consist not in the show but use of authority, and the true use of it is to insinuate itself into, not impose upon, men's reason.
His Coran acquainted those Hagarenes with the heavenly wisdom of their prophet and the spectacle convinced them of his sovereign prudence and conduct. The Hagarenes were astonished when they saw the motto of this incomparable leader to be "Dominion belongs to God alone" and observed their deportment when dismounted; and discerned that they equaled the most pious monks in their devotions and the most liberal princes in their alms; and that their affability, humility, with a detestation of all riot, luxury, and vainglory was such that the world was to receive examples from them since no age had produced any for the Muslemin to act by, and they were their precedents. But that which most endeared this people to them was that, at their first approach, they saluted the people of Agra with that exclamation so well known to the Arabians: _"Allah Allah Howa Cobar Allah God, God the **< 71>** great God,"_ and in their standard they observed the lunulet or half moon, the ancient arms of the Ismaelites which they had seen placed on the head of Astarte or Ashtareth, the great goddess of Arabia and Syria, to which these countries had been immemorially devoted; with which lunulets the Ismaelitish kings as Zebah and Zalmunna (Judges 8: 21. 24) did usually deck themselves and their camels. And thus the Jews expound the Shahoronim in that place.
They were received with all the expressions of joy and welcome imaginable. But the Muslemin declared a greater satisfaction in the sense that they were welcome than in any empty or luxurious expressions of it. They exercised themselves upon this deportment that it was not any scorn of their entertainment, or that they did not think themselves happy to have given so good a beginning to their negotiations, but that the wisdom and felicity of man chiefly consisted in serving the great God; that joy of this world was but imposture; that a man was to consider there was a time when he was not in being, and there would be a time when he should die; that the interval between both was so short and so uncertain the only difference between men was their good actions in this life, and their rewards in the future; that he mistakes his course who places his confidence in any but God, and misplaces his delights who takes pleasure in anything but what is agreeable to his will.
After these and suchlike discourses, the sun beginning to decline, they desired the opportunity of a retirement and water to wash themselves, wherein it was particularly observed with what care they washed their eyes, ears, nose, mouth, and hands for which they gave this reason: that though our knowledge was bred in us by our senses which were as the windows to let in the light, and as those senses, if duly employed, were the instruments of the soul to discover the wonderful works of God, yet human nature was so inclined to misapply them and to be led by them into an excess of sensuality that men ought to have a diligent watch over them. And to remind the Muslemin hereof, the prophet had appointed them such washings always before their prayers, and undoubtedly such memorials are not only efficacious to restrain men from vice but very acceptable to God, being silent testimonies of an aversion from sin and of a resolution to be cautelous for the future. After this, they went to their _sallah_ or prayers. They began with the solemn introduction of _"Allah Allah Howa Cobar Allah,"_ then they proceeded thus in the words of the first chapter of the Alcoran: "Glory be to the Lord of all creatures, the king of the last judgment. We honor thee, we invoke thee: assist us in our necessities, lead us in thy ways, bring us into the path of those to whom thou hast done good, and not into the way of those upon whom thou hast poured out thy wrath, nor into that **< 72>** of such as thou sufferest to go astray. Amen."
Those and other prayers made up of sentences out of the Alcoran, repeated with show of great zeal and inward sincerity, added much to the good opinion the people had of them and their prophet. The multitude either employed their eyes in observing each action of theirs or their tongues in inquiring after the discourses and deportments of these illustrious persons. At their prayers, they turned their faces toward Mecca and the Caaba as if they paid as much reverence to the temple of Mecca built by Abraham and Ismael as every Jew did to that of Jerusalem. This was interpreted by the Arabians as a novelty and liable to exception till they were assured by Ali that the _Reblah_ was changed from Jerusalem to the Caaba and that it was the pleasure of heaven that all Muslemin should pray towards the temple of Ismael. This reason was satisfactory because it made for the glory of their progenitor, and every man readily believes pleasing news. The prayers the Muslemin went to again before bed time, and those which they renewed in the morning at break of day, administered fresh cause of admiration and discourse.
And the more for that: it was observed that the Muslemin did express some signs of reverence to Venus or the Morning Star, which had been an ancient deity of the Arabians, and that they did also in the subsequent week observe as a kind of Sabbath the _Guimia_ or Friday which had been always a day of weekly adoration to the goddess Urania (portrayed as corniculate or with a crescent on her head) in Arabia. It is most certain that in the whole Alcoran there is not any precept for the observation of this _Guimia_ , but that Mahomet, understanding the wonted superstition of the Arabians, continued the solemnity of the day but altered the object of their worship to that of the one great God. So he retained the lunulets upon the mosques and in their ensigns, suggesting new reasons for a custom grown sacred, inveterate, and not to be abolished without hazard to his main design. "Non institutum sed relatum a Mahummde ut festud Guimia quod Oraniæ corniculatæ sacrum; a corniculata ejus & vetustissima effigie lunularum apud eos honere manasse videtur."
It is agreed by the most knowing in the oriental transactions that Mahomet resolved to make no greater change in Arabia than was necessary to his purpose and did ingeniously accommodate to his ends those superstitious usages which were imprinted in the breasts of the Ismaelites. It was a saying of that great personage: "He that knows not how to get up the ladder shall never get to the top of the house; that many things might be wished for which a wise man cannot hope for or pursue; that great designs are often frustrated when the authors discover **< 73>** their greatness. Let others cry this is good, this is gallant: the prudent consider what is possible, what is requisite, and how to turn to advantage the successes of each day."
These circumstances having sufficiently prepared the minds of the Hagarenes, the principals among them resort to Ali and his companions to be informed concerning the desires and commands of the prophet. Ali first gives them an account of his original, the prodigies at his birth, the miracles done by him, his conversation with the angel Gabriel, the austerity of his life, the incredible prudence of his deportment, the quality, virtue, and numbers of those that adhered to him, and whatever else might endear the prophet to them, omitting nothing that might gain upon their reason or credulity. He added that the Coran which Mahomet brought was in general no other doctrine than all the prophets had taught, the sum of whose documents was that God alone was to be worshipped and idols exterminated; that this was the religion of Noah which he preached and for contempt whereof the world was drowned, seventy-two persons (so the Arabians say) only escaping in the ark; that this was the doctrine which Salehus came to preach long ago to the Arabians betwixt Hijaz and Syria in the country of Heir (one of the tribes held to be totally extinguished).
The prophet Salehus came from God to the tribe of Thamud or Thomud, commanding them to desist from the worship of idols and associate gods and to worship only the true great God. Few believed the message; the rest demanded as a miracle that he would produce a camel out of such a rock, which Salehus did and that camel foaled another. Yet they persisted in their infidelity and hamstringed the camel, whereupon the almighty caused a thunder to arise and destroyed them all, the houses wherein they dwelt yet remaining, which the Hagarenes were not strangers to. That this was the doctrine Abraham taught when he was banished their country, and which Ismael afterwards settled there when he planted himself in Arabia, Ali doubted not but they had a traditional knowledge preserved in their songs of the time when idols were first brought into the Caaba; that it was an innovation in the true religion planted by Ismael who, together with Abraham, built that temple; that besides the introduction of a multitude of idols and associate gods they had by the mixture of lies and fables depraved, and in a manner, abolished the religion of Ismael; and that God had now at length been pleased to extend his mercy to the Arabians; that the Jews having lost their Coran which Moses gave them and made Ozair or Ezra an associate with God, receiving a Coran (this is the canon of the scripture and the Cabbala which the Jews derive from Ezra) of this as it were from the great God, and destroying the prophets, persecuting Isa when he was **< 74>** sent to reform them.
Also, the followers of Isa had lost the Coran sent to them, and associated Isa and Mary his mother with God, and in most places introduced idols into their churches and houses—that now God had raised a prophet out of the lineage of Ismael to publish the truth and restore the doctrine of Ismael to its purity. The Caaba we reverence more than any Coreischite at Mecca, and since it hath pleased God by His prophet to remove the _Reblah_ thither, towards that we direct our faces when we pray. The pool of Zamzam we hold no less sacred than they though not upon an idolatrous account, because we now know that when our mother Hagar was delivered of Ismael, he, dancing with his little feet, made way for a spring to break forth. But the water of the spring coming forth in such abundance as also with such violence that Hagar could make no use of it to quench her thirst, which was then very great, Abraham coming to the place commanded the spring to glide more gently and to suffer the water might be drawn out if it to drink. And having thereupon staid its course with a little bank of sand, he took of it to make Hagar and the child drink. The same spring is to this day called Zamzam from Abraham's making use of that word to stay it. We honor those stones which they so idolatrously worship: they are neither Mars, Bacchus, nor Venus, though upon the one they can observe the portrait of Venus, the last within the cloisters or court of the Caaba. On the ground, enclosed in an iron grate called _Makam Ibrahim_ , or the place of Abraham: upon that he stood when the Caaba was built. And there are the impressions of his feet in it, the print of the right foot being deeper than that of the left.
That other, called the Black Stone, which is riveted to the wall in a corner of the Caaba on Basra side is no idol either, but one of the precious stones in paradise brought thence by Adam, carried up to heaven again at the deluge and brought to Mecca again by the angel Gabriel when Abraham built the Caaba. It was as white as milk at first, but the sins of men have caused its color to degenerate into black. We are so far from detesting the Caaba and from dehorting them from going on pilgrimage thither that it is a fundamental article of our religion to undertake it, **< 75>** and none can be a Musulman who thinks himself not absolutely obliged to go thither to perform the usual rites and do his devotion at those stones, as now it is done but out of piety to the Great God only and reverence to our holy progenitors. The stones are blessed memorials of Abraham, Hagar, Ismael, not objects of our devotion, and we must worship not them. But there we doubt not but there is a benediction attends such as piously kiss the Black Stone and stepping do pass under it. There is a heap of stones near the way betwixt Medina and Mecca where the idolaters do now cast each three stones in their peregrination in honor of Merkolis or Mercury.
"Behold," said Ali, "to what a height of idolatry the true Muslemitical religion of our father Ismael is corrupted. After that the Caaba was built, and Ismael grown a stripling, the angel Gabriel appeared to Abraham and told him that God intended to make the highest trial of his affection and gratitude, and that he would have in acknowledgment of so many favors to sacrifice his son to him. Abraham immediately consented and being returned home bid Hagar call up her son and put on his best clothes that he might be better looked on at a wedding to which he intended to carry him. They departed the next day in the morning betimes and took their way towards mount Arafat, Abraham carrying along with him a good, sharp knife and some cords. But as soon as they were gone, _Sceithan_ , that is to say, the devil represented himself to Hagar in the shape of a man, reproached her with the easiness wherewith she had consented that her son Ismael should go from her, and told her that what Abraham had related unto her concerning the wedding was to which he was to bring him was pure forgery, and that he was carrying him to the shambles. Hagar asked him why Abraham would use her so since he had always expressed a great tenderness to her son. The devil made answer that God had commanded it should be so, whereto Hagar replied since it was God's good pleasure that it should be so to make that disposal of him, it was but fit he should comply therewith. Whereupon the devil, pressing harder upon her and treating her as an unnatural mother, endeavoring by those aggravations to bring into rebellion against God, she pelted him away with stones. **< 76>**
The devil's endeavor, proving unsuccessful that way, and too weak to overcome the obstinacy of a woman, he applied himself to Abraham, revived in him the tenderness and affection of a father, represented to him the horribleness of the murder he was going to commit and remonstrated unto him the little likelihood there was that God should be the author of so abominable an action. But Abraham, who was acquainted with the subtlety and artifices of that wicked spirit, sent him away and, to be the sooner rid of him, cast also a stone at him. The last attempt the devil had to make was to represent to Ismael the horror of death and the unnatural procedure of his father, but he found the same treatment from him as he had from the other two and had a good stone flung at his head. The father and son, being come up to the top of the mountain, Abraham said to his son: 'Ismael, my son, I cannot imagine thou knowest the occasion of our journey and the reason why I have brought to this place. It is only this that God hath commanded me to sacrifice thee.' Where to Ismael made answer that since it was God's pleasure it should be so his will be done. 'Only let me entreat thee father to grant me three things: the first is that thou have a care to bind me so fast that the pains of death may not engage me to attempt anything against thee. The second is that thou whet the knife well and after thou hast thrust into my throat that thou hold it very fast and shut thy eyes out of fear lest the cruelty of the action should dishearten thee from going through with it, and so leave me to languish a long time. And the third that when thou returnest home, thou remember my duty to my mother.'
Abraham, having promised to observe all those things, and whetted his knife, binds his son, directs his knife to his throat, and shutting his eyes holds it as fast as he could. But finding when he opened his eyes again that the knife had made no entrance, he is extremely troubled and tries the edge of it against a stone, which he cuts in two. He was so astonished thereat that he addressed himself to the knife, **< 77>** and asking it why having so good an edge as to cut a stone it could not as well cut his son's throat, the knife made answer that God would have it so. Whereupon the angel Gabriel took Abraham by the hand and said to him: 'Hold a little. God would only make trial of thy faith. Unbind thy son and sacrifice this he-goat.' And immediately, there came into the place a he-goat which Abraham offered to God for a burnt offering. The three stones which Hagar, Abraham, and Ismael threw at the devil are yet to be seen near the highway betwixt Arafat and Mecca, and these two great heaps of stones there have been made partly by our Muslemitical ancestors, partly by the deluded of the first. Of each of them brought three stones to be cast at the devil at the same place where these heaps are, to the end he may not distract them at their devotions at the Caaba or mount Arafat."
"I know," added Ali, "that the Jews and the pretended followers of Isa do say that it was not Ismael but Isaac that was to be sacrificed. But this is one of the corruptions of their Coran, for the intendments of God were greater towards Ismael than Isaac. Therefore, Sarah was made to be barren till our father was born. This Sarah foresaw and therefore hated him. This Abraham understood and therefore took such care (as also did the Angel Gabriel) of him. In Ismael was circumcision first celebrated. It was concerning Ismael that the promise was made to Hagar: 'I will multiply thy seed exceedingly; it shall not be numbered for multitudes.' The generation of Hagar was greater than the generation of Sarah: it shall reign unto the east and to the west, and God shall let them to rule over all the nations of the earth. Behold renowned Hagarenes your illustrious ancestors; view the country that you are possessed of: the three Arabia and the rich appanage in Mesopotamia and Syria, and compare them with the narrow and barren land of promise designed for the promise of Sarah. Inquire how often they have been totally conquered and carried away captive into foreign countries and their temple destroyed whilst you retain the ancient habitations. Nor can any monarch boast of an entire conquest by the most valiant **< 78>** Hagarenes. Your Caaba hath been profaned with idols but never destroyed, nor totally alienated from the worship of the great God."
These discourses raised in the Hagarenes not only a great attention but in one instant seemed to have gained them to the party of the prophet. They heard with a great deal of pleasure the glories of their extraction, the share that their progenitors had in the love of the great God that so high promises were made to their tribes and that was so mindful of the deserts of Arabia as to design it to be the seat of the most potent and renowned empire in the world. These fellows that understood no other delicacies than sour milk and parched peas or beans, no better array than what the hair of their goats or camels, and that coarsely spun and coarse woven did yield them no other beds or pallets than the ground, no other riches almost than a few camels, a lean horse or two, a bow and arrows, no other deity than a few mistaken stones which at a pilgrimage to the Caaba they or their ignorant ancestors had brought home and devoutly worshipped; or if any had been more illuminated, their religion mounted no further than to make some ill-favored cringes to the moon and mumble an orison to the Morning Star, crying: _"Allah, Allah, Howa Cabar."_ For this was their old form of prayer or doxology, which Mahomet most subtly turned into _Cobar_.
These fellows, I say, now began to imagine themselves the darlings of heaven, the heirs of paradise, and monarchs of the universe. And since they now comprehended the true original of the present religion and what it was whence they had degenerated, they resolved to be as good Muslemin as their father Ismael, and to own that worship (especially it being more facile and easy than their idolatry and present superstition) which all the prophets had preached and adhered unto. It is one of the most difficult parts of a prince to adjust employments unto **< 79>** their ministers and to make choice of suitable instruments for carrying on each affair.
The youth, the spirit, the fire of Ali did not a little contribute to the happy success of his negotiation. His success was the less suspected because he was not arrived at the years of dissimulation. His good mien, his prudence, and other virtues made the greater impression upon them because they were set off by an age in which they were extraordinary. His courtesy was able to compel his enemies to quell their passion and rendered his friends his slaves; his eloquence and his reason, which he could form according to the persons he dealt with, and he was of opinion that true eloquence and solid reason were but relative names and did not depend upon select words, numerous periods, apt cadence or arguments concluding in modes and figures, but in being operative and efficacious upon the persons he was to deal with; that there were times when the greatest artifices was to abandon all art, the greatest prudence to neglect its severe rules and where a wise... might have drawn the greatest advantages from untruths and fables without endamaging his reputation.
Such, I say, was his eloquence and reason that he seemed to have charmed their senses and possessed himself of all the affections of their souls. He prepared their courages as he pleased, infused boldness into the most fearful ambition, into spirits incapable of it, and which even then did not apprehend what they were instigated by, and persuaded the most impetuous and undisciplined to such a moderation and regularity of military discipline as might be subservient to their great ends. The example of his small and well-trained retinue did conduce not a little to this last point, and if the prophet had only commanded it from God and not introduced the practice of it in his domestics, the Coran would have been ineffectual and the design become abortive.
The Hagarenes were all eyes and all ears and their souls distracted between what they saw and what they heard. But the approach of noon gave Ali and his companions **< 80>** occasion to withdraw from their presence to prayer and so they had the greater liberty to recollect themselves. Dinner being brought in, which was served with more plenty and neatness than is usual amongst the Arabians, the illustrious pilgrim and his associates declined to taste of anything that appeared to be more delicate than ordinary, and the viands which were there added nothing to the entertainment of the Muslemin but as they testified their welcome and the kindness of their friends. They said that it was the command of their prophet that the Muslemin should not indulge themselves in such sensual pleasures in this life, that God had reserved them for the divertisements of paradise and the future world, that here our bodies are frail, our senses easily glutted so that such momentary delights were not worth our serious thoughts and regards, that they did but effeminate and intenerate the body and beset the soul, that courage and luxury were inconsistent, that since the great God did by the prophet call forth the Muslemin to extirpate idolatry and propagate Ismaelisme or Islamism by arms, they ought to prepare themselves for that holy but laborious militia, and that nothing did more conduce thereto than that a man should live at home as he did abroad, in the town as in the field, in the court as in the camp, that this was most healthy and withal would preserve their minds (which sympathized with their bodies) in an equality of temper and uniformity of disposition, and would render them more firm in their religion, fixed in their friendships, equal in their humors and tractable in their passions; that luxury was the seminary of all mischief, that even the first approaches thereto were dangerous, that if a man once indulged himself therein, the evil would become remediless, that our desires are apt to grow boundless when they transcend the limits of what is absolutely necessary, that inordinate desires were a perpetual torment never satisfied but always spurring **< 81>** men onto new projects, that content was the greatest felicity which was only attained by extinguishing our desires, and a Musulman was happy enough if he did not want.
The Hagarenes astonished at their parsimony and abstinence, inviting them to drink some Persian wine, adding that persons who fed so would stand in need of some refreshment by a draught thereof, which would recruit their strength and renew their vigor impaired by a laborious discipline and slender diet. But Ali declined their offer saying that their prophet had severely interdicted the Muslemin to taste any wine, that there was more pleasure in obeying the commandments of God than in the flavor of that generous liquor; that he who made man best understood his fabric and would not prohibit him anything without which he could not subsist; that Adam, Seth, Edris, lived to the greatest age without it; that it was but in imagination of our weakness which put us upon the pursuit of such cordials the sense whereof is only in opinion. It was true that the God of wine, Bacchus, was worshipped in Arabia and that foreigners held he was fostered there, but this was a corruption of the true Ismaelism: that both the idol and the liquor entitled to him were now to be banished and the Arabians to know that _Baccha_ signified no more than great and renowned, and however depraved now, was only a religious exclamation in praise of the great God; that whatever pleasure there was in wine, those sensual pleasures are inconsiderate in this life and therefore God had reserved the entire satisfaction of our senses till we come to paradise where all such delights will have their perfect relish and gusto, our immortal bodies being qualified with senses never to be dulled with satiety.
Hereupon he related the dialogue in the Alcoran between Mahomet and Abdias, a Jew. Abdias demanded of him what use there would be of wine in paradise. The prophet answered: "Your question is so subtle, that I must return a double answer to one interrogatory. I shall therefore satisfy why it may be drunk there and why not here. There were two angels, Azot and Marot, sent down from heaven by God into this world to instruct and govern mankind with this caution: that they should never judge unrighteously or **< 82>** drink wine. This being known, many repaired to them for justice, which they impartially administered amongst others. Appeared before them a very beautiful woman to complain against her husband. To incline them to favor her case, she invited them to dinner and treated them magnificently, charging her servants to ply them with wine to the drinking whereof she also frequently urged them. In short, they were made exceeding drunk and their feeling those impressions from her beauty which before they were not satisfied of, they importuned her to that compliance which the most amorous sigh after. She promised to consent provided that one of them should acquaint her with the way whereby they came down from heaven and the other with the passage up thither. They did so and she having disengaged herself, mounted straightways to heaven which when God perceived and inferring Himself of the manner of her arrival, He turned her into the Morning Star that she might there shine with as great luster as ever she did on earth. The two angels, being called to an account, were commanded to choose whether they would suffer torments in this world or in the world to come. They elected the first and remain hung to this day in iron chains with their heads downwards in the abyss of _Babilli_. What say you now, Abdias? Is it not reasonable that wine should be prohibited here on earth and yet allowed hereafter?"
The Hagarenes hereupon fell into admiration of the Coran and did not doubt but he who published such divine things must be the apostle of God and an intimate of heaven. They were convinced that it was not fitting for men on earth to drink wine since it had so evil effects upon those pure and angelical bodies. They then perceived the reason why their first progenitors paid a reverence to the Morning Star: that they did not worship the star as the idolaters did since but uttered an _"Allah Allah howa Cabar"_ to the honor of God who had placed that bright star in the firmament to put them in mind of the inconveniencies of drinking wine on earth where our life is an errand to serve and glorify God (not pamper our selves). And, to acquaint them of the future pleasures of the celestial paradise, all the topics that rhetoric itself yields should not have persuaded them so powerfully as this single apologue.
But the abstinence from wine being of such importance to the preserving of civil and military discipline, mutual friendship, obedience, dispatch, secrecy without which the Arabian **< 83>** monarchy could not be achieved, Ali thought fit to enforce that point by a second relation: that their prophet, being invited by a friend to an entertainment at his house, chanced in his way thither to be detained at a nuptial where he admired to see the innocent cheerfulness and mirth of each guest, how friendly they embraced and kissed each other and rendered mutual testimonies of unfeigned love. And, inquiring of the master of the house what it was that created in them so debonair and complacent a humor, he was told that this was the usual effect of wine and that they had drunk some, whereupon he pronounced a blessing upon that liquor which did produce so amicable a disposition in the breasts of mortals. The prophet departed thereupon, and as he returned the next day called there again, but he found things in another condition than he left them. Here lay a scattered leg, and there lay an arm cut off in one place: he saw a cripple lie in the other, some bereaved of one or both eyes. Hereupon he demanded whence that passion and mutual animosity betwixt friends, what occasioned the fray. The landlord told him that this was the usual consequence of drinking wine: that after they had drunk hard, they became mad and, from misunderstanding one another, proceeded to blows and so had killed some and maimed others. Upon this, Mahomet changed his benediction into a curse and prohibited his followers that they should never drink wine here.
Ali put a period to those kinds of discourses. And dinner being concluded, now that he found the Hagarenes sufficiently at the devotion of Mahomet, he determined to accomplish the utmost ends of his negotiation by an additional harangue to their purpose.
"Valiant Sons of Hagar and Ismael: If I thought I needed to speak any more to you, either to convince you of the truth of the religion our prophet teacheth, of the divine authority of the Coran each line whereof is a durable miracle and will always appear to be so as long as the language of Ismael doth continue upon earth, since no human wit or learning can produce anything equal to the least _surat_ or chapter thereof. If this were necessary, I would insist upon further arguments and add new motives to persuade you to Islamism, the sum whereof is avowed to be this by **< 84>** the testimony of the angel Gabriel himself: that a man confess there is no god besides the great God, and that Mahomet is the apostle of God, that a man observe strictly the five times of prayer daily, that he give alms, that he fast during the month of Ramadan, and that, if he can possibly, he make a religious pilgrimage unto Mecca and the Caaba.
I proceed to another point which makes as much for all your interest as it doth for the glory of God: that life, which heaven hath given you and which God may at any moment deprive you of, he is pleased to give an occasion particularly now to serve Him. And if you lose that upon this occasion, which a fever, a fall, the least casualty might otherwise bereave you of, the supreme joys of paradise are ascertained unto you. It is the divine pleasure that idolatry should be destroyed out of the earth and the progeny of Ismael are these whom God designs this high favor for, to compel all men to the true worship. But such is the divine goodness that besides the future rewards allotted for Muslemin, he hath annexed to this difficult and laborious employment empire and glory on earth! It is hereby that the promise concerning Ismael must take place, and it is by this means that the valiant Hagarenes must give laws to the utmost ends of the earth and extend the dominions as far as there is any habitable region.
The work is happily begun. The great God hath sent you a prophet to conduct you such as the sun never beheld. Edris, Noah, Abraham, Ismael, Moses, Isa had these characters imprinted on them that we reverence their memories and esteem each of them truly great. But none ever equaled Mahomet the last and chiefest apostle of God, nor was there enjoined to the world this double testimony viz. that God is the great God, and Mahomet the apostle of God. No Coran ever equaled his for subject or eloquence, and, to make way for him, you see that the Coran of Musay or Moses and the Coran of Isa are perished or so corrupted by the wickedness or negligence of their followers that there is no affiance to be placed therein. It is by a belief herein that Muslemin shall obtain the highest glory in heaven and on earth. No monarchy was ever parallel to what God by His prophet doth summon you.
If you behold the condition of the Greek monarchy and Christendom, all things **< 85>** will appear facile unto you. The subjects are so exasperated by oppression, so debauched in their manners, so indifferent in their religion, and after so many quick revelations by the death of the Emperors Mauritius and Phocas and the usurpation of Heraclius, so unconcerned for their prince who governs them, that you need no more than attempt their conquest to effect. There is no unity in their councils, no duty, no obedience in the soldiery so defrauded of their pay during the reigns of Mauritius and Phocas. There is no conduct or prudence in their generals or commanders, no union in their church. You shall no sooner advance your standard but the Arians will become your friends. Nay Muslemin, the numerous Jacobites (and the historians will enlarge their divisions) and rather live peaceably under your protection than anathematized, scorned, hated, persecuted, and depressed under the Melkites. It is natural for mankind to endure more patiently and willingly the rule of a foreigner and of one differing in profession from them than to be tyrannized over and trampled upon by one of their fellows of the same religion and of no better extraction than themselves. And those potent sects will bear with some content a yoke under which the domineering Melkites shall groan.
They are not unacquainted with the Arabian force. Your armies have lately carried terror over all Syria, Palestine, and Egypt: this a parcel of you heretofore did under Queen Maria in the reign of Queen Mavia, in the reign of Gratian and Valence. You vanquished their armies and enforced them to sue for peace by their solemn embassy to you in the time of the Emperor Justinus who preceded Justinian. King Almonder made the like conquest and enforced the emperor to send an embassy to him for peace. And, lately, did not a party of the Saracens under Chosroes, in the time of Phocas and Heraclius, overrun Egypt and added to the victories of the Persian King? I must tell you, renowned Hagarenes, though others know your puissance, you never understood it yourselves. You have always been the stipendaries and appendage to the Roman and Persian Empires; you have fought to make them great, not yourselves; you have, as it were, been subjects to the one sometimes, to the other sometimes. We find that Arabia has been divided betwixt both, and Aretas hath fought in favor of **< 86>** Justinian while Almondar hath fought against him, and Chosroes hath appointed princes to be one party and the Greek emperor to the other.
And what have you acquired by all the victories you have gained and the services you have rendered? Lo the Greeks: have they continued unto any of you the usual pay? Are they not indebted unto you by long arrearages, and what answer have they made to your just demands? And is it not that they have no money to spare for Hagarene dogs? Certainly you deserved a more civil and obliging a return, and you need not that heaven should excite you by a prophet to revenge this indignity, make them to feel your power once more, convince them how necessary your friendship is to them by letting them see you can be their masters? Bostra or Vostra gave birth and original to Marcus Julius Philippus, and an Arabian swayed over the Roman Empire but deprived him of the empire. To effect this, let us not live divided under more petty princes than we have tribes; let us all unite into one monarchy as we are all of one language and one parentage. We are all Hagarenes, all Ismaelites. The same Hegira will suit with all, the same crescents is our common standard. It is a pitiful thing to see into what necessity the petty princes are reduced to maintain themselves and to how many real evils they are exposed to, to conserve that vain image of liberty and that sweet delusion of sovereign authority that doth bewitch them. In expenses they consume themselves for their defense and almost give all that may be taken from them. That nothing may be taken from them, they are obliged to observe all the fancies and notions of the enemies and friends, and if they subsist it is not by their strength, for they have none but by their weakness. And because their countries are of so little concernment that they beget not a desire in ambitious persons, or their justice should be violated in the conquest of them, or they are under shelter from the enterprises of the one by the jealousy of the other, and conserve their liberty by this only reason. Let the ambitious hinder one another to seize upon them and to become their **< 87>** masters.
Consider with yourselves how often your divisions and subdivisions have made you a prey to the invading Persian or Roman. Have you not seen the Roman armies and been almost reduced to desolation under Trojan, Severus, and others? Has there ever been a war betwixt these two potent empires under which Arabia has not been harassed and the blood of Ishmaelites shed on one or both sides? Think of the calamities you have endured and examine from whence they have sprung; inquire what renown Arabia was arrived unto under Odenatus and Zenobia, but that some of you were inveigled and brought of to combat the others. They confess it: they confess they owe more unto your petty princes that could be mercenary than to their forces which you singly baffled. Independency is an empty name if poverty, weakness, and contempt are the consequences of it, and a commodious subjection is to be preferred before a shadow of sovereignty and a precarious insignificant power. The liberty is greater, the repute greater, the riches greater, and all are more secure if a small principality become the accessional of a puissant monarchy than if it subsist of itself.
I speak not this that our prophet demands or that God enjoins that you should lay down your power at the feet and submit it to the disposal of Mahomet. No, he is designed our prophet, not our emperor, and brings us no laws but what are to guide us to heaven or which God enjoins to be observed here when a nearer view shall have convinced your eyes. As fame no doubt hath filled your ears that he is altogether averse to the concerns of this world, that he is so far from depriving any Ishmaelite of his liberty, that he would set even a bird free if he saw him encaged and so remote from ambition and avarice that the greatest pleasure he takes in having anything is that he may give it away to some more indigent Muslemin. You will then lay aside all suspicions should the Coreischites of Mecca instill any into you now.
No, no, it is unity amongst the Ishmaelites; it is Islamism amongst all that our prophet is sent to promote. Learn but from him to worship one God, to reverence and pray towards one Caaba, to advance one pure religion and leading your forces. And whilst you yourselves dispose for the happy success of affairs, we pilgrims beg the honor of a precedency **< 88>** in the most laborious, perilous, and troublesome."
This speech being ended, Ali found the Hagarenes wholly bent to adhere to Mahomet and resolute to adventure their lives for the propagation and defense of the religion of Islamism. Nothing was to be heard but the _Allah Ekbar_ or exclamation of God the God and Mahomet His apostle.
He left two of his companions to instruct them in the _sallah_ or prayers and otherwise to form them into a convenient discipline that they might be ready upon any urgency, it being the determination of the prophet not to draw any greater forces together as yet, partly because the country about Medina was so very barren that even the Scenites or most wild and hardy Arabians could not be accommodated thereabouts. So excessive were the heats, so scorching the sands, so sterile the soil, and so great the scarcity of fresh and wholesome water, partly also because that his were Muslemin being as yet novices in their religion might be drawn into faction and mutiny or otherwise relinquish him. And it was prudential for him not to embody others than such as were firm to him than to be deserted by any that had been his follower. He knew the nature of the Arabians: how prone they were to listen to novelties, and how obstinate to maintain even with their lives whatsoever they were prepossessed with. Nor did he doubt by his emissaries to inveigle as many as should most resolutely support him against any that should come to attack him there. And it was not his intention to take Mecca by force, but by surrender, since he could not choose but profane or violate the respect which he had for the sacred Caaba by assaulting and storming the city.
Ali now prepares to depart from Agra and on the morrow hastens to Saraka which was the chief city of the Saracens. I find St. Jerome and Sozomen and many of the ancients to have believed that the Saracens were denominated from Sarah the wife of Abraham and that they took that name to conceal their descent from the handmaid Hagar. But this is so ridiculous a conceit that Scaliger, Fuller, Hottinger, Pococke, and all the intelligent moderns laugh at it. Nor did the Saracens ever claim kindred with Sarah or renounce Ismael and Hagar, but vowed that the majesty and greatness of Hagar was to transcend that of Sarah. **< 89>** I could willingly assent to Fuller (that since Saracens, usually in writers, is taken for all the Scenites and inhabitants of Arabia the Desert) that they were so called from _sarak_ , which in the Syriac tongue signifies empty and barren, their country being such. But since Hottinger thinks it strange that the Arabians should give themselves a Syriac name and not Arabic, I shall decline that.
But withal, I can as little think they would admit of a name from the Arabic _sarak_ , to _shark_ , to steal privately, which yet is the opinion of Scaliger, Hottinger, and Valesius, though Dr. Pococke dislike it (since they were public robbers, not private). He thinks therefore they were called Saracens from _sharkion_ which signifies the East because they lived eastward of Judæa, which reason had been better if the name had been of Jewish extraction. But in that or the Syriac language it signifies no such thing, nor could the Saracens call themselves so, there being others more easterly than they; nor could they do it in reference to the western Arabians—they being thus termed in history before any Arabians were settled in the western world contradistinct from them. I believe they were one province only of Arabia which was called Saraka and lies beyond the Nabatheans or Arabia Petræ, the inhabitants of which are called Saracens. Ptolemy calls the country Σαρακηνοί. I am persuaded of the truth hereof because the geographers, in distinguishing the Arabians, denominate them from their particular region as the _Cedrei, Agareni, Saraceni_ , et cetera.
And therefore I suppose Ali to have gone to the city (I have authors who style it so) of Saraka, and that in his journey and reception there happened nothing that need relation after what I have said of the Hagarenes, the Saracens being no less prepossessed by emissaries with the fame of Mahomet and his apostleship than those of Agra, and their customs and manners being the same. The most remarkable accident in the journey, and which contributed much to the veneration of Ali, was that toward the dawning of the day when the morning _sallah_ or prayer was to be said by the Muslemin, and they had begun their devotions, an unexpected fire consumed the cabin wherein two of the followers of Ali were lodged. The one of them chose rather to be burnt than to preserve his life by discontinuing his prayers, upon which he was so intent that neither the sight of the fire nor the noise and concourse of people, nor the importunities of such as called to and plucked him could any way divert his thoughts or make him express any sign **< 90>** that he heard or regarded them. The other escaped by a timely flight.
The news hereof coming to Ali, he immediately pronounced with extraordinary zeal the _Allah Ekber_ or Mahometan exclamation: "God, God, the great God." And, calling for the Musulman who had escaped, he told him that the man was happy who trusted in God; that the world was not constant or perpetual to anyone, but that our wisdom and felicity here consisted in resigning ourselves to the will of God and devoting our hearts entirely to him; that to serve God was our duty as his creatures and subjects to whom it was enjoined, our glory as Musulman and the way to eternal happiness. He declared the deceased to be a martyr and prayed that God's peace might be upon him, and his memory glorious who had expressed so great a devotion to God and so great a contempt of life that he would not interrupt his prayer to save himself: that prayer was the pillar of religion and key of paradise; that nothing ought to detain or divert a Musulman from his devotion; that he who could at such times think upon or mind anything else did not entirely resign his thoughts to the worship of God and did not merit the name of Musulman or true believer; that the value of life and of this world was inconsistent with a true faith concerning the felicity of the future; that this world was no other than a dead carcass or carrion; and they were dogs which pursued it.
This said, he commanded that he who had escaped should be severely bastinadoed, which chastisement he endured with a great deal of fortitude and cheerfulness, kissing afterwards the hands of him that chastised him and making him a present thereupon. The Saracens were astonished at this affair and to understand the patience with which the Muslemin underwent their punishments, though they received one hundred stripes and these so cruel that several pounds of flesh were to be cut afterwards from the bruised parts to effect their recovery; that they hold the first batons (such as were used to these purposes) to have been sent from heaven and that all of them were sacred, and those which were bruised or touched with that instrument of justice were exempt from torments after death; and that the party punished ought to kiss the hands of the lector **< 91>** and give him thanks and a present for every blow given.
The impressions which this spectacle made in the Saracens are not easily conjectured. Ali, who understood to derive advantages from any emergency without seeming to do so, omitted nothing that might engage that valiant nation to the service of the prophet. He instructed them in Islamism, made use of all those arguments which had prevailed on the Hagarenes: whatever might work upon their passion or reason, he urged unto them and drew motives as well from honor and interest as from piety. He desired they would do more for the great God than they had done in the behalf of the associate gods and show themselves as valiant under the apostle of God as under Odenatus and Zenobia, Almondar or Alhareth, Justinian, Heraclius, or Chosroes; that the Arabians were the same they always were, but that Persia and Christendom were so degenerated that they needed not to apprehend any difficulties in the conquest of either.
_CHAPTER 6_
THE RETURN OF ALI AND THE WARS OF MAHOMET
E SET ALL THINGS in such order here as he had done at Agra and retired back to Medina with a numerous retinue of volunteers who came of their own accord to attend and guard the prophet. They disciplined themselves there every day, and what time could be spared from their _sallah_ and their military exercises was employed in working upon some trade, the prophet teaching them that that food was most pleasant, nourishing, and blessed which every man gained by his particular industry and labor and that God delighted in those alms to be given which a man had gained himself. Mahomet received Ali with as much honor as became the gravity of the Apostle of God, and Ali prostrated himself before him with such reverence that he seemed really to believe what he usually professed—that the shadow of the prophet was as the shadow of God.
At the same time, there arrived news from the kingdom of the Abyssines, how Giafar, the son of Abutaleb and brother of Ali, had converted the Aluajash or Negush, called Aitshama, emperor of that kingdom, together with his subjects to Islamism. This Giafar, with other adherents of Mahomet at Mecca being **< 92>** persecuted by the Coreischites, desires Mahomet's permission to retire, which obtained, he withdrew into Ethiopia for protection. This Hegira or flight was some time before the pilgrimage of Mahomet to Medina. Now Giafar, residing in the court of Aitshama, did instruct him in the rise and doctrine of Mahomet. The Abyssines had always used circumcision not upon any religious but civil account and are said to have been converted to Christianity by Queen Candace's eunuch, and that Matthew and some others of the apostles preached there. Undoubtedly they were at first of the number of the Judaizing Christians and afterwards, as did the Arabians, they turned Jacobites. This affinity in religion, in circumcision, and in rejecting the Melkites (not to mention that the Abyssines had not long before reigned in Yemen for seventy years or more), and the general ignorance of the Christians whereby they did not understand the notions they had of the Trinity, did facilitate the conversion of the Negush who was soon convinced that it was impossible there should be three persons in the deity and that God should have a son. And hearing that Mahomet did not only style Isa prophet, but superlatively honored him as the word and spirit of God, he embraced Islamism.
Upon this intelligence, Mahomet commands Ibn Omar to prepare for an embassy to Ethiopia in the sixth year of the Hegira. The retinue had much of splendor and Ibn Omar carried a letter from the prophet which began thus: "In the Name of God, merciful and gracious, from Mahomet the Apostle of God to Negush Aitshama, King of the Abyssines, et cetera." His reception there was no less solemn than magnificent. Aitshama descends from his throne to receive the letter. He laid it presently on his eyes, and sitting on the ground, read the contents and returned a submissive answer which began with the _bismillah_ or form with which the strict Mahometans usually begin their discourses and letters and is the proem of every chapter in the Alcoran, viz: "In the Name of God, merciful and gracious. To the Apostle of God of glorious memory, from Aluajash Aitshama Ben Ahahar. **< 93>** Health, O Apostle of God, who are sent of God, et cetera."
This letter was carried by Giafar Aritha, the son of the Negush, and sixty of the princes of Abyssinia, who accompanied Giafar Ibn Omar and Aritha. The arrival of the Abyssines at Medina was attended with all the solemnity and splendor which became the prophet. There were in Medina several Christians. Mahomet commanded them, their presbyters and monks, to be present at the reception, and, after the first ceremonies were past, the prophet, having caused the people to be ranked with their faces towards Mecca and the Caaba, he commanded Giafar to read to them somewhat out of the Alcoran. He fixed upon the _surat_ of Mary (or Isa), and when they heard it they wept and publicly declared themselves Muslemin.
It is easy to apprehend what effects this embassy had upon all Arabia. Those of Yemen could not but call to mind their late subjection to the Abyssines and feared again a second conquest; those of Mecca were terribly affrighted and suspected the Islamism of Abyssines as a trick of state and rather feigned than real. They remembered the attempt which Abrahah Alashram had made upon the Caaba, which before related, which happened in the forty-second year of Chosroes or Anusherwan at which time Mahomet was born, and from which the Arabians made a new epoch, and to which there is a _surat_ in the Alcoran relating. The adherents and confederates for Mahomet were very much strengthened hereupon and more assured in the truth and success of his apostleship.
It became the Coreischites now to look to themselves and to prosecute him by a vigorous war. Mahomet, to amuse them the more and the better to discipline his few followers, had not embodied any such numbers nor made such show as might terrify the rest of the Arabians, the petty princes whereof might be thereby induced to hasten into a close league and confederacy to his destruction. In the end of the first year in which he came to Medina, he sent out his uncle Hamza (with white flag hallowed by himself), accompanied with thirty men. But this small troop produced no change anywhere, nor attempted anything memorable by reason of their small force, or that Mahomet would make no show of invading others till he was in a condition to defend himself.
SECOND YEAR OF THE HEGIRA
In **< 94>** the second year, his parties being better formed and more numerous, he resolved to distress Mecca and thereby render the people mutinous against their leaders. He sent out of Medina 319 to intercept a rich caravan belonging chiefly to the Coreischites and was going into Syria under the conduct of Abusofian and other principal captains of the Coreischites, guarded with 900 or 1,000 soldiers. Abusofian, the son of Hareth, finding himself in danger to be attacked, and knowing the discipline and valor of the Muslemin and the difficulty of protecting a large caravan and at the same time fight the enemy, designed to retreat. But in Beder, or Bader, the Mahometans reduced him to a necessity of fighting. Neither the number, valor, nor policy of the Coreischites could resist the fortune and prowess of the Muslemin. Seventy of the stoutest and bravest of the Coreischites fell that day, most of them commanders and as many more were taken prisoners, with the loss only of fourteen Muslemin who were declared martyrs. The fame of the victory and the spoil added much to the renown of the prophet. All of the riches were brought into the public treasury for the general benefit. For Mahomet had so principled his followers that they regarded nothing beyond a mere subsistence and the propagation of Islamism.
Mahomet was resolved to take Mecca and the Caaba by surrender and therefore would no further prosecute his victories. Besides, he did not think it prudence to grasp at more than he could securely manage. He knew that young converts are not so fixed to their profession and party, but easily become factious and mutinous or revolt again, and that a nearer approach to Mecca might render the Coreischites desperate and so alarm their neighbors that they might join in their defense. He knew that in the territories he had already acquired, as also in Yemen and the neighboring provinces of Persia, were a multitude of Jews who were not so obliged by the protection given them but that they would upon any opportunity advance the interest of their own nation and endeavor to resettle themselves in their old monarchy at Jerusalem as they had lately attempted under Chosroes and formerly upon diverse occasions. Nor was he ignorant that the Jews did hate him for magnifying Isa and advancing him (above Moses) to the dignity of a prophet whom they had put to death as a seditious person and esteemed the son of a whore. Whereupon he resolved to secure them before they should make any head and, by subduing them, increase the number of his victories, it being certain that the Coreischites would be glad to see him **< 95>** otherwise employed than in distressing them and would wait the issue of these new troubles.
I know not what open cause of jealousy the Jews gave to the prophet. But he dispatched Abubacr to them to demand that they should embrace Islamism, repeat the _sallah_ or Mahometan prayers, pay the tenth of their estates, and lend to God a considerable sum of money. No man was so fit for this employment as Abubacr, for, besides his great courage, he was exceeding passionate. No Jew could be a greater bigot in his way than Abubacr was in Islamism. They could not believe so little concerning Mahomet, but on the contrary he believed as much. He believed all that Mahomet said and all that was said of him. When the prophet reported that he had been carried in one night from Mecca to Jerusalem, the Coreischites laughed at it as a bold figment and imposture and asked Abubacr if he believed it. He readily answered that he did not only give credit to that but would believe and justify matters more incredible than these. He came to the Jews and pressed them to receive the commands of the prophet, urging them with the miracles of Mahomet, that being the most prevalent argument with that nation. But they were not moved thereby. How confidently so ever he reported them, they said they expected a Messiah of their own, the son of David, whose dominion should extend far and near, and, as to the lending any sum of money to God, Phinhas the son of Azura demanded if their God were so poor that he needed to take up money at interest.
The insolence of this question did so enrage Abubacr that he gave him a box on the ear protesting with all that he would have slain him but that the prophet had given them a cartel of security. He departed and acquainted Mahomet with their refusal, but mentioned not the words of Phinhas, protesting he durst not repeat their blasphemies. Mahomet was not displeased with the ill success of the negotiation, but presently curses the Jews in a particular _surat_ , declares that they are enemies to the Muslemin and aim at a distinct monarchy of their own, that they had always persecuted the prophets of their own nation, and were so arrogant as to imagine that God could not raise a prophet but from among them.
THIRD YEAR
In the third year therefore, he sets out against them, and in fifteen days destroys their castles, plunders them of their riches, and reduceth them to his mercy, killing Caabas the son of Alaszasi who was his most bitter enemy. It was well for him that he distressed the Jews in so short a time. For the Coreischites, thinking to find him so busied against the Jews that he would not be able to defend **< 96>** Medina against their powerful forces, sent Abusofian with three thousand foot and two hundred horse and three thousand camels (the milk whereof was their victuals) to attack the town. Mahomet draws forth his army to fight them. A bloody battle ensued thereupon in which the Muslemin were at first victors, but were at last overcome with the loss of Hamza, Mahomet's uncle, and seventy others of their party. Mahomet acted all the parts of a good commander and a valiant soldier and, since he despaired of conquest, determined to make good his retreat to Medina, the neighborhood whereof preserved from any great damage that day. The Coreischites, discovering him rallying his men and bravely fighting in the rear of his flying forces, bent all their power to destroy him, and Ochas, the son of Abumugid, wounded him in the lip with a javelin and shook out some of his fore teeth. Abdalla, the son of Sidhab, hurt him in the forehead. He was also wounded in the jaws. Notwithstanding all which, he escaped safe into Medina, and Abusofian, being distressed for want of water and necessaries, was obliged to withdraw his army to Mecca.
A man of less courage than Mahomet would hardly have subsisted after the ignominy of this discomfiture. But the prophet was undaunted, and casting the blame upon the Jews who had so unhappily diverted the Muslemin to flesh his vanquished troops with their slaughter, he fell upon the sons of Nadir, a tribe of Judaizing Arabians, routs them and pursues them into Syria, showing the country far and near his victorious troops when they thought him in a manner ruined. This happened in the fourth year of his stay at Medina.
All Arabia was alarmed at this last success, and the Jews and several other nations combined with the Coreischites against Mahomet and drew out ten thousand choice men to fight him. The prophet musters what forces he could from Medina, Agra, and Sarak, and determines not to be enclosed in any town, nor engage too near Medina lest the vicinity of a refuge might take of the courage of his soldiers. He marches forward to encounter them, but finding a consternation in his army, he declined the engagement and put in practice a stratagem which was new in Arabia. He caused his men to encamp and drew a strong line of circumvallation betwixt the Coreischites and the Jews, and gaining over one Naimus of the tribe of Gatfan, a potent man in the army of his enemies. By his means, those of Mecca and the Jews so quarreled that they broke up their army and departed without doing anything. Six of the Muslemin were slain in this expedition, and three of the infidels, two of which were **< 97>** slain by Ali in duel in the view of both camps, to the great encouragement of the Mahometans and terror of their enemies, one of them being a very valiant captain and as it were the soul of their army.
Great was the renown of Mahomet, who had, without any loss or hazard, caused so powerful an army thus to dislodge and dissolve. He pursued a brigade of them, besieged them five and twenty days, reduces them, cuts off the heads of 670 of the men, and distributes the women and children among the Muslemin for slaves. This happened in the fifth year of his stay at Medina.
SIXTH YEAR
In his sixth year he resolved to carry the war into the country of the Coreischites, gains several considerable victories as he goes, and marches to Hadibia, a place near Mecca. The Muslemin had their courages inflamed by this succession of victories and by the sight of the Caaba. The inhabitants of Mecca were disconsolate to see the danger so near them, and that so many expeditions against Mahomet had proved fruitless, and, being divided and distracted among themselves, knew not what to do. But the generous prophet, continuing in his resolution of taking Mecca rather by surrender than fight, comes to a treaty with the Coreischites which ended in a cessation of hostility for ten years to come. One cause of this truce was that if Mahomet or any of his followers had a pious intention to visit the Caaba, they might come without arms and perform their devotions.
Hereby the prophet gained many advantages. The inhabitants of Mecca were convinced of his strength and of his generosity to them and devotion to the Caaba. He had the opportunity of sending in emissaries under pretence of devotion and the glory of having faced and brought to composition the capital city of Arabia. His army hereupon inaugurates him solemnly of their own accord (without any solicitation of his) to be their _xeriff_ or prince.
He retires from Hadibia and marches against several little territories that had been in arms against him who, being excluded the cessation and deserted by the Coreischites, were easily subdued and forced to pay him an annual tribute of half their dates and to hold their lands at the pleasure of the conqueror. His armies now seemed invincible throughout Arabia, and the Jews, as well as others, were subjected upon the same terms. In this expedition Ali signalized himself at the battle of Chaibar where he seized on the gates of the town and managed them on his arm like a target. This was the event of the seventh year.
EIGHTH YEAR
In the eighth the Coreischites, finding the prejudice of this truce and that whilst they stood neuters their allies were destroyed, renounced the cessation and drew upon them the forces of Mahomet. He marcheth thitherward by **< 98>** easy journeys, and many of the great men, being no less sensible than the inhabitants of their weakness, turned, some really, and some out of fear, to Islamism. His uncle Abbas and Abusofian were of that number: the first withdrew out of the town to Mahomet, the other remained behind to render the prophet more important service by his stay. Mahomet entered Mecca without any opposition, having first proclaimed that all who retired to the house of Abusofian, all who shut their doors and offered no injury to the Muslemin, and all who fled for refuge to the Caaba should be secure. His entry seemed rather a procession to the Caaba than a triumph. Aljanabus tells us that upon his approach to the temple all the idols (even the great Hobal) did prostrate themselves unto him.
In the Caaba he broke in pieces the wooden pigeon which was there and cast it away, Ali being busied (as were the rest) in demolishing those idols. And, not being able to reach one that stood aloft, the prophet suffered him to stand upon his shoulders till it was done. The inhabitants immediately became Muslemin, but the rulers and chiefs who had been his enemies, and either scorned to believe him or despaired of mercy though they should do it, he put all to death, it being inconsistent with the absoluteness of the monarchy which he designed to permit a hereditary nobility. The people, thinking themselves happy in their own safety, did the less mourn for those which were slaughtered, and, whilst their minds were set upon a peaceable enjoyment of their own, they forgot revenge, nor did they think of remote consequences, and that whilst every one singly courted their prophet and emir, they did introduce an universal servitude.
Now we see Mahomet possessed of the metropolis of Arabia the Desert. Yet doth not all this power and series of prosperous attempts infuse into him new pride or outward grandeur: whatsoever fortune hath put into his hands doth only enable him to do more good, to bestow more alms and more to advance the glory of God. His Muslemin seem to be all animated with the same spirit, nor do the inhabitants of Mecca find themselves governed by an emperor and an army but by a prophet. It is not arbitrary power but the ostentation and abuse of it that makes it odious and tyrannical.
Whether it were the consummate wisdom of Mahomet that continued him in this equable temper, or the source of the mutability of human affairs in a man who had tried such vicissitudes, or whether the custom of dissembling was become natural to him, or that old **< 99>** age had secured him from those sallies which indiscreet youth is subject to, I know not. But it was very advantageous for him that this prosperous revolution did make no change for the worse in his demeanor. For there happened an insurrection within a few months after the destruction of the idols in the Caaba, which reduced his new religion and government into jeopardy and would have dethroned any but the prophet. Those who would not stir in behalf of Mecca and the Coreischites: whether it be that vulgar heads will not believe things till they fall out or out jealousy that they should run into a certain inconvenience of being subject to the Coreischites to avoid the uncertain apprehensions of Mahomet.
When they were ascertained that the idols were destroyed and the substance of the Arabian religion changed, some shadows and circumstances only remaining, the Thakifii and Hawazine Arabians take arms under one Melic, the son of Aufus. Their number was not formidable, being but four thousand in all, but they were Arabians and animated with zeal and revenge. Mahomet was determined to act securely and not obscure the last actions of his life by rashness or want of foresight. He drew forth an army of twelve thousand men, whereof ten thousand were veterans and the rest captives of the Hawazines and people of Taiph, and prepared to encounter them in the valley of Horam. Whether it were that fortune is seldom constant to her greatest darlings or that the Muslemin acted too securely presuming upon their numbers, tried valor, and discipline, so it happened that the idolaters totally routed them and pursued them to the gates of Mecca, where the gallant prophet, accompanied only by Abbas and Abusofian, made a stand and, leaning on his javelin, persuaded some of the Muslemin to rally and by the appearance of a new charge to put an end to the furious chase of the idolaters. The prophet must be owned to have showed the highest courage and prudence who could retain his judgment and valor amidst so universal a consternation, disorder, and flight.
The idolaters withdrew their forces, either not having strength enough to besiege the town (which is as big as Amsterdam), or that few know how to improve advantages when they transcend expectation, or that an undisciplined army is not fit to prosecute a tedious siege, or that they were forced to it by the sterility of the country and want of water (whereof there is little good and potable thereabouts), Melic retreated and gave Mahomet respite to infuse fresh resolution into his dismayed troops. The angel **< 100>** Gabriel presently brings him a _surat_ that no enemy could be safely despised, nor any human strength presumed upon; that albeit infidelity and idolatry were things odious to God, yet pride and presumption were also abominable to Him; that the Muslemin were puffed up with a conceit of their own strength; and that God had defeated them now to convince them of the necessity of His aid and blessing in all designs; that, if they would abandon the opinion of their own worth and puissance, He would repair their loss by sending an invisible legion of angels who should fight in their behalf.
This oracle gave new life and vigor to the Muslemin and made them more punctual in their obedience to their emir and prophet. He determines to fight Melic before the noise of this rout should run through Arabia, knowing that new conquests are always unsettled and that the minds of men, quickly reduced to obedience, are as soon lost and that since prosperity was the foundation of his apostleship, adversity would overthrow it. His success was such in the second engagement that with the loss only of four Musilmen and the slaughter of ninety infidels, he gained a complete victory and made himself master of all their riches: six thousand head of cattle, twenty-four thousand goats, forty thousand sheep and four thousand ounces of silver, and their wives and children taken prisoners. The infidels yielded themselves tributaries and vassals on condition to have their wives restored, and Melic rendered himself to the prophet and became a Musulman whereupon Mahomet restored to him all his possessions.
NINTH YEAR
In the ninth year of the Hegira he had no great difficulties to contest with: the remaining wars did rather exercise than endanger his forces. His followers became more fixed and endeared to him, and they who had embraced his religion out of fear persisted in it out of affection and conscience. They no longer resented the destruction of their idols, seeing that success attended on the followers of the great God. Many that had been his obstinate enemies became converts, and the princes of Dauma and Eila became his tributaries. He disarmed those which he suspected, leaving his forces in excellent discipline under good and faithful commanders, and placed a confidant of his governor of Mecca. He went to visit his old friends and assistants at Medina.
In the tenth year of Hegira he received no tidings but such as confirmed the daily growth and progress of Islamism. He received continual **< 101>** addresses and submissions from the new converts, and all Arabia seemed at his devotion. But that in Yemen there arose one Mosalleina who pretended to be his associate and partner in the apostleship and found many followers. But the sage Mahomet despised this new imposture, either because he thought the same cheat was not to be acted twice with success in so short a time, or that he fancied the renown of his valor and conduct was enough to retain all men in his obedience, or that he despised the luxury and effeminacy of the inhabitants of Yemen and imagined that such who had much to lose would not hazard their estates nor endure the hardships of war for a new religion, or whether he did suppose that petty insurrections contribute to the establishment of an absolute sovereignty. It is certain he despised Mosalleina.
_CHAPTER 7_
MAHOMET'S LAST PILGRIMAGE, HIS DEATH, AND BURIAL
E DID NOT SUPPRESS this year nor the next, which was the last of his life, but to testify his veneration to the Caaba, to return thanks to God for his success, and to give an example to the Arabians in what manner they should continue their pilgrimages of devotion to Mecca. He, in the company of all those who had followed him to Medina, began the most illustrious procession that ever was in the world. As you might at any time have seen these conquerors of Arabia at work, mending or making their cloths, cultivating rice, picking of oats, or selling parched peas, so now you will see them marching a long pilgrimage with a train of seventy-two thousand persons, with a garb and postures that would better suit with madmen than those who were to give laws to the universe. But there are different ways of getting esteem, and, in a diversity of circumstances, some lose their empires, glory, and riches by the same courses wherewith others acquire them.
Medina was the place whence Mahomet determined to begin his pilgrimage, and the fame thereof soon gathered together seventy-two thousand persons (men and women) to complete the train of the apostle. They say that Noah, when he went into the ark, took along with him seventy-two persons, and, for that reason, it is requisite the pilgrims of Mecca should amount to the number of seventy-two thousand, and that number ought to be so exactly observed that more must not be received as such that year, but that number must be complete. Otherwise (they say) the angels would be obliged to make up what were wanting, and it were want of respect to those spirits to put them to that trouble. When they set forth they are covered only with a shirt, nay, some go naked down **< 102>** to the waist. They march continually and after a particular fashion, for they are obliged to go after the rate of a trotting horse or rather of a camel galloping, and that with such earnestness that they hardly take the leisure to eat, drink, or sleep, out of an imagination that the sweat forced out of their bodies by that violent motion carries away with it all their sins and cleans them of all impurity. The women who might not be able to bear the inconveniences of such a march have the privilege to swathe up their breasts with a scarf.
The illustrious pilgrims continued thus their procession from Medina till they arrived at Mount Arafat, which is half a day's journey from Mecca, by the foot whereof lies the ordinary road to that city. Here the prophet related to his followers the sacrifice of Ismael and Hagar, adding that they must divert out of the way, as was usual of old in those pilgrimages, and fulfill a commandment of Abraham's to cast several stones in a certain place at the devil, in imitation of their father, and to drive him away that he might not instill any wickedness into them or divert their thoughts from the pure worship of God during this sacred pilgrimage: that though they seemed only to cast the stones in Akabah (the name of the place), yet indeed each stone did hit the devil and either wounded his face or broke his back; that not only Abraham, Hagar, and Ismael had found this way to avenge themselves on him, but also Adam in the Valley of Mine did pelt the devil with stones and put him to flight.
From hence, Mahomet conducted them up to Mount Arafat to view the place where Ismael was to have been sacrificed, and here they spent that night in a great devotion. At the foot of Mount Arafat, the people of Mecca had prepared a multitude of white rams: every one of the pilgrims bought one and carried up with him which there they slew (every one his own) in imitation and as a memorial of their fathers, Abraham and Ismael. After each hath killed and dressed his oblation, they eat a little thereof and give the rest to the poor (which flock thither in great numbers to receive alms) without reserving so much as the skin, saying that neither did Abraham when he performed his sacrifice. Towards the dawning of the day, they come down and go to the city of Mecca, where the high priest or chief imam of the Caaba makes procession conducting through the chief streets a camel which is appointed for sacrifice. The hair of the camel they account a very precious relic, which so that the pilgrims all throng as near as they can to the beast to snatch some **< 103>** of his hair which they fasten to their arms. After the priest hath walked the camel sufficiently, he leads him to the market place where the _Daroga_ or judge of the town, attended with other officers, kills him with an ax. As soon as the camel is dead, all the pilgrims endeavor to get a piece of him and throng so confusedly with knives in their hands that these devotions are never concluded but many pilgrims are hurt and killed which are reckoned as martyrs.
After this ceremony, the prophet leads the pilgrims to the Caaba. At their first approaches, they with a great shout proclaim the _Allah Ekber_ or "God, God, the great God," and then the double testimony of " _la Illah Mahumed resul Allah_ : the God and Mahomet his Apostle." They went seven times round the precincts of the Caaba, but with a variety of postures and some difference of pace. For thrice, they went a good round trot, and four times they walked gravely about it, agitating their bodies and shrugging their shoulders in a strange but Arabian manner, especially as they passed between the two stones of Safa and Meriah, the sight whereof reminded them of the sacredness of the Caaba and the judgments of God against impiety and irreverence. After all this, they all came to the _Hagiar Alasvad_ , or Black Stone, which he told them was brought from paradise and changed its color by reason of the sins of men. Here the prophet devoutly said his prayers, kissed the stone, and begged pardon for his sins and prosperity for the future, enjoining all his followers for ever so to do. It is advanced from the stone seven handfuls or a cubit and a half, being fixed in the wall, and under it every pilgrim did most submissively creep, weeping and deploring his sins and praying that he might arrive at paradise. Then they went to the other stone whereon Abraham stood when the Caaba was building. It lies in the midst of the court of the Caaba enclosed within an iron grate and into the prints of his feet they pour some water fetched from the pool Zamzam. And, having said their prayers, they drink it up thence and depart carrying home with them in a vessel some of the sacred water of Zamzam.
Thus Mahomet performed the pilgrimage and left an example for his followers how to continue it. It was the policy of the prophet not to reject all rites that had been abused to idolatry, nor to affright the Arabians into a rebellion or irreligion by making a total change in the substance and ceremonies of their devotion. The casting of stones was an usual rite in the honor of Mercury; to run naked or with no other garment than a loose linen covering was a part of the worship **< 104>** of Camosh. The other ceremonies appertained to Baal-Peor which there had been the deities of the Ismaelites. Nay, he continued the pilgrimage upon the same day that it had always been held, upon (viz.) the tenth of the month of Dulhagiiah. So he retained circumcision which was nationally used there always, as also the fast of Ashura was of ancient observance among the Coreischites. In like manner, their washings and rites of cleanliness were old usages which he confirmed. I am confident that I have read that the fast Ramadan, which lasts a month, was of an original more ancient than Mahomet; at least the Sabii kept one of thirty days. I meet with a particular reason, given by Ebnol Ehir, why Mahomet did undertake this difficult and laborious pilgrimage: and that was to convince the idolaters that his followers were as hardy and able of body as the other Arabians and that neither the scorching heats of Medina nor strict diet and discipline had any way enfeebled them.
For my part, I am apt to believe that he retired to Medina that he might not seem to relinquish and scorn his old assistants, or thinking himself safer there than in his acquests at Mecca where he had put several eminent persons to death, or to keep his men in their former discipline which might be corrupted at Mecca, or to show the Arabians that neither his victories nor his power had altered him, that he designed no monarchy but to conserve the repute of being apostle of God. For he did not abandon his deity dignity, nor the real exercise of it, but the show and appearance thereof. And, although devotion might engage him to the pilgrimage or a design to fix the people in a good opinion of him, by resettling after so many broils the usage and freedom of pilgrimages to the Caaba, yet I do believe that he thought it also a part of wisdom to visit again the dominions and not to invite the envious or ambitious to create new disturbances by his being absent so long. Besides that, the solemnity of this procession did manifest unto the people of Mecca that the demolishing of the idols had neither abated the resort, emoluments, or honor of their city which was now assured to be in a manner the metropolis of all the Mahometans.
Mahomet, having performed the pilgrimage and instructed the Muslemin in all points of their religion and worship, retires again to Medina, either out of pretended modesty or self-denial or to keep those countries **< 105>** which bordered upon Yemen in obedience. For not only Mosalleina had formed a considerable party thereabouts, but in the eleventh year of the Hegira one Aswad Absites declared himself a prophet of several provinces and cities in Arabia Fælix and had become very formidable. But a Musulman called Firus of Dailam assassinated him in his own home and put an end to those troubles. These little insurrections could not create any trouble to the prophet who was now possessed of the Caaba and had at his devotion so good an army well commanded and a rich exchequer.
But the end of his life drew on, and he who had begun so late those vast attempts, and had proceeded therein so leisurely, had only the satisfaction to see all Arabia, part of Syria, and Mesopotamia reduced under the obedience and become subjects to the religion and empire. A slow fever seizeth this great spirit and bereaves him of life. It was originally occasioned by poison given to him in a shoulder of mutton hashed at the taking of Chaibar in the seventh year of the Hegira by one Zeinaban, a young maiden, the daughter of Alhareth a Jew, who being asked why she did so horrid an action, briskly replied: because thereby she should either discover him to be a true prophet and apostle of God or free the Jews from the persecution of a fierce tyrant. No sooner had the prophet eaten of the mutton but he said this shoulder of mutton tells me that it is empoisoned and Bashar, a Musulman who did also eat of it, died presently.
Mahomet lived above three years after it and rendered great testimonies of his ability of body as well as prowess. Yet he did frequently say, and particularly in his last sickness, that the morsel which he ate at Chaibar still molested his stomach, and, when the mother of Bashar made him a visit as he was dying, the prophet said to her: "O mother of Bashar, I now feel my heartstrings break by the poison I swallowed with your son at Chaibar." His sickness continued thirteen days only. He died in the climacterical year of his age being sixty-three years old. As writers differ about his age so they disagree about the day of his death, which some say was the same with his birthday. It is confessed it was about that time.
When the prophet lay upon his deathbed, he called for pen, ink, and paper and told them he was very sensible how the Coran was written and that he was afraid lest they might fall into **< 106>** sundry errors and mistakes after his decease. And therefore he purposed to write them a treatise to preserve them from the danger of such an inconvenience, and (either I am strangely deceived in my memory or I have read that he told them then that were three thousand errors in the Coran, which it was the will of God at first to have divulged and now it was his will the Muslemin should be undeceived). But Omar and some others being by, and either tender of the repute of the prophet (and their own credit and safety), lest he should reveal anything which might derogate the Alcoran, or being apprehensive that the prophet was become frantic with his fever, they forced any to bring him pen, ink, and paper. And so he died leaving no such _surat_.
No sooner was he dead and the report thereof spread abroad in the city but the multitude flocked to his house and cried: "Do not ye bury the apostle of God. He is not dead. How could he die who is to witness for us to God? No, no, he is but withdrawn aside as Isa was. Forbear to bury him." With these did Omar join saying: "If any one aver that Mahomet is dead, I will kill him presently. He is not dead but assuredly conveyed away as Isa was, or rather as Moses was, when the people missed him forty days." But Abubacr wisely composed the tumult shewing out of the Alcoran that it behoved Mahomet to die as well as the other prophets, adding that if any purposed to worship, the prophet was certainly dead, but the God of Mahomet was the living God.
There arose a further contest where the prophet should be buried. Some said at Jerusalem, where most of the prophets were buried. Others said at Mecca, but the inhabitants of Medina prevailed to have him buried there. So they buried him in his own house, the house of Ayesha, under his bed, he having formerly told them that the prophets were constantly buried in the place where they died. There is a stately temple since built by the Mahometans upon the place and richly adorned within and without. Within this temple there is a chapel with a roof contrived by an extraordinary architect. In it there is a tombstone called _Haijar Monaour_ , or the bright stone, said to have appertained to Ayesha, the wife of Mahomet, and within that is lodged the body of the prophet. Nor is the tomb suspended in the air by loadstones or any other contrivance, but is placed in the floor and hung about with rich hangings of silk and **< 107>** gold and environed with rails of iron sumptuously gilded. The Mahometans do make pilgrimages to it as to the Caaba, the prophet having promised happiness to such as do this. None can approach the tomb, but they devoutly kiss the bars enclosing it.
He was forty years of age when he first began to pretend to the apostleship at Mecca. Thirteen years he continued there, and ten years of the Hegira were passed when he died. How far he carried his conquests in his lifetime is uncertain: some say he subdued not only three Arabias but Egypt, Antioch, Syria, Armenia, and all Palestine (except Jerusalem), that he vanquished the Emperor Heraclius in a battle slaying fifteen thousand of his men, others that he only secured himself of Arabia and part of Mesopotamia and Gaza together with the passages of Mount Sinai. Some would even subject Africa to him also. But this is certain, that, according as he by his travels had informed himself of the weakest parts of Christendom, he directed his successors to trace their steps with their victorious armies and first to Palestine, then Egypt, then Africa and Spain, and, within about fourscore years after his first appearance, you may read how his Musulmen were possessed of all those countries, and as much more on the side of Persia and Indostan, and which would seem incredible did not all agree to it.
Within twenty years after his first rise, AD 643, you read of nothing but of vast armies of the Greeks being beaten, Damascus, Jerusalem, Antioch, Egypt taken, and seventeen hundred sail of ships of theirs covering the seas, and terrifying Constantinople, Cyprus, and Rhodes. For discipline, the Spartans or old Romans never equaled the Saracens, and their empire did not grow by slow degrees. No sooner was their sun elevated above the horizon but it was in its meridian. And when it will decline no man knows.
CONCERNING THE JUSTICE OF THE MAHOMETAN WARS, AND THAT MAHOMET DID NOT PROPAGATE HIS DOCTRINE BY THE SWORD, WITH A VINDICATION OF MAHOMET'S CARRIAGE TOWARDS THE CHRISTIANS
T IS A VULGAR OPINION that Mahomet did propagate his doctrine by the sword and not only compelled the Arabians at first to receive his doctrine but obliged his successors by a perpetual vow or precept to endeavor the extirpation of Christianity and all other religions. But how generally so ever this be believed, and how great men so ever they be who assist it, this is no other than a plain mistake. It is most true that Mahomet did levy war in Arabia, but it was **< 108>** under the pretense of restoring an old religion, not to introduce a new one. He taught his followers to abolish idolatry everywhere and that all the world was obliged to the profession of these truths: that there was one God, that He had no associates, that there was a providence and a retribution hereafter proportionate to the good or evil actions of men. But that all mankind were to be enforced to the profession of his religion or that he compelled any thereto is a falsehood.
It is plain that many Christian doctors have held that Christianity may be enforced and that it is a just cause for one prince to invade and conquer another's territories to propagate the true religion. Thereby, they say that if one king may chastise and reduce under obedience the subjects of another, they may do as much for the Lord paramount of the universe. That kings are the ministers of God to execute wrath upon such as do evil, and that, being the viceroys of the almighty, they ought to assert the glory, cause, and sovereignty of God to be everywhere submitted to, there want not precedents for such actings. Amongst the Jews, Hyrcanus compelled the Idumeans to be circumcised and turn Jews, and all Christian annals furnish us with instances of the like nature. It is likewise true that several Christian doctors teach that a nation guilty of enormous sins may be invaded and that it is a just pretense of war to reclaim them from notorious wickedness, it being lawful to compel them to observe those laws of nature, whereto God hath originally subjected them.
These and sundry other titles of war are treated of and maintained by the divines who write concerning the Spanish rights over the Indians. But though the Christian doctors and some popes have urged them, and thereby prepared apologies for the Mahometans, I do not find that Mahomet proceeded any further in Arabia the Desert than to exterminate idolatry everywhere, but not to force men to the profession of Islamism. He himself did give letters of security and protection to the Jews and Christians in Arabia and never used any violence to them upon the account of religion. At Medina such Jews as peaceably paid tribute continued unmolested, though mortally hated, until the days of Omar, the son of Alchittabi, who expelled them out of Arabia, **< 109>** he being told that Mahomet had prohibited that two different religions should be tolerated within that country, the seat of his empire. But though they were thereupon expelled thence, which was rather an act of civil prudence than religion, yet were they never compelled to Mahometanism, nor banished his other territories.
The same Omar did give the Christians his security upon the taking of Jerusalem: "In the name of God, merciful and gracious, from Omar the son of Alchittabi to the inhabitants of the city Elia: Security and protection is granted as to their persons, children, wives, estates, and all their churches, that they be neither destroyed, alienated, nor prohibited the Christians to resort to." And when Amurcus the Saracen general under Abubacr did besiege Gaza, he made this declaration to the Christian governor thereof:
Our Lord hath commanded us to fight and conquer you except you will embrace our religion and so become our friends and brethren pursuing the same common interest with us, so you will have us your faithful allies. But if you will not accept of these terms, then submit to pay us tribute yearly, yourselves and your posterity forever, to us and our successors, and we will protect you against all opposition whatsoever, and you shall be in league with us. If you agree not hereunto, then the sword must decide our rights, and we will not desist until we have subdued you and put in execution the will of God.
By this declaration, it is manifest that the Mahometans did propagate their empire, but not their religion, by force of arms, and, albeit they did not permit others than Musulmen to enjoy any military or civil commands in their territories or entire conquests, yet the Christians and other religions might peaceably subsist under their protection if they paid the tribute demanded. In Spain the _Mozarabick_ Christians always lived quietly and safely under them, and others in their other kingdoms and dominions, an inviolate justice being preserved towards them; and though the rich and potent nobility and rulers were destroyed or reduced to nothing, which was done to prevent future rebellions, yet it is observed by Scaliger—and it is an assured truth—that the vulgar Greeks live in a better condition under the Turk at present than they did under their own emperors when there were perpetual murders practiced on their princes and tyranny on their people. But they are now secure from **< 110>** injury if they pay their taxes, and it is more the interest of the princes and nobles than of the people at present which keeps all Europe from submitting to the Turks.
The decree of Mahomet in his Alcoran concerning the Moslemicall wars does run thus: Make war upon those which do not believe in God, nor that there is a day of judgment, nor that those things are forbidden them which God and His prophet have forbidden unto them. Nor do administer due justice unto them who have taken cartels of security and, being subdued, do readily pay the appointed tribute. Hereby, such as have taken cartels of security he understands such Jews and Christians as had yielded themselves to Mahomet and taken protection under him. Thus Mahomet Ben Achmed expounds him. Elmomin, who collected his "History of the Saracens" out of the best Mahometan writers and was himself secretary of state to one of their princes, avowed that Mahomet did give protection and security to the pagans, Magicians, and Jews, and Christians also which swore fealty to him and paid him yearly tribute.
Moreover, that he sent Omar to the Christians to assure them that they should live securely under his dominion, and that he would esteem their lives as the lives of his Moslemin and of their goods as the goods of those others: to this purpose, there is extant a compact or league betwixt Mahomet and the Christians, published in France by Gabriel Sionita and reprinted by Johannes Fabricius a Dantzicker, which is by him affirmed to be most authentic, and mentioned by Selden, though Grotius takes it to be but a figment of the Christians that they might gain favor with the Moselmin. I shall not transcribe it because I think it to be suppositious. It is published in English by Mr. Rycaut in his "Relation of the Turkish Government" liber 2, caput 2. The sum of it is that the Christians submitting to him and paying their tributes duly shall live and enjoy the liberty of their religion without any molestation, and that there shall be a perpetual amity betwixt the Musulmen and them.
There are also sundry passages in the Alcoran wherein he permits the unbelievers to hold their own religion and declares that every of them—Jew, Christian, or other—might be saved if he hold that there was one God creator, a day of judgment, and lived justly and uprightly. In fine, it is **< 111>** manifest that Mahomet and his followers do make war not to enforce others to their religion but to enlarge their empire and reduce all under their subjection. This is the direct injunction of the Alcoran in the place already mentioned, and is avowed by Selden and Salmasius, and albeit they do call the territories of the Christians _Dar Elharb_ , or the enemy's country, and think they have a perpetual right to make war upon such, yet it is only upon the grounds aforesaid. So that the controversy betwixt them and the Christians is not whether religion may be propagated by arms, but whether it be lawful to make war on others nearly for the enlargement of empire. And herein Mahomet, as in other cases, hath the Jews for his defenders whose opinion for the affirmative is generally the same with his as to this matter. And it was heretofore the sense of most emperors. And if the Christians do not own the same sentiments, few princes do believe other, though they cloak their own ambition with different and more specious pretences. Maimonides and the Jews call such wars "praelia majestatis contra gentes alias, ut dilatet terminos regni & augeat magnitudinem ejus una cum famâ."
As Mahomet seems to have deduced his laws for an offensive war from the Jewish doctors, so did he derive his laws for conquest out of them. If he did everywhere destroy idolatry and not only within Arabia and the Caaba, this is agreeable to the Jewish rulers who did oblige the conquered nations to become stranger proselytes and submit to the seven commandments of Noah, one whereof was against idolatry. But if he did only condemn and not extirpate idolatry, if he only conquered the Gentiles and left them tributaries with permission to continue idolaters (which is probable), since the Turks and Moguls do so, and they would not do it were there any passage in the Alcoran or any _Assoniah_ or tradition to the contrary, then likewise, had he the Mosaical Laws for his precedent. For by the Mosaical direct law against idolatry, Deuteronomy 12:2, Exodus 34:13, such false worship was only to be extirpated out of the land of promise, and not in other acquests. It was by force only of an intervenient and prudential law of the Jewish Sanhedrin that idolatry was well to be exterminated out of the conquered territories lest thereby the Jews should be perverted from the worship of **< 112>** the true God thereunto.
So that Mahomet, in that he commanded all other religions to be excluded Arabia the Desert (for they are not prohibited elsewhere), did imitate the divine institution for the holy land, and in permitting variety in other places had the same great example for his precedent. But this is not all. The demeanor of the Saracens upon a victory is entirely consonant to the Jewish laws of war, as a Jew taken in war did not become a slave to a Jew, but all other captives did become slaves and were at the disposal of the conqueror, to be sold or employed in what service he pleased. So neither can a Mahometan be enslaved by a Mahometan, but all other captives are at the disposal of the conqueror and he may employ or make sale of them as he pleases. As to those that were not taken captive in war, but yielded themselves by surrender on articles, it was a perpetual law among the Jews that all places subdued by them should be reduced under bondage and the inhabitants be retained in servitude, though not absolute slavery. They were to live in an abject manner, paying a great submission and an arbitrary tribute to the victor, and never to bear any command in Israel, and were also liable to sundry personal services, occasionally in the building of public edifices, fortifications, the temple, et cetera. So the nations that, upon invasion, did render themselves were to be brought into servitude and made tributaries (Deuteronomy 20:10–11). Thus upon the children of the Ammonites and who had yielded to the victorious Israelites upon terms did Solomon levy a tribute of bond service, and they wrought personally at his public buildings (1 Kings 9:21, 2 Chronicles 8:7). So did David (1 Chronicles 22:2).
As the Mahometans do herein conform to the Mosaical Law so that they do nothing herein that is repugnant to the most exact rules of war, I find Grotius to justify. And if there be no injury offered thereby to them that are made captives, or subdued, I do not comprehend any reason that the Christians should complain of their hard usage under the Mahometans.
But to proceed. How equitable that Mahometan tenet is that wars are just for the enlargement of empire I will not determine. **< 113>** It is an opinion not so barbarous or uncouth but that old Greece and Rome, as well as Jewry, will avow it. But most certain it is that Mahomet did render a great testimony of his wisdom by introducing it among his followers, for it conduced much to the vastness of that empire which he designed, since the Mahometans could never want a pretense for war against others. It conduced much to the public tranquility of that empire which, being erected upon a military prudence, would run into civil broil and confusion were not there that Mahometan precept enjoined them to be continually solicitous to enlarge their territories. It brings also this additional benefit that since great kingdoms ought to know no period of their growth (it being therein as in natural bodies which when once arrived to a determinate pitch immediately decline and go to ruin), their opinion contributes to the subsistence and perpetuity of their monarchy. And how repugnant so ever the continuance of slavery be unto Christian charity, it is not absolutely unlawful or any more repugnant thereto than is war itself.
Nay, it is a moderation in the effects and rights of war, and I am sure Christian statesmen such as Busbecq have condemned the European policy for relinquishing so wise and so beneficial a practice. For the advantage to particular soldiers doth add to their courage, as also doth the continual sight of those they by servitude lead in a continual triumph. It makes the same enemies always despicable to them, whose ancestors and kindred they see every day to be their slaves. It makes them more desperate in fight, the indignities of a lasting slavery seeming worse than death to a valiant person, besides the profit that accrues to the public thereby, it being in vain to expect that such extraordinary pieces of architecture and fortification should ever be performed by the moderns as the ancients effected by their slaves.
To conclude: though the principles of the Christians seem to condemn slavery, yet in Portugal and other places it is frequently practiced, and perhaps the Christian laws and customs against such usage had no higher rise than ecclesiastical and civil policy, which the successors have indiscreetly (and not out of conscience duly informed) retained still. **< 114>**
CONCERNING THE CHRISTIAN ADDITIONS
HAVE ALREADY SHOWN that Arabia was the common receptacle for the persecuted Jews and Christians of all sorts and sects to retire unto. I must add thereto this limitation: that persecution works this effect upon all religions that such as then do continue firm and constant are more pious and devout than at other times, the indifferent and the vicious usually adhering to that religion which prevails and prospers. And the martial spirit seldom persists in that profession which excludeth him from employment. Upon this account, the Nestorians, Jacobites, the Arians, Arabians, and Judaizing Christians were of more exemplary piety in those days than the Trinitarians, either out of policy to ingratiate their partisans by such speciousness, and to inodiate those who oppressed so innocent and religious persons, or out of a real conscientiousness. The Judaizing Christians had the fairest pretences and were most tenacious of their customs and traditions, and by a series of persecutions from their first conversion until the days of Mahomet, and by the principles of their Christianity which was conformable to the self-denying doctrine of Isa and the discipline of the Essenes, they were a harmless religion such as might win much up the melancholy temper and suspicious humor of the Arabians. They were diffused from Pella in Judea to Cochab in the confines of Arabia toward Syria and so on to Yemen or Arabia the Happy and Chaibar or Mesopotamia.
The kings of the Arabians and their tribes, several of them, had been Christians, as I related: first Trinitarians, then Jacobites, and amongst several of their tribes as that of Gassan in Syria and its confines and of Rabia and Codan, and many other tribes had their churches and converts as the Jews had also in others towards Arabia the Happy and elsewhere. As Mahomet seems himself to have been a Judaizing Christian and a great honorer of Isa, so he always expressed a great reverence for them. And it is concerning them that he says that Isa their prophet should save them in the last day. But as to the Trinitarians (that is those Christians who advanced Christ and the Holy Ghost or the Virgin Mary into the deity whom Mahomet calls _Almushrikuna_ or Associators), these he universally judges to hellfire in his Alcoran as well for their tenets as wicked lives. That the Arabian Christians were men of just and strict deportment appears hence: that Mahomet **< 115>** saith of them that one might safely entrust them with any sum of money and they would restore it again. But the Jews were such villains that no man could trust them with a penny.
It is known how severe enemies the Christians were against all images and pictures. The Mahometans were so likewise, insomuch that upon their coins they stamp some pious sentence and not any effigies. And so bigoted were they as to this point as that not only the former Saracens but the Turks, till of late, would not receive as current any Christian money with an effigy stamped thereon. And they universally demolish all pictures and images where they conquer. Mahomet, though he did not force the Christians to his religion, yet he told them that such as believed in Isa ought to live according to his precepts with great humility, piety, and unconcernedness for the pomp and vanities of this world, that they ought neither to seek nor retain honor nor riches or go to war, or intermeddle with state affairs—these things being inconsistent with the doctrine of Isa; such as pursued those courses not being really Christians, since Christianity lies not so much in open profession of reverence and worship as in the practice of a holy life.
I shall here insert the opinion of Grotius concerning the Eastern Christians when Mahomet arose that it may appear with how much justice Mahomet did exclude them from the number of Christians:
Grotius: _De Veritate Religionis Christianæ, liber 6_. The true and simple piety which flourished among the Christians during their oppressions and persecutions began to grow out of fashion and to be disused after that Constantine and his successors had not only indemnified but honored Christianity, the world being as it were crowded into the Church. The Christian emperors ceased not to multiply needless wars, the bishops raised great quarrels and tumults in order to their particular advancement, and as at first, it proved calamitous to mankind that the tree of knowledge was preferred before the tree of life, so then was it almost fatal that curiosity of speculations was more valued than true godliness and religion because modeled and methodized into an art. The consequence whereof was that as heretofore in the building of Babel, the erection of too high a structure gave an occasion to the confusion of **< 116>** languages, so the elevated disquisitions of Christian doctors produced now a multitude of strange terms and unknown forms of speech, and afterwards discord. The commonalty, being amazed and perplexed with these inexplicable difficulties, and not knowing which to adhere unto or what to believe, and seeing each party establish its opinion upon scriptural interpretations and allegations, they began to complain of the obscurity and ambiguity, and at last turned their indignation into a perfect hatred of the Holy Writ.
Religion now was no longer terminated in the purity of the mind, but in the performance of certain outward ceremonies, as if the law, rites, and Judaism had been only varied not abolished, and men testified their Christianity not so much by amending their lives as by exterior and bodily gestures and humiliations and a zealous adherence to the party they owned insomuch that at last there were few real Christians, notwithstanding the multitude of professors. God did not connive at those sins and enormities but raised up the remote nations in Germany and Scythia to invade Christendom, which overran all like a torrent. And since the damages they did would not serve to instruct and reform the surviving Christians, Mahomet in Arabia began to preach a new doctrine which was directly opposite to the Christian religion, yet did, in words, contain a great part of the Christian practice. He was first credited by the Saracens, who revolted from the Emperor Heraclius, they having carried their conquests in a short time over Arabia, Syria, Palestine, Egypt, Persia, and afterwards into Africa and Spain. With their empire, they propagated their religion.
Nor is it a despicable plea of the Mahometans which is urged by Ismael ibn Ali, a Mahometan historian, that at the Nicene Council in the twentieth year of the reign of Constantine, there were assembled 2,048 bishops, out of which he chose 318 and adhered to their judgment in anathematizing Arius and so published a Christianity different from what had been taught before in the church. It is granted by the oriental historians of the church that such a number of bishops was summoned, as the Mahometans specify. So saith Eutychius and Josephus, an Egyptian presbyter, in his preface to the Arabic version of the councils. And it is confessed that all, excepting the **< 117>** 318, did vary from the Nicene faith. Nor is it plain that these 318 were all bishops. Some accounts represent it as if there were but 232 bishops and the rest presbyters and monks. If that council were so managed as it is most likely, the pretences of Mahomet are not despicable. And that Mahomet might with better color deprive the Christians of their honor, powers, and riches, and reduce them to their primitive condition of piety and poverty, it is by these sayings that he and his followers describe Isa.
Isa, the Son of Mary, peace be upon him, said thus: "He that greedily accumulates riches is like him that drinketh much salt water, the more he drinks the more his thirst increaseth and he desists not to drink until he bursts." Isa the Son of Mary said to John the Son of Zachariah: "If any man say anything concerning thee that is true, praise God for it; if he say any untruth, give unto God a greater thanks for that, and He will reward that disposition of mind in thee and add it to the catalogue of thy virtues, and thus, without any trouble, shalt thou multiply thy good actions." The world did once appear unto Isa in the shape of an old decrepit woman unto whom Isa said: "'How many husbands hast thou had?' And she replied, 'The number of them hath been such that I cannot tell them to you.' 'And are they all dead,' then added he, 'and have left thee?' The old woman said, 'Nay they did not leave me. I destroyed them all.' Isa rejoined: 'It is a wonder therefore that any should be such fools as to dote upon thee and not consider how thou hast used all before them to be cautioned thereby against the love of thee.'"
In the time of Isa, three men traveled together who having found a great treasure said: "We want sustenance. Let one of us go and buy food." So one of them went and upon the way reflected also to buy some poison and mix it with the meat and by the death of his companions to render himself sole possessor of the treasure. So he empoisoned the meat. In like manner, those two which stayed behind did purpose to kill him at his return and appropriate to themselves the benefit of their discovery; whereupon they slew him and also died themselves by the poison which he had mingled with the meat. At that **< 118>** time Isa came by with his apostles and said: "This is the condition of the world. Behold how these three have suffered by it. These are deceased and left behind what they thought to be owners of. Woe unto him that seeks riches in this world."
These pretenses were first made use of by Julian the Apostate thereby to depress the Christians and, by this stratagem, did Mahomet gain to his party all the ambitious and military who could not satisfy their inclinations unless they embraced the Saracenical religion. And withal, by the same means, he rendered Christianity contemptible and weak and fit only for subjection or slavery. He did also hereby give a just color to the pretense of his followers that they were the _Elmumenin_ or true believers, and that their enemies of the Christians were not indeed Christians so that they seem in their wars not to oppose but vindicate Isa.
In reference to him, the Christians were not altogether so candid. You will find as little integrity in the Christian narratives of him as in any before, so that the most dissolute Christians published as great untruths in their times as they who passed for saints. It is acknowledged by all the learned, after a severe inquiry into the Arabian writers, as well Christians as Mahometans, that Mahomet was descended of the principal tribe of the Arabians both by his father's side and mother's side, as I have represented it. Notwithstanding this, "Mahometis Arabis vitam qui descripserunt, multi fuerent qui esse non uno modo illius res tradunt, in eo tamen convenient omnes, (sed male, ut habet Erpen. de Ling. Arab. p. 42.) quod eum à plebejo vilíque genere ortum, pauperibus parentibus, patre Ethnico, matre Judæa affirmant."
It is certain that the Christians which lived under the Mahometans as Elmacin and others do mention Mahomet with great respect as "Mahomet of glorious memory" and "Mahometes super quo pax et benedictio, et cetera." Whereas others have proceeded so far as to say that he was even Antichrist and found out the number of the beast 666 in his name, writing it—than which nothing can be more ridiculous. Is it not mere folly to spell a man's name wrong, and then to imagine mysteries in **< 119>** it: some Greeks _Μαχουμ έθ_ others _Μωαμμ έθ_, the Latins _Machumet, Machomet, Magmed, Maomethes_ , in Arabic, if rightly pronounced, _Muhammed_ or _Mohammed_ , which signifies much desired.
I find that his father's name was Abdalla and that, he dying, he was educated by his grandfather Abdolmutleb and, upon his decease, by his uncle Abutaleb; and that he travelled twice to Jerusalem, besides his expeditions into Egypt, Africa, and Spain; that he conversed with the Christians of all sects. And it appears that he understood very well all their tenets and the most solid foundations which they went upon. Nor was he unacquainted with the Jewish principles and Talmudical learning, as is likewise manifest by his Alcoran, the state of Arabia being divided into Jews, Judaizing Arabians, Judaizing Christians (at Cochab there was one of the seats of the Nazarene Christians), Jacobites, Nestorians, Arians, Trinitarians, Manicheans, Montanists, Sabeans, and idolaters. This gave him occasion and opportunity to examine and try all sects and sorts of religions. But the Christians have given him two assistants, one Abdalla a Jew and Sergius a Nestorian monk, and represent Mahomet himself as an ignorant fellow who could not judge of what they instilled into him, which is the reason of such gross errors in his Alcoran.
But I cannot find any Abdalla besides his father who was a Coreischite, nor any tutor or companion of his called Sergius. And if Sergius were a Nestorian, why did not Mahomet adhere to Nestorianism and teach that Isa was true God and true man under a double personality? Why did he not mention Nestorius and Theodorus Mopsuestersis or Diodorus Tarsoris as holy men or saints and condemn Cyril of Alexandria? Neither he nor any of his followers have done this as I know of. Had any such thing been, this would have happened. Nor would he have declined to cajole so great an interest as the Nestorians then were, being so strengthened by the decree of Chosroes and so extended that even the Christians of Saint Thomas in the Indies are of that profession. No, he was undoubtedly a great admirer of Isa as a prophet or apostle of God, and of this he makes so great and **< 120>** frequent declarations, and that Isa was his predecessor and taught the same doctrine that it is but justice to style him a Christian.
Nor do I believe that he did cajole or love the Jews at all and consequently would not form his Alcoran so as to please them. It is recorded that he did so detest them albeit out of compliance to the Coreischites he had retained the fast of Ashura on the tenth day of the month of Moharam. Out of hatred to the Jews he would have altered the observation thereof to the ninth day of the same month had he lived but one year longer. I believe that he was a convert to the religion of the Judaizing Christians and did form his religion as far as possible in resemblance of theirs. They and the Arians were his principal instructors, but not any of them had any hand in the penning of the Alcoran. For that was not made before he began his apostleship in any desert, but, published upon several emergencies, most at Medina where that he should have had any such assistance, it is unimaginable. For it would have been suspicious in that city and amongst his followers and so near confidants and secretaries as he there retained about him.
As for the Arians, it is manifest that the Saracens have always retained a veneration for their Saint George, bishop of Alexandria, whom yet they do not allow for a prophet but one of their saints or fathers, and his life is written by Kessaeus a Mahometan as if he were such. And Ahmad Ben Edris, passing over the Nestorians as a foolish sort of Christian heretics, brings in a fable concerning Paul as if he had deluded the world into an opinion of the deity of Isa and given a beginning to the heresy of Eutychius and the Jacobites; and that an Arian, or else a Judaizing Christian, whom he calls an _Elmunim_ , or true believer, did anathematize Paul thereupon saying: "We were the companions of Isa, we saw him, we are descended from him, he was the servant of God and his Apostle, he never told us otherwise." Also that Mahomet did meet with thirty of the descendants of this _Elmunim_ or orthodox person, they being retired into an hermitage, and that they owned his doctrine and professed Moslenisme.
Let us not then give credit to the aforesaid stories **< 121>** but inscribe another original to the Mahometical doctrines. The same Christians say that he was troubled with the falling sickness, that he took advantage from thence to pretend to raptures, and that upon his recovery out of those paroxysms he would repeat the _Bismillah_ , "In the name of God, merciful and gracious," and published the _surats_ of the Alcoran. I have refuted this story already of which there is no mention among the Arabians nor do they magnify their prophet for any such raptures. I add that the Alcoran was not given out in that manner that each _surat_ was penned according to particular emergencies. And it would have been an important piece of intelligence for the Christians to have published the way of contriving epileptic fits according each emergency of his and deferring them at other times. No less vain is the story of Mahomet's pigeon which used to eat peas out of his ear, and therefore, as a representation of the Holy Ghost, would resort to his shoulder and seem to inspire or commune with him. I will not ask whether it be possible to breed a pigeon to that work, so as that it should be kept or fed invisibly (for otherwise it would have bred suspicion in his followers and watchful enemies), or do the feat without prejudice to the drum of the ear, or without discovering what it swallowed or what it sought after.
Nor whether Mahomet did own the Holy Ghost any more than the Arians and Judaizing Christians did. But I would be informed what ground there is for this fable seeing neither Mahomet nor his followers do speak of any such apparition of any pigeon. Nor doth any Christian of the Arabians mention it. Grotius indeed speaks of it, and Doctor Pococke thereupon consulted him desiring to know what grounds he had for such a relation. The reply was that he did not therein follow the narration of the Mahometans or Arabic Christians, but of the European Christians, and particularly of Scaliger (in his notes upon Manilius where this is reported). And this is all that can be said for the relation. But it is to no purpose for any to say that the Mahometans have ever since preserved a veneration or extraordinary respect for **< 122>** pigeons. They give another reason for this viz.: because that the prophet Noah did pronounce a blessing upon the pigeon and said she should be forever beloved and regarded by men for returning to the ark with an olive branch, whilst the raven stayed to prey upon the dead bodies, which appeared after the retiring of the waters. I am apt to believe this story concerning the pigeon was by some ignorant person transferred from Athanasius to Mahomet. For it is reported concerning that father how a pigeon in the street flew to him and settled on his shoulder by his ear. This the Trinitarians interpreted as a miracle but the Arians as magical. And indeed, in the legend of St. George, I find Athanasius reputed to be a magician.
I shall add another Christian fable concerning Mahomet: that he privately bred up a bull which was constantly fed out of his hand and thereby accustomed to run to him as soon as he came sight. And to the horns of this bull one day he fastened the Alcoran, and, as he was discoursing to the people concerning his new religion and laws, this bull was contrived into his sight, which immediately rushed through the crowd to the prophet and presented him with the Alcoran, which he received with much ostentation of piety as sent to him by God and read some of it to the Arabians there. And, at the same time, a pigeon came and brought a schedule in which was written: "He shall be king of the Arabians who yoketh this bull." Whereupon Sergius the Nestorian monk brought him a yoke, which he easily put on the neck of the bull aforesaid, and thereupon was saluted as king and the Alcoran received as being of divine authority.
With such stories as these have the Christians represented him to be the vilest imposter in the world and transformed the wisest legislator that ever was into a simple cheat. But neither was the Alcoran written or published all at one time, nor ever reduced into one volume entirely by him, but by Abubacr and Othman as I have shewed. Nor is there any mention of this miracle of the bull in the Saracen records when they speak of the wonders of the prophet. Behold the simplicity of the Christians then who were deluded and thought to delude by such fopperies as these. As little credit is there to be given to that other fable that Mahomet should promise the people of **< 123>** Mecca (they demanding a miracle) that he would cause a mountain to remove to him at his sermons. But it, not obeying his call, he briskly said if the mountain will not come to Mahomet, Mahomet will go to the mountain. Was there ever a greater foppery imagined yet in the legends of the Christian saints?
There are such tales that one will not admire if they made no better romances of their enemies. A greater objection is raised against him for his ignorance: this they say is acknowledged by himself in his Alcoran, where he brings in God saying that he had sent to the ignorant a prophet amongst them, that is to illiterate persons an illiterate prophet. Moreover that the inhabitants of Mecca were so illiterate at that time that they could neither write nor read, so that it seems enough to prove him illiterate that it is confessed he was born there. In fine he is generally acknowledged by the Arabians to be _Nabian Ommian_ , that is the illiterate prophet.
But none of those arguments are of any validity. Not the first: for though God should say that He had sent to the ignorant a prophet from amongst them, doth it therefore follow that he must be as ignorant as they? Is not the saying verified if it appear that he was of their lineage and country? Not the second: for what necessity is there that every one that was born and lived in Mecca should be illiterate? May there not be an Anacharsis in Scythia? Must the seven sages be as ignorant as the residue of Greece? Must not Ezra be able to read because the people could not (Nehemiah 8:8)? As to his being styled _Nabian Ommian_ , though that title be given him by the Arabians generally, yet they do not agree upon the meaning thereof, some of them saying that it was not by reason of his ignorance, but of his being born at Mecca which is termed _Ommal koras_ , or the mother city. And how can they call him truly the ignorant prophet since they believe he knew all things, which is acknowledged by Doctor Pococke? The same doctor doth acknowledge that one Warakah, a kinsman (or rather uncle according to Abunazarus), taught him to write Hebrew and the Arabic speech, may be as well penned in that character which was newly propagated at Mecca by one that had married the sister of Abusofian some years before Mahomet pretended to an apostleship. And who will imagine that so subtle **< 124>** a man as he and who had framed to himself so vast a design would omit the learning of a thing so important to his ends and so subservient to the promulgation of his doctrine as the writing of Arabic was? I add that since Othman and Ali could write and were his scribes, and so were several of his other followers, why should we think it impossible for Mahomet to write?
It is further considerable that in all this controversy as it is managed the question is only whether Mahomet could write and read the Arabic characters newly introduced there and not whether he were so learned as to understand the religion and rites, customs and histories of his country, the religion and learning of the Jews, and of the several sects of Christians. Concerning this, there can be no question amongst any that know the Alcoran and his constitutions which are such as demonstrate a profound insight into these matters and a knowledge of many minute particularities extending even to legendary stories and fables. And if this be notorious, I should not see any just grounds to conclude him illiterate though it were more demonstrable than it is that he could not write or read the Arabic characters.
Another fable of the Christians against Mahomet is that he should promise his followers to revive again in three days, or some such time, and that they expected his return there so long till his carcass grew noisome, and that they still expect his return, that his body is enclosed in an iron tomb and hangs in the air suspended by the force of two opposite loadstones. But these are such figments as there the Mahometans laugh at and deride the Christians for relating them. Doctor Pococke refutes them more than once. And I will here repeat his words as he delivered them in his speech at Oxford where he excites his scholars to study Arabic:
Historia quibus curae est ut Arabum eorum monumenta diligentiis evolverent, quam persuasum cuperem; ut ita tollerentur tot ineptæ, quas ignorantia istius linguæ nobis obtrusit, fabulæ. Ita fieret, ut non ultra Mahometis tumulum in aere pendulum somniaremus: nec falsum illius de reditu suo promissum urgentes, Assedis ipsius, quos absurdè haec credere dicimus... impingendo veris refutandis ineptos nosmet redderemus, nec amplius illos Saracenorum appellatione à Sara se oriundos jactare nugaremur. Hujusmodi sexcenta sunt, quibus occurri sine linguarum studio non potest.
This last passage of Doctor Pococke puts me in mind of some other **< 125>** forgeries with which the Christians did formerly asperse the Mahometans and proceeded so far therein that they obliged the Saracenical converts in their catechisms to pronounce anathemas against such practices as were never used by the Musulmen viz.: that they worshipped the Morning Star and Venus, by the name of Chabar or Chubar, and that they cried in honor thereto, _Cabar_ or _Cubar_ , signifying in Arabic the great goddess. Which assertion how great authors so ever it hath as Euthymius Zegabenus, the Saracen catechism, Constantius, Porphyrogenitus, Cedrenus, et cetera, is most false and is rejected by Selden, Pococke, Hottinger, et cetera: this exclamation among the Mahometans, being no other than their usual doxology of _Allah Ekbar Allah_ or _Allah Allah Howa Cobar Allah_. God, God the great God. No less false is that report that they worship Venus under the stone called Brachtan, in which stone, if you look exactly, you may see lineaments of that goddess portrayed, though Euthymius Zegabenus relates this: that the Saracens in their catechism are taught to anathematize the worship of it.
Yet it is a great untruth. This stone is no other than the Black Stone I have given an account of, which they do not worship at all, but kiss devoutly as a relic of paradise (or for other reasons alleged by them) which may have been adored perhaps by the old Arabians but not by Mahomet or his sect, nor was it called Brachtan. Our linguists are in as great a perplexity to divine whence the Christians received this appellation (except it be because they kiss it, "tabarracan behi, boni ominis seu benedictionis captandi gratia") as they are to find them place the stone Brachtan on the ground in the middle of the Caaba, which is really raised above two cubits from the ground and fixed in the wall and is a plain stone, having no effigies in it. That which carries the show of any sculpture in it is the other stone (not in the middle of the Caaba but in the midst of the Court or _Almesjad Alharan_ ) and retaineth the impressions of Abraham's feet, not the face of Venus:
Quod autem à Damasceno, & Euthymio asseritur, si quis accuratiùs inspectet videri in eo figuram capitis scalpro expressam, quod Veneris esse volunt, ex Arabum scriptis puto probari non posse. Alius illis lapis est sacer, cui insculptam, vel potius impressam, tradunt figuram, sed tantum à figura capitis quantum caput à pedibus distantem, nisi **< 126>** oculi superstitione lippi caput à pedibus distinguere nesciant. Lapis scil quo vestigia pedum Abrahami impressa, cùm vel illis inter aedificandum Caabam insisteret, ut innuit Abulfeda, vel dum ipse caput Ismaelis, quem visum venerat, uxor (ut Ahmed Ebn Yusef, & Safioddinus) laveret. Unde & illi nomen Makam Abrahim Locus (scilicet) Abrahami, vel, quo stetit Abrahamus.
As for the _Τζ ίτζαφα Μαρουά Τζάφα_ mentioned also as idols of the Mahometans by the said Euthymius Zegabenus, these are nothing else but the Safa and Marwa of the Mahometans between which they run in their pilgrimages at Mecca and are said to be two stones into which a man and a woman were metamorphosed for committing adultery together in the Caaba Nor are they the objects of any Mahometan devotion as I have showed. But it is most pleasant to read how Euthymius doth further aggravate the idolatry of the Saracens. "Ibidem esse illis ait - - - - - - Simulacrum Baccha Ismaketh dictum, quod ipse Mohammedes sc. - - - - - - Adoramen observationis appellat, & ut miseri barbari adorarent praecepit." The original of this barbarous idol or rather mistake was that in the Alcoran there is once found the word _Bebecca_ , which Beidani, the Arabian commentator, expounds in Mecca, there being a metathesis of B for M. And this _Baccha Ismaketh_ , if it be pronounced according to the Arabian manner, signifies now more than _Bacca_ or _Becca_ , and is the name of Macca or Mecca, and this Mecca called by the said Euthymius - - - - - - is to be explained _adoramen observationis_ , or the place towards which the Mahometans pray (or their _Keblah_ ).
Do not such writers as these, and such were all the Christians that writ of Mahomet (as well the Greeks as Latins), deserve much credit? And can we blame those Mahometans who despise the foolish relations our authors give of their prophet and religion? Certainly no people are more remote from idolatry than the Saracens, and, whatever name you give to their errors and follies, Maimonides (who was scholar to Averroes and traveled through Arabia and Egypt), and Doctor Pococke, will tell you they ought not to be thus stigmatized. That they were once idolaters and until the days of Heraclius did worship the star Venus by the name of _Cabar_ or the great goddess is yielded: but Mahomet put an end to all such idolatrous worship and the rites **< 127>** which he retained are continued to a different intention than they were first practiced upon. It must be avowed that they adore no other than the true God and err rather in the manner than in the object of their devotion, so that the Emperor Manuel Comnenus seems (in the judgment of Doctor Pococke) to have had good reason to have altered the form of abjuration which was imposed on the converted Saracens viz.: "I do anathematize the God of Mahomet" into another kind of renunciation.
But that we may the better understand the Mahometan religion, I shall give a brief account of the articles of their faith, not of the Alcoran, nor its formal words, but the formal words in which the Mahometans express themselves, as the Christians recite their creed and not the whole scripture, nor a verbal deduction thence. The Mahometans divide their religion into fundamentals and super structures, and though they have a great diversity of opinions in the explication of their law, yet they esteem him orthodox who observes these five articles where to every Musulman is obliged. So Algazali: and Mahomet himself is said to have determined the case.
These are:
First: The confession, there is no god beside God, and Mahomet is the apostle of God; In a constant saying of the prayers according to appointment; In giving alms; In performing a religious pilgrimage to Mecca; And in observing the fast of the month of Ramadan.
Of these five, two may seem to be established upon prudential reasons: the pilgrimage and the fast of Ramadan. For though Mahomet retained many Arabian usages (pretending only to reform the old and not introduce a new religion), and such I have showed this pilgrimage and fast to have been; yet I believe he wisely discerned that the retaining these two points would be of great use, since he designed a military empire to the support whereof valiant and hardy soldiers were necessary. And nothing could more conduce to the successive generation and education of such than these two institutions. For how active, laborious, and abstinent must the women as well as the men render themselves to be able to endure the pilgrimage and the fast aforesaid. The pilgrimage I have described largely. The fast is moveable, and every year happens **< 128>** a month later than before, so that it falls sometimes in summer, other times in winter, in the hottest and coldest seasons, the longest as well as shortest days. The fast is observed so strictly that from daybreak till starlight, not one (except travelers or sick persons) doth eat or drink anything or so much as wash his mouth with a little water. And the most dissolute persons who adventure at other times to drink wine will not then so much as smell to it. It is true that at night, when the _Emaum_ or priest declares it to be time to eat, they feed plentifully even to excess. And even this has its reason upon account of health. For this fasting and feasting in such extremes contributes to their health during the whole year. It is recommended by Celsus to the healthy not to live by rule but pursue a variety in diet lest they contract a custom the change whereof would be as dangerous as the continuance inconsistent with daily action and business.
I believe also the third precept concerning alms was political in its original. For having persuaded his followers into such a parsimony as was requisite to the making of them hardy, and to the making of them welcome in their quarters, that they might not lapse from his institutions and be debauched by riches. It was a sort of a Grecian law or leveling to oblige them to so extraordinary alms as he did. Mahomet calls it _Zacot_ which signifies as much as increase as if the giving alms to the needy were the principal means to augment their revenue. And this he inculcated to them that they might not grow effeminate through luxury or mutinous by means of their riches. Neither was there less prudence in the precept concerning prayers, for the injunction of the _salah_ five times in twenty-four hours obliged them to a diligence and sobriety, which perhaps no other contrivance could have engaged them to, and doth also imprint in them a sense of their religion which, without apostasy, nothing could obliterate. Besides, it is a part of that precept never to mention any prophet or person reverenced by the Mahometans but with this eulogy: "God's peace be with him of glorious memory, et cetera"; or any enemy thereto but thus: "God's curse be upon him, God keep him from hurting us, et cetera"—which sayings did fix them more and more in their religion and estranged them more and more from their adversaries.
AS TO THEIR OPINIONS CONCERNING GOD, PURGATORY, JUDGMENT, AND PARADISE, THEY ARE THESE:
HAT GOD IS ONE GOD; that there is none other; that He hath no equal, no son nor associate; that His eternity hath neither beginning nor end; that it is impossible to explain properly His attributes, and that no intellect can comprehend the extent of His dominion; that contemplative men may conjecture at His being by the daily occurents on earth, but never understand His essence; that the heavens are His throne, the earth His footstool, but that the government of both is no trouble to Him; that He is omnipotent, omniscient, omnipresent, who sits upon the universal throne by His essence, and by His understanding penetrates into all things; that His providence disposeth of all affairs below, neither doth anything fall out, not the corn grow, not the grass wither but according to the decrees of His eternal predestination; that whatsoever man doth ascribe to Him or imagine to be in Him, it is eternal, and those attributes do not argue any composition or distinction in His being; that all things in this world, good or evil, befall us according to His will; that the beginnings, progress, and conclusion of all emergencies depend absolutely upon Him: and that He determined from all eternity whatsoever should come to pass; that His knowledge extends to the deepest secrets; that nothing happens against or not according to His pleasure; that in all matters to think, to will, to do, depends upon Him; that the souls of men are immortal; that those who are preserved by faith and the intercession of the apostles of God—Moses, Isa, Mahomet—from sin do upon death live in happiness until the resurrection and day of judgment; that those who are more or less wicked must in the grave and in a kind of purgatory undergo some torments until the last day, and there with more or less difficulty they shall be saved, but that nothing of evil, how little so ever, shall escape unpunished, nor any thing of good, how small so ever, pass unrewarded.
This is the sum of the Mahometans' religion.
As to the manner whereby the Alcoran doth explicate their purgatory or paradise: it is not to be censured by the Christians. It is evident that the Jews, Judaizing Christians, and other sects of Christians were diffused through Arabia **< 130>** and Mahomet, as he continued the ancient usages, so he retained those principles which the nation had imbibed and which he had the Christians and Jews to depose for. Such were those, not only concerning the torments in the grave, praying for the dead, and purgatory, but also of paradise and its joys, as he expresseth in his Alcoran. Doth not Isa (Luke 22:30) speak of eating and drinking at his table in his kingdom and of drinking wine there (Mark 14:25)? I profess that I cannot distinguish betwixt the paradise described by the Jews (and consequently avowed by the Judaizing Christians) and that which Mahomet doth propose to his followers, and if the words be the same I cannot see why they are not as liable to any equitable construction, according as the reason or prejudices of men do sway, as anything delivered by the Jews or Christians. Since all the descriptions we use concerning the senses and nature of glorified bodies are equivocal and deduced from what we are accustomed to upon earth; since God himself in scripture is described with the parts, actions, and passions of a man, I do not comprehend wherein lies the fault or ignorance in giving the like account of paradise.
Much might be said for the Mahometan doctrine of predestination, did I think any doubted but that it was the general tenet of the Jews and primitive Christians. And, in reference to the soldiery, none venture their lives in battle like those who suppose they cannot die before their appointed time, that all the contrivances of men depend upon the sovereign will of God, that there is no such thing as chance or any mistakes in the management of human affairs but are all swayed by destiny. In the late wars of England it was an observation of O. Cromwell's that the best fighters were of this opinion and he gave no encouragement to such preachers as taught the contrary.
As to the rites and ceremonies used in their prayers, pilgrimages, and other occasions: many of them if not all were anciently used by the Arabians and prudentially accommodated. These are not reckoned as fundamentals or principal points of their religion, but as trials of their obedience to the more necessary law they teach. it adds to the majesty of the author of their religion that there be some parts thereof, some institutions, which transcend our reason: that where our intellect doth comprehend and assent to write or command as good **< 131>** or wise, albeit our compliance be most exact and ready, yet our devotion is less than when we entirely obey upon the command of God since in the other case, we seem rather to follow our own judgment than that of the legislator. And that as in great empires the reverence to the prince is best secured and established where some capricious and arbitrary decrees intervene and amaze rather than inform the minds of the subjects, so in religion, our obedience becomes more perfect when we know that the divine intellect and will is not subordinate or conformable to ours, but transcends it.
The Mosaical constitutions give much countenance to this plea and an allegorical brain which knows how to dive into mysteries may undoubtedly find out rich mines of knowledge, types, and figures in Mahometanism. Amongst these trials of obedience, they reckon the observation of Friday, circumcision, abstinence from swine's flesh and blood, etcetera. But as to circumcision and those other ceremonies purely Arabian, I take them to be extremely necessary to such an empire as he designed. And, his country not yielding numbers sufficient for the pursuance thereof, this obliging all to circumcision, et cetera, was no less wisdom than old Rome practiced in denizening foreigners or the Jews in their proselytes. And Mahomet, by prohibiting his Alcoran to be translated into other languages, did (as far as in him lay) oblige all his followers to an unity of language. And, certainly, an unity of language, religion, and customs conduceth very much to the strength and peace of a monarchy.
Concerning some particular institutions of Mahomet, it may not be amiss to treat because they seem to evince his great prudence as a legislator. One is the permission of polygamy. The Alcoran gives liberty to each Musulman to take to himself as many wives as he pleaseth: two, three, four, or more, except he fear he is unable to render all due benevolence. Neither is there any positive restraint in their law to a determinate number, wherein the doctrine of Mahomet doth exactly agree with the Law of Nature as Grotius, St. Austin, and all the Jewish rabbins even to Maimonides, whose saying exactly agrees with Mahomet's. As to **< 132>** the Law of Nature, I do aver, as you may see in Selden. But what the Law of Nature doth so indefinitely permit, the Mosaical Law hath somewhat moderated, for the kings of Israel are forbidden to multiply unto themselves wives (Deuteronomy 17:17). It is evident that David had several wives besides his concubines, even to the number of six or eight, and the rabbins tell us that the Jewish king might have eighteen wives, notwithstanding that precept. And that David did not sin therein doth appear hence, that God upbraideth him as with a particular favor that he had done him in giving him sundry wives (2 Samuel 12:8). And where his sins are reckoned up, it is said that David turned not aside from all that the Lord had commanded him, except in the case of Uriah's wife (1 Kings 15:5). As to private persons, there are rules fixed in the Levitical Law concerning such as have two wives how to demean themselves (Deuteronomy 21:15). And the precept of the brother marrying with his brother's wife is believed generally to conclude married persons also, so that we cannot imagine polygamy to be interdicted to the Jews.
If we consult Christianity, whether polygamy be thereby prohibited to all or only to bishops, who ought to be the husband of one wife (1Timothy 3: 2) may be a question. The Emperor Valentinian made a law that any man might have two wives and married two himself. It doth not seem to be a part of the ceremonial law, nor politically confined to the Jews only. How then comes it to be abrogated? Besides it was indubitably practiced by the Judaizing Christians, from whom Mahomet derived much of his religion and is practiced by the Jews in the east to this day: "Judei orientales plures ducunt uxores; occidentalibus quidem licet, sed honoris gratia non faciunt. Paulus noluit Christianos plures ducere et praecipue episcopos, ut sic Judaeis os obturaret qui Christianis hoc objiciebant. Judaeis non praecipit ut cum tres habeant, duas repudient, unam servent."
So that we may conjecture that Mahomet was led by the Judaizing Christians that polygamy was not prohibited by their gospel, and that he esteemed that tenet to be a corruption in the vulgar gospels and inconsistent with his doctrine who came not to abolish but to fulfill the law (Matthew 5:17), being rather **< 133>** a paganical tenet derived from the Roman constitutions and complied with by the degenerate Christians. It is indeed remarkable that the Mahometans do upon tradition permit but four wives, and since this tradition is conformable to the Jewish doctrines, why may we not think it consonant to that of the Judaizing Christians?
As for concubines, it seems they are not repugnant to the Law of Nature since Abraham, Nachor, Jacob, Eliphaz retained them; nor to the Mosaical Law since Gideon had one and Saul, David, Solomon, and Rehoboam many. It was not held inconsistent with Christianity in the days of Justinian for a single Christian to have a concubine, nor is it prohibited by the Canon Law. By the Mahometan laws and usages a Musulman hath no stint as to concubines, but they must not be other than slaves (between which a Musulman, by their civil law, no marriage can intervene). Upon inquiry, I find that polygamy and the use of concubines were most ancient and inveterate practices of the ancient world. And Mahomet might thereupon comply with them: they were both exceedingly subservient to the multiplying of subjects which is the sinew of empire and therefore prudential. They were requisite upon another score because that in the east and south it is observed that there are far more women than men: and he who pretended to be a prophet and a follower of Abraham, Moses, Isa, et cetera had their precedents and the Law of Nature to justify him in the allowance.
I do not find that polygamy is a piece of sensuality in the Mahometan religion or any argument thereof, nor one sentence in their whole religion, either Alcoran or traditions, tending that way. You may sooner hope to find such suggestions in the Old and New Testament. Yet there is as much luxury in the discipline of their prophet as in the constitutions of Lycurgus. I have in the embassy of Ali represented the sense of the Mahometans concerning the pleasures of this life. As to those of the future, I know not why he should be so blamed for representing the joys of paradise by sensual delights. For if our souls must rise with the same bodies which we have here, excepting that our mortality shall put on immortality (for pleasure ariseth as **< 134>** knowledge doth from our senses), and though the stoical and Christian (I add and Mahometan) arguments are conclusive against placing of happiness in sensuality here on earth, yet if we imagine our bodies to be of the same kind and only glorified, the state of the question varieth, and they all come to nothing. The four rivers of paradise of pure water, excellent milk, rich wine, and pure honey are the same with those of the Jews, saving that the rabbins would have their wine spiced, and that they think the Leviathan and Behemoth will make as good dishes there as caviar and botargo or sturgeon here. And they will have a river of oil and balsam, also viands of fruit, and bread, and butter, and thirty-seven tables made of pearl. And the description of the New Jerusalem (Revelation 21:22) doth so much resemble the paradise of Mahomet that one would hardly imagine that any should condemn the latter as ridiculous and gross and yet approve the former as spiritual truth. And the condemnation of the wicked to a lake of brimstone fire hath as much of folly as the paradise of Mahomet.
But this is a digression.
If I may conjecture another reason besides what policy suggested to Mahomet, I would deduce it from hence: that he held (as did the Jews and Judaizing Christians) that all men were absolutely obliged by that first precept of increasing and multiplying which could not be fulfilled by the sterile or those who left no issue behind them. And as the Mahometan marriages have an aspect this way, so there is nothing therein or in their divorces (which last were allowed by the ancient Christian emperors and the laws of the Goths and Franks in several cases) which the Jews and other oriental nations will not justify, as you may read in Selden's treatise concerning an Hebrew wife.
Another prudent law of Mahomet was that whereby he prohibited usury to his Musulmen. It was an ancient Arabian law that every man ought to improve his estate and that every one that did should be honored and who did not should be punished. This prudent legislator, knowing of what importance it was to an empire that was to be great and lasting, that the subjects be not **< 135>** too poor and needy, lest that incline them to rebellion and revolt against their prince, nor exasperated against each other (whence arises revenge, unjust self-preservation, and all its evil consequences) as frequently happens upon usury, amended this former law. And, besides his general obligation to deeds of charity, he did enjoin all his Musulmen to follow some trade or vocation whence he desired this other benefit, that his people had their thoughts and bodies perpetually employed (which is a great secret in government and which perhaps was the reason of the public shows among the Romans, et cetera) and the poorer tradesmen were the better satisfied, their employments not being accounted dishonorable.
Since the prince and the basket maker were of one trade (behold another political mystery), and to prevent any inconvenience that his Musulmen might not hold usury as lawful as trade and object that the bartering and exchange of goods was a sort of usury, he declares in a _surat_ that God had permitted the one but not the other. By his law it is prohibited any Musulman to practice usury not only with a Musulman but with any Christian living under the protection of their monarchy. Nay, if he comes out of any territory not subject to them, usury with him is unlawful. Any man that considers the civil laws of Mahomet will find perpetual reasons to admire his subtlety and wisdom. It was a secret of Moses that no Israelite should practice usury with an Israelite: and this was very well, considering that Moses did not design to enlarge the Jewish empire but to preserve his people entire and unanimous in their narrow precincts.
But Mahomet, proposing vaster designs, prohibits it to all strangers living under the protection of the Musulmen. For, had this been permitted, there being such a multitude of Christians and others among them, this _questuosa segnities_ (as Pliny calls it), the facile way of growing rich by usury would have effeminated the Mahometans. The frequent and great quarrels and tumults arising upon usury would have endangered their empire and rendered the government odious and oppressive to the stranger subjects: whereof he prohibited all usury with them. But yet it is a rule in their law that a Musulman may practice usury with a Christian living in a foreign territory and, by such means, extort or cheat out of him his estate, the reason of which is because such persons **< 136>** are, as it were, in a perpetual hostility with them, and where a war is always deemed lawful against them, usury may be put in execution. The Mahometan reason: "Quia facultates eorum patent veluti ad praedam et direptio earum licita est quacunque via" is the same which St. Ambrose gives: "Cui jure inferuntur arma, huic legitime indicantur usure: quem bello vincere potes, de hoc [cito] potes [centesima] vindicare te. Ab hoc usuram exige, quem non fit crimen occidere. Sine ferro dimicat qui usuram flagitat. Sine gladio se de hoste ulciscitur qui fuerit usuraruis exactor inimici. Ergo ubi jus belli, ibi etiam jus usuræ." Let then that please censure Mahomet as ignorant and brutish. Some discourses of politics, considering what evils arise from the Levitical permission of usury with strangers, would prefer the wisdom of Mahomet before that of Moses.
This puts me in mind of another Mahometan constitution against gaming and accumulating riches by any kind of lottery, which he prohibits, as also wine, as introducing discord and poverty and a neglect of their duty to God. From this law it appears how prudently Mahomet did weigh the least and most remote consequences and would not allow of those distinctions betwixt abuse and use or those sophisms by which the Christians delude themselves into practices that terminate at last in the ruin of their commonwealth. He knew how much it imported a Musulman to pray to God and have him reverentially before his eyes always. He foresaw that such as were given to gaming and engaged by the hopes of further winning or sense of losing would be apt to forget their _sallah_ or daily prayers and so lapse into irreligion. He foresaw that gaming and the profits arising thence would induce men to cheat each other and that the first principle of cheating was a contempt of God, a disregard of other men, and an inordinate desire of wealth. It was his usual saying that he that was professed with a vehement thirst after riches and temporal advantages is in a fair way to commit all manner of wickedness. He knew that private quarrels often occasioned public damages and involved families, towns, and kingdoms in an universal ruin. Nor did the inconveniences seem less which follow private losses, which not only include the small detriment of a few, but many, and excite the desperate and needy to the highest and most pernicious attempts in which the public suffers. He knew that the examples of some gamesters infect others, **< 137>** that men are naturally more prone to hope than fear, to be idle than work, to neglect than attend the service of God, to desire trials of enriching themselves suddenly though with great hazards, rather than stay the tedious procedure which industry and wisdom puts them on, and therefore made this severe prohibition, the strictness whereof is such that he permits them not so much as to draw lots who shall pay a shot.
Whether it were his great prudence or care for the worship of the true God, I shall not determine. But certainly his legislative care extended far when he prohibited all observation of omens and all divination by lots, as debates to do or forbear an action by opening the Alcoran as the Romans did Virgil, or shooting an arrow into the air, or drawing an arrow out of the sheaf wherein should be written: "It is not the pleasure of God." This great prophet would not suffer his Musulmen to employ anything but reason in their debates. He imprinted in their minds that there is not any such thing as chance, no mistakes in providence whereby that befalls one which God intended for another; and that it was a sort of atheism to imagine that God would reveal that by the flight or cry of a bird which He would conceal from human prudence; or to conceive that a man's hand could discover more than His judgment; that the Alcoran and conduct of prudence were less available to our direction than the blind drawing out or shooting up of an arrow.
It were an endless task to descant upon the particular motives upon which depends the excellency of his laws. What a discourse might be made upon his uniting the civil and ecclesiastical powers in one sovereign, upon his rejecting all the Christian scripture rather than decide amidst so great uncertainty of books and so difficult rules to judge of the right and to reconcile the different sects and tenets. Was it not prudently foreseen that it would be more easy to introduce a new religion than to reform such a one, and well conjectured that all interested parties would more willingly submit to a novel doctrine than yield themselves to have been all in an error except one party?
It may perhaps be urged as an argument of Mahomet's ignorance that he denies all contagion in the pest and other diseases in man or beast. But if we consider that Hippocrates speaks nothing thereof, and that much may be said for it out of physic and the doctrine of predestination, this objection will have less force than is usually imagined and in reference to the wisdom of it and the successes which his followers have **< 138>** gained by that opinion, whilst the Christians yield up their towns, break up their camps, and upon contrary apprehensions these things plead highly for it.
Alcoran—As to the Alcoran, I have shewed that it was not all written at one time but by parcels and upon several occasions. And it was no small obstacle to his progress that for want of paper the prophet was forced to write the scattered _surats_ at first upon the bones of sheep and other cattle from which occasion perhaps some _surats_ received their appellations, as that of "The Cow." It was never reduced into one volume by Mahomet, but by the care of Abubacr, his immediate successor, who made a collection of all that had been scatteredly written by the Prophet upon the bones or skins of animals, or leaves of palms, or preserved in the memory of his auditors, and deposited this entire copy with Hapsa, the daughter of Omar and wife of Mahomet.
And, after him, Othman, the third successor of the caliphs, suppressed all the spurious copies which either the ignorance or malice of men had diffused and ordered all for the future to be transcribed out of the copy of Hapsa. This Alcoran is written in Arabic verse. It is not one continued poem, but a collection of sundry _surats_ or poems which Mahomet published occasionally, the language, the numbers, the style are all so exquisite, inimitable, that Mahomet himself doth frequently urge this as the ground authentic testimony of his apostleship, that the Alcoran doth surpass all human wit and fancy, and offered to be counted an impostor if any man could but write ten verses equal to any therein. The Mahometans esteem each line of it as an entire miracle and say that, if miracles do attest to the reality of a prophet, the author of the Alcoran brought three thousand demonstrations of his legislative power; that other miracles, being once performed in the sight of a few, lose much of their evidence and certainly when they are communicated to posterity. But God by Mahomet took a better course by leaving to mankind one lasting miracle, the truth whereof should in all ages be satisfactory and convincing.
This is the assertion of Bredani and of Ahmed Ben Edris and the words of Algazali are these:
The Alcoran, a transcendent miracle, and which is more, one that is permanent, from generation to generation. Nor is there any lasting miracle of the prophet, excepting that whereunto he appealed, challenging all the wits of Arabia (and Arabia did then abound with thousands whose chief study was eloquence and poetry) to make one chapter or more that might compare therewith **< 139>** and thereby demonstrated to the most incredulous, the truth of his prophesy. And God said concerning it, that if all men and angels should combine to write anything like it, they should fail in their enterprise.
I do not find any understanding author who doth controvert the elegancy of the Alcoran, and it hath this advantage over the Christian Bible, that being a poem there is a greater liberty allowed to fiction, figurative expressions, and allegories than is allowed of in prose. Also defects in chronology and errors in history are here tolerable, though, for my part, I believe that many of the incoherencies and chronological and historical defaults are voluntary, partly because the vulgar, being prepossessed with them (in many cases this is evident to have dissented thereform) would have been prejudicial to his aims, the universal credit of the errors being likely to overbear the real truth of things, partly because it was a received tradition among the Jews and Judaizing Christians (and it is now made use of as an apology for our scripture) that the Spirit of God in the prophets is not confined to the grammatical rules' ordinary methods.
It is further observable that the Alcoran, being such a poem, is not to be judged of by any translation into prose, much less such as is formed in Christendom. Our English doth follow the French, and the French is very corrupt, altering and omitting many passages. There are so many stories alluded unto: such idioms of Arabic poetry and of the Arabian tongue that it is impossible to explicate it without the help of the Arabic, Persian, and Turkish commentaries, which our translators, not knowing (or for their interest) not regarding, they have obtruded to the world such figments as Mahomet never uttered. I have often reflected upon the exceptions made by the Christians against the Alcoran and find them to be no other than what may be argued with the same strength against our Bible, and what the Christians say for themselves will fully justify the Alcoran. Therefore I will not excuse him by comparing the errors of Mahomet with those of the Talmud and our ecclesiastical history, or the popish legends or the fables recorded in our Fathers and believed by the primitive Christians.
As for miracles, I do not find but that Mahomet constantly rejected their authority as impertinent and unnecessary since so many were obtruded on the world (especially by the Christians) that he scorned the pretense. And he had this further reason: that true miracles and **< 140>** false cannot be distinguished by any human test, that the wicked may do real miracles, that the original of miracles might be derived from magic, or are the effect of some celestial constellation ruling the nativity of particular persons. This last opinion was common among the Arabians and Chaldeans and the oriental astrologers so that for him to have insisted on miracles among them would have been to little purpose or advantage. Whatsoever of miracles befell him was ascribed to magic by the Coreischites. And yet did they importune him to transfer mountains, raise the dead, produce an angel visibly, to all which he replied that the greatest of miracles was the Alcoran, that such was their unbelief that they would be obstinate even against miracles and evade them by sundry pretensions.
That miracles were the works of God, not of man, that they never were arbitrary to the prophets, but God wrought them when, where, and how He pleased, and not only to confirm the truth, but sometimes to try His people; and that some prophets never wrought any: the Protestants in the beginning of the Reformation excused themselves handsomely as to this point. It is written of John Baptist expressly that he did no miracles nor do we read that Amos and others did any, that Antichrist may do some, and that the Papists have some real miracles. It would be tedious to transcribe their defense. Yet do I find those miracles recorded of Mahomet though it were his modesty or policy not to insist on them:
First. That he showed the people publicly the moon so cleft into two parts that a mountain was visible in it. Whether he had the use of any telescope or by what chance such an aperture discovered itself in the moon, I know not. But of this miracle there is no mention in the Alcoran, but it is said that before the end of the world such a rupture shall happen, and it bears a resemblance with the predictions of Christ concerning the end of the world. I find that some Mahometans do relate this miracle as if it had happened, but others deny it to have been done. And the Christians needed not to have spent themselves in refuting as impossible what is not pretended to have been done and what it is possible to have seen without the help of our modern glasses.
Secondly. That he, being afflicted with scorching heat, did once cause two trees to remove together thereby to shade him. **< 141>**
Thirdly. That the stones saluted him in the streets and cried: "Peace unto thee O Apostle of God," and it is reported of Mahomet that he should say that to his remembrance no stone saluted him in Mecca until God commissioned him for His prophet.
Fourthly. That twice when his army was in great distress for want of water, the prophet putting his hand into a little vessel of water, it issued thence betwixt his fingers in so great a quantity as to supply all their necessities.
Fifthly. That he fed multitudes with a little food, as eighty persons once with four measures of barley and one kid; at another time, also eighty persons with a little bread which multiplied as he brake it into pieces.
Lastly. That he relieved his whole army with a few dates which a damsel brought him in her hand, and there remained many after they were satisfied.
Sixthly. That an old beam or tree did groan audibly as loud as any camel when he removed his station where he preached (which was near to it and made use of a pulpit) and that when he returned to his usual place it desisted to groan.
Seventhly. That a camel came to him and complained of his master that he put him upon hard work and yet made slender provision to feed him.
Eighthly. That the sheep spake to him when he was eating the impoisoned shoulder of mutton and bid him beware to eat it because it was impoisoned.
These are the miracles which are related concerning him whereupon the Mahometans do not much rely. Yet do they say that though they were not done but once or seldom, yet is the concurrent testimony of so many miracles joined together a pregnant evidence of the truth of his apostleship since they are related by credible witnesses who would not conspire to cheat mankind in such a matter; which is also the plea of the Christians and therefore not to be rejected lightly. But the most judicious of them do principally insist upon the Alcoran as a standing miracle, since many persons of singular eloquence did attempt to write the like but never could equal it.
As to the pretenses out of our scripture which were urged in his behalf by his followers, they are those: that God (in the law) is said to have come from Sinai, to have appeared in Seir and manifested himself in Paran: that is, the Mosaical Law came from Mount Sinai, the Gospel from Seir, which are mountains adjacent to **< 142>** Jerusalem, and Paran are the mountains about Mecca where Mahomet arose.Also in the Psalms: it is said that God did manifest in Sion _Ectilan Mahumidan_ , which words import in the Syriac version a glorious crown, but they mystically or by way of allusion to Mahmud and Mahomet did accommodate to their prophet. Lastly whereas it is said in the Gospel, "Except I go hence, the comforter shall not come," this they interpret about Mahomet, and it is one of the names of Mahomet among the Saracens viz., the Comforter.
They say also that the Christians have corrupted their gospels and expunged many passages which gave credit to Mahomet, and that a Christian priest showed them in a true copy to that purpose and said there was another unsophisticated preserved at Paris.
FINIS
Other arguments which the Turks make use of against the Christians are such as follow: that whereas the Christians believe in one God, they also believe that He is the Father, the Son, and the Holy Ghost, which opinion is so contradictory in itself that it wants no other refutation.
For no human intellect can comprehend how one and the same can be Father, Son, and Holy Ghost, in one sole essence, and at one and the same time. And God did never desire a man to believe what he cannot understand or conceive, but rather adapted man's understanding to assent to and conceive what is necessary and possible to deny, and not conceive what is impossible. This distinguishing faculty, being its excellency, without which can be no true understanding, nor can a man form a right judgment of anything, to be obliged to anything anymore than a newborn child. It is to be confessed that there are many abstruse and hidden things of which our understandings cannot have a perfect notion, of which we can nevertheless judge whether they are possible or impossible and repugnant. We do not perfectly understand the future state, the joys of heaven and pains of hell: we cannot comprehend the methods of the almighty in the working of miracles, nor even His government in the ordinary course.
NOTES
INTRODUCTION
. The title continues: _ac doctrina omnis, quae & Ismahelitaru, lex, & Alcoranvm dicitur, ex Arabica lingua ante CCCC annos In Latinam translata_. That same year witnessed the publication in Paris of Guillaume Postel's _Alcorani seu legis Mahometi et Evangelistarum concordiae liber_ , which, notwithstanding its title ("concord"), stated that Islam was a heresy similar to Protestantism that could easily be defeated. See, however, Nancy Bisaha, who argues that Renaissance humanists examined Islam from a cultural and political rather than a religious perspective: _Creating East and West: Renaissance Humanists and the Ottoman Turks_ (Philadelphia, 2004). See also Matthew Dimmocle, _Mythologies of The Prophet Muhammad in Early Modern English Culture_ (Cambridge, 2013).
. John Tolan, "European Accounts of Muhammad's Life," in Jonathan E. Brockopp, ed., _The Cambridge Companion to Muhammad_ (New York, 2010), 226–250. See also Gunny, _The Prophet Muhammad_ , ch. 1.
. Shireen Khairallah, "Arabic Studies in England in the Late Seventeenth and Early Eighteenth Centuries" (Ph.D. diss., University of London, 1972), 82.
. P. M. Holt, "The Study of Arabic Historians in Seventeenth-Century England," in _Studies in the History of the Near East_ (London, 1973), 37, in 27–49.
. All folio references are to the University of London manuscript. I have retained the spelling of the manuscript in the course of this introduction.
. The only biography and study of Stubbe remains Jacob, _Henry Stubbe_. See also Wood, _Athenae Oxonienses_ ; Hill, _The Experience of Defeat_ , 252–277; and Feingold, "Stubbe, Henry (1632–1676)," in _ODNB_.
. G. J. Toomer, _Eastern Wisedome and Learning: The Study of Arabic in Seventeenth-Century England_ (Oxford, 1996), 224. Judging from their transliterations of Arabic words and phrases, neither Stubbe nor Prideaux knew Arabic. For a biography of Prideaux, see Khairallah, "Arabic Studies," 132–157.
. BL MS 32553, Letters from Stubbe to Hobbes, fols. 5 and 25v (8 July 1656 and 13 January 1657 respectively). The first letter in the collection is dated 8 July 1656 and the last is 6 May 1657. See also Nicastro, _Lettere di Henry Stubbe a Thomas Hobbes_. In that same year, 1657, Stubbe wrote in defense of Hobbes against Wallis, _A Severe Enquiry into the late Oneirocritica; or, An Exact Accovnt of the Grammatical Part of the Controversy betwixt Mr. Hobbes and J. Wallis D.D_. (London, 1657). Either Stubbe did not finish the translation or he never sent a copy to Hobbes, who, in 1668, translated and published a condensed version of _Leviathan_ in Latin.
. Thomas Hobbes, _The Correspondence_ , ed. Noel Malcolm (Oxford, 1994), 2:782; Wood, _Athenae_ , 3:1069.
. Ibid., 3:1071.
. Stubbe, _The Common-wealth of Israel_ , 3.
. Wood, _Athenae_ , 3:1071.
. See the preface to _An Essay in Defence of the Good old Cause_ and _A vindication of that most prudent and honourable knight_ , 1. This treatise was welcomed by John Locke, who wrote a letter to Stubbe in which he discussed the latter's views on toleration. See Locke, _Two Tracts on Government_ , 242–244.
. Cook, "Physicians and the New Philosophy," 251.
. Stubbe, _Legends no histories_ , 1—the second part. See the essay by Cook (in the previous note) on this confrontation.
. Ibid., 154.
. Stubbe, _Campanella revived_ , 17.
. In May 1670 he wrote to Secretary of State Arlington hoping that his quarrel with the Royal Society would not "displease your lordship or any other English patriot, since it has no other design than to support the monarchy, the Protestant religion, and the peace and welfare of the nation; and to vindicate the two universities and my own family," _Calendar of State Papers Domestic Series, Charles II, with Addenda, 1660 to 1670_ , 10:224.
. Stubbe, _Campanella revived_ , 62.
. Ibid., 10.
. Samuel Butler's "The Elephant in the Moon," in John Wilders and Hugh de Quehen, eds., _Hudibras Parts I and II and Selected Other Writings_ (Oxford, 1973), 206, l.431. Because of these attacks, Westfall, in _Science and Religion in Seventeenth-Century England_ , unfairly describes Stubbe as "a wholly venal scoundrel," 237n17.
. Stubbe, _Campanella revived_ , 7.
. Ibid., 13.
. See my study of Ross in _Islam in Britain_ (Cambridge, 1999), ch. 3. For the earlier Latin translations of the Qur'ān, see Hartmut Bobzin, "Latin Translations of the Koran: A Short Overview," _Der Islam_ 70 (1993): 193–206.
. Henry Stubbe, "To the Reader," in _A justification of the present war against the United Netherlands_.
. John Spurr, _England in the 1670s: "This Masquerading Age"_ (Oxford, 2000), 34.
. For a detailed discussion of these two treatises and Stubbe's political stance, see chapter 6 in Jacob, _Henry Stubbe_.
. The Edict of Milan, AD 313, in _A further iustification of the present war against the United Netherlands_ , 34, 43.
. Ibid., 83.
. Stubbe, _Rosemary & Bayes_, 4–5. See also James R. Jacob, "The Authorship of 'An Account of the Rise and Progress of Mahometanims,'" _N &Q_ 26 (1979): 10–11.
. Wood, _Athenae_ , 3:1071.
. Even nonconformist contemporaries recognized the difference between the two: see the manifesto of fifteen nonconformists, including Richard Baxter, _The Judgment of Non-conformists, of the Interest of Reason in matters of Religion_ (London, 1676), especially 6: "We deny not but some Non-conformists, and Conformists did cast out their suspitions of two very Learned rational Men, Mr. Hales, and Mr. Chillingworth, as if they had favoured Socinianisme, because they so much used, and Ascribed to Reason, in Judging of matters of Religion."
. Stubbe uses the new English term, _Islamism_ , in a positive sense. Contrast its use with Thomas Warmstry's _The Baptized Turk; or, a Narrative of the happy Conversion of Signioer Rigep Dandulo_ (London, 1658), 117. See also Thomas Bedwell who used "Alesalem," in "The Arabian Trudgman," in _Mohammedis Imposturae: That is, a Discovery of the Manifold Forgeries, Falshoods, and horrible impieties of the blasphemous seducer Mohammed_ (London 1615).
. The National Archives (TNA): SP 29/275/276. See also the reference to two other short pieces on Queen Elizabeth and in defense of the king's suspension of the "laws against conventicles by his declaration of March 15, 1672," _CSPD Charles II, October, 1672, to February, 1673_ , 14:350.
. See Stubbe's letter to Williamson on 8 July 1672 describing what he plans to write against the Dutch, _CSPD, Charles II, May 18th to September 30_ , 1672, 13:319. But Stubbe was so eager to proclaim his royalism that he aligned himself with Samuel Parker—who, like him, had conformed to the Anglican Church after the Restoration and become a mouthpiece for the monarchy.
. James and Mary were married by proxy in a Catholic ceremony on 20 September 1673. Stubbe's piece argued against marriage by proxy and wondered "What benefits may accrue, or be justly expected from the Farreigner with whom such Alliance and Marriage is to be Contracted."
. Actually, a few months earlier, in _A further justification_ , he had praised the "prudence of His Royal Highness" and criticized those "Inferiours" who "foment even just quarrels or resentments against their Suepriours" and "Men in Authority," 83–85 (irregular pagination).
. _CSPD Charles II, March 1st to October 31st, 1673_ , 15:599.
. Wood, _Athenae_ , 3:1082.
. BL MS 35835, 18 July 1674, fol. 276r in a cluster of fols. 269–276.
. "This day there came in A Tun of Madeira wine, which I think will be excellent for your Lady: I have tasted it & presume to prescribe a pipe of it for physick for her Stomach. It is not hot in the mouth, but warm in the Stomach, it is as of good a body as Sack, and better tasted than Bourdeaux, but paler coloured, it is two years old. It will come for about 12 d a bottle. It is good at Meal, but not for a Debauch. It bears 272v water in a general proportion and so will be better for her than Beer. It is an excellent Table Wine, & will seem extraordinary in the Country. My Lord of Ossery hath half a Tun & Sir Hugh Cholmly a Tun. This wine seldom or never comes into England, but in the West Indies they could not live without it, or digest any meat." BL MS 35835, fol. 272r–v.
. BL Sloane 35 is a posthumous inventory of his books.
. Joseph Glanvill, "To the Reader," in _A further discovery of M. Stubbe_ (London, 1671).
. Volunteers at the abbey kindly showed me the recent and "comprehensive" inventory of all the names on memorials and gravestones inside the church (26 July 2012). There was no record of Henry Stubbe.
. For studies of Stubbe, see Jacob, _Henry Stubbe_ ; Kaplan, "Greatrakes the Stroker; Champion, "Legislators, Imposters" and _The Pillars of Priestcraft Shaken_ , ch. 4; Kontler, "'Mahometan Christianity'"; Birchwood, "Vindicating the Prophet"; Rose, "Royal Ecclesiastical Supremacy and the Restoration Church"; Garcia, "A Hungarian Revolution in Restoration England" and _Islam and the English Enlightenment_.
. Jacob, _Henry Stubbe_ , 2.
. Ibid., 6, 129, and 139–160.
. Hill, _The Experience of Defeat_ , 263.
. Champion. _The Pillars of Priestcraft Shaken_ , ch. 4. See also idem, "'I remember a Mohometan Story of Ahmed ben Idris'," which focuses on Stubbe and Toland.
. See also Garcia, _"Islam and the English Protestant Imagination, 1660–1830"_ (Ph.D. diss., University of Illinois at Urbana-Champaign, 2007).
. Holt, _A Seventeenth-Century Defender of Islam_ , 29.
. See my "A Note on Alexander Ross and the English Translation of the Qur'ān," _Journal of Islamic Studies_ 23 (2011): 76–84.
. For a study of Milton's familiarity with Islam, see Eid Abdallah Dahiyat, _John Milton and the Arab-Islamic Culture_ (Amman, 1987), revised as _Once Upon the Orient Wave: Milton and the Arab Muslim World_ (London, 2012). See also William G. Kenton III, "English Liberty and Turkish Tyranny: The Symbolic Function of the East in Milton's Poetry and Prose" (Ph.D. diss., New York University, 2005), 119–134 ("Satan as Sultan").
. In so doing, Stubbe advanced the Islamic interpretation, which Johann Hottinger had called "errore plane intolerabili," _Historia Orientalis_ (Tiguri, 1660), 38.
. See C. Edmund Bosworth, "The Prophet Vindicated: A Restoration Treatise on Islam and Muhammad," _Religion_ 6 (1976): 11 in 1–12.
. H. John McLachlan, _Socinianism in Seventeenth-Century England_ (Oxford, 1951), 55. As he continues, Chillingworth and other Laudian clergy were accused of Socinianism as early as 1643, 164. See also John Marshall, _John Locke, Toleration and Early Enlightenment Culture_ (Cambridge, 2006) for a discussion of Chillingworth and anti-Trinitarianism in ch. 7.
. Champion, _The Pillars of Priestcraft Shaken_ , 107, 110.
. Quoted by Christopher Hill, _Milton and the English Revolution_ (London, 1977), 295, from _A Censure upon Certain Passages Contained in the History of the Royal Society_ (1670). See also Hill's discussion of "Anti-Trinitarianism," ibid., 285–296.
. Jacob, _Henry Stubbe_ , 75.
. See the English translation, Giovanni Paolo Marana, _The eight volumes of letters writ by a Turkish spy_ , trans. Daniel Saltmarsh (London, 1694).
. Cited by Jacob, _Henry Stubbe_ 159. Humerto Garcia states that Stubbe's work was "implicit" in John Toland's _Nazarenus_ because Magney compared Toland's "'Mahometan Christianity' to _The Rise and Progress of Mahometanism_." Garcia, _Islam and the English Enlightenment_ , 52. Magney mentions the "Physician of some note" (Stubbe), but he does not quote him at all, relying instead on Hottinger, Warner, and Reeland, the first two of whom were widely used by Stubbe. The reason he ignores Stubbe is because the latter had praised Muḥammad, while Hottinger and Warner had not. And, of course, Magney is vitriolic in his description of the Prophet and of Islam, as that chapter shows.
. Jacob invokes this episode in support of his view of Stubbe's "Socinian" leanings. Jacob, _Henry Stubbe_ , 155. But there is no evidence that Stubbe belonged to the group that, five years after the death of Stubbe, approached the ambassador. See my discussion of this episode in _Britain and Barbary, 1589–1689_ (Gainesville, 2005), 158–159.
. Wood, _Athenae_ , 3:1071.
. Quoted by Humberto Garcia, "Islam in the English Radical Protestant Imagination, 1660–1830," 52.
. Ibid., 51. The view had been advanced by Jacob, _Henry Stubbe_ , 139–142, and repeated by Hill, _The Experience of Defeat_ , 277.
. Jacob, _Henry Stubbe_ , 121.
. For a general discussion of Restoration theological arguments, see Gerard Reedy, S. J., _The Bible and Reason: Anglicans and Scripture in Late Seventeenth-Century England_ (Philadelphia, 1985). See the sermon by Barrow, "Of the Impiety and Imposture of Paganism and Mahometanism," _Theological Works of Isaac Barrow_ , 8 vols. (Oxford, 1830), esp. 5:24–31. Interestingly, his Latin treatise on "Epitome fidei et religionis Turcicae, a Muhameto Kureischita, arabum propheta" is less hostile. Barrow uses the term _Islam_ rather than _Mahometanism_ , ibid., 8:145 ff.
. He actually lumped the Socinians with "Atheists" and "Papists": _A Specimen of some animadversions upon a book entituled, Plus ultra_ (London, 1670), 13.
. None of these names appear in Jacob's study. Stubbe kept a 1649 copy of Salmasius's _Defensio Regia, pro Carolo I_ till the day he died (BL MS Sloane 35, fol. 18r). See also Champion, who counted the number of references to some of these writers in Stubbe's 1701 manuscript: Hottinger, thirty-six times and Pococke fifty-six times: "'I remember a Mahometan Story of Ahmed Ben Edris,'" 464 and note 59.
. See G. J. Toomer, "John Selden, the Levant and the Netherlands," in Alastair Hamilton, Maurits H. Van Den Boogert, and Bart Westerweel, eds., _The Republic of Letters and the Levant_ (Leiden, 2005), 53–76.
. Hobbes, _The Correspondence_ , 2:759–763.
. Gunny believes that Prideaux may have intended his account as a refutation of Stubbe ( _The Prophet Muhammad_ , 51). The propinquity between Stubbe's and Prideaux's accounts lies in the common sources that they used, especially Hottinger, Pococke, Rycaut, and Erpenius/al-Makīn. For references to Stubbe after his death, see Jacob, _Henry Stubbe_ , ch. 8.
. See my discussion of this treatise in _Europe Through Arab Eyes, 1578–1727_ (New York, 2009), 96–98. See also Adrian Reeland, _A Defence of the Mahometans from Several Charges fully laid against them by Christians_ in _Four treatises concerning the doctrine, discipline and worship of the Mahometans_ (London, 1712), 185, note c.
. Rijk Smitskamp, _Philologia Orientalis: A Description of Books Illustrating the Study and Printing of Oriental Languages in Sixteenth- and Seventeenth-Century Europe_ (Leiden, 1992), 287.
. As Loop shows, in his "Johann Heinrich Hottinger (1620–1667) and the _Historia Orientalis_ ," Hottinger used his extensive study of the Qur'ān and of various Arabic manuscripts to present a Protestant interpretation of church history. Jan Loop, _Church History and Religious Culture_ 88 (2008): 169–203.
. In _The Present State of the Ottoman Empire_ , Rycaut stated that as "Mahometanism" was first revealed, "it found a great part of the World illuminated with Christianity, endued with active Graces, Zeal and Devotion, and established within it self with purity of Doctrine, Union, and firm profession of Faith," _The Present State of the Ottoman Empire_ (London, 1668), 98.
. G. J. Toomer's _Eastern Wisedome and learning: the Study of Arabic in Seventeenth-Century England_ remains the most important book on the subject. See also the Panizzi Lectures of 1996 by Charles Burnett, _The Introduction of Arabic Learning into England_ (London, 1997) for a detailed discussion of the medieval period; and Karl H. Dannenfeldt, "The Renaissance Humanists and the Knowledge of Arabic," _Studies in the Renaissance_ 2 (1955): 96–117.
. _The first printed Catalogue of the Bodleian Library, 1605: a facsimile_ (Oxford, 1986).
. William Bedwell, "To the Christian Reader," in _Mohammedis imposturae: that is, A discovery of the manifold forgeries, falsehoods, and horrible impieties of the blasphemous seducer Mohammed_ (London, 1615). For the Arabic manuscripts at Oxford, see Colin Wakefield, "Arabic Manuscripts in the Bodleian Library: the Seventeenth-Century Collections," in G. A. Russell, ed., _The "Arabick" Interest of the Natural Philosophers in Seventeenth-Century England_ (Leiden, 1994), 128–146.
. The Hartlib Papers, Sheffield University, Great Britain: Samuel Hartlib Electronic Project, CD-ROM, 29/3/64 A. I am grateful to Professor Donald Dickson for his help.
. Mordechai Feingold, "Oriental Studies," in Nicholas Tyacke, ed., _The History of the University of Oxford_ (Oxford, 1997), 4:481, 490.
. Quoted by Toomer, _Eastern Wisedome_ , 108.
. Leonard Twells, _The Theological Works of the Learned Dr. Pocock_ , 2 vols. (London, 1740), 1:35.
. Brian Walton, _Polyglot_ (London, 1657), 1:95. Their topics, he explained, ranged from mathematics to Aristotle's physics and Ptolemy's geography, with collections, in the library of Fez alone, reaching thirty-two thousand volumes.
. Bodleian MS Or 298, 1. See also the 1669 _Lexicon Heptaglotton_ of Edmund Castell, professor of Arabic at Cambridge. In the _History of the Royal Society_ (1667), Thomas Sprat praised "the Learned Age of the Arabians" and mentioned how "some worthy and industrious Men of our Nation, who have search'd into their Monuments" believe the Arabians "almost compar'ed to Rome, and Athens." Thomas Sprat, _History of the Royal Society_ , ed. Jackson I. Cope and Harold Whitmore Jones (London, 1959), 45.
. Peter Rietbergen, "A Maronite Mediator Between Seventeenth-century Mediterranean Cultures," _LIAS_ 16 (1989), 25 in 13–41.
. Cited in Adrian Reeland, _A Defence of the Mahometans from Several Charges fully laid against them by Christians_ in _Four treatises_ , 61.
. _Synodikon, sive, Pandectae canonum ss, apostolorum, et conciliorum ab ecclesia Graeca receptorum_ (Oxford, 1672).
. _Purchas his Pilgrimage_ (London, 1617), 284. It is not clear whom Purchas had in mind, but Nicholas of Cusa had mentioned that a Christian Arab had refuted a Muslim who had tried to convert him, "Prologues to the Examination of the Koran" (1461) in _Toward a New Council of Florence: "On the Peace of Faith" and Other Works by Nicolaus of Cusa_ , trans. with introd. William F. Wertz (Washington, DC, 1993), 387. For a full translation of _Cribratio Alkorani_ , see Jasper Hopkins, _Nicholas of Cusa's De Pace Fidei and Cribratio Alkorani: Translation and Analysis_ (Minneapolis, 1990). As Hopkins observes, the Christian Arab is al-Kindī.
. See Edward Pococke's translation of Abū al-Faraj, _Historia Compendiosa Dynastiarum_ (Oxford, 1663), 93. For other early Christian Arabic views on Muḥammad, see Samir K. Samir, "The Prophet Muhammad as Seen by Timothy I and Some Other Arab Christian Authors," in David Thomas, ed., _Syrian Christians Under Islam: The First Thousand Years_ (Leiden, 2001): 75–106. For Arabic scientific material, see Francis J. Carmody's critical bibliography of _Arabic Astronomical and Astrological Sciences in Latin Translation: A Critical Bibliography_ (Berkeley, 1956), part 2.
. _Geographia nubiensis id est accuratissima totius orbis in septem climata divisi descriptio_ (Paris, 1619).
. This treatise later appeared separately in a small volume about _Arabia, seu Arab ūm vicinarumq[ue] gentium orientalium leges, ritus, sacri et profani mores, instituta et historia_ (Amsterdam, 1630), 1–90. It is not likely that Stubbe bothered with the translation of a summary of "De Nonnvllis Orientalivm Vrbibvs," which was published by Samuel Purchas in the _Pilgrimes_ of 1625 as "Mosleman superstitions and rites." Purchas explained that he had used the "Arabicke Bookes, by the said Maronites Gabriel and John," _Purchas His Pilgrimes_ (Glasgow, 1905), 9:162. Indicative of Purchas's hostile attitude to Islam is his selective translation of Sionita's text, his focus on the negative passages, his omission of the chapter on Christian sects in the East, and, most significantly, his contraction of the section on "Viri illustres qui Arab: lingua scripserunt," mentioning only a few of the names that the two Maronites had proudly included. Most egregiously, he omitted the last chapter in which Sionita had shown, by reference to the Qur'ān, how much Mary was venerated in Islamic belief—a celebration that Sionita's Catholic readers would have appreciated. It bears noting that Purchas referred at the end of the translation/adaptation to "my learned Friend Master Bedwell [and his book] called Mahomeds imposture" (9:118). As the title shows, the book was deeply hostile to Muḥammad.
. _Geographia nubiensis_ , 55 (the second part).
. Ibid., 41.
. Having been trained at the Maronite College in Rome, the two priests had been exposed to numerous Arabic texts: see the list of books at the college: Nasser Gemayel, _Les Échanges culturels entre les Maronites et l'Europe: Du Collège Maronite de Rome (1584) au Collège de 'Ayn Warqa (1789)_ , 2 vols. (Beirut, 1984), 1:180–190.
. As Camille Aboussouan noted, the aim was to show Europeans "la connaissance du monde des Arabes," _Exposition: Le livre et le Liban jusq'à 1900_ (Paris, 1982), 254.
. Gemayel, _Les Échanges culturels_ , 1:326–327.
. See A. S. Tritton, _The Caliphs and Their Non-Muslim Subjects_ (London, 1930), 5–17, and an extended discussion in Maher Y. Abu Munshar, _Islamic Jerusalem and Its Christians_ (London, 2007), chs. 2 and 3.
. Twells, _The Theological Works of the Learned Dr. Pocock_ , 1:18.
. A collection of geographical writings in Arabic by John Gagnier shows how, even in the first half of the eighteenth century, Idrīsī was still used, and how Gagnier was collating the Arabic of al-Idrīsī with the manuscript translation of Idrīsī by Pococke; see Bodleian MS Or 318.
. The title page of _Purchas his Pilgrimage_ mentions that an account of "the Saracenicall Empire Translated out of Arabike by T. Erpenius" is included in the fourth edition. But there is no mention of al-Makīn anywhere in the account about Islam. In 1657 a French translation appeared in Paris: _L'histoire mahometane; ou, Les qurante-neuf chalifes dv Macine_ , trans. Pierre Vattier. See also Rijk Smitskamp, _Philolgia Orientalis_ (Leiden, 1992), entry 85a.
. It is interesting that Ibn Khaldūn, who used al-Makīn in his _Kit āb al-'ibar_, consulted the first part that dealt with Jewish and early Christian history and not the part on Islam, which Erpenius translated. See Ayman Fu'ād Sayyid, "Maṣādir Ibn Khaldūn 'an tarīkh ghayr al-Muslimīn fī _Kit āb al-'ibar_," in Muḥammad Zakariya 'Anāni, ed., _Dir āsāt adabiya wa lughawiya_ (Cairo, 2011), 539–550. I am grateful to Professor Wadad Kadi for this reference.
. Edward Pococke, _Specimen Historiae Arabvm_ (Oxford, 1650), 383.
. For John Gregory's Arabic studies, see my "Some Notes on John Gregory and Islam," _Discoveries_ 14 (1997): 1–2, 6–7.
. See P. S. Van Koningsveld's discussion of its use in Muslim Spain, "Christian Arabic Literature from Medieval Spain: An Attempt at Periodization," in Samir Khalil Samir and Jørgen S. Nielsen, eds., _Christian Arabic Apologetics During the Abbasid Period (750–1258)_ (Leiden: 1994), 216–217.
. _Historia Saracenica, qua res gestae Muslimorum_ (Leiden, 1625), 2.
. "Erat autem optimae indolis, voce suavi, visitans &excipiens suos ut ipsum visitabant, &excipiebant, pauperis munerans, Magnates laudans; conversans cum infimatibus; & petentem à se aliquid, non repellens sine eo, aut sermone facili." _Historia Saracenica_ , 10.
. "Cumque venisset ad eum magnus quidam Christianus; surrexit honorem ei exhibens qua de re cum cum alloquerentur quidam, respondit; Cum venerit ad vos Primarius populi alicjus, honorate eum: Atque hic vir maximus est in populo suo. Dixit quoque: Benefacite Cophitis Aegypti: sunt enim vobis genere &affinitate juncti. Item: Qui Christianum opprimit, adversarium cum habebit die Juidicii. Et, Qui Christiano nocet, mihi nocet." _Historia Saracenica_ , 11.
. _Eutychii Agyptii, patriarchae orthodoxorum Alexandrini... Ex ejusdem Arabico nunc primùm typis edidit ac versione & commentario auxit Ioannes Seldenus_ (London, 1642). For a study of Selden's Eutychius, see Toomer's magisterial _John Selden_ , 2:600–614.
. Twells, _The Theological Works of the Learned Dr. Pocock_ , 1:53.
. This latter edition included a picture of John Selden along with the following explanation on the title page: "Illustriss: Johanne Seldeno" and "Interprete Edwardo Pocockio."
. Humphrey Prideaux wrote that Selden was thought to be the translator, but that it was Pococke who had done the work, although Selden had "born the Expences of this Chargeable Edition, the most Worthy and Learned Author of that Version acknowledged it by those words in the Title-page, which several having mistaken to the robbing him of the honour of his Work, as if Mr. Selden had begun the Translation, and Dr. Pocock finished it." Prideaux, _The True Nature of Imposture_ (London, 1697), 165.
. Cited in Toomer, _John Selden_ , 2:606.
. "Vita Auctoris," _Contextio Gemmarum_ (Oxford, 1658).
. Selden, _Eutychii Agyptii_ (1642), 70–76.
. See Eutychius of Alexandria, _The Book of Demonstration_ , ed. Pierre Cachia (Louvain, 1960), 1:12–13.
. See, for instance, Pococke, _Eutychii Patriarchae Alexandrini_ (Oxford, 1654), 1:307–309.
. Ibid., 2:279.
. Ibid. 2:284: "In nomine Dei misericordis, miseratoris, Ab Omaro Ebnil Chetabi, urbis AEliae incolis. Securos fore ipsos quod ad vitas suas, & liberos, opes, & Ecclesias suas; illas scil. nec dirutum iri, nec habitatum: testéque adhibuit." It is not clear if Stubbe knew that al-Makīn had relied on Ibn al-Baṭrīq.
. Ibid., 2:295.
. _Arabicae Linguae Tyrocinium id est Grammatica Arabica_ (1656), 250–263.
. Edward Pococke, "Prefatio ad Lectorem," in _Specimen_ , 3v. The original Arabic version was an abridgement of a Syriac version—but Pococke worked from the former version. See Lawrence L. Conrad, "On the Arabic Chronicle of Bar Hebraeus," _Parole de L'Orient_ 19 (1994): 319–378, especially part A. For the discussion of Islam, see 338–339.
. Prideaux, _The True Nature of Imposture_ , 153.
. It is interesting that John Gagnier, the French-born orientalist who lived in England until his death in 1740, copied from Pococke's translation of Abū al-Faraj as he compiled general geographical information: Bodleian MS Or 318, 239–266. For the importance and influence of Pococke's translation, see Hans Daiber, "The Reception of Islamic Philosophy at Oxford in the Seventeenth Century: The Pococks' (Father and Son) Contribution to the Understanding of Islamic Philosophy in Europe," in Charles E. Butterworth and Blake Andrée Kessel, eds., _The Introduction of Arabic Philosophy Into Europe_ (Leiden, 1994), 69 in 65–82.
. Pococke, _Historia Compendiosa_ (Oxford, 1663), 101.
. "Futurum est (inquit) ab hoc Puero magnum aliquid, cujus fama per Orientem & Occidentem se diffundet, nam cum approprinquaret nube obumbratus apparuit." Pococke, _Specimen_ , 9.
. Ibid.
. Interestingly, the same verb was used, "azharū al-dīn al-mustaqīm"/manifestabant, by Eutychius (in Selden's edition) to describe the Nicene position on Christology: _Eutychii Agyptii_ , 75.
. Pococke, _Specimen_ , 9, 12, 13.
. Ibid., 14.
. Stubbe does not state explicitly that Islam was a reformation of Christianity (fol. 137), but, in 1690, Arthur Bury was forthright: " _Mahomet_ professed all the articles of the Christian faith, and declared himself not an Apostate, but a Reformer." Arthur Bury, "The Preface," in _The Naked Gospel_ (London, 1690).
. See also Johann Hottinger, _Historia Orientalis quae, ex variis Orientalium Monumentis Collecta_ (Tiguri, 1651), 103–107. I will also use the 1660 edition (also published in Tiguri).
. Reeland, _A Defence of the Mahometans_ , 196. See the earlier use of Warner by Thomas Warmstry, who relied on him for the Qur'ānic view of Jesus, _The Baptized Turk_ , 101ff.
. _Compendium Historicum eorum quae Mahammedani de Christo et praecipuis aliquot religionis Christianae tradiderunt_ (Leiden, 1643), 15–16.
. It is not surprising that Daniel writes that Warner's approach was "polemic, but not primarily so." Norman Daniel, _Islam and the West: The Making of an Image_ (Edinburgh, 1960), 287. Hadith is the collection of sayings and determinations by the Prophet Muḥammad.
. Selden, _Eutychii Agyptii_ , 59.
. For the Muslim Jesus, see Tarif Khalidi, _The Muslim Jesus_ (Cambridge, Mass., 2001).
. (Leiden, 1644), 30–31. It was also included in Hottinger, _Historia Orientalis_ (1660), 516.
. Matar, _Europe Through Arab Eyes_ , 33.
. There is a brief mention of Stubbe in Abdelwahab El-Affendi, ed., _About Muhammad: The Other Western Perspective on the Prophet of Islam_ (Richmond, Surrey, 2010), xxvi.
. Marcus Zeuerius Boxhornius, _Historia universalis sacra et profana a Christo nato ad annum usque_ (Lugduni, 1652), 397. Hottinger mentions that Muḥammad's parents were poor, "paupers habūit parens," _Historia Orientalis_ (1651), 136.
. One source for this information is found in Bibliander, _Machvmetis Sarracanorvm principis vita_ , part 2, "De Haeresi Herachii et Principatu ac Lege Machvmeti."
. It is interesting that Gibbon in _Decline and Fall of the Roman Empire_ , introduction by Christopher Dawson (New York, 1978), compared the Arabs to the "Medici of Florence," 5:217.
. Sionita, _De Nonnvullis_ , in _Geographia nubiensis_ , 17.
. Thomas Erpenius, _Orationes Tres, De Linguam Ebraeae atque Arabicae Dignitate_ (Leiden, 1621), 42.
. See H. R. McAdoo, _The Spirit of Anglicanism_ (London, 1965), ch. 5.
. _The Alcoran of Mahomet_ (London, 1649), sig d r.
. But first published in 1691: _The Works of Sir William Temple_ (London, 1720), 2:221–222. See also Lancelot Addison, _The Life and Death of Mahumed_ (London, 1679), which appeared three years after Stubbe's death, ch. 9, and p. 52: "the Alcoran is a very rude Poem."
. For a study of Du Ryer, see Alastair Hamilton and Francis Richard, _André du Ryer and Oriental Studies in Seventeenth-Century France_ (London, 2004); and, for the Qur'ān, chapter 3 in my _Islam in Britain, 1558–1685_ (Cambridge, 1998).
. Pococke does not take a stand, although he advances evidence regarding the literacy of the Prophet. Edward Gibbon mentioned a certain "Mr. White" who argued in a sermon for the literacy of the Prophet, but used it to confirm "imposture." I have not been able to identify the _Sermons_ to which he alludes, _Decline and Fall_ , 5:233, n. 1. See the discussion of "ummiyy" as illiterate in Pockocke, _Specimen_ , 156.
. _Historia Josephi Patriarchae, ex Alcorano_ (Leiden, 1617).
. Edward Pococke, _Porta Mosis_ (Oxford,1655), 244. Five years earlier, in _Specimen_ , Pococke had stated that Psalm 50:2 referred to Muḥammad, 17. Selden believed that the Qur'ān had been composed by Muḥammad, but, as Toomer notes, more correctly, it was "God speaking to and through Muhammad." Toomer, _John Selden_ , 2:749n421.
. Pococke, _Specimen_ , 17.
. See Bibliander, _Machvmetis Sarracenorvm principis vita_ , 189–200. It also appears in Purchas, _Pilgrimage_ (1626), bk. 3, ch. 5, 259–263, and Boxhornius, _Historia universalis_ , 398–399. Another dialogue was translated into English by John Greaves, again between the Prophet and "Abdalla Ebn Salem the Jew," but with different content: Bodleian, MS. Locke c. 27, fols. 3–9.
. In regard to "Venery": while the manuscript mentions the Prophet's ability to satisfy "forty women" in one night (fol. 67), the BL Harleian 6189 manuscript mentions "two" (fol. 11).
. Pococke, _Specimen_ , 16.
. Richard Baxter, _The Reasons of the Christians Religion_ (London, 1667), 202–203.
. Hobbes too had used "poem" in his discussion of prophecy, but not in the context of Islam. Thomas Hobbes, _Leviathan_ , ed. C. B. Macpherson (Harmondworth, 1971), 457.
. Pococke, _Specimen_ , 274–292 (Arabic and Latin).
. Although it is not likely that he would have been familiar with it. See the study and the translation of the Morisco "epic" in Ṣalāḥ Faḍl, _Mal ḥamat al-Maghāzi al-Moriskiyya_ (Cairo, 1989).
. Part 2, ch. 10. The spelling "Hali" had been popular since Bibliander. Stubbe never uses it.
. As the author of "The Life and Death of Mahomet" in the English translation of the Qur'ān argued, "Haly" sought to inherit the" Power" of Muḥammad. _The Alcoran of Mahomet_ , xii.
. Twells, _The Theological Works of the Learned Dr. Pocock_ , 9.
. Sionita _De Nonnvllis_ , 24. Hottinger changes Sionita's words to "primus Moselmannorum" but without applying them to 'Ali, since the words are from Q 6:163. Hottinger, _Historia Orientalis_ (1660), 5.
. Erpenius, _Historia Saracenica_ , 43.
. Pococke, _Historia Compendiosa Dynastiarum, autore Gregorio Abul-Pharjio_ (Oxford, 1663), 102–103.
. Hottinger, _Historia Orientalis_ (1651), 370.
. Petrus Ibn Rahib, _Chronicon Orientale_ , ed. P. L. Cheikho (Beirut, 1903), 53
. Adam Olearius, _The Voyages and Travells of the Ambassadors Sent by Frederick Duke of Holstein, to the Great Duke of Muscovy, and the King of Persia. Begun in the year M.DC.XXXIII. and finish'd in M.DC.XXXIX_ , trans. John Davies (London, 1669), 161.
. Ibid., 276.
. Ibid., 278.
. Twells, _The Theological Works of the Learned Dr. Pocock_ , 1:9. See also Toomer, "Arabic Learning After the Restoration," where there is mention of Pococke's early lectures on the proverbs, "Proverbia Quaedam Alis" in 1636, and a manuscript with notes on these lectures, 215.
. P. M. Holt, "An Oxford Arabist: Edward Pococke," in _Studies in the History of the East_ (London, 1973), 6 in 1–26.
. "Alis his Arabick proverbs with my translation," quoted by Toomer, _Eastern Wisedome_ , 199.
. Hottinger, _Historia Orientalis_ (1660), 507ff.
. Ockley, Sentences of Ali (London, 1717), B 2r–v.
. _The Koran_ , 430–431n. For a biography of Sale, see Khairallah, "Arabic Studies," 203–263.
. Gibbon, _Decline and Fall of the Roman Empire_ , 5:278.
. See the section on eastern Christians in chapter 5 of Gerald MacLean and Nabil Matar, _Britain and the Islamic World: 1558–1713_ (Oxford, 2011).
. A3v–A4r.
. See references to the first two in the notes to the text below. Stubbe owned a London 1669 edition of Blount's travels (BL MS Sloane 35, fol. 19r).
. Especially after the publication of Bartolomé de las Casas, _The Tears of the Indians: being an historical and true account of the cruel massacres and slaughters of above twenty millions of innocent peoples_ (trans. J. P. 1656) as well as William Davenant's _The Cruelty of the Spaniards in Peru_ (1658) and John Dryden's _The Indian Emperour, or the Conquest of Mexico_ (1667). For an excellent discussion of the difference in imperial policies between Euro-Christians and Muslims, see Anouar Majid, _Freedom and Orthodoxy: Islam and Difference in a Post-Andalusian Age_ (Stanford, 2004).
. William Biddulph, _The Travels of certaine Englishmen_ (London, 1609), 84.
. In 1678 and 1679, after Stubbe's death, Thomas Smith and Paul Rycaut published accounts of the plight about the eastern Christians. Stubbe either did not know, or ignored, Smith's second epistle about the oriental Christians in _Epistolae Quae Quarum altero de Moribvs ac Institvtis Tvrcarvm agit: Altera Septem asiae Ecclesiarvm notitiam continent_ (1672).
. John Selden, _De Jure Naturali et Gentium_ (London, 1640), 734.
. Olearius, _The Voyages_ , 158. For a reference to al-Makīn, see 196.
. Pococke, _Porta Mosis_ , 260.
. Stubbe, _A Further iustification_ , 23.
. See the various references to Stubbe in the introduction to Locke, _Two Tracts on Government_ ; and a discussion of Stubbe's toleration in W. K. Jordan, _The Development of Religious Toleration in England_ , 4 vols. (Cambridge, MA., 1940), 3:335–340.
. Hill, _The Experience of Defeat_ , esp. 262.
. See my discussion in "John Locke and the 'Turbanned Nations,'" _Journal of Islamic Studies_ 2 (1991): 67–77.
. For Selden, see Toomer's quotations, _John Selden_ , 1:156; Hottinger's Liber II of _Historia Orientalis_ (both editions) is about the "Pseudopr."
. Although it was a merchant who endowed the Thomas Adams Chair of Arabic at Cambridge, the first to fill it was Abraham Wheelock, who held a bachelor of divinity from Cambridge (1624); while the chair at Oxford was endowed by Archbishop Laud, to be filled by the chaplain to the Levant Company, Edward Pococke. The first account of the Ottomans written by an English eyewitness was by the chaplain to the British ambassador to the Porte, Paul Rycaut.
THE PRINTED AND MANUSCRIPT SOURCES
. For a brief description of the life and political orientation of Hornby, see Justin Champion, "'I remember a Mahometan Story of Ahmed Ben Edris': Free-thinking Uses of Islam from Stubbe to Toland," _Al-Qantara_ 31 (2010): 449–51 in 443–480.
. John Gregory, _The Works of the reverend and learned Mr. John Gregory Master of Arts of Christ's Church, Oxon: in two parts_ (London, 1665, 1671, 1684). The first works that were published after his death were _Gregorii posthuma_ in 1649 (London) and _Notes and Observations on some passages of scripture_ in 1650 (London, and again in 1655, 1671). The _Notes_ also appeared as part of _The Works_.
. For Pococke and other English orientalists, see P. M. Holt, _Studies in the History of the Near East_ (London, 1973), chs. 1 and 2. Robert Wakefield was the first English scholar to write about the three languages of Hebrew, Arabic, and Aramaic: _Oratio de laudibus & utilitate trium linguarum: Arabicae, Chaldaicae & Hebraicae_ (London, 1524). See the translation by G. Lloyd Jones published in 1989, _On the Three Languages_ (Binghampton, NY).
. See, for instance, Hottinger, who discusses the Qur'ānic verses about Jesus and retains "Jesu": Johann Hottinger, _Historia Orientalis_ (Tiguri, 1660), 142–144.
THE ORIGINAL AND PROGRESS OF MAHOMETANISM
. Theodosius died in AD 395.
. The opening of the _Originall_ suggests a dedication that Stubbe may have been preparing for a patron or to the reader, as it echoes the opening of Newton's translation of _A Notable Historie of the Saracens_ : "I Am purposed to write an Historie concernyng the Actes of the Saracens, atchieued aswel in the East as in the West partes of the world: first because they were greate and renoumed over the face of the whole Earth and brought all things out of good state into tumultuous broyle and confuse disorder, and also because this power of theirs encreased, through the discorde and dissention of the Christians." Cello Augustino Curioni, _A Notable Historie of the Saracens_ , trans. Thomas Newton (London, 1575), 1.
. The four empires are mentioned in the book of Daniel: Babylonian, Persian, Macedonian, and Roman (Daniel 2-11). Contemporary with the rise of Islam were the Byzantine and the Sassanid empires.
. A reference to a German tribe that invaded Gaul—mentioned by Julius Caesar in his _Gallic Wars_.
. For the description of Muḥammad that follows, see Marcus Zeuerius Boxhornius, _Historia universalis sacra et profana a Christo nato ad annum usque MDCL_ (Lugduni Batavorum/henceforth Leiden, 1652), 401. Stubbe sometimes translates word for word.
. In this paragraph Stubbe borrows from Jirjis ibn al-'Amīd al-Makīn, _Historia Saracenica_ , trans. Thomas Erpenius (Leiden, 1625), 10; see also Johann Hottinger, _Historia Orientalis_ (Tiguri, 1651), bk. 2, ch. 3. Stubbe owned a copy of the latter. I will also use the 1660 edition, published in Tiguri.
. In the margin: "Blount has copied this in his Oracles of reason fol. 159," Hand A.
. Stubbe admired Jean Bodin's _Six Bookes of a Commonweale_ (London, 1606). The reference to the change of aristocracy to democracy and vice versa occurs on pp. 421–422. Stubbe owned a copy of the 1597 edition.
. Ibid., 417.
. Lycurgus and Solon were the lawmakers in Sparta and Athens, respectively.
. See Virgilio Malvezzi, _Discovrses upon Cornelius Tacitus_ , trans. Sir Richard Baker (London, 1642). But Stubbe differs in his interpretation of political change from Malvezzi, especially in "The second Discourse" where there is a discussion of the reasons for the political changes in Rome. Stubbe owned a copy of this edition.
. See Christopher Hill, "The Word 'Revolution,'" in _A Nation of Change and Novelty_ (New York, 1990), 82–102. In BL Harleian 6189, the word is replaced by "Resolutions," fol. 5.
. A reference to Antiochus IV (reg. 175–164 BC), who, in reaction to a revolt by the Jews, sacked Jerusalem, imposed the worship of Zeus, and gave support to the Hellenized Jews (who, as Stubbe suggests, later became the Sadducees) over the traditionalists (later the Pharisees). For the history of the revolt and its aftermath, see the fifth and sixth books of 2 Maccabees, and Josephus, _Antiquities_ , trans. William Whiston _The Life and Works of Flavius Josephus_ , intro. H. Stebbing (Philadelphia, n.d.), bk. 13, ch. 3.
. BL Harleian 6189 has "justify" instead of "honest," fol. 6, which also appears in BL Harleian 1876, fol. 5. Sadoc and Baithos were influenced by Epicureanism (denying the resurrection and reward and punishment). Their disciples became the Sadducees. Eleazar was the son of Moses (Exodus 18:4). According to Maimonides, the Kabbalah was secretly revealed to Moses and his son: see Walter Farquhar, _A Church Dictionary_ (Philadelphia, 1854), 78.
. Stubbe used the King James Bible.
. See John Lightfoot, _In Evangelium Matthæi, Horæ Hebraicæ et Talmudicæ_ (Cambridge, 1658), 31–32. Stubbe owned a copy of this edition.
. BL Harleian 6189 has "Rights," fol. 7, where BL Harleian 1876 has "Rites," fol. 6. The reference is to Antiochus IV. See Isaac Casaubon, _De Rebus Sacris et Ecclesiasticis, exercitationes XVI: ad Cardinalis Baronii prolegomena in annales et primam eorum partem_ (London, 1614), 6–22, where there is a discussion of the great calamities that befell the Jews.
. A numerology used to interpret the Hebrew scriptures.
. BL Harleian 6189 has "indifferently" instead of "indefinitively." There is the following note in BL Harleian 1876: "Concerning the weeks of Daniel & the visions in him yt they contain but the same thing repeated four times over; & terminate in the destruction of Jerusalem by Antiochus Epiphanes, & that they are but Tipicall & by way of parode accomodated to the last destruction of Jerusalem, See Jo. Masham's chronolog. Diatribe," fol. 6. I am unable to locate this text.
. Pompey conquered Palestine in 63 BC. BL Harleian 1876 has "possesseth," fol. 7, while BL Harleian 6189 has "possessed" and a note " _or possessed_ ," fol. 9.
. Acts 2:11.
. Lightfoot, _In Evangelium Matthæi_ , 21–22. Born in Babylonia, in the late first century BC, Hillel went to Palestine where he was instrumental in the development of the Talmud and the Mishnah.
. Archelaus was the son of Herod the Great and was banished in AD 6.
. It is possible that Stubbe is thinking here of Sabbatai Sevi who proclaimed himself a messiah in Izmir, and assembled a large following, until he converted to Islam. See John Evelyn, _The history of the three late famous impostors : viz. Padre Ottomano, Mahomed Bei, and Sabatai Sevi_ (Savoy, 1669), 41–103.
. Both _asinego_ and _intrado_ were new words in English, used by Thomas Herbert in _Some Years Travels into divers parts of Africa and Asia the Great_ (London, 1665). See _OED_ entries.
. Claudii Salmasii/Salmasius, _De Hellenistica Commentarius, Controversiam De Lingua Hellenistica dicidens, & plenissimè pertractans Originem & Dialectos Graæae Linguæ_ (Leiden, 1643), 199–200. Stubbe owned a copy of this edition.
. In the margin: "thus far Blount," Hand A. Charles Blount, _The Oracles of Reason_ (London, 1693), 164, the letter to Rochester.
. Caius Caligula, Roman Emperor (reg. AD 12–41); Herod Agrippa (reg. AD 33–44). See Solomon Zeitlin, "Did Agrippa Write a Letter to Gaius Caligula?" in the _Jewish Quarterly Review_ 56 (1965–66): 22–31.
. The Land of Onias was near Heliopolis in Egypt, where a large Jewish community lived. The temple was destroyed ca. AD 73. See the reference to Philo in _Evsebii Pamphili, Rvffini, Socratis, Theodoriti, Sozomeni, Theodori, Evagrii, et Dorothei Ecclesiastica Historia_ (Basel, 1570), 18–19.
. See for these terms _Eusebii Pamphili Ecclesiasticæ Historiæ Libri Decem. Ejusdem de Vita Imp. Constantini, Libri IV... Henricus Valesius_ (Paris, 1672), 27, in _Annotationes_.
. In Ezra 1:11, there is mention of the vessels of gold and silver that were taken from Babylon to Jerusalem.
. See John Lightfoot, _Horae Hebraicæ et Talmudicæ Impensæ in epistolam primam. S. Pauli ad Corinthios_ (Cambridge, 1664), 125–127. Stubbe owned a copy of this edition.
. Followed by "in his Jewish Antiquities gives this relation of the Condition of the Babylonish Jews at the Ascent of Esdras to Jerusalem & the remaining party, even in his days." The reference is from Josephus's _Antiquities_ , bk. 11, ch. 5.
. Stubbe read Manasseh ben Israel's _The Hope of Israel_ , which appeared in London in 1650 (rep. 1652) and described the various Jewish communities around the world.
. The Epistle of James is a Christianization of Jewish wisdom literature, which is why Stubbe associated its recipients with the Jews. See Lightfoot, _Horæ Hebraicæ et Talmudicæ_ , 130.
. The reference is to the Septuagint.
. Benjamin of Tudela traveled in the twelfth century. His account was published in Latin in 1633, but Stubbe derives his information from Edward Brerewood, _Enqviries Tovching the Diversity of Langvages, and Religions throughout the cheife parts of the World_ (London, 1614), 105. Stubbe owned a copy of the 1622 edition by Brerewood. Salmonasar (reg. 731–713 BC) was a Persian ruler mentioned in 2 Kings 17:6.
. In the right margin, Hand A: "[... ]ad Chro. 11. 16th that many of the Israelites came to Jerusalem with the Levites upon the [... ] in the time of Rehoboam," 2 Chronicles 11:16.
. Cf. John Selden, _De Iure Naturali & Gentium, Iuxta Disciplinam Ebraeorum, libri septem_ (London, 1640), 239–248, esp. 244. "Soria" is Selden's spelling of Syria; Nehardea and Pumbeditha are in Iraq and are mentioned by Benjamin of Tudela.
. Salmasius, _De Hellenistica Commentarius_ , 241.
. Josephus, _Antiquities_ , bk. 13, ch. 9.
. Under the Hasmonean ruler Hyracnus (reg. 134–104 BC), the Idumeans had been forcibly converted. Selden mentioned Nero as a convert in _De Iure Naturali_ , bk. 2, ch. 3.
. _Funus Linguæ Hellenisticæ sive Confutatio Exercitationis de Hellenistis et Lingua Hellenistica_ (Leiden, 1643), 72 ff. This book was a refutation of Salmasius. Stubbe owned a copy of this edition.
. Salmasius, _De Hellenistica Commentarius_ , 230. The Bar Kokhba revolt occurred in AD 132–136.
. Stubbe's use of the phrase "universal monarchy" in the early 1670s may be in reference to the French monarchy of Louis XIV. See his _A further iustification_ , 18.
. Acts 11:26.
. In BL Harleian 6189, the scribe adds after this word: "There is a Blank" fol. 19. BL Harleian 1876 has "Hobs Leviathan" in that space, fol. 14.
. In the margin: "here Blount go's on again in this same letter," Hand A.
. In the margin: "thus far" in regard to Blount's copying of the letter to Hobbes, Hand A.
. Joseph Mead (1586–1639) was author of the influential interpretation of the Book of Revelation, _Clavis Apocalyptica_ (1627), which had a tremendous influence on the millenarianism of the civil wars and the Interregnum. See Brian Ball's classic study, _A Great Expectation: Eschatological Thought in English Protestantism to 1660_ (Leiden, 1975). Stubbe may have been drawn to the writings of the Great Tew circle by Thomas Barlow, who visited there in the early 1640s: see Noel Malcolm, ed., _Thomas Hobbes, The Correspondence_ (Oxford, 1994), 2:785. Stubbe dedicated his _Deliciæ poetarum Anglicanorum in Græcum versæ quibus accedunt elogia Romae & Venetiarum_ (Oxford, 1658) to Barlow. William Chillingworth (1602–1644) was the chaplain of Lord Falkland and published his celebrated _The Religion of Protestants a Safe Way to Salvation_ (London, 1638): see ch. 3. The _Discourse of Infallibility_ by Lord Falkland, Lucius Cary (1610–1643), was published in 1646 (with frequent reprints). Cary showed how the millenarian tradition was rejected by the early church, ch. 4, and in "The Lord of Faulklands Reply," he focused on the history of early Christian heresies (including millenarianism and chiliasm). Stubbe owned a copy of Falkland's book.
. "Concluded" in the sense of "enclosed," as in Romans 11:32.
. This paragraph draws on Selden, _De Iure Naturali_ , bk. 2, ch. 7.
. Cf. Romans 1:16.
. Salmasius, _De Hellenistica Commentarius_ , 71.
. From the Greek, δεσπόσυνοι (the ones belonging to a master), but here referring to the relatives of Jesus. See Eusebius, _Ecclesiastica Historia_ , 10. Stubbe was using the Greek text of Eusebius (bk. 1, ch. 7); in the Latin, it is bk. 8. Sulpicius Severus was a Christian historian, ca. AD 363–425.
. "All these things which are handed down in the Gospel in regard to the rites of the Jews are taught in the same way without any discrepancy." Jospeh Scaliger, _De emendatione temporum_ (Frankfurt, 1593), where in bk. 6, Scaliger examines the differences in calculating the dates of the birth, baptism, and other events in the life of Christ.
. Johann Buxtorf (1564–1629), author of _De Synagoga Judaica_ (Hanover, 1604, English translation by A.B. in London, 1663) was professor of Hebrew at Basel. Chapters 20 and 21 describe the feast of Reconciliation, to which Stubbe refers. Stubbe owned a copy of the 1604 Latin edition.
. BL Harleian 6189 has "procreation" instead of "generation," fol. 18. See John Lightfoot, "Utcunque Iudæi Filium Dei negant eo sensu," in _In Evangelium Matthæi_ , 316. See also Buxtorf, _The Jewish Synagogue_ , ch. 36, for the Jewish understanding of the Messiah.
. The paragraph draws on Selden, _De Iure Naturali_ , 260–266.
. Thebeth is the tenth month in the religious year: see the reference in Claudii Salmasius, _Ossilegium Hellenisticæ sive Appendix ad Confutationem Exercitationis de Hellenistica_ (Leiden, 1643), 337.
. _Funus Linguæ Hellenisticæ_ , 89ff. There is no paragraph or unit division in this book and the discussion continues for pages about hellenized synagogues. For Tertullian, see 92.
. Also known as the Palestinian Talmud. In BL Harleian 1876, there is the following note: "Dr. Lightfoot (in addend. ad 1. Cor. c. 14 v.10) saith they did not there in Cæsarea read ye Scripture, but repeat yrr Phylacteries in Greek: which is as inconsistent with yr tenets as the other," fol. 20. See Lightfoot, _Horæ Hebraicæ et Talmudicæ_ , 145.
. The first entry in the _OED_ for "sophisticated" as falsified appears in Dryden's 1673 comedy, _The Assignation_.
. For the "Gospel of the Nazarenes" and the "Gospel according to the Hebrews," see the texts in Bart D. Ehrman and Zlatko Pleše, _The Aporcyphal Gospels: Texts and Translations_ (New York, 2011). Both gospels are apocryphal and belong to the second and third centuries. Epiphanius was bishop of Constantia (ca. 310–403) and author of _Contra Octaginta haereses opus Panarium_ , trans. F. Williams, in _The Panarion of Epiphanius of Salamis, Book 1_ , 2d ed. (Leiden, 2009): see his reference to these texts, 131. Stubbe's reference to the apocryphal gospels coincides with the "discovery" and publication of various other books of the apocrypha such as the _Protoevangelion and the Epistle of Barnabas_. The _Codex Alexandrinus that_ was given to King Charles I by Cyril Lucaris included the (apocryphal) _First Epistle of Clement to the Corinthians. Archbishop James Ussher and Gerard Vossius printed some of the letters of Ignatius in Oxford (1644) and Amsterdam (1646) respectively_.
. Acts 17:10–15. It is interesting that Stubbe uses the name "Cephas" instead of "Simon Peter." Epiphanius describes the Nazarenes of Pella in _Panarion_ , 123–130.
. In BL Harleian 1876 there is the following note: "It is said yt some of ye Corrinthians were of Christ: those were like ye disciples of John at Ephesus. Act. 19 who believed a Messiah, but not yt Jesus was he. See Dr. Lightfoot on yt place," fol. 21. John Lightfoot, _The Harmony, chronicle and order of the New Testament the text of the four evangelists_ (London, 1655), 107.
. The _aurum coronarium_ was mentioned in the Theodosian Code ( _CTh_.16.8.14) and was in "process of time a mere tribute in gold or in silver, which the Roman potentate received from those placed under his government." <http://www.forumancientcoins.com/numiswiki/view.asp?key=aurum%20coronarium>. (accessed 26 February 2012). See also Walo Messalinus, _De Episcopis et Presbyteris contra D. Petavium Loiolitam Dissertatio Prima_ (Leiden, 1650), 381. It is not clear if Stubbe recognized Messalinus as a pseudonym used by Salmasius.
. Ibid., 379–383.
. In BL Harleian 1876: "spectibiles, as the other grand Patriarchs Illustres, and Clarissimi, as also Primates," fol. 31/ notables, as the other grand illustrious patriarchs, most renowned.
. In BL Harleian 1876, there is a reference to "Dr. Thorndike's treatise of the service of God of religious assemblies," fol. 22: See Herbert Thorndike, _Of religious assemblies, and the publick service of God: a discourse according to the apostolicall rule and practice_ (Cambridge, 1642). The discussion of church offices is in chapter 7, from where Stubbe borrows "Sacredotes, Presbyteri, Antistites."
. See Leo Modena (1571–1648), _History of the rites, customes, and manner of life, of the present Jews, throughout the world_ , trans. Edmund Chilmead (London, 1650). Stubbe owned a copy of this edition and may have been thinking of the opening words of the treatise: "The Rites which are at this day observed, and in Use, among the Jewes, are not all of them of equall Authority, nor equally practiced by all, after one and the same Manner," 1. See also Selden, _De Iure Naturali_ , 365 ff. The 1650 Modena text was bound with _The Hope of Israel_ (see note 34).
. Ioannes Drusius, _De Sectis Ivdaicis commentarii... Iosephi Scaligeri I.C.F. Elenchus Trihaeresii ejusdem_ (Arnheim, 1619), 341.
. In BL Harleian 1876, there is the following note: "That there were Sadducee-Christians appears out of Justin Martyr's discourse with Tryphon: & such were Hymenæus & Philetus, who denied ye resurrection: tho' their names be Greek, they might be original Hebrews, or Jews, & have also Hebrew names, as Dr. Lightfoot shews upon 1. Cor. 1.1," fol. 23. The reference to Lightfoot is from _Horæ Hebraicæ et Talmudicæ_ , 1; Hymanæus and Philetus are mentioned in 2 Timothy 2:17–18.
. The first part of Epiphanius's _Panarion_ includes description of twelve heretical groups of Jewish Christians.
. In BL Harleian 6189 "Ceremoniall" and "certaine" are corrected over "criminal" and "coercive," fol. 33.
. See Buxtorf, _The Jewish Synagogue_ , 188. "Succedaneous" means substitute, a medical term first used by Sir Thomas Browne in 1646 ( _OED_ ).
. _Origen against Celsus_ , trans. James Bellamy (London, 1660), chiefly bks. 2 and 5.
. The Latin quotations are corrected and added at the bottom of the page by Hand A. The verse from Ovid is translated in L. P. Wilkinson, _Ovid Recalled_ (Cambridge, 1955), 287: "Ah, men too lax, who think that the gloomy crime of murder can be washed away by river-water." The quotation from Virgil is from the _Aeneid_ , 2: "Now you, Father, take up the gods of our ancestral home, our holy symbols. I cannot touch them without sin, until I have washed my hands in a living spring, for coming as I do straight from the fury of war, I have fresh blood still on them." Virgil, _Aeneid_ , trans. W. F. Jackson Knight (Penguin, 1958), 72.
. Gerard Vossius, _De baptismo disputationes XX_ (Amsterdam, 1648), 24, from where "aspersione."
. In BL Harleian 6189, the following sentence comes after "scripture" and is crossed out: "any more than [for these words follow again the next line but two] and not condemned in the Greek Church." In the margin, and in the same hand: "The words are interlined by the Corrector, over the words & _not condemned by the Gr. Ch_.," fol. 35. See 1 Corinthians 15:29 and Tertullian (ca. AD 160–225). Ernest Evans, ed. and trans., _Homily on Baptism_ (London, 1964), section 18.
. Sozomen in _Evsebii Pamphili, Rvffini, Socratis_ , 549. Vossius wrote about Gregory of Nazianzen (325–389) and the baptism of blood, _De baptismo_ , 24.
. Vossius, _De baptismo_ , 27, thesis 2.
. BL Harleian 1876 has the following note: "Or from ye use of Baptism in some pagan-worships. In multis Idolarum sacrilegis sacris baptizari himines perhibentur. Augustinni de baptism.adu.Donat.lib. 6.c.25," fol. 25.
. L Harleian 1876 has the following note: "Montacut.orig.sacri. part. 1.p.103.104. ye Days were called Nominalia or dies lustrici," fol. 25. See Richardi Montacuti (Richard Montague), _Episcopi Cissacestriensis, De originibus ecclesiasticis_ (London, 1636), 103: "Romani Nominalia appellabant... Lustrici dies infantium appellantur," and then citing Macrobius, "Nundina Romanorum Dea." The application to Christian children comes a page later, 104.
. BL Harleian 6189 has "Ember Week and days" after "previgils," fol. 36.
. "& μύσται & ἐπόπται into Paganos Græcos," in the margin, Hand A/ catechumens to petitioners and then to the faithful/believers.
. BL Harleian 1876 has the following note: "See this largely illustrated by Casaubon. exercit. adu. Baron. exercit. 16.§.43," fol. 26. The reference is to Casaubon's _De Rebus Sacris et Ecclesiasticis_ , 541–566.
. Imperial meal—"paganical" meals.
. Common meals, feasts/confraternities, meals/festivals, solemn feasts.
. Congregation and community.
. Lordly meal and community. BL Harleian 1876 has the following note: "This is further confirmed in yt Pliny, by comand from Trajan, prohibited Christian meetings as ἑταιρεĩαι"/congregations, fol. 27.
. 1 Corinthians 13.
. Followed by "fellowship," in BL 6189, fol. 39.
. BL Harleian 1876 adds "dis" above "honour" as a correction, fol. 28.
. Both are mentioned by Epiphanius in the _Panarion_.
. Discussion.
. Justin Martyr, _Dialogue with Trypho_ , ed. Michael Slusser, trans. Thomas B. Falls (Washington, DC, 2003), ch. 80.
. See John Selden, _De synedriis & praefecturis iuridicis veterum Ebraeorum_ (London, 1655), bk. 1, ch. 8. In BL Harleian 6189, "Decision" is followed by _"aut nescio quid,"_ and in the margin, "Interlined by the corrector of the Or.," fol. 41. Stubbe owned the 1653 edition of Selden's book. For a detailed study of Selden and Stubbe, see ch. 8 in Rosenblatt, _Renaissance England's Chief Rabbi_.
. Selden, _De Iure Naturali_ , 844.
. The discussion by Paul of idolatrous food occurs in 1 Corinthians 8:9–10 and 1 Corinthians 10:1–11. A marginal note in BL Harleian 1876 refers to Johann Heinrich Heidegger (1633–1698), "libert a lege. c.7. l. 3," fol. 29. I was unable to locate this text, but Stubbe owned a copy of Heidegger's _Rashi aboth, sive, de historia sacra patriarcharum. Exercitationes selectae_ (Amsterdam, 1667).
. In BL Harleian 6189, "paramount" is followed by "vel nescio," and in the margin, "Interlined by the Corrector," fol. 44. The story of Cornelius is in Acts 10.
. Selden, _De Iure Naturali_ , 180–181.
. See the discussion of this Trinitarian conundrum in Pliny and Tertullian: <http://fosterheologicalreflections.blogspot.com/2010/03/lightfoot-and-tertullians-mention-of.html>; accessed September 30, 2012.
. Artemon was a third-century Antitrinitarian in Rome; Pope Victor (AD 189–199); Pope Zephyrinus (AD 199–217). Stubbe is drawing on bk. 5, chs. 27 and 28 (Greek version) of Eusebius, _Ecclesiastica Historica_ , 73–74 (chs. 24 and 25 in the Latin version).
. See this discussion in Casaubon, _De Rebvs Sacris et Ecclesiasticis_ , 477–499.
. The Apostles' Creed contains nothing about the controversies regarding the human-divine nature of Christ—which is why it was acceptable to the Arians.
. St. Athanasius (AD 296–372), bishop of Alexandria, upheld the doctrine of the Incarnation against the Arians.
. In the margin, BL Harleian 1876, fol. 33: "In Ecclesia πολιτεία ita ordo à plebe vel Laicis olim distinguebatur, ut in civili gentium, ordo & plebs. Ordo est magistratus vel senatus. Walo Messalin. P.388." See Messalini, _De Espiscopis_ , 388. Waldonis Messalin was a pseudonym that Salmasius used: _De Episcopis et Presbyteris_ (Leiden, 1641).
. Stubbe is reading Buxtorf, _The Jewish Synagogue_ , 314–315. The story of Balaam and his prophecy to Balak appears in the book of Numbers, ch. 22.
. Rabbi Akiva (fl. early second century AD), whose writings furnished the basis for the Mishnah. See the discussion of Casaubon on Akiva and Bar Kochba in Anthony Grafton and Joanna Weinberg, with Alastair Hamilton, _"I have always loved the Holy Tongue": Isaac Casaubon, the Jews, and a Forgotten Chapter in Renaissance Scholarship_ (Cambridge, 2011), 319–320.
. BL Harleian 6189 has "four hundred," fol. 48.
. Followed by " _Idolls_ " in BL Harleian 6189, fol. 49, and a note, " _Interlined by the Corrector_."
. "The worshippers of Serapis are Christians, and those are devoted to the god Serapis, who (I find) call themselves the bishops of Christ. There is here no ruler of a Jewish synagogue, no Samaritan, no Presbyter of the Christians, who is not either an astrologer, a soothsayer, or a minister of obscene pleasures. The very Patriarch himself, should he come into Egypt, would be required by some to worship Serapis, and by others to worship Christ." From the letter of the Emperor Adrian in AD 134, translated in Robert Taylor, _Diegesis_ , 3d ed. (London, 1845), 386.
. BL Harleian 1876 includes the following note: "Tho' ye dignity of a Patriarch was not settled in ye Church till long after ye Nicene Council, is certain: yet yt ye word Patriarch was analogically used in ye Church. & yt ye Alexandrine was such: See Valles. in Socrat. Hist. Eccles.l.5.c.8 Hottinger. Hist. Orient. Pag.101," fol. 35. See _Evsebii Pamphili, Rvffini, Socratis_ , 311; and Hottinger, _Historia Orientalis_ (1651), 102.
. See the discussion of Eutychius/Ibn al-Baṭrīq in the introduction.
. The name given to Christians who offered incense to idols in order to escape persecution.
. "It follows that the eagles are the gods of the legions," Tacitus, _Annals_ , 2:17.
. Labarum was the military standard used by Constantine with the first two letters of the Greek spelling of "Christ."
. Office of chief priest.
. _CTh_.12.1.112: Florentio praefecto Augustali. In consequenda archierosyne ille sit potior, qui patriæ plura praestiterit nec tamen a templorum cultu observatione christianitatis abscesserit. Quippe indecorum est, immo ut verius dicamus, illicitum ad eorum curam templa et templorum sollemnia pertinere, quorum conscientiam vera ratio divinæ religionis imbuerit et quos ipsos decebat tale munus, etiamsi non prohiberentur, effugere. Emissa XVI kal. iul. Constantinopoli Honorio n. p. et Evodio conss. (386 iun. 16). <http://ancientrome.ru/ius/library/codex/theod/liber16.htm#8>; accessed September 30, 2012.
. Quintus Aurelius Symmachus (ca. AD 345–402). The pontifices maximi occupied the highest position in Roman religion.
. "Hathur" is the third month of the Coptic year: see Selden, _De synedriis_ , where, in chapter 15, there is a detailed description of Coptic feasts and fast days, along with the story about Alexander, 345. The story was told by Eutychius in _Contextio Gemmarum_ (Oxford, 1658), 435. BL Harleian 6189 has "Captives" for "Coptites," fol. 53.
. Amandus and Ӕlianus were insurgents in France, ca. AD 285.
. Aemilius Papinianus (AD 142–212), a Roman jurist.
. _Eusebii Pamphili Ecclesiasticæ Historiæ Libri Decem_., 251–252, in _Annotationes_.
. Selden, _De synedriis_ , 342ff.
. BL Harleian 1876 includes the following note: "Cod. Theodos. lib.16. tit.8.leg.29 cum notis Gothofredi," fol. 42. The code describes "the annual tribute that is to be collected from rulers of Jewish synagogues." _Imperial Laws and Letters Involving Religion, AD 395–431_ (Fourth Century Christianity, Wisconsin Lutheran College), <http://www.fourthcentury.com/index.php/imperial-laws-chart-395>; accessed September 30, 2012.
. "Jews will be restricted in their ceremonies; let us, meanwhile, follow the ancients in the preservation of their privileges, by whose law and the assent of our divinity it is ordained that those who are subject to the authority of the illustrious patriarchs (i.e. the archsynagogues, the patriarchs, the elders, and the rest who are involved in the sacrament of that religion) shall preserve those privileges which were conferred upon the first clerics of the venerable Christian law. The deified emperors Constantine, Constantius, Valentinian, and Valens decreed this with a divine order." Stubbe elaborates on the code: 1 July 397, _CTh_ 16.8.13, Arcadius, Honorius: "Jewish clergy are allowed to retain their own laws and rituals and are exempt from service as in municipal senates. They are to have the same privileges as Christian clergy." _Imperial Laws and Letters Involving Religion, AD 395–431_ (Fourth Century Christianity, Wisconsin Lutheran College), <http://www.fourthcentury.com/index.php/imperial-laws-chart-395>; accessed September 30, 2012.
. "Ethnic" is used to refer to Gentile, specifically Greek. BL 1876 has "Metaphysicks" instead of "Mathematicks," fol. 42.
. BL Harleian 1876 has "two hundred," fol. 44.
. In _A further iustification_ Stubbe thinks of "Mahometanism" immediately after discussing the Donatists, 65. Justinian reigned until AD 565 and therefore, appropriately, leads Stubbe to Muḥammad, born a few years later.
. Novatus (ca. AD 200–258); Maximinus Thrax (reg. AD 235–238); Diocletian (reg. AD 284–305).
. BL 6189 has "remitted" instead of "ruined," fol. 65.
. See the same use in _A further iustification_ , 41. Andrew Marvell also used this analogy in _A Short Historical Essay Concerning General Councils, Creeds, and Impositions, in Matters of Religion_ (1676), in _The Prose Works of Andrew Marvell_ , ed. Annabel Patterson (New Haven, 2003), 2:129.
. A church historian of the fifth century.
. Denius Petau/Petavius (1583–1652), Jesuit theologian and historian.
. Contrast Stubbe's treatment of this choice of bishops in _A further iustification_ , 39, where the tone is favorable.
. See Bodin, _Six Bookes_ , 537 where the marginalia state: "Religion not to be enforced," and "How a prince wel assured of the truth of his religion is to draw his subiects thereunto, being the fore diuided into sects and factions."
. The other sites which Stubbe does not mention were Milan, Seleucia, Nice, Tarsis/Tarsus, and Ariminum/Rimini. In BL Harleian 6189 and 1876, the number is 600, fols. 67 and 48 respectively.
. Salvian, Bishop of Marseilles, fifth-century author of _De gubernatione Dei_ (Oxford, 1629), especially bk. 5. Hilary was bishop of Poitiers (d. AD 368).
. In the margin: "Blount has copied this in a letter to Hobbs and ye 2 pages & a half Next," Hand A. Blount made a few changes in copying from Stubbe: see especially Malcolm's notes 31 and 40, _Thomas Hobbes_ , 2:763–766.
. BL Harleian 6189 has a note: "f. layd aside the Exercise of the Power," fol. 70.
. Curiously, the title nostrum numen was applied to Constantine. Circensian relates to circus.
. Eutychius of Constantinople (ca. AD 512–582); Dioscorus of Alexandria (d. AD 454); Severus of Antioch (AD 465–538); Jacob Baradaeus (ca. AD 500–578), bishop of Edessa; Nestorius (ca. AD 386–451).
. Followed by "part" in BL Harleian 6189, fol. 73.
. Council of Jerusalem (described in Acts 15), AD 48; (First) Council of Ephesus, AD 431; Council of Chalcedon, AD 451.
. "It is Greek, and therefore not understood," in Guillaume Ranchin, _A Review of the Councell of Trent_ , trans. G. L. (Oxford, 1638), 152. Stubbe owned a copy of this edition. The pope is Gregory the Great (ca. AD 540–604).
. Valentinus (reg. AD 364–375), Justinius II (reg. AD 565–578), Mauritius Tiberius (reg. AD 582–602). Phocas, Byzantine Emperor (reg. AD 602–610); Heraclius, Byzantine Emperor (reg. AD 610–641).
. Martianus/Marcian/Marcianus (AD 390–457), was emperor of the Eastern Empire.
. Brerewood, _Enqviries Tovching the Diversity of Langvages_ , 140–141.
. Chosroes I (reg AD 531–579); King Hormisdas (reg. AD 579–590); Chosroes II (reg. AD 590–628).
. "unimaginable" is followed by "effects, and" in BL Harleian 6189, fol. 57.
. See David Blondel, _Treatise of the sibyls, so highly celebrated, as well by the antient heathens, as the Holy Fathers of the Church_ , trans. J. D. (London, 1661), 3. Stubbe owned a copy of this edition. _Evsebii Pamphili, Rvffini, Socratis_ , 22, from which Stubbe borrows the phrase "Simoni Deo Sancto."
. Antoninus Pius, one of the "Five Good Emperors" (reg. AD 138–161). Justin was beheaded in AD 165 during the reign of Marcus Aurelius.
. The battle was between Emperor Marcus Aurelius and the Quadi in AD 174. Appolinaris was an apologist (fl. ca. AD 170).
. Cf. Casaubon, _De Rebvs Sacris et Ecclesiasticis_ , 70–87; _Eusebii Pamphili Ecclesiasticæ Historiæ Libri Decem_., 225, in _Annotationes_ ; and the earlier reference to Blondel.
. In both Bodleian 6189 and 1876, the word "postnate" precedes "authority."
. It is significant that Stubbe does not mention Anglicans, obviously thinking them the only true Christians: "I defend Truth, and the Church of England," he wrote to the dean of Christ Church, John Fell ( _A Censure_ (Oxford, 1670), dedication. Nor does he mention the Quakers whom, in 1659, he had mildly defended _A light shining out of darknes:... with a brief apologie for the Quakers, that they are not inconsistent with a magistracy_ (London, 1659). BL Harleian 1876 has the following note: "The Lieflanders are so ignorant, that it may be said Baptism excepted, they have not any character of Christianity. See Olearius, _The voyages_ , p. 30," fol. 60: _The voyages and travels of the ambassadors sent by Frederick, Duke of Hostein, to the Great Duke of Muscovy and the King of Persia_ , trans. John Davies (London, 1669), 32. Stubbe copied word for word about the inhabitants in Livonia. He owned a copy of this edition.
. Cf. Casaubon, _De Rebvs Sacris et Ecclesiasticis_ , 6–21.
. In the margin, Hand A: "v. Blunt fo. 104." John Dale was the author of _The analysis of all the Epistles of the New Testament: Wherein the chiefe things of every particular chapter are reduced to heads, for the memory_ (Oxford, 1652).
. _Eusebii Pamphili Ecclesiasticæ Historiæ, Libri Decem_., 34–36 in _Annotationes_. See also Philo on _The Contemplative Life_ in which he describes the _therapeutae_ , trans. Frank William (Bloomington, 1922), 3–26.
. _Evsebii Pamphili, Rvffini, Socratis_ , 23. The Latin uses the term _cultores_ ; therapeutae appears in the margin, in Greek, in association with the Essenes, 55. Epiphanius relied heavily on Eusebius.
. Salmasius, _De Hellenistica Commentarius_ , 181: "Inde etiam Hebræ lingua appellate est Syriaca." BL Harleian 1876, fol. 62, adds in the margin: "That Mark was also such is most probable." Eusebius wrote that Mark's gospel was written in Egypt, _Evsebii Pamphili, Rvffini, Socratis_ , 23.
. Salmasius, _De Hellenistica Commentarius_ , 254, 258, 250: "It must be rightly understood for a truth that nearly all of Christ's disciples and Apostles, being uneducated and common, evidently fishermen, sailors, and boatmen, understood no other language than the vernacular, that is, the Galilean and Syrian parlance which prevailed in that region. For even if many in Syria and Judea knew Greek, it did not reach at all the men of the basest class who knew only the vernacular and were entirely ignorant of Greek.... Therefore the Apostles wrote in their own idiom and the tongue and in the vernacular that was familiar to them, which was immediately translated into Greek by either hellenized Syrians or Greeks, who spoke Greek and who rendered it faithfully, and who were with the preaching Evangelists as supporters and assistants. In certain cases, this has been verified for a fact; in others, it is not known, since it is not apparent: nevertheless, concerning all of them, there is the resemblance of truth, since there is truth in some of them. For there is no difference between these men, who were equals in respect to tribe and station, as well as vocation and function.... This is a thing which holds for the books of the New Testament; for this reason it is also possible to convey why it is that they are written in a manner greatly different from the more elegant and pure type of spoken Hellenism. As you see, one may say that they were composed partly by the uneducated, partly by translators, who were even themselves not entirely familiar with Greek speech." In the margin of the manuscript, there are the following words: "Predicare non scribere precepti fuere."
. BL Harleian 1876 has "carry age" instead of "carriage," fol. 64.
. Lactantius, rhetorician who converted to Christianity (ca. AD 250–325); Annobius the Elder was his teacher; Minicius Felix, Christian apologist (fl. ca. AD 160–300). The Third Council of Carthage, AD August 397, specified the books of the Old and New Testaments: see B. F. Westcott, _A General Survey of the History of the Canon of the New Testament_ , 5th ed. (Edinburgh, 1881), 440, 541–542. Pope Clement's (alleged) letter was published in 1647: _Clement, the blessed Paul's fellow-labourer in the Gospel, his first epistle to the Corinthians_ (London, 1652 [1647]), 2–3.
. Stubbe may have been thinking of Pococke's _Specimen Historiæ Arabica_ , where Avicenna is associated with "Philosophiam Saracenicam recte inscripseris, non quod Barbara." Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 35. As a physician, Stubbe read the works of Avicenna, whose _Canons_ had been published as early as 1555 in Paris, _Avicennae Arabis Medicorvm_. Stubbe owned a copy of the Louvain edition of 1658.
CHAPTER 3
The title is written in the margin, Hand B: "A Breif Account of Arabia and the Saracens." Henceforth, all odd-numbered pages have "CHAP 3" in the upper left corner. In BL 1876, the title "The History of ye Saracens and of Mahomet" is followed by a blank page and then, in large letters: "A generall Preface to the account of the originall & progress of Muhammadanisme," fol. 57.
. Scribal variants include "Hagarens," "Hagarites," and "Agarens."
. See Gabriel Sionita, _De Nonnvullis Orientalivm Vrbibvs_ (Paris, 1619), 2, which is bound with the translation of al-Idrīsī's _Geographia Nubiensis_. Stubbe owned a copy of this edition. Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 33. The second part of Pococke's _Specimen_ consists of _Notæ in quibus aliqvam mvlta qvæ ad historiam orientalivm apprimè illustrandam faciunt_ , which had been printed in Oxford two years earlier, 1648, by the same printer.
. Pococke, _Specimen_ , 76–78, probably from Abū al-Fida who had mentioned that the Arab clan of Ṣālih was the first to establish a kingdom in Syria. In BL Sloane 1709 there is "Banu Salih filis Salhi" in the margin, fol. 94r.
. The Ghassānid Arabs (Banū Ghassān) were Christians from the tribe of Azd.
. Julius Solinus's third-century AD _De memorabilibus mundi_ was published frequently and translated into English by Arthur Golding as _The worthie work of Iulius Solinus Polyhistor: containing many noble actions of humaine creatures_ (London, 1587). But Stubbe is borrowing directly from Johann Hottinger, _Historia Orientalis_ (Tiguri, 1660), 211: "Solino & aliis Ayman dicitur." The scribe uses "Yaman."
. See Johann Hottinger, _Historia Orientalis_ (Tiguri, 1651) on "De Mvhammedis parentibvs," 134–137.
. Pococke, _Specimen_ , 150–152.
. Sionita, _Nonnvllis_ , 2. Joktan is the biblical name for Qaḥtān, Genesis 10:25–29. Stubbe copied "Mota-Arabes"/Mostaarabs from Pococke, _Specimen_ , 39; p. 45 has "Most Arabes." In the translation of _Eutychii Patriarchæ Alexandrini Annalium_ , the term is "Most Arabibus" (Oxford, 1654), 2:272. The first use of "Mozarab" appears in 1615 ( _OED_ ). The manuscript has "the Son of Heber the Son of Saleh" with "of Heber" crossed out. In BL Sloane 1709, it is not, fo. 94v. Sionita has "Heber," 2.
. This use of "Coreischites" is very early in English. The variants in the manuscript are "Koreischites" and "Coreishites."
. Pococke writes Jorham/Jorhamum in _Specimen_ , 38. The _OED_ records the first use of "Coreis"/Quraysh by Alexander Ross in his English translation of the Qur'ān (1649).
. Pococke, _Specimen_ , 4, 150–51. Pococke's spelling is "Hamyar," which Stubbe uses. Ḥimyar was a tribe in southwestern Arabia. "Ismailites" had been used as early as 1571 ( _OED_ ).
. Pococke, _Specimen_ , 40.
. Ibid., 138. See also Hottinger, _Historia Orientalis_ (1660), "Henoch, qui vocatur Adris," 30.
. The scribe wrote, "Maimonides apud Hottinger.hist.or.l.1.c.8." In _Historia Orientalis_ (1651), bk. 1, ch. 8, there is discussion of the religions of the Sabeans and the Nabateans, along with other religions of "veterum Arabum." There are numerous references to Maimonides.
. Pococke spelled the name, "Abulfeda," ignoring its patronymic construction. There was no published edition of Abū al-Fida's geography covering the Arabian Peninsula, although as Toomer states, Selden owned a manuscript of the geography ( _John Selden_ , 2:619 and also "Arabic Learning after the Restoration," in _Eastern Wisedome_ , 227). John Gregory mentions a manuscript of Abū al-Fida at Cambridge, in _The Works of the Reverend and learned Mr. John Gregory_ (London, 1665), 73 margin, but later adds: "For the Arabick-Nubian Geographie, translated into Latine by the Maronites, though otherwise of a rare and pretious esteem, yet it is not commended for this, That the Distances of Places are there set down by a gross Mensuration of miles: and John Leo's Africa is not so well. But when the Learned and long-promised Geographie of Abulfeda the Prince shall come to light, there can be nothing done there without this Meridian," 266. It is unclear what is meant by "long-promised," unless Gregory had known (before his death in 1646) about John Greaves who published in 1650 an Arabic and Latin selection by Abū al-Fida about the region "extra fluvium Oxum," _Chorasmiae et Mawaralnahrae_ (London, 1650). There may well have been a longer manuscript by Greaves that was circulating since in 1712 John Hudson published the earlier account by Greaves with a unit on Arabia, specifically describing the Islamic holy cities and many other locations, _Geographiae veteris scriptores Græci minores Accedunt Geographica Arabica_ (Oxford, 1712), 3:1–66, new pagination. At the same time, there are twelve manuscripts of Abū al-Fida's _Taqw īm al-buldān_ in the BnF, one of which is a copy made by G. Schikhart, professor of Hebrew at Tubingen: see MS Arabe 2241 in _Catalogue des Manuscrits Arabes_ , M. Le Baron de Slane (Paris, 1883–1895).
. God almighty.
. From Pococke, _Specimen_ , 108.
. Ibid., 139–140.
. Claudii Salmasius, _De Annis Climactericis et Antiqua Astorlogia diatribæ_ (Leiden, 1648), 578–579, where there is a discussion of the Greek origin of the Arabic word, ṭalṣam, "vulgo Talisman." Stubbe owned a copy of this edition. See also the discussion of "Tilsemat" in Hottinger, _Historia Orientalis_ (1660), 284–289.
. Pococke, _Specimen_ , 140–141.
. Ibid., 148–149.
. All these passages, from "They say that Noah" until here, follow Pococke, _Specimen_ , 144. Manuscript variants for 'Ka'ba and Mecca are the following: "the Caab," "Alcaab," "Caaba," "Kabe," 'Cabea," and "Meccah," "Mecha," "Macca," "Mecca."
. For references to the Sabians/al-Ṣābi'a in the Qur'ān, see 2:62; 5:69; 22:17.
. Pococke, _Specimen_ , 274. The reference is to the Nabateans, whose kingdom included the trading center in Petra.
. Thomas Erpenius, _Orationes tres, de linguarum Ebraeæ, atque Arabicæ dignitate_ (Leiden, 1621), 41; Hottinger, _Historia Orientalis_ (1651), bk 1, ch. 4.
. Pococke, _Specimen_ , 39.
. 'Adnān and Qaḥtān are the two ancestors from whom western and central Arabs and eastern Arabs are respectively descended.
. Hottinger, _Historia Orientalis_ (1651), 142–143.
. The tribe of Khuzā'a.
. Pococke, _Specimen_ , 42 where he writes, "akhsar min abi Gabshan." Pococke writes "Chozaah," which Stubbe borrows.
. Ibid., 114, 115.
. Ibid., 115. Stubbe borrows the translation of "Domum interdictam" from Pococke.
. In BL Harleian 1876, there is the following note: "It was always a part of Arabick devotion, to go about, & compass ye temple where they worshipped: to some such devotion perhaps David alludes when he says ps.77. so shall the congregation of ye people compass thee about. & Ps.26:6 so will I compass thine Altar," fol. 112.
. The references are to the tribe of Khuzā'a, and to King 'Amr ibn Luḥay.
. As'ad abū Karib was a Ḥimyarite king.
. Pococke, _Specimen_ , 80, 60, 81.
. Ibid., 98 ff.
. Ṣafā and Marwā are the two mountains between which Hagar ran to find sustenance for her son.
. The names are taken from Pococke, _Specimen_ , 118: "Abu Kobau," and 128 "Koaikaban." The latter might be a reference to Kawkaban in northern Yemen.
. The tribes of Ṭayy and Qaḥtān; Al-Ḥārith ibn Ka'b. Stubbe lifts this name from Pococke's notes, _Specimen_ , 109, where it refers to "Bani Hareth."
. Ibid., 60–64. Yūsuf Dhū Nuwās (reg. ca. AD 517–525), the last king of Ḥimyarite Yemen, converted to Judaism.
. Maḥmūd.
. Pococke, _Specimen_ , 63–64, where the whole episode, along with the Qur'ānic verse about it, is described.
. Pococke's spelling, _Specimen_ , 71. Anū Sherwān, king of Persia (reg. AD 488–513).
. Al-Nu'mān and al-Mundhir. Stubbe's spelling is from Pococke, ibid., 72.
. _Evsebii Pamphili... Ecclesiastica historia_ , 544, "De Mauia Saracenorum regina." For Mavia and the Christian Kings of Arabia, see Glen W. Bowersock, "Mavia, Queen of the Saracens," in _Studies on the Eastern Roman Empire_ (Goldbach, 1994), 127–140.
. Pococke, _Specimen_ , 85.
. There is a note referring to "Evagrius bk 4, ch. 12," the same words that appear in Pococke, _Specimen_ , 85.
. Al-Mūsil in Iraq.
CHAPTER 4
The title is written in the margin, Hand B. Henceforth, all odd numbered pages have "CHAP 4" in the upper left corner.
. The date 580 appears in Thomas Erpenius, _Orationes tres, de linguarum Ebraeæ, atque Arabicæ dignitate_ (Leiden, 1621), 42, but there is a discussion of other dates in Johann Hottinger, _Historia Orientalis_ (Tiguri, 1660), 217–218.
. 'Abdallah and Amīna.
. Roderici Ximenez, _Historia Arabum_ , ed. Thomas Erpenius (Leiden, 1625), 2. The treatise is bound with Jirjis ibn al-'Amīd al-Makīn, _Historia Saracenica_ , trans. Thomas Erpenius (Leiden, 1625).
. There is a note in BL Harleian 1876, fol. 117, to "Gregor.Abūlfarai.p.101," but it does not correspond with any of the three editions that had been published of Abū al-Faraj. Johann Hottinger, _Historia Orientalis_ (Tiguri, 1651), bk 2, ch. 1, writes about "Muhammedis Educatione, Itinere Syriaco," but there is no mention of travel beyond Syria "& alia loca," 210.
. BL Harleian 1876: "Others say yt he was an Hermit, & yt he cry'd out, having view'd Mahomet well: There is no God, but God & Mahomet, ye Apostle of God" (fol. 117). See Hottinger, _Historia Orientalis_ (1651), 203, where there is mention of a hermit who exclaims: "Non est Deus, nisi Deus, & Muhammed, Apostolus ejus."
. The month of Muḥarram. Stubbe is borrowing from Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 174–176, but the imperfect spelling of the month is the scribe's.
. Abū Bakr al-Ṣiddīq was not Muḥammad's uncle (d. AD 634), but he was the father of one of the Prophet's wives. In Arabic, however, a father-in-law is also an "uncle." The name is also spelled "Abubecr" in the manuscript.
. In the seventeenth century, "obnoxious" meant "exposed to harm" ( _OED_ ).
. The words in parentheses are in the margin, BL Harleian 1876, fo. 118, with a sign after "Princes" to add them into the text.
. Hottinger, _Historia Orientalis_ (1651), bk 2, ch. 1.
. Pococke, _Specimen_ , 170.
. Gaius Julius Priscus, born in Syria, rose from soldier to brother of Emperor Philip the Arab in the third century AD; Pertinax, Roman Emperor who started his life as a teacher and later as a soldier (AD 126–186); Don Ambrogio Spinola Doria (1569–1630), Italian aristocrat who served the Spanish crown. See also André Tiraqueau, _Commentarij de nobilitate, et ivre primigeniorvm, hac postrema editione ab autore ipso diligentissimè recogniti, & tertia ampliùs parte locupletati_ (Venice, 1574), chapter 33, "An mercatura derogit Nobilitati," especially paragraph 17 where there is a discussion of emperors and kings who were "mercatores."
. Boxhornius mentions the travels of Muḥammad to Egypt and Palestine, but not Spain. Marcus Zuerius Boxhornius, _Historia universalis sacra et profana a Christo nato ad annum usque MDCL_ (Leiden, 1652), 397. King Ricaredus was the first king of the Goths who was not Arian.
. Purchas repeats this story about rice, "Foolish and blasphemous traditions." _Purchas his Pilgrimage_ (London, 1626), 232. Purchas relied on the introduction of Sionita's 1619 translation of al- Idrīsī.
. Khadīja bint Khuwaylid, d. AD 619. Another variant: "Chadijah."
. Pococke, _Specimen_ , 170–171.
. Hottinger, _Historia Orientalis_ (1651), 206, for the dream.
. Tiraqueau, _Commentarij de nobilitate_ , chapter 25, "An nobilitas perdatur ob paupertatem," but the emphasis is on the poverty of Graeco-Roman, rather than biblical, figures.
. Waraqa ibn Nawfal, the paternal cousin of Khadīja, is mentioned in Bukhāri's rendition of the Hadith, narrated by 'Āisha (I, 1, 3): "Khadija then accompanied him [Muḥammad] to her cousin Waraqa bin Naufal bin Asad bin 'Abdul 'Uzza, who, during the pre-Islamic period became a Christian and used to write the writing with Hebrew letters. He would write from the Gospel in Hebrew as much as Allah wished him to write. He was an old man and had lost his eyesight." He died a few days after Muḥammad received the first revelation of the Qur'ān. There is no reference in the Hadith that he was a teacher to the Prophet. Toward the end of _Originall_ the name is transliterated correctly as "Warakeh."
. Ibn Muqla (AD 886–940), Persian calligrapher in Abbasid Baghdad; and, probably, al-Qāsim ibn Abī al-Bazza (d. AD 741), an exegete.
. Pococke, _Specimen_ , 157.
. Ximenez, _Historia Arabum_ , 2.
. Pococke, _Specimen_ , 169–170, 153.
. Another variant in the manuscript is "Aly."
. Al-Makīn, _Historia_ , 43.
. Genethlia is birthday celebration.
. The reference to al-Makīn is from Hottinger: "Undoubtedly Muhammad was born at the end of the night, when Libra was in the middle of the sky; in the middle of the night, to be sure, the constellation Taurus had crossed the meridian, for otherwise the prophet and leader would not have been able to coincide with it." Hottinger, _Historia Orientalis_ (1651), 146. Abū Ma'shar (AD 787–886) was an astrologer.
. Another variant of the Qur'ān is "Alkoran." I shall retain both of Stubbe's spellings as they appear in the manuscript.
. As Purchas shows, the term derives from Strabo, "Scenites vel Nomades," _Pilgrimage_ , 224. See J. Spencer Trimingham, _Christianity Among the Arabs in Pre-Islamic Times_ (London, 1979), appendix A, "Greek and Latin Terms designating Arab Nomads," for a discussion of the term.
. In AD 626.
. BL Harleian 1876 has "miserably" while BL Harleian 6189 has "universally," fols. 125 and 110 respectively.
. 'Umar ibn al-Khaṭṭāb (d. AD 644) and 'Uthmān ibn 'Affān (d. AD 656), second and third of the Righteous Caliphs. Other variants for "Othman" in the manuscript are "Otsman," "Otsmin," and "Osman." The spelling of "Omar" was consistent throughout the manuscript although there is a later variance in reference to "Ibn Omer."
. Pandects: a unit of laws in Justinian jurisprudence.
. Pococke, _Specimen_ , 158–161, esp. 160.
. The reference is from Nicholas Fuller, _Miscellaneorum Theologicorum_ , 2d ed. (Strasbourg, 1650), 448, but to a pseudo prophet, and not specifically to "Mahomet." See also Pococke, _Specimen_ , 374.
. Purchas makes this point, but without citing a source for it, _Purchas_ , 232. In BL Harleian 6189, the number is "two," fol. 111.
. See Bodin, _Six Bookes_ , 500–501.
. Geronimo Cardan (1501–1576), mathematician, and his father Fazio Cardan.
. Ximénes de Cisneros (1436–1517), Spanish statesman and scholar. See William Hickling Prescott, _History of the Reign of Ferdinand and Isabella the Catholic_ (London, 1885), 436.
. Pococke's spelling, _Specimen_ , 121.
. Sūrat al-An'ām (the Cattle) is one of the early Meccan revelations. The spelling _surat_ is taken from Pococke. The first _OED_ entry, however, is from Robert Boyle in 1661.
. Pococke, _Specimen_ , 108–109.
. The Qur'ānic 'Ozayr (9:30), revered by the Jews as "son of God." For the veneration of Mary, see Epiphanius, _The Panarion of Epiphanius of Salamis, Book 1_ , trans. F. Williams, 2d ed. (Leiden, 2009), 30.3.7, p. 122 and the discussion in Hottinger, _Historia Orientalis_ (1660), 343, and for 'Ozayr 334–36.
. Luqmān al-Ḥakīm/the Wise, after whom it is very likely that Sura 31 in the Qur'ān is named. Erpenius had translated "Fabūlae Locmani Sapientis" which appeared in _Arabicae Linguae Tryocinium, id est Grammatica Arabica_ (Leiden, 1656), 1–172. The book was edited by Jacob Golius. See also Hottinger, _Historia Orientalis_ (1660), 101–103, where there is a discussion of the sura. The association with Aesop is in _Historia Orientalis_ (1651), 69.
. He died in 619, the same year as the Prophet's wife, Khadīja.
. 'Aws and Khazraj were two tribes of Yathrib with connections to Jewish communities.
. Stubbe borrows "Muslimin" from Hottinger, _Historia Orientalis_ (1660), who uses it early on in the book, 4, "Moslemin."
. "Specious" used in its Latin derivation: beautiful, plausible.
. The previous four paragraphs are from al-Makīn. The material in the last paragraph was also developed from al-Makīn by Hottinger, _Historia Orientalis_ (1660), 424–425.
. Abū 'Abdallah Muḥammad al-Idrīsī, _Geographia Nubiensis_ , trans. Gabriel Sionita (Paris, 1619), 44–45.
. BL Harleian 6189 has "evince" instead of "revive."
. Stubbe takes some liberty with his source (as he does on numerous other occasions): al-Makīn does not state that the Prophet had favored Jesus, but had asked that the message of all "Prophetarum & Apostolorum" be accepted. Al-Makīn, _Historia_ , 3.
. Q 3:45.
. Most likely Khālid Abū Ayyūb (al-Anṣārī).
CHAPTER 5
The title is written in the margin, Hand B. Henceforth, all odd numbered pages have "CHAP 5" in the upper left corner.
. Other variants are "Islanisme," "Islanism," and "Islamisme."
. Sionita, _Geographia Nubiensis_ (Paris, 1619), ch. 1.
. See _Purchas his Pilgrimage_ (London, 1626), 231, where there is a similar description.
. BL Harleian 6189 has "two," fol. 122.
. See al-Makīn, _Historia_ , bk. 1, ch. 1.
. In the sense of a grammatically complete sentence _(OED)_.
. Fātima.
. Other variants are "Moslemin" and "Mosslemin."
. The noun is used in the singular and in the plural with "Mossleman," "Mussulman," "Mosleman," and "Mussulmen," as variants. The emphasis that Stubbe places on it is important since he challenged seventeenth-century misusages. In 1611 John Floria stated that "A Pagan beleeuer is a Mussulman," _Queen Annas New Worlde of Words_ (London, 1611), 195, and toward the end of the century, in 1695, and during his controversy with John Edwards (1637–1716), Locke ridiculed his opponent's ignorance of Arabic and the use of "Musselmen" instead of "in plain English, the Mahometans." Locke, _The Reasonableness of Christianity_ in _Works_ (London, 1963 [1833]), 7:282.
. Verbatim from Johann Hottinger, _Historia Orientalis_ (Tiguri, 1660), 438. Bilāl ibn al-Ḥārith, Suhayl ibn'Ammār, possibly Ṣuhayb al-Rūmi, and Khabbāb ibn al-Aratt. It is difficult to identify the others.
. The paragraph is from Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 102.
. The scribe split the noun in two. It should be _al-Muhajir ūn_.
. Paranomasia is pun.
. 1 Chronicles 5–18 describe the wars between the Hagarites and the Reubenites.
. Nicholas Fuller, _Miscellaneorum Theologicorum_ , 2d ed. (Strasbourg, 1650), 255.
. In BL Harleian 6189, there is "* _victory_ " rather than "history," but the asterisk reads: "* _f. History_ ," fol. 129.
. Trajan, Roman Emperor (reg. AD 98–117); Severus, Roman Emperor (reg. AD 193–211). See also Pococke, _Specimen_ , 53.
. Olearius, _The voyages and travels of the ambassadors sent by Frederick, Duke of Hostein, to the Great Duke of Muscovy and the King of Persia_ , trans. John Davies (London, 1669), 196.
. In BL Sloane 1709, the words are "ostentation but courage," fol. 102r.
. BL Harleian 1876 has the following note: "The Arabians amongst ye blessings God bestowed on ym reckon these: yt instead of Crowns each of ym hath his Tulipant: & yt yr swords are to ym what walls & bulwarks are to others," fol. 140.
. Stubbe takes liberty with the episode of "Arbitration," during the battle of Ṣiffīn when the slogan was first raised. It was described by al-Makīn, _Historia_ , 39.
. Fuller, _Miscellaneorum Theologicorum_ , bk. 1, ch. 13, on "De Astroarchæ & Vrania apud Herodianum: De Astarte: Deque Lunæ imaginæ in Saracenorum insignibus."
. Shahoronim: chains around the camel's neck with small moon-like or crescent-like ringlets, Judges 8:26; Isaiah 3:18. Stubbe's use of "lunulets" is before the first 1826 entry in _OED_. Stubbe is borrowing from Selden, _De Dis Syris_ , Syntagma II, 289.
. BL Harleian 1876 has the following note: "These are all Mahometan sayings, & some ascribed to Ali," fol. 141. The note also appears in BL Sloane 1709, fol. 102v.
. Verbatim from Olearius, _The voyages_ , 279. The note also mentions Lancelot Addison, _West Barbary, or, A short narrative of the revolutions of the kingdoms of Fez and Morocco_ (Oxford, 1671), 160, where there is another version of the Sura.
. In BL Harleian 1876, there is the following note: "Keblah is ye place of Heaven or Earth towards wch they praied: wch was Jerusalem, till Muhammad changed it to ye Caab," fol. 143. The scribe in MS 537 wrote "Reblah" instead of Pococke's "Keblah." Pococke, _Specimen_ , 113. In BL Sloane 1709, "Keblah" is replaced by "Kaabah," fol. 103r.
. John Selden, _De Dis Syris Syntagmata_ (Leipzig, 1662), 292, 294.
. In the margin, Hand A: "v. Selden de Dii Syrii in venere."
. Selden, _De Dis Syris Syntagmata_ , 291: "It was not established but related by Muhammad that the Guimia [Friday] feast seemed to have flowed from the rites of horned Urania and the ancient effigy of little moons near to them..it seemed to have flowed."
. See Fuller, _Miscellaneorum Theologicorum_ , bk. 1, ch. 13.
. In BL Harleian 1876, there is the following note: "This story is in the Alcoran: Dr. Pocock ubi supra. p. 87," fol. 145. In BL Harleian 6189, there is the following alteration: "... escaping in the Ark [* _that besides the Introduction of a multitude of Associate Gods_ ] that this..." The asterisk points to the following note: "* _Sic. This seems to be misplaced. It follows in the next page_ ," fol. 139. For the Prophet Ṣālih, see Q 7:73.
. The tribe of Thammūd. See Hottinger's discussion of Ṣālih and the episode on which Stubbe elaborates, _Historia Orientalis_ (1660), 44–48.
. In BL Harleian 1876, there is the following annotation: "This is avowed by Sharestanius, in Dr. Pocock ubi supra. p.54" (fo. 146). The reference to Shahrastānī is in Pococke, _Specimen_ , 53.
. Ibid., 119.
. Ibid., 118–121.
. Ibid., 120. See Hottinger, _Historia Orientalis_ (1660) who quotes the same passage, 356–357.
. The scribe wrote "fund a Mentall." BL Harleian 1876 has the following note: "Olearius & others give another reason for its blackness, & how it come there: but 'tis not strange amongst the Muhammadans to find many reasons for ye same thing," fol. 147.
. In BL Harleian 1876, there is the following note: "Selden, Syntag bk 2, ch. 14," fol. 148. There is a discussion of Mercury/Merkolis and the throwing of stones in chapter 15. Selden, _De Dis Syris Syntagmata_ , 353–354.
. Another variant is "Muslimittical."
. Stubbe relies on Olearius, _The voyages_ , 173, from where he borrows "sceithan." "that it should be soe" is underlined, but is omitted in BL Sloane 1709, fol. 104r.
. Genesis 17:23–27.
. Genesis 16:10.
. BL Harleian 1876 has this note: "Ahmed Ben Edris in scripto Elenetic.l.c.38," fol. 151, which Stubbe copied from Hottinger, _Historia Orientalis_ (1660), "Ahmed ben Edris, scripti sui, Elenchitici l. 1, c. 38," 12.
. Pococke, _Specimen_ , 110. In BL Sloane 1709 the sentence starting "crying Allah" and ending "turned into Cobar" is in the margin, fol. 105r. Is the scribe of the University of London manuscript copying from Sloane and integrating material into the text proper? Or was there another (lost) manuscript?
. There is "man" in Sloane 1709, fol. 105r, and in the other manuscripts.
. In BL Harleian 6189, "objects" replaces "projects," fol. 154.
. Pococke, _Specimen_ , 105–107. "Ismaelism" is followed by "which God was now resolved to put an end unto," BL Sloane 1709, fol. 106r.
. In BL Sloane 1709, "paradise" is followed by "& if it is lawfull to drink it there why not here," fol. 106r.
. See Marcus Zeuerius Boxhornius, _Historia universalis sacra et profana a Christo nato ad annum usque MDCL_ (Leiden, 1652), 398, for this dialogue. BL Sloane 1709 continues: Abdias replyed There is very good reason for it. "The Agareans... "fol. 106 r. There were numerous versions of this story: J. Kritzeck, _Peter the Venerable and Islam_ (Princeton, 1964), 89.
. In BL Harleian 1876, there is the following note: "All ys discourse at wine is alleged by me upon supposal yt in Arabia they had wine: wch is probable from yr worship of Bacchus: but if they had none, as Amianus Marcellinus (who was amongst ym) says he met with none that knew wt wine was: then did Mahomet comply with the constitution of ye Saracens in forbidding it: wisely foreseeing yt yir conquests & travells would acquaint them with it: & so he by those stories endeavoured to prevent the inconveniences of drinking it. But I believe they had wine in some places," fol. 157. The use of "apologue" is the first before the 1699 entry in the _OED_.
. _Augerii Gislenii Busbequii d. Legationis turcicae epistolae quatuor_ (Frankfurt, 1595), 106–107.
. The first use of "complacent" in this manner had appeared in 1660 ( _OED_ ).
. Pococke, _Specimen_ , 52–53, but the earlier part is taken from the Qur'ān.
. A note in BL Harleian 1876 reads: "This was written on Muhammad's seal: In duplex testimonium," fol.160.
. Also spelled "Melchite" in the manuscript.
. Gratian (AD 359–383) was a Roman emperor who continued the fight against the Arians. His father was Valentinian I, and his mother was Marina. His uncle was Valens.
. Justin I (reg. AD 518–527), Byzantine emperor and founder of the Justinian dynasty; Sozomen, in _Evsebii Pamphili, Rvffini, Socratis, Theodoriti, Sozomeni, Theodori, Evagrii, et Dorothei Ecclesiastica Historia_ (Basel, 1570), 544, "Saraceni cū Romanis federe conjuncti." There is mention of Palestine, Phoenicia, and Egypt, but not Syria.
. A note mentions "Abu al faraj 93," which corresponds to the same page in Edward Pococke, _Historia Compendiosa Dynastiarvm_ (Oxford, 1672).
. Fuller, _Miscellaneorum Theologicorum_ , 314. See also Pococke, _Specimen_ , 74.
. "Aretas" derives from the Arabic, Ḥāritha: al-Mundhir ibn al-Ḥārith was the king of the Ghassānid Arabs from AD 569 to ca. 581. It is not clear whom Stubbe had in mind in regard to "Aretas," unless it is to the NT reference, 2 Corinthians 11:22.
. BL Harleian 1876 has "Emperour" instead of "Empire," fol. 162. Philip the Arab (reg. AD 244–249). The reference to "Bostra" could be to Busra in Syria, since Philip was born in the province of Arabia.
. Odenatus and his wife Zenobia ruled Palmyra and fought the Romans in the third century.
. Sozomen in _Evsebii Pamphili, Rvffini, Socratis_ , 545. Another variant, rare, for Hagar is "Agar."
. Fuller, _Miscellaneorum Theologicorum_ , bk. 2, ch. 12, "De nomine Saracenorum, De Ismaëlitis, Cedrais, & Agarenis"; Pococke, _Specimen_ , 33; Hottinger, _Historia Orientalis_ (1651), 4–9.
. Hottinger, _Historia Orientalis_ , 7–8. For a discussion of the term _Saracens_ in English seventeenth-century usage, see Katharine Scarfe Beckett, _Anglo-Saxon Perceptions of the Islamic World_ (Cambridge, 2003), ch. 9. See especially her references to Thomas Newton, 205–207.
. BL Harleian 1876 has the following note: "This saying is related to have been ye most frequent of any in the mouth of Ali: Hottinger.hist.or.l.2.c.5," fol. 167.
. BL Harleian 1876 has the following note: "That ys is ye Muhammadan tenet concerning praier, see L. Addison of Westbarbary.c.9," fol. 167, Addison, _West Barbary_ , "Of the Moresco Church Government," 155–165.
. BL Harleian 1876 has the following note: "These are all Muhammadan sayings," fol. 168.
. Another example of Stubbe taking liberties with his sources. The episode is from _Augerii Gislenii Busbequii d. Legationis turcicæ epistolæ quatuor_ , 107–108, but it has nothing to do with 'Ali.
. BL Harleian 1876 has the following note: "Orotal ye Great God: Alilia ye Associate Goddess of ye Arabians," fol. 168.
CHAPTER 6
The title is written in the margin, Hand B. Henceforth, all odd numbered pages have "Chap 6" in the upper left corner. This chapter division is absent from BL Sloane 1709 where there is just a new paragraph.
. Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 172.
. See Acts 8:27ff. Henceforth the scribe used "Abyssines" instead of "Abyssinians."
. Edward Brerewood, _Enqviries Tovching the Diversity of Langvages, and Religions throughout the cheife parts of the World_ (London, 1614), 155.
. BL Harleian 1876 has the following note: "Gregor.Abulfarai.p.93. say's ye Arabian Christians were Jacobites," fol. 170, Edward Pococke, _Historia Compendiosa Dynastiarum_ (Oxford, 1663), 93–94. Also repeated in BL Sloane 1709, fol. 110v.
. Pococke, _Specimen_ , 63–64.
. John Selden, _Uxor Ebraica_ (Frankfurt, 1673), 395–396. Stubbe closely followed Selden, but he spelled the names differently. In BL Harleian 1876 there is the following note: "I am sorry yt Selden did not publish the letters entire," fol. 170.
. Johann Hottinger, _Historia Orientalis_ (Tiguri, 1660), 461. But Stubbe was confused about the names: Ja'far ibn abī Tālib went on the migration to Abyssinia; Zayd ibn Ḥāritha was the adopted son of the Prophet. There is no record of 'Abdallah ibn 'Umar.
. Jirjis ibn al-'Amīd al-Makīn, _Historia Saracenica_ , trans. Thomas Erpenius (Leiden, 1625), 4.
. The title of this section is written in the margin, Hand A.
. Al-Makīn, _Historia_ , 5. The reference is to "Abusofianum f. Harithi," Abū Sufyān ibn al-Ḥarith, a cousin of the Prophet and one of the Companions. Another variant in the manuscript is "Abusophian."
. The Battle of Badr—AD 624.
. Hottinger, _Historia Orientalis_ (1660), 329, "Phinhas, filius Azura," Finḥās, a rabbi in Medina. See also p. 358 where Hottinger contrasts the trustworthiness of the Christians with the deceit of the Jews.
. The title is written in the margin, Hand A.
. Hottinger, _Historia Orientalis_ , 328.
. Most probably 'Utba ibn Waqqāṣ at the battle of Uḥud in AD 625.
. The paragraph is from al-Makīn, _Historia_ , 4–5. The references are to "Ka'baum fil. Alasrafi"/Ka'b ibn al-Ashraf, "Ochas filius Abumugidi"/'Uqba ibn abī Mujīd, and "Abdalla quoqs filius Sjehabi"/ 'Abdallah ibn Shihāb.
. The reference is to "Naimus f.Masudae Gatfanites"/Na'īm ibn Mas'ūd al-Ghatfani.
. The title is written in the margin, Hand A.
. Al-Makīn, _Historia_ , 6–7, Hudaybiyya.
. The Treaty of Hudaybiyya, AD 628.
. Edward Pococke, _Historia Compendiosa Dynastiarvm_ (Oxford, 1672), 102.
. The title is written in the margin, Hand A.
. Pococke, _Specimen_ , 98, "AlJannabium."
. Ibid.
. The image of "vulgar heads" recalls Sir Thomas Browne's _Psuedodoxia Epidemica_ (London, 1646), ch. 5, where Browne is denouncing the Muslims for their credulity. Stubbe owned a copy of "Brown's vulgar Error et Religio Medici London 1672," BL MS Sloane 35, fol. 8r.
. Al-Makīn, _Historia_ , 8, "Melicum filium Aufi"; Hottinger, _Historia Orientalis_ (1651), 271. The tribes are Thaqīf and Hawāzin.
. The opposition of the Hawāzin tribe to Muḥammad was led by Mālik ibn 'Awf al-AnṣārĪ. "Taiph" is Ṭā'if, the city of the tribe of Thaqīf. "Horam" is Ḥunayn.
. Erpenius, _Orationes Tres, De Linguam Ebraeae atque Arabicae Dignitate_ (Leiden, 1621), 42ff.
. Hottinger, _Historia Orientalis_ (1651), 271–272.
. Qur'ān 9:25.
. Al-Makīn, _Historia_ , 8–9.
. The title is written in the margin, Hand A.
. Sihan ad-Dauma, in Yemen.
. Pococke, _Historia Compendiosa Dynastiarvm_ , 103. In the margin, there is "Mosaleima" which is similar to Pococke's "Mosailema." In this manuscript another variant is "Moseilina." Musaylima ibn Ḥabīb claimed to be a prophet in the ninth year after the _hijra_.
CHAPTER 7
The title is written in the margin, Hand B. Henceforth, all odd numbered pages have "CHAP 7" in the upper left corner.
. Quoting verbatim from Olearius, _The voyages and travels of the ambassadors sent by Frederick, Duke of Hostein, to the Great Duke of Muscovy and the King of Persia_ , trans. John Davies (London, 1669), 172.
. Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 315–316. The place is "al-'Aqaba."
. Gabriel Sionita, _De Nonnvullis Orientalivm Vrbibvs_ (Paris, 1619), 20.
. Olearius, _The voyages_ , 172.
. "Daroga" is a word that Stubbe must have taken from Olearius, 1662 ( _OED_ ). The passage continues in BL Sloane 1709, "giving him many blows on ye head neck & heart," fol. 114r.
. Pococke, _Specimen_ , 114.
. See Olearius, _The voyages_ , 173; Pococke, _Specimen_ , 120, 121. In BL Sloane 1709 the sentence is different: "advanced from the ground about 7 handfulls or 2 cubits & an half," fol. 114r.
. Pococke describes the pilgrimage, _Specimen_ , 310–316.
. Both are Moabite deities. See Judges 11:24, Numbers 25:3.
. The celebration of Eid al-Aḍha takes place between 10–12 of the month of Dhul Ḥijja.
. "Ashura" is 'Āshurā', the tenth day of the month of Muḥarrahm. The Jews fasted on that day in remembrance of the exodus, and so the Prophet instituted it as a fast day, as Stubbe notes.
. Ibn al-Athīr (1160–1233) was a historian, frequently mentioned by Pococke.
. For "accquests," BL Harleian1876 has "conquests" and BL Harleian 6189 has "acquaintance," fols. 188 and 209 respectively. In BL Sloane 1709, "power" is followed by "nor his inauguration to be Xerirriffs," fol. 114r.
. The scribe was careless. In both BL Harleian 1876 and Harleian 6189, the word is "dignity." And "broils" should be "brawls."
. Stubbe anglicized the name from Hottinger, _Historia Orientalis_ (1660), 416–417, "Firus, Dailamatia," Fayrūz al-Daylamī. In BL Sloane 1709, "prophet" is followed by "in Yaman, & multiplied his followers, & possessed himself of power, provinces," fol. 114v.
. Pococke, _Specimen_ , 189–190.
. Ibid., 189.
. Ibid., 178. Stubbe misunderstood (his memory failed him?) the reference in Hottinger, _Historia Orientalis_ (1660), where the quotation reads as follows: "the Prophet had received what none other had received, over 3,000 verses ('éaque varia ad tria millia, et amplius'). Had he received only the Qur'an, it would have been a sufficient demonstration of the miraculous" (488).
. BL Harleian 1876 has the following note: "The Muhammadans believe, & so he told ym, yt Isa was not crucify'd, but conveyed to heaven, & an imaginary body crucify'd in his stead," fol. 191. The reference is to Q 4:157. In BL Sloane 1709 the note reads: "Muhammadans believe {or so he told ym}[added above the line] yt Isa was not crucified but conveyed to heaven & an image... crucified in his stead," fol. 115r.
. Pococke, _Specimen_ , 14. See also 179–180, where Pococke cites al-Shahrastanī.
. Edward Pococke, _Historia Compendiosa Dynastiarvm_ (Oxford, 1672), 103.
. Sionita, _De Nonnvullis_ , 19. BL 6189 has "double Stone" instead of "tombstone," fol. 214.
. "Pretend" as in "offer, present, or put forward for consideration" ( _OED_ 1655 entry).
. The first entry of "Indostan" in the _OED_ refers to language rather than country, as with Stubbe ( _OED_ , E. Terry, 1655).
. In the margin: "Vide Blue Book 123 Chap. 8eg," Hand B.
CONCERNING THE JUSTICE OF THE MAHOMETAN WARS
In the margin, "fo. 158," Hand A.
. At the beginning of the second century AD, Hyrcanus forced the Idumeans to convert to Judaism. Stubbe had made the same point earlier.
. BL Harleian 1876 has the following note: "There are a multitude of decrees in ye Theodosian Code, for ye enforcing men to turn Christians ye like occur in the German & Spanish Chronicles," fol. 196.
. BL Harleian 1876 has a note about the work of Francisco de Vitoria, which contained a long section on "De Indis" immediately followed by a section on "De Indis, siue de iure belli Hispanororum in barbaros." The first title in this section is about "Christianis licet militare, & bella gerere," 129–173, 174–199: Francisco de Vitoria, _Relectiones undecim_ (Salamanca, 1565).
. BL Harleian 1876 has the following note: "This practise of his in yt Arabia, had ys political ground, yt whosoever will rule a diversity of nations, & religions, must somewhere have an united force to overballance any opposite power; & a few being unanimous will retain many, if divided sufficiently in subjection," fol. 196.
. Jirjis ibn al-'Amīd al-Makīn, _Historia Saracenica_ , trans. Thomas Erpenius (Leiden, 1625), bk. 1, ch. 1 where there is a description of Muḥammad's friendship with the Christians.
. Ibid., 28. Al-Makīn uses " _Ælia_ " for Jerusalem.
. 'Amr Ibn al-'Āṣ conquered Gaza from the Byzantines in AD 634. The paragraph draws on al-Makīn, _Historia_ , 19–20.
. Cf. John Selden, _De Iure Naturali & Gentium, Iuxta Disciplinam Ebraeorum, libri septem_ (London, 1640), 732. Selden presents numerous verses from the Qur'ān about the security that was afforded the Christians, 732–734.
. The reference is to al-Makīn, _Historia_ , 3. The reference to "Mahomet Ben Achmed" is very likely to Muḥammad ibn Jarrīr al-Ṭabari, the historian on whom al-Makīn relied in writing his chronicle.
. Ibid., 11.
. Hugo Grotius, _Epistolæ ad Gallos_ (Leiden, 1648), 239–240. Stubbe owned a copy of this edition. For Selden's view, see Toomer, _John Selden_ , 2:620.
. Paul Rycaut, _The Present State of the Ottoman Empire_ (London, 1670), 99–102.
. "Wars of sovereignty against other peoples, so that he might expand the borders of his realm and augment its greatness alone with fame." Wilhelm Schickard, _Jus regium hebraeorum, E tenebris Rabbinicis erutum, & luci donatum_ (Leipzig, 1674), 112.
. Cf. Selden, _De Iure_ , bk. 6, ch. 14: "De Federis ineundi ratione & captibus, ubi hostes se dederent."
. BL Harleian 1876 has the following note: "The principles of a narrow Monarchy, such as yt of Moses, do not become a great one, like unto yt of Muhammads," fol. 200.
. BL Harleian 1876 has the following note: "The Saracens have for yr rule of faith ye Alcoran: then ye Assonnah, or Tradition, wherein ye sayings & actions of yr prophet are recorded, as examplary & directive: then ye decrees & actions of his four Successours: & then lastly Reason. So Gregor Abulfaraj de mosr. Arab Sect.ult. cum notis Pocockii. p. 298," fol. 200. Cf. Pococke, _Speicmen_ , 298ff.
. Selden, _De Iure_ , 744.
. Ogier Gislain Busbecq, _Augerii Gislenii Busbequii d. Legationis turcicæ epistolæ quatuor_ (Frankfurt, 1595), 67r–68r.
. Hugo Grotius, _De Jure Belli ac Pacis_ (Amsterdam, 1631), 443.
. In the margin, "fol 164," Hand B.
. Jean Bodin, _Six Bookes of a Commonweale_ (London, 1606), "The disposition of the people is greatly to be obserued in the gouernment," 560.
. _Augerii Gislenii Busbequii d. Legationis turcicae epistolae quatuor_ , 66r–66v.
. BL Harleian 6189 has "explicable" instead of "despicable," fol. 229.
. BL Harleian 6189 has "never" instead of "ever," fol. 230.
. In margin: "h... usq Cap. 10," in Hand A. Under it at the bottom of the page, "to fol 165 Wanting here from fol 165 to fol 169. in blew book" in Hand B. BL Harleian 6189 has "performed" instead of "inform'd," fol. 230.
CONCERNING THE CHRISTIAN ADDITIONS
. Epiphanius, _The Panarion of Epiphanius of Salamis, Book 1_ , trans. F. Williams, 2d ed. (Leiden, 2009), 131, "Khokhabe in Hebrew."
. James Windet, _Minhah belulah, sive, Stromateus epistolikos_ (London, 1664), 222. Stubbe owned a copy of this edition.
. See Johann Hottinger, _Historia Orientalis_ (Tiguri, 1651), on the status of Christians (and Jews) at the time of Muḥammad, 212–238, although the section on the Christians focuses on early schisms, 228ff. The reference to Christians and money draws on al-Makīn and on Q 3:75.
. In margin: "fo. 135," Hand A.
. John Selden, _De Iure Naturali & Gentium, Iuxta Disciplinam Ebraeorum, libri septem_ (London, 1640), 238. Muḥammad ibn Aḥmad was a Qur'ānic exegete (d. 1459). Selden cites him often in his _Uxor Ebraica_.
. Hugo Grotius, _De Veritate religionis Christianæ_ (Amsterdam, 1662), 375–376. Stubbe did not use the English translation of 1632 and avoided the anti-Islamic emphasis in that chapter (and all the rest of the treatise). In margin: "fo. 166," Hand A. It is not clear what these references by the hands mean: there is a bracket from "that no man" until "universally" next to which is "fo. 135"; and from "demolish" to "confusion of" next to which is "fo. 166."
. Abū al-Fida.
. John Selden, _Eutychii Ægyptii, Patriarchæ Orthodoxorum Alexandrini_ (London, 1642), 74.
. Ibid., 16 "per totum." The reference is to Yusuf al-Misrial-Gibti's introduction in _Synodikon, sive, Pandectae canonum ss, apostolorum, et conciliorum ab ecclesia Graeca receptorum_ (Oxford, 1672), xix–xxi, where there is mention of "Arabicam Josephi Ægyptii." The Arabic accounts of the councils are 681–727.
. In the margin, an asterisk.
. The story appears in Levinus Warner, _Proverbiorum et sententiarum Persicarum Centuria collecta_ (Leiden, 1644), 30, and in Hottinger, _Historia Orientalis_ (1660), 515.
. Both BL Harleian 1876 and BL Harleian 6189 end with this reference to "Isa," although they differ in the ordering of the chapters. See the tables of contents in the discussion of the "Printed and Manuscript Sources: Editorial Policy," this volume.
. "There have been many who have described the life of the Arab Muhammad, which they relate in this manner; on this however all agree. But it is a bad thing that Erpenius, de Lingua Arabica p. 42 holds, that they affirm that he was born from a plebian and base stock, from impoverished parents—a pagan father and a Judean mother." Stubbe copies verbatim from Hottinger, _Historia Orientalis_ (1651), 136. Erpenius mentions that the Prophet's parents were "obscuris." Thomas Erpenius, _Orationes tres, de linguarum Ebraeæ atque Arabicæ dignitate_ (Leiden, 1621), 42, but it was Boxhornius who mentioned that the parents were idolaters and that the mother was "gente Iudaea." Marcus Zeuerius Boxhornius, _Historia universalis sacra et profana a Christo nato ad annum usque MDCL_ (Leiden, 1652), 397, 401.
. Hottinger discusses this number, _Historia Orientalis_ (1651), 146. BL Harleian 1876 has "Μαομέτις" in the blank space, fol. 66.
. Ibid., 9. It is interesting that Prideaux also refers to the correct spelling and enunciation of the name, but then retains "Mahomet." Humphrey Prideaux, _The True Nature of Imposture_ (London, 1697), xxii.
. Theodore of Mopsuestia (ca. AD 350–428); Diodorus of Tarsus (d. AD 392).
. In the right margin: an asterisk.
. BL Harleian 6189 has "undeniably false" instead of "unimaginable," fol. 258.
. "Kessaeus" is Muḥammad 'Abdalla al-Kisā'ī, mentioned by Henrys Sike in his 1697 edition of an apocryphal gospel, "The Gospel of Thomas," the manuscript of which had formerly been in the possession of Jacob Golius, "Praefatio ad lectorem," _Evangelium Infantiae_ , 2v. There must have been another manuscript by al-Kisā'ī about the lives of the prophets, _Qisas al-anbiy ā'_ which was heavily used by Hottinger in his _Historia Orientalis_. While the reference to "Casus Effendus in his Commentaries upon the Alcoran" suggests that John Gregory, _The Works_ (London, 1665), 9, may have seen a manuscript by this Muslim scholar, the reference by Stubbe points to his indebtedness to Hottinger.
. See Hottinger, _Historia Orientalis_ (1660), 352–354 from the manuscript by Ibn Idrīs al-Qarafī, _Al-Ajwiba al-f ākhira 'ala al-as'ila al-fājira_, ed. Bakr Zakī 'Awaḍ (Cairo, 1986), 318–322. See also Champion, "'I remember a Mahometan Story of Ahmed ben Edris.'"
. BL Harleian 1876 has "Arrians" instead of "Arabians," fol. 70. Actually, there is mention of it in Gabriel Sionita, _De Nonnvullis Orientalivm Vrbibvs_ (Paris, 1619), 21.
. Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 186, 187. The Scaliger reference is to the improvement on Marcus Manilius: _M. Manill Astonomicon a Iosepho Scaligero ex vetusto codice Gemblacensi infinitis mendis repurgatum_ (Leiden, 1600).
. In BL Harleian 1876, fo. 70, there is the following note: "See the prodigies of Ӕgypt lately translated." The reference to Noah's pigeon is from the medieval text by Murtaḍā ibn al-'Afīf, _The Egyptian history: treating of the pyramids, the inundation of the Nile, and other_ prodigies _of Egypt, according to the opinions and traditions of the Arabians. Written originally in the Arabian tongue by Murtadi, the son of Gaphiphus, rendered into French by Monsieur Vattier, Arabick professor to the king of France. And thence faithfully done into English by J. Davies of Kidwelly_ (London, 1672), esp. 86–87. For Stubbe's reference to Athanasius and his pigeon, see Champion, "Legislators, Imposters, and the Politic Origin of Religion," 346.
. There is a note referring to "Wierus de præstig. Dæmonum.l.1.c.19." See Johann Veyer, _De praestigiis daemonum, et incantantionibus ac ueneficijs, Libri V_ (Basel, 1564). But there is no direct mention of Muḥammad in that chapter.
. Henceforth, the scribe uses "Osman." Was he or the scribe correcting the earlier mistake?
. Pococke, _Specimen_ , 156, 157.
. Anacharsis was a Scythian philosopher who, after training in Athens, returned to his own people. According to Herodotus, he was killed because of his foreign ways.
. The seven pre-Socratic philosophers, seventh to sixth centuries BC.
. For "Abunazarus," see Hottinger, _Historia Orientalis_ (1651), 205.
. Al-Makīn, _Historia_ , 10, where there is reference to letters sent to the Prophet.
. Pococke, _Specimen_ , 180–181.
. "For those to whom history is a concern, I would desire that they pursue their Arabic works diligently, rather than persuasively; so that many foolish stories might be removed, which our ignorance of this language throws in our way. Thus it would come about that we might no longer dream that Muhammad's tomb is a pendulum in the air: nor would we continually propagate the falsehood of his promise concerning his return to his followers, whom we absurdly believe.... We might pay back those fools by striking against them with truths aimed to refute them; nor would we any longer talk nonsense in saying that the Saracen people boast descent from Sarah by way of their appellation. There are six hundred such things which are impossible to oppose without a study of their languages." Edward Pococke _Lamiato'l Ajam_ (Oxford, 1661), "Oratio in Auditorio Arabica habita," B4v. See also an excerpt from his oration "in auditorio Arabico 10. Augusti, 1636," 234.
. Euthymius Zigabenus (fl. 12th ca.), Byzantine theologian.
. John Selden, _De Dis Syris Syntagmata_ (Leipzig, 1662), 286; Pococke, _Specimen_ , 120–21; Hottinger, _Historia Orientalis_ (1651), bk. 1, ch. 7. But, as Toomer points out, Pococke disagreed with Selden's view that "Cobar" was the name of a goddess. Toomer, _John Selden_ , 1:241.
. Stubbe took this reference to Venus from Selden, _De Dis Syris Syntagmata_ , 216. Selden, citing Euthymius, concurred, unlike Stubbe who disagreed. See also Hottinger, _Historia Orientalis_ (1651), bk. 1, ch. 7, for discussion of the stone.
. Pococke, _Specimen_ , 114, 118 /for the sake of good omens or for obtaining blessing.
. Ibid., 120. "This is a thing, however, asserted by the Damascene and the Euthynian. If anyone should inspect more carefully the apparent figure of a head carved out with a scraper, which they want to be Venus's, I think it impossible to prove it by Arab carvers. Another of these [claims is] that the stone is a holy one, to which they offered the carved, or rather imprinted, figure, but it is as distant from the figure of the head as the head is from the feet, unless their eyes were so blurred by religious awe that they did not know how to distinguish the head from the feet. Of course, there is the stone in which the footprints of Abraham were imprinted when he stood on it during his building of the Ka'ba, as Abu al-Fida assents to, or (as Aḥmad ibn Yusef and Safioddinus say), when Ishmael's wife washed his head, which she revered when she saw it. Thence the place takes its name, the Maqām Ibrahīm, that is, the Place of Abraham, or where Abraham stood." The reference to the "Damascene" is probably to St. John of Damascus who wrote about this topic: D. J. Sahas, _John of Damascus on Islam: The 'Heresy of the Ishmaelites_ " (Leiden, 1972), 132-41.
. Professor Perale was unable to decipher the Greek word before Safa and Marwa. The spelling of "Marwa" is the accurate transliteration, unlike the "Meriah" of earlier pages.
. Pococke, _Specimen_ , 111. The scribe did not copy the Greek. "Elsewhere he says to these that it is.... called the likeness Baccha Ismaketh, which Mohammed himself named the Adoramen observationis [object of their prayer], and he instructed that the wretched barbarians adore it."
. 'Abdalla ibn 'Umar al-Bayḍāwi (1126–1160), a major commentator on the Qur'ān.
. Pococke, _Specimen_ , 112–113. The scribe did not copy the Greek names: Μάκε and Μάκχε.
. See Pococke, _Specimen_ , 128–129.
. In the margin:" Mahometan Religion w." Hand A.
. Pococke, _Specimen_ , 301. BL Harleian 6189 has "Originals, and" before "religion," fol. 273.
. Ogier Gislain Busbecq, _Augerii Gislenii Busbequii d. Legationis turcicae epistolae quatuor_ (Frankfurt, 1595), 105r–106r; and Paul Rycaut, _The Present State of the Ottoman Empire_ (London, 1670), 160–161. The note in BL Harleian 1876, fol. 79, mentions "Bartholom. Georgivita Peregrin. de morib. Turcar.c. de quadragesim": there is no title that includes the two words, "peregrin." and "quadrogesim." But see the two accounts by Bartolomej Georgijević published in 1598 and 1671, each of which includes one of the two words in the title. The account was very popular and reprinted regularly. English translations appeared in 1569 and 1661. Stubbe owned a copy of the latter edition.
. Aulus Cornelius Celsus (ca. 25 BC–ca. AD 50) was a Roman encyclopedist.
. Zakāt. The first use in English occurs in 1802, "zecchat" ( _OED_ ).
AS TO THEIR OPINIONS CONCERNING GOD
. In the right margin: "fol 144," Hand B.
. BL Harleian 6189 has "Holy Ghost" before "apostles," fol. 278.
. Johann Hottinger, _Historia Orientalis_ (Tiguri, 1651), 255.
. Johann Hottinger, _Dissertationem miscellanearum pentas: De abusu patrum_ (Tiguri, 1654), where there is a discussion of Muḥammad in comments by Maimonides and Gedaliam, 302–334.
. Because of the war against Holland, in _Further justification_ , Stubbe quietly praised Cromwell for having faced up to the naval rival. Although he described him as "HYPOCRITE" and mere "Oliver," he also showed how Cromwell defended England's right to the seas, 79. See the discussion of article 15.
. Paul Rycaut, _The Present State of the Ottoman Empire_ (London, 1670), 98, quoted verbatim by Stubbe. BL Harleian 1876 has "types" instead of "trials," fol. 83.
. Edward Pococke, _Specimen Historiæ Arabica_ (Oxford, 1650), 312–313.
. In the right margin: "nolitic institutions," Hand A. In BL Sloane 1786, "mysteries" is followed by "as an Indian into ye sea for pearl," fol. 181r.
. Wilhelm Schickard, _Jus regium hebraeorum, E tenebris Rabbinicis erutum, & luci donatum_ (Leipzig, 1674), 179–180.
. Hugo Grotius, _De Jure Belli_ (Amsterdam, 1667), bk.2, ch. 9, parags. 8 and 9; John Selden, _De Iure Naturali & Gentium, Iuxta Disciplinam Ebraeorum, libri septem_ (London, 1640), 564–565. For the agreement between Maimonides and Muḥammad, see p. 565. A note refers to Johannes Frischmuth (1619–1687), but the title of the work is not clear, BL Harleian 1876, fol. 85.
. Schickard, _Jus regium hebræorum_ , ch. 3, theor. 9, "Sed non tot a) uxores quot placebant," 173, where there is a reference to the verse in the Qur'ān about polygamy from the Sura on "Women," 4:3. In BL Sloane 1786 the sentence starts with "not so as that they were confined to one, for not to mention Solomon who is notorious," fol. 181v.
. Deuteronomy 25:5.
. Socrates in _Eusebii Pamphili, Rvffini, Socratis, Theodoriti, Sozomeni, Theodori, Evagrii, et Dorothei Ecclesiastica Historia_ (Paris, 1672), 306. Stubbe accepts this view, while Selden had doubted its authenticity: Toomer, _John Selden_ , 2:651 _n_ 181.
. "Eastern Jews take many wives; indeed it is permitted to western ones, but they do not do so out of respect. Paul did not want the Christians to take multiple wives and especially bishops, so that in this way he might silence the mouths of the Jews who were using it as an insult toward Christians. He did not warn the Jews that when they had three, they should repudiate two, and keep one." The quotation is translated in an article by Henk Jan de Jonge, "Scaliger's De LXXXV Canonibus Apostolorum Diatribe," _LIAS_ 2 (1975): 115–124, where de Jonge writes in n. 25: "A somewhat different Interpretation of Tit. i.6 (and I Tim. iii.2/12) is preserved in the _Secunda Scaligerana_ ed. Des Maizeaux, p. 402." Pierre Desmaizeaux was born a few years before Stubbe died and published two volumes of _Scaligerana_ (Amsterdam, 1740).
. John Selden, _Uxor Ebraica_ (Frankfurt, 1673), bk. 1, ch. 9 on "Polygamia; Monita & Consilia de uxorum numero Quaternario non excedendo."
. Selden, _De Iure_ , 570. In BL Sloane 1786, Saul is omitted and Samson is added, fol. 182 r.
. François Hotman, _De Castis incestisve nvptiis disputatio_ (Basel, 1594), 327, mentions Hagar, Bala, Zilpa "& illa Gedeonis, quæ non nominator."
. Selden, _Uxor Ebraica_ , 46.
. Lycurgus of Sparta was the legendary lawgiver.
. Pococke contested the view that Muslims place the supreme happiness of the afterlife in bodily pleasures: _Porta Mosis_ , 301. BL Sloane 1786 has "the Musselmans" instead of "Mahomet," fol. 182 v.
. Selden, _De Iure_ , 545. "Bredani" is most likely the exegete al-Bayḍāwi.
. Hottinger, _Historia Orientalis_ (1651), the description of Mecca, 142–144.
. Claudii Salmasius, _De vsvris liber_ (Leiden, 1638), 666–673, where Salmasius discusses usury in Islamic law between Muslims and non-Muslims, both enemies and friends. In BL Sloane 1786, "he" is replaced by "any Christian," fol. 182v.
. Profitable laziness.
. Salmasius, _De vsvris liber_ , 666–667.
. Ibid, 667. The passage from Ambrose is on p. 666. "Since their resources lie open as though for plunder and the pillage of these is lawful on every road" is the same which St. Ambrose gives: "For one against whom arms are justly borne, against this man let interest be lawfully declared: he whom you are able to conquer in war, regarding him, you can swiftly avenge yourself with the one percent. Exact interest from him whom it would not be a crime to kill. The man who demands interest fights without a sword. Without a sword the man who is the exactor of his enemy's interest avenges himself on the enemy. Therefore where there is a right to war, there is a right to interest."
. BL Harleian 1876 continues with a quotation from the Qur'ān: "His words are these, they will aske you if you will drink wine or adventure upon any sort of Gaming in wch there is Lottery. But do you answer, yt there is in ym something yt is exceeding sinfull & also somewhat yt is beneficial to man: but the benefit yt accrues from thence is over ballanced much by ye evil yt accompanies ym," fol. 91. In the margin there is a reference to "Mahometes in Alcoran c. 2." See Q 2:219.
. Pococke, _Specimen_ , 327.
. Hottinger, _Historia Orientalis_ (1651), 249.
. Pococke, _Specimen_ , 322.
. Ibid., 157.
. Ḥafṣa.
. Pococke, _Specimen_ , 362.
. "Publish" as in to "make public or generally known," "announce," _OED_ entry, J. Davies, 1662.
. Ibid., 190–191. See also Pococke's reference to al-Ghazāli as "cognomintu Hojjatol Eslam," 371.
. An English translation of the Qur'ān had appeared in 1649: see my "A Note on Alexander Ross and the English Translation of the Qur'an," _Journal of Islamic Studies_ , 23 (2011): 76–84. The French translation was made by André du Ryer in 1647.
. James Windet, _Minhah belulah, sive, Stromateus epistolikos_ (London, 1664), 223, where Windet also denounces the poor quality of the French and the English translations of the Qur'ān. Stubbe could be referring either to the 1543 Bibliander edition of the Latin translation of the Qur'ān by Robert of Ketton or to the Italian 1547 translation by Andrea Arrivabene who claimed to have translated from the Arabic, but actually used the Latin version: see _John Selden on Jewish Marriage Law: The "Uxor Hebraica,"_ translated with a commentary by Jonathan R. Ziskind (Leiden, 1991), 82n185. See Levinus Warner, _Epistola valedectoria in qua inter alia de stylo hisotriæ Timuri_ (Leiden, 1644), 9–10, where there is mention of the need for consulting Arabic, Turkish, and Persian interpretations of the Qur'ān.
. See the discussion of miracles in Hottinger, _Historia Orientalis_ (1651), 291ff.
. BL Harleian 6189 has "great Test" instead of "greatest," fol. 303.
. The whole section with the eight headings is taken from Pococke's notes, _Specimen_ , 187–188, 190–194. It is interesting that in BL Sloane 1786 miracles 6, 7, and 8 are copied and then crossed out, fol. 185v.
. All these "miracles" are listed by Abū al-Faraj in Pococke, _Specimen_ , 17.
. Mount Seir is specifically noted as the place that Esau made his home (Genesis 36:8). It was named for Seir the Horite, whose sons inhabited the land (Genesis 36:20). The children of Esau battled against the Horites and destroyed them (Deuteronomy 2:12). Mount Seir is also given as the location where the remnants "of the Amalekites that had escaped" were annihilated by five hundred Simeonites (1 Chronicles 4:42–43). The mountain is also mentioned in Ezekiel 35:10 ("A Prophecy Against Edom"). See also Pococke, _Specimen_ , 183–186, and _Historia Compendiosa Dynastiarum_ (Oxford, 1663), 103–104. A similar passage appears in the treatise by Aḥmad ibn Abdallaḥ, inscribed after Stubbe's treatise in the University of London Senate House manuscript: "God had Said in the holy Scriptures that hee will come from Ierocina that he would Shine in Zagar & bee exalted in Paran which is Meccha where Mahomet our Most Holy Prophet & Messenger of God did appear from hence these three things are to be noted. 1st that Ierocina is Mount Sinai where God gave to Moses the tables of the Law, that by Zagar is Mount Jerusalem where this Law of God did shine by Jesus Christ our Lord, who was alwaies accounted a prophet of God, who also Said that hee came not to destroy the Law of Moses but to confirm it, & as a testimony thereof was himself circumcised, And lastely that by Paran is understood Meccha where this Law of United shall be exalted illustrated & magnified. And it is manifest that this is Spoken & understood of our most holy prophet Mahomet who is the fullness of the Law of God & the Paraclete or Comforter himself who was promised in the Gospel (fol. 2).
. Stubbe borrows from Hottinger, _Historia Orientalis_ (1660), 13–14, where the Qur'ānic verse from Sura 49 is discussed.
. See the reference to the gospel mentioned by "Ahmed ben Edris, p. m. 305... Evengelium quintum... vocatur Evangelium infantiæ, quia in eo commemorantur res à Messia, super quo pax, profecta in juventute. Ad scribitur autem is Petro, à Maria, super qua pax." Hottinger, _Historia Orientalis_ (1660), 332.
. This paragraph is crossed out in the manuscript and does not appear in any other manuscript.
BIBLIOGRAPHY OF HENRY STUBBE
Henry Stubbe published his first work when he was nineteen years old in 1651. A few years later, and s eager to show off his mastery of Greek, he translated, among others, Donne's' "A Valediction: Forbidding Mourning" and Herrick's "Upon Julia Weeping" ( _Deliciae Poetarum_ , Oxford, 1658). In the dedications to his subsequent publications, Stubbe remembered men whom he admired or who had advanced his career: Richard Busby (his teacher), Thomas Barlow (his superior at the Bodleian), and Sir Henry Vane (his patron). Because of his Independent leanings, he wrote against Presbyterian authors, such as William Prynne and Richard Baxter, but he ranged widely in his interests, from discussing church government and medical cures to chocolate and political theory. He admired Hobbes and was "much esteemed" by him, thought Harrigton's _Oceana_ "light," praised Milton, corresponded with Locke (who signed his 1659 letter to him as "Admirer"), served John Owen, defended the Quakers, and attacked the Dutch. He was involved in a bitter exchange with supporters of the Royal Society, including Joseph Glanvill, and at one time caught the censorious eye of the Cambridge Platonist Henry More. He wrote pamphlets, treatises, and one-page broadsheets, original pieces as well as translations, rebuttals of other treatises and apologias for his actions. Many of his works proved popular and appeared in second and enlarged editions. After Stubbe's death, the physician James Cook found some of Stubbe's notes on the "Ars Cosmetica" and published them at the end of his book. Some of Stubbe's advice was on how to make gloves that whiten the hands and how to make teeth white. That Stubbe was writing his treatise on Islam while concocting cosmetic tricks highlights the intriguing paradoxes in the personality of the first English biographer of the Prophet Muḥammad.
Notwithstanding his attachment to the university and his voluminous output, the 1674 catalogue of books at the Bodleian included reference to only three of his many works: _Deliciæ Poëtarum Anglicanorum in Græcum versæ_ (1658), _The Savilian Professors case stated_ (1658), and _The Indian Nectar_ (1662) in Thomas Hyde, _Catalogus impressorum librorum bibliothecæ Bodlejanæ in academia Oxoniensi_ (Oxford, 1674), 184. In 1825 John Britton wrote that Stubbe's works were "now almost forgotten," _The History and Antiquities of Bath Abbey Church_ (London, 1825), 196, but in 1829, Edmund Oldfield, made a list of all his publications, including those written against him, _A topographical and historical account of Wainfleet and the Wapentake_ (London, 1829), 345–352.
Only three of his books have been published recently (2011): _A Censure Upon Certaine Passages Contained in the History of the Royal Society, Campanella Revived_ , and _Rosemary & Bayes_, all by the United Methodist Publishing House.
MANUSCRIPT
Letters to Hobbes: BL MS 32553, fols. 5–25.
Nicastro, Onofrio. _Lettere di Henry Stubbe a Thomas Hobbes_. Sienna, 1973.
"An enquiry into the Supremacy spiritual of the Kings of England, occasioned by a proviso in the late Act of Parliament against conventicles," TNA: SP 29/275/fos 276–284.
Letters from Bath: BL MS 35835, fols. 269–276.
For the manuscripts of the "Originall & Progress," see the list in "The Printed and Manuscript Sources: Editorial Policy."
PRINT
_Horae subsecivae, seu, Prophetiae Jonae et Historiae Susannae paraphrasis Graeca versibus heroicis_. London, 1651.
_Illustrissimo, summaeque spei juveni Henrico Vane Armigero, honoratissimi, & à me blurimùm observandi viri, Dni D.D. Henrici Vane de Raby, equitis aurati, filio primogenitor_. London, 1656.
_Clamor, rixa, joci, mendacia, furta, cachini, or A severe enquiry into the late oneirocritica published by John Wallis, grammar-reader in Oxon_. London, 1657.
_A Severe Enquiry into the late Oneirocritica: or, An Exact Accovnt of the Grammatical Part of the Controversy betwixt Mr. Hobbes and J. Wallis D.D_. (London, 1657).
_Deliciae poetarum Anglicanorum in Graecum versae quibus accedunt elogia Romae & Venetiarum / authore H. Stubbe_. Oxford, 1658.
_The Savilian professours case stated_. Oxford, 1658.
_A light shining out of darknes; or, Occasional queries submitted to the judgment of such as would enquire in to the true state of things in our times_. London, 1659.
_A vindication of that most prudent and honourable knight, Sir Henry Vane, from the lyes and calumnies of Mr. Richard Baxter, minister of Kidderminster: In a monitory letter to the said Mr. Baxter. / By a true friend and servant of the Commonwealth of England, &c_. London, 1659.
_The common-wealth of Israel; or, A brief account of Mr. Prynne's anatomy of the good old cause. By H.S_. London, 1659.
_An essay in defence of the good old cause; or, A discourse concerning the rise and extent of the power of the civil magistrate in reference to spiritual affairs_. London, 1659.
_The common-wealth of Oceana put into the ballance, and found too light; or, An account of the republick of Sparta with occasional animadversions upon Mr. James Harrington and the Oceanistical model / by Henry Stvbbe_. London, 1659.
_A letter to an officer of the Army concerning a select senate mentioned by them in their proposals to the late Parliament_. London, 1659.
_A vindication of that prudent and honourable knight, Sir Henry Vane, from the lyes and calumnies of Mr. Richard Baxter, minister of Kidderminster_. London, 1659.
_The common-wealth of Oceana put into the ballance, and found too light, or, An account of the republick of Sparta_. London, 1660.
_The Rota; or, News from the Common-wealths-mens club_. London, 1660.
_The Indian nectar; or, A discourse concerning chocolata the nature of cacao-nut_. London, 1662.
_The arts of grandeur and submission; or, A discourse concerning the behaviour of great men towards their inferiours, and of inferiour personages towards men of greater quality_. London, 1665.
_The miraculous conformist; or, An account of severall marvailous cures performed by the stroking of the hands of Mr. Valentine Greatarick with a physicall discourse thereupon_. Oxford, 1666.
_Legends no histories; or, A specimen of some animadversions upon the history of the Royal Society_. London, 1670.
_Lex talionis; sive, Vindiciae pharmacoporum_. London, 1670.
_Campanella revived; or, An enquiry into the history of the Royal Society_. London, 1670.
_A censure upon certain passages contained in the History of the Royall Society_. Oxford, 1670, expanded ed. 1671.
_An epistolary discourse concerning phlebotomy in opposition to G. Thomson pseudochymist, a pretended disciple of the Lord Verulam_. Np, 1671.
_The Lord Bacons relation to the sweating-sickness examined, in a reply to George Thomson, pretender to physick and chymistry_. London, 1671.
_Medice cura teipsum! or, The apothecaries plea_. London, 1671.
_A justification of the present war against the United Netherlands wherein the declaration of His Majesty is vindicated, and the war proved to be just_. London, 1672.
_Rosemary & Bayes; or, Animadversions upon a treatise called, The rehearsall transprosed_ [sic]. London, 1672.
_A further iustification of the present war against the United Netherlands illustrated with several sculptures_. London, 1673.
_The Paris gazette_. London? 1673.
_The history of the United Provinces of Achaia_. London, 1673.
_Select observations on English bodies of eminent persons in desperate diseases... In the Close is added, Directions for drinking of the Bath-Water, and Ars Cosmetica, or Beautifying Art: By Henry Stubbe, Physitian at Warwick_. London, 1679.
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Wood, Anthony à. _Athenae oxonienses: An exact history of all the writers and bishops who have had their education in the University of Oxford. To which are added the Fasti, or Annals of the said University_ , 3:1067–1083. 4 vols. New York, 1967.
INDEX
**Page numbers refer to the print edition but are hyperlinked to the appropriate location in the e-book.**
'Abbās, , ,
'Abdallah, Father of Muḥammad/ _Abdalla_ , , , 240 _n_ 2
'Abdallah ibn Shihāb/ _Abdalla, son of Sidhab_ , , 247 _n_ 16
'Abdallah ibn 'Umar, 247 _n_ 7
_Abdalla the Jew_ , , 223 _n_ 154
'Abd al-Muttalib, grandfather of Muḥammad/ _Abdolmutleb_ , ,
Aboussouan, Camille, 220 _n_ 96
Abraham, prophet, , , , , , ; banishment of, , ; and the Ka'ba, 117–18, , , 140–41, , , 253 _n_ 37; lineage of, , ; as proselytizer, ; sacrifice of Ishmael by, 141–43,
Abū al-Fida/ _Abulfeda_ , geographer, , , , 237 _n_ 3, 238 _n_ 15, 253 _n_ 37
Abū Bakr al-Ṣiddīq/ _Abubacr_ , , , , , , , , , 240 _n_ 7; and death of Muḥammad, ; as military commander, , ; Muḥammad, marriage of daughter to, ; as proselytizer, 160–61; and the Qur'ān, ,
_Abu Gabshan_ , , 239 _n_ 30
Abū Karib, As'ad/ _Abu Corb Assad_ , , 239 _n_ 35
Abū Ma'shar/ _Abu Maasor_ , astrologer,
Abū Sufyān ibn al-Ḥārith/ _Abusofian_ , , , 247 _n_ 10; conversion of, , ; as military commander, , ,
Abū Tālib / _Abutaleb_ , Uncle of Muḥammad, , , ; death of, ; in early life of Muḥammad, , ,
Adam, 116–17, ,
Adams, Thomas, 225 _n_ 193
Addison, Lancelot, , , 222 _n_ 148, 244 _n_ 25
'Adnān/ _Adnanus_ , , , 239 _n_ 27
Adrian, Roman emperor, , ,
al-'Afīf, Murtaḍā ibn, 252 _n_ 23
Agrippa, Herod, , , , 227 _n_ 28
Aḥmad ibn 'Abdallah / _Achmet Ben-Abdala_ , Andalusian author, , 257 _n_ 42
Aḥmad ibn Yusef, 253 _n_ 37
_Aitshama, Negush_ , Abyssinian king, 157–58
Akiva, Rabbi/ _Akibba_ , , 233 _n_ 110
Alashram, Abrahah, ,
Alexander, Patriarch,
Alexandrinus, Clement,
Alexandrinus, Dionysius,
'Ali ibn abī Ṭālib/ _Aly_ , , , , , , ; and alcohol, 146–48; characteristics of, ; , ; as military commander, , ; as negotiator with Hagarenes, , 139–40, 144–45, as preacher of Islam, , 39–43; and sacrifice of Ishmael, 140–43; in Saraka, 152–55; as secretary of Muḥammad, ,
_Aljanabus_ ,
Ambrose, St., , , , 255 _n_ 26
Amīna, mother of Muḥammad/ _Amena/Emena_ ,
Amos,
'Amr Ibn al-'Āṣ/ _Amurcus_ , , 250 _n_ 7
'Amr ibn Luḥay, , 239 _n_ 34
'Amr ibn al-Mundhir, King/ _Amrus_ ,
Anacharsis, , 252 _n_ 27
Annobius the Elder, , 236 _n_ 166
al-Anṣārī, Khālid Abū Ayyūb/ _Chalid Abiol_ , , 243 _n_ 54
al-Anṣārī, Mālik ibn 'Awf/ _Melic_ , , , 248 _n_ 27
Antiochus, Seleucid king, , 226 _n_ 13, 227 _n_ 6
Antoninus, Marcus, _see_ Aurelius, Marcus, Roman emperor
Antony, Mark,
Anū Sherwān/ _Anusherwan_ , , , , , 239 _n_ 44; _see also_ Chosroes I
Apollinaris, bishop of Hierapolis,
Apollo,
Apollo, Alexandrian Jew,
Apollonius Tyaneus,
Apolonides,
Arcadius, Byzantine emperor, ,
Archelaus, , 227 _n_ 23
Aretas, , , 246 _n_ 50
Arian(s), , , , , , ; and Christ's divinity, , , 232 _n_ 106, 232 _n_ 107; as alleged instructors of Muḥammad, , ; and Nicene Council, , ; oppression of, , , 122–23, ; persecution by, ; piety of, ; and pigeon story,
Aristotle, 218 _n_ 84
Arius, Theologian, ; _see also_ Arians
Arlington, First Earl of, secretary of state, , 214 _n_ 18
Arrivabene, Andrea, 256 _n_ 37
Artemon, , 232 _n_ 104
Ashley-Cooper, Anthony, _see_ Shaftesbury, Lord
_Ashur_ , son of Shem,
'Ashūrā'/ _Ashur/Ashura_ /Month of, , ,
Athanasius, St., , , , , 252 _n_ 23
Augustine/ _Austin_ , St.,
Augustus, Roman emperor,
Aurelius, Marcus, Roman emperor, , 235 _n_ 154, 235 _n_ 155
Averroes, , ,
Avicenna, , , 237 _n_ 167
Bacchus, , , , 245 _n_ 50
Baḥīrā, monk,
Baithos, 226 _n_ 14
Balaam,
Baradaeus, Jacob, bishop of Edessa, , 235 _n_ 144; _see also_ Jacobite(s)
Bar Kochba/ _Bencochab_ , , , 233 _n_ 110
Barlow, Thomas, , 229 _n_ 50,
Barnabas, St.,
Barrow, Isaac, , , 217 _n_ 67
_Bashar_ , ,
Basil, St.,
Baxter, Richard, , , 215 _n_ 32,
Baybars, Mumluk Sultan,
al-Bayḍāwi, Abdalla ibn 'Umar/ _Beidani/Bredani_ , , , 253 _n_ 40
Bedwell, William, , , 215 _n_ 3, 219 _n_ 92
Bellarmin, Cardinal,
Benjamin of Tudela/ _Tudelensis_ , , 228 _n_ 37
Bennet, Henry, First Earl of Arlington, _see_ Arlington, First Earl of
Bibliander, Theodor, , 222 _n_ 142, 223 _n_ 154, 256 _n_ 37
Bilāl ibn al-Ḥārith/ _Balal_ , , 243 _n_ 10
Bisaha, Nancy, 213 _n_ 1
Blondel, David, , 235 _n_ 153, 236 _n_ 156
Blount, Charles, , , , , 235 _n_ 141
Blount, Henry, , 224 _n_ 181
Bodin, Jean, 226 _n_ 8, 234 _n_ 138, 242 _n_ 37, 250 _n_ 21
Bosworth, C. Edmund,
Boxhornius, Marcus Zeuerius, , 222 _n_ 141, 223 _n_ 154, 226 _n_ 5, 241 _n_ 13, 251 _n_ 13
Boyle, Robert,
Britton, John,
Browne, Sir Thomas, 231 _n_ 76, 247 _n_ 25
Brutus,
Bury, Arthur, , 222 _n_ 131
Busbecq, Ogier Gislain, , 244 _n_ 51, 250 _n_ 18, 250 _n_ 22, 245 _n_ 45, 246 _n_ 60, 254 _n_ 45
Busby, Richard, ,
Butler, Samuel,
Butler, Thomas, Lord of Ossory, 215 _n_ 41
Buxtorf, Johann, , , 229 _n_ 57, 231 _n_ 76, 233 _n_ 109
Caiaphas/ _Caiaphus_ , 80–81
Caligula, Caius, Roman emperor, , 227 _n_ 28
Candace, Queen,
Cardan, Fazio, , 242 _n_ 38
Cardan, Geronimo, , 242 _n_ 38
Cary, Lucius, Lord Falkland, , , 229 _n_ 50
Casaubon, Isaac, , , , , , , 232 _n_ 87
Cedrenus,
Celestine, Pope,
Celsus, Aulus Cornelius, , 254 _n_ 46
Cephas, _see_ Peter, St.
Chalcedon, Council of, , , , , , 235 _n_ 146
Champion, Justin A. I., , , , , 258 _n_ 44
Charles II, king of England, , , , , ,
Cherbury, Lord Herbert of,
Chillingworth, William, , , , 215 _n_ 32, 216 _n_ 56, 229 _n_ 50
Cholmly, Sir Hugh, 215 _n_ 41
Chosroes I/ _Cosroes_ , Persian king, _see_ Anū Sherwān
Chosroes II/ _Cosroes_ , Persian king, , , 235 _n_ 151; conquest of Egypt by, ; death of, 126–27; persecution of Christians by, ,
Christ, _see_ Jesus
Cisneros, Ximénes de, cardinal, , 242 _n_ 39
Clarke, Samuel,
Claudius, Roman emperor,
Clement, Pope, , , 230 _n_ 64, 237 _n_ 166
Comnenus, Manuel, Byzantine emperor,
Constantine, Roman emperor, , , , , , , 234 _n_ 128; baptism of, , ; and Christ's divinity, ; and Nicene Council, , , ; religious authority of, , , 93–94,
Constantius, Roman emperor, , , , 234 _n_ 128
Cook, James,
_Cosroes, see_ Chosroes II
Cowley, Abraham,
Cromwell, Oliver, , 254 _n_ 5
Cyril of Alexandria, St.,
Dale, John, , 236 _n_ 160
Daniel, Norman, 222 _n_ 135
Davenant, William, 224 _n_ 182
David, , ; Christ as descendant of, ; lineage of, , , ; and polygamy, 202–3; throne of, ,
Dawson, John,
de las Casas, Bartolomé, 224 _n_ 182
Desmaizeaux, Pierre, 255 _n_ 14
de Vitoria, Francisco, 249 _n_ 3
Dimmock, Matthew, 213 _n_ 1
Dio Cassius,
Diocletian, Roman emperor, , , 234 _n_ 132
Diodorus Tarsoris/Diodorus of Tarsus, , 252 _n_ 16
Dioscorus of Alexandria, , 235 _n_ 144
Disney, Reverend John,
Donne, John,
Doria, Don Ambrogio Spinola, 240 _n_ 12
Dryden, John, , , 224 _n_ 182
Ecchellensis, Abraham, _see_ al-Ḥaqilānī, Ibrahīm
Edwards, John, 243 _n_ 9
Eleazar, nephew of Moses, , 226 _n_ 14
Elias,
Eliphaz,
Elizabeth I, queen of England, , 215 _n_ 34
Enoch, _see_ Idrīs
Epiphanes, Antiochus, Seleucid emperor, , , 227 _n_ 19
Epiphanius of Salamis, , , , , 230 _n_ 64, 231 _n_ 74
Erpenius, Thomas, , , , ; and Arabic language, , ; translation of al-Makīn by, 21–22, , 217 _n_ 72, 220 _n_ 101, 220 _n_ 102
Eusebius of Caesaria, , ,
Eutychians, , , , , _see also_ Eutychius of Constantinople
Eutychius of Alexandria, _see_ Ibn al-Baṭrīq, Sa'īd
Eutychius of Constantinople, , , , 235 _n_ 144
Ezra, , , ,
Fabricius, Johannes, ,
al-Faraj, Abū/Gregorios/Ibn al-'Ibrī, , , , , 221 _n_ 124, Arabic history of, 26–29; and the Qur'ān, ; as source for Stubbe, , ,
Fātima, daughter of Muḥammad/ _Phatemia_ ,
Felix, Minicius, 236 _n_ 166
Fell, John, 235 _n_ 158
Fayrūz al-Daylamī/ _Firus of Dailam_ ,
Finḥaṣ/ _Phinhas_ , , 247n12
Floria, John, 243 _n_ 9
Fraizer, Sir Alexander,
Freke, William,
Frischmuth, Johannes, 254 _n_ 10
Fuller, Nicholas, , 242 _n_ 35, 243 _n_ 15, 244 _n_ 22, 246 _n_ 59
Gagnier, John, 220 _n_ 100, 221 _n_ 124
Gaius Julius Priscus, , 240 _n_ 12
Garcia, Humberto, , 216 _n_ 61
George, St., bishop of Alexandria, , , ,
Georgijević, Bartolomej, 254 _n_ 45
Ghassanids/ _Gassan_ , , 237 _n_ 4, 246 _n_ 60
al-Ghatfānī, Na'īm ibn Mas'ūd/ _Naimus f.Masudae Gatfanites_ , , 247 _n_ 17
al-Ghazāli, Abu Hāmid/ _Algazali_ , , , , , 254 _n_ 44
Gibbon, Edward, , , 222 _n_ 143, 223 _n_ 150
_al-Gibti, Yusuf al-Misri_ , , 251 _n_ 9
Gideon,
Glanvill, Joseph, , , ,
Godefroy, Jacques,
Golius, Jacob, ,
Gratian, Roman emperor, , , 246 _n_ 56
Greaves, John, , 238 _n_ 15
Gregory, John, , , , , 238 _n_ 15
Gregory the Great, St., , 235 _n_ 147
Gregory of Nazianzen, St., , ,
Grotius, Hugo, , , , 55–56, , , , ,
Gunny, Ahmad, 217 _n_ 72
Hadrian, Roman emperor,
Ḥafṣa/ _Hapsa_ , wife of Muḥammad,
Hagar, , 141–43, , , , 239 _n_ 38
Hales, John, 215 _n_ 32
Ḥamza/ _Hamza_ , uncle of Muḥammad, ,
al-Ḥaqilānī, Ibrahīm, ,
al-Ḥārith ibn Ka'b/ _Alhareth Eb. Caab_ , , , 239 _n_ 40
Harrington, James,
Hartlib, Samuel,
al-Ḥaṣrūnī, Yūḥanna, , , ,
Heidegger, Johann Heinrich/ _Heideggerus_ ,
Heraclius, Byzantine emperor, , , , , 235 _n_ 148; aid against Abyssinians from, ; revolt against Phocas of, , 126–27, ; revolt by Saracens against, ; vanquishing by Muḥammad of,
Herbert, Thomas, 227 _n_ 25
Hermophilus,
Herod, King, _see_ Agrippa, Herod
Herrick, Robert,
Hesronita, John, _se_ al-Ḥaṣrūnī, Yūḥanna
Hilary, bshop of Poitiers, , 235 _n_ 140
Hill, Christopher, ,
Hillel, Rabbi,
Hippocrates,
Hobbes, Thomas, 213–14 _n_ 8, 223 _n_ 158, ; correspondence with Charles Blount of, , , 228 _n_ 49, 235 _n_ 141; correspondence with Henry Stubbe of, , , ; and secularism, ,
Honorius, Roman emperor, ,
Hormisdas, Persian king, , 235 _n_ 151
Hornby, Charles, , , , 56–57, , ,
Hottinger, Johann Heinrich, , , , 216 _n_ 54; on 'Ali, , 223 _n_ 164; on Arabians, , ; on church history, , 218 _n_ 75; on Jesus, 225 _n_ 4; on Muḥammad, , ; on Saracen catechism, ; as source for Magney, 216–17 _n_ 61; as source for Stubbe, , , , , , , 217 _n_ 69, 217 _n_ 72
Hudson, John, 238 _n_ 15
Hulago,
Hymenæus, 231 _n_ 73
Hyrcanus, King, , , 249 _n_ 1
Ibn Abī al-Bazza, al-Qāsim/ _Ali Ebn Bozea_ ,
Ibn al-Athīr/ _Ebnol Ehir_ , , 248 _n_ 12
Ibn al-Baṭrīq, Sa'īd/Eutychius, , , 24–26, , ; on Alexandria, , ; Arabic history of, 24–26; as source for Stubbe, , ; translation by Selden of,
Ibn Ḥunayn, Abū _Zayd/Ben-Honain_ ,
Ibn Idrīs al-Qarāfi, Aḥmad/ _Achmed Ben Edris_ , , , , 217 _n_ 69, 252 _n_ 20
Ibn Khaldūn, 220 _n_ 102
Ibn Muqla/ _Ebn Mooklah_ , , 241 _n_ 20
Ibn Qāsim al-Thaqafī, Muḥammad,
Ibn al-Rāhib, Petrus,
Idrīs/ _Edris_ , , , , , , ,
al-Idrīsī, Abū 'Abdallah Muḥammad, , , 220 _n_ 100
Irenæus, St., ,
_Isa, see_ Jesus
Isaac, , ,
Isidore, St.,
_Ismael, see_ Ishmael
Ishmael/ _Ismael_ , , , , , , , 253 _n_ 37; birth of, ; and the Ka'ba, , , , ; circumcision of, , ; descendants of, , , , , , 148–49; ejection by Abraham of, ; glory of, , ; and Medina, , ; reformed dialect of, , ; sacrifice of, 141–43, ; seat of,
_Ismael ibn Ali_ , ,
Israel, Manasseh ben, 228 _n_ 34
Jacob, , , ,
Jacob, James R., , , , 217 _n_ 62
Jacobite(s), , , ; Abū al-Faraj as, , ; Al- Rāhib as, ; as Christian faction, , , , , , ; conversion to, ; heresies of, , ; as instructors of Muḥammad, ; piety of, ; practices of, ; _see also_ Baradaeus, Jacob, Bishop of Edessa
Ja'far ibn abī Tālib, 247 _n_ 7
Ja'far ibn 'Omar/ _Giafar Ibn Omar_ , ,
James, duke of York 10–11, 215 _n_ 36
James, St., , , , ,
James, Thomas,
Jeremiah,
Jerome, St., , 83–84, ,
_Jesse, Rabbi_ ,
Jesus/'Īsa/ _Isa_ , , , , , , , , , ; crucifixion of, , 249 _n_ 19; distinctions in worship of, ; divinity of, ; Islamic respect for, , , ; Jewish belief in, ; persecution of, ; poverty and piety of, , 185–88; as prophet, , , , , , , , , ; in the Qur'ān, 29–31; reception of, ; reformation by, , ; ritual honoring of, , ; sacred writ of, ; as savior, , ; and Serapis, , 233 _n_ 113; as true Messiah, , , , , , ,
John the Baptist, , , , , ,
John Chrysotom, bishop of Constantinople,
Joseph,
Josephus, , , ,
Judaizers/Judaizing Christians, , , 108–9, , , ; and divinity of Christ, ; and Holy Ghost, ; Muḥammad as alleged, , ; and paradise, ; piety of, ; and polygamy, , ; as sect of Judaism, ,
Julian the Apostate, ,
Julius Caesar, , , 226 _n_ 4
Justin I, Roman emperor, , , , , , 234 _n_ 131, 246 _n_ 57; ejection of Goths by, 97–98, ; Trinitarian religion enforced by,
Justinian II, Roman emperor, , , 235 _n_ 148
Ka'b ibn al-Ashraf/ _Ka'baum fil. Alasrafi_ , , 247 _n_
Kant, Emmanuel,
Kérouaill Louise de, _see_ Portsmouth, Duchess of
Ketton, Robert of, , 256 _n_ 37
Khabbāb ibn al-Aratt/ _Cabbab_ , , 243 _n_ 10
Khadīja bint Khuwaylid, wife of Muḥammad/ _Chadija_ , , , , 241 _n_ 15, 242 _n_ 45
Khālid ibn al-Walīd/ _Chaledo Ebn Walid_ ,
al-Kindī, philosopher, 219 _n_ 89
al-Kissā'ī, Muhammad/ _Casus Eff endus/Kessaeus_ , , 252 _n_ 19
Lactantius, , 236 _n_ 166
Laud, William, archbishop of Canterbury, , 225 _n_ 193
Leo Africanus/ _John Leo_ , 238 _n_ 15
_Levi, Rabbi_ ,
Lightfoot, John, 226 _n_ 16, 227 _n_ 22, 228 _n_ 35, 229 _n_ 59, 229 _n_ 62, 230 _n_ 66, 231 _n_ 73
Locke, John, , 45–46, 214 _n_ 13, 243 _n_ 9,
Luke, St.,
Luqmān al-Ḥakīm/ _Lockman_ , , 242 _n_ 44
Lycurgus, Spartan lawgiver, , , 226 _n_ 10, 255 _n_ 19
Magney, Thomas, , 216–17 _n_ 61
_Mahomet, see_ Muḥammad
Maimonides, , , , , 226 _n_ 14, 238 _n_ 14
al-Makīn, Jirjis ibn al-'Amīd/ _Elmain/Elmomin_ , , , , , 220 _n_ 101, 220 _n_ 102, 241 _n_ 27; Arabic history of, 21–24, ; as source for Stubbe, , , , , , 217 _n_ 72
Malvezzi, Virgilio, 226 _n_ 11
Manilius, Marcus,
Marana, Giovanni Paolo,
Marcellinus, Amianus, 245 _n_ 50
Mark, St., ,
Mars,
Marshall, John, 216 _n_ 56
Martianus/Marcian, Roman emperor, , , , 235 _n_ 149
Martyr, Justin, , 231 _n_ 73; apology of, 104–5; beheading of, 235 _n_ 154; belief in the Messiah of, 77–78, 87–88
Marvell, Andrew, , ,
Matthew, St. ,
Maundrell, Henry,
Mauritius, Roman emperor, , , , , , 235 _n_ 148
Mavia, Queen, , 239 _n_ 46
Maximinius, _see_ Thrax
McLachlan, H. John,
Mead, Joseph, , 228 _n_ 50
Medorses, son of Chosroes II,
Melkite(s)/ _Melchite(s)_ , , , ; al-Baṭrīq as, , ; and Council of Chalcedon, ; oppression by, 149–50; persecution by Chosroes II of,
Mercury, ,
Messalin, Waldonis/Messalinus, Walo, _see_ Salmasius, Claude
Michael, St.,
Millenarian(s), , , 228–29 _n_ 50
Milton, John, , , , , , , 216 _n_ 53,
Minucius Felix,
Modena, Leo, 230 _n_ 71
Modena, Mary of, wife of King James II of England, , 215 _n_ 36
Montague, Richard, 231 _n_ 84
Montesquieu, Charles-Louis,
More, Henry,
Moses, , , , , , , , 226 _n_ 14; comparisons with Muḥammad of, , ; historicization of, ; law of, , , , , 257 _n_ 42; poverty of, ; and the Qur'ān, , ; and usury,
Muḥammad/ _Mahomet_ , , , , , , , , , , , , , ; and Abū Tālib, ; and Abyssinians, 157–59; and Africa, ; against idolatry, , ; as apostle of God, , ; and the Arabians, 138–39; birth of, , 234 _n_ 131; characteristics of, 69–70; Christian misconceptions about, 127–28, , 191–97, 253 _n_ 32; and Christian reform, 186–89; Christian respect of, , ; and circumcision, , ; and the Coreischites, , , 134–35, 159–63; death of, 173–74; departure for Medina of, ; destruction of idols by, 164–65; education of, 124–25; and empire, 181–82; entry into Mecca of, ; friendship with 'Ali of, , ; and Hadith, 222 _n_ 135; and the Hagarenes, 135–36, , , , ; and imposters, , ; as Judaizing Christian, , , ; last pilgrimage of, 169–72; literacy/illiteracy of, 33–34, 193–94, ; and Luqmān, ; marriage of, 123–24; military endeavors of, , , 160–65; and miracles, 209–11; and polygamy, 201–4; prohibition of alcohol by, 146–48, 245 _n_ 50; prohibition of gambling by, ; prohibition of usury by, 204–5; propagation of doctrine by, 177–78, ; and the Qur'ān, , , , , , , , , 223 _n_ 152, 241 _n_ 19; retirement out of Mecca of, , ; rise of,
al-Mundhir ibn al Nu'mān, King/ _Almondar/Almonder_ , , , , 246 _n_ 60
Musaylima ibn Ḥabīb (the Liar)/ _Mosalleina_ , , , 248 _n_ 34
Nachor,
al-Nāṣir, al-Mālik,
Nebuchadnezzar, ,
Nehemiah,
Nero, Roman emperor, , 228 _n_ 42
Nestorian(s), , , , , ; and Chosroes II, , , ; on divinity of Christ, , ; as heretics, ; as alleged instructors of Muḥammad, ; piety of, ; monk, Sergius the, , , ; _see also_ Nestorius
Nestorius, , , , 235 _n_ 144
Newton, Thomas, 225 _n_ 2
Nicholas of Cusa, , , 219 _n_ 89
Noah, , , ; commandments of, , , ; and flood, , ; idolatry and, ; and pigeon story, , 252 _n_ 23; poverty of, ; progeny of,
Novatians, , , 98–99
Novatus, St., , 234 _n_ 132
al-Nu'mān/ _Alnooman_ , ; _see also_ al-Mundhir
Nye, Stephen,
Ockley, Simon, 42–43, 224 _n_ 176
Odenatus of Palmyra, , , 246 _n_ 62
Oldfield, Edmund,
Olearius, Adam, 40–41, , , 236 _n_ 158, 244 _n_ 37, 245 _n_ 40, 248 _n_ 5
Origen, , ,
Ovid,
Owen, John, ,
_Ozair, see_ Ezra
Pagitt, Ephraim,
Pantaenus, church historian,
Papinianus, Aemilius, Roman jurist, , 234 _n_ 124
Parker, Samuel, , 215 _n_ 35
Pasor, Matthias,
Paul, St., , , , , , , , , , , ; apprehension of by Aretas, 111–12; ecstacies of,
Pertinax, Roman emperor, , 240 _n_ 12
Petavius, Denius, , 234 _n_ 136
Peter, St./Cephas/Simon Peter, , , , , , , , , , , , 230 _n_ 65
Philetus, 231 _n_ 73
Philip the Arab, Roman emperor, 240 _n_ 12, 246 _n_ 61
Philippus, Marcus Julius, Roman emperor,
Philo Judaeus, , 227 _n_ 29
Phocas, Byzantine emperor, , , , 149–50, 235 _n_ 148
Pilate, Pontius,
Pius, Antoninus, Roman emperor, , , 235 _n_ 154
Pliny, , , , 232 _n_ 91
Pococke, Edward, , , , , , , 255 _n_ 20; on 'Ali, , ; on Arab historians, , 26–27; as Arabic professor, , , ; as Arabic translater, , 221 _n_ 112, 221 _n_ 122, 221 _n_ 124; as chaplain to the Levant Company, 225 _n_ 193; on Islamic eschatology, ; on Muḥammad, , , , , , 194–95, 223 _n_ 150, 223 _n_ 152; on the Qur'ān, ; on the Saracens, , ; as source for Stubbe, , , , , , 217 _n_ 69, 217 _n_ 72
Pompey, , 227 _n_ 20
Porphyrogenitus, Byzantine title,
Portsmouth, Duchess of/Louise de Kérouaille,
Postel, Guillaume, 213 _n_ 1
Prideaux, Humphrey, , , 213 _n_ 7, 221 _n_ 112; as classmate of Stubbe, , ; on Muḥammad, , 217 _n_ 72; remarks against Islam by,
Prynne, William, ,
Ptolemy, , , , , 218 _n_ 84
Purchas, Samuel, , 219 _n_ 89, 219 _n_ 92, 241 _n_ 14
Qaḥtān/ _Joktan/Kahtan_ , , , , 237 _n_ 8, 239 _n_ 27
Qur'ān/ _Alcoran/Coran_ , the, , , , , , , , , , , ; and alcohol, 255 _n_ 27, ; and 'Ali, , ; attacks against, 33–34, , , ; biography of Muḥammad from, , ; and church history, , 218 _n_ 75; alleged corruptions of, , , ; divine revelation of, , 34–35, , , 223 _n_ 152, 241 _n_ 19; and Hagarenes, , , , ; and idolatry, ; and Jesus, 29–31; and miracles, , 209–11, 249 _n_ 18; and paradise, ; and polygamy, 201–3, 254 _n_ 11; and prayer, ; refutation of Trinitarianism in, ; religious tolerance and, , , , 250 _n_ 8; Stubbe on, , 13–14, ; translations of, , , , , , 223 _n_ 162, 256 _n_ 36, 256 _n_ 37; on Trinitarians, ; and Virgin Mary, , 219 _n_ 92; writing of, , , 173–74, , 191–93, 207–8
al-Raḍī, al-Sharīf/al-Balāgha, Nahj,
Reeland, Adrian, , 217 _n_ 61
Rehoboam,
Ricaredus, King, 122–23, 241 _n_ 13
Richelieu, Cardinal,
Rochester, John Wilmot, Second Earl of, , ,
Ross, Alexander, , , ,
Rycaut, Paul, , , , 225 _n_ 193; on Muḥammad, , ; on relationship between Christianity and Islam, 218 _n_ 76; as source for Stubbe, , 217 _n_ 72
Ryer, André du,
Sadoc, , 226 _n_ 14
Saffiodinus, 253 _n_ 37
Sale, George,
Ṣālih/ _Salehus_ ,
Salmasius, Claude (Claudius), , , 230 _n_ 67, 232–33 _n_ 108; on Greek language, 107–8; on Jews, 76–77; on Muḥammad, ; as source for Stubbe, , , , 217 _n_ 69
Salmonasar, , 228 _n_ 37
Salvian, bishop of Marseilles, , 235 _n_ 140
Sarah, , ,
Saturn,
Saul,
Scaliger, Jospeh, , , , , , , 229 _n_ 56
Scaliger, Pacifique,
Schickard, Wilhelm, , 250 _n_ 13, 254 _n_ 9, 254 _n_ 11
Selden, John, , , , , , ; as Arabic translator, , , 221 _n_ 112; on Hebrew wives, ; on Muḥammad, , , , ; on the Qur'ān, ; as source for Stubbe, , ,
Serapis, Egyptian deity,
Sergius, Monk, , ,
Servianus,
Seth, , , , ,
Severus, Roman emperor, , 243 _n_ 17
Severus, Sulpicius,
Severus of Antioch, , 235 _n_ 144
Sevi, Sabbatai, 227 _n_ 24
Shaftesbury, Lord/Anthony Ashley-Cooper,
al-Shahrastānī, Muḥammad ibn 'abd al-Karīm, , 244 _n_ 33, 249 _n_ 20
Shairani, Hafiz Mahmud Khan, , , ,
Shem,
Sidney, Algernon,
Sike, Henry, 252 _n_ 19
Simon, Richard,
Simon Peter, _see_ Peter, St.
Simon Magus,
Sionita, Gabriel, _see_ al-Ṣuhyūnī, Jibrā'īl
Sirces, son of Chosroes II,
Siroes II/ _Syroes_ , Persian king,
Smith, Thomas, ,
Socrates, ,
Solinus, Julius, , 237 _n_ 5
Solomon, , ,
Solon, , 226 _n_ 10
South, Robert,
Sozomen, Salminius Hermias,
Sprat, Thomas, , , 218 _n_ 85
Stephen, St.,
Stillingfleet, Edward, ,
Strabo, ,
Ṣuhayb al-Rūmi/ _Zohaib_ , , 243 _n_ 10
Suhayl ibn 'Ammār/ _Ammar_ , , 243 _n_ 10
al-Ṣuhyūnī, Jibrā'īl/Gabriel Sionita, , 19–21, , , 219 _n_ 92
Symmachus, Quintus Aurelius, , 233 _n_ 121
al-Ṭabarī, Muḥammad ibn Jarrīr, , , , , 250 _n_ 9
Tarquin, Roman king, ,
Temple, Sir William,
Tertullian, , , , ,
Thābit ibn Qurra/ _Thabet Ben-Corra_ ,
Theodore of Mopsuestia, , 252 _n_ 16
Theodosius, Roman emperor, , , , , 225 _n_ 1; and Arians, ; baptism of, ; and Christ's divinity, ; religious authority of, , ,
Theodosius the Younger,
Theodotus,
Thomas, St.,
Thorndike, Herbert, 230 _n_ 70
Thrax, Maximinus, Roman emperor, , 234 _n_ 132
Tiberius, Roman emperor,
Tillotson, John, ,
Timothy, St.,
Titus, Roman emperor,
Tolan, John,
Toletanus, Rodericus,
Toomer, G. J., 218 _n_ 77, 223 _n_ 152; 224 _n_ 172, 238 _n_ 15, 253 _n_ 34
Traherne, Thomas, ,
Trajan, Roman emperor, , 232 _n_ 91, 243 _n_ 17
Treglown, Jeremy,
Trinitarian(s), , , ; and Arians, , ; canon of, ; forced conversion to, ; Jesus and, ; Justinian and, ; Muḥammad on, ; piety of, ; and pigeon story, ; Stubbe against, , ,
Trypho/ _Tryphon_ ,
Twells, Leonard,
'Umar ibn al-Khaṭṭāb/ _Alchittabi/Omar_ , , , , , 241 _n_ 32; Pact of, , , ; and the Qur'ān, 173–74; as second caliph, ; and treaty of peace,
'Uqba ibn abī Mujīd/ _Ochas_ , , 247 _n_ 16
'Utba ibn Waqqās, 247 _n_ 15
'Uthmān ibn 'Affān/ _Osman/Othman_ , , , 241 _n_ 32; as secretary of Muḥammad, , ; and the Qur'ān, ; as third caliph,
Valens, Roman emperor, , , , , 234 _n_ 128
Valentinian, Roman emperor, , , 234 _n_ 128, 235 _n_ 148, 246 _n_ 56; and Arians, ; baptism of, ; and polygamy,
Valois, Henry, ,
Vane, Sir Henry, , , ,
Venus, ,
Vespasian, Roman emperor, ,
Victor, Pope, , 232 _n_ 104
Virgil, ,
Virgin Mary, , , , , , 219 _n_ 92
Vossius, Gerard, , ,
Wakefield, Robert, 225 _n_ 3
Walton, Brian,
Waraqa ibn Nawfal/ _Warakeh bin Naufal_ , , 241 _n_ 19
Warmstry, Thomas, 215 _n_ 33
Warner, Levinus, , , 216–17 _n_ 61
Wheelock, Abraham, 225 _n_ 193
Wilmot, John, _see_ Rochester, John Wilmot, Second Earl of
Windet, James, 256 _n_ 37
Wolsey, Cardinal,
Wood, Anthony à, , , ,
al-Yūsi, al-Ḥasan,
Yūsuf Dhū Nuwās al- Ḥimyarī/King _Du Nowas_ , , 239 _n_ 41
Zayd ibn Ḥāritha, 247 _n_ 7
Zaynab/ _Zeinaban_ ,
Zegabenus, Euthymius, 195–96, 253 _n_ 33
Zenobia, Queen, , , 246 _n_ 62
Zephyrinus, Pope, , 232 _n_ 104
Zoroastar/ _Zaradast_ ,
Zorobabel,
Zosimus, ,
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/**
* Creates a new list box control.
*
* @-x-less ListBox.less
* @class tinymce.ui.ListBox
* @extends tinymce.ui.MenuButton
*/
define("tinymce/ui/ListBox", [
"tinymce/ui/MenuButton"
], function(MenuButton) {
"use strict";
return MenuButton.extend({
/**
* Constructs a instance with the specified settings.
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break;
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* Getter/setter function for the control value.
*
* @method value
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var self = this, active, selectedText, menu;
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"redpajama_set_name": "RedPajamaGithub"
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| 1,658
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\section{Introduction}\label{sec1}
It is expected that collisions between two nuclei at
ultra-relativistic energies will lead to a phase transition
from hadrons to the fundamental constituents, quarks
and gluons, usually referred to as Quark-Gluon-Plasma
(QGP). Experiments at the Relativistic Heavy Ion Collider (RHIC) at $\sqrt{s}_{NN}$=200 GeV Au+Au collisions \cite{BRAHMSwhitepaper} \cite{PHOBOSwhitepaper}\cite{PHENIXwhitepaper} \cite{STARwhitepaper} and at the Large Hadron Collider (LHC) at $\sqrt{s}_{NN}$=2.76 TeV Pb+Pb collisions \cite{Aamodt:2010pb}\cite{Collaboration:2010cz}\cite{Aamodt:2010jd}\cite{Aamodt:2010pa} had provided compelling evidences for production of QGP.
One of the experimental observables of QGP is
the azimuthal distribution of produced particles.
It is best studied by decomposing it in a Fourier series,
\begin{equation} \label{eq1}
\frac{dN}{d\phi}=\frac{N}{2\pi}\left [1+ 2\sum_n v_n cos(n\phi-n\psi)\right ], n=1,2,3...
\end{equation}
\noindent $\phi$ is the azimuthal angle of the detected particle and
$\psi$ is the plane of the symmetry of initial collision zone. In $\sqrt{s}_{NN}$=200 GeV Au+Au collisions, second flow harmonic ($v_2$), usually referred to as the elliptic flow, has been extensively studied experimentally as well as theoretically. Experimentally observed finite, non-zero $v_2$ is now regarded as definite proof of collective QCD matter creation in Au+Au collisions.
Qualitatively, elliptic
flow is naturally explained in a hydrodynamical model,
rescattering of secondaries generates pressure and drives
the subsequent collective motion. In non-central collisions, the reaction zone is asymmetric (almond shaped),
pressure gradient is large in one direction and small in the
other. The asymmetric pressure gradient generates the
elliptic flow. As the fluid evolve and expands, asymmetry in the reaction zone decreases and a stage arises when the
reaction zone become symmetric and system no longer
generates elliptic flow. Elliptic flow is early time phenomena. It is a sensitive probe to, (i) degree of thermalisation,
(ii) transport coefficient and (iii) equation of state of the
early stage of the fluid.
Ideal and viscous hydrodynamic models have been extensively used to analyze the experimental data at RHIC and LHC energy collisions.
Most of the analyses were performed with smooth
initial matter distribution obtained from geometric overlap of density distributions of the colliding nuclei.
For smooth matter distribution, the plane of symmetry of the collision zone coincides with the reaction plane (the plane containing the impact parameter and the beam axis).
The odd Fourier coefficients are zero by symmetry. One of the important realization in recent years, is that the participating nucleons, rather than the reaction plane,
determines the symmetry plane of the initial collision zone \cite{Manly:2005zy}. The realization is the results of analysis of various experimental data, e.g. the two particle correlation in $\Delta \phi$-$\Delta \eta$ plane \cite{Adams:2005ph}\cite{Putschke:2007mi}\cite{Abelev:2008ac}. The peculiar structure in two particle correlations known as 'ridge' and 'shoulder', observed both in STAR and PHENIX experiments have most compelling explanation provided the third flow harmonic, the triangular flow $v_3$ develops in the collisions.
Specifically, if initial condition is parameterized with quadrapole and triangular moments, response of the medium to these anisotropies is reflected in the two body correlation as ridge and shoulder \cite{Alver:2010gr},\cite{Alver:2010dn}.
Importance of the higher order flow harmonics in explaining the peculiar structures in two body correlation was also argued by Sorensen \cite{Sorensen:2010zq}.
The ridge structure in $p{\bar p}$ collisions \cite{Khachatryan:2010gv} \cite{Velicanu:2011dp} also has a natural explanation if odd harmonic flows develop. Recently, ALICE collaboration has observed odd harmonic flows in Pb+Pb collisions \cite{:2011vk}. In most central collisions, the elliptic flow ($v_2$) and triangular flow ($v_3$) are of similar magnitude. In peripheral collisions however, elliptic flow dominates. More recently, PHENIX collaboration \cite{Adare:2010ux}\cite{Adare:2011tg}\cite{Lacey:2011av} measured triangular flow in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions.
In the present paper, in event-by-event hydrodynamics, with Monte-Carlo Glauber model initial energy density distribution, we have simulated $\sqrt{s}_{NN}$=200 GeV Au+Au collisions in 0-10\% to 40-50\% collision centralities.
Simulation results compare well with the existing experimental data on charged particles multiplicity, transverse momentum spectra at $p_T\leq 1 GeV$, integrated and differential elliptic flow. Higher flow coefficients however are over predicted.
We have also studied the centrality dependence of
the correlation between the (integrated) flow coefficients with the initial spatial asymmetry measures.
In 0-10\%-40-50\% collisions, elliptic flow remains strongly correlated with the initial eccentricity. Triangular flow is strongly correlated with initial triangularity only in very central collisions. The correlation is reduced significantly in peripheral collisions. Higher flow coefficients $v_n$, n=4-5, even in central collisions, is only weakly correlated with initial asymmetry measures, $\epsilon_n$, n=3-5 and the correlation is more reduced in peripheral collisions.
The paper is organized as follows; in section \ref{sec2}, the Monte-Carlo Glauber model for initial energy density for use in hydrodynamic simulations is briefly discussed. In section \ref{sec3}, hydrodynamic equations, initial conditions, equation of state used in the simulations are described. Results of the simulations are described in section \ref{sec4}. Finally, all the results are summarized in section \ref{sec5}.
\section{Monte-Carlo Glauber model of initial energy density distribution}\label{sec2}
In theoretical simulations of event-by-event hydrodynamics, one generally uses the Monte-Carlo Glauber model to obtain the initial energy density distribution in an event. Details of the Monte-Carlo Glauber model can be found in \cite{Alver:2008aq}.
In a Monte-Carlo Glauber model, according to the density distribution of the colliding nuclei, two nucleons are randomly chosen. If the transverse separation
between the two nucleons is below $\sqrt\frac{\sigma_{NN}}{\pi}$, they are assumed to interact. Where $\sigma_{NN}$ is the nucleon-nucleon interaction cross section, taken here as 42 mb for Au-Au collisions at $\sqrt{s_{NN}}$ = 200 GeV.
Transverse position of the participating nucleons is then known in each event. The positions will fluctuate from event-to-event. If a particular event has $N_{part}$ participants, participants positions in the transverse plane can be labeled as, $(x_1,y_1), (x_2,y_2)....(x_{N_{part}},y_{N_{part}})$. Energy density distribution in the particular event can be obtained by assuming that each participant deposit energy $\varepsilon_0$ in the transverse plane,
\begin{equation}\label{eq2}
\varepsilon(x,y) \approx \varepsilon_0 \sum_{i=1}^{N_{part}} \delta(x-x_i,y-y_i)
\end{equation}
Fluid dynamical models require continuous density distribution and discrete distribution as in Eq.\ref{eq2} cannot be evolved in a hydrodynamical model. To use in a hydrodynamic model, the discrete density distribution has to be converted into a smooth energy-density distribution. This can be done by smearing the discrete participating nucleon positions by some smoothing function, $\delta(x-x_i,y-y_i) \rightarrow g(x-x_i,y-y_i,\zeta_1,\zeta_2..)$,
$\zeta_i$ being parameters of the smoothing function $g$.
\begin{equation} \label{eq3}
\varepsilon(x,y)=\varepsilon_0 \sum_{i=1}^{N_{part}} g(x-y,x_i,y-y_i,\zeta_1,\zeta_2....)
\end{equation}
One generally uses a Gaussian smoothing function. However, there can be other choices, e.g. in \cite{RihanHaque:2012wp}, a Woods-Saxon distribution function was used for the smoothing. In the present simulations, we have used a Gaussian distribution
\begin{equation}
g_{gauss}(x-x_i,y-y_i,\sigma) \propto e^{-\frac{{(x-x_i)^2+(y-y_i)^2}}{2\sigma^2} }, \label{eq6}
\end{equation}
\noindent of width $\sigma$=0.5 fm.
\begin{figure}[t]
\center
\resizebox{0.35\textwidth}{!}
\includegraphics{dnch2.eps}
}
\caption{(color online) The black circles are the PHENIX data for the centrality dependence of charged particles multiplicity in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions. The red symbols are the multiplicity in simulated events. The lines are to guide the eye. }
\label{F1}
\end{figure}
\section{Hydrodynamic equations, equation of state, initial conditions}\label{sec3}
With the Monte-Carlo Glauber model initial condition for
initial energy density, space-time evolution of the fluid, in each event, is obtained by solving the energy-momentum conservation equations,
\begin{eqnarray}\label{eq7}
T^{\mu\nu}&=&(\varepsilon+p)u^\mu u^\nu -P g^{\mu\nu}, \\
\partial_\mu T^{\mu\nu}&=&0,
\end{eqnarray}
\noindent where $\varepsilon$ and $p$ are the energy density and pressure respectively, $u$ is the hydrodynamic 4-velocity. We have assumed ideal fluid formation and disregarded any dissipative effect.
Assuming boost-invariance, hydrodynamic equations are solved in $(\tau=\sqrt{t^2-z^2},x,y,\eta_s=\frac{1}{2}\ln\frac{t+z}{t-z})$ coordinate system, with the code AZHYDRO-KOLKATA \cite{Chaudhuri:2008sj}.
Hydrodynamics equations are closed with an equation of state (EoS) $p=p(\varepsilon)$.
Currently, there is consensus that the confinement-deconfinement transition is a cross over. The cross over or the pseudo critical temperature for the quark-hadron transition is
$T_c\approx$170 MeV \cite{Aoki:2006we,Aoki:2009sc,Borsanyi:2010cj,Fodor:2010zz}.
In the present study, we use an equation of state where the Wuppertal-Budapest \cite{Aoki:2006we,Borsanyi:2010cj}
lattice simulations for the deconfined phase is smoothly joined at $T=T_c=174$ MeV, with hadronic resonance gas EoS comprising of all the resonances below mass $m_{res}$=2.5 GeV. Details of the EoS can be found in \cite{Roy:2011xt}.
In addition to the initial energy density for which we use the Monte-Carlo Glauber model, solution of hydrodynamic equations requires to specify the thermalisation or the initial time $\tau_i$ and fluid velocity ($v_x(x,y),v_y(x,y)$) at the initial time. A freeze-out prescription is also needed to convert the information about fluid energy density and velocity to invariant particle distribution. We assume that the fluid is thermalized at $\tau_i$=0.6 fm and the initial fluid velocity is zero, $v_x(x,y)=v_y(x,y)=0$. The freeze-out temperature is fixed at $T_F$=130 MeV.
We use Cooper-Frye formalism to obtain the invariant particle distribution of $\pi^-$ from the freeze-out surface. Resonance production is included. Considering that pions constitute $\sim$ 20\% of all the charged particles, $\pi^-$ invariant distribution is multiplied by the factor $2\times 1.2$ to approximate the charged particle's invariant distribution.
From the invariant distribution, harmonic flow coefficients are obtained as \cite{arXiv:1104.0650},
\begin{figure}[t]
\center
\resizebox{0.35\textwidth}{!}
\includegraphics{ptspec.eps}
}
\caption{(color online) Charged particles transverse momentum distribution with and without multiplicity fluctuations. }
\label{F2}
\end{figure}
\begin{eqnarray}
v_n(y,p_T)e^{in\psi_n(y,p_T)}&=&\frac{\int d\phi e^{in\phi} \frac{dN}{dy p_Tdp_T d\phi}} {\frac{dN}{dy p_Tdp_T}} \label{eq8}\\
v_n(y)e^{in\psi_n(y)}&=& \frac{ \int p_T dp_T d\phi e^{in\phi} \frac{dN}{dy p_T dp_T d\phi} } { \frac{dN}{dy} } \label{eq9}
\end{eqnarray}
In a boost-invariant version of hydrodynamics, flow coefficients are rapidity independent.
Present simulations are suitable only for central rapidity, $y\approx$0, where boost-invariance is most justified. Hereafter, we drop the rapidity dependence. $\psi_n$ in Eqs.\ref{eq8},\ref{eq9} is the participant plane angle for the n-th flow harmonic.
We characterise the asymmetry of the initial collision zone in terms of various moments of the eccentricity \cite{Alver:2010gr},\cite{Alver:2010dn},\cite{Teaney:2010vd},
\begin{figure}[t]
\vspace{0.3cm}
\center
\resizebox{0.35\textwidth}{!}
\includegraphics{v2int.eps
}
\caption{(color online) Black circles are PHOBOS measurements
for the centrality dependence of elliptic flow in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions. The simulation results for event averaged elliptic flow are shown as the red circles. The red squares, up triangles and down triangles are simulation results for triangular flow ($v_3$), rectangular flow ($v_4$) and pentangular flow ($v_5$) respectively. }
\label{F3}
\end{figure}
\begin{eqnarray}
\epsilon_n e^{in\psi_n} &=&-\frac{\int \int \varepsilon(x,y) r^n e^{i2\phi}dxdy}{\int \int\varepsilon(x,y) r^n dxdy} \label{eq11}, n=2,3,4,5
\end{eqnarray}
\noindent where $x=rcos\phi$ and $y=rsin\phi$. Eq.\ref{eq11} also determine the participant plane angle $\psi_n$. Asymmetry measures,
$\epsilon_2$ and $\epsilon_3$ are called eccentricity and triangularity. $\epsilon_4$ and $\epsilon_5$ essentially measures the squareness and five-sidedness of the initial distribution. In the following,
$\epsilon_4$ will be called rectangularity. In the same vein, $\epsilon_5$ will be called pentangularity. Fourth flow coefficient $v_4$ is generally referred as hexadecpolar flow. In following, we refer it as the rectangular flow, which is more appropriate. $v_5$ will be referred as the pentangular flow.
\section{Results}\label{sec4}
\subsection{Centrality dependence of charged particles multiplicity and $p_T$ spectra}
We have simulated 0-10\%, 10-20\%, 20-30\%, 30-40\% and 40-50\% Au+Au collisions at
$\sqrt{s}_{NN}$=200 GeV. In each collision centrality, we have simulated
$N_{event}$=1000 events. The constant $\epsilon_0$ is fixed to reproduce experimental charged particles multiplicity in 0-10\% collision. It was then kept fixed for all the other collision centralities. In Fig.\ref{F1}, simulated charged particles multiplicities are compared with the PHENIX data \cite{Adler:2004zn}. Once the model parameters are fixed to reproduce experimental multiplicity in 0-10\% collision, event-by-event simulations well reproduces the experimental multiplicity in other collision centralities.
We do note that in peripheral collisions, simulated multiplicity overestimate the experimental multiplicity by $\sim$10-15\%.
Even though charged particles multiplicities are well reproduced, the model simulations
failed to reproduce charged particles $p_T$ spectra, in particular in the high $p_T$ region. In Fig.\ref{F2}, model simulations for charged particles $p_T$ spectra, in 0-10\%, 10-20\%, 20-30\%, 30-40\% and 40-50\% collision centralities are compared with the PHENIX measurements \cite{Adler:2003au}. Simulated spectra explains the experimental data only up to $p_T\approx$1 GeV. In all the collision centralities, at higher $p_T$, model produces less particles than in experiment. The results
are interesting. It is well known that, compared to smooth hydrodynamics, in event-by-event hydrodynamics, $p_T$ spectrum is hardened \cite{arXiv:1104.0650}. Still the hardening is not enough to produce requisite number of particles at large $p_T$.
It is also well known that $p_T$ spectra is hardened in viscous fluid. Better fit to
charged particles $p_T$ spectra at large $p_T$ is expected if viscous rather than ideal fluid is formed in the collisions.
We do note that in the present simulations, we have
not made any conscientious attempt to fit the $p_T$, spectra. Only the
charged particles multiplicity in 0-10\% collision was fitted. Varying other parameters e.g. initial time, initial fluid velocity, freeze-out temperature etc. fit to charged particles $p_T$ spectra may be improved.
\begin{figure}[t]
\center
\resizebox{0.4\textwidth}{!}
\includegraphics{envn.eps
}
\caption{Event averaged flow coefficients ($v_n$) against the asymmetry measures,
($\epsilon_n$) for n=2-5. }
\label{F4}
\end{figure}
\subsection{Centrality dependence of flow coefficients}
\begin{figure}[t]
\center
\resizebox{0.40\textwidth}{!}
\includegraphics{vnpt2.eps}
}
\caption{(color online) In five panels, event-by-event simulations for flow coefficients $v_n$, n=2-5 in Au+Au collisions are shown. The black, red, green and yellow lines are respectively for elliptic flow $v_2$, triangular flow $v_3$, rectangular flow $v_4$ and pentangular flow $v_5$. The black circles, red squares and green triangles are PHENIX measurements for elliptic, triangular and rectangular flow in Au+Au collisions at RHIC.}
\label{F5}
\end{figure}
\subsubsection{Integrated flow coefficients}
Integrated flows are one of the important observables in heavy ion collisions. As discussed earlier, initial spatial symmetry is converted into momentum asymmetry, which is quantified in terms of different flow coefficients.
For example, elliptic flow ($v_2$) is response of an initial eccentricity ($\epsilon_2$) of the collision zone. Triangular flow ($v_3$) is response of initial triangularity ($\epsilon_3$) of the medium. Similarly, higher flow coefficients $v_4$ and $v_5$ are response of initial rectangularity ($\epsilon_4$) and pentangularity ($\epsilon_5$)of the initial medium.
In Fig.\ref{F3}, the black circles are PHOBOS measurements \cite{Back:2004mh} for centrality dependence of elliptic flow ($v_2$). $v_2$ increases rapidly as the collisions become more and more peripheral. Present simulations for $v_2$ in event-by-event hydrodynamics are shown as red circles. The simulations results agree well with the experimental data. In Fig.\ref{F3},
simulation results for (integrated) triangular flow ($v_3$), rectangular flow ($v_4$) and pentangular
flow ($v_5$) are also shown. Triangular flow also increases as the collisions become more and more peripheral. However, rate of increase is much slower than that for elliptic flow. $v_4$ and $v_5$ on the otherhand appears to be approximately independent of the collision centrality. From central 0-10\% to peripheral 40-50\% collisions, they change by less than a few percent.
\begin{figure}[t]
\center
\resizebox{0.3\textwidth}{!}
\includegraphics{e2v2.eps}
}
\caption{Correlation between elliptic flow ($v_2$) and initial eccentricity ($\epsilon_2$). Simulation results for $v_2$ is plotted against the initial eccentricity for $N_{event}$=1000 events. For a perfect correlation $v_2 \propto \epsilon_2$, all the points should lie on a straight line.}
\label{F6}
\end{figure}
In smooth hydrodynamics, elliptic flow in Au+Au collisions has been investigated in detail. Approximately, elliptic flow is proportional to initial eccentricity $\epsilon_2$. Dependence of the event averaged flow coefficients ($\la v_n \ra$,n=2-5) on the asymmetry measures ($\la \epsilon_n \ra$)
in event-by-event hydrodynamics is shown in Fig.\ref{F4}. The symbols, from left to right corresponds to 0-10\%, 10-20\%, 20-30\%, 30-40\% and 40-50\% Au+Au collisions. As expected, asymmetry measures increases with collision centralities. The increase is most in $\epsilon_2$, by a factor of $\sim$3.5 from 0-10\% collision centrality to 40-50\% centrality. In other asymmetry measures, $\epsilon_n$, n=3-5, the increase is more modest, factor of $\sim$ 2-2.5 only. As it is in smoothed hydrodynamics, in event-by-event hydrodynamics also,
elliptic flow increase, approximately linearly, with the initial eccentricity,
$\la v_2 \ra \propto \la \epsilon_2 \ra$. Higher flow coefficients, $v_3$ also increase with initial triangularity, however, the increase evidently is not linear.
Still higher flow coefficients $v_4(v_5) $, approximately remains the same in all the collision centralities (as already shown in Fig.\ref{F3}), they
appear to be independent of the asymmetry measures, $\epsilon_4(\epsilon_5)$.
Approximate centrality independence of higher flow coefficients, $v_4$ and $v_5$ in event-by-event hydrodynamics indicate that unlike the elliptic or triangular flow, rectangular flow $v_4$ or pentangular flow $v_5$ may not be related to initial asymmetry measure of the collision zone. Later, we will discuss the issue in more detail.
\subsection{Differential flow coefficients}
Differential flow coefficients are very sensitive observables and a model is well tested by comparing its predictions against experimental differential flow data. In Fig.\ref{F5}, in five panels (a)-(e), event-by-event hydrodynamic simulations for the differential flow coefficients, in 0-10\%, 10-20\%, 20-30\%, 30-40\% and 40-50\% Au+Au collisions are shown. In each panel, the black, red, green and yellow lines are the simulation results for elliptic flow, triangular flow, rectangular flow and pentangular flow respectively. We have shown only the event averaged values. In each collision centralities, PHENIX measurements \cite{Adare:2010ux}\cite{Adare:2011tg}\cite{Lacey:2011av} for the elliptic, triangular and rectangular flow are shown as the black circles, red squares and yellow triangles. Simulations do reproduce the trend of the data,
$v_2>v_3>v_4$. Event-by-event hydrodynamic simulations for the differential elliptic flow in Au+Au collisions agree well with the PHENIX data in all the collision centralities. In peripheral collisions, at $p_T>$ 2 GeV, experimental data are marginally over predicted. We have simulated Au+Au collisions in the ideal fluid approximation.
If instead of ideal fluid, viscous fluid is produced, better agreement with data is expected. Indeed, explicit event-by-event hydrodynamic simulations \cite{Qiu:2011hf}\cite{Schenke:2011zz}\cite{Chaudhuri:2011qm} do indicate that the event averaged flow coefficients reduces with viscosity.
In any case the agreement with data for elliptic flow measurements is much better for event-by-event ideal hydrodynamics compared to ideal hydrodynamic calculations with smooth CGC/Glauber model initial conditions \cite{Roy:2012jb}\cite{Chaudhuri:2009hj}.
Even though simulation results for elliptic flow reasonably well agree with the PHENIX experiment, simulation results for triangular ($v_3$) and rectangular flow ($v_4$) appear to over predict the PHENIX data for the same. Interestingly, triangular flow is more over predicted than the rectangular flow. Also, the discrepancy between simulations and experiment is more in peripheral collisions than in central collisions. For example, in 0-10\% collision, simulated $v_3$,
in the $p_T$ range 1-2 GeV, over predict the PHENIX data by $\sim$ 30\%. In 30-40\% collision, the data are over predicted by $\sim$60\% or more. In 30-40\% collisions, rectangular flow, in the $p_T$ range 1-2 GeV, is overpredicted by 5-10\% only. Here again, better agreement with data is expected if instead of ideal fluid, viscous fluid is formed in Au+Au collisions.
\begin{figure}[t]
\center
\resizebox{0.3\textwidth}{!}
\includegraphics{e3v3.eps}
}
\vspace{0.4cm}
\caption{Same as in Fig.\ref{F6} but for triangular flow ($v_3$) and initial triangularity ($\epsilon_3$).}
\label{F7}
\end{figure}
In Fig.\ref{F5}, the yellow lines represents the pentangular flow $v_5$. We did not find any experimental data for the pentangular flow. Following the trend of the simulation results for higher harmonic $v_3$, $v_4$, which are overpredicted in simulations, we do expect that the present simulation also over predict $v_5$. Present simulations then suggest that in experiments, in a peripheral 40-50\% Au+Au collisions, in the $p_T$ range 1-2 GeV, $\sim$2-5\% or less pentangular flow may be expected.
\subsection{Correlation between (integrated) flow coefficients and initial asymmetry measures}
Recently, in \cite{Chaudhuri:2011pa}\cite{Chaudhuri:2012wn}, correlation between integrated flow coefficients ($v_n$) and initial asymmetry measures ($\epsilon_n$) of the collision zone was studied in event-by-event hydrodynamics. It was shown that while elliptic flow remain strongly correlated with initial eccentricity, correlations between the higher flow coefficients $v_n$ and initial asymmetry measures $\epsilon_n$, n=3,4,5, are much more weak. In \cite{Chaudhuri:2011pa}\cite{Chaudhuri:2012wn} correlations between flow coefficients and asymmetry measures, in a single collision centrality, were studied. How the correlations are affected, as a function of collision centralities were not studied.
\begin{figure}[t]
\center
\resizebox{0.3\textwidth}{!}
\includegraphics{e4v4.eps}
}
\caption{Same as in Fig.\ref{F6} but for rectangular flow ($v_4$) and initial rectangularity ($\epsilon_4$).}
\label{F8}
\end{figure}
In Fig.\ref{F6}, we have plotted the simulated elliptic flow ($v_2$) against the initial eccentricity ($\epsilon_2$) in $N_{event}$=1000 events. If $v_2$ is perfectly correlated with $\epsilon_2$, all the points should lie on a straight line. One observes that in central collisions, elliptic flow is strongly correlated with eccentricity. The correlation is marginally reduced in more peripheral collisions. One can conclude that in event-by-event hydrodynamics also, the correlation between elliptic flow and initial eccentricity is strong, irrespective of the collision centrality.
In Fig.\ref{F7}-\ref{F9},
results obtained for higher flow harmonics are shown. In central collisions, correlation between triangular flow ($v_3$) and initial triangularity ($\epsilon_3$) is strong, though degree of correlation appear to be less than that in elliptic flow. In more peripheral collisions however, correlation is significantly reduced. Correlation between higher flow harmonics, $v_4(v_5)$ and initial asymmetry measure $\epsilon_4(\epsilon_5)$ even in central collisions is visibly much weaker than the corresponding correlation between elliptic flow and initial eccentricity. The correlations deteriorate as the collisions become more and more peripheral. Indeed, from the scatter plot of $v_4$ and $v_5$ in peripheral collisions, it is difficult to claim that the flow coefficients are correlated with the asymmetry measures.
In \cite{Chaudhuri:2011pa} a quantitative measure was defined to quantify the correlation between flow coefficients and initial spatial asymmetry measure. A modified form is used here to quantify the correlation.
For a perfect correlation, $v_n\propto \epsilon_n$ and simulated flow coefficients will fall on a straight line. Dispersion of the
flow coefficients around the best fitted straight line then gives a
measure of the correlation. We thus define a correlation measure
function $C_{measure}$,
\begin{figure}[t]
\center
\resizebox{0.3\textwidth}{!}
\includegraphics{e5v5.eps}
}
\caption{Same as in Fig.\ref{F6} but for pentangular flow ($v_5$) and initial pentangularity ($\epsilon_5$).}
\label{F9}
\end{figure}
\begin{equation}
C_{measure}(v_n)=1-\frac{\sum_i [ v_n^i(\epsilon_n) -v_{n,st.line}(\epsilon_n) ]^2}{\sum_i [ v^i_{random}(\epsilon) -v_{st.line}(\epsilon) ]^2}
\end{equation}
$C_{measure}$
essentially measures the dispersion of the simulated flow coefficients from the best fitted straight line, relative to completely random flow coefficients. It varies between 0 and 1.
If flow coefficients are perfectly correlated then $v_n \propto \epsilon_n$ and $C_{measure}$ is identically unity. For completely random flow coefficients, $C_{measure}$=0. To obtain an even ground for comparison of $C_{measure}$ for different flow coefficients, the flow coefficients ($v_n$) and the asymmetry parameters ($\epsilon_n$) are scaled to vary between 0 and 1. In Fig.\ref{F10}, we have shown the correlation measures for the flow coefficients as a function of collision centrality.
The elliptic flow remain strongly correlated with initial eccentricity ($c_{measure}(v_2)\approx 0.95-0.99$) in 0-50\% collisions. In central, 0-10\%, 10-20\% collisions, triangular flow ($v_3$) is also strongly correlated with initial triangularity ($\epsilon_3$), ($c_{measure}(v_2)\approx 0.95$). Correlation is significantly reduced in more peripheral collisions and in 40-50\% collisions,
$c_{measure}(v_2)\approx 0.75$. In higher flow coefficients, correlation is even less
in peripheral collisions.
If departure of $C_{measure}$ from unity is interpreted as a measure of flow uncorrelated with the initial
asymmetry measure, for elliptic flow $v_2$, in 0-50\% collisions, less than $\sim$ 5\% of the flow is uncorrelated with initial eccentricity. In higher flow coefficients, $v_n$, n=3-5, uncorrelated flow grow with collision centrality. For example, in rectangular flow $v_4$, uncorrelated flow grows from $\sim$10\% in 0-10\% collisions to $\sim$40\% in 40-50\% collision.
As it was discussed previously, present analysis of Au+Au data also indicate that better description to the data will be obtained if viscous fluid rather than ideal fluid is formed in Au+Au collision. In \cite{Chaudhuri:2011pa} viscous effects on the correlation between elliptic flow and initial eccentricity and between triangular flow and initial triangularity were studied. It was shown that the correlations reduce significantly in viscous fluid. In more realistic event-by-event hydrodynamic simulation of Au+Au collisions, with viscous fluid, correlation between higher flow coefficients and asymmetry measures then expected to reduce even more than obtained presently.
Compartively low $C_{measure}$ or equivalently, large uncorrelated higher flow harmonics is difficult to understand.
In ideal hydrodynamics, final flow coefficients are related to the initial transverse energy density, or more appropriately on the pressure gradients only. We have even assumed zero initial velocity. Yet, the flow coefficients, $v_3$, $v_4$ and $v_5$ in peripheral collisions are largely unrelated to the initial asymmetry measures. Elliptic flow on the other hand is perfectly correlated with initial asymmetry measure. Source of the uncorrelated flows in higher harmonics can not be discerned presently. Possibly, other aspects of the
initial density (higher moments or products of moments) are important in development of higher harmonics.
\begin{figure}[t]
\center
\resizebox{0.35\textwidth}{!}
\includegraphics{cmeasure.eps}
}
\caption{(color online) collision centrality dependence of the correlation measure (see text) for $v_2$, $v_3$, $v_4$ and $v_5$ }
\label{F10}
\end{figure}
\section{Summary and conclusions}\label{sec5}
To summarize, in event-by-event hydrodynamics, we have studied the centrality dependence of charged particles multiplicity, $p_T$ spectra, and flow coefficients (integrated and differential) in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions. The initial energy density distributions are obtained from Monte-Carlo Glauber model. The Monte-Carlo Glauber model participant positions are smoothed with a Gaussian distribution of width $\sigma$=0.5 fm and normalized to reproduce experimental charged particles multiplicity in 0-10\% collision. We have simulated a large number of events, $N_{event}$=1000 in each collision centrality. Once the initial transverse energy density is fixed to reproduce multiplicity in 0-10\% collision, the model reproduces the experimental multiplicity in other collision centralities within reasonable accuracy.
Experimental charged particles transverse momentum spectra, however are reproduced in the model, only in a limited $p_T$ range, $p_T \leq $1GeV. At $p_T >$ 1 GeV, simulated spectra under predict the experiment. In the simulations, dissipative effects are neglected. Dissipative effect like (shear) viscosity, will enhances particle production, more at high $p_T$ than at low $p_T$. Better description to the $p_T$ spectra is expected if instead of ideal fluid, viscous fluid is produced in Au+Au collisions.
We have also compared the model simulations for integrated and differential flow coefficients with experimental data. Experimental (integrated) elliptic flow in Au+Au collisions are correctly reproduced in simulations. The model also reasonably well reproduces the experimental differential elliptic flow in 0-10\%-40-50\% collisions.
In peripheral collisions, elliptic flow data however is overpredicted at high $p_T$. Higher flow coefficients $v_3$ and $v_4$ however are over predicted, more in peripheral than in central collisions. Here again, better description to the data is expected if instead of ideal fluid, viscous fluid is produced in Au+Au collisions.
We have also studied correlation between (integrated) flow coefficients and initial asymmetry measures of the collision zone. In all the collision centralities (0-10\% to 40-50\%) elliptic flow is strongly correlated with
the initial asymmetry measure, the eccentricity of the collision zone. The higher flow
coefficients however show much less correlation with the corresponding asymmetry measures. We have quantified the correlation and observe that with the exception for
elliptic flow, which remain strongly correlated in all the collision centralities, for the higher flow coefficients, $v_3$, $v_4$ and $v_5$, correlation reduces significantly in more peripheral collisions. It appears that in higher flow coefficients, a significant part of the flow is unrelated to the initial asymmetry measures.
The reason for the
flow unrelated to the initial asymmetry can not be discerned presently. One can only conclude that apart from the initial density distribution of collision zone, other aspects, e.g. higher moments or product of higher moments are also important for the development of higher harmonics.
\section*{Acknowledgments}
RH, VR and BM are supported by DAE-BRNS project Sanction No. 2010/21/15-BRNS/2026.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 439
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\section{Introduction}
The approach of this paper is motivated by the modern interpretation of the
Heine-Stieltjes multiparameter spectral problem as presented in \cite{Sh} and
\cite{ShSh}. Let us recall some relevant results in the matrix set-up.
Given integers $m>0$ and $n\ge 0$ consider the space ${ Mat}(m,m+n)$ of
complex-valued $m \times (m+n)$-matrices. For $s=0,\dots, n$
define the $s$-th unit matrix
$${\mathcal{I}}_s := (\delta_{s+i-j})\in { Mat}(m,m+n).$$
(In what follows the sizes of matrices can be infinite.)
\begin{definition}[see \cite{ShSh}]
Given a matrix $A\in{ Mat}(m,m+n)$ define its \defin{eigenvalue locus}
${\mathcal{E}}_A$ as
\[
{\mathcal{E}}_A := \left\{ (x_0,x_1,\dotsc,x_n) \in \mathbb{C}^{n+1} \colon \rank \left( A
- \sum_{s=0}^n x_s {\mathcal{I}}_s \right) < m \right\}.
\]
For $n=0,$ ${\mathcal{E}}_A$ coincides with the usual set of eigenvalues of a
square matrix $A$.
\end{definition}
\begin{proposition}[ Lemma~1 of \cite{ShSh}]\label{prop:eig}
For arbitrary $A\in { Mat}(m,m+n)$ the eigenvalue locus ${\mathcal{E}}_A$ consists of
$\binom{m+n}{n+1}$ points
counting multiplicities. In other words, counting multiplicities there exist
$\binom{m+n}{n+1}$ eigenvalue tuples $(x_0,x_1,\dotsc,x_n)$ such that
$A-\sum_{s=0}^{n}x_s {\mathcal{I}}_s$ has rank smaller than $m$.
\end{proposition}
\begin{remark}
Notice that for $n>0,$ the locus ${\mathcal{E}}_A$ is not a complete
intersection since it is given by the vanishing of all maximal minors of $A$.
(A similar phenomenon can be observed for common zeros of
multivariate Schur polynomials, since Schur polynomials are given by determinant
formulas.)
\end{remark}
\begin{notation}\label{deff}
Given an infinite matrix ${\mathcal{A}}=(a_{ij}),\; i=1,\dotsc,\infty;
j=1,\dotsc,\infty$, an integer $n\geq 0$, and an $m$-tuple of positive integers
$I=(i_1,i_2,\dots, i_m)$ satisfying $1\le i_1<i_2<\dotsc <i_m\leq m+n$,
consider the submatrix $A_I$ of ${\mathcal{A}} - \sum_{s=0}^n x_s {\mathcal{I}}_s$ formed
by the first $m$ rows and $m$ columns indexed by $I.$
Define
\begin{equation}\label{eq:main}
P^I_{\mathcal{A}}(x_0,x_1,\dotsc,x_n):=\det A_I.
\end{equation}
\end{notation}
Evidently, $P^I_{\mathcal{A}}(x_0,\dotsc,x_n)$ is a maximal minor of the principal $m
\times (m+n)$ submatrix of ${\mathcal{A}}- \sum_{s=0}^n x_s {\mathcal{I}}_s$ formed by
its $m$ first rows and $m+n$ first columns. Therefore
$P^I_{\mathcal{A}}(x_0,\dotsc,x_n)$ is a polynomial in $x_0,\dotsc,x_n.$
\begin{proposition}\label{prop:basis}
In the above notation the
following holds:
\begin{enumerate}
\item[(i)] for any multiindex $I$ with
$|I|=m$, $\deg P^{I}_{\mathcal{A}}(x_0,\ldots, x_n)=m$;
\noindent
\item[(ii)] all $\binom{m+n}{m}$ polynomials
$P^I_{\mathcal{A}}(x_0,...,x_n)\in \mathbb{C}[x_0,\dots,x_n]$ with $|I|=m$ are linearly
independent which implies that the totality of all polynomials
$P^I_{\mathcal{A}}(x_0,...,x_n)$ is a linear basis of $\mathbb{C}[x_0,\dots,x_n]$;
\noindent
\item[(iii)] the set ${\mathcal{E}}_{\mathcal{A}}^{(m)}$ of
common zeros of all $P^I_{\mathcal{A}}(x_0,...,x_n)$ with $|I|=m$
is a finite subset of $\mathbb{C}^{n+1}$ of cardinality $\binom{m+n}{n+1}$ counting
multiplicities.
\end{enumerate}
\end{proposition}
\begin{remark}\label{rem:2} Notice that for $\binom {m+n}{m}$ randomly chosen
polynomials in $\mathbb{C}[x_0,x_1\ldots, x_n]$ of degree $m$ the set of their common
zeros is typically empty.
\end{remark}
Proposition \ref{prop:basis} together with our
numerical experiments motivate the following question.
\medskip
Given an arbitrary infinite matrix ${\mathcal{A}}$ as above, associate to each
${\mathcal{E}}_{\mathcal{A}}^{(m)}$ its ``root-counting'' measure $\mu_{\mathcal{A}}^{(m)}$
supported on ${\mathcal{E}}_{\mathcal{A}}^{(m)}\subset \mathbb{C}^{n+1}$ by assigning to every
point $p\in {\mathcal{E}}_{\mathcal{A}}^{(m)}$ the point mass ${\kappa(p)}/{\binom
{m+n}{n+1}}$
where $\kappa(p)$ is the multiplicity of $p$.
(Obviously, $\mu_{\mathcal{A}}^{(m)}$ is a discrete probability measure.)
\medskip\noindent
{\bf Main Problem.} \label{prob:1} Under which assumptions on ${\mathcal{A}}$ does the
weak limit $\mu_{{\mathcal{A}}}=\lim_{m\to\infty}\mu_{{\mathcal{A}}}^{(m)}$ exists?
In case when $\mu_{{\mathcal{A}}}$ exists,
is it possible to describe the support and density of the measure?
\medskip
In the classical case $n=0$, the above problem was intensively studied by many
authors. The main focus has been when ${\mathcal{A}}$ is either a Jacobi or a Toeplitz matrix
(or their generalizations such as block-Toeplitz matrices etc.), see e.g.
\cite{BS, BG, Wi1, Wi2}.
The main goal of this note is to present a multivariate analogue of the
well-known theorem by P.~Schmid and F.~Spitzer \cite{SS},
where they describe $\mu_{{\mathcal{A}}}$ for an arbitrary banded Toeplitz matrix ${\mathcal{A}}$
in the case $n=0.$
Namely, let ${\mathcal{A}} = (c_{i-j}),$ with $i,j=1,2,\dotsc$ be an infinite,
banded Toeplitz matrix, where $c_i=0$ if $i<-k$ or $i>h.$
Fixing $n\ge 0$ as above, we obtain for each positive integer $m$ the
eigenvalue locus ${\mathcal{E}}_{\mathcal{A}}^{(m)}$ of the
principal $m \times (m +n)$ submatrix $A^{(m)}$ of ${\mathcal{A}}$.
\medskip
Define the \defin{limit set $B_{{\mathcal{A}}}$ of eigenvalue loci} as
\begin{equation}\label{eqn:Bdef2}
B_{\mathcal{A}} = \left\{ \mathbf{x} \in \mathbb{C}^{n+1} \colon \mathbf{x}=\lim_{m \rightarrow \infty}
\mathbf{x}_m, \mathbf{x}_m \in {\mathcal{E}}_{\mathcal{A}}^{(m)
\right\},\; \mathbf{x} = (x_0,\dotsc,x_n).
\end{equation}
In other words, $B_{\mathcal{A}}$ is the set of limit points of the sequence
$\{{\mathcal{E}}_{\mathcal{A}}^{(m)}\}$. Thus $B_{\mathcal{A}}$ is the support of the
limiting measure $\mu_{\mathcal{A}}$ if it exists.
(For a general infinite matrix ${\mathcal{A}}$ as above,
its limit set $B_{\mathcal{A}}$ might be empty.)
Set
\begin{align}\label{eq:symbol}
Q(t,\mathbf{x}) = t^k\left(\sum_{j=-k}^h c_j t^j -\sum_{j=0}^n x_j t^j \right),
\end{align}
and let
$\alpha_1(\mathbf{x}),\alpha_2(\mathbf{x}),\dots,\alpha_{k+h}(\mathbf{x})$ be the roots of
$Q(t,\mathbf{x})=0,$
ordered according to their absolute values, i.e. $|\alpha_i(\mathbf{x})|\leq
|\alpha_{i+1}(\mathbf{x})|$ for all $0<i<k+h.$
Let $C_{\mathcal{A}} $ be the real semi-algebraic set given by the condition:
\begin{equation}\label{eqn:Cdef2}
C_{\mathcal{A}} = \{ \mathbf{x} \in \mathbb{C}^{n+1} \colon
|\alpha_k(\mathbf{x})|=|\alpha_{k+1}(\mathbf{x})|=\dots=|\alpha_{k+n+1}(\mathbf{x})| \}.
\end{equation}
Our main conjecture is as follows.
\begin{conjecture}\label{conj:main}
For any banded Toeplitz matrix ${\mathcal{A}}$,
if $B_{\mathcal{A}}$ is defined by \eqref{eqn:Bdef2} and $C_{\mathcal{A}}$
defined by \eqref{eqn:Cdef2} then $B_{\mathcal{A}}=C_{\mathcal{A}}$.
\end{conjecture}
By Conjecture~\ref{conj:main} the set $B_{\mathcal{A}}$ is a
real semi-algebraic $(n+1)$-dimensional subset of $\mathbb{C}^{n+1}.$
In the classical case $n=0$,
Conjecture~\ref{conj:main} is settled by P.~Schmidt and F.~Spitzer in
\cite{SS}.
Another important case when Conjecture~\ref{conj:main} has been proved follows from
some known results on multivariate Chebyshev polynomials,
which is is presented in Example~\ref{ex:A} below.
Namely, when $k=1$ and $h=n+1$ with $c_{-1}$ and $c_{n+1}$ non-zero,
we may do a affine change of the variables and a scaling of ${\mathcal{A}}$.
This reduces to the latter case to $c_{-1}=c_{n+1}=1$ and all other $c_i=0.$
For these particular values, the family $\{P_{\mathcal{A}}^I(\mathbf{x})\}$
becomes the multivariate Chebyshev polynomials of the second kind, see e.g.
\cite{DL,KO, BE,Xu}.
These polynomials also have a close connection to another well-known family of
polynomials, namely the Schur polynomials.
\noindent
\begin{example}\label{ex:A}
{\rm For the above matrices corresponding
to the multivariate Chebyshev polynomials the eigenlocus
${\mathcal{E}}_{\mathcal{A}}^{(m)}$ can be described explicitly,
see for example \cite{GE}.
More precisely, the points in ${\mathcal{E}}_{\mathcal{A}}^{(m)}$ lie on
a real, $n$-dimensional surface $C_{\mathcal{A}}\subset \mathbb{C}^{n+1}$
which is naturally parametrized by an $(n+1)$-dimensional torus $T^{n+1}$.
This parametrization is given by
\begin{align}\label{eq:chebylocus}
C_{\mathcal{A}} = \left\{ \mathbf{x} \in \mathbb{C}^{n+1} |
x_j =
-e_{j+1}\left(\exp(i\theta_1),\dotsc,\exp(i\theta_{n+1}),\exp(i\theta_{n+2}
)\right)\right\}
\end{align}
where $(\theta_1,\dotsc,\theta_{n+1})$ lie on $T^{n+1},$ $\sum_{j=0}^{n+2}
\theta_j =0,$ and
$e_j$ is the $j$-th elementary symmetric function in $n+2$ variables.
Notice that for $\mathbf{x} \in C_{\mathcal{A}},$
\begin{align}
Q(t,\mathbf{x}) &= 1+x_0 t + x_1 t^2 + \dotso + x_n t^{n+1} + t^{n+2}= \\
&= \prod_j (t+ e^{i\theta_{j}})
\end{align}
by the Vieta formulas. Thus, for $\mathbf{x} \in C_{\mathcal{A}},$
\emph{all} roots of $Q$, (as a polynomial in $t$)
have absolute value equal to 1 when the $x_j$ are parametrized as in
\eqref{eq:chebylocus}.
Furthermore, the points in ${\mathcal{E}}_{\mathcal{A}}^{(m)}$
are also expressed by \eqref{eq:chebylocus},
with the parameters $(\theta_1,\dots, \theta_{n+2})$ being certain rational
multiples of $\pi,$
distributed in a regular lattice.
The mapping from the $2$-torus to the eigenlocus is illustrated in
Fig.~\ref{fig1}.
Another interesting aspect of Example~\ref{ex:A} is that all the points
$\mathbf{x}=(x_0,\dots, x_n)$
in the sets ${\mathcal{E}}_{\mathcal{A}}^{(m)}$
satisfy the conditions $x_j = \overline{x_{n-j}}$, $j=0,1,\dots,n,$
which explains why we can draw $C_{\mathcal{A}} \subset \mathbb{C}^2$ in Fig.~\ref{fig1}a as
a 2-dimensional set.
For larger $n$, $C_{\mathcal{A}}$ is a $(n+1)$-dimensional analogue of the
two-dimensional deltoid, shown in Fig.~\ref{fig1}a.}
\end{example}
For general ${\mathcal{A}}$, we do not have the inclusion ${\mathcal{E}}_{\mathcal{A}}^{(m)}
\subseteq C_{\mathcal{A}}$ for arbitrary finite $m$.
However, if ${\mathcal{A}}$ has an additional extra symmetry, this seems to be case.
\begin{definition}
A banded Toeplitz matrix such that its $Q(t,\mathbf{x})$ in \eqref{eq:symbol}
satisfies
\[
\overline{Q(t,x_0,x_1,\dotsc,x_n)} =
\overline{t}^{h+k-1}Q(1/\overline{t},\overline{x}_n,\overline{x}_{n-1},\dotsc,
\overline{x}_0)
\]
is called \defin{multihermitian} of order $n$.
\end{definition}
\begin{conjecture}\label{conj:conjugates}
If ${\mathcal{A}}$ is multihermitian of order $n$, then
each point $\mathbf{x}=(x_0,x_1,\dotsc,x_n) \in {\mathcal{E}}_{\mathcal{A}}^{(m)}$
satisfies $x_j = \overline{x_{n-j}}$ for $j=0,1,\dotsc,n.$
\end{conjecture}
Conjecture~\ref{conj:conjugates} obviously holds for the case $n=0,$
as it reduces to the fact that hermitian matrices have real eigenvalues.
It is also straightforward to check that Conjecture \ref{conj:conjugates}
is true for the Chebyshev case of Example~\ref{ex:A} above.
We have extensive numerical evidence for this conjecture.
Another strong indication supporting Conjecture~\ref{conj:conjugates} is that if ${\mathcal{A}}$ is multi-hermitian,
then every point
$\mathbf{x} \in C_{\mathcal{A}}$ (which by Conjecture \ref{conj:main} is in the limit
eigenlocus) satisfies
the required symmetry $x_j = \overline{x_{n-j}}$ for $j=0,1,\dotsc,n.$
\begin{figure}[ht!]
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=1\textwidth]{deltoidpoints}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=1\textwidth]{hexagonpoints}
\end{subfigure}
\caption{
The eigenvalue locus ${\mathcal{E}}_2^{(20)}$ and its pull-back to $T^2$.
The torus $T^2$ is covered with a hexagon,
where each triangle is mapped to the eigenlocus.
The 6-fold symmetry is due to the $S_3$-action by permutation of the
arguments $\theta_1,\theta_2,\theta_3$ in \eqref{eq:chebylocus}.
(Notice $\theta_1+\theta_2+\theta_3=0$ and this is the subspace which is
illustrated in the figure to the right.)
}
\label{fig1}
\end{figure}
The next group of examples are bivariate analogues of
special univariate cases originally studied in \cite{SS},
and later in \cite{BG}, where they are referred to as ``star-shaped curves'':
\begin{example}\label{ex:B}
{\rm The bivariate case when
$Q(t,\mathbf{x}) = 1 + t^d x_0 + t^{d+1}x_1 + t^{2d+1}$, $d\geq 1$
gives sets in $\mathbb{C}^2$ where $x_0 = \overline{x_1}$,
by Conjecture \ref{conj:conjugates}. They correspond to Toeplitz matrices of the
form
\[
\begin{pmatrix}
x_0 & x_1 & 1 & 0 & 0 & \cdots \\
1 & x_0 & x_1 & 1 & 0 & \cdots \\
0 & 1 & x_0 & x_1 & 1 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{pmatrix}, \;
\begin{pmatrix}
x_0 & x_1 & 0 &1 & 0 & 0 & \cdots \\
0 & x_0 & x_1 & 0 & 1 & 0 & \cdots \\
1 & 0 & x_0 & x_1 & 0 & 1 & \cdots \\
\vdots & \vdots & \vdots & \vdots& \vdots & \ddots
\end{pmatrix}, \dotsc
\]
The above two matrices represent $d=1$ and $d=2.$
Figures~\ref{fig2} and \ref{fig3} present the distributions of $x_0 \in
\mathbb{C},$ for $d=2,3,4$. (Recall that $x_1=\bar{x}_0$.)
The points shown on these figures belong to ${\mathcal{E}}_{\mathcal{A}}^{(m)}$ for
$m=13,14,15$,
and the curves are certain hypocycloids, parametrizing the boundary of
$C_{\mathcal{A}}$.
More explicitly, for a given integer $d\ge 1$ the hypocycloid boundary for $x_0
\in \mathbb{C}$ is given by
\[
x_0 = (-1)^d e^{-i (d+2) \theta} \left((d+2) e^{i (2 d+3) \theta}+d+1\right)
\text{ where } \theta \in [0,2\pi],
\]
which is one of the implications of Conjecture \ref{conj:main}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{cycloid5.eps}
\includegraphics[scale=0.45]{cycloid7.eps}
\end{center}
\vskip 0.3cm
\caption{5-edged star, when $d=2$ and 7-edged star, when $d=3$}
\label{fig2}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{cycloid9.eps}
\end{center}
\vskip 0.3cm
\caption{9-edged star, when $d=4$.}
\label{fig3}
\end{figure}}
\end{example}
\medskip
Finally, the main result of this note is as follows;
\begin{theorem}\label{th:main}
For any banded Toeplitz matrix ${\mathcal{A}}$,
where $B_{\mathcal{A}}$ is defined by \eqref{eqn:Bdef2} and $C_{\mathcal{A}}$ is defined by
\eqref{eqn:Cdef2},
one has $B_{\mathcal{A}}\subseteq C_{\mathcal{A}}$.
\end{theorem}
\medskip\noindent
\textit{Acknowledgements.} The authors are sincerely grateful
to Professor M.~Tater who actively participated in the consideration of some
initial examples related to multivariate Chebyshev polynomials for his help
and to the Nuclear Physics Institute in \v Re\v z near
Prague for the hospitality in March 2011.
We want to thank Professor A.~Gabrielov of Purdue University
for his help with the proof of Proposition~\ref{cor:main}.
\section{Proofs}
\begin{proof}[Proof of Proposition~\ref{prop:basis}] We shall prove items
\rm{(i)} and \rm{(ii)} simultaneously by calculating the leading homogeneous
part of
$P^I_{\mathcal{A}}(x_0,...,x_n)$. Let us order the set of all admissible indices
$I=(1\le i_1<\ldots <i_m\le m+n)$ lexicographically. We can also order
lexicographically all monomials of degree $m$ in $x_0,\ldots, x_n$. By
equation~\eqref{eq:main} $P^I_{\mathcal{A}}(x_0,...,x_n)=\det A_I$ where the columns
of $A_I$ are indexed by $I$. Let $\widetilde P^I_{\mathcal{A}}(x_0,...,x_n)$ be the
homogeneous part of $P^I_{\mathcal{A}}(x_0,...,x_n)$ of degree $m$. One can easily
see that the product of all entries on the main diagonal of $A_I$ contains the
monomial $\mathfrak{m}_I$ of degree $m$ given by $\mathfrak m_I=\prod_{j=1}^m
x_{i_j-j+1}$. Moreover it is straight-forward that $\widetilde
P^I_{\mathcal{A}}(x_0,...,x_n)=\mathfrak{m}_I+\ldots$ where $\ldots$ stands for the
linear combination of monomials $\mathfrak{m}_{I^\prime}$ of degree $m$ coming other
$I^\prime$ which are lexicographically smaller than $I$. In other words, the
matrix formed by $\widetilde P^I_{\mathcal{A}}(x_0,...,x_n)$ versus monomials is
triangular in the lexicographic ordering with
unitary main diagonal which proves items \rm{(i)} and \rm{(ii)}.
Item \rm{(iii)} is just a reformulation of Proposition~\ref{prop:eig} above.
\end{proof}
Throughout the rest of the paper,
we use the convention ${\boldsymbol\alpha}=(\alpha_1,\dotsc,\alpha_{h+k}).$
We will also assume that $c_h=1$, which corresponds to a rescaling of the original matrix ${\mathcal{A}}.$
This is equivalent to the assumption that $Q(t,\mathbf{x})$ is monic.
By shifting the variables in $\mathbf{x}$, we may also assume, without loss of generality,
that $c_0=c_1\dotso=c_n=0$ in ${\mathcal{A}}.$
In the above notation, it is convenient to work with the roots of $Q(t,\mathbf{x}).$
This motivates the following definitions.
Let $\Gamma_j\subset \mathbb{C}^{h+k},\; j=k,\dots, k+n$ denote the
real semi-algebraic hypersurface
consisting of all ${\boldsymbol\alpha}=(\alpha_1,\dotsc, \alpha_{h+k})$ such that
when the $\alpha_j$ are ordered with increasing modulus, $|\alpha_j|=|\alpha_{j+1}|$.
Similarly, let $G_j$ be defined as the real semi-algebraic set
\[
\{ \mathbf{x} \in \mathbb{C}^{n+1} \colon Q(t,\mathbf{x}) = (t-\alpha_1)\dotso (t-\alpha_{h+k}) \text{ where } {\boldsymbol\alpha} \in \Gamma_j \}.
\]
Then, by definition, $C_{\mathcal{A}} = \bigcap_{j=k}^{k+n} G_j.$
\begin{proposition}\label{prop:compact}
For any banded Toeplitz matrix ${\mathcal{A}}$ and any
non-negative $n<h$, the set $C_{\mathcal{A}}$ defined by \eqref{eq:symbol}-\eqref{eqn:Cdef2}
is compact.
\end{proposition}
\begin{proof}
As discussed above,
we may without loss of generality assume that $c_h=1$ and
$c_0 = c_1 = \dotso = c_n=0.$
Since $Q$ may be assumed to be monic,
we have $c_j = e_{h-j}(-{\boldsymbol\alpha})$ for $-k\leq j<0$ and $n<j\leq h,$
and $x_j = -e_{h-j}(-{\boldsymbol\alpha})$ when $0\leq j \leq n.$
Thus, it suffices to show that the set of ${\boldsymbol\alpha} \in \mathbb{C}^{h+k}$
that satisfies the conditions \eqref{eq:symbol}-\eqref{eqn:Cdef2},
is compact.
It is also evident that the set $C_{\mathcal{A}}$ is closed,
so we only need to show that it is bounded. We show this fact by contradiction.
\medskip
Assume we have a sequence of roots $\{{\boldsymbol\alpha}^m\}_{m=1}^\infty$ of \eqref{eq:symbol}
such that $\|{\boldsymbol\alpha}^m\| \rightarrow \infty$
where \eqref{eqn:Cdef2}
is satisfied for each ${\boldsymbol\alpha}^m.$
We assume that the modulus of the roots are always ordered increasingly.
There are two cases to consider.
\medskip\noindent
\textbf{Case 1:} Assume that for some $0\leq b < k$, a sequence of individual
roots satisfies the condition $|\alpha_{b+1}^m|\rightarrow \infty$ but
$|\alpha_j^m|$ are bounded for all $m$ and $j\leq b.$
Then consider $e_{h+k-b}({\boldsymbol\alpha}).$
Since $b < k,$ in our notation $e_{h+k-b}({\boldsymbol\alpha})$ equals the
coefficient $c_{b-k}.$
Notice that $e_{h+k-b}$ contains the term
$\alpha_{b+1} \alpha_{b+2} \dotsm \alpha_{h+k}$
which grows quicker than all
other terms in the expansion of $e_{h+k-b}({\boldsymbol\alpha}).$
This contradicts the assumption $e_{h+k-b}({\boldsymbol\alpha})=c_{b-k}.$
\medskip\noindent
\textbf{Case 2:}
Assume that for some $b$ with $k+n\leq b < h+k,$ we have a sequence of
individual roots $|\alpha_{b+1}^m|\rightarrow \infty$ but $|\alpha_j^m|$ are
bounded for all $m$ and $j\leq b.$ Consider
\[
e_{b}({\boldsymbol\alpha})=e_{b}(\alpha_1,\dotsc,\alpha_{h+k}) = \sum_{\sigma \in \binom{[h+k]}{b}}
\frac{e_0}{\alpha_{\sigma_1}\alpha_{\sigma_2} \dotsm \alpha_{\sigma_{b}}}.
\]
This contains an expression with the denominator $\alpha_1 \alpha_2 \dotsm
\alpha_{b}$, i.e. the product of all bounded roots.
Now, since $h+k-b$ roots among all $h+k$ roots grow in absolute value,
and the product $\alpha_1 \dotsc \alpha_{h+k}$ equals $c_h,$
it follows that $|\alpha_1 \alpha_2 \dotsm \alpha_{b}| \rightarrow 0$ as
$m\rightarrow \infty,$
and this term converges to 0 quicker than any other product
$\alpha_{\sigma_1}\alpha_{\sigma_2} \dotsm \alpha_{\sigma_{b}}.$
Thus, $|e_{b}|$ should grow.
This contradicts the assumption $e_b({\boldsymbol\alpha}) = c_{h-b}.$
Notice that under our assumptions, the above cases cover all possible
ways for a sequence of roots to diverge.
Since both cases yield a contradiction, it follows that any sequence of roots of \eqref{eq:symbol}
satisfying \eqref{eqn:Cdef2}
must be bounded. Thus, $C_{\mathcal{A}}$ is compact.
\end{proof}
The following result is multivariate analog of a known fact in the case $n=0$, see \cite[Prop. 11.18 and 11.19]{BS}.
\begin{proposition}\label{cor:main}
In the notation of \eqref{eq:symbol}--\eqref{eqn:Cdef2},
for any $\mathbf{x}$ belonging to the boundary $\partial C_{\mathcal{A}}$ of $C_{{\mathcal{A}}}$,
at least one of the following three conditions is satisfied:
\begin{enumerate}
\item[(i)] the discriminant of $Q(t,\mathbf{x})$ with respect to $t$ vanishes, i.e. $Q(t,\mathbf{x})$ has (at least) a double root in $t$.
\item[(ii)] $|\alpha_{k-1}(\mathbf{x})|=|\alpha_{k}(\mathbf{x})|=|\alpha_{k+1}(\mathbf{x})|=\dots=|\alpha_{k+n+1}(\mathbf{x})|$.
\item[(iii)] $|\alpha_{k}(\mathbf{x})|=|\alpha_{k+1}(\mathbf{x})|=\dots=|\alpha_{k+n+1}(\mathbf{x})|=|\alpha_{k+n+2}(\mathbf{x})|$.
\end{enumerate}
\end{proposition}
\begin{proof}
We need the following two simple statements.
\begin{lemma}\label{lm:triv} Let $Pol_d$ be the set of all monic polynomials of degree $d$ with complex coefficients.
Let $\Sigma_{p,q}\subset Pol_d$ be the subset of polynomials
satisfying
\begin{equation}\label{eq:pq}
|\alpha_p|=|\alpha_{p+1}|=\cdots =|\alpha_{q}|,
\end{equation}
where $1\le p<q\le d$ and $\alpha_1,\alpha_2,\dots,\alpha_d$ being the roots of
polynomials ordered according to their increasing absolute values.
Then $\Sigma_{p,q}$ is a real semi-algebraic set of codimension $q-p$
whose boundary is the union of three pieces: $\Sigma_{p-1,q}$, $\Sigma_{p, q+1}$
and the intersection of $\Sigma_{p,q}$ with the standard discriminant in $Pol_d$,
i.e. the set of polynomials having multiple roots.
(Notice that if $p=1$ then $\Sigma_{p-1,q}$ is empty, and if $q=d$ then $\Sigma_{p,q+1}$ is empty by definition.)
\end{lemma}
\begin{proof}
$\Sigma_{p,q}$ is obtained as the image under the Vieta map of an obvious
semi-algebraic set $|\alpha_1|\le|\alpha_2|\le\dots\le|\alpha_p|=|\alpha_{p+1}|=\dots =|\alpha_q|\le|\alpha_{q+1}|\le \dots \le |\alpha_d|$.
Notice that the Vieta map is a local diffeomorphism outside the preimage of the standard discriminant.
Therefore the boundary of $\Sigma_{p,q}$ must either belong to the standard discriminant
or to one of $\Sigma_{p-1,q}$ or $\Sigma_{p,q+1}$.
The former is the common boundary between $\Sigma_{p,q}$ and $\Sigma_{p-1,q-1}$ and the
latter is the common boundary between $\Sigma_{p,q}$ and $\Sigma_{p+1,q+1}$.
\end{proof}
Given a closed Whitney stratified set $X$
(for example, semi-analytic) we say that $X$ is \emph{a $k$-dimensional stratified set without boundary} if
\begin{enumerate}
\item[(i)] the top-dimensional strata of $X$ have dimension $k$;
\item[(ii)] for any point $x$ lying in any stratum of dimension $k-1$,
one can choose orientation of the (germs of) $k$-dimensional
strata of a sufficiently small neighborhood of $x$ in $X$
so that $\partial X=0$.
\end{enumerate}
\begin{lemma}\label{lm:inter}
The boundary of the intersection of any closed
semi-algebraic set $\Gamma$ with any closed algebraic
set $\Theta$ is included in the intersection of the boundary $\partial \Gamma$ with $\Theta$.
\end{lemma}
\begin{proof}
Observe that any real algebraic variety $X$ of dimension $k$ is a stratifiable set without boundary.
Indeed, the fact we are proving is local,
and it suffices to prove it for generic $x$ on $(k-1)$-dimensional strata.
Consider an embedding of $X$ in
some high-dimensional linear space,
take the Whitney stratification with $x$ on its stratum $Y\subset B$
of dimension $k-1,$ and a transversal to $Y$ of codimension $k-1$ at $x.$
Therefore, we may now assume that the germ of $X$ near $x$
is topologically a product of a germ of algebraic curve
and a germ of a smooth manifold of dimension $k-1$.
Furthermore, a germ of any real algebraic curve $\Gamma$ can be always
oriented so that $\partial\Gamma=0$ which follows from the existence of
Puiseux series for an arbitrary branch of algebraic curve.
This argument shows that any point in the intersection $\Gamma\cap \Theta$
which does not belong to the boundary of $\Gamma$ can not lie on the boundary of
this intersection which settles Lemma~\ref{lm:inter}.
\end{proof}
Lemmas~\ref{lm:triv} and \ref{lm:inter} immediately imply Proposition~\ref{cor:main} since every $C_{{\mathcal{A}}}$ is the intersection of an appropriate $\Sigma_{p,q}$ with an appropriate affine subspace in $Pol_{k+h}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:main}]
In our notation, let $D^m_j(\mathbf{x})$ be the determinant of the $m \times m$-matrix $A_I$
with $I=\{j+1,j+2,\dotsc,j+m\}$ for $0\leq j \leq n.$ It is evident that
${\mathcal{E}}_{\mathcal{A}}^{(m)}$ is a subset of the set
$\widetilde{{\mathcal{E}}}_{\mathcal{A}}^{(m)}$ of solutions to the system of
polynomial equations
\begin{align}\label{eq:determinantsystem}
D^m_0(\mathbf{x})=D^m_1(\mathbf{x})=\dotso=D^m_n(\mathbf{x})=0.
\end{align}
We will show a stronger statement that, in notation of Theorem~\ref{th:main},
\[
\lim_{m\to\infty} \widetilde{{\mathcal{E}}}_{\mathcal{A}}^{(m)}\subseteq C_{\mathcal{A}}.
\]
Although each individual $ \widetilde{{\mathcal{E}}}_{\mathcal{A}}^{(m)}$ (considered
as a points set with multiplicities) is strictly bigger than
${{\mathcal{E}}}_{\mathcal{A}}^{(m)}$ the limits $B_{\mathcal{A}}=\lim_{m\to\infty}
{{\mathcal{E}}}_{\mathcal{A}}^{(m)}$ and $\lim_{m\to\infty}
\widetilde{{\mathcal{E}}}_{\mathcal{A}}^{(m)}$ seem to coincide as infinite sets.
The next proposition accomplishes the proof of Theorem \ref{th:main}.
\end{proof}
In Theorem 4 of \cite{Al} it was shown that
each sequence of determinants $\{D^m_j(\mathbf{x})\}_{m=1}^\infty$ as above satisfies a linear
recurrence relation with coefficients depending on $\mathbf{x}.$
The characteristic polynomial $\chi_j(t)$
of the $j$-th recurrence can be factorized as
\begin{align}\label{eq:chareq}
\chi_j(t,\mathbf{x})=\prod_\sigma (t- r_{j\sigma} ), \text{ where } r_{j\sigma} &= (-1)^{k+j}(\alpha_{\sigma_1}
\cdots \alpha_{\sigma_{k+j}})^{-1}, \sigma \in \binom{[k+h]}{k+j}.
\end{align}
\begin{proposition}\label{pr:main}
Suppose that $\{\mathbf{x}_m\}_1^\infty$, is a sequence of
points in $\mathbb{C}^{n+1}$ satisfying the system of equations:
\begin{align}\label{eq:system}
D^m_j(\mathbf{x}_m) &=0 \text{ for } j = 0,1,\dotsc,n \text{ and } m=1,2,\dotsc
\end{align}
and such that the limit $\lim_{m \rightarrow \infty} \mathbf{x}_m =:\mathbf{x}^*$ exists.
Then for all $j=0,\dotsc,n$
$ |\alpha_{k+j}(\mathbf{x}^*)|=|\alpha_{k+j+1}(\mathbf{x}^*)|$
when the $\alpha_i$ are indexed with increasing order of their modulus.
\end{proposition}
\begin{proof}
Provided that all the roots of $\chi_j(t,\mathbf{x})$ are distinct,
by using a version of Widom's formula, (see \cite{Al,BS}) we have
\begin{equation}\label{eq:widomform}
D^m_j(\mathbf{x}) = \sum_\sigma \prod_{l \in \sigma, i\notin \sigma} \left( 1 - \frac{\alpha_l(\mathbf{x})}{\alpha_i(\mathbf{x})} \right)^{-1}
\cdot r_{j\sigma}(\mathbf{x})^m.
\end{equation}
We may assume that for $\mathbf{x}^*$ and fixed $j$, the $r_{j\sigma}(\mathbf{x}^*)$ are
ordered decreasingly with respect to their modulus (for some ordering $\sigma=1,2,\dotsc$).
The goal is to prove that $|r_{j1}(\mathbf{x}^*)|=|r_{j2}(\mathbf{x}^*)|$
since this implies $|\alpha_{k+j}(\mathbf{x}^*)|=|\alpha_{k+j+1}(\mathbf{x}^*)|.$
We show this fact by contradiction.
\medskip
Assume that $|r_{j1}(\mathbf{x}^*)|>|r_{j2}(\mathbf{x}^*)|\geq \dotso \geq |r_{jb}(\mathbf{x}^*)|,$
i.e. that the largest root is simple and has modulus
strictly larger than any other root of the characteristic equation \eqref{eq:chareq}.
By examining \eqref{eq:widomform},
it is evident that $r_{j1}(\mathbf{x}_m)^m$ is the dominating term for sufficiently large $m,$
that is, $D^m_j(\mathbf{x}_m)/r_{j1}(\mathbf{x}_m)^m \rightarrow L \neq 0$ as $m \rightarrow \infty.$
By standard properties of linear recurrences, this holds
even when there are multiple zeros among the smaller roots;
remember that our assumption was that $r_{j1}(x_m)$ is a simple zero of \eqref{eq:chareq}
when $m$ is large enough.
Hence, for sufficiently large $m$, $D^m_j(\mathbf{x}_m) \approx L r_{j1}(x_m)^m,$
which is non-zero for sufficiently large $m$.
This contradicts the condition that $\mathbf{x}_m$ satisfies \eqref{eq:system}.
Consequently, $|r_{j1}(\mathbf{x}^*)|= |r_{j2}(\mathbf{x}^*)|$ for $j=0,1,\dotsc,n$
and this implies Proposition \ref{pr:main}.
\end{proof}
Proposition~\ref{pr:main} implies that $\mathbf{x}$ lies in $B_{\mathcal{A}}$ only if $\mathbf{x}$ is a
limit of solutions to \eqref{eq:system},
but such limit $\mathbf{x}$ must satisfy that $|\alpha_{k}(\mathbf{x})|=|\alpha_{k+1}(\mathbf{x})|=\dotso =
|\alpha_{k+n+1}(\mathbf{x})|.$ Therefore, $B_{\mathcal{A}} \subseteq C_{\mathcal{A}}.$
\section{Further directions}
\noindent
\textbf{1.} It seems relatively easy to describe the stratified structure
of $C_{\mathcal{A}}$ at least in case of generic ${\mathcal{A}}$.
In particular, in the Chebyshev case of Example \ref{ex:A} the set $C_{\mathcal{A}}$
has the same stratification as a simplex of corresponding dimension.
One can also understand the stratified structure of the sets $\Sigma_{p,q}$
introduced in Lemma~\ref{lm:triv}. Since each $C_{\mathcal{A}}$ is obtained from a
corresponding $\Sigma_{p,q}$ by intersecting it with an affine
subspace the stratified structure of the former for generic ${\mathcal{A}}$ is
also describable.
On the other hand, our Example~\ref{ex:B} seems to show more
complicated stratified structure due to the presence of additional symmetry.
\medskip
\noindent
\textbf{2.} We say that an (infinite) complex-valued matrix ${\mathcal{A}}$ has a {\it
weak univariate orthogonality property} if the sequence of characteristic
polynomials of its principal minors obeys the standard $3$-term recurrence
relation with complex coefficients. There is a straightforward version of this notion
for finite square matrices. Obviously, any Jacobi matrix has this property. However,
it seems that for any $m\ge 3$ the set $WO_m\subset Mat(m,m)$ of all $m\times
m$-matrices with the latter property has a bigger dimension than the set $Jac_m
\subset Mat(m,m)$ of all Jacobi $m\times m$-matrices.
\begin{problem}
Find the dimension of $WO_m$?
\end{problem}
\medskip
\noindent
\textbf{3.} Analogously, given a non-negative integer $n$, we say that an
(infinite) complex-valued matrix ${\mathcal{A}}$ has a {\it weak $n$-variate
orthogonality property} if the above family
$\{P^I_{\mathcal{A}}(x_0,x_1,\dotsc,x_n)\}$ (see Definition~\ref{deff}) satisfies the
3-term recurrence relation \rm{(2.2)} of Theorem~2.1 of \cite{Xu} with complex
coefficients.
There are many similarities between families
$\{P^I_{\mathcal{A}}(x_0,x_1,\dotsc,x_n)\}$ and families of multivariate orthogonal
polynomials which by one of the standard definitions of such polynomials also
satisfy \rm{(2.2)} of Theorem~2.1 of \cite{Xu} with real coefficients.
Our computer experiments show that in this aspect the case $n>0$ is quite
different from the classical case $n=0$. In particular, we believe that the
following conjecture holds.
\begin{conjecture} Given $n>0$, a banded matrix ${\mathcal{A}}$
has a \textit{weak $n$-variate orthogonality property}
if it is of the form
\[
{\mathcal{A}}=
\begin{pmatrix}
a_0 & a_1 & a_2 & \dots & a_{n+1} & 0 & 0 & 0 & \dots \\
d_{-1} & d_0 & d_1 & \dots & d_n & d_{n+1} & 0 & 0 & \dots \\
0 & d_{-1} & d_0 & \dots & d_{n-1} & d_{n} & d_{n+1} & 0 & \dots \\
0 & 0 & d_{-1} & \dots & d_{n-2} & d_{n-1} & d_{n} & d_{n+1} & \dots \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots \\
\end{pmatrix},
\]
where $a_0,\dotsc,a_{n+1},d_{-1},\dotsc,d_{n+1} \in \mathbb{C}.$
\end{conjecture}
|
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{"url":"https:\/\/www.tutorialspoint.com\/how-to-plot-a-single-line-in-matplotlib-that-continuously-changes-color","text":"# How to plot a single line in Matplotlib that continuously changes color?\n\nMatplotlibPythonData Visualization\n\nTo plot a single line that continuously changes color, we can take the following steps\u2212\n\n\u2022 Set the figure size and adjust the padding between and around the subplots.\n\u2022 Create random x and y data points using numpy.\n\u2022 Create a figure and a set of subplots.\n\u2022 Iterate the index in the range of 1 to 100.\n\u2022 Plot x and y data points with random color in a loop.\n\u2022 To display the figure, use show() method.\n\n## Example\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport random\n\nplt.rcParams[\"figure.figsize\"] = [7.50, 3.50]\nplt.rcParams[\"figure.autolayout\"] = True\n\nx = np.linspace(1, 10, 100)\ny = np.sin(x)\nfig, ax = plt.subplots()\n\nfor i in range(0, 100, 5):\nr = random.random()\nb = random.random()\ng = random.random()\ncolor = (r, g, b)\nax.plot(x[i:i+5+1], y[i:i+5+1], c=color, lw=7)\n\nplt.show()\n\n## Output\n\nPublished on 05-Jun-2021 08:15:43","date":"2022-05-22 18:11:04","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\": 0.1940123587846756, \"perplexity\": 4219.0352811480425}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662545875.39\/warc\/CC-MAIN-20220522160113-20220522190113-00540.warc.gz\"}"}
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{"url":"http:\/\/nag.com\/numeric\/MB\/manual64_24_1\/html\/E04\/e04ujf.html","text":"Integer type:\u00a0\u00a0int32\u00a0\u00a0int64\u00a0\u00a0nag_int\u00a0\u00a0show int32\u00a0\u00a0show int32\u00a0\u00a0show int64\u00a0\u00a0show int64\u00a0\u00a0show nag_int\u00a0\u00a0show nag_int\n\nChapter Contents\nChapter Introduction\nNAG Toolbox\n\n# NAG Toolbox: nag_opt_nlp1_sparse_option_string (e04uj)\n\n## Purpose\n\nTo supply individual optional parameters to nag_opt_nlp1_sparse_solve (e04ug).\n\n## Syntax\n\n[lwsav, iwsav, rwsav, inform] = e04uj(str, lwsav, iwsav, rwsav)\n[lwsav, iwsav, rwsav, inform] = nag_opt_nlp1_sparse_option_string(str, lwsav, iwsav, rwsav)\n\n## Description\n\nnag_opt_nlp1_sparse_option_string (e04uj) may be used to supply values for optional parameters to nag_opt_nlp1_sparse_solve (e04ug). It is only necessary to call nag_opt_nlp1_sparse_option_string (e04uj) for those parameters whose values are to be different from their default values. One call to nag_opt_nlp1_sparse_option_string (e04uj) sets one parameter value.\nEach optional parameter is defined by a single character string, of up to 72$72$\u00a0characters, consisting of one or more items. The items associated with a given option must be separated by spaces, or equals signs [ = ]$\\left[=\\right]$. Alphabetic characters may be upper or lower case. The string\n`Print Level = 1`\nis an example of a string used to set an optional parameter. For each option the string contains one or more of the following items:\n \u2013 a mandatory keyword; \u2013 a phrase that qualifies the keyword; \u2013 a number that specifies an integer or double value. Such numbers may be up to 16$16$\u00a0contiguous characters in Fortran's I, F, E or D formats, terminated by a space if this is not the last item on the line.\nBlank strings and comments are ignored. A comment begins with an asterisk (*) and all subsequent characters in the string are regarded as part of the comment.\nFor nag_opt_nlp1_sparse_option_string (e04uj), each user-specified option is normally printed as it is defined, on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)), but this printing may be suppressed using the keyword Nolist. Thus the statement\n```\n[lwsav, iwsav, rwsav, inform] = e04uj('Nolist', lwsav, iwsav, rwsav);\n```\nsuppresses printing of this and subsequent options. Printing will automatically be turned on again after a call to nag_opt_nlp1_sparse_solve (e04ug) and may be turned on again at any time using the keyword List.\nFor nag_opt_nlp1_sparse_option_string (e04uj) printing is turned off by default, but may be turned on at any time using the keyword List.\nOptional parameter settings are preserved following a call to nag_opt_nlp1_sparse_solve (e04ug) and so the keyword Defaults is provided to allow you to reset all the optional parameters to their default values before a subsequent call to nag_opt_nlp1_sparse_solve (e04ug).\nA complete list of optional parameters, their abbreviations, synonyms and default values is given in Section [Optional Parameters] in (e04ug).\n\nNone.\n\n## Parameters\n\n### Compulsory Input Parameters\n\n1: \u00a0\u00a0\u2002 str \u2013 string\nA single valid option string (as described in Section [Description] and in Section [Optional Parameters] in (e04ug)).\n2: \u00a0\u00a0\u2002 lwsav(20$20$) \u2013 logical array\n3: \u00a0\u00a0\u2002 iwsav(550$550$) \u2013 int64int32nag_int array\n4: \u00a0\u00a0\u2002 rwsav(550$550$) \u2013 double array\nThe arrays lwsav, iwsav and rwsav must not be altered between calls to any of the functions nag_opt_nlp1_sparse_option_string (e04uj), (e04ug), (e04uh) or nag_opt_init (e04wb).\n\nNone.\n\nNone.\n\n### Output Parameters\n\n1: \u00a0\u00a0\u2002 lwsav(20$20$) \u2013 logical array\n2: \u00a0\u00a0\u2002 iwsav(550$550$) \u2013 int64int32nag_int array\n3: \u00a0\u00a0\u2002 rwsav(550$550$) \u2013 double array\n4: \u00a0\u00a0\u2002 inform \u2013 int64int32nag_int scalar\nContains zero if a valid option string has been supplied and a value > 0$\\text{value}>0$ otherwise (see Section [Error Indicators and Warnings]).\n\n## Error Indicators and Warnings\n\nINFORM = 5${\\mathbf{INFORM}}=5$\nThe supplied option is invalid. Check that the keywords are neither ambiguous nor misspelt.\n\nNot applicable.\n\nNone.\n\n## Example\n\n```function nag_opt_nlp1_sparse_option_string_example\nn = int64(4);\nm = int64(6);\nncnln = int64(3);\nnonln = int64(4);\nnjnln = int64(2);\niobj = int64(6);\na = [1e25;1e25;1e25;1;-1;1e25;1e25;1e25;-1;1;3;-1;-1;2];\nha = [int64(1);2;3;5;4;1;2;3;5;4;6;1;2;6];\nka = [int64(1);6;11;13;15];\nbl = [-0.55;-0.55;0;0;-894.8;-894.8;-1294.8;-0.55;-0.55;-1e25];\nbu = [0.55;0.55;1200;1200;-894.8;-894.8;-1294.8;1e25;1e25;1e25];\nstart = 'C';\nnames = {'Varble 1'; 'Varble 2'; 'Varble 3'; 'Varble 4'; 'NlnCon 1'; ...\n'NlnCon 2'; 'NlnCon 3'; 'LinCon 1'; 'LinCon 2'; 'Free Row'};\nns = int64(0);\nxs = [0;0;0;0;0;0;0;0;0;0];\nistate = zeros(10, 1, 'int64');\nclamda = [0;0;0;0;0;0;0;0;0;0];\nleniz = int64(1000);\nlenz = int64(1000);\n[cwsav,lwsav,iwsav,rwsav,ifail] = nag_opt_init('nag_opt_nlp1_sparse_solve');\n[lwsav,iwsav,rwsav,ifail] = ...\nnag_opt_nlp1_sparse_option_string('Major Print level = 1', ...\nlwsav, iwsav, rwsav);\n[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, ...\nobj, user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...\nnag_opt_nlp1_sparse_solve(@confun, @objfun, n, m, ncnln, nonln, njnln, ...\niobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, clamda, ...\nlwsav, iwsav, rwsav, 'lenz', lenz, 'leniz', leniz);\naOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz\nninf, sinf, obj, ifail\n\nfunction [mode, f, fjac, user] = ...\nconfun(mode, ncnln, njnln, nnzjac, x, fjac, nstate, user)\nf = zeros(ncnln, 1);\n\nif (mode == 0 || mode == 2)\nf(1) = 1000*sin(-x(1)-0.25) + 1000*sin(-x(2)-0.25);\nf(2) = 1000*sin(x(1)-0.25) + 1000*sin(x(1)-x(2)-0.25);\nf(3) = 1000*sin(x(2)-x(1)-0.25) + 1000*sin(x(2)-0.25);\nend\n\nif (mode == 1 || mode == 2)\n% nonlinear jacobian elements for column 1.\nfjac(1) = -1000*cos(-x(1)-0.25);\nfjac(2) = 1000*cos(x(1)-0.25) + 1000*cos(x(1)-x(2)-0.25);\nfjac(3) = -1000*cos(x(2)-x(1)-0.25);\n% nonlinear jacobian elements for column 2.\nfjac(4) = -1000*cos(-x(2)-0.25);\nfjac(5) = -1000*cos(x(1)-x(2)-0.25);\nfjac(6) = 1000*cos(x(2)-x(1)-0.25) + 1000*cos(x(2) -0.25);\nend\n\nfunction [mode, objf, objgrd, user] = ...\nobjfun(mode, nonln, x, objgrd, nstate, user)\n\nif (mode == 0 || mode == 2)\nobjf = 1.0e-6*x(3)^3 + 2.0e-6*x(4)^3\/3;\nend\n\nif (mode == 1 || mode == 2)\nobjgrd(1) = 0;\nobjgrd(2) = 0;\nobjgrd(3) = 3.0e-6*x(3)^2;\nobjgrd(4) = 2.0e-6*x(4)^2;\nend\n```\n```\n\n*** E04UGA\n\nParameters\n----------\n\nFrequencies.\nCheck frequency......... 60 Expand frequency....... 10000\nFactorization frequency. 50\n\nQP subproblems.\nScale tolerance......... 9.00E-01 Minor feasibility tol.. 1.05E-08\nScale option............ 1 Minor optimality tol... 1.05E-08\nPartial price........... 1 Crash tolerance........ 1.00E-01\nPivot tolerance......... 2.04E-11 Minor print level...... 0\nCrash option............ 0 Elastic weight......... 1.00E+02\n\nThe SQP method.\nMinimize................\nNonlinear objective vars 4 Major optimality tol... 1.05E-08\nFunction precision...... 1.72E-13 Unbounded step size.... 1.00E+20\nSuperbasics limit....... 4 Forward difference int. 4.15E-07\nUnbounded objective..... 1.00E+15 Central difference int. 5.56E-05\nMajor step limit........ 2.00E+00 Derivative linesearch..\nDerivative level........ 3 Major iteration limit.. 1000\nLinesearch tolerance.... 9.00E-01 Verify level........... 0\nMinor iteration limit... 500 Major print level...... 1\nInfinite bound size..... 1.00E+20 Iteration limit........ 10000\n\nHessian approximation.\nHessian full memory..... Hessian updates........ 99999999\nHessian frequency....... 99999999\n\nNonlinear constraints.\nNonlinear constraints... 3 Major feasibility tol.. 1.05E-08\nNonlinear Jacobian vars. 2 Violation limit........ 1.00E+01\n\nMiscellaneous.\nVariables............... 4 Linear constraints..... 3\nNonlinear variables..... 4 Linear variables....... 0\nLU factor tolerance..... 5.00E+00 LU singularity tol..... 2.04E-11\nLU update tolerance..... 5.00E+00 LU density tolerance... 6.00E-01\neps (machine precision). 1.11E-16 Monitoring file........ -1\nCOLD start.............. Infeasible exit........\n\nWorkspace provided is IZ( 1000), Z( 1000).\nTo start solving the problem we need IZ( 628), Z( 758).\n\nconfun sets 6 out of 6 constraint gradients.\nobjfun sets 4 out of 4 objective gradients.\n\nCheap test on confun...\n\nThe Jacobian seems to be OK.\n\nThe largest discrepancy was 4.41E-08 in constraint 2.\n\nCheap test on objfun...\n\nThe objective gradients seem to be OK.\nGradient projected in two directions 0.00000000000E+00 0.00000000000E+00\nDifference approximations 1.74111992322E-19 4.48742248252E-21\n\nVariable State Value Lower Bound Upper Bound Lagr Mult Residual\n\nVarble 1 BS 0.118876 -0.55000 0.55000 -1.2529E-07 0.4311\nVarble 2 BS -0.396234 -0.55000 0.55000 1.9243E-08 0.1538\nVarble 3 BS 679.945 . 1200.0 1.7001E-10 520.1\nVarble 4 SBS 1026.07 . 1200.0 -2.1918E-10 173.9\n\nConstrnt State Value Lower Bound Upper Bound Lagr Mult Residual\n\nNlnCon 1 EQ -894.800 -894.80 -894.80 -4.387 3.3643E-09\nNlnCon 2 EQ -894.800 -894.80 -894.80 -4.106 6.0049E-10\nNlnCon 3 EQ -1294.80 -1294.8 -1294.8 -5.463 3.3549E-09\nLinCon 1 BS -0.515110 -0.55000 None . 3.4890E-02\nLinCon 2 BS 0.515110 -0.55000 None . 1.065\nFree Row BS 4091.97 None None -1.000 4092.\n\nExit E04UGA - Optimal solution found.\n\nFinal objective value = 5126.498\n\naOut =\n\n1.0e+03 *\n\n-0.9327\n1.9565\n-0.7213\n0.0010\n-0.0010\n-0.9893\n-0.9651\n1.5197\n-0.0010\n0.0010\n0.0030\n-0.0010\n-0.0010\n0.0020\n\nnsOut =\n\n1\n\nxsOut =\n\n1.0e+03 *\n\n0.0001\n-0.0004\n0.6799\n1.0261\n-0.8948\n-0.8948\n-1.2948\n-0.0005\n0.0005\n4.0920\n\nistateOut =\n\n3\n3\n3\n2\n0\n0\n0\n3\n3\n3\n\nclamdaOut =\n\n-0.0000\n0.0000\n0.0000\n-0.0000\n-4.3870\n-4.1056\n-5.4633\n0\n0\n-1.0000\n\nminiz =\n\n628\n\nminz =\n\n758\n\nninf =\n\n0\n\nsinf =\n\n0\n\nobj =\n\n5.1265e+03\n\nifail =\n\n0\n\n```\n```function e04uj_example\nn = int64(4);\nm = int64(6);\nncnln = int64(3);\nnonln = int64(4);\nnjnln = int64(2);\niobj = int64(6);\na = [1e25;\n1e25;\n1e25;\n1;\n-1;\n1e25;\n1e25;\n1e25;\n-1;\n1;\n3;\n-1;\n-1;\n2];\nha = [int64(1);2;3;5;4;1;2;3;5;4;6;1;2;6];\nka = [int64(1);6;11;13;15];\nbl = [-0.55;-0.55;0;0;-894.8;-894.8;-1294.8;-0.55;-0.55;-1e25];\nbu = [0.55;0.55;1200;1200;-894.8;-894.8;-1294.8;1e25;1e25;1e25];\nstart = 'C';\nnames = {'Varble 1'; 'Varble 2'; 'Varble 3'; 'Varble 4'; 'NlnCon 1'; 'NlnCon 2'; ...\n'NlnCon 3'; 'LinCon 1'; 'LinCon 2'; 'Free Row'};\nns = int64(0);\nxs = [0;0;0;0;0;0;0;0;0;0];\nistate = zeros(10, 1, 'int64');\nclamda = [0;0;0;0;0;0;0;0;0;0];\nleniz = int64(1000);\nlenz = int64(1000);\n[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04ug');\n[lwsav,iwsav,rwsav,ifail] = e04uj('Major Print level = 1', lwsav, iwsav, rwsav);\n[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, ...\nobj, user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...\ne04ug(@confun, @objfun, n, m, ncnln, nonln, njnln, ...\niobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, clamda, ...\nlwsav, iwsav, rwsav, 'lenz', lenz, 'leniz', leniz);\naOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz\nninf, sinf, obj, ifail\n\nfunction [mode, f, fjac, user] = ...\nconfun(mode, ncnln, njnln, nnzjac, x, fjac, nstate, user)\nf = zeros(ncnln, 1);\n\nif (mode == 0 || mode == 2)\nf(1) = 1000*sin(-x(1)-0.25) + 1000*sin(-x(2)-0.25);\nf(2) = 1000*sin(x(1)-0.25) + 1000*sin(x(1)-x(2)-0.25);\nf(3) = 1000*sin(x(2)-x(1)-0.25) + 1000*sin(x(2)-0.25);\nend\n\nif (mode == 1 || mode == 2)\n% nonlinear jacobian elements for column 1.\nfjac(1) = -1000*cos(-x(1)-0.25);\nfjac(2) = 1000*cos(x(1)-0.25) + 1000*cos(x(1)-x(2)-0.25);\nfjac(3) = -1000*cos(x(2)-x(1)-0.25);\n% nonlinear jacobian elements for column 2.\nfjac(4) = -1000*cos(-x(2)-0.25);\nfjac(5) = -1000*cos(x(1)-x(2)-0.25);\nfjac(6) = 1000*cos(x(2)-x(1)-0.25) + 1000*cos(x(2) -0.25);\nend\n\nfunction [mode, objf, objgrd, user] = objfun(mode, nonln, x, objgrd, nstate, user)\n\nif (mode == 0 || mode == 2)\nobjf = 1.0e-6*x(3)^3 + 2.0e-6*x(4)^3\/3;\nend\n\nif (mode == 1 || mode == 2)\nobjgrd(1) = 0;\nobjgrd(2) = 0;\nobjgrd(3) = 3.0e-6*x(3)^2;\nobjgrd(4) = 2.0e-6*x(4)^2;\nend\n```\n```\n\n*** E04UGA\n\nParameters\n----------\n\nFrequencies.\nCheck frequency......... 60 Expand frequency....... 10000\nFactorization frequency. 50\n\nQP subproblems.\nScale tolerance......... 9.00E-01 Minor feasibility tol.. 1.05E-08\nScale option............ 1 Minor optimality tol... 1.05E-08\nPartial price........... 1 Crash tolerance........ 1.00E-01\nPivot tolerance......... 2.04E-11 Minor print level...... 0\nCrash option............ 0 Elastic weight......... 1.00E+02\n\nThe SQP method.\nMinimize................\nNonlinear objective vars 4 Major optimality tol... 1.05E-08\nFunction precision...... 1.72E-13 Unbounded step size.... 1.00E+20\nSuperbasics limit....... 4 Forward difference int. 4.15E-07\nUnbounded objective..... 1.00E+15 Central difference int. 5.56E-05\nMajor step limit........ 2.00E+00 Derivative linesearch..\nDerivative level........ 3 Major iteration limit.. 1000\nLinesearch tolerance.... 9.00E-01 Verify level........... 0\nMinor iteration limit... 500 Major print level...... 1\nInfinite bound size..... 1.00E+20 Iteration limit........ 10000\n\nHessian approximation.\nHessian full memory..... Hessian updates........ 99999999\nHessian frequency....... 99999999\n\nNonlinear constraints.\nNonlinear constraints... 3 Major feasibility tol.. 1.05E-08\nNonlinear Jacobian vars. 2 Violation limit........ 1.00E+01\n\nMiscellaneous.\nVariables............... 4 Linear constraints..... 3\nNonlinear variables..... 4 Linear variables....... 0\nLU factor tolerance..... 5.00E+00 LU singularity tol..... 2.04E-11\nLU update tolerance..... 5.00E+00 LU density tolerance... 6.00E-01\neps (machine precision). 1.11E-16 Monitoring file........ -1\nCOLD start.............. Infeasible exit........\n\nWorkspace provided is IZ( 1000), Z( 1000).\nTo start solving the problem we need IZ( 628), Z( 758).\n\nconfun sets 6 out of 6 constraint gradients.\nobjfun sets 4 out of 4 objective gradients.\n\nCheap test on confun...\n\nThe Jacobian seems to be OK.\n\nThe largest discrepancy was 4.41E-08 in constraint 2.\n\nCheap test on objfun...\n\nThe objective gradients seem to be OK.\nGradient projected in two directions 0.00000000000E+00 0.00000000000E+00\nDifference approximations 1.74111992322E-19 4.48742248252E-21\n\nVariable State Value Lower Bound Upper Bound Lagr Mult Residual\n\nVarble 1 BS 0.118876 -0.55000 0.55000 -1.2529E-07 0.4311\nVarble 2 BS -0.396234 -0.55000 0.55000 1.9243E-08 0.1538\nVarble 3 BS 679.945 . 1200.0 1.7001E-10 520.1\nVarble 4 SBS 1026.07 . 1200.0 -2.1918E-10 173.9\n\nConstrnt State Value Lower Bound Upper Bound Lagr Mult Residual\n\nNlnCon 1 EQ -894.800 -894.80 -894.80 -4.387 3.3643E-09\nNlnCon 2 EQ -894.800 -894.80 -894.80 -4.106 6.0049E-10\nNlnCon 3 EQ -1294.80 -1294.8 -1294.8 -5.463 3.3549E-09\nLinCon 1 BS -0.515110 -0.55000 None . 3.4890E-02\nLinCon 2 BS 0.515110 -0.55000 None . 1.065\nFree Row BS 4091.97 None None -1.000 4092.\n\nExit E04UGA - Optimal solution found.\n\nFinal objective value = 5126.498\n\naOut =\n\n1.0e+03 *\n\n-0.9327\n1.9565\n-0.7213\n0.0010\n-0.0010\n-0.9893\n-0.9651\n1.5197\n-0.0010\n0.0010\n0.0030\n-0.0010\n-0.0010\n0.0020\n\nnsOut =\n\n1\n\nxsOut =\n\n1.0e+03 *\n\n0.0001\n-0.0004\n0.6799\n1.0261\n-0.8948\n-0.8948\n-1.2948\n-0.0005\n0.0005\n4.0920\n\nistateOut =\n\n3\n3\n3\n2\n0\n0\n0\n3\n3\n3\n\nclamdaOut =\n\n-0.0000\n0.0000\n0.0000\n-0.0000\n-4.3870\n-4.1056\n-5.4633\n0\n0\n-1.0000\n\nminiz =\n\n628\n\nminz =\n\n758\n\nninf =\n\n0\n\nsinf =\n\n0\n\nobj =\n\n5.1265e+03\n\nifail =\n\n0\n\n```","date":"2015-04-26 10:27:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 11, \"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\": 0.8956988453865051, \"perplexity\": 7557.724049911127}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246654264.98\/warc\/CC-MAIN-20150417045734-00263-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/doc.cfd.direct\/notes\/cfd-general-principles\/under-relaxation","text":"## 5.6Under-relaxation\n\nSec.\u00a05.5 concludes that the matrix of a typical transport equation is not guaranteed to be diagonally dominant. Some action may therefore be required to ensure a convergent solution.\n\nUnder-relaxation is a general method used to improve solution convergence by limiting the amount a variable changes during a solution step.\n\nDuring a solution step, assume a single value of a \ufb01eld in one cell changes from its current value to the new value . Under-relaxation would limit the change by a fraction , , so that the value taken from that solution step is\n\n (5.13)\nIn some situations, Eq.\u00a0(5.13 ) is applied after a solution step. This simple approach is known as \ufb01eld under-relaxation, which has one notable disadvantage that it requires additional storage of the intermediate \ufb01eld in computer memory.\n\nWhen a solution step involves solving a matrix equation, the new values come from an iterative method like Gauss-Seidel. Combining the under-relaxation of Eq.\u00a0(5.13 ) with the Gauss-Seidel calculation of Eq.\u00a0(5.4 ) gives:\n\n (5.14)\nRearranging Eq.\u00a0(5.14 ) gives the following relation:\n (5.15)\nEquation\u00a05.15 is simply the matrix equation modi\ufb01ed by:\n\u2022 increasing the diagonal coe\ufb03cients by division by ;\n\u2022 multiplying the di\ufb00erence between the new and original coe\ufb03cients by the current and adding it to the source .\n\nModifying the matrix equation this way, known as equation under-relaxation, provides an alternative to Eq.\u00a0(5.13 ) for under-relaxing a solution of , without the temporary storage of .\n\n### Ensuring diagonal dominance\n\nThe modi\ufb01cation to expressed by Eq.\u00a0(5.15 ) inspires a strategy to ensure diagonal dominance of the matrix as follows.\n\nEach diagonal coe\ufb03cient which does not satisfy Eq.\u00a0(5.9 ) is increased until it is diagonally equal. The change to the coe\ufb03cient is multiplied by the current and added to .\n\nThis approach to ensure diagonal dominance is e\ufb00ective since it only modi\ufb01es matrix coe\ufb03cients where necessary. Otherwise, if the discretisation schemes, and and are favourable, then no changes to the matrix are necessary.\n\nNotes on CFD: General Principles - 5.6 Under-relaxation","date":"2022-08-17 12:49:12","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8276270031929016, \"perplexity\": 1351.4233976001178}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572908.71\/warc\/CC-MAIN-20220817122626-20220817152626-00640.warc.gz\"}"}
| null | null |
\section{Introduction to Theta Functions of Curves}
Let $\mathcal X}%\def\X{\mathfrak X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\mathbb C.$ We denote
the moduli space of genus $g$ by ${\mathcal M}_g$ and the hyperelliptic locus in ${\mathcal M}_g$ by $\mathcal H_g.$ It is well known that dim
${\mathcal M}_g = 3g-3$ and $\mathcal H_g$ is a $(2g-1)$ dimensional subvariety of ${\mathcal M}_g.$ Choose a symplectic homology basis for $\mathcal X}%\def\X{\mathfrak X$,
say
\[ \{ A_1, \dots, A_g, B_1, \dots , B_g\}\]
such that the intersection products $A_i \cdot A_j = B_i \cdot B_j =0$ and $A_i \cdot B_j= {\delta }_{i j}$.
We choose a basis $\{ w_i\}$ for the space of holomorphic 1-forms such that $\int_{A_i} w_j = {\delta }_{i j},$ where ${\delta }_{i
j}$ is the Kronecker delta. The matrix $\Omega= \left[ \int_{B_i} w_j \right] $ is the \textbf{period matrix} of $\mathcal X}%\def\X{\mathfrak X$ .
The columns of the matrix $\left[ I \ | \Omega \right]$ form a lattice $L$ in $\mathbb C^g$ and the Jacobian of $\mathcal X}%\def\X{\mathfrak X$ is $\mbox{Jac }(\mathcal X}%\def\X{\mathfrak X)
= \mathbb C^g/ L$. Let $$\mathfrak H_g =\{\tau : \tau \,\, \textit{is symmetric}\,\, g \times g \, \textit{matrix with positive definite
imaginary part} \}$$ be the \textbf{Siegel upper-half space}. Then $\Omega \in \mathfrak H_g$. The group of all $2g \times 2g$
matrices $M \in GL_{2g}(\mathbb Z)$ satisfying
\[M^t J M = J \,\,\,\,\,\,\,\, \textit{with} \, \,\,\,\,\,\, J = \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix} \]
is called the \textbf{symplectic group} and denoted by $Sp_{2g}(\mathbb Z)$.
Let $M = \begin{pmatrix} R & S \\ T & U
\end{pmatrix} \in Sp_{2g}(\mathbb Z) $ and $\tau \in \mathfrak H_g$ where $R,$ $S,$ $T$ and $U$ are $g \times g$ matrices. $Sp_{2g}(\mathbb Z) $ acts transitively on $\mathfrak H_g$ as
\[ M(\tau) = (R \tau + S)(T \tau + U)^{-1}. \]
\noindent Here, the multiplications are matrix multiplications. There is an injection
\[ {\mathcal M}_g \hookrightarrow \mathfrak H_g/ Sp_{2g}(\mathbb Z) =: \mathcal A_g \] where each curve $C$ (up to isomorphism) goes to its Jacobian in $\mathcal A_g.$
If $\ell$ is a positive integer, the principal congruence group of degree $g$ and of level $\ell$ is defined as a
subgroup of $Sp_{2g}(\mathbb Z)$ by the condition $M \equiv I_{2g} \mod \ell.$ We shall denote this group by
$Sp_{2g}(\mathbb Z)(\ell).$
For any $z \in \mathbb C^g$ and $\tau \in \mathfrak H_g$ the \textbf{Riemann's theta function} is defined as
\[ \theta (z , \tau) = \sum_{u\in \mathbb Z^g} e^{\pi i ( u^t \tau u + 2 u^t z ) }\]
where $u$ and $z$ are $g$-dimensional column vectors and the products involved in the formula are matrix products. The
fact that the imaginary part of $\tau$ is positive makes the series absolutely convergent over every compact subset of
$\mathbb C^g \times \mathfrak H_g$.
The theta function is holomorphic on $\mathbb C^g\times \mathfrak H_g$ and has quasi periodic properties,
$$\theta(z+u,\tau)=\theta(z,\tau)\quad \textit{and}\quad
\theta(z+u\tau,\tau)=e^{-\pi i( u^t \tau u+2z^t u )}\cdot \theta(z,\tau)$$
where $u\in \mathbb Z^g$; see \cite{Mu1} for details.
The locus $ \Theta: = \{ z \in \mathbb C^g/L : \theta(z, \Omega)=0 \}$ is called the \textbf{theta divisor} of $\mathcal X}%\def\X{\mathfrak X$.
Any point $e \in \mbox{Jac } (\mathcal X}%\def\X{\mathfrak X)$ can be uniquely written as $e = (b,a) \begin{pmatrix} 1_g \\ \Omega \end{pmatrix}$ where $a,b
\in \mathbb R^g$ are the characteristics of $e.$
We shall use the notation $[e]$ for the characteristic of $e$ where $[e] = \ch{a}{b}.$ For any $a, b \in \mathbb Q^g$, the
theta function with rational characteristics is defined as a translate of Riemann's theta function multiplied by an
exponential factor
\begin{equation} \label{ThetaFunctionWithCharac} \theta \ch{a}{b} (z , \tau) = e^{\pi i( a^t \tau a + 2 a^t(z+b))} \theta(z+\tau a+b ,\tau).\end{equation}
\noindent By writing out Eq.~\eqref{ThetaFunctionWithCharac}, we have
\[ \theta \ch{a}{b} (z , \tau) = \sum_{u\in \mathbb Z^g} e^{\pi i ( (u+a)^t \tau (u+a) + 2 (u+a)^t (z+b) ) }. \]
The Riemann's theta function is $\theta \ch{0}{0}.$ The theta function with rational characteristics has the following
properties:
\begin{equation}\label{periodicproperty}
\begin{split}
& \theta \ch{a+n} {b+m} (z,\tau) = e^{2\pi i a^t m}\theta \ch {a} {b} (z,\tau),\\
&\theta \ch{a} {b} (z+m,\tau) = e^{2\pi i a^t m}\theta \ch {a} {b} (z,\tau),\\
&\theta \ch{a} {b} (z+\tau m,\tau) = e^{\pi i (-2b^t m -m^t \tau m - 2m^t z)}\theta \ch {a} {b} (z,\tau)\\
\end{split}
\end{equation}
where $n,m \in \mathbb Z^n.$ All of these properties are immediately verified by writing them out.
A scalar obtained by evaluating a theta function with characteristic at $z=0$ is called a \emph{theta constant} or
\emph{thetanulls}. When the entries of column vectors $a$ and $b$ are from the set $\{ 0,\frac{1}{2}\}$, then the
characteristics $ \ch {a}{b} $ are called the \emph{half-integer characteristics}. The corresponding theta functions
with rational characteristics are called \emph{theta characteristics}.
Points of order $n$ on $\mbox{Jac }(\mathcal X}%\def\X{\mathfrak X)$ are called the $\frac 1 n$-\textbf{periods}. Any point $p$ of $\mbox{Jac }(\mathcal X}%\def\X{\mathfrak X)$ can be written
as $p = \tau \,a + b. $ If $\ch{a}{b}$ is a $\frac 1 n$-period, then $a,b \in (\frac{1}{n}\mathbb Z /\mathbb Z)^{g}.$ The $\frac 1
n$-period $p$ can be associated with an element of $H_1(\mathcal X}%\def\X{\mathfrak X,\mathbb Z / n\mathbb Z)$ as follows: Let $a = (a_1,\cdots,a_g)^t,$ and $b
= (b_1,\cdots,b_g)^t.$ Then
\[
\begin{split}
p & = \tau a + b \\
& = \big(\sum a_i \int_{B_i} \omega_1,\cdots, \sum a_i \int_{B_i} \omega_g \big)^t + \big(b_1 \int_{A_1} \omega_1,\cdots,b_g
\int_{A_g} \omega_g \big) \\
\end{split}
\]
\[
\begin{split}
& = \big(\sum (a_i \int_{B_i} \omega_1 + b_i\int_{A_i} \omega_1) ,\cdots, \sum (a_i \int_{B_i} \omega_g + b_i\int_{A_i} \omega_g) \big)^t\\
& = \big( \int_C \omega_1, \cdots, \int_C \omega_g \big)^t
\end{split}
\]
where $C = \sum a_i B_i + b_i A_i. $ We identify the point $p$ with the cycle $\bar{C} \in H_1(\mathcal X}%\def\X{\mathfrak X,\mathbb Z / n\mathbb Z)$ where
$\bar{C} =\sum \bar{a_i} B_i + \bar{b_i} A_i,$ $\bar{a_i} = n a_i$ and $\bar{b_i} = n b_i$ for all $i.$
\subsection{Half-Integer Characteristics and the G\"opel Group} In this section we study groups of
half-integer characteristics. Any half-integer characteristic $\mathfrak m \in\frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g}$ is given by
\[
\mathfrak m = \frac{1}{2}m = \frac{1}{2}
\begin{pmatrix} m_1 & m_2 & \cdots & m_g \\ m_1^{\prime} & m_2^{\prime} & \cdots & m_g^{\prime} \end{pmatrix},
\]
where $m_i, m_i^{\prime} \in \mathbb Z.$ For $\mathfrak m = \ch{m ^\prime}{m^{\prime \prime}} \in \frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g},$ we
define $e_*(\mathfrak m) = (-1)^{4 (m^\prime)^t m^{\prime \prime}}.$ We say that $\mathfrak m$ is an \emph{even} (resp. \emph{odd})
characteristic if $e_*(\mathfrak m) = 1$ (resp. $e_*(\mathfrak m) = -1$). For any curve of genus $g$, there are $2^{g-1}(2^g+1)$ (resp.,
$2^{g-1}(2^g-1)$ ) even theta functions (resp., odd theta functions). Let $\mathfrak a$ be another half-integer characteristic.
We define
\[
\mathfrak m \, \mathfrak a = \frac{1}{2} \begin{pmatrix} t_1 & t_2 & \cdots & t_g \\ t_1^{\prime} & t_2^{\prime} & \cdots &
t_g^{\prime}
\end{pmatrix}
\]
where $t_i \equiv (m_i\, + a_i) \mod 2$ and $t_i^{\prime} \equiv (m_i^{\prime}\, + a_i^{\prime} ) \mod 2.$
For the rest of the thesis we only consider characteristics $\frac{1}{2}q$ in which each of the elements
$q_i,q_i^{\prime}$ is either 0 or 1. We use the following abbreviations:
\[
\begin{split}
&|\mathfrak m| = \sum_{i=1}^g m_i m_i^{\prime}, \quad \quad \quad \quad \quad \quad \quad \quad \quad
|\mathfrak m, \mathfrak a| = \sum_{i=1}^g (m_i^{\prime} a_i - m_i a_i^{\prime}), \\
& |\mathfrak m, \mathfrak a, \mathfrak b| = |\mathfrak a, \mathfrak b| + |\mathfrak b, \mathfrak m| + |\mathfrak m, \mathfrak a|, \quad \quad {\mathfrak m\choose \mathfrak a} = e^{\pi i \sum_{j=1}^g m_j
a_j^{\prime}}.
\end{split}
\]
\indent The set of all half-integer characteristics forms a group $\Gamma$ which has $2^{2g}$ elements. We say that two
half integer characteristics $\mathfrak m$ and $\mathfrak a$ are \emph{syzygetic} (resp., \emph{azygetic}) if $|\mathfrak m, \mathfrak a| \equiv 0 \mod
2$ (resp., $|\mathfrak m, \mathfrak a| \equiv 1 \mod 2$) and three half-integer characteristics $\mathfrak m, \mathfrak a$, and $\mathfrak b$ are syzygetic if
$|\mathfrak m, \mathfrak a, \mathfrak b| \equiv 0 \mod 2$.
A \emph{G\"opel group} $G$ is a group of $2^r$ half-integer characteristics where $r \leq g$ such that every two
characteristics are syzygetic. The elements of the group $G$ are formed by the sums of $r$ fundamental characteristics;
see \cite[pg. 489]{Baker} for details. Obviously, a G\"opel group of order $2^r$ is isomorphic to $C^r_2$. The proof of
the following lemma can be found on \cite[pg. 490]{Baker}.
\begin{lem}The number of different G\"opel groups which have $2^r$ characteristics is
\[
\frac{(2^{2g}-1)(2^{2g-2}-1)\cdots(2^{2g-2r+2}-1)}{(2^r-1)(2^{r-1}-1)\cdots(2-1)}.
\]
\end{lem}
If $G$ is a G\"opel group with $2^r$ elements, it has $2^{2g-r}$ cosets. The cosets are called \emph{G\"opel systems}
and are denoted by $\mathfrak a G$, $\mathfrak a \in \Gamma$. Any three characteristics of a G\"opel system are syzygetic. We can find a
set of characteristics called a basis of the G\"opel system which derives all its $2^r$ characteristics by taking only
combinations of any odd number of characteristics of the basis.
\begin{lem}
Let $g \geq 1$ be a fixed integer, $r$ be as defined above and $\sigma = g-r.$ Then there are
$2^{\sigma-1}(2^\sigma+1)$ G\"opel systems which only consist of even characteristics and there are
$2^{\sigma-1}(2^\sigma-1)$ G\"opel systems which consist of odd characteristics. The other $2^{2\sigma}(2^r-1)$ G\"opel
systems consist of as many odd characteristics as even characteristics.
\end{lem}
\proof The proof can be found on \cite[pg. 492]{Baker}. \qed
\begin{cor}\label{numb_systems}
When $r=g,$ we have only one (resp., 0) G\"opel system which consists of even (resp., odd) characteristics.
\end{cor}
Let us consider $s=2^{2\sigma}$ G\"opel systems which have distinct characters. Let us denote them by
\[\mathfrak a_1 G,\mathfrak a_2 G,\cdots,\mathfrak a_s G.\] We have the following lemma.
\begin{lem}
It is possible to choose $2\sigma+1$ characteristics from $\mathfrak a_1, \mathfrak a_2,\cdots, \mathfrak a_s,$ say $\bar{\mathfrak a}_1,$
$\bar{\mathfrak a}_2,$ $\cdots,$ $\bar{\mathfrak a}_{2\sigma+1}$, such that every three of them are azygetic and all have the same
character. The above $2\sigma+1$ fundamental characteristics are even (resp., odd) if $\sigma \equiv 1,0 \mod 4$
(resp.,$\equiv 2,3 \mod 4$).
\end{lem}
\noindent The proof of the following lemma can be found on \cite[pg. 511]{Baker}.
\begin{lem}
For any half-integer characteristics $\mathfrak a$ and $\mathfrak h,$ we have the following:
\begin{equation}\label {Bakereq1}
\theta^2[\mathfrak a](z_1,\tau) \theta^2[\mathfrak a \mathfrak h](z_2,\tau) = \frac{1}{2^{g}} \sum_\mathfrak e e^{\pi i |\mathfrak a \mathfrak e|} { \mathfrak h \choose \mathfrak a \mathfrak e}
\theta^2[\mathfrak e](z_1,\tau)\theta^2[\mathfrak e \mathfrak h](z_2,\tau).
\end{equation}
\end{lem}
We can use this relation to get identities among half-integer theta constants. Here $\mathfrak e$ can be any half-integer
characteristic. We know that we have $2^{g-1}(2^g+1)$ even characteristics. As the genus increases, we have multiple
choices for $\mathfrak e.$ In the following, we explain how we reduce the number of possibilities for $\mathfrak e$ and how to get
identities among theta constants.
First we replace $\mathfrak e$ by $\mathfrak e \mathfrak h$ and $z_1=z_2= 0$ in Eq.~\eqref{Bakereq1}. Eq.~\eqref{Bakereq1} can then be written
as follows:
\begin{equation}\label {Bakereq2}
\theta^2[\mathfrak a] \theta^2[\mathfrak a \mathfrak h] = 2^{-g} \sum_\mathfrak e e^{\pi i |\mathfrak a \mathfrak e \mathfrak h|} { \mathfrak h \choose \mathfrak a \mathfrak e \mathfrak h} \theta^2[\mathfrak e] \theta^2[\mathfrak e \mathfrak h].
\end{equation}
We have $e^{\pi i |\mathfrak a \mathfrak e \mathfrak h|}{ \mathfrak h \choose \mathfrak a \mathfrak e \mathfrak h} = e^{\pi i |\mathfrak a \mathfrak e|}{ \mathfrak h \choose \mathfrak a \mathfrak e} e^{\pi i |\mathfrak a
\mathfrak e, \mathfrak h|}.$ Next we put $z_1=z_2= 0$ in Eq.~\eqref{Bakereq1} and add it to Eq.~\eqref{Bakereq2} and get the following
identity:
\begin{equation}\label {Bakereq3}
2\theta^2[\mathfrak a] \theta^2[\mathfrak a \mathfrak h] = 2^{-g} \sum_\mathfrak e e^{\pi i |\mathfrak a \mathfrak e|} (1 + e^{\pi i|\mathfrak a \mathfrak e, \mathfrak h|}) \theta^2[\mathfrak e] \theta^2[\mathfrak e \mathfrak h].
\end{equation}
If $|\mathfrak a \mathfrak e, \mathfrak h| \equiv 1 \mod 2$, the corresponding terms in the summation vanish. Otherwise $1 + e^{\pi i|\mathfrak a \mathfrak e,
\mathfrak h|} = 2.$ In this case, if either $\mathfrak e$ is odd or $\mathfrak e \mathfrak h$ is odd, the corresponding terms in the summation vanish
again. Therefore, we need $|\mathfrak a \mathfrak e, \mathfrak h| \equiv 0 \mod 2$ and $|\mathfrak e| \equiv |\mathfrak e \mathfrak h| \equiv 0 \mod 2,$ in order to
get nonzero terms in the summation. If $\mathfrak e^*$ satisfies $|\mathfrak e^*| \equiv |\mathfrak e^* \mathfrak h^*| \equiv 0 \mod 2$ for some
$\mathfrak h^*,$ then $\mathfrak e^*\mathfrak h^*$ is also a candidate for the left hand side of the summation. Only one of such two values
$\mathfrak e^*$ and $\mathfrak e^* \mathfrak h^*$ is taken. As a result, we have the following identity among theta constants
\begin{equation}\label {eq1}
\theta^2[\mathfrak a] \theta^2[\mathfrak a \mathfrak h] = \frac{1}{2^{g-1}} \sum_\mathfrak e e^{\pi i |\mathfrak a \mathfrak e|} { \mathfrak h \choose \mathfrak a \mathfrak e} \theta^2[\mathfrak e]\theta^2[\mathfrak e
\mathfrak h],
\end{equation}
where $\mathfrak a, \mathfrak h$ are any characteristics and $\mathfrak e$ is a characteristics such that $|\mathfrak a \mathfrak e, \mathfrak h| \equiv 0 \mod 2,$
$|\mathfrak e| \equiv |\mathfrak e \mathfrak h| \equiv 0 \mod 2$ and $\mathfrak e \neq \mathfrak e \mathfrak h.$
By starting from the Eq.~\eqref{Bakereq1} with $z_1 = z_2$ and following a similar argument to the one above, we can
derive the identity,
\begin{equation}\label{eq2}
\theta^4[\mathfrak a] + e^{\pi i |\mathfrak a, \mathfrak h|} \theta^4[\mathfrak a \mathfrak h] = \frac{1}{2^{g-1}} \sum_\mathfrak e e^{\pi i |\mathfrak a \mathfrak e|} \{ \theta^4[\mathfrak e] + e^{ \pi
i |\mathfrak a, \mathfrak h|} \theta^4[\mathfrak e \mathfrak h]\}
\end{equation}
where $\mathfrak a, \mathfrak h$ are any characteristics and $\mathfrak e$ is a characteristic such that $|\mathfrak h| + |\mathfrak e, \mathfrak h| \equiv 0 \mod 2,$
$|\mathfrak e| \equiv |\mathfrak e \mathfrak h| \equiv 0 \mod 2$ and $\mathfrak e \neq \mathfrak e \mathfrak h.$
\begin{rem}
$|\mathfrak a \mathfrak e ,\mathfrak h| \equiv 0 \mod 2$ and $|\mathfrak e \mathfrak h| \equiv |\mathfrak e| \equiv 0 \mod 2$ implies $|\mathfrak a, \mathfrak h| + |\mathfrak h| \equiv 0 \mod
2.$
\end{rem}
We use Eq.~\eqref{eq1} and Eq.~\eqref{eq2} to get identities among thetanulls in Chapter 2 and in Chapter 3.
\subsection{Hyperelliptic Curves and Their Theta Functions}
A hyperelliptic curve $\mathcal X}%\def\X{\mathfrak X,$ defined over $\mathbb C,$ is a cover of order two of the projective line $\mathbb P^1.$ Let $z$ be the
generator (the hyperelliptic involution) of the Galois group $Gal(\mathcal X}%\def\X{\mathfrak X / \mathbb P^1).$ It is known that $\langle z \rangle $ is
a normal subgroup of the $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X)$ and $z$ is in the center of $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X)$. A hyperelliptic curve is ramified in $(2g
+2)$ places $w_1, \cdots, w_{2g+2}.$ This sets up a bijection between isomorphism classes of hyperelliptic genus g
curves and unordered distinct (2g+2)-tuples $w_1,\cdots,w_{2g+2} \in \mathbb P^1$ modulo automorphisms of $\mathbb P^1.$ An unordered
$(2g+2)$-tuple $\{w_i\}_{i=1}^{2g+2}$ can be described by a binary form (i.e. a homogenous equation $f(X,Z)$ of degree
$2g+2$). To describe $\mathcal H_g,$ we need rational functions of the coefficients of a binary form $f(X,Z),$ invariant under
linear substitutions in X and Z. Such functions are called absolute invariants for $g = 2$; see \cite{S7} for their
definitions. The absolute invariants are $GL_2(\mathbb C)$ invariants under the natural action of $GL_2(\mathbb C)$ on the space of
binary forms of degree $2g + 2.$ Two genus $g$ hyperelliptic curves are isomorphic if and only if they have the same
absolute invariants. The locus of genus $g$ hyperelliptic curves with an extra involution is an irreducible
$g$-dimensional subvariety of $\mathcal H_g$ which is denoted by $\mathcal L_g.$ Finding an explicit description of $\mathcal L_g$ means
finding explicit equations in terms of absolute invariants. Such equations are computed only for $g = 2;$ see \cite{S7}
for details. Writing the equations of $\mathcal L_2$ in terms of theta constants is the main focus of Chapter 2. Computing
similar equations for $g \geq 3$ requires first finding the corresponding absolute invariants. This is still an open
problem in classical invariant theory even for $g = 3.$
Let $\mathcal X}%\def\X{\mathfrak X \longrightarrow \mathbb P^1$ be the degree 2 hyperelliptic projection.
We can assume that $\infty$ is a branch point.
Let
\[ B := \{\alpha_1,\alpha_2, \cdots ,\alpha_{2g+1} \}\]
be the set of other branch points. Let $S = \{1,2, \cdots, 2g+1\}$ be the index set of $B$ and $\eta : S \longrightarrow
\frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g}$ be a map defined as follows:
\[
\begin{split}
\eta(2i-1) & = \begin{bmatrix}
0 & \cdots & 0 & \frac{1}{2} & 0 & \cdots & 0\\
\frac{1}{2} & \cdots & \frac{1}{2} & 0 & 0 & \cdots & 0\\
\end{bmatrix}, \\
\eta(2i) & =\begin{bmatrix}
0 & \cdots & 0 & \frac{1}{2} & 0 & \cdots & 0\\
\frac{1}{2} & \cdots & \frac{1}{2} & \frac{1}{2} & 0 & \cdots & 0\\
\end{bmatrix}
\end{split}
\]
where the nonzero element of the first row appears in $i^{th}$ column.
We define $\eta(\infty) $ to be $
\begin{bmatrix}
0 & \cdots & 0 & 0\\
0 & \cdots & 0 & 0\\
\end{bmatrix}$.
For any $T \subset B $, we define the half-integer characteristic as
\[ \eta_T = \sum_{a_k \in T } \eta(k) .\]
Let $T^c$ denote the complement of $T$ in $B.$ Note that $\eta_B \in \mathbb Z^{2g}.$ If we view $\eta_T$ as an element of
$\frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g}$ then $\eta_T= \eta_{T^c}.$ Let $\triangle$ denote the symmetric difference of sets, that is $T
\triangle R = (T \cup R) - (T \cap R).$ It can be shown that the set of subsets of $B$ is a group under $\triangle.$ We
have the following group isomorphism:
\[ \{T \subset B\, |\, \#T \equiv g+1 \mod 2\} / T \sim T^c \cong \frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g}.\]
For $\gamma = \ch{\gamma ^\prime}{\gamma^{\prime \prime}} \in \frac{1}{2}\mathbb Z^{2g}/\mathbb Z^{2g}$, we have
\begin{equation}\label{parityIdentity}
\theta [\gamma] (-z , \tau) = e_* (\gamma) \theta [\gamma] (z , \tau).\end{equation}
It is known that for hyperelliptic curves, $2^{g-1}(2^g+1) - {2g+1 \choose g}$ of the even theta constants are zero.
The following theorem provides a condition for the characteristics in which theta characteristics become zero. The
proof of the theorem can be found in \cite{Mu2}.
\begin{thm}\label{vanishingProperty}
Let $\mathcal X}%\def\X{\mathfrak X$ be a hyperelliptic curve, with a set $B$ of branch points. Let $S$ be the index set as above and $U $ be the
set of all odd values of $S$. Then for all $T \subset S$ with even cardinality, we have $ \theta[\eta_T] = 0$ if and only if
$\#(T \triangle U) \neq g+1$, where $\theta[\eta_T]$ is the theta constant corresponding to the characteristics $\eta_T$.
\end{thm}
When the characteristic $\gamma$ is odd, $e_* (\gamma)=1.$ Then from Eq.~\eqref{parityIdentity} all odd theta constants
are zero. There is a formula which satisfies half-integer theta characteristics for hyperelliptic curves called
Frobenius' theta formula.
\begin{lem} [Frobenius]
For all $z_i \in \mathbb C^g$, $1\leq i \leq 4$ such that $z_1 + z_2 + z_3 + z_4 = 0$ and for all $b_i \in \mathbb Q^{2g}$, $1\leq i
\leq 4$ such that $b_1 + b_2 + b_3 + b_4 = 0$, we have
\[ \sum_{j \in S \cup \{\infty\}} \epsilon_U(j) \prod_{i =1}^4 \theta[b_i+\eta(j)](z_i) = 0, \]
where for any $A \subset B$,
\[
\epsilon_A(k) =
\begin{cases}
1 & \textit {if $k \in A$}, \\
-1 & \textit {otherwise}.
\end{cases}
\]
\end{lem}
\proof See \cite[pg. 107]{Mu1}. \qed
A relationship between theta constants and the branch points of the hyperelliptic curve is given by Thomae's formula.
\begin{lem}[Thomae]\label{Thomae}
For all sets of branch points $B=\{\alpha_1,\alpha_2, \cdots ,\alpha_{2g+1} \},$ there is a constant $A$ such that for all $T\subset B,$ $\# T$ is even, \\
\[\theta[\eta_T](0;\tau)^4 =(-1)^{\#T \cap U} A \prod_{i<j \atop i,j \in T \triangle U} (\alpha_{i} - \alpha_{j}) \prod_{i<j \atop i,j \notin T \triangle U} (\alpha_{i} -
\alpha_{j}) \] \\
\noindent where $\eta_T$ is a non singular even half-integer characteristic corresponding to the subset $T$ of branch
points.
\end{lem}
See \cite[pg. 128]{Mu1} for the description of $A$ and \cite[pg. 120]{Mu1} for the proof. Using Thomae's formula and
Frobenius' theta identities we express the branch points of the hyperelliptic curves in terms of even theta constants.
\subsection{Cyclic Curves and Their Theta Functions}
A cyclic cover $\mathcal X}%\def\X{\mathfrak X \longrightarrow \mathbb P^1$ is defined to be a Galois cover with cyclic Galois group $C.$ We call it a
normal cyclic cover of $\mathbb P^1$ if $C$ is normal in $G=Aut(\mathcal X}%\def\X{\mathfrak X)$ where $Aut(\mathcal X}%\def\X{\mathfrak X)$ is the automorphism group of the curve
$\mathcal X}%\def\X{\mathfrak X.$ Then $\bar{G} = G/C$ embeds as a finite subgroup of $PGL(2,\mathbb C)$ and it is called the reduced automorphism group
of $G.$
\noindent An affine equation of a cyclic curve can be given by the following:
\begin{equation}\label{cyclic}
y^m = f(x) = \prod_{i=1}^s (x-\alpha_i)^{d_i} , \, m = |C|, \,\, 0 < d_i < m.
\end{equation}
Note that when $d_i>0$ for some $i$ the curve is singular.
Hyperelliptic curves are cyclic curves with $m=2$. After Thomae, many mathematicians, for example Fuchs, Bolza, Fay,
Mumford, et al., gave derivations of Thomae's formula in the hyperelliptic case. In 1988 Bershdaski and Radul found a
generalization of Thomae's formula for $Z_N $ curves of the form
\begin{equation} \label{Nakayashiki}
y^N = f(x) = \prod_{i=1}^{Nm}(x-a_i).
\end{equation}
In 1988 Shiga showed the representation of the Picard modular function by theta constants. He considered the algebraic
curve in the $(x,y)$ plane which is given by
\begin{equation}\label{picardcurve}
C(\epsilon) : y^3 = x (x-a_0) (x-a_1) (x-a_2)
\end{equation}
where $\epsilon = [a_0, a_1,a_2]$ is a parameter on the domain
$$\Lambda = \{ \epsilon : a_0 a_1 a_2 (a_0-a_1) (a_0-a_2) (a_1-a_2) \neq 0 \}.$$
He gave a concrete description of the Picard work \cite{picard}. His result can be considered an extension of the
classical Jacobi representation $\lambda = \frac{\theta_2^4}{\theta_3^4}$, where $\theta_i(z,\tau)$ indicates Jacobi's theta function
and $\theta_i$ is the convention for $\theta_i(0,\tau)$, for the elliptic modular function $\lambda(\tau)$ to the special case of
genus 3.
In 1991, Gonzalez Diez studied the moduli spaces parameterizing algebraic curves which are Galois covering of $\mathbb P^1$
with prime order and with given ramification numbers. These curves have equation of the form
\begin{equation}\label{Gabino}
y^p = f(x) = \prod_{i=1}^{r}(x-a_i)^{m_i} ; \textit {p prime and } p \nmid \sum m_i.
\end{equation}
He expresses $a_i$ in terms of functions of the period matrix of the curve.
Farkas (1996) gave a procedure for calculating the complex numbers $a_i$ which appear in the algebraic equation
\begin{equation}\label{Farkas}
y^p = \prod_{i=1}^{k}(x-a_i) \, \, \, \, \textit{with} \,\,\, p|k
\end{equation}
in terms of the theta functions associated with the Riemann surface of the algebraic curve defined by the
Eq.~\eqref{Farkas}. He used the generalized cross ratio of four points according to Gunning. Furthermore he considered
the more general problem of a branched two-sheeted cover of a given compact Riemann surface and obtained the relations
between the theta functions on the cover and the theta function to the original surface.
Nakayashiki, in 1997, gave an elementary proof of Thomae's formula for $Z_N$ curves which was discovered by Bershadsky
and Radul.
Enolski and Grava, in 2006, derived the analogous generalized Thomae's formula for the $Z_N$ singular curve of the form
\begin{equation} \label{Enolski}
y^N = f(x) = \prod_{i=1}^{m}(x-\lambda_{2i})^{N-1} \prod_{i=1}^{m} ( x-\lambda_{2i+1}).
\end{equation}
\noindent We summarize all the results in the following theorem.
\begin{thm}\label{CyclicCurvesThm}
Consider the algebraic curve $ \mathcal X}%\def\X{\mathfrak X : y^n = f(x)$ defined over the complex field $\mathbb C.$
\noindent\textbf{Case 1:} If $\triangle_f \neq 0,$ say $f(x) = \prod_{i=1}^k (x- \lambda_i)$ then,
\underline {i)} If $n | k,$ say $k = mn$ for some $m \in \mathbb N$ then,
for an ordered partition $\Lambda = (\Lambda_0, \cdots, \Lambda_{n-1})$ of $\{1,2,\cdots, nm\},$ we have
\[ \theta[e_\Lambda](0)^{2n} = C_\Lambda (det A)^n \prod_{i<j}(\lambda_i -
\lambda_j)^{2 n \sum_{\ell \in L} q_\ell(k_i) q_{\ell}(k_j) + \frac{(n-1)(2n-1)}{6}}\]
where $k_i = j$ for $i \in \Lambda_j$ and $e_\Lambda \equiv \Lambda_1 + 2 \Lambda_2 +\cdots + (n-1)\Lambda_{n-1} - D -
\varsigma$ is the associated divisor class of the partition $\Lambda,$ $L = \big\{-\frac{N-1}{2}, -\frac{N-1}{2} +1,
\cdots, \frac{N-1}{2}\big\},$ $q_\ell(i) = \frac{1-N}{2N} +\,\, \textit{fraction part of}\,\,\big(\frac { \ell + i +
\frac{N-1}{2}}{N}\big) \, \, \, \, \textit{for} \, \, \ell \in L,$ $\varsigma$ is Riemann's constant and $C_\Lambda$
depends on the partition $\Lambda$ having the property that for two different partitions $\Lambda$ and $\Lambda^\prime$
we have $C_\Lambda ^{2N} =C_{\Lambda ^\prime} ^{2N}.$
Moreover if $n $ is a prime $p,$ the branch points $\lambda_i$ of the curve $y^n = x(x-1)(x-\lambda_1) \cdots
(x-\lambda_{k-3})$ can be given by
\[ E_i^n \lambda_i = (\lambda(P_k, Q_0, Q_1, Q_\infty))^n\]
where
$\lambda(P_k, Q_0, Q_1, Q_\infty) = \frac{\theta(e+\phi_{Q_0}(P_k))\theta(e+\phi_{Q_{\infty}}(Q_1))}
{\theta(e+\phi_{Q_\infty}(P_k))\theta(e+\phi_{Q_{0}}(Q_1))},$ while $Q_0,$ $Q_1,$ and $Q_\infty$ denote the points in
the curve corresponding to the points $0,$ $1,$ and $\infty$ in $\mathbb P^1$ respectively, $P_i$'s are points in the curve
corresponding to the points $\lambda_i,$ $E_i$ is a constant depending on the point $P_i$
and $\phi_P $ is an injective map from $\mathcal X}%\def\X{\mathfrak X$ to $\mathbb C^g/G.$
\underline {ii)} If $ n \nmid k,$ then,
if $n=3$ and $k=4$, then the parameters $\lambda_1, \lambda_2, \lambda_3$ can be given as follows:
$$\lambda_1 = \theta^3 \chs{0}{\frac{1}{6}}{0}{0}{\frac{1}{6}}{0}, \quad \quad \lambda_2 = \theta^3
\chs{0}{\frac{1}{6}}{0}{\frac{1}{3}}{\frac{1}{6}}{\frac{1}{3}}, \quad \quad \lambda_3 = \theta^3
\chs{0}{\frac{1}{6}}{0}{\frac{2}{3}}{\frac{1}{6}}{\frac{2}{3}}.$$
\noindent \textbf{Case 2:} If $\triangle_f = 0,$ let $f(x) = \prod_{k=0}^m (x- \lambda_{2k+1})\prod_{k=1}^m (x-
\lambda_{2k})^{n-1}.$ Then,
\[
\begin{split}
\theta[e_m] (0; \Omega)^{4N} = &\frac{\prod_{i=1}^{N-1} \mbox{det } A_i^{2N}}{(2 \pi i )^{2mN(N-1)} } \prod_{1 \leq i < k \leq m
}(\lambda_{2i}-\lambda_{2k})^{N(N-1)} \\
& \times \prod_{0 \leq i < k \leq m
}(\lambda_{2i+1}-\lambda_{2k+1})^{N(N-1)} \\
& \times (\frac{\prod_{i \in I_1 , j \in J_1 }(\lambda_i - \lambda_j) \prod_{i \in I_2 , j \in J_2 }(\lambda_i - \lambda_j)}
{\prod_{i \in I_1 , k \in I_2 }(\lambda_i - \lambda_k) \prod_{j \in J_1 , k \in J_2 }(\lambda_i -
\lambda_j)})^{2(N-1)},
\end{split}
\]
where
$e_m = \nu ((N-1) \sum_{i \in I_1}P_i +(N-1) \sum_{j \in J_1} P_j - D- \triangle )$ is a nonsingular
$\frac{1}{N}$ characteristic,
$J_1 \subset J_0 = \{2,4,\cdots,2m+2\}$ and $I_1 \subset I_0 =
\{1,3,\cdots,2m+1\}$ with
$|J_1| + |I_1| = m+1$ and $I_2 = I_0 - I_1 , J_2 = J_0 - J_1 - 2m+2,$ and
$\triangle = (N-1) \sum_{k=1}^m P_{2k} - P_{\infty}$ is the Riemann divisor of the curve $\mathcal X}%\def\X{\mathfrak X.$
\end{thm}
\begin{proof}
For proof of the part $i)$ of case 1, see \cite{NK}. When $n$ is prime, the proof can be found in \cite{Farkas}. The
main point of \cite{SHI} is to prove part $ii)$ of case 1. The proof of case 2 can be found in \cite{Enolski}.
\end{proof}
\subsubsection{Relations Among Theta Functions for Algebraic Curves with Automorphisms}
In this section we develop an algorithm to determine relations among theta functions of a cyclic curve $\mathcal X}%\def\X{\mathfrak X$ with
automorphism group $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X)$. The proof of the following lemma can be found in \cite{RF}.
\begin{lem}\label{Shiga}
Let $f$ be a meromorphic function on $\mathcal X}%\def\X{\mathfrak X,$ and let $$(f) = \sum_{i=1}^{m}b_i - \sum_{i=1}^{m}c_i $$ be the divisor
defined by $f.$ Take paths from $P_0$ (initial point) to $b_i$ and $P_0$ to $c_i$ so that $\sum_{i=1}^{m}
\int_{P_0}^{b_i} \omega = \sum_{i=1}^{m} \int_{P_0}^{c_i} \omega .$
For an effective divisor $P_1 + \cdots + P_g,$ we have
\begin{equation}
f(P_1) \cdots f(P_g) = \frac{1}{E} \prod_{k=1} \frac{\theta(\sum_{i} \int_{P_0}^{P_i} \omega - \int_{P_0}^{b_k} \omega -
\triangle , \tau )} {\theta(\sum_{i} \int_{P_0}^{P_i} \omega - \int_{P_0}^{c_k} \omega - \triangle , \tau )}
\end{equation}
where $E$ is a constant independent of $P_1, \dots , P_g,$ the integrals from $P_0$ to $P_i$ take the same paths both
in the numerator and in the denominator, $\triangle$ denotes the Riemann's constant, and $\int_{P_0}^{P_i} \omega = \left(
\int_{P_0}^{P_i} \omega_1, \dots, \int_{P_0}^{P_i} \omega_g \right)^t.$
\end{lem}
This lemma gives us a tool that can be used to find branch points in terms of theta constants. By considering the
meromorphic function $f = x$ on $\mathcal X}%\def\X{\mathfrak X$ and suitable effective divisors, we can write branch points as ratios of
thetanulls. We present some explicit calculations using the Lemma ~\ref{Shiga} in Chapter 3 and 4. The hard part of
this
\noindent method is the difficulty of writing complex integrals in terms of characteristics.
\begin{alg}
Input: A cyclic curve $\mathcal X}%\def\X{\mathfrak X$ with automorphism group $G$, $\sigma \in G$ such that $| \sigma| =n$, $g (\mathcal X}%\def\X{\mathfrak X^\sigma)
=0$ and $\< \sigma \> \triangleleft G$. \\
Output: Relations among the theta functions of $\mathcal X}%\def\X{\mathfrak X$\\
Step 1: Let $\Gamma = G/\<\sigma\>$ and pick $\tau \in \Gamma$ such that $\tau$ has the largest order $m$.
Step 2: Write the equation of the curve in the form \[ y^n = f(x^m) \,\,\, \textit{or} \,\,\, y^n = x f(x^m).\]
Step 3: Determine the roots $\lambda_1, \dots , \lambda_r$ of $f(x^\tau)$ in terms of the theta functions.
Step 4: Determine relations on theta functions using Gr\"obner basis techniques.
\end{alg}
For step 3, we can use Lemma ~\ref{Shiga}. If the curve in step 3 falls into one of the categories given in Theorem
~\ref{CyclicCurvesThm}, we can use the corresponding equation to invert the period map without worrying about the
complex integrals.
\section{Genus 2 curves}
Let $k$ be an algebraically closed field of characteristic zero and $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve defined over $k$. Consider
a binary sextic, i.e. a homogeneous polynomial $f(X,Z)$ in $k[X,Z]$ of degree 6:
$$f(X,Z)=a_6 X^6+ a_5 X^5Z+\dots +a_0 Z^6.$$ The polynomial functions of the coefficients of a binary sextic $f(X,Z)$ invariant under
linear substitutions in $X,Z$ of determinant one. These invariants were worked out by Clebsch and Bolza in the case of
zero characteristic and generalized by Igusa for any characteristic different from 2.
{\it Igusa $J$-invariants} $\, \, \{ J_{2i} \}$ of $f(X,Z)$ are homogeneous polynomials of degree $2i$ in $k[a_0,
\dots , a_6]$, for $i=1,2,3,5$; see \cite{S7} for their definitions. Here $J_{10}$ is the discriminant of $f(X,Z)$.
It vanishes if and only if the binary sextic has a multiple linear factor. These $J_{2i}$ are invariant under the
natural action of $SL_2(k)$ on sextics. Dividing such an invariant by another invariant with the same degree, gives an
invariant (eg. absolute invariant) under $GL_2(k)$ action. The absolute invariants of $\mathcal X}%\def\X{\mathfrak X$ are defined in terms of
Igusa invariants as follows: \[ i_1 := 144 \frac{J_4}{J_2^2}, \quad \quad i_2:= -1728 \frac{J_2 J_4 - 3 J_6}{J_2^3},
\quad \quad i_3:= 486 \frac{J_{10}}{J_2 ^5}. \]
Two genus 2 fields (resp., curves) in the standard form $Y^2=f(X,1)$ are isomorphic if and only if the corresponding
sextics are $GL_2(k)$ conjugate.
\subsection{Half Integer Theta Characteristics} For genus two curve, we have six odd theta
characteristics and ten even theta characteristics. The following are the sixteen theta characteristics where the first
ten are even and the last six are odd. For simplicity, we denote them by $\theta_i(z)$ instead of $\theta_i \ch{a} {b} (z ,
\tau)$ where $i=1,\dots ,10$ for the even functions and $i=11, \dots, 16$ for the odd functions.
\[
\begin{split}
\theta_1(z) &= \theta_1 \chr {0}{0}{0}{0} (z , \tau), \qquad \qquad \theta_2(z) = \theta_2 \chr {0}{0}{\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_3(z) &= \theta_3 \chr {0}{0}{\frac{1}{2}}{0}(z , \tau), \qquad \qquad
\theta_4(z) = \theta_4 \chr {0}{0}{0}{\frac{1}{2}} (z , \tau)\\
\theta_5(z) &= \theta_5 \chr{\frac{1}{2}}{0} {0}{0}(z , \tau), \qquad \qquad
\theta_6(z) = \theta_6 \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} (z , \tau)\\
\theta_7(z) &= \theta_7 \chr{0}{\frac{1}{2}} {0}{0} (z , \tau), \qquad \qquad
\theta_8(z) = \theta_8 \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0} (z , \tau)\\
\end{split}
\]
\[
\begin{split}
\theta_9(z) &= \theta_9 \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0}(z , \tau), \qquad \qquad
\theta_{10}(z) = \theta_{10} \chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} (z , \tau)\\
\theta_{11}(z) &= \theta_{11} \chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} (z , \tau), \qquad \qquad
\theta_{12}(z) = \theta_{12} \chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} (z , \tau)\\
\theta_{13}(z) &= \theta_{13} \chr{\frac{1}{2}}{0} {\frac{1}{2}}{0} (z , \tau), \qquad \qquad
\theta_{14}(z) = \theta_{14} \chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{0} (z , \tau)\\
\theta_{15}(z) &= \theta_{15} \chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}} (z , \tau), \qquad \qquad
\theta_{16}(z) = \theta_{16} \chr{\frac{1}{2}}{\frac{1}{2}} {0}{\frac{1}{2}} (z , \tau)\\
\end{split}
\]
\begin{rem}
All the possible half-integer characteristics except the zero characteristic can be obtained as the sum of not more
than 2 characteristics chosen from the following 5 characteristics:
\[ \left\{\chr {0}{0}{\frac{1}{2}} {\frac{1}{2}}, \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}}, \chr{0}{\frac{1}{2}} {0}{0}, \chr{\frac{1}{2}}{0}
{\frac{1}{2}}{\frac{1}{2}},\chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}.\] The sum of all 5 characteristics in the
set determines the zero characteristic.
\end{rem}
Take $\sigma = g - r =0 $. Then a G\"opel group $G$ contains four elements. The number of such G\"opel groups is 15.
Let $G = \{0, \mathfrak m_1, \mathfrak m_2, \mathfrak m_1 \mathfrak m_2\}$ be a G\"opel group of even characteristics (we have six such groups). Let
$\mathfrak b_1, \mathfrak b_2, \mathfrak b_1 \mathfrak b_2 $ be the characteristics such that the $G , \mathfrak b_1 G , \mathfrak b_2 G , \mathfrak b_1 \mathfrak b_2 G $ are all the
cosets of the group $G.$ Then each of the systems other than $G$ contains two odd characteristics and two even
characteristics. Consider equations given by Eq.~\eqref{eq1} and Eq.~\eqref{eq2}. If $\mathfrak h$ denotes any one of the 3
characteristics $\mathfrak m_1,\mathfrak m_2,\mathfrak m_1 \mathfrak m_2$, then we have 6 possible characteristics for $\mathfrak e$, which satisfy $|\mathfrak e,\mathfrak h|
\equiv | \mathfrak h| \equiv 0$. They are $0, \mathfrak n, \mathfrak b, \mathfrak h, \mathfrak n \mathfrak h, \mathfrak b \mathfrak h$ where $\mathfrak n$ is a characteristic in the G\"opel
group other than $\mathfrak h$, and $\mathfrak b$ is an even characteristic chosen from one of the systems $\mathfrak b_1 G, \mathfrak b_2 G, \mathfrak b_1
\mathfrak b_2 G$. The following three cases illustrate the possible values for characteristic $\mathfrak h$ and for characteristic
$\mathfrak e$. Without loss of generality, we can take only three values for $\mathfrak e$ which give rise to different terms on the
right hand side of Eq.~\eqref{eq1} and Eq.~\eqref{eq2}.
\noindent \textbf{Case 1:} $\mathfrak h= \mathfrak m_1.$
Take $\mathfrak e \in \{ 0, \mathfrak m_2, \mathfrak b_1 \}$ and take $\mathfrak a = \mathfrak b_1$. Then from Eq.~\eqref{eq1} and Eq.~\eqref{eq2} we have
\[
\begin{split}
&{ \mathfrak m_1 \choose \mathfrak b_1} \theta^2[ 0] \theta^2[ \mathfrak m_1] + e^{\pi i |\mathfrak b_1 \mathfrak m_2|} { \mathfrak m_1 \choose \mathfrak b_1 \mathfrak m_2} \theta^2[\mathfrak m_2] \theta^2[\mathfrak m_1
\mathfrak m_2] -
\theta^2[\mathfrak b_1] \theta^2[\mathfrak b_1 \mathfrak m_1] = 0, \, \, \\
&\theta^4[ 0] + \theta^4[ \mathfrak m_1] + e^{\pi i |\mathfrak b_1 \mathfrak m_2|} [ \theta^4[ \mathfrak m_2] + \theta^4[ \mathfrak m_2 \mathfrak m_1]] - [ \theta^4[ \mathfrak b_1] + \theta^4[
\mathfrak b_1 \mathfrak m_1]] = 0. \\
\end{split}
\]
\noindent \textbf{Case 2:} $\mathfrak h= \mathfrak m_2.$
Take $\mathfrak e \in \{ 0, \mathfrak m_1, \mathfrak b_2 \}$ and take $\mathfrak a = \mathfrak b_2$. Then from Eq.~\eqref{eq1} and Eq.~\eqref{eq2} we have
\[
\begin{split}
&{ \mathfrak m_2 \choose \mathfrak b_2} \theta^2[ 0] \theta^2[ \mathfrak m_2] + e^{pi i |\mathfrak b_2 \mathfrak m_1|} { \mathfrak m_2 \choose \mathfrak b_2 \mathfrak m_1} \theta^2[ \mathfrak m_1] \theta^2[\mathfrak m_1
\mathfrak m_2] -
\theta^2[ \mathfrak b_2] \theta^2[\mathfrak b_2 \mathfrak m_2] = 0, \,\, \\
&\theta^4[ 0] + \theta^4[ \mathfrak m_2] + e^{pi i |\mathfrak b_2 \mathfrak m_2|} [ \theta^4[ \mathfrak m_1] + \theta^4[ \mathfrak m_1 \mathfrak m_2]] - [ \theta^4[ \mathfrak b_2] + \theta^4[
\mathfrak b_2 \mathfrak m_2]] = 0. \\
\end{split}
\]
\vspace*{.4cm}
\noindent \textbf{Case 3:} $\mathfrak h= \mathfrak m_1 \mathfrak m_2.$
Take $\mathfrak e \in \{ 0, \mathfrak m_1, \mathfrak b_1 \mathfrak b_2 \}$ and take $\mathfrak a = \mathfrak b_1 \mathfrak b_2$. Then from Eq.~\eqref{eq1} and Eq.~\eqref{eq2}
we have
\[
\begin{split}
&{ \mathfrak m_1 \mathfrak m_2 \choose \mathfrak b_1 \mathfrak b_2} \theta^2[ 0] \theta^2[ \mathfrak m_1 \mathfrak m_2] + e^{pi i |\mathfrak b_1 \mathfrak b_2 \mathfrak m_1|} { \mathfrak m_1 \mathfrak m_2 \choose \mathfrak b_1
\mathfrak b_2 \mathfrak m_1} \theta^2[ \mathfrak m_1] \theta^2[ \mathfrak m_2] -\\
&\theta^2[ \mathfrak b_1 \mathfrak b_2] \theta^2[ \mathfrak b_1 \mathfrak b_2 \mathfrak m_1 \mathfrak m_2] = 0, \,\, \\
&\theta^4[ 0] + \theta^4[ \mathfrak m_1 \mathfrak m_2] + e^{pi i |\mathfrak b_1 \mathfrak b_2 \mathfrak m_1|} [ \theta^4[ \mathfrak m_1] + \theta^4[ \mathfrak m_2]] - [ \theta^4[ \mathfrak b_1 \mathfrak b_2] +
\theta^4[ \mathfrak b_1 \mathfrak b_2 \mathfrak m_1 \mathfrak m_2]] = 0. \\
\end{split}
\]
The identities above express the even theta constants in terms of four theta constants; therefore, we may call them
fundamental theta constants,
$$\theta[0] ,\theta[ \mathfrak m_1], \theta[ \mathfrak m_2], \theta[\mathfrak m_1 \mathfrak m_2].$$
\subsection{Identities of Theta Constants}
We have only six G\"opel groups such that all characteristics are even. The following are such G\"opel groups and
corresponding identities of theta constants.
\begin{description}
\item [i)] $G = \left\{0 = \chr{0}{0}{0}{0}, \mathfrak m_1 = \chr {0}{0}{0}{\frac{1}{2}}, \mathfrak m_2 = \chr
{0}{0}{\frac{1}{2}}{0}, \mathfrak m_1 \mathfrak m_2 = \chr {0}{0}{\frac{1}{2}} {\frac{1}{2}} \right\}$ is a G\"opel group. If $\mathfrak b_1 =
\chr{\frac{1}{2}}{0} {0}{0}, \mathfrak b_2 = \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0}$, then the corresponding G\"opel systems are
given by the following:
\[
\begin{split}
G &= \left\{ \chr{0}{0}{0}{0}, \chr {0}{0}{0}{\frac{1}{2}}, \chr {0}{0}{\frac{1}{2}}{0},
\chr {0}{0}{\frac{1}{2}} {\frac{1}{2}} \right\}, \\
\mathfrak b_1 G &= \left\{\chr{\frac{1}{2}}{0} {0}{0}, \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}},
\chr{\frac{1}{2}}{0} {\frac{1}{2}}{0}, \chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}} \right\}, \\
\mathfrak b_2 G &= \left\{\chr{0}{\frac{1}{2}} {\frac{1}{2}}{0}, \chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}},
\chr{0}{\frac{1}{2}} {0}{0},
\chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}, \\
\end{split}
\]
\[
\begin{split}
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{0}, \chr{\frac{1}{2}}{\frac{1}{2}}
{\frac{1}{2}}{\frac{1}{2}}, \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}, \chr{\frac{1}{2}}{\frac{1}{2}} {0}{\frac{1}{2}}
\right\}.
\end{split}
\]
Notice that from all four cosets, only $G$ has all even characteristics as noticed in Corollary~\ref{numb_systems}.
Using Eq.~\eqref{eq1} and Eq.~\eqref{eq2}, we have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_5^2 \theta_6^2 & = & \theta_1^2 \theta_4^2 - \theta_2^2 \theta_3^2 ,\\
\theta_5^4 + \theta_6^4 & =& \theta_1^4 - \theta_2^4 - \theta_3^4 + \theta_4^4, \\
\theta_7^2 \theta_9^2 & = & \theta_1^2 \theta_3^2 - \theta_2^2 \theta_4^2, \\
\theta_7^4 + \theta_9^4 &= & \theta_1^4 - \theta_2^4 + \theta_3^4 - \theta_4^4, \\
\theta_8^2 \theta_{10}^2 & = & \theta_1^2 \theta_2^2 - \theta_3^2 \theta_4^2, \\
\theta_8^4 + \theta_{10}^4 & = & \theta_1^4 + \theta_2^4 - \theta_3^4 - \theta_4^4. \\
\end{array}
\right.
\]
These identities express even theta constants in terms of four theta constants. We call them fundamental theta
constants $\theta_1, \, \theta_2, \, \theta_3, \, \theta_4$.
Following the same procedure, we can find similar identities for each possible G\"opel group.
\item [ii)] $G = \left\{0 = \chr {0}{0}{0}{0}, \mathfrak m_1 = \chr {0}{0}{\frac{1}{2}}{0}, \mathfrak m_2 = \chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}},
\mathfrak m_1 \mathfrak m_2 = \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}\right\}$
is a G\"opel group. If $\mathfrak b_1 = \chr {0}{0}{\frac{1}{2}}{0} , \mathfrak b_2 =
\chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} $, then the corresponding G\"opel systems are given by the following:
\[
\begin{split}
G &= \left\{ \chr {0}{0}{0}{0}, \chr {0}{0}{\frac{1}{2}}{0}, \chr{\frac{1}{2}}{\frac{1}{2}}
{\frac{1}{2}}{\frac{1}{2}}, \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}\right\}, \\
\mathfrak b_1 G &= \left\{\chr {0}{0}{\frac{1}{2}}{0}, \chr {0}{0}{0}{\frac{1}{2}}, \chr{\frac{1}{2}}{\frac{1}{2}}
{0}{\frac{1}{2}},
\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{0} \right\}, \\
\end{split}
\]
\[
\begin{split}
\mathfrak b_2 G &= \left\{\chr {\frac{1}{2}}{0}{0}{\frac{1}{2}}, \chr{\frac{1}{2}}{0} {\frac{1}{2}}{0}, \chr{0}{\frac{1}{2}}
{\frac{1}{2}}{0}
, \chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}, \\
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}}, \chr{\frac{1}{2}}{0} {0}{0}, \chr{0}{\frac{1}{2}}
{0}{0} \chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} \right\}.
\end{split}
\]
We have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_3^2 \theta_4^2 & =& \theta_1^2 \theta_2^2 - \theta_8^2 \theta_{10}^2, \\
\theta_3^4 + \theta_4^4 & = &\theta_1^4 + \theta_2^4 - \theta_8^4 - \theta_{10}^4, \\
\theta_6^2 \theta_9^2 & = & -\theta_1^2 \theta_{10}^2 + \theta_2^2 \theta_8^2, \\
\theta_6^4 + \theta_9^4 & = & \theta_1^4 - \theta_2^4 - \theta_8^4 + \theta_{10}^4, \\
\theta_5^2 \theta_7^2 & = & \theta_1^2 \theta_8^2 - \theta_2^2 \theta_{10}^2, \\
\theta_5^4 + \theta_7^4 & = & \theta_1^4 - \theta_2^4 + \theta_8^4 - \theta_{10}^4. \\
\end{array}
\right.
\]
\item [iii)] $G = \left\{0 = \chr {0}{0}{0}{0}, \mathfrak m_1 = \chr {0}{0}{\frac{1}{2}}{0}, \mathfrak m_2 = \chr{0}{\frac{1}{2}}{0}{0},
\mathfrak m_1 \mathfrak m_2 = \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0} \right\}$ is a G\"opel group. If $\mathfrak b_1 = \chr {0}{0}{\frac{1}{2}}
{\frac{1}{2}} , \mathfrak b_2 = \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0} $, then the corresponding G\"opel systems are given by
the following:
\[
\begin{split}
G &= \left\{ \chr {0}{0}{0}{0}, \chr {0}{0}{\frac{1}{2}}{0}, \chr{0}{\frac{1}{2}}{0}{0},
\chr{0}{\frac{1}{2}} {\frac{1}{2}}{0} \right\}, \\
\mathfrak b_1 G &= \left\{\chr {0}{0}{\frac{1}{2}} {\frac{1}{2}}, \chr {0}{0}{0}{\frac{1}{2}}, \chr{0}{\frac{1}{2}}
{\frac{1}{2}}{\frac{1}{2}},
\chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}, \\
\mathfrak b_2 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}, \chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{0},
\chr{\frac{1}{2}}{0} {0}{0},
\chr{\frac{1}{2}}{0} {\frac{1}{2}}{0} \right\}, \\
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}}, \chr{\frac{1}{2}}{\frac{1}{2}}
{0}{\frac{1}{2}}, \chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}}, \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} \right\}.
\end{split}
\]
We have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_2^2 \theta_4^2 & = &\theta_1^2 \theta_3^2 - \theta_7^2 \theta_{9}^2, \\
\theta_2^4 + \theta_4^4 & = & \theta_1^4 + \theta_3^4 - \theta_7^4 - \theta_{9}^4, \\
\theta_8^2 \theta_5^2 & = & \theta_1^2 \theta_{7}^2 - \theta_3^2 \theta_9^2, \\
\theta_8^4 + \theta_5^4 & = & \theta_1^4 - \theta_3^4 + \theta_7^4 - \theta_{9}^4, \\
\theta_6^2 \theta_{10}^2 & = & -\theta_1^2 \theta_9^2 + \theta_3^2 \theta_{7}^2, \\
\theta_6^4 + \theta_{10}^4 & = & \theta_1^4 - \theta_3^4 - \theta_7^4 + \theta_{9}^4. \\
\end{array}
\right.
\]
\medskip
\item [iv)] $G = \left\{0 = \chr {0}{0}{0}{0}, \mathfrak m_1 = \chr {0}{0}{0}{\frac{1}{2}}, \mathfrak m_2 = \chr{\frac{1}{2}}{0} {0}{0},
\mathfrak m_1 \mathfrak m_2 = \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} \right\}$ is a G\"opel group. If $\mathfrak b_1 = \chr {0}{0}{\frac{1}{2}}
{\frac{1}{2}}, \mathfrak b_2 = \chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}$, then the corresponding G\"opel systems are given by the
following:
\[
\begin{split}
G &= \left\{ \chr {0}{0}{0}{0}, \chr {0}{0}{0}{\frac{1}{2}}, \chr{\frac{1}{2}}{0} {0}{0},
\chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} \right\},\\
\mathfrak b_1 G &= \left\{\chr {0}{0}{\frac{1}{2}} {\frac{1}{2}}, \chr {0}{0}{\frac{1}{2}}{0}, \chr{\frac{1}{2}}{0}
{\frac{1}{2}}{\frac{1}{2}},
\chr{\frac{1}{2}}{0} {\frac{1}{2}}{0} \right\}, \\
\mathfrak b_2 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}, \chr{\frac{1}{2}}{\frac{1}{2}}
{0}{\frac{1}{2}},\chr{0}{\frac{1}{2}} {0}{0},
\chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}, \\
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} ,\chr{\frac{1}{2}}{\frac{1}{2}}
{\frac{1}{2}}{0}, \chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}}, \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0} \right\}.
\end{split}
\]
We have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_2^2 \theta_3^2 & = & \theta_1^2 \theta_4^2 - \theta_5^2 \theta_{6}^2, \\
\theta_2^4 + \theta_3^4 & = & \theta_1^4 + \theta_4^4 - \theta_5^4 - \theta_{6}^4, \\
\theta_8^2 \theta_7^2 & = & \theta_1^2 \theta_{5}^2 - \theta_4^2 \theta_6^2, \\
\theta_8^4 + \theta_7^4 & = & \theta_1^4 - \theta_4^4 + \theta_5^4 - \theta_{6}^4, \\
\theta_9^2 \theta_{10}^2 & = & -\theta_1^2 \theta_6^2 + \theta_4^2 \theta_{5}^2, \\
\theta_9^4 + \theta_{10}^4 & = & \theta_1^4 - \theta_4^4 - \theta_5^4 + \theta_{6}^4. \\
\end{array}
\right.
\]
\medskip
\item [v)] $G = \left\{0 = \chr {0}{0}{0}{0}, \mathfrak m_1 = \chr {\frac{1}{2}}{0} {0}{0}, \mathfrak m_2 = \chr {0}{\frac{1}{2}} {0}{0},
\mathfrak m_1 \mathfrak m_2 = \chr {\frac{1}{2}}{\frac{1}{2}}{0}{0} \right\}$ is a G\"opel group. If $\mathfrak b_1 = \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}} , \mathfrak b_2 =
\chr {0}{0}{\frac{1}{2}}{0} $, then the corresponding G\"opel systems are given by the following:
\[
\begin{split}
G &= \left\{ \chr {0}{0}{0}{0}, \chr {\frac{1}{2}}{0} {0}{0}, \chr {0}{\frac{1}{2}} {0}{0},
\chr {\frac{1}{2}}{\frac{1}{2}}{0}{0} \right\}, \\
\mathfrak b_1 G &= \left\{\chr {\frac{1}{2}}{0}{0}{\frac{1}{2}}, \chr {0}{0}{0}{\frac{1}{2}}, \chr{\frac{1}{2}}{\frac{1}{2}}
{0}{\frac{1}{2}},
\chr{0}{\frac{1}{2}} {0}{\frac{1}{2}} \right\}, \\
\end{split}
\]
\[
\begin{split}
\mathfrak b_2 G &= \left\{\chr {0}{0}{\frac{1}{2}}{0},\chr{\frac{1}{2}}{0} {\frac{1}{2}}{0}, \chr{0}{\frac{1}{2}}
{\frac{1}{2}}{0},
\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{0} \right\}, \\
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}}, \chr {0}{0}{\frac{1}{2}} {\frac{1}{2}},
\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}}, \chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}}\right\}.
\end{split}
\]
We have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_4^2 \theta_6^2 & = & \theta_1^2 \theta_5^2 - \theta_7^2 \theta_{8}^2, \\
\theta_4^4 + \theta_6^4 & = & \theta_1^4 + \theta_5^4 - \theta_7^4 - \theta_{8}^4, \\
\theta_3^2 \theta_9^2 & = & \theta_1^2 \theta_{7}^2 - \theta_5^2 \theta_8^2, \\
\theta_3^4 + \theta_9^4 & = & \theta_1^4 - \theta_5^4 + \theta_7^4 - \theta_{8}^4, \\
\theta_2^2 \theta_{10}^2 & = & \theta_1^2 \theta_8^2 - \theta_5^2 \theta_{7}^2, \\
\theta_2^4 + \theta_{10}^4 & = & \theta_1^4 - \theta_5^4 - \theta_7^4 + \theta_{8}^4. \\
\end{array}
\right.
\]
\medskip
\item [vi)] $G = \left\{0 = \chr {0}{0}{0}{0}, \mathfrak m_1 = \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}}, \mathfrak m_2 = \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0},
\mathfrak m_1 \mathfrak m_2 = \chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} \right\}$ is a G\"opel group. If $\mathfrak b_1 =
\chr{\frac{1}{2}}{0} {0}{0} , \mathfrak b_2 = \chr{0}{\frac{1}{2}} {0}{0}$, then the corresponding G\"opel systems are given by
the following:
\[
\begin{split}
G &= \left\{ \chr {0}{0}{0}{0}, \chr {\frac{1}{2}}{0}{0}{\frac{1}{2}}, \chr{0}{\frac{1}{2}} {\frac{1}{2}}{0},
\chr{\frac{1}{2}}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} \right\}, \\
\mathfrak b_1 G &= \left\{\chr{\frac{1}{2}}{0} {0}{0}, \chr {0}{0}{0}{\frac{1}{2}} , \chr{\frac{1}{2}}{\frac{1}{2}}
{\frac{1}{2}}{0},
\chr{0}{\frac{1}{2}} {\frac{1}{2}}{\frac{1}{2}} \right\}, \\
\mathfrak b_2 G &= \left\{\chr{0}{\frac{1}{2}} {0}{0}, \chr{\frac{1}{2}}{\frac{1}{2}} {0}{\frac{1}{2}}, \chr
{0}{0}{\frac{1}{2}}{0},
\chr{\frac{1}{2}}{0} {\frac{1}{2}}{\frac{1}{2}}\right\}, \\
\mathfrak b_3 G &= \left\{\chr{\frac{1}{2}}{\frac{1}{2}} {0}{0}, \chr{0}{\frac{1}{2}} {0}{\frac{1}{2}}, \chr{\frac{1}{2}}{0}
{\frac{1}{2}}{0}, \chr {0}{0}{\frac{1}{2}} {\frac{1}{2}} \right\}.
\end{split}
\]
We have the following six identities for the above G\"opel group:
\[
\left \{ \begin{array}{lll}
\theta_2^2 \theta_3^2 & = & \theta_1^2 \theta_4^2 - \theta_5^2 \theta_{6}^2, \\
\theta_2^4 + \theta_3^4 & = & \theta_1^4 + \theta_4^4 - \theta_5^4 - \theta_{6}^4, \\
\theta_8^2 \theta_7^2 & = & \theta_1^2 \theta_{5}^2 - \theta_4^2 \theta_6^2, \\
\theta_8^4 + \theta_7^4 & = & \theta_1^4 - \theta_4^4 + \theta_5^4 - \theta_{6}^4, \\
\theta_9^2 \theta_{10}^2 & = & -\theta_1^2 \theta_6^2 + \theta_4^2 \theta_{5}^2, \\
\theta_9^4 + \theta_{10}^4 & = & \theta_1^4 - \theta_4^4 - \theta_5^4 + \theta_{6}^4. \\
\end{array}
\right.
\]
\end{description}
From now on, we consider $\theta_1, \, \theta_2, \, \theta_3,$ and $\theta_4$ as the fundamental theta constants.
\medskip
\subsection{Inverting the Moduli Map} Let $\lambda_i,$ $i=1, \dots, n,$ be branch points of the genus
$g$ smooth curve $\mathcal X}%\def\X{\mathfrak X.$ Then the moduli map is a map from the configuration space $\Lambda$ of ordered $n$ distinct
points on $\mathbb P^1$ to the Siegel upper half space $\mathfrak H_g.$ In this section, we determine the branch points of genus 2
curves as functions of theta characteristics. The following lemma describes these relations using Thomae's formula. The
identities are known as Picard's formulas. We will formulate a somewhat different proof for Picard's lemma.
\begin{lem}[Picard] Let a genus 2 curve be given by
\begin{equation} \label{Rosen2}
Y^2=X(X-1)(X-\lambda)(X-\mu)(X-\nu).
\end{equation} Then, $\lambda, \mu, \nu$ can be written as follows:
\begin{equation}\label{Picard}
\lambda = \frac{\theta_1^2\theta_3^2}{\theta_2^2\theta_4^2}, \quad \mu = \frac{\theta_3^2\theta_8^2}{\theta_4^2\theta_{10}^2}, \quad \nu =
\frac{\theta_1^2\theta_8^2}{\theta_2^2\theta_{10}^2}.
\end{equation}
\end{lem}
\begin{proof}
There are several ways to relate $\lambda, \mu, \nu$ to theta constants, depending on the ordering of the branch points
of the curve. Let $B = \{\nu, \mu,\lambda, 1,0\}$ be the branch points of the curve in this order and $U = \{\nu,
\lambda, 0\}$ be the set of odd branch points. Using Lemma~\ref{Thomae}, we have the following set of equations of
theta constants and branch points:
\begin{equation}\label{Thomaeg=2}
\begin{array}{ll}
\theta_1^4 = A \, \nu \lambda (\mu -1) (\nu - \lambda), &
\theta_2^4 = A \, \mu (\mu -1) ( \nu - \lambda), \\
\theta_3^4 = -A \, \mu \lambda (\mu - \lambda) (\nu - \lambda), &
\theta_4^4 = -A\, \nu (\nu - \lambda) (\mu - \lambda), \\
\theta_5^4 = A \, \lambda \mu (\nu - 1) ( \nu - \mu),&
\theta_6^4 = -A \, (\nu - \mu) (\nu -\lambda) ( \mu -\lambda),\\
\theta_7^4 = -A \, \mu (\nu -1) ( \lambda -1) (\nu - \lambda), &
\theta_8^4 = -A \, \mu \nu (\nu - \mu) (\lambda -1), \\
\theta_9^4 = A \, \nu ( \mu -1) (\lambda - 1) (\mu - \lambda), &
\theta_{10}^4 = -A \, \lambda ( \lambda - 1) (\nu - \mu) \\
\end{array}
\end{equation}
where $A$ is a constant.
By choosing appropriate equations from the set Eq.~\eqref{Thomaeg=2} we have the following:
\[ \lambda^2 =\left(\frac{\theta_1^2\theta_3^2}{\theta_2^2\theta_4^2}\right)^2, \quad \mu^2 = \left(\frac{\theta_3^2\theta_8^2}{\theta_4^2\theta_{10}^2}\right)^2, \quad \nu^2
=\left(\frac{\theta_1^2\theta_8^2}{\theta_2^2\theta_{10}^2}\right)^2.
\]
Each value for $(\lambda, \mu, \nu )$ gives isomorphic genus 2 curves. Hence, we can choose
\[ \lambda = \frac{\theta_1^2\theta_3^2}{\theta_2^2\theta_4^2}, \quad \mu = \frac{\theta_3^2\theta_8^2}{\theta_4^2\theta_{10}^2}, \quad \nu =
\frac{\theta_1^2\theta_8^2}{\theta_2^2\theta_{10}^2}.\]
This completes the proof.
\end{proof}
\subsection{Automorphism Groups of Curves}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve defined over an algebraically closed field $k$ of characteristic zero. We denote its
function field by $K:=k (\mathcal X}%\def\X{\mathfrak X)$ and $Aut(\mathcal X}%\def\X{\mathfrak X)=Aut(K/k)$ is the automorphism group of $\mathcal X}%\def\X{\mathfrak X$. In any characteristic different
from 2, the automorphism group $Aut(\mathcal X}%\def\X{\mathfrak X)$ is isomorphic to one of the groups given by the following lemma.
\begin{lem}\label{thm1} The automorphism group $G$ of a
genus 2 curve $\mathcal X}%\def\X{\mathfrak X$ in characteristic $\ne2$ is isomorphic to \ $C_2$, $C_{10}$, $V_4$, $D_8$, $D_{12}$, $C_3 \rtimes D_8$,
$ GL_2(3)$, or $2^+S_5$. The case $G {\, \cong\, } 2^+S_5$ occurs only in characteristic 5. If $G {\, \cong\, } \mathbb Z_3 \rtimes D_8$ (resp., $
GL_2(3)$), then $\mathcal X}%\def\X{\mathfrak X$ has equation $Y^2=X^6-1$ (resp., $Y^2=X(X^4-1)$). If $G {\, \cong\, } C_{10}$, then $\mathcal X}%\def\X{\mathfrak X$ has equation
$Y^2=X^6-X$.
\end{lem}
For the proof of the above lemma and the description of each group see \cite{S7}. For the rest of this chapter, we
assume that $char(k)=0.$
One of the main goals of Section 2.4 is to describe each locus of genus 2 curves with fixed automorphism group in terms
of the fundamental theta constants.
We have the following lemma.
\begin{lem}\label{possibleCurve}
Every genus two curve can be written in the form:
\[
y^2 = x \, (x-1) \, \left(x - \frac {\theta_1^2 \theta_3^2} {\theta_2^2 \theta_4^2}\right)\, \left(x^2 \, - \frac{\theta_2^2 \, \theta_3^2 +
\theta_1^2 \, \theta_4^2} { \theta_2^2 \, \theta_4^2} \cdot \alpha \, x + \frac {\theta_1^2 \theta_3^2} {\theta_2^2 \theta_4^2} \, \alpha^2 \right),
\]
where $\alpha = \frac {\theta_8^2} {\theta_{10}^2}$ can be given in terms of $\, \, \theta_1, \theta_2, \theta_3,$ and $\theta_4$,
\[
\alpha^2 + \frac {\theta_1^4 + \theta_2^4 - \theta_3^4 - \theta_4^4}{\theta_3^2 \theta_4^2 - \theta_1^2 \theta_2^2 } \, \alpha + 1 =0.
\]
Furthermore, if $\alpha = {\pm} 1$ then $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$.
\end{lem}
\proof Let us write the genus 2 curve in the following form: $$Y^2 = X (X-1) (X-\lambda) (X-\mu) (X-\nu)$$ where $\lambda ,\mu
,\nu$ are given by Eq. \eqref{Picard}. Let $\alpha := \frac {\theta_8^2} {\theta_{10}^2}$. Then,
\[
\begin{array}{ll}
\mu = \frac{\theta_3^2}{\theta_4^2}\, \alpha, & \nu = \frac{\theta_1^2}{\theta_2^2} \, \alpha.
\end{array}
\]
Using the following two identities,
\begin{equation}\label{Frobenius}
\begin{split}
\theta_8^4 + \theta_{10}^4 &= \theta_1^4+\theta_2^4-\theta_3^4-\theta_4^4, \\
\theta_8^2 \theta_{10}^2 &= \theta_1^2 \theta_2^2 - \theta_3^2 \theta_4^2
\end{split}
\end{equation}
we have
\begin{equation}\label{rootof}
\alpha^2 + \frac {\theta_1^4 + \theta_2^4 - \theta_3^4 - \theta_4^4}{\theta_3^2 \theta_4^2 - \theta_1^2 \theta_2^2 } \, \alpha + 1 = 0.
\end{equation}
If $\alpha=\pm 1$ then $\mu \nu = \lambda$. It is well known that this implies that the genus 2 curve has an elliptic
involution. Hence, $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$.
\endproof
\begin{rem} {i)} From the above we have that $\theta_8^4=\theta_{10}^4$ implies that $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$. Lemma \ref{lemma1}
determines a necessary and equivalent statement when $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$.
{ii)} The last part of Lemma 2.4 shows that if $\theta_8^4=\theta_{10}^4$, then all coefficients of the genus 2 curve are
given as rational functions of the four fundamental theta functions. Such fundamental theta functions determine the
field of moduli of the given curve. Hence, the curve is defined over its field of moduli.
\end{rem}
\begin{cor}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve which has an elliptic involution. Then $\mathcal X}%\def\X{\mathfrak X$ is defined over its field of moduli.
\end{cor}
This was the main result of \cite{Ca}.
\subsection{Describing the Locus of Genus Two Curves with Fixed Automorphism Group by Theta Constants}
The locus $\mathcal L_2$ of genus 2 curves $\mathcal X}%\def\X{\mathfrak X$ which have an elliptic involution is a closed subvariety of $\mathcal M_2$. Let
$W= \{\alpha_1, \alpha_2, \beta_1, \beta_2, \gamma_1, \gamma_2 \}$ be the set of roots of the binary sextic, and $A$ and $B$ be
subsets of $W$ such that $W=A \cup B$ and $| A \cap B | =2$. We define the cross ratio of the two pairs $z_1,z_2 ;
z_3,z_4 $ by
$$(z_1,z_2 ; z_3,z_4) = \frac{z_1 ; z_3,z_4}{z_2 ; z_3,z_4} = \frac{z_1-z_3}{z_1-z_4} : \frac{z_2-z_3}{z_2-z_4}.$$
Take $ A = \{ \alpha_1, \alpha_2, \beta_1 , \beta_2\}$ and $B = \{ \gamma_1, \gamma_2, \beta_1, \beta_2\}$.
Jacobi \cite{Krazer} gives a description of $\mathcal L_2$ in terms of the cross ratios of the elements of $W$:
$$ \frac {\alpha_1-\beta_1} {\alpha_1-\beta_2} : \frac {\alpha_2-\beta_1} {\alpha_2-\beta_2}= \frac {\gamma_1-\beta_1} {\gamma_1-\beta_2} : \frac
{\gamma_2-\beta_1} {\gamma_2-\beta_2}.
$$
We recall that the following identities hold for cross ratios:
\[
(\alpha_1,\alpha_2\,;\beta_1,\beta_2)=(\alpha_2,\alpha_1;\beta_2,\beta_1)=(\beta_1,\beta_2;\alpha_1,\alpha_2)=(\beta_2,\beta_1;\alpha_2,\alpha_1)
\]
and
\[
(\alpha_1,\alpha_2;\infty,\beta_2)=(\infty,\beta_2;\alpha_1,\alpha_2)=(\beta_2;\alpha_2,\alpha_1).
\]
Next, we use this result to determine relations among theta functions for a genus 2 curve in the locus $\mathcal L_2$. Let $\mathcal X}%\def\X{\mathfrak X$
be any genus 2 curve given by the equation
$$Y^2=X(X-1)(X-a_1)(X-a_2)(X-a_3).$$
We take $\infty \in A \cap B$. Then there are five cases for $\alpha \in A \cap B $, where $\alpha $ is an element of the set $
\{0,1, a_1, a_2, a_3\}$. For each of these cases there are three possible relationships for cross ratios as described below:\\
\noindent {i)} $A \cap B = \{ 0, \infty\}$: The possible cross ratios are
\begin{description}
\item $(a_1,1;\infty,0) = (a_3,a_2;\infty,0), \quad \quad (a_2,1;\infty,0) = (a_1,a_3;\infty,0),$ \\
\item $(a_1,1;\infty,0) =(a_2,a_3;\infty,0).$
\end{description}
\noindent {ii)} $A \cap B = \{ 1, \infty\}$: The possible cross ratios are
\begin{description}
\item $(a_1,0;\infty,1)=(a_2,a_3;\infty,1),\quad \quad (a_1,0;\infty,1)=(a_3,a_2;\infty,1),$\\
\item $(a_2,0;\infty,1)=(a_1,a_3;\infty,1).$
\end{description}
\noindent {iii)} $A \cap B = \{ a_1, \infty\}$: The possible cross ratios are
\begin{description}
\item $(1,0;\infty,a_1)=(a_3,a_2;\infty,a_1),\quad \quad (a_2,0;\infty,a_1)=(1,a_3;\infty,a_1),$\\
\item $(1,0;\infty,a_1)=(a_2,a_3;\infty,a_1).$
\end{description}
\noindent {iv)} $A \cap B= \{ a_2, \infty\}$: The possible cross ratios are
\begin{description}
\item $(1,0;\infty,a_2)=(a_1,a_3;\infty,a_2),\quad \quad (1,0;\infty,a_2)=(a_3,a_1;\infty,a_2),$\\
\item $(a_1,0;\infty,a_2)=(1,a_3;\infty,a_2).$
\end{description}
%
\noindent {v)} $A \cap B = \{ a_3, \infty\}$: The possible cross ratios are
\begin{description}
\item $(a_1,0;\infty,a_3)=(1,a_2;\infty,a_3),\quad \quad (1,0;\infty,a_3)=(a_2,a_1;\infty,
a_3),$\\
\item $(1,0;\infty,a_3)=(a_1,a_2;\infty,a_3).$
\end{description}
We summarize these relationships in Table 2.1.
\begin{lem}\label{lemma1}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve. Then $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } V_4$ if and only if the theta functions of $\mathcal X}%\def\X{\mathfrak X$ satisfy
\begin{equation}\label{V_4locus1}
\begin{split}
(\theta_1^4-\theta_2^4)(\theta_3^4-\theta_4^4)(\theta_8^4-\theta_{10}^4)
(-\theta_1^2\theta_3^2\theta_8^2\theta_2^2-\theta_1^2\theta_2^2\theta_4^2\theta_{10}^2+\theta_1^4\theta_3^2\theta_{10}^2+ \theta_3^2\theta_2^4\theta_{10}^2)\\
(\theta_3^2\theta_8^2\theta_2^2\theta_4^2-\theta_2^2\theta_4^4\theta_{10}^2+\theta_1^2\theta_3^2\theta_4^2\theta_{10}^2-\theta_3^4\theta_2^2\theta_{10}^2)
(-\theta_8^4\theta_3^2\theta_2^2+\theta_8^2\theta_2^2\theta_{10}^2\theta_4^2\\
+\theta_1^2\theta_3^2\theta_8^2\theta_{10}^2
-\theta_3^2\theta_2^2\theta_{10}^4)(-\theta_1^2\theta_8^4\theta_4^2-\theta_1^2\theta_{10}^4\theta_4^2+\theta_8^2\theta_2^2\theta_{10}^2\theta_4^2+\theta_1^2\theta_3^2\theta_8^2\theta_{10}^2)\\
(-\theta_1^2\theta_8^2\theta_3^2\theta_4^2+\theta_1^2\theta_{10}^2\theta_4^4
+\theta_1^2\theta_3^4\theta_{10}^2-\theta_3^2\theta_2^2\theta_{10}^2\theta_4^2)(-\theta_1^2\theta_8^2\theta_2^2\theta_4^2+\theta_1^4\theta_{10}^2\theta_4^2\\
-\theta_1^2\theta_3^2\theta_2^2\theta_{10}^2+\theta_2^4\theta_4^2\theta_{10}^2) (-\theta_8^4\theta_2^2\theta_4^2
+\theta_1^2\theta_8^2\theta_{10}^2\theta_4^2-\theta_2^2\theta_{10}^4\theta_4^2+\theta_3^2\theta_8^2\theta_2^2\theta_{10}^2)\\
(\theta_1^4\theta_8^2\theta_4^2-\theta_1^2\theta_2^2\theta_4^2\theta_{10}^2-\theta_1^2\theta_3^2\theta_8^2\theta_2^2+\theta_8^2\theta_2^4\theta_4^2)
(\theta_1^4\theta_3^2\theta_8^2-\theta_1^2\theta_8^2\theta_2^2\theta_4^2\\
%
-\theta_1^2\theta_3^2\theta_2^2\theta_{10}^2+\theta_3^2\theta_8^2\theta_2^4)(\theta_1^2\theta_8^4\theta_3^2-\theta_1^2\theta_8^2\theta_{10}^2\theta_4^2+\theta_1^2\theta_3^2\theta_{10}^4
-\theta_3^2\theta_8^2\theta_2^2\theta_{10}^2)\\
(\theta_1^2\theta_8^2\theta_4^4-\theta_1^2\theta_3^2\theta_4^2\theta_{10}^2+\theta_1^2\theta_3^4\theta_8^2-\theta_3^2\theta_8^2\theta_2^2\theta_4^2)
=0.
\end{split}
\end{equation}
\end{lem}
However, we are unable to determine a similar result for cases $D_8$ or $D_{12}$ by this argument. Instead, we will
use the invariants of genus 2 curves and a more computational approach. In the process, we will offer a different proof
for the lemma above.
\begin{center}
\begin{table}\label{tab_1}
\caption{Relation of theta functions and cross ratios}
\begin{tabular}{|c|c|c|c|}
\hline & Cross ratio & $f(a_1,a_2,a_3)=0$ & theta constants \tabularnewline[8pt]
\hline
1& $(1,0;\infty,a_1)=(a_3,a_2;\infty,a_1)$ & $a_1a_2+a_1-a_3a_1-a_2$ &
$-\theta_1^2\theta_3^2\theta_8^2\theta_2^2-\theta_1^2\theta_2^2\theta_4^2\theta_{10}^2+$\\
& & &$\theta_1^4\theta_3^2\theta_{10}^2+ \theta_3^2\theta_2^4\theta_{10}^2$ \tabularnewline[4pt]
\hline
2 & $(a_2,0;\infty,a_1)=(1,a_3;\infty,a_1)$ & $a_1a_2-a_1+a_3a_1-a_3a_2$ &
$\theta_3^2\theta_8^2\theta_2^2\theta_4^2-\theta_2^2\theta_4^4\theta_{10}^2+$\\
& && $\theta_1^2\theta_3^2\theta_4^2\theta_{10}^2-\theta_3^4\theta_2^2\theta_{10}^2$ \tabularnewline[4pt]
\hline
3& $(1,0;\infty,a_1)=(a_2,a_3;\infty,a_1)$ & $a_1a_2-a_1-a_3a_1+a_3$ & $-\theta_8^4\theta_3^2\theta_2^2+\theta_8^2\theta_2^2\theta_{10}^2\theta_4^2+$ \\
& &&$\theta_1^2\theta_3^2\theta_8^2\theta_{10}^2-\theta_3^2\theta_2^2\theta_{10}^4$ \tabularnewline[4pt]
\hline
4& $(1,0;\infty,a_2)=(a_1,a_3;\infty,a_2)$ & $a_1a_2-a_2-a_3a_2+a_3$ & $-\theta_1^2\theta_8^4\theta_4^2-\theta_1^2\theta_{10}^4\theta_4^2+$ \\
& &&$\theta_8^2\theta_2^2\theta_{10}^2\theta_4^2+\theta_1^2\theta_3^2\theta_8^2\theta_{10}^2$ \tabularnewline[4pt]
\hline
5& $(1,0;\infty,a_2)=(a_3,a_1;\infty,a_2)$ & $a_1a_2-a_1+a_2-a_3a_2$&$-\theta_1^2\theta_8^2\theta_3^2\theta_4^2+\theta_1^2\theta_{10}^2\theta_4^4+$\\
&&&$\theta_1^2\theta_3^4\theta_{10}^2-\theta_3^2\theta_2^2\theta_{10}^2\theta_4^2$ \tabularnewline[4pt]
\hline
6 & $(a_1,0;\infty,a_2)=(1,a_3;\infty,a_2)$ & $a_1a_2-a_3a_1-a_2+a_3 a_2$ &
$-\theta_1^2\theta_8^2\theta_2^2\theta_4^2+\theta_1^4\theta_{10}^2\theta_4^2-$ \\
& &&$\theta_1^2\theta_3^2\theta_2^2\theta_{10}^2+\theta_2^4\theta_4^2\theta_{10}^2$ \tabularnewline[4pt]
\hline
7&$(a_1,0;\infty,a_3)=(1,a_2;\infty,a_3)$ & $a_1a_2-a_3a_1-a_3a_2+a_3$ &
$-\theta_8^4\theta_2^2\theta_4^2+\theta_1^2\theta_8^2\theta_{10}^2\theta_4^2-$ \\
&&&$\theta_2^2\theta_{10}^4\theta_4^2+\theta_3^2\theta_8^2\theta_2^2\theta_{10}^2$ \tabularnewline[4pt]
\hline
8&$(1,0;\infty,a_3)=(a_2,a_1;\infty, a_3)$&$a_3a_1-a_1-a_3a_2+a_3$ & $\theta_8^4-\theta_{10}^4$\tabularnewline[4pt]
\hline
9&$(1,0;\infty,a_3)=(a_1,a_2;\infty,a_3)$ &$ a_3a_1+a_2-a_3-a_3a_2$ &
$\theta_1^4\theta_8^2\theta_4^2-\theta_1^2\theta_2^2\theta_4^2\theta_{10}^2-$ \\
&&&$\theta_1^2\theta_3^2\theta_8^2\theta_2^2+\theta_8^2\theta_2^4\theta_4^2$ \tabularnewline[4pt]
\hline
10&$(a_1,0;\infty,1)=(a_2,a_3;\infty,1)$ & $-a_1+a_3a_1+a_2-a_3 $&$\theta_1^4\theta_3^2\theta_8^2-\theta_1^2\theta_8^2\theta_2^2\theta_4^2-$\\
&&&$\theta_1^2\theta_3^2\theta_2^2\theta_{10}^2+\theta_3^2\theta_8^2\theta_2^4$ \tabularnewline[4pt]
\hline
11& $(a_1,0;\infty,1)=(a_3,a_2;\infty,1)$ &$ a_1a_2-a_1-a_2+a_3$&$\theta_1^2\theta_8^4\theta_3^2-\theta_1^2\theta_8^2\theta_{10}^2\theta_4^2+$\\
&&&$\theta_1^2\theta_3^2\theta_{10}^4-\theta_3^2\theta_8^2\theta_2^2\theta_{10}^2$ \tabularnewline[4pt]
\hline
12& $(a_2,0;\infty,1)=(a_1,a_3;\infty,1)$&$a_1-a_2+a_3a_2-a_3$&$\theta_1^2\theta_8^2\theta_4^4-\theta_1^2\theta_3^2\theta_4^2\theta_{10}^2+$ \\
&&&$\theta_1^2\theta_3^4\theta_8^2-\theta_3^2\theta_8^2\theta_2^2\theta_4^2$ \tabularnewline[4pt]
\hline
13&$(a_1,1;\infty,0) = (a_3,a_2;\infty,0)$ & $a_1a_2-a_3$ & $\theta_8^4-\theta_{10}^4 $ \tabularnewline[4pt]
\hline
14& $(a_2,1;\infty,0) = (a_1,a_3;\infty,0)$ & $a_1-a_3a_2$ &$ \theta_3^4-\theta_4^4$ \tabularnewline[4pt]
\hline
15&$(a_1,1;\infty,0) = (a_2,a_3;\infty,0)$ & $a_3a_1-a_2$ & $ \theta_1^4-\theta_2^4 $ \tabularnewline[4pt] \hline
\end{tabular}
\end{table}
\end{center}
\begin{lem}
{i)} The locus $\mathcal L_2$ of genus 2 curves $\mathcal X}%\def\X{\mathfrak X$ which have a degree 2 elliptic subcover is a closed subvariety of
$\mathcal M_2$. The equation of $\mathcal L_2$ is given by
\[
\begin{split}\label{eq_L2_J}
8748J_{10}J_2^4J_6^2- 507384000J_{10}^2J_4^2J_2-19245600J_{10}^2J_4J_2^3-6912J_4^3J_6^{34}\\
-592272J_{10}J_4^4J_2^2 +77436J_{10}J_4^3J_2^4 -3499200J_{10}J_2J_6^3+4743360J_{10}J_4^3J_2J_6\\
-870912J_{10}J_4^2J_2^3J_6
+3090960J_{10}J_4J_2^2J_6^2 -78J_2^5J_4^5-125971200000J_{10}^3\\
-81J_2^3J_6^4+1332J_2^4J_4^4J_6 +384J_4^6J_6+41472J_{10}J_4^5+159J_4^6J_2^3 &\\
-47952J_2J_4J_6^4
+104976000J_{10}^2J_2^2J_6-1728J_4^5J_2^2J_6+6048J_4^4J_2J_6^2\\
\end{split}
\]
\begin{equation}
\begin{split}
%
-9331200J_{10}J_4^2J_6^2 -J_2^7J_4^4 +12J_2^6J_4^3J_6+29376J_2^2J_4^2J_6^3-8910J_2^3J_4^3J_6^2\\
%
-2099520000J_{10}^2J_4J_6+31104J_6^5 -5832J_{10}J_2^5J_4J_6 -54J_2^5J_4^2J_6^2 \\
%
-236196J_{10}^2J_2^5-80J_4^7J_2 +108J_2^4J_4J_6^3 +972J_{10}J_2^6J_4^2 = & 0.
\end{split}
\end{equation}
{ii)} The locus of genus 2 curves $\mathcal X}%\def\X{\mathfrak X$ with $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } D_8$ is given by the equation of $\mathcal L_2$ and
\begin{equation}
\label{D_8_locus} 1706J_4^2J_2^2+2560J_4^3+27J_4J_2^4-81J_2^3J_6-14880J_2J_4J_ 6+28800J_6^2 =0.
\end{equation}
{iii)} The locus of genus 2 curves $\mathcal X}%\def\X{\mathfrak X$ with $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } D_{12}$ is
\begin{equation}
\label{D_12_locus}
\begin{split}
-J_4J_2^4+12J_2^3J_6-52J_4^2J_2^2+80J_4^3+960J_2J_4J_6-3600
J_6^2 &=0, \\
864J_{10}J_2^5+3456000J_{10}J_4^2J_2-43200J_{10}J_4J_2^3-
2332800000J_{10}^2\\
-J_4^2J_2^6 -768J_4^4J_2^2+48J_4^3J_2^4+4096J_4^5 &=0.\\
\end{split}
\end{equation}
\end{lem}
Our goal is to express each of the above loci in terms of the theta characteristics. We obtain the following result.
\begin{thm}\label{theorem1}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve. Then the following hold:
{i)} $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } V_4$ if and only if the relations of theta functions given Eq.~\eqref{V_4locus1} holds.
{ii)} $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } D_8$ if and only if the Eq. I in \cite{web} is satisfied.
{iii)} $Aut(\mathcal X}%\def\X{\mathfrak X){\, \cong\, } D_{12}$ if and only if the Eq. II and Eq. III in \cite{web} are satisfied.
\end{thm}
\proof Part {i)} of the theorem is Lemma~\ref{lemma1}. Here we give a somewhat different proof. Assume that $\mathcal X}%\def\X{\mathfrak X$ is a
genus 2 curve with equation
$$Y^2=X(X-1)(X-a_1)(X-a_2)(X-a_3)$$
whose classical invariants satisfy Eq.~\eqref{eq_L2_J}. Expressing the classical invariants of $\mathcal X}%\def\X{\mathfrak X$ in terms of $a_1,
a_2, a_3$, substituting them into \eqref{eq_L2_J}, and factoring the resulting equation yields
\medskip
\begin{equation}
\begin{split}\label{L2_factored}
(a_1a_2-a_3)^2 (a_1-a_3a_2)^2 (a_3a_1-a_2)^2 (a_1a_2-a_2-a_3a_2+a_3)^2\\
(a_3a_1+a_2-a_3-a_3a_2)^2(-a_1+a_3a_1+a_2-a_3)^2(a_1a_2-a_1-a_2+a_3)^2\\
(a_1a_2-a_1+a_3a_1-a_3a_2)^2(a_1a_2-a_3a_1-a_3a_2+a_3)^2\\
(a_3a_1-a_1-a_3a_2+a_3)^2(a_1a_2+a_1-a_3a_1-a_2)^2\\
(a_1a_2-a_1-a_3a_1+a_3)^2 (a_1a_2-a_1+a_2-a_3a_2)^2\\
(a_1-a_2+a_3a_2-a_3)^2(a_1a_2-a_3a_1-a_2+a_3 a_2)^2 =&\, 0.
\end{split}
\end{equation}
It is no surprise that we get the 15 factors of Table 2.1. The relations of theta constants follow from Table 2.1.
{ii)} Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve which has an elliptic involution. Then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with the
equation
$$Y^2=X(X-1)(X-a_1)(X-a_2)(X-a_1 a_2).$$
If $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X) {\, \cong\, } D_8$ then the $SL_2 (k)$-invariants of such curve must satisfy Eq.~\eqref{D_8_locus}. Then, we get
the equation in terms of $a_1$ and $a_2$.
By writing the relation $a_3 = a_1 a_2$ in terms of theta constants, we get $\theta_4^4 = \theta_3^4$. All the results above
lead to part ii) of the theorem.
{iii)} The proof of this part is similar to part ii).
\endproof
We express the conditions of the previous lemma in terms of the fundamental theta constants only.
\begin{lem}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve. Then we have the following:
\noindent {i)} $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$ if and only if the fundamental theta constants of $\mathcal X}%\def\X{\mathfrak X$ satisfy
\begin{equation}\label{V_4locus2}
\begin{split}
( \theta_{{3}}^4-\theta_{{4}}^4 ) (\theta_{{1}}^4 -\theta_{{3}}^4 ) ( \theta_{{2}}^4-\theta_{{4}}^4 ) ( \theta_{{1}}^4
-\theta_{{4}}^4 ) ( \theta_{{3}}^4-\theta_{{2}}^4 ) ( \theta_{{1}}^4- \theta_{{2}}^4 ) \\
( -\theta_{{4}}^2+\theta_{{3}}^2 +\theta_{{1}}^2-\theta_{{2}}^2 )(
\theta_{{4}}^2-\theta_{{3}}^2+\theta_{{1}}^2-\theta_{{2}}^2
) ( -\theta_{{4}}^2-\theta_{{3}}^2+\theta_{{2}}^2+\theta_{{
1}}^2 ) \\
( \theta_{{4}}^2+\theta_{{3}}^2+\theta_{{2}}^2+\theta_ {{1}}^2 ) ( {\theta_{{1}}}^{4}{\theta_{{2}}}^{4}+
{\theta_{{3}}}^{4}{\theta_{{2}}}^{4} +{\theta_{{1}}}^{4}{\theta_{{3}}}^{4} -2\,\theta_{{1}}^2\theta_{{2}}^2\theta
_{{3}}^2\theta_{{4}}^2 )\\
\left( -{\theta_{{3}}}^{4}{\theta_{{2}}}^{4}-{
\theta_{{2}}}^{4}{\theta_{{4}}}^{4}-{\theta_{{3}}}^{4}{\theta_{{4}}}^{4} + 2\,\theta_{{1}}^2\theta_{{2}}^2\theta_
{{3}}^2\theta_{{4}}^2 \right)( {\theta_{{2}}}^{4}{\theta_{{4}}}^{4} +{\theta_{{1}}}^{4}{\theta _{{2}}}^{4} +{
\theta_{{1}}}^{4}{\theta_{{4}}}^{4}\\
-2\,\theta_{{1}}^2\theta_{{2}}^2\theta_{{3}}^2\theta_{{4}}^2 ) \left(
{\theta_{{1}}}^{4}{\theta_{{4}}}^{4}+{\theta_{{3}}}^{4}{\theta_{{4}}}^{4}+{\theta_{{1}}}^{4}{\theta_{ {3}}}^{4}
-2\,\theta_{{1 }}^2\theta_{{2}}^2\theta_{{3}}^2\theta_{{4}}^2\right) = & \, 0.\\
\end{split}
\end{equation}
\noindent {ii)} $D_8 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$ if and only if the fundamental theta constants of $\mathcal X}%\def\X{\mathfrak X$ satisfy Eq.(3) in
\cite{Sh}.
\noindent {iii)} $D_6 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$ if and only if the fundamental theta constants of $\mathcal X}%\def\X{\mathfrak X$ satisfy Eq.(4) in
\cite{Sh}.
\end{lem}
\proof Notice that Eq.~\eqref{V_4locus1} contains only $\theta_1, \theta_2, \theta_3, \theta_4, \theta_8$ and $\theta_{10}.$ Using
Eq.~\eqref{Frobenius}, we can eliminate $\theta_8$ and $\theta_{10}$ from Eq.~\eqref{V_4locus1}.
The $J_{10}$ invariant of any genus two curve is given by the following in terms of theta constants:
\[ J_{10} = \frac{\theta_1^{12} \theta_3^{12}}{\theta_2^{28} \theta_4^{28} \theta_{10}^{40}} \, (\theta_1^2\theta_2^2 - \theta_3^2 \theta_4^2)^{12} (\theta_1^2\theta_4^2 - \theta_2^2
\theta_3^2)^{12} (\theta_1^2\theta_3^2 - \theta_2^2 \theta_4^2)^{12}.\] Since $J_{10} \neq 0,$ the factors $(\theta_1^2\theta_2^2 - \theta_3^2 \theta_4^2),
(\theta_1^2\theta_4^2 - \theta_2^2 \theta_3^2)$ and $(\theta_1^2\theta_3^2 - \theta_2^2 \theta_4^2)$ cancel in the equation of the $V_4$ locus. The
result follows from Theorem ~\ref{theorem1}. The proof of part ii) and iii) is similar and we avoid details.
\endproof
\begin{rem}
For part ii) and iii), the equations are lengthy and we don't show them here. But by using the extra conditions $\theta_4^2
= \theta_3^2$ or $\theta_4^2 = - \theta_3^2$, we could simplify the equation of the $D_8$ locus as follows:
\noindent {i)}When $\theta_4^2 = \theta_3^2$, we have
\begin{equation}
\begin{split}
( \theta_{{1}}^4-\theta_{{2}}^4 ) ( \theta_{{1}}^2\theta_{{2}}^2-{\theta_{{3}}}^{4} ) ( {
\theta_{{2}}}^{2}+{\theta_{{1}}}^{2}+2\,{\theta_{{3}}}^{2} ) (
{\theta_{{2}}}^{2}+{\theta_{{1}}}^{2}-2\,{\theta_{{3}}}^{2}
) (
2\,{\theta_{{1}}}^{4} \\
-2\,{ \theta_{{1}}}^{2}{\theta_{{2}}}^{2} +{\theta_{{3}}}^{4} )
( -2\,{\theta_{{ 2}}}^{4}-{\theta_{{3}}}^{4}+2\,{\theta_{{1}}}^{2}{\theta_{{2}}}^{2}
) ( -10\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{12}{\theta
_{{3}}}^{8}\\
+206\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{4}{\theta_{{3}}}^{ 16} +8\,{\theta_{{1}}}^{8}{ \theta_{{2}}}^{16}
-34\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{8}{\theta_{{3}}}^{12} -126\,{
\theta_{{1}}}^{2}{\theta_{{2}}}^{6}{\theta_{{3}}}^{16}\\
+18\,{\theta_{{1 }}}^{2}{\theta_{{2}}}^{10}{\theta_{{3}}}^{12}+27\,{\theta_{{1}}}^{8}{\theta_{{3}}}^{16}
-132\,{\theta_{{1}}}^{8}{\theta_{{2}}}^{8}{\theta_{{3}}}^{8}
-34\,{\theta_{{1}
}}^{8}{\theta_{{2}}}^{4}{\theta_{{3}}}^{12}\\
-16\,{\theta_{{1}}}^{8}{ \theta_{{3}}}^{4}{\theta_{{2}}}^{12} -16\,{\theta_{{1}}}^{6}{\theta_{{2
}}}^{14}{\theta_{{3}}}^{4} -126\,{\theta_{{1}}}^{6}{\theta_{{2}}}^{2}{ \theta_{{3}}}^{16}
+24\,{\theta_{{1}}}^{6}{\theta_{{2}}}^{6}{\theta_{{3 }}}^{12}\\
+68\,{\theta_{{1}}}^{6}{\theta_{{2}}}^{10}{\theta_{{3}}}^{8} -
24\,{\theta_{{1}}}^{12}{\theta_{{2}}}^{12}+8\,{\theta_{{2}}}^{8}{\theta _{{1}}}^{16} -10\,{\theta_{{1}}}^{12}{
\theta_{{3}}}^{8}{\theta_{{2}}}^{4} \\
-16\,{\theta_{{1}}}^{12}{\theta_{{3 }}}^{4}{\theta_{{2}}}^{8} +88\,{\theta_{{1}}}^{10}{\theta_{{3}}}^{4}{
\theta_{{2}}}^{10} +18\,{\theta_{{1}}}^{10}{\theta_{{2}}}^{2}{\theta_{{ 3}}}^{12}
+68\,{\theta_{{1}}}^{10}{\theta_{{3}}}^{8}{\theta_{{2}}}^{6} \\
+27\,{\theta_{{2}}}^{8}{\theta_{{3}}}^{16}
-16\,{\theta_{{1}}}^{14}{\theta_{{2}}}^{6}{\theta_{{3}}}^{ 4} ) = & 0.
\end{split}
\end{equation}
\noindent {ii)} When $\theta_4^2 = -\theta_3^2$, we have
\begin{equation}
\begin{split}
( \theta_{{1}}^4-\theta_{{2}}^4 ) ( {\theta _{{3}}}^{4}+{\theta_{{1}}}^{2}{\theta_{{2}}}^{2} ) (
-{\theta_{{2}}}^{2}+{\theta_{{1}}}^{2}-2\,{\theta_{{3}}}^{2} ) (
-{\theta_{{2}}}^{2}+{\theta_{{1}}}^{2}+2\,{\theta_{{3}}}^{2 } )\\
({\theta_{{3}}}^{ 4} +2\,{\theta_{{1}}}^{2}{\theta_{{2}}}^{2} +2\,{\theta_{{1}}}^{4} )
( 2\,{ \theta_{{2}}}^{4}+{\theta_{{3}}}^{4}+2\,{\theta_{{1}}}^{2}{\theta_{{2} }}^{2}) (
206\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{4}{\theta_{{3
}}}^{16}\\
-10\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{12}{ \theta_{{3}}}^{8}+27\,{\theta_{{2}}}^{8}{\theta_{{3}}}^{16}
-34\,{\theta_{{1}}}^{4}{\theta_{{2}}}^{8}{\theta_{{3}}}^{12}
+ 126\,{\theta_{{1}}}^{2}{\theta_{{2}}}^{6}{\theta_{{3}}}^{16}\\
-18\,{ \theta_{{1}}}^{2}{\theta_{{2}}}^{10}{\theta_{{3}}}^{12} -68\,{\theta_{{1}}}^{10}{\theta_{{3}}}^{8}{\theta_{{
2}}}^{6} +8\,{\theta_{{1 }}}^{8}{\theta_{{2}}}^{16} +27\,{\theta_{{1}}}^{8}{\theta_{{3}}}^{16} \\
-132\,{\theta_{{1}}}^{8}{\theta_{{2}}}^{8}{\theta_{{3}}}^{8}-34\,{
\theta_{{1}}}^{8}{\theta_{{2}}}^{4}{\theta_{{3}}}^{12} -16\,{\theta_{{1 }}}^{8}{\theta_{{3}}}^{4}{\theta_{{2}}}^{12}
+16\,{\theta_{{1}}}^{6}{ \theta_{{2}}}^{14}{\theta_{{3}}}^{4} \\
+126\,{\theta_{{1}}}^{6}{\theta_{{ 2}}}^{2}{\theta_{{3}}}^{16} -24\,{\theta_{{1}}}^{6}{\theta_{{2}}}^{6}{
\theta_{{3}}}^{12} -68\,{\theta_{{1}}}^{6}{\theta_{{2}}}^{10}{\theta_{{
3}}}^{8}-24\,{\theta_{{1}}}^{12}{\theta_{{2}}}^{12} \\
+16\,{\theta_{{1}}}^{14}{\theta_{{2}}}^{6}{\theta _{{3}}}^{4} -10\,{\theta_{{1}}}
^{12}{\theta_{{3}}}^{8}{\theta_{{2}}}^{4} -16\,{\theta_{{1}}}^{12}{ \theta_{{3}}}^{4}{\theta_{{2}}}^{8}
-88\,{\theta_{{1}}}^{10}{\theta_{{3 }}}^{4}{\theta_{{2}}}^{10}\\
-18\,{\theta_{{1}}}^{10}{\theta_{{2}}}^{2}{ \theta_{{3}}}^{12} +8\,{\theta_{{2}}}^{ 8}{\theta_{{1}}}^{16} ) = & 0.
\end{split}
\end{equation}
\end{rem}
Define the following as\\
\[A = (\frac{\theta_2}{\theta_1})^4, \quad B = (\frac{\theta_3}{\theta_1})^4, \quad C=(\frac{\theta_4}{\theta_1})^4,
\quad D = (\frac{\theta_8}{\theta_1})^4, \quad E =(\frac{\theta_{10}}{\theta_1})^4.
\]
Using the two identities given by Eq.~\eqref{Frobenius}, we have
\[
\begin{split}
1+A-B-C-D-E & = 0, \\
{A}^{2}-2\, D EA+2\,BCA+{C}^{2}{B}^{2}-2\, D ECB+{D}^{2}{E}^{2} & =0.
\end{split}
\]
Then we formulate the following lemma.
\def\hookrightarrow{\hookrightarrow}
\begin{lem}
Let $\mathcal X}%\def\X{\mathfrak X$ be a genus 2 curve. Then $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X)$ if and only if the theta constants of $\mathcal X}%\def\X{\mathfrak X$ satisfy
\begin{equation}
\begin{split}
( B-A ) ( A-C ) ( B-C ) ( 1-A ) ( 1-B )( 1-C )( 1-2\,C+2\,A +{A}^{2}{C}^{2}
\\
%
-4\, D E
-AC -2\,{A}^{2}BC
+2\,A D EBC+A{B}^{2}+ D EBC +AD EB -{A}^{2}\\
%
+4\,ABC
-2\,A{B}^{2}{C}^{2}
-{A}^{2}B+A D E
-{B}^{2}{C}^{2}-2\,B{C}^{2}+{B}^{2}C ) ( - D EBC\\
%
-4\,ABC
+{B}^{2}{C}^{2} +AC+A{B}^{2}C -A DEB
+{A}^{2}
+{A}^{2}C
+AB{C}^{2}\\
%
- D EC
-2\,A DEC-{A}^{2}{C}^{2}-{A}^{2}BC-A{C}^{2} -A D E ) =& 0.
\end{split}
\end{equation}
\end{lem}
\section{Genus 3 curves}
\subsection{Introduction to Genus 3 Curves} In this section, we focus on genus 3 cyclic curves. The
locus $\mathcal L_3$ of genus $3$ hyperelliptic curves with extra involutions is a $3$-dimensional subvariety of $\mathcal H_3.$ If $\mathcal X}%\def\X{\mathfrak X
\in \mathcal L_3$ then $V_4 \hookrightarrow Aut(\mathcal X}%\def\X{\mathfrak X).$ The normal form of the hyperelliptic genus $3$ curve is given by $$y^3 =
x^8 + a_3 X^6+a_2 x^4 + a_1 x^2 + 1$$ and the dihedral invariants of $\mathcal X}%\def\X{\mathfrak X_3$ are $u_1 = a_1^4 + a_3 ^4, u_2 =(a_1^2 +
a_3^2) a_2 , u_3 = 2 a_1 a_3.$ The description of the locus of genus $3$ hyperelliptic curves in terms of dihedral
invariants or classical invariants is given in \cite{S2}. We would like to describe the locus of genus $3$
hyperelliptic curves with extra involutions and all its sub loci in terms of theta functions.
The list of groups that occur as automorphism groups of genus $3$ curves has been computed by many authors. We denote
the following groups by $G_1$ and $G_2$: $$G_1 = \langle x,y | x^2, y^6 ,x y x y^4 \rangle, \quad \quad \quad G_2 =
\langle x,y |x^4, y^4, (xy)^2, (x^{-1} y)^2 \rangle .$$
In Table 2, we list all possible hyperelliptic genus 3 algebraic curves; see \cite{MS} for details. In this case
$\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X)$ has a central subgroup $C$ of order 2 such that the genus of $\mathcal X}%\def\X{\mathfrak X^C$ is zero. In the second column of the
table, the groups which occur as full automorphism groups are given, and the third column indicates the reduced
automorphism group for each case. The dimension $\delta$ of the locus and the equation of the curve are given in the
next two columns. The last column is the GAP identity of each group in the library of small
groups in GAP. Note that $C_2, C_4$ and $C_{14}$ are
the only groups which don't have extra involutions. Thus, curves with automorphism group $C_2, C_4$ or $C_{14}$ do not
belong to the locus $\mathcal L_3$ of genus 3 hyperelliptic curves with extra involutions.
In Table 3, we list the automorphism groups of genus $3$ nonhyperelliptic curves. In the table, the second column
represents the normal cyclic subgroup $C$ such that $g(\mathcal X}%\def\X{\mathfrak X^C) = 0.$ For the last 3 cases in the table, the automorphism
groups of the curves are not normal homocyclic covers of $\mathbb P^1.$ The only cyclic curves are curves with automorphism
groups $C_4^2 \rtimes S_3,$ $C_3,$ $C_6,$ $C_9$ and two other groups given by $(16,13)$ and $(48,33)$ in GAP identity.
In this chapter we write the equations of the cyclic curves of genus 3 by using theta constants.
\medskip
\begin{center}
\begin{table}[t!]\label{tablehyperelliptic}
\caption{Genus 3 hyperelliptic curves and their automorphisms}
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline & & & & &\\
&$\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X)$ & $\mbox{$\overline{Aut}$ } (\mathcal X}%\def\X{\mathfrak X)$ & $\, \delta \, $ &equation $y^2= f(x) $ & Id.\\
& & & & &\\
\hline \hline &&&& & \\
1 & $C_2$ &$\{1\}$ & 5&$x(x-1)(x^5+ax^4+bx^3+cx^2+dx+e)$& $(2,1)$ \\
& & & & &\\
2 & $C_2 \times C_2$ &$C_2$& 3&$ x^8 + a_3 x^6 + a_2 x^4 + a_1 x^2 + 1$ & $(4,2)$\\
3 & $C_4$ & $C_2$ &2&$x(x^2-1)(x^4+ax^2+b)$& $(4,1)$\\
4 & $C_{14}$ &$C_7$ &0&$x^7-1$ & $(14,2)$ \\
& & & & &\\
5 & $C_2^3$ &$D_4$ &2&$(x^4+ax^2+1)(x^4+bx^2+1)$ &$(8,5)$\\
6 & $C_2 \times D_8$ &$D_8$ &1&$x^8+ax^4+1$ & $(16,11)$ \\
7 & $C_2\times C_4$ &$D_4$ & 1&$(x^4-1)(x^4+ax^2+1)$& $(8,2)$\\
8 & $D_{12}$ &$D_6$ &1&$x(x^6+ax^3+1)$ &$(12,4)$\\
9 & $G_1$ &$D_{12}$ & 0&$x(x^6-1)$ & $(24,5)$\\
10 & $G_2$ &$D_{16}$ &0& $x^8-1$& $(32,9)$\\
& & & & &\\
11 & $C_2 \times S_4$ & $S_4$ &0 & $x^8+14x^2+1$ & $(48,48)$ \\
& & & & &\\
\hline\hline
\end{tabular}
\end{table}
\end{center}
\begin{center}
\begin{table}[h!]\label{tableNonHyper}
\caption{Genus 3 non hyperelliptic curves and their automorphisms}
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline
&&& &&\\
$\#$ & $Aut(\mathcal X}%\def\X{\mathfrak X)$ & $C$ & $Aut(\mathcal X}%\def\X{\mathfrak X)/C$& equation & Id. \\
&&&&& \\
\hline \hline
&&&&&\\
1& $V_4$ & $ V_4$ & $\{1\}$& $x^4+y^4+ax^2y^2+bx^2+cy^2+1=0$&(4,2) \\
2& $D_8$ & $ V_4$ & $C_2$& take\ $b=c$& (8,3)\\
3 & $S_4$ & $ V_4$ & $S_3$& take\ $a=b=c$ &(24,12) \\
4& $C_4^2 {\Bbb o} S_3$ & $ V_4$ & $S_4$& \ take \, $a=b=c=0$ \, or\, $y^4=x(x^2-1)$ & (96,64) \\
\hline
5 & $16$ & $C_4$& $V_4$& $y^4=x(x-1)(x-t)$&(16,13) \\
6& $48$ & $C_4$& $A_4$& $y^4=x^3-1$ &(48,33) \\
\hline
7& $C_3$ & $C_3$& $\{1\}$& $y^3=x(x-1)(x-s)(x-t)$&(3,1) \\
8& $C_6$ & $C_3$& $C_2$& take\ $s=1-t$& (6,2) \\
9& $C_9$ & $C_3$& $C_3$ & $y^3=x(x^3-1)$&(9,1)\\
\hline &&&&&\\
10& $L_3(2)$ & & & $x^3y+y^3z+z^3x=0$ &(168,42) \\
\hline
& &&&&\\
11& $S_3$ & & & $a(x^4+y^4+z^4)+b(x^2y^2+x^2z^2+y^2z^2)+$&(6,1)\\
& & & &$c(x^2yz+y^2xz+z^2xy)=0$&\\
\hline& && &&\\
12& $C_2$ & & & $ x^4+x^2(y^2+az^2) + by^4+cy^3z+dy^2z^2$&(2,1)\\
&&&& $ +eyz^3+gz^4=0$, \ \ either $e=1$ or $g=1$ &\\
\hline \hline
\end{tabular}
\end{table}
\end{center}
Figure ~\ref{figgenus=3} describes the inclusions among all subloci for genus 3 curves. In order to study such
inclusions, the lattice of the list of automorphism groups of genus 3 curves needs to be determined. Let's consider the
locus of the hyperelliptic curve whose automorphism group is $V_4 = \{1,\alpha,\beta,\alpha \beta\}.$ Suppose $\alpha$
is the hyperelliptic involution. Since the hyperelliptic involution is unique, the genus of the quotient curve
$\mathcal X}%\def\X{\mathfrak X^{\langle \beta \rangle}$ is 1. Also we have $\langle \alpha \rangle \cong C_2 \hookrightarrow V_4$ and $\langle
\beta \rangle \cong C_2 \hookrightarrow V_4.$ Therefore the locus of the hyperelliptic curve with automorphism group
$V_4$ can be embedded into two different loci with automorphism group $C_2.$ One comes from a curve that has
hyperelliptic involution and the other comes from a curve which does not have hyperelliptic involution. Similarly we
can describe the inclusions of each locus. The lattice of the automorphism groups for genus 3 curves is given Figure
1.
\subsection{Theta Functions for Hyperelliptic Curves} For genus three hyperelliptic curves, we have 28
odd theta characteristics and 36 even theta characteristics. The following shows the corresponding characteristics for
each theta function. The first 36 are for the even functions and the last 28 are for the odd functions. For simplicity,
we denote them by $\theta_i(z)$ instead of $\theta_i \ch {a} {b} (z , \tau)$ where $i=1,\dots ,36$ for the even functions and
$i=37, \dots, 64$ for the odd functions.
\[
\begin{split}
\theta_1(z) &= \theta_1 \chs {0}{0}{0}{0} 0 0 (z , \tau), \qquad \qquad
\theta_2(z) = \theta_2 \chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_3(z) &= \theta_3 \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 0 (z , \tau), \qquad \qquad
\theta_4(z) = \theta_4 \chs 0 0 0 {\frac{1}{2}} 0 0 (z , \tau)\\
\theta_5(z) &= \theta_5 \chs {\frac{1}{2}} 0 0 0 {\frac{1}{2}} 0 (z , \tau), \qquad \qquad
\theta_6(z) = \theta_6 \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 0 {\frac{1}{2}} (z , \tau)\\
\theta_7(z) &= \theta_7 \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 (z , \tau), \qquad \qquad
\theta_8(z) = \theta_8 \chs 0 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 (z , \tau)\\
\theta_9(z) &= \theta_9 \chs 0 0 0 0 0 {\frac{1}{2}}(z , \tau),\qquad \qquad
\theta_{10}(z) = \theta_{10} \chs {\frac{1}{2}} 0 0 0 0 0 (z , \tau)\\
\theta_{11}(z) &= \theta_{11} \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{12}(z) = \theta_{12} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau)\\
\theta_{13}(z) &= \theta_{13} \chs 0 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{14}(z) = \theta_{14} \chs 0 {\frac{1}{2}} 0 0 0 0 (z , \tau)\\
\theta_{15}(z) &= \theta_{15} \chs 0{\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{16}(z) = \theta_{16} \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau)\\
\theta_{17}(z) &= \theta_{17} \chs 0 0 0 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{18}(z) = \theta_{18} \chs 0 0 {\frac{1}{2}} 0 0 0 (z , \tau)\\
\end{split}
\]
\begin{figure}[h]
\begin{center}
\begin{picture}(485,410)(-65,-50)
\thicklines
\qbezier(-10,-40)(-10,150)(40,315)
\put(40,185){\line(0,1){130}}
\qbezier(40,185)(115,213)(190,241)
\dashline{4}[1](190,241)(298,190)
\dashline{4}[1](300,168)(290,110)
\dashline{4}[1](190,240)(240,110
\qbezier(40,315)(77,211)(115,110)
\qbezier(35,165)(44,137)(53,110)
\qbezier(35,165)(35,100)(35,35)
\qbezier(35,165)(95,130)(100,35)
\qbezier(115,93)(133,64)(151,35)
\dashline{4}[1](159,93)(210,35
\qbezier(57,93)(104,64)(151,35)
\dashline{4}[1](159,93)(100,35
\dashline{4}[1](240,93) (100,35)
\dashline{4}[1](288,93) (151,35)
\dashline{4}[1](190,240)(210,35)
\dashline{4}[1](255,35)(288,93)
\dashline{4}[1](288,93)(310,35)
\dashline{4}[1](240,93)(310,35)
\qbezier(35,18)(48,-11)(61,-40)
\qbezier(100,18)(80,-11)(61,-40)
\qbezier(100,18)(120,-11)(140,-40)
\qbezier(90,-40)(120,-11)(151,18)
\dashline{4}[1](151,18)(140,-40)
\dashline{4}[1](310,18)(140,-40)
\dashline{4}[1](189,-40)(159,93
\dashline{4}[1](232,-40)(210,18)
\dashline{4}[1](232,-40)(255,18)
\dashline{4}[1](277,-40)(310,18)
\dashline{4}[1](325,-40)(310,18)
\put(30,175){\small{$ V_4$}}
\put(30,325){\small{ $C_2$}}
\put(185,250){\small{ $C_2$}}
\put(295,175){\small{$V_4$}}
\put(285,100){\small{$D_8$}
\put(235,100){\small{$S_3$}}
\put(154,100){\small{$C_3$}
\put(110,100){\small{$C_4$}}
\put(48,100){\small{$(C_2)^2$}}
\put(18,25){\small{$C_2 \times C_2$}}
\put(95,25){\small{$D_{12}$}}
\put(133,25){\small{$C_2 \times D_8$}}
\put(205,25){\small{$C_6$}}
\put(250,25){\small{$16$}}
\put(305,25){\small{$S_4$}}
\put(-15,-50){\small{$C_{14}$}
\put(56,-50){\small{$G_1$}}
\put(85,-50){\small{$G_2$}}
\put(125,-50){\small{$C_2 \times S_4$}}
\put(184,-50){\small{$C_{9}$}}
\put(227,-50){\small{$48$}}
\put(272,-50){\small{$96$}}
\put(315,-50){\small{$L_3(2)$}}
\put(-40,-50){\small{0}}
\put(-40,25){\small{1}}
\put(-40,100){\small{2}}
\put(-40,175){\small{3}}
\put(-40,250){\small{4}}
\put(-40,325){\small{5}}
\put(-50,355){{Dimension}} \put(-50,340){ of Loci}
\qbezier(205,328)(175,328)(245,328) \put(250,325){{hyperelliptic}}
\dashline{4}[1](199,313)(245,313) \put(250,310){{non hyperelliptic}}
\end{picture}
\caption{Inclusions among the loci for genus $3$ curves with automorphisms.} \label{figgenus=3}
\end{center}
\end{figure}
\[
\begin{split}
\theta_{19}(z) &= \theta_{19} \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad
\qquad
\theta_{20}(z) = \theta_{20} \chs 0 {\frac{1}{2}} 0 0 0 {\frac{1}{2}} (z , \tau)\\
\theta_{21}(z) &= \theta_{21} \chs 0 0 0 0 {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{22}(z) = \theta_{22} \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 0 (z , \tau)\\
\theta_{23}(z) &= \theta_{23} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau),\qquad
\qquad
\theta_{24}(z) = \theta_{24} \chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau)\\
\end{split}
\]
\[
\begin{split}
\theta_{25}(z) &= \theta_{25} \chs {\frac{1}{2}} 0 0 0 0 {\frac{1}{2}}(z , \tau),\qquad \qquad
\theta_{26}(z) = \theta_{26} \chs 0 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_{27}(z) &= \theta_{27} \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 (z , \tau),\qquad \qquad
\theta_{28}(z) = \theta_{28} \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau)\\
\theta_{29}(z) &= \theta_{29} \chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 0 (z , \tau),\qquad \qquad
\theta_{30}(z) = \theta_{30} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_{31}(z) &= \theta_{31} \chs{\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{32}(z) = \theta_{32} \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 (z , \tau)\\
\theta_{33}(z) &= \theta_{33} \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad
\qquad
\theta_{34}(z) =\theta_{34}\chs 0 0 0 {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau) \\
\theta_{35}(z) &= \theta_{35} \chs {\frac{1}{2}} 0 0 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{36}(z) = \theta_{36}\chs {\frac{1}{2}} {\frac{1}{2}} 0 0 0 0 (z , \tau) \\
\theta_{37}(z) &= \theta_{37} \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 0 (z , \tau),\qquad \qquad
\theta_{38}(z) = \theta_{38}\chs {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 (z , \tau) \\
\theta_{39}(z) &= \theta_{39} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{40}(z) = \theta_{40} \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau) \\
\theta_{41}(z) &= \theta_{41} \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{42}(z) = \theta_{42}\chs 0 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}}(z , \tau) \\
\theta_{43}(z) &= \theta_{43} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 (z , \tau),\qquad \qquad
\theta_{44}(z) =\theta_{44} \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 (z ,\tau) \\
\theta_{45}(z) &= \theta_{45} \chs 0 0 {\frac{1}{2}} 0 0 {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{46}(z) = \theta_{46} \chs 0 {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau) \\
\theta_{47}(z) &= \theta_{47} \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{48}(z) = \theta_{48} \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau)\\
\theta_{49}(z) &= \theta_{49} \chs{\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{50}(z) = \theta_{50}\chs {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 {\frac{1}{2}} (z , \tau) \\
\theta_{51}(z) &= \theta_{51} \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}}(z , \tau),\qquad \qquad
\theta_{52}(z) = \theta_{52}\chs 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau) \\
\theta_{53}(z) &= \theta_{53} \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}}(z , \tau),\qquad \qquad
\theta_{54}(z) = \theta_{54}\chs 0 {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 (z , \tau) \\
\theta_{55}(z) &= \theta_{55} \chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{56}(z) = \theta_{56} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_{57}(z) &= \theta_{57} \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 (z , \tau),\qquad \qquad
\theta_{58}(z) = \theta_{58} \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 (z , \tau)\\
\theta_{59}(z) &= \theta_{59} \chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 (z , \tau),\qquad \qquad
\theta_{60}(z) = \theta_{60} \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\theta_{61}(z) &= \theta_{61} \chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} (z , \tau),\qquad \qquad
\theta_{62}(z) = \theta_{62} \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}}(z , \tau)\\
\theta_{63}(z) &= \theta_{63} \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 (z , \tau),\qquad \qquad
\theta_{64}(z) = \theta_{64} \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} (z , \tau)\\
\end{split}
\]
\begin{rem}
Each half-integer characteristic other than the zero characteristic can be formed as a sum of not more than 3 of the
following seven characteristics:
\[
\begin{split}
& \left\{ \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 0, \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} 0, \chs
{\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}}, \chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}}
{\frac{1}{2}}, \chs
0 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}},\right.\\
& \left. \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0, \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}}
{\frac{1}{2}} {\frac{1}{2}}\right\}.\\
\end{split}
\]
The sum of all characteristics of the above set gives the zero characteristic. The sums of three characteristics give
the rest of the 35 even characteristics and the sums of two characteristics give 21 odd characteristics.
\end{rem}
It can be shown that one of the even theta constants is zero. Let's pick $S = \{1,2,3,4,5,6,7\}$ and $U = \{1,3,5,7\}.$
Let $T = U.$ Then By Theorem ~\ref{vanishingProperty} the theta constant corresponding to the characteristic $\eta_T
=\chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} $ is zero. That is $\theta_{12} = 0.$ Next,
we give the relation between theta characteristics and branch points of the genus 3 hyperelliptic curve in the same
way we did in the genus 2 case. Once again, Thomae's formula is used to get these relations. We get 35 equations with
branch points and non-zero even theta constants. By picking suitable equations, we were able to express branch points
in terms of thetanulls similar to Picard's formula for genus $2$ curves. Let $ B = \{a_1 , a_2 , a_3 , a_4 , a_5 , 1 ,
0\}$ be the finite branch points of the curves and $U = \{a_1, a_3, a_5, 0\}$ be the set of odd branch points.
\begin{thm}
Any genus 3 hyperelliptic curve is isomorphic to a curve given by the equation
\[Y^2=X(X-1)(X-a_1)(X-a_2)(X-a_3)(X-a_4)(X-a_5), \] where
\[ a_1 =\frac{\theta_{31}^2\theta_{21}^2}{\theta_{34}^2\theta_{24}^2}, \, \, a_2=
\frac{\theta_{31}^2\theta_{13}^2}{\theta_{9}^2\theta_{24}^2}, \, \, a_3 = \frac{\theta_{11}^2\theta_{31}^2}{\theta_{24}^2\theta_{6}^2}, \, \, a_4 =
\frac{\theta_{21}^2\theta_{7}^2}{\theta_{15}^2\theta_{34}^2}, \, \, a_5= \frac{\theta_{13}^2\theta_{1}^2}{\theta_{26}^2\theta_{9}^2}.
\]
\end{thm}
\proof Thomae's formula expresses the thetanulls in terms of branch points of hyperelliptic curves. To invert the
period map we are going to use Lemma~\ref{Thomae}. For simplicity we order the branch points in the order of $a_1,
a_2, a_3, a_4, a_5, 0, 1,$ and $\infty$. Then the following set of equations represents the relations of theta
constants and $a_1,$ $\dots,$ $a_5.$ We use the notation $(i,j)$ for $(a_i-a_j)$.
\[
\begin{split}
{\theta_{{1}}}^{4} & = A \,\left(1,6 \right) \left(3,6 \right) \left(5,6 \right) \left( 1 ,3 \right) \left( 1 ,5 \right)
\left( 3, 5 \right) \left( 2 ,4
\right) \left( 2 ,7 \right) \left( 4 ,7 \right) \\
{\theta_{{2}}}^{4} & =- A \, \left( 3,6 \right) \left( 5,6 \right) \left( 3 ,5 \right) \left( 1 , 2 \right) \left( 1 ,4
\right) \left(2 ,4 \right) \left( 3 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{3}}}^{4} & = A \, \left( 3,6 \right)\left( 4,6 \right) \left( 3 , 4 \right) \left( 1, 2 \right) \left( 1 , 5
\right) \left( 2 ,5 \right) \left( 1 ,7 \right) \left( 2,7 \right) \left( 5 ,7 \right) \\
{\theta_{{4}}}^{4} & = -A \, \left( 2,6 \right)\left( 3,6 \right)\left(5,6 \right) \left( 2 ,3 \right) \left( 2 ,5 \right)
\left( 3 ,5 \right) \left(1, 4 \right) \left( 1 ,7 \right) \left( 4 ,7 \right) \\
{\theta_{{5}}}^{4} & = A \, \left(4,6 \right)\left( 5,6 \right) \left( 4,5 \right) \left( 1 ,2 \right) \left(1, 3 \right)
\left( 2 ,3\right) \left( 1 , 7 \right) \left( 2 ,7 \right) \left( 3 ,7 \right) \\
{\theta_{{6}}}^{4} & = A \, \left( 1,6 \right) \left( 2,6 \right) \left( 3 ,4 \right) \left( 3 ,5 \right) \left( 4 ,5
\right) \left( 1 ,2
\right) \left( 1 , 7 \right) \left( 2 ,7 \right) \\
\end{split}
\]
\[
\begin{split}
{\theta_{{7}}}^{4} & = A \, \left( 2,6 \right)\left( 3,6 \right)\left( 4,6 \right) \left( 1 ,5 \right) \left( 2 ,3 \right)
\left( 2 ,4 \right) \left( 3 ,4
\right) \left( 1 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{8}}}^{4} & = A \, \left( 2,6 \right)\left( 3,6 \right) \left( 2 ,3 \right) \left( 1 , 4 \right) \left( 1 ,5
\right) \left( 4 ,5
\right) \left( 1 ,7 \right) \left( 4 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{9}}}^{4} & = -A \, \left( 1,6 \right)\left( 3,6 \right) \left( 1 ,3 \right) \left( 2 ,4 \right) \left( 2 ,5
\right) \left( 4 ,5
\right) \left( 1, 7 \right) \left( 3 ,7 \right) \\
{\theta_{{10}}}^{4} & = -A \,\left( 3,6 \right) \left( 5,6 \right) \left( 3 ,5 \right) \left( 1 ,2 \right) \left( 1, 4
\right) \left( 2 ,4
\right) \left( 1 ,7 \right) \left( 2 ,7 \right) \left( 4 , 7 \right) \\
{\theta_{{11}}}^{4} & = -A \, \left(3,6 \right)\left( 4,6 \right) \left( 5,6 \right) \left( 3 ,4 \right)
\left( 3 ,5 \right) \left( 4 ,5 \right) \left(
1 ,2 \right) \left( 1, 7 \right) \left( 2 ,7 \right) \\
{\theta_{{13}}}^{4} & = A \, \left( 2,6 \right)\left( 4,6\right)\left( 5,6\right) \left( 1 ,3 \right) \left( 2 ,4 \right)
\left( 2 ,5 \right) \left( 4 ,5
\right) \left( 1 ,7 \right) \left( 3 ,7 \right) \\
{\theta_{{14}}}^{4} & = A \, \left( 2,6 \right)\left( 5,6 \right) \left( 2 ,5 \right) \left( 1 ,3 \right) \left( 1 , 4
\right) \left( 3 ,4
\right) \left( 1 ,7 \right) \left( 3 ,7 \right) \left( 4 ,7 \right) \\
{\theta_{{15}}}^{4} & = -A \, \left( 1,6 \right) \left( 5,6 \right) \left( 1 ,5 \right) \left( 2 ,3 \right) \left( 2 ,4
\right) \left( 3 ,4
\right) \left( 1 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{16}}}^{4} & = A \, \left( 1,6 \right) \left( 2 ,3 \right) \left( 2 ,4 \right) \left( 2 ,5 \right) \left( 3 ,4
\right) \left( 3 ,5 \right) \left( 4 , 5 \right) \left( 1 ,7 \right)
\\
{\theta_{{17}}}^{4} & = A \, \left( 1,6 \right) \left( 4,6 \right) \left( 2 ,3 \right) \left( 2 ,5 \right) \left( 3 ,5
\right) \left( 1 , 4
\right) \left( 1 ,7 \right) \left( 4 ,7 \right) \\
{\theta_{{18}}}^{4} & = -A \, \left(2,6 \right) \left( 4,6 \right) \left( 1 ,3 \right) \left( 1 ,5 \right) \left( 3 ,5
\right) \left( 2 ,4
\right) \left( 1 ,7 \right) \left( 3 ,7 \right) \left( 5, 7 \right) \\
{\theta_{{19}}}^{4} & = A \, \left( 3,6 \right)\left( 4,6 \right) \left( 1 ,2 \right) \left( 1 ,5 \right) \left( 2 ,5
\right) \left( 3 ,4
\right) \left( 3 , 7 \right) \left( 4 , 7 \right) \\
{\theta_{{20}}}^{4} & = -A \, \left( 2,6\right) \left( 1 ,3 \right) \left( 1 , 4 \right) \left( 1 , 5 \right) \left( 3 ,4
\right) \left( 3 ,5 \right) \left( 4 , 5 \right) \left( 2 , 7 \right)
\\
{\theta_{{21}}}^{4} & = -A \, \left( 1,6 \right)\left( 4,6 \right)\left( 5,6 \right) \left( 1 ,4 \right) \left( 1 ,5
\right) \left( 4 ,5 \right) \left( 2 ,3
\right) \left( 2 , 7 \right) \left( 3 ,7 \right) \\
{\theta_{{22}}}^{4} & = -A \, \left( 1,6 \right)\left( 3,6 \right)\left( 4,6 \right) \left( 1 , 3 \right) \left( 1 , 4
\right) \left( 3 ,4 \right) \left( 2 ,5
\right) \left( 2 , 7 \right) \left( 5 , 7 \right) \\
{\theta_{{23}}}^{4} & = A \, \left( 1,6 \right)\left( 2,6 \right) \left(3 ,4 \right) \left( 3 ,5 \right) \left( 4 , 5
\right) \left( 1 ,2
\right) \left( 3 , 7 \right) \left( 4, 7 \right) \left( 5, 7 \right) \\
{\theta_{{24}}}^{4} & = A \, \left( 4,6 \right)\left( 5,6 \right) \left( 1 ,2 \right) \left( 1 ,3 \right) \left( 2, 3
\right) \left( 4 ,5
\right) \left( 4 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{25}}}^{4} & = A \, \left( 3,6 \right) \left( 1 ,2 \right) \left( 1 ,4 \right) \left( 1 ,5 \right)
\left( 2 ,4 \right) \left( 2 ,5 \right) \left( 4 ,5 \right) \left( 3 ,7 \right) \\
{\theta_{{26}}}^{4} & = -A \, \left( 2,6 \right)\left( 4,6 \right) \left( 1 ,3 \right) \left( 1 ,5 \right) \left( 3 ,5
\right) \left( 2 ,4
\right) \left( 2 ,7 \right) \left( 4 ,7 \right) \\
{\theta_{{27}}}^{4} & = -A \, \left( 1,6 \right)\left( 5,6 \right) \left( 1 ,5 \right) \left( 2 ,3 \right) \left( 2 ,4
\right) \left( 3 ,4
\right) \left( 2 ,7 \right) \left( 3 ,7 \right) \left( 4 , 7 \right) \\
{\theta_{{28}}}^{4} & = -A \, \left( 1,6 \right)\left( 3,6 \right) \left( 1 ,3 \right) \left( 2 ,4 \right) \left( 2 ,5
\right) \left( 4 ,5
\right) \left( 2 ,7 \right) \left( 4 ,7 \right) \left( 5 ,7 \right) \\
{\theta_{{29}}}^{4} & = A \, \left( 1,6 \right)\left( 2,6 \right)\left( 4,6 \right) \left( 3 ,5 \right) \left( 1 ,2 \right)
\left( 1 ,4 \right) \left( 2, 4
\right) \left( 3, 7 \right) \left( 5 ,7 \right) \\
{\theta_{{30}}}^{4} & = A \, \left(5,6 \right) \left( 1 ,2 \right) \left( 1,3 \right) \left( 1 ,4 \right) \left(
2, 3 \right) \left( 2 ,4 \right) \left( 3,4 \right) \left( 5 , 7 \right) \\
{\theta_{{31}}}^{4} & = -A \, \left( 1,6 \right)\left( 2,6 \right)\left( 3,6 \right) \left( 1, 2 \right) \left( 1 ,3
\right) \left( 2 ,3 \right) \left( 4 ,5
\right) \left( 4 ,7 \right) \left( 5 ,7 \right) \\
{\theta_{{32}}}^{4} & = A \, \left( 1,6 \right)\left( 4,6 \right) \left( 2 ,3 \right) \left( 2, 5 \right) \left( 3 ,5
\right) \left( 1 , 4
\right) \left( 2 , 7 \right) \left( 3, 7 \right) \left( 5 , 7 \right) \\
{\theta_{{33}}}^{4} & = A \, \left( 2,6 \right)\left( 5,6 \right) \left( 1 ,3 \right) \left( 1 ,4 \right) \left( 3, 4
\right) \left( 2 ,5
\right) \left( 2 , 7 \right) \left( 5 ,7 \right) \\
{\theta_{{34}}}^{4} & = A \, \left( 2,6\right)\left( 3,6 \right) \left( 1 ,4 \right) \left( 1 , 5 \right) \left( 4, 5
\right) \left( 2 ,3
\right) \left( 2 ,7 \right) \left( 3 , 7 \right) \\
{\theta_{{35}}}^{4} & = -A \, \left( 4,6 \right) \left( 1 ,2 \right) \left( 1 ,3 \right) \left( 1, 5 \right) \left( 2, 3
\right) \left( 2 ,5 \right) \left( 3, 5 \right) \left( 4 ,7 \right)
\\
{\theta_{{36}}}^{4} & = -A \, \left( 1,6\right)\left( 2,6 \right)\left( 5,6 \right) \left( 1 ,2 \right) \left( 1 ,5
\right) \left( 2 ,5 \right) \left( 3 ,4
\right) \left( 3, 7 \right) \left( 4 ,7 \right) \\
\end{split}
\]
\noindent Our expectation is to write down the branch points as quotients of thetanulls. By using the set of equations
given above we have several choices for $a_1,\dots,a_5$ in terms of theta constants.
\begin{center}
\begin{tabular}{c c c c}
Branch Points & \multicolumn {3}{c} {Possible Ratios}\\[5pt]
$a_1^2$ & $\left(\frac{\theta_{36}^2\theta_{22}^2}{\theta_{33}^2\theta_{19}^2}\right)^2$, &
$\left(\frac{\theta_{31}^2\theta_{21}^2}{\theta_{34}^2\theta_{24}^2}\right)^2$, &
$\left(\frac{\theta_{29}^2\theta_{1}^2}{\theta_{26}^2\theta_{2}^2}\right)^2$ \\
$a_2^2$ & $\left(\frac{\theta_{4}^2\theta_{29}^2}{\theta_{2}^2\theta_{17}^2}\right)^2$, &
$\left(\frac{\theta_{36}^2\theta_{7}^2}{\theta_{15}^2\theta_{19}^2}\right)^2$,
&
$\left(\frac{\theta_{31}^2\theta_{13}^2}{\theta_{9}^2\theta_{24}^2}\right)^2$ \\
$a_3^2$ & $\left(\frac{\theta_{4}^2\theta_{22}^2}{\theta_{33}^2\theta_{17}^2}\right)^2$, &
$\left(\frac{\theta_{11}^2\theta_{31}^2}{\theta_{24}^2\theta_{6}^2}\right)^2$, &
$\left(\frac{\theta_{7}^2\theta_{1}^2}{\theta_{26}^2\theta_{15}^2}\right)^2$ \\
$a_4^2$ & $\left(\frac{\theta_{11}^2\theta_{29}^2}{\theta_{2}^2\theta_{6}^2}\right)^2$, &
$\left(\frac{\theta_{21}^2\theta_{7}^2}{\theta_{15}^2\theta_{34}^2}\right)^2$, &
$\left(\frac{\theta_{22}^2\theta_{13}^2}{\theta_{9}^2\theta_{33}^2}\right)^2$ \\
$a_5^2$ & $\left(\frac{\theta_{4}^2\theta_{21}^2}{\theta_{34}^2\theta_{17}^2}\right)^2$, &
$\left(\frac{\theta_{11}^2\theta_{36}^2}{\theta_{19}^2\theta_{6}^2}\right)^2$, &
$\left(\frac{\theta_{13}^2\theta_{1}^2}{\theta_{26}^2\theta_{9}^2}\right)^2$ \\
\end{tabular}
\end{center}
Let's select the following choices for $a_1, \cdots, a_5$:
\[
a_1 = \frac{\theta_{31}^2\theta_{21}^2}{\theta_{34}^2\theta_{24}^2}, \, \,
a_2=
\frac{\theta_{31}^2\theta_{13}^2}{\theta_{9}^2\theta_{24}^2}, \, \, a_3 =\frac{\theta_{11}^2\theta_{31}^2}{\theta_{24}^2\theta_{6}^2}, \, \, a_4 =
\frac{\theta_{21}^2\theta_{7}^2}{\theta_{15}^2\theta_{34}^2}, \quad a_5= \frac{\theta_{13}^2\theta_{1}^2}{\theta_{26}^2\theta_{9}^2}.
\]
This completes the proof.
\endproof
\begin{rem}
{i)} Unlike the genus 2 case, here only $\theta_1,$ $ \theta_6,$ $ \theta_7,$ $\theta_{11},$ $ \theta_{15},$ $ \theta_{24},$ $ \theta_{31}$ are
from the same G\"opel group.
{ii)} For genus 2 case such relations are known as Picard's formulae. The calculations proposed by Gaudry on genus 2
arithmetic on theta function in cryptography is mainly based on Picard's formulae.
\end{rem}
\subsection{Theta Identities for Hyperelliptic Curves}
Similar to the genus 2 case we can find identities that hyperelliptic theta constants are satisfied. We would like to
find a set of identities that contains all possible even theta constants. A G\"opel group, Eq.~\eqref{eq1} and
Eq.~\eqref{eq2} all play a main role in this task. Now consider a G\"opel group for genus 3 curves. Any G\"opel group
$G$ contains $2^3=8$ elements. The number of such G\"opel groups is $135.$ We have $24$ G\"opel groups such that all of
the characteristics of the groups are even. The following is one of the G\"opel groups which has only even
characteristics:
\[
\begin{split}
G = & \left\{ c_1 = \chs {0}{0}{0}{0} 0 0, c_2 = \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 0,c_3 \chs
{\frac{1}{2}} 0 0 0 0 0, c_4 = \chs 0 {\frac{1}{2}} 0 0 0 0, c_5 = \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 0, \right. \\
& \left.c_6 = \chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 0, c_7 = \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 0 0 , c_8 = \chs 0 0
{\frac{1}{2}} 0 0 0 \right\}.
\end{split}
\]
\noindent By picking suitable characteristics $\mathfrak b_1,$ $\mathfrak b_2,$ and $\mathfrak b_3$ we can find the G\"opel systems for group
$G.$ Let's pick $\mathfrak b_1 = \chs 0 0 0 {\frac{1}{2}} {\frac{1}{2}} 0,$ $\mathfrak b_2 = \chs 0 0 0 0 {\frac{1}{2}} 0,$ and $\mathfrak b_3
= \chs 0 0 0 {\frac{1}{2}} 0 {\frac{1}{2}},$ then the corresponding G\"opel systems are given by the following:
\[
\begin{split}
G = & \left\{ \chs {0}{0}{0}{0} 0 0, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 0, \chs
{\frac{1}{2}} 0 0 0 0 0, \chs 0 {\frac{1}{2}} 0 0 0 0, \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 0, \right. \\
& \left.\chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 0, \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 0 0 , \chs 0 0 {\frac{1}{2}} 0 0
0 \right\},\\
\mathfrak b_1 G = & \left\{ \chs 0 0 0 {\frac{1}{2}} {\frac{1}{2}} 0, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}
0,\chs {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}} 0,\chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0,
\chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0, \right.\\
%
& \left. \chs{\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}
0, \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0, \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 \right\},\\
%
\mathfrak b_2 G = & \left\{ \chs 0 0 0 0 {\frac{1}{2}} 0, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 ,\chs {\frac{1}{2}} 0 0 0 {\frac{1}{2}}
0, \chs 0 {\frac{1}{2}} 0 0 {\frac{1}{2}} 0, \chs 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0,\right.\\
%
& \left.\chs{\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}}
0,\chs {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} 0, \chs 0 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 \right\},\\
\mathfrak b_3 G = & \left\{\chs 0 0 0 {\frac{1}{2}} 0 {\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0
{\frac{1}{2}},\chs {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 {\frac{1}{2}}, \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0
{\frac{1}{2}}, \chs 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}}, \right.\\
%
& \left.\chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0
{\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} 0 {\frac{1}{2}}, \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}}\right\},\\
\mathfrak b_1 \mathfrak b_2 G = & \left\{\chs 0 0 0 {\frac{1}{2}} 0 0, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0
0, \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} 0 0, \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} 0 0, \chs 0 {\frac{1}{2}}
{\frac{1}{2}} {\frac{1}{2}} 0 0 , \right.\\
%
& \left.\chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} 0 0, \chs {\frac{1}{2}}
{\frac{1}{2}} 0 {\frac{1}{2}} 0 0 , \chs 0 0 {\frac{1}{2}} {\frac{1}{2}} 0 0 \right\},\\
\mathfrak b_1 \mathfrak b_3 G = & \left\{\chs 0 0 0 0 {\frac{1}{2}} {\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0
{\frac{1}{2}} {\frac{1}{2}}, \chs {\frac{1}{2}} 0 0 0 {\frac{1}{2}} {\frac{1}{2}}, \chs 0 {\frac{1}{2}} 0 0
{\frac{1}{2}} {\frac{1}{2}}, \chs 0{\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}}, \right.\\
%
& \left.\chs {\frac{1}{2}} 0
{\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} , \chs {\frac{1}{2}} {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}}, \chs 0
0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}}\right\},\\
\mathfrak b_2 \mathfrak b_3 G = & \left\{ \chs 0 0 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}}
{\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}, \chs {\frac{1}{2}} 0 0 {\frac{1}{2}} {\frac{1}{2}}
{\frac{1}{2}}, \chs 0 {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}, \chs 0 {\frac{1}{2}} {\frac{1}{2}}
{\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}},\right.\\
%
& \left. \chs {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}
{\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}} 0 {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}, \chs 0 0 {\frac{1}{2}}
{\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}}\right\},\\
\mathfrak b_1 \mathfrak b_2 \mathfrak b_3 G = & \left\{ \chs 0 0 0 0 0 {\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0
{\frac{1}{2}}, \chs {\frac{1}{2}} 0 0 0 0 {\frac{1}{2}}, \chs 0 {\frac{1}{2}} 0 0 0 {\frac{1}{2}}, \chs 0 {\frac{1}{2}}
{\frac{1}{2}} 0 0 {\frac{1}{2}}, \right.\\
& \left.\chs {\frac{1}{2}} 0 {\frac{1}{2}} 0 0 {\frac{1}{2}}, \chs {\frac{1}{2}} {\frac{1}{2}}
0 0 0 {\frac{1}{2}}, \chs 0 0 {\frac{1}{2}} 0 0 {\frac{1}{2}}\right\}.\\
\end{split}
\]
The above G\"opel systems contain all 64 characteristics for genus 3. Except for the G\"opel group, each of the systems
contains 4 odd characteristics and 4 even characteristics. If $\mathfrak h$ denotes one of the characteristics from the G\"opel
group other than $\chs {0}{0}{0}{0} 0 0,$ then $|\mathfrak e \mathfrak h| \equiv |\mathfrak e| \equiv 0 \mod 2$ has $20$ solutions.
\begin{exa}
If $\mathfrak h = \chs {\frac{1}{2}} {\frac{1}{2}} {\frac{1}{2}} 0 0 0 $, then all the characteristics of $G$ and all the even
characteristics of the G\"opel systems of $\mathfrak b_1 G,$ $\mathfrak b_3 G$ and $\mathfrak b_1 \mathfrak b_3 G$ are the possible characteristics for
$\mathfrak e.$ There are 20 of them.
\end{exa}
Without loss of generality, take the $10$ possible choices for $\mathfrak e$ which give rise to different terms in the series
Eq.~\eqref{eq1} and Eq.~\eqref{eq2}. For each $\mathfrak h$ in the G\"opel group other than $\chs {0}{0}{0}{0} 0 0,$ we can
choose $\mathfrak a$ such that $|\mathfrak a,\mathfrak h| + |\mathfrak h| \equiv 0 \mod 2.$ Take $\mathfrak a$ to be respectively $\mathfrak b_1,$ $\mathfrak b_2,$ $\mathfrak b_3,$
$\mathfrak b_1 \mathfrak b_2,$ $\mathfrak b_1 \mathfrak b_3,$ $\mathfrak b_2 \mathfrak b_3,$ and $\mathfrak b_1 \mathfrak b_2 \mathfrak b_3$ to the cases when $\mathfrak h$ is equal to the
characteristics $c_2,$ $c_3,$ $c_4,$ $c_5,$ $c_6,$ $c_7,$ and $c_8$ respectively. By picking $\mathfrak a$ and $\mathfrak h$ with
these characteristics, we can obtain formulas which express the zero values of all the even theta functions in terms of
8 theta nulls: $\theta_1,$ $\theta_3,$ $\theta_{10},$ $\theta_{14},$ $\theta_{18},$ $\theta_{22},$ $\theta_{29},$ $\theta_{36}.$
We obtain the following 14 equations. The first set is obtained by using Eq.~\eqref{eq1}; all the computations are done
by using Maple 10,
\[
\begin{split}
& 3\,{\theta_{{13}}}^{2}{\theta_{{23}}}^{2}-{\theta_{{28}}}^{2}{\theta_{{11}}}^{2}-{\theta_{{34}}}^{2}{\theta_{{12}
}}^{2}+{\theta_{{35}}}^{2}{\theta_{{15}}}^{2}+{\theta_{{
24}}}^{2}{\theta_{{16}}}^{2}-{\theta_{{30}}}^{2}{\theta_{{17}}}^{2}={\theta_{{3}}}^{2}{\theta_{{1}}}^{2}\\
&-{\theta_{{22}}}
^{2}{\theta_{{10}}}^{2}-{\theta_{{29}}}^{2}{\theta_{{14}}}^{2}+{\theta_{{36}}}^{2}{\theta_ {{18}}}^{2},\\
&3\,{\theta_{{21}}}^{2}{\theta_{{5}}}^{2}+{\theta_{{20}}}^{2}{\theta_{{6}}}^{2}-{\theta_{{31}}}^{2}{\theta_{{8}}}^
{2}-{\theta_{{25}}}^{2}{\theta_{{9}}}^{2}+{\theta_{{30}}
}^{2}{\theta_{{15}}}^{2}-{\theta_{{35}}}^{2}{\theta_{{17}}}^{2}={\theta_{{10}}}^{2}{\theta_{{1}}}^{2}\\
& -{\theta_{{22}}}
^{2}{\theta_{{3}}}^{2}-{\theta_{{36}}}^{2}{\theta_{{14}}}^{2}+{\theta_{{29}}}^{2}{\theta_{{18 }}}^{2},\\
&3\,{\theta_{{34}}}^{2}{\theta_{{16}}}^{2}-{\theta_{{27}}}^{2}{\theta_{{4}}}^{2}+{\theta_{{25}}}^{2}{\theta_{{6}}}
^{2}+{\theta_{{32}}}^{2}{\theta_{{7}}}^{2}-{\theta_{{20}}}^{2}{\theta_{{9}}}^{2}-{\theta_{{24}}}^{2}{\theta_{{12}}}^{2}={\theta_{{14}}}^{2}{\theta_{{1}}}^{2}\\
&+{\theta_{{29}} }^{2}{\theta_{{3}}}^{2}-{\theta_{{36}}
}^{2}{\theta_{{10}}}^{2}-{\theta_{{22}}}^{2}{\theta_{{18 }}}^{2},\\
&3\,{\theta_{{4}}}^{2}{\theta_{{32}}}^{2}+{\theta_{{31}}}^{2}{\theta_{{5}}}^{2}-{\theta_{{27}}}^{2}{\theta_{{7}}}^
{2}-{\theta_{{21}}}^{2}{\theta_{{8}}}^{2}+{\theta_{{23}}
}^{2}{\theta_{{11}}}^{2}-{\theta_{{28}}}^{2}{\theta_{{13}}}^{2}={\theta_{{18}}}^{2}{\theta_{{1}}}^{2}\\
&-{\theta_{{36}}}
^{2}{\theta_{{3}}}^{2}-{\theta_{{29}}}^{2}{\theta_{{10}}}^{2}+{\theta_{{22}}}^{2}{\theta_{{14 }}}^{2},\\
&3\,{\theta_{{17}}}^{2}{\theta_{{15}}}^{2}+{\theta_{{19}} }^{2}{\theta_{{2}}}^{2}-{\theta_{{7}}}^{2}{\theta_{{4}}}^
{2}-{\theta_{{33}}}^{2}{\theta_{{26}}}^{2}+{\theta_{{32} }}^{2}{\theta_{{27}}}^{2}-{\theta_{{35}}}^{2}{\theta_{{
30}}}^{2}={\theta_{{22}}}^{2}{\theta_{{1}}}^{2}\\
&+
{\theta_{{10}}}^{2}{\theta_{{3}}}^{2}-{\theta_{{18}}}^{2}{\theta_{{14}}}^{2}-{\theta_{{36}}}^{2}{\theta_{{29}}}^{2},\\
&3\,{\theta_{{26}}}^{2}{\theta_{{2}}}^{2}+{\theta_{{8}}}^{2}{\theta_{{5}}}^{2}+{\theta_{{16}}}^{2}{\theta_{{12}}}^{2}-{\theta_{{33}}}^{2}{\theta_{{19}}}^{2}-{\theta_{{31}
}}^{2}{\theta_{{21}}}^{2}-{\theta_{{34}}}^{2}{\theta_{{ 24}}}^{2}={\theta_{{29}}}^{2}{\theta_{{1}}}^{2}\\
&-{\theta_{{14}}}
^{2}{\theta_{{3}}}^{2}-{\theta_{{18}}}^{2}{\theta_{{10}}}^ {2}+{\theta_{{36}}}^{2}{\theta_{{22}}}^{2},\\
&3\,{\theta_{{9}}}^{2}{\theta_{{6}}}^{2}+{\theta_{{33}}}^
{2}{\theta_{{2}}}^{2}-{\theta_{{13}}}^{2}{\theta_{{11}}}^{2}-{\theta_{{26}}}^{2}{\theta_{{19}}}^{2}-{\theta_{{25}
}}^{2}{\theta_{{20}}}^{2}+{\theta_{{28}}}^{2}{\theta_{{ 23}}}^{2}={\theta_{{36}}}^{2}{\theta_{{1}}}^{2}\\
&+{\theta_{{14}}}^{2}{\theta_{{10}}}^ {2}-{\theta_{{18}}}^{2}{\theta_{{3}}}^{2}-{\theta_{{29}}}^{2}{\theta_{{22}}}^{2}.\\
\end{split}
\]
By using Eq.~\eqref{eq2} we have the following set of equations:
\[
\begin{split}
&3\,{\theta_{{13}}}^{4}+3\,{\theta_{{23}}}^{4}-{\theta_{{28}}}^{4}-{\theta_{{11}}}^{4}-{\theta_{{34}}}^{4}-
{\theta_{{12}}}^{4}+{\theta_{{35}}}^{4}+{\theta_{{15}}}
^{4}+{\theta_{{24}}}^{4}+{\theta_{{16}}}^{4}-{\theta_{{30}}}^{4}\\
&-{\theta_{{17}}}^{4}={\theta_{{3}}}^{4}+{\theta_{{1}}}^{4}-{\theta_{{
22}}}^{4}-{\theta_{{10}}}^{4}-{\theta_{{29}}}^{4}-{\theta_{{14}}}^{4}+{\theta_{{ 36}}}^{4}+{\theta_{{18}}}^{4},\\
&3\,{\theta_{{21}}}^{4}+3\,{\theta_{{5}}}^{4}+{\theta_{{20}}}^{4}+{\theta_{{6}}}^{4}-{\theta_{{31}}}^{4}-{\theta
_{{8}}}^{4}-{\theta_{{25}}}^{4}-{\theta_{{9}}}^{4}
+{\theta_{{30}}}^{4}+{\theta_{{15}}}^{4}-{\theta_{{35}}}^{4}\\
&-{\theta_{{17}}}^{4}={\theta_{{10}}}^{4}+{\theta_{{1}}}^{4}-{\theta_{{
22}}}^{4}-{\theta_{{3}}}^{4}-{\theta_{{36}}}^{4}-{\theta_{{14}}}^{4}+{\theta_{{29}} }^{4}+{\theta_{{18}}}^{4},\\
\end{split}
\]
\[
\begin{split}
&3\,{\theta_{{34}}}^{4}+3\,{\theta_{{16}}}^{4}-{\theta_{{27}}}^{4}-{\theta_{{4}}}^{4}+{\theta_{{25}}}^{4}+{
\theta_{{6}}}^{4}+{\theta_{{32}}}^{4}+{\theta_{{7}}}^{4}-{\theta_{{20}}}^{4}-{\theta_{{9}}}^{4}
-{\theta_{{24}}}^{4}\\
& -{\theta_{{12}}}^{4}={\theta_{{14}}}^{4}+{\theta_{{1}}}^{4}+{\theta_{
{29}}}^{4}+{\theta_{{3}}}^{4}-{\theta_{{36}}}^{4}-{\theta_{{10}}}^{4}-{\theta_{{22}} }^{4}-{\theta_{{18}}}^{4},\\
&3\,{\theta_{{4}}}^{4}+3\,{\theta_{{32}}}^{4}+{\theta_{{31}}}^{4}+{\theta_{{5}}}^{4}-{\theta_{{27}}}^{4}-{\theta
_{{7}}}^{4}-{\theta_{{21}}}^{4}-{\theta_{{8}}}^{4}
+{\theta_{{23}}}^{4}+{\theta_{{11}}}^{4}-{\theta_{{28}}}^{4}\\
&-{\theta_{{13}}}^{4}={\theta_{{18}}}^{4}+{\theta_{{1}}}^{4}-{\theta_{{
36}}}^{4}-{\theta_{{3}}}^{4}-{\theta_{{29}}}^{4}-{\theta_{{10}}}^{4}+{\theta_{{22}} }^{4}+{\theta_{{14}}}^{4},\\
&3\,{\theta_{{17}}}^{4}+3\,{\theta_{{15}}}^{4}+{\theta_{ {19}}}^{4}+{\theta_{{2}}}^{4}-{\theta_{{7}}}^{4}-{\theta
_{{4}}}^{4}-{\theta_{{33}}}^{4}-{\theta_{{26}}}^{4 }+{\theta_{{32}}}^{4}+{\theta_{{27}}}^{4}-{\theta_{{35}
}}^{4}\\
&-{\theta_{{30}}}^{4}={\theta_{{22}}}^{4}+{\theta_{{1}}}^{4}
+{\theta_{{10}}}^{4}+{\theta_{{3}}}^{4}-{\theta_{{18}}}^{4}-{\theta_{{14}}}^{4}-{\theta_{{36}}}^{4}-{\theta_{{29}}}^{4},\\
&3\,{\theta_{{26}}}^{4}+3\,{\theta_{{2}}}^{4}+{\theta_{{8}}}^{4}+{\theta_{{5}}}^{4}+{\theta_{{16}}}^{4}+{\theta_{{12}}}^{4}-{\theta_{{33}}}^{4}-{\theta_{{19}}}^{4
}-{\theta_{{31}}}^{4}-{\theta_{{21}}}^{4}-{\theta_{{34} }}^{4}\\
& -{\theta_{{24}}}^{4}={\theta_{{29}}}^{4}+{\theta_{{1}}}^{4}-{\theta_{{ 14}}}^{4}-{\theta_{{3}}}^{4}-{\theta_{{18}}}^{4}-{\theta_
{{10}}}^{4}+{\theta_{{36}}}^{4}+{\theta_{{22}}}^{4},\\
&3\,{\theta_{{9}}}^{4}+3\,{\theta_{{6}}}^{4}+{\theta_{{
33}}}^{4}+{\theta_{{2}}}^{4}-{\theta_{{13}}}^{4}-{\theta_{{11}}}^{4}-{\theta_{{26}}}^{4}-{\theta_{{19}}}^{
4}-{\theta_{{25}}}^{4}-{\theta_{{20}}}^{4}+{\theta_{{28
}}}^{4}\\
&+{\theta_{{23}}}^{4}={\theta_{{36}}}^{4}+{\theta_{{1}}}^{4}-{\theta_{{18}}}^{4}-{\theta_{{3}}}^{4}+{\theta_{{14}}}^{4}+{\theta
_{{10}}}^{4}-{\theta_{{29}}}^{4}-{\theta_{{22}}}^{4}.
\end{split}
\]
\begin{rem}
Similar to the genus 2 case we can consider all the G\"opel groups and obtain all possible relations among thetanulls
by following the above procedure. It is tedious and quite long so we don't do it here.
\end{rem}
\subsection{Genus 3 Non-Hyperelliptic Cyclic Curves} Using formulas similar to Thomae's formula for
each family of cyclic curve $y^n=f(x),$ one can express the roots of $f(x)$ in terms of ratios of theta functions as in
the hyperelliptic case. In this section we study such curves for $g=3$. We only consider the families of curves with
positive dimensions since the curves which belong to 0-dimensional families are well known.
Notice that the definition of thetanulls is different in this
part from the definitions of thetanulls in the hyperelliptic case. We define the following three theta constants:
\[
\theta_1 = \theta\chs{0}{\frac{1}{6}}{0}{\frac{2}{3}}{\frac{1}{6}}{\frac{2}{3}}, \quad \theta_2 = \theta
\chs{0}{\frac{1}{6}}{0}{\frac{1}{3}}{\frac{1}{6}}{\frac{1}{3}}, \quad \theta_3 = \theta
\chs{0}{\frac{1}{6}}{0}{0}{\frac{1}{6}}{0}.
\]
Next we consider the cases 7, 8 and 5 from Table 3.2.
\noindent \textbf{Case 7:} If the group is $C_3$, then the equation of $\mathcal X}%\def\X{\mathfrak X$ is given by $$y^3=x(x-1)(x-s)(x-t).$$ Let
$Q_i$ where $i= 1..5$ be ramifying points in the fiber of $0,1,s,t,\infty$ respectively. Consider the meromorpic
function $f = x$ on $\mathcal X}%\def\X{\mathfrak X$ of order 3. Then we have $(f) = 3 Q_1 - 3 Q_5.$ By applying the Lemma~\ref{Shiga} with $P_0 =
Q_5$ and an effective divisor $2Q_2 + Q_3,$ we have the following:
\begin{equation} \label{Shiga1}
E s = \prod_{k=1}^3 \frac{\theta( 2\int_{Q_5}^{Q_2} \omega + \int_{Q_5}^{Q_3} \omega - \int_{Q_5}^{b_k} \omega - \triangle , \tau
)} {\theta( 2\int_{Q_5}^{Q_2} \omega + \int_{Q_5}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
Once again, we apply Lemma~\ref{Shiga} with an effective divisor $Q_2 + 2Q_3$ and we have the following:
\begin{equation} \label{Shiga2}
E s^2 = \prod_{k=1}^3 \frac{\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_3} \omega - \int_{Q_5}^{b_k} \omega - \triangle ,
\tau )} {\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
By dividing Eq.~\eqref{Shiga2} by Eq.~\eqref{Shiga1} we have
\begin{equation} \label{s}
\begin{split}
s =& \prod_{k=1}^3 \frac{\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_3} \omega - \int_{Q_5}^{b_k} \omega - \triangle
, \tau )} {\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_3} \omega - \triangle , \tau )} \\
& \times \prod_{k=1}^3 \frac {\theta( 2\int_{Q_5}^{Q_2} \omega + \int_{Q_5}^{Q_3} \omega - \triangle , \tau )}{\theta( 2\int_{Q_5}^{Q_2} \omega
+ \int_{Q_5}^{Q_3} \omega - \int_{Q_5}^{b_k} \omega - \triangle
, \tau )}.
\end{split}
\end{equation}
By a similar argument, we have
\begin{equation} \label{t}
\begin{split}
t =& \prod_{k=1}^3 \frac{\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_4} \omega - \int_{Q_5}^{b_k} \omega - \triangle
, \tau )} {\theta( \int_{Q_5}^{Q_2} \omega + 2\int_{Q_5}^{Q_4} \omega - \triangle , \tau )} \\
& \times \prod_{k=1}^3 \frac {\theta( 2\int_{Q_5}^{Q_2} \omega + \int_{Q_5}^{Q_4} \omega - \triangle , \tau )}{\theta( 2\int_{Q_5}^{Q_2} \omega
+ \int_{Q_5}^{Q_4} \omega - \int_{Q_5}^{b_k} \omega - \triangle , \tau )}.
\end{split}
\end{equation}
Computing the right hand side of Eq.~\eqref{s} and Eq.~\eqref{t} was one of the main points of \cite{SHI}. As a result
we have $s = \frac{\theta_2^3}{\theta_1^3}$ and $ r = \frac{\theta_3^3}{\theta_1^3}.$
\noindent \textbf{Case 8:} If the group is $C_6$, then the equation is $y^3=x(x-1)(x-s)(x-t)$ with $s = 1-t.$ By using
the results from Case 7, we have $\theta_2^3 = \theta_1^3 - \theta_3^3.$
\noindent \textbf{Case 5:}
If $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X){\, \cong\, } (16, 13)$, then the equation of $\mathcal X}%\def\X{\mathfrak X$ is given by $$y^4=x(x-1)(x-t).$$ This curve has 4 ramifying
points $Q_i$ where $i= 1..4$ in the fiber of $0,1,t,\infty$ respectively. Consider the meromorpic function $f = x$ on
$\mathcal X}%\def\X{\mathfrak X$ of order 4. Then we have $(f) = 4 Q_1 - 4 Q_4.$ By applying Lemma~\ref{Shiga} with $P_0 = Q_4$ and an effective
divisor $2Q_2 + Q_3$, we have the following:
\begin{equation} \label{Shiga3}
E t = \prod_{k=1}^4 \frac{\theta( 2\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle , \tau
)} {\theta( 2\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
Once again, we apply Lemma~\ref{Shiga} with an effective divisor $Q_2 + 2Q_3$ and we have the following:
\begin{equation} \label{Shiga4}
E t^2 = \prod_{k=1}^4 \frac{\theta( \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle ,
\tau )} {\theta( \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
We have the following by dividing Eq.~\eqref{Shiga4} by Eq.~\eqref{Shiga3}:
\begin{equation}\label{GP16}
\begin{split}
t =& \prod_{k=1}^4 \frac{\theta( \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega -
\triangle , \tau )} {\theta( \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \triangle , \tau )} \\
& \times \prod_{k=1}^4 \frac{\theta( 2\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \triangle , \tau )} {\theta( 2\int_{Q_4}^{Q_2} \omega
+ \int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle
, \tau )}.
\end{split}
\end{equation}
In order to compute the explicit formula for $t,$ one has to find the integrals on the right-hand side. Such
computations are long and tedious and we intend to include them in further work.
\begin{rem}
In case 5 of Table 3, the parameter $t$ is given by $$\theta[e]^4 = A (t-1)^4 t^2,$$
where $[e]$ is the theta characteristic corresponding to the partition $(\{1\}, \{2\}, \{3\}, \{4\})$ and $A$ is a
constant; see \cite{NK} for details. However, this is not satisfactory since we would like $t$ as a rational function
in terms of theta constants. The method in \cite{NK} does not lead to another relation among $t$ and the thetanulls,
since the only partition we could take is the above.
\end{rem}
Summarizing all of the above, we have
\begin{thm}
Let $\mathcal X}%\def\X{\mathfrak X$ be a non-hyperelliptic genus 3 curve. The following statements are true:
\begin{description}
\item [i)] If $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X) {\, \cong\, } C_3$, then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation
\[ y^3 = x (x-1) \left(x-\frac{\theta_2^3}{\theta_1^3}\right) \left(x- \frac{\theta_3^3}{\theta_1^3}\right).\]
\item [ii)] If $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X) {\, \cong\, } C_6$, then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation
\[ y^3 = x (x-1) \left(x-\frac{\theta_2^3}{\theta_1^3}\right) \left(x- \frac{\theta_3^3}{\theta_1^3}\right) \,\,\textit{with}\,\, \theta_2^3 = \theta_1^3 -
\theta_3^3.\]
\item [iii)] If $\mbox{Aut} (\mathcal X}%\def\X{\mathfrak X)$ is isomorphic to the group with GAP identity (16, 13), then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation
\[y^4 = x (x-1)(x-t)\,\, \]
where $t$ is given by Eq.~\eqref{GP16}.
\end{description}
\end{thm}
\section{Genus 4 curves}
In this section we focus on genus 4 curves. For the genus 4 curves, the complete set of all possible full
automorphism groups and the corresponding equations are not completely calculated yet. In this chapter we consider a
few of the cyclic curves of genus 4. Let us first consider the genus 4 hyperelliptic algebraic curves. For these
curves, we have $2^{g-1} (2^g +1) = 136$ even half-integer characteristics and $2^{g-1} (2^g -1) = 120$ odd
half-integer characteristics. Among the even thetanulls, 10 of them are 0. We won't show the exact information here.
Following the same procedure as for $g=3,$ the branch points of genus 4 hyperelliptic curves can be expressed as
ratios of even theta constants and identities among theta constants can be obtained. The following Table ~\ref{tab_4}
gives some genus 4 non-hyperelliptic cyclic curves; see Table 2 of \cite{SH-KM} for the complete list.
\begin{center}
\begin{table}[h]\label{tab_4}
\caption{Some genus 4 non hyperelliptic cyclic curves and their automorphisms} \centering
\begin{tabular}{||c|c|c|c||}
\hline \hline
$\#$ & dim & Aut($\mathcal X}%\def\X{\mathfrak X$) & Equation\\
\hline \hline
1 & 3 & $C_2$ & $y^3 = x(x-1)(x-a_1)(x-a_2)(x-a_3)$\\
2 & 2 &$C_3 \times C_2$ & $y^3 = (x^2-1)(x^2 - \alpha_1)(x^2- \alpha_2)$\\
3 & 1 & $C_5$ & $y^5 = x(x-1)(x-\alpha)$\\
4 & 1& $C_3 \times C_2 $ & $y^3 = (x^2-1)(x^4 - \alpha x^2 +1)$ \\
\hline \hline
\end{tabular}
\end{table}
\end{center}
The Figure ~\ref{figg=4} shows the inclusions of loci of the genus 4 curves.
\subsection{Inverting the Moduli Map} In this section we will express branch points of each cyclic curve
in Table 4.1
as ratios of theta nulls.
\noindent\textbf{Case 1:} $C : y^3 = x (x-1) (x-a_1) (x-a_2) (x-a_3).$ In this curve $\infty$ is a branch point. We can
use result of \cite{NK} to find out $a_1, a_2, a_3$ in terms of thetanulls. First we need to find the partitions of
the set $\{1,2,3,4,5,6\}.$ The Table 5 shows all possible partitions of $\{1,2,3,4,5,6\}$ into 3 sets and the labeling
of the corresponding thetanulls.
For each partition we can apply the generalized Thomae's formula to obtain an identity. According to this labeling of
theta constants and the generalized Thomae's formula we have the following relations:
\[
\begin{split}
&{\theta_{{1}}}^{6} = c_1 \left( a_{{1}}-a_{{2}} \right) ^{3} \left( a_{{1} }-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right) a_{{1}}a_{{2}}a_{{3} } \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1
\right) ^{3},\\
%
&{\theta_{{2}}}^{6} = c_2 \left( a_{{1}}-a_{{2}} \right) ^{3}
\left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}} \right) a_{{1}
}a_{{2}}{a_{{3}}}^{3} \left( a_{{1}}-1 \right) \left( a_{{2}}-1
\right) \left( a_{{3}}-1 \right), \\
&{\theta_{{3}}}^{6} =c_3 \left( a_{{1}} -a_{{2}} \right) ^{3} \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a _{{3}}
\right) a_{{1}}a_{{2}}a_{{3}} \left(
a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right), \\
&{\theta_{{4}}}^{6} = c_4 \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) ^{3} \left( a_{{2}}-a_{{3}}
\right)
a_{{1}}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) ^{3} \left( a_{{3}}-1 \right), \\
&{ \theta_{{5}}}^{6} = c_5 \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{ 3}} \right) ^{3} \left( a_{{2}}-a_{{3}}
\right) a_{{1}}{a_{{2}}}^{3}a_ {{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3 }}-1 \right),\\
&{\theta_{{6}}}^{6} = c_6 \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) ^{3} \left( a_{{2}}-a_{{3}}
\right)
a_ {{1}}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right),\\
%
\end{split}
\]
\begin{table}[ht]
\caption{Partitions of $\{1,2,3,4,5,6\}$ into 3 sets}
\begin{tabular}{||c|c||}
\hline \hline
Theta constant & Corresponding partition\\
\hline \hline
$\theta_1$ & $[1,2],[3,4],[5,6]$\\
$\theta_2$ & $[1,2],[3,5],[4,6]$\\
$\theta_3$ & $[1,2],[3,6],[4,5]$\\
$\theta_4$ & $[1,3],[2,4],[5,6]$\\
$\theta_5$ & $[1,3],[2,5],[4,6]$\\
$\theta_6$ & $[1,3],[2,6],[4,5]$\\
$\theta_7$ & $[1,4],[2,3],[5,6]$\\
$\theta_8$ & $[1,4],[2,5],[3,6]$\\
$\theta_9$ & $[1,4],[2,6],[3,5]$\\
$\theta_{10}$ & $[1,5],[2,3],[4,6]$\\
$\theta_{11}$ & $[1,5],[2 ,4],[3,6]$\\
$\theta_{12}$ & $[1,5],[2,6],[3,4]$\\
$\theta_{13}$ & $[1,6],[2,3],[4,5]$\\
$\theta_{14}$ & $[1,6],[2,4],[3,5]$\\
$\theta_{15}$ & $[1,6],[2,5],[3,4]$\\
\hline
\end{tabular}
\end{table}
\[
\begin{split}
&{\theta_{{7}}}^{6} = c_7 \left( a_{{1}} -a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3 }}
\right)
^{3}a_{{1}}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) ^{3} \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right),\\
&{\theta_{{8}}}^{6 } = c_8 \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right)
a_{{1}}{a_{{2}}}^{3}a_{{3}} \left( a_{ {1}}-1 \right) ^{3} \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right),\\
&{\theta_{{9}}}^{6} = c_9 \left( a_{{1}}-a_{{2}} \right) \left( a _{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right)
a_{{1}}a_{{2}}{ a_{{3}}}^{3} \left( a_{{1}}-1 \right) ^{3} \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right),\\
&{\theta_{{10}}}^{6} = c_{10} \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right) ^{3}{a_{{1}}}^{3}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right), \\
%
&{\theta_{{11}}}^{6} = c_{11} \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}} \right)
{a_{{1}}}^{3}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) ^{3} \left( a_{{3}}-1 \right),\\
&{\theta_{{12}}}^{6} = c_{12} \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right) {a_{{1}}}^{3}a _{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1
\right)^{3},\\
&{\theta_{{13}}}^{6} = c_{13} \left( a_{{1}}-a_{ {2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right)
^{3}a_{{1}}a_{{2}}a_{{3}} \left( a_{{1}}-1 \right) \left( a_ {{2}}-1 \right) \left( a_{{3}}-1 \right),\\
&{\theta_{{14}}}^{6} = c_{14} \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right)
a_{{1}}a_{{2}}{a_{{3}}}^{3} \left( a_{ {1}}-1 \right) \left( a_{{2}}-1 \right) ^{3} \left( a_{{3}}-1 \right),\\
&{\theta_{{15}}}^{6} = c_{15} \left( a_{{1}}-a_{{2}} \right) \left( a_{{1}}-a_{{3}} \right) \left( a_{{2}}-a_{{3}}
\right) a_{{1}}{a_{{2} }}^{3}a_{{3}} \left( a_{{1}}-1 \right) \left( a_{{2}}-1 \right) \left( a_{{3}}-1 \right) ^{3}
\end{split}
\]
where $c_i$'s are constants and depend on the partition $\Lambda_i.$ From the above set of equations we can write
$a_1,a_2,a_3$ in terms of theta constants:
\begin{equation}\label{case1_g4}
a_1 ^2 = \delta_1(\frac{\theta_{10}}{\theta_{13}})^6, \quad \quad a_2 ^2 = \delta_2(\frac{\theta_{5}}{\theta_{6}})^6, \quad \quad
a_3^2 = \delta_3 (\frac{\theta_{2}}{\theta_{3}})^6
\end{equation}
where $\delta_1 = \frac{c_{10}}{c_{13}}, \quad \quad \delta_2 =\frac{c_{5}}{c_{6}}, \quad \quad \delta_3 =
\frac{c_{2}}{c_{3}}.$
Using the result of case 1 we can write the equations of cases 2 and case 4 in terms of thetanulls.
\begin{figure}
\begin{center}
\includegraphics[width=1.2\textwidth]{wijesiri_fig4.pdf}
\caption{Inclusions among the loci for genus 4 curves.}
\label{figg=4}
\end{center}
\end{figure}
\clearpage
\noindent\textbf{Case 2:} In this case the curve can be written as
\begin{equation}\label{case3}
y^3 = (x-1)(x+1)(x - \sqrt{\alpha_1})(x + \sqrt{\alpha_1})(x- \sqrt{\alpha_2})(x+ \sqrt{\alpha_2}).
\end{equation}
Consider the transformation given by $$ x \longrightarrow \frac{x-1}{2x-1}.$$
Under this transformation we obtain a curve that is isomorphic to the given curve and
\noindent the new curve is given by the equation $$y^3 = x(x-\frac{2}{3})(x - \gamma_1)(x - \gamma_2)(x - \gamma_3)(x -
\gamma_4)$$
where $\gamma_1 = \frac{\sqrt{\alpha_1} -1 }{2\sqrt{\alpha_1} -1 },$ $\gamma_2 = \frac{-\sqrt{\alpha_1} -1
}{-2\sqrt{\alpha_1} -1 },$ $\gamma_3 = \frac{\sqrt{\alpha_2} -1 }{2\sqrt{\alpha_2} -1 }$, and $\gamma_4 =
\frac{-\sqrt{\alpha_2} -1 }{-2\sqrt{\alpha_2} -1 }.$ Using this transformation we map the branch point 1 of the curve
given by the Eq.~\eqref{case3} to 0.
Again by using the transformation $$x \longrightarrow \frac {-2x +1 } {3x -2}, $$ we can find another curve isomorphic
to the above two curves. This transformation maps $\frac{2}{3}$ to $\infty.$ With this transformation the curve is
given by the equation
$$y^3 = x(x - \delta_1)(x - \delta_2)(x - \delta_3)(x-\delta_4)$$
where $\delta_i = \frac{-2\gamma_i + 1}{3 \gamma_i-2}.$ By using the transformation given by $$x \longrightarrow
\frac {x+1} {\frac{\delta_1}{\delta_1+1}x+ \frac{2\delta_1 +1}{\delta_1+1}},$$
we can find the curve $$y^3 = x(x-1)(x - \beta_1)(x - \beta_2)(x - \beta_3)$$
where $\beta_i =\frac{(\delta_1+1)(\delta_{i+1}+1)}{\delta_1 \delta_{i+1} +2 \delta_1 +1},$ which is isomorphic to the previous 3
algebraic curves. Now we are in case 1. From the result of case 1, we can write the $\beta_i,$ $i=1,2,3$ as ratios of
thetanulls. But we like to have $\alpha_1$ and $\alpha_2$ as functions of theta constants. Notice that we have the
following 3 relations on $\alpha_1,$ $\alpha_2,$ $\beta_1,$ $\beta_2,$ and $\beta_3$:
\[
\begin{split}
\beta_1 & = \frac{\alpha_1}{\alpha_1 -2 -2(\sqrt{\alpha_1} -1)},\\
\beta_2 & = \frac{\sqrt{\alpha_1 \alpha_2}} {\sqrt{\alpha_1 \alpha_2} + \sqrt{\alpha_1} - \sqrt{\alpha_2}}, \\
\end{split}
\]
\[
\begin{split}
\beta_3 & = \frac{\sqrt{\alpha_1 \alpha_2}} {\sqrt{\alpha_1 \alpha_2} - \sqrt{\alpha_1} - \sqrt{\alpha_2}}. \\
\end{split}
\]
Using these relations, $\alpha_1$ and $\alpha_2$ can be written as rational functions of $\beta_1,$ $\beta_2,$ and
$\beta_3$ given by the following:
\begin{equation}
\begin{split}
{\alpha_1} &= \frac { 2\, \beta_{{1}}\beta_{ {2}}\left( -\beta_{{3}}+\beta_{{2}} \right)
}{2\,\beta_{{1}}\beta_{{3}}+2\,\beta_{{1}}\beta_{{2}}+{\beta_{{2}}
}^{2}\beta_{{3}}-6\,\beta_{{1}}\beta_{{2}}\beta_{{3}}-2\,\beta_{{1}}{
\beta_{{2}}}^{2}+3\,\beta_{{1}}{\beta_{{2}}}^{2}\beta_{{3}}},\\
{\alpha_2}& = {\frac {2\,\beta_{{1}}(\beta_{{3}}-\beta_{{2}})}{-4\,\beta
_{{1}}-\beta_{{2}}\beta_{{3}}+4\,\beta_{{1}}\beta_{{3}}+4\,\beta_{{1}}
\beta_{{2}}-3\,\beta_{{1}}\beta_{{2}}\beta_{{3}}}},
\end{split}
\end{equation}
\noindent with the condition of $\beta_1, \beta_2$ and $\beta_3$
\[
\begin{split}
( \beta_{{1}}{\beta_{{3}}}^{2}+2\,\beta_{{1}}\beta_{{2}
}\beta_{{3}}+\beta_{{1}}{\beta_{{2}}}^{2}+{\beta_{{2}}}^{2}{\beta_{{3}
}}^{2}-4\,\beta_{{1}}\beta_{{2}}{\beta_{{3}}}^{2}-4\,\beta_{{1}}{\beta
_{{2}}}^{2}\beta_{{3}}+3\,\beta_{{1}}{\beta_{{2}}}^{2}{\beta_{{3}}}^{2 } ) \\
( -\beta_{{3}}-\beta_{{2}}+2\,\beta_{{2}}\beta_{{3}}
) &=0.
\end{split}
\]
The branch points of the curve given by Eq.~\eqref{case3} can be expressed as ratios of theta constants by using all of
the above information.
\noindent\textbf{Case 4:} In this case the curve is given by
\begin{equation}\label{case1}
y^3 = (x^2-1)(x^4 - \alpha x^2 +1).
\end{equation}
This is a special case of case 2. By writing out the equation of case 2, we have $y^3 = (x^2-1)(x^4 - (\alpha_1
+\alpha_2) x^2 + \alpha_1 \alpha_2).$ Take $\alpha = \alpha_1 +\alpha_2 $ and $\alpha_1 \alpha_2 = 1.$
\noindent\textbf{Case 3:} In this case, the equation is given by $y^5 = x(x-1)(x-\alpha).$ This curve has 4 ramifying
points $Q_i$ where $i= 1..4$ in the fiber of $0,1,t,\infty$ respectively. The meromorpic function $f = x$ on $\mathcal X}%\def\X{\mathfrak X$ of
order 4 has $(f) = 4 Q_1 - 4 Q_4.$ By applying Lemma~\ref{Shiga} with
\noindent $P_0 = Q_4$ and an effective divisor $4Q_2 + Q_3,$ we have the following:
\begin{equation} \label{case2-1}
E \alpha = \prod_{k=1}^5 \frac{\theta( 4\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle ,
\tau )} {\theta( 4\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
Again by applying Lemma~\ref{Shiga} with an effective divisor $3Q_2 + 2Q_3,$ we have the following:
\begin{equation} \label{case2-2}
E \alpha^2 = \prod_{k=1}^5 \frac{\theta(3 \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle
, \tau )} {\theta( 3\int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \triangle , \tau )}.
\end{equation}
We have the following by dividing Eq.~\eqref{case2-2} by Eq.~\eqref{case2-1}:
\begin{equation}\label{case3_g4}
\begin{split}
\alpha =& \prod_{k=1}^5 \frac{\theta(3 \int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega -
\triangle , \tau )} {\theta( 3\int_{Q_4}^{Q_2} \omega + 2\int_{Q_4}^{Q_3} \omega - \triangle , \tau )} \\
& \times \prod_{k=1}^5 \frac{\theta( 4\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \triangle , \tau )} {\theta( 4\int_{Q_4}^{Q_2} \omega + \int_{Q_4}^{Q_3} \omega - \int_{Q_4}^{b_k} \omega - \triangle
, \tau )}.
\end{split}
\end{equation}
By calculating integrals on the right-hand side in terms of thetanulls, we can write the branch point $\alpha$ as a
ratio of thetanulls. Summarizing all of the above, we have
\begin{thm}
\begin{description}
\item [i)] If $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X) \cong C_3,$ then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation $$y^3 =
x(x-1)(x-a_1)(x-a_2)(x-a_3),$$ where $a_1$, $a_2,$ and $a_3$ are given in case (1) in terms of thetanulls.
\item [ii)] If $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X) \cong C_3 \times C_2,$ then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation $$y^3 = (x^2-1)(x^2 - \alpha_1)(x^2-
\alpha_2),$$ where $\alpha_1,$ and $\alpha_2$ are given in case (2) in terms of thetanulls.
\item [iii)] If $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X) \cong C_5,$ then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation $$y^5 = x(x-1)(x-\alpha),$$ where
$\alpha$ is given in case (4) in terms of thetanulls.
\item [iv)] If $\mbox{Aut}(\mathcal X}%\def\X{\mathfrak X) \cong C_6 \times C_2,$ then $\mathcal X}%\def\X{\mathfrak X$ is isomorphic to a curve with equation $$y^3 = (x^2-1)(x^4 - \alpha x^2
+1),$$ where $\alpha$ is given in case (3) in terms of thetanulls.
\end{description}
\end{thm}
\section{Concluding Remarks}
In Sections 2, 3, and 4, the main idea was to write down the branch points as quotients of thetanulls explicitly for
cyclic curves of genus 2, 3, and 4 with extra automorphisms. For hyperelliptic algebraic curves, we can use Thomae's
formula to express branch points as ratios of thetanulls. We used Maple 10 for all computations. For non-hyperelliptic
cyclic curves, we used various methods in order to invert the period map. The method described in Lemma ~\ref{Shiga} in
Chapter 1 gives the general method to find branch points in terms of thetanulls. The main drawback of this method is
the difficulty of writing complex integrals as functions of theta characteristics. Some of the results in Chapter 2 and
Chapter 3 already appeared in \cite{Previato}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Marie-Christine Umdenstock (* 29. Februar 1964 in Deutschland) ist eine ehemalige französische Fußballspielerin.
Vereinskarriere
Die in Deutschland geborene Marie-Christine Umdenstock zog mit ihren Eltern früh ins Elsass, wo sie bei der ASPTT Strasbourg Fußball spielte, und zwar spätestens ab 1983 für deren erste Frauenelf. Als Umdenstocks Team zu Beginn der Saison 1984/85 mit Michèle Wolf einen hochkarätigen Zugang verzeichnen konnte, verfehlte es den Einzug ins Halbfinale der damals noch in einer Mischung aus Gruppenspiel- und K. o.-Modus ausgetragenen Landesmeisterschaft nur um Haaresbreite.
Die Abwehrspielerin wechselte gleich anschließend zum FC Lyon, für den sie die nächsten 15 Jahre spielte und wo sie auch zur Nationalspielerin wurde (siehe unten). Mit Lyon gewann sie insgesamt vier französische Meistertitel, den ersten 1991 und die weiteren 1993, 1995 und 1998, nachdem der Landesverband eine eingleisige oberste Frauenliga, das Championnat National 1 A, geschaffen hatte. Für Marie-Christine Umdenstock, die während der 1990er Jahre auch Spielführerin des FCL war, war für diese Erfolge die Eingespieltheit der Frauen, aber "vor allem die außerordentliche Solidarität innerhalb des Kaders" verantwortlich. Zu ihren Mitspielerinnen gehörten französische Nationalspielerinnen wie Emmanuelle Sykora, Cécile Locatelli, Jocelyne Gout und Hoda Lattaf, aber für eine aus Verletzungsgründen nicht einmal komplette Saison auch die Russin Irina Olegowna Grigorjewa. Neben dem Fußball, der im Frauenbereich seinerzeit noch ein reiner Amateursport war, arbeitete Umdenstock als Bürokauffrau im Autohandel eines Vereinssponsors.
Mit 36 wollte sie sportlich etwas kürzertreten und schloss sich im Jahr 2000 dem SC Caluire Saint-Clair an, einem Zweitligisten aus der Nachbarschaft Lyons. Zehn Monate später stand ihre Mannschaft als Aufsteiger fest, und Marie-Christine Umdenstock kehrte noch einmal in die höchste Spielklasse zurück. Diese
Saison 2001/02 beendete der SCCSC abgeschlagen als Schlusslicht; auch im erstmals ausgetragenen Landespokalwettbewerb schied Umdenstocks Elf schon im Sechzehntelfinale gegen die Ligakonkurrentinnen des HSC Montpellier aus. Anschließend beendete sie ihre Karriere.
Stationen
ASPTT Strasbourg (bis 1985)
FC Lyon (1985–2000)
SC Caluire Saint-Clair (2000–2002, davon 2000/01 in D2)
In der Nationalelf
Marie-Christine Umdenstock debütierte im März 1986 bei einem 3:1-Sieg gegen die gastgebenden Belgierinnen in der französischen A-Nationalmannschaft, als Trainer Francis Coché sie knapp zehn Minuten vor dem Abpfiff für ihre ehemalige Vereinskameradin Michèle Wolf einwechselte. Es dauerte dann gut anderthalb Jahre, ehe sie unter Cochés Nachfolger Aimé Mignot zu ihrem zweiten Einsatz kam, erneut in Belgien und diesmal in der Startformation. Ab diesem Zeitpunkt war sie aus der Nationalelf nicht mehr wegzudenken und absolvierte bis zum Mai 1992 insgesamt 36 Länderspiele ohne einen eigenen Torerfolg. Dabei spielte sie ganz überwiegend in der zentralen Abwehr; neben ihr in der Innenverteidigung standen besonders häufig Nathalie Tarade oder Sophie Ryckeboer-Charrier, dazu als Außenverteidigerin ihre Lyoner Mitspielerin Véronique Nowak. Weshalb Mignot Umdenstock, die in den Jahren danach im Verein ja noch höchst erfolgreich spielte, ab Mitte 1992 nie mehr berief, sondern auf die Achse Diacre–Locatelli setzte, ist nicht bekannt.
Während dieser Jahre hatten sich die Französinnen nicht für die Endrunde eines kontinentalen Turniers qualifizieren können. In der Qualifikation zur Europameisterschaft 1989 in Deutschland war Marie-Christine Umdenstock diesem Ziel noch am nächsten gekommen, als die Französinnen Gruppensieger geworden waren, dann aber in den beiden Play-offs an Italien scheiterten, wobei ihre Gegenspielerin Carolina Morace drei der vier italienischen Tore erzielte. Gegen Frauschaften aus dem deutschsprachigen Raum hat sie lediglich eine Partie bestritten – das war im März 1991, als Frankreich durch Treffer von Heidi Mohr und Silvia Neid mit 0:2 gegen Deutschland unterlag.
Palmarès
Französische Meisterin: 1991, 1993, 1995, 1998 (und Vize 1994)
36 A-Länderspiele, kein Tor für Frankreich
Literatur
Pascal Grégoire-Boutreau: Au bonheur des filles. Cahiers intempestifs, Saint-Étienne 2003, ISBN 2-911698-25-8
Weblinks
Umdenstocks Datenblatt auf der Seite des französischen Verbands
Datenblatt bei footofeminin.fr
Anmerkungen und Nachweise
Fußballnationalspieler (Frankreich)
Fußballspieler (FC Lyon)
Franzose
Geboren 1964
Frau
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,990
|
\section{Introduction}
Building new asymmetric encryption schemes has always been one of the main goals of cryptographers. After the idea of public key cryptography was presented in \cite{DH}, only few more public key encryption schemes were developed such as the RSA \cite{RSA}, the El Gamal \cite{ ELG}, the McEliece cryptosystem \cite{MCE}, the NTRU \cite{NTRU} or the HFE \cite{HFE} (for an overview \cite{OV,SAL}). Some new ideas for building new cryptographic schemes based on semigroup actions can also be found in \cite{ROS}, while in the context of knapsack quantum cryptographic schemes we refer for instance to \cite{QC}. {What D. Naccache and J. Stern built in \cite{NSK} was a proposal for an asymmetric protocol (NSK) following the earlier ideas of Morii and Kasahara in \cite{MK1}, further developed by Kasahara et al. in \cite{MK2,MK3}. The NSK protocol consists of a shuffling modulo $p$ of an easy problem over the integers, i.e. the factorization of a composite integer where the prime factors are chosen among a fixed set of small size.}
Given $p$ a prime and $\mathbb{Z}/p\mathbb{Z}$ the finite field of remainder classes, the NSK protocol is based on the unique factorization property of $\mathbb{Z}$, which guarantees the uniqueness of the encryption.
This approach can be generalized to the case of multiplicative monoids (Section \ref{sec1}), and the NSK protocol is just a particular instance for the monoid $(\mathbb{Z},\cdot)$ of the general framework (subsection \ref{particularcase}).
Using this new general setting we are able to construct an analogous of the NSK protocol relying on the unique factorization properties of $\mathbb{F}_q[x]$, instead of $\mathbb{Z}$, where $\mathbb{F}_q$ is the finite field of order $q$ (Section \ref{poly}).
The security of our particular proposal will rely on the arithmetic structure of the finite field $\mathbb{F}_q[x]/(h(x))$ for some $h(x)\in \mathbb{F}_q[x]$, irreducible of suitable degree (instead of the finite field of remainder classes $\mathbb{Z}/p\mathbb{Z}$). One of the main advantages of this kind of setting is that the security is based on an exponentiation over a finite field in such a way that it will be unfeasible for an attacker even to set up a discrete logarithm problem (DLP). Indeed, as we will show in the following, since the optimal version of the NSK protocol requires that the chosen prime be next to $\prod_i p_i$, the factorization of $p-j$ for some small $j$ could allow for a reduction to a DLP.
In our case, instead, we choose a set of irreducible polynomials and fix the degree of the reducing polynomial. By doing so there is no information leakage.
Our new structural conditions will be related only to the degree of the carrier polynomials used for the encryption, avoiding any kind of DLP reduction.
In subsection \ref{security} some issues concerning the security of the protocol will be addressed, in particular to avoid \emph{subgroup attacks}, that could possibly lead to information.
This new setting will lead to some advantages in terms of computational costs of encryption and decryption. In fact, arithmetics over finite fields $\mathbb{F}_{q^m}$ is considered to be preferrable than arithmetics over $\mathbb{Z}_{p}$ when $p\simeq q^m$ and $q\ll p$ in terms of computations. We will analyse the key features of our protocol, such as the number of parameters involved for the setting up of the public key, and this will allow us to show a greater deal of flexibility, in comparison with the NSK protocol.
{In subsection \ref{compare} we will analise the asymptotics of the information rate of our protocol, showing that it is equal to that of \cite{NSK}. An exact formula for the information rate will also be provided.}
As a subproduct, we present in Section \ref{CRT} a variation of the polynomial protocol where the irreducibility of $h(x)$ is dropped. The encryption is performed over a suitable direct sum of fields, and a decryption is available thanks to the Chinese Remainder Theorem.
\section{The new class}\label{sec1}
In this section we will present a generalized version of the protocol presented in \cite{NSK}.
Let $S$ be a monoid and $\sim$ a finite index congruence on $S$. We will denote the class of an element $s\in S$ with respect to $\sim$ as $[s]$.
\begin{definition}
A morphism $\psi$ will be said to be $\sim$proper, if
\begin{itemize}
\item $\psi\colon S\longrightarrow S$ is injective
\item $\psi$ is compatible with $\sim$ (i.e. $\psi(x)\sim\psi(y)$ iff $x\sim y$)
\item the induced application $\widetilde{\psi}\colon {S}/{\sim}\longrightarrow {S}/{\sim}$ is invertible.
\end{itemize}
\end{definition}
\begin{definition}\label{defcryptable}
Given $L\in\mathbb{N}$ we will say that $S$ is $L$-\emph{cryptable} under $\sim$ if there exists a $\sim$proper morphism $\psi$ and elements $s_1,\dots,s_L\in S$ such that
\[\alpha_\sim^\psi: \mathbb{Z}_2^L \longrightarrow S/\sim\]
\[m=(m_1,\dots,m_L)\mapsto \left[\prod^L_{i=1}\psi(s_i)^{m_i}\right]\]
is an injective application.
\end{definition}
The following proposition will be useful later on
\begin{prop}\label{injalpha} Given a monoid $S$ that is $L$-cryptable under $\sim$, the following maps are also injective:
\[\alpha^\psi: \mathbb{Z}_2^L \longrightarrow S\]
\[(m_1,\dots,m_L)\mapsto \prod^L_{i=1}\psi(s_i)^{m_i}\]
\[\alpha_\sim: \mathbb{Z}_2^L \longrightarrow S/\sim\]
\[(m_1,\dots,m_L)\mapsto \left[\prod^L_{i=1}s_i^{m_i}\right]\]
\[\alpha: \mathbb{Z}_2^L \longrightarrow S\]
\[(m_1,\dots,m_L)\mapsto \prod^L_{i=1}s_i^{m_i}.\]
\end{prop}
\begin{proof}
The proof follows by observing that, since $\psi$ is $\sim$proper morphism, then also $\alpha_\sim$ is injective. Also $\alpha_\sim^\psi$ injective implies that $\alpha^\psi$ is injective.
Again, since $\psi$ is an injection, also $\alpha$ is injective.
\end{proof}
{As we have already pointed out}, this properties are necessary to keep the encryption meaningful. In the following we will see how it is possible to find non trivial examples of this construction.
Now, denote the image of any map $f$ between sets by $\Im(f)$, and consider the following problems:
\begin{problem}\label{problem1}
Given $c\in \Im(\alpha^\psi_\sim)$ find $m$ such that $\alpha^\psi_\sim(m)=c$
\end{problem}
\begin{problem}\label{problem2}
Given $c'\in \Im(\alpha_\sim)$ find $m$ such that $\alpha_\sim(m)=c'$.
\end{problem}
Let now $S$, be an $L$-cryptable monoid under a congruence $\sim$. Whenever a given triple $(S,\sim,\psi)$ is such that {Problem \ref{problem1}} is difficult, {Problem \ref{problem2}} is easy we define a cryptosystem as follows. Let
\[(S,\sim,L,\widetilde{\psi}([s_1]),\dots,\widetilde\psi([s_L]))\]
be the public key and
\[(\widetilde{\psi}^{-1}, s_1,\dots,s_L)\]
be the secret key, the main operations are given by
\begin{itemize}
\item{\emph{Encryption}: $E(m):=\alpha^\psi_\sim(m) = \prod^L_{i=1}\widetilde{\psi}([s_i])^{m_i}=: c$}
\item{\emph{Decryption}: $D(c)$ is given by solving Problem \ref{problem2} for $c'=\widetilde{\psi}^{-1}(c)$.}
\end{itemize}
\begin{remark}
The reader should observe that in the definition of the protocol we did not use the injectivity of $\psi$ nor the fact that $S/\sim$ is a quotient of a monoid $S$. This is nevertheless the case in all the examples of this protocol we could find, where Problem \ref{problem2} is easy since a \emph{suitable} lift to $S$ is given. Indeed, in practical situations the problem will be solved computing $(\alpha^{-1}\circ\Gamma)(c')$ where $\Gamma$ is a lift $S/\sim\longrightarrow S$ such that the following diagram
\begin{align}\label{liftdiagramma}
\xymatrix{
\mathbbm{Z}_2^L \ar[r]^\alpha \ar[rd]_{\alpha_\sim} & \Im(\alpha) \\
& \Im (\alpha_\sim) \ar[u]_{\widehat{\Gamma}} }
\end{align}
commutes when $\widehat{\Gamma}:=\Gamma |_{\Im(\alpha_\sim)}$
\end{remark}
\begin{remark}\label{definforate}
Notice that the information rate is given by $L/b$ where $b$ is the number of bits that are needed to represent an element of $S/\sim$
\end{remark}
In what follows we will show how the NSK protocol fits in this rather general framework, as well as brand new protocols involving polynomials over finite fields.
\subsection{NSK as a particular instance}\label{particularcase}
In this section we will show how the Naccache-Stern (NSK) protocol fits in our general framework, in the case $S=(\mathbb{Z},\cdot)$.
Consider the prime ideal $P=\left<p\right>$ generated by a prime number $p\in\mathbb{Z}$. Let us denote by $\sim$ the congruence induced by the ideal $P$. Such a congruence is obviously of finite index. Let $v$ be a positive integer with $u=v^{-1}\mod{p-1}$, and let
\[
\begin{aligned} \psi\colon\mathbb{Z}&\longrightarrow \mathbb{Z}\\ a&\longmapsto a^v.\end{aligned}
\]
It can be easily checked that $\psi$ is a $\sim$proper morphism of $\mathbb{Z}$.
Now choose $L$ distinct prime numbers $p_i$ such that $\prod_{i=1}^L p_i<p$.
\begin{prop}The map
\begin{equation}\begin{aligned}
\alpha_\sim^\psi\colon \mathbb{Z}_2^L&\longrightarrow\mathbb{Z}/p\mathbb{Z}\\
(m_1,\dots,m_L)&\longmapsto\left[\prod\limits_{i=1}^Lp_i^{m_iv}\right]
\end{aligned}\end{equation}
is an injection and $(\mathbb{Z},\cdot)$ is therefore $L$-cryptable under the relation induced by the ideal generated by $p$.
\end{prop}
\begin{proof}
Assume that there exist two $L$-tuples $(m_1,\dots,m_L),(n_1,\dots,n_L)$ such that $\alpha_\sim^\psi(m_1,\dots,m_L)=\alpha_\sim^\psi(n_1,\dots,n_L)$, then:
\[
\left[\prod\limits_{i=1}^Lp_i^{m_iv}\right]=\left[\prod\limits_{i=1}^Lp_i^{n_iv}\right]\Rightarrow \left[\prod\limits_{i=1}^Lp_i^{m_iv}\right]^u=
\left[\prod\limits_{i=1}^Lp_i^{n_iv}\right]^u\Leftrightarrow
\left[\prod\limits_{i=1}^Lp_i^{m_i}\right]=\left[\prod\limits_{i=1}^Lp_i^{n_i}\right]
\]
in $\mathbb{Z}/p\mathbb{Z}$. Since $\prod_{i=1}^Lp_i^{m_i}$ and $\prod_{i=1}^Lp_i^{n_i}$ are smaller than $p$ we also have
\begin{equation}\label{canlift1}
\prod\limits_{i=1}^Lp_i^{m_i}=\prod\limits_{i=1}^Lp_i^{n_i}
\end{equation}
in the unique factorization domain $\mathbb{Z}$, which implies $m_i=n_i\ \forall i$.
\end{proof}
\begin{remark}
Notice that we are able to express equation \eqref{canlift1} because we can always consider the canonical representative $x\in\{0,\dots,p-1\}$ in the remainder class modulo $p$. This representative is also the only representative in $\Im(\alpha)$ by construction, and therefore we have a canonical lift satisfying \eqref{liftdiagramma}.
\end{remark}
\begin{remark}\label{NSKleak}
The reader should observe that when $p=t+\prod_i p_i $ for $t$ small, than the information rate is maximal. Unfortunately in this case factoring $p-t$ is easy because $p-t$ is $p_L$-smooth and $p_L\ll p $, and this gives informations about the bare carriers $p_i$'s. Indeed in this case breaking the NSK protocol is not harder than solving the DLP for the $p_i$'s. Nevertheless the protocol remains interesting for additional features like \cite[Section 3]{NSK}.
\end{remark}
\section{A polynomial version}\label{poly}
In this section we give a version of the protocol that works over $\mathbb{F}_{q^d}$ instead of $\mathbb{Z}/p\mathbb{Z}$ in such a way that $q^d$ will be
of the same order of magnitude than the size $p$ of the field $\mathbb{Z}/p\mathbb{Z}$ in the NSK but $q\ll p$.
In this case the specific difficult problem we want to rely on is the following
\begin{problem}
Let $\mathbb{F}$ be a finite field and $L\in\mathbb{N}$.Given $y_1,\dots, y_L\in \mathbb{F}$,
\[\alpha: \mathbb{Z}_2^L \longrightarrow \mathbb{F}\]
\[\alpha(m)=\prod_i y_i^{m_i}\]
and $c\in \Im(\alpha)$, find $m$ such that $\alpha(m)=c$.
\end{problem}
Let now $k=\mathbb{F}_q$ and $k[x]$ the polynomial ring in one variable over $k$.
Let $h(x)$ be an irreducible element in $k[x]$ of degree $d$.
Set $\sim$ to be the congruence associated to the ideal $H=\left<h(x)\right>$ generated by the irreducible polynomial $h(x)$. An efficient algorithm to find irreducible polynomials of fixed degree is given, for instance in \cite{POLLEG}.
Set \[S=(k[x],\cdot)\] and \[S':=S/\sim\;=\left((k[x]/H)^*,\cdot\right)\]
where $(k[x]/H)^*=(k[x]/H) \setminus \{0\}$.
Fix $v,u\in \mathbb{N}$ such that $\gcd{(v, |S'|) }=\gcd{(v, q^d-1) }=1$ and $uv\equiv 1 \mod |S'| $.
Set
\[\begin{aligned}
\widetilde{\psi}\colon S'&\longrightarrow S'\\
[s]&\longmapsto [s^v].
\end{aligned}\]
\begin{remark}$\,$
\begin{itemize}
\item{$\widetilde{\psi}^{-1}\colon [z]\longmapsto [z]^u$}
\item{$k[x]/H\cong \mathbb{F}_{q^d}$ is again a finite field}
\end{itemize}
\end{remark}
Let now $L\in \mathbb{N}$ such that there exist $L$ distinct irreducible monic polynomials
{$p_1,\dots, p_L\in\mathbb{F}_q[x]$} with the property
\begin{equation}\label{degreeproperty}
\sum_{i=1}^L \deg{p_i}<d.
\end{equation}
Notice that in the present description of the protocol there are several different strategies to choose the polynomials; we will analyse the properties of some interesting choices in the following sections.
Again, we have the encryption map:
{\begin{prop}
$(k[x],\cdot)$ is an $L$ cryptable monoid with the map
\begin{equation}\begin{aligned}
\alpha_\sim^\psi: \mathbb{Z}_2^L &\longrightarrow S'\\
m=(m_1,\dots,m_L)&\longmapsto \left[\prod^L_{i=1}p_i^{vm_i}\right].
\end{aligned}\end{equation}
\end{prop}
\begin{proof} Definition \ref{defcryptable} requires that the map $\alpha_\sim^\psi$ be an injection. Assume
\[\alpha_\sim^\psi(m_1,\dots,m_L)=\alpha_\sim^\psi(n_1,\dots,n_L)\]
\[ \left[\prod^L_{i=1}p_i^{vm_i}\right]= \left[\prod^L_{i=1}p_i^{vn_i}\right].\]
It follows
\[ \left[\prod^L_{i=1}p_i^{vm_i}\right]^u= \left[\prod^L_{i=1}p_i^{vn_i}\right]^u\]
\[ \left[\prod^L_{i=1}p_i^{m_i}\right]= \left[\prod^L_{i=1}p_i^{n_i}\right]\]
where, in the last equation, we can assume no reduction has happened, since property
(\ref{degreeproperty}) holds. Indeed
\begin{equation}\label{canlift2} \prod^L_{i=1}p_i^{m_i}= \prod^L_{i=1}p_i^{n_i}.\end{equation}
Recalling that $k[x]$ is a unique factorization domain we have $m_i=n_i\ \forall i$.
\end{proof}
}
So our cyphered text is given by $c(x)=\alpha_\sim^\psi(m_1,\dots,m_L)$. The explicit decryption for this protocol is simply given by the polynomial division of the decyphered code $(c(x))^u$, that is to say
\begin{equation}\label{decryption}
m_i=1 \Longleftrightarrow (c(x))^u=0 \mod p_i(x) .
\end{equation}
\begin{remark}
We stress once again the fact that in obtaining equation \eqref{canlift2} we used the canonical lift
\begin{align*}
\Gamma\colon &S/\sim\,\longrightarrow S \\
& [f(x)]\longmapsto g(x)
\end{align*}
where, for any representative $l(x)\in[f(x)]$, $g(x)$ is the remainder of the division of $l(x)$ by $h(x)$ in $k[x]$, and it is obviously independent of the choice of $l(x)$.
The decryption is effectively performed in $\Im(\alpha)$ and the solution to Problem \ref{problem2} is then given by $(\alpha^{-1}\circ \Gamma)(c(x)^u)$.
\end{remark}
The information rate $\mathcal{I}=L/\deg(h)\log_2(q)$ depends on the choice of the carrier polynomials. We will explain later how to maximise this value.
\begin{remark}
Once the $p_i$'s are fixed the top information rate for this protocol is obtained when we choose $h(x)$ such that
\begin{equation}\label{topinfocondition}
\sum\limits_{i=1}^L\deg{p_i}=\deg{h}-1.
\end{equation}
Indeed the information rate can always be maximised since it is always possible to choose $h(x)$ in $k[x]$ such that \eqref{topinfocondition} is satisfied (cf. Remark \ref{NSKleak}) without allowing for a straightforward reduction to a DLP. This case will be analysed in detail in \ref{efficiency}.
\end{remark}
\subsection{A simple example}
We now give an example in which $k[x]= \mathbb{F}_2[x]$ and the space of messages has size $2^9$. In order to reach a message size of $9$ bits, we need exactly $9$ keys, that is to say monic irreducible polynomials in $\mathbb{F}_2[x]$. From finite field theory, we know that there are exactly $q$ monic polinomials of degree 1, and
\[
\frac{q^d-q}{d}
\]
irreducible monic polynomials of prime degree $d$. So, for $q=2$ we have two polynomials of degree 1, one polynomial of degree 2, two polynomials of degree 3 and six polynomials of degree 5. For the sake of simplicity, even if the example is non optimal as we will explain, let us choose all the irreducible monic polynomials of degree 1,2 and 5, summing up to exactly 9 keys, namely:
\begin{gather}
p_1=x\\
p_2=1 + x\\
p_3=1 + x + x^2\\
p_4=1 + x^2 + x^5\\
p_5=1 + x^3 + x^5\\
p_6=1 + x + x^2 + x^3 + x^5\\
p_7=1 + x + x^2 + x^4 + x^5\\
p_8=1 + x + x^3 + x^4 + x^5\\
p_9=1 + x^2 +x^3 + x^4 + x^5.
\end{gather}
Then, the public key $h(x)$ must be of degree
\[d=\mathrm{deg}(h(x))=\sum_{i=1}^9\mathrm{deg}(p_i(x)) + 1 = 35\]
and irreducible. For instance we may take
\begin{equation}
h(x)=1+x^2+x^{35}
\end{equation}
and set our protocol onto $\mathbb{F}_{2^{35}}\cong(\mathbb{F}_2[x]/H)^*$, whose order is $2^{35}-1$ when $H=\left<h(x)\right>$.
We choose the secret key and the decryption exponent, accordingly, to be $v=3821$ and $u=25169564954$, so that $uv=1\ \mathrm{mod}(2^{35}-1)$.
Then we may publish the 9 carrier keys $p_i^v \mod {(h(x),2)}$:
\begin{align}
p_1^v=&1 + x^2 + x^4 + x^{10} + x^{12} + x^{18} + x^{22} \\
&+ x^{23} + x^{24} + x^{26} + x^{27} + x^{29} + x^{32}\notag\\
p_2^v=&x + x^3 + x^5 + x^6 + x^7 + x^{10} + x^{12}+ x^{13}\\
& + x^{17} + x^{20} + x^{21} + x^{22} + x^{24} + x^{28} + x^{30} + x^{32}\notag\\
p_3^v=&x + x^4 + x^5 + x^7 + x^{13} + x^{20}+ x^{22} \\
& + x^{28} + x^{29} + x^{30} + x^{31} + x^{32} + x^{33} + x^{34}\notag\\
p_4^v=&1 + x^2 + x^3 + x^4 + x^{11} + x^{14} + x^{15} + x^{17} + x^{18}\\
& + x^{19} + x^{20} + x^{21} + x^{24} + x^{28} + x^{30} + x^{34}\notag\\
p_5^v=&1 + x + x^2 + x^3 + x^4 + x^7 + x^8 + x^9 + x^{10} + x^{11} + x^{15}\\
& + x^{18} + x^{20} + x^{21} + x^{22} + x^{24} + x^{26} + x^{29} + x^{32} + x^{33}\notag\\
p_6^v=&1 + x + x^2 + x^4 + x^7 + x^{12} + x^{13} + x^{15} + x^{16} +\\
& x^{18} + x^{21} + x^{22} + x^{23} + x^{24} + x^{30} + x^{34}\notag\\
p_7^v=&1 + x^4 + x^8 + x^9 + x^{10} + x^{15} + x^{19} + x^{28} + x^{30} + x^{32} + x^{33}\\
p_8^v=&x + x^3 + x^4 + x^5 + x^8 + x^{10} + x^{12} + x^{13} + x^{15} + x^{16} \\
&+ x^{17} + x^{25} + x^{26} + x^{27} + x^{28} + x^{30}\notag\\
p_9^v=&x + x^4 + x^6 + x^7 + x^{10} + x^{11} + x^{12} + x^{13} + x^{14} + x^{15} + x^{16}\\
& + x^{17} + x^{18} + x^{20} + x^{23} + x^{24} + x^{30} + x^{31} + x^{32} + x^{33}.\notag
\end{align}
Suppose we want to send the message $m=111000111\in{\mathbb{Z}_2^9}$, we encode it into
\begin{align}
c=&\prod\limits_{i=1}^9p_i^{vm_i} \mod {(h(x),2)}=\notag\\
=&\ x^2 + x^3 + x^6 + x^{10} + x^{15} + x^{16} + x^{17} + x^{18}\notag\\
+&\ x^{20} + x^{21} + x^{23} + x^{26} + x^{27} + x^{30} + x^{31} + x^{33} + x^{34}.
\end{align}
Once the message has been received, it is sufficient to take the {$u$-th} power, and the result is as follows:
\begin{align}
c^u=&\prod\limits_{i=1}^9p_i^{vum_i} \mod {(h(x),2)}=\prod\limits_{i=1}^9p_i^{m_i} =\notag\\
=&\ x + x^3 + x^4 + x^6 + x^{11} + x^{12} + x^{14} + x^{15} + x^{16} + x^{19}
\end{align}
whose factorization yields:
\begin{equation}\begin{aligned}
\mathsf{Factor}_{2}(c^u)=&x (1 + x) (1 + x + x^2) (1 + x + x^2 + x^3 + x^5)\\
& (1 + x + x^3 + x^4 + x^5) (1 + x^2 + x^3 + x^4 + x^5).
\end{aligned}\end{equation}
We used the factorization algorithm in this simple example because we are working with small messages. The decryption algorithm presented in \eqref{decryption} is to be considered preferential.
The information rate associated to this encryption protocol is
\begin{equation}
\mathcal{I}=\frac{L}{\deg(h)}=\frac{9}{35}\cong 25,7\%
\end{equation}
with the size of the space of messages being $2^9$.
\begin{remark}
A similar example is presented in \cite{NSK}, with $2^8$ messages. In the cited example the information rate is slightly higher than ours, yet comparable, but the space of messages is smaller.
If we wanted to match the size of space of messages it would be sufficient to remove one polynomial of degree 5, obtaining an information rate of $\mathcal{I}=8/30\sim 26,7\%$.
Remarkably enough, as in the NSK-protocol there is apparently no key leakage, our protocol preserves the security of the carrier keys. As a matter of fact, factoring the cyphertext $c$, one gets no information whatsoever on the cleartext, as it can be seen in the given example:
\begin{multline}
\mathsf{Factor}_2(c)=x^2 \left(x^4+x^3+1\right) (x^{28}+x^{25}+x^{24}+x^{23}+x^{22}+x^{21}+x^{20}+x^{18}+\\
x^{17}+x^{15}+x^{14}+x^{12}+x^{11}+x^{10}+x^8+x^6+x^5+x^4+x^3+x+1)
\end{multline}
\end{remark}
{
\begin{remark}
More generally, let $g(x)$ be the public modulus and
\[p_1^{vm_1} p_2^{vm_2}\cdots p_L^{vm_L}\equiv c(x) \mod g(x)\] a cyphetext. Observe that over $\mathbb{F}_{q}[x]$ we have \[P(x)=p_1^{vm_1} p_2^{vm_2}\dots p_L^{vm_L}=t(x)g(x)+c(x)\] for some $t(x)\in\mathbb{F}_{q}[x]$. Now notice that infering on the factorization of $P(x)$ from the data of $c(x)$ in terms of the factor basis \[\{p_1^{vm_1},\dots ,p_L^{vm_L}\}\] is the difficult problem on which the protocol relies, since the factorization of polynomials behaves badly with respect to reductions modulo irreducible polynomials.
As a matter of fact, we base the security of our protocol on the randomness of the factorization of elements in the image of the map
\[\Gamma_{g,c}: \mathbb{F}_q[x] \longrightarrow \mathbb{F}_{q}[x]\]
\[\Gamma_{g,c}(t(x))=t(x)g(x)+c(x)\]
In general, the usual security one expects using prime numbers as carriers (NSK) can be extended to monic irreducible polynomials.
\end{remark}
}
As we already { pointed out}, we are using here a non-optimal setting for our example, in that we skipped the polynomials of degree 3 and 4, and used all those of degree 5 instead. If we decided to optimize the information rate, we could take the two polynomials of degree 1, the single polynomial of degree 2, two of degree 3 and three of degree 4, for an overall encoding power of $2^8$ messages. Notice that the space of messages is again equal to the example given in \cite{NSK}.
Choosing polynomials of degree 3 and 4 instead of 5 allows us to reduce the degree of $h(x)$, that is to say the number of bits that are needed to encrypt a message. So, if we compute the information rate in this case we obtain a much better result:
\begin{equation}
\mathcal{I}=\frac{\log_2{m}}{\log_2{c}}=\frac{8}{23}\cong 34,78\%
\end{equation}
which is slightly higher than the information rate presented in \cite{NSK} for the same message size.
The procedure works exactly the same when we change the ground field from $p=2$ to $p=3$. This time we may choose three polynomials of degree 1, three of degree 2 and two of degree 3, all monic and irreducible, allowing us to reduce the overall degree of $h(x)$ to $\mathrm{deg}(h(x))= 16$. In this case, for the same message size, we get an information rate of
\begin{equation}
\mathcal{I}=\frac{8}{16\log_2 3}\cong 31,55\%
\end{equation}
which is not better than the information rate in \cite{NSK}, for a space of messages of the same size, yet comparable.
\subsection{Flexibility of the protocol}
We have already pointed out in the previous sections that the important condition \eqref{degreeproperty} can be fulfilled in several different ways according to the strategy we use in choosing the carrier polynomials $p_i$'s.
In what follows we will present a strategy that optimises the information rate and one that, to our analysys, improves security.
{We will give a detailed analysis of the asymptotics of the information rate of our protocol and of NSK, showing that they have the same behaviour. In what follows our finite field $k$ will be $\mathbb{F}_q$ for some prime power $q$.
}
\subsubsection{Optimization of the information rate}\label{efficiency}
The optimization of the information rate is ensured by the following:
\begin{prop}
There exists a strategy that maximises the information rate $\mathcal{I}$ for any choice of $q$ and $L$. Moreover, in this strategy the information rate is determined by the closed formula
\begin{equation}\label{inforate}
\mathcal{I}(q,N)=\frac{\sum\limits_{n=1}^N\frac{1}{n}\sum\limits_{k\mid n}\mu\left(\frac{n}{k}\right)q^{{k}}}{\left( \sum\limits_{n=1}^N\sum\limits_{k\mid n}\mu\left(\frac{n}{k}\right)q^{{k}}+1\right)\log_2q}
\end{equation}
where $\mu(x)$ is the M\"obius function.
\end{prop}
\begin{proof}
We defined the information rate to be $\mathcal{I}=L/(\deg{h}\log_2q)$ and we know that the degree of $h$ depends on the particular choice of carrier polynomials. The strategy we will consider is simply given by choosing \emph{all} irreducible polynomials of all degrees up to a given degree $N$. Denote the number of degree-$n$ irreducible polynomials in $\mathbb{F}_q[x]$ by $D^q_n$, we have the formula
\begin{equation*}
D_n^q=\frac{1}{n}\sum_{k\mid n}\mu\left(\frac{n}{k}\right)q^k
\end{equation*}
where $\mu(x)$ is the M\"obius function. The overall number of chosen polynomials, that is the number of bits that the plain text is composed by, as well as the sum of the degrees of the $p_i$'s are given by a closed formula, namely:
\begin{align}
L&=\sum\limits_{n=1}^N D_n^q= \sum\limits_{n=1}^N\sum\limits_{k\mid n}\mu\left(\frac{n}{k}\right)\frac{q^{{k}}}{n}\\\label{degh}
\deg(h(x))&=\sum_{n=1}^N nD_n^q+1=\sum\limits_{n=1}^N\sum\limits_{k\mid n}\mu\left(\frac{n}{k}\right)q^{{k}}+1
\end{align}
for some maximal degree $N$ (which is dependent on $L$ if we consider $L$ to be the fundamental parameter). Then, the information rate $\mathcal{I}$ as a function of the prime power $q$ and (implicitly) the parameter $L$ has the desired closed expression.
It is easy to gather that such a choice of the polynomials guarantees maximal information rate, in that we are lowering as much as possible the degree of $h(x)$ and as a result the number of bits of the encrypted message.
\end{proof}
\begin{remark}
The obvious disadvantage of the strategy above is that one can always assume that the bare carrier polynomials are known, for we take all of them progressively up to degree N. As a matter of fact, the strategy above gives us a clear upper bound for the information rate, for all different combinations of $L$ and $q$. Notice, however, by comparison with the tables of \cite{NSK}, that this is the same strategy adopted by Naccache and Stern, where the chosen prime $p$ has the same size of $\mathsf{NextPrime}(\prod p_i)$.
\end{remark}
Within this strategy it is important to notice that all the variations proposed in \cite[Section 2.3]{NSK} are importable in the present context. For example, it is possible to express the message $m$ in a basis different from 2, and this would lead to some modification to the suitable degrees for our carriers. Moreover, it is possible to restrict the space of messages to constant-weight strings. This last choice increases the information rate since it allows to lower the degree of $h(x)$. In fact, if $w$ is the constant weight, the bound on the degree of $h$ is:
\[\deg{h}>wN\]
where N is the highest degree of the chosen carriers.
Apart from these extensions, the standard NSK protocol is summarized in the table presented in \cite[Section 2.2]{NSK}, where the information rate for 512, 1024 and 2048 bits-sized $p$'s is given. The strategy we have just outlined to reach the maximal information rate, allows us to obtain the exact values presented in \cite{NSK} matching the degree of our polynomial $h$ with the size of their prime $p$ and $L$ with the size $\mathcal{M}$ of the message. So we are able to obtain the same information rate.
The matching procedure works as follows: compute the degree of $h$ obtained by choosing all polynomials up to a given degree, say 9 to obtain $\deg{h}=977$. Then, top it to the next block, in this case 1024 bits, choosing \emph{some} polynomials of one degree higher, in this case 11. This leads to an increase in the number $L$ of carrier polynomials from 127 to 131, and the information rate is then given by the ratio $L/\deg{h}$.
\begin{table}
\centering
\begin{tabular}{c|c|c}
$L$ (bits) & $\deg{h}$ (bits) & $\mathcal{I}$\\\hline
131 & 1024 & 12,8 \%\\
233 & 2048 & 11,4\% \\
418 & 4096 & 11,2\%
\end{tabular}\caption{{Information rate matching with \cite[Section 2.2]{NSK}}}\label{tab1}
\end{table}
{ In Table \ref{tab1} we show how to match the examples presented in \cite{NSK}, and the last row is obtained by extending their calculations to 4096 bits. If we go further and compute the relevant figures in the case of 8192 bits we find almost perfect agreement also in this case (cf. Table \ref{tab2}). It will be clear in what follows why this happens. }
\begin{table}
\centering
\begin{tabular}{c|c|c|c}
$L$ (bits)& $\mathcal{M}$ (bits) & Size of $p$ \& $\deg{h}$ (bits) & $\mathcal{I}$\\\hline
759& 758 & 8192 & 11,4\%
\end{tabular}\caption{{Extension to next block and matching of the information rate}}\label{tab2}
\end{table}
{
\subsubsection{Asymptotics comparison with previous works}\label{compare}
We will prove in this section that our protocol has the same asymptotic information rate of \cite{NSK}. A naive explanation of this fact is given by arguing that the number of primes below a certain number of bits has the same behaviour as the number of irreducible polynomials in $\mathbb{F}_q[x]$ below a certain degree.
Let us fix the notation
\[
a_N\sim \,b_N\ \ \Longleftrightarrow \lim_{N\rightarrow\infty} \frac{a_N}{b_N}=1.
\]
}
{
We will make use of the following
\begin{lemma}\label{lemlem}
\begin{equation}\label{lemma}
\sum\limits_{n=1}^N D_n^q \sim \frac{q}{q-1} D^q_N
\end{equation}
\end{lemma}
\begin{proof}
First recall that \cite[Theorem 2.2]{Poly}
\[
D_n^q \sim \frac{q^n}{n}
\]
and therefore the sums behave asymptotically as
\[
\sum\limits_{n=1}^N D_n^q \sim \sum\limits_{n=1}^N \frac{q^n}{n}.
\]
Then we have \eqref{lemma} if and only if
\begin{equation}
\lim_{N\rightarrow \infty} \frac{\sum\limits_{n=1}^N \frac{q^n}{n}}{\frac{q^N}{N}} = \frac{q}{q-1}.
\end{equation}
Now, denote by $S_N\mathrm{:=}\sum\limits_{n=1}^N \frac{N}{n}q^{n-N}$ and observe that it might be expressed in terms of the recursive sequence
\begin{equation}\label{recursion}
S_{N+1} = \frac{1}{q}\frac{N+1}{N} S_N +1.
\end{equation}
for the initial value $S_1=1$. Consider $S_-=\liminf_{N\rightarrow\infty}S_N$ and $S_+=\limsup_{N\rightarrow\infty}S_N$. Passing to the $\limsup$ and $\liminf$ in \eqref{recursion} we get the same equation for $S_\pm$:
\[
S_\pm=\frac{S_\pm}{q} +1
\]
provided that they are both finite. Assuming that they are, we conclude that
\begin{equation}
\lim_{N\rightarrow\infty}S_N=S_\pm=\frac{q}{q-1}
\end{equation}
This assumption is legitimate since $S_N\geq 0$ for all $N\in \mathbb{N}$, thus $S_-\geq 0$, and for $S_+$ we observe that
\begin{itemize}
\item When $x\in\mathbb{R}^+$ we have that $\frac{q^x}{x}$ is increasing for $x\geq \frac{1}{\log{q}}\geq 2$, since $q\geq 2$, and in particular this is true for $x\in\mathbb{N}^*$;
\item $\limsup_{N\rightarrow \infty} \frac{N}{q^N}\sum\limits_{n=1}^N \frac{q^n}{n}=\limsup_{N\rightarrow \infty} \frac{N}{q^N}\sum\limits_{n=2}^N \frac{q^n}{n}$.
\end{itemize}
It follows that
\[
\limsup_{N\rightarrow \infty}\frac{N}{q^N}\sum\limits_{n=1}^N \frac{q^n}{n} = \limsup_{N\rightarrow \infty} \frac{N}{q^N}\sum\limits_{n=2}^N \frac{q^n}{n} \leq \limsup_{N\rightarrow \infty}\frac{N}{q^N}\int\limits_{2}^{N+1}\frac{q^x}{x}dx
\]
where the last inequality comes from the fact that $\sum\limits_{n=2}^N \frac{q^n}{n}$ are the lower sums of $\int\limits_{2}^{N+1}\frac{q^x}{x}dx$, since $\frac{q^x}{x}$ is increasing for $x\geq 2$. Moreover
\[
\lim_{N\rightarrow \infty}\frac{\int\limits_{2}^{N+1}\frac{q^x}{x}dx}{\frac{q^N}{N}} = \lim_{t\rightarrow \infty}\frac{\int\limits_{2}^{t+1}\frac{q^x}{x}dx}{\frac{q^t}{t}}=\lim_{t\rightarrow \infty}\frac{\frac{q^{t+1}}{t+1}}{\frac{q^t}{t}(\log{q} - \frac{1}{t})}=\frac{q}{\log{q}}
\]
where the second equality follows from the De L'H\^{o}pital rule. This proves that
\[
0\leq \liminf_{N\rightarrow \infty} S_N \leq \limsup_{N\rightarrow \infty} S_N \leq \frac{q}{\log{q}}
\]
and yields the claim.
\end{proof}
}
{ We are now ready to prove
\begin{prop}\label{MSKprop}
\begin{equation}\label{asympt}
\mathcal{I}(q,N)\sim \frac{1}{\log_2{q}}\frac{1}{N}
\end{equation}
\begin{proof}
Observe that $nD_n^q\sim q^n$ and therefore, from \eqref{inforate}
\[
\mathcal{I}(q,N)\sim\frac{\sum\limits_{n=1}^N \frac{q^n}{n}}{\log_2{q}\sum\limits_{n=1}^N q^n}
\]
Now, it is easy to gather that
\begin{equation}\label{geoseries}
\sum\limits_{n=1}^N q^n \sim \frac{q}{q-1} q^N
\end{equation}
then, plugging the results of \eqref{geoseries} and of Lemma \ref{lemlem} into \eqref{inforate}, we obtain
\begin{equation}
\mathcal{I}(q,N)\sim \frac{1}{\log_2{q}}\frac{\frac{q}{q-1}\frac{q^N}{N}}{\frac{q}{q-1}q^N} =\frac{1}{\log_2{q}} \frac{1}{N}.
\end{equation}
\end{proof}
\end{prop}
{
We would like to compare this result with the information rate of the NSK protocol. Notice that in order to make a consistent comparison we must understand the role of our parameter $N$ in the NSK.
Once $q$ is fixed, bounding the degree of the carrier polynomials by $N$ is the same as bounding
the number of bits required to represent any of them by the quantity $M=\left\lfloor N \log_2(q)\right\rfloor$.
The analogous bound for the NSK is then given by bounding the number of bits of the prime carriers by $M$. This is the same as bounding the prime carriers themselves by $2^M\simeq q^N$.
In the following proposition the comparison is made explicit.}
{
\begin{prop}\label{NSKprop}
Let $N$ be the bound on the degree of the carrier polynomials and $M=\left\lfloor N \log_2(q)\right\rfloor$ the analogous bound for the bits of the prime carriers in the NSK.
The information rate for the NSK protocol is asymptotically given by
\begin{equation}
I_{NSK}\sim \frac{1}{\log_2{q}}\frac{1}{N}.
\end{equation}
\begin{proof}
It is known \cite[Equation 2]{Erd} that for large $m\in\mathbb{N}$
\[
\prod\limits_{p<m}p \sim e^{m}.
\]
Let us consider $m=2^M\simeq q^N$, then $\prod_{p<q^N}p \sim \exp{q^N}$. Now, the number of prime numbers up to $q^N$ asymptotically goes, by the prime number theorem, as
\[
\pi(q^N)\sim \frac{q^N}{N\ln{q}}.
\]
In our case this will be the number of carrier prime numbers up to $q^N$. On the other hand $\exp{q^N}$, which is the size of the prime modulus of \cite{NSK}, has $\left\lfloor q^N\log_2{e}\right\rfloor$ digits, and therefore the information rate is computed as
\begin{equation}
I_{NSK}\sim \frac{\frac{q^N}{N\ln{q}}}{q^N\log_2{e}}=\frac{1}{\log_2{q}}\frac{1}{N}
\end{equation}
\end{proof}
\end{prop}}
{
By comparing Propositions \ref{MSKprop} and \ref{NSKprop} it is now clear that the two information rates have the same behaviour. This explains that the matching procedure we perform at the end of the previous section will attain the information rate of NSK also in the asymptotic limit. Moreover it justifies the claim on the large-$N$ behaviour of irreducible polynomials with respect to prime numbers. }
}
\subsubsection{Some precautions to avoid subgroup-like attacks}\label{security}
The security of this protocol is strictly related to the size of the degree of $h$ and, as a consequence, to the range of degrees that the carriers can have. Indeed, when the carriers are chosen within a large set, the attacker will not have chances (in terms of a brute force attack) to find the $p_i$'s to set up a discrete logarithm problem for the pair $(p_i,p_i^s)$ for any $i$.
As a matter of fact, the knowledge of $h$ will only lead to the following information on the degrees:
\[\deg(h)=\sum_i{\deg(p_i)}+1.\]
This is not the case when working with integers and primes in $\mathbb{Z}/p\mathbb{Z}$, where we can always assume that the prime factors are known
when $p\simeq \prod_i p_i$.
We first sketch a subgroup like attack in the most \emph{unsafe} case.
Let $G$ be an abelian group and $p^v_1,\dots,p^v_L$ be carriers, as in Section \ref{poly}.
Let the order of $p^v_i$ in $G$ be $n_i$ and suppose $\gcd{(n_i,n_j)}=1$ for $i\neq j$.
Let now
\[M_j=n_1\cdots n_{j-1}\cdot n_{j+1}\cdots n_L.\]
It is easy to observe that, for a generic cyphertext $c$, $m_j=1$ if and only if $c^{M_j}\neq 1$. As it is elementary to observe, this leads to decryption in $L$ steps. Moreover, it can also be adapted to work when the condition $\gcd{(n_i,n_j)}=1$ is just partially fulfilled. {In this case, indeed, only partial information on the text can be extracted.}
Consider now the decomposition in cyclic subgroups of the multiplicative group of the finite field $(\mathbb{F}_{q^d})^*$. In order to avoid subgroup-like attacks on the cyphertext we will require all the $p_i$'s to be generators of the same subgroup of large order. This will lead to certain requirements on $q^d-1$.
The most natural choice to solve this problem is asking that the degree $d$ of the reducing polynomial $h(x)$ be constrained by the following:
\begin{equation}\label{chone}
r:=\frac{q^d-1}{q-1}\ {\mathit{is\ prime}}.
\end{equation}
Now we have to choose the $p_i$'s such that
\begin{equation}\label{chtwo}
p_i(x)^r\neq 1 \mod{h(x)}\quad \forall i\in\{1,\dots,L\}.
\end{equation}
When these conditions are satisfied, all the $p_i$'s are generators for $(\mathbb{F}_{q^d})^*$.
\subsection{``Chinese remainder'' version}\label{CRT}
In what follows we will present another example of a protocol that fits the general picture, which stems on the well known chinese remainder theorem. To do this, let us introduce a large prime power $q$ and a natural number $L\in \mathbb{N}$. Consider now the monoid $S=(\mathbb{F}_q^{L+1})^*$, with the multiplication defined componentwise, and the set $R=\{r_1,\dots,r_{L+1}\}\subseteq\mathbb{F}_q$.
Let $\alpha_i\in \mathbb{F}_q\backslash R\quad\forall i\in\{1,\dots,L\}$ and choose two large integers $u,v$ such that $uv=1\mod{(q-1)}$. Compute the following list of vectors $p_i\in(\mathbb{F}_q^{L+1})^*$ as
\[
(g_i)_j:=(r_j-\alpha_i)
\]
\[
(p_i)_j:=(g_i)^v.
\]
Let
\[((\mathbb{F}_q^{L+1})^*, \{p_1,\dots,p_L\})\]
be the public key and
\[(\{g_1,\dots, g_L\}, \{r_1\dots r_L\})\]
be the secret key.
Let
\[F:\mathbb{Z}_2^L \longrightarrow S\]
\[(m_1,\dots,m_L)\mapsto \prod_{i=1}^L p_i^{m_i}\]
be the encryption map.
\begin{remark}
Observe that the information rate is
\[\frac{L}{(L+1)\log_2(q)}.\]
\end{remark}
\begin{prop}
$F$ is an injection.
\end{prop}
\begin{proof}
We define a polynomial on $\mathbb{F}_q[x]$ by
\[
h_R(x):=\prod_{i=1}^{L+1}(x-r_i)
\]
whose set of zeros coincide with $R$.
We will prove the proposition by showing how to compute the inverse over the image of $F$ using $h(x)$, i.e. we will show how to uniquely decrypt any cyphertext $c\in \Im(F)$ using the secret key.
Let
\[\psi: S \longrightarrow S\]
\[x\mapsto x^v\]
\[G: \mathbb{F}_q[x]/h_R(x) \stackrel{\mathrm{CRT}}{\longrightarrow} \mathbb{F}_q^L\]
\[k(x)\mapsto (k(r_1),\dots,k(r_L))\]
and
\[\Gamma: \mathbb{F}_q[x]/h_R(x) \longrightarrow \mathbb{F}_q[x]\]
be the canonical lift.
The decryption map $D$ is given by checking $\Gamma( G^{-1}(\psi^{-1}(x)))$ modulo $g_i(x)=(x-\alpha_i)$: whenever it is zero it means $m_i=1$, where $ \psi^{-1}(x)= x^u$. Observe that the decryption is well defined: the map
\[\alpha_\sim^\psi: \mathbb{Z}^L_2\longrightarrow \mathbb{F}_q[x]/(h_R(x))\]
is clearly injective (and then $\alpha_\sim$ is, by Proposition \ref{injalpha}) since the product of all the $g_i(x)$ has degree $L<L+1$.
Observe that $\sim$ is as usual the relation induced by the ideal of $h_R(x)$.
\end{proof}
\section{Outlook and further research}
In the present communication we have given a new setting to produce many examples of knapsack encryption schemes, showing also how a remarkable example such as \cite{NSK} perfectly fits our framework. We have proposed a next-to-simplest example when the monoid is chosen to be $(k[x],\cdot)$, one realization of which is given by $\mathbb{F}_q[x]$ reduced by the ideal of an irreducible polynomial of suitable degree.
{
This brand new application of the knapsack idea reproduces the key results presented in \cite{NSK} in terms of information rate, but allows us to improve some important features such as
\begin{itemize}
\item the information rate is shown to be deterministic by providing an exact formula for it (cf. \cite[Section 2.2]{NSK}).
\item it reduces the computations over $\mathbb{F}_{q^d}$ with $p\sim q^d$ but $q\ll p$, where $\mathbb{F}_q$ is a field of small characteristic.
\end{itemize}
A non trivial variation of this scheme has been found, by taking into account a polynomial which splits over the base field and applying the chinese remainder theorem, allowing the computations to be performed over a direct sum of fields.
}
In \cite{NSK} Naccache and Stern conjectured that it might be possible to elliptic curve their scheme, and the new general framework we have presented might be of some help to address this problem.
Moreover, it would be interesting to see how the recent improvements to the NSK protocol presented in \cite{CHEV} may apply to our polynomial instance. This will be matter of further studies.
\section*{Acknowledgements}
The authors would like to thank Patrik K\"uhn, G\'erard Maze, Joachim Rosenthal and Davide Schipani for helpful discussions and suggestions. M.S. acknowledges partial support from SNF grant 200020\_149150/1.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,015
|
package com.mundane.androidtechniqueapply.view.widget;
import android.content.Context;
import android.graphics.Bitmap;
import android.graphics.BitmapFactory;
import android.graphics.Canvas;
import android.graphics.Color;
import android.graphics.Paint;
import android.graphics.PorterDuff;
import android.graphics.PorterDuffXfermode;
import android.util.AttributeSet;
import android.util.Log;
import android.view.View;
import com.mundane.androidtechniqueapply.R;
/**
* Created by fangyuan.zhou on 2017/3/27 11:27
*/
public class MyXfermode extends View {
private int width;
private int height;
private Paint mPaint;
private Bitmap mbitmap;
private Bitmap moutbitmap;
public MyXfermode(Context context) {
super(context);
}
public MyXfermode(Context context, AttributeSet attrs) {
super(context, attrs);
initBitmap();
}
public void initBitmap(){
//禁用硬件加速器,因为有些硬件加速器不支持
setLayerType(LAYER_TYPE_SOFTWARE, null);
//设置抗锯齿
mPaint =new Paint(Paint.ANTI_ALIAS_FLAG);
mPaint.setColor(Color.YELLOW);
mbitmap= BitmapFactory.decodeResource(getResources(), R.drawable.a1);
moutbitmap= Bitmap.createBitmap(mbitmap.getWidth(), mbitmap.getHeight(), Bitmap.Config.ARGB_8888);
Log.d("图片信息", "宽: " + mbitmap.getWidth()+ ", 高: "+mbitmap.getHeight());
//以后的绘制都将显示在moutbitmap上面
Canvas canvas = new Canvas(moutbitmap);
//Dst
canvas.drawCircle(mbitmap.getWidth() / 2, mbitmap.getHeight() / 2, mbitmap.getWidth() / 2, mPaint);
PorterDuffXfermode mode = new PorterDuffXfermode(PorterDuff.Mode.SRC_IN);
mPaint.setXfermode(mode);
//Src
canvas.drawBitmap(mbitmap, 0, 0, mPaint);
}
@Override
protected void onDraw(Canvas canvas) {
canvas.drawBitmap(moutbitmap, 0, 0, null);
}
@Override
protected void onMeasure(int widthMeasureSpec, int heightMeasureSpec) {
super.onMeasure(widthMeasureSpec, heightMeasureSpec);
width = getDefaultSize(getSuggestedMinimumWidth(), widthMeasureSpec);
height = getDefaultSize(getSuggestedMinimumHeight(), heightMeasureSpec);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,148
|
{"url":"https:\/\/www.nature.com\/articles\/sdata201666?error=cookies_not_supported&code=3692a3a6-5b30-4d97-af56-63000760c00f","text":"Data Descriptor | Open | Published:\n\n# Mapping internal connectivity through human migration in malaria endemic countries\n\n## Abstract\n\nHuman mobility continues to increase in terms of volumes and reach, producing growing global connectivity. This connectivity hampers efforts to eliminate infectious diseases such as malaria through reintroductions of pathogens, and thus accounting for it becomes important in designing global, continental, regional, and national strategies. Recent works have shown that census-derived migration data provides a good proxy for internal connectivity, in terms of relative strengths of movement between administrative units, across temporal scales. To support global malaria eradication strategy efforts, here we describe the construction of an open access archive of estimated internal migration flows in endemic countries built through pooling of census microdata. These connectivity datasets, described here along with the approaches and methods used to create and validate them, are available both through the WorldPop website and the WorldPop Dataverse Repository.\n\nDesign Type(s) data integration objective \u2022 observation design Human Migration digital curation Homo sapiens \u2022 Angola \u2022 Burundi \u2022 Benin \u2022 Burkina Faso \u2022 Botswana \u2022 Central African Republic \u2022 Cote d'Ivoire \u2022 Cameroon \u2022 Democratic Republic of the Congo \u2022 Republic of Congo \u2022 Comoros \u2022 Djibouti \u2022 Eritrea \u2022 Ethiopia \u2022 Gabon \u2022 Ghana \u2022 Guinea \u2022 Gambia \u2022 Guinea-Bissau \u2022 Equatorial Guinea \u2022 Kenya \u2022 Liberia \u2022 Madagascar \u2022 Mali \u2022 Mozambique \u2022 Mauritania \u2022 Malawi \u2022 Mayotte \u2022 Namibia \u2022 Niger \u2022 Nigeria \u2022 Rwanda \u2022 Sudan \u2022 Senegal \u2022 Sierra Leone \u2022 Somalia \u2022 South Sudan \u2022 Sao Tome and Principe \u2022 Swaziland \u2022 Chad \u2022 Togo \u2022 Tanzania \u2022 Uganda \u2022 Republic of South Africa \u2022 Zambia \u2022 Zimbabwe \u2022 Afghanistan \u2022 Azerbaijan \u2022 Bangladesh \u2022 Bhutan \u2022 China \u2022 Georgia \u2022 Indonesia \u2022 India \u2022 Iran \u2022 Cambodia \u2022 South Korea \u2022 Laos \u2022 Sri Lanka \u2022 Myanmar \u2022 Malaysia \u2022 Nepal \u2022 Pakistan \u2022 The Philippines \u2022 Papua New Guinea \u2022 North Korea \u2022 Saudi Arabia \u2022 Solomon Islands \u2022 Thailand \u2022 Tajikistan \u2022 Timor-Leste \u2022 Viet Nam \u2022 Vanuatu \u2022 Yemen \u2022 Argentina \u2022 Belize \u2022 Bolivia \u2022 Brazil \u2022 Colombia \u2022 Costa Rica \u2022 Dominican Republic \u2022 Ecuador \u2022 Guatemala \u2022 Guyane \u2022 Guyana \u2022 Honduras \u2022 Haiti \u2022 Mexico \u2022 Nicaragua \u2022 Panama \u2022 Peru \u2022 Paraguay \u2022 El Salvador \u2022 Suriname \u2022 Venezuela \u2022 anthropogenic habitat\n\nMachine-accessible metadata file describing the reported data (ISA-tab format)\n\n## Background & Summary\n\nAccording to the International Organization for Migration1 and The World Bank2, without accounting for seasonal and temporary migrants, more than 1 billion people are currently living outside their places of origin, with about 740 million of them classified as internal migrants. Additionally, in 2014 around 67 million passengers travelled on international and domestic flights every week3 and hundreds of millions are estimated to commute daily by public transport and private vehicles4. Human mobility is expected to continue rising in volume and reach, producing increasing global connectivity that has a range of impacts, including rising numbers of invasive species, the spread of drug resistance, and disease pandemics. In this context, quantifying human mobility across multiple temporal and spatial scales, becomes crucial for quantifying its effects on society5,\u200b6,\u200b7, evaluating its relationship with the environment8,9, better understanding human-related processes such as urbanization and land use change10,\u200b11,\u200b12,\u200b13, and providing a strong evidence base to support both development14,\u200b15,\u200b16 and public health17,\u200b18,\u200b19 applications and policies.\n\nIn public health, the role of human mobility in the spread of infectious diseases is exemplified by the presence of HIV\/AIDS in areas outside where it first emerged at the beginning of the twentieth century20,\u200b21,\u200b22, the 2003 SARS epidemic23, the 2007 Chikungunya outbreaks in Italy and France24,25, the 2009 H1N1 pandemic26, the 2014 Ebola outbreak in Western Africa27, the resurgence of malaria cases in areas where the disease was once eliminated28, and the worldwide spread of drug resistant pathogens29. Consequently, it is clear that to provide better informed guidelines, both at the national and international level, the effects of human mobility and connectivity in driving disease dynamics need to be better understood and accounted for refs 30,\u200b31,\u200b32,\u200b33.\n\nLocal malaria elimination and global malaria eradication are rising up the international agenda34,\u200b35,\u200b36. Evidence from the previous global malaria eradication program37, as well as from recent studies, control campaigns, and elimination efforts38,\u200b39,\u200b40,\u200b41 highlight the importance of accounting for human mobility in designing elimination plans. Infected people may unknowingly transport malaria parasites (potentially including antimalarial-resistant strains42) into new areas. Parasites can be imported either from other countries43 or from other areas within the same country44. Thus, because of the flow of imported cases from high- to low-transmission settings, the latter will face difficulties in achieving elimination and maintaining malaria-free status if it is achieved43. Nevertheless, despite the importance of these dynamics being long recognized45,46, attempts to translate human mobility model outputs into malaria policy are still rare47.\n\nAs detailed in Tatem7, sources of human mobility data potentially useful for modelling pathogen movements include: air and sea travel data records (including open access modelled versions of them); census migration data; travel history and displacement surveys; GPS tracking data and volunteered geographic information (with the latter including geolocated social media data), and even satellite night-time light data. In particular, patient travel history data, containing detailed demographic information and travel motivations, are traditionally used to understand malaria parasite importation patterns48,\u200b49,\u200b50. Recently, mobile phone call detail records (CDRs) have been increasingly used for measuring short-term human movements51,52 and thus, either alone38,53,54 or in combination with travel history data55 and malaria case data, for supporting malaria control and elimination strategic planning.\n\nHowever, because of difficulties in sharing and accessing CDRs (mostly due to commercial and privacy concerns)7,56,57, alternative datasets are required in order to quantify and map internal connectivity across continental scales. To this end, using CDRs, Wesolowski et al.56 and Ruktanonchai et al.58 demonstrated that widely-available and easy-to-obtain census-derived internal migration flow data can serve as reliable proxies for the relative strength of within-country human connectivity across multiple temporal scales.\n\nWithin the framework of the WorldPop Project (www.worldpop.org), and following the approaches described in Henry et al.59 and Garcia et al.60 (Fig. 1), internal census-based migration microdata available through the online IPUMS-International (IPUMSI) database61, along with a number of other ancillary datasets, were assembled and processed to produce an open access archive of estimated 5-year (2005\u20132010) internal human migration flows for every Plasmodium falciparum and Plasmodium vivax (hereafter simply referred as Pf and Pv, respectively) endemic country62,63 (Supplementary Table 1).\n\n## Methods\n\n### Estimating internal migration flows between administrative units\n\nFollowing Garcia et al.60 a gravity model-based approach was used to estimate the total number of people migrating from one administrative unit to any other administrative unit, between 2005 and 2010, within each malaria endemic country located in Africa, Asia, Latin America and the Caribbean62,63 (Supplementary Table 1).\n\nThe simplest gravity-type spatial interaction model, proposed by Zipf64, considers the total population in a location of origin i and in a location of destination j (henceforth simply indicated as i and j), and the distance between the two locations to predict the migration flow (MIGij) between them. Thus, migration flows between administrative units can be estimated using the following function: $(1)MIGij=Pi\u03b1Pj\u03b2dij\u03b3$ where $Pi\u03b1$ and $Pj\u03b2$ represent the populations in the location of origin i and of destination j, respectively, and $dij\u03b3$ represents the distance between i and j; with \u03b1, \u03b2, and \u03b3 being parameters, used to indicate the magnitude of the effect for each covariate, that are typically estimated in the statistical modelling framework.\n\nIn this study, following the notation from Henry et al.59 and Garcia et al.60, the basic gravity-type spatial interaction in equation (1) was extended in order to include additional geographical and socioeconomic factors described in detail in the Data collection and preparation subsection below. Since the census-based migration microdata extracted from the IPUMSI database61 represent only a sample of the total census, a logistic regression was used to model the proportion of people migrating between administrative units65. In particular, the logistic regression was used to model the proportion of people residing in j in the census year who were in i \u2018n\u2019 years prior to the census. Thus, the proportion of migrants in j in the census year that were previously residing in i was estimated using the following logistic regression function: $(2)pij=e\u03b20+\u03b21Pi+\u03b22Pj\u2212\u03b23dij1+e\u03b20+\u03b21Pi+\u03b22Pj\u2212\u03b23dij$ where $pij=MIGij\/TOTj$; with MIGij and TOTj representing the number of people residing in j in the census year that were in i \u2018n\u2019 years prior to the census and the total population residing in j in the census year, respectively.\n\nInitially, a separate vector \u03b2=(\u03b20, \u03b21, \u2026, \u03b2n) of coefficients was used in the linear predictor of the gravity model for each country (including malaria non-endemic countries), in Africa, Asia, Latin America and the Caribbean, for which migration data were available in the IPUMSI database61 (hereafter referred as IPUMSI countries; Table 1).\n\nHowever, since the main aim of this study was to estimate internal human migration flows for malaria endemic countries for which migration data are not available, ultimately, models where the linear predictors were common across all countries located in the same continent were constructed (under the assumption of homogeneity of the process along the space). To investigate possible nonlinear relationships, models where linear predictors were replaced by additive predictors, using a Generalized Additive Modelling (GAM) framework66, were also explored.\n\nGAM is a type of regression that, while preserving the functionality of using linear terms, allows covariates to have different and possibly opposite effects on the response variable by incorporating regression coefficients with smooth nonlinear form (Fig. 2).\n\nThus, all possible combinations of covariates (listed in Table 2 and Supplementary Table 2) were tested in a logistic regression model and then only the linear predictors of all continuous covariates of the best predictive logistic regression model were also modelled using a GAM.\n\nFor each continent, the overall combinations of covariates and model types were explored using a multi-step approach to identify the model with the greatest predictive power in countries for which migration data were not available. The best model was then selected using a leave-one-out cross-validation approach67 in which the observed proportion of migrants in j previously residing in i for all countries except one were used for fitting models, that were subsequently used to predict the proportion of migrants in j previously residing in i in the withheld country. The correlation coefficient (R2) was selected to measure the variance explained after verifying homoscedasticity and testing overdispersion using a chi-squared test. This process was then repeated through iteratively withholding one country at the time. For each model, the R2 values for all withheld countries were averaged and used to rank each models according to their predictive power averaged across all withheld countries (Fig. 3). The overall best predictive model for each continent (Supplementary Table 3) was then used to predict the proportion of migrants residing in j who were previously residing in i for every malaria endemic country located in the corresponding continent (refer to Supplementary Table 4a,b and c for summary statistics of each best predictive model for Africa, Asia, and Latin America and the Caribbean, respectively).\n\nFinally, in order to estimate the total number of people that migrated from i to j between 2005 and 2010 (Figs 4,5,6), for each country the predicted proportion of migrants residing in j was multiplied by the 2010 total population in j; with the latter calculated using either the corresponding WorldPop68,\u200b69,\u200b70 or the Gridded Population of the World version 4 (GPWv4)71 dataset adjusted to match United Nations Population Division (UNPD) estimates for 2010 (ref. 72). Refer to the Data collection and preparation subsection section below for a detailed description of how the population datasets mentioned above were identified and used.\n\nBoth model selection and prediction were performed using an R73 script contained in the WorldPop-InternalMigration-v1 code74 briefly described in the Code availability subsection below.\n\n### Response variable and covariates\n\nFor each country, the response variable, or the proportion of migrants residing in j in the census year that were residing in i 5 years prior to the census, was obtained by dividing the number of migrants residing in j in the census year that were residing in i 5 years prior the census by the total population residing in j in the census year; with both numbers based only on the information contained in IPUMSI census samples.\n\nThe administrative units spatially matching the IPUMSI migration microdata were used to calculate the distance between each pair of administrative units, their area, total population, and proportion of urban population. These main covariates (Table 2), along with other covariates derived from them (Supplementary Table 2), represent the pull and push migration factors, known to influence internal migration59,60,77, that were used to extend the basic gravity model proposed by Zipf64.\n\nOther factors, including environmental factors59,60, and country-specific factors, such as literacy and percentage of male population59 or infrastructure and transportation78, were not used because (i) the factors listed in the previous paragraph alone proved to be able to explain most of the variance in the gravity models of Garcia et al.59, and (ii) only globally available datasets were explored in order to consistently model internal migration across all countries.\n\n### Calculating response variable and covariates\n\nFor each country, the total population in each administrative unit was calculated using the corresponding WorldPop79 (Data Citation 1: Harvard Dataverse http:\/\/dx.doi.org\/10.7910\/DVN\/PUGPVR) or GPWv4 (ref. 80) population count raster dataset adjusted to match UNPD estimates for 201072. The GPWv4 datasets were resampled to the spatial resolution of the WorlpPop datasets and used only for countries for which the WorldPop datasets were not available (Supplementary Table 1).\n\nThe area of each unit was calculated using each country vector administrative unit dataset projected to the most appropriate country-specific projected coordinate system, in order to minimize areal distortion, and ultimately reprojected to GCS WGS84.\n\nThe proportion of people in urbanized areas in each unit was calculated using the MODIS 500\u2009m Global Urban Extent raster dataset81,82. The latter was converted to vector polygons, using the ArcGIS \u2018Raster to Polygon\u2019 tool83, and intersected with the reprojected country vector administrative unit dataset using the ArcGIS \u2018Intersect\u2019 tool83. Then, both the intersect output (containing polygons representing the total urban area within each unit uniquely identified by its \u2018IPUMSID\u2019) and the country vector administrative unit dataset were rasterized, at the resolution of the corresponding raster population dataset (i.e., 3 arc seconds 3 arc equals to approximately 100\u2009m at the equator), and co-registered with it.\n\nThe two raster outputs, along with the population count raster dataset, were then input to the ArcGIS \u2018Zonal Statistics as Table\u2019 tool83 to generate two tables containing the total population and urban population in each unit (with the rasterized administrative units and thus their \u2018IPUMSIDs\u2019 used to define the zones). Subsequently, both tables were joined to the attribute table of the vector administrative unit dataset, using the \u2018IPUMSID\u2019 field to perform the join operation, and the proportion of urban population in each unit was calculated simply dividing its urban population by its total population.\n\nThe geodesic distance between each pair of administrative units, with the latter represented by their centroids, was calculated using the ArcGIS \u2018Generate Near Table (Analysis)\u2019 tool83. The \u2018IN_FID\u2019 and \u2018NEAR_FID\u2019 fields (identifying the administrative unit of origin and destination, respectively) in the output \u2018distance\u2019 table were then used for joining twice the \u2018centroid attribute\u2019 table using the centroid \u2018ID\u2019 field to perform the join operation. Since the \u2018centroid attribute\u2019 table contains the attributes of each administrative unit represented by the corresponding centroid, the join operation allowed to generate a \u2018distance\u2019 table containing all pairs of origin and destination administrative units along with their \u2018IPUMSIDs\u2019 and attributes including the unit\u2019s area, total population, and proportion of urban population. Origin and destination \u2018IPUMSID\u2019 fields were then renamed \u2018NODEI\u2019 and \u2018NODEJ\u2019, respectively.\n\nA \u2018contiguity\u2019 table containing information about spatial contiguity of administrative units (defined based on polygons sharing an edge) was generated using the ArcGIS \u2018Generate Spatial Weights Matrix\u2019 tool83 and subsequently joined with the \u2018distance\u2019 table to obtain a new table containing all main covariates, listed in Table 2, calculated at the unit level. This join operation (based on both the \u2018NODEI\u2019 and \u2018NODEJ\u2019 field the in the \u2018distance\u2019 table and the corresponding \u2018IPUMSID\u2019 and \u2018NID\u2019 field in the \u2018contiguity\u2019 table) was performed through two different R scripts depending on whether the country is an IPUMSI or a non-IPUMSI countries. In particular, the R script for the IPUMSI countries added to the new table a \u2018MIGIJ\u2019 field containing the number of people that migrated from each \u2018NODEI\u2019 to each other \u2018NODEJ\u2019 according to the IPUMSI migration microdata and calculated the response variable.\n\nFinally, on a continent basis, all IPUMSI country tables were merged together and input to an R73 script that generated the additional covariates listed in Supplementary Table 2, identified the best predictive model for each continent, as described in the previous section, and was used to estimate the 5-year (2005\u20132010) internal human migration flows for every malaria endemic country using the best predictive model selected for the corresponding continent.\n\nAll operations described above, excluding the reprojection of the vector administrative unit datasets and the calculation of their surface areas, for all IPUMSI and non-IPUMSI countries, were performed using the WorldPop-InternalMigration-v1 code74 briefly described in the Code availability subsection below.\n\n### Code availability\n\nThe WorldPop-InternalMigration-v1 code74, used to produce the open access archive of estimated 5-year (2005\u20132010) internal human migration flows described in this article, is publicly available through Figshare. It consists of 1) a Microsoft Visual Studio 2010 user interface allowing users to upload the IPUMSI census microdata to a PostgreSQL database; 2) example SQL queries that were used to match the spatial detail of the IPUMSI migration data to spatial detail of the corresponding administrative unit dataset and to identify internal migrants within the IPUMSI census samples 3) an ArcToolbox geoprocessing tool82 that assigns a unique ID to each administrative unit and calculates the corresponding total population and proportion of urban population; 4) a Python84\/ArcPy83 script that creates two tables, one containing spatial contiguity information between each pair of administrative units (\u2018contiguity.csv\u2019) and another one containing the ISO country code, the continent in which the country is located, the distance between each pair of administrative units, their total population, proportion of urban population, surface area, and the geographic coordinates (GCS WGS84) of their centroid (\u2018distance.csv\u2019); 5) two R73 scripts, one for the IPUMSI countries used to query the IPUMSI migration microdata loaded in the PostgreSQL database, calculate the response variable, and join the query result with the two output tables of the python script, and another one for the non-IPUMSI countries used just to join together the two output tables of the python script; and 6) an R73 script that performs the model selection and estimates the 5-year (2005\u20132010) internal human migration flows between subnational administrative units.\n\nAll available sets of code are named progressively and must be run sequentially according to the order in which they are presented above. They are also internally documented in order to both briefly explain their purpose and, when required, guide the user through their customization.\n\n## Data Records\n\nAll datasets described in this article, referring to all Pf and Pv endemic countries listed in Supplementary Table 1, are publicly and freely available both through the WorldPop Dataverse Repository (Data Citation 2: Harvard Dataverse http:\/\/dx.doi.org\/10.7910\/DVN\/RUWQQK) and the WorldPop website (http:\/\/www.worldpop.org.uk\/data\/data_sources\/). However, it is important to note that while the datasets stored in the Dataverse Repository represent the datasets produced at the time of writing, and will be preserved in their published form, the datasets stored on the WorldPop website may be updated as more recent IPUMSI migration data for the countries listed in Table 1, become available. Similarly, the datasets stored on the WorldPop website may be updated as IPUMSI census-based migration microdata become available for additional malaria endemic and non-endemic countries located in Africa, Asia, Latin America and the Caribbean. Indeed, the availability of migration data for additional countries may enable further improvements of the predictive power of the gravity models used to estimate the internal migration flows. For each county, the corresponding internal migration dataset, along with a point dataset showing the nodes of the migration network, (Table 3) can be obtained by downloading the corresponding zipped archive associated with the continent in which the country of interest is located.\n\n## Technical Validation\n\n### Goodness of fit and error p-value\n\nAll countries available in the IPUMSI database were used to assess the accuracy of the predicted proportion of migrants in j in the census year that were previously residing in i 5 years prior to the census by comparing them with the corresponding observed values from the IPUMSI migration microdata. For each country, the goodness of fit (R2) between predicted and observed values and the corresponding error P-value, representing the average probability that predicted migration values lay outside the distribution of the observed values, are reported in Table 4 below. Both metrics were derived using (i) the observed IPUMSI migration flows from each i to any other j and (ii) the predicted IPUMSI-based migration flows calculated by multiplying the predicted proportion of migrants residing in j in the census year by the IPUMSI-based total number of people residing in j in the census year.\n\n## Usage Notes\n\nThe estimated internal human migration flows between subnational administrative units can be used to support a range of applications from planning interventions, to measuring progress, designing strategies, and predicting response variables that are intrinsically dependent on migration flows and internal connectivity.\n\nOngoing work involves the integration of these datasets with malaria prevalence raster datasets85,\u200b86,\u200b87 in order to inform local elimination and global eradication planning by identifying subnational communities of malaria movement and sources and sinks of transmission within them36,43,58. Similarly, these datasets could be used to better model the spread and improve understanding of the drivers of the distributions of other infectious diseases, such as West Nile Virus, schistosomiasis, river blindness, and yellow fever, which are endemic in some of the countries listed in Supplementary Table 1. Additionally there are many uses of these data beyond infectious disease dynamics, in the fields of trade, demography, transportation and economics, for example.\n\nThere are a number of limitations, caveats, and assumptions inherent in the approach that should be considered when using the datasets outlined here. For consistency, internal migration flows were estimated using a fixed set of pull and push factors common to all countries and thus only a limited number of covariates were used to fit the gravity-type spatial interaction models and to create predictions. For this reason, as is a trade-off in the production of generalizable models, the model fit varied between countries and for some of them, such as Malawi, China, Cambodia, India, and Venezuela (Table 4), poor fits could be improved by considering additional, locally-specific migration drivers that could help to increase the percentage of variance explained60,78. Other limitations are the fact that migration models were fitted using only a small sample (ranging between 0.07 and 10%) of the full census for each country, and that in each sample a small number of households were swapped across administrative units. Moreover, the spatial detail at which migration is captured and summarized varies by country. Because of this, for some countries, the modelled role of some of the pull and push factors, may not have been captured at the spatial level at which they influence migration as recorded in the census. It is also important to consider that the underlying migration data are based only on permanent movements captured by the census and other types of migrations, such as seasonal movements and forced displacements, may be not captured by the model88,\u200b89,\u200b90.\n\nThe two main assumptions behind the approach presented here are that for each country (i) the census samples are considered to be representative at the administrative unit level at which migration was recorded and (ii) the percentage of people migrating between administrative units is considered to be constant over time. Regarding the second assumption, it is important to highlight that the use of census data from many years ago for some countries may have generated inaccurate estimates for the period considered in this study (i.e., 2005\u20132010), for example because of major changes in the countries\u2019 socio-economic conditions from the time period covered by the census (e.g., the rapid economic development and urbanization that has occurred in China during the last two decades91,92). Similarly, in some other countries, either the presence of conflicts93 or the occurrence of natural disasters88,89 during the specific time period covered by the census may have produced fluctuations in the number of internal migrants and consequently biased results for the period considered in this study.\n\nFinally, the estimated internal flows represent modelling outputs generated using ancillary covariate datasets, and thus, to avoid circularity they should not be used to make predictions or explore relationships with any of these ancillary datasets. It is also important to note that these ancillary datasets are modelling outputs in themselves and thus they have a degree of uncertainty that will carry over into the migration estimates.\n\nHow to cite this article: Sorichetta, A. et al. Mapping internal connectivity through human migration in malaria endemic countries. Sci. 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Global Administrative Unit Layers. Available at (2015).\n\n77. 77.\n\nEstimating flows between geographical locations: \u2018get me started in\u2019 saptial interaction modelling. UCL working paper series 181, 1\u201324 (2012).\n\n78. 78.\n\n& African migration: trends, patterns, drivers. Comp. Migr. Stud 4, doi:10.1186\/s40878-015-0015-6 (2016).\n\n79. 79.\n\nWorldPop. Population\u2014individual countries. Available at (2015).\n\n80. 80.\n\nCenter for International Earth Science Information Network, Columbia University. Gridded Population of the World, Version 4 (GPWv4): Population Count Adjusted to Match 2015 Revision of UN WPP Country Totals. Available at (NASA Socioeconomic Data and Applications Center, 2015).\n\n81. 81.\n\n, & MODIS 500m Global Urban Extent. Available at (2009).\n\n82. 82.\n\n, & A new map of global urban extent from MODIS satellite data. Environ. Res. Lett. 4, 044003 (2009).\n\n83. 83.\n\nESRI. ArcGIS Desktop: Release 10.1 (ESRI, 2012).\n\n84. 84.\n\n& Interactively Testing Remote Servers Using the Python Programming Language. CWI Quarterly 4, 283\u2013303 (1991).\n\n85. 85.\n\net al. A new world malaria map: Plasmodium falciparum endemicity in 2010. Malar. J 10, 1\u201316 (2011).\n\n86. 86.\n\net al. A long neglected world malaria map: Plasmodium vivax endemicity in 2010. PLoS Negl. Trop. Dis. 6, e1814 (2012).\n\n87. 87.\n\net al. The effect of malaria control on Plasmodium falciparum in Africa between 2000 and 2015. Nature 526, 207\u2013211 (2015).\n\n88. 88.\n\net al. Rapid and near Realtime Assessments of Population Displacement Using Mobile Phone Data Following Disasters: the 2015 Nepal Earthquake. PLOS Currents Disasters 1, doi: 10.1371\/currents.dis.d073fbece328e4c39087bc086d694b5c (2016).\n\n89. 89.\n\n, , , & Improved response to disasters and outbreaks by tracking population movements with mobile phone network data: a post-earthquake geospatial study in Haiti. PLoS Med. 8, e1001083 (2011).\n\n90. 90.\n\net al. Explaining seasonal fluctuations of measles in Niger using nighttime lights imagery. Science 334, 1424\u20131427 (2011).\n\n91. 91.\n\net al. A new urban landscape in East-Southeast Asia, 2000-2010. Environ. Res. Lett. 10, 034002 (2015).\n\n92. 92.\n\net al. Spatiotemporal patterns of population in mainland China, 1990 to 2010. Sci. Data 3, 160005 (2016).\n\n93. 93.\n\n, , & The journey to safety: conflict-driven migration flows in Colombia. Int. Regional Sci. Rev 33, 157\u2013180 (2010).\n\n94. 94.\n\nInternational Organization for Standardization. 3166-1 alpha-3 (three-letter) country codes. Available at (2015).\n\n## Data Citations\n\n1. 1.\n\nSorichetta, A Harvard Dataverse http:\/\/dx.doi.org\/10.7910\/DVN\/PUGPVR (2015)\n\n2. 2.\n\nSorichetta, A Harvard Dataverse http:\/\/dx.doi.org\/10.7910\/DVN\/RUWQQK (2016)\n\n## Acknowledgements\n\nThe authors wishes to acknowledge the statistical offices that provided the underlying data making this research possible: Central Bureau of Census and Population Studies, Cameroon; Ghana Statistical Services; National Statistics Directorate, Guinea; National Statistical Office, Malawi; National Directorate of Statistics and Informatics, Mali; National Agency of Statistics and Demography, Senegal; Statistics South Africa; Bureau of Statistics, Uganda; Central Statistics Office, Zambia; Central Agency for Public Mobilization and Statistics, Egypt; Department of Statistics, Morocco; National Statistical Service, Armenia; National Statistical Committee, Kyrgyzstan; Ministry of Statistics and Programme Implementation, India; BPS Statistics Indonesia; National Statistical Office, Thailand; National Institute of Statistics, Cambodia; National Bureau of Statistics, China; Department of Statistics, Malaysia; National Statistics Office, Philippines; General Statistics Office, Vietnam; National Statistical Office, Mongolia; Bureau of Statistics, Fiji; National Institute of Statistics and Censuses, Argentina; National Institute of Statistics, Bolivia; Institute of Geography and Statistics, Brazil; National Administrative Department of Statistics, Colombia; National Institute of Statistics and Censuses, Costa Rica; National Statistics Office, Dominican Republic; National Institute of Statistics and Censuses, Ecuador; Department of Statistics and Censuses, El Salvador; Institute of Statistics and Informatics, Haiti; National Institute of Statistics, Geography, and Informatics, Mexico; National Institute of Information Development, Nicaragua; National Institute of Statistics and Informatics, Peru; National Institute of Statistics, Venezuela; Office of National Statistics, Cuba; Statistical Institute, Jamaica; National Institute of Statistics, Uruguay. A.S. is supported by funding from the Bill & Melinda Gates Foundation (OPP1106427, 1032350). A.J.T. is supported by funding from NIH\/NIAID (U19AI089674), the Bill & Melinda Gates Foundation (OPP1106427, 1032350, 1134076), the Clinton Health Access Initiative, National Institutes of Health, and a Wellcome Trust Sustaining Health Grant (106866\/Z\/15\/Z). This work forms part of the outputs of WorldPop (www.worldpop.org) and the Flowminder Foundation (www.flowminder.org). The funders had no role in study design, data collection and analysis, decision to publish, and preparation of the manuscript.\n\n## Affiliations\n\n1. ### WorldPop, Geography and Environment, University of Southampton, Highfield Campus, Southampton SO17 1BJ, UK\n\n\u2022 Alessandro Sorichetta\n\u2022 , Tom J. Bird\n\u2022 , Nick W. Ruktanonchai\n\u2022 , Elisabeth zu Erbach-Schoenberg\n\u2022 , Carla Pezzulo\n\u2022 , Natalia Tejedor\n\u2022 \u00a0& Andrew J. Tatem\n2. ### Flowminder Foundation, Roslagsgatan 17, Stockholm SE-11355, Sweden\n\n\u2022 Alessandro Sorichetta\n\u2022 , Tom J. Bird\n\u2022 , Nick W. Ruktanonchai\n\u2022 , Elisabeth zu Erbach-Schoenberg\n\u2022 , Carla Pezzulo\n\u2022 , Natalia Tejedor\n\u2022 \u00a0& Andrew J. Tatem\n3. ### Institute for Life Sciences, University of Southampton, Highfield Campus, Southampton SO17 1BJ, UK\n\n\u2022 Alessandro Sorichetta\n4. ### GeoData, University of Southampton, Highfield Campus, Southampton SO17 1BJ, UK\n\n\u2022 Natalia Tejedor\n\u2022 , Ian C. Waldock\n5. ### Bill and Melinda Gates Foundation, 440 5th Ave N., Seattle, Washington 98109, USA\n\n\u2022 Andres J. Garcia\n6. ### CHICAS, Lancaster Medical School, Lancaster University, Lancaster LA1 4YG, UK\n\n\u2022 Luigi Sedda\n\n## Authors\n\n### Contributions\n\nA.S. coordinated the study, undertook data collection and assembly, and drafted the manuscript. A.S. and T.J.B. undertook data analyses, generated the internal migration datasets, and prepared the tables and figures. A.S. and L.S. performed the technical validation of the internal migration datasets. A.S., T.J.B., N.R., and A.J.T. conceptualized the study. A.S., T.J.B., N.R., E.z.E.S., C.P., N.T., I.C.W., J.D.S., A.J.G., and L.S. developed the WorldPop-InternalMigration-v1 code. T.J.B. implemented the R script used for model selection and prediction. C.P., A.J.G., I.C.W., and J.D.S. aided with data assembly. A.J.T. aided drafting the manuscript and conceived the study. All authors read and approved the final version of the manuscript.\n\n### Competing interests\n\nThe authors declare no competing financial interests.\n\n## Corresponding author\n\nCorrespondence to Alessandro Sorichetta.\n\n## PDF files\n\n1. 1.\n\n2. 2.\n\n### DOI\n\nhttps:\/\/doi.org\/10.1038\/sdata.2016.66\n\n\u2022 ### Utilizing general human movement models to predict the spread of emerging infectious diseases in resource poor settings\n\n\u2022 M. U. G. Kraemer\n\u2022 , N. Golding\n\u2022 , D. Bisanzio\n\u2022 , S. Bhatt\n\u2022 , D. M. Pigott\n\u2022 , S. E. Ray\n\u2022 , J. S. Brownstein\n\u2022 , N. R. Faria\n\u2022 , D. A. T. Cummings\n\u2022 , O. G. Pybus\n\u2022 , D. L. Smith\n\u2022 , A. J. Tatem\n\u2022 , S. I. Hay\n\u2022 \u00a0&\u00a0R. C. Reiner\n\nScientific Reports (2019)\n\n\u2022 ### Local and regional dynamics of chikungunya virus transmission in Colombia: the role of mismatched spatial heterogeneity\n\n\u2022 Sean M. Moore\n\u2022 , Quirine A. ten Bosch\n\u2022 , Amir S. Siraj\n\u2022 , K. James Soda\n\u2022 , Guido Espa\u00f1a\n\u2022 , Alfonso Campo\n\u2022 , Sara G\u00f3mez\n\u2022 , Daniela Salas\n\u2022 , Benoit Raybaud\n\u2022 , Edward Wenger\n\u2022 , Philip Welkhoff\n\u2022 \u00a0&\u00a0T. Alex Perkins\n\nBMC Medicine (2018)\n\n\u2022 ### Mapping road network communities for guiding disease surveillance and control strategies\n\n\u2022 Emanuele Strano\n\u2022 , Matheus P. Viana\n\u2022 , Alessandro Sorichetta\n\u2022 \u00a0&\u00a0Andrew J. Tatem\n\nScientific Reports (2018)\n\n\u2022 ### Travel patterns and demographic characteristics of malaria cases in Swaziland, 2010\u20132014\n\n\u2022 Natalia Tejedor\u2010Garavito\n\u2022 , Nomcebo Dlamini\n\u2022 , Deepa Pindolia\n\u2022 , Nick W. Ruktanonchai\n\u2022 , Victor Alegana\n\u2022 , Arnaud Le Menach\n\u2022 , Nyasatu Ntshalintshali\n\u2022 , Bongani Dlamini\n\u2022 , David L. Smith\n\u2022 , Andrew J. Tatem\n\u2022 \u00a0&\u00a0Simon Kunene\n\nMalaria Journal (2017)\n\n\u2022 ### Mapping multiple components of malaria risk for improved targeting of elimination interventions\n\n\u2022 Justin M. Cohen\n\u2022 , Arnaud Le Menach\n\u2022 , Emilie Pothin\n\u2022 , Thomas P. Eisele\n\u2022 , Peter W. Gething\n\u2022 , Philip A. Eckhoff\n\u2022 , Bruno Moonen\n\u2022 , Allan Schapira\n\u2022 \u00a0&\u00a0David L. Smith\n\nMalaria Journal (2017)","date":"2019-04-19 06:23:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 6, \"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\": 0.5887080430984497, \"perplexity\": 10178.200133927288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578527148.46\/warc\/CC-MAIN-20190419061412-20190419083412-00086.warc.gz\"}"}
| null | null |
Independent News and Politics
Patriot Act fails fast track in house…
February 09th, 2011 Comments(1)
But the bill will be back before too long, when it will be able to pass with a simple majority.
Ron Paul gives great arguments, from many key points, about why this particular piece of legislation is a waste of time and a threat to our American liberties:
If only we could give him a few more minutes to talk…
The rotten political center
March 16th, 2010 Comments(0)
In my short lifespan, I've heard a lot of talk and celebration for this concept of bipartisanship. With the possible exception of Fox News, most media outlets will applaud a so-called "moderate" politician for ignoring their platforms or the wishes of their constituents in the name of compromise.
So who are the idols of the "independent center?" We hear a lot of talk about people like McCain and Lieberman when it comes to this myth of bipartisan moderation – but check out this bill they are currently trying to sneak through Congress:
Whenever within the United States, its territories, and possessions, or outside the territorial limits of the United States, an individual is captured or otherwise comes into the custody or under the effective control of the United States who is suspected of engaging in hostilities against the United States or its coalition partners through an act of terrorism, or by other means in violation of the laws of war, or of purposely and materially supporting such hostilities, and who may be an unprivileged enemy belligerent, the individual shall be placed in military custody for purposes of initial interrogation and determination of status in accordance with the provisions of this Act.
An individual, including a citizen of the United States, determined to be an unprivileged enemy belligerent under section 3(c)(2) in a manner which satisfies Article 5 of the Geneva Convention Relative to the Treatment of Prisoners of War may be detained without criminal charges and without trial for the duration of hostilities against the United States or its coalition partners in which the individual has engaged, or which the individual has purposely and materially supported, consistent with the law of war and any authorization for the use of military force provided by Congress pertaining to such hostilities.
We've come a long way in this country, when the center of our political debate is owned by people like these – people who don't believe in the fundamental values of our nation or the rules of the constitution that they have sworn to uphold.
If this is the middle then, I am proud to be in the fringe element: left or right, wherever there are people who believe in freedom and justice, I will be there.
Terrorists Still Incredibly Unlikely to Hurt You
January 04th, 2010 Comments(0)
Despite all of the fear-mongering and the huge military & defense budgets we expand every year, terrorists pose a risk so small and insignificant that we may as well worry about the risk of lightning or asteroids. Actually, spending some of this money on science might not be a bad thing, and our efforts against the weather will probably have more of an effect than our efforts to change people's behavior and insulate ourselves perfectly from political violence.
FiveThirtyEight calculates the odds of being killed in an airborn terrorist attack at about 1 incident per 16,553,385 departures.
[T]he odds of being on given departure which is the subject of a terrorist incident have been 1 in 10,408,947 over the past decade. By contrast, the odds of being struck by lightning in a given year are about 1 in 500,000.
People have a hard time judging risks in relative terms – and these leads people to obsess over seemingly horrible disasters that are incredibly unlikely to happen. In the mundane world of reality and statistics, anyone concerned about security and mortality should be taking a look at our cars, roads, and medical system. Its not a 'maybe they'll kill us more in the future' situation there, its something that is guaranteed to kill a whole lot of Americans this year and every year following.
So forget the fact that our militarism kills innocent civilians, or ignore the fact that our massive spending does little to secure us, but at least accept the fact that there was really nothing to be afraid of in the first place. Let's redirect the violent efforts of our war on terror to more constructive purposes – and the progress we create will do more to keep us safe and respected around the world than anything our guns could accomplish now.
Keep an Eye on Pakistan…
December 19th, 2009 Comments(0)
The so-called "War on Terror" is hardly over now that Bush has left office – in fact, it is intensifying and spreading across the borders from its original locations.
In Iraq, all eyes are on the Iranian border, especially as Iranian forces appear to have seized a lucrative oil field near the often-disputed national boundaries. Turkey has made a few moves and is posturing to influence the Kurdish north, but they're also a bit more interested in maintaining amicable relations with the U.S.
In Afghanistan, the problems of terrorism and fundamentalism don't stop where the border with Pakistan was drawn. Tribal influences don't always respect political or cartographic lines.
As such, it shouldn't be too much of a surprise to hear that the U.S. military has been increasingly active in Pakistan, Yemen, and possibly even inside Iran as well (in relation to the protests against the latest elections).
Instead of spreading democracy and building stability, the military interventions have disrupted a delicate and vulnerable balance of powers across the Middle East. Yet while the pre-existing order was somewhat stable, prior to meddling, it was a fragile situation that forced conflicting cultural ideologies into a single political unit.
As the war continues in search of some new order, it will increasingly ignore the arbitrary borders of the Middle East, and more nations will come to be involved. Intended or not, the ultimate result of our actions abroad may lead to a new united Islamic super-state – a sort of regional power not unlike the European Union or the United States.
More great political blogs
Underground Politics
Reddit World Politics (Uncensored)
Booker opens lead against McGrath
The week of national police riots
Florida Surrenders to Covid-19
Boris Johnson and Coronavirus: A story in three acts
And it's Biden's to lose
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Deliverance on Tough on crime: FBI busts KC SWAT team
Newsbusters: Stephanopoulos Blames 2011 Budget Deficit On Bush | Katy Pundit on Bernanke Blames the Blogs
Independent News and Politics » That didn't take long… on Patriot Act fails fast track in house…
Independent News and Politics » Anarchists at a Tea Party on Its Getting Hard to Talk about Politics
Independent News and Politics » Localism and Spontaneous Order in the Commons on Quietly Invading Africa
Independent News and Politics © admin 2022 | Freedom-Black & Widgetized by Tina Silva
Freedom Blue Plus improved by Eyoung. Kudos to Frank Helmschrott, Michael and Fredrik for the original Design.
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{
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| 5,419
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Gilbert X. Drendel, Jr. began representing the Village of North Aurora in 1967. He continued to represent North Aurora until his retirement from the practice of law. Kevin Drendel came alongside in 1991 and learned from his example. Gilbert (Gib) took pride in his service to the community, not just the work he performed as part of his practice, but in every way. He was an active participant in the community and considered it an honor to serve the community in representing the local governmental body.
Gib Drendel and Rich Horwitz were in on ground the floor in getting the legislation passed that established the Illinois Mathematics and Science Academy and represented IMSA for years. Over the years, the Firm has represented many governmental bodies as general council and on an ad hoc basis, including the continuous representation of the Village of North Aurora from 1967 to the present day. Most recently, the Firm has been retained by the City of Batavia.
In addition to representing municipalities, school districts, fire protection districts and other local governmental bodies over the years, Kevin G. Drendel has gotten involved in telecommunications and cable issues. He has negotiated cable franchise agreements for a number of communities, has negotiated many water tower and other leases for governmental bodies leasing to telecommunications providers, has drafted zoning provisions to deal with the unique issues triggered by the proliferation of telecommunication towers and facilities, has represented communities setting up local cable access television stations, has represented local cable access entities and has been involved in other ways telecommunications and cable issues as they dovetail with local governmental bodies.
"I've known Kevin for 15 years and have worked with him on numerous telecommunications projects. In my role as a telecommunications consultant, I have sat on a number of municipal teams where Kevin has been legal counsel. Kevin is a skilled negotiator and has been very successful in using a methodical and reasoned approach in negotiating agreements having a complex subject matter.
Over the years, Kevin's expanding knowledge and experience in blending municipal and telecommunications law has made him one of a handful of experts in the metropolitan Chicago area that have demonstrated the legal and technical ability to help municipalities, community colleges, school districts, and other public bodies develop working relationships with telecommunications providers. For those reasons, I would strongly recommend Kevin Drendel as legal counsel on matters involving municipal and telecommunications law." – Stu Chapman, Municipal Services Associates
In addition to Kevin Drendel, Carolyn D. Jansons, and Roman J. Seckel have years of experience in the representation of municipalities and local governmental work.
The list of governmental bodies for which the Firm has provided legal services includes:
Village of North Aurora
City of Batavia
City of Aurora
The Illinois Mathematics and Science Academy
Batavia School District
East Aurora School District
West Aurora School District
Montgomery & Countryside Fire Protection District
Wasco Sanitary District
|
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| 8,191
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Sounds interesting. Might be worth putting a post on the forum and may be a bit more detail. I know .Charles posts on here regularly, his profile says that he is a Costs Draftsman for a Solicitors firm. He may be able to help as a starting point.
I've challenged some legal fees twice and got some of my costs back. Once with a solicitor and once a barrister. But my complaints involved other aspects as well as costs and I settled before needing to involve the LO. So money I got backs was not the challenged costs alone.
I expect the Legal Ombudsman publishes its decisions or has some case studies which you can look through.
Hi, I am in the process of challenging my legal fees, I believe I was given bad advise and as a result the divorce was rushed through, even through I would have been happy to wait to years, also I said I couldnt afford half the cost of divorce and my solicitor said ok I'll put a footnote on the form to say ur in a weaker finanicial position, well this was ignored and I ended up paying half of his divorce costs, also any claim I could of had on his Mums estate. So I signed the decree nisie and she said she was "surprised" that they then weren't for the absolute, dispite her asking them to wat until financial settlement, well being being "surprised" wasn't good enough, I'm suffering now before of that poor advise. Do you think I should ask for all my money back I still owe £1200 from a £4,000 bill ? Feeling cross and disappointed.
Sorry you've had this bad experience and it's sounding as though you may have received poor/bad advice, however, we would have to know a lot more about the details of your divorce to be able to comment.
Many wikis never read the blogs, so if you post your query in the Forum with details of the divorce, (length of marriage etc) people may be able to advise you from their own experience. A visit to the local CAB would be helpful too.
|
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| 4,110
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Wheaton, MD - On November 26, Montgomery County Executive Ike Leggett, Montgomery County Department of Transportation Director Al Roshdieh, and District 4 Montgomery County Councilmember Nancy Navarro joined Clark Construction and StonebridgeCarras to celebrate the completion of the Wheaton Revitalization Project's underground parking structure and infrastructure and to acknowledge the hard work and dedication of the project's trades people. This transit-oriented development, which is now back-to-grade, will help to transform Wheaton's central business district.
As part of the Wheaton Revitalization Project, Clark is constructing a 308,000 square-foot government office building. In addition to the 14-story office building, the project will provide 397 underground parking spaces, first-floor retail space, and a new town square that will serve as a gathering space for the community.
The office building, which is targeting LEED platinum certification, will house the Maryland-National Capital Park and Planning Commission and numerous Montgomery County municipal offices including Environmental Protection, Health and Human Services, Permitting Services, Recreation and Community Use of Public Facilities as well as the Mid-County Regional Services offices.
Construction completion is slated for May 2020.
|
{
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| 315
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Q: How to deploy Owin Self Hosted Signalr Service to production I am trying to publish Owin Self Hosted Signalr Service(using TopShelf) to production server but it doesn't seem that clients can connect to the server after it's published. The settings and codes are pretty much the same as what you can find in the internet.
Startup.cs
app.Map("/service", map =>
{
map.UseCors(CorsOptions.AllowAll);
var hubConfiguration = new HubConfiguration
{
EnableJSONP = true,
EnableDetailedErrors = true
};
map.RunSignalR(hubConfiguration);
});
}
ServiceClass.cs (Server)
public bool StartService(HostControl hostControl)
{
//const string hostingUrl = "http://localhost:9000/";
const string hostingUrl = "http://production-server:9000/";
SignalR = WebApp.Start<Startup>(hostingUrl);
}
Default.html (Client)
<head>
<script src="Scripts/service.js"></script>
<script src="Scripts/bootstrap.min.js"></script>
<script src="Scripts/jquery-3.0.0.min.js"></script>
<script src="Scripts/jquery.signalR-2.2.3.min.js"></script>
<!--<script src="http://localhost:9000/service/hubs"></script>-->
<script src="http://production-server:9000/service/hubs"></script>
</head>
service.js (Client)
//$.connection.hub.url = "http://localhost:9000/service";
$.connection.hub.url = "http://production-server:9000/service";
// Declare a proxy to reference the hub.
var sample = $.connection.service;
$.connection.hub.logging = true;
$.connection.hub.start().done(function() {
}).fail(function () { console.log('Cannot Connect!'); });;
});
I have tried to
*
*make both server and client use localhost url -> code above Works
*make both server and client use production url -> not working
*make server code use localhost and client use production url -> not working
*Allow port 9000 in firewall setting as answered here
How to Deploy Owin Self Hosted Application In Production?
Error Message in Chrome console
Failed to load resource: net::ERR_CONNECTION_TIMED_OUT
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 316
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Q: SQL do not insert duplicates I have been looking for a solution to best implement a do not insert if the line already exists. I've read a lot of answers which al differ in some way and it's a little above my level to make any sense.
For example the following post is the top result:
SQL Server Insert if not exist
But I cannot understand why the variables are used and how I define table1 and table2. Would somebody be able to explain this a litle further so that I can apply it to my situation?
In my example I have about 5 columns (Date, Number1, Number2, Text1, Text2) that I'd like to insert from table1 to table2, but only if the do not exist. Update is not necessary. (Therefore merge is out of the question)
A: An alternative to using NOT EXISTS, would be to use the EXCEPT keyword to do a "difference" between the data in the two result sets, selected from both Table1 and Table2:
INSERT INTO table2 (Date, Number1, Number2, Text1, Text2) values
SELECT A.Date, A.Number1, A.Number2, A.Text1, A.Text2
FROM Table1 A
EXCEPT
SELECT B.Date, B.Number1, B.Number2, B.Text1, B.Text2
FROM Table2 B;
A: A not exists should work using a correlated subquery.
The sub select identifies all records in Table1 which do not exist in Table2 by joining on all the fields we would be inserting. This ensures we only insert records which do not already exist in table2.
INSERT INTO table2 (Date, Number1, Number2, Text1, Text2)
VALUES
(SELECT A.Date, A.Number1, A.Number2, A.Text1, A.Text2
FROM Table1 A
WHERE NOT EXISTS (SELECT 1
FROM Table2 B
WHERE A.Date = B.Date
AND A.Number1 = B.Number1
AND A.Number2 = B.Number2
AND A.Text1 = B.Text1
AND A.Text2 = B.Text2))
Since we don't care what value is selected as it's throwaway from the subquery, I simply select 1 from the not exists correlated subquery.
Now if there are other columns we need to consider, then we would have to add those to our query, or if there's a primary key would could consider instead of all the columns, then we would only need to join on it.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: mysql select from one server insert to another mysql server? I wont to do insert to one mysqll server to and get the data or do the select part
of the insert into select from anther mysql server
not just copy entire tables is this possible in my sql
I know that in MS Sql server you can do a linked server
thanks
A: The excellent, FOSS, cross-platform SQL Workbench query tool supports exactly this, for any database you have a JDBC driver for, as documented in this manual page.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: fastest way to insert just a few rows over thousands of querys MYSQL I have thousands of insert rows for products however I also need to insert suppliers information at the same time.
There is no way to query just what the suppliers information is out of the system I am exporting the information from (name, supplier number etc) apart from getting it attached to every single product the product export.
Unless I go through every product information export, there is no way to be sure that all suppliers have been updated/ inserted.
To put it in perspective there is only about 5 suppliers to about 2000 products.
What is the most optimized way to insert the suppliers, rather than check every single time if the supplier exists because almost all of the time it will?
What is the best way to do this?
I'm using pdo and prepared statements and transactions...
A: *
*Select all the suppliers
*Go with inserting the product data
*
*check whether you have the particular supplier
*
*if not, then insert it
*insert the product
*Collect supplier data for all the product inserts
*Upon finishing inserting products, loop over suppliers and update them
A: Your request it's not very clear (at least for me).
You mean that, every time you insert a new product on DB you want an easier way to check if the product's supplier already exists on database?
If it is so, every time you insert a new Product you can insert/update supplier's info with the following statement:
REPLACE INTO suppliers (name, number, ...) VALUES ([name], [number], ...);
This statement does the following:
*
*If current supplier:
*
*does not exists on DB it will save it on suppliers table.
*if already exists on DB it will update supplier's DB data with provided info.
I think it's the quicker way for you to do what you want.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Black "Oscars Boycott" Hypocrisy
TNO Staff — January 21, 2016 25 comments
Black activists who have attacked the 2016 Oscars awards for being "too white" ignore the fact that there are hundreds, if not thousands, of blacks-only organizations all across America dedicated exclusively to promoting black entertainment, music, professions, recreation, education, and even beauty pageants.
Even more bizarrely, the allegation that the Oscars are "too white" ignores the fact that the award process is controlled by Hollywood insiders—the vast majority of whom are Jewish liberals who are on the far left of the political spectrum—and the most "pro-black," as their film output demonstrates year after year.
Cheryl Boone Isaacs, the black head of the Board of Governors of the Academy of Motion Picture Arts and Sciences—which runs the Oscars—said she was going to "take action to alter the make-up" of the organization's membership after producer Spike Lee said he was going to boycott the 2016 awards because it was "lily white."
Lee was joined in his protest by actress Jada Pinkett Smith, wife of black actor Will Smith.
Lee—most infamous for his vicious anti-white films such as Malcolm X and CSA: Confederate States of America—is possibly the most hypocritical of all.
Lee's first college was the all-black, all-male, Morehouse College in Atlanta, Georgia, where alumni include Martin Luther King, Jr. Somehow, Lee did not have a problem attending that racially-exclusive educational institution—because it was for blacks only, no doubt.
Even more hypocritically, Lee has won awards from two blacks-only movie organizations. The Black Reel Awards (set up to "annually recognize and celebrate the achievements of black people in feature, independent, and television films") awarded several prizes for his Love and Basketball film.
Lee also won prizes from the "Black Movie Awards" (BMA)—an annual ceremony held to recognize achievements of black film actors and films that "stand out in their portrayal of Black experience."
The BMA was founded in 1997, with an inaugural event at the American Black Film Festival (ABFF), which is a "four-day event dedicated to bringing awareness of entertainment content made by and about people of African descent to a worldwide audience."
Once again, Lee seems to have no trouble at all with all these blacks-only and whites-not-allowed/welcome organizations—but seems to have a serious problem with Hollywood, which has—for the first time in many years—not nominated a black actor or film producer for an Oscar.
For her part, Pinkett Smith also has a long history of blacks-only activism: in 1997, she was emcee of the black "Million Woman March" protest march in Philadelphia, Pennsylvania. Speakers at the event included Winnie Mandela (a convicted kidnapper and fraudster whose criminal activities were so numerous that Nelson Mandela divorced her to distance himself from her), Pinkett Smith, and Attallah and Ilyasah Shabazz (the daughters of infamous anti-white racist Malcolm X).
Like Lee, Pinkett Smith seems not to have any problem with gatherings when they are for blacks only.
This hypocrisy is not limited to the acting world, however.
No one, for example, complains about the blacks-only Black Entertainment Television (BET), a cable and satellite television channel (owned by the BET Networks division of Viacom) which specifically targets black audiences—and exclusively uses black actors, producers, directors etc.—such as Pinkett Smith and Lee.
Also, no one seems to object to the National Black Farmers Association, the National Black Law Students Association, the National Black MBA Association, the National Black Nurses Association, the National Black Police Association, the National Conference of Black Lawyers, the National Organization for the Professional Advancement of Black Chemists and Chemical Engineers, the National Organization of Black Women in Law Enforcement, the National Society of Black Engineers, the National Association for Black Veterans, the National Association of Black Accountants, the National Association of Black Geologists and Geophysicists, the National Association of Black Journalists, the National Association of Blacks in Criminal Justice, the National Bar Association, the National Black Chamber of Commerce, the National Association for the Advancement of Colored People, the National Action Network, the National Afro-American League, the National Alliance of Black School Educators, the National Black Justice Coalition, the Miss Black America competition, the Miss Black USA competition, and many, many more, too numerous to name here.
This list does not even address the equal number of Hispanic only organizations, Asian only organizations, American-Indian organizations, etc., all of which are formed exclusively on a racial basis.
Of course, if any white person had dared to form even one of these organizations listed above, and called it, for example, the "National Association of Whites in Criminal Justice," the controlled media—and doubtless people like Pinkett-Smith and Lee—would have been among the first to attack them as "racists."
Yet the double standard to which all white people are held, still reigns supreme: all ethnic groups are allowed to organize on racial grounds—except whites.
The fake flare-up over the Oscars is but one more symptom of this knee-jerk—and controlled media endorsed—anti-white campaign.
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anti-white racism blacks-only organizations Breaking Featured Hollywood Oscars race
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forbes mag says:
Apparently it doesn't occur to Jada Pinkett-Smith and her fellow negroes that this year, perhaps those who were nominated turned in better performances than the negro actors. That would match the common trait of negroes that they have a grossly inflated sense of self-worth and are over the top in a disproportional self confidence.
The alternative is that the negro boycotters simply don't care, and they are suggesting that negroes should be nominated for awards, not on merit, but simply for being negroes which is about what our country has come to anyway under jewish control for the past hundred years.
Charles Wagon says:
Welcome to Obamas Amerika. Everyone gets a participation trophy no matter how little effort they put into it. I hate to say it but I'm beginning to believe the Islam's will win. They don't play, they mean business. Here we pander and pamper our children into believing life is rosey and all you need is a $125,000 college degree and everything will fall in your lap. The west may very well be doomed…..
Charlie, tie your shoe laces and pull up your pants. I'm tired of penitentiary fashion.
edd says:
The usual 'White anything' haters are gaining too much credibility, they are forgetting the simple fact that maybe they did not have any films/actors that were good enough,and as the excellent article points out they are far to over-weighted with their own puffed up racists mono organizations to even have the neck to make these complaints, how about a white athlete complaining they should share out world records for say the one hundred metres, because it is always non whites that hold it most of the time,and not always fairly either,you wouldn't,the thing is like all things in life it's the same highly privileged moaners.
oldgodger says:
no conspiracy here they just wasn't good enough suck it up.
Virtually every film or TV show and it does not matter if it is Historical or a period drama it has to include Black Actors.
They winge far too mu ch and should be ignored.
CDminico says:
Mate, they are running a version of Beowulf – with BLACK Vikings – you really could not believe this carp!
I read from another source that there is a black actor in the television series 'Beowulf' here in the UK! Ridiculous!
Everyone see the Oscars this time!!!
joburg says:
here we go again playing the racist card again.let them form a black actors federation.
let them form a black actors federation instead off playing the racist all the time
Hypocrisy is the right word. There is no support now for white people any more.
Regardless of motivations or race baiting why on earth should anyone care about a group of spoiled pampered entitled hollywood actor types having a beef with other spoiled pampered entitled hollywood actor types and there spoiled pampered entitled hollywood executive types? Give me a break! Has making a living by pretending to be someone else made them lose touch with the challenges people in the real world are facing. Please can someone pull the plug on the MSM and plug it into the common sense channel
Fortunately according to the law of averages.I won't be around in twenty years time to see the pending Apocolypse.The world will never be the same after that.I foresee brother battling against brother.Light against dark.Black against white.Black against black will cause the most damage.Religion against religion.
Rerevisionist says:
I wonder how many members there are of (e.g.) Black Accountants? I note the list includes the NAACP, notably for being Jewish-staffed, Jewish-run, Jewish-funded, Jewish-scripted, and er, Jewish.
HardWeenie says:
Remember, also, how the libs scoff whenever the right-wing is made to feel indignant by the obvious snubbing of nominations of films which have a conservative nature. They skipped over Gran Torino and American Sniper and the response to rage from the right was the predictable "the academy has no duty to pick your favorite films", "this is about awarding the best films, not pleasing everybody", "maybe conservatives just don't make good movies?", etc.
Where are these types of arguments when the nogs don't get their affirmative actions Oscars?
Isn't entitlement wonderful.
Serbiana says:
Everybody should boycott "Oscar" and Hollywood. It is political platform, not art. When USA attacked Serbia all the bad boys in Hollywood productions were Serbs. After us, it was Arab turn, now it is Whites that are bad boys. Of course, USA marines and soldiers are always there to act as heroes and save the day (but just in Hollywood movies). BS.
There is no art is USA. Truly, there is NO ART in the USA- all what they have is the "best sellers" such as "Fifty shades of grey" (soft porn), music industry saturated with vulgar hip-hop where Snoopy Dog is a "star", movies such as Hateful8 or The Reverent (mind polluting) and some silly and profane "performances" usually about feminism, free sex and LGBT.
Filth!
dedelafleche says:
In the UK, they have MOBO awards (music of black origin), just imagine the stink if we had MOWO awards, or maybe, Whites are no good at music?
Let's have White Oscars and let them have Black Whatever.
dedelafleche, Now you are entering the world of 'no 'no' if you're white ,not allowed,they can have thousands of 'Black only' institutions or organizations but not one 'white only' allowed, even with the MOBO awards they usually have a few token whites,for their entertainment,presenting or talking for them, strange when you think what race invented the word 'Apartheid'.
The USA has more organisations for all the various non-White American people than anywhere else. The people behind all this garbage are 100% hypocrites – and it is plainly obvious that they want to eliminate all white people from Earth just as the mass invasion of Europe indicates.
I loved this truthful statement: "Even more bizarrely, the allegation that the Oscars are "too white" ignores the fact that the award process is controlled by Hollywood insiders—the vast majority of whom are Jewish liberals who are on the far left of the political spectrum—and the most "pro-black," as their film output demonstrates year after year."
So what realm of HollyWeird would one get the idea that only Islamists would win? That is if the electorate were prejudice. As stated opinionated earlier: these people really are out of touch with reality.
Select a good script and director and ACT, then you might be in the running for the award.
wwwest says:
Nelson Mandela was a terrorist. Winnie Mandela had a small army of murderers who she would send out every night. Is it any wonder that Nelson Mandela was willing to forgive the whites when he knew that if he pursued them he would be found out about all the crimes he had committed. What a pair of crooks they were. How dare Pinkett Smith invite Winnie Mandela to appear at a function in the USA. Smith and his missus obviously never read the news. What a pair of hypocrites. Smith made the movie 'Concussion' about a Nigerian Doctor who moved to the States and discovered that a good many NFL players developed severe brain damage due to receiving hits on their heads. Apparently Will Smith played the part of the doctor but he could not even get the accent right – what does that show you about him being an actor good enough to get nominated for an Oscar.
Phillip stelk says:
And think of all the 'Afro American' actresses and actors that had no problem taking the money (and getting a start on their current status) acting and starring in those 'Black' films of the '70's.
Jeff Schaefer says:
Don't you know you are not supposed to talk about these groups, it's a secret.
Leave a Reply to joburg Cancel reply
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Il existe différents prieuré Saint-Louis:
Prieuré Saint-Louis-de-la-Rougemare à Rouen, qui accueille aujourd'hui le théâtre de la Rougemare.
Prieuré Saint-Louis de Poissy, devenu aujourd'hui le musée du jouet.
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Bangles is the eponymous first EP by The Bangles. It was released in 1982 by Faulty Products and reissued in 1983 by I.R.S. Records when Faulty Products went out of business. The songs remained widely unavailable thereafter, with only occasional rereleases of individual songs. The whole five-song EP was eventually reissued as part of the Bangles' 2014 compilation, Ladies and Gentlemen... The Bangles!.
This would be the group's only release to feature original bassist Annette Zilinskas, who left in early 1983 and was replaced by Michael Steele, and did not record with them again until the 2018 multigroup album 3 x 4.
A three-song CD Mini Single CD3 version of the EP was released on A&M/I.R.S. Records in 1988.
Background
The Bangles began in Los Angeles as a garage rock band, popularly associated with similar bands from the area in the Paisley Underground music scene. After self-releasing a well-received debut single, "Getting out of Hand" (1981), the group was signed by music industry executive Miles Copeland to his new record label Faulty Products, an independent U.S.-based subsidiary of I.R.S. Records. The band quickly recorded a five-song mini-album which was released in June 1982.
Composition
The EP includes four original songs penned by bandmembers, as well as "How Is The Air Up There?", a cover of the 1960s single by New Zealand band The La De Das.
Track listing
"The Real World" (S. Hoffs, V. Peterson)
"I'm in Line" (D. Peterson, V. Peterson, S. Hoffs)
"Want You" (V. Peterson)
"Mary Street" (S. Hoffs, V. Peterson)
"How Is the Air Up There?" (Steve Duboff, Artie Kornfeld)
1988 three-song CD Mini Single CD3 Release "Bangles"
"The Real World" (S. Hoffs, V. Peterson) Remixed by David Kahne. Previously unreleased.
"Mary Street" (S. Hoffs, V. Peterson)
"I'm in Line" (D. Peterson, V. Peterson, S. Hoffs)
Personnel
The band's full musical credits are listed on the EP's back cover:
Susanna Hoffs – rhythm guitar, vocals
Debbi Peterson – drums, vocals
Vicki Peterson – lead guitar, vocals
Annette Zilinskas – bass guitar, harmonica, vocals
Vocals are credited to Vicki and Debbie Peterson and Susanna Hoffs on every track. Vicki Peterson sings the lead vocal on "Want You" and "How Is the Air Up There?" while Debbie Peterson sings "I'm In Line"; Susanna Hoffs sings "The Real World", and she and Vicki Peterson share the lead on "Mary Street". Annette Zilinskas provides vocals only once, as one of the backing harmonists on "Want You".
Music industry veteran Craig Leon served as the record producer. Leon was already well known in rock and indie circles for his production work with the Ramones and Blondie. Leon also played piano on "Mary Street" and "The Real World", and the latter song includes additional piano work by Ethan James. The album cover art was designed by Ewa Wojciak with photography by Bob Seideman.
Shortly after the EP's release, Annette Zilinskas left the band and was replaced by former Runaways bassist Michael Steele.
Musical style
The Bangles' early years were informed by a 1960s garage rock sensibility, and the 1982 EP maintains a stylistic link between the "Getting out of Hand" debut single and the band's first full-length album, the critically acclaimed All Over the Place (1984). Music critics often note the irony of their subsequent rise from guitar-based rock devotees to "one of the most successful chart groups of the '80s with their slickly produced synth pop". In his book Music: What Happened?, Scott Miller names "The Real World" as one of the top songs of the 1980s, and remarks of the EP: "Those who know only 'Eternal Flame' might be amazed at how inventive and together they were in their relative infancy". The Bangles themselves consider the early material, lesser-known though it was, to be vital to their own story: as Susanna Hoffs told Billboard in 2014, "I think it's as representative of who we really are and as authentic as anything the Bangles have ever done. There's a kind of architecture to those songs – three-part harmonies, guitar-driven, jangly over a kind of garage rock rhythm is who we are now, still, as much as we were back then."
Release
The original EP was released on vinyl in 1982 by Faulty Products (catalog #FEP 1302). The label folded at the end of the year, and Copeland's major label, I.R.S. Records, rereleased it again in 1983 (catalog #SP-70506).
In 1988, A&M/I.R.S. Records released a three-song CD Mini Single CD3 with "Mary Street" and "I'm in Line" from the original vinyl EP release, and a very different mix of "The Real World" remixed and remastered by David Kahne prior to his producing the first Bangles album.
The full five-song set, however, remained out of print since its initial vinyl release until 2014, when it was included on the album Ladies and Gentlemen... The Bangles!. Featuring numerous early Bangles rarities, this compilation also includes a previously unreleased demo version of "The Real World", shorter and sung in a different key.
References
External links
1982 debut EPs
The Bangles albums
I.R.S. Records EPs
Illegal Records EPs
Albums produced by Craig Leon
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{
"redpajama_set_name": "RedPajamaWikipedia"
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This is a Planet Earth parody animation created by Nicolas King with the help of fellow artists Vojislav Milanovic, Ramtin Ahmadi and Nawaz Ahmed imagining a not-so-distant future where a massive herd of Boston Dynamics SpotMinis try to migrate across the desolate wasteland that was once our beautiful earth. I imagine in a hundred years this will probably be accurate. Or I'll be declared King Of Earth after heralding a new age of global consciousness and respect for our planet. "I don't see that happening." Well you can't say I didn't offer. "Just out of curiosity, how would you go about that anyways?" A disease that wipes out humanity so we can return earth to the plants and animals but BEFORE the robots gain consciousness and establish dominance over everything. "So you would die too." That's just the sort of kind and benevolent leader I am. "A real martyr." The martyest martyr that ever was or will be.
Thanks to Luc, who agrees humanity's road keeps forking and we keep taking the wrong way.
Maybe BostonDynamics can take pity on us and design the robots to help remediate the environment.
Create an epidemic of consciousness with love... wake them up!!!
I'm on the fence about whether you're a spammer or not.
It comes after the migration to their birth spots in the mystical land of Boston.
Only way I can think of to make it better: Sir David Attenborough voiceover reading binary, in full emotive intonation.
"Next time, on Black Mirror"
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{
"redpajama_set_name": "RedPajamaC4"
}
| 1,402
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NAME = 'eEye Digital Security SecureIIS'
def is_waf(self):
# credit goes to W3AF
detected = False
r = self.normalrequest()
if r is None:
return
response, responsebody = r
if response.status == 404:
return
headers = dict()
headers['Transfer-Encoding'] = 'z' * 1025
r = self.normalrequest(headers=headers)
if r is None:
return
response, responsebody = r
if response.status == 404:
detected = True
return detected
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,912
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